INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING Int. J. Adapt. Control Signal Process. 2007; 21:391–414 Published online 30 October 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/acs.928
Finite-model adaptive control using an LS-like algorithmz Hongbin Ma1,*,y 1
Temasek Laboratories, National University of Singapore, 117508 Singapore
SUMMARY Adaptive control problem of a class of discrete-time nonlinear uncertain systems, of which the internal uncertainty can be characterized by a finite set of functions, is formulated and studied by using an least squares (LS)-like algorithm to design the feedback control law. For the finite-model adaptive control problem, this algorithm is proposed as an extension of counterpart of traditional LS algorithm. Stability in sense of pth mean for the closed-loop system is proved under a so-called linear growth assumption, which is shown to be necessary in general by a counter-example constructed in this paper. The main results have been also applied to parametric cases, which demonstrate how to bridge the non-parametric case and parametric case. Copyright # 2006 John Wiley & Sons, Ltd. Received 12 August 2005; Revised 5 April 2006; Accepted 2 May 2006 KEY WORDS:
finite-model adaptive control; LS; feedback; nonlinear; stability; uncertainty
1. INTRODUCTION 1.1. Motivation In this paper, we study finite-model adaptive control problem, which will be explained later. This problem is motivated by the research initiated by Lei Guo on the capability and limitation of feedback mechanism (see [1–10]), which appeared in the last decade. To make the ideas clear, we just introduce two typical existing results. Guo [1] investigated the following first-order discrete-time nonlinear system: ytþ1 ¼ yybt þ ut þ wtþ1 ;
b>0
ð1Þ
where y is the unknown parameter, the parameter b characterizes the nonlinearity of the system, and fwt g is the Gaussian noise. It has been found and proved that for system (1) there is a *Correspondence to: Hongbin Ma, Temasek Laboratories, National University of Singapore, 117508, Singapore. y E-mail:
[email protected] z This work was done in Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences; Graduate School of Chinese Academy of Sciences. Contract/grant sponsor: National Natural Science Foundation of China; contract/grant number: 60504037
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critical stability phenomenon: the above system is not a.s. globally stabilizable if and only if b54: Here, we can regard that the internal uncertainty of system (1) is described by the following set of functions: 4 Fb ¼ ff : R ! Rjf ðxÞ ¼ yxb ; y 2 Rg Impossibility part of this result is extended to more general nonlinear systems (see [5, 11]) and similar results for corresponding system with bounded noise can be found in [4]. In [7], a non-parametric discrete-time nonlinear system ytþ1 ¼ f ðyt Þ þ ut þ wtþ1
ð2Þ
is studied, where the unknown function f belongs to the following set: 4
FðL; cÞ ¼ ff : R ! Rjjf ðxÞ f ðx0 Þj4Ljx x0 j þ c; 8x; x0 2 Rg It has been shown that system (2) is not globally stabilizable if and only if L54: Impossibility part of this result has been also extended to more general nonlinear systems (see [3, 10]). Obviously, systems (1) and (2) are so simple that they are in fact fully actuated in the absence of internal uncertainty; however, the results listed above show that the presence of internal uncertainty makes stabilization of them much difficult and sometimes even impossible in case that the size of uncertainty set (Fb or FðL; cÞ) is large enough. So the internal complexity of a plant, which in mathematics can be abstracted by a set F of functions, is usually the primary source of difficulties in designing control laws; and it is possible that any controller cannot stabilize the uncertain system if the ‘size’ of F; which characterizes the complexity of internal uncertainty, is so large that it ‘exceeds’ the capability of the whole feedback mechanism. We can also notice that in all the previous research listed above, without exception the considered uncertainty set F contains infinite many functions. Thus, naturally an interesting question arises: how about the capability or limitation of feedback mechanism when the uncertainty set F just contains (essentially) finite many functions? This motivates the following finite-model adaptive control problem: Problem Consider a general uncertain discrete-time plant ytþ1 ¼ Hðyt ; ut ; wtþ1 Þ
ð3Þ
where the unknown function H belongs to a given set H which has just finite number of members or has ‘essentially’ finite number of members, fwt g is the noise sequence and fut g is the sequence of control signals. Here by ‘essentially’, we mean that we can choose a set H0 with finite (known) members H0 ¼ fH1 ; H2 ; . . . ; HM g such that every member H of set H lies in the neighbourhood of one certain model HK in some sense, where the index K need not to be unique or known a priori. The purpose of designing control laws is to make the closed-loop system stable, or in other words, the control law is used to deal with both internal (structure) uncertainty and external (disturbance) uncertainty. We can use the information provided from the history up to time t to design control signal ut ; that is to say, ut can be a causal function of fyt ; yt1 ; . . . ; y0 ; ut1 ; . . . ; u0 g: ut ¼ ht ðyt ; yt1 ; . . . ; y0 ; ut1 ; . . . ; u0 Þ Copyright # 2006 John Wiley & Sons, Ltd.
ð4Þ
Int. J. Adapt. Control Signal Process. 2007; 21:391–414 DOI: 10.1002/acs
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We hope to know: can we always find a feedback control law which guarantees the stability of closed-loop system? Or what conditions can be imposed on H to guarantee existence of stabilizing controller for system (3)? Though the proposed finite-model adaptive control problem is motivated by research on capability or limitation of feedback mechanism, we can see many instances of such kind of problems in practice. For example, in many plants we do not know the exact internal structure (or settings), however, by some physical or experimental knowledge we know that the internal structure (or settings) has just finite possibilities (they can be viewed as known ‘models’). So the study of finite-model adaptive control problem may help us not only understand the capability or limitation of feedback mechanism much deeper, but also provide insights on practical or potential applications.
1.2. Other related researches Two existing areas, robust control and conventional adaptive control, are relating to the mentioned finite-model adaptive control problem. In fact, if we have known K first, we can then try to solve the mentioned problem in framework of robust control regarding HK as a nominal model. However, since K is usually unknown, in general a fixed controller corresponding to a certain model Hi cannot stabilize all possible cases of the plant, and consequently some techniques other than robust control are required in our problem except those cases that K can be identified first completely. In conventional adaptive control, usually parametric uncertain systems are studied, which means that all the cases of the plant have common or similar structure and the only differences are the values of parameters. In our problem, if the members of H can be parameterized by some parameters, the problem may be possibly solved in the framework of adaptive control; however, we remark that great differences among members of H are allowed here, which will bring many difficulties in theoretical analysis, so conventional adaptive control techniques may be inapplicable for our problem. Thus, our problem has features of both robust control and adaptive control, and can be regarded as a special robustadaptive control problem with property of finiteness available. To make the aim of this paper more clear, we list several related existing researches here. Many other papers related to them can be found in the vast literature, and we do not list them all to save space. For linear systems with unknown parameters, there has been great amount of literature. For example, in [12], a linear system with white noise and unknown parameter vector taken from a finite set is studied, and the LQG problem for this system is solved by a so-called cost-biased approach, which consequently gives proof of stability of closed-loop system as a by-product. Some nonlinear systems with special parametric structure are also studied in the literature (see, e.g. [13–15]). For example, [14] studies the adaptive control of continuous-time systems with convex or concave parametrization, however, there are some limitations in this approach: only deterministic (or disturbance-free) systems are studied; the nonlinear parameterized part is assumed to be bounded; and some strong conditions like ‘convex’ or ‘concave’ are required. Later in [16] a so-called min–max estimator for a class of discrete-time systems with nonlinear parameterizations is studied and some conditions are found to guarantee parameter convergence, where boundedness of nonlinear part, compactness of parameter domain and bounded noise are required. Copyright # 2006 John Wiley & Sons, Ltd.
