Model Based and Robust Control Techniques for Internal Combustion Engine Throttle Valves

Transcription

1 Model Baed and Robut Control Technique for Internal Combution Engine Throttle Valve Jacob Lykke Pederen A thei ubmitted in partial fulfilment of the requirement of the Univerity of Eat London for the degree of Doctor of Philoophy October 3 Reviion:..

2 i Acknowledgment I am greatly thankful to my upervior, Profeor Stephen Dodd, whoe valuable technical upport, knowledge and experience enabled me to finih thi work. On a peronal level I would like to thank, Profeor Stephen Dodd and hi wife, Margaret, for the many talk and lunche. I thank Dr W. Hony for being my Director of tudie ince Profeor Dodd became Emeritu. I would alo like to thank my colleague at Delphi for the upport and in particular Johan Dufberg for reviewing the work. Alo, I would like to take thi opportunity to thank my manager, Anthony Potter and Gary Forward, for upporting thi work. I owe many thank to my wife, Inger Lie, for all the upport and help he ha given me through the year of thi reearch.

3 ii Summary The performance of poition controller for a throttle valve ued with internal combution engine of heavy good vehicle i invetigated uing different control technique. The throttle valve i modelled including the hard top and tatic friction (ticklip friction), which are nonlinear component. Thi include a new imple approach to the modelling of tatic friction. Thi nonlinear model wa validated in the time domain uing experimental reult, parameteried by experimental data uing a Matlab baed parameter etimation tool. The reulting tate pace model wa linearied for the purpoe of deigning variou linear model baed controller. Thi linearied model wa validated uing experimental data in the frequency domain. The correct deign of each model baed controller i firt confirmed by imulation uing the linear throttle valve model, the pecified tep repone being expected. Then the robutne i aeed in the frequency domain uing the Matlab Control Sytem Deign Toolbox and in the time domain by imulation uing Monte Carlo baed plant parameter mimatching between the imulated real plant and it model ued for the control ytem deign. Once atifactory performance of a pecific controller i predicted by imulation uing the linear model, thi i replaced by the nonlinear model to acertain any deterioration in performance. Controller exhibiting atifactory performance in imulation with the nonlinear plant model are then invetigated experimentally. The et of controller invetigated in thi work include type that are not currently employed commercially, a well a traditional one, coniting of the IPD, PID, DPI controller and the linear tate feedback controller with and without an integrated oberver. The other controller are the liding mode

4 iii controller, oberver baed robut controller (OBRC) and the polynomial controller. The traditional controller are deigned uing partial pole placement with the derived linear plant model. The other controller have tructure permitting full pole placement, of which robut pole placement i an important option. In the pole placement deign, the location of the cloed loop pole are determined uing the ettling time formula. Depite the ue of robut pole placement, the tatic friction caued a limit cycle, which led to the ue of an anti-friction meaure known a dither. The 4 different controller were invetigated for their ability to control the throttle valve poition with nonlinear friction, parameter variation and external diturbance. Thi information wa gathered, together with qualitative information regarding eae of deign and practicability to form a performance comparion table. The original contribution emanating from the reearch programme are a follow: The ucceful application of new control technique for throttle valve ubject to ignificant tatic friction The firt time invetigation of partial and robut pole placement for throttle valve ervo ytem. A implified tatic friction model which can be ued for other application.

5 iv Content ACKNOWLEDGMENTS... I SUMMARY... II LIST OF FIGURES... IX LIST OF SYMBOLS...XXI LIST OF ACRONYMS...XXIII INTRODUCTION ENGINE SYSTEM THROTTLE VALVE Hardware Decription Ideal Control Sytem Specification Current Control Technique MOTIVATION CONTRIBUTION STRUCTURE OF THE THESIS MODELLING INTRODUCTION THE ELECTRICAL MODEL MECHANICAL MODEL Linear Dynamic Model The Mechanim of Friction Preliminary Experiment to Ae the Randomne of the Friction... 48

6 v.3.4 The Friction Model Hard Stop LINEAR SYSTEM MODEL State Space Model Tranfer Function NONLINEAR SYSTEM MODEL REDUCED ORDER LINEAR SYSTEM MODEL MODEL PARAMETERISATION PARAMETER MEASUREMENT Gear Ratio DC Motor Voltage Contant DC Motor Reitance DC motor inductance DC Motor Torque Contant DC motor moment of inertia and the kinetic friction Throttle Valve Sytem Moment of Inertia The Coil Spring Hard Stop Throttle Valve Sytem Kinetic Friction Static and Coulomb Friction PARAMETER ESTIMATION Introduction DC Motor Model Parameter... 78

7 vi 3..3 Throttle Valve Model Parameter MODEL VERIFICATION IN THE TIME DOMAIN MODEL VALIDATION IN THE FREQUENCY DOMAIN CONTROL TECHNIQUES AND PERFORMANCE ASSESSMENT INTRODUCTION THE EARLIER DEVELOPMENTS LEADING TO THE PID CONTROLLER METHODOLOGY Simulation Detail Experimental Setup Simulation and Experimental validation Senitivity and Robutne Aement COMMON FEATURES Introduction Pole Placement Deign uing the Settling Time Formula Nonlinear Friction and Control Dither Integrator Anti Windup TRADITIONAL CONTROLLERS Introduction Potential Effect of Zero Controller Deign Simulation and Experimental reult LINEAR STATE FEEDBACK CONTROL Baic Linear State Feedback Control... 74

8 vii 4.6. State Oberver Controller Deign Simulation and Experimental reult OBSERVER BASED ROBUST CONTROL Introduction and Brief Hitory Controller Deign Simulation and Experimental reult POLYNOMIAL CONTROL Introduction and Brief Hitory Baic Polynomial Controller Polynomial Control with Additional Integrator for Zero Steady State Error in the Step Repone Controller Deign Simulation and Experimental reult SLIDING MODE CONTROL AND ITS RELATIVES Introduction and Brief Hitory Baic Sliding Mode Control Method for Eliminating or Reducing the Effect of Control Chatter Controller Deign Simulation and Experimental reult PERFORMANCE COMPARISONS CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH... 3

9 viii 6. OVERALL CONCLUSIONS Modelling Control technique RECOMMENDATIONS FOR FURTHER RESEARCH REFERENCES APPENDIX A. ENGINE SYSTEMS OVERVIEW A.. The Natural Apirated Dieel Engine A.. The Turbo Charged Dieel Engine A. PARAMETERS USED FOR THE SIMULATION A.3 CALCULATIONS FOR LINEAR STATE FEEDBACK WITH INTEGRATOR FOR STEADY STATE ERROR ELIMINATION A.4 H-BRIDGE WITH OUTPUT CURRENT MEASUREMENT A.5 THROTTLE VALVE EXPLODED VIEW PUBLISHED WORK A COMPARISON OF TWO ROBUST CONTROL TECHNIQUES FOR THROTTLE VALVE CONTROL SUBJECT TO NONLINEAR FRICTION FORCED DYNAMIC CONTROL OF NON-MINIMUM-PHASE PLANTS VIA STUDY OF THE CLASSICAL INVERTED PENDULUM

10 ix Lit of Figure Figure.: An example of a chematic for a turbocharged Euro VI engine configuration with high preure EGR and throttle valve... 4 Figure.: A throttle valve... 6 Figure.3: Schematic of a throttle valve... 7 Figure.4: Throttle valve poition control ytem... 9 Figure.5: Deired throttle poition demand for a typical drive cycle during DPF regeneration... 3 Figure.6: Sytem ettling time definition... 3 Figure.7: Maximum / minimum control effort a function of deired ettling time... 3 Figure.8: A ection of the generation DPF cycle with different ettling time 33 Figure.: A diaembled throttle valve Figure.: Throttle valve chematic diagram Figure.3: Model of a bruh DC motor... 4 Figure.4: Electrical chematic of the DC motor... 4 Figure.5: State pace repreentation of equation (.3)... 4 Figure.6: Throttle body chematic without the DC motor Figure.7: Gear ytem Figure.8: Tranfer the moment of inertia and friction to the other ide of the gear Figure.9: Throttle plate ide of the gear ytem Figure.: Repreentation of lumped ytem Figure.: Surface interaction Figure.: The ame day friction repeatability experiment... 5 Figure.3: Different day friction repeatability experiment... 5 Figure.4: The experiment accumulated difference... 5 Figure.5: Claic friction model (Papadopoulo and Chapari, ) Figure.6: The new friction model and it component... 55

11 x Figure.7: Friction model implementation Figure.8: New friction model imulation Figure.9: Hard top model Figure.: Linear throttle model Figure.: State variable block diagram Figure.: Linear throttle model in control canonical form Figure.3: Nonlinear throttle model Figure.4: Second order throttle valve model Figure 3.: Simplified DC motor model... 7 Figure 3.: Meaured current and the value ia T Figure 3.3: DC motor friction and moment of inertia validation Figure 3.4: Meaure of the coil pring torque Figure 3.5: Parameter etimation uing the toolbox from Mathwork Figure 3.6: Meaurement data ued for the DC motor model parameter etimation Figure 3.7: Etimation of the DC motor parameter... 8 Figure 3.8: A ubet of the data et ued for the throttle valve model parameter etimation... 8 Figure 3.9: Nonlinear throttle valve model with parameter value... 8 Figure 3.: Comparion between the non-linear plant model and the plant (Blue dahed: Experimental data. Green: Simulated data) Figure 3.: An example of a peudo random binary equence Figure 3.: Bode plot Comparion between the non-linear plant model and the plant Figure 4.: Proportional controller applied to the throttle valve Figure 4.: Root locu of the ytem with unity gain feedback control Figure 4.3: Step repone with proportional controller adjuted for critical damping... 9 Figure 4.4: Experimental etup for teting the throttle valve poition control trategie... 93

12 xi Figure 4.5: PI controller implemented in dspace (implified diagram) Figure 4.6: Experimental hardware Figure 4.7: Throttle poition demand waveform for control tet Figure 4.8: Standard linear control ytem tructure Figure 4.9: Senitivity of the linear throttle valve control loop with a proportional controller... Figure 4.: A D matrix for parameter variation tet... Figure 4.: Illutration of D parametric variation for Monte-Carlo analyi (Standard deviation = 3 %, Mean value = )... Figure 4.: Frequency ditribution of parameter ued for the Monte Carlo analyi (Standard deviation = 3 %, Mean value = )... 3 Figure 4.3: Parameter ditribution ued for the Monte Carlo ( R for.:. a )... 4 Figure 4.4: Throttle poition operation envelope... 4 Figure 4.5: Settling time definition... 7 Figure 4.6: Linear high gain robut control ytem... 8 Figure 4.7: Root locu with repect to K... 9 Figure 4.8: Root locu of cloed loop ytem uing robut pole placement. Figure 4.9: Block diagram for ideal tep repone generation, a) third order, b) fourth order... 3 Figure 4.: Dither ignal generator... 4 Figure 4.: Dither ignal level... 5 Figure 4.: Experimental reult of a P-controller with and without dither... 6 Figure 4.3: Amplitude pectra... 7 Figure 4.4: PI controller with integrator anti-windup... 8 Figure 4.5: Integrator anti-windup performance... 9 Figure 4.6: The PID controller... Figure 4.7: The DPI controller with the throttle valve plant... Figure 4.8: The IPD controller... Figure 4.9: The zero effect on the cloed loop tep repone... 4

13 xii Figure 4.3: Cloed loop pole location for T. [ec]... 7 Figure 4.3: Cloed loop repone of a IPD controller with partial pole placement... 8 Figure 4.3: Throttle valve and IPD controller with differentiation filter... 8 Figure 4.33: The cloed loop pole and zero location for the IPD controller with a differentiating filter... 3 Figure 4.34: Simulated tep repone with/without noie filter compenation 3 Figure 4.35: DPI controller with differentiation filter... 3 Figure 4.36: The impact of the zero on the cloed loop ytem repone Figure 4.37: Cloed loop imulation of the DPI controller with a linear throttle valve plant model and limit on the controller output Figure 4.38: DPI controller with precompenator Figure 4.39: Cloed loop imulation of the DPI controller with precompenator cancelling both zero Figure 4.4: DPI controller with integrator anti-windup Figure 4.4: Cloed loop tep repone of the DPI controller with integrator antiwindup (Large tep) Figure 4.4: Cloed loop tep repone of the DPI controller with integrator antiwindup (Small tep) Figure 4.43: Simplified dicrete time DPI controller with feed forward Figure 4.44: Simplified continuou time DPI controller... 4 Figure 4.45: Example of a waveform ued for the tuning of the DPI... 4 Figure 4.46: PID controller with differentiation noie filter... 4 Figure 4.47: PID controller with differentiation filter, precompenator and integrator anti windup Figure 4.48: Cloed loop imulation of the PID controller with precompenator cancelling both zero Figure 4.49: Cloed loop tep repone, from. to.3 [rad] Figure 4.5: Experimental and imulated repone of the IPD controller... 46

14 xiii Figure 4.5: The difference between the deired and the experimental cloed loop repone Figure 4.5: IPD controller during a pring failure Figure 4.53: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: 5% ) Figure 4.54: Control tructure ued to analye the enitivity... 5 Figure 4.55: IPD enitivity... 5 Figure 4.56: Cloed loop tep repone, from. to.3 [rad]... 5 Figure 4.57: Experimental and imulated repone of the DPI controller Figure 4.58: The difference between the deired and the experimental cloed loop repone Figure 4.59: DPI controller during a pring failure Figure 4.6: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: 4% ) Figure 4.6: Control tructure ued to analye the enitivity Figure 4.6: DPI enitivity Figure 4.63: Cloed loop tep repone, from. to.3 [rad] Figure 4.64: Experimental and imulated repone of the DPI controller... 6 Figure 4.65: The difference between the deired and the experimental cloed loop repone... 6 Figure 4.66: DPI controller with feed forward during a pring failure... 6 Figure 4.67: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: % ) Figure 4.68: Control tructure ued to analye the enitivity Figure 4.69: Manually tuned DPI enitivity Figure 4.7: PID cloed loop tep repone Figure 4.7: Cloed loop tep repone, from. to.3 [rad], uing a precompenator Figure 4.7: Experimental and imulated repone of the PID controller... 68

15 xiv Figure 4.73: The difference between the deired and the experimental cloed loop repone Figure 4.74: PID controller during a pring failure... 7 Figure 4.75: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: 4% )... 7 Figure 4.76: Control tructure ued to analye the enitivity... 7 Figure 4.77: PID enitivity Figure 4.78: LSF control ytem Figure 4.79: LSF control ytem with a tate oberver Figure 4.8: A baic third order oberver tructure Figure 4.8: LSF plu integral control... 8 Figure 4.8: LSF control of the throttle valve with teady tate compenation 8 Figure 4.83: The cloed loop pole location of the LSF control loop Figure 4.84: Pole location of an LSF plu integral control loop with a robut pole-to-pole ratio of Figure 4.85: Oberver aided LSF control with integrator for teady tate error elimination Figure 4.86: Simplified control ytem block diagram for deign of the LSF controller Figure 4.87: Third order oberver tructure Figure 4.88: Oberver aided LSF with integrator anti-windup ued for the experiment and imulation... 9 Figure 4.89: Throttle valve tep repone with/without integrator anti-windup (K=.)... 9 Figure 4.9: Retructure a baic oberver to a ingle correction loop... 9 Figure 4.9: Single correction loop oberver with noie filter Figure 4.9: Single correction loop oberver aided LSF with integrator for teady tate compenation Figure 4.93: Oberver with ingle loop correction controller Figure 4.94: Cloed loop tep repone, from. to.3 [rad]... 96

16 xv Figure 4.95: Experimental and imulated repone of the LSF controller with teady tate compenation Figure 4.96: The difference between the deired and the experimental cloed loop repone Figure 4.97: Cloed loop tep repone, from. to.3 [rad] Figure 4.98: Experimental and imulated repone of the LSF controller with integrator uing robut pole placement... Figure 4.99: The difference between the deired and the experimental cloed loop repone... Figure 4.: LSF with integrator controller during a pring failure... Figure 4.: Simulated cloed loop repone difference done for a number of different robut pole placement ratio... 3 Figure 4.: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: % )... 4 Figure 4.3: Control tructure ued to analye the enitivity... 5 Figure 4.4: LSF with integrator enitivity... 6 Figure 4.5: Cloed loop tep repone, from. to.3 [rad]... 7 Figure 4.6: Experimental and imulated repone of the oberver aided LSF controller with integrator uing robut pole placement... 8 Figure 4.7: The difference between the deired and the experimental cloed loop repone... 9 Figure 4.8: Oberver aided LSF controller during a pring failure... Figure 4.9: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: 5% )... Figure 4.: Control tructure ued to analye the enitivity... Figure 4.: Oberver aided LSF control with integral term enitivity... 3 Figure 4.: Cloed loop tep repone, from. to.3 [rad]... 4 Figure 4.3: Experimental and imulated repone of the retructured oberver aided LSF controller with integrator uing robut pole placement... 5

17 xvi Figure 4.4: The difference between the deired and the experimental cloed loop repone... 6 Figure 4.5: Retructured oberver aided LSF with integrator during a pring failure... 7 Figure 4.6: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: 5% )... 8 Figure 4.7: Control tructure ued to analye the enitivity... 9 Figure 4.8: Retructured oberver aided LSF control with integrator enitivity... Figure 4.9: Plant and model mimatch... Figure 4.: Correction loop controller ued for etimating the diturbance... Figure 4.: Subtraction of U from control input to compenate ˆe e Ue U.... Figure 4.: Input converion block diagram... 3 Figure 4.3: Overall OBRC tructure for a ingle input, ingle output plant. 4 Figure 4.4: OBRC tructure with a LSF controller... 5 Figure 4.5: Individual pole placement ued for OBRC... 7 Figure 4.6: Cloed loop tep repone, from. to.3 [rad]... 3 Figure 4.7: Experimental and imulated repone of the OBRC... 3 Figure 4.8: The difference between the deired and the experimental cloed loop repone... 3 Figure 4.9: OBRC during a pring failure Figure 4.3: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: % ) Figure 4.3: Control tructure ued to analye the enitivity Figure 4.3: OBRC enitivity Figure 4.33: a) PID controller converted into the b) baic linear SISO controller form Figure 4.34: Digital R-S-T controller canonical tructure... 38

18 xvii Figure 4.35: The general tructure of the Polynomial control ytem... 4 Figure 4.36: Polynomial control of throttle valve with additional integrator Figure 4.37: Control ytem of Figure 4.36 howing controller polynomial46 Figure 4.38: Implementation of the Polynomial control with additional integrator Figure 4.39: Cloed loop tep repone with one fat pole Figure 4.4: Polynomial control with additional integrator and a econd order plant model ued for the controller deign... 5 Figure 4.4: Simulated cloed loop repone tep repone at Figure 4.4: Simulated cloed loop repone tep repone uing a precompenator with the ettling time..3.4 T ec Figure 4.43: Cloed loop tep repone, from. to.3 [rad] Figure 4.44: Experimental and imulated repone of the polynomial controller Figure 4.45: The difference between the deired and the experimental cloed loop repone Figure 4.46: Polynomial controller during a pring failure Figure 4.47: Maximum / minimum throttle poition and DC motor voltage envelope at a pole group ratio = 4 (Standard deviation: 4% )... 6 Figure 4.48: Control tructure ued to analye the external diturbance enitivity... 6 Figure 4.49: Polynomial control enitivity... 6 Figure 4.5: Implementation of the Polynomial control with additional integrator... 6 Figure 4.5: Cloed loop tep repone, from. to.3 [rad] Figure 4.5: Experimental and imulated repone of the polynomial controller Figure 4.53: The difference between the deired and the experimental cloed loop repone Figure 4.54: Polynomial controller during a pring failure p

19 xviii Figure 4.55: Maximum / minimum throttle poition and DC motor voltage envelope at a pole group ratio = 6 (Standard deviation: 5% ) Figure 4.56: Control tructure ued to analye the external diturbance enitivity Figure 4.57: Polynomial control enitivity Figure 4.58: Cloed loop tep repone, from. to.3 [rad] Figure 4.59: Experimental and imulated repone of the polynomial controller... 7 Figure 4.6: The difference between the deired and the experimental cloed loop repone... 7 Figure 4.6: Polynomial controller during a pring failure... 7 Figure 4.6: Maximum / minimum throttle poition and DC motor voltage envelope at a T. ec and a pole group ratio = 6 (Standard deviation: % ) Figure 4.63: A baic variable tructure control ytem Figure 4.64: Double integrator plant Figure 4.65: Phae portrait for a double integrator plant with b Figure 4.66: Block diagram of a Bang-Bang controller for a SISO plant Figure 4.67: Cloed loop phae portrait of a double integrator plant w, w... 8 Figure 4.68: Cloed loop repone and bang-bang controller output of a double integrator plant... 8 Figure 4.69: An example of a trajectory for the double integrator plant Figure 4.7: Plant output not directly linked to the plant tate... 8 Figure 4.7: Diplay of equivalent control for imulation of Figure Figure 4.7: An example of a baic SMC for a throttle valve plant Figure 4.73: Baic liding mode controller behaviour Figure 4.74: Baic SMC with a Control Smoothing Integrator Figure 4.75: DC-Motor current level. Sample frequency = 3 Hz for

20 xix Figure 4.76: Boundary Layer Sliding Mode Control... 9 Figure 4.77: Switching boundary SMC with meaurement noie filtering and integrator with aturation... 9 Figure 4.78: Practicable SMC with control moothing integrator and variable gain to minimie control chatter for mall poition error... 9 Figure 4.79: Boundary layer method SMC with meaurement noie filtering 94 Figure 4.8: Boundary layer method SMC with integrator in the forward path and meaurement noie filtering Figure 4.8: Cloed loop tep repone, from. to.3 [rad] Figure 4.8: Experimental and imulated repone of the SMC - control moothing integrator method Figure 4.83: The difference between the deired and the experimental cloed loop repone with a maximum gain of Figure 4.84: The difference between the deired and the experimental cloed loop repone with a fixed gain of Figure 4.85: SMC - Control moothing integrator method during a pring failure... 3 Figure 4.86: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: 8% )... 3 Figure 4.87: Structure ued for analying enitivity Figure 4.88: SMC - Control moothing integrator method enitivity with a fixed gain = Figure 4.89: SMC - Control moothing integrator method enitivity with a fixed gain = Figure 4.9: Cloed loop tep repone, from. to.3 [rad] Figure 4.9: Experimental and imulated repone of the SMC - Boundary layer method Figure 4.9: The difference between the deired and the experimental cloed loop repone Figure 4.93: SMC - Boundary layer method during a pring failure... 39

21 xx Figure 4.94: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: 5% )... 3 Figure 4.95: Structure ued for analying enitivity... 3 Figure 4.96: SMC - Boundary layer method enitivity... 3 Figure 5.: Step repone difference for comparion # Figure 5.: Step repone difference for comparion # Figure A: Baic chematic of a natural apirated Dieel engine Figure B: Turbo charger from Cummin Turbo Technologie Figure C: Baic chematic of a turbo charged Dieel engine Figure D: Schematic of a turbo charged Dieel engine with EGR Figure E: An example of a chematic for Euro VI engine configuration with high and low preure EGR Figure F: LSF with integrator and plant model Figure G: H-bridge chematic Figure H: H-bridge and current meaurement board Figure I: Throttle valve exploded view

22 xxi Lit of Symbol L Inductance [Henry - H] L DC motor armature inductance [Henry - H] a R Reitance [Ohm - Ω] R DC motor armature reitance [Ohm - Ω] a I Current [Ampere - A] V k e k t Voltage [Volt] DC motor voltage contant [V/(rad/ec)] DC motor torque contant [Nm/A] J Moment of inertia [kg m ] J x θ ω Throttle ytem inertia (lumped) Angle [rad] Angle peed [rad/ec] k Coiled pring contant [Nm/rad] pring N Gear ize [m] r kkinetic n T To Gear radiu [m] Kinetic (vico) friction contant [Nm ec/rad] Sytem order, if nothing ele i tated Control ettling time [ec] Oberver ettling time [ec] 3 Volume [ m ] p Preure [ N / m ] m Sec S Ma flow [kg/ec] Second Switching function r Pole-to-pole ratio [-] pp r Minimum pole-to-pole ratio [-] ppmin h Sampling time interval [ec] Sigma

23 xxii Laplace variable Torque [Newton Meter - Nm] Cloed loop pole Open loop pole zero

24 xxiii Lit of Acronym ADC DAC DC DPF DPI e.m.f. ECU EGR FFT HGV I/O IPD LCR LSF LTI Matlab MAXB NOx OBRC PID PRBS PWM SCR Analogy-to-Digital Converter Digital-to-Analogy Converter Direct Current Dieel Particulate Filter Derivative Proportional Integral controller Electromotive force Electronic controller unit Exhaut Ga Recirculation Fat Fourier Tranform Heavy Good Vehicle Input / Output Integral Proportional Derivative controller Inductance (L), capacitance (C) and reitance (R) Linear State Feedback Linear Time Invariant Mathematical tool from Mathwork, Inc. Micro Auto Box Nitrogen Oxide Oberver Baed Robut Control Proportional Integral Derivative controller Peudo random binary equence Pule Width Modulation Selective Catalytic Reduction Simulink Diagram imulation tool from Mathwork, Inc. SISO Single Input Single Output SMC Sliding Mode Control VGT Variable Geometry Turbine VSC Variable Structure Control

25 . Introduction 4 Introduction. Engine Sytem Figure. how an overview of the engine ytem that thi reearch programme upport, a detailed decription of which i given in Appendix A.. Air filter Ga flow direction Intercooler Throttle valve Air Compreor Intake manifold Engine block Fuel injector Exhaut ga Turbo haft Cylinder Exhaut manifold Muffler VGT EGR valve EGR cooler Figure.: An example of a chematic for a turbocharged Euro VI engine configuration with high preure EGR and throttle valve The throttle valve, decribed in more detail in the following ection, i the focu of thi reearch programme but thi will be equally ueful for the other valve employed in the ytem a each of thee ha imilar characteritic.

26 . Introduction 5. Throttle Valve On a petrol engine the throttle valve i ued to control the air-to-fuel ratio by applying a variable contraint to the air path, which will reduce the air flow. On Dieel engine (Figure.) the throttle valve i ued a a mean to increae the EGR rate and reduce the air-to-fuel ratio, in a low power operating range. In thi range, the operation of the VGT vane ha no effect and therefore the throttle ha to be ued. The amount of air into the engine can be controlled by cloing the throttle valve, creating a lower preure in the intake manifold. Thi lower preure can alo be ued to induce more EGR flow through it high preure path, auming that the exhaut manifold preure tay contant. For the EURO VI regulation, a high EGR rate i needed in the low power range making ue of the throttle valve. During the DPF regeneration, the air-to-fuel ratio need to be controlled to within a pecific range, which will require the ue of the throttle valve. Furthermore, the throttle valve can be ued to damp engine haking following key off by cloing it. Throttle valve uffer from coniderable nonlinear friction in their mechanim that make it difficult to control accurately, particularly a it i ubject to ignificant variation due to change in temperature and wear over the engine lifetime. The tatic (tick-lip) friction component i particularly troubleome and can caue controller limit cycling (Townend and Salibury, 987) which can compromie the engine emiion performance. Thi alo preent a challenge for the control ytem deigner when it i important to obtain a precribed dynamic repone to reference input poition change.

27 . Introduction 6.. Hardware Decription The throttle ytem (Figure.) conit of a pring loaded throttle plate which i driven by a direct current (DC) motor through a gear ytem. A pre-windup coil pring applie a reidual torque which make the valve open in the cae of an electrical failure. The throttle plate poition i meaured by a potentiometer type enor attached to the plate, where fully open = [rad] and fully cloed =.57 [rad]. Throttle plate Poition enor DC motor Gear ytem Throttle body Figure.: A throttle valve The air flow through the throttle valve i a function of the air-to-fuel ratio, EGR rate and after-treatment demand. In the normal operation mode, the throttle reference poition i calculated by uing the deired throttle air flow, preure and the temperature. The ma flow paing the throttle valve illutrated in Figure.3 can be modelled by the ientropic (contant entropy) flow equation for a convergingdiverging nozzle (Wallance et al., 999, Schöppe et al., 5).

28 . Introduction 7 t u p t m throttle p u pl Figure.3: Schematic of a throttle valve For the ga ma flow through the throttle valve: p u p t mthrottle Athrottle pl C RT p u u (.) For the non-choked flow (ub-onic): p p p p, pu pu p u p u For the choked flow (onic): where R: Ga contant c p v t t t t p t p t, pu pu / c (.4 for air) (.) (.3) C : Throttle valve flow coefficient (dimenionle) A throttle : Geometrical effective valve area p u : Uptream preure p t : Throat preure m throttle : Ma flow through the throttle valve pl : Throttle plate poition The geometrical area for flow paage of an elliptical throttle plate:

29 . Introduction 8 where r : Pipe radiu r : Maximum throttle plate radiu A r co throttle pl pl r r (.4) In theory, equation (.) to (.4) could be rearranged to get the deired throttle poition, uing the deired ga flow, the ga temperature and it preure. In practice, however, it i difficult to get an accurate throttle poition by thi mean due to model parametric error. Thi could be circumvented by creating two function, one baed on the phyic of the ytem and a econd one baed on empirical data, a follow. mthrottle f mthrottle, Tu, pu, pt p u p t RT pu u (.5) where pl f f mthrottle, Tu, pu, pt (.6) deired f : A function which convert the corrected flow demand f into a throttle poition The aim of the throttle valve poition control ytem (Figure.4) i to control the poition (i.e., the angle) of the plate inide the throttle valve. In the normal mode, the deired throttle poition demand from equation (.6) i fed into the throttle poition controller, but in pecial circumtance thi can be overridden by other demand uch a the DPF regeneration mode or engine hut down (key off). The controller meaure the throttle plate poition and adjut thi to achieve the deired poition demand. The throttle plate poition i adjuted uing the torque produced by the DC motor and gear ytem.

30 . Introduction 9 Throttle poition demand Throttle valve poition controller Armature voltage Meaured throttle poition Throttle valve ytem Figure.4: Throttle valve poition control ytem The torque i proportional to the DC motor armature current. The current i controlled by the throttle valve poition controller output driver... Ideal Control Sytem Specification The aim of the throttle valve poition control ytem deign i to achieve the following:. The poition reference mut be followed with a minimum delay and teady tate error.. The control ytem mut exhibit robutne to minimie the impact of the tatic (tick-lip) friction, change in friction parameter due to wear & tear. 3. The control ytem mut be able to compenate for failure of the retention pring (pre-windup coil pring). Figure.5 how an example of a deired throttle valve poition demand ( = fully open,.57 = fully cloed) for a typical drive cycle during DPF regeneration (ource: Delphi Dieel Sytem). In the DPF regeneration mode the valve i ued to control the narrow air/fuel ratio operation range for the DPF to heat up and tay in the regeneration mode. If the air-to-fuel ratio exceed the

31 . Introduction 3 boundarie it can caue the DPF to top regenerating or in the wort cae damage the DPF ceramic tructure due to the generated heat..5 Deired throttle poition demand [rad] Time [ec] Figure.5: Deired throttle poition demand for a typical drive cycle during DPF regeneration The dynamic of the deired cloed loop ytem repone ha to be choen in uch a way that it track the demand without too much lag. The lag can caue the air flow to differ from that demanded for ignificant period and impact the engine output emiion. Chooing too fat a deired cloed loop ytem repone can, however, wear down the actuator, in thi cae the DC motor and gear ytem in the throttle valve. In addition the peed of repone i retricted by the DC motor voltage upply. Figure.6 how a claic definition of the ytem ettling time T (Franklin et al., ). Thi i defined a the time it take from applying an ideal intantaneou tep input to the time at which the ytem output ha entered

32 . Introduction 3 and remained within a pecified range of the expected teady tate value. In thi cae the range ha been choen to /- 5%. Steady tate value /- 5% T Time [ec] Figure.6: Sytem ettling time definition Figure.7 how the minimum and maximum peak value of the controller output during a imulation of a cloed loop pule repone (pule duration econd). The imulation i done on a nonlinear throttle valve plant model, with variou deired cloed loop ettling time. The controller i in thi cae implemented a a tate pace controller with an integrator in the outer loop to remove the teady tate offet. Thi will be decribed in more detail in Chapter 4. The aturation level for the throttle actuator ued for thi project i /- volt. Figure.7 indicate that peak control effort needed to achieve a deired ettling time of. econd for a poition demand change nearly at the maximum value. Thi will bring the control ytem into aturation, but only at the maximum level. The control aturation reulting from attempting to reduce thi further would eriouly deteriorate the control ytem performance.

33 . Introduction 3 Maximum Minimum Maximum actuator aturation level 5 Controller output amplitude level [volt] 5-5 Minimum actuator aturation level Deired ettling time [ec] Figure.7: Maximum / minimum control effort a function of deired ettling time Figure.8 i a ample of the deired throttle valve poition demand from the DPF regeneration cycle. It how the imulated impact of the ettling time on the ytem repone during the regeneration cycle. When the ettling time increae the lag between the deired repone (throttle poition demand) and the imulated cloed loop ytem repone ha an increaed dynamic lag, a expected. A tated before, too much lag can caue the emiion output level to increae, while controller aturation indicate too fat a repone which can prematurely wear the throttle ytem. The deired cloed loop ettling time hould be choen a a compromie between lag and aturation of the control ytem, in thi cae. econd.

34 . Introduction 33.5 Ideal cloed loop ytem repone (T=.5) Throttle poition demand Ideal cloed loop ytem repone (T=.) [rad] Time [ec] Figure.8: A ection of the generation DPF cycle with different ettling time..3 Current Control Technique The throttle valve poition i currently governed by a PID/DPI controller with feed forward and meaure to overcome the tatic friction. Depite the preence of the integral term, the feed forward i ued to counteract the coil pring torque, which alo avoid prolonged tranient behaviour due to relying olely on the integral action for thi. To minimie the effect of the tatic friction an additional ocillatory ignal can be added to the control variable that produce a correponding torque jut ufficient to overcome the tatic friction. Thi i known a control dither (Leonard and Krihnapraad, 99). The amplitude and frequency of thi ignal are adjuted at the commiioning tage. Thi, however, i quite difficult and time conuming.

35 . Introduction 34 In mot cae the traditional controller, i.e., PID/DPI controller and their variant are employed and tuned by certain procedure, uch a Zeigler-Nichol (Mehram and Kanojiya, ) or trial and error..3 Motivation The aim of thi reearch i to tudy the control technique ued in vehicle power train for the poition control of the throttle valve and to eek new control technique taking advantage of the flexibility of modern proceing technology to achieve an improved performance and reduce the commiioning time. The adoption of model baed control technique and increaed robutne againt parametric uncertaintie and external diturbance are expected to play major role in reaching thi goal. Model baed control ytem deign mean the derivation of deign formulae for the adjutable parameter of a controller baed on a mathematical model of the plant (in thi cae the throttle valve) and a deign pecification. Major benefit of thi approach are a follow. a) The control ytem deign can be validated by comparing the repone of a imulation of the control ytem with the expected repone et by the deign pecification. b) The robutne can be aeed by introducing external diturbance and parametric mimatche in the imulation and oberving the reulting deviation of the control ytem repone from the nominal repone determined in (a). Model baed control ytem deign, i adopted throughout thi thei, a it form a firm bai for comparion of the different control technique.

36 . Introduction 35.4 Contribution The original contribution emanating from the reearch programme, that entailed the comparion of fourteen different controller, are a follow: The ucceful application of new control technique for throttle valve ubject to ignificant tatic friction The firt time invetigation of partial and robut pole placement for throttle valve ervo ytem. A implified tatic friction model which can be ued for other application..5 Structure of the thei A the reearch focue on control technique ued for the intake throttle valve in heavy duty (HD) vehicle power train, chapter give a general engine ytem overview and a more detailed decription of the intake throttle valve. The firt tak in the development of a model baed control algorithm i the etablihment of a mathematical model of the particular plant to be controlled. So Chapter preent a detailed electrical and mechanical nonlinear model of the intake throttle valve. Then a linear model i derived on the bai of the nonlinear one. In chapter 3 the nonlinear model i parameteried by the ue of meaurement, calculation and a parameter etimation tool from Mathwork. An introduction to the PID controller i given in the beginning of chapter 4. Then the methodology of the controller performance aement i introduced. Thi i followed by explanation of the common feature of the controller under invetigation, compriing pole placement, control dither and integrator antiwindup. The different control trategie are then introduced and the correponding control law for the throttle valve are derived. Then the performance of each i aeed. In Chapter 5 the performance of all 4 different controller are compared. Finally, Chapter 6 preent the overall concluion and recommendation for further reearch.

