Ilchmann, Achim: Nonidentifierbased highgain adaptive control


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1 lchmann, Achim: Nonidentifierbased highgain adaptive control Zuerst erschienen bei: London : Springer, 1993 SBN (Lecture notes in control and information sciences ; 189). Zugl.: Hamburg, Univ., FB Mathematik, Habil.Schr. : 1993
2 Lecture Notes in Control and nformation Sciences 189 Editors: M. Thoma and W. Wyner
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4 r Achim lchmann NondentifierBased HighGain Adaptive Control SpringerVerlag ondon Berlin Heidelberg New York Paris Tokyo Hong Kong Barcelona Budapest
5 Series Advisory Board L.D. Davisson' M.J. Grimble' H. Kwakernaak' A.G.J. MacFarlane J.L. Massey 'J. Stoer Y. Z. Tsypkin ' A.J. Viterbi Author Achim lchmann, PhD nstitut frr Angewandte Mathematik, Universität Hamburg, Bundesstrasse 55, Hamburg, Germany SBN 354G'19t458 SpringerVerlag Berlin Heidelberg New York SBN G SpringerVerlag New York Berlin Heidelberg Apart from any fair dealing for the purposes of research or private surdy, or criticism or review, as permined under the Copyright, Designs and Patents Act 19E8, this publication may only be rcproduced, storcd or transmitted, in any form or by any me8ns, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning rcproduction outside those terms strould be sent to the publistrers. G SpringerVerlag London Limited 1993 Printed in Great Britain The publistrermakes no rcprcscnbtion, expreer or implicd, with regard to the accuracy of the information contained in thie book and cannot accept any legal responsibility or liability for any emors or omissions that may be made. Tlpesetting: Camen ready by author hinted and bound by Antony Rowe Ltrl., Chippentram, Wiltstrirp 69/383G' Printed on acidfree paper
6 'r\ P rrerace '[']rc sr:rninal work of Morse, Nussbaurn and Willems & l3yrrres irr 1983/4 has initiated the study of adaptive controllers for dyramical systems in which the adaptation strategy does not invoke any identification mechanism. Over the last clecaclc, this field of adaptive control has become a rnajor research t,opic. ''he present work gives a rather complete'state of the arrt'of the following more specific area: The system classes under consideration contain linear (possibly nonlinearly perturbed), finite dimensional, continuous tirne systems which are stabilizable by highgain output feedback, therefore, in particular the system is rninimum phase. Simple aclaptive controllers involving a simple switching strategy in the feedback are designed. The switching strategy is mainly tuned by a one parameter controller based on output data alone. Control objectives consiclcrcd are stabilization, tracking, and servomechanisnr action. n addition, robustness with respect to nonlinear perturbations and perforrnance improvements are investigated. wrote the present text during a two years research visit to the Centre for Systerrrs :rnd Control Engineering at the University of Exeter, U.K., frorn October September The hospitality of the centre, with its stimulating environrnent, made a big contribution and thank especially Dave Owens and Stuart, 'lownley, of the centre, for numerous helpful discussions and suggestions. [ :rlso benefitted frorn many stimulating discussions rn'ith Gene Ryan of the Univcrsity of Bath and Hartmut Logemann of the University of Bremen. am indebted to leinrich Voß of the University of larnburg who read the mzrnuscript ancl rnade several critical and helpful comments. Finally, thanks are due to Dir:t,er Neuffer frorn the University of Stuttgart who spent considerable tirne an<l pat,iencc on introclucing me to SMULNK and MATLA. My visit was nr:rclr: possible by o two years research grzrnt from Deutsche Forschutrgsgt:rrte,ittschzrft (DFC;) ancl additional support carr]e frorn the University of Exeter zrnd thc iic SCFNC prograrrrne, whicir zrre hereby gratefrrlly acknowledged. arnl;urg, tebrrtary, 1993 Ar:hzrn ilr:htnann
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8 Contents Nornenclature ix rrtroduction HighGain Stabilizability 2.1 Minimurn phase systems 2.2 llighgain stabilizable systems Notes and References 28 Ahnost Strict Positive Realness 3.1 Almost strictly positive real systems 3.2 Minimum phase systems 3.3 Notes and References Jniversal Adaptive Stabilization Switching functions Stabilization via Nussbaumtype switching Stabilizatiorr via switching decision function FJxponential stabilization via exponential weighting factor Exponential stabilization via piecewise constant gain Notes and References 101 fjniversal Adaptive T\'acking 5.1 Asymptotic tracking 5.2 )'lracking 5.3 Notes ancllteferences l1'2 t.)9 L.)
