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signi®cant advances in adaptive and robust control in recent years, control of systems with large-size uncertainty remains a dicult task. Not only are the control problems complicated, so is the analysis of stability and performance. It is well known [12,14] that classical adaptive algorithms prior to 1980 were all based on the following set of standard assumptions or variations of them: (i) An upper bound on the plant order is known. (ii) The plant is minimum phase. (iii) The sign of high frequency gain is known. (iv) The uncertain parameters are constant, and the closed loop system is free from measurement noise and input/output disturbances. Classical adaptive algorithms are known to suer from various robustness problems [34]. A number of attempts have been made since 1980 to relax the assumptions above. A major breakthrough occurred in the mid-1980s [17, 21, 35] for adaptive control of LTV plants with suciently small in the mean parameter variations. Later attempts were made for a broader class of systems. Fast varying continuous-time plants were treated in [36], assuming knowledge of the structure of the parameter variations. By using the concept of polynomial dierential (integral) operators the problem of model reference adaptive control was dealt with in [32] for a certain class of continuous-time plants with fast time-varying parameters. An interesting approach based on some internal self-excitation mechanism was considered in [7] for a general class of LTV discrete-time systems. The global boundedness of the state was proved. However, it must be noted that the presence of such self-excitation signals in a closed loop system is often undesirable. In another research line, a number of switching control algorithms have been proposed recently by several authors [2, 6, 8, 20, 23, 24, 31], thus signi®cantly weakening the assumptions in (i)±(iv). Both continuous and discrete linear time-invariant systems were considered. Research in this direction was originated by the pioneering works of Nussbaum [31] and Martensson [20]. Nussbaum considered the problem of ®nding a smooth stabilizing controller ztf gt; zt utgyt; zt & 5:1 for the one-dimensional system xtaxtqut ytxt & 5:2 with both q T 0anda > 0 unknown. In [31] Nussbaum describes a whole family of controllers of the form (5.1) which achieve the desired stabilization of Adaptive Control Systems 81 the system (5.2). For example, it was shown that every solution xt; zt of x ax qxz 2 1 osz=2 exp z 2 z xz 2 1 & 5:3 has the property that lim t3I xt0 and lim t3I zt exists and is ®nite. We note that the structure of the adaptive controller is explicitly seen from (5.3). Another important result proved in [31] is that there exists no stabilizing controller for the plant (5.2) expressed in terms of polynomial or rational functions. A more general result was presented by Martensson [20]. In particular, it was shown in [20] that the only a priori information which is needed for adaptive stabilization of a minimal linear time-invariant plant is the order of a stabilizing controller. This assumption can even be removed if a slightly more complicated controller is used. Consider the following dynamic feedback problem. Given the plant x Ax Bu; x P R n ; u P R m ; y Cx; y P R r & 5:4 and the controller z Fz Gy; z P R l u Hz Ky @ 5:5 where m, r are known and ®xed, and n is allowed to be arbitrary. It is easy to see that this is equivalent to the static feedback problem x A x B u y C x; u K y @ 5:6 where x x T z T T , u u T z T T , y y T z T T and A, B, C, and K are matrices of appropriate dimensions. Let the regulator be u ghkNhk y k jj yjj 2 jj ujj 2 @ 5:7 where Nh is an `almost periodic' dense function and h and g are continuous, scalar functions satisfying a set of four assumptions (see [20] for more details). Martensson's result reads: `Assume that l is known so that there exists a ®xed stabilizing controller of the form (5.5), and that the augmentation to the form (5.6) has been done. Then the controller (5.7) will stabilize the system in the sense that xt; zt; kt 3 0; 0; k I as t 3I 5:8 where k I < I'. 