Multi-Layer Switching Control

Transcription

1 5 American Conrol Conference June 8-1, 5. Porland, OR, USA FrC7.3 Muli-Layer Swiching Conrol Idin Karuei, Nader Meskin, Amir G. Aghdam Deparmen of Elecrical and Compuer Engineering, Concordia Universiy Monréal, Quebec, Canada H3G 1M8 i karuei@ece.concordia.ca,n meskin@ece.concordia.ca,aghdam@ece.concordia.ca Absrac In his paper, adapive conrol of sysems using swiching echniques is invesigaed. I is assumed ha he plan model belongs o a known finie se of models. I is also assumed ha a se of conrollers which solve he robus servomechanism problem for he family of plan models and a se of simulaneous sabilizers for cerain subses of plan models are given. I is shown ha by using he above se of conrollers and simulaneous sabilizers and choosing a proper swiching sequence, one can minimize he number of swichings o desabilizing conrollers. This can significanly improve he ransien response of he sysem, which is one of he common weak poins in mos swiching conrol schemes. Simulaion resuls show he effeciveness of he proposed mehod in improving he ransien response. Keywords: Swiching conrol, Adapive conrol, Family of plans, Simulaneous sabilizer, Transien response. I. INTRODUCTION The conrol of a parially known plan has received considerable aenion in he adapive conrol lieraure. One of he relaively new lines of research in his area is swiching conrol which was moivaed o weaken he classical a priori informaion required in convenional adapive conrol and can be raced back o [1]. During he pas several years, swiching conrol schemes have been developed o accomplish a wide variey of asks which would no have been possible using radiional adapive conrol mehods [], [3], [4], [5], [6], [7], [8], [9], [1]. Swiching conrol of sysems using family of plans was firs inroduced by Miller and Davison [4]. In his approach, i is assumed ha a high performance conroller is designed for each plan in he family. Then, by using a proper swiching mechanism which moniors he norm of error in he oupu, he sysem locks ono a sabilizing conroller afer a number of swichings. One of he advanages of swiching conrol compared o he convenional adapive conrol is is effeciveness for highly uncerain sysems. However, he main disadvanage of swiching conrol is bad ransien response in general. Several mehods have been proposed o reduce he magniude of he ransien response [11], [1]. One of he main reasons for undesirable ransien response in swiching conrol sysems is ha in he ransiion from he iniial conroller o he final one, he sysem may swich o several desabilizing conrollers. In his paper, a mehod is proposed o improve he ransien response of he swiching conrol sysem by reducing This work has been suppored by he Naural Sciences and Engineering Research Council of Canada under gran RGPIN /5/$5. 5 AACC 477 he number of swichings o desabilizing conrollers. The proposed mehod uilizes differen layers of conrollers wih differen properies. This is an exension of he swiching mehod inroduced in [4], which assumes ha he se of plans models {P i : i =1,,...,p} is given, and upperbounds on he disurbance and reference inpu magniudes are available. I is also assumed ha he plan is conrollable and observable. Swiching in he sysem occurs when he norm of he error signal becomes greaer han or equal o he corresponding upper-bound signal. In our proposed mulilayer scheme, p layers of conrollers are designed, where p denoes he number of models in he family of plans. Layer k {,...,p } consiss of a se of conrollers which have he propery ha each one sabilizes k plans in he family and desabilizes he remaining p k plans. Layer 1 consiss of a se of p conrollers, where each one solves he robus servomechanism problem for one of he models in he family. The main difference beween he previous swiching mehods and he proposed one is he addiional conrollers which represen layers,...,p and are used o improve he ransien response. Throughou his paper, he previous swiching conrol mehods will be referred o as single-layer swiching. This paper is organized as follows. The problem formulaion is given in secion II. A muli-layer swiching sysem is proposed in secion III and in secion IV he mehod is compared o a single-layer counerpar. II. PROBLEM FORMULATION I is assumed ha he curren plan P() belongs o a known finie se of plan models given by : P() Π:={P i : i =1,,...p} (1) I is also assumed ha each plan model in he above se is described by he following sae-space equaions ẋ = A i x + B i u + E i w y = C i x + F i w (a) (b) where x() R n i, i p = {1,,...,p} is he sae, u() R m is he conrol inpu, y() R r is he oupu, y ref () R r is he reference inpu and w() R v is he disurbance signal. Reference inpu and disurbance are bounded piecewise coninuous funcions. As in he previous works, i is assumed ha for each i p here exiss a high performance conroller K i of he

2 form ż = G i z + H i u + J i y ref (3a) u = K i z + L i y + M i y ref (3b) This se represens he firs layer of conrollers in our proposed muli-layer archiecure and is denoed by Φ 1. Φ 1 = {K i : i p}, N(Φ 1 )=N(p) (4) On he oher hand, he se of conrollers of layer k, k =,...,p is denoed by Φ k, as follows Φ k = {K i1 i...i k i 1,i,...,i k p} (5) where i j,j =1,...,k are disinc inegers and he indices of each conroller represen he plans ha can be sabilized by ha conroller, e.g. K i1 i...i k only sabilizes plan models P i1, P i,...,p ik, and desabilizes he oher plans in he se Π. According o he above definiion, in each layer k p, here exis N(Φ k ) k combinaions of sable closed-loop configuraions. The oal number of all sable configuraions corresponding o all conroller layers is given by p φ = N(Φ k ) k k=1 The closed-loop conrol law corresponding o he conroller K i1 i...i k can be wrien in he following form [4] ũ = K i1 i...i k ỹ (6) which resuls in a sable sysem corresponding o he conrollable and observable plan P j : j {i 1,i,...i k } as follows x = Ãj x + B j ũ + Ẽjw (7a) where and à j = [ x x = z [ Aj D j = I ỹ = C j x + D j y ref + F j w ] [ u, ũ = ż ] [ Bj, Bj = I, Ẽ j = [ Ej ], ỹ = y z y ref ], Cj = ], Fj = F j (7b) (8) C j I, [ ] Li1i K i1 i...i k =...i k K i1i...i k M i1i...i k H i1 i...i k G i1 i...i k J i1 i...i k, According o he above formulaion, here are differen layers of conrollers which are used in he muli-layer swiching mehod. Definiion 1: Throughou his paper, a swiching o a desabilizing conroller will be called an unsable swiching. I is desired now o find a swiching pah which consiss of a mos one unsable swiching in general, beween he conrollers of differen layers. Figure 1 shows a family of 6 plan models and he archiecure of conroller layers, where he plan models are represened by black circles and conrollers of layer 1,, 3 and 4 are represened by riangles, squares, penagons and hexagons, respecively. Layer 4 Layer 3 K613 K345 K4561 K13 K34 K456 Layer K3 K45 Layer 1 Plans Fig. 1. K1 K K3 K4 K5 K6 P 1 P P 3 P 4 P 5 P 6 Four layers of conrollers for six plan models. III. MAIN