翻译一ggA pseudoinverse-based iterative learning control.pdf

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002 831A Pseudoinverse-Based Iterative Learning ControlJayati Ghosh and Brad PadenAbstractLearning control is a very effective approach for tracking con-trol in processes occuring repetitively over a fixed interval of time. In thisnote, an iterative learning control (ILC) algorithm is proposed to accom-modate a general class of nonlinear, nonminimum-phase plants with distur-bances and initialization errors. The algorithm requires the computation ofan approximate inverse of the linearized plant rather than the exact inverse.An advantage of this approach is that the output of the plant need not bedifferentiated. A bound on the asymptotic trajectory error is exhibited viaa concise proof and is shown to grow continuously with a bound on the dis-turbances. The structure of the controller is such that the low frequencycomponents of the trajectory converge faster than the high frequency com-ponents.Index TermsIterative learning control (ILC), nonlinear tracking,pseudo inverse.I. INTRODUCTIONIterative learning control (ILC) refers to a class of self-tuning con-trollers where the system performance of a specified task is graduallyimproved or perfected based on the previous performances of identicaltasks. The most common applications of learning control are in the areaof robot control in production industries, where a robot is required toperform a single task, say pick-and-place an object along a given trajec-tory, repetitively. With a feedback controller alone, the same trackingerror would persist in every repeated trial. In contrast, a learning con-troller can use the information from the previous execution to improvethe tracking performance in the next execution. While in some appli-cations, the need to repeat a trajectory multiple times for learning maybe a disadvantage, we focus our attention on those many others wherelearning control is a natural solution.In this note, we propose a modification of the iterative learning con-trol algorithm presented in 1 so that it can be applied to a more genericclass of nonlinear nonminimum phase plants with input disturbanceand output sensor noise. In Section II, a learning controller is proposedby formulating a pseudo-inverse of the linearized plant at the origin. InSection III, simulation examples are presented to show the performanceof the proposed learning controller. Finally, Section IV concludes thenote.II. NONLINEAR NONMINIMUM PHASE PLANT WITH DISTURBANCESIn this section, we present a robust iterative learning algorithm fornonlinear systems. We consider only square (same number of inputsand outputs) time-invariant nonlinear systems.A. System DescriptionConsider a nonlinear system which is stable-in-first-approximationat 120 6148(i.e., the linearized plant has all the eigenvalues in the openManuscript received September 28, 1999; revised February 27, 2001 and Oc-tober 26, 2001. Recommended by Associate Editor S. Hara. This work was sup-ported by the National Science Foundation under Grant CMS-9800294.J. Ghosh was with the University of California, Santa Barbara, CA 93106USA. She is now with Agilent Technologies, Inc., Bio-Research Solutions Unit,Santa Clara, CA 95051 USA (e-mail: jayati_).B. Paden is with the Department of Mechanical and Environmental Engi-neering, University of California, Santa Barbara, CA 93106 USA (e-mail:).Publisher Item Identifier S 0018-9286(02)04754-2.left half complex plane) and also input-to-state stable95120105401164161102 40120105401164141 43 103 401201054011641411171054011641439840120105401164141119105401164159120105404841 61 4812110540116416110440120105401164141 43 1181054011641404941where 105 is the index of iteration of ILC, 102117105103491056148is a family of inputsequence, 1201054011641 50110, 1171054011641 and 1211054011641 50109and 102 5811033110,103 58110331102109, 104 5811033109, 98 58110331102112, 11910550112, 11810550109. The function 1191054011641 represents both repetitive and random boundeddisturbances of the system; it may be stiction, nonreproducible friction,state-independent modeling errors, etc., 1181054011641 represents sensor noise.A desired trajectory 1211004011641 is supported on finite interval (116 50 9148598493)of the time axis. The objective of learning is to construct a sequenceof input trajectories 102117105103491056149such that 11710533 1173and 11734011641 causes thesystem to track a trajectory 1211004011641 “as closely as possible” on 0, 84Wemake the following assumptions:(6549) The functions 10240141, 10340141, 10440141 are continuously differentiableand 9840141 is continuous.