Int. J. Adapt. Control Signal Process. 2007; 21:391–414 DOI: 10.1002/acs
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Problems similar to our problem are also discussed in the literature, especially in the recently emerged area of switching supervisory control. For example, in [17], a supervisory switching logic approach is proposed to make a nonlinear parameterized system practical input-to-state stable, and perfect mathematical results can be obtained provided that the noise is bounded and a bank of robust input-to-state stabilizing controllers with corresponding associated ISS-Lyapunov functions are available. By using the concepts of input-to-state stability, their approach can cover very wide class of nonlinear systems, though the so-called ISS-Lyapunov functions and other related functions should be constructed first to apply this approach in practice. However, their idea is based on ‘controller falsification’ (see, e.g. [18, 19]), which requires that the noise must be bounded so that the designed switchings can stop after finite steps. Thus, by this approach, a fixed controller will be adopted eventually and adaptability will be lost after finite steps, which is a very good property in analysis but restricts this approach also in cases of unbounded noise or time-varying plants (see [18] for more discussion). To deal with time-varying plants, other techniques such as periodic resetting (see [17] for some discussion) must be employed also when this approach is applied, and corresponding theoretical analysis remains to be open now as far as we know. 1.3. About this paper This paper and companion papers [18, 20] together try to give some partial answers to the finitemodel adaptive control problem by different approaches, ideas and methods of theoretical analysis. The approaches in this paper and [20] can also be reinspected in the following general framework of switching supervisory control. Assume that there exists a stabilizing controller Ki for each model Hi ; uðiÞ t ¼ Ki ðyt Þ i.e. the closed-loop system for model Hi ytþ1 ¼ Hi ðyt ; Ki ðyt Þ; wtþ1 Þ is stable, we hope to construct an adaptive controller for plant H ut ¼ utðit Þ ¼ Kit ðyt Þ where fit g is a switching sequence to be designed, such that the closed-loop system for plant H ytþ1 ¼ Hðyt ; Kit ðyt Þ; wtþ1 Þ
ð5Þ
is stable, or in other words, the following general switched system ytþ1 ¼ Fit ðyt ; wtþ1 Þ;
4
Fj ðx; wÞ ¼ Hðx; Kj ðxÞ; wÞ
is stable for a specially designed sequence fit g: Then the problems left are how to design the sequence fit g and how to impose conditions on fF1 ; F2 ; . . . ; FM g; which are consequently imposed on sets H and H0 implicitly, to guarantee the stability of the closed-loop system. This paper uses an least squares (LS)-like algorithm to design the sequence fit g; while a similar WLS-like algorithm is used in [20]; see also [18] for other approaches different from switching supervisory control. In our approaches, the noise sequence need not be bounded in general and the total number of adaptive switchings is not necessary to be finite, which enables it possible to deal with time-varying plants without modifying the controller. To this end, a general concept of stability in sense of pth mean will be employed to study the closed-loop Copyright # 2006 John Wiley & Sons, Ltd.
Int. J. Adapt. Control Signal Process. 2007; 21:391–414 DOI: 10.1002/acs
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system. And we hope our approaches can also be directly applied without constructing ISSLyapunov functions, which will be one advantage of our approaches; however, usually more difficulties will be encountered in the mathematical analysis, which may limit us in studying special class of discrete-time systems. One significant example is that the convergence and stability of Astro¨m-Wittenmark (minimum variance) self-tuning regulator (AW-STR), which was proposed for linear control systems and based on the idea of least squares, was ever a long standing open problem (see [21–24]) until about a decade ago. We study the LS-like algorithm here mainly because the LS algorithm is a kind of widely used algorithm in system identification and adaptive control and it often has good performance in practice. In fact, the idea used in this algorithm is similar to the one used in AW-STR except that no explicit closed formula like recursive Riccati equation in AW-STR can be derived because no unified special structure information is assumed for the plants and models. The only benefit we gain in our problem is the finite number of models, which plays important role to help us make theoretical analysis. In this paper, we first assume that at time t we can use the information fyt ; yt1 ; . . . ; y0 ; ut1 ; ut2 ; . . . ; u0 g to compute Fk ðyj ; wjþ1 Þ for any integer j5t and k ¼ 1; 2; . . . ; M; which will be used to design the LS-like algorithm later. For example, we consider the following system (cf. [17]): ytþ1 ¼ f ðyt ; wtþ1 Þ þ gðyt ; ut Þ where f is an unknown function and g is a known function, which makes Fk ðyj ; wjþ1 Þ (j5t) available at time t since we can use yjþ1 ; yj and uj to obtain that Fk ðyj ; wjþ1 Þ ¼ f ðyj ; wjþ1 Þ þ gðyj ; Kk ðyj ÞÞ ¼ yjþ1 gðyj ; uj Þ þ gðyj ; Kk ðyj ÞÞ For ease of presentation, we only consider cases of additive noise, that is to say, we assume that Fk ðx; wÞ has the following decomposition: Fk ðx; wÞ ¼ Fk ðxÞ þ w which will not make confusion by number of parameters. For non-decomposable cases, some more conditions on Fk ðx; wÞ should be imposed based on the same idea in this paper, so we do not discuss these cases to make the idea more simple and clear. In this paper we mainly consider scalar cases ðyt 2 RÞ; see Remark 2.3 for some discussion on vector cases ðyt 2 Rn Þ: A very simple (yet non-trivial) system is ytþ1 ¼ f ðyt Þ þ ut þ wtþ1
ð6Þ
where f ðÞ 2 F is an unknown function characterizing the internal uncertainty of system; however, we have M known models ytþ1 ¼ fi ðyt Þ þ ut þ wtþ1 uðiÞ t
ð7Þ
4
and naturally we can take ¼ Ki ðyt Þ ¼ fi ðyt Þ; and consequently Fj ðx; wÞ ¼ f ðxÞ fj ðxÞ þ w: For system (6), let F0 ¼ ff1 ; f2 ; . . . ; fM g; obviously, the set F0 represents the known models. This paper is organized as follows: In Section 2, an LS-like algorithm will be given first, then the stability of this algorithm will be given. Later we try to bridge non-parametric cases and parametric cases and apply our main results into parametric nonlinear systems in Section 3. Then in Section 4 a counter-example is constructed to show the necessity of linear growth condition in our results, and several simulation examples are given to demonstrate applicability of the proposed algorithm. Further, in Section 5 we present the proof of main theorem. Finally, some concluding remarks are given in Section 6. Copyright # 2006 John Wiley & Sons, Ltd.
