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# Optimal and Robust Control ECE 7360

Utah State University

GPA 3.86

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This 34 page Class Notes was uploaded by Seth Gibson on Wednesday October 28, 2015. The Class Notes belongs to ECE 7360 at Utah State University taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/230400/ece-7360-utah-state-university in ELECTRICAL AND COMPUTER ENGINEERING at Utah State University.

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Date Created: 10/28/15

Chapter 15 H00 Controller Reduction 219 0 problem formulation 0 additive reduction 0 coprime factor reduction 220 Problem Formulation All stabilizing controllers satisfying Tzwcgt0 lt 7 K 75M007Q Q E 737100 HQHOO lt 7 where JWCgt0 is of the form M11lt M21lt8gt M22lt8gt CI E V MOO such that 21 321313101 and fl 311323102 are both stable ie M131 and M231 are both stable Find a controller K with a minimal order such that HfAG K OOlt7 Find a stable Q such that K fgMOO Q has minimal order and Qcgt0 lt m 221 Additive Reduction Consider the class of reduced order controllers fr K0 Wgnwl A e RHOO W1W1 1W2W2 1 e RHOO such that fgG K0Cgt0 lt 7 K and K0 have the same right half plane poles Then We fr El Q E R7100 with QCgt0 lt 7 such that K fgMOO Q ll 0 IO 0 O K K712 a Q fdkil 01 a 7 HQHOO lt 1 lt2 WEEK 00 lt 1 ltgt HfgltKgl K0 W2AW1gtHOO lt 7 ltgt Wamllm lt1 where 7712 0 311312 7712 0 0 W1 R21 R22 0 W2 R R 7 K0 I 11 12 8ltK1 gt 321 322 I O Redheffer7s Lemma 1 00 g 1 and AOO lt1 Wamllm lt 1 222 Suppose W1 and W2 are stable minimum phase and invertible transfer matrices such that 1 is a contraction Let K0 be a stabilizing controller such that fgG K0 lt 7 Then K is also a stabilizing controller such that W0 fol lloo OOltyif MANGO Hw k Kuwilm lt1 1 can always be made contractive for suf ciently small W1 and W2 We would like to select the largest W1 and W2 Assume Rggcgt0 lt y and de ne 0 R11l 0 RH Rf1 0 leg 0 0 R21l 0 1 322 R5 0 l Rgg 0 L1 L2 Lg L3 L fylt Then 1 is a contraction if W1 and W2 satisfy WYWOA 0 0 W2W2Ngt71 L1 L2 Lg L3 An algorithm that maximizes detWfW1 detW2W2 has been devel oped by Goddard and Glover 1993 223 Coprime Factor Reduction All controllers such that TzwCgt0 lt 7 can also be written as Klt5gt fZltMooy Q 911C 912gtlt921Q 922V1 3 UV l C2912 2271ltQ911 921 I VilU where Q E R7100 QCgt0 lt 7 and UV 1 and 1740 are respectively right and left coprime factorizations over R7100 and 21 311623165 32 3116231152 3115231 l i 1191 i A A A71A A A A71A A A71 21 22 i 01 1311132102 D12 P11921132 D1A1D21 D2 1102 D2 11D22 D231 can 912 l 21 213916 l A31 215913 213351 6 621 622 02 P22125101 D21 AD22ADf21D11 DA22Df21 D301 D lpll D1721 ng 1 31 BgD lDll l A 971 Df216ll Dbl Df21D11 02 1322ng101 1322ng1 D21 1322131311311 l A BlDil g 31D 11 BQ BlDingg l 971 D5162 DEE Dilpgg A A AAA A A71 A A AAA 01 D11D2102 D11D21 D12 D11D21D22 224 Let ISO A 1265 be the central H00 controller fgG K0 OO lt 7 Let U V E R7100 with clet Voo 75 0 be such that e U 12 lt1 7 1I O 922 A O I Then K UV l is also a stabilizing controller and fgG lt 7 71 V 00 Note that K is a stabilizing controller such that Tzwcgt0 lt 7 if and only if there exists a Q E RHOO with QCgt0 lt 7 such that U G G 11Q 12 i Q 014 V 21Q 22 I and K2UV4 De ne 410 e U A I 7 91 12 A O I 922 V and partition A as A A U AV Then A U G I O A A 12 G 7 A G 7 U V 922 O I I AV and A Av 1 i G AUltI Avgt71 vu Am4 I De ne A rUu Am1LNVu Am4 225 Q at1 AV 1 Then fr if1 UV1 and Q 7AU1 Av 1 7 olAltr l0 Ilnfl 0 1 ol W m 0 m M Again by Redheffer7s