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1 !"#! $ %!&'(! &% * +!%,&% * "-%&-!&+ "-%-!" &'&&' ". /0. 3

2 Dedcated to my beloved parents and sster! "

3 TABLE OF CONTENTS ACKNOWLEDGEMENTS OBJECTIVES OF THE PROJECT ABSTRACT v LIST OF FIGURES v LIST OF TABLES AND FLOW DIAGRAMS V LIST OF SYMBOLS V ABBREVIATIONS AND ACRONYMS x CHAPTER : INTRODUCTION TO CONTROL SYSTEMS, UNCERTAINTY AND ROBUST CONTROL..... Control Systems Uncertanty and Robust Control..... CHAPTER : QUANTITATIVE FEEDBACK THEORY The Development of QFT Desgn method (Problem Defnton Desgn specfcatons (Tracng Models Translaton of Parameter space Uncertanty nto nto Uncertanty Templates Stablty Bound or U-contour Tracng Bounds (Horowtz Templates Dsturbance Rejecton Dsturbance Bounds Loop Shapng CHAPTER 3: CONTROLLER DESIGN Introducton Graphcal desgn usng Phase lead and Phase lag controllers Phase lead controller......

4 3.. Phase lag controller User Graphcal program Controller Desgn usng Optmsaton Technques Optmsaton methods Proportonal plus Dervatve (PD controller Proportonal plus Integral plus Dervatve (PID controller CHAPTER 4: PRE-FILTER DESIGN Introducton Pre-Flter Desgn Types of Pre-Flter Desgn example (for PID controller CHAPTER 5: SIMULATION RESULTS Introducton Tracng Performance-Smulatons Dsturbance rejecton-smulatons CONCLUSIONS APPENDIX A: TRACKING SPECIFICATIONS A APPENDIX B: M-CIRCLES D APPENDIX C: SINGULAR VALUE DECOMPOSITION..... J APPENDIX D: LIST OF MATLAB FILES (M-FILES.... L LIST OF REFERENCES M

5 ACKNOWLEDGEMENTS I would le to express my grattude to my supervsor, Dr G D Halas for all hs help, gudance encouragement and nterest throughout my wor. I extend my thans to all my close frends also for ther patence and support. I am grateful to the Unversty of Leeds, whch provded me wth the necessary support for completng the fnal year project. I should also le to than my parents for ther fnancal support throughout my tme at Leeds.

6 OBJECTIVES OF THE PROJECT - To develop software tools for robust control desgn of hghly uncertan systems nvolvng the QFT method. - To carry out a comprehensve case study and llustrate the method va approprate examples. - To automate the control desgn (loop shapng stage usng optmsaton technques

7 ABSTRACT Ths report outlnes the full wor on the fnal year project. The project s ttle s CAD for robust control usng the QFT desgn method. The am of the project s to develop software tools, sutable for the robust control desgn of hghly uncertan SISO systems. The desgn of these systems s based on I Horowtz s QFT method. Ths s a frequency doman loop-shapng desgn technque, whch s fully descrbed n Chapter. The report s a step-by-step gude to the desgn. It ncludes an ntroducton to control and robust systems, an explanaton of the QFT method, and the problem defnton of the desgn usng an llustratve example. It contnues wth desgns of phase lead and lag compensators va graphcal technques. Next the applcaton of optmsaton methods for the desgn of optmal PD and PID controllers s dscussed. Desgnng an approprate pre-flter completes the desgn procedure. Fnally, a number of smulatons show that the desgn technque was successful and meets the gven specfcatons. The report concludes wth a summary of the project wor and ts results and suggests future drectons, whch can be followed n order to mprove certan aspects of the desgn. The Appendx summarses aspects of the theores used for the purposes of the project and a lst of Matlab fles created and used.

8 LIST OF FIGURES Fgure..A OL and CL systems Fgure..B OL and CL systems wth external dsturbance Fgure..A TDF control system plus external dsturbance at the output of the plant 3 Fgure..B Plant Uncertanty or Parameter Space 4 Fgure.3.A Tme doman step responses of upper and lower tracng models 7 Fgure.3.B Frequency response of upper and lower tracng bounds 7 Fgure.4.A Uncertanty template and Nomnal Plant 8 Fgure.4.B Convex Hull of Uncertanty template 9 Fgure.4.C NC wth Uncertanty templates 0 Fgure.4.D NC wth Nomnal OL 0 Fgure.5.A NC wth U-contour (llustraton Fgure.5.B NC wth U-contour (m-value M. Fgure.6.A NC wth Horowtz templates 3 Fgure.7.A nd order Dsturbance rejecton Model 6 Fgure.7.B nd order Dsturbance rejecton Model Bode plot (magntude 6 Fgure.8.A NC wth Dsturbance bounds 8 Fgure.8.B NC wth Horowtz templates, dsturbance bounds, U-contour and nomnal OL 9 Fgure.8.C NC wth maxmum bounds, U-contour, and nomnal OL 9 Fgure 3...A Bode plots (magntude and phase of a phase lead controller Fgure 3...B Loop shapng usng one phase lead controller 3 Fgure 3...A Bode plots (magntude and phase of a phase lag controller 4 Fgure 3...B Loop shapng usng one phase lead controller 5 Fgure 3...C Loop shapng (fnal usng a number of phase lead/lag controllers 6 Fgure 3.3..A Bode plots of a PD controller 9 Fgure 3.3..B NC wth Horowtz templates and U-contour dscrete phases ϕ (PD controller 3 Fgure 3.3..C a PD optmal desgn for G( s s( s + a 37 Fgure 3.3..D Bode plots of optmal PD controller 38

9 Fgure 3.3..E a PD optmal desgn for G ( s s 39 Fgure A Bode plots of a PID controller 40 Fgure B a PID optmal desgn for G( s s( s + a 50 Fgure C Bode plots of optmal PID controller 5 Fgure D PID optmal desgn for G( s ( s + a( s + b 5 Fgure 4..A Frequency responses of CL system wthout the pre-flter and the desred range of acceptable CL frequency responses 53 Fgure 4..A Desred range of bounds and maxmum spread of CL system responses wthout the pre-flter 54 Fgure 4..B Allowable frequency response (magntude range of pre-flter 55 Fgure 4.4.A Magntude frequency response of desgned pre-flter F(s 57 Fgure 4.4.B CL bounds after the ntroducton of the desgned pre-flter F(s 57 Fgure 5..A Step responses of CL system (phase lead/lag controllers a wth pre-flter for G( s s( s + a 59 Fgure 5..B Fgure 5..C Step responses of CL system (PD controller wth pre-flter a for G( s 60 s( s + a Step responses of CL system (PID controller wth pre-flter a for G( s 60 s( s + a Fgure 5.3.A Dsturbance rejecton performance of system (phase lead/lag controllers 6 Fgure 5.3.B Dsturbance rejecton performance of system (PD controller 6 Fgure 5.3.C Dsturbance rejecton performance of system (PID controller 6 Appendces Fgure A. Comparson of spread before and after the ntroducton of poles/zeros n the (tracng models B Fgure B. Feedbac (negatve system D

10 Fgure B. Nyqust plane wth constant M-crcles F Fgure B.3 NC wth a range of M-crcles G Fgure B.4 Converson of M-crcle (M> from NP to NC H Fgure B.5 Converson of M-crcle (M< from NP to NC I Fgure B.6 Converson of M-crcle (M from NP to NC I LIST OF TABLES AND FLOW DIAGRAMS Table..a Reasons of naccurate representaton of a practcal Physcal system, by a plant model 3 Table..b Characterstcs of Robust Control System 3 Table.3.a Tracng specfcatons n dbs 8 Table.7.a Dsturbance rejecton gan 7 Flow Dagram 3..3.f Procedure of grahcal desgn 7