82 Adaptive stabilization of uncertain discrete-time systems One such set of functions given by Martensson is hklog k 1=2 ; k ! 1; ghsin h 1=2 1h 1=2 5:9 Martensson's method is based on a `dense' search over the control parameter space, allows for no measurement noise, and guarantees only asymptotic stability rather than exponential stability. These weaknesses were overcome in [8] where a ®nite switching control method was proposed for LTI systems with uncertain parameters satisfying some mild compactness assumptions. Dierent modi®cations of Martensson's controller aimed at achieving Lyapunov stability, avoiding dense search procedures, as well as extending this approach to discrete-time systems have been reported recently (see, e.g., [2, 8, 19, 23]). However, the lack of exponential stability might result in poor transient performance as pointed out by many researchers (see, e.g., [8,19] for simulation examples). Below we present a simple example of a controller based on a dense search over the parameter space. This controller is a simpli®ed version of that presented in [19]. Example 1.1 The second order plant xt 1a 1 xta 2 xt À 1butt; x; u P R 5:10 with a 1;2 P R; b T 0 being arbitrary unknown constants and sup t!t 0 j tj < I has to be controlled by the switching controller utktxt5:11 where k0h1 and kthi, t Pt i ; t i1 and hi is a function dense in R de®ned so that it successively looks at each interval Àp; p; p P N and tries points 1=2 p apart, namely, h11 h4À0:5 h71:75 h20:5 h5À1 etc: h30 h62 The system performance is monitored using a function ÀtMt iÀ1 t iÀ1 tÀt iÀ1 jxt iÀ1 j t iÀ1 5:12 For each i > 1 such that t iÀ1 TI, the switching instant is de®ned as t i minft X t> t iÀ1 ; jxtj>Mt iÀ1 t iÀ1 tÀt i jxt iÀ1 jt iÀ1 g if this exists I otherwise 5:13 @ where 0 < Mt,0<t < 1 and 0 <t are strictly positive increasing functions satisfying the following conditions lim t3I MtI, Adaptive Control Systems 83 lim t3I t1, lim t3I tI. The behaviour of the closed loop system with a 1 À2:2, a 2 0:3 and b 1 is illustrated in Figure 5.1(a)±(b). A dierent switching control approach, called hysteresis switching, was reported in a number of papers [22, 27, 37] in the context of adaptive control. In these papers, the hysteresis switching is used to swap between a number of `standard' adaptive controllers operating in regimes of the parameter space. The switching, in these cases, is used to avoid the `stabilizability' problem in adaptive controllers. Conventional switching control techniques are all based on some mechanism of an exhaustive search over the entire set of potential controllers (either a continuum set [20] or a ®nite set [8]). A major drawback is that the search may converge very slowly, resulting in excessive transients which renders the system `unstable' in a practical sense. This phenomenon can take place even if the closed loop system is exponentially stable. To alleviate this problem, several new switching control schemes have been proposed recently. The so-called supervisory control of LTI systems for adaptive set-point tracking is proposed by Morse [25, 26] to improve the transient response. A further extension of Morse's approach is given in [13]. A very similar, in spirit, supervisory control scheme for model reference adaptive control is analysed in [29]. The main idea of supervisory control is to orchestrate the process of switching into feedback controllers from a pre-computed ®nite (continuum) set of ®xed controllers based on certain on-line estimation. This represents a signi®cant departure from traditional estimator based tuning algorithms which usually employ recursive or dynamic parameter tuning schemes. This approach has apparently signi®cantly improved the quality of regulation, thus demonstrating that switching control if properly performed is no longer just a nice theoretical toy but a powerful tool for high performance control systems design. However, several issues still remain unresolved. For example: 84 Adaptive stabilization of uncertain discrete-time systems 0 50 100 −1 −0.5 0 0.5 1 x 10 9 output (a) 0 50 10 0 −2 −1 0 1 2 controller gain (b) Figure 5.1 Example of a dense search (i) a ®nite convergence of switching is not guaranteed. This aspect is especially important in situations when convergence of switching is achievable. It seems intuitively that in adaptive control of a linear time- invariant system it is desirable that the adaptive controller `converges' to a linear time-invariant controller; (ii) the analysis of the closed loop stability is quite complicated and often dependent on the system architecture. Without a simpler proof and better understanding of the `hidden' mechanisms of supervisory switching control its design will remain primarily a matter of trial and error. In this chapter, we present a new approach to switching adaptive control for uncertain discrete-time systems. This approach is based on a localization method, and is conceptually dierent from the supervisory control schemes and other switching schemes. The localization method was initially proposed by the authors for LTI systems [39]. This method has the unique feature of fast convergence for switching. That is, it can localize a suitable stabilizing controller very quickly, hence the name of localization. Later this method was extended to LTV plants in [40]. By utilizing the high speed of localization and the rate of admissible parameter variations exponential stability of the closed loop system was proved. The main contribution of this chapter is a uni®ed description of the method of localization. We show that this method is also easy to implement, has no bursting phenomenon, and can be modi®ed to work with or without a known bound on the exogenous disturbance. To highlight the principal dierences between the proposed framework and existing switching control schemes, in particular, supervisory switching control, we outline potential advantages of localization based switching control: (i) The switching controller is ®nitely convergent provided that the system is time invariant. Depending on how the switching controller is practically implemented the absence of this property could potentially have far reaching implications. (ii) Unlike conventional switching control based on an exhaustive search over the parameter space, the switching converges rapidly thus guaranteeing a high quality of regulation. (iii) The closed loop stability analysis is comparatively simple even in the case of linear time-varying plants. This is in sharp contrast to supervisory switching control where the stability analysis is quite complicated and depends on the system architecture. (iv) Localization based switching control is directly applicable to both linear time-invariant and time-varying systems. (v) The localization technique provides a clear understanding of the control mechanism which is important in applications. The rest of this chapter is organized as follows. Section 5.2 introduces the class Adaptive Control Systems 85 of LTI systems to be controlled and states the switching adaptive stabilization problem. Two dierent localization principles are studied in Sections 5.3 and 5.4. We also study a problem of optimal localization, which allows us to obtain guaranteed lower bounds on the number of controllers discarded at each switching instant and adaptive stabilization in the presence of unknown exogenous disturbance. Simulation examples are given in Section 5.5 to demonstrate the fast switching capability of the localization method. Conclusions are reached in Section 5.6. 5.2 Problem statement We consider a general class of LTI discrete-time plants in the following form: Dz À1 ytNz À1 utt À 1t À 15:14 where ut is the input, yt is the output, t is the exogenous disturbance, t represents the unmodelled dynamics (to be speci®ed later), z À1 is the unit delay operator: Nz À1 n 1 z À1 n 2 z À2 FFFn n z Àn 5:15 Dz À1 1 d 1 z À1 FFF d n z Àn 5:16 Remark 2.1 By using simple algebraic manipulations, measurement noise and input disturbance are easily incorporated into the model (5.14). In this case, yt, ut, and t represent the measured output, computed input and (generalized) exogenous disturbance, respectively. For example, if a linear time-invariant discrete-time plant is described by yz Nz À1 Dz À1 uzdz qz where dz