Lernrna HAUU A l oo lt 1 since 0 I 0 IN 0 Nil is a contraction and lt 1 2 Hence Him AWle lt 7 Therefore fgG lt 7 Let 0 A 6231621 be the central H00 controller fgG K0 OO lt 7 Let 017 6 RHOO with det l7oo 7t 0 be such that 7 1 0 O I 621 eml ll we lt1 A c 1 a A Then K V U is also a stabilizing controller and fgG K OO lt 7 suf cient conditions H00 controller reduction gt frequency weighted H00 model reduction H2 and H00 Optimal Control 83 Introduction to the Problems Consider a stabilizable and detectable linear timeinvariant system 2 with a proper controller 2 C where xAxBuEw 2 vA0VBCy Z yC1x 0uD1w c uCcVDCy zC2xD2u0w xeiRquotQ statevariable ueiRmQ controlinput y e 91quot Q measurement amp w 6 911 Q disturbance z E SW Q controlled output v 6 ERquot Q controller state 84 Prepared by Ben M Chen The problems of H2 and H00 optimal control are to design a proper control law 20 such that when it is applied to the given plant with disturbance ie 2 we have The resulting closed loop system is internally stable this is necessary for any control system design The resulting closedloop transfer function from the disturbance w to the controlled output 2 say Tms is as small as possible ie the effect of the disturbance on the controlled output is minimized H2 optimal control the H2norm of Tms is minimized H00 optimal control the HOGnorm of T2s is minimized Note A transfer function is a function of frequencies ranging from 0 to 00 It is hard to tell if it is large or small The common practice is to measure its norms instead H2norm and H00 norm are two commonly used norms in measuring the sizes of a transfer function 85 Prepared by Ben M Chen The Closed Loop Transfer Function from Disturbance to Controlled Output R ecallthat vZAchchy xAxBuEw 28 C 1 2 CV Cy Z yC1x 0uDlw and gt 1391 2 Acv BCC1x Dlw Acv BcClxBchw zC2xD2u0w I 39A B D E 39 x x Ccv cy WE 5CAxBCcvBDcC1xDlwEw D 3 y C1 x 1w 2 sz DZCcv D2D8C1x Dlw z C2xD2CcvDcy i I i x A BDcClx BCcv E BDCD1w I i z C2 DZDCC1x DZCCV DZDCDIW I XAxBCcvBDcyEw z C2xD2CcvD2Dcy x ABDC1 BC x EBDD1 N C C w Acl x Bclw v BECl A v 3ch c gt x z C1 DZDCC1 D2C5 DZDCD1 w Ccl 7 Dclw v 86 Prepared by Ben M Chen Thus the closedloop transfer function from w to z is given by Tzw S CclSI Acl7chl Dcl The resulting closedloop system is internally stable if and only if the eigenvalues of A BD 0C1 BC 0 cl BCCI A C are all in open left half complex plane Remark For the state feedback case C1 I and D1 0 Le all the states of the given system can be measured 20 can then be reduced to u F x and the corresponding closedloop transfer function is reduced to Tms C2 D2F sI A BF 1E The closedloop stability implies and is implied thatA B F has stable eigenvalues 87 Prepared by Ben M Chen H2norm and Honnorm of a Transfer Function Definition HZnorm Given a stable and proper transfer function Tzws its H2norm is de ned as 2 trace T2wjwTmjwHde Graphically lT jw I i H2norm Note The H2norm is the total energy corresponding to the impulse response of Tzws Thus minimization of the H2norm of Tzws is equivalent to the minimization of the total energy from the disturbance w to the controlled outputz 88 Prepared by Ben M Chen Definition Hmnorm Given a stable and proper transfer function Tzws its HOGnorm is de ned as TN 00 Osgng Gmax Tzw where omax Tzwja denotes the maximum singular value of Tzwgw For a singleinput single output transfer function Tzws it is equivalent to the magnitude of TZWQw Graphically szwaN Note The HOGnorm is the worst case gain in Tzws Thus minimization of the Hoenorm of Tzws is equivalent to the minimization