11 LIST OF SYMBOLS K(s controller (s-doman G(s plant (s-doman G o (s nomnal plant (s-doman F(s of pre-flter (s-doman L(s open loop (s-doman L o (s nomnal open loop (s-doman R(s reference nput (s-doman D(s dsturbance (s-doman Y(s output sgnal (s-doman P parameter space T R T D T R (s δ R (j HF M p t s, t x t p B u B l M D n p u x x db A A T R(A, R(A R n m R n nput-output tracng bounds dsturbance bounds tracng control rato spread of tracng bounds at hgh frequences pea overshoot settlng tme pea tme upper tracng bound lower tracng bound -th frequency element dsturbance rejecton model natural frequency percent undershoot absolute value of x, magntude of x magntude of x n dbs matrx A[a j ] n m transpose of matrx A range of matrx A real vector space of real n m vectors real vector space of real n-vectors 0 zero scalar, matrx or vector a j -th, j-th element of a matrx

12 ran(a ran of matrx A S S perp. (perpendcular to S ζ dampng factor σ (A -th sngular value of matrx A

13 ABBREVIATIONS AND ACRONYMS CL ES GUI HF LHP LHS LTI MATLAB MIMO MISO NC NP OL PD PI PID QFT RHP RHS RI SISO SVD TDF TF UHFB Closed Loop Error Sgnal Graphcal User Interface Hgh Frequency(-es Left Hand Plane Left Hand Sde Lnear Tme Invarant MAtrx LABoratory Mult-nput, Mult-output Mult-nput, Sngle-output Nchols Chart Nyqust Plane Open Loop Proportonal plus Dervatve Proportonal plus Integral Proportonal plus Integral plus Dervatve Quanttatve Feedbac Theory Rght Hand Plane Rght Hand Sde Reference Input Sngle-nput, Sngle-output Sngular Value Decomposton Two Degrees of Freedom Transfer Functon Unversal Hgh Frequency Bound

14 CHAPTER ONE INTRODUCTION TO CONTROL SYSTEMS, UNCERTAINTY AND ROBUST CONTROL. Control Systems There are two types of Control Systems, Open Loop & Closed Loop Systems llustrated n fgure..a: ( Input R Controller Plant Output Y Input R + Controller Plant Output Y ( Feedbac Element Fgure..A: ( Open loop and ( Closed Loop Systems In the case of open loop systems, the output has no effect on the nput sgnal. In the case of closed loop systems, however the output through the feedbac element affects the nput sgnal, deally n such a manner as to mantan the desred output value [Ref.]. The feedbac element provdes the means for feedng bac the output sgnal, n order to compare t wth the reference nput sgnal [Ref.]. Often, undesrable external nput sgnals can enter the feedbac loop and they have effects on the output (Open-Loop systems or on both the output and the nput (Closed-Loop systems. One type of such sgnals, are external dsturbances. Fgure..B, shows an external step dsturbance sgnal enterng at the plant s output:

15 Chapter One Introducton to Control Systems, Uncertanty and Robust Control D ( Input R Controller Plant + + Output Y ( Input R + Controller Plant D + + Output Y Feedbac Element The Dsturbance nput s usually a step nput. Fgure..B: ( Open loop and ( Closed loop systems, wth external dsturbance From the above fgure, n the Open loop system, the dsturbance appears drectly on the output and as a result t cannot be attenuated by means of the controller. In the Closed loop system however, the dsturbance affects both the output and the nput sgnals. It s thus possble to reduce ts effect on the output by desgnng a controller K(s approprately. In addton to dsturbance rejecton the control system should also have good tracng propertes (the output should follow the reference nput fast and accurately wth small steady state errors {s.s.e} and also good stablty margns.. Uncertanty and Robust Control The desgn of a control system accordng to classcal methods assumes full nowledge of the plant and the controller. In practcal systems, however, the plant model wll always be an naccurate representaton of the actual physcal system due to:

16 Chapter One Introducton to Control Systems, Uncertanty and Robust Control - Parameter changes - Unmodeled dynamcs - Unmodeled tme delays - Changes n equlbrum pont (operatng pont Table..a: Reasons of naccurate representaton of a practcal physcal system, by a plant model Robust Control methods address the problem of uncertanty systematcally. They am to mantan adequate performance and stablty margns despte the presence of uncertanty n the dynamcs of the plant. For the purposes of ths project QFT was used n the desgn and analyss of control systems charactersed by sgnfcant uncertanty and undesred external nputs [Ref.,,3,4]. A robust control system has the followng characterstcs: - Low senstvtes to parameter changes - Closed loop stablty s mantaned wthn the range of parameter change - Its performance does not deterorate rapdly wth parameter change Table..b: Characterstcs of a Robust Control System The meanng of the above terms s llustrated n the example of fgure..a: R(s ES RI Pre-Flter F(s System Dynamcs + + Controller Plant K(s G(s + D(s Y(s Feedbac sgnal Feedbac Element Fgure..A: TDF control system plus external dsturbance at the output of the plant. 3

17 Chapter One Introducton to Control Systems, Uncertanty and Robust Control The man example of a plant model that consdered n ths report s taen as the followng second order system: ( a G s, where [,0] and [,0] s( s + a a ( Ths model s smply used to llustrate the desgn method. The software developed was wrtten wth general SISO plants n mnd [Ref.]. In equaton (, and a are uncertan but constant parameters varyng n the gven ranges. The regon of plant uncertanty s shown n fgure..b: 0 P Parameter Space 0 a Fgure..B: Regon of plant uncertanty (nown as Parameter Space Note that ths plant has sgnfcant uncertanty due to the range of ts uncertan parameters. Snce the parameters a and can vary smultaneously, the plant exhbts both gan and phase uncertanty n the frequency doman. The man objectve of robust control n ths case s to mantan the stablty and performance propertes of the closed loop system, as a and vary n the ndcated ranges. In Chapter Four, other plant models are used to llustrate the performance of the desgn methods. 4

18 CHAPTER TWO QUANTITATIVE FEEDBACK THEORY. The Development of QFT QFT s a frequency doman loop shapng desgn method [Ref.,,3,4]. I M Horowtz proposed ths method n 969 and snce then t has been developed for SISO, MISO and MIMO LTI uncertan plants. In addton, the method has also been extended to non-lnear and tme-varyng systems. QFT s a systematc desgn methodology for systems charactersed by sgnfcant parameter uncertanty. 3. Desgn method (Problem Defnton QFT emphasses the use of feedbac n order to acheve adequate robust system performance tolerances despte the presence of plant uncertanty and dsturbance sgnals. It formulates, by quanttatve means, the desgn objectves n terms of the followng sets [Ref.]:. T R {T R } and T D {T D }, acceptable tracng nput-output bounds and acceptable dsturbance output bounds, respectvely, typcally defned n the frequency doman (.e. n terms of magntude Bode plots.. P {P}, possble uncertan plants, defned ether n the frequency doman ( uncertanty templates or n the parameter-space. The objectve s to guarantee that the control rato T R (between the reference nput and the system s output wll exst wthn the bounds T R ; and that the rato T D (between the dsturbance nput and the system s output wll exst wthn the bounds T D, for all P n P. Thus, the technque proposes that the desgned system has to meet the tracng specfcatons and has to reject the dsturbance nput for all possble plants. For the purposes of ths project, QFT theory s appled to a SISO system, whch ncludes external dsturbance nput. 4 For llustraton purposes the TDF unty feedbac cascade compensated system of fgure..a wll be used. The plant model s gven by equaton ( and the parameter space s shown n fgure.. The reference and the dsturbance sgnals are both unt step nputs {R o, D o }. 3 There are alternatve technques wth the same am,.e. H optmal control 4 The system can be descrbed as a pseudo-miso system due to the exstence of the external dsturbance, although MISO systems are normally charactersed by the presence of multple reference nputs 5

19 Chapter Two Quanttatve Feedbac Theory.3 Desgn specfcatons (Tracng Models The closed loop system has to satsfy a number of specfcatons. These correspond to the closed loop tme responses to specfc reference or dsturbance nput sgnals. For the purpose of QFT these are frst translated n the frequency doman. The tracng control rato models are based on a smple second order system and are synthessed n order to specfy tme and gan tolerance responses (under-damped and over-damped condtons [Appendx A] [Ref.]. In ths case the upper and lower bounds of the tracng control ratos {T R (s} are shown below: Upper tracng bound, under-damped wth pea gan M P. (approx. 0% overshoot and wth settlng tme t s sec. TR U ( s + 30 ( s ( s + ± j3.969 ( Lower tracng bound, over-damped wth settlng tme t s sec. TR L 8400 ( s ( s + 3( s + 4( s + 0( s + 70 (3 The zero at s-30 {term (s+30} n ( and the pole at s-70 {term (s+70} n (, were ntroduced because the dfference between the upper and the lower tracng bounds at hgh frequences δ R (j HF needs to be wder compared to the hgh frequency model uncertanty. Fgures.3.A and.3.b show the tme doman step responses and the frequency doman responses, of the upper and lower bounds, respectvely. 6

