Direct Modeling and Robust Control of a Servo-pneumatic System. Ph.D. Thesis

Transcription

1 Bdapest University of Technology and Economics Department of Mechatronics, Optics and Engineering Informatics Direct Modeling and Robst Control of a Servo-pnematic System Ph.D. Thesis Károly Széll Spervisor: Péter Korondi D.Sc., Professor Bdapest, 2015.

2 Acknowledgements I wold like to express my gratitde to everyone who helped me in my stdies leading to this thesis work. Otmost thank to my spervisor, Professor Péter Korondi (Department of Mechatronics, Optics and Mechanical Engineering Informatics, Bdapest University of Technology and Economics), who provided an excellent environment to accomplish my aspirations and for everyone at the department. Special thanks to my family spporting me in my stdies all the way to reach here. ii

3 Contents 1 INTRODUCTION Motivation Goal of the dissertation Strctre of the dissertation NOMENCLATURE Abbreviations Roman letters Greek letters Position control Pressre control... 4 PART I Theoretical Backgrond STATE OF THE ART Overview of the literatre SERVO-PNEUMATIC Pressre change in the cylinder Pressre change de to volme change Pressre change de to mass flow Pressre change de to temperatre change Flow coefficient of the pnematic valve State space model BASIC FRICTION MODELS TENSOR PRODUCT MODEL TRANSFORMATION Tensor prodct transformation Redction of the dimension of the measred data SLIDING MODE CONTROL History of sliding mode control Sliding srface design Control Law Chattering free implementation, Sector Sliding Mode Sliding Mode Based Model Reference Adaptive Compensation Discrete-Time Implementation PART II Contribtion iii

4 6 DIRECT TENSOR PRODUCT MODEL TRANSFORMATION Modeling Stribeck-friction Model A Model B Model C Comparison of the Stribeck-friction models Systematic design of sliding sector Modeling friction with hysteresis loop Model D Model E Model F Comparison of the models with hysteresis loop SLIDING MODE BASED MODEL REFERENCE ADAPTIVE COMPENSATION Decomposition Position control Pressre control Identification of the pnematic valve Experimental reslts PART III Conclsion THESES Thesis 1: [P4] Thesis 2: [P4] Thesis 3: [P2, P5, P6, P7, P8, P9, P10, P11] Thesis 4: [P1, P3, P5, P6, P11] AUTHOR S PUBLICATIONS BIBLIOGRAPHY iv

5 List of Figres Figre 2.1 Two chambers with state variables... 7 Figre 2.2 Pressre change de to temperatre change after inflation Figre 2.3 Pressre change de to temperatre change after deflation Figre 2.4 Flow coefficient Figre 2.5 Real flow coefficient Ψ and the approximation Ψ Figre 2.6 Servo-pnematic system Figre 3.1 Colomb friction Figre 3.2 Colomb and viscos friction Figre 3.3 Colomb and viscos friction with breakaway force Figre 3.4 Stribeck friction Figre 3.5 Friction force as fnction of the velocity at different nominal chamber pressres [65] Figre 3.6 Velocity dependent friction model [65] Figre 4.1 TP transformation based design algorithm Figre 4.2 Basic idea of TP transformation Figre 4.3 HOSVD decomposition Figre 5.1 Sliding sectors (in case n 2 ) Figre 5.2 Sliding mode based feedback compensation Figre 5.3 Discrete-time chattering phenomenon Figre 5.4 Controller scheme for position control Figre 6.1 Experimental setp for friction identification Figre 6.2 Schematic of the experimental setp Figre 6.3 Experimental excitation Figre 6.4 Excitation for a single stick-slip v

6 Figre 6.5 Acceleration dring stick-slip Figre 6.6 Velocity dring stick-slip Figre 6.7 Friction hysteresis Figre 6.8 Friction characteristic of Model A Figre 6.9 The weighting coefficients as the fnction of velocity for Model B Figre 6.10 Friction characteristic of Model B Figre 6.11 The weighting coefficients as the fnction of velocity for Model C Figre 6.12 Friction characteristic of Model C Figre 6.13 MATLAB Simlink simlation model Figre 6.14 Acceleration of Model A Figre 6.15 Velocity of Model A Figre 6.16 Acceleration of Model B Figre 6.17 Velocity of Model B Figre 6.18 Acceleration of Model C Figre 6.19 Velocity of Model C Figre 6.20 The sliding sector Figre 6.21 Simlated friction-hysteresis Figre 6.22 Simlated friction-hysteresis (magnified) Figre 6.23 Friction characteristic of Model D Figre 6.24 The weighting coefficients of Model E for accelerating piston Figre 6.25 The weighting coefficients of Model E for slowing piston Figre 6.26 Friction characteristic of Model E for accelerating piston Figre 6.27 Friction characteristic of Model E for slowing piston Figre 6.28 Friction characteristic of Model E Figre 6.29 The weighting coefficients of Model F for accelerating piston vi

7 Figre 6.30 The weighting coefficients of Model F for slowing piston Figre 6.31 Friction characteristic of Model F for accelerating piston Figre 6.32 Friction characteristic of Model F for slowing piston Figre 6.33 Friction characteristic of Model F Figre 6.34 Acceleration of Model D Figre 6.35 Velocity of Model D Figre 6.36 Acceleration of Model E Figre 6.37 Velocity of Model E Figre 6.38 Acceleration of Model F Figre 6.39 Velocity of Model F Figre 7.1 Servo-pnematic system Figre 7.2 Servo-pnematic Figre 7.3 Decomposition of the servo-pnematic system Figre 7.4 Control schematic of the servo-pnematic system Figre 7.5 Position control schematic Figre 7.6 Position control schematic with inner pressre controller loop Figre 7.7 Position control schematic for discrete-time controller Figre 7.8 Pressre control schematic Figre 7.9 Linearization via inverse fnction Figre 7.10 Pressre control schematic with static and dynamic parts Figre 7.11 Pressre control schematic for discrete-time controller Figre 7.12 Servo-valve Figre 7.13 Drawing of the servo-valve Figre 7.14 Strctre of the servo-valve Figre 7.15 Experimental setp for the identification of the valve vii

8 Figre 7.16 Inflation pressre crve Figre 7.17 Deflation pressre crve Figre 7.18 Flow coefficient Figre 7.19 Inflation pressre crve Figre 7.20 Mass flow-rate for inflation Figre 7.21 Exponential pressre crve for deflation Figre 7.22 Linear fnction of pressre and pressre gradient for deflation Figre 7.23 Servo-valve characteristic Figre 7.24 Pressre gradient as fnction of voltage for 5 bar Figre 7.25 Overlapping of the borings Figre 7.26 Srface of the inlet orifice as the fnction of the inpt voltage Figre 7.27 Control schematic of the servo-pnematic system Figre 7.28 Experimental reslts of pressre control Figre 7.29 Experimental reslts of pressre control (magnified) Figre 7.30 Control voltage of the proportional valve Figre 7.31 Filtered estimated distrbance signal Figre 7.32 Experimental reslts of position control Figre 7.33 Pressre signal of the inner pressre control loop Figre 7.34 Filtered estimated distrbance signal compared to external load Figre 8.1 TP model transformation Figre 8.2 Stribeck-friction trajectory of the analytical TP model Figre 8.3 Stribeck-friction trajectory of the direct TP model Figre 8.4 The weighting coefficients for Stribeck-friction with analytical TP model.. 74 Figre 8.5 The weighting coefficients for Stribeck-friction with direct TP model Figre 8.6 The sliding sector viii

9 Figre 8.7 Friction trajectory of the analytical hysteretic TP model Figre 8.8 Friction trajectory of the direct hysteretic TP model Figre 8.9 Stribeck-friction trajectory of the analytical hysteretic TP model Figre 8.10 Stribeck-friction trajectory of the direct hysteretic TP model Figre 8.11 The weighting coefficients for Stribeck-friction with analytical hysteretic TP model Figre 8.12 The weighting coefficients for Stribeck-friction with direct hysteretic TP model Figre 8.13 Colomb-viscos-friction trajectory of the analytical hysteretic TP model 78 Figre 8.14 Colomb-viscos-friction trajectory of the direct hysteretic TP model Figre 8.15 The weighting coefficients for Colomb-viscos-friction with analytical hysteretic TP model Figre 8.16 The weighting coefficients for Colomb-viscos-friction with direct hysteretic TP model Figre 8.17 Control schematic of the servo-pnematic system Figre 8.18 Typical pressre crve of inflation Figre 8.19 Pressre gradient Figre 8.20 Typical pressre crve of deflation Figre 8.21 Pressre gradient as the fnction of pressre for deflation Figre 8.22 Pressre gradient of the pnematic system ix

10 List of Tables Table 6.1 Elements of the experimental setp Table 6.2 RMSD of the Stribeck-friction models Table 6.3 RMSD of the models with hysteresis loop Table 7.1 Elements of the experimental setp x