RESULT One of he shorcomings of swiching conrol mehods is he bad ransien response which is mainly conribued by swiching o desabilizing conrollers before he sysem locks ono he correc conroller. When he plan dynamics change from one model o anoher one, he sysem may swich o a series of desabilizing conrollers unil i finds he correc conroller. Addiion of new layers of conrollers can poenially reduce he number of unsable swichings as will be shown laer. Definiion : Any conroller whose indices include all bu one of he indices of anoher conroller is called a paren of ha conroller. For insance, K i1i...i k 1 is a paren of K i1i...i k. On he oher hand, any conroller in a layer oher han layer 1 is called a child o is paren in he lower layer. Definiion 3: A child-paren swiching roue is a swiching pah from a conroller in one of he higher layers o a conroller in he firs layer ha consiss of only childparen conrollers. A. Muli-layer Swiching Algorihm Assume ha he plan is sabilized by conroller K i1 in he firs layer. Once a change in he model occurs, he new plan model is known o be one of he remaining p 1 models p {i 1 } = {i,i 3,...,i p }, which resuls in insabiliy of he closed-loop sysem. The following assumpion is made for he developmen of he proposed algorihm. Assumpion 1: i) There exiss a conroller in layer p which desabilizes wo of plan models and sabilizes all oher models. 4773

3 ii) Each of he plan models is desabilized by a leas one conroller in layer p. iii) There exiss a child-paren roue from any of he conrollers in layer p o a leas one conroller in he firs layer. Algorihm 1 1) Swich o a conroller in layer p which can sabilize all of he models excep P i1 and any one of he oher models in he se. Le his conroller be denoed by K i3 i 4...i p.if he closed-loop sysem becomes unsable, he acual plan model is idenified o be P i. Swich o K i and sop. Oherwise, go o he nex sep. ) A his poin, i is known ha he acual plan model belongs o he se {P i i = i 3,i 4,...,i p }. Swich o one of he paren conrollers of K i3 i 4...i p in layer p 3. Le his conroller be denoed by K i4i 5...i p. If he closed-loop sysem becomes unsable, he acual plan model is idenified o be P i3. Swich o K i3 and sop. Oherwise, go o he nex sep. 3) A his poin, i is known ha he acual plan model belongs o he se {P i i = i 4,i 5,...,i p }. Swich o one of he paren conrollers of K i4 i 5...i p in layer p 4. Le his conroller be denoed by K i5i 6...i p. If he closed-loop sysem becomes unsable, he acual plan model is idenified o be P i4. Swich o K i4 and sop. Oherwise, go o he nex sep.. p-) A his poin, i is known ha he acual plan model belongs o he se {P i i = i p 1,i p }. Swich o one of he wo conrollers K ip 1 or K ip. Assume ha we swich o K ip 1. If he closed-loop sysem becomes unsable, he acual plan model is idenified o be P ip ; swich o K ip and sop. Oherwise, he acual plan model is P ip 1 ; swich o K ip 1 and sop. I can be easily verified ha using he swiching sequence described in Algorihm 1, i is guaraneed ha he sysem will evenually swich o he correc conroller wih a mos one unsable swiching, provided all required conrollers in differen layers exis. Example 1: Assume ha here is a family of 6 plans as shown in Figure. Iniially, he acual plan model is P 4 which is sabilized by conroller K 4. Assume ha he plan model changes o P a ime which is he new unknown plan. The sysem becomes unsable and i swiches o K 613 which sabilizes P 1, P, P 3, P 6 and desabilizes P 4 and P 5. The sysem becomes