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Proof (a) Suppose at time t 1, (5.89) is violated. This can occur in one of two ways which we consider separately: (i) zt 1zt 1 i t > Ákxtk c 0 5:91 In this case, because of the ordering of ut i in (5.87), and the de®nition of zt 1 i in (5.83), then zt 1 i > Ákxtk c 0 5:92 for all i P k; FFF; j sÀ1 ; j s fg (ii) zt 1zt 1 i t < À Ákxtk c 0 5:93 In this case zt 1 i < À Ákxtk c 0 5:94 for all i j 1 ; j 2 ; FFF; k fg . In either case, we see that if (5.91) is violated at time t, then s t1 1 2 s t 5:95 from which the result follows. (b) First, we note that the control is well de®ned, that is, S t is never empty. This follows since there is at least one index, namely the index of the set i which contains the true plant, which is always an element of S t . Next, we note that although we cannot guarantee that we converge to the correct control, from (a) we know (5.85) is satis®ed all but a ®nite number of times. Since s T is a stabilizing inclusion, then by de®nition the states and all signals will be bounded. Furthermore, since s t is a stabilizing inclusion, there exist 0 ; 0 and P0; 1 such that if the inclusion (5.89) is satis®ed, for t Pt 0 ; t 0 T, then kxt 0 Tk 0 T kxt 0 k 0 5:96 (Note that if this is not the case, then from the de®nition, s t is not a stabilizing inclusion.) Also, there exist " and " such that when (5.89) is violated: kxt 1k " kxtk " 5:97 If we de®ne 1 2 0 " and 1 0 " 0 0 0 " , then after some algebraic manipulations we can show that for any t 0 ; T > 0 such that (5.89) is violated not more than once in the interval, t 0 ; t 0 T, then kxt 0 Tk 1 T kxt 0 k 1 5:98 Also, we can show that with 2 " 0 1 " 3 0 2 ,and 2 0 " 0 " 1 106 Adaptive stabilization of uncertain discrete-time systems 1 0 " 0 " 2 0 1 0 " " , provided (5.89) is not violated more than twice in the interval t 0 ; t 0 T, then kxt 0 Tk 2 T kxt 0 k 2 5:99 Repeating this style of argument leads to the conclusion that with N " 0 0 N ; N 0 " N 0 0 " N À 1 0 " À1 45 0 " then if there are not more than N switches in t 0 ; t 0 T, then kxt 0 Tk N T kxt 0 k N 5:100 The desired result follows from (a) since we know that there are at most N log 2 s times at which (5.89) is violated. Case 2: L > 1 Suppose that we do not know a single C such that I t is a stabilizing inclusion, and CB is of known sign, then using ®nite covering ideas [8], as in Remark 4.3 let L l1 l L l1 s l m1 l m 5:101 where for each l, we know C l ; Á l ; c l 0 such that s t is a stabilizing inclusion on l and the sign of C l B is constant for all plants in l . At this point, one might be tempted to apply localization, as previously de®ned, on the sets l individually and switch from l should the set of valid indices, S l , become empty. Unfortunately, this procedure cannot be guaran- teed to work. In particular, if l does not contain the true plant, s t need not be a stabilizing inclusion, and so divergence of the states may occur without violating (5.89). To alleviate this problem, we use the exponential stability result, (5.90), in our subsequent development. Algorithm D We initialize ti0; R 0 f1; 2; FFF; Lg and take any l 0 P R 0 . We then perform localization on l , with the following additional 2 steps: If at any time kxtk > tÀti kxtik 5:102 (where ; ; are the appropriate constants for l from Theorem 4.1), then we set S l fg. If at any time t; S l becomes empty, we set R t R tÀ1 Àflg; tit, and we take a new l from R t . Adaptive Control Systems 107 2 In fact, we can localize simultaneously within other O i ; i T l: however, for simplicity and brevity we analyse only the case where we localize in one set at a time. With these modi®cations, it is clear that Theorem 4.1 can be extended to cover this case as well: Corollary 4.1 The control algorithm (5.86)±(5.88) with the above modi®ca- tions applied to a plant with decomposition as in (5.101) satis®es: (a) There are no more than: L À 1 L l1 log 2 s l instants such that jzt 1 l t j!