37 . Modelling 36 Modelling. Introduction The throttle ytem (Figure.) conit of a pring loaded throttle plate which i mechanically connected to a bruhed DC motor through a gear ytem (Scattolini et al., 997). The pre-treed coil pring i a afety meaure preventing the engine talling in cae of an electric fault, in which the motor i not energied by making the plate go to it open poition. The plate poition i meaured by a potentiometer with an output range between.5 and 4.5 [V] with total poition accuracy of /- %. The non-zero output range i to inure that the control ytem can detect if the poition ignal wire break. A pictorial view of the throttle valve component i hown in Figure.. A more detailed exploded view of the throttle valve may be found in appendix A.5. DC motor Throttle plate Poition enor Gear ytem Tooth wheel with coil pring (inide) Figure.: A diaembled throttle valve The throttle valve ytem model comprie two part: an electrical and mechanical model. The electrical model conit of the equation of the

38 . Modelling 37 armature circuit of the DC motor while the mechanical model conit of the equation modelling the mechanical load, including the moment of inertia, the gear ytem, the pring and friction. Figure. how a chematic diagram of the throttle valve ytem, tarting from the left with DC motor. The DC motor i modelled a an electric load (reitance R a and inductance L a ) and back electromotive force (e.m.f.) which depend on the haft peed. The output torque from the DC motor i proportional to the current. The mechanical ytem i modelled a a gear ytem with a moment of inertia and kinetic friction component on both ide. The gear i ued to amplify the DC motor torque. Ra L DC motor a V in - ek e m J m m k i m t a k kinetic N m Gear train i a Coil pring k pring J pl Valve plate k kinetic 5V pl V N pl Hard top Throttle poition feedback Figure.: Throttle valve chematic diagram Table. define all the parameter of the model.

39 . Modelling 38 Table.: Parameter of throttle valve model Quantity Decription Unit V DC motor input voltage V in i DC motor armature current A a R DC motor armature reitance Ohm a L DC motor armature inductance H a k DC motor torque contant Nm/A t k DC motor voltage contant V/(rad/ec) e e DC motor back e.m.f. generated voltage V Torque generated by the DC motor Nm m DC motor poition rad m m DC motor peed rad/ec Throttle valve plate poition rad pl J Moment of inertia for the valve plate kg*m^ pl J Moment of inertia for the DC motor kg*m^ m k Coiled pring contant Nm/rad pring kkinetic kkinetic The lumped kinetic (vico) friction contant including the DC motor bearing and half of the gear train friction The lumped kinetic (vico) friction contant including the throttle plate bearing and half of the gear train friction Nm ec/rad Nm ec/rad N DC motor wheel diameter m m N Throttle plate wheel diameter m pl N / N Gear ratio - pl m

40 . Modelling 39 The throttle valve i a butterfly valve type, which mean that air flow paing through the throttle valve will not create a load torque and thi i therefore not included in the model. The main load torque come from a linear coiled pring which increae it torque in proportion with the cloing angle of the throttle valve. The mechanical and electrical model of the throttle valve are combined in ubection.4 to form a complete linear tate pace model and the correponding tranfer function model. The linear tate pace model i ued for the controller and oberver deign in Chapter 4. The linear model i extended to form a more comprehenive model in ubection.5 that include hard top, tatic and Coulomb friction, making the model nonlinear. The nonlinear model i ued to develop the control trategie and to validate them before they are teted on the experimental etup. In ubection 3 the model i parameteried by uing meaurement, experiment and a parameter etimation tool from Mathwork. The parameteried model i then validated in the time and frequency domain.. The Electrical Model The DC motor in the throttle valve i mechanically commutated (bruhed) with permanent tator magnet (Figure.3). The electric current, i a, which run through the armature winding, with the reitance R a and inductance L a, generate a torque k t m equal to the current amplitude multiplied by the contant k i (.) m t a

41 . Modelling 4 The armature voltage, V in, upplied by a DC voltage ource, drive the armature current, i a, through the motor. In thi cae the input voltage ource i the throttle valve poition controller. The torque i generated by the current through the armature winding interacting with the magnetic field from the tator magnet. The maximum torque i generated when the angle between the conducting winding and magnetic field i 9. Thi angle i maintained by the commutator that witche the different et of armature winding on and off a they rotate under the magnet. Bruh N Stator magnet i a Armature / Rotor winding θ m Stator magnet Shaft Bearing S Bruh Commutator Figure.3: Model of a bruh DC motor The torque force the armature/haft to pin. The peed of the DC motor, m, generate a back e.m.f. proportional to the peed via the contant k e. Thu e k (.) e m The fater the DC motor pin the larger the armature voltage needed to maintain the required armature current, a hown in Figure.4.

42 . Modelling 4 Ra La V in - i a ek e m Figure.4: Electrical chematic of the DC motor If the DC motor i upplied by a contant voltage, the motor peed will ettle to a contant equilibrium value. The armature current will be limited by the back e.m.f. and the reulting electromagnetic torque will be jut enough to drive the mechanical friction and load at the contant equilibrium peed. With reference to Figure.4, the differential equation modelling the electrical part i a follow. di t d t V t L R i t k dt dt a m in a a a e dia t dm t Vin t ia t Ra ke dt La dt (.3) The armature current generating the electromagnetic torque, i a, i found by integrating (.3). Figure.5 how the tate variable block diagram model of the electrical part of the DC motor correponding to (.3). vin - L a Ia R a k e m Figure.5: State pace repreentation of equation (.3)

43 . Modelling 4.3 Mechanical Model The mechanical model of the throttle valve i plit into three ection: the dynamic model, the friction model and the hard top. The dynamic model comprie the moment of inertia of the mechanical component, the gear train and the friction. Initially, a linear model will be developed for the model baed linear control ytem deign. Thi include the kinetic friction, ometime called the vicou friction, in which the friction torque i directly proportional to the relative velocity between the moving urface. A more detailed friction model, however, i needed later, that include nonlinear friction which i ignificant in thi application. Thi i developed in ubection.3.4, by adding tatic and Coulomb friction component to the kinetic friction. Another ytem nonlinearity conit of the two hard top of the butterfly valve that limit the movement of the throttle valve plate poition. Thi i modelled uing a high gain linear feedback loop approximation explained in ubection Linear Dynamic Model The dynamic model, depicted in Figure.6, contain the linear part of the mechanical throttle model coniting of the load pring, the moment of inertia, the gear train and the kinetic friction.

44 . Modelling 43 N m m m J m Coil pring k kinetic pl Gear train k pring J pl Valve plate k kinetic N pl Figure.6: Throttle body chematic without the DC motor The gear ytem in the throttle valve conit of three part (Figure.), a tooth wheel directly mounted on the DC motor haft, a middle wheel with two different tooth wheel diameter and a tooth wheel mounted on the valve plate haft. The model, however, i implified to jut two wheel with a ingle gear ratio a hown in Figure.7. r r Figure.7: Gear ytem Here, r, are the radii of the toothed wheel,, are the tooth wheel torque and, are the tooth wheel angle of rotation. The following relationhip hold for thi gear ytem

45 . Modelling 44 r r (.4) N r N r (.5) The mechanical model for Figure.6 (diregarding the gear train moment of inertia) i baed on the following torque balance equation. where the term are defined in Table.. m i, m f, m i, pl f, pl ping, pl (.6) Table.: Torque component of mechanical model J Torque from the DC motor moment of inertial i, m m m k Kinetic friction torque for the DC motor ide f, m kinectic m J Torque from valve plate moment of inertial i, pl pl pl k Kinetic friction torque for valve plate ide f, pl kinectic pl k Spring torque (valve plate ide) ping, pl pring pl Relationhip (.5) i ued to implify the mechanical model by referring the moment of inertia and kinetic friction from the DC motor to the valve plate ide. Alo the lumped moment of inertia J (DC motor, J m, and half of the gear train) and the kinetic friction contant, k kinetic, are referred to the other ide of the gear by uing relationhip (.5), a hown in Figure.8.

46 . Modelling 45 N m J k kinetic N J N pl m k kinetic N N pl m N pl Figure.8: Tranfer the moment of inertia and friction to the other ide of the gear Figure.9 how a chematic of the plate ide of the gear ytem. pl k kinetic J k pring Figure.9: Throttle plate ide of the gear ytem Here, J i the lumped moment of inertia including the valve plate, J pl, and half of the gear train. On the valve plate ide i the torque acting on the plate haft from the gear ytem. The load on thi ubytem i the linear pring

47 . Modelling 46 torque which i a function of the valve plate poition, the kinetic friction torque and the inertial torque due to the moment of inertia. Thu d pl dpl J k kinetic pl kpring (.7) dt dt The DC motor torque i tranferred to the valve plate ide by uing (.5). Thu Then (.7) can be rewritten a N pl m (.8) Nm N d d (.9) pl pl pl m J k kinectic Nm dt dt pl kpring Combining the two part, i.e., the DC motor part which ha been tranferred to the valve plate ide (Figure.8) and the mechanical load repreented by equation (.9), yield where the lumped moment of inertia i and the lumped kinetic friction i d d (.) N pl pl m Jx k kinetic pl kpring N dt dt N Jx J J pl Nm k k N k pl kinetic kinetic kinetic Nm Figure. how the lumped ytem on the throttle plate ide..

48 . Modelling 47 N pl m N m k kinetic J x pl k pring Figure.: Repreentation of lumped ytem A tated previouly, the coil pring i pre-treed in the factory to keep the throttle open in the cae of an electrical failure. To model thi, an offet torque i added by mean of an angle offet, Initial pring, a follow: pl pl Initial pring pring k (.) It hould be noted that thi initial pring torque i only to be included in the nonlinear model including the end top..3. The Mechanim of Friction To move a mechanical part that ha cloe contact with another mechanical part require a level of force (Figure.). Thi force level i known a the mechanical friction force. Thi friction come from the interaction between the roughne on the two urface, where moother urface will decreae the friction force (Popov, ).

49 . Modelling 48 Moving urface Friction force Fixed urface Figure.: Surface interaction Through time, the throttle valve on a vehicle will be expoed to moiture and dirt that infiltrate the mechanical ytem. Thi will reult in an increae in the friction between relatively moving component. The amount of friction will change during the day due to temperature change of the mechanical component, but alo throughout the lifetime of the throttle valve due to wear. A mentioned before, the friction can caue problem for the controller and even make it limit cycle (Townend and Salibury, 987) (Sanjuan and He, 999) (Radcliffe and Southward, 99). Thi point out how important it i to imulate a control ytem deign with a friction model included, prior to implementation..3.3 Preliminary Experiment to Ae the Randomne of the Friction It would appear from the decription in ubection.3. that the tochatic friction force i a function of the diplacement between the two mechanical urface, giving friction force repeatability if the mechanical motion i repeated. Random friction force variation would, however, be produced by quai-freely moving foreign bodie (dirt). To tet the repeatability of the friction effect in the

50 . Modelling 49 throttle valve application, a et of preliminary experiment wa carried out uing a tandard throttle valve and an exiting proportional-integral (PI) poition controller with a low amplitude dither. Dither i ued a an anti-friction meaure which i explained in a later chapter. Eentially good repeatability would indicate that the final control ytem could be deigned to directly compenate for the friction force. On the other hand, bad repeatability would indicate the need for the final control ytem to exhibit robutne againt unknown friction effect. The ytem wa teted uing a ramp function a the poition reference input to the controller. A low ramp i particularly good for teting friction due to it low relative velocity, which exaggerate the effect of tatic friction, to be explained hortly. The ame experiment wa repeated firtly four time on one day to detect relatively hort term friction variation and then on two different day to detect any longer term friction variation. Thee experiment are repreented in Table.3. Table.3: Preliminary friction experiment Day Day t t pl pl t t pl pl3 t pl - pl4 t - Note: On day only two experiment were performed. Figure. how the uperimpoed reult for day.

51 . Modelling 5. Zoom B pl t pl t pl3 t pl4 t Throttle poition [rad] Zoom A Zoom A Time [ec] Zoom B Figure.: The ame day friction repeatability experiment Thi indicate very little experiment-to-experiment variation (Zoom A-B). Figure.3 how the uperimpoed reult of two experiment carried out on two eparate day.

52 . Modelling 5. pl t pl t Throttle poition [rad] Zoom A Zoom B Zoom A Time [ec] Zoom B Figure.3: Different day friction repeatability experiment Thi indicate more variation than in Figure.. Thi experiment wa repeated with a higher level of dither but thi indicated no improvement. To examine the enemble variation in error from one experimental run to the next, i.e., the randomne of the error, accumulative error for the above experiment were calculated. Thee are defined a t ea i, j t pl, i pl, j d, j,, k,, i j (.) Note that i the relative time in the ene that data from everal data experimental run, taken at different abolute time, are compared on the ame time cale tarting at. The rationale behind thi i that the larger the enemble variation between experimental run, i, and experimental run, j, the larger the mean lope of ea, i j t ha to be, which mut be poitive. Only the firt two experiment are included here, ince four experiment have ix aociated

53 . Modelling 5 accumulative error, which are conidered ufficient. Figure.4 how the reult. Accumulative error [rad] pl t t t t t t : : pl pl pl : pl pl : pl pl : pl pl : pl pl t t t t t t Time [ec] Figure.4: The experiment accumulated difference The figure reflect the obervation made by comparing Figure. with Figure.3 that the day-to-day experimental error are greater than thoe for experiment performed on the ame day. All the graph of Figure.4 indicate coniderable enemble variation from one run to another, in all ix combination. Thi confirm the randomne of the friction in the throttle valve which make it difficult to produce an accurate friction model for the purpoe of direct compenation in the controller. The concluion i that controller robutne mut be relied upon to counteract the effect of friction in conjunction with added control dither. The control dither i explained in more detail in a later chapter.

54 . Modelling The Friction Model Friction i conidered to have three different component: the kinetic, Coulomb and tatic friction (Henen, ) The kinetic friction i linear and dependent on the velocity, which i in many cae caued by the roughne of the urface. The Coulomb friction (teady friction) i contant but direction dependent. The tatic friction (tick-lip friction) i the meaure of the friction force required to jut tart the relative motion commencing at zero velocity. Thi nonlinear friction can be ubtantial compared to the other friction component. The tatic friction can, in ome cae, be dependent on the poition of the mechanical component due to the randomne of the aperitie on the urface (Henen, ), but the modelling of thi ha not been included in thi work. The claical friction model of a bi-directional mechanical ytem, uch a the throttle valve, illutrated by the velocity to torque tranfer characteritic hown in Figure.5. Figure.5: Claic friction model (Papadopoulo and Chapari, ) It comprie the three component already introduced: i) Kinetic friction torque ii) Coulomb friction torque k (.3) kinetic kinetic

55 . Modelling 54 coulomb ign k (.4) coulomb iii) Static (tick-lip) friction torque e, e,, tatic ign e, e,, (.5) where e i the externally applied torque and i the breakaway torque. The propoed tatic friction model (Papadopoulo and Chapari, ), ha the drawback of inaccuracy around zero velocity. A generic friction model wa propoed by (Haeig and Friedland, 99) (Majd and Simaan, 995) which include a more realitic continuou tranition between the breakaway torque and the um of the kinetic and Coulomb torque. The nonlinear function ued, however, i relatively complicated. To circumvent thi, a new approach i preented by the author (Pederen and Dodd, ) which i impler and impoe a lower computational demand, a follow: y (.6) total kinetic Coulomb tatic t where coulomb, kinetic are defined by equation (.3) and (.4). A tranition ection, defined by y t, i introduced to mooth the zero croing y t,, (.7) The tatic friction component ha been modelled around a rectangular hyperbola function to form a imilar hape to the one ued in the claical friction model. Thu tatic A B ign (.8)

56 . Modelling 55, B where A B Figure.6. and together with are defined in Tranition ection total friction kinetic friction coulom b friction tatic friction Figure.6: The new friction model and it component In Figure.6, at zero velocity the friction i modelled a zero to increae the model tability. The lope in the tranition ection i large to decreae the impact on the tatic friction model. To implify the parameteriation of the friction model the two parameter and are et initially to contant value of. [rad/ec] and. [Nm]. Figure.7 how a implified implementation of the new friction model in block diagram form. Thi can be implemented in Simulink by uing tandard block. In general it ha to be emphaied that to get an accurate imulation reult uing Simulink it i required to run the model in variable tep mode and with zero croing enabled.

57 . Modelling 56 y t t k kinectic kinetic ign k coulom b coulom b total A B ign tatic Figure.7: Friction model implementation An output reult from imulating the new friction model i hown in Figure.8. Thi i only an example and i not yet parameteried for the throttle valve total friction Nm rad / ec Figure.8: New friction model imulation

58 . Modelling Hard Stop The throttle plate ha a limited operation range, normal from to about 9. Thee mechanical poition contraint are called hard top, ee Figure.. Thi can be modelled by a high gain control loop applying a torque ufficient to retrain the ytem between two fixed poition., hard top khard top pl pl, max pl plmax khard top pl pl, min pl plmin plmin pl plmax (.9) Figure.9 how how equation (.9) can be implemented plm ax pl plm in max min - H ard top - k hard top Figure.9: Hard top model Thi will apply a force equal to the um for torque acting on the mechanical ytem. The poition will obviouly go beyond the minimum pl min and maximum pl max poition but by a negligible amount for k hard top ufficiently large (). A imilar trategy i ued by Mathwork in ome of their Simulink model.

59 . Modelling 58.4 Linear Sytem Model The mechanical and electrical model are combined in thi ubection to form a linear tate pace model (ubection.4.) and a tranfer function (ubection.4.) for the throttle ytem. The linear throttle model i baed on the equation of the previou ection but doe not include the hard top and the extended friction model. Firt, uing equation (.3) and (.5), t when uing equation (.) di d t N a dt L dt N Vin ia Ra ke pl pl a m pl pl pl m Jx k kinetic Nm dt dt pl kpring (.) N d d t (.) d pl Npl dpl t m kkinetic pl kpring dt J x Nm dt (.) A model of the complete ytem, hown in Figure., can be obtained uing equation (.), (.) and (.) Vin - L a R a Ia k t m N pl pl pl N - m J x pl k kinetic k e m N N pl m k pring Figure.: Linear throttle model

60 . Modelling State Space Model Since the block diagram model of Figure. comprie three interconnected firt order ubytem, when expreed in the time domain, it become a tate pace model, the general form of which i hown in Figure.. u b x dt x C i a pl pl A Figure.: State variable block diagram Figure. lead directly to the phyical tate repreentation in which the tate variable are phyical variable of the plant, i.e., x x x i (.3) a (.4) pl (.5) 3 pl The tate pace equation correponding to Figure. are a follow: x Ax b u (.6) y Cx (.7) A hown in Figure., the meaurement variable are the phyical tate variable, yi x, y x and y x and therefore (3) 3 C I, where I (3) i the unit matrix of dimenion, 3 3. Note that the meaurement equation i often hown a y Cx D u, but in thi cae, D, ince none of the tate variable can repond intantaneouly to a tep change in the control input, u t v t in. Thi i true for nearly all phyical plant. Subtituting for i, a pl and pl in equation (.) and (.) uing equation (.3), (.4), (.5) and (.) yield

61 . Modelling 6 and x v x R k x N pl in a e La Nm x k x N k x x k pl t kinetic 3 pring Jx Nm (.8) (.9) and the third tate differential equation follow from equation (.4) and (.5) a x x (.3) 3 pl The plant matrix correponding to the tate differential equation, (.8), (.9) and (.3) i R ke N a pl La La Nm k N k k J N J J t pl kinetic pring A (.3) x m x x and the input matrix i L a b. (.3) A tated above, the output matrix i C (.33) The complete tate pace model i therefore a follow

62 . Modelling 6 R ke Npl a La La Nm x x L a kt Npl k k kinetic pring x x u Jx Nm Jx J x x 3 x 3 i a x x pl pl x 3 (.34) (.35).4. Tranfer Function The plant tranfer function i needed for the deign of ome of the controller. Maon rule can be ued to find the tranfer function from the block diagram of Figure.. Maon rule tate that the tranfer function of the ignal flow graph i (Franklin et al., ) where and G Y Gi i (.36) U G i i the forward path gain, the ytem determinant given by loop gain product of non-touching loop taken two at a time i (.37) i are the forward path determinant (due to loop that do not touch the path). Since the block diagram convey the ame information a the ignal flow graph, it will be ufficient to refer to Figure.. The plant determinant may then be found a Ra k k kinetic pring ke k N t pl La Jx Jx La Jx Nm R k R k L J L J a kinetic a pring 3 a x a x (.38)

63 . Modelling 6 In thi cae, there i only one forward path gain, N G k (.39) pl t La Nm Jx and all four loop touch the forward path, giving. In thi cae equation (.36) reduce to G Y G (.4) U Subtituting for G and in equation (.4) uing equation (.38) and (.39) then yield the plant tranfer function in the form, in where the coefficient are given by a R k L J a pring / a x Y b V a a a 3 (.4) a k / J k k / L J N / N R k / L J pring x e t a x pl m a kinetic a x a R / L k / J a a kinetic x b k N L N J t pl / a m x Another tate pace model may be formed directly from the tranfer function equation (.4) with the control canonical tate repreentation, a hown in Figure. in the Laplace domain.

64 . Modelling 63 Vin b - Y a a a Figure.: Linear throttle model in control canonical form.5 Nonlinear Sytem Model In thi ubection, the linear model developed in the previou ubection i extended to include three ignificant mechanical nonlinearitie compriing a) the hard top b) the tatic friction c) the Coulomb friction While the linear model i convenient for the control ytem deign, modelling thee nonlinearitie make it poible to model throttle valve behaviour more accurately. The reulting nonlinear model i ued for predicting the performance yielded by the different control trategie before they are teted on the experimental etup. A block diagram of the complete plant model including thee nonlinearitie i hown in Figure.3.

65 H ard top. Modelling 64 Ia Vin Time delay - L a k t m N pl pl N - m J x pl R a total friction pl teady friction kinetic friction tatic friction Friction model pl k e m N N pl m Hard top k pring Initial pring Figure.3: Nonlinear throttle model The hard top model repreent the limit of the throttle plate movement impoed by the mechanical deign of the throttle. A decribed in ubection.3.5, the hard top model only introduce a retraint torque if the throttle plate poition exceed the predefined minimum and maximum poition. Thi feature i eential to keep the throttle plate within limit when the pre-treed coil pring torque i applied. In addition to the nonlinearitie, two more feature are introduced to make the model more realitic, a follow: i) The pre-treed coil pring offet torque, already introduced in ubection.3., i modelled by adding a contant angle, Initial pring, to the variable throttle angle, pl, before the pring contant, k pring. ii) A pure time delay block i introduced at the input to model the delay introduced by the pule width modulation of the H-bridge driver circuit in the armature circuit of the DC motor, and the ampling proce.

66 . Modelling 65 Certain nonlinear phenomena have not been modelled, a the experimental work i conidered ufficient to ae the effect on the control ytem performance. The firt i the cogging torque from the gear train between the DC motor output haft and the throttle plate. Thi i only predicted to have a minor effect on the ytem during tranient and no great effect on the poition control accuracy. The econd i poition dependent tatic friction. A thi will vary ignificantly from one throttle valve to the next due to manufacturing tolerance, it i unpredictable. The approach adopted i therefore to et the parameter of the poition independent friction model decribed above to yield friction level at leat equal to the maximum one expected in practice..6 Reduced Order Linear Sytem Model The third order linear throttle valve model, hown in Figure., can be reduced to a econd order model by eliminating the inductance L a. Thi i poible ince the time contant, La R a, i relatively mall. With reference to 4 Appendix A., L 8.37 H and R.795 Ohm, giving L a R a 4 a 3 [ec]. Thi i o mall compared with the required control loop ettling time, which i et to. [ec] in thi work, that the exponential mode aociated with thi electrical time contant will not have a ignificant effect. With reference to Figure., it hould be noted that the preence of the back e.m.f. loop of the model will mean that the electrical time contant will not have preciely the value, Ra L a, but it will be of the ame order. The reduced order model i then Figure. with the inductance removed, a hown in Figure.4. a

67 . Modelling 66 Vin - R a Ia k e k t m m N N N pl N - m pl m J x pl pl pl k kinetic k pring Figure.4: Second order throttle valve model Uing Maon rule to find the tranfer function give where the coefficient are given by a kpring Jx Y b Vin a a (.4) a k R k J N N k J t a e x pl m kinetic x b k N R N J t pl a m x

68 3. Model Parameteriation 67 3 Model Parameteriation The lack of dataheet for the throttle valve made it neceary to meaure and etimate all the parameter needed to form an accurate model. The approach taken i referred to a grey box parameter etimation. Thi i a cro between white box and black box parameter etimation. In white box parameter etimation, the individual component of the plant are modelled uing baic phyical principle. At the other extreme, black box parameter etimation completely ignore the internal phyical tructure of the plant and ue obervation of the meaured output repone to given input to fit a mathematical model by calculation of the contant coefficient. Grey box etimation i uually applied where white box etimation i preferred but i only applied to a ubet of the phyical component for which it i practicable. A et of ubytem i then identified that contain all the remaining component and the black box approach i applied to thee ubytem. In puruance of the white box approach, the throttle valve wa diaembled to meaure parameter uch a the diameter of the gear wheel and the DC motor voltage contant, k e. Some parameter uch a the moment of inertia and tatic friction could not be meaured directly due to lack of appropriate equipment or lack of ufficient preciion even with the bet available equipment. To extract thoe parameter value from the plant, the parameter etimation tool wa ued from the Simulink Deign Optimization toolbox from Mathwork. The parameter etimation tool ue meaurement of the real plant input and output. The tool adjut the model parameter uch, with the ame input, that the model output repone follow that of the real plant within certain tolerance. A et of meaurement from the aembled throttle valve wa collected, each with a different input voltage waveform to improve the parameter etimation.

69 3. Model Parameteriation Parameter Meaurement The meaured parameter are defined below together with the value obtained. 3.. Gear Ratio With reference to Figure., the following dimenion and ratio were determined: DC-motor tooth wheel: Ø.5 mm Gear link tooth wheel (): Ø 4.5 mm Gear link tooth wheel (): Ø 6 mm Throttle plate tooth wheel: Ø 5 mm Ratio (DC-motor / gear link ): 4.5/.5 ~ 3.6 Ratio (gear link / throttle plate): 5/6 ~ 3. Total gear ratio between the DC-motor and the throttle plate N / N :.5 pl m 3.. DC Motor Voltage Contant The implet approach to determine the contant, k e [V/(rad/ec)], i to drive the DC motor at a contant peed (by an external electric motor), and meaure the output voltage of the unloaded motor. Unfortunately thi wa not poible with the equipment available and a different approach had to be adopted. The DC motor i diconnected from the gear ytem and a DC voltage upply i connected to it terminal. Before tarting the tet a reflection pad i attached to the DC motor haft to meaure it peed by a light reflection peed meter. The tet i done with two different voltage,,.5 and 3 [V] and the peed recorded: (66.46, 89.4, 4.6) [rad/ec]. The value of k e i found to be to. [V/(rad/ec)], on the aumption that the meaurement lie on a traight line and the DC motor current i mall. Further the kinetic friction for the DC

70 3. Model Parameteriation 69 motor i aumed to be mall. Thi value i ued in the initiation of the parameter etimation tool DC Motor Reitance A DC voltage i applied to the motor terminal and the current i meaured. Then the reitance R i calculated ( R 3. [Ohm]) by uing Ohm law. Thi a tet ha been done at zero peed to eliminate the effect of the back e.m.f. The value obtained wa ued in the initiation of the parameter etimation tool. a 3..4 DC motor inductance The inductance i meaured uing by a digital LCR meter (Wheattone bridge). The meaured inductance (Table 3.) depend on the frequency ued by LCR meter due to the eddy current loe in the ferromagnetic inductor core material. The LCR meter voltage i et to [V] peak-to-peak. Table 3. Meaured DC motor inductance Meaurement frequency [khz] Meaured inductance [mh] The PWM witch frequency ued in the experiment for the parameteriation wa khz and therefore the value.7 [mh] wa choen. The value found i ued a initial value for the parameter etimation tool DC Motor Torque Contant

71 3. Model Parameteriation 7 The DC motor torque contant k t, can be found by a load tet where the DC motor i run with contant load and it current i meaured. It wa not poible to do thi tet due to the lack of equipment. Intead it wa aumed to be equal to the DC motor voltage contant (Mohan et al., 995)(Page ), i.e., k e already etimated in ubection DC motor moment of inertia and the kinetic friction Due to limitation in the meaurement etup it i not poible to meaure the DC motor peed when diconnected from the throttle valve ytem. The DC motor current i the only available meaurement which can be logged. It i neceary to find a way to meaure/calculate the DC motor moment of inertia and the kinetic friction which only depend on the DC motor current. A implified verion of the DC motor model preented in ection.4 i hown in Figure 3. in which the armature inductance ha been ignored on the bai that the electrical time contant, La R a, i much maller than the mechanical time contant, J / k. m kinetic Vin - R a Ia k e J k m kinetic m k e Figure 3.: Simplified DC motor model Here, R a i the armature reitance, k e i the motor contant, J m i the armature moment of inertia and k kinetic the DC motor kinetic friction contant. The parameter, k t, R a and The tranfer function from k e were found from the previou ubection. V to in I i a

72 3. Model Parameteriation 7 I R V a in ke R J k a a m kinetic Jm k kinetic Ra Ra kkinetic ke / Ra J k k R m kinetic m / a (3.) The DC motor kinetic friction, k kinetic, i aumed to be the predominant load in the teady tate. Uing equation (3.) in teady tate ( ) and olve for k kinetic kkinetic ke Ia Ra Ra Ia V in k kinetic k k / R V / R I kinetic e a in a a (3.) where I a and V in are the DC motor current and input voltage in the teady I V, without tate. The tranient repone of the fictional ubytem, a in zero i, will be in X V J k k R m kinetic e / a, (3.3) / x t V e tt in, (3.4) where T Jm / kkinetic ke / R a with unit [ec]. Uing the above, kkinetic Jm d ia t x t x t R k k / R R k k / R dt a kinetic e a a kinetic e a (3.5) Uing equation (3.5) and (3.4)

73 3. Model Parameteriation 7 At t T V in kkinetic t T J ia t e e kkinetic ke / Ra Ra RaT / m t / T V in kkinetic t / T kkinetic ke / R a t / T e e kkinetic ke / Ra Ra Ra (3.6) V in kkinetic kkinetic ke / R a ia T e e (3.7) kkinetic ke / Ra Ra Ra The calculated value, i a T, i ued to log the point in time where the meaured DC motor current, from the experiment, pae the value i a T. The logged point in time i then ued to calculate the DC motor moment of inertia by T Jm / kkinetic ke / R a J T k k / R m kinetic e a (3.8) The DC motor i diconnected from the gear ytem and a voltage tep, V, i applied to it input terminal while the input current i in a t i meaured. Figure 3. how the meaured current, the calculated teady tate current i t and the value of i a a T found by equation (3.7). The ripple on the current i caued by the power electronic driving the DC motor.

74 3. Model Parameteriation Meaured current ia ia t T.4 Current [A].3. Steady tate current T=.84 Time [ec] Figure 3.: Meaured current and the value i T a Firt the teady tate current and the voltage i ued to find the kinetic friction, 5 k kinetic 5.6 [ / ( / ec)] Nm rad from equation (3.). The value for i a T i found from equation (3.7) and the point in time where the meaured current pae ia T i found by viual inpection in Figure 3., for T.84 [ec]. Thi time i inerted in equation (3.8) from which the DC motor moment of inertia i calculated J m [ Kg m ]. The calculated value of value of k kinetic and J m were then ued to find a calculated i t by uing equation (3.6), and a imulated value of i a the ytem in Figure 3.. The reult i hown Figure 3.3. a t by uing

75 3. Model Parameteriation Meaured Calculated Simulated.5 DC motor current [A] Time [ec] Figure 3.3: DC motor friction and moment of inertia validation Thi confirm the accuracy of the method. Hence thee value are ued a the initial value for the parameter etimation tool Throttle Valve Sytem Moment of Inertia The throttle valve mechanical ytem contain different component uch a the plate and gear wheel, which all contribute to the moment of inertia. Etimation of thi part of the mechanical ytem i impracticable with the procedure of ubection 3..6, due to the ytem hard top. Intead, approximate calculation of the moment of inertia contribution of the plate, haft and gear wheel were carried out on the bai of identifying ma element and their radii of gyration from their centre of rotation. The net moment of inertia contribution i ued in the initiation of the parameter etimation tool The Coil Spring

76 3. Model Parameteriation 75 An etimate of the coil pring contant ( k pring ) and the coil pring initial torque ( k ) i done by uing a hanging cale [kg] a a torque meter and Initial pring pring the throttle poition enor (Figure 3.4). The DC motor i mechanically diconnected from the gear train to minimie the tatic friction and cogging torque. Two piece of heet metal with hole in the end are clamped onto the valve plate. Thi i ued a an extenion for the hanging cale to be attached. A piece of tring i fatened to the extenion and attached to the hanging cale in the other end. The hanging cale i ued to meaure the force needed to move the throttle plate away from the initial poition and then from one poition to another. The poition i meaured by attaching a voltage meter to the poition enor output. Gear train Coil pring k pring String Valve plate r Screw Hanging cale pl Hard top Metal clamp 5V V N pl Throttle poition feedback Figure 3.4: Meaure of the coil pring torque The initial torque needed to move the plate r F r m a (3.9) Moving the plate to a poition (= 7 ) before the metal clamp extenion touche the throttle body

77 3. Model Parameteriation 76 r F r m g (3.) where F i force in the tring and g the tandard gravity. The pring contant kpring. Nm / Rad (3.) where and 7 / 8 The coil pring initial torque in unit of Rad Initial pring.5rad (3.) k pring Thee value are ued a the initial value for the parameter etimation tool Hard Stop The mechanical propertie of the hard top were not regarded a being in cope of thi project. It wa conidered ufficient to et an initial value of 5 for the gain of the poition retraint loop in the imulation repreenting the hard top. Thi wa ued in the initiation of the parameter etimation tool. The maximum throttle poition wa meaured to be 9. The minimum ha been et at. to enable linear operation in the imulation around, without actuating the hard top retraint. 3.. Throttle Valve Sytem Kinetic Friction Etimation of the kinetic friction of the mechanical ytem driven by the DC motor uing the method preented in ubection 3..6 i impracticable due to the ytem hard top. The parameter etimation tool wa therefore entirely relied upon for etimation of thi kinetic friction component. 3.. Static and Coulomb Friction

78 3. Model Parameteriation 77 For the reaon explained in ubection 3.., the tatic and Coulomb friction wa etimated uing the parameter etimation tool, noting that thi i not retricted to linear ytem. 3. Parameter Etimation 3.. Introduction The Simulink Deign Optimization toolbox from Mathwork wa ued to etimate the parameter offline and improve the accuracy of the calculated and meaured parameter. Firt the meaured input and output data from tet done on the real plant are imported into the tool. The meaured input data i automatically applied to a plant model, running in Simulink, by the tool. Before the firt run an initial parameter et i loaded, which i defined by the uer. After each imulation run the tool adjut the model parameter to minimie the error between the meaured real plant output and the model output a illutrated in Figure 3.5. Thi proce i repeated a number of time until the error i lower than a pecified value. The parameter etimation tool can run the model with variou et of meaured data in order to include variation from different running condition.

79 H ard top 3. Model Parameteriation 78 Parameter etimation tool Meaurement Data et # Vin t Time delay - L a Throttle valve model k t N pl N - m J x Parameter etimation algorithm R a total friction kinetic friction teady friction tatic friction Friction model Meaurement Data et # Meaurement Data et # pl t ia t error error k e N N pl m pl Hard top k pring Initial pring Parameter etimate Figure 3.5: Parameter etimation uing the toolbox from Mathwork To maximie the accuracy, the etimation wa plit into two part:. The DC motor model parameter etimated uing meaurement of the DC motor current and the correponding armature voltage. The reulting parameter etimate were ued in the overall throttle valve etimation of part two.. The throttle valve model parameter etimated with variou et of meaurement elected to make the etimation more precie and fit variou running condition. 3.. DC Motor Model Parameter The DC motor parameter were etimated uing three et of data, a hown in Figure 3.6. Thee were choen to get a good etimate of k e, L a, R a and the time delay. To get a good etimate of the parameter k e (and k t ) the DC motor peed ha to be greater than zero, while an input voltage tep will excite the

80 3. Model Parameteriation 79 dynamic of the current giving a better etimate of L a. The initial parameter et ued for thi i pecified in ubection 3.. a) b) c) Motor voltage [V] Motor current [A] Throttle poition [rad] time [ec] time [ec] time [ec].5.5 Figure 3.6: Meaurement data ued for the DC motor model parameter etimation The data logged DC motor armature voltage and velocity were ued a input ignal to the DC motor model while the DC motor current wa ued a an output reference, a hown in Figure 3.7. The DC motor velocity wa derived by differentiating the low pa filtered throttle poition ignal. The three data et were repeatedly et to run by the parameter etimation tool to get a good etimate of k e, L a, R a and the time delay.

81 3. Model Parameteriation 8 Meaurement Data et # Vin t Time delay DC motor model - L a iâ t Meaurement Data et # t m k e R a kˆe tˆd Lˆa Rˆa Parameter etimation algorithm ia - error t Meaurement Data et # Parameter etimation tool Figure 3.7: Etimation of the DC motor parameter The reult of thi parameter fit wa ued to initiate the etimation of all the throttle valve model parameter Throttle Valve Model Parameter Etimating all the throttle valve parameter i accomplihed uing 7 different data et with variou input voltage waveform. A ubet of the data et ued for thi parameter etimation can be een in Figure 3.8. The motor voltage wa ued a input ignal to the throttle valve model while the throttle valve poition and DC motor current were ued a output reference. The parameter etimation tool allow weighting factor to be impoed on all the data ued a reference. Thi feature wa ued to down-cale the impact of the DC motor current on the etimation reult. Thi wa done to obtain a better fit with the throttle valve poition, ince thi i the controlled output and therefore more important.