sable and swiches o K 13 which is he paren of he previous conroller K 613. The sysem remains sable and should swich o a paren of he curren conroller. The curren conroller has hree parens K 1, K 13, and K 3. Assume ha he sysem swiches o K 3. The sysem becomes sable and should swich o one of he parens of K 3. Assume ha i swiches o K 3. This conroller desabilizes he sysem and i is he only ime he sysem becomes unsable. A his poin, he new acual plan model is idenified o be P and he sysem swiches o K. Layer 4 Layer 3 K613 K345 K4561 K13 K34 K456 Layer K3 K45 Layer 1 Plans K1 K K3 K4 K5 K6 P 1 P P 3 P 4 P 5 P 6 Fig.. Swiching in four layers. Solid arrows represen sable swichings and dashed arrow denoe an unsable swiching. B. Srucure of layers A muli-layer srucure can conain several conrollers for a family of plans. I is no necessary o have all hese conrollers in order o reduce he number of unsable swichings o less han or equal o one. Assume ha he condiions of Assumpion 1 are me. Then, once he plan model changes wihin he known se of plan models, i is guaraneed ha he sysem will swich o he correc conroller afer a mos one unsable swiching. Remark 1: I is o be noed ha he number of all possible conrollers for layer k ha can sabilize k plan models and desabilize he remaining p k models is p! equal o. Thus, he oal number of all possible k!(p k)! conrollers in he proposed muli-layer srucure is equal o. More specifically, he number of all possible p k=1 p! k!(p k)! conrollers for layer p is equal o p(p 1).However,i can be easily verified ha only p conrollers for layer 1 and fix( p+1 ) conrollers for oher layers would suffice, where fix(.) represens he neares ineger owards zero. Since designing simulaneous sabilizers for layers, 3,..., p can be difficul in pracice, one may design a mos (p ) simulaneous sabilizers. Example : Assume ha a family of seven plans Π= {P 1, P, P 3, P 4, P 5, P 6, P 7 } is given. The proposed mulilayer archiecure consiss of five layers. There can be 7 6 = 1 conrollers on he highes layer heoreically bu only 4 of hem are necessary. The condiions of Assumpion 1 can be saisfied in differen ways. For insance, one can choose he se of conrollers {K 1345, K 34567, K 5671, K 7134 } o represen Φ 5. The nex layers should hen be designed in a way ha child-paren roues exis from all conrollers of Φ 5 o hose of layer 1. Φ 4 = {K 345, K 3456, K 671, K 713 } is one of he several choices for layer four. Since he pair K 345 and K 3456 and he pair K 671 and K 713 have a common paren, a smar choice for conrollers of he hird layer would be 3)fix( p+1 Φ = {K 345, K 71 }. Layer wo can hen be chosen as Φ = {K 34, K 71 }. Finally, he firs layer should have seven conrollers as in a single-layer archiecure.

4 C. Swiching Mechanism The swiching insans will be obained by using he same approach as in [4]. The mehod consiss of wo phases. Firs, a bound on iniial condiion is obained and hen he desired conroller is found by swiching beween differen conrollers. Le he -norm of a vecor x R n be denoed by x. Similarly, for a marix A R n m, A will denoe he corresponding induced norm of A. 1) Finding a Bound on Iniial Condiion: The following resul from Lemma 1 in [4] provides an upper-bound for he iniial condiion wih u(.) = where W i = T T x() α i1 y(τ) dτ + α i b (9) e A i τ C Ce A iτ dτ, α i3 = he smalles singular value of W i, α i1 =/α i3, T [ ] α i =(/α i3 ) C ie Ai( τ) E i dτ + F i d b = upper-bound on norm of disurbance. ) Searching for he correc Conroller: In his