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Date Created: 09/23/15
Robust Control 9 Design Summary amp Examples Harry G Kwatny Departinth of Mechanical Engineering amp Mechanics Drexel University Outline The HOO Problem Solution Via riteration Robust stability Via coprime factorization Robust stability with loop shaping Example noncollocated revisited The H Problem We will formulate a problem similar to above but with two changes 1 the disturbances are not stochastic wt may be any member of the class WI2 1 2 Choose K so that the maximum of quot21quot2 over all admissible disturbances wt is a minimum ie min max 21 2 K lel The H2 Problem Cont d Recall zTzzzdz ZT jm2jm day 1 WT jwFT jwijwowdw The maximum performance energy over all disturbances with unit norm occurs When W is alligned with the maximum eigenvalue of F F I zTzzzdz i j Wow FowWowmw mgxaz Fmw F 2 mgnan Solution of the Hm Problem A stabilizing controller that satis es 75 lt 1 is Asl eZsLs 7 1 0 where A AB LD2W BZFZLC2 ZLD22F F7RR BZTX L 7YCZTVVV 1 T 1 W ZB X Z 17 2121 7 X 2 012 2 0 satisfy the Riccati equations Ks If yis xed these Riccati equations are decoupled 1 XJ 15 r X 32123327 7 23ij X 7R RijRfu 7 r r 1 1 r 7 1 r AKYA 7YC2V C277CC 12714 vava Solution of the H Problem Cont d And the following 3 conditions are satis ed 1 The Hamiltonial matrix Aisznxr 7 mainly ixmmmxz 7A782R3R2Y has no eigenvalues on the imaginary axis Equivalently ABW BZF is stable 2 The Hamiltonian matrix 7 T 7 1 Aerwa Cz CZVW Cw TCi 7 r r 7V W Va my C has no eigenvalues on the imaginary axis Equivalently 1 A LC 21CTC ls stable 7 3 pYX lt 72 wherep max is the spectral radius Strategy for Solving H2 Typically there is a minimum value of yfor Which the above conditions are satis ed This is the optimal control The usual method is to nd the optimal control by yiteration Begin With a large yand reduce it until one of the conditions fail Then use a bisection method until Within an acceptable tolerance Usually we don t need to get that close a few iterations Will suf ce Robust Controller Design with Coprime Uncertaingl Consider the family of perturbed plants GP Dl AD391Nl AN AN ADHL lt5 with some stability margin 5 gt 0 The system is robustly stabilized by the controller u Ky if and only if 117GKVD Robust Stabilizer Design Problem Find the lowest achievable A y s l 5 7 and the corresponding maximum stability margin 5 and the corresponding controller K Solution to RSDP mm 6 1 pXZ1Z WhereX Z are the unique positive definite sol39ns of the ARE39s A BSquotDTCZ Z A BSquotDTCT ZCTR 1CZ BSquotBT 0 A BS 1DTCT X X A BS IDTC XBSquotBTX CTR 1C 0 and Gs 4 minimal realization ABCD R 1 DDT s 1 DTD Solution to RSDP Cont d a solution that guarantees K 1 G1lt 1 Df I Syfor some jgtjmin is given by K ABFy2LT IZCTCDF y2L7 IZCT BTX D7 F S 1DTCBTX L1 y2IXZ Loop Shaping for Performance Closed Loop Transfer Functions Assume the closed loop is stable Then 1 disturbance rejection gt E S ltltl 2 noise attenuation cgt 5T ltltl reference tracking cgt 5T z gT zl control energy gt 5 KS ltltl robust stability with additive uncertainty gt 5 KS ltltl cxylbw robust stability with mult output uncertainty gt E T ltltl Loop Shaping for Performance Open Loop Transfer Function Assume the closed loop is stable Then 1 disturbance rejection cgt gL gtgt1 2 noise attenuation cgt 5L ltlt1 3 reference tracking cgt gL gtgt1 4 control energy gt E K ltlt1 5 robust stability with additive uncertainty gt E K ltlt 1 if E L ltlt1 ON robust stability with mult output uncertainty gt E L ltlt1 Exam le Collocated vs Non collocated The goal is to position the load A position sensor can be placed on motor or motor load load39 exible shaft Example 51 0 1 0 0 91 0 i 01 7 ikJl wlJ1 kJ1 0 01 0 dt 62 0 0 0 1 Hz 0 w2 kJ2 0 ikJZ fezJ2 w2 ilJz 51 51 z0 010 1 or z10 0 0 1 HI HI 2 2 y z J1J2 1k1cl 02 0 Exam le Sim li ed W amp 2 212 Resvansi magxnaw Axs a m Am Simpli ed Noncollocated m Lucas x magman AXE n m m Noncollocated Ends mgrm Gm 3 45 an a 1 2 radsac Pm 7335 vies st 