9 vltt CONTENt'S 6 R.obustness L Additive nonlinear state and input perturbations 6.2 Sector bounded inputoutput nonlinearities 6.3 )Tracking controller r25 t34 r Notes and References Perforrnarlce mproved output and gain behaviour Arbitrary good transient and steadystate response Notes and References 183 Exponential Stability of the Terrninal Systern Rootloci of minimum phase systems 8.2 Topological aspects 8.3 Notes and References 186 r Referencers t97 ndex 203
10 NorrrenclatLlre l/ ltr+ (rr_ ) a+(a) R[.r]'' "" R(") "  G L,,(K) r].r (o),(a) /r"'i" (1)) /r,r* (Ä) lal l'l l" ll'll LrQ) {1,..., i/} the set of nonnegativc (nonposit,ive) real open right (left) half conrplex plarre the set of rn x rrl rnatrices ovcr the polynornials ring of real the set of rn x m matrices over the fielcl of rational real functions general linear group of invertible n x n rnatrices with entries in 1{ {* e l lllrll < Ä} for rn N, ) > 0 the spectrurn of the matrix,4 g (ln x't rnininral singular value of the rnatrix A C,"^" maxinral singular value of thc rnatrix A e C,"*'n the determinant of the matrix A e C"^" numbers positive \t< r,p'r > for r [t',p  pt' ]R'"' definite ll'llrtlre vector space of rnea,srrrable firnctions / :  U1.", 1C R an interval, z being defirrecl by the context,, srrctr that l/( ) lt4,1(,):), wht:re
11 NOMENC4TURE ll/( ) llr"1'; c, (1, R) wr,* (1, R) d.r( ), Dr(')  ( r rr/p l/ll/(")ll"a"l { Lr es sup llf (") ll lr s for p [l,oo) f"r p  oo the vector space of ptimes continuously differentiable functions/ : * R*,p NU{oo} the Sobolev space of functions / :, * R which are absolutely continuous on compact intervals and /('), /(') g.["" (R) ' 'distance' functions defined in (5.12) respectively (518)
12 Chapter 1 ntroduction A wide range of control theory deals with the design of a feedback controller for a. knoutn plant so tirat certain control objectives are achieved. The fundarnental clifference between this approach and that of adaptiue controlis that in adaptive corrtrol the plant is ttot knoun exactly, only structural information is available, e.g. nrinirnality, rninimum phase, or known relative degree. The aim is therefore to clesign a single controller which achieves prespecified control objectives for every rnember of a given class. The controller has to learn from the output data and, b:rsed on this information, to adjust its parameters. The first, attempts in adaptive control go back to the late 1950's, but it was only in the 1970's that a breakthrough was made. Subsequently, during the 1980's the field of adaptive control has matured. For a survey see Aströln (1987) and Narendra (1991). Up to the end of the 1970's, adaptive controllers were a combination of identification or estimation mechanisms of the plant paranreters together with a feedback controller. An area of nonidentzfierbased adaptive control was initiated by Mareels (1984), Märtensson (1985), Morse (1983), Nussbaum (1983), and Willems and Byrnes (1984). n their approach, the adaptive feedback strategy is not based on any identification or estirnation of the process to be controlled. This seminal work opened up an intensively studied specialised field within adaptive control, where the class of systems under consideration are either minirnum phase or, more generally, only stabilizable and detectable. See lchmann (1991) for a survey. n the present text, nonidentifierbased adaptive controllers for minimum phase systems are studied, thus all controllers are designed according to the highgazn properties of the system class. No assumptions are made on the upper bound of the order of the process, nor on the upper bound of the sign of the highfrecluency gain, no injection of probing signals is required, and the control strat,egy is rnore efficient than for nonminimum phase systems. 'he objective is to provide a single controller (consisting of a feedback law ancl a pararrreter zrdaptation l:rw) which can control each system belonging to a certain cla.ss of