(6550) 1174850 76499267489266114, where 66114is a closed subset of a Banachspace.(6551) The system is stable-in-first-approximation and input-to-statestable.(Note: If the system is not stable, it may be stabilized prior to appli-cation of our methods).(6552) The disturbances 11910540141 and 11810540141 are bounded by 98119and 98118,respectively (i.e., 10711910540116411072098119and 10711810540116411072098118).(6553) The desired trajectory 121100lies sufficiently close to a trajectory94121100which satisfies the following equations:95941201004011641611024094120100401164141 43 103409412010040116414194117100401164194120100404841 61 4894121100401164161104409412010040116414159 56116 50 914859849358 (2)For such a system, an ILC is proposed as shown in Fig. 1.B. Formulation of Learning ControllerIn this section, a good candidate for the learning controller 7667 ofFig. 1 is derived by first linearizing the plant and then a “pseudoinverse”of the linearized plant is used as the learning controller.The update law of the ILC is written in terms of the operators 80, thelinearized plant 688010648, its adjoint 6880106348and 84 for 116 50 9148598493 is117105434940116416184 4011710543 14117105416184 11710543401173 43 6880106348688010648410496880106348pseudoinverse59688010640121100080 4011710541416184 11710543401173 43 688010634868801064841049688010634840121100012110541 58 (3)Note that if 1174861 84 401174841,84 4011710540116414314117105401164141 61 11710540116414384 4014117105401164141 forall 105. (Note that in Fig. 1, the truncation operator 84 is placed before thesumming junction).Defining 688010648: Since the nonlinear system (1) is input-to-state-stable (6553) and 104 is continuous (6549), it defines a causal nonlinearinput-to-output map 80 as: 80 58 1171055533 12110559 76499148594941 33 7649914859494158Since 80 is stable-in-first-approximation (6553), we define a stabletime-invariant input-to-output linear operator 688010648by linearizing thesystem (1) around (1201056148, 1171056148, 1191056148, 1181056148) as follows:951412040116416165141204011641436614117401164161 48141214011641616714120401164158 (4)0018-9286/02$17.00 2002 IEEE832 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002Fig. 1. Nonlinear learning control system 80: nonlinear plant, 7667. Learningcontroller 84 : truncation operator.where 65 102120404841, 66 103404841, 67 104120404841. Hence, 68801064858 14117 55331412159 76499148594941 33 7649914859494158 Since 14117 50 76499148598493 and 65 is Hurwitzin (4) we can replace61 48 with 1412040649416148and not alterthe inputoutput (IO) map defined by (4), and hence, the only mapprovided is 49 0 49.Defining 6880106348: Consider the IO map for the adjoint system149522120 6106584142212006784142211759 1422120406494161481422121 616684142212058 (5)Since 65 is Hurwitz, 06584is hyperbolic (i.e., none of the eigen-values have zero real part). Further, (5) defines a unique noncausalmapping as shown by Devasia et al. 2 (see the Appendix ).688010634858 1422117 5533 142212159 764933 764958 The adjoint system satisfies theproperty 10468801064811759118105 61 104117596880106348118105 3.Defining 6880106117: Neglecting higher order terms we obtain a lin-earized plant around the solution 1201054011641 to (1)95141261201054011641611021204012010541141261201054011641431031204012010541141261201054011641117105401164143 103 401201054114117105401164143984012010541119105401164114126120105404841 6148141261211054011641611041204012010541141261201054011641431181054011641 (6)where 10212040120105401164141 6410261641204012010540116414159 10312040120105401164141 64103616412040120105401164141.Since (4) is stable, it can be proved by Lyapunov methods that(6) is also bounded-inputbounded-output (BIBO) stable if 120105lies within a certain bound. Note that, here also we can re-place 14126120105404841 61 48 (as in (4) with 1412612010540649416148and not alterthe IO map. Define 651054011641 10212040120105401164141 43 103120401201054011641411171054011641,661054011641 10340120105401164141, 671054011641 10412040120105401164141, 981054011641 9840120105401164141. Thestable linear system (6) has a solution and defines a linear IO map:688010611758 141171055533 1412110559 764933 7649.Defining 6880106121591148: Motivated by the concept of a pseudo-inverse 