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2. MAIN RESULTS 2.1. An LS-like algorithm The algorithm listed below can be used to deal with both bounded and unbounded noise and it can overcome some shortcomings of algorithms based on the idea of controller falsification (see, e.g. [17–19]). We try to study the following LS-like algorithm: Algorithm 2.1 Take control law as follows: ut ¼ uðit t Þ where
( it ¼
ð8Þ
arg mink St ðFk Þ if jyt j5d otherwise
it1
For 14k4M; recursively define St ðFk Þ as ðS0 ðFk Þ ¼ 0Þ St ðFk Þ ¼ St1 ðFk Þ þ jFk ðyt1 ; wt Þjp Here d50 and p51 can be arbitrarily taken. Remark 2.1 From the definition of St ðFk Þ; we have St ðFk Þ ¼
t X
jFk ðyj1 ; wj Þjp
ð9Þ
j¼1
Especially when d ¼ 0; we have it ¼ arg mink St ðFk Þ: When p ¼ 2; d ¼ 0; this algorithm becomes LS algorithm. Parameter d is introduced to tune the performance and the speed of the system: taking proper d can improve the efficiency of this algorithm; however, when d is taken too large, though the algorithm will need less computation, the control performance of the system may become poor. Parameter p can be used to tune the robustness of the system: smaller p may bring better robustness, for example, the estimate in case of p ¼ 1 is often called one kind of ‘robust estimate’ in statistics. Remark 2.2 Tracking problem can be solved a sequence of deterministic signals fynt g; Pt similarly. To track n p we need only take St ðFk Þ ¼ j¼1 jFk ðyj1 ; wj Þ yj j ; and the controllers uðiÞ t ði ¼ 1; 2; . . . ; MÞ should be constructed to track yntþ1 for corresponding models. Results for tracking problem can be given easily based on main results in this section, thus we do not discuss it any more. Remark 2.3 Algorithm 2.1 only considers cases where yt are scalars. Naturally, we may ask: is this algorithm still applicable to high dimensional cases? In fact, when yt are vectors, we can define Fk ðyj1 ; wj Þ similarly and use some norm of vectors instead of j j: However, the main result of this paper and its proof should be revised correspondingly and the choice of the norm and verification of assumptions will become a bit more difficult. In principle, the ideas and results are similar, hence Copyright # 2006 John Wiley & Sons, Ltd.
Int. J. Adapt. Control Signal Process. 2007; 21:391–414 DOI: 10.1002/acs
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we only give a simulation example later to illustrate the use of this algorithm in these cases. Similar discussion can be applied to plant model with a (linear or nonlinear) regression on the control input values since corresponding state-space model can be obtained by some transformation. Remark 2.4 Simulations show that the LS-like algorithm can deal with slow time-varying systems. However, we should remark that this algorithm is not suitable for long-term fast time-varying systems since all the history data are used by the same weight but we know that recent data should be more relevant in this case. So naturally similar (forgetting factor) WLS-like algorithm can be introduced, which is studied in companion paper [20]. Other techniques such as time window, restarting, etc. can also be introduced to enhance sensitivity of the LS-like algorithm. 2.2. Preliminary concepts Before we state the main results, we introduce several concepts of stability. Definition 2.1 System yt ¼ Fðyt1 ; wt Þ is said to be bounded input bounded output stable (BIBO stable) if the output sequence fyt g is bounded provided that the noise sequence fwt g is bounded. Definition 2.2 System yt ¼ Fðyt1 ; wt Þ is said to be stable in sense of pth mean if the output sequence fyt g of system is bounded in sense of pth mean, i.e. lim sup T!1
T 1 X jyt jp 51 T t¼1
ð10Þ
provided that the noise sequence fwt g satisfies lim sup T!1
T 1 X jwt jp 51 T t¼1
ð11Þ
Throughout this paper, we always assume that (A0) all functions (Fi ; Ki ; etc.) involved are locally bounded. This basic assumption is not restrictive, and we will not mention this assumption later. 2.3. Stability of Algorithm 2.1 For convenience, we introduce the following notations: rðhÞ * ¼ lim sup05jxj!1 jhðxÞj=jxj; rðg; * g0 Þ ¼ rðg * g0 Þ: Obviously, rðg; * g0 Þ can be viewed as a distance measure between gðxÞ and g0 ðxÞ: Then we need the following assumptions to study Algorithm 2.1 for system (3). AL. For any 14i4M; rðF * i Þ51: That is to say, 4
jFi ðxÞj4Ajxj þ A0
8x 2 R
for some constants A50 and A0 50: (Note: here we have no restrictions for the values of A and A0 :) Copyright # 2006 John Wiley & Sons, Ltd.
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AK. There exists a model HK 2 H0 such that rðF * K Þ5e; where e 2 ð0; 1Þ is a constant small 4 enough such that ae ¼ MCep 51; here C is a constant depending on p; H0 and H: AW. The noise sequence is bounded in sense of pth mean, i.e. lim sup T!1
T 1 X jwt jp 51 T t¼1
ð12Þ
Theorem 2.1 For system (3), suppose that assumptions AL, AK and AW hold. Then, under Algorithm 2.1, the closed-loop system (5) is stable in sense of pth mean. Remark 2.5 Assumption AL requires that functions Fi ðxÞ (i=1, 2,. . .,M) have linear growth rate. Assumption AK guarantees at least one model HK can be used to approximate the real plant H well and FK ðxÞ can be viewed as ‘approximation error’ or ‘unmodelled dynamics’. Assumption AK is relatively easy to verify for special system (6), for example, in case of f 2 F0 ; there exists a K such that f ¼ fK ; consequently FK ðxÞ ¼ 0: Remark 2.6 For bounded noise or i.i.d random noise sequence with finite absolute moment of any order (such as Gaussian noise), we only require that e51; then we can always take p > 1 large enough such that ae 51; so the condition ae 51 in Assumption AK can be removed in these special cases. Remark 2.7 For bounded noise, in general, we expect the output sequence fyt g is bounded; however, it is not true for the LS-like algorithm. A counter-example in Section 4 shows that the output sequence fyt g can even be unbounded under bounded noise if Assumption AL is not satisfied. However, under some weak assumptions, we can prove that BIBO stability holds for the WLS-like algorithm in [20] and some algorithms in [18]. See also the companion papers [18, 20] for details. Remark 2.8 We have mentioned some difficulties in applying Algorithm 2.1 in Section 1. For analysis of traditional LS algorithm in system identification and adaptive control, many efforts have been done and considerable progress has been made in the past decades (see, e.g. [25, 26] and references therein). For stochastic linear systems, by making full use of special structural characteristics of recursive LS algorithm, basic properties of LS estimator are obtained by Guo and Chen (see, e.g. [23, 26, 27]). However, for Algorithm 2.1, basic properties of LS algorithm are not applicable due to lack of formulae like Riccati equation, so we must explore new approaches to study the LS-like algorithm proposed before. We can see that from the proof in Section 5, the characteristic of finite models and the assumption of linear growth rate will play important role in the stability analysis. A counter-example in Section 4 shows that in general the linear growth assumption cannot be removed, which is also required for traditional LS algorithm (see also [5]). Copyright # 2006 John Wiley & Sons, Ltd.