13 2 systems. The control objectives are stabilization, tracking or servornechanism action, partly under performance requirements and in the presence of nonlinear perturbations. We illustrate the idea by the simplest example we can think of:''he system to be st,abilized belongs to the class of scalar systems describecl by b(t) y(t) ar(t) + öu(l) cx(t), z(0)  xs) (1 t) where e,b,c,ro R are unknown and the only structural assumption is cö > 0, i.e. the sign of the highfrequency gain is known to be positive. lf we apply the feedback law u(t)  ky(t) to (1.1), then the closedloop system has the form i(t) la kcblr(t),r(0)  x:s (12) Clearly, if aflcbl < lßl and sfgn(ß) : sign(cb), then (1 2) isexponentiallyst,able. {owever, a,b,c are not known and thus the problem is to find adaptively an appropriate k so that the motion of the feedback system tends to zero. Now a timeuarying feedback is built into the feedback law u(t)  k(t)y(t), (i.3) where,t(t) has to be adjusted so that it gets large enough to ensure stability but also remains bounded. This can be achieved by the adaptation rule Ä1ry  y(t)2, e(0) R (1 4) The nonlinear closedloop system (1.1), (1.3),(1.4), i.e. J'(t)  la  k(t)cbfu(t), k(t)  "' l: "(")'ds * e(0), (*(0),'(0)) R2 (1 5) has at least a solution on a small interval [0,r), and the nontrivial solution r(t) _ "flt"ft( s )crjas r19; is rnonotonically increasing as long as c  k(t)cb ) 0. lence k(l) 2 l(cr(o))' + ft(o) increases as well. Therefore, there exists at* ) 0 such that o k(t. )cä  g and (1.5) yields o  ft(f)cö < 0 for all t > t*. lence the solution r(t) decays exponentially and liml*"" e(r)  ßoo R exists. This is a special example for the following concept of universal adaptive control. Suppose E denotes a certain class of linear, finite dimensional, tinrcinvariant systems of the form i(t) vu) Ar(t) + Bu(t), z(0)  C r(t) + Du(t) ' g \ t (1 6)
14 3 wlrere: (tl, B,C,D) [R'xn x R'',x'2 x R,'"" x R'r'x') are unknown, nt is usually fixecl, t,he state dirnension n is an arbitrary and unknown nurnber. 'fhe aim is to design a single adaptive output feedback mechanisrn,rr(l)  f (y( )lto,rl) which is a universal stabilizer for the class. i.e. if "(l)  f (y( )llo'l) is applied to any systenr (1.6) belonging to, then the output 37(l) of the closedloop system tends to zero as tends to infinity and the intcrtral variables are bounded. n the present t,ext, most of the adaptive stabilizers are of the following sinrple forrn (cf. F igtrre 1.1): A 'tuning' parameter A(l), generated by an aclaptation law k(t)  g(y(l)),,t(0),lo, (r 7 t whcrc g :R'' * R is continuous and locally Lipschitz, is irnplementecl into thc fc,'rlback law via u(t)  F(,L(t))y(t), whert: l/ :R r trr"* is piecewise continuous ancl locally Lipschitz. (t 8) r t Adaptive Controller Adaptation Law Process i:arbu a:cr*du L Figure 1.1: Universal adaptive stabilizer Definitiorr A controller, consisting of the aclaptation law (1 7) ancl the feedback rule (1.8), is c.af lr:rl a ttniue.rsul adaptiue stabilizerfor the class of systems X, if for arbitrary init,iäl condition ro l" and any system (1.6) belonging to, tiie closc.rlloop systenr (1 6)(1.8) has a solution the properties (i) t,lrere cxists zr unique solution (r( ),Ä'()):[U, x)  R'.'+1, (ii) e,( ), y(.), rr( ), t'(.) are boundccl,
15 (iii) lirnl*oo y(t)  0, (i") liml*oo k(r)  ßoo R exists. The concept of adaptive tracking is similar. Suppose a class./rer of reference signals is given. t is desired that the error between the output y(t) of (1.6) and the reference signal y."r(l) "(t) : y(t)  y..r(t) is forced, via a simple adaptive feedback mechanism, either to zero or towards a ball around zero of arbitrary small prespecified radius Ä > 0. The latter is called )tracking. To achieve asymptotic tracking, an internal model itr) u(t)  A e (t) * B* u(t), (0) : 40 : c e (t) * D* u(t) ( 1.e) wlrere (A*, B*, C*, D") e [t"'" n' y Tf7.n'Xtnx