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Relaxing convergence assumptions for continuous adaptive control
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Relaxing convergence assumptions for continuous adaptive control
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RELAXING CONVERGENCE ASSUMPTIONS FOR CONTINUOUS ADAPTIVE CONTROL by Mubarak Alharashani ||||||||||||||||||||||||{ A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) August 2010 Copyright 2010 Mubarak Alharashani Dedication I dedicate this thesis in memory of my father who started this study with me and in honor of my mother for her prayers also this thesis dedicated to my wife and my kids for their understanding ii Acknowledgments First and foremost, I would like to express my sincere gratitude towards my Ph.D. advisor and committee chairman, Professor Michael G. Safonov, for his inspira- tional instruction, guidance, understanding and continuous support throughout my Ph.D. years at USC. Professor Safonov was always there to listen and to give advice. I would also like to extend my sincere gratitude to other two members of my defense committee: Professor Edmond Jonckheere and Professor Firdaus Udwa- dia for their invaluable support and advise. Special thanks go to Professor Petros Ioannou and Professor Si-Zhao Qin for serving on my qualication examination committee and for their valuable advice and discussions. Lastly and perhaps most importantly, I am heartily thankful to who made this thesis possible such as my mother, brothers, sisters, wife, kids and rest of my family for their encouragement, prayers and continuous support. iii Table of Contents Dedication ii Acknowledgments iii List of Figures vi Notation vii Abstract ix Chapter 1. Introduction 1 1.1 Brief Review of Adaptive Control Methods 1 1.2 Stability of Switched Systems 6 1.3 Motivation 9 1.4 Outline of the Thesis 12 Chapter 2. Basic Concepts 13 2.1 Preliminaries 13 Chapter 3. Unfalsied Adaptive Control 21 3.1 Denitions in Unfalsied Adaptive Control Theory 21 Chapter 4. Results 25 4.1 Problem Formulation and Preliminary Results 25 4.2 Main Result 32 Chapter 5. Performance Criterion 34 5.1 Performance Criterion Example 34 iv Chapter 6. Comparison 43 6.1 Compactness Property 46 6.2 Optimality 48 6.3 Convergence 50 6.4 Performance Improvement 58 6.4.1 Performance In the Presence of Disturbances 60 6.4.2 Performance In the Absence of Disturbances 60 6.5 Summary 61 Chapter 7. Conclusion and Future Direction 62 7.1 Conclusion 62 7.2 Future Directions 63 Bibliography 65 v List of Figures Figure 1.1. Gain adaption MRAC. (From Ref. [Cha87]) 3 Figure 1.2. Switched closed-loop system 7 Figure 2.1. General adaptive control system (P; ^ k L ) 14 Figure 4.1. Unfalsied adaptive control system (P; ^ k L ) 26 Figure 5.1. Unfalsied adaptive control system (P; L ) 35 Figure 5.2. Control conguration 35 Figure 5.3. Fictitious reference generator 37 Figure 5.4. Control conguration example 39 Figure 6.1. Supervisory control block diagram 52 vi Notation P : Linear truncation operator. R + : (0 ,1): R: (-1;1): x : truncated signal. y p : plant output in MIT algorithm. y m : model reference output in MIT algorithm. r: reference signal. u: plant input. y: plant output. z : (u;y). Z: set of all possible z in the interval time 0 to1: z : experiment data in time interval (0,): e: error signal. K D : disturbance gain. K C : adjustable control gain. K m : model reference gain. vii : discrete switching events. k: a controller. k f : nal controller in the closed-loop. ^ k L : the controller in the feedback loop. K: controller set. V : performance criterion. ^ k: optimal controller. t: time. (P; ^ k L ): closed-loop system. viii Abstract Adaptive control convergence has been proved for long time by using slow switch- ing schemes through separating the two successive switching events by a posi- tive time interval (e.g., dwell-time, average dwell-time, hysteresis switching tech- nique). This thesis addresses the inherent limitations of some logic-based switch- ing among innite (i.e. continuum) set of candidate controllers. In this thesis, we examine adaptive control convergence in the context of well-known hysteresis switching algorithm by relaxing the usual requirement that the hysteresis con- stant is strictly positive. Relaxing this constraint allows the adaptive controller to converge to a unique optimum in the case of an innite (continuum) candidate controller set, provided that at least one controller in the controller set has the ability to satisfy adaptive control performance. ix Chapter 1. Introduction 1.1 Brief Review of Adaptive Control Methods In recent years adaptive control has become a topic of active research. The concept of adaptive control is not new; control techniques based on switching between dierent controllers have been used since the 1950s [DL51, WYK58]. Adaptive control has a rich and varied literature; for more details, the reader can refer to textbooks such as [AW94, IS96, Cha87, NA89], which contain additional explanations