phase conrol acion is applied and he upper-bound signal inroduced in [4] is given by ṙ j,i1 i...i 1 () =λ j r j,i1 i...i k () + γ (j,i1i...i k ) ũ() K i1 i...,i k (ỹ() D j y ref ) + γ (j,i1i...i k ) 3 b (13) wih iniial condiion r j,i1 i...i k (T + )=r j,i1 i...i k (T ) + γ (j,i1 i...i k ) 1 e λ jr j,i1 i...i T k µ j (14) Each closed-loop conroller-plan pair has an upper-bound signal which is a funcion of he norm of he error. I is ofen desired o use a smooh error signal by applying a filer as follows r = λ r()+(λ λ) ỹ() Dy ref, r(t )= (15) where λ <min{λ i : i p}. Each ime he filered error signal mees he upper-bound signal corresponding o he curren conroller K i1i...i k and plan P j (j {i 1,i,...,i k }), insabiliy is deeced. In oher words, he sysem will swich o anoher candidae conroller when Wih ũ() =for [,T] and z() =, find θ := T y(τ) dτ and define he following φ upper-bound signals for all sable closed-loop configuraions: ṙ j,i1i...i k () =λ j r j,i1i...i k () + γ (j,i1 i...i k ) K i1i...,i k (ỹ D j y ref ) + γ (j,i1 i...i k ) 3 b, [,T] r j,i1 i...i k () = (1) where from Lemma of [4] here exis λ j,i1 i...i k < and > such ha γ (j,i1i...i k ) 1 and e (Ãj+ B j Ki1 i...i k Cj ) γ (j,i1 i...i k ) 1 e λ j,i 1 i...i k (11) γ (j,i1 i...i k ) = γ (j,i1 i...i k ) 1 B j (1a) γ (j,i1 i...i k ) 3 Define = γ (j,i1 i...i k ) 1 Ẽj + B j Ki1i...i k Fj (1b) µ j =[α j1 θ + α j b ] 1 Assuming ha w() b, if he acual plan model is P j, i follows from (9) ha x() µ j r() = C j r j,i1 i...i k ()+ F j b + ε, (16) where ε is an arbirary posiive value [4]. The swiching sequence of Algorihm 1 requires ha he sysem swiches from he higher layer conrollers o he lower layer ones even if he sysem is sabilized in a higher layer. Unlike unsable swichings, sable swichings canno be idenified hrough he upper-bound signals. In order o deec sabiliy, a sufficienly long ime-inerval will be used such ha if he norm of error does no mee he upperbound signal, he sysem is sable. This ime-inerval can be obained by considering worse case scenario associaed wih iniial condiions, reference inpu and disurbance signal. I can also be obained experimenally. This ime duraion will be referred o as safey ime and will be denoed by d. Lemma 1: The conroller K i1i...i k desabilizes he sysem iff he filered error signal mees any of he upperbound signals corresponding o P i1, P i,...p ik and conroller K i1 i...i k. Proof of Lemma 1: Suppose ha he curren conroller is K i1i...i k which can sabilize boh P i1 and P i. For simpliciy and wih no loss of generaliy, i will be assumed ha Ci : i p has a uni norm. The upper-bound signals r i1,i 1 i...i k and r i,i 1 i...i k are used as in [4] and are obained from (1) and (13). Two new upper-bound signals are defined as follows ṙ i 1,i 1i...i k () =λ i1 i...i k r i 1,i 1i...i k () + γ (i1 i...i k ) ũ() K i1 i...i k (ỹ() D j y ref ()) (17) + γ (i1i...i k ) 3 b,

5 ṙ i,i 1 i...i k () =λ i1 i...i k r i,i 1 i...i k () + γ (i1 i...i k ) ũ() K i1i...i k (ỹ() D j y ref ()) + γ (i1 i...i k ) 3 b where (18) γ (i1i...i k ) 1 = max(γ (i1,i 1i...i k ) 1,γ (i,i 1i...i k ) 1 ), (19a) γ (i1 i...i k ) = max(γ (i1,i 1 i...i k ),γ (i,i 1 i...i k ) ), (19b) γ (i1i...i k ) 3 = max(γ (i1,i 1i...i k ) 3,γ (i,i 1i...i k ) 3 ), (19c) λ i1i...i k = max(λ i1,i 1i...i k,λ i,i 1i...i k ). (19d) Since (11) holds for he new upper-bound signals, eiher r i1,i 1i...i k or r i 1,i 1i...i k can be chosen as he upper-bound signal for P i1. A similar discussion can be made for he upper-bound signals of