82 3. Model Parameteriation 8 a) b) c) d) Motor voltage [V] Motor current [A] Throttle poition [rad] time [ec] time [ec] time [ec] time [ec] Figure 3.8: A ubet of the data et ued for the throttle valve model parameter etimation The nonlinear model i hown in Figure 3.9 together with the final parameter etimate. Vin Time delay =.6 [m ec] -4 =8.37 [H] - L a =.795 [ohm] R a k e k t m =.57 [Nm/A] m Figure 3.9: Nonlinear throttle valve model with parameter value N N N pl N - m k kinetic =. [Nm ec /rad] k coulomb =.836 [Nm] ω =.[rad/ec] Γ =.539 [Nm ec /rad] ω =.47[rad/ec] Γ =.[Nm ec /rad] =.57 [V ec/rad] =.5 [-] -4 =8.5 [kgm ] pl m J x Hard top k pring pl Ia pl total friction pl teady friction kinetic friction tatic friction Friction model H ard top =.58 [Nm/rad] pl Initial pring = 4.47 [rad] k hard top =[Nm/rad]

83 3. Model Parameteriation 8 A full lit of the parameter ued for the imulation work can be found in Appendix. 3.3 Model Verification in the Time Domain The validation i done by a applying the ame time varying DC motor voltage to both the non-linear plant model and the real plant. The validation were done with no poition controller applied (open loop). Figure 3. how four of the mot ignificant validation experiment where the blue ignal i the experimental data and the green ignal the data from the non-linear model. In the bottom of Figure 3. the difference between the real plant and imulated plant i hown. a) b) c) d) Motor voltage [V] Motor current [A] Throttle poition [rad] Experiment - Simulated [rad] time [ec] time [ec] time [ec] time [ec].. -.

84 3. Model Parameteriation 83 Figure 3.: Comparion between the non-linear plant model and the plant (Blue dahed: Experimental data. Green: Simulated data) The four experiment hown are deigned to validate the friction model parameter. In Figure 3. b) the voltage i ramped up lowly until the throttle valve cloe and then ramped down again to the point jut before it will open. The tatic friction applie a force great enough to keep the throttle valve cloed. The experiment hown in Figure 3. c) and d) are deigned to make the throttle valve poition operate over a limited range, again with focu on the friction. 3.4 Model Validation in the Frequency Domain A known way of ytem identification i to create a frequency repone model of the plant (Ljung, 998). Thi can be ued to validate the parameter found by creating a frequency repone model for the real plant and the model, and then do a comparion between the two. Thi ha been carried out on thi reearch programme a an additional validation tet, and to find the throttle valve ytem bandwidth. The bandwidth i needed to enable a ufficiently high throttle poition ampling frequency to be et. To etimate a frequency repone model for a plant the input and output ignal are ampled and proceed uing fat Fourier tranform. Thi information i then ued in the Sytem Identification Toolbox from Mathwork to create a frequency repone model. The quality of the reult obtained from the Sytem Identification Toolbox depend on a number of factor uch a the form of the input ignal, the ampling period of the data acquiition and the ignal-to-noie ratio. The

85 3. Model Parameteriation 84 experiment on the real plant, to capture the input and output ignal, can be intruive or nonintruive. In the intruive experiment the input ignal, from the controller, i replaced by a tep, impule or inuoidal ignal. In the nonintruive experiment, a mall ignal i added to the controller output before entering the input on the real plant. Common type of ignal ued are inuoidal one with different frequencie panning the intended bandwidth of the control ytem to be ultimately deigned. Another type i the peudo random binary equence (PRBS) hown in Figure 3. (Pintelon and Schouken, 4) with a Fourier pectrum panning thi bandwidth. Changing the type of ignal ued for the ytem identification can reult in lightly different frequency repone model. It i therefore a good idea to imulate the method on a known plant model of imilar form to the one expected from the identification proce uing different input ignal type and elect the one that produce the cloet approach to the known plant model..5 PRBS ignal [V] Time [ec] Figure 3.: An example of a peudo random binary equence

86 3. Model Parameteriation 85 The PRBS i uually a good choice and thi i elected for the throttle ytem identification. A illutrated in Figure 3. the PRBS i a pule train ignal with a variable mark-pace ratio which yield a relatively flat frequency pectrum over it bandwidth. Since it i generated by digital regiter fed by a clocked combinatorial Boolean function of the regiter tate, producing a maximal but finite length equence, the ignal repeat with a fixed period and it i thi property that give rie to the term peudo-random rather than jut random. Thi effect, however, doe not eriouly colour the power pectrum and therefore doe not impair the ytem identification. The PRBS ignal i added to the output from the controller and it amplitude level hould be low compared to the controller output. Thi will minimie the PRBS ignal impact on the control loop. The frequency band of the PRBS can be choen by electing the PRBS update rate (ampling frequency) and it equence length. To ummarie, the factor that are important when uing the PRBS are: - The amplitude of the ignal - The update rate - The equence length - The ampling frequency of the output ignal from the plant - The number of the time the PRBS equence i repeated - The power electronic witch frequency (H-Bridge) For the PRBS experiment on the throttle valve ytem the following parameter are et - Data ampling frequency = 5 Hz - PRBS update rate: T [ec] 5 - PRBS equence length: N 3 eq

87 3. Model Parameteriation 86 o Minimum injected frequency: f min N T eq.49[ Hz] o Maximum injected frequency: f max 5[ Hz] T - PRBS cycle length = econd - PRBS amplitude = /-.5 [V] - Experiment length = 4 econd ( repetition) During the experiment, the throttle poition i kept at around 45 (open loop) which i about midway between the end top to allow the maximum amplitude of movement for nominally linear operation. From the experimental input and output data a tranfer function i generated uing the Sytem Identification Toolbox from Mathwork. In fact, the ame procedure i ued to generate a tranfer function from the nonlinear throttle valve model. Although, trictly, the tranfer function i a notion applying only to linear ytem, the reult obtained with a nonlinear plant i, arguably, imilar to that obtained analytically by the method of lineariation about the operating point. Thi i certainly true for continuou nonlinearitie but the tick lip friction, which i ignificant in the throttle valve application, i dicontinuou. Depite thi there i no other known way to obtain a better tranfer function model for control ytem deign. The retriction of continuou nonlinearitie doe mean that the tranfer function model cannot be heavily relied upon. Thi i only being ued for the initial controller deign with the poibility of having to make controller adjutment following the firt experimental trial. The bode plot of the real plant and the identified plant model are plotted together in Figure 3.. The plot are imilar from to about.5 [rad/ec] but a difference between them of around -5 [db] i evident up to [rad/ec] and increae to about -8 [db] at the upper limit of frequency. The throttle ytem bandwidth i omewhat le than [Hz].

88 3. Model Parameteriation 87 Magnitude [db] Real plant Model plant Phae [ ] Frequency [Rad/ec] Figure 3.: Bode plot Comparion between the non-linear plant model and the plant

89 4. Control Technique and Performance Aement 88 4 Control Technique and Performance Aement 4. Introduction In thi chapter, the different control technique to be invetigated are explained. The controller are then deigned. Thi i followed by imulation and correponding experimental reult that for a bai for the performance aement and the comparion of the following chapter. The correct procedure in the etablihment of any control ytem deign i to firt conider the implet poible controller and identify any hortcoming, thereby etablihing a need, if neceary, to introduce pecific feature. In thi way, unneceary complexity i avoided. In the cae of the throttle valve application under tudy, thi led everal year ago to the PID controller, which ha been employed in many ytem to thi date. Modern digital proceor, however, permit more ophitication in the controller without increaing the complexity and cot of the hardware and therefore reducing it reliability. Under thee circumtance it i very much worth conidering more ophiticated control technique that might offer advantage in performance improvement or eae of commiioning. Thi i the motivation of the reearch programme. In ubection 4., the proce leading to the etablihment of the PID controller i briefly reviewed, tarting with the imple proportional controller, o that the ubequent work of thi reearch programme i et in context.

90 4. Control Technique and Performance Aement The Earlier Development Leading to the PID Controller The implet controller that can be conidered i the proportional controller a hown applied to the linear throttle valve model defined by tranfer function (.4). Thi i hown in Figure 4. Yr - k U b Y 3 a a a Throttle valve Figure 4.: Proportional controller applied to the throttle valve where output and Y i the controller reference input, r valve DC motor. Y i the throttle valve poition U i the controller output which i a voltage driving the throttle Uing the parameter value found in Chapter 3, the plant ha three open loop pole at,,3 33, 35.,, and Figure 4. how the root locu. Im Re Figure 4.: Root locu of the ytem with unity gain feedback control

91 4. Control Technique and Performance Aement 9 It i evident that there i an upper limit on the gain, K, beyond which intability reult, indicated by the point where the complex conjugate loci cro the imaginary axi of the -plane. At lower gain for which the ytem i table, the damping ratio would be too low due to the relatively mall negative real part of the complex conjugate pole. At the critical value of K at the break-away point, the two dominant pole are ituated at,3 8, 8 yielding a ettling time (5% criterion) given (Dodd, 8) by the ettling time formula, T c.5 n T (4.) where n and Tc / 8, yielding T.5 [ec]. Any attempt to reduce thi ettling time by increaing K would caue overhooting. Alo, the ytem i ubject to a teady tate error due to the plant being of type, i.e., containing no pure integrator, due to the preence of the retention pring decribed in ubection.. Thi teady tate error i evident in Figure y r t y t Throttle poition [rad] Time [ec] Figure 4.3: Step repone with proportional controller adjuted for critical damping

92 4. Control Technique and Performance Aement 9 Thi teady tate error will impair the performance of an engine management ytem, ince it ha to operate in teady tate for a ubtantial proportion of it lifetime. Thi neceitate the introduction of an integral term in the controller but, alone, render difficult the tak of obtaining an acceptable ettling time and minimal overhoot of the tep repone. Hence the derivative term i called for that enable the overhooting to be reduced when increaing the proportional gain in an attempt to reduce the ettling time. The reult i the PID controller. Before moving on, it hould be mentioned that the PID controller applied to a third order plant model yield a cloed loop ytem of fourth order. Since there are only three adjutable controller parameter on the PID controller, only three of the four cloed loop pole can be placed a deired. Thi retriction i removed in ome of the controller conidered in thi reearch programme, enabling deign by the method of pole placement a decribed in ubection Methodology 4.3. Simulation Detail Throughout the reearch the control trategie are imulated, uing Matlab and Simulink, with both the linear and nonlinear throttle valve model from Chapter. The control trategie and throttle valve model are implemented uing block diagram a in Simulink (referred a Simulink model). The Simulink model are hown in the continuou domain uing the Laplace operator to make the invetigation work eaier. Mot of the imulation are run in variable tep mode, which can make the execution fater and enure relatively high preciion. The liding mode control, however, in it baic form, incorporate a dicontinuou element that can caue variable tep algorithm to become tuck in an effort to

93 4. Control Technique and Performance Aement 9 maintain preciion by reducing the tep-length. In thee cae a fixed tep algorithm i employed which, in any cae i needed for the real time implementation uing the rapid prototype hardware from company dspace ( It i important to note that the zero croing detection i enabled in the nonlinear friction model, wherever poible, to maximie the accuracy. All the control trategie to be implemented on dspace are firt validated uing Simulink to avoid iue that might damage the throttle valve Experimental Setup The control trategie are teted by uing a throttle valve ytem connected to the MicroAutoBox (MAXB) from dspace through a power amplifier, hown in Figure 4.4. A control trategy block diagram (model) i formed by uing Simulink. The model i compiled and downloaded into the MAXB. The MAXB i a rapid prototype ytem which conit of a powerful main proceor and ome peripheral hardware including ADC, PWM, DAC and digital input/output (I/O).

94 4. Control Technique and Performance Aement 93 Control Dek & Simulink Laptop - Throttle poition demand - Throttle poition feedback - Throttle velocity - DC motor current Other ignal and variable volt PWM ignal volt Amplified PWM output to the DC motor MAXB Power amplifier (H-bridge) DC motor current feedback Throttle poition feedback Motor direction ignal Throttle valve ytem DC motor current meaurement enor Figure 4.4: Experimental etup for teting the throttle valve poition control trategie The downloaded code run in real time, according to the pecified ampling frequency, on the main proceor in the MAXB. MAXB upport the implementation of Laplace operator but only with a fixed ampling time. The MAXB will meaure the pecified input like the ADC, run the control trategy and update the output at each ample point, hown in Figure 4.5. y r t Controller DC motor voltage demand u t u dspace block PWM output Digital output dspace block Plant dspace block ADC ADC dspace block Ia t pl t Motor rotation direction demand Figure 4.5: PI controller implemented in dspace (implified diagram)

95 4. Control Technique and Performance Aement 94 The meaured input to the MAXB are the DC motor current and the throttle plate angular poition. The plate angular velocity i calculated from the meaured poition. The output are a DC motor voltage demand and the motor rotational direction. The DC motor voltage i in the form of a pule width modulation (PWM) ignal. The motor rotational direction i a digital logic ignal whoe tate determine the ign of the armature voltage applied to the DC motor. The MAXB output are connected to the power amplifier (H-Bridge) which i decribed in Appendix A.4. The H-Bridge output i connected to the DC motor on the throttle ytem. The meaured DC motor current and throttle plate poition are fed back to the MAXB. When the model i running in the MAXB, all the variable can be monitored, changed and logged with the ue of a program running on the attached computer called ControlDek from dspace. Figure 4.6 how the experimental hardware ued for the invetigation. Two power upplie are ued to make it poible to adjut the H-Bridge voltage level.

96 4. Control Technique and Performance Aement 95 dspace ytem H-bridge Power upply H-bridge & Current enor Throttle valve dspace Power upply Figure 4.6: Experimental hardware Simulation and Experimental validation Three reference input function have been choen to tet the control performance, a hown in Figure 4.7.

97 4. Control Technique and Performance Aement 96.5 Ramp.5 Deired throttle poition demand [rad] Pule Drive cycle.5 Slow ramp: Pule: Figure 4.7: Throttle poition demand waveform for control tet Drive cycle: 5 5 Time [ec] Ued to tet the controller capabilitie of handling tatic friction. Thi will tet the dynamic of the ytem uch a under- overhoot and the ettling time. Thi i a ample of the poition demand data hown in Figure.5 taken from an operating throttle valve control ytem, which enable the ability of the control ytem to follow fat and low poition demand to be aeed. The normal operational range of the throttle valve i between and.5 [rad]. The reference input have been caled o a to avoid the mechanical ytem hard top. Thi allow the ytem to operate in the continuou mode in which the control ytem under invetigation are intended to operate.

98 4. Control Technique and Performance Aement Senitivity and Robutne Aement Senitivity The enitivity i defined here a a meaure of how much the tranient and teady tate repone differ from thoe pecified in the preence of plant modelling uncertaintie and external diturbance (Dodd, 3). Converely, robutne i defined a the ability of a control ytem to maintain it pecified tranient and teady tate performance depite plant modelling uncertaintie and external diturbance. So an additional performance aim i to minimie the enitivity, which i equivalent to maximiing the robutne. Any linear control tructure can be tranformed into the tandard form hown in Figure 4.8, where function while D i an external diturbance, K and Yr - H are controller tranfer function. Controller E K H U D - V G i the plant tranfer Plant G Y Note that the error i noted Figure 4.8: Standard linear control ytem tructure from the error, E Y Y tranfer function, r E (not meaning a derivative!) to ditinguih it, the difference being due to the feedback H. The cloed loop tranfer function i then G cl Y K G (4.) Y K H G r Then the enitivity of the cloed loop tranfer function with repect to the plant tranfer function i defined a

99 4. Control Technique and Performance Aement 98 where S C p / cl / Gcl G lim (4.3) G G K H G G G G i the cloed loop tranfer function and G G G cl i the aumed plant tranfer function, i.e., the tranfer function model of the plant. The robutne i then C C K H G Rp Sp. (4.4) K H G The Matlab Simulink Control Sytem Deign toolbox i ued throughout thi reearch to extract the enitivity for variou controller by realiing, with reference to Figure 4.8, that K H G V D Yr S C p, (4.5) Thi i a meaure of what proportion of the diturbance get through to the plant while the controller i trying to cancel it. It i therefore a direct meaure of enitivity with repect to external diturbance. It i alo known that deviation of the plant parameter from the nominal one in the plant model ued for the control ytem deign may be repreented by equivalent (but not phyically preent) external diturbance applied to the model. Thi further explain the ue of tranfer function (3.5) to repreent the enitivity derived on the bai of plant parameter variation through (4.3). The tranfer function, E Y r, i alo the enitivity tranfer function given by equation (4.5). The reader i warned, however, that taking E Yr for any linear control ytem, where E Y Y r i the control error, will yield the required enitivity tranfer function only if H after converting the control ytem block diagram to the form of Figure 4.8. If H, then C E and E Y S E r. p

100 4. Control Technique and Performance Aement 99 The control error tranfer function, G E Y e r, ha a meaning eparate from the enitivity, which i imply an indication of how the control error behave. In the frequency domain, it would be poible to plot what could be termed the error frequency repone, G db G S db C p C S. In cae where H log p e log C of GedB could lie ignificantly above that of p e, alongide, it i poible that the plot S db but thi would not necearily indicate poor performance a it could reult from a nonoverhooting monotonic tep repone, while an overhooting tep repone, which i, arguably, le deirable, could yield a lower lying GedB plot. Thi ha been confirmed from reult obtained from the throttle valve control ytem but ince the value of GedB in aeing the performance of a control ytem i doubtful, only the enitivity reult are preented. To ummarie, the enitivity of a control ytem may be aeed by creating a Bode magnitude plot with input, D, and output, V, uing a Simulink block diagram with a umming junction inerted at the plant input a hown in Figure 4.8. The reult from the toolbox i a Bode plot where the magnitude i in Decibel C C SpdB log Sp j (4.6) To illutrate the enitivity an example i hown of the linear throttle valve model with a proportional controller. The control tructure hown in Figure 4.8 i ued for thi where H, K.9, D and G i given by the linear throttle valve model a tranfer function (.4). Figure 4.9 how the enitivity plot for thi uing the Simulink toolbox.

101 4. Control Technique and Performance Aement 5 Senitivity [db] Frequency [rad/ec] Figure 4.9: Senitivity of the linear throttle valve control loop with a proportional controller To check thi reult, equation (4.3) i applied with the throttle valve model (.4) and the value given in Appendix, a follow. S C p 3 3 K H G V a a a D a a a K b At low frequencie. Hence (4.7) C a a a a Sp lim 5.95[ db] a a a K b a K b At high frequencie 3 3 C a a a Sp lim [ db] a a a K b Thee calculation are in agreement. 3 3 (4.8) (4.9)

102 4. Control Technique and Performance Aement Parameter Variation uing Monte Carlo Analyi Variation in the true plant parameter with repect to thoe ued for a controller deign can caue coniderable departure from the pecified performance or even cloed loop intability. Thee parametric uncertaintie can come from a number of ource uch a product tolerance and parameter etimation. For the throttle valve ued in thi reearch there are even phyical parameter ued in the model upon which the controller deign are baed, which are R a, L a, k e, t k, k kinetic, k pring and J x, defined in Chapter. To tet a ytem enitivity with repect to jut two parameter for all poible parameter variation in (/-5%) tep up to (/-%) extreme would entail 5 combination and a many tet, a illutrated in Figure 4.. Variation of parameter # -% -5% % 5% % -% -5% % 5% % Variation of parameter # Figure 4.: A D matrix for parameter variation tet The black dot indicate the different combination of parameter variation. Intead, a Monte Carlo analyi implement a normal tatitical ditribution of each parameter, which i more realitic, with the maximum limit replaced by 3 value. In thi way, more tet are carried out for parameter value that are more likely to occur, thereby achieving a maller number of tet run than would

103 4. Control Technique and Performance Aement be needed with a flat tatitical ditribution, which i implied by the imple cheme of Figure 4.. The concentration of parameter combination about the nominal value given by Monte-Carlo analyi i illutrated in Figure Variation of parameter # [%] Variation of parameter # [%] Figure 4.: Illutration of D parametric variation for Monte-Carlo analyi (Standard deviation = 3 %, Mean value = ) A normally ditributed random et of number (pecific number of obervation) with a pecific tandard deviation and a mean value of zero are generated for each parameter (Figure 4.). Seven different et are generated, one for each parameter teted, with a et length =.

104 4. Control Technique and Performance Aement 3 Normally ditributed peudorandom number R a L a Number of obervation k t k e k kinetic k pring J x Ditribution interval [%] Figure 4.: Frequency ditribution of parameter ued for the Monte Carlo analyi (Standard deviation = 3 %, Mean value = ) All the random number data ued for the parameter variation tet wa generated in the early tate of the reearch. The ame random number et have been ued for each imulation invetigation of robutne to inure that fare comparion are made. Figure 4.3 how the number of ample for each variation interval a a function of the tandard deviation in percent, in thi cae for %. R a. The tandard deviation range generated i from % to 3% in tep of

105 4. Control Technique and Performance Aement 4 Number of ample Standard deviation [%] Variation [%] Figure 4.3: Parameter ditribution ued for the Monte Carlo ( R for.:. a ) The parameter variation validation i done by repeating the imulated cloed loop repone of a control ytem, changing the imulated plant parameter from one run to the next while keeping the model parameter fixed at the nominal value, i.e., the controller gain/parameter are fixed. The output repone are aeed againt predefined boundarie in Figure 4.4, which cannot be exceeded. Throttle poition [rad].. /- 5%.5 Time [ec] Figure 4.4: Throttle poition operation envelope

106 4. Control Technique and Performance Aement 5 If a imulation i about to penetrate a boundary it i topped and the level of the parameter change i logged. The boundarie in Figure 4.4 are deigned with room for a controller overhoot of % and a teady tate error of maximum /- 5%. The aement procedure, common for each controller, i a follow. Each controller i deigned uing the nominal linear plant model parameter and a pecified ettling time of. [ec]. An initial cloed loop imulation i carried out with a tep reference poition applied at t [ec]. The plant model ued i the nonlinear model, from ubection.5, with nominal parameter. The cloed loop output repone, yt, i teted againt the defined boundarie of Figure 4.4.If the output repone croe a boundary at thi tage the imulation i topped and the controller i deemed unuitable a it would be unable to cope with the nonlinear friction effect even with the imulated linear plant parameter equal to the nominal value. Having paed thi initial tet, a et of normally ditributed value i loaded for each of the even plant parameter referred to in Figure 4. and prepared for the Monte-Carlo analyi imulation, commencing with % tandard deviation. Then imulation are run, the even plant parameter being loaded in equence from the etablihed et at the beginning of each run. Then the whole et of imulation i repeated for tandard deviation increae by % until a boundary of Figure 4.4 i croed, whereupon the imulation equence for the controller concerned i terminated and the tandard deviation noted a a performance meaure, the higher the better Failure Analyi A decribed in ubection.., in a typical DPF regeneration drive cycle the throttle poition could be 95-97% fully cloed a hown in Figure.5. Thi will make the engine ytem very enitive to a udden change in the throttle valve

107 4. Control Technique and Performance Aement 6 poition. In the wort cae, the engine could tall due to air tarvation. Thi udden change in poition could be caued by breakage of the pre-loading pring with the conequent tep change in the load torque to zero. In thi reearch, the control trategie are teted for the impact of uch a pring failure. Thi i only imulated due to the difficulty of implementing a pring collape on the real throttle valve ytem. In the imulation the deired throttle poition i et to.47 [rad] (~94% maximum) and at t [ec] the pring torque i removed. 4.4 Common Feature 4.4. Introduction The purpoe of thi ubection i to preent the common deign feature that the different controller under invetigation hare, to minimie repetition in the ubection dealing with the pecific controller Pole Placement Deign uing the Settling Time Formula Multiple Pole Placement The ettling time T, Figure 4.5, i defined a the time it take from applying an ideal intantaneou tep input to the time at which the ytem output ha entered and remained within a pecified range, in thi cae it i choen to /- 5%.

108 4. Control Technique and Performance Aement 7 /- 5% Steady tate value Deired cloed loop tep repone [-] T Time [ec] Figure 4.5: Settling time definition The Dodd 5% ettling time formula (Dodd, 8), T.5 n T, (4.) will be ued throughout to deign each controller to meet a ettling time pecification with zero overhoot. The coincident cloed loop pole location i then where Yr,,, n c.5 n (4.) T T c n.5 n Y T C TS Y.5 n r T C TS n (4.) i the reference input, T c i the cloed-loop time contant and n i order of the control ytem.

109 4. Control Technique and Performance Aement Robut Pole Placement An example of a linear high gain robut control ytem i ued to introduce the robut pole placement method (Dodd, 3). The plant i econd order, a hown in Figure 4.6, with a gain of c and two pole at a control gain, K and repone. The input. The two,, T c, are calculated to yield the deired cloed loop D i an arbitrary external diturbance ignal. Y r - K Tc D - c Y a Plant Figure 4.6: Linear high gain robut control ytem The cloed loop tranfer function relationhip of Figure 4.6 i Then Y Thi indicate ideal robutne a K c c Yr D a a K c T c a Yr D K a T K c c (4.3) lim Y Yr (4.4) K T c Y i completely independent of the plant parameter and the external diturbance but in practice, K cannot be infinite but can be made ufficiently large for the cloed loop dynamic to be made nearly independent of the plant parameter and the diturbance input. In the

110 4. Control Technique and Performance Aement 9 ideal cae, the deired cloed loop repone i dictated by one cloed loop pole at / Tc. To analye the ytem with finite K, a Root locu plot i hown in Figure 4.7 where the open loop tranfer function i K c Tc a and the cloe loop pole are at / T c and / Tf, and a K, T c Tc and Tf. T c Im T f T c ' a Re Figure 4.7: Root locu with repect to K Thi mean that a K i increaed, the fat cloed loop pole at / Tf become very large and the cloed loop pole approaching / T c become dominant. A more detailed examination of high gain control can be found in the liding mode control of ubection 4.9. The magnitude of the fat pole at / Tf will, in practice, be limited by the ampling period h of the ytem. The only rigorou way of determining the lower limit of T f below which intability occur would be to determine the root of the characteritic polynomial in the z-plane and enuring they lie within the unit circle but a guideline that i fairly reliable i T h. It i poible that ome ytem would remain table for Tf h but it mut be realied that a T f i reduced there will alway be a threhold, dependent upon the particular ytem, below which intability will occur. Here, the ytem will be deigned o that f h Tf Tc (4.5) to be reaonably ure of avoiding intability due to the ampling proce and give the ytem adequate robutne againt parameter change and

111 4. Control Technique and Performance Aement diturbance. In any cae, confirmation of the correct behaviour by imulation i advied. Intuitively, the robutne can be thought of a being produced by a high value of the gain, K, that give the control loop a high degree of tiffne. The quetion then arie of how the maximum value of K can be calculated for (4.5) to be atified. While thi i relatively traightforward for the econd order example above, it i le o for higher order example. Thi problem, however, can be overcome by the method of robut pole placement. In the econd order example, thi conit of firt chooing the cloed loop pole poition, one preciely at / Tc and the other to atify (4.5). So thi ha the advantage over the high gain approach of enuring a pecified cloed loop dynamic, while previouly thi wa determined by T c Tc. Then the controller mut have at leat two adjutable parameter that can be calculated to yield thee cloed loop pole location. A imilar approach can be ued to deign any linear control ytem of order, n, uing n adjutable controller parameter. If a nonoverhooting tep repone i required, then n of the cloed loop pole can be placed at T c to yield a pecified ettling time of T (5% criterion) uing the ettling time formula (4.) with n replaced by n. Thu T c T.5 n (4.6) If the fat pole T f, however, i not located far enough away from the dominant pole() the cloed loop repone may depart ignificantly from the ideal repone. Hence a minimum pole-to-pole domination ratio i ued to inure a minimum ditance between the fat pole, dominant pole() at ratio into inequality (4.5). Thu r ppmin (Dodd, 3) T f, and the / Tc. Inerting the minimum pole-to-pole dominance f c pp min h T T / r (4.7)

112 4. Control Technique and Performance Aement Figure 4.8 how an example of the deired pole location for a controller deign uing robut pole placement. / h / Tf / Tf max Deign range rpp min T c Im / Tc Re Figure 4.8: Root locu of cloed loop ytem uing robut pole placement minimum fat pole magnitude Tf max Note that rppmin rppmin.. dominant pole magnitude T T T c f max c Hence the diplacement along the real axi of the -plane between the dominant pole poition and the cloet poition that the fat pole can occupy i r a hown in Figure 4.8. T T T pp min f max c c Although, in theory, the notion of cloed loop pole ceae to exit upon loop cloure around a nonlinear plant, robut pole placement i known to be ucceful with plant containing continuou nonlinearitie. A rudimentary explanation i that the real implementation work in the time domain and the robutne i attained via relatively high gain. It will be een, for example, in ection 4.9 that liding mode control with a boundary layer force the tate trajectory to reide in a cloe neighbourhood of a fixed boundary in the tate pace that realie a precribed tranient repone of the cloed loop ytem. Thi boundary eparate the tate pace into two region, one for control aturation at the poitive limit and the other for control aturation at the negative limit. Plant nonlinearitie affect the form of the trajectorie approaching the fixed boundary but do not affect the tate trajectorie within the boundary layer that traddle the fixed boundary. The trajectorie are trapped within the boundary layer by the high gain.

113 4. Control Technique and Performance Aement Partial Pole Placement If the number, n, of independently adjutable gain of a linear control ytem i le than it order, n, then complete pole placement a decribed in ubection cannot be carried out. It i poible, however, to place n of the cloed loop pole and calculate the reulting location of the remaining n n pole. If the n cloed loop pole are placed uing the ettling time formula to yield a pecified ettling time of T, then thi will be achieved in reality if the n n.5 n T, ince pole have coniderably larger negative real part than the deired pole will be dominant. The advantage of pole placement i the ue of a impler controller to achieve the ame pecified performance to that achievable by a more ophiticated controller that can be deigned by complete pole placement. Thi may, however, not be poible, the real ettling time being greater than T and the tep repone poibly containing undeirable ocillatory mode if nn. In thi cae, it would be poible to find a lower value of T that would be realied. If thi i too long, then another control technique permitting complete pole placement would be needed Model for Performance Aement In order to ae the imulated and experimental tep repone, thee are compared with the repone of ideal model implemented in Simulink baed on the 5% ettling time formula. In view of equation (4.), thee can take the form of a chain of identical firt order ubytem, a hown in Figure 4.9.

114 4. Control Technique and Performance Aement 3 r Y T 6 T 6 T 6 YIdeal a) n=3 r Y T 7.5 T 7.5 T 7.5 T 7.5 YIdeal b) n=4 Figure 4.9: Block diagram for ideal tep repone generation, a) third order, b) fourth order Nonlinear Friction and Control Dither In motion control, tatic friction can caue the controller to limit cycle due to the nonlinear peed-torque profile (Armtrong-Helouvry and Amin, 994) (Leonard and Krihnapraad, 99). The limit cycle can be minimied by injecting an alternating ignal, known a dither, into the control input cauing an alternating torque or force from the actuator intended to operate the ytem beyond the peak tatic friction torque (Figure.8). Thi entail mechanical movement that ha to be maintained by a ufficiently high dither ignal amplitude. The mechanical movement i kept within acceptable limit by etting the dither frequency to a ufficiently high value, which make ue of the inertia in the mechanim. Dither modifie the nonlinear velocity-torque characteritic by effectively eliminating the tatic friction (Zame and Shneydor, 976) (Iannellia et al., 5) (Zame and Shneydor, 977). In mot cae the frequency of the dither ignal i above the cut-off frequency of the cloed loop ytem in which the ignal i injected. In thi cae the dither ignal will not have a ignificant effect on the controlled output due to the filtering propertie of the plant.

115 4. Control Technique and Performance Aement 4 A variety of different dither ignal can be found throughout the indutry including random, inuoidal and puled waveform. A pule train of contant frequency i ued for thi reearch but the amplitude i witched between two level dependent on the control error magnitude, yr y, a hown in Figure 4.. When the abolute poition difference i maller than % of the maximum angular excurion of radian, the dither ignal i turned off to inure that it doe not unnecearily move the controlled output away from the etpoint once it i cloe to it. y t r y t - u / Dither enable level [%] Level A Level B Enable Level A Level B Pule generator Level A Level B Dither output ignal Figure 4.: Dither ignal generator A hown in Figure 4. (and Figure 4.) the dither ignal witche between zero and a poitive level, A or a negative level, B that oppoe the ign of the error, y y. Thi automatically help the controller to achieve a mall teady r tate error within the % full cale band.

116 4. Control Technique and Performance Aement 5 Level A e[%] e yr y y max [%] Level B Figure 4.: Dither ignal level Figure 4. how experimental data in which a proportional controller (Figure 4.) i applied to the throttle valve, with and without added dither. The gain of the proportional controller i et to kp 7 and a low ramp reference input (deired throttle poition) i applied. The experimental repone without dither how a taircae type of repone which i the effect of the nonlinear tatic (tick-lip) friction. The repone with dither i improved with a much maller teady tate error and maller hort-term variation (deviation from a mean ramp function).

117 4. Control Technique and Performance Aement 6.5 Deired throttle poition Throttle valve poition with dither Throttle valve poition without dither Poition [rad] Time [ec] Figure 4.: Experimental reult of a P-controller with and without dither It hould be noted that even with only linear kinetic friction, a proportional controller, in theory, yield a teady tate error proportional to the lope of the reference input ramp, o the reidual teady tate error viible with the dither in Figure 4. i not due to the dither not operating correctly.