and qz are the input disturbance and plant noise, respectively, the plant can be rewritten as Dz À1 yzNz À1 uzNz À1 dzDz À1 qz À1 Consequently, the exogenous input z is Nz À1 dzDz À1 qz À1 . We will denote by the vector of unknown parameters, i.e. n n ; FFF; n 2 ; Àd n ; FFF; Àd 1 ; n 1 T 5:17 Throughout the chapter, we will use the following nonminimal state-space description of the plant (5.14): xt 1AxtButEtt 5:18 86 Adaptive stabilization of uncertain discrete-time systems where xtut À n 1ÁÁÁut À 1jyt À n 1ÁÁÁyt T 5:19 and the matrices A, B and E are constructed in a standard way A 01 0ÁÁÁ 0 ÁÁÁ 00 F F F F F F F F F F F F 0 ÁÁÁ 01 0ÁÁÁ 00 0 ÁÁÁ 00 0ÁÁÁ 00 00ÁÁÁ 0010ÁÁÁ F F F F F F F F F F F F 00ÁÁÁ 00ÁÁÁ 01 n n n nÀ1 ÁÁÁ n 2 Àd n ÁÁÁ Àd 2 Àd 1 P T T T T T T T T T T T T T T T R Q U U U U U U U U U U U U U U U S 5:20 B 0 F F F 0 1 C 0 F F F 0 n 1 P T T T T T T T T T T T T T T T T T R Q U U U U U U U U U U U U U U U U U S Y E 0 F F F 0 0 C 0 F F F 0 1 P T T T T T T T T T T T T T T T T T R Q U U U U U U U U U U U U U U U U U S 5:21 We also de®ne the regressor vector t xt ut ! 5:22 Then, (5.14) can be rewritten as yt T t À 1t À 1t À 15:23 The following assumptions are used throughout this section: (A1) The order n of the nominal plant (excluding the unmodelled dynamics) is known. (A2) A compact set P R 2n , is known such that P : Adaptive Control Systems 87 (A3) The plant (5.14) without unmodelled dynamics (i.e. t0) is stabiliz- able over . That is, for any P , there exists a linear time-invariant controller Cz À1 such that the closed loop system is exponentially stable. (A4) The exogenous disturbance is uniformly bounded, i.e. for all t 0 P N sup t!t 0 j tj " 5:24 for some known constant " : (A5) The unmodelled dynamics is arbitrary subject to jtj " t" sup 0 k t tÀk jjxkjj 5:25 for some constants ">0 and 0 <1 which represent the `size' and `decay rate' of the unmodelled dynamics, respectively. Remark 2.2 Assumption (A1) can be relaxed so that only an upper bound n mx is known. Assumption (A4) will be used in Sections 5.3±5.4 and will be relaxed to allow " to be unknown in Sections 5.3.2 and 5.4.1 where an estimation scheme is given for " . Remark 2.3 We note that the assumptions outlined above are quite standard and have been used in adaptive control to derive stability results for systems with unmodelled dynamics (see, e.g., [7, 16, 21, 30] for more details). The switching controller to be designed will be of the following form: utK it xt5:26 where K it is the control gain applied at time t, and it is the switching index at time t, taking value in a ®nite index set I. The objective of the control design is to determine the set of control gains K I fK i ; i P Ig5:27 and an on-line switching algorithm for it so that the closed loop system will be `stable' in some sense. We note that switching controllers can be classi®ed according to the logic governing the process of switching. Here are some typical examples. 1. Conventional switching control The switching index is de®ned as it it À 1 if G t 0 it À 11 otherwise & 5:28 where G t is some appropriately chosen performance index. This type of 88 Adaptive stabilization of uncertain discrete-time systems switching control is ®nitely convergent and based on an exhaustive search over the parameter space (see, e.g., [8], [9]). 2 Supervisory switching control The switching index is de®ned as it it À 1 if t À st < t d arg min iPI je i tj otherwise & 5:29 where st is the time of the most recent switching, t d is a positive dwell time, and e i t, Vi P I is a weighted prediction error computed for the ith nominal system. This type of switching control has been extensively studied recently by a number of researchers (see, e.g., [25, 26]). The proof of the closed loop stability in this case is not dependent on ®nite convergence of the switching process, furthermore, supervisory switching control is not ®nitely convergent in general. 