20 Chapter Two Quanttatve Feedbac Theory Upper tracng model Lower tracng model Fgure.3.A: Tme doman step responses of upper and lower tracng models Bu Bl Fgure.3.B: Frequency response of upper and lower tracng bounds Bu and Bl, respectvely 7

21 Chapter Two Quanttatve Feedbac Theory The dfference between the upper and lower bounds δ R (j n fgure 3..B, for eght chosen frequences s gven n Table.3.a. Frequences (rads/sec Dfference δ R (j (dbs Table.3.a: Tracng specfcatons δ R (j,,,3, (n dbs Functon trac_ul( was used to produce the frequency and tme doman step responses of the upper and lower models..4 Translaton of Parameter Space Uncertanty nto Uncertanty Templates QFT assumes that the plant uncertanty can be represented by a set of templates on the complex plane, nown as uncertanty templates [Ref.]. Each template encloses all the possble frequency responses of the plant for a specfc frequency. As an example, eght frequences are consdered {0.5,, 3, 5, 0, 30, 60} rads/sec and as a result eght uncertanty templates wll be formed. The complex plane, on whch the uncertanty templates are dsplayed, s nown as the Nchols Chart (NC. It represents the open loop gan of a system n dbs versus the open loop phase of the system n degrees, n the frequency doman. One uncertanty template, n ths case for 5 5 rads/sec, s shown n fgure.4.a: Nomnal Plant Fgure.4.A: Uncertanty template for 5 5 rads/sec and Nomnal plant, on the NC. 8

22 Chapter Two Quanttatve Feedbac Theory Generally, uncertanty templates on the NC can be translated to the open loop transmsson functon of the system, L(jG(jK(j va the controller K(s. The varaton of L(j can be assumed to be the same as the varaton of G(j for all ts possble values at any specfc frequency. Ths s because the compensator K(j s assumed to be fxed. The nomnal L(j, ndcated by L o (j, arses from a controller K(s and the substtuton for and a whch corresponds to our choce of nomnal plant. Although ths choce s arbtrary (any fxed a and combnaton would be acceptable t s consdered good practce to select the nomnal plant whch has ts NC template pont at the lower left corner for all frequences for whch the templates were obtaned. To fnd the uncertanty templates for the model gven n (, functon templ_cs( was used. To reduce the computatonal burden of the algorthm, functon c_hull( was used to reduce the number of the ponts representng each template. Fgure.4.B shows the convex hull of the template of fgure.4.a. Nomnal Plant Fgure.4.B: Convex Hull of uncertanty template for 5 5 rads/sec The eght uncertanty templates (one for each of the eght selected frequences on the NC are shown n fgure.4.c. Moreover, the nomnal open loop L o (j for K(s s shown n fgure.4.d. 9

23 Chapter Two Quanttatve Feedbac Theory Fgure.4.C: NC wth uncertanty templates for {0.5,,, 3, 5, 0, 30, 60} rads/sec Fgure.4.D: NC wth nomnal open loop L o (j 0

24 Chapter Two Quanttatve Feedbac Theory.5 Stablty Bound or U-Contour The stablty bound or U-contour (sometmes referred to as Unversal Hgh Frequency Boundary UHFB, defnes the regon of the NC whch should not be penetrated by the nomnal open loop system L o (jg o (jk(j at hgh frequences. The reason for ths s as follows: To establsh mnmum dampng for the nomnal closed loop system t s well nown that L o (jg o (jk(j should not enter an approprately chosen M-crcle (n ths case M. [Appendx B]. Snce, (a at hgh frequences the uncertanty template becomes a vertcal lne and (b we want to enforce the mnmum dampng requrement for all plants, the lowest pont of the M-crcle must be extended downwards by a gan V whch s equal to the gan uncertanty spread at hgh frequences. For example, consder a general system as the one n fgure.5.a. In ths example the uncertanty template s a vertcal lne of length V dbs and the nomnal open loop response s represented by ts lowest pont. Then for all uncertan L s to le outsde the M. crcle at hgh frequences, the nomnal open loop response must le on or below the U- contour. M-crcle V Uncertanty Template U-contour Nomnal Open Loop L o Fgure.5.A: NC wth U-contour, example for llustraton

25 Chapter Two Quanttatve Feedbac Theory In general, gan V s obtaned as: V log or V log 0 0 L( j G( j [ G ( j 0log G ( j ] lm 0log 0 max 0 mn (4 For the model gven n (, V s found usng (4 as: V log thus V log 0 0 whch mples V 0log 0 G( j lm 0log G( j lm 0log (0 0 0log [ G ( j 0log G ( j ] [ ( a 0log ( j 0log ( a + 0log ( j ] max max 0 0 mn ( V 40 0V 40dBs 0 mn 0 Functon hf_bound( was used to produce the U-contour for the system n (, whch s shown n fgure.5.b Fgure.5.B: NC wth U-contour (m-crcle value, M.

26 Chapter Two Quanttatve Feedbac Theory.6 Tracng bounds (Horowtz templates In order to acheve the closed loop specfcatons, the plant templates must le on or above specfc areas on the NC. These areas are n the forms of contours and they are nown as Tracng bounds or Horowtz templates [Ref.,,3]. They specfy the mnmum open loop gan for the system to acheve the desred robust performance. In the case of the model gven n ( and for the selected eght frequences, eght tracng bounds wll be plotted on the NC. Functons hr_bnds( and tr_bnds(. Fgure.6.A dsplays the tracng contours for our system. Fgure.6.A: NC wth Horowtz templates (for eght frequences The Horowtz templates can be obtaned from the followng consderatons: The control rato of the system wthout the presence of the pre-flter s: L( j T R ( j, where L( j G( j K( j + L( j (4 3

27 Chapter Two Quanttatve Feedbac Theory The locus of all ponts T R M, s nown as an M-crcle [Appendx B] [Ref.5]. The grd lnes shown n fgure.6.a, represent the M-crcles for dfferent values of M. For each uncertanty template, the correspondng tracng specfcaton requre that: max p P { T ( j T ( j } δ ( j max dbs mn dbs (5 Wth the plant templates placed at any locaton on the NC, the above condton s equvalent to M max M mn δ ( j (6 where M max and M mn denote the maxmum and the mnmum M-value among all ponts of the templates, respectvely. Consequently, f at ts current locaton condton (6 s not satsfed, the open loop gan must be ncreased. The ponts defnng the Horowtz template (for each phase represent the gan at whch condton (6 s met wth equalty. In practce, the Horowtz templates are calculated n software va a smple bsecton algorthm over a fnte phase grd..7 Dsturbance Rejecton In the presence of external dsturbance nputs, the system must not only satsfy the tracng specfcatons but also the dsturbance rejecton specfcatons n order to have the desred performance. The dsturbance rejecton models used are ether frst or second order systems [Ref.]. In ths case a second order dsturbance rejecton model s needed to reject the unt step dsturbance nput at the output of the plant n fgure..a. The requrement s: y( t a, for t p t x (7 where a p and t x are desgn parameters. For a second order model, the dsturbance response n the tme doman s of the form, y( t D o e at cosbt (8 The correspondng dsturbance rejecton model s: s( s + a M D ( s ( s + a + b (9 4