11 1 INTRODUCTION 1.1 Motivation The research work follows the traditions of Department of Mechatronics, Optics and Engineering Informatics. The reslts presented in this dissertation are based on practical observations, which raised problems with the need for the application of higher mathematical theories. The reslts discssed in this work are intended to introdce new methods which facilitate the tilization of higher theories for pnematic systems, ths to constitte a bridge between the mathematical soltions and the engineering applications. Pnematic actators are widely sed in indstrial field de to their advantages like low cost, drability, high power-to-weight ratio. However their modeling and controlling is challenging as they are strongly nonlinear: air compressibility, heat transfer, friction etc. Ths, the pnematic systems belong typically to the grop of variable strctre systems. In the field of control theory one of the most recent topics is the robst control of variable strctre systems. The dissertation introdces sch a soltion for pnematic systems. The theses are based on the characteristics of pnematics, and they are searching control soltions to these characteristics both in theory and in practice. Ths the research work belongs to the field of mechatronics. 1.2 Goal of the dissertation Lots of theoretical models have been elaborated sing tensor prodct dring the last decade, however few applications have been practically implemented yet sing tensor prodct model transformation. This dissertation introdces a practical application of tensor prodct model transformation. Friction is a permanent problem in case of engineering applications. Both modeling and controlling of friction is really challenging. Therefore my first goal dring my research work was to develop a model which can simplify the identification of friction phenomenon. Servo-pnematic systems are complex and strongly nonlinear ths they are hard to handle in case of accrate positioning tasks. My second goal was to rearrange the description of the servo-pnematic system in sch a way, that it becomes easier to handle. 1

12 The pressre gradient of the chamber of a pnematic cylinder is nonlinear for many reasons. My third goal was to develop a new measrement evalation method for the identification of a pnematic servo-valve. 1.3 Strctre of the dissertation The dissertation is divided into three parts. Part I discsses the theoretical backgrond of the dissertation. In Part II the new reslts are stdied. Part III concldes the dissertation. Part I smmarizes all the preliminaries which are concerned dring Part II. Chapter 2 gives a general description of pnematic systems. Chapter 3 presents the basic friction models. In Chapter 4 Tensor Prodct Model Transformation is introdced. In Chapter 5 the basic steps of sliding mode control design can be fond. Part II is devoted to the own contribtion. Chapter 6 introdces a measrement method for the identification of friction hysteresis for a pnematic cylinder. The friction is modeled both the conventional way and based on the newly proposed direct tensor prodct model transformation. Chapter 7 discsses the identification and control of the servo-pnematic cylinder. It defines a new method for the identification of the proportional pnematic valve and tilizing this reslt a block-oriented regroping of the pnematic system is shown. The blocks are arranged according the so-called Drazenovic condition which facilitates the tilization of the sliding mode based model reference adaptive compensation approach. Part III smmarizes the new reslts and achievements of the dissertation in form of for theses. 1.4 NOMENCLATURE Abbreviations qlpv LTI LMI SVD HOSVD CHOSVD RHOSVD TP SMC HOSMC qasi Linear Parameter Varying Linear Time Invariant Linear Matrix Ineqality Singlar Vale Decomposition Higher-Order Singlar Vale Decomposition Compact Higher-Order Singlar Vale Decomposition Redced Higher-Order Singlar Vale Decomposition Tensor Prodct Sliding Mode Control Higher-Order Sliding Mode Control 2

13 DSMC TSMC Dynamic Sliding Mode Control Terminal Sliding Mode Control Roman letters A i [m 2 ] srface of the piston A vi [m 2 ] flow srface of the valve F fr [N] friction force l [m] stroke length p i [Pa] pressre of the chamber p [Pa] pressre of pstream flow p d [Pa] pressre of downstream flow p crit [-] critical pressre ratio q m [kg/s] mass flow rate R [J/(mol*K)] specific gas constant of air T [K] absolte temperatre v [m/s] piston velocity V [m 3 ] dead volme at the end of the piston 0 x [m] piston position Greek letters [-] correction coefficient [-] heat capacity ratio [kg/m 3 ] density [-] flow coefficient Position control x [µm] piston position v [µm/s] piston velocity x ref [µm] reference position p idm [Pa] ideal control pressre p [Pa] real inpt pressre ˆp [Pa] inpt pressre of the observer ˆv [µm/s] observer velocity [-] switching srface 3

14 p [Pa] estimated distrbance signal SMC p [Pa] filtered estimated distrbance signal SMC,eq Pressre control p [Pa] real inpt pressre p [Pa] reference control pressre ref idm [V] ideal control voltage [V] real inpt voltage û [V] inpt voltage of the observer ˆp [Pa] observer pressre [-] switching srface [V] estimated distrbance signal SMC [V] filtered estimated distrbance signal SMC,eq 4

15 PART I Theoretical Backgrond STATE OF THE ART Overview of the literatre 5

16 2 SERVO-PNEUMATIC Pnematic actators are widely sed in indstrial field de to their advantages like low cost, drability, high power-to-weight ratio etc. However their modeling and controlling can be very challenging [1], [2], [3], [4], [5], de to their strong nonlinearity: nonlinear overlapping of valve sections, air compressibility, leakage, heat transfer [6], friction etc. There are several related works in the technical literatre for modeling [7], [8], [9] and controlling [10], [11], [12], [13], [14], [15] pnematic systems. Even modeling only parts of a pnematic system can be very challenging. Different approaches for modeling of pnematic valves can be fond in [16], [17], [18], [19]. Friction is also a crcial part of pnematic positioning [20], [21], [22], [23]. There are also a wide range of control design methods for pnematic systems based on PID controller [24], [25], [26], [27], fzzy controller [28], [29], sliding mode controller [30], [31], [32], [33], observer [34], [35] and friction compensation [36], [37], [38], [39], [40]. To achieve nanoaccracy even piezoelectric elements are proposed [41]. The qestion dring modeling and controlling of pnematic actators is which phenomenon is worth modeling. If we carry ot a deeper investigation of the system it can be highlighted that the inflence of the thermal effects is minimal. The behavior of the servo-pnematic system depends on electronic, mechanic, flid and thermal effects. Comparing these effects we can see that the time constant of the thermal phenomenon is at least one order of magnitde higher, ths the heat transfer has no significant inflence on the dynamic behavior of the pnematic system. Based on the above considerations or model is handled as adiabatic, reversible, ths an isentropic process. The stick-slip effect of friction can be well demonstrated by a spring-mass system. The mechanical part of the pnematic system can also be interpreted as a simple springmass system, where the chamber volmes act like springs and pressre gives the spring stiffness. This stiffness is relatively small for pnematic systems. In case of a pnematic cylinder friction appears between the piston and the chamber-wall and between the piston-rod and the cylinder-cover. It has a strong inflence dring position control de to the relatively small stiffness. Ths a pnematic cylinder is ideal for the investigation of the stick-slip phenomenon. 2.1 Pressre change in the cylinder The following section discsses the pressre change based on [42]. The position of the piston can be controlled by the pressre acting on the srface of the piston, ths by the chamber pressres. These chambers can be modeled separately. The case of the pressre change can be the following: change of the chamber volme (piston motion) 6

17 mass flow in the chamber (inflation and deflation process) heat flow between the gas in the chamber and the air as environment throgh the wall of the chamber Dring the investigation of the system the following simplifications have been taken: adiabatic processes the gas in the chamber is ideal the flows are one-dimensional and stationary V 1 V 2 p 1 T 1 m T 2 p 2 Figre 2.1 Two chambers with state variables As the literatre [1], [43] gives detailed description of the corresponding physical laws, the necessary eqations will be presented only briefly. Combined gas law: p V p V T T (2.1) State eqation (m is the mass of the gas): pv m R T (2.2) p RT (2.3) Adiabatic reversible, i.e. isentropic state change: pv konst (2.4) If the state change is adiabatic: 0 Q1 2 Wf,12, (2.5) where dq cm dt (2.6) 7

18 c c c p v heat capacity in case of isobar state change heat capacity in case of isochor state change and W Vdp f Specific gas constant: R c c (2.7) p v Heat capacity ratio: c (2.8) c The pressre conditions of the gas in the chamber can be described by (2.9): p v,, p m t V t T t p p p V m T t V m T (2.9) As shown in the eqation (2.9) the pressre change of the chambers depends on the volme change of the chamber, the mass flow and the heat transfer. These correlations will be discssed in the following sbsections Pressre change de to volme change The pressre change cased by the change of the chamber s volme can be described by (2.10), which is cased by the displacement of the piston: p V p V V (2.10) Dring the derivation of the eqation we assme that there is no heat transfer, i.e. the process is isentropic (2.4). The system is closed; particles cannot enter or exit the system. Ths the effects of the pressre and the volme change acting on each other can be derived by total differential: Rearranging (2.11): dp dv 0 (2.11) p V 8