of the worst case gain situation on the effect from the disturbance w to the controlled outputz 89 Prepared by Ben M Chen In ma and Optimal Controllers Definition The in mum of H2 optimization The in mum of the H2 norm of the closedloop transfer matrix Tzws over all stabilizing proper controllers is denoted by yzquot that is Tm M2 20internallystabilizesZ y 1nf Definition The H2 optimal controller A proper controller 20 is said to be an H2 optimal controller if it internally stabilizes 2 and Tm II2 3 Definition The in mum of H00 optimization The in mum of the Hoenorm of the closedloop transfer matrix Tzws over all stabilizing proper controllers is denoted by ywquot that is y inf TN quot00 26internallystabilizesZ De nition The H00 y suboptimal controller A proper controller 20 is said to be an H00 y suboptimal controller if it internally stabilizes 2 and TN quot00 ltygt x2 90 Prepared by Ben M Chen Critical Assumptions Regular Case vs Singular Case Most results in H2 and H00 optimal control deal with a socalled a regular problem or regular case because it is simple An H2 or H00 optimal problem is said to be regular if the following conditions are satisfied 1 D2 is of maximal column rank ie D2 is a tall and full rank matrix 2 The subsystem ABC2Dz has no invariant zeros on the imaginary axis 3 D1 is of maximal row rank ie D1 is a fat and full rank matrix 4 The subsystem AEC1D1 has no invariant zeros on the imaginary axis An H2 or H00 optimal problem is said to be singular if it is not regular ie at least one of the above 4 conditions is satisfied 91 Prepared by Ben M Chen Solutions to the State Feedback Problems the Regular Case The state feedback H2 and H00 control problems are referred to the problems in which all the states of the given plant 2 are available for feedback That is the given system is 5c 2 A x B u EW 2 y x z C2 x D2 u where A B is stabilizable 02 is of maximal column rank and A B CZ DZ has no invariant zeros on the imaginary axis In the state feedback case we are looking for a static control law 92 Prepared by Ben M Chen Solution to the Regular H2 State Feedback Problem Solve the following algebraic Riccati equation HZARE ATP PA CZTCZ PB CZTD2D2TD2 1D2TC2 BTP 0 for a unique positive semidefinite solution P 2 0 The H2 optimal state feedback law is then given by uzF xz DZTDZ 1D2TC2BTPx It can be showed that the resulting closedloop system Tzws has the following property lsz w I2 r It can also be showed that y trace ETPE Note that the trace of a matrix is defined as the sum of all its diagonal elements 93 Prepared by Ben M Chen Example Consider a system characterized by A B E 39 i5 2i ii iii x x u W 3 4 1 2 2 yx zzh Hx1u C2 D2 Solving the following H2ARE using MATLAB we obtain a positive de nite solution 144 40 P 40 16 F 41 17 and The closedloop magnitude response from the disturbance to the controlled output 0 0 Z 10 10 10 10 Frequency radsec The optimal performance or in mum is given by y 191833 94 Prepared by Ben M Chen Classical Linear Quadratic Regulation L QR Problem is a Special Case of H2 Control It can be shown that the wellknown LQR problem can be reformulated as an H2 optimal control problem Consider a linear system 9 C AxBu x0 X0 The LQR problem is to find a control law u Fx such that the following index is minimized 00 T T J IO x Qxu Rudt where Q 2 0 is a positive semidefinite matrix and R gt 0 is a positive definite matrix The problem is equivalent to finding a static state feedback H2 optimal control law u Fx for XAxBuXOw yzx 0 M u Q4 0 N l 95 Prepared by Ben M Chen Solution