of the dierent types of adaptive control theory. An adaptive control technique was rst proposed by Draper and Li in 1952 [DL51]. The next main step in adaptive control theory was taken in 1958 by Whitaker et al. [WYK58]. They were planning to design an aircraft ight con- trol system when they recognized that a xed gain feedback control does not help in this situation. Since the dynamics of the aircraft is changing from one operating point to another, it needs an advanced control system that has the ability to learn and tune its own parameters. At that time, model reference 1 adaptive control (MARC) was proposed by Whitaker [WYK58] to deal with the varying system dynamics of the aircraft system. Their further work was reported in [Whi59a, Whi62, OWK61]. One of the earliest denitions of the term \adaption" was introduced by Drenick and Shahbender [DS57] in 1957: adaptive systems in control theory are control systems that monitor their own performance and adjust their parameters in the direction of better performance. Adaptive control is usually used to control imprecisely known plants. The main goal of adaptive control is to achieve improved performance by choosing among a given set of candidate controllers using real-time data and prior infor- mation. Many studies have been published to achieve this goal. The MIT rule was suggested by Osbourne [OWK61] and Whitaker [Whi59b]. The general idea behind the MIT rule is to control a stable system with unknown gain by using a gradient descent algorithm to adjust a scalar parameter to reach zero output dierence between the modeled linear system and the actual plant (system to be controlled), which is simply illustrated in Fig. 1.1. Both systems (reference model and actual plant) used with the MIT rule are derived from the same ref- erence signal, r. The key idea is to minimize the integral of the square of error between y m and y p by adjusting K c such that K c K D will eventually be equal to the model reference gain K m , more detail about this algorithm can be found in [Cha87]. 2 H(s) KD KC Adaptive Control Mechanism H(s) Disturbance + - e e y P y m Reference Signal r Km Figure 1.1. Gain adaption MRAC. (From Ref. [Cha87]) Even with this straightforward method, which requires much prior informa- tion about the plant (e.g., knowing the exact transfer function of the plant, the plant has stable transfer function, knowing the gain's sign, etc.). The method may yield unpredictable poor performance or even instability in some circumstances, as shown in [Par66, HP73]. In the 1960s, which are considered to be a golden period for adaptive control, several studies and developments in adaptive con- trol were introduced (e.g., stochastic control, state space techniques, Lyapunov stability theory, dual control, etc.). These studies played a crucial role in under- standing and improving the concept of an adaptive control system. However, the stability and convergence of these developments were proved based on restricted plant assumptions, like linear time-invariance, minimum phase plant, no noise, no time delays, known upper bound of the plant order, no external disturbances, 3 and so on. When one or more of these assumptions fail to hold, which is the case in most practical situations, traditional adaptive control may not be able to cope with the system, as shown by Rohrs [RVAS85]. Since this list of \unrealis- tic" assumptions rarely holds in practice, a powerful and successful method was needed to deal with the lack of instability and ll the gaps in understanding the adaptive control weakness. By the mid 1980s, a signicant development eort had led to a new frame- work known as robust adaptive control. The robust adaptive control method successfully coped with a system in the presence of bounded disturbance charac- teristics and ensured system robustness under the assumption of some bounded disturbances. In the late 1980s and early 1990s, several studies were published on control theory that contributed to the development of robust adaptive control (e.g., [IS88, ID91, DI91]). The basic idea of robust adaptive control is to design a controller for a known nominal plant based on a given \small" bound of uncer- tainties around the nominal model by choosing the worst case scenario controller. Although robust adaptive control theory has been used successfully in sev- eral applications, it has inherent limitations: in order to ensure the robustness properties by using this method, the system may fail to achieve the optimal per- formances. The other drawback of robust adaptive control is that the method requires prior information about the plant and assumes a suciently small bound on the uncertainties, while such information may not be available in real time or 4 large uncertainties might arise in practice, which will cause the real system to lie outside the predicted uncertainty bound. Since we are dealing with an unknown plant in most cases, these assumptions about the plant were easily violated, with the missing information making