28 Chapter Two Quanttatve Feedbac Theory where aζ n and b n (-ζ -/, n whch n s the natural frequency and ζ s the dampng factor of the model. The model parameters can be obtaned from the ntal value of the dsturbance nput D o, the percent undershoot p u, the settlng tme t x and the maxmum gan α p of the dsturbance response. It can be easly shown that for ths model: n ζ t x a ln D p o (0 ζ can be obtaned from a loo up table and s related to ts percent undershoot. The model n equaton (9 also contans a non-domnant zero, whch ensures that the fnal model s compatble wth the degree of the control rato of the system. Note also that the zero s0 n (9 enforces asymptotc rejecton to step dsturbance nputs. I the example consdered n ths project, a second order model s needed wth D o, p u 5%, α p 0. and t x sec. The followng parameters were obtaned usng functon dop_mda(: n.65 rad/sec ζ0.43 α.5 b.39 The dsturbance model obtaned s: M D s( s +.5 ( s ( s ( The above model rejects the dsturbance nput (the step response of the dsturbance model stays wthn the regon [-0.,0.] after sec wth an undershoot of approxmately 5%, as verfed from the smulaton shown n fgure.7.a. Fgure.7.B shows the magntude-frequency response of M D (s. 5

29 Chapter Two Quanttatve Feedbac Theory Fgure.7.A: Second order Dsturbance rejecton model In the frequency doman the magntude bode plot of the model s the followng: Fgure.7.B: Magntude Bode plot of the second order dsturbance rejecton model 6

30 Chapter Two Quanttatve Feedbac Theory The blue crcles n fgure.7.b represent the gan at each of the eght specfed frequences {0.5,,, 3, 5, 0, 30, 60} rads/sec. The results are summarsed n table.7.a. Frequences (rads/sec Gan A (dbs (n absolute terms Table.7.a: Dsturbance rejecton gan (A,,,3,..,8.8 Dsturbance Bounds When an external dsturbance nput s present, the desgn must satsfy both tracng and dsturbance rejecton specfcatons n order to be charactersed as successful. For dsturbance rejecton the nomnal open loop L o (j must le on or above specfc regons on the NC to ensure that the closed loop dsturbance rejecton specfcatons wll be successfully met. These areas are nown as the dsturbance bounds and they are formed n a smlar way to the tracng bounds. They specfy the mnmum open loop gan of the system requred to meet the dsturbance rejecton specfcatons for all uncertan plants. These are obtaned as follows: The dsturbance-modellng rato of the system n fgure..a s: T ( j YD + G( j K( j + L( j ( For the dsturbance bounds on the NC the requrement for fulfllng the dsturbance rejecton specfcatons s that: T YD ( j A, for all (3 where A s the correspondng gan n dbs of the Bode plot n table.7.a, at each frequency. Accordng to the above, the dsturbance bounds for the example consdered here were found usng functons ds_bnds( and dst_bnd(.fgure.8.a shows the dsturbance bounds on the NC: 7

31 Chapter Two Quanttatve Feedbac Theory Fgure.8.A: NC wth dsturbance bounds The nomnal open loop must le on or above the dsturbance bounds n fgure.8.a for the correspondng frequences, n order to meet the dsturbance rejecton specfcatons. The system must le above the dsturbance bounds and the tracng bounds n order to meet smultaneously the dsturbance rejecton and the tracng specfcatons. Moreover, t must not penetrate the U-contour n order to establsh a mnmum dampng for the nomnal closed loop system. Fgure.8.B shows the NC wth the Horowtz templates, the dsturbance bounds, the U- contour, and the nomnal open loop. In addton, fgure.8.c shows the maxmum bounds that the nomnal open loop must le on or above, so that the system wll meet the dsturbance and tracng specfcatons smultaneously. 8

32 Chapter Two Quanttatve Feedbac Theory Fgure.8.B: NC wth Horowtz templates, dsturbance bounds, U-contour, and nomnal open loop Fgure.8.C: NC wth maxmum bounds, U-contour, and nomnal open loop 9

33 Chapter Two Quanttatve Feedbac Theory From fgure.8.c, note that n the left regon of the NC (regon of phases from 360 o untl the frst phase at whch the U-contour starts formng the nomnal open loop (f t exsts n ths regon must le on or above the bound at 0 dbs (straght lne to ensure stablty. It s also a good practce to consder the U-contour as a performance bound; ths helps wth the mplementaton of the algorthms gven later n ths project..9 Loop shapng So far we have seen that n order to attan the desred closed loop specfcatons the followng requrements must be fulflled: (a L(j must not penetrate the U-contour to ensure stablty and mnmum dampng. (b The nomnal open loop must le on or above the tracng and dsturbance bounds on the NC (for each correspondng frequency to acheve robust tracng performance and dsturbance rejecton. If t happens to exst n the left regon of the NC, t must also le on or above the 0 db bound to ensure stablty. Therefore the open loop response wll be shaped wth the am to satsfy (a and (b above. Open loop shapng can be acheved usng compensator desgn. A compensator or controller s a dynamc system that s used n cascade wth the plant to acheve the desred open loop characterstcs. In the next chapter four types of controllers are dscussed, phase lead, phase lag, PD, and PID. The frst two controllers are used for loop shapng usng a graphcal desgn. The PD and PID controllers are desgned va optmsaton methods to fnd the best soluton to the desgn problem. 0

34 CHAPTER THREE CONTROLLER DESIGN 3. Introducton Ths chapter dscusses the desgn of approprate controllers, whch are used so that the system wll acheve desred open loop characterstcs. Four types of controllers are consdered, phase lead/phase lag, PD, and PID [Ref.5,7]. The frst two types are used to shape the loop sutably va a graphcal desgn. The PD and PID controllers are desgned to gve the optmal soluton to a sutably defned optmsaton problem. The process and the results are llustrated usng approprate examples. 3. Graphcal desgn usng Phase Lead and Phase lag controllers In order to shape the loop approprately (see Secton.9, gan, phase lead or phase lag need to be njected at each frequency of nterest so that the desgn wll gve an acceptable result accordng to the gven specfcatons [Ref.5,7]. 3.. Phase Lead Controller A phase lead controller has a transfer functon of the form: s + K( s s + α (4 where α>. The phase lead controller njects postve phase n the system for all frequences. The maxmum phase shft s gven by: φ m sn α (5 α + and occurs at frequency: m α (6

35 Chapter Three Controller Degn (Graphcal Fgure 3...A, shows the Bode plots of the followng phase lead controller: s + K( s 5 s + 30 where 5 rads/sec, α and, usng formulae (5 and (6, ϕ m 9.47 o at m. rads/sec. m Fgure 3...A: Bode plots (magntude and phase of phase lead controller As t can be seen from the Bode plots, the phase lead controller acts as a hgh pass flter because t ncreases the gan at hgh frequences relatve to the gan at low frequences. As an example, fgure 3...B shows a loop shapng usng one phase lead controller:

36 Chapter Three Controller Degn (Graphcal Fgure 3...B: NC wth overall bounds, ntal nomnal open loop (green lne and new nomnal open loop usng one phase lead controller (blue lne From fgure 3...B we can see that the use of a phase lead controller ntroduces hgh gan at hgh frequences and attenuates low frequences. Moreover, the nomnal open loop moves outsde the U- contour at the frequency at whch maxmum phase was njected. 3.. Phase Lag Controller The transfer functon of a phase lag controller s: s + α K( s s + (7 where α>. The phase lag controller njects negatve phase to the system for a specfc range of frequences. The maxmum phase shft s gven by equaton (5 and t occurs at m gven by 3

37 Chapter Three Controller Degn (Graphcal equaton (6. Fgure 3...A shows the Bode plots of a phase lag controller wth the followng transfer functon: s + K( s 30 s + 5 where 5 rads/sec, α and, usng formulae (5 (we also have to ntroduce a mnus sgn due to phase lag and (6, ϕ m o at m. rads/sec. m Fgure 3..A: Bode plots (magntude and phase of phase lag controller As can be seen from the above fgure, the phase lag controller acts as a low pass flter as t ncreases gan at low frequences relatve to the gan at hgh frequences. Therefore t can be used n cascade wth a phase lead controller to gve a reasonable desgn. Fgure 3...B shows the shapng of the loop after the use of one phase lead controller (from fgure 3...B and one phase lag controller n cascade. 4

38 Chapter Three Controller Degn (Graphcal Fgure 3...B: NC wth ntal nomnal open loop, and new nomnal open loop after usng one phase lead controller (from fgure 3...B and one phase lag controller. Thus, by usng approprate desgned phase lead and phase lag controllers n cascade, a soluton to the loop-shapng problem can be obtaned. Ths s shown n fgure 3...C (next page, where a number of phase lead and phase lag controllers are used to gve the best performance. 5