19 1 dp dv p V (2.12) Investigating the temporal change of the variables: dp dv 1 p dt V dt (2.13) This finally leads to (2.14): dp p p pv V A x (2.14) dt V V This eqation describes the pressre change cased by the chamber s volme change Pressre change de to mass flow (2.15) describes the pressre change de to the change of the mass cased by the actation of the pnematic valve or leakage, while the volme of the chamber and the gas temperatre is constant. p m p m m (2.15) Since the volme of the chamber is constant, the change of the mass will change the density. dmvd (2.16) Rearranging eqation (2.15), simplifying with the time derivate and with the sbstittion of (2.16): p dp 1 m d V (2.17) Expressing p from the state eqation (2.3) and sbstitting it into the adiabatic eqation (2.4), the change of the pressre can be defined as the fnction of the density: p 0 p 0 (2.18) Differentiating with respect to ρ 9

20 dp p 0 d 0 (2.19) Expressing 0 from eqation (2.18), and sbstitting it into eqation (2.19): dp d p (2.20) And with the sbstittion of the gas law (2.3) : dp d RT (2.21) Ths, the change of the pressre de to mass flow can be described with the following eqation: p m m RT V (2.22) Pressre change de to temperatre change (2.23) describes the pressre change cased by temperatre change. p T p T T (2.23) The temperatre in a cylinder chamber changes when the piston is moving, inflation or deflation takes place [44], [45]. In Figre 2.2 and Figre 2.3 experimental measrements are shown of pressre change de to temperatre change after a short period of inflation or short period of deflation respectively. Figre 2.2 Pressre change de to temperatre change after inflation 10

21 Figre 2.3 Pressre change de to temperatre change after deflation It depends on the task whether the effort is worth spending on modelling heat transfer. The state space model applied in this dissertation does not inclde the modelling of heat transfer. Instead of modelling it is handled as a distrbance and is compensated based on the state space model. 2.2 Flow coefficient of the pnematic valve To determine the pressre changes described by (2.9) it is necessary to know the mass flow, which depends for example on pstream and downstream pressres or the flow srface of the pnematic valve. For modeling the valve the correlations of an ideal nozzle will be applied: The Bernolli-eqation in order to describe the flow of gases. pd 2 2 vd v dp 2 (2.24) ρ p The eqation of the adiabatic state change in form of (2.25). p T T d p p d κ1 κ (2.25) With the help of (2.24) the mass flow of the deflation can be determined assming that the gas in the chamber is in qasi-static state, ths its velocity is zero. v 2 d 2 pd dp (2.26) ρ p p 11

22 By expressing p from the state eqation (2.3), and sbstitting it into the eqation of the adiabatic state change (2.25), the change of the density can be calclated as the fnction of the pressre. ρ p 1 ρ κ p (2.27) where p 0, ρ 0 is the atmospheric pressre, and the density of the air on atmospheric pressre. If we sbstitte (2.27) into (2.26) the integration gives: p 0 1 κ pd κ κ p 1 κ p ρ p 2 vd 2 κ 1 (2.28) The average velocity of the downstream gas after sbstittions and simplifications: v d 2 κ ρ p κ1 κ κ p p d 1 1 (2.29) Assming that the density of the downstream gas is constant in time (stationary flow) at the otflow point, the volmetric flow rate can be calclated by (2.30): V A v (2.30) v And the mass flow rate can be calclated by (2.31): q m A v ρ (2.31) v After sbstitting (2.29): κ1 κ 2κ p p d m v d 1 κ 1 ρ p q A ρ (2.32) Applying (2.27) to d and sbstitting the reslt into (2.32): (2.33) 2 κ1 κ κ κ p d p d qm Av 1 2ρ p κ 1 p p 12

23 The literatre calls the expression nder the first sqare root flow coefficient, which will be denoted by Ψ. 2 κ1 κ κ κ p d p d 1 κ 1 p p (2.34) Figre 2.4 represents the flow coefficient as the fnction of p d /p pressre ratio. Figre 2.4 Flow coefficient The maximm of (2.34) can be defined by the derivative (2.35). This maximm will be the so-called critical pressre ratio. p d dψ p 0 p d d p (2.35) After the derivation: p κ κ d 1 κ p d 1 p p κ p κ κ d κ p d p d 1 κ p κ 1 p p 0 (2.36) Only the second member of the eqation can be zero, which reslts (2.37): 1κ κ p d 1 κ p 2 (2.37) 13

24 Ths, the critical pressre ratio in case of an ideal gas (κ=1.4) is: p d 2 p κ 1 crit κ κ (2.38) The velocity of the downstream gas at critical pressre ratio calclated by (2.29) after sbstittion of the gas-law (2.3): κ κ1 2κ 2 κ1 κ 2κ κ 1 vd R T 1 RT κ 1 κ 1 κ 1 κ 1 (2.39) The temperatre of the pstream gas can be obtained, if the vale of the critical pressre ratio (2.37) is sbstitted into the eqation of the adiabatic state change (2.25) T κ 1 Td (2.40) 2 If we sbstitte (2.40) into (2.39) the flow velocity of the downstream gas will reslt the sonic speed: 2κ κ 1 κ 1 vd R T d κ RTd (2.41) κ1 2 κ1 When the critical pressre ratio is reached the downstream velocity and the state variables of the gas do not change. The reason for this phenomenon is that the pressre change in gases is actally a pressre wave, the propagation velocity of which is eqal to the sonic speed. Ths, when the velocity of the downstream gas reaches the sonic speed, the pressre change is not able to step over the flow srface of the valve; ths, the change of the pressre ratio cannot modify the pressre distribtion along the chamber and the orifice. Ths, above the critical pressre ratio the so-called flow coefficient remains constant. In order to simplify the calclations, the parameter can be approximated: 2 pd pcrit ' p d p pd Ψ Ψ 0 1 if pcrit 0,528 (2.42) p 1 pcrit p ' p d pd Ψ Ψ 0 0, 484 if pcrit 0,528 p p (2.43) 14

25 Figre 2.5 Real flow coefficient Ψ and the approximation Ψ Eschmann [46] showed dring his measrements that the relative error of the approximated and the accrate vale of the flow coefficient is maximm 0,3%. In case of the sal work domain p d /p =0,2 0,8 the relative error is less than 0,2%. The mass flow rate sbstitting the flow coefficient: p d qm Av 2 ρ p Ψ (2.44) p The density dependence can be eliminated with the help of the gas law (2.3): 2 q A p Ψ m p d v ( ) (2.45) RT p Dring the calclation of mass flow a correction coefficient is necessary de to losses (friction, heat) and the geometrical characteristics of the orifice, which is denoted by α. q α A p 2 p Ψ d m v RT p (2.46) The literatre gives experimental reslts for the correction coefficient in case of different systems based on simlation and measrements [16]. 2.3 State space model Consider a servo-pnematic system according to Figre 2.6. On the schematic there is a pnematic cylinder controlled by two independent proportional valves based on the feedback signal of the encoder and the pressre sensors. 15

26 U P A 1 A 2 V2 V 1 U P p 1 p 2 M M Reference position Controller Figre 2.6 Servo-pnematic system Let s bild p a state space model for the system above. The state variables are the piston position x, the piston velocity v, the pressre of the left chamber p 1 and the pressre of the right chamber p 2, the control signals 1 and 2 are the inpt voltages of the proportional valves. where x x 0 0 A A v 0 a v 0 0 A ( ), (2.47) v1 1 m m p 1 p1 b31 0 Av 2( 2) 0 a p2 p2 0 b41 0 a a a a F () v vm Fr 22() v A1p1 ( x, p ) V A x A2p2 ( x, p ) V A ( l x) RT pd 2 b31 ( x, p1 ) p V0 A1 x p RT RT pd 2 b41( x, p2) p V0 A2 ( l x) p R T (2.48) (2.49) (2.50) (2.51) (2.52) 16

27 3 BASIC FRICTION MODELS Engineers enconter the problem of friction in any mechanical system. Friction force is strongly nonlinear and varies considerably while the system is working. In the case of high-precision applications friction makes the sitation even more complex, as the stick-slip effect occrs near the target position. This overview is neither intended to be exhastive nor detailed. It is only to briefly review some of the most widely applied friction models [P-4]. Probably the most simple and most straightforward way of modeling friction is to assme friction is constant and opposite to the direction of motion (see Figre 3.1). This makes friction independent from the vale of the velocity and size of the contact area: F F sign( v), (3.1) where F c friction force is proportional to the normal load: Fr c F c F (3.2) N This is the Colomb friction model. One of the main shortcomings of the Colomb model is the absence of zero velocity friction force, which in reality is very present. Also the independence from the velocity is contrary to what has been experienced with real systems. To overcome these isses, the Colomb model can be completed for instance with the viscos friction model which states: F Fr F v (3.3) v The model is sed for the friction cased by the viscosity of the flids, specifically lbricants. A combination with Colomb friction yields (see Figre 3.2): F F sign( v) F v (3.4) Fr c v The model can be refined by adding the inflence of an external force for the friction at rest (see Figre 3.3). This, however, leads to a discontinos fnction. Here, an important contribtion has been made by Stribeck, who proposed a model which involves a nonlinear, bt continos fnction: v vs FFr ( v) FC Fs FC e sign( v) Fv v, (3.5) 17