R"'' x Rt, i, implemented in series interconnection with an universal The precompensator resp. internal model reference signals. An internal model is not adaptive stabilizer, cf. Figure 1.2. (1.9) contains the dynamics of the necessary if Atracking is desired. Adaptive Tracking Controller nternal Model Process 9."f : A' * B'u u:c'4* D' i:ar*bu gcr*du Figure 1.2: Universal Adaptive Tlacking with nternal Model Definition A controller, consisting of an adaptation lau,t(1.7), a feedback lau (1 8), and an internal model (1 9) is called a uniuersal adaptiue tracking controller for the class of systems! and reference signals.)y'..r, if for every U."r(.).)."r. ts R', o R"', and every system (1.6) belonging to X, the closedloop systern (1 6)(1.9) satisfies
16 5 (i) t,here exists a unique solution (r( ), ( ),f(.))'[0,m) * R72+n'+i (ii) the variables r(/),y(t), r(r), (l) 'blow up' no faster than 9.er(l), (iii) limt*o,[y(t)  9..r(l)]  0, (ir') liml*"o fr(l)  k.'. R exists. F'or prespecifiecl Ä > 0, (1 7), (1,8) is called a un,zaersal adapt.iue \stabzlizer resp. ),tracking controllerif (i),(ii), and (iv) hold true, but instead of (iii) the weaker condition [v(t)  e.er(l)]  E^(o) as l*oo is satisfied, i.e. the error e(l) approacires the closed as tcnds to oo. f zr universal adaptive controller is applied to ( 1.6) then we call the systern L the termznol systern, provided it is well defined. ball of radius ) around zero for some (ro, Ao) R." x R, '(,t,))tcl r(r) J Many results presented in this text fit into the framework described above. We will also consider linear systerns subjected to nonlinear perturbations in the state, input and output, corrupted input and output, noise, and nonclillerentiable gain adaptation. Due to the nonlinearities, the solution of the closedloop systern of t,he norninal system and ttre aclaptive feedback rnechanisrn is no longer unique, but all solution will meet the desired control objectives. Some feedback strategies contain nondifferential gain adaptation, other have multiparameter gain adaptation, i.e. g(.) i" (1 7) is a mapping from Rto R' for some n{ > 1. Many results can be extended, by no mearrs trivially, to classes of infinitedimensional systems. This will not be treated here, but the zrva.ilable literature is discussed in the sections'notes and References'ending each chapter. ''tre outline of the chapters is as follows. n Chapter 2 ancl 3 we do not deal with any adaptation rnechanisrns, inst,ead the systern classes under consi<ieration are analysecl ancl results to be used later are prr:parcd. n ('Lrapter 2, we study irr detail the piropi:rtics of nrult,ivzrri:rblc rnininlunr plrzrse sysl,errrs, give corrvenit:nt, slat,e spac.e forrns, prove the so called highgain lcmrnat,a for relative clegree ancl 2 systr:rns, and clr:rive irnl>ortant, incclualitit:s relaling past inputs ;rncl out,1>uts with t]re present, output. T'lris k:ads t,o a ral, her cornplete understanding of highgain stzrbilizable systerns and toc,ls which
17 will be used throughout the remaining chapters. n Chapter 3, the class of strictly positive real ancl alrnost strictly positivc real systems is investigated. Although this class is more restrictive than minimunr phase systems it turns out that, by simplc input and output transformations, every minimum phase system is equivalent to an almost strictly positive real system. The result will become important for the stability proofs of adaptive stabilizers given in Chapter 4. Recently published results on multivariable strictly positive real systems, in particular its relationsphip to the Lur'e equations, are used to understand the effect of highgain control, and simplify the proofs for adaptive stabilizers. Moreover, it allows consideration not only of strictly proper, but of proper linear systems. n the first section of Chapter 4, the concept of switching functions is studied. Different switching functions are introduced and it is