118 4. Control Technique and Performance Aement Without dither With dither 4.5 x U f.5 Y f Frequency (Hz) Frequency (Hz) a) Plant input b) Plant output Figure 4.3: Amplitude pectra A mentioned before, the dither ignal frequency hould be higher than the plant bandwidth. Figure 4.3 demontrate thi by howing the amplitude pectra of the plant input and output ignal. The input and output amplitude pectra without the dither are inerted a reference. Figure 4.3 a) how a high amplitude with the dither at the fundamental frequency of Hz and the odd harmonic frequencie of 3 and 5 Hz which are related to the dither ignal quare waveform. Mot of the dither i filtered out by the throttle valve a hown in Figure 4.3 b) Integrator Anti Windup A invetigated in ubection.., a hort ettling time, le than. [ec], will make the output from the controller aturate if a tep reference input of ubtantial amplitude i applied, or if, a occur in normal operation on a vehicle, very rapid change of the reference input occur. The aturation i caued by the limitation of the throttle ytem hardware. The aturation may make the integrator of the controller ramp up indefinitely, referred to a integrator windup, if the poition reference input i above a critical value beyond

119 4. Control Technique and Performance Aement 8 which the ytem remain in aturation and i unable to drive the teady tate error to zero (Franklin et al., ). With maller reference poition input, the ytem i able to come out of aturation and drive the teady tate error to zero but the aturation can caue an undeirable overhooting and underhooting that would not occur with linear operation under the ame controller gain etting. To circumvent thee problem, a trategy for integrator anti-windup can be applied, which will top the integral action during the aturation. There are multiple way of implementing an integrator anti-windup trategy (Atrom and Rundqwit, 989, Franklin et al., ). The anti-windup trategy need to keep the output of the controller within, or cloe to, the operational limit of the actuator. The trategy ued for thi reearch i hown in Figure 4.4. Yr - k p u m in u m ax Plant Y k i U ' - K Integrator anti-windup U Figure 4.4: PI controller with integrator anti-windup The eential element i the aturating element with unity gain. During normal operation (unaturated) min ' u u ' t u u t u t max and therefore the actuation error, u t u ' t rendering thi loop inactive. If the ytem attempt to aturate, u 't umax or u ' t umin and therefore the actuation error i u t u t, of the anti-windup loop i zero, '. For a ufficiently high value of the gain, K, thi error i kept to very mall proportion by the anti-windup u ' t u t u. Thi inhibit the integral action and limit the loop, o max

120 4. Control Technique and Performance Aement 9 demanded controller output to a value exceeding the aturation limit by a negligible amount. A an example, Figure 4.5 how the performance of a PI controller applied to a plant with tranfer function Y U with and without the integrator anti-windup. (4.8).6.4. Without anti-windup With anti-windup 3.5 Without anti-windup With anti-windup Without anti-windup With anti-windup u t u ' t Plant output - y t.8.6 Controller output Time [ec] Time [ec] a) Plant output y t b) Controller output u t Figure 4.5: Integrator anti-windup performance

121 4. Control Technique and Performance Aement 4.5 Traditional Controller 4.5. Introduction The well-known workhore of indutrial control ytem i the proportional integral derivative (PID) controller hown in the general control loop of Figure 4.6. PID controller Y r - k p k d k i U N D Plant Y Figure 4.6: The PID controller Uing Maon rule on Figure 4.6, the cloed loop tranfer function i D ki N kp kd Y D Yr k N D k i d kp ki N kp kd kd kp ki N (4.9) It i immediately evident, however, that thi controller introduce two zero into the cloed loop tranfer function that are the root of k k k. Even if d p i the gain, k p, k i and k d, are et to yield real negative cloed loop pole, thee zero can caue a ingle overhoot in the tep repone and poibly an underhoot too. Thi will be explained hortly. One variant of the PID controller, that i available in ome indutrial controller, i to change the derivative term to act only on the controlled output, Y,

122 4. Control Technique and Performance Aement rather than the error, Y Y Figure 4.7. r. Thi yield the DPI controller hown in DPI controller Y r - k p k i U - k d N D Plant Y Figure 4.7: The DPI controller with the throttle valve plant In thi cae the cloed loop tranfer function i ki N kp Y D Yr k N i D kd kp ki N kp kd D kp kin (4.) Thu only one zero i introduced at ki kp by the controller but thi can till caue a ingle overhoot in the tep repone, even if all the cloed loop pole are real and negative, a will be een hortly. The idea leading from the PID to the IPD controller can be extended further by changing the proportional term to be fed only by the controlled output, leaving only the integral term to act on the error, Y Y IPD controller hown in Figure 4.8. r Y,, reulting in the

123 4. Control Technique and Performance Aement Y r - IPD controller k i - k p U - k d N D Plant Y Figure 4.8: The IPD controller The cloed loop tranfer function i then k N i Y D kin Yr ki N D kd kp ki N kp kd D (4.) In thi cae the controller introduce no finite zero and if all the cloed loop pole are negative and real, the tep repone cannot contain overhoot. The following ubection explain how finite zero can caue overhoot and underhoot Potential Effect of Zero If the plant ha no finite zero and the gain of the IPD controller of ubection 4.5. are et to yield real negative cloed loop pole, then the tep repone will be a monotonically increaing function reaching a contant teady tate value equal to the reference input tep value. Let thi tep repone be yipd t. Now let the correponding tep repone from the DPI and PID control loop, with the ame reference input, be denoted, of (4.), y t and y DPI PID t. Next, by inpection

124 4. Control Technique and Performance Aement 3 where K N (4.) Q i YIPD Yr Q D k k k N. (4.3) d p i Then by inpection of (4.9) and (4.), with a common Y, N kp kin kp YDPI kp ki Yr Yr YIPD Q ki Q ki and N k k p d YPID kd kp ki Yr YIPD Q ki ki Uing equation (4.4) and (4.5) the correponding function in time are p dpi ipd ipd ki dt r (4.4) (4.5) k d y t y t y t (4.6) k d k d y t y t y t y t (4.7) p d pid ipd ipd ipd ki dt ki dt Figure 4.9 how the imulated tep repone of. In thi cae the plant, N / y t, y t and y t dpi D, i the linear throttle valve model and the ettling time ued to deign the controller gain are. [ec]. The IPD controller achieve the deired non-overhooting tep repone with the right ettling time. However, both the DPI and PID controller produce an overhoot. pid ipd

125 4. Control Technique and Performance Aement 4. IPD controller DPI controller PID controller Cloed loop repone [rad] Time [ec] Figure 4.9: The zero effect on the cloed loop tep repone In fact, the PID controller ha the potential of producing an underhoot a well, due to the econd derivative term in (4.7). In thi cae, it actually reduce the overhoot relative to that of the DPI controller but i inufficient to produce an underhoot. It would therefore appear that the IPD controller would be the bet choice. On the other hand, an external pre-compenator i introduced in ection and to cancel the zero. Thi give preciely the ame performance and reulting enitivitie, which will be the ame in all three cae.

126 4. Control Technique and Performance Aement Controller Deign IPD Controller The deign of the IPD controller gain are baed on the knowledge of the linear throttle valve model, which in thi work i referred a model baed control. The characteritic equation for the IPD cloed loop ytem i then given by equation 3 (4.3) with N b and D a a a from equation (.4). Thu a b k a b k a b k (4.8) 4 3 d o p i Thi i of the fourth order but ha only three deign parameter ( k, k, k ) which p i d make it impoible to do a full pole aignment. Thi can be circumvented by doing partial pole aignment a decribed in ection The IPD loop can be deigned by placement of the three pole p, p and p 3, to achieve the deired tep repone. Thee pole can be choen freely. Thu d d d p p p (4.9) 3 3 Comparing the characteritic polynomial (4.8) with that of (4.9) how there i ha to be one dependent cloed loop pole at q, which require 3 d d d q 4 3 d q d q d d q d d q (4.3) Equating the characteritic polynomial of (4.3) and (4.8), and making the controller gain and q the ubject of the reulting equation yield q a d (4.3) k k d p d q d a b (4.3) d q d a b (4.33) k dq i (4.34) b

127 4. Control Technique and Performance Aement 6 The ettling time formula from ubection 4.4. will now be ued, auming a third order cloed loop repone on the aumption that the three coincident cloed loop pole at p,,3 pc are dominant with repect to the pole at q. Uing the ettling time formula, equation (4.), for the deired cloe loop ettling time, T, with n 3 yield pc 6 c T. Then T T T T d d d p Hence 3 (4.35) d 8 / T, d 8 / T and d 6 / T (4.36) Since the dependent pole i located at d q d a 8 / T a, and then According to (Dodd, 3), where the pole at Re,,3 Re 6 c (4.37) T 3 a (4.38) d c d pp min c d pp min c r r (4.39) r ppmin i the minimum pole-to-pole dominance ratio for which the effect of d can be regarded a negligible. Thi i baed on the well-known fact that the larger the real part of a pole in the left half of the -plane, the fater the impule repone of the mode with which the pole i aociated will decay. In the cae under tudy, the number of dominant pole i 3 and the number of dominated pole i, and the table given in (Dodd, 3) give rppmin 5.4 (4.4) It i evident from equation (4.38) that equation (4.4) can only be atified if 3 a 5.4 a.4 a.4 (4.4) c c c c

128 4. Control Technique and Performance Aement 7 There i therefore an upper limit of c determined by the plant parameter, a, beyond which the deired performance cannot be attained. In view of equation (4.37) and equation (4.4), there i a lower limit on the deired ettling time of a.4 T (4.4) T min min a The value obtained for a from the experimental tet of Chapter i 3338, giving T min.43 [ec]. Since thi i far maller than poible in the throttle valve application, due to control aturation with any ignificant change in the reference poition input and thi actually ha a lower limit in the region of.5 [ec], the partial pole placement i not retrictive. Figure 4.3 how the relative poition of the dominant and dependent cloed loop pole for T. [ec]. Im p q 36 4 Re p c 6,,3 Figure 4.3: Cloed loop pole location for T. [ec] Figure 4.3 how the imulated tep repone uing the IPD controller with the linear throttle valve model from Chapter.4.. It i evident that the dependent pole increae the ettling time by a negligible amount beyond the nominal value.

129 4. Control Technique and Performance Aement 8 /- 5% Deired throttle poition Cloed loop repone [rad] Deired cloed loop ettling time Time [ec] Figure 4.3: Cloed loop repone of a IPD controller with partial pole placement It i known from the literature (Atröm and Hägglund, 995) that meaurement noie will be amplified by the differentiator in the feedback path of the controller (Figure 4.8). To avoid the amplification of high frequency component of meaurement noie by the differentiator that would be applied to the plant control input, a firt order low-pa filter (noie filter) can be added in conjunction with the differentiator a hown in Figure 4.3. Thi low-pa filter ha a flat frequency repone in the pa-band. Yr - k i k p - U b Y - k d 3 a a a Throttle valve f f Figure 4.3: Throttle valve and IPD controller with differentiation filter

130 4. Control Technique and Performance Aement 9 By uing Maon rule on Figure 4.3, the cloed loop tranfer function i Y k b k b Y a a a a a b k k b i f i r f f f d f p f i p f f i a k b k b k b (4.43) The order increae by one, compared to tranfer function (4.), due to the firt order filter ued for the differentiator. If the filter cut-off frequency, f, i not included a an adjutable parameter in the pole placement procedure, then partial pole placement of three cloed loop pole uing the three controller gain leave two dependent cloed loop pole requiring a factor q q, of the cloed loop characteritic polynomial. Then the polynomial (4.3) i replaced by 3 d d d q q d q d q d q d q d q d q d q d d q (4.44) Equating the denominator of tranfer function (4.43) to polynomial (4.44), and olving the reulting equation for the three controller gain and the coefficient, q and q, yield q a d (4.45) f q a a d d q (4.46) f i / k d q b (4.47) f / k d q d q a k b b (4.48) p f i f / k d d q d q a a k b b (4.49) d f p f Without the noie filter, and auming no plant modelling error, the three dominant cloed loop pole lie preciely at,,3 6 / T 6 for T.. The introduction of the filter, however, will caue thee pole to hift from thi location, but not ignificantly if

131 4. Control Technique and Performance Aement 3 6 / T 6. (4.5) f Otherwie the cloed loop pole will be forced to move ignificantly from the deired location and conequently impair the performance of the control ytem. f i et to 5 [rad/ec] (~8 [Hz]) to atify inequality (4.5) and lower than the free pole at ~36 [rad/ec], hown in Figure 4.3. A plot of the cloed loop pole and zero location for the IPD controller with a firt order filter and the linear throttle valve plant model i hown in Figure The filter introduce a cloed loop zero and pole between the dominant pole group and the dependent pole. p 33 5 Introduced by the filter Im p c 6,,3 { Re z 5 p 34 4 Figure 4.33: The cloed loop pole and zero location for the IPD controller with a differentiating filter Figure 4.34 how the impact of the filter on the imulated cloed loop tep repone and the error between the deired and the actual tep repone. The dahed blue line indicate a precie repone when the filter coefficient f i included in the calculation of the gain for the IPD controller, equation (4.45) to (4.49). The dahed red line how a 4 time higher difference between the deired and the actual tep repone, when the filter coefficient included in the calculation, equation (4.3) to (4.34). f i not

132 4. Control Technique and Performance Aement 3 Deired throttle poition Throttle poition [rad] Deired repone Actual repone with filter compenation Actual repone with no filter compenation.5..5 Difference between deired repone and actual repone [rad] Time [ec] Figure 4.34: Simulated tep repone with/without noie filter compenation Although taking the filter into account in the partial pole placement reduce the error by a factor of four, it i not really needed in thi cae, a from a practical viewpoint, both of the tep repone of Figure 4.34 are ufficiently cloe to the ideal one to be acceptable DPI Controller A DPI controller, firt introduced in 4.5., i hown in Figure 4.35 with a noie filter for the differentiator,

133 4. Control Technique and Performance Aement 3 Yr - k p k i - U b Y 3 a a a k d Throttle valve f f Figure 4.35: DPI controller with differentiation filter The tranfer function in thi cae i ki b kp 3 Y a a a Yr b ki kdf k 3 p a a a f (4.5) bk p kib f kpb f kib kib fa f kpb kibf Y Y a a a a k b a k b r f f p f d f (4.5) The partial pole placement deign equation for thi DPI controller are identical to thoe for the IPD controller with the noie filter derived in ubection , becaue the denominator of tranfer function (4.5) i identical to the cloed loop tranfer function (4.43). Hence equation (4.45) to (4.49), incluive yield the required controller gain, k d, polynomial coefficient, q and q. k p and k i together with the dependent pole The degree of the numerator polynomial of tranfer function (4.5), however, ha increaed by one relative to tranfer function (4.43) for the IPD controller, equation (4.43). Thi introduce a cloed loop zero. A decribed in ubection 4.5., and the in literature (Franklin et al., ), thi zero can caue the tep repone to overhoot. Figure 4.36 how the imulated cloed loop repone

134 4. Control Technique and Performance Aement 33 of the DPI and IPD controller applied to the linear throttle valve model with a ettling time of T. [ec]. In thi cae there i no actuator aturation limit applied in the imulation model. Throttle poition [rad].5.5 Deired repone DPI repone IPD repone Controller output [V] Time [ec] Figure 4.36: The impact of the zero on the cloed loop ytem repone A well a the overhoot being larger than acceptable, it introduce the further unwanted ide effect of exceive initial level of the control, ut, a hown in Figure 4.36 during the initial phae t.5 [ec]. Thi can caue a problem on a real ytem due to the control aturation limit, a invetigated in ubection A imulation of the DPI control ytem with aturation, at /- Volt, i hown in Figure The controller output i aturated at Volt from t = to.5 [ec], which caue integrator windup and the tep repone overhoot to increae from about.5 to.5 [rad], a evident by comparing Figure 4.36 with Figure 4.37.

135 4. Control Technique and Performance Aement 34 Throttle poition [rad].5.5 Deired repone Simulated repone Controller output [V] Time [ec] Figure 4.37: Cloed loop imulation of the DPI controller with a linear throttle valve plant model and limit on the controller output To circumvent the overhoot problem, two different trategie have been applied and imulated: ) A precompenator i ued to cancel the cloed loop zero, and ) An anti-integrator windup trategy i applied. ) Figure 4.38 how the DPI controller baed ytem with the precompenator attached. The tranfer function, R( ) / Z, can be deigned to remove the effect of the cloed loop zero and pole by pole-zero cancellation. Thi i done without changing the characteritic dynamic of the cloed loop ytem.

136 4. Control Technique and Performance Aement 35 Yr R( ) Z Yr Precompenator ' - k p k i - k d U b Y 3 a a a Throttle valve f f Figure 4.38: DPI controller with precompenator The precompenator will in thi cae be ued to cancel both of the zero. Thi i achieved by making the denominator of the precompenator cancel the numerator, b k k b k b k b, of the cloed loop tranfer function p i f p f i (4.5), but firt thi may be implified by normaliation with repect to the contant coefficient to yield bk p kib f kpb f kib kp f kib f k i f (4.53) Since the control loop already ha a unity DC gain due to the integral term, then the pre-compenator mut alo have a unity DC gain. Hence the precompenator tranfer function i imply the reciprocal of polynomial (4.53). Thu R ( ) Z kp f k i f (4.54) A imulation of a cloed loop tep repone where both zero are cancelled in thi way i hown in Figure An important obervation i that the controller output doe not reach the aturation limit due to the lower reference input, yr ' t produced by the precompenator. In thi cae the precompenator make the need for an integrator anti windup trategy unneceary.

137 4. Control Technique and Performance Aement 36 Throttle poition [rad].5.5 Deired repone Simulated repone Controller output [V] Time [ec] Figure 4.39: Cloed loop imulation of the DPI controller with precompenator cancelling both zero ) Minimiing the overhoot caued by the controller output aturation can be achieved by the integrator anti-windup trategy introduced in ubection Thi will limit the aturation by manipulating the integrator input to hold the primary plant control input, u, within limit approximately equal to u max and u min a hown in Figure 4.4, by etting the gain, K, to a ufficiently large value.

138 ' 4. Control Technique and Performance Aement 37 Yr - Integrator anti-windup K - k i k p U - k d u m in u m ax U b Y 3 a a a Throttle valve f f Figure 4.4: DPI controller with integrator anti-windup Figure 4.4 how the imulated cloed loop tep repone of the implemented trategy, hown in Figure 4.4, with the integrator anti-windup gain et to K. Throttle poition [rad].5 Deired repone Simulated repone Controller output [V] Time [ec] Figure 4.4: Cloed loop tep repone of the DPI controller with integrator antiwindup (Large tep)

139 4. Control Technique and Performance Aement 38 In thi cae, the overhoot of the tep repone ha been eliminated due to the action of the anti-windup loop but it i important to note that it cannot be eliminated by thi mean for reference input tep that are not large enough to caue control aturation, a hown in Figure 4.4. Throttle poition [rad] Deired repone Simulated repone Controller output [V] Time [ec] Figure 4.4: Cloed loop tep repone of the DPI controller with integrator antiwindup (Small tep) DPI Controller with Feed Forward and Manual Tuning Throughout indutry it i common for uitable controller gain to be found manually by trial-and-error and ometime by pecial ytematic procedure dependent upon the application. Thi approach i therefore included in thi work to benchmark the other control trategie againt, not regarding performance, which would not be expected to be optimal in any ene, but regarding comparion of the peronnel effort needed.

140 4. Control Technique and Performance Aement 39 For the tuning of controller a number of different tool and procee are available uch a the Ziegler-Nichol tuning rule (Shahrokhi and Zomorrodi, 3) (Atröm and Hägglund, 995). For thi work the majority of the controller tuning i undertaken with the aid of a toolbox from Mathwork called Deign Optimiation. The controller elected for the manual tuning i one typical for throttle valve poition control in the automotive indutry. Thi i the DPI controller with a feed forward loop to compenate for the pre-treed coil pring torque. The original DPI controller employed wa deigned in the dicrete time domain (z-domain) with dicontinuou function. The dicontinuou function are ued for improving the control performance e.g. applying different gain. A implified model, ued for thi work, without the dicontinuou function i hown in Figure y r z - pring k p k i z k z d u pring z pring u z k out y z Figure 4.43: Simplified dicrete time DPI controller with feed forward Here, the feed forward ignal, counteract the torque from the coiled pring and u z y z, i ued to pring r pring pring k out i a percent to duty cycle converion factor, ince the coil drive i implemented uing pule width modulated witched power electronic. By uing the feed forward compenation, the feedback part of the controller i only required to move the

141 4. Control Technique and Performance Aement 4 throttle valve from one poition to another with the required cloed loop dynamic. The feed forward compenation i ueful epecially if the coiled pring characteritic i nonlinear, in which cae the linear compenation ha to be replaced by a nonlinear mapping function. The dicrete time model, Figure 4.43, i converted into a continuou time model ued for invetigation of enitivity later on, hown in Figure 4.44, Yr - k h i pring k p U pring - k h d pring k out U Y Figure 4.44: Simplified continuou time DPI controller where h i the ampling time interval. A Simulink model of the dicrete time IPD controller with the feed forward and the nonlinear throttle model of ection.5 with ample and hold unit at the input and output i ued for the Mathwork toolbox tuning. The nonlinear throttle model i ued becaue it include the pre-treed coil pring model. The controller reference input yr t i a predefined ignal, in thi cae the taircae haped waveform of Figure 4.45, hown in green. The Deign Optimiation toolbox tune the controller by adjuting the controller gain until yt track the deired cloed loop repone (Figure 4.45, hown in red), while running the Simulink model. The deired cloed loop repone i obtained a the output of a chain of four identical firt order block with time contant et according to the 5% ettling time formula with T. [ec].

142 4. Control Technique and Performance Aement 4 Other ignal were ued for the tuning uch a ramp and inuoidal waveform. The Deign Optimiation toolbox work in a imilar manner to the Deign Optimization toolbox ued in chapter Controller ignal [rad].5 Deired cloed loop repone Controller reference input Simulated throttle valve poition Time [ec] Figure 4.45: Example of a waveform ued for the tuning of the DPI The black waveform in Figure 4.45 i the imulated throttle valve poition, uing the final tuned gain, found by the toolbox and manual tuning. yt, PID Controller A PID controller, firt introduced in 4.5., with a noie filter for the differentiator i included a it i till arguably the mot commonly ued controller in indutry. Thi i hown in Figure 4.46 applied to the throttle valve.

143 4. Control Technique and Performance Aement 4 k p Yr - k d f f U b Y 3 a a a k i Throttle valve Figure 4.46: PID controller with differentiation noie filter The cloed loop tranfer function in thi cae i ki kdf b kp 3 Y f a a a Yr b ki kdf k 3 p a a a f bk p f kdb kib f kpb f kib f f p f df kib f a f kpb kibf a a a a k b a k b (4.55) A for the DPI controller, the denominator of the tranfer function, which i the cloed loop characteritic polynomial, i identical to that of the IPD controller with the meaurement noie filtering, given by tranfer function (4.43). In thi cae, the deign equation for the controller gain, k p, k i and k d together with the coefficient, q and q, of the dependent pole polynomial factor in the partial pole aignment are given by equation (4.45) to (4.49) incluive. The numerator polynomial of the cloed loop tranfer function (4.55), i of econd degree, indicating the preence of cloed loop zero with a potential overhooting problem. To remove the effect of the zero both the technique () and () invetigated in ubection are applied together, i.e., a precompenator to cancel both of the zero and an integrator anti windup loop. The complete control ytem i hown in Figure 4.47.

144 4. Control Technique and Performance Aement 43 k p Yr R( ) Z - Precompenator k d k i f f - K Integrator anti-windup U ' u m in u m ax U b 3 a a a Throttle valve Y Figure 4.47: PID controller with differentiation filter, precompenator and integrator anti windup With reference to the numerator of the cloed loop tranfer function (4.55), the precompenator tranfer function loop zero i given by R( ) / Z ued to cancel the two cloed R ( ) fkb i Z b k k b k b k b k b p f d i f p f i (4.56) A imulation of a cloed loop tep repone where both zero are cancelled in thi way i hown in Figure A the DPI controller in ubection , the PID controller output doe not reach the aturation limit due to the lower reference input, yr ' t produced by the precompenator.

145 4. Control Technique and Performance Aement 44 Throttle poition [rad].5 Deired repone Simulated repone Controller output [V] Time [ec] Figure 4.48: Cloed loop imulation of the PID controller with precompenator cancelling both zero The reult of an experimental baed comparion of the PID controller with and without a precompenator and integrator anti-windup can be een in ubection (Figure 4.7) Simulation and Experimental reult IPD Controller The IPD controller gain are determined a decribed in ubection , equation (4.47) to (4.49) with T. [ec] and f 5 [rad/ec]. A dither ignal i added to the control ignal to reduce the effect of the tatic friction a decribed in ubection The integrator anti-windup trategy i enabled to minimie the duration of the control aturation following the application of a tep reference input. The IPD controller i teted experimentally with three different

146 4. Control Technique and Performance Aement 45 reference input function a decribed in ubection The dspace ytem i ued for the experiment a explained in ubection The imulation reult preented in thi ubection are obtained with the full nonlinear plant model preented in ection.5. Firt, imulated and experimental repone to a tep reference poition change within the throttle valve top limit are preented in Figure Throttle poition [rad] Reference input Deired cloed loop repone Simulated Experimental. T Time [ec] Figure 4.49: Cloed loop tep repone, from. to.3 [rad] The vertical black line mark the nominal ettling time of T.[ec]. It i evident that both the experimental and imulated repone come cloe to thi at.4 [rad]. Figure 4.5 how uperimpoed imulated and experimental cloed loop repone uing the IPD controller with three different reference input function, the reaon for which are tated in ubection In all cae, the imulated and experimental repone are remarkably cloe.

147 4. Control Technique and Performance Aement 46 Throttle poition [rad] Throttle poition [rad] Throttle poition [rad] Ramp Simulated Experiment Pule Drive cycle 5 5 Time [ec] Figure 4.5: Experimental and imulated repone of the IPD controller Figure 4.5 how the difference between the deired and the experimental cloed loop repone correponding to Figure 4.5. The relatively mall difference could be attributed to the accuracy of the plant model but they could alo be due to the robutne of the control loop, which would give nearly the ame repone depite mimatching between the plant and it model.

148 4. Control Technique and Performance Aement 47. Ramp Difference between deired repone and actual repone [rad] Pule Drive cycle Time [ec] Figure 4.5: The difference between the deired and the experimental cloed loop repone A explained in ubection a pring failure can caue the engine to tall due to air tarvation. The behaviour of the IPD controller during a pring break, at t [ec], i imulated uing the nonlinear throttle valve model, and the reult i hown in Figure 4.5. It how a good robutne againt the diturbance, with an acceptable deviation from the throttle poition demand. The ocillation on the control ignal at ignal that increae in amplitude when the control error, t. [ec] are the added dither yr y, exceed a preet threhold of % of the full cale movement range a decribed in ubection

149 4. Control Technique and Performance Aement 48 Throttle poition [rad] Throttle poition Throttle poition demand Throttle fully cloed Controller output [V] Time [ec] Figure 4.5: IPD controller during a pring failure The robutne againt plant parameter deviation away from the nominal value i teted uing the Monte Carlo method decribed in ubection The reult of the parameter variation imulation i hown in Figure 4.53 for the maximum poible tandard deviation of 5%. The figure how the operational envelope for imulation run, where the blue and red line are the minimum and maximum value. The nominal parameter cloed loop controller repone and controller output are hown in black.

150 4. Control Technique and Performance Aement 49 Throttle poition [rad] Nominal Minimum Maximum DC motor voltage [v] Nominal Minimum Maximum Time [ec] Figure 4.53: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: 5% ) Thi indicate that the plant parameter can deviate ignificantly from the nominal value before having an advere impact on the performance of the controller. The enitivity for the IPD controller i analyed in the frequency domain by uing the relationhip of equation (4.3) in ubection Thi i done with the aid of the Matlab Control Sytem Analyi Toolbox and the block diagram of Figure 4.54 implemented in Simulink with S db. C output to obtain p D a the input and V a the

151 4. Control Technique and Performance Aement 5 Y r - E k i - k p U - k d D - V b a a a 3 Throttle valve Y f f Figure 4.54: Control tructure ued to analye the enitivity Ignoring the differential filtering term, a bandwidth, the enitivity tranfer function i f i well outide the control loop V C Sp D b ki. k 3 d kp a a a a a a a a b k a b k b k d p i (4.57) The enitivity reult i hown in Figure 4.55.

152 4. Control Technique and Performance Aement 5 - Senitivity [db] Frequency [rad/ec] Figure 4.55: IPD enitivity Thi indicate a relatively low enitivity, equivalent to high robutne. Thi correpond well with the time domain reult found by the above pring failure analyi and the Monte Carlo parameter variation imulation DPI Controller The cloed loop characteritic polynomial for the DPI controller i the ame a that for the IPD controller and therefore the gain are calculated a decribed in ubection , equation (4.47) to (4.49) with T. [ec] and f 5 [rad/ec]. The precompenator i deigned to cancel both zero, a decribed in ubection A dither ignal i added to the control ignal to reduce the effect of the tatic friction a decribed in ubection The integrator antiwindup trategy i enabled to minimie the duration of any control aturation occurring following application of a tep reference input. The DPI controller i teted experimentally with three different reference input function, the

153 4. Control Technique and Performance Aement 5 decription and purpoe of which are given in ubection The dspace ytem i ued for the experiment a explained in ubection The imulation reult preented in thi ubection are obtained with the full nonlinear plant model preented in ection.5. Firt, imulated and experimental repone to a tep reference poition change within the throttle valve top limit are preented in Figure Throttle poition [rad] Reference input Deired cloed loop repone Simulated Experimental. T Time [ec] Figure 4.56: Cloed loop tep repone, from. to.3 [rad] The vertical black line mark the nominal ettling time of T.[ec]. It i evident that both the experimental and imulated repone come cloe to thi at.4 [rad]. Figure 4.55 how the imulated and experimental cloed loop repone of the DPI controller.

154 4. Control Technique and Performance Aement 53 Ramp Throttle poition [rad] Throttle poition [rad] Throttle poition [rad].5 Simulated Experiment Pule Drive cycle Time [ec] Figure 4.57: Experimental and imulated repone of the DPI controller The relatively mall difference could be attributed to the accuracy of the plant model but they could alo be due to the robutne of the control loop, which would give nearly the ame repone depite mimatching between the plant and it model. Figure 4.58 how the difference between the deired and the experimental cloed loop repone, indicating good tracking and negligible teady tate error.

155 4. Control Technique and Performance Aement 54. Ramp Difference between deired repone and actual repone [rad] Pule Drive cycle Time [ec] Figure 4.58: The difference between the deired and the experimental cloed loop repone A decribed in ubection a pring failure can caue the engine to tall due to air tarvation. The behaviour of the DPI controller during a pring break, at t [ec], i imulated uing the nonlinear throttle valve model, and the reult i hown in Figure It how a good robutne againt the diturbance, with little deviation from the throttle poition demand. The ocillation on the control ignal at that increae in amplitude when the control error, t. [ec] are the added dither ignal yr y, exceed a preet threhold of % of the full cale movement range a decribed in ubection

156 4. Control Technique and Performance Aement 55 Throttle poition [rad] Throttle poition Throttle poition demand Throttle fully cloed Controller output [V] Time [ec] Figure 4.59: DPI controller during a pring failure The robutne againt plant parameter deviation away from the nominal value i teted uing the Monte Carlo method decribed in ubection The parameter variation imulation ue the nonlinear throttle valve model with the controller output aturation, integrator anti-windup, dither and the precompenator. The reult of the parameter variation imulation i hown in Figure 4.6 for the maximum poible tandard deviation of 4%. The figure how the operational envelope for imulation run, where the blue and red line are the minimum and maximum value. The nominal parameter cloed loop controller repone and controller output are hown in black.

157 4. Control Technique and Performance Aement 56 Throttle poition [rad] Nominal Minimum Maximum DC motor voltage [v] Nominal Minimum Maximum Time [ec] Figure 4.6: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: 4% ) Thi indicate that the plant parameter can deviate ignificantly from the nominal value before having an advere impact on the performance of the controller. The enitivity for the DPI controller i analyed in the frequency domain by uing the relationhip of equation (4.3) in ubection Thi i done with the aid of the Matlab Control Sytem Analyi Toolbox and the block diagram of Figure 4.6 implemented in Simulink with S db. C output to obtain p D a the input and V a the

158 4. Control Technique and Performance Aement 57 Yr E - k p k i k d D U - b Y - V 3 a a a Throttle valve f f Figure 4.6: Control tructure ued to analye the enitivity Ignoring the differential filtering term, a bandwidth, the enitivity tranfer function i f i well outide the control loop V C Sp D b ki. k 3 d kp a a a a a a a a b k a b k b k d p i (4.58) The enitivity reult i hown in Figure 4.6.

159 4. Control Technique and Performance Aement 58 - Senitivity [db] Frequency [rad/ec] Figure 4.6: DPI enitivity Thi indicate a relatively low enitivity, equivalent to high robutne. Thi correpond well with the time domain reult found by the above pring failure analyi and the Monte Carlo parameter variation imulation DPI Controller with Feed Forward and Manual Tuning The gain ued for the imulation and experiment were found a decribed in ubection , where: k k k k pring pring p i d out 95 /

160 4. Control Technique and Performance Aement 59 A dither ignal i added to the control ignal to reduce the effect of the tatic friction a decribed in ubection The integrator anti-windup trategy i enabled to minimie the aturation during the tep reference input. The DPI controller with feed forward i teted experimentally with three different reference input function a decribed in ubection The dspace ytem i ued for the experiment a explained in ubection The imulation reult preented in thi ubection are obtained with the full nonlinear plant model preented in ection.5. Firt, imulated and experimental repone to a tep reference poition change within the throttle valve top limit are preented in Figure Throttle poition [rad] Reference input Deired cloed loop repone Simulated Experimental. T Time [ec] Figure 4.63: Cloed loop tep repone, from. to.3 [rad] The vertical black line mark the nominal ettling time of T.[ec]. It i evident that both the experimental and imulated repone come cloe to thi at.4 [rad].

161 4. Control Technique and Performance Aement 6 Figure 4.64 how the imulated and experimental cloed loop repone of the DPI controller with three different reference input function which indicate a good correlation between the two. Throttle poition [rad] Throttle poition [rad] Throttle poition [rad] Ramp.5 Simulated Experiment Pule Drive cycle Time [ec] Figure 4.64: Experimental and imulated repone of the DPI controller The relatively mall difference could be attributed to the accuracy of the plant model but they could alo be due to the robutne of the control loop, which would give nearly the ame repone depite mimatching between the plant and it model. Figure 4.65 how the difference between the deired and the experimental cloed loop repone, indicating good tracking and a mall teady tate error.

162 4. Control Technique and Performance Aement 6. Ramp Difference between deired repone and actual repone [rad] Pule Drive cycle Time [ec] Figure 4.65: The difference between the deired and the experimental cloed loop repone A decribed in ubection a pring failure can caue the engine to tall due to air tarvation. The behaviour of the DPI controller with feed forward during a pring break, at t [ec], i imulated uing the nonlinear throttle valve model, and the reult i hown in Figure The low throttle poition repone would caue the throttle valve to remain cloed for econd, which would reult in an engine tall. The ocillation on the control ignal at t [ec] are the added dither ignal that increae in amplitude when the control error, yr y, exceed a preet threhold of % of the full cale movement range a decribed in ubection

163 4. Control Technique and Performance Aement 6 Throttle poition [rad] Throttle poition Throttle poition demand Throttle fully cloed Controller output [V] Time [ec] Figure 4.66: DPI controller with feed forward during a pring failure The low recovery i due to the tuning giving the integral gain a low value of, ki.4. During the tuning the feed forward pring compenation wa active and therefore there wa no need for a fat integration action. The robutne againt plant parameter deviation away from the nominal value i teted uing the Monte Carlo method decribed in ubection The parameter variation imulation ue the nonlinear throttle valve model with the controller output aturation, integrator anti-windup, dither and the precompenator. The reult of the parameter variation imulation i hown in Figure 4.67 for the maximum poible tandard deviation of %. The figure how the operational envelope for imulation run, where the blue and red line are the minimum and maximum value. The nominal parameter cloed loop controller repone and controller output are hown in black.

164 4. Control Technique and Performance Aement 63 Throttle poition [rad] Nominal Minimum Maximum DC motor voltage [v] Nominal Minimum Maximum Time [ec] Figure 4.67: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: % ) The reult how that the ytem i very enitive to parameter variation. Thi reult i difficult to compare directly with the model baed control deign in thi work due to the way the Monte Carlo tet i performed. The enitivity for the DPI controller i analyed in the frequency domain by uing the relationhip of equation (4.3) in ubection Thi i done with the aid of the Matlab Control Sytem Analyi Toolbox and the block diagram of Figure 4.68 implemented in Simulink with S db. C output to obtain p D a the input and V a the

165 4. Control Technique and Performance Aement 64 Yr E - k h i pring k p U pring - k h d pring k out U D - V b 3 a a a Throttle valve Y Figure 4.68: Control tructure ued to analye the enitivity The tranfer function i V C Sp D bk k i h out. k 3 dh kp pring a a a 4 3 a a a 4 3 a a bkout kdh a bkout kp pring ki bk outki h (4.59) The enitivity reult i hown in Figure 4.69.

166 4. Control Technique and Performance Aement 65 - Senitivity [db] Frequency [rad/ec] Figure 4.69: Manually tuned DPI enitivity Thi indicate a relatively high enitivity, equivalent to low robutne. Thi correpond well with the time domain reult found by the above pring failure analyi and the Monte Carlo parameter variation imulation PID Controller The cloed loop characteritic polynomial for the PID controller i the ame a that for the IPD controller and therefore the gain are determined a in ubection , equation (4.47) to (4.49) with T. [ec] and f 5 [rad/ec]. The precompenator i deigned to cancel both zero, a decribed in ubection A dither ignal i added to the control ignal to reduce the effect of the tatic friction a decribed in ubection

167 4. Control Technique and Performance Aement 66 The imulation reult preented in thi ubection are obtained with the full nonlinear plant model preented in ection.5. Figure 4.7 how three experimental tep repone with ) the baic PID controller (black line), ) a () with integrator anti-windup introduced (green line) and 3) a () with a precompenator introduced (blue line). The PID controller without integrator anti-windup and precompenator ha an overhoot a expected, which correlate well with the imulation in ubection and PID PID with a precompenator PID with integrator anti windup. Throttle poition [rad] Time [ec] Figure 4.7: PID cloed loop tep repone Firt, imulated and experimental repone to a tep reference poition change within the throttle valve top limit are preented in Figure 4.7.

168 4. Control Technique and Performance Aement Throttle poition [rad] Reference input Deired cloed loop repone Simulated Experimental. T Time [ec] Figure 4.7: Cloed loop tep repone, from. to.3 [rad], uing a precompenator The vertical black line mark the nominal ettling time of T.[ec]. It i evident that both the experimental and imulated repone come cloe to thi at.4 [rad]. The PID controller i teted experimentally with three different reference input function a decribed in ubection Figure 4.7 how the imulated and experimental cloed loop repone which indicate a good correlation between the two. The integrator anti-windup trategy i enabled to minimie the duration of any aturation following a tep reference input.

169 4. Control Technique and Performance Aement 68 Ramp Throttle poition [rad] Throttle poition [rad] Throttle poition [rad] Pule Drive cycle 5 5 Time [ec] Simulated Experiment Figure 4.7: Experimental and imulated repone of the PID controller The relatively mall difference could be attributed to the accuracy of the plant model but they could alo be due to the robutne of the control loop, which would give nearly the ame repone depite mimatching between the plant and it model. Figure 4.73 how the difference between the deired and the experimental cloed loop repone, indicating good tracking and negligible teady tate error.