28 where v s is the Stribeck velocity, is an empirical parameter, F S is the static friction force. A similar model was employed by Hess and Soom [47]. ( Fs F ) C FFr ( v) FC sign( v) F 2 vv 1 ( v/ vs ) (3.6) The Stribeck crve is an advanced model of friction as a fnction of velocity (see Figre 3.4). Althogh it is still valid only in steady state, it incldes the model of Colomb, static and viscos friction as bilt-in elements. Figre 3.1 Colomb friction Figre 3.2 Colomb and viscos friction Figre 3.3 Colomb and viscos friction with breakaway force Figre 3.4 Stribeck friction There are several more advanced models in the technical literatre [48], [49], [50], [51] like the static friction models, the Karnopp model [52] and the Armstrong model [53], [54] and the dynamic friction models, the Dahl model [55], the LGre model [56], [57], [58], [59], [60] and the Leven model [61]. There are also hysteretic friction models in the literatre [62], [63], [64]. The friction model applied in the dissertation is based on the research reslts of Nori [65]. Based on experimental measrements shown in Figre 3.5 Nori proposed a hysteretic model with Stribeck behavior for increasing speed and Colomb-viscos behavior for decreasing speed (see Figre 3.6). 18

29 Figre 3.5 Friction force as fnction of the velocity at different nominal chamber pressres [65] Figre 3.6 Velocity dependent friction model [65] 19

30 v vs FC 1 FS FC 1 e sign( v) 1v, if sign( v) sign( v) Ff () v FC 2sign( v) 2v, if sign( v) sign( v) (3.7) where F f is the friction at gross sliding, F C1 and F C2 are the Colomb friction for increasing and decreasing speed respectively, and 1 and 1 are viscos the friction coefficients for increasing and decreasing speed respectively [65]. This dissertation does not intend to introdce a new friction model. Only a new representation of the existing models is proposed which is sitable for controller design. The main contribtion of this new model is that the effect of the hysteresis applied in the simlation and the model is constrcted in sch a way that can be handled by controller design algorithms sitable for nonlinear systems. 20

31 4 TENSOR PRODUCT MODEL TRANSFORMATION The following chapter discsses the tensor prodct model transformation based on [66], [67].The tensor prodct (TP) model form is a dynamic model representation wherepon Linear Matrix Ineqality (LMI) based control design techniqes [68], [69], [70], [71] can immediately be exected. It describes a class of Linear Parameter Varying (LPV) models by the convex combination of linear time invariant (LTI) models, where the convex combination is defined by the weighting fnctions of each parameter separately. The TP model transformation is a recently proposed nmerical method to transform LPV models into TP model form [72], [73], [74]. An important advantage of the TP model forms is that the convex hll of the given dynamic LPV model can be determined and analyzed by one variable weighting fnctions. Frthermore, the feasibility of the LMIs can be considerably relaxed in this representation via modifying the convex hll of the LPV model [75]. In [76] and [77] TP model transformation based soltions are proposed for pnematic systems. 4.1 Tensor prodct transformation Consider a parametrically varying dynamical system [78] : x( t) A( p( t)) x( t) B( p( t)) ( t) y( t) C( p( t)) x( t) D( p( t)) ( t), (4.1) p q with inpt () t, otpt y () t and state vector x () t matrix is a parameter-varying object, where t n. The system p is a time varying N-dimensional parameter vector, and is an element of the closed hypercbe a, b a, b... a, b N N N. The parameter p(t) can also inclde some elements of x(t). Given the LPV system description in (4.1), it can be reformlated sing: A( p( t)) B( p( t)) Sp ( ( t)) C( p( t)) D( p( t)) ( n p) ( nq) (4.2) ths: x( t) x( t) Sp ( ( t)) y( t) ( t) (4.3) Expression (4.2) may be approximated over any p(t) sing a nmber of R LTI system matrices (S r, r=1,..., R). These S r matrices are also known as LTI vertex models. 21

32 The convex combination can be bilt sing the weighting fnctions r 0,1 w p t in sch manner, that S(p(t)) fits into the convex hll formed by S r, that is S(p(t)) = co{s 1,S 2,..., S R } w(p(t)). The explicit form of the tensor prodct then becomes: x(t) x(t) 1 2 IN I I N... wn, i ( p ( )) n n t Si 1, i2,..., i N y(t) i11 i22 i 1 1 (t) N n (4.4) The fnction w n,j (p n (t)) is the j th basis fnction belonging to the n th dimension of Ω and p n (t) is the n th element of the p(t) vector. I n denotes the nmber of weighting fnctions sed in the n th dimension. The mltiple indices i 1, i 2,..., i N point at the LTI system associated with the i th n weighting fnction. There are R N I LTI systems denoted S i1,i2,,in which reslts in the TP model representation: n n1 S ( p( t)) R wr ( p( t)) Sr r 1 (4.5) TP model to be a convex combination, the weighting fnctions mst satisfy: I n n, p : w ( p ) 1 (4.6) n n, i n i1 The main steps of the Tensor Prodct Model Transformation as shown in Figre 4.1 are: first we need a discretized model in p t. As shown in Figre 4.1, the discretized model can be obtained from measrement in direct way or sing a nonlinear analytical S(p(t)) model and the comptation of system matrix S(g) over the grid points g of a hyper-rectanglar grid net defined in. the second step extracts the singlar vale based orthonormal strctre of the system, namely, this step determines the minimal nmber the LTI systems in orthonormal position according to the ordering of the singlar vales and defines the orthonormal discretized weighting fnctions of the searched polytopic model. the LTI systems and the discretized weighting fnctions can be modified, in order to satisfy frther conditions for the weighting fnctions: for instance, this step can ensre the convexity of the weighting fnctions (4.6). 22

33 DIRECT METHOD ANALYTICAL MODEL- BASED METHOD Measred Data High Dimension Parametric Nonlinear Analytical Model (loss of information) Identification Discretization Dimension redction Redced Dimension Parametric Model Figre 4.1 TP transformation based design algorithm 4.2 Redction of the dimension of the measred data This method is sed as a software tool [79], [72], [78]. The basic idea is described here see Figre 4.2. The p(t) is sampled in n points by vectors v k. Between the sampled points, p(t) is approximated by interpolation. It is well known that p(t) can be described by two orthogonal base vectors v b1 and v b2 in a properly selected coordinate system. This simple idea is generalized for TP transformation. Figre 4.2 Basic idea of TP transformation It is known from matrix algebra, that each matrix can be written in the form: A = U Λ V, (4.7) 23

34 n m Where A is an arbitrary matrix, U is a matrix that contains the eigenvectors of the matrix A A T, Λ contains the so called singlar vales in its diagonal. V contains the eigenvectors of the matrix A T A again. Λ is a diagonal matrix, often denoted as a vector. The occrrence of zeros in matrix Λ allows s to decrease the size of matrix A. In case of a tensor, it has to be nfolded into bidimensional space, to form an ordinary matrix (first step in Figre 4.3), then the singlar vale decomposition (SVD) can be applied, ths obtaining a simplified system. Finally the matrix mst be packed back into its original tensor form. The above operations can be performed along every dimension (Figre 4.3), ensring the best possible redction of the system, reslting finally in a higher order singlar vale decomposition (HOSVD). Figre 4.3 HOSVD decomposition 24

35 5 SLIDING MODE CONTROL 5.1 History of sliding mode control The variable strctre systems (VSS) have some interesting characteristics in control theory [P-10]. A VSS might also be asymptotically stable if all the elements of the VSS are nstable itself. Another important featre that a VSS - with appropriate controller - may get in a state in which the dynamics of the system can be described by a differential eqation with lower degree of freedom than the original one. In this state the system is theoretically completely independent of changing certain parameters and of the effects of certain external distrbances (e.g. non-linear load). This state is called sliding mode and the control based on this is called sliding mode control (SMC) which has a very important role in the control of power electronic devices. The theory of variable strctre system and sliding mode has been developed decades ago in the Soviet Union. The theory was mainly developed by Vadim I. Utkin [80] and David K. Yong [81]. According to the theory sliding mode control shold be robst, bt experiments show that it has serios limitations. The main problem by applying the sliding mode is the high freqency oscillation arond the sliding srface, the so-called chattering, which strongly redces the control performance. Only few cold implement in practice the robst behavior, predicted by the theory. Many have conclded that the presence of chattering makes sliding mode control a good theory game, which is not applicable in practice. In the next period the researchers invested most of their energy in chattering free applications, developing nmeros soltions. The sliding mode control has a niqe place in control theories. First, the exact mathematical treatment represents nmeros interesting challenges for the mathematicians [82], [83], [84]. Secondly, in many cases it can be relatively easy to apply withot a deeper nderstanding of its strong mathematical backgrond and is therefore widely sed in engineering practice [85]. The design of a sliding mode controller consists of three main steps: 1. Design of the sliding srface. 2. Design of the control law. (It forces towards and holds the system trajectory on the sliding srface.) 3. Chattering free implementation. 25