plan P i. In oher words, here exis ime insans τ >τ 1 and τ 4 >τ 3 such ha r(τ 1 )=r i1,i 1 i...i k (τ 1 )+ F i1 b + ε r(τ )=r i 1,i 1 i...i k (τ )+ F () i1 b + ε r(τ 3 )=r i,i 1i...i k (τ 3 )+ F i b + ε r(τ 4 )=r i,i 1i...i k (τ 4 )+ F (1) i b + ε Subracing (18) from (17) resuls in: r i 1,i 1,i 1i...i k r i,i 1,i 1i...i k = k e λ i 1 i...i k () where k > for r i 1,i 1,i 1 i...i k >r i,i 1,i 1 i...i k. I follows from () ha if r(τ )=r i,i 1i...i k (τ )+ F i b + ε ε >: ε = ε + k e λi 1 i...i k τ4 + F i1 F i b > (3) and τ 5 >τ 4 such ha r(τ 5 )=r i,i 1,i 1 i...i k (τ 5 )+ F i b + ε = r i,i 1,i 1i...i k (τ 5 )+k e λ i 1 i...i k τ 4 + F i b + F i1 F i b + ε (4) since F i + F i1 F i F i1 and e λi 1 i...i k τ4 e λ i 1 i...i k τ 5, hen r(τ 5 ) r i,i 1,i 1 i...i k (τ 5 )+k e λi 1 i...i k τ5 + F i1 b + ε = r i 1,i 1,i 1i...i k (τ 5 )+ F i1 b + ε (5) I follows from (3) and (5) ha r(τ 3 )=r i,i 1 i...i k (τ 3 )+ F i b + ε τ 6 <τ 5 : r(τ 6 )=r i 1,i 1i...i k (τ 6 )+ F i1 b + ε (6) Subsiuing τ = τ 6 ino () resuls in r(τ 6 )=r i,i 1 i...i k (τ 6 )+ F i b + ε r(τ 1 )=r i1,i 1i...i k (τ 1 )+ F (7) i1 b + ε This implies ha if he filered upper-bound signal mees he smaller boundary signal corresponding o he closedloop pair (K i1 i...i k, P i ), i will definiely mee he oher boundary signal corresponding o he closed-loop pair (K i1i...i k, P i1 ). Define i o be min(τ 1,τ 3 ). According o Lemma 1, for he higher layer conrollers only he smalles 4776 upper-bound signal associaed wih each conroller and is corresponding plan models needs o be compared o he filered signal as i resuls in smaller ime insans. Theorem 1: Consider he sysem (). Using he swiching sequence of Algorihm 1, and he swiching insans s = min( i 1 + d, i ) where i represens he ime insans given in Lemma 1 ( := ) and d is he safey ime, he sysem will evenually swich o he correc conroller wih no more han one unsable swiching. Proof of Theorem 1: The proof follows immediaely from Lemma 1 and he resuls of Theorem 1 in [4]. IV. NUMERICAL EXAMPLE Example 3: Consider he following unsable nonminimum phase plan model used in [9] and [13] : s 1 P = λ (s )(s +1), 1 <λ() < 6 A family of four plan models P i = {P 1, P, P 3, P 4 } is hen considered as follows s 1 P 1 = (s )(s +1), P =P 1, P 3 =4P 1, P 4 =6P 1 The high-performance conrollers of firs layer are obained as follows K 1 = 448s + 45s 18, 31s(s 9) K = 1 K 1, K 3 = 1 4 K 1, K 4 = 1 6 K 1. The second layer consiss of hree conrollers which are designed using he simulaneous sabilizaion mehod presened in [14] K 1 = 45.31s s s s s s s 515.6s 67.1 K 3 =.65s s s s s s s 515.6s 67.1 K 34 =.758s s s + 45s s s s 47.9s 87.1 I can be easily verified ha conroller K 1 sabilizes he plan models P 1, P and desabilizes P 3, P 4. Similarly, (K 3, P ), (K 3, P 3 ), (K 34, P 3 ) and (K 34, P 4 ) are sable closed-loop pairs while (K 3, P 1 ), (K 3, P 4 ), (K 34, P 1 ) and (K 34, P ) are unsable pairs. Assume ha iniially he acual plan model is P 1 and a some poin of ime i changes o P 4. In he single-layer approach he sysem will swich from K 1 o K, hen o K 3, and finally o K 4. The firs wo swiching insans are unsable. In he muli-layer approach he sysem will swich from K 1 o K 3, and hen o K 4. The only unsable swiching corresponds o K 3. Figures 3 and 4 show he closed-loop simulaion resuls using he proposed muli-layer algorihm and he single-layer mehod of [4], respecively. These figures show a 9% reducion in he magniude of he ransien response using he proposed muli-layer algorihm.