39 Chapter Three Controller Degn (Graphcal Fgure 3...C: NC wth ntal nomnal open loop and fnal nomnal open loop (the controller obtaned helps the system to satsfy all gven specfcatons 3..3 User Graphcal Program Snce a number of st order controllers are requred to shape L o approprately, a user frendly graphcal program was desgned to mae ths desgn tas easer. For ths purpose functon gr_des( was wrtten. The procedure for the desgn s the followng: By clcng on a pont (desred frequency at whch the max phase shft wll be njected on the NC, gr_des( returns the pont s co-ordnates (magntude and phase. From these co-ordnates, gr_des( estmates the nearest pont on the L o (nearest exact frequency of nterest and the true magntude and phase are obtaned. Then by clcng on a second pont, the requred phase (lead or lag and gan are estmated and the approprate controller s selected va the functons ph_lead( and ph_lag(. Ths procedure has been mplemented n such a way (wth the use of a loop, whch offers the desgner the followng capabltes: 6

40 Chapter Three Controller Degn (Graphcal. Desgn new controller. Connect new controller n cascade wth old controller 3. Remove the last desgned controller 4. Ext when desgn s completed and save all data nformaton The loop s shown by means of the followng logc dagram: LK agr *G Contnue? NO Ext (save data YES Desgn YES K new Store new controller and shape the loop YES NO Contnue? Keep controller? NO Remove last controller Contnue? YES Ext (save data NO Flow Dagram 3..3.f: Procedure of graphcal desgn The fnal desgn s llustrated usng functon mantest(. Ths functon returns the overall controller obtaned from the desgn. 7

41 Chapter Three Controller Degn (Graphcal 3.3 Controller Desgn usng Optmsaton Methods In the prevous secton the desgn of a controller consstng of phase lag and lead cascade terms was dscussed. Ths secton presents optmsaton methods for desgnng proportonal and dervatve (PD and proportonal, ntegral and dervatve (PID controllers. The frst part s an ntroducton to optmsaton methods n general, whle the next two parts dscuss the applcaton of these methods to the desgn of a PD and a PID controller, respectvely. Two smulatons one for each type of controller show that the methods wor approprately Optmsaton methods Optmsaton accordng to Fletcher [Ref.6] s the scence of determnng the best solutons to certan mathematcally defned problems, whch are often models of physcal realty. Optmsaton and ts hybrd methods can be appled to a great range of problems and practcal applcatons. The soluton obtaned by these methods s typcally smpler, more accurate, and cost effectve. For the purposes of ths project, optmsaton s used n order to determne algorthmc solutons to the problem of desgnng optmal PD and PID controllers for hghly uncertan plants. The procedure s descrbed below and the smulatons are manly based on the plant gven n equaton ( Proportonal plus Dervatve (PD Controller The frst type of controller, whch can be used to gve a soluton to the desgn problem, s the PD controller. Here, both the error and the dervatve of the error are used for control, and ts transfer functon s the followng: K ( s pd p + d s (8 The PD controller, effectvely adds a zero at s- p / d to the OL TF. Both the transent response and the steady state error (s.s.e are mproved, because the PD antcpates large errors and tres to correct them before they occur [Ref.5,7]. In frequency doman terms, the PD controller adds phase lead to the nomnal open loop, reachng a maxmum addtonal phase lead of 90 o at hgh frequences (ths s due to the domnant nature of the dervatve at hgh frequences. At low frequences the domnant part s the proportonal term whch only adds gan to the nomnal OL. The phase s always postve due to the dervatve term (the proportonal term has 0 o phase. For example, fgure 3.3..A shows the Bode plots of a PD controller wth the followng transfer functon: 8

42 Chapter Three Controller Degn (Graphcal K pd ( s + s Fgure 3.3..A: Bode plots of a PD controller In order to optmse the controller the followng consderatons were made: The frequency response (sj of the PD controller (from equaton (8, n terms of magntude and phase s: K ( j j pd p + d (9 Magntude (lnear: m ( K ( j + ( pd p d (0 Phase: ( tan d φ 0 p ( Suppose that ϕ( o s fxed at any specfed frequency o,.e. ϕ( o ϕ o. Then by usng equaton (, ϕ o s gven by: 9

43 Chapter Three Controller Degn (Graphcal 30 ( ( o o o p d o p d o p o d o o γ φ φ φ φ : tan tan tan ( where γ o s a constant whch fxes the relaton between p and d. Moreover, usng ( the phase of the controller at any frequency s: ( tan tan( tan tan ( γ φ φ o o o p d It follows from the analyss above that f the phase of the compensator s fxed at any specfed frequency (.e. ϕ( o ϕ o then the phase ϕ of the compensator at all other frequences s also fxed. Under these condtons the magntude (lnear of the controller can be expressed as: ( ( γ o d d p d d p j K m.e. m( d. Now consder the nomnal plant model G o and the nomnal OL system L o G o K. Then the phase of the nomnal OL s: ( tan ( ( ( ( ( γ β ψ o o o j K j G j L + + where the phase of the nomnal plant model β( s fxed and nown. Also, the magntude of the nomnal open loop n dbs s: ( (3 (4 (5 whch mples or,

44 Chapter Three Controller Degn (Graphcal L ( j o db G ( j o db + K( j db.e., L ( j o db A db + 0log 0 ( d + 0 log 0 γ o + (6 where, A db s the gan of the nomnal plant G o at any frequency, n dbs. Now, consder the Nchols chart wth Horowtz templates {f(, ϕ, f(, ϕ,,f( N, ϕ} and U-contour B: Mag(dBs f(,ϕ ϕ ϕ ϕ n ϕ f(,ϕ B (U-contour f( N,ϕ Fgure 3.3..B: NC wth Horowtz templates and U-contour, dscrete phases ϕ (PD controller The desgn constrants that the PD controller has to satsfy are:. L(j db f(, for all,,3,,n. L(j B, for all,,3,,n 3

45 Chapter Three Controller Degn (Graphcal Accordng to QFT theory [Ref.,3], the asymptotc gan of the open loop system must be a mnmum (subject to satsfyng the Horowtz and U-contour constrants. Ths s to avod overdesgnng the system (e.g. resultng n a closed loop bandwdth larger than s absolutely necessary whch could mply measurement nose amplfcaton and possble nstablty due to un-modelled dynamcs. The asymptotc gan of the nomnal open loop system s gven by: lm { ( + j G ( j } p d o (7 Suppose that: G ( j o A p at very hgh frequences. (Assume that p, where p s the pole/zero excess of the nomnal plant. Then the asymptotc gan of the nomnal open loop s: A L o ( j p d at hgh frequences. Snce A and p are fxed from the plant, L o (j hgh s mnmsed by mnmsng d. Hence the followng optmsaton problem s formulated: Mnmse d subject to the followng constrants: L( j f (, ψ, for all,,3,,n (where ψ L(j AND L( j B, for all,,3,,n 3

46 Chapter Three Controller Degn (Graphcal Ths suggests an algorthm for mnmsng d. Frst assume that the axs representng the phase angle of the NC has been dscretsed nto a lnearly spaced array {ψ,ψ,ψ 3,,ψ n } (n can tae any value but 00 s sutable 6. Suppose that we fx the phase at the nomnal open loop at the frst frequency of nterest,.e. L o (j ψ. Then: K( j L o ( j G o ( j or K( j ψ β ( tan d p (8 Hence d p tan ( tan( ψ β ( (9 s fxed. Thus K ( j d + γ o or K( j + + 0log 0 ( d 0 log 0 db γ o (30 and hence 6 The result depends on the dscretsaton of the phases, f the dscretsed phases comply wth the phases of the bounds then the result s nearly accurate. 33