Á l t kxtk c 0;l t 5:103 (where l t denotes the value of l at time t). (b) All signals in the closed loop are bounded. In particular, there exist constants " ; " <I; " P0; 1 such that for any t 0 ; T > 0 kxt 0 Tk " " T kxt 0 k " 5:104 Proof Follows from Theorem 4.1. 5.4.1 Localization in the presence of unknown disturbance In the previous section the problem of indirect localization based switching control for linear uncertain plants was considered assuming that the level of the generalized exogenous disturbance t was known. This is equivalent to knowing some upper bound on t. The ¯exibility of the proposed adaptive scheme allows for simple extension covering the case of exogenous distur- bances of unknown magnitude. This can be done in the way similar to that considered in Section 5.3.2. Omitting the details we just make the following useful observation. The control law described by Algorithms C and D is well de®ned, that is, R t Tfgfor all t ! t 0 if c l 0 ! sup t!t 0 jC l Etj; Vl 1; FFF; L. This is the key point allowing us to construct an algorithm of on-line identi®cation of the parameters c l 0 ; l 1; FFF; L. 5.5 Simulation examples Extensive simulations conducted for a wide range of LTI, LTV and nonlinear systems demonstrate the rapid falsi®cation capabilities of the proposed method. We summarize some interesting features of the localization technique observed in simulations which are of great practical importance. (i) Falsi®cation capabilities of the algorithm of localization do not appear to be sensitive to the switching index update rule. One potential implication of this observation is as follows. If not otherwise speci®ed any choice of a new switching index is admissible and will most likely lead to good transient performance; (ii) The speed of localization does not appear to be closely related to the total 108 Adaptive stabilization of uncertain discrete-time systems number of ®xed controllers obtained as a result of decomposition. The practical implication of this observation (combined with the quadratic stability assumption) is that decomposition of the uncertainty set can be conducted in a straightforward way employing, for example, a uniform lattice which produces subsets i , i 1; 2; FFF; L of an equal size. Example 5.1 Consider the following family of unstable (possibly nonmini- mum phase) LTV plants: yt1:2yt À 1À1:22yt À 2b 1 tut À 1b 2 tut À 2t 5:105 where the exogenous disturbance t is uniformly distributed on the interval À0:1; 0:1, and b 1 t and b 2 t are uncertain parameters. We deal with two cases which correspond to constant parameters and large-size jumps in the values of the parameters. Case 1: Constant parameters The a priori uncertainty bounds are given by b 1 tPÀ1:6; À0:150:15; 1:6; b 2 tPÀ2; À11; 25:106 i.e. fÀ1:6; À0:150:15; 1:6ÂÀ2; À11; 2g. To meet the require- ments of the localization technique, we decompose into 600 nonintersecting subsets with their centres i b 1i ; b 2i ; i 1; FFF; 600 corresponding to Adaptive Control Systems 109 0 50 10 0 −5 0 5 output (b) 0 50 100 −5 0 5 control (c) 0 50 100 0 200 400 600 switching index (a) Figure 5.4 Example of localization: constant parameters b 1i PfÀ1:6; À1:5; FFF; À0:3; À0:2; 0:2; 0:3; FFF; 1:5; 1:6g b 2i PfÀ2; À1:9; FFF; À1:1; À1; 1; 1:1; FFF; 1:9; 2g respectively. Figures 