6 Now assume ha he plan model changes from P 4 o P 3. The single-layer mehod swiches from K 4 o K 1, hen o K, and finally o K 3. Two unsable swichings occur using single-layer mehod. However, he proposed muli-layer algorihm will swich from K 4 o K 3. The sysem becomes sable and hen swiches o K 3. I is o be noed ha in his case no unsable swiching occurs using muli-layer algorihm. I can be verified ha he maximum ampliude of he ransien response is 4.. I is much smaller han ha of he single-layer mehod of [4] which is V. CONCLUSION In his paper, a swiching conrol algorihm for a family of known plan models is proposed which has a good ransien response compared o he exising mehods. The algorihm uses differen layers of conrollers, where layer 1 consiss of one high-performance conroller for each plan model and oher layers consis of simulaneous sabilizers for cerain subses of family of plan models. The proposed algorihm gives a sequence of conrollers of differen layers for swiching and can be used wih any swiching scheme. In his paper, he swiching scheme of [4] is considered. I is guaraneed ha while he sysem searches for he correc conroller in he firs layer, i swiches o a mos one desabilizing conroller and all oher conrollers in he given sequence sabilize he sysem. The sysem swiches o he nex conroller if he curren conroller desabilizes he sysem, or he curren conroller sabilizes he sysem bu does no belong o layer 1. The closed-loop sysem is idenified o be unsable if he norm of he error his is corresponding upper-bound, and is idenified o be sable if i does no hi he upper-bound for a sufficienly long ime. The simulaion resuls show significan improvemen in he magniude of he ransien response compared o ha of he single layer mehod proposed in [4]. ACKNOWLEDGEMENT The auhors graefully acknowledge he useful commens of Shauheen Zahirazami in developmen of he algorihm. y() K4 Swiching insans K a K b Fig. 3. Closed-loop simulaion resuls for Example 3, using he mulilayer scheme, when he plan models change from P 1 o P 4. (a) Oupu signal; (b) swiching insans. 4 y() a K4 Swiching K3 insans K K b Fig. 4. Closed-loop simulaion resuls for Example 3, using he singlelayer scheme of [4], when he plan models change from P 1 o P 4. (a) Oupu signal; (b) swiching insans. REFERENCES [1] A. S. Morse, Recen problems in parameer adapive conrol, In I. D. Landau, edior, CNRS Colloquium on Developmen and Uilizaion of Mahemaical Models in Auomaic Conrol, pp , [] B. Marensson, The order of any sabilizing regulaor is sufficien a priori informaion for adapive sabilizaion, Sys. Conr. Le., pp , [3] D. E. Miller and E. J. Davison, An adapive conroller which can sabilize any sabilizable and deecable li sysem, In C.I. Byrnes, C. F. Marin & R. E. Saek,(Eds.), Pro. of he eighh inernaional symposium on mahemaics of nework and sysems,az, [4] D. E. Miller and E. J. Davison, Adapive conrol of a family of plans, In D. Hinrichsen,& B. Marensson(Eds.), Conrol of uncerain sysems: Proc. In. Workshop, Berman, Wes Germany, Progress in Sysems and Conrol heory, Vol. 6, pp , [5] D. E. Miller and E. J. Davison, The self-uning robus servomechanism problem, IEEE Trans. Auoma. Conr., vol. 34, pp , [6] J. Balakrishman and K. S. Narendra, Inelligen conrol using fixed and adapive models., Proc. American Conr. Conf., pp , [7] M. S. Branicky, Analyzing coninuous swiching sysems: Theory and examples, Proc. American Conr. Conf., pp , [8] M. Chang and E. J. Davison, Conrol of unknown sysems using swiching conrollers: he self-uning robus servomechanism problem, Proc. 33rd IEEE Conf. Decision Conr., pp , [9] A. S. Morse, Supervisory conrol of families of linear se-poin conrollers-par 1: Exac maching, IEEE Trans. Auoma. Conr., pp , [1] A. G. Aghdam and E. J. Davison, Pseudo-decenralized swiching conrol, Auomaica, pp , 3. [11] K. S. Narendra and J. Balakrishnan, Improving ransien response of adapive conrol sysems using muliple models and swiching, IEEE Trans. Auoma. Conr., vol. 39, no. 9, pp , Sep [1] D. E. Miller and E. J. Davison, An adapive conroller which provides an arbirarily good ransien and seady-sae response, IEEE Trans. Auoma. Conr., vol. 36, no. 1, pp , [13] B. F. J. Doyle and A. Tannenbaum, Feedback conrol heory, Macmillan Publishing Company, New York, 199. [14] M. Vidyasagar, Conrol sysem synhesis: A facorizaion approach, MITPress, Cambridge, Mass, 1985.