47 Chapter Three Controller Degn (Graphcal Lo ( j 0log ( d + 0 log log db + 0 γ o ( G ( j 0 o (3 d should be large enough to satsfy the Horowtz specfcatons (ncludng dsturbance bounds f applcable,.e.: Lo ( j f (, ψ db (3 where ψ L o (j, for all,,3,,n. Ths mples (from equaton (3 that 0log 0 ( d f (, ψ 0 log0 ( G + o ( j 0log 0 γ o (3a The above s equvalent to: or ( 0log 0 ( d max f (, ψ 0log 0 G o ( j 0 log0 + {,,3,..., } N γ o d 0^ max f (, ψ 0 log {,,3,..., N} 0 ( G ( j 0 o 0log 0 γ o + (33 In order to mnmse the asymptotc gan of L o (j we must choose d equal to the RHS of equaton (33. Note, that ψ L o (j s fxed for all,,3,,n. Specfcally: ψ, s fxed by assumpton o ( o o ψ L j G ( j + tan ( γ, where γ o [tan(ϕ / ] ψ N L ( j G o N o ( j + tan N ( γ o N The followng algorthm was mplemented n MATLAB: 34

48 Chapter Three Controller Degn (Graphcal Algorthm PD. Intalse an array to store the local mnmum varable d for each value of L o (j.. Intalse an array to store the relevant values of p. 3. Outer loop (runs n tmes, where n s the number of dscretsed fxed phases for the frst frequency of nterest 3. Fnd constant γ o usng the followng expresson: γ o tan ( φ ( j G o where, ϕ s the value of the fxed phases for,,3,,n for the frst frequency. 3. Intalse an array to store phases of nomnal open loop for all frequences of nterest (t ntalses every tme the value of fxed phase s changed. 3.3 Intalse an array to store the values of d obtaned for all frequences each tme the outer loop runs (t also ntalses every tme the outer loop runs, and ths helps to dentfy the mnmum local d for each 3.4 Inner loop (runs a total of N tmes, where N s the number of frequences of nterest. fxed by assumpton For the frst frequency (when, the phase of the nomnal open loop s 3.4. For the rest of the frequences (when runs from untl N fnd the phase of the nomnal open loop usng the followng expresson: ψ G ( j + tan o ( γ o 35

49 Chapter Three Controller Degn (Graphcal whch s fxed because every tme the loop runs the phase of the nomnal plant s fxed and nown, the frequency s nown and the constant γ o s also nown (obtaned from the outer loop. The OL phase for the frst frequency s exactly the same as the phase of the bounds. For all other frequences the magntude of the bounds s obtaned approxmately va lnear nterpolaton For each fnd d usng formula (33 (met wth equalty that satsfes the requrements and temporary store t. End of Inner loop. 3.5 Fnd the local mnmum d that satsfes the requrements for each, and store t. 3.6 Fnd the relevant p usng formula ( (frst solve for p and also store t. End of Outer loop 4. Obtan the value of the global mnmum d (optmal soluton and the correspondng ndex. 5. Use the ndex obtaned n step 4 to fnd the relevant value of p. END OF ALGORITHM The above algorthm (whch was mplemented n Matlab functon cpd_opt(, wors as expected and fnds the optmal controller whch gves the soluton to the desgn problem. In order to llustrate the procedure, consder the followng two examples: - Example : Consder the plant gven n equaton ( and the specfcatons for tracng requrements and dsturbance rejecton gven n Sectons.3 and.7. Fgure 3.3..C shows the nomnal open loop wth the frequences of nterest, the U-contour and the correspondng maxmum bounds. It also shows the new nomnal open loop correspondng to the desgned optmal PD controller. 36

50 Chapter Three Controller Degn (Graphcal Fgure 3.3..C: PD optmal desgn (old nomnal open loop (green lne and new nomnal open loop after the desgn of the controller (blue lne. The crcles represent the frequences of nterest The optmal PD controller obtaned s gven by: K pd _ opt ( s s From fgure 3.3..C t can be clearly seen that at low frequences the addtonal phase lead ntroduced by K pd_opt s small whle at hgh frequences rses to almost 90 o (the phase of the nomnal open loop reaches asymptotcally 90 o n ths example. The shaped open loop system satsfes the tracng/dsturbance specfcatons and s outsde the U-contour. At three frequences ( 4, 5, 7, L o (j les exactly on the correspondng bounds. Fgure 3.3..D shows the Bode plots of the controller obtaned above, where the gan and the phase ntroduced can be clearly seen (the frequences of nterest are mared wth a blue crcle: 37

51 Chapter Three Controller Degn (Graphcal Fgure 3.3..D: Bode plots of optmal PD controller (gan and phase ntroduced at each frequency -Example : Consder the system: a G ( s, where a [,0] s (34 An optmal PD controller needs to be desgned so that the above system s stablsed and satsfes the tracng specfcatons gven n the prevous example (n ths case there s no dsturbance rejecton. Followng the procedure of Secton 3.3., the optmal controller was obtaned as: K pd _ opt ( s s Fgure 3.3..E shows the NC wth the nomnal open loop, the U-contour, and the correspondng bounds and the new nomnal open loop after the desgn of the optmal PD. In ths case the phase range s from 80 o untl 90 o because the controller ntroduces only postve phase wth a maxmum of 90 o : 38

52 Chapter Three Controller Degn (Graphcal Fgure 3.3..E: PD optmal desgn for G(s a/s, where a 0 (old nomnal open loop and frequences of nterest n green, and nomnal open loop after desgnng the controller n blue The above examples show that for a realsable system, the desgn of an optmal PD controller followng the procedure dscussed here s successful. A smlar procedure can be followed n order to desgn a PI controller whch has the followng transfer functon: K p ( s p + s (35 Both the error and ts ntegral are used for control. The acton of a PI controller s to reduce the steady state errors by ncreasng the type of the system by, and ts use s very common n process control or regulatng systems. The next secton presents the procedure to desgn an optmal PID controller. 39

53 Chapter Three Controller Degn (Graphcal Proportonal plus Integral plus Dervatve (PID Controller PID controllers, also nown as three term or process controllers, are one of the most common type of controllers used commercally. The transfer functon of a PID controller s the followng: K pd ( s p + + s d s (36 The am s to adjust the three gan factors (proportonal, ntegral and dervatve control accordng to the dynamcs of the plant, so that both the degree of error reducton (f not error elmnaton and the dynamc response wll be acceptable. In the frequency doman the PID controller ntroduces phase lag to the nomnal open loop (reachng almost 90 o at low frequences due to the domnance of the ntegral term and phase lead (reachng almost 90 o at hgh frequences due to the domnance of the dervatve term. In ntermedate frequences the ntroduced phase s ether negatve (due to ntegral term or postve (due to dervatve term, the proportonal term havng 0 o phase ntroduces only gan [Ref.5,7]. To llustrate, fgure A shows the Bode plots of a PID controller wth the followng transfer functon: K pd ( s + + 3s s Fgure A: Bode plots of a PID controller 40

54 Chapter Three Controller Degn (Graphcal In the context of ths project, the desgn constrants that the PID controller has to satsfy are:. L(j db f(, for all,,3,,n. L(j B, for all,,3,,n As n the case of the PD controller, the open loop gan must be a mnmum (subject to satsfyng the above constrants, see Secton The asymptotc gan of the nomnal open loop s gven by: lm p + + j d j G o ( j (37 Suppose that: G ( j o A p at very hgh frequences. (Assume that p, where p s the pole/zero excess of the nomnal plant. Then the asymptotc gan of the nomnal open loop s: A L o ( j p d at hgh frequences. Snce A and p are fxed from the plant, L o (j hgh s mnmsed by mnmsng d. Hence the followng optmsaton problem s formulated: Mnmse d subject to the followng constrants: L ( j f (, ψ, for all,,3,,n (where ψ L(j AND L( j B, for all,,3,,n 4

55 Chapter Three Controller Degn (Graphcal 4 The frequency response (sj of the controller (from equaton (36 s: j j j K d p pd + + ( or j j j K d p pd + ( In terms of magntude (lnear: ( + d p pd j K In terms of phase: p d pd j K φ tan ( : ( Note that the sgn of the phase ϕ( can be ether postve or negatve (always ϕ [-90,90] and ths depends on the three terms (.e. t depends on the domnance of the dervatve or the ntegral term, provded that all terms have the same sgn. In contrast to the PD controller, the phase of the PID controller wll be fxed at any two specfed frequences.e. and. The phase of the controller at all other frequences wll then also be fxed accordng to the combnaton of the two fxed phases at the two specfed frequences. Usng equaton (40, ϕ r (fxed phase for s gven by: p d r tan : ( φ φ Then, (38 (39 (40