5.4(a)±(c) illustrate the case where is constant. The switching sequence fi1; i2; FFFg depicted in Figure 5.4(a) indicates a remarkable speed of localization. Case 2: Parameter jumps The results of localization on the ®nite set f i g 600 i1 are presented in Figures 5.5(a)±(e). Random abrupt changes in the values of the plant parameters occur every 7 steps. In both cases above the algorithm of localization in Section 5.2 is used. However, in the latter case the algorithm of localization is appropriately modi®ed. Namely, It is updated as follows It It À 1 It if It À1 It Tfg It otherwise @ 5:107 110 Adaptive stabilization of uncertain discrete-time systems 0 50 10 0 −40 −20 0 20 output (b) 0 50 100 −50 0 50 control (c) 0 50 100 0 200 400 600 switching index (a) 0 50 10 0 −2 0 2 parameter b1 (d) 0 50 100 −2 0 2 parameter b2 (e) Figure 5.5 Example of localization: parameters jump every 7 steps Once (or if) the switching controller, based on (5.107) has falsi®ed every index in the localization set it disregards all the previous measurements, and the process of localization continues (see [40] for details). In the example above a pole placement technique was used to compute the set of the controller gains fK i g 600 i1 . The poles of the nominal closed loop system were chosen to be (0, 0.07, 0.1). Example 5.2 Here we present an example of indirect localization considered in Section 5.4. The model of a third order unstable discrete-time system is given by yt 1a 1 yta 2 yt À 1a 3 yt À 2utt5:108 where a 1 ; a 2 ; a 3 are unknown constant parameters, and t 0 sin0:9t represents exogenous disturbance. The a priori uncertainty bounds are given by a 1 PÀ1:6; À0:10:1; 1:6; b 2 PÀ1:6; À0:10:1; 1:6; a 3 P0:1; 1:6 5:109 i.e. fÀ1:6; À0:10:1; 1:6ÂÀ1:6; À0:10:1; 1:6Â0:1; 1:6g. Choosing Adaptive Control Systems 111 0 20 40 60 80 100 120 140 160 180 20 0 −4 −2 0 2 4 Output (a) 0 20 40 60 80 100 120 140 160 180 20 0 −6 −4 −2 0 2 4 6 Control (b) Figure 5.6 Example of indirect localization the vector C and the stabilizing set s as prescribed in Section 5.4, we obtain s X jzt 1j Ájjxtjjc 0 5:110 where C 0; 0; 1 and Á 0:6. We decompose into 256 nonintersecting subsets with their centres i a 1i ; a 2i ; a 3i ; i 1; FFF; 256 corresponding to a 1i PfÀ0:3; À0:7; À1:1; À1:5; 0:3; 0:7; 1:1; 1:5g5:111 a 2i PfÀ0:3; À0:7; À1:1; À1:5; 0:3; 0:7; 1:1; 1:5g5:112 a 3i Pf0:3; 0:7; 1:1; 1:5g5:113 respectively. This allows us to compute the set of controller gains fK i g 256 i1 , K i k 1i ; k 2i ; k 3i . Each element of the gain vector k ij , i Pf1; 2; 3g, j Pf1; FFF; 256g takes values in the sets (5.111), (5.112), (5.113), respectively. The results of simulation with 0 0:1, a 1 À1:1, a 2 À0:7, a 3 1:4, are presented in Figure 5.6(a)±(b). Algorithm C has been used for this study. 5.6 Conclusions In this chapter we have presented a new uni®ed switching control based approach to adaptive stabilization of parametrically uncertain discrete-time systems. Our approach is based on a localization method which is conceptually dierent from the existing switching adaptive schemes and relies on on-line simultaneous falsi®cation of incorrect controllers. It allows slow parameter drifting, infrequent large parameter jumps and unknown bound on exogenous disturbance. The unique feature of localization based switching adaptive control distinguishing it from conventional adaptive switching controllers is its rapid model falsi®cation capabilities. In the LTI case this is manifested in the ability of the switching controller to quickly converge to a suitable stabilizing controller. We believe that the approach presented in this chapter is the ®rst design of a