4we define learning controller by the following linear operator:6880106121591148401173 43 6880106348688010648410496880106348(7)for 11 54614858 We call this “approximate inverse” the 11-pseudo inverseof 688010648. For simplicity the 11 pseudoinverse is referred to as simplya pseudo-inverse in the rest of this note. In time-domain using (4) and(5) 401173 43 688010634868801064841: 14117 33 14126121 is14 951201495221206165 480678467 065841412014221204366481411714120406494114221204064941614814126121 611114117 43 1422121 61 1114117 43 6684142212058 (8)Since 688010648is stable, (8) is hyperbolic with eigenvalues21406541 91 21400658441. Hence, by 2 401173 43 6880106348688010648415814117 55331412612159 764933 7649and is noncausal. Solving (8) for 14117, we see that theinverted operator 401173 43 688010634868801064841049is14 951201495221206165 0661104966840678467 0658465141201422120436611049481412612114120406494114221204064941614814117 611104914126121 066841422120 58 (9)The eigenvalues of the above system are continuous functions of 11.Inthe limit 11 3349, 6511is hyperbolic (since 65 is Hurwitz). Thus, wecan always choose an 11 for which 6511is hyperbolic. The system (9) issolved by the stable-noncausal-solution approach of Devasia et al. 2.Hence, 401173 43 68801063486880106484104958 14126121 5533 1411759 764933 764958The learning controller is the pseudoinverse401173 43 6880106348688010648410496880106348and is given in time-domain by14 9512014952212014 951226165 066110496684661104966840678467 06584484848065846514120142212014122120434848067841412159884064941614814117 61 48 01104966841104966841412014221201412258 (10)6599is block diagonal, therefore the eigenvalues of 6599are eigenvalues of6511(9) and06584. Since 6511is hyperbolic for some 11, 6599is hyperbolic.Hence, 401173 43 688010634868801064841049688010634858 14121 5533 14117, 764933 7649andthe solution of the linear controller described by (10) can be obtainedusing stable-noncausal-solution approach 2. (Using initial conditionsat 116 61 049 rather than 116 6148allows us to control 88404841 via 1412140049594893.Thus tracking performance can be improved relative to assumptions of14121 17 48 on 40049594893 and 88404841 61 48).C. Convergence AnalysisDefinition 1: We define the 21 norm for a function 120 589148598493 33107by1071204014110721115117112116509148598493101021116107120401164110758 (11)Note that 10712010721201071201074920 101218410712010721for 216248, implying that10712010721and 10712010749are equivalent norms. Thus convergence results canbe proved using either norm.Induced 21-norm: 107651072161 115117112107117107 614959107117107 5461481076511710721.Define the Fourier transform of 688010648by 70406880106484161586880106484010241.Condition 1: 1066880106484010241106 62126248 56102 (i.e., no finite or infinitezeros on 10633 axis). Observe that 76499148598493 26 7650.Theorem 1: If the assumptions (65496553) and Condition 1 are satis-fied, then the algorithm (3) produces a sequence of inputs which con-verges to 1173, if there are no disturbances (i.e., 1191056148and 1181056148) andno initialization errors (120105404841 61 120100404841 56105). If 119105and 118105are boundedand the initial state error is bounded (107120100404841 0 120105404841107 609812048), 117105converges to 664011735911441,as105 3349. The radius 114 of the ball 664011735911441depends continuously on the bounds on the disturbances 119105and 118105andthe initialization error. If there exists a 11710050 764992 67489148598493 with80401171004161121100, then 117105converges to the desired input solution 117100.IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002 833Proof: The proof relies on the application of a variant of the con-traction mapping theorem 5 to the input sequence. The main ideaof the proof is to show that 107221411710543491072120 2610722141171051072143 98100where 48 202660494022141171051171000 11710541. This implies that 108105109105334911511711210722141171051072133496140490264198100, where 98100is a continuous function of the bounds on dis-turbances and initialization error. Construct the sequence 10211710540141103491056148by defining:117486184 40117484159 117105434961 8440120 4014159119 40141419111710540141935984 11710543401173 43 688010634868801064841049688010634840121100080 401171054141 588440120 4014159119 4014141911171054014193 is denoted by 841054011710541 for simplicity in the rest ofthis note. Now, from (3) and the linearity of the truncation operator,(12), shown