170 4. Control Technique and Performance Aement 69. Ramp Difference between deired repone and actual repone [rad] Pule Drive cycle Time [ec] Figure 4.73: The difference between the deired and the experimental cloed loop repone A decribed in ubection a pring failure can caue the engine to tall due to air tarvation. The behaviour of the DPI controller during a pring break, at t [ec], i imulated uing the nonlinear throttle valve model, and the reult i hown in Figure It how a good robutne againt the diturbance, with little deviation from the throttle poition demand. The ocillation on the control ignal at that increae in amplitude when the control error, t. [ec] are the added dither ignal yr y, exceed a preet threhold of % of the full cale movement range a decribed in ubection

171 4. Control Technique and Performance Aement 7 Throttle poition [rad] Throttle poition Throttle poition demand Throttle fully cloed Controller output [V] Time [ec] Figure 4.74: PID controller during a pring failure The robutne againt plant parameter deviation away from the nominal value i teted uing the Monte Carlo method decribed in ubection The parameter variation imulation ue the nonlinear throttle valve model with the controller output aturation, integrator anti-windup, dither and the precompenator. The reult i hown in Figure 4.75 for the maximum poible tandard deviation of 4%. The figure how the operational envelope for imulation run, where the blue and red line are the minimum and maximum value. The nominal parameter cloed loop controller repone and controller output are hown in black.

172 4. Control Technique and Performance Aement 7 Throttle poition [rad] Nominal Minimum Maximum DC motor voltage [v] Nominal Minimum Maximum Time [ec] Figure 4.75: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: 4% ) Thi indicate that the plant parameter can deviate ignificantly from the nominal value before having an advere impact on the performance of the controller. The enitivity for the PID controller i analyed in the frequency domain by uing the relationhip of equation (4.3) in ubection Thi i done with the aid of the Matlab Control Sytem Analyi Toolbox and the block diagram of Figure 4.76 implemented in Simulink with S db. C output to obtain p D a the input and V a the

173 4. Control Technique and Performance Aement 7 k p D Yr - E k d k i f f U - b Y V 3 a a a Throttle valve Figure 4.76: Control tructure ued to analye the enitivity Ignoring the differential filtering term, a bandwidth, the enitivity tranfer function i V D b ki. k 3 d kp a a a f i well outide the control loop a a a a a b k a b k b k d p i (4.6) The enitivity reult i hown in Figure 4.77.

174 4. Control Technique and Performance Aement 73 - Senitivity [db] Frequency [rad/ec] Figure 4.77: PID enitivity Thi indicate a relatively low enitivity, equivalent to high robutne. Thi correpond well with the time domain reult found by the above pring failure analyi and the Monte Carlo parameter variation imulation.

175 4. Control Technique and Performance Aement Linear State Feedback Control 4.6. Baic Linear State Feedback Control Let the general linear, SISO, time invariant (LTI) plant be repreented by the tate pace equation, T x Ax bu, y c x (4.6) where n x i the tate vector, u i the calar control input, y i the calar meaured output and y r i the correponding reference input. Here, A i the plant matrix, b i the input matrix, and T c i the output matrix, linear tate feedback (LSF) control i a feedback control technique in which the tate vector, x, i fed back to the control input according to the control law, u T yrr gx (4.6) Since the component of the gain vector, g, are independently adjutable, it i poible, in theory, to place the full et of cloed loop pole at predetermined location in the -plane (Franklin et al., ). The plant mut be controllable in order to implement thi method, or any other control ytem that can be deigned by pole placement. Provided there i full acce to all the tate in the real plant, a hown in Figure 4.78, the control engineer i free to deign the ytem to achieve a deired cloed loop tranient repone. y r LSF control law r g u b Real Plant x dt A x T c y Figure 4.78: LSF control ytem Thi can be done by determining the characteritic polynomial of the cloed loop ytem uing Figure 4.78 and equating thi with the polynomial of the ame order

176 4. Control Technique and Performance Aement 75 found by uing the ettling time formula a in ubection The gain r i adjuted uually to yield zero teady tate error of the tep repone, but the accuracy of thi depend upon the accuracy of the aumed plant tate pace equation State Oberver The LSF of ubection 4.6. require that all the tate variable are available from the plant. In ome cae it i difficult to gain acce to ome tate due to the cot of the required intrumentation or the fact that it might not be phyically poible to meaure all of them, a in ome chemical procee. However, method to recontruct the miing tate or all the tate variable have been developed during the lat 5 year. The firt paper to invetigate the general oberver theory i in the paper On the General Theory of Control Sytem by R.E. Kalman (Kalman, 96), but thi had an accent on the tochatic apect, i.e., minimiing the random error in the tate etimate due to plant noie and meaurement noie. A complete theory for the non tatitical tate etimation problem wa developed by David Luenberger (Luenberger, 964) and followed by a more general paper (Luenberger, 97). Figure 4.79 how the block diagram of the generic SISO linear tate feedback control ytem employing a tate oberver producing the tate etimate, ˆx, needed for the control.

177 4. Control Technique and Performance Aement 76 d b x dt x T c y y r LSF control law r g u Real Plant b k ˆx dt A ˆx e Model correction loop T c ŷ ˆx A Oberver Figure 4.79: LSF control ytem with a tate oberver In the notation of thi ubection, the etimate of a contant parameter, P, i denoted by P. The tate oberver i baed on a model of the real plant driven by the ame control input a applied to the real plant, whoe tate i controlled to follow that of the real plant by mean of a correction loop actuated by the error, e y yˆ, and applied to the integrator input of the plant model via a gain vector, k. By chooing a uitable et of correction loop gain, the error, et, i made to converge toward zero uch that ˆx loop ha ettled, a required. x, once the correction The complete et of equation obeyed by the ytem in Figure 4.79 i a follow. T x Ax bu, y c x (4.63) T, xˆ Axˆ bu k y yˆ yˆ c x ˆ (4.64) u y r gx ˆ (4.65) r

178 4. Control Technique and Performance Aement 77 where k i the oberver correction loop gain vector and T A, b, c are the matrice of the plant model parameter correponding to T A, b, c aumed for the real plant. The need for the correction loop indicated in Figure 4.79 i bet undertood by conidering the ituation that would occur without it. Suppoe that the model correction loop i opened by etting will occur if k k... k k n. Then an error, e y yˆ, ) the model parameter are mimatched, i.e., A, b, c A, b, c ) the initial model and plant tate are different, i.e., x x ˆ 3) a diturbance ignal i preent, i.e., d. Without the correction loop, thi error may grow, but cloure of the correction loop will drive the error to negligible proportion, in the preence of condition (), () and (3), enuring xˆ x if the oberver i deigned correctly. Thi can be achieved by determining the correction loop gain vector, k, uing pole placement but only if the plant i obervable (Dodd, 3). It i traightforward to how that the tate etimation error, ε xˆ x, may be made to converge toward zero from an arbitrary initial condition on the aumption that A A and b b. Then ubtracting equation (4.63) from equation (4.64) yield T T y y xˆ x A xˆ x k ˆ A xˆ x kc x xˆ A xˆ x kc xˆ x, i.e, ε T A kc ε (4.66) If the gain matrix, k, i choen o that the eigenvalue of the matrix, have negative real part, then ε a t. T A kc,

179 4. Control Technique and Performance Aement 78 The gain vector, k, can be choen independently from the control law gain vector, T g, a the eparation principle applie (Franklin et al., ) if A, b, c,, A b c and in practice thi may be aumed. The dynamic behaviour of the correction loop depend on it characteritic equation, which can be found uing Maon formula on Figure 4.79 to obtain det I A Ck (4.67) The pole placement can then be carried out uing the ettling time formula a follow det I A Ck.5 n / T o (4.68) where T o i the oberver ettling time and n i ytem order. To achieve the deired cloed loop tep repone with an arbitrary initial tate etimation error, the ettling time for the oberver, T o, ha to be choen coniderably horter than the ettling time, T o T T, for the control law. It i uual to chooe n T o, to atify /5. In (Franklin et al., ), thi condition i tated in term of the oberver correction loop pole and the main control loop pole. In ome cae the ratio, T / T, i increaed coniderably, beyond the minimum value of 5, uch o a T / T 5 to obtain a atifactory cloed loop ytem repone in the o preence of ignificant plant modelling error. Reducing T o, however, increae the noie content of the tate etimate due to meaurement noie, i.e., noie that originate in the meaurement intrumentation, through an aociated general increae in the element of k. An example of a third order oberver tructure for a plant without finite zero i hown in Figure 4.8, the tructure of the plant model i the ame a that of the aumed plant.

180 4. Control Technique and Performance Aement 79 Y U b k - a k k ˆX ˆX ˆX 3 E Yˆ - a a Figure 4.8: A baic third order oberver tructure The characteritic equation of the oberver i a a a k 3 k k 3 k a k a k a 3 a k a k k a ak ak k (4.69) Uing the Dodd ettling time formula to obtain a non-overhooting cloed loop repone for a third order ytem, n 3, with the oberver ettling time T o yield the deired cloed loop characteritic polynomial, To To To To (4.7) Equating the characteritic polynomial (4.69) and (4.7), and iolating the three control parameter yield, k : 8 8 a k k a (4.7) To To

181 4. Control Technique and Performance Aement 8 k : k : 8 8 a k k k a k (4.7) To To 6 6 a a k a k k k a a k a k (4.73) 3 3 To To Controller Deign Linear State Feedback with Integrator for Steady State Error Elimination The LSF controller hown in Figure 4.78 will have a teady tate error for a contant reference input if there i ) a parameter difference between the plant model and the plant or ) an external diturbance. Thi can be circumvented by adding an integrator in the forward path, hown in the general SISO control ytem of Figure 4.8. Alo the adjutable gain, aignment. LSF controller with teady tate compenation k i, permit deign by pole Plant Y r U x x Y T k i b dt c - g A Figure 4.8: LSF plu integral control Figure 4.8 how an LSF control ytem with a linear model of the throttle valve, for the determination of the controller gain.

182 4. Control Technique and Performance Aement 8 Y r k i - U - - L a Ia Throttle valve k t N pl N - m J x R a k e m N N pl m k kinetic k pring g 3 g Ia g Y Figure 4.8: LSF control of the throttle valve with teady tate compenation Uing Maon rule on Figure 4.8 to get the cloed loop tranfer function (detailed calculation in Appendix A.3) where: b k N t L N pl ki J a m x r Y b Y a a a a (4.74) a k N t L N pl ki J a m x a R k k N g g k a pring t pl 3 pring La Jx La Nm Jx La Jx a a 3 k k k N R k k N g g k J L J N L J L N J L J pring t e pl a kinetic t pl 3 kinetic x a x m a x a m x a x R k g L J L a kinetic a x a 3

183 4. Control Technique and Performance Aement 8 Uing the deired tranfer function (4.) baed on the ettling time formula to obtain a non-overhooting cloed loop tep repone for a fourth order ytem, n 4, the characteritic polynomial i T T 4T 8T 6T (4.75) Equating the characteritic polynomial (4.75) and the denominator from equation (4.74) give g 3 : g : g : k i : 3 R k g 3 R k g T L J L T L J a kinetic 3 a kinetic 3 a x a a x L 35 k k k N R k k N g g k 4T J L J N L J L N J L J pring t e pl a kinetic t pl 3 kinetic x a x m a x a m x a x 35 kpring kt k N e pl Ra kkinetic k kinetic La Nm g g 3 J 4T Jx La Jx Nm La Jx La J x kt N pl 35 R k k N g g k 8T L J L N J L J g a pring t pl 3 pring 3 a x a m x a x 35 R k g k L N a pring 3 pring a m 3 8T La Jx La Jx kt Npl J x a x (4.76) (4.77) (4.78) 565 k N k 565 J N k La (4.79) 6T L N J 6T k N t pl i x m 4 i 4 a m x t pl Figure 4.83 how the cloed loop pole location for the LSF controller and linear throttle valve with a ettling time of T. [ec].

184 4. Control Technique and Performance Aement 83 Im p 75,,3,4 Figure 4.83: The cloed loop pole location of the LSF control loop Re LSF Controller with Integrator for Steady State Error Elimination and Robut Pole Placement To deign a robut et of cloed loop pole, one pole can be placed at a location away from the dominant pole group, invetigated in ubection The minimum pole-to-pole dominance ratio, r ppmin, i ued to inure that the ingle fat pole i located a minimum ditance away from the dominant pole. Thi i alo needed to obtain the robutne. For a fourth order ytem, n 4, with three dominant pole,,,3, all at the ame location, and one fat pole q, the deired characteritic polynomial i 3 p q (4.8) 3 3 3p 3p p q p q p pq p p q p q (4.8) Uing the multiple pole location, p,,3 given by equation (4.) baed on the ettling time formula (4.), for n 3, equation (4.8) become where q q q q 3 3 T T T T T T q T r. 6/ pp (4.8) Equating the denominator from equation (4.74) and the characteritic polynomial (4.8), and olving the reulting equation for the four control parameter yield,

185 4. Control Technique and Performance Aement 84 g 3 : g : 8 Ra kkinetic g 3 8 Ra k kinetic q g3 q L T La Jx La T La Jx a (4.83) 8 8 k k k N R k k N g g k T T J L J N L J L N J L J pring t e pl a kinetic t pl 3 kinetic q x a x m a x a m x a x 8 8 kpring kt k N e pl Ra kkinetic k kinetic La Nm g q g 3 J T T Jx La Jx Nm La Jx La J x kt N pl g : 8 6 R k k N g g k T T L J L N J L J a pring t pl 3 pring q 3 a x a m x a x 8 6 R k g k L N a pring 3 pring a m g q 3 Jx T T La Jx La Jx kt Npl x (4.84) (4.85) k i : 6 k N k 6 J N q k q La (4.86) T L N J T k N t pl i x m 3 i 3 a m x t pl Uing equation (4.6) and inequality (4.7) T f T.5 nrpp min (4.87) For a ytem of n 3 and with 3 dominant pole r min 5.4 (Dodd, 3). Uing thi for T. [ec] pp T f T T.4.5 nrpp min f (4.88) Figure 4.84 how the cloed loop pole location for the LSF controller with a robut pole-to-pole ratio of and a ettling time of T. [ec].

186 4. Control Technique and Performance Aement 85 Im p 4 p 6,,3 Re Figure 4.84: Pole location of an LSF plu integral control loop with a robut pole-to-pole ratio of There are three pole located at,,3 6 and the fat one at, indicating that the fat pole i further away than the recommended threhold of f ( h) 5, according to inequality (4.7). Depite thi, a will be een in ubection , the ytem perform correctly. Thi emphaie the fact that inequality (4.7) i not a rigorou tability criterion but jut a general guideline. f Oberver Aided LSF Control with Integrator for Steady State Error Elimination and Robut Pole Placement A Figure 4.8 how, the LSF controller need to have acce to all the plant tate. In the cae of the throttle valve, the throttle poition and DC motor current can be meaured, and the velocity can be calculated by differentiating the throttle poition. However if the DC motor current meaurement could be eliminated from the control trategy it could ave on the cot of the electronic controller unit (ECU), in which the control trategie are implemented. Figure 4.85 how the LSF controller from ubection combined with the oberver of Figure 4.8.

187 4. Control Technique and Performance Aement 86 Yr - k i - U b Y 3 a a a Throttle valve k k E k Yˆ - b g 3 g - a ˆX ˆX ˆX 3 a a g Figure 4.85: Oberver aided LSF control with integrator for teady tate error elimination The real plant poition output, Y, i ued directly for the integral outer loop. The other option, however, i to ue the etimated poition, Yˆ, from the oberver for thi loop. Thi would reduce the impact of meaurement noie but there could be a difference between the two tranient repone. Simulation and experiment uing both ignal have been done, however, without finding a notable difference in the reult. Thi i due to a) the noie from the throttle meaurement potentiometer being relatively mall and b) the initial oberver and plant tate being well matched. The deign of the LSF plu integral controller and the oberver are carried out eparately, auming that the eparation principle applie (Franklin et al., ). Figure 4.86 how the LSF plu integral controller, ued for the determination of the LSF gain.

188 4. Control Technique and Performance Aement 87 Yr - k i U - b - Y a a g 3 g a g Figure 4.86: Simplified control ytem block diagram for deign of the LSF controller Here a R k L J a pring / a x a k / J k k / L J N / N R k / L J pring x e t a x pl m a kinetic a x a R L k J a / a kinetic x b k N L N J t pl / a m x Uing Maon rule to get the cloed loop tranfer function of Figure 4.86 yield r Y kb i Y a b g a b g a b g k b i (4.89) A et of robut cloed loop pole will be determined a in ubection by uing the ettling time formula for the dominant pole. Equating the characteritic polynomial (4.8) and the denominator from equation (4.89), and olving the reulting equation for the four control parameter yield, g 3 :

189 4. Control Technique and Performance Aement 88 g : g : k i : 8 8 q a b g g q a / b T 3 3 T q a b g g q a / b T T T T q a b g g q a / b T T T T 3 3 (4.9) (4.9) (4.9) 6/ pp where 6 6 q k b k q / b T 3 i i 3 T (4.93) q T r. The minimum ratio for r pp i the ame a for the controller deigned in ubection The oberver gain are determined by uing the ettling time formula a in ubection 4.6. and the working i repeated here for convenience (Figure 4.87). Y U b k - a k k ˆX ˆX ˆX 3 E Yˆ - a a Figure 4.87: Third order oberver tructure

190 4. Control Technique and Performance Aement 89 The oberver characteritic equation (4.69) i 3 a k a k k a ak ak k (4.94) Uing the Dodd ettling time formula to obtain a non-overhooting cloed loop repone for a third order ytem, n 3, with the oberver ettling time T o yield the deired cloed loop characteritic polynomial, To To To To (4.95) Equating the characteritic polynomial (4.94) and (4.95), and olving the reulting three equation for the oberver gain yield, k : k : k : 8 8 a k k a (4.96) To To 8 8 a k k k a k (4.97) To To 6 6 a a k a k k k a a k a k (4.98) 3 3 To To The oberver ettling time, T o, i choen time fater than the ettling time, T, which i ued for the deign of the controller gain. Integrator anti-windup If the controller ettling time, T, i choen much lower than. [ec] the control output will aturate for a ignificant period of time following a large tep reference input change, cauing the controller to overhoot, due to the integrator wind-up. To circumvent thi problem the integrator anti-windup trategy from ubection i added. The final block diagram i hown in Figure 4.88.

191 4. Control Technique and Performance Aement 9 Integrator with Anti-windup Yr - K k i - - U ' u m in u m ax U Y Plant Throttle valve b k - E k k Yˆ - g 3 g a a a g Figure 4.88: Oberver aided LSF with integrator anti-windup ued for the experiment and imulation A uitable value of K. wa firt found by repeatedly imulating the cloed loop ytem tep repone with reducing ettling time and afterward validating the reult experimentally. Figure 4.89 how the reult. The overhoot i reduced by 4% with K., but can be reduce further by increaing the value of K.

192 4. Control Technique and Performance Aement Throttle poition [rad] Deired cloed loop repone Experimental cloed loop repone Deired cloed loop repone Experimental cloed loop repone Time [ec] Time [ec] a) Without integrator anti-windup b) With integrator anti-windup Figure 4.89: Throttle valve tep repone with/without integrator anti-windup (K=.) Retructured Oberver Aided LSF Control with Integrator for Steady State Error Elimination and Robut Pole Placement The idea of retructuring the oberver originate from the oberver baed robut control (OBRC) tructure invetigated in ection 4.7. Thi new tructure i invetigated in view of the particular way the OBRC work but i included here to find out if it enhance the performance of a control ytem employing conventional linear tate feedback control. Figure 4.9 a) how a conventional oberver tructure for a triple integrator plant with an input gain, b m, and the correction loop implemented in three part with the error applied to each integrator input via a eparately adjutable gain, a in ubection The three correction loop part, however, can be combined into one thereby forming a ingle loop tructure a hown in Figure 4.9 (b).

193 4. Control Technique and Performance Aement 9 U b m k k k ˆX ˆX ˆX 3 E Y Yˆ - U b m k k k ˆX ˆX ˆX 3 E Y Yˆ - a) Baic oberver b) Retructured oberver Figure 4.9: Retructure a baic oberver to a ingle correction loop A they tand, the two oberver are mathematically equivalent, and in the OBRC application, the output of the block with tranfer function, k k k, i ued. In practice the differentiator in thi block will amplify any meaurement noie from Y with accentuation of high frequency component. Thi problem can be circumvented by adding a low pa noie filter to the correction loop (Dodd, 3). Figure 4.9 how an oberver having the ingle loop correction with the added third order noie filter, having denominator polynomial coefficient, f, f and f. Provided the correction loop i table in that all the ix pole lie in the left half of the -plane then the error, et, will decay toward zero a required. By chooing the filter pole ufficient large, they will have an inignificant impact on the oberver ability to drive the error et to zero, but in the extreme the meaurement noie problem will reurface through too high a cut-off frequency. Suitable filter gain ( f, f, f ) are found a a part of the oberver gain deign, which i hown later on in thi ection.

194 4. Control Technique and Performance Aement 93 U b m k k k 3 f f f ˆX ˆX ˆX 3 E Y - Yˆ Figure 4.9: Single correction loop oberver with noie filter The retructured oberver combined with LSF controller i hown in Figure 4.9 applied to the throttle valve model intead of the triple integrator. Yr - k i - U b Y 3 a a a Throttle valve k k k 3 f f f E - g 3 g b - a a Yˆ g a Figure 4.9: Single correction loop oberver aided LSF with integrator for teady tate compenation A previouly, the LSF controller and oberver are deigned eparately, auming that the eparation principle applie (Franklin et al., ). Thi enable the LSF controller deign of ubection to be ued. A block diagram of the oberver i hown in Figure 4.93.

195 4. Control Technique and Performance Aement 94 Y k k k 3 f f f E - U b - Yˆ a a a Figure 4.93: Oberver with ingle loop correction controller It will be enured that all ix of the correction loop pole lie in the left half of the -plane by pole placement uing the filter coefficient, f, f and f, a well a the gain, k, k and k. Thi i actually imilar to the approach in the polynomial control of ection 4.8. The oberver characteritic polynomial i given by b k k k a a a f f f 3 3 f a f a f a f a f a f a a f a f a f b k a f a f b k a f b k (4.99) Uing the ettling time formula to deign a non-ocillatory correction loop repone for a ixth order ytem, n 6, 6 6q 5q q 5q 6q q T o where q T (4.) / o and T o i the ettling time for the combined oberver correction loop and the noie filter. It i important to note that with thi approach the effective cut-off frequency will be of the ame order a the correction loop

196 4. Control Technique and Performance Aement 95 bandwidth, which will achieve more effective filtering than by the approach of deigning the filter eparately with a much higher bandwidth than the correction loop o a to avoid intability. Equating the characteritic polynomial equation (4.99) and (4.), and olving the reulting ix equation for the correction loop gain and filtering coefficient yield f : f : f : f a 6q f 6q a (4.) f a f a 5q f 5q a f a (4.) f a f a f a q f q a f a f a (4.3) 3 3 k : k : k : 4 4 a f a f a f b k 5q k 5 q a f a f a f / b (4.4) 5 5 a f a f b k 6q k 6 q a f a f / b (4.5) 6 6 a f b k q k q a f / b (4.6) Simulation and Experimental reult Linear State Feedback with Integrator for Steady State Error Elimination The LSF controller gain are deigned a decribed in ubection , equation (4.76) to (4.79) with T. [ec]. A dither ignal i added to the control ignal to reduce the effect of the tatic friction a decribed in ubection The integrator anti-windup trategy i enabled to minimie the duration of the control aturation following the application of a tep reference

197 4. Control Technique and Performance Aement 96 input. The LSF controller i teted experimentally with three different reference input function a decribed in ubection The dspace ytem i ued for the experiment a explained in ubection The imulation reult preented in thi ubection are obtained with the full nonlinear plant model preented in ection.5. Firt, imulated and experimental repone to a tep reference poition change within the throttle valve top limit are preented in Figure Throttle poition [rad] Reference input Deired cloed loop repone Simulated Experimental. T Time [ec] Figure 4.94: Cloed loop tep repone, from. to.3 [rad] The vertical black line mark the nominal ettling time of T.[ec]. It i evident that both the experimental and imulated repone do not match the deired ettling time and do not exhibit the intended cloed loop dynamic. It hould be mentioned that a imulation with the linear plant model and perfectly matched controller yielded the ideal tep repone. It wa oberved that control aturation did not occur during thee ocillatory repone. Thi behaviour i therefore attributed to tatic friction.

198 4. Control Technique and Performance Aement 97 Figure 4.95 how the imulated and experimental cloed loop repone of the LSF controller with three different reference input function, a decribed in ubection 4.3.3, which indicate ocillation in both the experimental and imulated repone, more o in the experimental one. Throttle poition [rad] Throttle poition [rad] Throttle poition [rad] Ramp Simulated Experiment Pule Drive cycle 5 5 Time [ec] Figure 4.95: Experimental and imulated repone of the LSF controller with teady tate compenation Figure 4.96 how the difference between the experimental and imulated repone. In view of the poor ocillatory experimental performance of thi deign, it i not taken further.

199 4. Control Technique and Performance Aement 98 Ramp Difference between deired repone and actual repone [rad] Pule Drive cycle Time [ec] Figure 4.96: The difference between the deired and the experimental cloed loop repone Since thee difference are coniderable, they are attributed to plant modelling error, particularly in the area of the tick-lip friction, which caue a limit cycle with an integral term included in the controller, and the fact that the LSF controller i not robut with thi particular pole placement LSF Controller with Integrator for Steady State Error Elimination and Robut Pole Placement The LSF controller gain are determined a decribed in ubection , equation (4.83) to (4.86) with T. [ec] and a robut pole-to-pole ratio of 5. A dither ignal i added to the control ignal to reduce the effect of the tatic friction a decribed in ubection The integrator anti-windup trategy i enabled to minimie the aturation during the tep reference input. The LSF controller i teted experimentally with three different reference input function

200 4. Control Technique and Performance Aement 99 a decribed in ubection The dspace ytem i ued for the experiment a explained in ubection The imulation reult preented in thi ubection are obtained with the full nonlinear plant model preented in ection.5. Firt, imulated and experimental repone to a tep reference poition change within the throttle valve top limit are preented in Figure Throttle poition [rad] Reference input Deired cloed loop repone Simulated Experimental. T Time [ec] Figure 4.97: Cloed loop tep repone, from. to.3 [rad] The vertical black line mark the nominal ettling time of T.[ec]. It i evident that both the experimental and imulated repone come cloe to thi at.4 [rad]. Figure 4.98 how the imulated and experimental cloed loop repone.

201 4. Control Technique and Performance Aement Ramp Throttle poition [rad] Throttle poition [rad] Throttle poition [rad] Pule Drive cycle 5 5 Time [ec] Simulated Experiment Figure 4.98: Experimental and imulated repone of the LSF controller with integrator uing robut pole placement Figure 4.99 how very mall difference between the deired and the experimental cloed loop repone (Figure 4.98) Hence the robut pole placement ha been very effective.

202 4. Control Technique and Performance Aement. Ramp Difference between deired repone and actual repone [rad] Pule Drive cycle Time [ec] Figure 4.99: The difference between the deired and the experimental cloed loop repone A decribed in ubection a pring failure can caue the engine to tall due to air tarvation. The behaviour of the LSF controller during a pring break, at t [ec], i imulated uing the nonlinear throttle valve model, and the reult i hown in Figure 4.. It how a good robutne againt the diturbance, with little deviation from the throttle poition demand. The ocillation on the control ignal at that increae in amplitude when the control error, t. [ec] are the added dither ignal yr y, exceed a preet threhold of % of the full cale movement range a decribed in ubection

203 4. Control Technique and Performance Aement Throttle poition [rad] Throttle poition Throttle poition demand Throttle fully cloed Controller output [V] Time [ec] Figure 4.: LSF with integrator controller during a pring failure Figure 4. how the difference between the deired (with ideal cloed loop tranfer function) and imulated cloed loop tep repone with different poleto-pole ratio. The nonlinear throttle valve plant model i ued for thee imulation with a ettling time of T. [ec]. The plot how that the difference i getting maller when the pole-to-pole ratio rie, indicating a better robutne for rpp 5. For r, the pole placement i coincident (non-robut) pp a in ubection and the unacceptable ocillation are again evident.

204 4. Control Technique and Performance Aement 3 Difference between deired repone and imulated repone [rad] Time [ec] Pole ratio = Pole ratio = Pole ratio = Pole ratio = 3 Pole ratio = 4 Figure 4.: Simulated cloed loop repone difference done for a number of different robut pole placement ratio The robutne againt plant parameter deviation away from the nominal value i teted uing the Monte Carlo method decribed in ubection The parameter variation imulation ue the nonlinear throttle valve model with the controller output aturation, integrator anti-windup and dither. The LSF gain are deigned uing a robut pole-to-pole ration of. The reult of the parameter variation imulation i hown in Figure 4. for the maximum poible tandard deviation of %. The figure how the operational envelope for imulation run, where the blue and red line are the minimum and maximum value. The nominal parameter cloed loop controller repone and controller output are hown in black.

205 4. Control Technique and Performance Aement 4 Throttle poition [rad] Nominal Minimum Maximum DC motor voltage [v] Nominal Minimum Maximum Time [ec] Figure 4.: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: % ) Further imulation were performed to find the tandard deviation for other robut pole-to-pole ratio and the reult are hown in Table 4.. Table 4. Pole ratio [%] A tandard deviation from to 6%, Table 4., indicate that the etimated parameter ued for the control gain deign can vary ignificantly before having an impact on the performance of the controller. The enitivity for the LSF controller i analyed in the frequency domain by uing the relationhip of equation (4.3) in ubection Thi i done with the aid of the Matlab Control Sytem Analyi Toolbox and the block diagram of

206 4. Control Technique and Performance Aement 5 Figure 4.3 implemented in Simulink with C output to obtain S db. p D a the input and V a the Y r E k i - U - D - V - L a R a k e Ia m Throttle valve k t Npl N - N N m pl m J x k kinetic k pring g 3 g Ia g Y Figure 4.3: Control tructure ued to analye the enitivity The enitivity of the LSF controller i hown in Figure 4.4 for a robut poleto-pole ratio of. The figure indicate a relatively low enitivity, equivalent to high robutne. Thi correpond well with the time domain reult found by the above pring failure analyi and the Monte Carlo parameter variation imulation, in Table 4.. Increaing the robut pole-to-pole ratio a per Table 4. only enhance the robutne with a mall amount.

207 4. Control Technique and Performance Aement 6 - Senitivity [db] Frequency [rad/ec] Figure 4.4: LSF with integrator enitivity Thi indicate a relatively low enitivity, equivalent to high robutne. Thi correpond well with the time domain reult found by the above pring failure analyi and the Monte Carlo parameter variation imulation Oberver Aided LSF Control with Integrator for Steady State Error Elimination and Robut Pole Placement The LSF controller gain are determined a decribed in ubection , equation (4.9) to (4.93) with T. [ec] and a robut pole-to-pole ratio of 5. The oberver gain are deigned uing equation (4.96) to (4.98) with T o T /. A dither ignal i added to the control ignal to reduce the effect of the tatic friction a decribed in ubection The integrator anti-windup trategy i enabled to minimie the aturation during the tep reference input. The oberver aided LSF controller i teted experimentally with three different

208 4. Control Technique and Performance Aement 7 reference input function a decribed in ubection The dspace ytem i ued for the experiment a explained in ubection The imulation reult preented in thi ubection are obtained with the full nonlinear plant model preented in ection.5. Firt, imulated and experimental repone to a tep reference poition change within the throttle valve top limit are preented in Figure Throttle poition [rad] Reference input Deired cloed loop repone Simulated Experimental. T Time [ec] Figure 4.5: Cloed loop tep repone, from. to.3 [rad] The vertical black line mark the nominal ettling time of T.[ec]. It i evident that both the experimental and imulated repone come cloe to thi at.4 [rad]. Figure 4.6 how the imulated and experimental cloed loop repone.

209 4. Control Technique and Performance Aement 8 Throttle poition [rad] Throttle poition [rad] Throttle poition [rad] Ramp Simulated Experiment Pule Drive cycle 5 5 Time [ec] Figure 4.6: Experimental and imulated repone of the oberver aided LSF controller with integrator uing robut pole placement Figure 4.7 how very mall difference between the deired and the experimental cloed loop repone (Figure 4.6), again indicating that the robut pole placement ha been very effective.

210 4. Control Technique and Performance Aement 9. Ramp Difference between deired repone and actual repone [rad] Pule Drive cycle Time [ec] Figure 4.7: The difference between the deired and the experimental cloed loop repone A decribed in ubection a pring failure can caue the engine to tall due to air tarvation. The behaviour of the oberver aided LSF controller during a pring break, at t [ec], i imulated uing the nonlinear throttle valve model, and the reult i hown in Figure 4.8. It how a good robutne againt the diturbance, with little deviation from the throttle poition demand. The ocillation on the control ignal at ignal that increae in amplitude when the control error, t. [ec] are the added dither yr y, exceed a preet threhold of % of the full cale movement range a decribed in ubection

211 4. Control Technique and Performance Aement Throttle poition [rad] Throttle poition Throttle poition demand Throttle fully cloed Controller output [V] Time [ec] Figure 4.8: Oberver aided LSF controller during a pring failure The robutne againt plant parameter variation away from the nominal value i teted uing the Monte Carlo method decribed in ubection The parameter variation imulation ue the nonlinear throttle valve model with the controller output aturation, integrator anti-windup and dither. The LSF controller and the oberver gain are deigned a decribed in the tart of thi ubection. The reult of the parameter variation imulation i hown in Figure 4.9 for the maximum poible tandard deviation of 5%. The figure how the operational envelope for imulation run, where the blue and red line are the minimum and maximum value. The nominal parameter cloed loop controller repone and controller output are hown in black.

212 4. Control Technique and Performance Aement Throttle poition [rad] Nominal Minimum Maximum DC motor voltage [v] Nominal Minimum Maximum Time [ec] Figure 4.9: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: 5% ) Thi indicate that the plant parameter can deviate ignificantly from the nominal value before having an advere impact on the performance of the controller. The enitivity for the oberver aided LSF controller i analyed in the frequency domain by uing the relationhip of equation (4.3) in ubection Thi i done with the aid of the Matlab Control Sytem Analyi Toolbox and the block diagram of Figure 4. implemented in Simulink with the input and C V a the output to obtain S db. p D a

213 4. Control Technique and Performance Aement D Yr - E k i - U b Y - V 3 a a a Throttle valve k k E k Yˆ - b g 3 g - a ˆX ˆX ˆX 3 a a g Figure 4.: Control tructure ued to analye the enitivity The enitivity of the oberver aided LSF controller i hown in Figure 4. for a robut pole-to-pole ratio of 5 and T T /. The figure indicate a relatively low enitivity, equivalent to high robutne. Thi correpond well with the time domain reult found by the above pring failure analyi and the Monte Carlo parameter variation imulation. o

214 4. Control Technique and Performance Aement 3 - Senitivity [db] Frequency [rad/ec] Figure 4.: Oberver aided LSF control with integral term enitivity Retructured Oberver Aided LSF Control with Integrator for Steady State Error Elimination and Robut Pole Placement The LSF controller gain are determined a decribed in ubection , equation (4.9) to (4.93) with T. [ec] and a robut pole-to-pole ratio of 5. The oberver gain are deigned uing equation (4.) to (4.6), ubection , with T T /. A dither ignal i added to the control ignal to reduce o the effect of the tatic friction a decribed in ubection The integrator anti-windup trategy i enabled to minimie the aturation during the tep reference input. The oberver aided LSF controller i teted experimentally with three different reference input function a decribed in ubection The dspace ytem i ued for the experiment a explained in ubection 4.3..

215 4. Control Technique and Performance Aement 4 The imulation reult preented in thi ubection are obtained with the full nonlinear plant model preented in ection.5. Firt, imulated and experimental repone to a tep reference poition change within the throttle valve top limit are preented in Figure Throttle poition [rad] Reference input Deired cloed loop repone Simulated Experimental. T Time [ec] Figure 4.: Cloed loop tep repone, from. to.3 [rad] The vertical black line mark the nominal ettling time of T.[ec]. It i evident that both the experimental and imulated repone come cloe to thi at.4 [rad]. Figure 4.3 how the imulated and experimental cloed loop repone.

216 4. Control Technique and Performance Aement 5 Throttle poition [rad] Throttle poition [rad] Throttle poition [rad] Ramp Simulated Experiment Pule Drive cycle 5 5 Time [ec] Figure 4.3: Experimental and imulated repone of the retructured oberver aided LSF controller with integrator uing robut pole placement Figure 4.4 how very mall difference between the deired and the experimental cloed loop repone (Figure 4.3), indicating yet again that the robut pole placement ha been very effective.

217 4. Control Technique and Performance Aement 6. Ramp Difference between deired repone and actual repone [rad] Pule Drive cycle Time [ec] Figure 4.4: The difference between the deired and the experimental cloed loop repone A decribed in ubection a pring failure can caue the engine to tall due to air tarvation. The behaviour of the oberver aided LSF controller during a pring break, at t [ec], i imulated uing the nonlinear throttle valve model, and the reult i hown in Figure 4.5. It how a good robutne againt the diturbance, with little deviation from the throttle poition demand. The ocillation on the control ignal at ignal that increae in amplitude when the control error, t. [ec] are the added dither yr y, exceed a preet threhold of % of the full cale movement range a decribed in ubection

218 4. Control Technique and Performance Aement 7 Throttle poition [rad] Throttle poition Throttle poition demand Throttle fully cloed Controller output [V] Time [ec] Figure 4.5: Retructured oberver aided LSF with integrator during a pring failure The robutne againt plant parameter deviation away from the nominal value i teted uing the Monte Carlo method decribed in ubection The parameter variation imulation ue the nonlinear throttle valve model with the controller output aturation, integrator anti-windup and dither. The LSF controller and the oberver gain are deigned a decribed in the tart of thi ubection. The reult of the parameter variation imulation i hown in Figure 4.6 for the maximum poible tandard deviation of 5%. The figure how the operational envelope for imulation run, where the blue and red line are the minimum and maximum value. The nominal parameter cloed loop controller repone and controller output are hown in black.