falsi®cation based switching controller which is applicable to a wide class of linear time-invariant and time-varying systems and which exhibits good transient performance. Appendix A Proof of Theorem 3.1 First we note that it follows from Lemma 3.1 and the switching index update rule (5.37) that the total number of switchings made by the controller is ®nite. Let ft 1 ; t 2 ; FFF; t l g be a ®nite set of switching instants. By virtue of (5.31)±(5.33) the behaviour of the closed loop system between any two 112 Adaptive stabilization of uncertain discrete-time systems consecutive switching instants t s ; t j ; 1 s; j l; t j ! t s is described by xt 1ABK it s xtEtt A it s B it s K it s xtE t 5:114 where j tj r it s jjtjj " " t. Therefore, taking into account the structure of the parameter dependent matrices A and B, namely the fact that only the last rows of A and B depend on the last equation can be rewritten as xt 1A it s Át B it s ÁtK it s xtE t5:115 for some Át X jjÁtjj r it s q and j tj " " t. This is a direct consequence of the fact that the last equation in (5.114) can be rewritten as yt 1 T it s t t and that mx jjÁjj 1 jjÁ T tjj jjtjj holds for any t. By De®nition 3.2 and condition (C3 H ) the system (5.115) is quad- ratically stable with t0 and t s being ®xed; moreover, there exists a positive de®nite matrix H T t s H t s such that P t s mx jjÁtjj r it s q jjA it s Át B it s ÁtK it s jj H t s < 1 5:116 Here jjxjj H x T Hx 1=2 and for any matrix A P R nÂn , jjAjj H denotes the corresponding induced matrix norm. The equation (5.115) along with the property of quadratic stability guarantee that between any two consecutive switchings the closed loop system behaves as an exponentially stable LTI system subject to small parametric perturbations Át and bounded dis- turbance t and this property holds regardless of the possible evolution of the plant parameters. This is the key point making the rest of the proof transparent. Assume temporarily that " t0, then it follows from (5.115), (5.116) that jjxt s 1jj H t s P t s jjxt s jj H t s t s 5:117 jjxt s 2jj H t s P 2 t s jjxt s jj H t s P t s 1 t s 5:118 FFF jjxt s kjj H t s P k t s jjxt s jj H t s k i1 P iÀ1 t s 5:119 jjxt s kjj max H t s = min H t s 1=2 P k t s jjxt s jj t s k i1 P iÀ1 t s = min H t s 1=2 5:120 where t s mx jj " jjEjj H t s . Adaptive Control Systems 113 Denote M mx t 1 i t l max H t s = min H t s 1=2 ; mx t 1 i t l P i < 1; 5:121 M " mx t 1 i t l i = min H i 1=2 I j1 P jÀ1 i < I5:122 Since it P I t 0 ; K it PfK i g L i1 for all t P N; itPI t 0 there exist constants 0 < M 0 < I; 0 mx jj " jjEjj < I such that jjxt s jj M 0 jjxt s À 1jj 0 5:123 for any switching instant t 1 t s t l . Hence, t 0 ; t l X jjxt 1 jj M 0 jjxt 1 À 1jj 0 M 0 M t 1 Àt 0 À1 jjxt 0 jj M 0 M " 0 5:124 jjxt 2 jj M 0 jjxt 2 À 1jj 0 M 2 0 M 2 t 2 Àt 0 À2 jjxt 0 jj M 2 " 5:125 where M 2 " M 0 MM 0 M " 0 M " 0 ; ::: t l ; I X jjxtjj M l 0 M l tÀt 0 Àl jjxt 0 jj M l " 5:126 Having denoted M 1 M 0 M= l ; M 2 " M l " < I we obtain (5.42). To conclude the proof we note that the result above can be easily extended to the case " t T 0, provided that the `size' of unmodelled dynamics " is suciently small. Indeed, let t T 0. First, we note that due to the term " t in the algorithm of localization (5.33)±(5.37) the process of localization cannot be disrupted by the presence of small unmodelled dynamics. In view of (A5), (5.117)±(5.126) it is easy to show that provided that " is suciently small t l ; I X jjxtjj M l 0 M l tÀt 0 Àl jjxt 0 jj M l " M "jjxt 0 jj 5:127 with M being a positive constant independent of xt 0 . Therefore jjxtjj M 1 tÀt 0 M "jjxt 0 jj M l " 5:128 