1 asvawsec Magrimde a 2nu Phase m qn x 1 m u39 quu nw raisin Robu t tabilig gammin 38202 Zeropolegain s02363 5 2 7 016635 2002 s3125 s2235 s 2 070845 3806 gammin 7946 Zeropolegain 7527 s2001 S00l715 5 2 7 081545 3005 s001998 s 2 32945 2744 5 2 060545 2733 Step Respon e Magnrluda 8 pm my mu i 4 on i E m am 1 JED Shaped Robust Stability and Dlagvam sm 11 5 an n as vsdisec Pm 573 day at am veutsec mar J Mamma as Phase meg Robust Control 1 Introduction amp Linear State Equations Harry G Kwatny Department of Mechanical Engineean amp Mechanics Drexel University Outlme Introduction 1 Linear System Theory Basics 2 I State space models controllab ityobservabllity Nominal Control Design State Space Perspective 3 Pole placement LQG Separation Principle Transfer Function Models Open amp Closed Loop 4 5 I ssth t ss poles amp zeros welliposedness Performance in the Frequency Domain 6 I Sensitivity functions fundamental trades performance limim Nominal Control Design Frequency Domain Perspective 7 I L Grevisi ed Minimax Modern paradigm Robust Stability amp Nyquist Analysis 89 I Nyquist small gain thm MIA structure Robust Performance 10 Example Jet Engme Compressor A 331322 M g mi w v astwwa s g s I Iowamwuwnn r 7 v i s m xgg a mm mm sMlnvo mum rm mum mmmsms Example MG3 Model 5 1 Pc 11 A 1 CDT 1 z 11 7A Y LM I L gtAsin9d9 A Z 012 214 I gtAjquot gtAsin9sin9d9 the annulusaveraged ow coef cient IIthe pressure rise coef cient A the amplitude of the first harmonic of annular ow distribution TC The function is the compressor characteristic Tthe throttle characteristic Example Computational Data tMwpga q aln ltIgtT 1 N17 11 Ai 3A2 1 2 3 j2 41150 IZ A AAZ 4 2 Example Axisymmetric Eguilibria gAOg I 3 e uns able stab a 25 2 15 l 05 05 1 l 5 2 2 5 3 Exam 1e NonAxis mmetric Stall Linearization 5c fxuu y hxuu equilibria 0 fxuu 0 xii41 linearizal ion AxxutAxxufAu AAxBAu Ay xiuifAxxiui Au CAxDAu Solving Linear State Equations icAtxBtut xERquotuERquot39 givenxt0xoutf0rt2t0 ndxt fortZtO c A t x b t b t B tu I forced or nonhomogeneous c A t x homogeneous Aggroach Basic Properties Fundamental System of Sol ns Variation of Parameters Formula State Transition Matrix for Constant A 39 Laplace Transform 39 Matrix Exponential 39 Eigenvalueeigenvector Basic properties x1I xZ I sol39ns of homog 0102 consth U xI 01x1 I czxZ I is a sol n of homog x1I xZ I sol ns of forced U xI x1 I 7x2 I is a sol n of homog xp I any sol n of forced xh I any sol n of homog U xI xh I xp I is a sol n of forced Basic Properties Continued x1 Atx1Btult x2 Atx2Btu2t U xtwx1t x2t isasol39nof xAzxBzauz u2z Atbt Btut piecewise continuous on 1112D exists a unique sol n of forced passing through xt0x0gt10611gt12 Fundamental System of Solutions xI a sol n of homog and xI0 0 2 xI E 0 x1 Ix39 I sol ns of homog 2 clxlIc39x39I is also a solution the set of sol ns x1 I x I is linearly dependent if there is a nontrivial set of constants cl c such that clxlI c39x39 I 20 Otherwise the set of sol ns is linearly independent Fundamental System Cont d Note if the system is linearly independent then for no value oft can the vectors x t be linearly dependent the systems ofsolutions x1 txn t is a lndamental systems of sol s if it is linearly independent a fundamental solutions always eXists every sol n of homog can be expressed as a linear combination of x1 I xn t Fundamental Matrix De ne the lndamental matriXX I from the fundamental set of solutions x1t x t Xtixlt 9W Noticethat 5c A9q 3X AX To ndXt solve Xt AtXt withXt0 X0 detX0 0 xn 0 State Transition Matrix The state transition matrix is 13010 XtX 1tO Notice that q ZJO satisfies hAO hJDUwI State Transition Property h haa10neh Variation of Parameters Formula Recall any sol n of forced satis es xt xh t xp 1 where xh t X tc for constant vector 0 satis es homog xp 1 is any particular sol n of forced We seek xp Assume the form xp 1 Xtct jcp Axp Bu andX Ax c39tX 1tBtut CI MLX ITB7u7d7 Variations of Parameters Formula Cont d Thus we have xp t j mel TB7u7d7 xt XtcJtXtX 1TB7u7dr