This paper extends some recent results on the parameterization of all admissible pairs in a class of 2-parameter current-cycle-feedback ILC algorithms. In addition, a necessary and sufficient condition is given under which the associated set of equivalent controllers coincides with the set of all stabilizing controller

Meinsma, G.

Verwoerd, M.H.A.

Vries, T.J.A. de

English

M.H.A. Verwoerd

G. Meinsma

T.J.A. de Vries

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On the parameterization of all admissible pairs in a class of CCF-ILC algorithms

de Vries, T.J.A.

University of Twente Research Information

Proceeding of the 2004 American Control Conference Boston, Massachusetts June 3 0 .  July 2,2004 FrM18.5 On the parameterization of all admissable pairs in a class of CCF-ILC algorithms M.H.A. Verwoerd, G. Meinsma, T.J.A. de Vnes Faculty of Electrical Engineering, Mathematics, and Computer Science, University of Twente, The Netherlands. {m.h.a.verwoerd,g.meinsma,t.j.a.devries}@utwente.nl Absfrac-This paper extends some recent results on the parameterization of all admissable pairs in a class of 2- parameter cnrrent-cycle-feedback ILC algorithms. In addi- tion, a necessary and sufficient condition is given under which the associated set of equivalent controllers coincides with the set of dl stabilizing controllers. I .  INTRODUCTION A N D  PROBLEM STATEMENT Given a plant P : U + Y ,  y = Pu, along with some desired output Y d ,  the objective in Iterative Learning Control is to construct a sequence of inputs {uo,ul,. . .} such that the corresponding sequence of outputs { y o ,  y1 , .  . .} converges to some limit value 1 := limb,, Yk that is close to Y d  in some sense. More specifically, the aim is to define an algorithmic procedure prescribing how future inputs may be constructed from recordings of past in- and outputs. The analysis in this section in concerned with one such family of procedures, namely a class of 2-parameter iterations uk+1 = QUk + L e k  + C e k + i  (1) Here ek := Y d  - y k  denotes the current tracking error. The parameters Q and L are constrained to be causal, bounded linear operators, that is Q,L E RH,. The feedback term C is assumed to be stabilizing. The class of iterations is parameterized by the free pa- rameters Q and L. Of all conceivable combinations (Q, L), those that will generate a converging sequence of inputs are of particular interest. Such pairs are called admissable. The set of all admissable pairs is denoted by A [4]. For every (Q,  L )  E A, one can define the associated equivalent controller [l], [Z], [3], [4], [5 ] ,  see Figure 1.  K = ( I - Q ) - ' ( L t C )  (2) Fig. I .  Equivalent Feedback Controller (dashed). This equivalent controller has the property that it is al- ways stabilizing. Now let K A  denote the set of all equivalent 0-7803-8335-4/04/$17.00 02004 AACC controllers K.4 := { K : K = ( I - Q ) - ' ( L + C ) ;  ( Q , L ) E d }  and let the set of all stabilizing controllers be denoted by K .  Clearly X.4 c IC. Recently it was shown [4] that in case C = 0 (in which case P is assumed to be stable) both sets coincide, but that this is not true for all C. In the next section it is shown that the above result is just a special case of a general result which says that I C A  = IC if and only if C E EH,. 11. WHEN I C A  AND IC COINCIDE Theorem I: Given the class of iterations (1) and the associated sets K.4 and IC. Then, for any stabilizing C, K.4 C IC with equality (=) iff C is stmngly stabilizing, i.e. iff C is stable (C E 'RN,) and stabilizing. Pmot (Sufficiency) Suppose C E RHH, and let K E K be any stabilizing controller. To prove that K E Ka, define Q = ( K - C ) ( I + P K ) - ' P  L = ( K - C ) ( I + P K ) - '  (3) The controller K being stabilizing, by definition all closed- loop transfer matrices are stable. Since C is also stable, so are Q and L. To prove that ( Q J )  E d note that (Q - LP) ( I  + CP)-' = 0, which is a sufficient condition for admissability [41. And with (Q,L)  E A, by definition K E x.4. m To prove necessity, a few more intermediate results are required. Lemma 2: Let K E K.4 be a controller induced by some admissable pair (Q,L)  E A. Then there exists an " e q u i v y  141 pair (QO,Lo) E A such that K = (I - Qo)-  (Lo + C )  and (Qo - LOP) ( I +  CP)-' = 0. There exists such (&Lo) E A iff if the following set of equations has a solution (Q, L )  E A. Proof: ( L + C )  = ( I - Q ) K  (4) (Q-LP) ( I+CP)- '  = 0 The unique solution to the above set of equations is given by Q o  = ( K - C ) P ( I + K P ) - '  Lo = ( K - c ) ( I + P K ) - '  (5) 51 56 From (4) we obtain K - C  = L + Q K  (6) with Q ,  L E RH, by assumption of admissability. After substituting (6) into ( 5 ) ,  inspection shows that Qo,Lo E RH,. Hence, by construction (Qo, LO) is admissable. This Lemma 2 says that among all admissable pairs defining the same equivalent controller, there is one and only one pair (Qo, LO) which satisfies the additional constraint. Let do denote the set of all such pairs, i.e. A0 = { ( Q , L ) E d :  ( Q - L P ) ( I + C P ) - ' = O }  concludes the proof. This set allows for an efficient parameterization A0 := {(Q, L )  = ( Z N ,  Z M ) ;  Z E RH,} (7) where P = M-'N is a left-coprime factorization over RH-. Through (7) one arrives at a parameterization of the set ICA. Lemma 3: Let C = V-'U and P = M-'N be any left-coprime factorization of the plant and the controller respectively. Then the set of all equivalent controllers ICA is parameterized by P p o t  (of Theorem 1, necessity) Suppose C RH,. Take Z = I and let K be the corresponding stabilizing con- troller (Lemma 4). By uniqueness of the Youla parameter it is clear that the corresponding controller K belongs to IC, iff there exists Z E RH, such that V Z  = I .  This however implies that V is bistable and C = V-'U stable, which contradicts the starting assumption. This concludes the proof. m 111. CONCLUSION The results presented in this paper build on and make use of a framework for analysis that was put forward in a series of recent papers by the same authors. An important conclusion to be drawn from this work is that the choice of the current cycle parameter most definately affects (constrains) the achievable performance. REFERENCES [ I ]  Peter B. Goldsmith The fallacy of causal iterative leaming control. In Pmc. Of rhc 40th Conference on Decision and Contmi, pages 4475- 4480, Orlando. Florida, USA. 2001. [2] Peter B. Goldsmith. On the equivalence of causal LTI iterative leaming control and feedback control. Automtica. 38(4):703-708, 2002. 131 M.H.A. Venvaerd, 0. Meinsma. and T.J.A. de V i e s .  On the use of noncausal LTI operaton in iterative leaming control. Pmceedings of thc 41sl Conference on Decision and Conlmi. pages 3362-3366,2002. [4] M.H.A. Venvoerd. G. Meinsma, and T.J.A. de Vies .  On equivalence classes in iterative leaming control. Pmceedings of the American Conrml Confcrcnce, Denver: Colomdo, USA, mges 3632-3637.2C03. [SI M.H.A. Vekwrd. G. Meinsma and T.J.A. de Vies. A class of non- contractive. Uial-dependent update mles for iterative leaming eonml. Submitted for presentation (11 the Americnn Conrml Conference. 2004. 161 Kemin Zhau. John C. Doyle. and Keith Glover. Robut and Optimol ' A  = { = (' - vzM)-' (' + V Z N )  ; ' ' RH-} Pro08 The equivalent controller is given by K = ( I - Q ) - ' ( L + C )  Conrml. F'rentice Hall, 1996. with (Q,  L )  E & this evaluates to K = ( I  - ZM)-'  (V-'U + Z N )  = (V - VZM)- '  (U + V Z N )  (8) This concludes the proof. m Although the above parameterization seems to depend on specific factorizations, in actual fact the choice of coprime factors is immaterial. This is immediate from the fact that left-coprime factors are unique up to a left multiplication with a bistable transfer function. The next lemma restates an important result on the parameterization of all stabilizing controllers. Lemma 4 (Youla-Kufera): Given C = V-'U and P = M-'N with U,V  and M , N  left-coprime. Assume C is stabilizing. Then the set IC of all stabilizing controllers is given by 16, Theorem 12.71 Inspection IC = { K = ( V - ~ M ) - ~ ( U + ~ N ) ; ~ E R H ,  shows that the respective parameterizations 1 ofI C A  (Lemma 3) and IC (Lemma 4) are equivalent if and only if for every 2 E RH, there exists 2 E RH, such that 2 = V Z .  The condition for equality is clearly satisfied in case C is strongly stabilizing (C E RH,) since then V is bistable and Z can be selected as Z = V- 'Z .  The proof of Theorem 1 is now straightforward. 5157 

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Goldsmith The fallacy of causal iterative leaming control.

On the equivalence of causal LTI iterative leaming control and feedback control.

On the use of noncausal LTI operaton in iterative leaming control.

RH-} Pro08 The equivalent controller is given by K = (I-Q)-'(L+C) Conrml. F'rentice Hall,

Submitted for presentation (11 the Americnn Conrml Conference.

On the parameterization of all admissible pairs in a class of CCF-ILC algorithms

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