is valid for all t 0 P N, t ! t l . From (5.128) and assumption (A5) exponential stability of the closed loop system (if M l " 0) or exponential convergence of the states to the residual set (if M l " > 0) can be easily established. Indeed, in this case it is always possible to specify a suciently large integer T such that M 1 T M " < 1. This, in turn, trivially implies stability. The ®nite number of the controller switchings follows directly from the switching index update rule (5.37). This also implies the ®nite convergence of switching; however, it is 114 Adaptive stabilization of uncertain discrete-time systems quite dicult, in general, to put an upper bound on t l . This obviously does not aect the stability properties of the closed loop. Appendix B Proof of Theorem 3.4 First we note that the property 1 follows directly from the structure of the algorithm of localization (5.66). It is straightforward to verify that relations (5.60)±(5.65) guarantee that the sequence of localization sets It is well de®ned. To prove 2 consider ®rst the case 0. It is clear that t kst Ik; " k À 1 Tfg 5:129 if min kPst;t f " k 1g ! " for all t > t 0 . Since, according to (5.64), the estimate " t is updated only if (5.129) does not hold, and taking into account the discrete nature of updating expressed by (5.65) we conclude that sup t!t 0 " t " 5:130 Let >0. Then it is easy to see that the arguments above remain valid for any ®nite interval of time st; stt d , provided that the rate of parameter variations is suciently small, namely, q=t d . To conclude the proof of (5.130) it suces to note that the estimate " t in (5.64) does not change if t À st!t d . Proof of statements 5:3; 5:4 follows closely those of Theorem 3.1. Here we present a brief sketch of the proof. Consider a ®nite time interval T st; stt d ; l < t d < I. Let " st ! " , then the total number of switchings s made by the controller over T satis®es the condition s l if q=t d . Therefore, the states are bounded by (5.126) with t 0 replaced by st. Moreover, (5.126) is valid for any time interval " T st; st " t; " t > t d such that " t kst Ik; " st Tfg 5:131 Relying on (5.126) and taking into account the fact that the index st is reset every time when (5.131) is violated for t À st!t d it is always possible to choose suciently large integer t d such as to guarantee exponential stability of the closed loop system. Let " be unknown, then for any " t 0 > 0 the inequality (5.126) can be possibly violated no more than " =1 times. Relying on this fact and using standard arguments exponential stability of the closed loop system is easily established. Adaptive Control Systems 115 [...]... Nonlinear Dynamic Systems ± A Survey on Input/Output Approaches', Automatica, 26, 4, 65 1 -67 7 [ 16] Ioannou, P A and Sun, J (19 96) Robust Adaptive Control Prentice-Hall [17] Kreisselmeier, G (19 86) `Adaptive Control of a Class of Slowly Time-varying Plants', Systems and Control Letters, 8, 97±103 [18] Kung, M C and Womack, B F (1983) `Stability Analysis of a Discrete-Time Adaptive Control Algorithm Having... Decision Control, 1732±1737 [7] Etxebarria, V and De La Sen, M (1995) `Adaptive Control Based on Special Compensation Methods for Time-varying Systems Subject to Bounded Disturbances', Int J Control, 61 , 3, 66 7 69 9 [8] Fu, M and Barmish, B R (19 86) `Adaptive Stabilization of Linear Systems via Switching Control' , IEEE Trans Auto Contr., AC-31, 12, 1097±1103 [9] Fu, M and Barmish, B R (1988) `Adaptive. .. −4 0 20 40 60 80 100 (a) 120 140 160 180 200 20 40 60 80 100 (b) 120 140 160 180 200 6 Control 4 2 0 −2 −4 6 0 Figure 5 .6 Example of indirect localization 112 Adaptive stabilization of uncertain discrete-time systems the vector C and the stabilizing set s as prescribed in Section 5.4, we obtain s X jz