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3. APPLICATIONS OF MAIN RESULTS In this section, we try to extend some results for non-parametric cases to parametric cases. To demonstrate the idea more clearly, we only consider special system (6) in this section. 3.1. Proper open cover To bridge non-parametric cases and parametric cases, we need study a functional space ðL; rÞ * first. Denote by L the set of functions with linear growth rate, and then we can define topology in L according to ‘distance’ rð * ; Þ: For any function g 2 L; we can define an open ball with radius r as follows: Bðg; rÞ ¼ fg0 2 L : rðg; * g0 Þ5rg 4
and consequently we can define a pseudo-metric space ðL; rÞ: * Now assume that the real system f 2 F; where F is a compact subset of L: Then there exist finite open balls fBðfi ; ri Þg such that Bðf1 ; r1 Þ; Bðf2 ; r2 Þ; . . . ; BðfM ; rM Þ cover the set F; i.e. M [ Bðfi ; ri Þ F i¼1
For such an open cover, let e ¼ maxðr1 ; r2 ; . . . ; rM Þ
ð13Þ
if the condition ‘ae 51’ of Theorem 2.1 holds for some constant p51; fBðfi ; ri Þ; i ¼ 1; 2; . . . ; Mg is said to be a proper open cover of F: By Theorem 2.1, the following theorem can be obtained. Theorem 3.1 Assume that the real system f 2 F; where F is a compact subset of L: If a proper open cover of compact set F exists for some constant p51; then under Algorithm 2.1, system (6) is stable in sense of pth mean. Given a compact subset F of L; does there always exist a proper open cover of F? The answer is affirmative in cases of bounded noise. In fact, for arbitrarily small 05e51; [ F Bðg; eÞ g2F
so by definition of compact set, there exists a finite sub-cover fBðgi ; eÞ; i ¼ 1; 2; . . . ; Mg of the open cover fBðg; eÞ; g 2 Fg such that M [ Bðgi ; eÞ ð14Þ F i¼1
where M is a constant depending on e: Fix e and M; then we can take p > 1 large enough to satisfy ae 51: That is to say, the following theorem holds: Theorem 3.2 Assume that the real system f 2 F; where F is a compact subset of L: Then in case of bounded noise, taking p > 1 large enough, under Algorithm 2.1 with F0 ¼ fg1 ; g2 ; . . . ; gM g defined in Copyright # 2006 John Wiley & Sons, Ltd.
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(14), the output sequence fyt g of system is bounded in sense of pth mean: T 1 X lim sup jyt jp 51 T T!1 t¼1
3.2. Results for parameterized nonlinear systems Based on results stated above, we can extend the results to parametric case. We omit the proof of Theorem 3.3 to save space.
Theorem 3.3 Consider the parametric uncertain system ytþ1 ¼ f ðy; yt Þ þ ut þ wtþ1 where the unknown parameter y lies in a known compact set D R; and function f ðy; xÞ has linear growth rate with respect to x for each y 2 D: jf ðy; xÞj4Ay jxj þ A0y If function f ðy; xÞ satisfies one of the following conditions: 1. f ðy; xÞ is equicontinuous with respect to y; 2. f ðy; xÞ is smooth, and @f ðy; xÞ=@y has linear growth rate with respect to x for any y 2 D; 3. jf ðy; xÞ f ðy0 ; xÞj4gðjy y0 jÞ jxj þ oðjxjÞ for any y; y0 2 D; where oðjxjÞ does not depend on y; y0 and gðÞ is a class-K function, i.e. gð0Þ ¼ 0 and g : R50 ! R50 is strictly increasing and continuous. Then in case of bounded noise, taking p > 1 large enough, we can find a proper open cover fBðfi ; ri Þ; i ¼ 1; 2; . . . ; Mg of compact set F: Consequently, under Algorithm 2.1, the output sequence fyt g of system is bounded in sense of pth mean: T 1 X lim sup jyt jp 51 ð15Þ T!1 T t¼1
Now we give some examples to use this theorem. Example 3.1 Consider function f ðy; xÞ ¼ y sinðyxÞ; where the unknown parameter y 2 D ¼ ½a; b; and a5b can be arbitrary real numbers. Obviously, f ðy; xÞ is neither convex nor concave in general. Noting that @f ðy; xÞ @y ¼ jsinðyxÞ þ yx cosðyxÞj4jyj jxj þ 1 for any y 2 D; by Theorem 3.3, discrete-time nonlinear system with unknown parameter y 2 D ytþ1 ¼ y sinðyyt Þ þ ut þ wtþ1 Copyright # 2006 John Wiley & Sons, Ltd.
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can be stabilized by Algorithm 2.1 (taking proper p > 1) in the sense of pth mean: lim sup T!1
T 1 X jyt jp 51 T t¼1
ð16Þ
Example 3.2 Consider function f ðy; xÞ ¼ a1 ðyÞa2 ðxÞ þ eðy; xÞ
ð17Þ
where a1 ðÞ is a smooth function, a2 ðxÞ ¼ OðjxjÞ and eðy; xÞ ¼ oðjxjÞ: It can be easily proved that jf ðy; xÞ f ðy0 ; xÞj4CD jy y0 j jxj þ oðjxjÞ for any y; y0 2 D; where CD is a constant depending on compact set D: So Theorem 3.3 can be applied. When in the special case a1 ðyÞ ¼ y; a2 ðxÞ ¼ x and eðy; xÞ is bounded, this example shows that the parametric linear system can be stabilized by using Algorithm 2.1 (taking proper p > 1) rather than traditional recursive LS algorithm.
4. NUMERICAL EXAMPLES 4.1. A counter-example In this section, we construct a counter-example to show that the output sequence fyt g of system (6) can be unbounded even in sense of mean when Assumption AL is removed. (In cases of bounded noise, we can also construct another counter-example in which Assumption AL is satisfied but the output sequence is unbounded. So, in general, the results in Theorem 2.1 cannot be enhanced for bounded noise. To save space, we omit discussion on this counterexample.) We should first point out that existence of such examples was never predicted in thousands of various simulations for Algorithm 2.1. The reason why counter-examples were not found in simulations is that, by theoretical analysis, counter-examples must have many ‘strange’ properties (omitted here to save space) which are rarely seen in practical control systems. In our simulations, many kinds of nonlinear functions are used (including functions commonly used and their combinations, functions with infinite oscillations or heavy jumps, and even functions randomly generated), but we fail to find counter-examples. Thus, the counter-example presented below may be just of value in theory, and we need not worry about encountering such kind of examples in practical use. Since the shapes of functions constructed here look very complex, their mathematical expressions will not be given explicitly any more. The other way round, we try to explain the construction procedure to show that such a counter-example really exists. For simplicity, we take p ¼ 1; d ¼ 0: By Algorithm 2.1, it ¼ arg min sk ðtÞ k
Copyright # 2006 John Wiley & Sons, Ltd.