exact model-following impossible, as shown by a well-known example [RVAS85]. This situation was the motivation for several studies that searched for a smarter way to deal with adaptive control problems without requiring numerous assumptions about the plant and its structure. Subsequent research was directed toward re- laxing many of the assumptions about the plant. Some of these studies succeeded to relax some, but not all, of the assumptions, as reported in [Mor96, ABLM01]. In 1986, proof of adaptive control stability under perhaps the weakest assump- tion in the history of adaptive control appeared in ([Mar86, FB86]). They showed that it is possible to design an adaptive controller that will converge under only a feasibility assumption (i.e., at least one controller in the candidate controller set has the ability to satisfy the adaptive control performance) using pre-routed switching among the candidate controllers until the stabilizing controller is found. Although this idea does not require many assumptions about the plant beyond feasibility it had few practical applications because of its shortcomings. This ap- proach works by switching the candidate controllers one by one into the feedback loop until the control performance is satised by one of them, which can cause poor transient response. In addition, in the case of a large number of candidate 5 controllers, the process may require a long time for the stabilizing controller to be switched into the loop. A similar approach called data-driven unfalsied adaptive control was pro- posed by Safonov in [ST97]. This approach can overcome the above pre-routed shortcomings through directly validating the candidate controllers by using exper- imental data only, with no assumptions about the plant beyond feasibility. Other unfalsied adaptive control studies can be found in ([SWS04, WS05, BBMT09, VHDJS05, VHDJS08, ISP08, BHMT, WHK99]) and elsewhere. This algorithm has the ability to detect whenever an active controller fails to achieve the per- formance and to switch it out of the loop when the given data prove this failure. Sucient conditions for the stability and convergence of unfalsied adaptive con- trol were proven in [SWS04, WSS04] under a feasibility assumption and the cost detectability property of performance criterion. It has been found that if the system is cost detectable and the feasibility assumption holds, the unfalsied adaptive control approach always converges to a stabilizing controller. 1.2 Stability of Switched Systems The main reason for introducing adaptive control is to ensure the satisfactory performance (e.g., regulation and tracking problems) of a closed-loop system by switching among a given set of candidate controllers when no single con- troller is capable of achieving the desired performance objectives. Therefore, this 6 Plant K1 K3 Kn K2 Supervisor u z y Figure 1.2. Switched closed-loop system algorithm requires a system served by a multi-controller set (nite or innite set of controllers) and we refer to such systems as multi-controller systems. If the switching between controllers happens to be in a discrete form, the multi- controller system is called a hybrid system because of the combination of discrete dynamics associated with switching events and the continuous dynamics sys- tem associated with the rest of the system. A well-dened performance criterion should be chosen to re ect the desired performance. The whole process is or- chestrated by a smart unit called a supervisor, which is responsible for making a decision, at each instant of time, about when to switch and which controller should be used next, based on the available plant input/output data and perfor- mance criterion. A switched closed-loop system is shown in Fig. 1.2. 7 One serious challenge facing switching systems is occurrence of innitely fast switching (chattering). Chattering phenomena can occur because of fast discon- tinuous switching and can cause unmodeled dynamics excitation and unaccept- able system dynamics behavior. Therefore, the key method for avoiding these phenomena is to separate the two successive switching events by a positive time interval. These undesirable phenomena were the motivation for several stud- ies (e.g., [Mor96, Mor97, HM99, MMG92, MGHM88]). The concept of dwell- time switching was studied in the context of supervisory control by Morse in [Mor96, Mor97]. In these studies, Morse showed how to introduce chatter-free switching by using a suciently large dwell time. Although successful applications of switching control techniques have been reported by using the dwell-time algorithm [Mor96, Mor97], it is not capable of coping with the control of nonlinear systems because of the nite escape time possibilities [HM98]. A new concept called average dwell-time was introduced by Hespanha in [HM99]; this concept is an extension of the dwell-time technique. In [HM99], Hespanha proved that the average time period between successive switches should be greater than a suciently large \specied" constant