to the Regular H State Feedback Problem Given y gt yo solve the following algebraic Riccati equation HmARE 71 ATP PA CZTCZ PEETPyz PB Csz2 DZTDZ DJC2 BTP 0 for a unique positive semidefinite solution P 2 0 The H00 y suboptimal state feedback law is then given by uzF xz DZTDZ 1D2TC2BTPx The resulting closedloop system Tzws has the following property TN 00 lt 9 Remark The computation of the best achievable H00 attenuation level ie ywquot is in general quite complicated For certain cases yo can be computed exactly There are cases in which yo can only be obtained using some iterative algorithms One method is to keep solving the HODARE for different values of y until it hits yo for which and any y lt ywquot the HoeARE does not have a solution Please see the reference textbook by Chen 2000 for details 96 Prepared by Ben M Chen Example Again consider the following system A B E 1521 101 ill x x u W 34 1 2 2 yx z1 lxlu C2 D2 It can be showed that the best achievable H00 performance for this system is Solving the following HOGARE using MATLAB with y 5001 we obtain a positive de nite solution 3301115 1100288 1100288 266791 and F 1100298 366801 The closedloop magnitude response from the disturbance to the controlled output 4 6 102 10 10 Frequency radsec 1039 100 Clearly the worse case gain occurred at the low frequency is roughly equal to 5 actually between 5 and 5001 97 Prepared by Ben M Chen Solutions to the State Feedback Problems the Singular Case Consider the following system again 5c 2 A x B u E w 2 y x z C2 x D2 u where A B is stabilizable 02 is not necessarily of maximal rank and A B CZ DZ might have invariant zeros on the imaginary axis Solution to this kind of problems can be done using the following trick or socalled a perturbation approach Define a new controlled output Z C2 D 2 3 8x 8 x small perturbations en 0 8 Clearly Z oc E ifs O 98 Prepared by Ben M Chen Now let us consider the perturbed system XAxBuEw C2 D2 i y 2 x where 52 1 and 132 0 E 2x52u 0 81 Obviously 52 is of maximal column rank and AB5252 is free of invariant zeros for any 8 gt 0 Thus E satis es the conditions of the regular state feedback case and hence we can apply the procedures for regular cases to i nd the H2 and H00 control laws Example r 2i x x 2ixnuriw 3 4 3 4 1 2 y 2 y 2 x EV Z 1 1 8 z1 1x0u z x 99 Prepared by Ben M Chen Solution to the Generale State Feedback Problem Given a small 8 gt O Solve the following algebraic Riccati equation HZARE A73 13A 252 1313 55252T527152T52 BT13 0 for a unique positive semide nite solution 13 2 0 Obviously 13 is a function of s The H2 optimal state feedback law is then given by u 15 x 552 152T52BT13 x It can be showed that the resulting closedloop system Tzws has the following property Tzw2 gty2 as e gtO It can also be showed that N 1 traceETPEA gt 3 as 8 gt O 100 Prepared by Ben M Chen Example Consider a system characterized by 39 is 21 i1 11 x x Ll W 3 4 1 2 2 y x z 1 1 x 0 u Solving the following H2ARE using MATLAB with s 1 we obtain N 1861968 462778 462778 182517 39801 F 462778 182517 N 212472 49311 F 493111 189748 49311 18975 8 00001 N 16701 00424 00424 00112 F 423742 112222 The closedloop magnitude response from the disturbance to the controlled output 25 U Ma g nitude 0 1039 100 102 10 Frequency radsec The optimal performance or in mum is given by 101 Prepared by Ben M Chen Solution to the General H State Feedback Problem C2 D2 5 Step1Givenaygtywchoosee1 523 81 and Rip Step 2 De ne the corresponding 52 and 132 Step 3 Solve the following algebraic Riccati equation HmARE AT I3A 672 