5.3 Direct localization principle The switching algorithms to be used in this section are based on a localization technique. This technique, originally used in [39] for LTI plants, allows us to falsify incorrect controllers very rapidly while guaranteeing exponential stab- ility of the closed loop system. In this section, we describe a direct localization principle (see, e.g., [40]) for LTI plants which is slightly dierent from [39] but is readily extended to LTV plants. The main idea behind this principle consists of simultaneous falsi®cation of potentially stabilizing controllers based ex- plicitly on the model of the controlled plant. That implies the use of some eective mechanism of discarding controllers inconsistent with the measurements. The speci®c notion of stability to be used in this section is described below. De®nition 3.1 The system (5.14) satisfying (A1)±(A5) is said to be globally " - exponentially stabilized by the controller (5.26) if there exist constants M 1 > 0, 0 <<1, and a function M 2 Á X R 3 R with M 2 00 such that jjxtjj M 1 tÀt 0 jjxt 0 jj M 2 " 5:30 holds for all t 0 ! 0, xt 0 , " ! 0, and Á and Á satisfying (A4)±(A5), respectively. The de®nition above yields exponential stability of the closed loop system provided that " 0 and exponential attraction of the states to an origin centred ball whose radius is related to the magnitude of the exogenous disturbance. First, we decompose the parameter set to obtain a ®nite cover f i g L i1 which satis®es the following conditions: Adaptive Control Systems 89 (C1) i & , i Tfg; i 1; FFF; L: (C2) L i1 i . (C3) For each i 1; FFF; L, let i and r i > 0 denote the `centre' and `radius' of i , i.e. i P i and jj À i jj r i for all P i . Then, there exist K i , i 1; FFF; L, such that j mx ABK i j < 1; Vk À i k r i ; i 1; FFF; L: 5:31 Conditions (C1)±(C2) basically say that the uncertainty set is presented as a ®nite union of non-empty subsets while condition (C3) de®nes each subset i as being stabilizable by a single LTI controller K i . It is well known that such a ®nite cover can be found under assumptions (A1)±(A3) (see, e.g., [8, 24, 25] for technical details and examples). More speci®cally, there exist (suciently large) L, (suciently small) r i , and suitable K i , i 1; FFF; L, such that (C1)±(C3) hold. Leaving apart the computational aspects of decomposing the uncertainty set satisfying conditions (C1)±(C3) we just note that decomposition can be conducted o-line, moreover, some additional technical assumptions (see, e.g., (C3 H ) below) make the process of decomposing pretty trivial. The computational complexity of decomposing the uncertainty set, in general, depends on many factors including the `size' of the set, its dimension and `stabilizability' properties, and has to be evaluated on a case-by-case basis. The key observation used in the localization technique is the following fact: given any parameter vector P j and a control gain K it for some it; j 1; FFF; L.Ifitj, then it follows from yt T t À 1t À 1t À 15:32 that j T j t À 1Àytj r j jjt À 1jj " " t À15:33 This observation leads to a simple localization scheme by elimination: If the above inequality is violated at any time instant, we know that the switching index it is wrong (i.e. it T j), so it can be eliminated. In identi®cation theory this concept is sometimes referred to as falsi®cation; see, e.g., a survey [15] and references therein. The unique feature of the localization technique comes from the fact that violation of (5.33) allows us not only to eliminate it from the set of possible controller indices, but many others. This is the key point! As a result, a correct controller can be found very quickly. We now describe the localization algorithm. Let It denote the set of `admissible' control gain indices at time t and initialize it to be It 0 f1; 2; FFF; Lg5:34 Choose any initial switching index it 0 PIt 0 . For t > t 0 , de®ne It j X 5:33 holds; j 1; FFF; L fg 5:35 Then, the localization algorithm is simply given by 90 Adaptive stabilization of uncertain discrete-time systems [...]... switching = instant, and controller Kj is discarded From (5. 35) we have j P I
t 1, equivalently " " T
t > y
t 1
rj qjj
tjj
t 1 j or 5: 45 " " T
t < y
t 1 À
rj qjj
tjj À À
t 1 j 5: 46 Taking z À
t=jj
tjj for (5. 45) , or z
t=jj
tjj for (5. 46) and using Adaptive Control Systems 95 (5. 43) we see that there are