to ensure the exponential stability of the switched system. Fundamental contributions were made by Morse [MMG92] and Middleton [MGHM88]. In [MMG92], Morse and his co-workers proved that the stability and convergence of adaptive control can be achieved with a nite number of 8 switches when using a hysteresis switching algorithm under certain plant as- sumptions. While, in [WPSS05], Wang showed that this algorithm can be more powerful when used with just the feasibility assumption, given that the used performance criterion has the cost-detectability property. Although the hystere- sis switching algorithm has been applied to several successful applications (e.g., [MMG92, MGHM88, LHM00, HLM + 01, SS08]), it has some drawbacks, espe- cially in the case of an innite set (e.g., containing a continuum) of candidate controllers, as we will discuss in later chapters, which is the main source of mo- tivation for this work. 1.3 Motivation According to Astr om and Wittenmark in [AW94]: In every language, `to adapt' means to change a behavior to conform to new circumstances. Intuitively, an adaptive controller is thus a controller that can modify its behavior in response to changes in the dynamics of the process and the character of the disturbances. When a xed controller is not capable of coping with unknown or time varying plant parameters, switching among a set of candidate controllers algorithms is needed to ensure satisfactory performance. Two dierent techniques for switching between controllers have been used: continuous adaptive tuning and logic-based switching. In both cases, a primary goal of adaptive control is to ensure the stability and convergence of a controller to reach optimum performance. The switching process in adaptive control is orchestrated by a supervisory unit based 9 on the given data and performance criterion. If there is no constraint on how the supervisor unit works, innitely fast switching (chattering) may occur, which could cause an unbounded signal (instability). Several techniques have been proposed to avoid this undesirable phenomenon. The main goal of these techniques is to ensures a non-zero dwell time by separat- ing the two successive switching events by a positive time interval length. One of the most famous technique is the hysteresis switching algorithm reported in [MMG92, MGHM88]. Under certain assumptions, the convergence and stability of adaptive control systems by a nite number of switches have been proven when using the hysteresis switching algorithm. The beauty of the hysteresis switching algorithm is that it can cope with nonlinear systems unlike, the dwell-time algo- rithm. Several successful applications have been reported in adaptive control sys- tems for both nite and innite sets of candidate controllers when using hys- teresis switching techniques, which can be found in ([SS08, MMG92, MGHM88, WPSS05, HLM + 01, HLM03, LHM00]) and elsewhere. Using an innite set of candidate controllers could create a better environment that would help the fea- sibility assumption to hold because of the cardinality dierence between nite and an innite sets of candidate controllers. Signicant progress has been made using such an innite set, with the help of the hysteresis switching algorithm [SS08, HLM03, LHM00, HLM + 01]. The authors of these studies succeeded in 10 proving the stability and convergence of adaptive control systems using a strictly positive hysteresis constant. Although the hysteresis switching algorithm can play an important role in the stability and convergence of adaptive systems, it has some drawbacks. One of the biggest obstacles facing this algorithm is that it does not ensure optimal performance, especially in the case of an innite set (e.g., containing a contin- uum) of candidate controllers, which means some performance may be sacriced. In this case, the hysteresis switching algorithm ensures the convergence of the adaptive controller to the neighborhood of the continuum controllers within a radius (hysteresis constant) far from the optimal controller. In this thesis, we will consider the case in which the candidate controller set K has continuum controllers. In Chapter 3.1, we propose the above problem and establish a theoretical proof of adaptive control convergence to a unique optimum controller. 11 1.4 Outline of the Thesis This thesis is organized as follows: Chapter 2 presents an overview of the preliminary denitions and notation. Chapter 3 introduces some unfalsied adaptive control denitions and con- cept needed in this thesis. Chapter 4 gives the problem formulation and results. Chapter 5 provides an example of a performance criterion satisfying su- cient conditions for convergence. Chapter 6 contains a comparison between the idea introduced in this thesis and local priority hysteresis switching logic [HLM + 01]. Conclusions follow in Chapter 7. 