Now impose xt to 3 c X 1t0x0 3 xt XtX 1t0x0 XIX 17B7u7d7 xtlt1gttt0x0 L ITBTurd7 Transition Matrix Via Laplace Transform Assume A constant matrix dgttt0 ACIgttt0 CIgtt0t0 1 U SCI st0 I ACID st0 U CIgtst0 2 S A CIgttt0 f1s1 A 1 Transition Matrix Via Matrix Exp1 Assume A constant matrix and qgtttuAU441ttuquot39Aktt kquot39 substitutein dgtttnAdgtttn A12AZ titnmla4ktitnk A AAU A AAu 2A2 AA1 A2 iAZA 1 quot39AAuAA1tit quot39AAktitnk39w Transition Matrix Via Matrix Exp2 zzo A0Alz zoAkz zok IAz zolZAZz zozAkz zok Ao The condition lolo 1 2 A0 1 so zzo IAz zoAZz IOZAkz zok By analogy with ea 1al lolZaZZ ZOZakl lok define the matrix exponential 2AM IAz zoLZAZz IOZAkz zok CaleyHamilton Theorem Cayley Hamilton Theorem Every square matrix satis es its own characteristic equation ie 111 A Aquot aH1H a0 U A A an71Aquot71a01 0 From this we obtain 8 010 tI a1tA 0n1tAquot391 Transition Matrix Via Eigensystem Assume A constant matrix and 114 21 1n is its eigenvalue set 1 quot to39 39 l 39 39 13911hnThen Xt M hle z line1 ltIgttt0XtX 1t0 hie IhzeMH hnekquotquot hi lhz M1 fear t tn 0 ltIgtrroIhi Me I I hi hi Ihz I I W 0 emrru Robust Control 6 The Regulator Problem Harry G Kwatny Departinth of Mechanical Engineering amp Mechanics Drexel University Part I 1 Introduction 2 Linear System Theory Basics 3 Nominal Control Design State Space Perspective 4 Transfer Function Models Open amp Closed Loop 5 Performance in the Frequency Domain 6 Nominal Control Design Frequency Domain Perspective Part III 7 The Regulator Problem 8 Nominal Control Design Modern Paradigm 9 Robust Stability SISO Systems 10 Robust Stability MIMO Systems 11 Robust Performance 12 Controller Design Outline Review Classical Nominal Design I The LQR Problem I A Generalization to LQR I MinMax Control I Solving the Ricatti Equation I Classical LQG I Stability Margins amp LTR Tracking amp Disturbance Rejection Internal Model Principle Disturbance Reiection Setup w disturbance z performance variables Controller y measuremenm 14 control state equations x Ax Ew Bu Eimrmoif rf mi 145 v39 m rm mtg d1sturbance w Zw VJ performance Z Cx Fw Du measurements y 5x I7w Eu xe R we Riue Rmze Rmye R39 Objectives Regulator Problem Find a controller to achieve the following 1 Regulation 2t gt 0 as t gt 0 2 Intemal Stability Achieve speci ed transient response Robust Regulator Problem Find a solution to the Regulator Problem that satis es 3 Robustness l and 2 should be maintained under speci ed small perturbations of plant andor control parameters S olut10n Part 1 Re gulat1on Consider the possibility of a control 17 t that produces a trajectory 7t for some unspeci ed initial state in and any initial disturbance vector w so that the corresponding 7t E 0 Then 717 w must satisfy A3 Ew B17 W Zw 0 C Fw D17 Assume a solution ofthe form 7 Xw17 Uw 5 XZw AXwEwBUw 0 CXw FwDUw Thus the hypothesized control 17 t exists if there areXU that satisfy Vw qu mlgr ms 1 113115 a Solution Part 2 Stability Define Eu 5x Now if AB is controllable it easy to choose 5L K5x so that the closed loop 55c A BK 5x has desired transient characteristics With K chosen the control can be written as a function of the system states x w u E5u Uw5u UwK5x u UwKxXW K U7KX KTOT Solutlon Part 3 Observatlon The control will be implemented using estimates of the composite state xw Consider the composite system d x A E x B u dt w 0 Z w 0 y 5x Fw Ifthe composite system is observable we can choose a matrix L so that the following observer has the desired dynamics d 56 A E fc B A A A A uLCxFw y dt w 0 Z w 0 Properties of the Loop M5 FJJEW DJW nib ng almmbmm Example mum torque 7 command y matar szzd controllzr 7194203 x918897 3J209 sv498393 Robust Control 7 Nominal Controller the Modern Paradigm Harry G Kwatny Department of Mechanical Engineean amp Mechanics Drexel University Outline Setup LQG in the Modern Paradigm The H2 Problem Definition The HoO Problem Definition Solution of the H2 amp HoO Problems MATLAB Problems
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