ISEETlSy2 133 132 1152f 1352 BTI3 0 for 13 Step 4 If 13 gt 0 go to Step 5 Otherwise reduce the value of e and go to Step 2 Step 5 Compute the required state feedback control law L12 x 5J52 15 2BT13x It can be showed that the resulting closedloop system Tzws has T N quot00 lt 7 102 Prepared by Ben M Chen Example Again consider the following system z1 1x0u It can be showed that the best achievable H00 performance for this system is Solving the following HOGARE using MATLAB with y 06 and g 0001 we obtain a positive de nite solution 151877 09874 09874 00981 and F 987363 981181 The closedloop magnitude response from the disturbance to the controlled output 08 06 04 02 Clearly the worse case gain occurred at the low frequency is slightly less than 06 The design specification is achieved 103 Prepared by Ben M Chen Solutions to Output Feedback Problems the Regular Case Recall the system with measurement feedback ie icAx BuEw Z yzClx Dlw zC2xD2u where A B is stabilizable and A C1 is detectable Also it satis es the following regularity assumptions 1 02 is of maximal column rank Le 02 is a tall and full rank matrix 2 The subsystem ABCZDZ has no invariant zeros on the imaginary axis 3 D1 is of maximal row rank ie D1 is a fat and full rank matrix 4 The subsystem AEC1D1 has no invariant zeros on the imaginary axis 104 Prepared by Ben M Chen Solution to the Regular H2 Output Feedback Problem Solve the following algebraic Riccati equation HZARE ATP PA CZTCZ PB CZTD2D2TD2 1D2TC2 BTP 0 for a unique positive semide nite solution P 2 0 and the following ARE QAT AQ EET QCIT EDJ111J 1D1ET CIQ 0 for a unique positive semide nite solution Q 2 0 The H2 optimal output feedback law is thengivenby 2 2ABFKC1v Ky C u F v where F D2TD2 1D2TC2 BTP and K QCJ EDITD1D1T 1 Furthermore y trace ETPE trace ATP PA CZTCZ Q 105 Prepared by Ben M Chen Example Consider a system characterized by x1 0111713 2 y0 1x z1 1x1u 1w Solving the following H2AREs using MATLAB we obtain 144 40 F 41 17 40 16 497778 233333 243333 233333 140000 160000 and an output feedback control law 223333 243333 v y 5 2 V 38 29 16 u 41 17 v The closedloop magnitude response from the disturbance to the controlled output 50 0 10392 100 102 10 Frequency radsec The optimal performance or in mum is given by 106 Prepared by Ben M Chen Solution to the Regular H Output Feedback Problem Given a y gt ywquot solve the following algebraic Riccati equation HmARE ATP PA CZTCZ PEETPy2 PB CZTD2D2TD271D2TC2 BTP 0 for a unique positive semide nite solution P 2 0 and the following ARE QAT AQ EET QCZTCZQ y2 QCJ EDITDIDJ 1DIET CIQ 0 for a unique positive semide nite solution Q 2 0 In fact these P and Q satisfy the socalled coupling condition pPQ lt 92 The H00 y suboptimal output feedback law is then given by f2 2 A v BC y u C C v where A A y zEETP BF 1 y ZQP 1KC1 y leETP BC 1 y2QP 1K CC F and where F DZTDZ 1D2TC2 BTP K QC EDITDIDIT 1 107 Prepared by Ben M Chen Example Consider a system characterized by xi 81818 2 y0 1x z1 1x1u 1w It can be showed that the best achievable H00 performance for this system is Solving the following HwAREs using MATLAB with y 97 we obtain 144353 401168 498205 233556 401168 160392 233556 140118 38814 184866 183658 v v y 25 59414 914112 894227 u 41116 47039 v The closedloop magnitude response from the disturbance to the controlled output 0 1039 100 102 10 Frequency radsec Clearly the worse case gain occurred at the low frequency is slightly less than 97 The design specification is achieved 108 Prepared by Ben M Chen

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