at the bottom of the page, holds. Following 6, we showthat the Frechet derivative of 80 is given by 6880106117. That is, 6880106117satisfies10810510910714117 1073348107804011710543 1411710541080 401171054106880106117911411710593107107141171051076148 (13)In (13), let 115401411710541 be defined by: 115401411710541 804011710543 1411710541 0 8040117105410688010611791141171059358 From (13), we can see 115 is 11140141171054158 Denoting 1411710591118105011710593, we can rewrite (12) as107841054011710541 0 84105401181054110761 84 0141171050 401173 43 688010634868801064841049688010634880 40117105411 43401173 43 68801063486880106484104916880106348401154014117105414380 40117105414368801061179114117105934161 84 01411710543401173 43 688010634868801064841049168801063484011540141171054143688010611791141171059341 58Since 115401411710541 is 111401411710541,10810510910714117 107334840107401173 43 6880106348688010648410496880106348107107115401411710541107416110714117105107 614858This implies that 5615624859 57 146248 such that 10714117105107201441107401173 43 68801063486880106484104968801063481071071154014117105411071071411710510720 15604958 (14)Boundingnd 14120: From assumptions 6549, 6551, 6552:10765105107201071021204012010541107 43 1071031204012010541107107117105107209865, 10766105107 61 10710340120105411072098103,10767105107 61 10710412040120105411072098104120, 10798105107209898. Therefore, 1076510720986559 10766107209810359 10767107209810412058 From (6), we can write1412612010540116416114126120105404841431164865105402841141261201054028414366105402841141171054028410981054028411191054028411002858 (15)Therefore, using the triangle-inequality and bounds on 98105and 119105,wehave107141261201051072010714126120105404841107431164810765105402841107107141261201054028411074310766105402841107107141171054028411074398989811910028. Using GronwallBellman inequality (see 5, p. 63) 107141261201051072010198 11610714126120105404841107431164810198 4011602841409810310714117105402841107439898981194110028. Multiplying (15)by 101021116, defining 754910997120409865599810341 and assuming 21627549we have10102111610714126120105107201014098 021411161071412612010540484110743 7549116481014075 02141401160284110102128107141171054028411071002843 98989811911648101021281014098 0214140116028411002858Note that for a constant 1071071072161 107. Taking 115117112 over 116 50 9148598493 we have1071412612010510721201071412612010540484110743754921075494901014075 021418410714117105402841107214398989811921098654901014098 021418458 (16)Similarly from (4), it can be proved1071412010721201071412040484110743754921075494901014075 02141841071411710540284110721(17)where 14117105is the input to (4).Defining 16880106117: Define a linear operator 1688010611768801061170688010648, so that1688010611758 141171055533 112110559 764933 764958 (18)From (6), the output of the operator 6880106117is: 1412612110540116416167105401164114126120105401164101181054011641 and from (4) the output of the operator 688010648is 1412140116416167141204011641.This implies, 1121105401164161141261211054011641 0 14121401164161671054011641141261201054011641 0 1181054011641 067141204011641. Therefore, using (16), (17), and the bound on 118105we can write1071121105107212010767105107211071412612010510721431076710721107141201072143 981182098104120107141261201051072143 107141201072143 98118205098104120754921075494901014075 02141841071411710510721439810412098989811921098654901014098 021418443 98104120107141261201054048411072143 107141204048411072143 9811858Showing Contraction Mapping: From (12), we have the equa-tion shown at the bottom of the next page. Define 107401173 4368801063486880106484104968801063481072161 7149. It can be shown in the fol-lowing way that if 688010648satisfies Condition 1, then 10711401173 436880106348688010648410499114117105931072120 15111071411710510721, where 48 60151160 4958 With thechoice of 11 sufficiently small, 1511can be made arbitrarily small.Let 22121 61 68801064822117,2289 4010241586170221214011641 and22854010241586170221174011641.107688010648221171075049049221214011641221213401164110011661490492289 4010241228934010241100102614904991688010648401024122854010241939168801064840102412285401024193310010258(70: Fourier Transform)1078410540117105410841054011810541107 61 84 11710543401173 43 688010634868801064841049688010634840121100080 401171054141084 11810543401173 43 6880106348688010648410491688010634840121100080 40118105414161 84 1171050401173 43 6880106348688010