shown how they relate to the Nussbaum conditions. These results are needed for the remaining sections of the chapter. n Section 4.2, various universal adaptive stabilizers based on Nussbaumtype switching are derived for systems (A, B,C,D) under milcl assumptions on the highfrequency gain matrix. A universal adaptive stabilizer for the class of all singleinput, singleoutput, positive or negative highgain stabilizable systems is given. Throughout the section the feedback strategy is continuous respectively piecewise continuous, i.e. f'(.) in (1.8) is continuous respectively piecewise constant. n Section 4.3, an alternative switching strategy based on a switching decision function is introduced. Apart from robustness properties shown in Chapter 6, the advantage of this different approach is tliat there is no need to implement a scalinginvariant Nussbaum function whicli is behaving very rapidly. n Sections 4.4 and 4.5, we deal with the problem of how to obtain exponential decay of the state r(l) of the closedloop system. n Section 4.4, this is achieved by introducing an exponential weighting factor in the gain adaptation, whereas in Section 4.5 a different approach uses piecewise constant gain implementation in the feedback strategy. n Chapter 5, the problem of how to track signals belonging to a class of reference signals is investigated. One solution to the problem of asymptotic tracking is presented in Section 5.1. We use series interconnection between arr internal model, representing the dynamics of the reference signals, and universal adaptive stabilizers studied in Chapter 4. This approach is restricted to sinusoid reference signals, whereas in Section 5.2, at the expense of Atracking, a controller using deadzones is introduced which does not invoke an internal model and works for a much larger class of reference signals. Well posedness and robustness properties of the universal adaptive stabilizers of Chapter 4 and the asymptotic and )tracking controllers of Chapter 5 are investigated in Chapter 6. n Section 6.1, it is proved that the problern of universal adaptive stabilization is well posed with respecr to nonlinearities in the state equation of the nominal system. Robustness with respect to other nonlinear perturbations is proved. n section 6.2, rve show that rnanluniversal adaptive stabilizers of singleinput, singleoutput systems tolerate sector bouncled inputoutput nonlinearities, for multiinput, multioutput almost strictly positive real systems ever multivariable sector bouncled inputoutput nonlinea
18 rities are allowed. n Section 6.3, we prove t,hat the Ästabilization respectively )tracking controller is capable of toierating a nruch larger class of nonlinear perturbations in the input ancl state as well as sect,or bounded inputoutput nonlinearities, even input and output corruptecl noise is allou'ecl. 'fhe purpose of Section 7.1. is to illustrate the qualitative dynarnical behaviour of nrany universal adaptivc stabilizers and to introcluce modifications of the previous universal adaptive stabilizers and )tracking controllers which lead to an improvement of the transient behaviour. n Section 7.2, a universal adaptive stabilizer is designed whic.h, at the expense of derivative feedback, achieves (prespecified) arbitrarily srnall overshoot, of the outprut, and, rrloreover, guarantees that the output is less thern an zrrbitrarily small, prespecified constant in an arbitrarily small, prespecified period of time. 'he results on rootloci of singleinput, singleoutjrut nrininrum phase systems rlerived in Section 8.1 are used in Section 8.2 to show that that the piecewise constant stabilizer introduced in Section 4.5'almost always'(w.r.t. tlie sequence of thresholds) yields an exponentially stable terminal system. Each chapter is finalized with a section on the literature and related problenrs, in particular, extendecl results for infinitedinrensional systems are quoted.