219 4. Control Technique and Performance Aement 8 Throttle poition [rad] Nominal Minimum Maximum DC motor voltage [v] Nominal Minimum Maximum Time [ec] Figure 4.6: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: 5% ) Thi indicate that the plant parameter can deviate ignificantly from the nominal value before having an advere impact on the performance of the controller. The enitivity for the retructured oberver aided LSF controller i analyed in the frequency domain by uing the relationhip of equation (4.3) in ubection Thi i done with the aid of the Matlab Control Sytem Analyi Toolbox and the block diagram of Figure 4.7 implemented in Simulink with the input and C V a the output to obtain S db. p D a

220 4. Control Technique and Performance Aement 9 D Yr - E k i U - b Y - V 3 a a a Throttle valve k k k 3 f f f - g 3 g b - a a Yˆ g a Figure 4.7: Control tructure ued to analye the enitivity The enitivity of the retructured oberver aided LSF controller i hown in Figure 4.8 for a robut pole-to-pole ratio of 5 and T T /. The figure indicate a relatively low enitivity, equivalent to high robutne. Thi correpond well with the time domain reult found by the above pring failure analyi and the Monte Carlo parameter variation imulation. o

221 4. Control Technique and Performance Aement - Senitivity [db] Frequency [rad/ec] Figure 4.8: Retructured oberver aided LSF control with integrator enitivity

222 4. Control Technique and Performance Aement 4.7 Oberver Baed Robut Control 4.7. Introduction and Brief Hitory The oberver baed robut control (OBRC) control technique wa intigated in (Dodd, 7) and further invetigated in (Stadler et al., 7) (Stadler, 8) (Fallahi, 3). In OBRC, an oberver i ued to etimate the external diturbance referred to the control input. Plant parametric uncertaintie can be repreented by part of uch an external diturbance. The etimate from the oberver i therefore a combination of plant parametric error and phyical external diturbance, if they exit. Thi diturbance etimate i added to the control ignal with the aim of cancelling both the phyical diturbance and the effect of the plant parameter error. Let G be a model of the plant with tranfer function, m U, exit o that Y Y e U m Ue a hown in Figure 4.9. Plant G Y G, where an input, Plant model G m Ym Y Figure 4.9: Plant and model mimatch The ignal Ue can be etimated by uing the retructured oberver from ubection , called the ingle correction loop oberver. Thi employ a correction loop controller, o H, a hown in Figure 4..

223 4. Control Technique and Performance Aement U Plant Y G ˆ Correction loop U e E controller H o - Plant model Gm Ym Figure 4.: Correction loop controller ued for etimating the diturbance Ue The cloed loop tranfer function, Ym U, depend on Gm and Ho well a a G, but if the correction loop controller i deigned to achieve E, then Y Y o that Y U Y U G m require Ho to embody relatively large gain. m. Thi, however, Figure 4. how Figure 4. with the plant repreented by it model and the diturbance input a in Figure 4.9. U' U - Ue Plant Plant model G m Y Uˆe Correction loop E controller H o - Figure 4.: Subtraction of Um U from control input to compenate ˆe Plant model G m Ym U. e

224 4. Control Technique and Performance Aement 3 Through the action of the correction loop controller, the error, with negligible proportion regardle of diturbance etimate, diturbance, Uˆe E, i maintained U. It then become apparent that the, may be ued to reduce the effect of the actual U, by forming a primary control input, U e the plant and it model, from which Uˆe that in the hypothetical, ideal cae of U U would be complete, giving a perfectly robut ytem. ˆe, applied to both i ubtracted, a hown. It i evident e, the diturbance cancellation Next, it i poible to implify the connection on the left hand ide of the block of Figure 4., reulting in the block diagram of Figure 4., which i functionally identical. U ' U Plant Y - G Uˆ e Correction loop controller H o E - Plant model Gm Ym Figure 4.: Input converion block diagram It i now evident that the primary control variable, U, i applied directly to the plant model. Since the tate variable of the plant model are available, they can be ued to complete a linear tate feedback model control loop, a hown in the complete OBRC block diagram of Figure 4.3.

225 4. Control Technique and Performance Aement 4 Yr LSF control law U ' U Plant Y - G Uˆ e Correction loop controller H o E - X m Plant model Gm Ym Figure 4.3: Overall OBRC tructure for a ingle input, ingle output plant 4.7. Controller Deign A third order OBRC with a LSF controller i hown in Figure 4.4 applicable to the throttle valve. Y r r - U' - g 3 g g U Uˆe b b a a a - 3 k k k f f f 3 a Throttle valve Correction loop controller a a Yˆ Y E -

226 4. Control Technique and Performance Aement 5 Figure 4.4: OBRC tructure with a LSF controller For tandard linear tate feedback controller uing oberver, the eparation principle applie (Dodd, 3) through the aumption of the plant and it model being identical. In OBRC, thi aumption cannot be made but if the correction loop controller i firt deigned to maintain E regardle of the real plant, then the deign of the LSF controller can be performed ubequently and eparately from the oberver a if the eparation principle did apply. In view of the forgoing dicuion, the complete cloed loop control ytem dynamic are thoe of the LSF applied to the plant model. With reference to Figure 4.4, the relevant cloed loop tranfer function i Yˆ r b Y a b g a b g a b g r 3 3, (4.7) where r i the reference input caling coefficient needed for a unity DC gain. The plant model tranfer function coefficient are calculated uing the etimate of the phyical plant parameter dicued in Chapter, a follow. a R k L J a pring / a x a k / J k k / L J N / N R k / L J pring x e t a x pl m a kinetic a x a R L k J a / a kinetic x b k N L N J t pl / a m x Deigning the control loop by the method of pole aignment uing the ettling time formula for n 3, the deired characteritic polynomial i

227 4. Control Technique and Performance Aement T T T T (4.8) where T i the ettling time of the tep repone (5% criterion). Equating the polynomial (4.8) and the denominator from equation (4.7) yield the following gain formulae. g 3 : g : g : 8 8 a b g g a / b T 3 3 T 8 8 a b g g a / b T T 6 6 a b g g a / b T 3 3 T (4.9) (4.) (4.) To find the value for r the final value theorem i ued with a unit-tep input ( U / ), Yˆ Yˆ lim Y r Yr r b lim 3 a bg 3 a bg a bg r b a r g a b g b (4.) For the oberver, the correction loop controller parameter are found a in ubection The correction loop characteritic polynomial i

228 4. Control Technique and Performance Aement 7 f a f a f a f a f a f a a f a f a f b k a f a f b k a f b k (4.3) An initial imulation wa carried out uing multiple pole placement for the oberver model correction loop and for the linear tate feedback control loop, uing the ettling time formula for each, with variou ratio between the two ettling time, the oberver ettling time being horter in all cae. The imulation reult proved atifactory with a linear plant model but unfortunately intability occurred with the full nonlinear model, thi poor reult being confirmed experimentally. The problem wa attributed to the tick-lip friction. Subequently the invetigation wa carried further by conidering other pole placement pattern. A the correct operation of the oberver model correction loop i critical, different pole placement were conidered for thi, the linear tate feedback pole being left a previouly. Satifactory performance wa found for the ditributed oberver correction loop pole pattern hown in Figure 4.5. Introduced by the correction loop controller Im Introduced by the LSF tructure Re Figure 4.5: Individual pole placement ued for OBRC In an attempt to achieve the deired robutne, the oberver correction loop pole were eparated from the multiple pole et of the linear tate feedback loop whoe ettling time wa T, uing the pole-to-pole dominance ratio, r pp i, (Dodd, 3), for the i th correction loop pole, p i, i,,,6. The reaon for thi i that the dominated pole (in thi cae the oberver correction loop pole) give the ytem more robutne through being produced by larger gain that

229 4. Control Technique and Performance Aement 8 give the control loop tiffne that would not be obtained without thi dominance. Given that no coincident correction loop pole location worked, the only choice wa to pread the pole. Thi wa done heuritically but with the contraint that r pp i r ppmin. Referring to Figure 4.5, there are three dominant pole due to the linear tate feedback loop and ix dominated pole, p i. Let them be arranged uch that pk pk, k,3,,6. The oberver correction loop pole with the mot influence i the one with the magnitude, p. Let the dominant (LSF loop) pole location be p d. Then according to (Dodd, 3), where rppmin 5.4. The LSF loop ettling time i given by p r pp min p (4.4) d T.5 n 6 6 pd (4.5) p p T d n3 d Then inequality (4.4) may be rewritten a p 6r pp min In thi cae, T. [ec]. Then p 34. The oberver correction loop pole are choen via r ppi a (4.6) T rppi 5.5,6,8.5,,4.5,57.5 i,,,6 The correponding pole magnitude are then choen a (4.7) 6r pp i pi 33,36,5,,55,345 T i,,,6 (4.8) Uing equation (4.7) the deired characteritic polynomial i then, p p... p (4.9) 6 The calculation of the characteritic polynomial i performed by uing Matlab numerically and the reult i given in the form,

230 4. Control Technique and Performance Aement 9 where n d d d d d d (4.) d are the calculated coefficient for n..5 Equating the polynomial (4.3) and (4.) yield f : f : f : f a d5 f d5 a (4.) f af a d4 f d4 af a (4.) f af af a d3 f d3 af af a (4.3) k : k : k : a f a f a f b k d k d a f a f a f / b (4.4) a f a f b k d k d a f a f / b (4.5) a f b k d k d a f / b (4.6) Simulation and Experimental reult The OBRC gain are determined a decribed in ubection The LSF tructure gain are found uing equation (4.9) to (4.) with T. [ec] and the OBRC correction loop gain are found by placing the ix pole individually with a robut pole ratio of equation (4.7). A dither ignal i added to the control ignal to reduce the effect of the tatic friction a decribed in ubection No integrator anti-windup trategy i enabled during the experiment. The OBRC i teted experimentally with three different reference

231 4. Control Technique and Performance Aement 3 input function a decribed in ubection The dspace ytem i ued for the experiment a explained in ubection The imulation reult preented in thi ubection are obtained with the full nonlinear plant model preented in ection.5. Firt, imulated and experimental repone to a tep reference poition change within the throttle valve top limit are preented in Figure Throttle poition [rad] Reference input Deired cloed loop repone Simulated Experimental. T Time [ec] Figure 4.6: Cloed loop tep repone, from. to.3 [rad] The vertical black line mark the nominal ettling time of T.[ec]. It i evident that both the experimental and imulated repone come cloe to thi at.4 [rad]. Figure 4.7 how the imulated and experimental cloed loop repone.

232 4. Control Technique and Performance Aement 3 Throttle poition [rad] Throttle poition [rad] Throttle poition [rad] Ramp Simulated Experiment Pule Drive cycle 5 5 Time [ec] Figure 4.7: Experimental and imulated repone of the OBRC Figure 4.8 how very mall difference between the deired and the experimental cloed loop repone (Figure 4.7), reulting from the robutne of thi control technique.

233 4. Control Technique and Performance Aement 3. Ramp Difference between deired repone and actual repone [rad] Pule Drive cycle Time [ec] Figure 4.8: The difference between the deired and the experimental cloed loop repone A decribed in ubection a pring failure can caue the engine to tall due to air tarvation. The behaviour of the OBRC during a pring break, at t [ec], i imulated uing the nonlinear throttle valve model, and the reult i hown in Figure 4.9. It how a good robutne againt the diturbance, with little deviation from the throttle poition demand.

234 4. Control Technique and Performance Aement 33 Throttle poition [rad] Throttle poition Throttle poition demand Throttle fully cloed Controller output [V] Time [ec] Figure 4.9: OBRC during a pring failure The robutne againt plant parameter variation away from the nominal value i teted uing the Monte Carlo method decribed in ubection The parameter variation imulation ue the nonlinear throttle valve model with the controller output aturation, integrator anti-windup and dither. The reult of the parameter variation imulation i hown in Figure 4.3 for the maximum poible tandard deviation of %. The figure how the operational envelope for imulation run, where the blue and red line are the minimum and maximum value. The nominal parameter cloed loop controller repone and controller output are hown in black.

235 4. Control Technique and Performance Aement 34 Throttle poition [rad] Nominal Minimum Maximum DC motor voltage [v] Nominal Minimum Maximum Time [ec] Figure 4.3: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: % ) Thi indicate that the plant parameter can deviate ignificantly from the nominal value before having an advere impact on the performance of the controller. The enitivity for the OBRC i analyed in the frequency domain by uing the relationhip of equation (4.3) in ubection Thi i done with the aid of the Matlab Control Sytem Analyi Toolbox and the block diagram of Figure 4.3 implemented in Simulink with D a the input and S db. C to obtain p V a the output

236 4. Control Technique and Performance Aement 35 Yr E - - D U - b Y V 3 a a a Throttle valve k k k 3 f f f Correction loop controller - g 3 g b - a a g a Figure 4.3: Control tructure ued to analye the enitivity The enitivity of the OBRC i hown in Figure 4.3. The figure indicate an average enitivity, equivalent to average robutne. Thi correpond well with the time domain reult found by the above pring failure analyi and the Monte Carlo parameter variation imulation.

237 4. Control Technique and Performance Aement 36 4 Senitivity [db] Frequency [rad/ec] Figure 4.3: OBRC enitivity

238 4. Control Technique and Performance Aement Polynomial Control 4.8. Introduction and Brief Hitory Any linear controller for SISO plant with two input, output Y, r Y and one U can be repreented by the general tranfer function relationhip, U G Y G Y (4.7) r r y Thi i illutrated pictorially in Figure 4.33 for a control ytem employing a PID controller. a) b) Y r - k p k d S U G( ) Plant Y Y r Gr - U G( ) Plant Y k i Gy PID controller Linear SISO controller Figure 4.33: a) PID controller converted into the b) baic linear SISO controller form Uing Maon formula on the PID control tructure in Figure 4.33 a) with G removed, kd kp k i kd kp k i U Yr Y (4.8) Comparing equation (4.7) and equation (4.8) then how that the two tranfer function of the general linear SISO form are identical and given by G G k k k /. r y d p i The polynomial controller ha a different tructure to the general SISO form of Figure 4.33 but i alo general in that it can repreent any other linear SISO controller. It name i due to the polynomial of it tranfer function being

239 4. Control Technique and Performance Aement 38 hown explicitly, it deign being via the choice of their coefficient. It i particularly intereting ince it only require Y and r Y a input but can be deigned by complete pole aignment for any linear plant if an accurate model i available. Before thi, the only available linear controller with thi capability wa the linear tate feedback controller upported by an oberver that had to be deigned eparately. Before proceeding further, it mut be tated that the polynomial controller ha preciely the ame tructure a the now well etablihed RST controller, which i a digital controller formulated in the z-domain (Landau and Zito, 6). The RST controller i already ued by the proce indutry in a wide range of different application. Figure 4.34 how the block diagram of a general control ytem employing thi controller. Yr z Tz - Sz Rz Uz Bz Az Plant Yz R-S-T controller Figure 4.34: Digital R-S-T controller canonical tructure The acronym, RST, jut conit of the ymbol ued for the polynomial of the controller. It i really a polynomial controller and thi more decriptive title i preferred in thi work, particularly a it will be conidered in the -domain. The purpoe of the component of the RST controller are a follow. Rz : Polynomial of a pecific degree whoe coefficient can be ued for deign of the controller by pole aignment.

240 4. Control Technique and Performance Aement 39 Sz : A polynomial of a certain minimum order that render the controller realiable by enuring that it i a caual ytem (i.e., it output can be calculated uing preent and pat known value of it input, in contrat to future value that are unknown) and whoe coefficient can alo be ued for deign of the controller by pole aignment. Thi alo provide a filtering function to alleviate the effect of meaurement noie. Tz : Reference input polynomial that can be ued to cancel the cloed loop pole for dynamic lag compenation. The tranfer function relationhip of the RST controller can be expreed in the form of equation (4.7), but in the z-domain, uing Figure 4.34, a follow. R z T z U z Y z Yr z S z S z (4.9) It i therefore poible to realie a wide range of different controller within the RST controller. The deign procedure for determining the polynomial of the RST controller given by (Landau and Zito, 6) i different from that developed here, which i a traightforward pole placement procedure baed on the ettling time formula of Dodd (3), that applie in the continuou -domain a well a the dicrete z-domain Baic Polynomial Controller The Polynomial Controller tructure, hown in Figure 4.35, i identical to that of the RST controller but a different notation i ued for the controller polynomial to avoid confuion with the RST controller, which i trictly formulated in the dicrete time domain.

241 4. Control Technique and Performance Aement 4 Y r R ( ) Yr Z Precompenator ' - F ( ) H ( ) U B A Plant Y Polynomial controller Figure 4.35: The general tructure of the Polynomial control ytem The polynomial controller i conidered in the -domain henceforth or the z- domain (Dodd, 3). The tranfer function relationhip of thi controller i obtained from Figure 4.35 a R U Yr H Y F Z (4.3) The purpoe of the component polynomial are imilar to thoe already tated in ubection 4.8. in the z-domain for the RST controller but, in part, have different interpretation in the continuou -domain and are therefore given again a follow. H : Feedback polynomial with a minimum number of coefficient equal to F : it degree, complete pole placement. n h, ufficient to enable deign of the controller by Filter polynomial with a minimum number of coefficient equal to it degree, n f, that avoid having to etimate the output derivative that would otherwie be required to implement function, H, the tranfer / F alo forming a low pa filter that avoid amplification of high frequency component of meaurement noie that would otherwie occur due to the preence of H.

242 4. Control Technique and Performance Aement 4 R : Z : Pre-compenator numerator polynomial for cancellation of the cloed loop pole, if needed, to achieve zero dynamic lag between and yt. Pre-compenator denominator polynomial for cancellation of any cloed loop zero, if neceary, to prevent over/under-hooting of the tep repone that would otherwie occur, or a part of the proce of achieving zero dynamic lag, in conjunction with yr t R. In the throttle valve application, error due to tick-lip friction are reduced by uing hort ettling time, while repecting the tability limit et by the ampling time and any poition enor lag, to tighten the control loop but thi can caue too much throttle activity in normal operation. To overcome thi problem, the hort control loop ettling time i maintained while the overall ettling time i increaed uing external pole placement via Z. The pre-compenator polynomial may be written a n r z i i i and (4.3) i R r Z z i i It hould be noted that to independently place all the zero of the precompenator, only n of the n coefficient of only n z of the nz coefficient of r r n R are needed. Similarly, Z are needed to independently place all the pole of the pre-compenator. To fix the DC gain, ytem to unity, which i uual, the DC gain, Y Y Y Y, of the control r r r, of the precompenator ha to be et equal to the reciprocal of the DC gain, Y Y, of the feedback control loop. For thi, one more coefficient i needed. Thi will be done by normaliation with repect to the coefficient of nz in Z, i.e., r

243 4. Control Technique and Performance Aement 4 Then the DC gain of the control ytem i r. z (4.3) The cloed loop tranfer function obtained from Figure 4.35 i r Y R B Y Z A F B H (4.33) The pole placement deign of the feedback control loop, however, may be carried out independently from that of the pre-compenator and thi ue it characteritic polynomial, BH A F (4.34) The coefficient of the polynomial, A will be een, n h f i i i and (4.35) i H h F f i i F may be normalied with repect to the coefficient of without preventing the ytem from being deignable by complete pole placement (Dodd, 3). Thu are ued for the pole placement. n nf f (4.36) n f The plant tranfer function polynomial coefficient are given by n a b i i i and (4.37) i A a B b i i where na nb. No lo of generality i uffered by normaliation w.r.t. the n coefficient of na. Hence a (4.38) n a The order of the feedback control loop i equal to the degree of the characteritic polynomial (4.34), which i

244 4. Control Technique and Performance Aement 43 a f b h N max deg A F,deg B H max n n, n n (4.39) A ytem deign contraint i that the degree of H i limited o a to avoid any algebraic loop, requiring that the relative degree of the loop tranfer function i poitive. By inpection of Figure 4.35, the loop tranfer function i The relative degree i therefore and ince r, it follow that L B H. A F (4.4) r n n n n (4.4) a f b h n n n n (4.4) a f b h In view of equation (4.39) and inequality (4.4), the ytem order ha to be N n n (4.43) For complete pole placement to be poible, then the total number of independently adjutable controller parameter ha to be equal to N. The only adjutable parameter are the nh coefficient of coefficient of F. Hence In view of (4.43) and (4.44), a f H and the n n N (4.44) h f n n n n n n (4.45) h f a f h a A final deign contraint i that the degree of n f F need to be ufficiently high to avoid amplification of the high frequency component of meaurement noie due to the differentiating action of H. With reference to Figure 4.35, thi require the relative degree of the tranfer function, H F, to be nonnegative. Thu Then from inequality (4.46) and equation (4.45), n f n (4.46) h

245 4. Control Technique and Performance Aement 44 nf nh na (4.47) Finally, to achieve a unity cloed loop DC gain the coefficient r of R can be ued. The required value for r can be found by etting in tranfer function (4.33) and then equating thi to unity. Thu, recalling z (equation (4.3), K Y r b a f b h r DC, CL Yr z af bh b (4.48) The detail of the pole placement deign will be preented in ubection applied to the throttle linear valve model Polynomial Control with Additional Integrator for Zero Steady State Error in the Step Repone The polynomial control in it original form will allow a teady tate error to occur in the tep repone due to the friction of the throttle valve. Thi problem can be circumvented by adding an extra integrator in the forward path of the control loop, a hown in Figure Y r R ( ) Yr Z Precompenator ' - F ( ) H ( ) ' U Additional integrator U b a a a 3 Throttle valve Y Polynomial controller Figure 4.36: Polynomial control of throttle valve with additional integrator It i evident from thi figure that the additional integrator and the throttle valve together can be conidered a the plant to control with control input, U, for the purpoe of determining the controller polynomial. Thi will be referred to a

246 4. Control Technique and Performance Aement 45 the augmented plant. Then the theory of ub-ection 4.8. can be applied directly. The augmented plant order i that for the throttle valve plu that of the the extra integrator, i.e., N 3 4, the augmented plant tranfer function being G' Y B b b U A a a a a a a in the tandard form, where b a a a a (4.49) a a, a a, a a and a (4.5) Controller Deign Polynomial Control with Additional Integrator for Steady State Error Elimination The deign of the control ytem of Figure 4.36 by pole aignment will now be preented together with the reult of a preliminary imulation that neceitated a change to robut pole aignment, which i explained within thi ubection. Firt, the degree of the controller polynomial, (4.47). Thu F, n n 4 3 n n 3 h a f a Hence for the minimum order ytem, nf 3. Hence H are determined by F f f f 3 (4.5) 3 H h h h h (4.5) 3

247 4. Control Technique and Performance Aement 46 Uing equation (4.48) to determine r yield r a f b h h (4.53) b a The detail of the polynomial controller are hown in Figure Y r r - f f f 3 Additional integrator U U b a a a 3 Throttle valve Y h h h h 3 3 Polynomial controller Figure 4.37: Control ytem of Figure 4.36 howing controller polynomial Equating the cloed loop characteritic polynomial (4.34) to the deired characteritic polynomial, which ha to be of the ame degree, yield B H A F a a a f f f b h3 h h h 3 a f a f a f bh3 a f a f bh af b h b h a f a f a f a a f a f f d6 d5 d4 d3 d d d (4.54) where d i i,,,6, are the deired polynomial coefficient. Figure 4.38 how the tructure ued to implement the polynomial controller of Figure 4.37 where the controller polynomial coefficient (i.e., the controller gain) are found by firt etting the deired characteritic polynomial uing equation (4.54).

248 4. Control Technique and Performance Aement 47 Y r r f f f h h h h Additional integrator U U b a a a 3 Throttle valve Y Figure 4.38: Implementation of the Polynomial control with additional integrator Initially the pole placement wa carried out uing the ettling time formula (5% criterion) with multiple placement of all even pole. Then d d d d d d d n p TS n n7 7p p 35p 35p p 7p p where p T. The deired polynomial coefficient are then d 7 p, d p, d 35 p, d 35 p, d p, d 7p and d p (4.55) Thi worked with the linear throttle valve model in the imulation. Unfortunately, however, the ytem immediately aturated when ubtituting the nonlinear model of ubection.5. Thi wa attributed to the tick-lip friction combined with the control aturation limiting. In view of the potential of robut control technique to accommodate plant nonlinearitie, the technique of robut pole aignment introduced in ubection wa applied. Thi entailed placing one pole with a relatively large value, implicitly introducing relatively high gain to give the robutne while the remaining pole, which dominate the cloed loop dynamic, are placed coincidently uing the ettling time formula. The deired characteritic polynomial i then N.5N 3 D T T rpp (4.56)

249 4. Control Technique and Performance Aement 48 Here, r pp i the pole-to-pole dominance ratio (Dodd, 3), which enure that the N pole placed uing the ettling time formula are dominant a well a enuring that the ingle dominated pole i ufficiently large to give the ytem robutne. Again, thi did not work correctly with the full nonlinear model of ubection.5, even with rpp 6. The relevant cloed loop tep repone imulation i hown in Figure 4.39, with Yr [rad]. It i evident that the controller output aturate and throttle valve hit the hard top. Throttle poition [rad].5.5 Throttle poition Throttle poition demand Controller output [V] Time [ec] Figure 4.39: Cloed loop tep repone with one fat pole In a further attempt to olve the problem, the robut pole placement principle wa extended to have more than one fat pole, with the idea that thi could produce more robutne. Here, the et of N cloed loop pole are plit into to two group of n d dominant pole and n r robut pole. Then the n d dominant pole are placed uing the ettling time formula and the n r robut pole are linked to the dominant pole by the pole-to-pole dominance ratio r pp cloed loop characteritic polynomial i. Then the deired

250 4. Control Technique and Performance Aement 49 n n n d n r.5.5 d r D T T rpp (4.57) where nd nr N. Good imulation reult were obtained with nd 4 and nr 3, thee are preented in ubection The deign equation for thi cae are a follow. The deired cloed loop characteritic polynomial i where p 7.5 T and q 6r pp T. Hence, Subtituting 4 3 D p q (4.58) q 4p 6p qp 3q p 8qp q p q p qp 8q p 4q p qp q p 6q p q p 4q p q p D p p p p q q q (4.59) D in equation (4.54) uing (4.59) yield the following equation for the controller polynomial coefficient. f : f : f : a f 3q 4p f 3q 4p a (4.6) a f a f 6p qp 3q f 6p qp 3q a f a (4.6) a a f a f f p qp q p q f 4p 8qp q p q a a f a f 3 3 (4.6)

251 4. Control Technique and Performance Aement 5 h 3 : h : af af af bh3 p qp 8q p 4q p h p qp 8q p 4 q p a f a f a f / b (4.63) h : h : a f a f b h qp q p q p h 3qp q p 6 q p a f a f / b (4.64) a f b h 3q p 4q p h 3q p 4 q p a f / b (4.65) b h q p h q p / b (4.66) r : r h (4.67) Reduced Order Polynomial Control with Additional Integrator for Steady State Error Elimination In thi ubection, a polynomial controller i deigned uing the reduced order throttle valve model from ubection.6, equation (.4). It will be recalled that thi wa obtained by removing the armature inductance reulting in a econd order model. A in ubection , an additional integrator i inerted at the plant input for the purpoe of avoiding a teady tate error in the tep repone. The complete control ytem block diagram i hown in Figure 4.4.

252 4. Control Technique and Performance Aement 5 Y r r - f f Additional integrator U U b a a Throttle valve Y h h h Polynomial controller Figure 4.4: Polynomial control with additional integrator and a econd order plant model ued for the controller deign Following the ame procedure a in ubection , the cloed loop characteritic polynomial, AF BH characteritic polynomial, D, to yield i equated to the deired 3 a a f f b h h h af b h b h f a f a f a a f a f b h d d d d d where d i i,,,4, are the deired polynomial coefficient. (4.68) Good imulation reult were obtained uing the deired cloed loop characteritic polynomial given by (4.57) for the robut pole placement with n 3 and n. Thee are preented in ubection The deign d r equation for thi cae are a follow. The deired cloed loop characteritic polynomial i where p 6 T and q 4.5r pp T. Hence, 3 D p q (4.69)

253 4. Control Technique and Performance Aement 5 Subtituting q 3p 3p 6qp q D p p p q q p qp q p p q qp q p (4.7) D in equation (4.68) uing (4.7) yield the following equation for the controller polynomial coefficient. f : f : f a q 3p f q 3p a (4.7) f af a p qp q 3 6 f 3p 6qp q af a (4.7) h : h : h : 3 af af bh p 6qp 3q p h p 6qp 3 q p af af / b (4.73) af b h 3p q qp h 3p q qp af / b (4.74) 3 3 b h q p h q p / b (4.75) r : r h (4.76) Obtaining Larger Settling Time uing the Precompenator Figure 4.4 how the reult of an invetigation by imulation of attempting to increae the ettling time through value likely to be pecified by uer of the control ytem.

254 4. Control Technique and Performance Aement T =.3 econd T =. econd T =.4 econd. Throttle poition [rad] Time [ec] Figure 4.4: Simulated cloed loop repone tep repone at T..3.4 [ec] A i evident, if a ettling time longer than. econd i required, the gain of the control loop are inufficient to produce enough actuator torque to overcome the tatic friction reulting in limit cycle ocillation. Thee increae in amplitude a T i increaed becaue a larger poition error i needed to generate the minimum torque needed to produce movement. In the extreme, for T.4 ec., the theoretical limit cycle amplitude exceed the end top limit o no limit cycle can occur, the ytem intead taying at the limit. Thi problem can be circumvented by introducing a precompenator with a dominant pole placed to yield the required ettling time, having previouly deigned the polynomial control loop to have a ufficiently hort ettling time and aociated high gain to overcome the tatic friction. The polynomial control loop wa deigned uing robut pole placement a decribed in ubection The pecified ettling time i now that of the

255 4. Control Technique and Performance Aement 54 precompenator which will be denoted by T p. Then the polynomial control loop ettling time (not now realied by the actual ytem) i et to a value horter than the minimum needed to overcome the tatic friction problem. Setting T T /5 wa found to be ufficient. p The precompenator i deigned uing ettling time formula (4.) with n. Thu r ' R Y r Z Y 3 Tp 3 T p (4.77) Figure 4.4 how a cloed loop tep repone imulation of the polynomial controller and the nonlinear throttle valve model, uing a precompenator with T..3.4, noting that, for comparion purpoe, they are the ame p a the value of T et to produce the reult of Figure Throttle poition [rad] Difference between deired repone and actual repone [rad] Time [ec] Figure 4.4: Simulated cloed loop repone tep repone uing a precompenator with the ettling time..3.4 T ec p

256 4. Control Technique and Performance Aement 55 It i evident that atifactory performance ha now been achieved Simulation and Experimental reult Polynomial Control with Additional Integrator for Steady State Error Elimination The polynomial controller gain are determined a decribed in ubection , equation (4.6) to (4.67) with T. [ec], two group of pole, n 4 and n 3, and a robut pole-to-pole ratio of 4. A dither ignal i added p q to the control ignal to reduce the effect of the tatic friction a decribed in ubection The integrator anti-windup trategy i enabled to minimie the aturation during the tep reference input. The polynomial controller i teted experimentally with three different reference input function a decribed in ubection The dspace ytem i ued for the experiment a explained in ubection The imulation reult preented in thi ubection are obtained with the full nonlinear plant model preented in ection.5. Firt, imulated and experimental repone to a tep reference poition change within the throttle valve top limit are preented in Figure 4.43.

257 4. Control Technique and Performance Aement Throttle poition [rad] Reference input Deired cloed loop repone Simulated Experimental. T Time [ec] Figure 4.43: Cloed loop tep repone, from. to.3 [rad] The vertical black line mark the nominal ettling time of T.[ec]. It i evident that both the experimental and imulated repone come cloe to thi at.4 [rad]. Figure 4.44 how the imulated and experimental cloed loop repone.

258 4. Control Technique and Performance Aement 57 Throttle poition [rad] Throttle poition [rad] Throttle poition [rad] Ramp Simulated Experiment Pule Drive cycle 5 5 Time [ec] Figure 4.44: Experimental and imulated repone of the polynomial controller Figure 4.45 how very mall difference between the deired and the experimental cloed loop repone (Figure 4.44), which i attributed to the robut pole placement.

259 4. Control Technique and Performance Aement 58. Ramp Difference between deired repone and actual repone [rad] Pule Drive cycle Time [ec] Figure 4.45: The difference between the deired and the experimental cloed loop repone A decribed in ubection a pring failure can caue the engine to tall due to air tarvation. The behaviour of the polynomial controller during a pring break, at t [ec], i imulated uing the nonlinear throttle valve model, and the reult i hown in Figure It how a good robutne againt the diturbance, with little deviation from the throttle poition demand.

260 4. Control Technique and Performance Aement 59 Throttle poition [rad] Throttle poition Throttle poition demand Throttle fully cloed Controller output [V] Time [ec] Figure 4.46: Polynomial controller during a pring failure The robutne againt plant parameter variation away from the nominal value i teted uing the Monte Carlo method decribed in ubection The parameter variation imulation ue the nonlinear throttle valve model with the controller output aturation, integrator anti-windup and dither. The polynomial controller gain are deigned uing a robut pole-to-pole ration of 4. The reult of the parameter variation imulation i hown in Figure 4.47 for the maximum poible tandard deviation of 4%. The figure how the operational envelope for imulation run, where the blue and red line are the minimum and maximum value. The nominal parameter cloed loop controller repone and controller output are hown in black.

261 4. Control Technique and Performance Aement 6 Throttle poition [rad] Nominal Minimum Maximum DC motor voltage [v] Nominal Minimum Maximum Time [ec] Figure 4.47: Maximum / minimum throttle poition and DC motor voltage envelope at a pole group ratio = 4 (Standard deviation: 4% ) Thi indicate that the plant parameter can deviate ignificantly from the nominal value before having an advere impact on the performance of the controller. The enitivity for the implemented polynomial controller i analyed in the frequency domain by uing the relationhip of equation (4.3) in ubection Thi i done with the aid of the Matlab Control Sytem Analyi Toolbox and the block diagram of Figure 4.48 implemented in Simulink with the input and C V a the output to obtain S db. p D a

262 4. Control Technique and Performance Aement 6 f f f Y - U - Y r r h h h h Additional integrator D V b 3 a a a Throttle valve Figure 4.48: Control tructure ued to analye the external diturbance enitivity The enitivity reult i hown in Figure Senitivity [db] Frequency [rad/ec] Figure 4.49: Polynomial control enitivity Thi indicate a relatively low enitivity, equivalent to high robutne. Thi correpond well with the time domain reult found by the above pring failure analyi and the Monte Carlo parameter variation imulation.

263 4. Control Technique and Performance Aement Reduced Order Polynomial Control with Additional Integrator for Steady State Error Elimination Figure 4.5 how the implementation verion of the polynomial controller of Figure 4.4 applied to the non-reduced third order throttle valve model to tet the ability of the impler controller deigned uing the reduced order model. f f Y U Y r r h h h Additional integrator 3 b a a a Throttle valve Figure 4.5: Implementation of the Polynomial control with additional integrator The polynomial controller gain are determined a decribed in ubection , equation (4.7) to (4.76) with T. [ec], two group of pole, n 3 and n, and a robut pole-to-pole ratio of 6. A dither ignal i added p q to the control ignal to reduce the effect of the tatic friction a decribed in ubection The integrator anti-windup trategy i enabled to minimie the aturation during the tep reference input. The polynomial controller i teted experimentally with three different reference input function a decribed in ubection The dspace ytem i ued for the experiment a explained in ubection The imulation reult preented in thi ubection are obtained with the full nonlinear plant model preented in ection.5.

264 4. Control Technique and Performance Aement 63 Firt, imulated and experimental repone to a tep reference poition change within the throttle valve top limit are preented in Figure Throttle poition [rad] Reference input Deired cloed loop repone Simulated Experimental. T Time [ec] Figure 4.5: Cloed loop tep repone, from. to.3 [rad] The vertical black line mark the nominal ettling time of T.[ec]. It i evident that both the experimental and imulated repone come cloe to thi at.4 [rad]. Figure 4.5 how the imulated and experimental cloed loop repone.

265 4. Control Technique and Performance Aement 64 Throttle poition [rad] Throttle poition [rad] Throttle poition [rad] Ramp Pule Drive cycle Time [ec] Simulated Experiment Figure 4.5: Experimental and imulated repone of the polynomial controller Figure 4.53 how very mall difference between the deired and the experimental cloed loop repone (Figure 4.5), which i attributed to the robut pole placement.

