The purpose of these notes is to state a series of simply stated questions in adaptive control

French, Mark

Megretski, A.

English

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Open problems in robust adaptive control

Southampton (e-Prints Soton)

Open problems in adaptive controlMark French¤ , Alexandre Megretski yMTNS 20041 IntroductionThe purpose of these notes is to state a series of simply stated questions in adaptive control.2 BackgroundWe consider causal mappings P : Ue ! Ye and C: Ye ! Ue, where P and C represent a plantand a controller, respectively, and U and Y are normed vector spaces such as L2(R+;Rm) andUe, Ye are the analogous extended spaces, for example L2;e(R+;Rm). Our central concern iswith the system of equations:[P;C] :y1 = Pu1; y0 = y1 + y2u2 = Cy2; u0 = u1 + u2;¾(2.1)where u0;u1;u2 2 U, y0;y1;y2 2 Y and which correspond to the classical feedback con¯gurationof a plant and controller as depicted in Figure 1. We will state our problems in terms of theu0u1 y1PC y0u2 y2¡++¡Figure 1: The closed-loop.operator:¦P==C : W ! W : w0 =µu0y0¶7!µu1y1¶= w1:We are interested in the following fundamental quantity: given a disturbance level d ¸ 0, anominal plant P and a controller C,BP;C(d) = supfr ¸ 0 j ±(P;P1) · r =) k¦P1==Cw0k < 1 for all kw0k · dg: (2.2)Note that BP;C is unde¯ned if ¦P==C is not BIBO stable.¤Department of Electronics and Computer Science, University of Southampton, SO17 1BJ, UK,mcf@ecs.soton.ac.uky...13 Global Limitations of Performance and Robustness1. Answers to the following questions are of interest in any sensible signal space setting, e.g.U;Y = L2(R+) or L1(R+). Suppose P(µ): dom(P) ! Y is de¯ned by:P(µ)(u1) = y1 where _ y1 = µy1 + u1; y1(0) = 0: (3.3)Then we know there exists a causal controller C and aµ ¸ 0, °µ 2 K 1 such that:k¦P(µ)==Cw0k · aµ + °µ(kw0k) 8µ 2 R: (3.4)(a) What is the minimal achievable rate of growth of °µ? Concretely, ¯nd:p = inffq ¸ 0 : 9 causal C s.t. °µ(r) = O(rq); 8µ 2 Rg: (3.5)(b) Is the above in¯mum attained?(c) Is there a control design which achieves a robustness margin independent of thedisturbance level? That is, ¯nd or prove the non-existence of, a controller C and aconstant bP(µ);C > 0 such that:BP(µ);C(d) = bP(µ);C > 0 8µ 2 R; 8d ¸ 0: (3.6)Observe that if there exists a causal C s.t. °µ(r) = O(r) 8µ 2 R then (c) follows, i.e. p = 1in (a) and (b) implies (c), or p < 1 in (a) implies (c).2. We ask the same questions in a discrete time setting, e.g. with U;Y = l2(Z+) or l1(Z+),andP(µ)(u1) = y1 where y1(k + 1) = µy1(k) + u1(k); y1(0) = 0: (3.7)3. As an alternative generalisation of the concept of (e.g. L2) gain stability, we ask the samequestions (a),(b) when (3.4) is replaced by (3.8):kT¿¦P(µ)==Cw0k · aµ + kT¿°µ ± (jw0(¢)j)k 8¿ > 0; 8µ 2 R; (3.8)and where T¿ denotes the truncation operator:T¿(v) :=½v(t); t 2 [0;¿]0; t 2 (¿;1):In the particular case when U = Y = L2(R+), this corresponds to the existence of 0 ¸M > ¡1 such that the following inequalityZ ¿0(°µ(jw0(t)j))2 ¡¡¦P(µ)==Cw0(t)¢2 dt ¸ M > ¡1 (3.9)holds for all ¿ > 0 (in the case U = Y = L1(R+), (3.4) and (3.8) are equivalent).4. A weaker version of questions (a),(b) is to allow C to be dependent of µmax, and to replace(3.4) with the requirement that the parameterised controller set fC(µmax)gµmax¸0 has theproperty that there exists aµ > 0, °µ 2 K such that for all µmax ¸ 0,k¦P(µ)==C(µmax)w0k · aµ + °µ(kw0k) 8jµj · µmax: (3.10)A similar replacement can also be made for (3.8). Note that it is critical that aµ, °µ areindependent of µmax. The corresponding weak version of question (c) becomes to ¯nd orto prove the non-existence of, a controller C and a constant bP(µ) > 0 such that:BP(µ);C(d) = bP(µ);C(µmax) > bP(µ) > 0 8µ 2 R; 8d ¸ 0: (3.11)1K denote the class of functions °: R+ ! R+ with the properties: °(0) = 0, ° is monotonically increasing.2We now make some remarks on the classes of uncertain nominal plants considered. The twoplant classes considered are deliberately simple; nevertheless the questions asked are not trivial.From a control perspective, the continuous-time class (3.3) is high gain stabilizable, hence anyrationale for adaptive control has to that of (suitably formulated) superior performance. Thediscrete plant (3.7) represents a simple class which captures many of the di±culties of traditionaladaptive control, including non-minimum phase behaviour; here the existence of stabilizingadaptive controllers already beats linear controllers.A more general problem is to ascertain which of the above properties are achievable in the moregeneral setting of appropriate classes of (LTI) plants P(µ) suitably parameterised by µ 2 Rn.4 Adaptive Control as H-In¯nity OptimizationLet P = fPµgµ2£ be a family of ¯nite order LTI plants (causal, discrete time), parameterized byparameter µ ranging over set £. Each Pµ is assumed to have two vector inputs (\disturbance"w = wµ and \control" u = uµ) and two vector outputs (\cost" z = zµ and \sensor" y = yµ). Thedimension of yµ is assumed to be independent of µ, and the same assumption is made about thedimension of uµ. In addition, assume that two functions °+ : £ 7! (0;1) and °¡ : £ 7! (0;1)are given.A general question of interest can be formulated as follows: for which families fPµ and functions°§ does there exist an e±cient algorithm for either ¯nding a single (in general, nonlinear) strictlycausal feedback law u(¢) = K(y(¢)) which makes the closed loop gain from wµ to zµ less than°+(µ) for all µ 2 £, or certifying that no single strictly causal feedback law u(¢) = K(y(¢)) iscapable of making the closed loop gain from wµ to zµ less than °¡(µ) for all µ 2 £.In the case when £ contains a single element, we get the standard suboptimal H-In¯nity op-timization problem, which has an elegant and e±cient solution. In the case when £ is ¯nite,one can inquire about existence of a polynomial time algorithm which solves this problem when°+(µ) = ½°¡(µ) for all µ, where ½ > 1 is a given constant.3

Open problems in robust adaptive control

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