56 Chapter Three Controller Degn (Graphcal 43 p d r tan( φ or tan( r p d φ whch mples 0 tan( r p d φ Thus 0 tan( φ r p d Smlarly, ϕ j (the fxed phase for s gven by: p d j tan : ( φ φ then 0 tan( φ j p d Equaton (4 and (4 can be arranged n the form of a matrx as A (x3 (3x 0: 0 tan( tan( p d j r φ φ Provded that the frst frequency s not equal to the second frequency, the ran of matrx A s equal to. Equaton (43 then mples that the vector [ d p ] T les n the (one dmensonal (4 (4 (43

57 Chapter Three Controller Degn (Graphcal Kernel of matrx A,.e. that the three gans d, and p are fxed (up to scalng. Asmple method of calculatng the Kernel of A s provded by the Sngular Value Decomposton (SVD [Appendx C]. Applyng the SVD to matrx A n equaton (43, we get A 0 0 V T σ 0 0 V [ U ] U T σ (44 where the range of A (R (A s equal to the range of U (R (U and the ernel of A ( er(a s equal to the range of V (R (V. It s also nown that, A 0 (45 where d p and er(a whch s equal to R (V. Then, vector can be expressed as: V λ d p V V V 3 λ where λ s an adjustable gan factor. Moreover, d, and p can be assocated wth V, V and V 3 respectvely. Thus the magntude and the phase of the controller from equatons (39 and (40 can be wrtten as: Magntude (lnear: 44

58 Chapter Three Controller Degn (Graphcal Phase: V K pd ( j λ V3 + V K pd ( j tan V V V3 (46 (47 Note that equaton (47 mples that the phase of the controller (and thus also the phase of the nomnal open loop s now fxed at every frequency. Clearly, d wll be mnmsed when λv s mnmum. The optmal value d (subject to the constrants gven n page 4 and ϕ( ϕ r and ϕ( ϕ j can be found from the followng consderaton. The robust performance objectves are satsfed f 7 where Thus Lo ( j f (, ψ, for all,,3,,n o db L ( j G ( j + K ( j db o db pd db K ( j f (, ψ G ( j for all,,3,,n pd db o db Here N s the number of frequences of nterest. Substtutng from equaton (46, 0log ( λ f (, ψ G ( j 0 0 o log db 0 for all,,3,,n, whch s equvalent to V 3 + V V or 0log 0 ( λ max f (, ψ Go ( j 0 log {,,3,.., N} db 0 V 3 + V V ~ 7 Note that constrant L(j B can also be formulated n the form L(j > f (, ψ. 45

59 Chapter Three Controller Degn (Graphcal log (, ( max 0^ 3 0 } {,,3,..., db o N V V V j G f ψ λ Multplyng both sdes of the above equaton wth V, log (, ( max 0^ 3 0 } {,,3,..., db o N V V V j G f V V ψ λ Ths equaton says that provded d s chosen to be larger than the RHS term, the constrants are satsfed. Hence the optmal d s gven by, log (, ( max 0^ 3 0 } {,,3,..., _ mn db o N d V V V j G f V ψ Note that throughout ths analyss, the phases of L o (j and L o (j are fxed as, ψ L o (j ϕ r and ψ L o (j ϕ j Clearly, the overall optmal value of d (.e. the mnmum value of d whch satsfes the robust performance constrants can be obtaned by tang the mnmum over all combnatons of phases L o (j and L o (j as these vary wthn ther allowable ranges. The followng algorthm, whch obtans the optmal PID controller, was mplemented n MATLAB: (48

60 Chapter Three Controller Degn (Graphcal Algorthm PID. Fnd the phase of the nomnal open loop (K(s, for the frst frequency.. Fnd the phase of the nomnal open loop, for the second frequency. 3. Set the range of the frst fxed phase ϕ r for (due to the acton of the PID controller n the nterval [-90 o + L o (j,90 o + L o (j ]. 4. Set the range of the second fxed phase ϕ j for (agan due to the acton of the PID controller n the nterval [-90+ L o (j,90+ L o (j ]. 5. Intalse arrays to store the optmal values of d,, p for all combnatons of ψ for fxed ψ. 6. Intalse arrays to store the optmal d,, p for all combnatons of phases (ψ and ψ. 7. Outer loop (runs n tmes, where n s the number of dscretsed ponts for the frst fxed phase ϕ r. 7. Intermedate loop (runs tmes, where s the number of dscretsed ponts for the second fxed phase ϕ j. 7.. Perform SVD for matrx A (for each combnaton of ϕ r and ϕ j Hold the values of vector V Intalse matrx to store the phase of the nomnal open loop for all frequences of nterest (t ntalses each tme a SVD s performed Intalse arrays to store local mnmum d and correspondng and p for each tme the nner loop runs (these ntalse each tme the combnaton of phases change Inner loop (runs N tmes, where N s the number of frequences of nterest. 47

61 Chapter Three Controller Degn (Graphcal For the frst two frequences (,, the phase of the nomnal open loop s fxed by assumpton For the rest of the frequences fnd the phase of the nomnal open loop usng: ψ G o (j + K pd (j, for all 3,4,,N These are fxed because every tme the loop runs the phase of the nomnal plant s fxed and nown and the phase of the controller s also nown (va the SVD. For all frequences of nterest the magntude of the bounds s obtaned approxmately va lnear nterpolaton For each fnd the gan λ usng equaton (48 that satsfes the requrements and temporary store t. Then fnd d, and p by multplyng them wth the assocated element of V (force p postve so that the phase of the controller wll always be n the nterval [-90 o,90 o ]. End of Inner loop Fnd the local mnmum of d for the current combnaton of phases and store t, store also the correspondng and p. End of Intermedate loop. 7. Fnd the mnmum d of all mnma obtaned n 7..7 and store t. Also store the correspondng and p. End of Outer loop. 8. Fnd the global mnmum d and also the correspondng and p. End of Algorthm 48

62 Chapter Three Controller Degn (Graphcal The above algorthm (was mplemented n Matlab functon pd_op(, wors as expected and fnds the optmal PID controller whch gves the soluton to the desgn problem. In order to get the best controller, whch gves the soluton the followng consderatons were made: (. The above procedure s followed only when the all elements of vector V have the same sgn. Ths mples that all controller gans ( d,, p wll be postve and the phase of the controller wll always be n the range [-90 o, 90 o ]. The controller wll ntroduce phase lag at low frequences and phase lead at hgh frequences as desred. (. If the elements have dfferent sgn, then all gans are set to nfnty (and so they are dscarded when the mnmum s chosen. (3. It s a good practce to fx the phase of the nomnal open loop at the frst and the last frequency of nterest. By followng ths approach the senstvty of the soluton s mproved. Ths also allows the phase n the last frequency vary n a range outsde the U-contour, whch means that the number of combnatons s reduced. (4. The two fxed phases are ndependent and as a result dfferent number of dscretsaton ponts can be used for each one. Ths helps to determne the magntude of the bounds more accurately va lnear nterpolaton. -Example : Consder the plant gven n equaton (. Here a PID controller s needed n order to help the system meet the tracng and dsturbance rejecton specfcatons gven n Sectons.3 and.7. Fgure B shows the nomnal open loop wth the selected frequences, the U-contour and the correspondng maxmum bounds. Moreover, t shows the new nomnal open loop correspondng to the optmal PID controller. 49

63 Chapter Three Controller Degn (Graphcal Fgure B: PID optmal desgn (ntal nomnal open loop (green lne and new nomnal open loop after the desgn of the controller (blue lne. The crcles represent the frequences of nterest The optmal PID controller obtaned s gven by: K 4.53 ( s.5 + s pd _ opt s From fgure B t can be seen that at low frequences due to the domnance of the ntegral term, phase lag s ntroduced to the nomnal open loop whle at hgh frequences due to the domnance of the dervatve term, phase lead s ntroduced. Note also that the shaped optmal open loop les on or above the bounds, wth four frequences of nterest (,3,4,6 lyng exactly on the bounds. To llustrate the acton of the PID controller, fgure C shows ts Bode plots (gan and phase ntroduced by the controller. The frequences of nterest are mared wth a crcle. 50