36 5.2 Sliding srface design The following linear time invariant (LTI) system is considered; first the reference signal is spposed to be constant and zero. A single inpt mltiple otpt system is discssed. The system is transformed to a reglar form [86]. x1 A11 A12 x1 0 x2 A21 A22 x2 B2 x1 y C D x2 x x 1 2 y n1 (5.1) with inpt t (), otpt yt (), state vector x () t and B2 0. The switching srface, of the sliding mode, where the control has discontinity, can be written in the following form [85], where λ is the "srface vector". x 2 λx 1 0 and ( n 1) λ (5.2) When sliding mode occrs (when 0 and x2 λx 1), the design problem of the sliding srface can be regarded as a linear state feedback control design for the following sbsystem: x1 A11x 1 Ax 12 2 (5.3) In (5.3) x 2 can be considered as the inpt of the sbsystem. A state feedback controller x2 λx 1 for this sbsystem gives the switching srface of the whole VSS controller. In sliding mode x ( A A λx ) (5.4) Varios linear control design methods based on state feedback (pole placement, LQ optimal, freqency shaped method, H ) were proposed for (5.4) to design the switching srfaces in the last decade [85]. The main problem is that these methods are not sitable for a non-linear system which is more challenging. The soltion can be the Tensor Prodct model transformation. 5.3 Control Law To ensre that the system remains in the sliding mode ( 0 ) the condition 0 (5.5) 26

37 shold hold. The simplest control law which can lead to sliding mode is the relay: M sign( ) (5.6) In case of pnematic control this can be realized by a monostable solenoid pnematic valve. The problem of the relay type of controller is that it does not ensre the existence of sliding mode for the whole state space, and relatively big vales of M are necessary which might case a severe chattering phenomenon. This control law is preferable if the controller's sample freqency is nearly eqal to the maximm switching freqency of the pnematic valve. If sliding mode exists then there is a continos control, so-called "eqivalent" control eq, which can hold the system on the sliding manifold. It can be calclated from 0. x 2 λx 1 0 (5.7) A x A x B λ( A x A x ) 0 (5.8) eq can be expressed from (5.7) eq ( ) ( A A ) x B A λa x λ (5.9) 2 In the practice, there is never perfect knowledge of the whole system and its parameters. Only ˆeq, the estimation of eq, can be calclated. Since eq does not garantee the convergence to the switching manifold in general, a discontinos term is sally added to ˆeq. ˆ M sign( ) (5.10) eq The control law (5.10) cannot be realized by monostable solenoid pnematic valve; proportional pnematic valve is needed. 5.4 Chattering free implementation, Sector Sliding Mode The chattering in the basic sliding mode control is essential de to the reqirement that the system state mst stick to the switching srface. There are several soltions for elimination of chattering [87], [88], [89]. The LMI based sliding mode control is discssed in [90] and [91]. Here the sector sliding mode [92], [93], [94] is 27

38 discssed since it can be extended for TP model based sliding mode control [95], [96], [78], [97], [98]. Obviosly this reqirement is too strict when only finite switching rate is available. Replacing the switching srface to the sliding sector may enable the system state to move continosly. To implement the proposed approach, two sliding srfaces are defined first x λx, r 1,2 (5.11) r 2 r 1 0 Then the two sliding srfaces divide the whole state space into three regions defined as Definition 1: R { x ( x) 0 and ( x) 0} R R { x ( x) 0 and ( x) 0} { x ( x) ( x) 0} Here the region R 3 is the sliding sector. The control strategy of the proposed modified sliding mode control method is (5.12) eq d Where eq is the continos "eqivalent" and d is defined as d 2 12 M Msign xr1r 2 x R 3 (5.13) As shown in Figre 5.1, let s represent the sector by the srface 1 2 x 2 λx 1 0 where λ λ λ (5.14) 2 28

39 Figre 5.1 Sliding sectors (in case n 2 ) 5.5 Sliding Mode Based Model Reference Adaptive Compensation There are several alternatives of the classical Model Reference Adaptive Control (MRAC) [99], [100], [101], [102]. Consider the following model with external distrbances and ncertain parameters satisfying the so-called Drazenovic [103] condition, written in the reglar state eqation form [104], [100]: x1 A11 A 22 x1 0 0 f t x2 A21 A21 A22 A22 x2 B2 B2 E2, (5.15) nm m m, where x1, x2,, A ij ( i, j1,2) and B 2 denote the nominal or desired (ideal) system matrices. The bar refers to the reference vale in this paper. A ( 1,2) 2 j j and B 2 are the respective ncertain pertrbations and f(t) is an nknown, bt bonded distrbance with bonded first time derivative. Now is defined as the ncertainties and the distrbance of the system. A x A x B E f t (5.16) The second line of (5.15) can be rewritten by. x2 A21x1 A22x2 B2 (5.17) According to [105], x 2 is modeled by a discontinos observer: ˆ ˆ SMC x A x A x B, (5.18) 29

40 where SMC is the discontinos feedback. The goal of the design is to find a feedback signal denoted SMC, sch that the motion of the system (5.15) is restricted to belong to the manifold S m S x x x ˆ 0, where. (5.19) In sliding mode ( x ˆ 2x 2 0), SMC contains information on the system's parametric ncertainties and the external distrbance, which can be sed for feedback compensation (compare (5.17) and (5.18)). It is tre again that the order of the differential eqation describing the sliding srface is lower, than the order of the original differential eqation describing the system The simplest control law, which can lead to sliding mode, is the relay [85]: SMC, i Misign i. (5.20) If is in the range of B2( range( B 2)), the ideal sliding mode occrs [101]. In the practice, there is no way to calclate the eqivalent control SMC, eq precisely, bt it can be estimated by a low pass filter (LPF) for 5.2. SMC as shown in Figre x Ref x Controller idm Real System x 1 x 2 û Observer ˆx 2 SMC SMC LPF SMC, eq Figre 5.2 Sliding mode based feedback compensation There are two loops in Figre 5.2. The observer sliding mode controller loop shold be as fast as possible to achieve an ideal sliding mode. This is granted becase this loop is realized in a comptation engine in the recent application. The real system compensator (consisting of sliding mode controller and LPF) loop shold be faster than the change of the distrbance. On the other hand, the smallest n-modeled resonant 30

41 freqency of the real system shold be ot of the bandwidth of that loop to avoid chattering. 5.6 Discrete-Time Implementation The robstness of continos-time sliding mode control is realized by high-gain obtained by the high-freqency switching of control signal SMC. To adapt the slidingmode philosophy for a digital observer, the sampling freqency shold be increased compared to other types of observer methods [101]. If the switching freqency of SMC is not high enogh, might chatter arond the manifold 0 as shown in Fig. 9, where k T denotes the time of k th sampling, and the system might be inflenced. Figre 5.3 Discrete-time chattering phenomenon In case of a discrete-time observer, the observer feedback signal SMC[ k ] may switch from positive to negative vale and vice versa reslting, 0 even if SMC, eq 0 SMC average. The role of the discontinos term [ ] in the observer (5.18) is to SMC k sppress the effect of the ncertain pertrbations and bonded distrbances. In order to redce the chattering, we shold redce the change of. Since SMC, eq is continos, only the change of SMC, eq dring the sampling period mst be estimated. The chattering can be redced by the following discrete-time observer law. SMC [ k] [ k 1] k sign( [ k]) (5.21) SMC SMC, eq 31

42 The control system is shown in Figre 5.4. The inpt signal of the system is the reference position. x Ref x Controller idm Real System x 1 x 2 Z -1 û Observer ˆx 2 SMC SMC LPF SMC, eq Figre 5.4 Controller scheme for position control 32

43 PART II Contribtion 33

44 6 DIRECT TENSOR PRODUCT MODEL TRANSFORMATION This chapter concentrates on the description of the friction F Fr, ths on the definition of a 22, the friction element of (2.47) the state space model of the pnematic system [P-4]. For this prpose an experimental setp was bilt focsing on the stick-slip effect. The experiments were carried ot on a standard DSNU PPV-A Festo pnematic cylinder eqipped with a 3/3 proportional valve, a pressre sensor and an accelerometer. The experimental setp is shown in Figre 6.1 and Figre 6.2. The hardware elements are listed in Table 6.1. Figre 6.1 Experimental setp for friction identification Table 6.1 Elements of the experimental setp Nr. Element Type 1 Control nit NI crio Pressre sensor FESTO SDE-10-10V/20mA 3 3/3 proportional valve HOERBIGER ORIGA SERVOTEC 4 Pnematic cylinder Festo DSNU PPV-A 5 Accelerometer Freescale MMA2260D 6 Interface 34