12 Chapter 2. Basic Concepts 2.1 Preliminaries Consider a general adaptive control system (P; ^ k L ) shown in Fig. 2.1 mapping r7! (u;y), where u and y are the measured plant input and output vector sig- nals respectively,r is reference signal,P is a plant and ^ k L is the controller in the feedback loop. The input signal of supervisor is the measured data z := 2 6 4 u y 3 7 5 and the output of the supervisor is the chosen ^ k L where the adaptive control law has the general form u := ^ k L (t;z) 2 6 4 u y 3 7 5 13 Plant Supervisor u z y k L Reference Signal r Controller Figure 2.1. General adaptive control system (P; ^ k L ) So at each time the supervisor will switch in the loop the best controller among the controller set K based in measured dataz and performance criterion. Let's denote the nal controller switched in the loop by k f (z) at the time t f (z). Comment 2.1.1 The controller set K may have nite or innite (e.g containing a continuum) number of controllers. We limit our consideration in this thesis to the case in which the candidate controller set K has innitely many controllers (typically, a continuum of controllers). 14 Denition 2.1.1 [Saf80] Linear truncation operator P :x!x is given by P x(t) = 8 > < > : x(t); if t2 [0;] 0; otherwise: and x refer to P x(t) as shown below x (t) = 8 > < > : x(t); if t2 [0;] 0; otherwise: Let one possible experimental plant data for the switching adaptive system shown in Fig. 2.1 be z = (u;y) and let Z represent the set of all possible z in the interval time 0 to1: z represent the truncated signal z. Thus, z is the experiment data in time interval (0,): Denition 2.1.2 ` 2 norm of a truncated signal P x is given as kxk = r Z 0 x(t) T x(t)dt: Denition 2.1.3 [Ber99] Let C R n be a convex set and let f : R n 7! R be dierentiable over C then, f is strictly convex over C if f(y)f(x) + (yx) 0 rf(x); 8x;y2C 15 Denition 2.1.4 [KS90] If the function f is twice continuously dierentiable, then f is strongly convex in k with parameter c if and only if r 2 k (V (k;z;t)) c> 0 for all k. Denition 2.1.5 [KS90] Let C R n be a convex set and let f : R n 7! R be dierentiable over C then, f is strongly convex (or uniformly convex) on C if and only if, for any x;y2C: f(y)f(x) +rf(x) T (yx) + 2 jjyxjj 2 where r 2 f(x)> 0: Comment 2.1.2 It is not necessary for a function to be dierentiable in order to be strongly convex. Denition 2.1.6 [Ber99] Let C R n be a convex set and let f : R n 7! R be twice continuously dierentiable over C. then, f is strictly convex over C if r 2 f(x) is positive denite for every x2C. Lemma 2.1.1 [Ber99] Let f be a strongly convex function then, the local mini- mum of f is also a global minimum and there exists at most one minimum of f. 16 Lemma 2.1.2 [Ber99] Let f be a strongly convex function in x and let x be a local minimum of f :R n 7!R, thenr x f(x ) = 0. Denition 2.1.7 A level set in R n is dened as L() =fx2R n jf(x)g for some 2R. Denition 2.1.8 (k;z;t) is an equi-quasi-positive denite (EQPD) function in k, that is, there exists a continuous nondecreasing scalar function such that (0) = 0 and (k;z;t) ( ^ k(t);z;t) (kk ^ k(t)k)> 0 for all t, all z2Z and all k ^ k(t)6= 0. Remark An equi-quasi-positive denite function has the same properties as posi- tive denite function except that the minimum of equi-quasi-positive denite func- tion occur at k = 0 (i.e. k = k ^ k) while the minimum of positive denite function occur at k = 0. Denition 2.1.9 [IS96] (Persistence of Excitation (PE)) A piecewise continu- ous signal vector :R + 7!R n is PE in R n with a level of excitation 0 > 0 if there exist constants 1 , T 0 > 0 such that 1 I 1 T 0 R t+T 0 t () T ()d 0 I, 8t 0 (I) 17 Although the matrix () T () is singular for each , (I) requires that (t) varies in such a way with time that the integral of the matrix () T () is uni- formly positive denite over any time interval [t;t +T 0 ]. If we express (I) in the scalar form, i.e., 1 1 T 0 R t+T 0 t (q T ()) 2 d 0 , 8t 0 where q is any constant vector in R n withjqj = 1, then the condition can be interpreted as a condition on the energy of in all directions. Denition 2.1.10 (second-order Taylor-theorem expansion) LetCR n and let f :R n 7!R be twice continuously dierentiable over C then, f(x) =f(a) +rf(a)(xa) +r 2 f() ax or =a + (1)x for 2 [0; 1] where the gradientrf(x) of the function f(x) is a row vector of size n, i.e., rf(x) = @f @x 1 (x); @f @x 2 (x); ; @f @xn (x) the Hessianr 2 f(x) is an nn matrix; r 2 f(x) = 0 B B B B B B B @ @ 2 f @x 2 1 (x) @ 2 f @x 1 @x 2 (x) @ 2 f @x 1 @xn (x) @ 2 f @x 2 @x 1 (x) @ 2 f @x 2 2 (x) @ 2 f @x 2 @xn (x) . . . . . . . . . . . . @ 2 f @xn@x 1 (x) @ 2 f @xn@x 2 (x) @ 2 f @x 2 n (x) 1 C C C C C C C A 18 and xa = 0 B B B B B B B @ x 1 a 1 x 2 a 2 . . . x n a n 1 C C C C C C C A Denitions of input-output stability related to the ` 2e -norm can be found in [Zam66], which pertains to the ratio between the norm of the output z to the norm of the input v. A slight generalization of the input-output stability of Zames [Zam66] has been developed by Willems [Wil73, Wil76]. The role of the and ~ is to prevent the denominator from assuming values too close to zero. Denition 2.1.11 (Stability and Gain) [WPSS05] We say a systemF with input v and output z is stable if for every input v2 ` 2e -norm there exist constants , 0 such that kz k<kv k +,8t> 0 (?) otherwise, it is said to be unstable. Furthermore, if the (?) equation holds with a single pair , 0 for all v2` 2e -norm, then the system F is said to be nite-gain stable, in which case the gain of F is the least such . 