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20 Chapter 2 High Gain Stab ilizability n this chapter, we derive several properties of highgain stabilizable and/or minimum phase systems which are essential for getting a deeper insight into these system classes, and wtrich will be used throughout the remaining chapters. 2.L Minirnurn phase systems We shall show that all nontrivial systems which are stabilizable by highgain output feedback are necessarily minimum phase. Therefore, it is important to study the class of minimum phase systems. We will also give simple and convenient state space forms for relative degree or 2 systems and prove a crucial integral inequality relating past input and output data to the present output. Definition Let G(.) e m(s;rrlxln be a rational matrix with SmilhMcMillan form r.. (s) diag {( "r(s)'',ä3,0',0)  U(r)tC(")tz(")' (2.r) where U(.),V(.) e R["]x are unimodular, r,bp1")g(.) : r, e;(.),,lrro R[s] are monic and coprime and satisfy et1.)le;+r(), rlrr+t(.)lrlrr(.) for i ,...,r. Set + + e (s): ll e;(s), ü(") ' ll ü;(') i=l.s6 is a (transntzssion) zero of G(.), if e(so) : 0, and a pole of G(), if y'(.so) : 0. f G()  g( ) tr[s], then d"gy'(.)  dege( ) is called Lhe relatzue degree of s () i
21 10 CHAP']:T:R 2. HGHGÄ.N STABLZABL'Y {;() is ptoper resp. strtctly propet' if deg r/,( ) > dege(.) resp. d"g,/,(.) > dege(.). 'he system i(t) = Ar(t)*Ru(t), v(t)  cr(t)+du(t) with (A,B.,C,D) R"' x m,'"^ x fli. n'*' x l.*'', i" called a minimal realization of G() R(s)*, if (A, B) is controllable and (,4, C) is observable ancl G(s)  C(s^  A)rB + D. G() is said to be minimum phase, if e(s) 0 for all.s e O1. A state space system (A,B,C,D) R'x'r x TR'z'^ x R*"' x R',* is called minimum phase, if it is stabilizable and detectable and G(") ha^s no zeros in c+. A characterization of the minimum phase condition for state space system is given in the following proposition. Proposition 2..2 (A, B, C, D) R'x' det xil RN i' K TfT all s O1 if, a.d only if, the (f following,win. three f l conditio.s lol ns are af( :e satisfied (i) rklslna,bl :n. n for all s 6i e ( C1, i.e.(a,b) isstabilizablebystate feedback,.["t_ Al  (tt) rk : t for all s C+, i.e. (,4,C) is detectable, L C J (iii) G(") has no zeros in O=,.. r, l"i L s,. C a) A rh xr ffl. B D X J T. J R 0 ftx' fi7 fo)r satisfies Proof: We trse the notation of Definition Coppel (1974), 'fheorenr 10, has proved that, if (Ä, B,C) is detectable ancl stabilizable, then so O+ is a zero of r/,( ) (including multiplicity) if and only if it is a zero of det(.1,  A). Using this result together with Schur's forrnula, see e.g. Gantmacher (195g), the proposition follows from s[,,  ' A B ', _ t^t Al rr /1/^r,rr1n, ls1r. Ä D
22 2.1. M{TMUN{ PHASE S}'S?EMS 1l ''he following lenrrna provides a useful state space fortn into which every systenr u (l) v(t) Ar(t) * Bu(t), C r(t), r(0)ro R' (2 2) witlr (,4,8,C) R'"'x.lRrlx x R"'with det(cb) f 0 can be converted. 'f5r state space transformation is representing the direct sum of the range of B alcl t,he kernel of C. t makes possible the separation of the inputs and orrtputs fronr the rest of the system states. Leurrna 2L.3 (lorrsicler the system (2.2) with det(c13) a basis rnatrix of ker C. t follows that g, lc', N')t, where l/ : (V'V) state space transformation (y',,')'  (2.2) into 0 and let,/ JR'x('.l clenote.9 : lb(cb)t, Y] has t,he inverse t v' lt.  B(CB)1cr]. lerrce the S 1c  ((C *)t., (Nr)t)' converts v(t) t (t) Ary(t) * Azz(t) + CBU(t) Asy(t) + A4z(t) ( v(0) '(0) J n1 L) & t). (23) lere At R*^,A, Rx(nrn),At e 6(nzn)*^,,An R("'n)x(n"n), so that Av A2 A3 Aa  Sr AS. f (,.4,8,C) is minimum phase, then,4a in (2 3) is asymptotically stable. Proof: The proof of the transformation is straightforward ancl therefore onrittecl. Stability of Aa is a consequence of det(c B) + 0, the minirnum phase assumption, and of the following equation which holds for all s e C1 slna C B 0 slrn  A1 As rn 42 snrn  Aq 0 CB lc B l.l"1, rn  Aol * 0.! ior strictly proper, scalar, nrinimurn pliase systerrrs of relative dcgree 2, a slightly rnore complicated brrt still very useful stat,e space clescription is also avail:rblr:.