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where 4
sk ðtÞ ¼ St ðFk Þ ¼
t X
jFk ðyj1 ; wj Þj ¼
j¼1
t X
jgk ðj 1Þj
j¼1
For convenience, the following notations are used herein and hereafter: 4
gk ðj 1Þ ¼ Fk ðyj1 ; wj Þ ¼ f ðyj1 Þ fk ðyj1 Þ þ wj ;
k ¼ 1; 2; . . . ; M
Consequently, yj ¼ f ðyj1 Þ þ uj1 þ wj ¼ Fij1 ðyj1 ; wj Þ ¼ gij1 ðj 1Þ
ð18Þ
Now we simply state the procedure of construction: 1. Obviously, for any fixed 14k4M; given an increasing sequence fsk ðtÞg1 t¼1 ; we can determine a sequence fjgk ðtÞjg1 : t¼1 1 2. Given sequences fsk ðtÞg1 t¼1 and fjgk ðtÞjgt¼1 for k ¼ 1; 2; . . . ; M; by definition of it ; it ¼ 1 arg mink sk ðtÞ; we can determine sequences fit g1 t¼1 and fyt gt¼1 via (18). 1 3. Since we have known the sequences fyt gt¼1 and fgk ðtÞg1 t¼1 ; the values of fk ðyt Þ; t ¼ 1; 2; 3; . . . can be determined provided that function f ðxÞ and the noise sequence fwt g are given. Then the values of function fk ðxÞ at other points in R can be defined by interpolation. 4. By the last three steps, we have defined function fk ðÞ for each k: Now we need to check whether each function fk ðÞ satisfies assumption A0. Two conditions should be verified: (a) The points in sequence fyt g1 t¼1 must be non-identical, which guarantees that fk ðxÞ is really a function, i.e. a map from R to R; 1 (b) For any bounded sub-sequence fytj g1 j¼1 of sequence fyt gt¼1 ; the corresponding sequence ffk ðytj Þg must be bounded. Now we construct this counter-example. Assume that we have three models ðM ¼ 3Þ; and the first model is the real system, i.e. f1 ðxÞ f ðxÞ: For simplicity, we take f ðxÞ 0; wt w ¼ 1: According to the construction procedure stated before, we need only give sequences fsk ðtÞg1 t¼1 for k ¼ 1; 2; 3: Obviously, we have g1 ðtÞ ¼ wtþ1 ¼ w for model 1, and consequently s1 ðtÞ ¼ wt: Now we give the construction of s2 ðtÞ and s3 ðtÞ: Let j0 ¼ 1: Define Dn ¼ n n! jn ¼ 1 þ
n X
Dl ¼ ðn þ 1Þ!
l¼1
en ¼ 12 n n! then define functions hn ðxÞ ¼ dn ðxÞ ¼ hn ðx þ 1Þ hn ðxÞ ¼ Copyright # 2006 John Wiley & Sons, Ltd.
x Dn þ 1 x xþ1 x Dn þ 1 ¼ Dn x Dn x þ 1 ðDn x 1ÞðDn xÞ Int. J. Adapt. Control Signal Process. 2007; 21:391–414 DOI: 10.1002/acs
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For every positive integer t; there must exist a corresponding integer m such that jm1 4t5jm If m is an odd number, then define s2 ðtÞ ¼ hm ðt þ 12 jm1 Þ þ jm1 12 s3 ðtÞ ¼
D m em ðt jm1 Þ þ jm1 þ em Dm
s2 ðtÞ ¼
D m em ðt jm1 Þ þ jm1 þ em Dm
otherwise, define
s3 ðtÞ ¼ hm ðt þ 12 jm1 Þ þ jm1 12 By detailed analysis (omitted to save space), we know that Assumption AL does not hold in this example. By computation, we have yjm ¼ 13Dm þ emþ1 þ 56 ¼ 13 m m! þ ðm þ 1Þ ðm þ 1Þ! þ 56 ! 1 Obviously, lim
m!1
as m ! 1
yjm m þmþ1 ¼1 ¼ lim m!1 3ðm þ 1Þ jm
and consequently lim sup t!1
Rt yj 5 lim m ¼ 1 m!1 jm t
So the output sequence fyt g of the system is not bounded in sense of mean. This indicates that Assumption AL in Theorem 2.1 cannot be removed in general. 4.2. Simulation examples Several simulation examples are given here to demonstrate applicability of the LS-like algorithm. Example 1 We consider the following plant: xtþ1 ¼ yxt sinðby þ 12cxt Þ þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jyðxt þ 1Þj þ ut þ wtþ1
ð19Þ
where y is an unknown parameter taking its value in interval ½0; 5: Here bxc is the largest integer no greater than x: The nonlinear way in which y appears in (19) makes most nonlinear adaptive control techniques inapplicable. We apply the LS-like algorithm to stabilize this system. For any y 2 Yk ¼ ½k 12; k þ 12Þ; we can take a robust control law utðkÞ ¼ kxt sinðkxt Þ
ð20Þ
Take the set of models F0 ¼ ff1 ; f2 ; . . . ; f5 g; p where fk ðxÞ ¼ kx sinðkxÞ: Obviously, for any y 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½1; 5 and function f ðx; yÞ ¼ yxt sinðbyþcxt Þ þ jyðxt þ 1Þj; there exists a corresponding model fK 2 F0 such that rðf * ; fK Þ51: Copyright # 2006 John Wiley & Sons, Ltd.