19 Denition 2.1.12 [WPSS05] (Incremental Stability and Incremental Gain) We say that F is incrementally stable if, for every pair of inputs v 1 , v 2 and outputs z 1 =Fv 1 , z 2 =Fv 2 , there exists constants ~ , ~ 0 such that k[z 2 z 1 ] k< ~ k[v 2 v 1 ] k + ~ ,8t> 0; (??) and the incremental gain of F , when it exists, is the least ~ satisfying (??) for some and all v 1 , v 2 2` 2e : 20 Chapter 3. Unfalsied Adaptive Control 3.1 Denitions in Unfalsied Adaptive Control Theory The beauty of using the Morse-Mayne-Goodwin [MMG92] hysteresis switching algorithm with unfalsied adaptive control is that it is possible to adjust the controller's parameters based only on the measured data without any assump- tions about the plant beyond feasibility, with convergence ensured for any strictly positive hysteresis constant [SS08]. In this method, the potential performance of every candidate controller is evaluated directly from the measured data using some suitably dened performance criterion, without trying to identify the ac- tual process. This algorithm is typically fast to converge because it does not require inserting controllers in the feedback loop to be falsied. 21 The main contribution of the unfalsied adaptive control algorithm is that, the algorithm does not require any assumption about the plant (i.e. plant- assumption-free method) in order to ensure the stability of the system, given the feasibility of the adaptive control problem and a cost detectable performance criterion. The feasibility is dened as the existence of at least one controller in the candidate controller set that has the ability to stabilize the system. The cost- detectability property is a condition of the performance criterion that ensures closed-loop stability for the switched multi-controller adaptive control (MCAC) system whenever stabilization is feasible. For this reason, an adaptive control system that employs cost-detectability has been called a \safe adaptive control systems" [WPSS05]. In this chapter, some important concepts and denitions of unfalsied adap- tive control are presented. Further details about unfalsied adaptive control can be found in [SS08, WPSS05, JS99]. Denition 3.1.1 [JS99] Given a set of measurements of I/O data (u;y) and a candidate controller k i 2 K a ctitious reference signal for k i is a hypothetical reference signal ~ r i that would have produced exactly the same measurements data (u;y) had the candidate controller k i been in the feedback loop with the unknown plant during the entire time period over which the measurements data (u;y) were collected. 22 Denition 3.1.2 [SS08] The adaptive control problem is said to be feasible if a candidate controller set K contains at least one controller that achieves stability and performance goals. Comment 3.1.1 It is unknown a prior which controller k in a controller set K that achieves the satisfactory performance. Denition 3.1.3 [SS08] A controller K is said to be a feasible controller if it satises given stability and performance constraints. Denition 3.1.4 [WPSS05] Given V, K and a scalar 2 R, we say that a controller k2 K is falsied at time with respect to cost level by past mea- surement information z if V (k;z;)> . Otherwise the control law k is said to be unfalsied by z . Denition 3.1.5 [WPSS05] Let r denote the input and let z := 2 6 4 u y 3 7 5 denote the resulting plant data collected while ^ k L (t;z) is in the loop. Consider the adap- tive control system (P; ^ k L ) of Fig. 2.1 with input r and output z := 2 6 4 u y 3 7 5 . The pair (V;K) is said to be cost detectable if, without any assumptions on the plantP and for every ^ k L (t;z)2K, the following statements are equivalent: 23 1). V (k f ;z;) is bounded as increases to innity; 2). The stability of the system (P; ^ k L (;z)) is unfalsied by (r;z). Denition 3.1.6 [SS08] Stability of the system : r7!z is said to be falsied by the data (r;z) if sup 2R + ;krk6=0 kzk krk =1 Otherwise, it is said to be unfalsied. 24 Chapter 4. Results 4.1 Problem Formulation and Preliminary Re- sults Consider an unfalsied adaptive control system (P; ^ k L ) shown in Fig. 4.1 map- ping r7! (u;y), where u and y are the measured plant input and output vector signals respectively, r is reference signal,P is unknown plant and ^ k L is the con- troller in the feedback loop. The input signal of supervisor is the measured data z := 2 6 4 u y 3 7 5 and the output of the supervisor is the chosen ^ k L where the adaptive control law has the general form u := ^ k L (t;z) 2 6 4 u y 3 7 5 25 Unknown Plant Unfalsified Control Algorithm u z y k L Reference Signal r Controller Figure 4.1. Unfalsied adaptive control system (P; ^ k L ) So at each instant of time the supervisor will switch in the loop the best con- troller among the controller set K based in measured data z and performance criterion. In this thesis, we call the scalar valued function,V :KZR + !R + [f1g, a performance criterion. It is used to evaluate candidate controllersk based on past data z . The performance criterion V (k;z;) assumed to be causally dependent of z, that is, for all > 0 and all z2` 2 , V (K;z;) =V (K;z ;) where z is the truncated signal z from initial to the current time . 