23 2 CHAP'ET 2. HGH.GÄN STAB,ZABLTY d('y,).\ 1 o1/ f: nlü(t)  1", #f a[ llüio l+l,e \ z(t)l Lo, 'öo ;")\",t>) L;"1 where dz R, 04, as R'', Au R('" 2)x(n2). lf (2.2) is minimum phase, then a(aa) C C. u(t,) (2.4) Proof: Choose V R'x('2) of full rank so thar k",l:")  V trt"2 t is easily verified, that the inverse of t si' : lcabv. V* : (V'V)tV', and hence e : cs1 and eäb LoAyields A t (SrSz)taSrSz Moreover, ö52  e, S;tb 6, and cazb wrth S:.9rS2. Using (2.4), we have,91 :: : c o ::  eä2b  [Ab, b,v](cab)1 ca _ n#" [r"  (ou [1,0,...,0], ä : 5, 1ö: J.:r., oj  cab. Applying elementrary row and o o si,,[t 1 t L: ;,:_,] L Lernrna 2.L.4 f the system (2.2) is singleinput, singleoutput, (A, B,C)  (A., b, c), and of relative degree 2,i.e. cb  0, cab f 0, then there exists a coordinate transforinto mation S e GL,(R), such that Srr : (y,ü,t')' converts (2.2) 4.. columm operation r o o * 1 * l 0 0 n_z J 1 0 a3 "[ bca(cab) s1.4s,  0 A e a3cab, l and is givcnby 'll S tt t, * 1 * + * + * * * 1 of the fornt hence (2 4) holds )1,, A b c 0 ) )os a[ 0 Ä,.*z  Ae cab 0 0 cab.l)/,z ,4ol
24 2. j. MNNUM PASE SYSTEMS 1.) Lr) ff (2.2) is nrinimum Phase, then arrd hetrce Ae is asymptotically it follorvs that länz  Aal + 0 for all ) stable. This completes the proof. O+, D Anot her itrrportant consequenceof strictly proper minimurn phase systerns rvitir linr,*.',, s(i(s) Gf'.(R) (for singleinput, singleoutput this simply means they are of relative degree 1) is, that a sirnple inputoutput description of the svsteln is Possible. Lernrna 2L5 f t,he syst,em (2.2) is mininrum phase r,r'ith det(cf)) f 0, then there exists a bounded and causal operator L : Lo(O, oo) * Lr(0, m), for all P [], rc], so that the inputoutput behaviour of (2.2) is described by y(t)  A1y(r) + L(a()) (t) * CBu(t) + u(t), y(0)  Cro (2 5) wittr,4r R", and (')'[0,)* R all exponentially decaying analytic function taking into account the initial condition of a part of the internal state. Figure 2.1: nputoutput Description Proof: Without lossof generality we nray in the form (2 3). Definethe causal opcrator assume that ttre systern (2.2) is L(a( ))tt)  Az.l o t ea,(t.') A*lG)dr.