45 A 1 A 2 V2 V 1 U P p 1 p 2 M Control nit Figre 6.2 Schematic of the experimental setp The identification of friction is based on the investigation of the stick-slip effect [106]. As mentioned before, this effect might be mch stronger in the case of pnematic cylinders, making the identification of the friction force easier. As excitation let s set a fixed voltage on the proportional valve, in other words start to load the left chamber with slowly rising pressre ( p 1 in Figre 6.2) while the right chamber is open to atmospheric pressre ( p 2 in Figre 6.2). In Figre 6.3 we can see the pressre difference of the two chambers dring or experimental measrement which gives s the force acting on the piston srface. Figre 6.3 Experimental excitation As we can see, when the excitation force reaches the vale of the static friction, the piston starts to move (slip), which motion rapidly increases the volme of the left chamber. De to the increasing volme the pressre of the left chamber drops, reslting 35

46 low pressre difference. Ths the excitation force will also drop, the piston starts to slow down, and at the end it will reach its new steady state (stick). As the excitation voltage of the proportional valve is continosly at the same voltage level, at steady state the pressre will start to increase again. Ths the process starts from the beginning with the slipping of the piston at an excitation force stepping over the static friction. This way measrements can be taken right in the area of the friction hysteresis. In Figre 6.4 and Figre 6.5 one single stick-slip process is ct from the measrement, showing the measred pressre and acceleration: Figre 6.4 Excitation for a single stick-slip Figre 6.5 Acceleration dring stick-slip Based on the measred acceleration we can calclate the velocity by integration. Figre 6.6 shows the velocity-profile dring stick-slip. We know the mass and the acceleration of the piston. Ths we know the force needed for the motion. Based on the measred pressre we indirectly measred the force acting on the piston. The difference of the force acting on the piston and the force needed for the motion was dissipated by the friction. Ths we have the friction force as the fnction of the velocity (see Figre 6.7). Figre 6.6 Velocity dring stick-slip Figre 6.7 Friction hysteresis 36

47 6.1 Modeling Stribeck-friction The models were derived based on the measrement of a single stick-slip phenomenon. The discssed reslts are not intended to be correct from the point of view of measrement technology bt to introdce a practical modeling method for control design. Three different Stribeck-friction models were derived to make a comparison based on the measrements: MODEL A: Nonlinear parametric model withot hysteresis loop based on (3.6) (conventional model based on literatre). MODEL B: TP model derived via analytical model-based method of TP model transformation (see Figre 4.1) based on nonlinear parametric model withot hysteresis loop based on (3.6). MODEL C: TP model derived via direct method of TP model transformation (see Figre 4.1) directly applying the measrement reslts of the experiments. The friction of the pnematic cylinder can be modeled as the fnction of the velocity as follows: F ( v) a ( v) v m (6.1) Fr a 22 Fr 22() v F () v, (6.2) vm system. Where a 22 is the friction element of the state space model of the pnematic Model A Based on the friction model (3.6) proposed by the literatre, the identification of the parameters has given the following reslt: FC Fv vs FS [ N] 3.7 [ N] 15[ Ns / m] 0.019[ m / s] 37

48 Model A describes only the Stribeck crve withot hysteresis, ths the friction force is the same for acceleration and for braking sitation (see Figre 6.8). Figre 6.8 Friction characteristic of Model A Model B Model B was designed sing the analytical model-based method of TP model transformation (see Figre 4.1) based on (3.6). The operation domain of or experiment defines the domain and grid size of the discretization process for TP model transformation. Ths the domain for the velocity is between and 0.05 m/s with a grid size 1000 which takes 1000 discrete vales in the defined domain (this domain and grid size is applied for every TP model transformation in this chapter). As in this case there is no hysteresis loop, the TP model transformation was applied only for the parametric eqation (3.6) and the reslts are denoted by Str. After applying the TP model transformation, the friction can be modeled sing the following linear combination: a w a w a (6.3) Str Str1 Str1 Str 2 Str 2 22, an an 22, an an 22, an Str1 4 a22, an (6.4) a, (6.5) Str 2 22, an 359 where the weighting coefficients as the fnction of the velocity can be seen in Figre

49 Figre 6.9 The weighting coefficients as the fnction of velocity for Model B The friction characteristic of Model B can be seen in Figre Figre 6.10 Friction characteristic of Model B As we can see the model gives a good approximation of the measrement reslts Model C Model C was designed sing the direct method of TP model transformation (see Figre 4.1). The same domain and grid size (velocity between and 0.05 m/s, grid size 1000) was applied as in the case of Model B bt for data inpt instead of the parametric eqation (3.6) directly the measrement data was sed. After applying the TP model transformation, the friction can be modeled sing the following linear combination: a w a w a (6.6) Str Str1 Str1 Str 2 Str 2 22, dir dir 22, dir dir 22, dir Str1 4 a22, dir (6.7) a, (6.8) Str 2 22, dir

50 where the weighting coefficients as the fnction of the velocity can be seen in Figre Figre 6.11 The weighting coefficients as the fnction of velocity for Model C The friction characteristic of Model C can be seen in Figre Figre 6.12 Friction characteristic of Model C Note that Model C has almost the same friction characteristic as derived from the measrement. In Figre 6.12 they are overlapping each other. The difference depends on the grid size of the TP model transformation. The more measrement data we se in the TP model transformation process the smaller the difference will be. There are two advantages of Model C. The first advantage is that it is not necessary to identify the parameters of the friction model (3.6), bt we can se directly or measrement data. The second is that we created a model sitable for TP control design, which provides already implemented control design tools for nonlinear systems. 40

51 6.1.4 Comparison of the Stribeck-friction models For model evalation prposes a MATLAB Simlink simlation model was implemented (see Figre 6.13). The friction models are compared to the measrement reslts by sing the same excitation ( p 1 in Figre 6.2 and Figre 6.13) measred dring a single stick-slip phenomenon shown in Figre 6.4. Ths as excitation the pressre difference from the measrement is applied in every simlation and the responses of the three friction models are investigated. The comparison is based on the investigation of the acceleration- and velocity-responses of the different models. In the figres below the measrement reslts shown in Figre 6.5 and Figre 6.6 are denoted as Measrement and they are compared to the simlated acceleration and velocity otpt of the model shown in Figre Figre 6.13 MATLAB Simlink simlation model 41

52 Figre 6.14 Acceleration of Model A Figre 6.15 Velocity of Model A Figre 6.16 Acceleration of Model B Figre 6.17 Velocity of Model B Figre 6.18 Acceleration of Model C Figre 6.19 Velocity of Model C 42

53 The comparison of the root-mean-sqare deviation (RMSD) of the model reslts compared to the measrement reslts is shown in Table 6.2. Table 6.2 RMSD of the Stribeck-friction models Deviation Model A Model B Model C Acceleration [m/s 2 ] Acceleration [%] Velocity [m/s] Velocity [%] As we can see the three models have almost the same reslts. For Model A and Model B it can be explained by the similarity of the friction trajectories of the models (see Figre 6.8 and Figre 6.10). In case of Model C the deviation to the measrement depends on, how mch noise the sensors have collected dring the measrement. The reslts conclde that all the three modeling methods can be applied for the same prpose bt Model B and Model C has advantages as compared to Model A. The main advantage of Model B is that the algorithms of TP control design for nonlinear systems can be applied for it. The same advantage can be mentioned in the case of Model C. Moreover, in case of Model C it is not necessary to identify the parameters of the friction model, bt the measrement data can be directly applied for modeling. 6.2 Systematic design of sliding sector The method described in section 5.4 is implemented and a systematic design method is developed. The inpt of the piston is a force F in. The dynamic eqation of the Str1 Str 2 system with the parameter a and a Stri Fin mv a22 vm i 1,2 (6.9) Let s assme that the position reference signal is a constant. Let s denote the position error by x 1 and its time derivative by x 2. If the reference signal is a constant than x2 v. The dynamic eqation of the error x a x F m Stri in (6.10) x x (6.11)