266 4. Control Technique and Performance Aement 65. Ramp Difference between deired repone and actual repone [rad] Pule Drive cycle Time [ec] Figure 4.53: The difference between the deired and the experimental cloed loop repone A decribed in ubection a pring failure can caue the engine to tall due to air tarvation. The behaviour of the LSF controller during a pring break, at t [ec], i imulated uing the nonlinear throttle valve model, and the reult i hown in Figure It how a good robutne againt the diturbance, with little deviation from the throttle poition demand. The ocillation on the control ignal at that increae in amplitude when the control error, t. [ec] are the added dither ignal yr y, exceed a preet threhold of % of the full cale movement range a decribed in ubection

267 4. Control Technique and Performance Aement 66 Throttle poition [rad] Throttle poition Throttle poition demand Throttle fully cloed Controller output [V] Time [ec] Figure 4.54: Polynomial controller during a pring failure The robutne againt plant parameter variation away from the nominal value i teted uing the Monte Carlo method decribed in ubection The parameter variation imulation ue the nonlinear throttle valve model with the controller output aturation, integrator anti-windup and dither. The polynomial controller gain are deigned uing a robut pole-to-pole ration of 6. The reult of the parameter variation imulation i hown in Figure 4.55 for the maximum poible tandard deviation of 5%. The figure how the operational envelope for imulation run, where the blue and red line are the minimum and maximum value. The nominal parameter cloed loop controller repone and controller output are hown in black.

268 4. Control Technique and Performance Aement 67 Throttle poition [rad] Nominal Minimum Maximum DC motor voltage [v] Nominal Minimum Maximum Time [ec] Figure 4.55: Maximum / minimum throttle poition and DC motor voltage envelope at a pole group ratio = 6 (Standard deviation: 5% ) Thi indicate that the plant parameter can deviate ignificantly from the nominal value before having an advere impact on the performance of the controller. The enitivity for the polynomial controller i analyed in the frequency domain by uing the relationhip of equation (4.3) in ubection Thi i done with the aid of the Matlab Control Sytem Analyi Toolbox and the block diagram of Figure 4.56 implemented in Simulink with C V a the output to obtain S db. p D a the input and

269 4. Control Technique and Performance Aement 68 f f Y U - Y r r h h h Additional integrator D V b 3 a a a Throttle valve Figure 4.56: Control tructure ued to analye the external diturbance enitivity The enitivity reult i hown in Figure Senitivity [db] Frequency [rad/ec] Figure 4.57: Polynomial control enitivity Thi indicate a relatively low enitivity, equivalent to high robutne. Thi correpond well with the time domain reult found by the above pring failure analyi and the Monte Carlo parameter variation imulation.

270 4. Control Technique and Performance Aement Obtaining Larger Settling Time uing the Precompenator The polynomial controller gain are determined a decribed in ubection , equation (4.6) to (4.67) with T T / 5 [ec], two group of pole, n 4 and n 3, a robut pole-to-pole ratio of 6 and a precompenator with p q a ettling time Tp. [ec]. A dither ignal i added to the control ignal to reduce the effect of the tatic friction a decribed in ubection The integrator anti-windup trategy i enabled to minimie the aturation during the tep reference input. The polynomial controller i teted experimentally with three different reference input function a decribed in ubection The dspace ytem i ued for the experiment a explained in ubection p The imulation reult preented in thi ubection are obtained with the full nonlinear plant model preented in ection.5. Firt, imulated and experimental repone to a tep reference poition change within the throttle valve top limit are preented in Figure Throttle poition [rad] Reference input Deired cloed loop repone Simulated Experimental. T Time [ec] Figure 4.58: Cloed loop tep repone, from. to.3 [rad]

271 4. Control Technique and Performance Aement 7 The vertical black line mark the nominal ettling time of T. [ec]. It i evident that both the experimental and imulated repone come cloe to thi at.4 [rad]. Figure 4.59 how the imulated and experimental cloed loop repone. Throttle poition [rad] Throttle poition [rad] Throttle poition [rad] Ramp.5 Simulated Experiment Pule Drive cycle Time [ec] Figure 4.59: Experimental and imulated repone of the polynomial controller Figure 4.6 how mall difference between the deired and the experimental cloed loop repone (Figure 4.59), which, again, attributed to the robut pole placement.

272 4. Control Technique and Performance Aement 7. Ramp Difference between deired repone and actual repone [rad] Pule Drive cycle Time [ec] Figure 4.6: The difference between the deired and the experimental cloed loop repone A decribed in ubection a pring failure can caue the engine to tall due to air tarvation. The behaviour of the polynomial controller during a pring break, at t [ec], i imulated uing the nonlinear throttle valve model, and the reult i hown in Figure 4.6. It how a good robutne againt the diturbance, with little deviation from the throttle poition demand.

273 4. Control Technique and Performance Aement 7 Throttle poition [rad] Throttle poition Throttle poition demand Throttle fully cloed Controller output [V] Time [ec] Figure 4.6: Polynomial controller during a pring failure The robutne againt plant parameter variation away from the nominal value i teted uing the Monte Carlo method decribed in ubection The parameter variation imulation ue the nonlinear throttle valve model with the controller output aturation, integrator anti-windup and dither. The polynomial controller gain are deigned uing a robut pole-to-pole ration of 6. The reult of the parameter variation imulation i hown in Figure 4.6 for the maximum poible tandard deviation of %. The figure how the operational envelope for imulation run, where the blue and red line are the minimum and maximum value. The nominal parameter cloed loop controller repone and controller output are hown in black.

274 4. Control Technique and Performance Aement 73 Throttle poition [rad] Nominal Minimum Maximum DC motor voltage [v] Nominal Minimum Maximum Time [ec] Figure 4.6: Maximum / minimum throttle poition and DC motor voltage envelope at a T. ec and a pole group ratio = 6 (Standard deviation: % ) Thi indicate that the plant parameter can deviate ignificantly from the nominal value before having an advere impact on the performance of the controller.

275 4. Control Technique and Performance Aement Sliding Mode Control and it Relative 4.9. Introduction and Brief Hitory Variable Structure Control (VSC) ytem were introduced by Emelyanov among other in 96, but only in Ruian. The VSC i a form of a dicontinuou nonlinear controller in which, effectively, the control variable of the plant i witched between the output of two controller connected permanently to the plant meaurement variable, a hown in Figure y r Controller with tructure Controller with tructure Switch u u Plant y Switching law x Variable Structure Controller Figure 4.63: A baic variable tructure control ytem The purpoe of uch a controller i to achieve robutne with repect to plant parametric uncertaintie and external diturbance, i.e., the dynamic repone of yt to y r t i a intended in the control ytem deign and i not ignificantly influenced by unknown plant parameter change and external diturbance. Whether or not thi i attained depend upon the tate repreentation, the only uitable one being T r x k y y y y (4.78)

276 4. Control Technique and Performance Aement 75 where r i the relative degree of the plant and k i a known contant. For a linear plant with tranfer function, N, r deg D deg N D y u. In more general term, and in the time domain, which applie to nonlinear a well a linear plant, r i the lowet order derivative of algebraically on change in y r ut, in the ene that a tep change in t at the ame intant. yt that depend ut caue a tep The intended operational mode of the VSC ytem i the liding mode. The witching law in Figure 4.63 i of the form gn, u t S x y (4.79) r The baic control objective i to drive S, y condition. The equation r x to zero and then maintain thi r S x, y (4.8) define a witching boundary of dimenion, n, in the n -dimenional tate pace for a SISO plant. In the literature (Utkin et al., 999) the term, witching manifold, i frequently ued, which refer to everal concurrent witching boundarie, one for each control variable of a multivariable plant. Since the reearch undertaken here i retricted to SISO plant, the term, witching boundary will be ued. If the tate trajectorie, t x for max u u and x t for u u max, under witching law equation (4.79) are directed toward the boundary equation (4.8) from both ide in a finite region on the boundary including the point where the tate trajectory firt meet the witching boundary, then after thi event, the control, ut, will witch at a high frequency and with a varying mark-pace ratio o a to hold the tate point on the boundary. Generally, the tate point i free to move in the boundary. The manner in which it move depend on the cloed loop ytem differential equation, which i determined by equation (4.8). If S, y x i deigned correctly, then the tate r

277 4. Control Technique and Performance Aement 76 will converge to a point at which y y and the behaviour of yt during thi r convergence will be a deired. Since during thi convergence, the tate point appear to lide in the boundary, then the ytem i defined to be operating in a liding mode. Specifically, if the tate repreentation i according to equation (4.78) then the cloed loop differential equation determined by equation (4.8) become r r S y, y, y,, y, y. (4.8) In the 97 a book from Itki (Referenced in (Spurgeon and Edward, 998)) and a paper (Utkin, 977) wa publihed in Englih invetigating the Sliding Mode Control (SMC). Since then a variety of publication have emerged; Book (Spurgeon and Edward, 998), (Sabanovic et al., 4), (Dodd, 3), and in particular one paper hould be mentioned A Control Engineer' Guide to Sliding Mode Control (Utkin et al., 999) which contain many good reference Baic Sliding Mode Control The claic double integrator plant i ued a an example for the introduction to liding mode control: ut b x t x t y t Figure 4.64: Double integrator plant x t where u t i the input to the plant and x, x are the two tate. The dynamic are x t x t (4.8) x t b u t (4.83)

278 4. Control Technique and Performance Aement 77 To viualie the olution for the double integrator plant the two tate variable can be plotted againt each other in a two dimenional pace called the tate plane, in thi cae referred to a the phae plane, a one tate variable i the derivative of the other. The olution can be found by forming the tate trajectory differential equation, dividing equation (4.8) by (4.83) x t dx / dt b u t x t dx / dt x t dx b u t dx x t (4.84) If ut i aumed contant, a olution can be found by integrating equation (4.84) where c i a contant. x dx b u t dx (4.85) x t b x t u t c The above olution indicate that a parabolic hape will form for different value of c, and dependent on the ign of u the parabola will be open to the right or left. State trajectory X X X -. X a ) u u m ax b ) u u m ax Figure 4.65: Phae portrait for a double integrator plant with b

279 4. Control Technique and Performance Aement 78 Figure 4.65 how a imulation of the double integrator plant differential equation in Matlab for the tate x. u umax and u umax, with different initial value of To apply a Bang-Bang controller to the double integrator plant the control law ha to be deigned in uch a way that it force the cloed loop ytem to have an equilibrium at,, x x for x x,. Looking at Figure 4.65 (a) or (b) how no ign of the plant coming to ret at,, r ref u u x x for which max, indicating that cloed loop control i needed. In liding mode control, the firt tep i to form a bang-bang tate control law for which u u Figure Bang-Bang Control law P lant S w itching y u r S m ax u y x g x, u Function S x, y r u m ax Signum x S y h x max, a hown in Figure 4.66: Block diagram of a Bang-Bang controller for a SISO plant The linear witching function for a SISO plant i where x : Plant tate w : Contant n : Number of plant tate S x, y w x y w x... w x (4.86) r r n n

280 4. Control Technique and Performance Aement 79 In the baic SMC a ignum function i ued to repreent the witching of u. Thu where u umax gn S x, x, y r (4.87) for S, gns for S, for S (4.88) The ignum function can be regarded a a high gain in view of it infinite lope at the origin a hown in Figure 4.66, which will make the control trategy very robut againt parameter variation and diturbance, uch a the failing pring in a throttle valve. The witching function for the double integrator plant i S x, x, yr w x yr w x (4.89) Figure 4.67 how a cloed loop phae portrait uing thi witching function with y r x and a lope of. The plot i done with different initial tate x x, r to how how they move toward the et point at the origin.,

281 4. Control Technique and Performance Aement 8 Sliding boudary X X Figure 4.67: Cloed loop phae portrait of a double integrator plant for w, w Figure 4.68 how the bang-bang controller output ut and the plant tate x t, x t a a function of time with the initial condition, x x..5 x, y x Plant tate ut Time [ec] Figure 4.68: Cloed loop repone and bang-bang controller output of a double integrator plant

282 4. Control Technique and Performance Aement 8 In Figure 4.69 one trajectory i hown to viualie the different tate the Bang-Bang controller goe through with the ame initial condition a Figure The initial condition for the ytem i at P (at time = [ec]) where u u max making the tate move toward the poitive part of x. There will be no change in the controller output from P till P. When the trajectory reache the point P (at time = [ec]) it enter the boundary layer and the controller output will witch to u u max. After point P i reached, the controller will witch u at an infinite frequency, in theory, with a continuouly varying markpace ratio to keep the plant tate on the witching boundary. Thi will, in thi cae, move the tate toward the centre of the phae portrait (for yr ). P X P3. -. P4 X P Figure 4.69: An example of a trajectory for the double integrator plant. When the ytem tate ( x, x ) are on witching boundary egment between P and P4 (which could be referred to a the liding boundary in view of the ytem behaviour), the ytem will be governed by equation (4.89) wx t yr t w x t x t x t yr t (4.9) T where Tc w / wand y r i the et point. c

283 4. Control Technique and Performance Aement 8 The cloed loop ytem i only of order of one while the plant order i two. Thi i due to the fact that the SMC force the tate to tay on the witching boundary, thereby removing one degree of freedom of motion in the tate pace. In general, if the plant order i n, the cloed loop ytem will be of order n. The general linear witching function i S y, x w x y w x w x... w x (4.9) r n r 3 3 n n The baic aim of any controller i to control the output of the plant, match the reference input yr yt, to t. Thi i not generally hard to achieve but the additional aim here i to attain a precribed cloed loop dynamic repone uch a pecified ettling time and zero overhoot. If the tate repreentation of the plant model upon which the controller deign i baed i uch that the output y t depend on the tate, x, x... x n, via plant parameter. A change in thee plant parameter can caue a difference in the repone to change in the reference input and poibly violate the control ytem performance pecification. Thi point i illutrated in Figure 4.7. Plant S w itching A u y r S m ax a u y Function S u S x, x, y r m ax x B Signum b x Figure 4.7: Plant output not directly linked to the plant tate Suppoe yr t i ufficiently lowly varying for liding motion to be maintained. Depite thi, if the plant parameter, A or B, change, then the repone of yt

284 4. Control Technique and Performance Aement 83 to a given yr t will change, thereby defeating the object of achieving robutne. To circumvent thi iue the tate repreentation of equation (4.78) can be ued, in which the tate conit of the plant output, yt, and it derivative. Replacing the tate in equation (4.9) with thee derivative yield n,... S y y w y y w y w y w y (4.9) r r 3 n The order of the liding boundary i n which make one of the gain in equation (4.9) redundant. Hence dividing equation (4.9) with w yield w w w, 3 n n S yr y y yr y y... y (4.93) w w w and therefore the contant are redefined a follow. n,... S y y y y w y w y w y (4.94) r r n Note that the order, n, i ued in equation (4.94) rather than the relative degree, r, ince the plant conidered in thi reearch programme. i.e., the throttle valve, ha no finite zero in the tranfer function. The Equivalent Control Method decribed in (Spurgeon and Edward, 998) (Trivedi and Bandyopadhyay,, Xinghuo et al., 8), can be ued to analye the behaviour of the ytem when it i on, or cloe to the liding manifold (line egment between P and P4 in Figure 4.69). It decribe the continuou fictitiou control variable, witching actual control output, ueq t, that i equivalent to the rapidly ut, in the ene that it would keep the tate trajectory in the witching boundary. It i, in fact, the hort term average value of

285 4. Control Technique and Performance Aement 84 the rapidly witching phyical control, whoe form i ueful in analying the ytem behaviour. The algebraic olution for ueq t i found by auming that the tate are on the witching boundary by etting the witching function S, implying S. For a linear SISO plant x A x B u (4.95) where u i the control input and x are the tate of the plant. For yr t the witching boundary i Hence T S x w x w x w x w x w x (4.96) n n w T w S T x AxBu T T ueq B A w w x (4.97) Figure 4.7 how again the imulation of Figure 4.68 but with ut () replaced u t. The value u t i a low pa filtered verion of by () eq filtered to check the correctne of u t. eq u t which i ued

286 4. Control Technique and Performance Aement 85 u [V] Plant tate x, y x u eq u filtered Time [ec] Figure 4.7: Diplay of equivalent control for imulation of Figure Method for Eliminating or Reducing the Effect of Control Chatter Two method of control chattering elimination are preented here with the throttle valve a the plant. Figure 4.7 how the baic liding mode control ytem a a tarting point. Y r - S - u max u max Switch w U b a a a 3 Throttle valve Y w Figure 4.7: An example of a baic SMC for a throttle valve plant

287 4. Control Technique and Performance Aement 86 The cloed loop ytem can be deigned to yield the deired repone uing the ettling time formula a in the previou ection. The characteritic equation of Figure 4.7, for S and Y uing equation (4.94). Thu r Y w w w w w (4.98) Uing the ettling time formula for n, T T 4T (4.99) Comparing equation (4.98) and (4.99) yield w 4 T / 8 and w 9/ T w. A already pointed out, the baic SMC will witch at an infinite frequency with variable mark-pace ratio to keep the tate on the witching boundary. In practice, however, the ampling frequency will be finite which will allow the tate trajectory to execute a zig-zag movement about the liding boundary a hown in Figure 4.73 (b). The aociated control witching can be damaging for the actuator or ytem. Thi phenomenon i known a control chatter.

288 4. Control Technique and Performance Aement Controller output u [v] 5-5 Throttle velocity [Rad/ec] Time [ec] Throttle poition [rad] a) Control variable b) State trajectory Figure 4.73: Baic liding mode controller behaviour Control Smoothing Integrator Method To avoid the control chatter an integrator can be inerted between the controller output and the plant to mooth out the witching (Dodd and Walker, 99) (Vittek et al., 8) (Sira-Ramirez, 993, Teng and Chen, ). Thi will be referred to a the control moothing integrator method. Figure 4.74 how the new controller with the extra integrator between the controller and the plant in which U will be a filtered value of achieve the cloed loop ytem repone i done in two tep. U'. The control ytem deign to a) The additional integrator i aumed to be a part of the plant which will increae it order by one. b) To accommodate for the increae in the plant order, the baic liding mode controller order ha to be increaed by one.

289 ' 4. Control Technique and Performance Aement 88 u Y m ax r U b Y a a a u m ax U New plant Switch Throttle valve w 3 w w Figure 4.74: Baic SMC with a Control Smoothing Integrator Figure 4.75 how a imulation reult of a throttle valve ytem controlled by a baic SMC with and without a control moothing integrator. Plot (a) how the coil current of the baic SMC without the integrator where the amplitude level of the current > /- 4 amp. Plot (b) how that the current amplitude ha now decreaed by more than a factor of, through adding the integrator on the output of the witching function.

290 4. Control Technique and Performance Aement 89 5 a) Without the control moothing integrator.4 b) With the control moothing integrator DC motor current [A] Time [ec] Time [ec] Figure 4.75: DC-Motor current level. Sample frequency = 3 Hz. The deired cloed loop repone for T. [ec] and a ampling time of /3 [ec] i the ame for both plot. In the cae with the extra integrator the witch level, u max, i a number in the oftware that can be made a large a poible to maximie the range of tate over which the robutne i retained. Thi, however, i limited ultimately by the aturation limit of the phyical control variable, which i /- Volt for the throttle valve ued Boundary Layer Method Another way to avoid the control chatter i to replace the ignum witching function in the forward path with a high gain, i.e., high lope, tranfer characteritic with aturation (Dodd, 4) (Dodd and Vittek, 9). Thu u umax Saturation K S y r, y (4.)

291 4. Control Technique and Performance Aement 9 where u will be aturated at when S u / K max umax. Thi introduce a region traddling the original witching boundary, between two aturation boundarie. Thi will make the tate move continuouly toward the et point on the liding urface while the control alo behave moothly, approximating the equivalent control decribed in ubection The global behaviour of thi high gain controller i imilar to the baic SMC. In theory, with K the boundary layer hrink to infiniteimal proportion and make the tate trajectory identical to that of the baic ideal SMC but with the equivalent control replacing the original control witching at infinite frequency. Figure 4.76 how thi high gain SMC which will be called the boundary layer SMC. u Y m ax U b r Y r 3 - a a a u m ax Gradient = K Throttle valve q q c c Figure 4.76: Boundary Layer Sliding Mode Control In thi cae there are three controller parameter, q c, q c and K, determined uing the ettling time formula and pole placement a in the previou chapter. The gain, K, i finite which reult in a teady tate error, a in the baic LSF, but thi i generally much maller than that of the LSF if the latter i deigned with a multiple cloed loop pole to achieve the ame ettling time. r i found by letting Y Y. / r in the tranfer function for

292 4. Control Technique and Performance Aement Controller Deign Control Smoothing Integrator Method It i well known that meaurement noie will make it impractical to implement differentiator without noie filtering. The differentiator are combined with imple low pa filter, a hown in Figure u Y m ax r U b Y a a a u m ax Sat Switch Throttle valve w 3 w f f f f f f w Figure 4.77: Switching boundary SMC with meaurement noie filtering and integrator with aturation The filter frequency, f, i choen to be higher than the bandwidth of the cloed loop ytem, in thi cae 5 [rad/ec]. An integrator anti windup cheme i needed to prevent large error excurion that would otherwie occur due to the hardware impoed aturation limit. In thi intance the integrator i connected directly to the plant of which the voltage limit are known. A imple integrator with aturation could be ued to perform thi function. A can be een in Figure 4.75 (b) the method reduce but doe not entirely eliminate the control chatter. To minimie the control chatter even more, therefore, a variable witching level could be ued on the input of the integrator a hown in Figure The idea i that the integrator output hould ideally be contant in teady tate condition. By making the witching level maller a

293 4. Control Technique and Performance Aement 9 the teady tate i approached, i.e., a the triangular integrator output, et reduce, the ocillation limit of ut, will be made ufficiently mall for the application in hand. For the tet with the throttle valve the variable witch level A i choen linear a a function of the et, a follow., et. A 5 et.,. et.5 7, et.5 (4.) Table E x ABS Y r U b Y X a a a Switch A Sat Throttle valve w 3 w f f f f f f w Figure 4.78: Practicable SMC with control moothing integrator and variable gain to minimie control chatter for mall poition error The minimum and maximum witch level depend on the application and have been choen to uite the throttle valve. The influence here i a relatively high level of tatic friction which caue a ignificant limit cycle of the cloed loop ytem if the minimum level i too low. If the maximum level i choen too high, the cloed loop ytem could be untable. The level have to be found

294 4. Control Technique and Performance Aement 93 empirically, but a imulation can be ued for determining preliminary etting that can be applied in the initial experiment with reaonable confidence. The characteritic equation for the ytem of Figure 4.78 in the liding mode for which S and Y i given by r Y w w w w w w w w (4.) The ettling time formula for n 3 yield the deired characteritic equation, T T T T (4.3) Equating the left hand ide of polynomial (4.) and (4.3) yield w 3 T 3 / 6 w 8 / T w 3 w 8 / T w 3 (4.4) Boundary Layer Method A tated before, noie filtering i needed with the output differentiator, a hown in Figure 4.79.

295 4. Control Technique and Performance Aement 94 Y U b r Y r 3 K - a a a Throttle valve q q c c q q f f Figure 4.79: Boundary layer method SMC with meaurement noie filtering By replacing the polynomial with a tranfer function it i poible to chooe a uitable filter characteritic via q f and q f. All the gain can be deigned uing the ettling time formula and pole placement or robut pole placement decribed in ubection The teady tate error that would be caued by finite gain, K, in the ytem of Figure 4.79 can be eliminated by an integrator added in the forward path a hown in Figure 4.8. r Y - K i K - K - u min u max U b a a a 3 Throttle valve Y qc qc qf qf Figure 4.8: Boundary layer method SMC with integrator in the forward path and meaurement noie filtering Note that integrator anti-windup ha been added a in the ytem of Figure 4.78.

296 4. Control Technique and Performance Aement 95 Uing Maon formula on Figure 4.8, without aturation, yield Y KiKb qf qf Y q a q a q a a q a q a Kb aq f aqf Kb qc Ki a q Kb q K q Kb K q r f f f f f f c i f i f (4.5) To enhance robutne the deired cloed loop characteritic equation i deigned by robut pole placement uing two group of pole, one for the deired ettling time and a fater one for the filter: p T /6 and p r min p c f pp c where rpp min 6., which i the minimum pole-to-pole ratio for dominance in a ixth order ytem with 3 dominant pole. Thu p p p p p p p p f c f c f c f c p p 9p p p p 3p p p p 9p p f c f c f c f c f c f c 3 3 3pf pc pf pc pf pc (4.6) Equating the denominator of equation (4.5) with the left hand ide of equation (4.6), to determine the three control parameter yield q : f q a 3 p p q 3 p p a f f c f f c q : f K : 3 9 q a q a p p p p f f f c f c q 3 p p 9p p a q a f f c f c f a q a q a Kb p p p p p p f f f c f c f c K p p 9p p p p a q a q a b 3 3 f c f c f c f f

297 4. Control Technique and Performance Aement 96 q : c q : c K i : 3 9 a q a q Kb q K p p p p p p f f c i f c f c f c q 3p p p p 9p p a q a q Kb K c f c f c f c f f I 3 a q Kb q K q p p p p f c i f f c f c q 3p p p p a q Kb K q c f c f c f I f Kb K q p p K p p b Kq i f f c i f c f Simulation and Experimental reult Control Smoothing Integrator Method The liding mode controller gain, w, w and w 3, are determined a decribed in ubection , equation (4.4) with T. [ec], f 5 [rad/ec] and a maximum gain of A 7. A dither ignal i added to the control ignal to reduce the effect of the tatic friction a decribed in ubection The integrator aturation limit are enabled during the tep reference input. The SMC i teted experimentally with three different reference input function a decribed in ubection The dspace ytem i ued for the experiment a explained in ubection The imulation reult preented in thi ubection are obtained with the full nonlinear plant model preented in ection.5. Firt, imulated and experimental repone to a tep reference poition change within the throttle valve top limit are preented in Figure 4.8.

298 4. Control Technique and Performance Aement Throttle poition [rad] Reference input Deired cloed loop repone Simulated Experimental. T Time [ec] Figure 4.8: Cloed loop tep repone, from. to.3 [rad] The vertical black line mark the nominal ettling time of T.[ec]. It i evident that both the experimental and imulated repone come cloe to thi at.4 [rad]. Figure 4.8 how the imulated and experimental cloed loop repone.

299 4. Control Technique and Performance Aement 98 Ramp Throttle poition [rad] Throttle poition [rad] Throttle poition [rad].5 Simulated Experiment Pule Drive cycle Time [ec] Figure 4.8: Experimental and imulated repone of the SMC - control moothing integrator method The experiment, Figure 4.8, were repeated without control dither added. The reult howed very little difference between the two et of experiment. Thi i due to the witching element in the control trategy, hown in Figure 4.78, which will generate control chatter, even with the moothing integrator in loop. Figure 4.83 how the difference between the deired and the experimental cloed loop repone (Figure 4.8), indicating that the tracking i adequate but not a good a would be expected uing a robut control technique.

300 4. Control Technique and Performance Aement 99. Ramp Difference between deired repone and actual repone [rad] Pule Drive cycle Time [ec] Figure 4.83: The difference between the deired and the experimental cloed loop repone with a maximum gain of 7 The experiment, Figure 4.8, wa repeated with a fixed gain of 3. The difference between the deired and the experimental cloed loop repone i een to increae in Figure 4.84.

301 4. Control Technique and Performance Aement 3. Ramp Difference between deired repone and actual repone [rad] Pule Drive cycle Time [ec] Figure 4.84: The difference between the deired and the experimental cloed loop repone with a fixed gain of 3 A decribed in ubection a pring failure can caue the engine to tall due to air tarvation. The behaviour of the SMC controller during a pring break, at t [ec], i imulated uing the nonlinear throttle valve model, and the reult i hown in Figure It how a poor robutne againt the diturbance and the throttle cloe for about. [ec]. The ocillation on the control ignal at amplitude when the control error, t. [ec] are the added dither ignal that increae in yr y, exceed a preet threhold of % of the full cale movement range a decribed in ubection

302 4. Control Technique and Performance Aement 3 Throttle poition [rad] Throttle poition Throttle poition demand Throttle fully cloed Controller output [V] Time [ec] Figure 4.85: SMC - Control moothing integrator method during a pring failure The robutne againt plant parameter deviation away from the nominal value i teted uing the Monte Carlo method decribed in ubection The parameter variation imulation ue the nonlinear throttle valve model with the controller output aturation, integrator aturation (moothing integrator) and dither. The reult of the parameter variation imulation i hown in Figure 4.86 for the maximum poible tandard deviation of 8%. The figure how the operational envelope for imulation run, where the blue and red line are the minimum and maximum value. The nominal parameter cloed loop controller repone and controller output are hown in black.

303 4. Control Technique and Performance Aement 3 Throttle poition [rad] Nominal Minimum Maximum DC motor voltage [v] Nominal Minimum Maximum Time [ec] Figure 4.86: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: 8% ) A tandard deviation of 8%, Figure 4.86, indicate that the etimated parameter ued for the control gain deign can vary lightly before having an impact on the performance of the controller. The enitivity for the liding mode controller i analyed in the frequency domain by uing the relationhip of equation (4.3) in ubection Before thi can be done, the witching element ha to be replaced by a high gain, k, a in the boundary layer method, to render the cloed loop ytem linear, a hown in Figure 4.87.

304 4. Control Technique and Performance Aement 33 Y E r U - b Y 3 k - - V a a a Throttle valve D w 3 w f f f f f f w Figure 4.87: Structure ued for analying enitivity The enitivity i then done with the aid of the Matlab Control Sytem Analyi Toolbox and the block diagram of Figure 4.87 implemented in Simulink with D a the input and C V a the output to obtain S db. p The enitivity of the liding mode controller i hown in Figure The figure indicate a relatively low enitivity, equivalent to high robutne. Thi correpond well with the time domain reult found by the Monte Carlo parameter variation imulation, but not o well with time domain pring failure analyi, hown in Figure Thi can partial be explained by the fixed gain = 7 ued for the enitivity analye, if the gain value i reduced to 3, hown in Figure 4.89, the enitivity rie by db.

305 4. Control Technique and Performance Aement 34 - Senitivity [db] Frequency [rad/ec] Figure 4.88: SMC - Control moothing integrator method enitivity with a fixed gain = 7

306 4. Control Technique and Performance Aement 35 - Senitivity [db] Frequency [rad/ec] Figure 4.89: SMC - Control moothing integrator method enitivity with a fixed gain = Boundary Layer Method The liding mode controller gain are determined a decribed in ubection , with T. [ec], two group of pole, np 3 and nf 3, and a robut pole-to-pole ratio of 4. A dither ignal i added to the control ignal to reduce the effect of the tatic friction a decribed in ubection The integrator anti-windup trategy i enabled to minimie the aturation during the tep reference input. The SMC i teted experimentally with three different reference input function a decribed in ubection The dspace ytem i ued for the experiment a explained in ubection 4.3..

307 4. Control Technique and Performance Aement 36 The imulation reult preented in thi ubection are obtained with the full nonlinear plant model preented in ection.5. Firt, imulated and experimental repone to a tep reference poition change within the throttle valve top limit are preented in Figure Throttle poition [rad] Reference input Deired cloed loop repone Simulated Experimental. T Time [ec] Figure 4.9: Cloed loop tep repone, from. to.3 [rad] The vertical black line mark the nominal ettling time of T.[ec]. It i evident that both the experimental and imulated repone come cloe to thi at.4 [rad]. Figure 4.9 how the imulated and experimental cloed loop repone.

308 4. Control Technique and Performance Aement 37 Throttle poition [rad] Throttle poition [rad] Throttle poition [rad] Ramp Simulated Experiment Pule Drive cycle 5 5 Time [ec] Figure 4.9: Experimental and imulated repone of the SMC - Boundary layer method Figure 4.9 how mall difference between the deired and the experimental cloed loop repone (Figure 4.9), which, in contrat to the reult with the control moothing integrator in ubection , which i what would be expected uing a robut control technique.

309 4. Control Technique and Performance Aement 38. Ramp Difference between deired repone and actual repone [rad] Pule Drive cycle Time [ec] Figure 4.9: The difference between the deired and the experimental cloed loop repone A decribed in ubection a pring failure can caue the engine to tall due to air tarvation. The behaviour of the SMC during a pring break, at t [ec], i imulated uing the nonlinear throttle valve model, and the reult i hown in Figure It how a good robutne againt the diturbance, with little deviation from the throttle poition demand. The ocillation on the control ignal at amplitude when the control error, t. [ec] are the added dither ignal that increae in yr y, exceed a preet threhold of % of the full cale movement range a decribed in ubection

310 4. Control Technique and Performance Aement 39 Throttle poition [rad] Throttle poition Throttle poition demand Throttle fully cloed Controller output [V] Time [ec] Figure 4.93: SMC - Boundary layer method during a pring failure The robutne againt plant parameter variation away from the nominal value i teted uing the Monte Carlo method decribed in ubection The parameter variation imulation ue the nonlinear throttle valve model with the controller output aturation, integrator anti-windup and dither. The reult of the parameter variation imulation i hown in Figure 4.94 for the maximum poible tandard deviation of 5%. The figure how the operational envelope for imulation run, where the blue and red line are the minimum and maximum value. The nominal parameter cloed loop controller repone and controller output are hown in black.

311 4. Control Technique and Performance Aement 3 Throttle poition [rad] Nominal Minimum Maximum DC motor voltage [v] Nominal Minimum Maximum Time [ec] Figure 4.94: Maximum / minimum throttle poition and DC motor voltage envelope (Standard deviation: 5% ) Thi indicate that the plant parameter can deviate ignificantly from the nominal value before having an advere impact on the performance of the controller. The enitivity for the liding mode controller i analyed in the frequency domain by uing the relationhip of equation (4.3) in ubection Thi i done with the aid of the Matlab Control Sytem Analyi Toolbox and the block diagram of Figure 4.95 implemented in Simulink with C V a the output to obtain S db. p D a the input and

312 4. Control Technique and Performance Aement 3 Y r E K U - b Y i K V a a a Throttle valve D q q c c q q f f Figure 4.95: Structure ued for analying enitivity The enitivity of the liding mode controller i hown in Figure 4.96 for a robut pole-to-pole ratio of 4. The figure indicate a relatively low enitivity, equivalent to high robutne. Thi correpond well with the time domain reult found by the above pring failure analyi and the Monte Carlo parameter variation imulation. - Senitivity [db] Frequency [rad/ec] Figure 4.96: SMC - Boundary layer method enitivity

313 5. Performance Comparion 3 5 Performance Comparion The information gained from the imulation and experimental reult ection for each controller above i condened here into a form that enable recommendation to be made for future throttle valve controller. Only the linear tate feedback plu integral controller with coincident pole placement, in ubection , could definitely be rejected on the bai of a poor tep repone. Firt preliminary comparion of the remaining controller are made by aembling graph howing a) the difference between the experimental tep repone and the imulated tep repone with the nonlinear plant model, and b) the difference between the experimental tep repone and the ideal tep repone (i.e., the imulated tep repone with the linear plant model). Thee are preented in Figure 5. and Figure 5.. Arguably, the mot robut controller are thoe that exhibit the mallet difference between the imulated and experimental tep repone, a the true plant parameter will alway be different from thoe aumed in the controller deign, which are common for all the controller (except the DPI controller with feed forward and manual tuning). On the other hand, the controller will all have been deigned uing the linear plant model for which the ideal tep repone i the one obtained by imulating the control ytem with thi linear model uing the nominal parameter. To avoid awarding high mark to controller that have mall difference between poor experimental and imulated tep repone, the difference between the ideal tep repone and the experimental tep repone i included in Figure 5. and Figure 5..