Int. J. Adapt. Control Signal Process. 2007; 21:391–414 DOI: 10.1002/acs
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H. MA
In this example, the noise sequence is taken from normal distribution Nð0; s2 Þ with s ¼ 5; and the parameter y is randomly taken from interval ½1; 5: For this example we give two simulations in cases of p ¼ 1 and 2. In both simulations, we take d ¼ 10 and the noise sequences are taken to be identical. The sequences xt ; ut ; wt and it in each simulation are depicted in sub-figures of Figures 1(a) and (b), respectively. From the simulations given here, we can see that taking p ¼ 1 may be better than p ¼ 2 since it needs less computation cost and it results in fewer switchings. Other simulations for different values of d will not be given here to save space. Example 2 We consider the same plant as described in [17], i.e. a double cart with uncertain elastic coupling, which is a continuous-time linear model x. 1 ¼ yðx1 x2 Þ bx’ 1 þ u x. 2 ¼ yðx2 x1 Þ bx’ 2
ð21Þ
where y is an unknown parameter and b is a known constant. As in [17], assuming the state x ¼ ½x1 ; x’ 1 ; x2 ; x’ 2 0 is available for feedback, this model can be rewritten in state-space form as x’ ¼ ðF0 þ yF1 ÞxðtÞ þ Gu uðtÞ þ Gd dðtÞ 2
0
6 60 6 F0 ¼ 6 60 4 0
1
0
b
0
0
0
0
0
0
3
7 0 7 7 7; 1 7 5 b
Gu ¼ ½0; 0; 0; 10 ;
2
0
6 6 1 6 F1 ¼ 6 6 0 4 1
0
0
0
1
0
0
0
1
0
3
7 07 7 7 17 5 0
ð22Þ
Gd ¼ ½0; 1; 0; 10
where the uncertain y takes on values in Y ¼ ½0:05; 10: For this plant, we just consider the stabilization problem here. Taking sampling period T ¼ 0:1 s; by using Euler’s method, we can obtain a discrete-time model xtþ1 ¼ Ay xt þ But þ wtþ1
ð23Þ
Obviously, system (23) is a linear model with single input and multiple states. Due to the large uncertainty in possible values of y; one common linear controller ut ¼ Fxt cannot always guarantee the stability of the plant. So we divide the interval Y into several sub-intervals Yi ; i ¼ 1; 2; . . . ; M; and then design corresponding robust stabilizing linear controllers ðiÞ uðiÞ be the centre of interval Yi and AðiÞ be the corresponding t ¼ Fi xt for all y 2 Yi : Let y Ay ; then traditional linear-quadratic regulator design method can be applied to solve Fi for LTI plant ðAðiÞ ; BÞ: In this example, we use seven intervals to cover Y: ½0:05; 0:1; ½0:1; 0:2; ½0:2; 0:5; ½0:5; 1; ½1; 2; ½2; 5; ½5; 10: (Note: The intervals are taken to guarantee good robust stability of ðiÞ controller uðiÞ t for all y 2 Yi : We can use other methods to design controllers ut and the method used here is just a simple one.) Copyright # 2006 John Wiley & Sons, Ltd.
Int. J. Adapt. Control Signal Process. 2007; 21:391–414 DOI: 10.1002/acs
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Figure 1. Simulation example 1: (a) p ¼ 1; and (b) p ¼ 2:
In this simulation, the disturbance sequence dt is taken from normal distribution Nð0; s2 Þ with s ¼ 0:5; and we apply the simple LS-like control law ut ¼ Fit xt with it ¼ arg mink St ðkÞ; Copyright # 2006 John Wiley & Sons, Ltd.
Int. J. Adapt. Control Signal Process. 2007; 21:391–414 DOI: 10.1002/acs
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H. MA
output seq
noise seq
3.5
0.5
3
0.4
2.5 2
0.3
1.5
0.2
1 0.1
0.5 0
0
100
200
300
0
0
100
control seq
200
300
switch seq
10
12
5
10 8
0
6 4 2 0
100
200
300
0
0
100
200
300
Figure 2. Simulation example 2.
where St ðkÞ ¼
t X
jjxj Buj1 BFk xj1 jj2
j¼1
here jj jj denotes the commonly used vector norm. And we let the plant be time-varying as follows: yðtÞ ¼ 7:5 þ 2et , where et is randomly taken from standard normal distribution Nð0; 1Þ. The sequences xt ; ut ; wt and yðit Þ are depicted in four sub-figures of Figure 2, respectively. Obviously, yðtÞ fluctuates around 7.5 significantly. From this simulation example, we can see that the LS-like algorithm can also deal with varying parameter yðtÞ well. 5. PROOF OF MAIN THEOREM 5.1. Several lemmas We first give a lemma, an extension of Cr -inequality, which is needed to yield appropriate estimate. Lemma 5.1 Assume constant p > 0; then for any constant c > 1; there exists a constant c0 > 1; such that ðx þ yÞp 4cjxjp þ c0 jyjp Copyright # 2006 John Wiley & Sons, Ltd.
8x; y 2 R
Int. J. Adapt. Control Signal Process. 2007; 21:391–414 DOI: 10.1002/acs
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FINITE-MODEL ADAPTIVE CONTROL
Proof When 05p41; because ðx þ yÞp 4jxjp þ jyjp ; we can take any c0 > 1: Now we consider the case of p > 1: In this case take constant c0 > 1 such that 1=ðp1Þ 1=ðp1Þ 1 1 þ 0 ¼1 c c
ð24Þ
Let z ¼ jyj=jxj; we need only prove that for any z50; 4
HðzÞ ¼ ð1 þ zÞp c c0 zp 40 Obviously, Hð0Þ ¼ 1 c50;
lim HðzÞ ¼ 150
z!1
Now we calculate the critical point(s) of HðzÞ: Notice that " # 1 þ z p1 0 p1 0 0 p1 p1 pc z ¼ pz c H ðzÞ ¼ pð1 þ zÞ z so obviously the critical point z ¼ z0 should satisfy that 1 þ z0 p1 ¼ c0 z0 Because H 0 ðzÞ > 0 when 05z5z0 and H 0 ðzÞ50 when z > z0 ; function HðzÞ reaches its maximum at point z ¼ z0 : By (24), we have c ¼ ð1 þ z0 Þp1 and consequently the maximum of HðzÞ is Hðz0 Þ ¼ ð1 þ z0 Þp c c0 zp0 ¼ ð1 þ z0 Þp ð1 þ z0 Þp1
1 þ z0 z0
p1
zp0 ¼ 0
This completes the proof of Lemma 5.1.
&
Lemma 5.2 Assume that rK ¼ rðF * K Þ5e; then there exist two constants cK ; c0K such that jFK ðx; wÞjp 4ep jxjp þ cK jwjp þ c0K Proof Take a number Z 2 ðrK ; eÞ; then by definition of rK ; there must exist a constant CZ 50 such that jFK ðxÞj4Zjxj þ CZ By Lemma 5.1, for fixed arbitrary c0 > 1; there exists corresponding c00 > 1 such that jFK ðx; wÞjp ¼ jFK ðxÞ þ wjp 4c0 Zp jxjp þ c00 ðCZ þ jwjÞp
8x 2 R
Since Z5e; we can choose c0 > 1 such that c0 Zp 5ep ; and by applying Lemma 5.1 again for ðCZ þ jwjÞp ; we can see that the lemma is true. & Copyright # 2006 John Wiley & Sons, Ltd.
Int. J. Adapt. Control Signal Process. 2007; 21:391–414 DOI: 10.1002/acs
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H. MA
Lemma 5.3 Under Assumption AL, for any fixed integer l > 0; we have jyt jp 4C1ðlÞ jytl jp þ a0 þ a1 jwt jp þ a2 jwt1 jp þ þ al jwtlþ1 jp where
C1ðlÞ ; a0 ; a1 ; . . . ; al
ð25Þ
are positive constants.