25 T4 CAPTER 2. {GGÄ{ STABLZAB.TY Since Aq is asymptotically stable, there exist Mr, ) 0, so that ll"o"'ll 1 M1e't for all t > 0. This yields, see e.g. Vidyasagar (1978) pp , that lll(y)( )llr p(o,t) L 'fherefore, f : Lo(0, oo) * Lr(0, oo) is well defined and bounded. By applying Variations of Constants to the second equatiorr in (2.3), and inserting z(l) into the first equation, we obtain l " ü0 = As(t)*Azl"o^'z(0) + *CBu(t). J"on(t')ery1";a"l L U J Setting u(t)  AzeAn'r(0), the proof of the lemma is complete. n For strictly proper minimum phase systems (A, B, C) with det(c B) 0, it is possible to relate the present output to past output and input data via the following inequality, where no information of the state variables is required. This will be an another important tool for the stability proofs of universal adaptive stabilizers in Chapter 4. Lerrrrna 2.L.6 Suppose the system (2.2) is rninimurn ptraseand satisfies del(cb) t' 0. Let P() : R^ * R, v,* lfu)  (., ) ffic if a+0 l. o if v:0. Then for every positivedefinite matrix P  Pr R"'' (depending only on A, B,C and P) such tlrat there exists ' > A r ^ ' r i = tlvt,)llä 2""' "' ' J" J, or, more general, for arbitrary p > 1, it holds that, i i _llrtt)llp"< tvlllrsllp+,lr / ttots)llpds+ ltrt')llot'(p(v(")),p('btt(s))d P t r t, (2 T) for arbitrary initial condition r(0) R', for arbitrary piecervise continuous u(') : * [0,c.,') R'', where cr (0,.c], a1d for all [0,.)
26 2.1. MNTMUM PHASE SYSTEMS 15 Proof: (.), We first consider the case p:2 (2 5) yields, for all s [0,c.,'), r d,,,,.rr1. t,ß(llv(")llf')  (v("), PAty(s) + l'u,(s) + PL(y) (") + PCBu(s)) ö o for + (v("), PCBU(s)) (2 8) Mz:llPAtll + llpll Applying ölder's inequality and using (26) gives r / r \ r l r /', \'/', \' J ttrt")il ill(yt(s)llds s l,lv(")il'd"l l,tctü(s)il'. \l / \i ),rs 0 t llv(")ll'd' (2 e) and, since tr(s)  A2eAn'z(0), S e t. rl r llly(s)ll lltr(s)llds o ( (, t r \ 6 ; \ ' tlrt'lll'a" 6 / M3llz(0)ll llv(")ll2ds + u3ll,(o)ll' (2 10) for Ms: sup r>0 (t,azea",'r") 2 0 ntegration of (2.8) over [0,1] and [0,c,,,), inserting (2.9) and (2.10) yields, for all )no...lt', 1 2 llv(o)ll? +M.,  llz )t z + t + MrMSll:10)ll2 + (v("), P C Bu(s))ds.l 0.l 0 t lly(") ll'd". ) 7),i ''his proves the first inequalit,lrt itv. (b), 'he map ** lly(r)l lp isnot differentiable but, since V(.) is differentiable, it is absolutely continuous. Tl T] fherefore, the set,11 : {t' = f0, [0, ). t^t ) llv(t )llp is nol cliflr:renl,iable]
27 16 CAPTER 2. GH.GÄN SA'ABLZABLTY is of measure zero. LeL,, Jz : "(') Now a routine calculation gives, for all s Thus it follows frorn : ; *(ttv(")ttä) where An application,() Lp(0,t), {t e [0, c,.,) fttrt"ltto = f (v(") { llv (2.5) that, for all s llv(")llä' ( ri is not continuous at t ) R+ \ (/r g ir),. Pü(s)) / \ ffi,v(')*o 0,y(s)  0. R+ \ (/t g Jz), ), P A1y(s) M4: llpll"+ illp1'll + llplll of Hölder's inequality gives, for q  FPu(s) + PL(y) (") + PCBU(s)l [llv(")l lo + lll(v) (")llllv(")llo' +l ly(") lloi' PCBU(s)), f,,o li J ttrtrllr' llu(s)llas L'. J L"t J f, +lly(")llo' ll(") lll 1 r 1  ' q ' p 1 and every.f',0 11) Since Jru Jz is of measure zero, integration of *llv(")llpo over [0,r] \ (./r u yields, by using (2.11), for all t [0,r,r), r  r  i s + ;ltatt)llä ittvtolllä lla{")llp'(p(y(s)), PCBU(s))ds Jz) t f *Ms 0 llv(")llo + llv(")llo' illc(v)(")ll + ll(")lll d" jttrroltt'" t +.l 0 lly(") llr' U3(y(")), PC Bu(s))ds +,v/s llv( )ill," (o,r) + llv(')tl?lä,'t llc(v) ( )llro(0,r) +llv( )ll?lä,,rllollr"1o,,.l],
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