t 1j Ájjx
tjj c0
5:110 where C
0; 0; 1 and Á 0 :6 We decompose into 2 56 nonintersecting... `Adaptive Stabilization of Linear Systems with Singular Perturbations', Proc IFAC Workshop on Robust Adaptive Control, Newcastle, Australia [10] Fu, M (19 96) `Minimum Switching Control for Adaptive Tracking', Proceedings 25th IEEE Conference on Decision and Control, Kobe, Japan, 3749±3754 [11] Furuta, K (1990) `Sliding Mode Control of a Discrete System' , Systems and Control Letters, 14, 145±152 [12]... discrete-time system is given by
5:108 y
t 1 a1 y
t a2 y
t À 1 a3 y
t À 2 u
t
t where a1 ; a2 ; a3 are unknown constant parameters, and
t 0 sin
0:9t represents exogenous disturbance The a priori uncertainty bounds are given by a1 P À1 :6; À0:1 0:1; 1 :6 ; b2 P À1 :6; À0:1 0:1; 1 :6 ; a3 P 0:1; 1 :6
5:109 i.e fÀ1 :6; À0:1 0:1; 1 :6 ÂÀ1 :6; À0:10:1; 1 :6 Â0:1; 1 :6 g... Multivariable Linear Systems', in Proceedings of the 23rd IEEE Conference on Decision and Control, Las Vegas, 1574±1577 [5] Byrnes, C I., Lin, W and Ghosh, B K (1993) `Stabilization of Discrete-Time Non-linear Systems by Smooth State Feedback', Systems and Control Letters, 21, 255± 263 [6] Chang, M and Davison, E J (1995) `Robust Adaptive Stabilization of Unknown MIMO Systems using Switching Control' , Proceedings...
t P À1 :6; À0:15 0:15; 1 :6 ; b2
t P À2; À1 1; 2
5:1 06 i.e fÀ1 :6; À0:15 0:15; 1 :6 Â À2; À1 1; 2g To meet the requirements of the localization technique, we decompose into 60 0 nonintersecting subsets with their centres i
b1i ; b2i ; i 1; F F F ; 60 0 corresponding to 110 Adaptive stabilization of uncertain discrete-time systems 400 0 output 20 200 0 0 control 50... Controllers For Nonlinear Systems', Automatica, 23, 209±214 [2] Bai, E W (1988) `Adaptive Regulation of Discrete-time Systems by Switching Control' , Systems and Control Letters, 11, 129±133 [3] Barmish, B R (1985) `Necessary and Sucient Conditions for Quadratic Stabilizability of an Uncertain System' , J Optimiz Theory Appl., 46, 4, 399±408 [4] Byrnes, C and Willems, J (1984) `Adaptive Stabilization of... S (1984) Adaptive Filtering Prediction and Control Prentice-Hall, Englewood Clis, N.J [13] Hocherman-Frommer, J., Kulkarni, S R and Ramadge, P (1995) `Supervised Switched Control Based on Output Prediction Errors', Proc 34th Conference on Decision and Control, 23 16 2317 [14] Hocherman-Frommer, S K J and Ramadge, P (1993) `Controller Switching Based on Output Prediction Errors, preprint, Department... the closed loop system Let be unknown, then for any
t0 > 0 the " 1 times inequality (5.1 26) can be possibly violated no more than
= Relying on this fact and using standard arguments exponential stability of the closed loop system is easily established 1 16 Adaptive stabilization of uncertain discrete-time systems References [1] Agarwal, M and Seborg, D E (1987) `Self-tuning Controllers For . by a 1 PÀ1 :6; À0:10:1; 1 :6 ; b 2 PÀ1 :6; À0:10:1; 1 :6 ; a 3 P0:1; 1 :6 5:109 i.e. fÀ1 :6; À0:10:1; 1 :6 ÂÀ1 :6; À0:10:1; 1 :6 Â0:1; 1 :6 g. Choosing Adaptive Control Systems 111 0. 26, 4, 65 1 -67 7. [ 16] Ioannou, P. A. and Sun, J. (19 96) Robust Adaptive Control. Prentice-Hall. [17] Kreisselmeier, G. (19 86) `Adaptive Control of a Class of Slowly Time-varying Plants', Systems. `Adaptive Control Based on Special Compensation Methods for Time-varying Systems Subject to Bounded Disturbances', Int. J. Control, 61 , 3, 66 7 69 9. [8] Fu, M. and Barmish, B. R. (19 86) `Adaptive
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