Proof By Assumption AL, we have jFi ðxÞj4Ajxj þ A0 for any 14i4M and x 2 R: Thus, jyt j ¼ jFit1 ðyt1 Þ þ wt j4Ajyt1 j þ ðA0 þ jwt jÞ
ð26Þ
Then apply Lemma 5.1 repeatedly, we can see that (25) is true for some constants C1ðlÞ ; a0 ; a1 ; . . . ; al : & 5.2. Proof of Theorem 2.1 By Algorithm 2.1, for any t > 0; St ðFk Þ ¼
t X
jFk ðyj1 ; wj Þjp
j¼1
Therefore, by Lemma 5.2 we have St ðFK Þ ¼
t X
jFK ðyj1 ; wj Þjp
j¼1
5 ep
t X
jyj1 jp þ cK
t X
j¼1
jwj jp þ c0K t
j¼1
¼ Wt0 4
For convenience, we introduce the following notations: Rn ¼
n X
jyj jp
ð27Þ
jwj jp
ð28Þ
j¼1
Wn ¼
n X j¼1
N ¼ f1; 2; 3; . . .g
ð29Þ
n% ¼ f1; 2; . . . ; ng
ð30Þ
J ¼ fj 2 N : jyj j5dg
ð31Þ
Jn ¼ J \ n% ¼ fj 2 n% : jyj j5dg
ð32Þ
Copyright # 2006 John Wiley & Sons, Ltd.
Int. J. Adapt. Control Signal Process. 2007; 21:391–414 DOI: 10.1002/acs
FINITE-MODEL ADAPTIVE CONTROL
409
L ¼ the limit set of fij : j 2 Jg
ð33Þ
IðkÞ ¼ fj 2 J : ij ¼ kg
ð34Þ
and we denote by jXj the number of elements in set X: Our objective is to prove 1 lim sup Rn 51 n!1 n For any j 2 N; by definition of ut ; we have yj ¼ Fij1 ðyj1 ; wj Þ
ð35Þ
By (31), for any j 2= J; jy Pj j4d: If the set J has just finite elements, then obviously there exists a constant C0 such that j2J jyj jp 4C0 : Therefore, X X Rn ¼ jyj jp þ jyj jp 4C0 þ d p ðn jJn jÞ4C0 þ d p n ð36Þ j2Jn
j2%n\Jn
And, consequently, lim supn!1 ð1=nÞRn 4d p 51: So, in the following we only consider the case that the set J has infinite elements. Since jJj ¼ 1; the limit set L of fij : j 2 Jg must be non-empty. By definition of L; for any k 2= L; the set IðkÞ has just finite elements; and for any k 2 L; the set IðkÞ has infinite elements. P P p Therefore, for some constant C1 ; k 2= L j2IðkÞ jyj j 4C1 : Thus, X X X X X X X Rn ¼ jyj jp þ jyj jp þ jyj jp 4 jyj jp þ C1 þ d p n ð37Þ k2L j2IðkÞ\%n
k 2= L j2IðkÞ\%n
So we need only study 4
rn ¼
X
jyj jp ¼
j2IL \%n
where IL ¼
[
j2%n\Jn
X
k2L j2IðkÞ\%n
jFij1 ðyj1 ; wj Þjp
j2IL \%n
IðkÞ ¼
k2L
[
fj 2 J : ij ¼ kg
k2L
Let L0 ¼ the limit set of fij1 : j 2 IL g and define I 0 ðk0 Þ ¼ fj 2 IL : ij1 ¼ k0 g Because IL has infinite elements, the set L0 must be non-empty. By the definition of ‘limit set’, for any k0 2= L0 ; I 0 ðk0 Þ has just finite elements; and for any k0 2 L0 ; I 0 ðk0 Þ has infinite elements. Therefore, there exists a constant C2 > 0 such that X X jFij1 ðyj1 ; wj Þjp 4C2 k0 2= L0 j2I 0 ðk0 Þ
Copyright # 2006 John Wiley & Sons, Ltd.
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H. MA
And, consequently, rn ¼
X
X k0
2= L0
4 C2 þ
jFij1 ðyj1 ; wj Þjp þ
j2I 0 ðk0 Þ\%n
X
X
X
X
k0 2L0
j2I 0 ðk0 Þ\%n
jFij1 ðyj1 ; wj Þjp
jFij1 ðyj1 ; wj Þjp
ð38Þ
k0 2L0 j2I 0 ðk0 Þ\%n
Now let J 0 ¼ fj 2 J : j 1 2 Jg: Noting also for any j 2 I 0 ðk0 Þ; ij1 ¼ k0 ; we have X 0 ð2 0 rn 4C2 þ ½vð1 n ðk Þ þ vn ðk Þ
ð39Þ
k0 2L0
where
X
0 vð1 n ðk Þ ¼ 4
jFk0 ðyj1 ; wj Þjp
j2I 0 ðk0 Þ\%n\J 0
X
4
0 vð2 n ðk Þ ¼
jFk0 ðyj1 ; wj Þjp
ð40Þ
j2I 0 ðk0 Þ\%n\J 0 0 = J 0 means j 2= J or j 1 2= J by definition First, we consider the term vð2Þ n ðk Þ: Obviously, j 2 of J 0 : Because j 2 I 0 ðk0 Þ IL J; we must have j 1 2= J: By definition of J; we must have jyj1 j4d: By basic assumption A0, there exists a constant Cd > 0 such that jf ðxÞ fi ðxÞj4 Cd ði ¼ 1; 2; . . . ; MÞ when jxj4d: Therefore, X X 0 vð2Þ jFk0 ðyj1 ; wj Þjp 4 ðCd þ jwj jÞp ð41Þ n ðk Þ ¼ j2I 0 ðk0 Þ\%n\J 0
Thus, we have rn 4C2 þ
X
j2I 0 ðk0 Þ\%n\J 0
0 ð2 0 ½vð1 n ðk Þ þ vn ðk Þ4C2 þ
k0 2L0
X
ðCd þ jwj jÞp þ
j2%n\J 0
Next, we need only consider
X
0 vð1Þ n ðk Þ ¼
X
0 vð1Þ n ðk Þ
ð42Þ
k0 2L0
jFk0 ðyj1 ; wj Þjp
j2I 0 ðk0 Þ\%n\J 0
If I 0 ðk0 Þ \ J 0 is a set with finite elements, then obviously there exists a constant C3 > 0 such that X jFk0 ðyj1 ; wj Þjp 4C3 j2I 0 ðk0 Þ\%n\J 0 0 0 0 and consequently, vð1Þ n ðk Þ4C3 : Therefore, for any k 2 L ; we need only consider the case that 0 0 0 I ðk Þ \ J is a set of infinite elements. To proceed to make analysis, we first prove L0 ¼ L: Let L0 be the limit set of fit g1 t¼1 : Obviously, L L0 ; L0 L0 : Let J ¼ f%j 1 ; %j 2 ; %j 3 ; . . .g; %j 1 5%j 2 5%j 3 5 : By definition of it ; for any positive integer t : %j m 4t5%j mþ1 ; we have it ¼ i%jm ; then by definitions of L and L0 ; we have L0 L; so consequently we have L ¼ L0 : Similarly, for any positive integer t : %j m1 4t5%j m ; we have it ¼ i%j m1 ¼ i%jm 1 ; then by definitions of L and L0 ; we have L L0 ; thus, consequently we have L0 ¼ L0 ¼ L: Let j0 ¼ 1: For every fixed k0 2 L0 ; suppose that I 0 ðk0 Þ \ J 0 ¼ fj1 ; j2 ; j3 ; . . .g; j1 5j2 5 j3 5 : Obviously, for any m; we have ijm 1 ¼ k0 : For any n > 0; there exists a corresponding
Copyright # 2006 John Wiley & Sons, Ltd.