314 5. Performance Comparion 33 An initial ranking ha been made by viual inpection of thee figure, the firt being the bet: ) SMC with boundary layer ) Retructured oberver aided LSFI control with robut pole placement 3) Oberver baed robut control 4) Polynomial controller with additional integrator 5) LSFI control with robut pole placement 6) IPD, DPI and PID controller 7) Oberver aided LSFI control with robut pole placement 8) Reduced order polynomial controller with additional integrator 9) DPI controller with feed forward and manual tuning ) SMC with control moothing integrator

315 5. Performance Comparion 34.5 IPD controller.5 DPI controller Difference [rad] Difference between the imulated and experimental tep repone Difference between the deired and experimental tep repone DPI controller with feed forward and manual tuning.5 PID controller.. Difference [rad] LSF Controller with Integrator for Steady State Error Elimination and Robut Pole Placement.5. Oberver Aided LSF Control with Integrator for Steady State Error Elimination and Robut Pole Placement Difference [rad] Time [ec] Time [ec] Figure 5.: Step repone difference for comparion #

316 5. Performance Comparion Retructured Oberver Aided LSF Control with Integrator for Steady State Error Elimination and Robut Pole Placement.5. Oberver Baed Robut Control.5.5 Difference [rad] Difference between the imulated and experimental tep repone Difference between the deired and experimental tep repone Polynomial Control with Additional Integrator for Steady State Error Elimination.5. Reduced Order Polynomial Control with Additional Integrator for Steady State Error Elimination.5.5 Difference [rad] SMC - Control Smoothing Integrator Method.5 SMC - Boundary Layer Method Difference [rad] Time [ec] Time [ec] Figure 5.: Step repone difference for comparion #

317 5. Performance Comparion 36 The reult are ummaried in Table 5. to give an overview of the different control technique invetigated and their individual performance uing the following, mainly qualitative, criteria: Overall performance regarding reference tracking, tranient repone and teady tate error G: Good, A: Acceptable, P: Poor Anti-friction needed (Dither ignal) Y: Ye, N: No 3 External diturbance tolerance (Spring failure) G: Good, A: Acceptable, P: Poor 4 Robutne againt plant parameter variation Maximum poible tandard deviation [%] 5 Senitivity indication L: Low, M: Moderate, H: High 6 Complexity of the deign procedure L: Low, M: Moderate, H: High 7 Practicability including ample rate and extra input (DC motor current) G: Good, A: Acceptable, P: Poor

318 7) Practicability 6) Complexity of the deign procedure 5) Senitivity indication 4) Robutne againt plant parameter variation 3) External diturbance tolerance ) Anti-friction needed ) Overall performance 5. Performance Comparion 37 Table 5.: Overall performance comparion data Controller IPD G Y A 5% L L G DPI G Y A 4% L L G DPI with Feed Forward and Manual Tuning A Y P % M L G PID G Y A 4% L L G Linear State Feedback plu Integrator Linear State Feedback plu Integrator with robut pole placement Oberver Aided LSF with Integrator with robut pole placement Retructured Oberver Aided LSF plu Integrator with robut pole placement Oberver Baed Robut Control Polynomial Control with Additional Integrator Reduced Order Polynomial Control with Additional Integrator Polynomial Control with Precompenator SMC with Control Smoothing Integrator SMC with Boundary Layer P Y L P G Y A to 6% L L P G Y A 5% L M G G Y A 5% L M G A Y G % M H G G Y A 4% L L G G Y A 5% L L G G Y A % L L G A N P 8% L L A 3 G Y A 5% L L G

319 5. Performance Comparion 38 Note for Table 5.: With k 7 A DC motor current meaurement i needed 3 Tet indicated that a higher ample rate would give the controller better performance. 4 Not teted due to poor performance. For the purpoe of obtaining a ranking from Table 5., the following numerical coring will be attached to the rating P ; A ; G ; H ; M ; L 3; Y ; N. (5.) In addition, the tandard deviation rating, %, i needed but it would be appropriate to introduce a weighting factor of le than unity for the following reaon: with reference to all the figure of ection 4.5, 4.6, 4.7, 3.8 and 4.9, howing the et of tep repone of the Monte Carlo run for each controller, although the tep repone are very cloely grouped in many cae they all, including the ideal tep repone, come very cloe to the upper left corner of the lower rectangular no go area. It therefore require very little deviation from the nominal ideal repone for the ytem to fail. The controller whoe nominal tep repone come the cloet to thi no go corner therefore tand at a diadvantage. Two example of thi are the OBRC and the polynomial controller with the precompenator. Thee, repectively, core only % and % on the maximum tandard deviation while other controller core much higher with a larger pread of tep repone (indicating poorer robutne) due to the ideal one being well clear of the no go corner. The tet, though an indutry tandard one i therefore harh and arguably a little unfair. In view of thi, the tandard deviation aement will be downcaled o that the maximum contribution i 3, which i no more than the maximum contribution 3 from the qualitative aement. Hence % i added to the qualitative 6 rating, the reult being preented in Table 5..

320 5. Performance Comparion 39 Table 5. Numerical rating and ranking Controller Rating Ranking SMC with Boundary Layer 3.8 ( t ) IPD 3.8 (6 th ) Reduced Order Polynomial Control with Additional Integrator 3.8 (8 th ) Polynomial Control with Additional Integrator 3.6 (4 th ) DPI 3.6 nd (6 th ) PID 3.6 (6 th ) Retructured Oberver Aided LSF plu Integrator with.8 ( nd ) robut pole placement 3 rd Oberver Aided LSF with Integrator & robut pole.8 (7 th ) placement Linear State Feedback plu Integrator with robut pole. 4 th (5 th ) placement SMC with Control Smoothing Integrator.5 5 th ( th ) Oberver Baed Robut Control th (3 rd ) DPI with Feed Forward and Manual Tuning th (9 th ) The ranking in parenthei on the right are the preliminary ranking obtained by viual inpection of Figure 5. and Figure 5.. Unfortunately thee preent ome anomalie but it mut be realied that mot of the qualitative apect of the aement are independent of the appearance of the tep repone error. It can be concluded, however, that manual tuning, which i currently practiced, i not recommended. The liding mode control with the control moothing integrator produced diappointing reult but further work on thi i encouraged ince not all avenue of thi technique have been explored. A far a the model baed control ytem deign i concerned, linear tate feedback with coincident pole placement i alo not recommended but robut pole placement with the ame control technique i worth conidering further. The traditional controller (PID, DPI and IPD) core high, but with model baed deign. It i clear that the polynomial controller and the boundary layer baed liding mode controller perform well and hould be conidered eriouly. t

321 6. Concluion and Recommendation for Further Reearch 3 6 Concluion and Recommendation for Further Reearch 6. Overall Concluion 6.. Modelling A generic third order nonlinear throttle valve plant model ha been developed including hard top and friction. The nonlinear friction model, which i an original contribution, include tatic friction, and ha been developed to be realitic without inordinately lowing down imulation around the zero velocity range. A practical obervation, however, ha been that the friction tranfer characteritic between the relative velocity and the friction force varie from day-to-day, particularly with the temperature rie following the ytem turn on, making it difficult to compenate for thi in model baed control trategie. The parameter of the plant model were identified by grey box methodology, i.e., by mean of a combination of phyical meaurement and mathematical model determination from experimental data, aided by the Simulink Deign Optimization toolbox from Mathwork. The model wa validated in both the time domain, hown in ubection 3.3, and in the frequency domain, hown in ubection 3.4, howing good correlation with the phyical plant. Thi wa further verified in the Simulation and Experimental reult of the individual control trategie under invetigation. 6.. Control technique Thi work focue on the angular poition control of the plate of a throttle valve ued for controlling the air flow to a Dieel engine.

322 6. Concluion and Recommendation for Further Reearch 3 A comprehenive range of different control trategie have been conidered, including the orthodox one, with a view to exploring the poibilitie of taking advantage of modern digital implementation to achieve performance level unattainable by the traditional method, from the point of view of eae of deign and commiioning a well a accuracy and robutne. The three traditional PID, DPI and IPD controller are included but model baed gain determination ha been exhautively explored for them (a well a the traditional manual tuning approach). The non-orthodox controller fall into two categorie: a) thoe baed on the robut control trategie of liding mode control and oberver baed robut control and b) thoe baed on the linear control trategie of linear tate feedback control (with and without oberver) and polynomial control (often recognied a the RST controller in the linear dicrete z-domain). The complete et of controller have been aeed and compared regarding their performance for poitioning accuracy, parametric uncertainty enitivity and diturbance rejection, a well a the deign effort and commiioning effort entailed. It i important to recall that any linear controller can be expreed by the baic tranfer function relationhip, U G Y G Y (6.) r r y Each controller, however, will have pecific order and relative degree of the tranfer function, G and G r y. Some of the controller may be baed on different control trategie but have the ame order and relative degree of thee tranfer function. Furthermore, it may be poible to adjut their parameter to make the tranfer function identical. In thi cae, although the two controller may have an entirely different tructure, in theory they may be deigned to have preciely the ame performance regarding parametric uncertainty, enitivity, diturbance rejection and control accuracy. Thi would explain the cloene of performance of ome of the controller examined in

323 6. Concluion and Recommendation for Further Reearch 3 Chapter 3. Thi would alo explain the polynomial controller with robut pole placement and liding mode controller with boundary layer performing imilarly, a both have eparated cloed loop pole that give robutne. Taking an overview of the work reported in Chapter 3, it i evident that for the throttle valve application the difference in performance between the bet and wort of the top ten (out of fourteen) controller identified in Table 5. i relatively mall o that none of thee can really be rejected for future throttle valve control ytem. All thee, however, were deigned uing model baed technique. Notably, the econd wort performer of the complete et wa a traditional controller tuned manually. A trong recommendation to the indutry, at leat for the throttle valve application, i therefore to adopt the more cientific approach of determining a mathematical model of the plant and deigning a controller on the bai of thi model. It i evident from ubection that coincident pole placement uing the polynomial controller yielded unatifactory reult and even conventional robut pole placement with one cloed loop pole of large magnitude did not produce ufficient improvement. Thi reult i attributed to the nonlinear tatic friction a imulation with linear kinetic friction predicted atifactory performance. Thi problem, however, did not occur with the other controller. The explanation i the higher order, i.e., even, of the polynomial control loop, when deigned on the bai of the third order throttle valve model. In contrat, the liding mode controller with boundary layer give a control loop order of only three and worked well with only one cloed loop pole of large magnitude. On the other hand, modifying the pole plit uing the polynomial controller to give four coincident dominant cloed loop pole and three cloed loop pole of large magnitude for robutne yielded even better reult than obtained with the lower order control loop, albeit by only a mall margin. Thee were imilar to the reult obtained with the oberver aided linear tate feedback (LSF) plu

324 6. Concluion and Recommendation for Further Reearch 33 integral control. Thi ytem i alo of eventh order. In thi cae, the deign of the main fourth order LSF loop wa baed on conventional robut pole placement, with only one cloed loop pole of large magnitude. The three oberver pole, however, were alo larger in magnitude than all the main LSF loop pole, which i tandard practice, reulting in three dominant pole and four large pole. Thi i not quite the ame plit a ued for the polynomial controller but would explain the imilarly good performance. Finally, it will be recalled from ubection that a pecial ytem wa introduced that enabled an extended ettling time, often requeted in the indutry, to be attained depite the tatic friction. Thi wa achieved by tightening up the feedback loop by ignificantly reducing it pecified ettling, in thi cae from. [ec] to.4 [ec] (no problem with meaurement noie being oberved) and then inerting a precompenator in the reference input that reduce the acceleration of the tep repone to level avoiding initial control aturation by introducing a dominant pole that et the real ettling time, in thi cae to. [ec]. Thi ytem performed better than all of the ytem compared in ection 5 but i not included in thi comparion becaue it i a pecial cae. It i important to mention, however, a it would be a practicable approach. The ame baic idea could be ued with all the other controller but then the difference in the controller performance would probably become even maller than oberved in ection 5. For the purpoe of finding the bet controller, the comparion made i conidered better. 6. Recommendation for Further Reearch At the outet of the reearch programme, the approach wa to obtain the mot accurate model of the throttle valve within practical contraint and then deign controller baed on thi model, which i of third order. Conequently it wa relatively late in the reearch programme that the potential of greatly implifying

325 6. Concluion and Recommendation for Further Reearch 34 the polynomial controller by uing a reduced order throttle valve model wa recognied. The model order reduction i carried out by ignoring the time contant of the plant mainly influenced by the inductance and reitance of the actuator coil ince it i at leat an order of magnitude le than the time contant aociated with the moment of inertia and kinetic friction of the mechanical aembly. Thi implified controller, in fact, produced the bet reult. It i therefore trongly recommended to redeign all the other nine controller (taken from the ten bet in Table 5.) and ae thee. Thi alo enable the traditional controller all to be deigned by complete pole aignment, rather than partial pole aignment, which wa neceary uing the third order model. In view of the good performance obtained uing the polynomial controller with the three-four robut multiple pole plit, it i recommended to undertake further reearch to find an optimal robut pole placement pattern, i.e., not necearily with multiple pole, and etablih how much further improvement in the robutne i attainable. Once an optimal pole pattern ha been etablihed, in which the ratio between the pole magnitude are fixed, then a pecified ettling time could be realied by imply caling the pole magnitude. In fact, the invere caling law between pole pattern and tep repone time cale could be ued to etablih a new ettling time formula for a pecific pole pattern that would render the control ytem deign traightforward. Since the throttle valve characteritic are known to lowly change during the lifetime of a vehicle, the performance of a controller with fixed parameter may deteriorate ignificantly given that ideal robutne i unattainable. Due to thi problem, periodic vehicle maintenance include time conuming in-itu controller retuning. Thi could be avoided if an on-line plant parameter etimation algorithm could be ued to continually update the model baed controller parameter, thereby realiing a form of adaptive control. Since the

326 6. Concluion and Recommendation for Further Reearch 35 plant model i fairly imple, it i recommended to carry out further reearch to arrive at a practicable on-line parameter etimation model. A problem are known to occur when attempting to etimate the parameter of a plant having widely differing pole magnitude with ignificant meaurement noie, it i trongly recommended to bae the parameter etimation on the econd order throttle valve model in which the electrical time contant of the actuator coil i ignored. Thi i another reaon for re-deigning the controller uing thi reduced order plant model. Some challenge would be expected, however, due to the variable tatic friction characteritic. Another integrator anti-windup trategy that ha been uccefully employed in the indutry with traditional controller could be invetigated to etablih whether an improvement in performance over the high gain control loop method i poible: The propoed trategy avoid the high gain control loop and i a follow. All controller containing an integral term produce a demanded control output given by u u u (6.) where u I i the output of the integral term and u c i the net contribution from all the other term in the controller. The phyical control input, u, i ubject to aturation and given by umax if u u u u if u u I c max max umax if u umax (6.3) For example, in a linear tate feedback plu integral controller applied to a third order plant uch a the throttle valve, and I I r (6.4) u K y y dt

327 6. Concluion and Recommendation for Further Reearch 36 u g xˆ g xˆ g xˆ (6.5) c 3 3 Whenever aturation occur, the integrator i reinitialied to I u u c (6.6) K Finally, it wa realied that the go/no-go tet ued to determine the maximum allowed parametric variance during the Monte Carlo run, though at the moment an indutry tandard, i le than ideal. It i propoed intead to carry out ome tet uing another criterion, which i the RMS control error of the tep repone relative to the ideal tep repone, the tep reference input level, time. Thu A maximum threhold, yideal t, taken a a proportion of Y r, taken over a duration equal to the ettling T e rm y t y ideal t dt Y, (6.7) r max e rm, would be decided in advance. Then the variance, %, would be tepped up and the Monte Carlo run continued until e rm e max rm whereupon the correponding maximum variance, max % figure of merit for the controller under tet, a previouly., would be noted a a

328 Reference 37 Reference ARMSTRONG-HELOUVRY, B. & AMIN, B. (994) PID control in the preence of tatic friction: exact and decribing function analyi. American Control Conference. ASTRÖM, K. J. & HÄGGLUND, I. (995) PID controller theory deign and tuning. ASTROM, K. J. & RUNDQWIST, L. (989) Integrator Windup and How to Avoid It. American Control Conference, 989. BANKS, A., NIVEN, M. & ANDERSSON, P. () Booting technology for Euro VI and Tier 4 final heavy duty dieel engine without NOx aftertreatment. Ricardo Conulting Engineer Ltd. DODDS, S. J. (4) A novel approach to robut motion control of electric drive with model uncertainty. Advance in Electrical and Electronic Engineering, 3. DODDS, S. J. (7) Oberver baed robut control. Advance in Computing and Technology Conference (AC&T), Univerity of Eat London. DODDS, S. J. (8) Settling time formulae for the deign of control ytem with linear cloed loop dynamic. Advance in Computing and Technology Conference (AC&T), Univerity of Eat London. DODDS, S. J. (3) Feedback Control: Technique, Deign and Indutrial application Springer (Planed for publication by the end of the year). DODDS, S. J. & VITTEK, J. (9) Sliding mode vector control of PMSM drive with flexible coupling in motion control. Advance in Computing and Technology Conference (AC&T), Univerity of Eat London. DODDS, S. J. & WALKER, A. B. (99) Three axi Sliding Mode Attitude Control of Rigid body Spacecraft with Unknown Dynamic Parameter. International Journal of Control, 54. FALLAHI, A. (3) Robut Control of Dieel Driveline. ACE (Thei). Univerity of Eat London. FRANKLIN, G. F., POWELL, D. P. & EMAMI-NAEINI, A. () Feedback Control Of Dynamic Sytem (Fourth Edition).

329 Reference 38 HAESSIG, D. A. & FRIEDLAND, B. (99) On the Modeling and Simulation of Friction. American Control Conference, 99. HENSEN, R. H. A. () Controlled Mechanical Sytem with Friction. DISC Thei (ISBN ). Techniche Univeriteit Eindhoven. HEYWOOD, J. B. (988) Internal Combution Engine Fundamental. IANNELLIA, L., JOHANSSONB, K. H., JÖNSSONC, U. T. & VASCAA, F. (5) Averaging of nonmooth ytem uing dither. Automatica. KALMAN, R. E. (96) On the General Theory of Control Sytem. Firt IFAC Mocow Congre. LANDAU, L. D. & ZITO, G. (6) Digital Control Sytem - Deign Identification and Implementation, Springer. LEONARD, N. E. & KRISHNAPRASAD, P. S. (99) Adaptive friction compenation for bi-directional low-velocity poition tracking. Proceeding of the 3t IEEE Conference on Deciion and Control. LJUNG, L. (998) Sytem Identification - Theory for the Uer, Pearon Education. LUENBERGER, D. (97) An introduction to oberver. IEEE Tranaction on Automatic Control, 6, LUENBERGER, D. G. (964) Oberving the State of a Linear Sytem. IEEE Tranaction on Military Electronic, 8, MAJD, V. J. & SIMAAN, M. A. (995) A continuou friction model for ervo ytem with tiction. Proceeding of the 4th IEEE Conference on Control Application. MESHRAM, P. M. & KANOJIYA, R. G. () Tuning of PID controller uing Ziegler-Nichol method for peed control of DC motor. Advance in Engineering, Science and Management (ICAESM),. MOHAN, N., UNDERLAND, T. M. & ROBBINS, W. P. (995) Power Electronic Converter, Application an Deign (Second Edition). PAPADOPOULOS, E. G. & CHASPARIS, G. C. () Analyi and modelbaed control of ervomechanim with friction. IEEE/RSJ International Conference on Intelligent Robot and Sytem.

330 Reference 39 PEDERSEN, J. L. & DODDS, S. J. () A comparion of two robut control technique for throttle valve control ubject to nonlinear friction. Advance in Computing and Technology Conference (AC&T), Univerity of Eat London. PINTELON, R. & SCHOUKENS, J. (4) Sytem Identification - A Frequency Domain Approach, John Wiley & Son. POPOV, V. L. () Contact Mechanic and Friction, Springer. RADCLIFFE, C. J. & SOUTHWARD, S. C. (99) A Property of Stick-Slip Friction Model which Promote Limit Cycle Generation. American Control Conference, 99. SABANOVIC, A., FRIDMAN, L. & SPURGEON, S. K. (4) Variable Structure Sytem: from Principle to Implementation Intitution of Electrical Engineer. SANJUAN, M. & HESS, D. P. (999) Undertanding dynamic of machinery with friction through computer imulation. ASEE Southeatern Section Conference. SCATTOLINI, R., SIVIERO, C., MAZZUCCO, M., RICCI, S., POGGIO, L. & ROSSI, C. (997) Modeling and identification of an electromechanical internal combution engine throttle body. Control Engineering Practice, 5, SCHÖPPE, D., GEURTS, D., BALLAND, J. & SCHREURS, B. (5) Integrated Strategie for Boot and EGR Sy-tem for Dieel Engine to achieve mot tringent Emiion Legilation.. Aufladetechnichen Konferenz in Dreden. SHAHROKHI, M. & ZOMORRODI, A. (3) Comparion of PID Controller Tuning Method. 8th National Iranian Chemical Engineering Congre SIRA-RAMIREZ, H. (993) On the dynamical liding mode control of nonlinear ytem. International Journal of Control, Special Iue on Variable Structure Control, 57. SPURGEON, S. K. & EDWARDS, C. (998) Sliding Mode Control: Theory and Application, Taylor & Franci. STADLER, P. A. (8) Modelling and Control of a Vacuum Air Bearing Linear Drive in the Nanometer Range. SCOT Thei. Univerity of Eat London. STADLER, P. A., DODDS, S. J. & WILD, H. G. (7) Oberver baed robut control of a linear motor actuated vacuum air bearing. Advance in Computing and Technology Conference (AC&T), Univerity of Eat London.

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332 Appendix 33 Appendix A. Engine Sytem Overview A.. The Natural Apirated Dieel Engine With reference to Figure A, the freh atmopheric air enter the ytem through the air filter which remove particle like dirt and and which would otherwie damage the engine ytem. The air enter the four-troke engine through the inlet valve into a cylinder (one at the time) during the intake troke. Once the intake valve are cloed, the compreion troke commence in which the engine piton move toward the top of the cylinder. The trapped air in the cylinder increae in temperature to everal hundred degree Celiu. Jut before the piton reache top dead centre, the fuel i injected. Thi tart to combut due to the high air temperature inide the cylinder. The preure force from the combution puhe the piton down on the power troke, to drive the crank haft. The crank haft i connected to the wheel of the vehicle through the drive train (gearbox, drive haft etc.). Air filter Ga flow direction Air Intake manifold Engine block Fuel injector Exhaut ga Cylinder Exhaut manifold Muffler Figure A: Baic chematic of a natural apirated Dieel engine

333 Appendix 33 At a certain point the exhaut valve i opened and the hot ga exit the cylinder through the exhaut manifold into the muffler/ilencer which remove the noie due to preure alternation from the combution. The air-to-fuel ratio inide the cylinder i very important for the combution proce. The toichiometric air-to-fuel ratio (Heywood, 988) AF ratio Air ma (A.) Fuel ma for a Dieel engine i 4.6, meaning that to burn kg of fuel, 4.6 kg of air i needed to burn all the fuel. Compared to the petrol engine proce where it i important to maintain a pecified toichiometric air-to-fuel ratio, the Dieel engine proce can run with a wider range of air-to-fuel ratio. In practice, the Dieel engine run with air-to-fuel ratio between 9 and 6. Too low an A/F ratio, however, will uffocate the combution reulting in a decreae in torque and an increae in the moke / particulate matter level. Some of the particulate matter (PM) i fine particle which can caue eriou health iue. The natural apirated Dieel engine ha a tendency to run with a low A/F ratio caued by the air retriction like the air filter, air duct etc. To circumvent thi problem the air preure in the intake manifold can be increaed letting more air into the cylinder making the A/F ratio higher. Thi can be done by adding a mechanical compreor (driven by the crank haft) or a turbo charger (driven by the exhaut ga flow) to the engine ytem. The turbo charger ha two component compriing a compreor and a turbine which are linked together with a rotating mechanical haft a hown in Figure B. The turbine i located on the exhaut ide where the hot ga pae through it. The ga expand through the turbine, which convert the energy from the ga into mechanical energy making the mechanical haft rotate.

334 Appendix 333 Shaft Figure B: Turbo charger from Cummin Turbo Technologie The haft drive the compreor which will increae the air preure on the output. The compreor i located between the air filter and the intake manifold a hown in Figure C. A.. The Turbo Charged Dieel Engine In a turbo charged engine ytem (Figure C) the freh atmopheric air enter the ytem through the air filter a in the natural apirated engine. The air i booted to a higher preure by the compreor, thi increae the air denity but alo it temperature. Thi i taken advantage of to increae the air denity further by paing thi heated air through an air-to-air cooler called the intercooler. The cooled air enter the engine through the inlet valve and from thi point the event that take place are the ame a in the naturally apirated engine decribed in ubection.

335 Appendix 334 Air filter Ga flow direction Intercooler Air Compreor Intake manifold Engine block Fuel injector Exhaut ga Turbo haft Cylinder Exhaut manifold Muffler Turbine Figure C: Baic chematic of a turbo charged Dieel engine The exhaut ga expand through the turbine, which pin typically in a range between 9, to, [rpm], and the air, which i cooled by thi proce, leave the engine ytem through a muffler/ilencer a it wa decribed in ubection. Increaing the A/F ratio will, however, alo increae the combution temperature leading to an increae in the production of nitrogen oxide ( NO ) in the exhaut gae. The NO x ha a damaging effect on the environment, forming a) particulate by reacting with ammonia and b) ozone by reacting with volatile organic compound. The ozone can have a damaging effect on the repiratory ytem. Thee conideration have led to enhanced emiion tandard around the world, in particular Japan, US, Europe, motly concerned with CO, NO x, HC and PM. Different region have, at preent, different maximum emiion threhold, but a world harmonied tandard i being developed. In Europe there are different emiion tandard for car, off road vehicle and HD Dieel engine, entailing different maximum threhold and x

336 Appendix 335 tet pecification. Today there are two emiion tet for HD Dieel engine, a teady-tate (Table A.) and a tranient tet cycle (not hown). The emiion tandard for HD Dieel engine i omewhat different from that of car, becaue the engine i teted in iolation from the vehicle. Thi implie that if more than one vehicle ue the ame engine model, the tet only ha to be carried out jut one time. Table A.: EU Emiion Standard for HD Dieel Engine: Steady-State Teting Standard Date CO [g/kwh] NOx [g/kwh] HC [g/kwh] PM [g/kwh] EURO none EURO I EURO II EURO III EURO IV EURO V EURO VI Source: The Internet (Dieelnet.com & Delphi.com ) In the teady-tate tet the engine i teted at different peed and load point in which the emiion i meaured and teted againt the threhold in Table A.. In the tranient tet, the engine i again teted at different peed and load but including engine top and tart. In thi cae, the emiion level are monitored continually during the tet and their peak value are not allowed to exceed a certain value. The firt emiion tandard EURO for HD Dieel engine came into place in the late 98. Through the year emiion level have decreaed dramatically, a it i evident in Table A., a pertinent example being the NO x

337 Appendix 336 level dropping from 5.8 [g/kwh] in 988 to.4 in (-385%). Thee decreaing emiion threhold have forced the automotive indutry to find new technologie to meet the lower emiion threhold. There are different way of achieving lower emiion outlet uch a piton and cylinder deign, Dieel injector deign, fuel propertie, exhaut ga after-treatment and exhaut ga recirculation (EGR). EGR help to reduce NO x by reducing the peak combution temperature. The idea i to mix inert ga (i.e., ga that cannot take part in the combution proce) with freh air to reduce the peak combution temperature. Thi inert ga i contained in the exhaut ga, which i re-circulated to the intake ide of the engine (Figure D). For thi, the exhaut ga recirculation rate i defined a rate EGR m / m m (A.) egr egr air where m air i the air ma flow rate and megr i the EGR ma flow rate. The EGR rate will moderate the NO x concentration in the exhaut gae, a higher EGR rate giving le NO x. Air filter Ga flow direction Intercooler Throttle valve m air m egr Compreor Air Intake manifold Engine block Fuel injector Exhaut ga Turbo haft Cylinder Exhaut manifold Muffler VGT EGR valve EGR cooler Figure D: Schematic of a turbo charged Dieel engine with EGR

338 Appendix 337 The EGR path (EGR valve and EGR cooler) in Figure D, i often referred to a a high preure EGR due to the high preure in the exhaut / intake manifold. Thi high preure i a reult of the operating range for the turbine and compreor. The EGR flow can be adjuted by the valve in the EGR path, but the maximum flow rate which can be achieved depend on the differential preure acro the EGR path (EGR valve, EGR cooler and pipe work). The differential preure i a function of factor uch a the turbo ytem operating point, inlet and outlet valve deign and the combution characteritic. In Figure D, the turbine ha been replaced by a variable geometry turbine (VGT). The operating point of the VGT can be adjuted by opening and cloing it vane. Thi varie the turbine peed, making it poible to adjut the compreor output preure. The preure can only be adjuted in a limited range which i a function of the engine operation point, uch a the exhaut flow and temperature. The preure in the intake manifold determine how much ga/air can go into the cylinder per intake troke. More engine torque, require more air which again require a higher intake manifold preure. If the EGR rate of equation A. ha to be kept contant, then m egr ha to increae a well making the preure even higher. One of the problem of uing EGR i that the re-circulated exhaut ga contain particulate matter from the combution which clog up the EGR cooler in time. A mentioned before more air in the cylinder give a cleaner combution which again lower the PM output, but it alo lead to an increae in the formation. A higher NOx EGR rate, however, decreae the NO x formation but increae the particulate output, caued by the increaed inert ga in the

339 Appendix 338 combution chamber. Thi conflict between the EGR and air reult in a compromie for the et point of the AF ratio and the EGR rate. The low level of emiion in the tandard have forced engine manufacturer to add more emiion decreaing device to the engine ytem to meet thee (Bank et al., ). A chematic example of a EURO VI engine configuration can be een in Figure E. There are two major difference between thi and the previou figure which are the low/high preure EGR loop and the exhaut ga after treatment. Air filter Flow meaurement Ga flow direction Valve actuator with poition feedback Intercooler Throttle valve T MAF Air Exhaut ga P = Preure enor Valve actuator with poition feedback T = Temperature enor Valve actuator with poition feedback SCR catalyt EGR valve EGR cooler Engine block Compreor Low preure EGR loop DPF DOC Intake manifold P T P T Exhaut manifold High preure EGR loop Muffler T Exhaut Exhaut throttle aftertreatment Exhaut aftertreatment VGT EGR valve EGR cooler Vane actuator with poition feedback Valve actuator with poition feedback Figure E: An example of a chematic for Euro VI engine configuration with high and low preure EGR The after treatment i a neceity for mot modern engine ytem to meet the trict emiion regulation. There are different after treatment ytem available on the market and till more are coming. They can be plit into the following two group:

340 Appendix 339 a) The NO x reduction ytem, which i uually baed on elective catalytic reduction (SCR), which convert nitrogen oxide into N and water. For thi ytem, either ammonia ( NH 3 ) or urea ( NH ) ha to be added to the exhaut ga before the catalyt. For mot SCR ytem on HGV, a water diluted urea olution i ued called AdBlue, tored in a eparate tank. Thi olution i doed (pumped) into the hot exhaut ga before the SCR unit for it to evaporate into ga form. The amount of AdBlue i determined by how much the engine NO x level ha to be reduced. b) The particulate reduction ytem, which require a Dieel oxidation catalyt (DOC) and a Dieel particulate filter (DPF) inerted in the exhaut ytem a cloe a poible to the engine outlet manifold, to maximie the temperature and hence the reaction rate. The DPF catche the particle from the exhaut ga in a ceramic filter tructure. When the engine run the DPF will lowly clog up with particulate matter, thereby impeding the exhaut ga. The DPF ued on HGV are of the regenerative type. During the regeneration, the trapped particulate matter i burned away in the DPF, creating ah. The ah leave the ceramic filter and into the urrounding environment through the exhaut ytem. The regeneration burn of the DPF tart when a minimum temperature of everal hundred C i reached and enough oxygen i available. The DOC i an oxidation catalyt which can create heat by oxidiing/burning the non-burned fuel and oxygen in the exhaut ga. The heat from the DOC can tart the regeneration proce in the DPF. Too high a temperature in the DPF can damage the filter tructure, and too low a temperature will top the regeneration. It i therefore important to keep the correct exhaut Air-to-fuel ratio during the regeneration of the DPF. The function of the low preure EGR i the ame a the high preure EGR, which i to lower the exhaut NOx level. The low preure EGR take the ga after it ha been filtered through after-treatment which it leave with le

341 Appendix 34 particulate matter. The low preure EGR path alo benefit from a higher level of cooling making the denity higher. The low preure EGR i ometime ued to ave money on the exhaut after-treatment ytem and can in ome cae remove the need for after-treatment. To control the flow through the low preure EGR path, two valve are ometime needed, a hown in Figure E. The EGR valve, in the low preure EGR path, i ued to control the flow through the low preure EGR cooler and the exhaut throttle valve to create a differential preure over the low preure EGR ytem. With reference to Figure E, there are five poition control loop indicated, required for four valve and the VGT. The throttle valve, decribed in more detail in Chapter, i the focu of thi reearch programme but thi will be equally ueful for the other application a each of thee ha imilar characteritic.

342 Appendix 34 A. Parameter ued for the Simulation Parameter ued throttle valve for the imulation:. DC motor wheel diameter [-] - N m =. Throttle plate wheel diameter [-] - N pl =.5 3. DC motor torque contant [Nm/A] - k t = DC motor peed contant [V ec/rad] - k e = DC motor reitance [ohm] - Ra Inductance [H] - L a Total ytem pure time delay (Electrical & mechanical) [ec] - Sytem_delay =.6 8. Coil pring contant [Nm/rad] - k pring = Throttle ytem moment of inertia (plate, DC motor and pindle) [kg*m^] - J x Initial pring poition (Simulate the initial pring torque) [Rad] - Initial pring = Hard top gain [Nm/rad] - k hard top =. Maximum poition hard top [Rad] - pl max = 9/36**pi

343 Appendix Minimum poition hard top [Rad] - pl min = -./36**pi 4. Kinetic friction contant [Nm ec/rad] - k kinetic =. 5. Coulomb friction contant [Nm] - k coulomb = Static friction model peed contant [rad/ec] - =. 7. Static friction model torque contant [Nm ec/rad] - = Static friction model peed contant [rad/ec] - = Static friction model torque contant [Nm ec/rad] - =.

344 Appendix 343 A.3 Calculation for Linear State Feedback with Integrator for Steady State Error Elimination Thi ection contain a detailed calculation for the Linear State Feedback with Integrator for Steady State Error Elimination in ubection Yr k i - U L a Ia 4 k t N pl N - m J x Y g 3 g R a k e m N N pl m k kinetic k pring g Figure F: LSF with integrator and plant model Uing Maon rule to calculate the cloed loop tranfer function of Figure F (number refer to encircled number) L: [3,4,3] L: [5,6,5] L3: [5,6,7,5] L4: [3,4,5,6,3] L5: [,3,4,] L6: [,3,4,5,6,] L7: [,3,4,5,6,7,] L8: [,,3,4,5,6,7,] Forward path gain (from y r to y ) [,,,3,4,5,6,7]

345 Appendix 344 k N t L N pl a m x k 4 i J (G.) Loop path gain L: Ra, L: L a kkinetic J, L3: kpring J, L4: g3 k N t pl L5:, L6: g L L N J a x a m x k N t pl k N t pl L7: g, L8: k L N J 3 4 i L N J a m x a m x x kt k N e pl La Jx Nm, None touching loop Ra kkinetic (L,L): L J, (L,L3): R k a pring 3 L J, (L,L5): g3 kkinetic L J, g k 3 pring (L3,L5): 3 L J a a x x a x a x G k N t pl ki 4 La Nm Jx L L L3 L4 L5 L6 L7 L8 L L L L3 L L5 L3 L5 (G.)

346 Appendix 345 G R k g L J L 4 3 a kinetic 3 a x a k N t L N pring t e pl a kinetic t pl 3 kinetic a pring t pl 3 pring t pl i pl ki J a m x k k k N R k k N g g k J L J N L J L N J L J x a x m a x a m x a x R k k N g g k k N k L J L N J L J L N J a x a m x a x a m x (G.3) Uing equation (G.3), and after ome manipulation where b k N t L N pl ki J a m x r Y b G Y a a a a (G.4) a a a a k N t L N pl ki J a m x R k k N g g k a pring t pl 3 pring La Jx La Nm Jx La Jx k k k N R k k N g g k J L J N L J L N J L J pring t e pl a kinetic t pl 3 kinetic x a x m a x a m x a x R k g a kinetic 3 3 La Jx La

347 Appendix 346 A.4 H-bridge with Output Current Meaurement The DC motor i driven by an H-bridge circuit acting like a power amplifier, hown in Figure G. The demand i converted by the dspace ytem into a pule width modulation (PWM) ignal with fixed witch frequency and a markpace ratio (duty cycle) which dependent on the voltage demand. The H- bridge driver convert the PWM and direction ignal, into pule which drive the four tranitor dependent on the direction. To drive the DC motor in the clockwie direction the tranitor T and T4 have to be on, and for the anticlockwie direction tranitor T and T3 have to activated. DC motor rotational direction PWM driver ignal Motor direction ignal H-bridge driver T T - - DC motor T3 T4 DC power upply - Current flow direction DC motor current Current meaurement enor Figure G: H-bridge chematic The output current from the H-bridge i meaured by a current enor. The current enor i a cloed loop Hall effect type with a high frequency bandwidth. Throughout the indutry the norm i to ample a PWM driven current ynchronou to the witching to eliminate the aliaing effect. Unfortunately the dspace cannot ample the ignal ynchronou due to limitation in the hardware. The output from the current enor i low pa filter by a paive filter acting a a high frequency antialiaing filter. To circumvent

348 Appendix 347 the problem with aynchronou ampling the current ignal i overampled (= khz) and a periodical average ignal i generated every time the control trategy run. The H-Bridge and current meaurement circuit are done on eparate board to keep the witching noie to a minimum. The witching noie i generated when the H-Bridge tranitor are rapidly turn on and off (witch frequency = khz). The H-Bridge and current meaurement circuit board are made by the author, hown in Figure H. DC motor current meaurement H-Bridge Figure H: H-bridge and current meaurement board

349 Appendix 348 A.5 Throttle Valve Exploded View Figure I: Throttle valve exploded view

350 Publihed Work 349 Publihed Work A comparion of two robut control technique for throttle valve control ubject to nonlinear friction

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