z; j; Â number of controller indices... plant is, using (5. 18) z
t 1i XCAx
t CBu
ti CE
t z
t 1 À CB u
t À u
ti 5: 83 Adaptive Control Systems 1 05 Note that if the true plant is in the set i , then from (5. 83) and (5. 81) CB i z
t 1 C A À Ai x
t CE
t 5: 84 CBi " and, therefore, if the true plant is in i , then from (5. 82), and with c0 jCE j jz
t 1i j Ákx
tk c0 5: 85 Our proposed control algorithm... 5: 74 From (5. 74) it is clear that if z and are bounded, then in view of (5. 70), x
t is bounded Therefore, s t is a stabilizing inclusion for any c0 Finally, we take the control 1
CAx
t 5: 75 u
t À CB which gives z
t 1 CE
t 5: 76 Therefore, for c0 ! jCEj supt j
tj; s t is satis®ed for all t > 0, and the proof is complete Remark 4.2 The control, (5. 75) , is a `one step ahead' control. .. f1 ; 3 g; f2 ; 4 g; f3 ; 4 gg; K3 ff1 ; 2 ; 5 g; f1 ; 3 ; 5 g; f3 ; 4 ; 5 g; f2 ; 4 ; 5 gg Since JPK3 J  we conclude that ind  3 and the optimal switching index is given by i
t 5 To compute a guaranteed lower bound on the index of 1 2 * * 5 * 3 4 * Figure 5. 3 Example of optimal localization * Adaptive Control Systems 99 localizaton ind  Algorithm B is used We have... 1.2 5: 88 FFF 1 .5 Apply the `median' control: We then have the following stability result for this control algorithm Theorem 4.1 The control algorithm, (5. 86)± (5. 88), applied to a plant where C is known, and where the decomposition (5. 80) has the properties that (5. 82) is satis®ed and s t is a stabilizing inclusion, has the following properties: (a) The inclusion: s t X jz
tj Ákx
t À 1k c0 5: 89... and so for any c0 > 0; z
t satis®es (5. 69) The equation for the closed loop system takes the form x
t 1 Ax
t Bu
t PAx
t 5: 73 which is not exponentially stable Therefore, (5. 72) implies that there is no c0 such that s t is a stabilizing inclusion We now establish the converse Suppose Adaptive Control Systems 103 (5. 70) is satis®ed Then we can rewrite (5. 18) as: x
t 1 PAx
t Bu
t... 2; F F F ; Lg 5: 34 Choose any initial switching index i
t0 P I
t0 For t > t0 , de®ne I
t f j X 5: 33 holds; j 1; F F F ; Lg Then, the localization algorithm is simply given by 5: 35 Adaptive Control Systems I
t I
t À 1 I
t; Vt > t0 91 5: 36 1 The switching index is updated by taking & i
t À 1 if t > t0 and i
t À 1 P I
t i
t any member of I
t otherwise 5: 37 A simple... Âg k 5: 51 then there must exist an element j P Â, such that ind
j ; Â m, moreover = j P HmÀ1 ; j P Â À HmÀ1 5: 52 since otherwise, by de®nition of separable sets ind
j ;  m À 1 But it follows from (5. 51) that  À HmÀ1 fg On the other hand by De®nition 3.3 and the properties of separable sets (b), (c) the index of localization of the set  cannot be smaller than that given by (5. 50) This... switching control are summarized in the following theorem Let subfÁg denote the set of subscripts of all the elements in fÁg Theorem 3.2 (i) The solution to the problem of optimal localization may not be unique and is given by the set 5: 53 Iopt subf À HmÀ1 g where m ind  1 arg mxfk X Hk T Âg k 5: 54 " (ii) For any ! 0; " ! 0, the total number of switchings l made by the Adaptive Control Systems... the algorithm of localization (5. 36) Lemma 3.1 Given the uncertain system (5. 14) satisfying assumptions (A1)± (A5), suppose the ®nite cover fi gL of satis®es conditions (C1)±(C3) Then, i1 the localization algorithm given in (5. 34)± (5. 37) applied to an LTI plant (5. 14) possesses the following properties: (i) I
t T f g; Vt ! t0 : 1 In fact, we will see in Section 5. 3.1 that there may be `clever' . class Adaptive Control Systems 85 of LTI systems to be controlled and states the switching adaptive stabilization problem. Two dierent localization principles are studied in Sections 5. 3 and 5. 4 unresolved. For example: 84 Adaptive stabilization of uncertain discrete-time systems 0 50 100 −1 −0 .5 0 0 .5 1 x 10 9 output (a) 0 50 10 0 −2 −1 0 1 2 controller gain (b) Figure 5. 1 Example of a dense. of uncertain discrete-time systems optimal switching controller (5. 48), (5. 49) applied to an LTI plant (5. 14) satis®es the relation lÀ1 p0 ind Ât p À2 4 L À 1 5: 55 where t p ; p 0; 1;
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