Int. J. Adapt. Control Signal Process. 2007; 21:391–414 DOI: 10.1002/acs
FINITE-MODEL ADAPTIVE CONTROL
411
T ¼ TðnÞ > 0 such that jT 4n5jTþ1 ; therefore, I 0 ðk0 Þ \ n% \ J 0 ¼ fj1 ; j2 ; . . . ; jT g; We will consider two subcases: (a) ijT ¼ ijT 1 ¼ k0 ; (b) ijT =ijT 1 ¼ k0 : (a) Since ijT ¼ k0 ; by definition of it ; we have k0 ¼ ijT ¼ arg min SjT ðFk Þ
ð43Þ
k
consequently, SjT ðFk0 Þ4SjT ðFK Þ4Wj0T 4Wn0 So we obtain that
X
0 vð1Þ n ðk Þ ¼
ð44Þ
jFk0 ðyj1 ; wj Þjp
j2I 0 ðk0 Þ\%n\J 0
4
jT X
jFk0 ðyt1 ; wt Þjp ¼ SjT ðFk0 Þ
t¼1
4 Wn0
ð45Þ
(b) Since ijT =k0 ; (44) does not hold and (45) is not true in general. However, since ijT 1 ¼ k0 ; similarly we have SjT1 ðFk0 Þ4SjT 1 ðFk0 Þ4SjT 1 ðFK Þ4Wj0T 1 4Wn0 Thus, by Lemma 5.3, X 0 vð1Þ n ðk Þ ¼
ð46Þ
jFk0 ðyj1 ; wj Þjp
j2I 0 ðk0 Þ\%n\J 0
4
jT1 X
jFk0 ðyt1 ; wt Þjp þ jyjT jp
t¼1
¼ SjT1 ðFk0 Þ þ jyjT jp 4 Wn0 þ ðC1ðlÞ jyjT l jp þ a0 þ a1 jwjT jp þ a2 jwjT 1 jp þ þ al jwjT lþ1 jp Þ
ð47Þ
Now we choose positive integer l4jLj as follows: (b.1) If there exists 15s4jLj such that jT s 2= J; then we take l ¼ s: In this case we have jyjT l j5d (b.2) Otherwise, for s ¼ 0; 1; 2; . . . ; jLj; we must have jT s 2 J: For sufficient large n; i.e. for sufficiently large T; we have ijT s 2 L ¼ L0 Obviously, there must exist 04s0 5s4jLj such that ijT s0 ¼ ijT s ¼ k 2 L Copyright # 2006 John Wiley & Sons, Ltd.
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so, consequently, SjT s0 ðFk Þ4Wj0T s0 Then taking l ¼ s 1; we obtain that jyjT l jp ¼ jFk ðyjT s ; wjT sþ1 Þjp 4SjT sþ1 ðFk Þ4SjT s0 ðFk Þ4Wj0T s0 4Wn0 So based on (b.1) and (b.2), we have ðlÞ p p p 0 0 p 0 vð1Þ n ðk Þ 4Wn þ C1 maxðd ; Wn Þ þ a0 þ a1 jwjT j þ a2 jwjT 1 j þ þ al jwjT lþ1 j Þ
4CWn0 þ C 0
ð48Þ
where the constant C can be taken as (in the worst case) C ¼ 1 þ C1ðjLjÞ þ maxða1 ; a2 ; . . . ; ajLj Þ So no matter in case (a) or (b), for any fixed k0 2 L0 ; we have X 0 0 0 0 0 vð1Þ n ðk Þ ¼ jL jCWn þ jL jC
ð49Þ
k0 2L0
Thus, rn 4C2 þ
X
0 ð2 0 ½vð1Þ n ðk Þ þ vn ðk Þ4
k0 2L0
X
ðCd þ jwj jÞp þ jL0 jCWn0 þ constant
ð50Þ
j2%n\J 0
By definitions of Wt0 and Rt ; we have Wn0 ¼ ep Rn1 þ cK Wn þ c0K n So finally we have Rn 4 rn þ C1 þ d p n X vn ðk0 Þ þ C2 þ C1 þ d p n 4 k0 2L0
4 jL0 jCep Rn1 þ ðjL0 jC cK Wn þ jL0 jC c0K n þ d p n þ constantÞ 4 ae Rn1 þ Vn
ð51Þ
where 4
Vn ¼ jLjC cK Wn þ jLjC c0K n þ d p n þ constant
ð52Þ
* þC * 0 for two positive constants C * and C * 0: It is obvious that Vn ¼ OðnÞ; i.e. we have Vn 4Cn Then we can show that by mathematical induction Rn 4
* *0 C C nþ 1 ae 1 a0e
ð53Þ
In fact, obviously it holds for n ¼ 0: Now we suppose it holds for n 1; i.e. Rn1 4 Copyright # 2006 John Wiley & Sons, Ltd.
* *0 C C ðn 1Þ þ 1 ae 1 ae Int. J. Adapt. Control Signal Process. 2007; 21:391–414 DOI: 10.1002/acs
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Then in case of n; by (51) we have * þC *0 Rn 4 ae Rn1 þ Cn * *0 C C * þC *0 4 ae ðn 1Þ þ þ Cn 1 ae 1 ae ¼
* *0 * C C C nþ ae 1 ae 1 ae 1 ae
4
* *0 C C nþ 1 ae 1 ae
Finally, by (53), we immediately obtain that * C 1 51 lim sup Rn 4 1 ae n!1 n This completes the proof.
&
6. CONCLUSION In this paper, we studied the adaptive control problem of a class of discrete-time nonlinear uncertain systems of which the internal uncertainty can be characterized by a finite set of functions. We propose an LS-like algorithm to design the feedback control law, which is tested to be effective and efficient in lots of simulations. Under a so-called linear growth assumption, we prove that the closed-loop system for this algorithm is stable in sense of pth mean. The main results have been applied to parametric cases. A counter-example is also constructed to show that the linear growth assumption cannot be removed.
ACKNOWLEDGEMENTS
I would like to express my sincere thanks to Prof. Lei Guo for his valuable advice. I would like to thank also the referees and the editors for their constructive and helpful comments and suggestions, which improve this paper much.
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