26 Denition 4.1.1 Consider a continuum controllers set K (e.g. KR n ): Denition 4.1.2 The optimal controller ^ k(t), if exists, at t is dened as ^ k(t) = argmin k 2 K V (K;z;t) Assumption 4.1.1 The performance criterion V (k;z;t) is continuous in k and t. Lemma 4.1.1 [Rud76] Every nonempty set of real numbers which bounded above has a supremum. Assumption 4.1.2 Adaptive control problem is feasible (Def. 3.1.2). Comment 4.1.1 (i:e:9M2R ; s:t: V ( ^ k(t);z;t)M <1) Let V L (z) = sup t V ( ^ k(t);z;t) Assumption 4.1.3 The performance criterion V (k;z;t) is monotonically in- creasing in t, V (k;z;t 2 )V (k;z;t 1 ) 8t 2 t 1 and all k. 27 Lemma 4.1.2 V ( ^ k(t 2 );z;t 2 )V ( ^ k(t 1 );z;t 1 ) for all t 2 t 1 and all k. Proof (by monotonicity property) V (k;z;t 2 )V (k;z;t 1 ) 8t 2 t 1 (1) (by denition 4.1.2) V ( ^ k(t);z;t)V (k;z;t) (2) (from (1)) V ( ^ k(t 2 );z;t 2 )V ( ^ k(t 2 );z;t 1 ) (from (2)) V ( ^ k(t 1 );z;t 1 ) ) V ( ^ k(t 2 );z;t 2 )V ( ^ k(t 1 );z;t 1 ) 8 t 2 t 1 ) V ( ^ k(t);z;t) monotonically increasing sequence. Lemma 4.1.3 [Rud76] If V ( ^ k;z;t) is monotonically increasing sequence in R. Then V ( ^ k;z;t) converges if and only if it is bounded above. Assumption 4.1.4 V ( ^ k(t);z;t) unique for each t2R + . 28 Lemma 4.1.4 Let V : KZR + ! R + [f1g be a continuous equi-quasi- positive denite function in k, k2R n , and continuous monotonic increasing in t. Assume that V ( ^ k(t);z;t) is bounded above. Then, there exists a time t M such that ^ k(t) lies in a compact subset L, LK, for all t>t M . Proof Dene a family of functionalsf t (k) =V (k;z t ;t)8tt M . Let = sup t V ( ^ k(t);z;t) (see lemma 4.1.1). Consider L() to be a level set inR n (Def. 2.1.7). SinceL()R n it is sucient to show thatL() is bounded and closed. Dene s(t) to be s(t) = 8 > < > : min !1 sup kkk V (k;z;t); if exists 1, otherwise: SinceV (k;z;t) is continuous equi-quasi-positive denite function ink and has a unique minimum ^ k(t) at eacht (assumption 4.1.4), thens(t)>V ( ^ k(t);z;t). Let (t) =s(t)V ( ^ k(t);z;t)> 0. By lemma 4.1.2,V ( ^ k(t);z;t) monotonic increasing sequence. SinceV ( ^ k(t);z;t) is monotonic increasing sequence bounded above by , then for every (t)> 0 there exists t M such that V ( ^ k(t);z;t)<(t) for all tt M (this is true by lemma 4.1.3). 29 We know that for all tt M the condition V ( ^ k(t);z;t)<(t) is satised. ) V ( ^ k(t);z;t)<s(t)V ( ^ k(t);z;t) 8tt M ) <s(t) 8tt M In what follows, we will show thatL() is bounded and closed for alltt M (i.e., f t (k) =V (k;z t ;t)). First, we show L() is bounded for t t M . Suppose to the contrary that L() is not bounded. Then there exists a sequencefk m g L() such that lim m!1 kk m k =1. Since f t (k) is equi-quasi-positive denite function, lim m!1 f t (k m ) s> (i:e:9N2N such that8`N f t (k ` )> ). Thenfk m g6L(), which contradicts the above assumption. Hence, L() is bounded. Next, we show that L() is closed for tt M : Letfk m gL() be a conver- gent sequence, and k f = lim m!1 k m . Since f t is continuous, f t (k f ) = lim m!1 f t (k m ). Also, f t (k m )8t. Then, f t (k f ) = lim m!1 k m lim m!1 =, so k f 2L(). Thus L() is closed and bounded, therefore compact. 30 Lemma 4.1.5 [Ber99](Weierstrass theorem) LetK be a non empty subset of R n and let V :K7!R be lower semicontinuous at all points of K. If K is compact, then ^ k(t) = argmin k 2 K V (K;z;t) exists. Lemma 4.1.6 If V (k;z;t)V ( ^ k(t);z;t) (kk ^ k(t)k) then,V ( ^ k(t 2 );z;t 2 ) V ( ^ k(t 1 );z;t 1 ) (k ^ k(t 2 ) ^ k(t 1 )k) 8 t 2 t 1 : Proof Since, V (k;z;t)V ( ^ k(t);z;t) (kk ^ k(t)k) ) V ( ^ k(t 2 );z;t 1 ) V ( ^ k(t 1 );z;t 1 ) (k ^ k(t 2 ) ^ k(t 1 )k) (by monotonicity property) V ( ^ k(t 2 );z;t 2 )V ( ^ k(t 2 );z;t 1 ) 8 t 2 t 1 ) V ( ^ k(t 2 );z;t 2 )V ( ^ k(t 1 );z;t 1 ) V ( ^ k(t 2 );z;t 1 )V ( ^ k(t 1 );z;t 1 ) (k ^ k(t 2 ) ^ k(t 1 )k) 8 t 2 t 1 ) V((t 2 );z;t 2 ) V ( ^ k(t 1 );z;t 1 ) (k ^ k(t 2 ) ^ k(t 1 )k) 8 t 2 t 1 : Lemma 4.1.7 [Rud76] InR n , every Cauchy sequence converges. 