64 Chapter Three Controller Degn (Graphcal Fgure C: Bode plots of optmal PID controller (gan and phase ntroduced -Example : Consder the system: G( s ( s + a( s + b, where [,4], a [,5] and b [,4] The desgn of an optmal PID s needed so that the system wll satsfy the tracng specfcatons gven n the prevous example (there s no dsturbance rejecton ncluded. Moreover, n ths case the chosen frequences are {0., 0.5,,, 3, 5, 0, 60} rads/s. The optmal controller was desgned followng the procedure of Secton Fgure D shows the NC wth the nomnal open loop, the U-contour, and the Horowtz templates (whch are the correspondng bounds and the new nomnal open loop after the desgn of the optmal PID controller. 5

65 Chapter Three Controller Degn (Graphcal Fgure D: PID optmal desgn for G(s/(s+a(s+b, where the nomnal open loop before the desgn of the controller s n green and the new nomnal open loop after the desgn of the controller s n blue, the frequences of nterest are represented by crcles. From the above examples, the desgn of an optmal PID controller for realsable systems followng the procedure dscussed n ths secton was successful. In order to complete the desgn procedure, an approprate pre-flter wll be desgned. Ths s dscussed n the next chapter. 5

66 CHAPTER FOUR PRE-FILTER DESIGN 4. Introducton The desgn of an approprate nomnal open loop L o (j guarantees only that the varaton ( spread of the magntude response of the control rato T R (j s wthn the allowed specfcatons (.e. less or equal than δ R (j [Ref.]. The role of a pre-flter n a control system s to place L( j LmT ( j + L( j wthn the gven specfcatons n the frequency doman (see Secton.3, fgure.3.b. That s, the varaton of T R (j must le wthn the bounds B u and B l. Fgure 4..A shows the bounds B u and B l wth the varaton of T R (j for the example consdered n equaton ( wth parameter range of {,5,0} and a{,5,0}. Note that n order to fnd the maxmum spread of the varaton of T R (j, the mnmum and maxmum values for each uncertanty parameter must be ncluded,.e. the best and the worst case-combnaton. Fgure 4..A: Frequency responses of CL system wthout pre-flter and the desred range of acceptable CL frequency responses 53

67 Chapter Four Pre-flter Desgn The next secton dscusses the desgn procedure of an approprate pre-flter, whch wll adjust the CL system responses wthn the desred range. 4. Pre-flter Desgn. Frst, the CL system responses wthout the pre-flter are determned by tang varous combnatons of the uncertanty parameters of the plant (ncludng at least the maxmum and the mnmum values of each of the parameters to ensure maxmum spread.. From the responses determned n step, the maxmum and mnmum bounds are obtaned (ther dfference gves the maxmum spread. Fgure 4..A, shows the desred range of bounds and the maxmum spread of the CL system responses. LmTmax LmTmn B u B l Fgure 4..A: Desred range of bounds and maxmum spread of CL system responses wthout the pre-flter 3. From steps and above, we obtan the dfferences [B u -LmT max ] and [B l -LmT mn ]. These dfferences represent the maxmum acceptable bound and the mnmum acceptable bound of the magntude frequency response of the pre-flter. Fgure 4..B shows the range n wthn the bound of the pre-flter must le n order to satsfy that the CL system responses wll le wthn the gven specfcatons. 54

68 Chapter Four Pre-flter Desgn Fgure 4..B: Allowable frequency response (magntude range of pre-flter 4. Usng straght-lne approxmatons (usually by nspecton from the graphs, F(s can be synthessed such that LmF(j wll le wthn the range of allowable bounds from step 3. Moreover for step forcng functons, lm s 0 { F( s } s enforced so that the s.s.e to step nputs s zero. 5. The pre-flter F(s obtaned from above procedure ensures that the CL system responses le wthn the specfed range shown n fgure 4..A (B u - B l, for all combnatons of the uncertan parameters [Ref.]. Note that for ths example, as can be seen from fgure 4..B the frequency response of F(s at certan frequences can vary more than at other. Ths can be verfed from fgure 4..A where we can see that at frequences 4,5 the CL system responses have a greater range of adjustment because the specfcatons are not tght. In addton n frequences,,6,7 the specfcatons are very tght and therefore the pre-flter bounds are lmted (.e. only one straght-lne approxmaton can be chosen. 55

69 Chapter Four Pre-flter Desgn 4.3 Types of Pre-flter Usually a second order pre-flter suffces; ths has the followng transfer functon: Ag F( s s + s + c c (49 where, A g s an adjustable gan (usually, c and c are the frst and second cut-off frequences, respectvely. These constants can be found va the procedure presented n Secton 4.. Hgher order pre-flters can be used n order to gve a more accurate result. Nevertheless, hgher order pre-flters are more complex and thus the mnmum-order possble pre-flter s desred. 4.4 Desgn example (for PID controller The desgn of an approprate pre-flter s the same for all types of controllers,.e. phase lead/lag cascade networs, PDs, PIDs. For llustraton purposes of the procedure, the system descrbed n equaton ( and the optmal PID controller, obtaned n Secton Fgure 4..A shows the desred range of tracng bounds and the CL system responses wthout the pre-flter, and fgure 4..A shows the maxmum spread of the CL responses wthout the pre-flter and the gven specfcatons. Moreover fgure 4..B, shows the allowable range of bounds, wthn whch the response of the desgned pre-flter has to le. By nspecton, an approprate nd order pre-flter for ths system was found to be: F ( s s + s (50 Its magntude frequency response can be seen n fgure 4.4.A. Clearly ths les wthn the allowable range gven n fgure 4..B. 56

70 Chapter Four Pre-flter Desgn Fgure 4.4.A: Magntude frequency response of desgned pre-flter F(s Fgure 4.4.B shows that by usng the desgned pre-flter -from equaton (50- the CL responses le wthn the range of allowed specfcatons. Fgure 4.4.B: CL bounds after the ntroducton of the desgned pre-flter F(s 57

71 Chapter Four Pre-flter Desgn From the above, t follows that the system wll meet both the gven tracng specfcatons and also the dsturbance rejecton specfcatons. Ths can be seen n the next Chapter, whch ncludes the smulaton results of the system, descrbed n equaton (, for the three types of controllers used n ths project. 58

72 CHAPTER FIVE SIMULATION RESULTS 5. Introducton Ths chapter presents the smulaton results of the overall system ncludng the phase lead/lag cascaded networs (from the graphcal desgn, the PD controller, and the PID controller, whch were obtaned n ths project. Note that the pre-flter for each one of the overall systems was desgned by followng the procedure of Secton 4.. The system, whch was used to llustrate the fnal results of the desgn, s gven n equaton (. 5. Tracng Performance-Smulatons Frst the tracng performance of the system was obtaned for each controller and approprate preflter. Fgure 5..A shows the tracng response (step responses for certan combnatons of the uncertan parameters of the system ncludng the phase lead/lag cascaded networs. Fgure 5..A: Step responses of CL system wth approprate pre-flter ncludng phase lead/lag cascaded networs. 59

73 Chapter Fve Smulaton Results Fgure 5..B shows the step responses of the CL system ncludng the PD controller. Fgure 5..B: Step responses of CL system wth approprate pre-flter ncludng PD controller. Fnally, fgure 5..C shows the step responses of the CL system ncorporatng the PID controller. Fgure 5..C: Step responses of CL system wth approprate pre-flter ncludng PID controller. 60

74 Chapter Fve Smulaton Results From the fgures above t can be seen that the step responses of the system for all controllers desgned were acceptable. The system responses to a unt reference nput le wthn the gven specfcatons n all cases. Thus, ts behavour wll be wthn the specfed allowable range. 5.3 Dsturbance rejecton-smulatons The system must also satsfy the dsturbance rejecton specfcatons (gven n secton.7. Fgure 5.3.A llustrates the response of the system wth the phase lead/lag cascaded networs to a unt step dsturbance at the output of the plant G(s. Fgure 5.3.A: Dsturbance rejecton of system wth phase lead/lag networs n cascade Fgure 5.3.B shows the response of the system usng the PD controller to a unt step dsturbance nput at the output of the plant. 6