54 The sliding sector is defined by two sliding lines (see Figre 6.20). (6.12) 1 x2 1 x1 0 (6.13) 2 x2 2x1 0 Figre 6.20 The sliding sector The eqivalent control can be calclated from 0 (see (5.7)-(5.9)) x (6.14) i 2 ix1 0 Sbstitting (6.10) to (6.14) Stri Fin i a22 x2 ix1 0 (6.15) m The eqivalent control can be expressed F eqi m a x x i 1,2 (6.16) Stri 22 2 i 2 cases i is selected in sch a way that the eqivalent control shold be the same in both F F a x x a x x (6.17) m eq1 eq2 Str1 Str m Since a a (6.18) Str 2 Str The width of the sector is described by the following parameter 1 2 (6.19) 1 44

55 The parameter of the sliding line can be expressed Str 2 Str1 a22 a (6.20) 6.3 Modeling friction with hysteresis loop In case of modeling friction with hysteresis loop the friction models (3.4) and (3.6) are applied, based on the sign of the acceleration: ( Fs F ) C FC sign( ), if sign( ) sign( ) 2 v Fvv v v FFr () v 1 ( v/ vs ) Fcsign( v) Fvv, if sign( v) sign( v) (6.21) To avoid discontinity the signm fnction is approximated by a sigmoid fnction: 2 sign( v) v 1 e (6.22) Figre 6.21 shows the simlated friction-hysteresis. In Figre 6.22 the hysteresis loop is magnified distingishing the acceleration and braking sitation. Figre 6.21 Simlated friction-hysteresis Figre 6.22 Simlated friction-hysteresis (magnified) Three different Stribeck-friction models with hysteresis loop were derived to make a comparison based on the measrements: MODEL D: Nonlinear parametric model with hysteresis loop based on (6.21). MODEL E: 45

56 TP model derived via analytical model-based method of TP model transformation (see Figre 4.1) based on nonlinear parametric model with hysteresis loop based on (6.21). MODEL F: TP model derived via direct method of TP model transformation (see Figre 4.1) directly applying the measrement reslts of the experiments Model D For Model D the same friction parameter set is applied as for Model A bt a friction with hysteresis loop is modeled based on the friction model described by (6.21). The friction-hysteresis of Model D is shown in Figre Figre 6.23 Friction characteristic of Model D The model is based on nonlinear parametric eqations Model E Model E was designed sing the analytical model-based method of TP model transformation (see Figre 4.1) based on (6.21). For the TP model transformation the same domain and grid size was applied, ths the domain for the velocity is between and 0.05 m/s with a grid size To specify the hysteresis loop, the TP model transformation was applied for both parametric eqation (3.4) and (3.6). (3.6) is denoted by Str and (3.4) is denoted by Vis. After applying the TP model transformation, the friction for accelerating piston can be modeled sing the following linear combination: a w a w a (6.23) Str Str1 Str1 Str 2 Str 2 22, an an 22, an an 22, an Str1 4 a22, an (6.24) 46

57 a. (6.25) Str 2 22, an 359 The friction for slowing piston can be modeled sing the following linear combination: a w a w a (6.26) Vis Vis1 Vis1 Vis2 Vis2 22, an an 22, an an 22, an Vis1 3 a22, an (6.27) a, (6.28) Vis2 22, an 376 where the weighting coefficients as the fnction of the velocity can be seen in Figre 6.24 and Figre Figre 6.24 The weighting coefficients of Model E for accelerating piston Figre 6.25 The weighting coefficients of Model E for slowing piston The friction characteristic for accelerating and slowing piston can be seen in Figre 6.26 and Figre Figre 6.26 Friction characteristic of Model E for accelerating piston Figre 6.27 Friction characteristic of Model E for slowing piston The shape of the weighting coefficients is qite straightforward to explain. The nonlinear friction terms are modeled sing a varying viscosity coefficient, which is represented by the a 22 element in the system matrix. A small viscos coefficient in a

58 dominates at high speed, where the Colomb friction is relatively small. A very large viscos coefficient in 1 a dominates at low speed, where the Colomb friction is 22 comparatively large. We can also see that the rate of Str1 a and 22 Vis1 a gives s the rate of 22 F S and F C. The condition determining the change between the two models is given as follows: F Fr Str a22, if sign( v) sign( v) () v Vis a22, if sign( v) sign( v) (6.29) The friction characteristic of Model E can be seen in Figre Figre 6.28 Friction characteristic of Model E As we can see the model gives a good approximation of the measrement reslts Model F Model F was designed sing the direct method of TP model transformation (see Figre 4.1). The same domain and grid size (velocity between and 0.05 m/s, grid size 1000) was applied as in the case of Model E bt for data inpt instead of the parametric eqation (6.21) directly the measrement data was sed. After applying the TP model transformation, the friction for accelerating piston can be modeled sing the following linear combination: a w a w a (6.30) Str Str1 Str1 Str 2 Str 2 22, dir dir 22, dir dir 22, dir Str1 4 a22, dir (6.31) a. (6.32) Str 2 22, dir

59 The friction for slowing piston can be modeled sing the following linear combination: a w a w a (6.33) Vis Vis1 Vis1 Vis2 Vis 2 22, dir dir 22, dir dir 22, dir Vis1 4 a22, dir (6.34) a, (6.35) Vis2 22, dir 383 where the weighting coefficients as the fnction of the velocity can be seen in Figre 6.29 and Figre Figre 6.29 The weighting coefficients of Model F for accelerating piston Figre 6.30 The weighting coefficients of Model F for slowing piston The friction characteristic for accelerating and slowing piston can be seen in Figre 6.31 and Figre Figre 6.31 Friction characteristic of Model F for accelerating piston Figre 6.32 Friction characteristic of Model F for slowing piston The condition determining the change between the two models is given as follows: F Fr Str a22, if sign( v) sign( v) () v Vis a22, if sign( v) sign( v) (6.36) 49

60 The friction-trajectory of Model F can be seen in Figre Figre 6.33 Friction characteristic of Model F Note that jst like in the case of Model C, Model F has also almost the same friction hysteresis as derived from the measrement. In Figre 6.33 they are overlapping each other. The difference depends again on the grid size of the TP model transformation. The more measrement data we se in the TP model transformation process the smaller the difference will be. There are two advantages of Model F. One is that it is not necessary to identify the parameters of the friction model (6.21), bt we can se directly or measrements. The second is that we created a model sitable for TP control design, which provides already implemented control design tools for nonlinear systems Comparison of the models with hysteresis loop As in the case of the previos comparison for model evalation prposes the same MATLAB Simlink simlation model was applied (see Figre 6.13). The friction models are compared to the measrement reslts by sing the same excitation ( 1 p in Figre 6.2 and Figre 6.13) measred dring a single stick-slip phenomenon shown in Figre 6.4. Ths as excitation the pressre difference from the measrement is applied in every simlation and the responses of the three friction models are investigated. The comparison is based on the investigation of the acceleration- and velocity-responses of the different models. In the figres below the measrement reslts shown in Figre 6.5 and Figre 6.6 are denoted as Measrement and they are compared to the simlated acceleration and velocity otpt of the model shown in Figre

61 Figre 6.34 Acceleration of Model D Figre 6.35 Velocity of Model D Figre 6.36 Acceleration of Model E Figre 6.37 Velocity of Model E Figre 6.38 Acceleration of Model F Figre 6.39 Velocity of Model F 51

62 The comparison of the root-mean-sqare deviation (RMSD) of the model reslts compared to the measrement reslts is shown in Table 6.3. Table 6.3 RMSD of the models with hysteresis loop Deviation Model D Model E Model F Acceleration [m/s 2 ] Acceleration [%] Velocity [m/s] Velocity [%] Model E is almost the same as Model D considering the friction model. This can also be noticed on the simlation reslts. The difference depends on the grid size of the TP model transformation. The more discretized data we se in the TP model transformation process the smaller the difference will be. The main advantage of Model E is that the algorithms of TP control design for nonlinear systems can be applied for it. Model F gives s a bit bigger difference as it is designed sing the measrement data directly with no filtering. Ths the noise collected by the sensors is also bilt in the model. This inaccracy can be redced sing the statistical reslts of repeated measrements. There are two advantages of Model F. The first advantage is that it is not necessary to identify the parameters of the friction model, bt the measrement data can be directly applied for modeling. The second is that Model F is sitable for TP control design, which provides already implemented control design tools for nonlinear systems. In most cases both the direct (directly from measrement) and the analytical model-based method can be applied, ths we can select the appropriate method based on their advantages and disadvantages. Advantages of the direct method are that the definition of the model-parameters is not necessary and there is no loss of information. On the other hand it has to be taken into accont that the distrbances dring the measrement are also modeled. Based on these attribtions of the investigated modeling tools it can be decided for a given modeling problem when the direct method can be applied. For example in the case of modeling of friction the identification of the parameters can take a lot of effort, which can be avoided sing the direct method. On the other hand for sch systems where the measrement has a lot of distrbances which cannot be handled by other elements of the system the analytical model-based method is the appropriate soltion. 52

63 7 SLIDING MODE BASED MODEL REFERENCE ADAPTIVE COMPENSATION This chapter concentrates on the implementation of the sliding mode based model reference adaptive compensation discssed in section 5.5. The experiments were carried ot on a standard Mecman pnematic cylinder eqipped with a 3/3 proportional valve, two pressre sensors and an incremental linear encoder. The experimental setp is shown in Figre 7.1. The hardware elements are listed in Table 7.1. Figre 7.1 Servo-pnematic system Table 7.1 Elements of the experimental setp Nr. Element Type 1 Control nit NI crio Interface 3 Pressre sensor FESTO SDE-10-10V/20mA 4 Incremental linear encoder Mittoyo AT115 5 Positioning table 6 Pnematic cylinder Mecman /3 proportional valve HOERBIGER ORIGA SERVOTEC 53

64 The motivation for this experimental setp was to investigate the control problematics of today s commercial vehicle gearboxes. In case of a pnematic gearbox only one chamber is controlled, which is working against a spring. In the case of the experimental setp the spring is sbstitted by a closed chamber acting as an air spring. The schematic can be seen in Figre 7.2. U P A 1 A 2 V2 V 1 U P p 1 p 2 M Reference position Control nit Figre 7.2 Servo-pnematic The state space model of the experimental setp: x x 0 A A v 0 a v 0 A ( ) p m m p 1 p 1 b 31 0 a v1 1 2 a a F () v vm Fr 22() v A1p1 ( x, p ) V A x RT pd 2 b31 ( x, p1 ) p V0 A1 x p RT (7.1) (7.2) (7.3) (7.4) 54

65 7.1 Decomposition The servo-pnematic system (2.47) does not satisfy the so-called Drazenovic condition (5.15). Ths the sliding mode based model reference adaptive compensation introdced in section 5.5 cannot be applied to this model. To resolve this conflict the system can be rearranged into sch sbsystems which satisfy the Drazenovic condition separately. Ths the sliding mode based model reference adaptive compensation can be applied to these sbsystems. The conditions for the sliding mode based model reference adaptive compensation are that the control inpt shold be able to work with high freqency, and the difference between the observer and the real system shold be less than the maximal control inpt. For the decomposition the same approach was applied as the conventional method for a DC-motor control design. In case of a DC-motor the control system is sally decomposed into two control loops: voltage and crrent control loops. In the case of the presented pnematic setp the system can be decomposed to position and pressre control [107] (see Figre 7.3). Pnematic sbsystem p Mechanical sbsystem x x Figre 7.3 Decomposition of the servo-pnematic system Both the position control sbsystem and the pressre control sbsystem satisfy the Drazenovic condition and the sliding mode based model reference adaptive compensation can be implemented for them (see Figre 7.4). The condition for sch a cascade control strctre is that the inner loop shold be faster than the oter loop. x Ref Position Controller p Ref Pressre Controller Pnematic sbsystem p Mechanical sbsystem x x Figre 7.4 Control schematic of the servo-pnematic system 55

66 7.2 Position control The state space model of the position control loop is: x x 0 A p 1 A2 2 f () t v 0 a 22 v E m m a F () v vm Fr 22() v (7.5) (7.6) p 1 (7.7) This redced model in contrast with the original model (2.47) satisfies the Drazenovic condition (5.15), which facilitates the tilization of the proposed sliding mode based model reference adaptive compensation. The control schematic based on Figre 5.4 is shown in Figre 7.5. x Ref PID Controller p idm p Piston x v Z -1 ˆp Observer vˆ SMC psmc LPF p SMC, eq Figre 7.5 Position control schematic where x [µm] piston position v [µm/s] piston velocity x ref [µm] reference position p idm [Pa] ideal control pressre p [Pa] real inpt pressre ˆp [Pa] inpt pressre of the observer ˆv [µm/s] observer velocity [-] switching srface p [Pa] estimated distrbance signal SMC 56

67 p [Pa] filtered estimated distrbance signal SMC,eq In this control design the controlled state variable is the position and the observed state variable is the velocity of the piston. For the realization of the position control loop an inner pressre control loop is also needed (see Figre 7.6). x Ref PID Controller p idm p Ref Pressre Controller p Piston x v Z -1 ˆp Observer vˆ SMC psmc LPF p SMC, eq Figre 7.6 Position control schematic with inner pressre controller loop As nowadays mostly discrete-time controllers are applied a discrete approach is also shown in Figre 7.7. x Ref PID Controller p idm p Ref Pressre Controller p Piston x v Z -1 ˆp Observer vˆ vˆ Z -1 SMC p SMC LPF p SMC, eq Figre 7.7 Position control schematic for discrete-time controller This discrete-time control schematic is applied in the experimental setp as well. 57

68 7.3 Pressre control The state space model of the pressre control loop for one chamber is: p 0 p b A ( ) E f ( t) (7.8) 11 v1 1 RT pd 2 b11 ( x, p1 ) p V0 A1 x p RT (7.9) This redced model satisfies the Drazenovic condition (5.15), which facilitates the tilization of the proposed sliding mode based model reference adaptive compensation. The reason why the original system does not satisfy the Drazenovic condition (5.15), while the sbsystems do satisfy it, is that dring the decomposition the a 32 element of (7.1) is neglected in the observers and handled as a distrbance. The control schematic based on Figre 5.4 is shown in Figre 7.8. p Ref PID Controller idm Valve and chamber p Z -1 û Observer ˆp SMC SMC LPF SMC, eq Figre 7.8 Pressre control schematic where p [Pa] real inpt pressre p [Pa] reference control pressre ref idm [V] ideal control voltage [V] real inpt voltage û [V] inpt voltage of the observer ˆp [Pa] observer pressre [-] switching srface [V] estimated distrbance signal SMC [V] filtered estimated distrbance signal SMC,eq 58

69 In this control design the controlled and the observed state variable is in both cases the pressre of the chamber. To compensate the nonlinear copling between the pressre gradient in the chamber and the inpt voltage of the proportional valve, the transfer fnction of this sbsystem is separated into static and dynamic parts. The 1 inverse fnction of the static part W can be tilized to linearize the sbsystem [107] (see Figre 7.9). Static y Ref Linear Controller linear 1 W Static real WStatic W Dynamic y Figre 7.9 Linearization via inverse fnction In the servo-pnematic system the dynamics of the proportional valve is mch faster than the pnematic phenomenon, ths the observer model W ˆV of the proportional valve can be described by static invertible algebraic eqation ignoring the state variables belonging to the proportional valve. The dynamic part of the transfer fnction is the capacity of the chamber which is represented by the integrator in the control schematic (see Figre 7.10). p Ref PID Controller p idm ˆ 1 W V idm Valve p 1 s p Z -1 û WˆV p ˆ 1 s ˆp SMC SMC LPF SMC, eq Figre 7.10 Pressre control schematic with static and dynamic parts In the experimental setp the discrete-time form of the pressre controller (see Figre 7.11) is applied. The definition of the inverse fnction is discssed in the next section. 59

70 p Ref PID Controller p idm ˆ 1 W V idm Valve p 1 s p Z -1 û Wˆ V pˆ ˆp Z -1 SMC SMC LPF SMC, eq Figre 7.11 Pressre control schematic for discrete-time controller 7.4 Identification of the pnematic valve The inverse fnction is defined based on the characteristics of the proportional valve. The aim of this section is to define the correlation between the inpt voltage of the proportional valve, the pressre in the chamber and the pressre gradient in the chamber (see Figre 7.23). There are several standards for the identification of the flow coefficient of a pnematic valve [16], [108], [109], [110], [111]. The dissertation proposes a simplified measrement evalation method, which is based on similar observations as the JFPS 2009:2002 [109] standard and the method proposed by de las Heras [110] bt in contrast with the standard methods the proposed soltion defines the pressre gradient of a pnematic system that consists of a proportional valve and a chamber instead of defining the flow coefficient of the proportional valve. The applied valve is a HOERBIGER ORIGA SERVOTEC proportional valve. The strctre of the valve is shown in Figre 7.12, Figre 7.13 and Figre Figre 7.12 Servo-valve Figre 7.13 Drawing of the servo-valve 60

71 As it is shown in Figre 7.14 the srface of the inlet orifice depends on the angle of the spool and not on the spool displacement. Figre 7.14 Strctre of the servo-valve The measrement setp bilt for the identification of the proportional valve is shown in Figre The measrement setp consists of two vessels, two pressre sensors and the proportional valve. One of the vessels has a bigger volme, the other one is in the order of magnitde of the chamber of a pnematic cylinder. The last one can be sbstitted by the cylinder itself with fixed position. The bigger vessel is applied as a pffer to smoothen the spply pressre, while the smaller one is the investigated chamber. The aim of the first pressre sensor is to spervise the spply pressre. The investigation is based on the pressre signal measred by the second pressre sensor. Investigated chamber P U M Control nit P U Pffer vessel Figre 7.15 Experimental setp for the identification of the valve 61