31 4.2 Main Result Theorem (Main Result) Consider the feedback adaptive control system (P; ^ k L ) in Fig. 2.1. Assume that the adaptive control problem is feasible, and the associ- ated performance criterionV (K;z;t) is monotone increasing int and continuous in k. Assume further that V (K;z;t) is equi-quasi-positive denite function in k (Def. 2.1.8) and V ( ^ k(t i );z;t i ) unique for each t i . Then, the adaptive control system converges to a unique controller as time proceeds. Proof (from lemma 4.1) V ( ^ k(t 2 );z;t 2 ) V ( ^ k(t 1 );z;t 1 ) (k ^ k(t 2 ) ^ k(t 1 )k) 8 t 2 t 1 Since, V L (z)V ( ^ k(t 2 );z;t 2 ) )V L (z)V ( ^ k(t 1 );z;t 1 )V ( ^ k(t 2 );z;t 2 )V ( ^ k(t 1 );z;t 1 ) (k ^ k(t 2 ) ^ k(t 1 )k) (from lemma 4.1.3) for each > 0 there exists t N such that jV ( ^ k(t 2 );z;t 2 ) V ( ^ k(t 1 );z;t 1 )j(k ^ k(t 2 ) ^ k(t 1 )k) 8 t 1 ;t 2 t N 32 Since, is nondecreasing continuous function and satises(0) = 0. Then8> 0 9 t N such that ()(k ^ k(t 2 ) ^ k(t 1 )k). Therefore, k ^ k(t 2 ) ^ k(t 1 )k 8 t 1 ;t 2 t N . By Cauchy criterion (lemma 4.1.7) ^ k(t) converges. Since ^ k(t) is unique (assump- tion 4.1.4), Hence the adaptive control system converges to a unique controller. 33 Chapter 5. Performance Criterion 5.1 Performance Criterion Example Consider an unfalsied adaptive control system (P; L ) shown in Fig. 5.1 map- ping r7! (u;y), where u and y are the measured plant input and output vector signals respectively, r is reference signal,P is unknown plant and L is the con- troller in the feedback loop. Consider a controller structure in Fig. 5.2, where ii is a constant parameter, ii 2 , is a parameter vectors, 2R 2n and is a set of parameter vectors. Then the control low has the form u(t) =r(t) + 1i f i1 u(t) + 2i f i2 y(t) where f i1 =L 1 (F i1 (s)) and f i2 =L 1 (F i2 (s)), 34 Unknown Plant P Unfalsified Control Algorithm u z y k L Reference Signal r Controller Figure 5.1. Unfalsied adaptive control system (P; L ) Controller Figure 5.2. Control conguration 35 0 1i and 0 2i are n 1 vectors, 0 1i = 0 B B B B B B B @ 11 12 . . . 1n 1 C C C C C C C A , 0 2i = 0 B B B B B B B @ 21 22 . . . 2n 1 C C C C C C C A and is 1 2n vector, = ( 11 12 1n 21 22 2n ) and F i1 (s) and F i2 (s) are n 1 vectors of stable lters as shown below F i1 = 0 B B B B B B B @ F 11 F 21 . . . F n1 1 C C C C C C C A , F i2 = 0 B B B B B B B @ F 12 F 22 . . . F n2 1 C C C C C C C A Consider the performance criterionV (;z;t). The optimal controller param- eters (t) at each instant of time is dened as (t) = argmin 2 V ( ;z;t) Fictitious reference signal ~ r is not the true signal (Def. 3.1.1). For each i there is a ctitious reference signal ~ r i that would have produced exactly the same 36 Controller Figure 5.3. Fictitious reference generator measurements data (u;y) had the candidate controller i been in the feedback loop with the unknown plant during the entire time period over which the mea- surements data (u;y) were collected. Given dataz = (u;y) and controller with the structure in Fig. 5.2. Its ctitious reference signal ~ r(;z) would be ~ r(;z) =T ()z =u 1i F i1 (s)u 2i F i2 (s)y where T () is a ctitious reference generator of the controller conguration in Fig. 5.2. The ctitious reference generator T () structure is illustrated in Fig. 5.3. 37 A ctitious error signal ~ e is the error between the ctitious reference signal and the actual plant output y, which can be written as ~ e i = ~ r i y The ctitious error signal for the data z = (u;y) and controller with the structure in Fig. 5.2 is ~ e(;z) = ~ r(;z)y =u 1i F i1 (s)u 2i F i2 (s)yy An example of the performance criterion and the conditions under which it ensures convergence according to the previous theorem may be constructed as follows. Consider a controller structure in Fig. 5.4. Its ctitious reference signal would be ~ r(;z) =uk 1 uy where 1i , 2i , f i1 , and f i2 in this example are 0 1i = 0 B B B B B B B @ k 1 0 . . . 0 1 C C C C C C C A , 0 2i = 0 B B B B B B B @ 1 0 . . . 0 1 C C C C C C C A 38 Controller Figure 5.4. Control conguration example and f i1 = 0 B B B B B B B @ 1 0 . . . 0 1 C C C C C C C A , f i2 = 0 B B B B B B B @ 1 0 . . . 0 1 C C C C C C C A And the associated ctitious error signal is ~ e(;z) =uk 1 uyy =u(1k 1 ) 2y 39 Consider the well-known performance criterion integral norms of estimation errors V (;z(t);) = Z 0 ke(t)k 2 dt In the unfalsied adaptive algorithm, the only available information about the actual plant is the data (u;y), so we will use the ctitious error signal instead of the true error. This ctitious error signal should converge to the true signal if the ctitious reference generator has a stable structure (See Reference [ST97]). V ( i ;z(t);) = Z 0 k~ e i (t)k 2 dt = Z 0 ku(t)(1k 1 ) 2y(t)k 2 dt Then, r k 1 V ( i ;z(t);) =2 Z 0 u(t)(u(t)(1k 1 ) 2y(t))dt and r 2 k 1 V ( i ;z(t);) = 2 Z 0 u(t) 2 dt Denition 5.1.1 We say that the system is persistently excitated if the hessian is strictly positive denite for all t suciently large. 40 Under the persistent excitation assumption, the functionV ( i ;z(t);t) is uni- formly convex function in k for suciently large time t. For this example, we can derive explicit conditions on ~ e i that guarantee parameter convergence by considering u 2 does not tend to zero. Therefore, whenever the systems is persistently excited, this performance criterion has the the uniform convexity property. The persistent excitation (PE) property de- ned by us is crucial in many adaptive schemes where parameter convergence is one of the objectives and is closely related to the persistent excitation of [NA87, Eyk74, BS86, Bit84, AB66, And77]. The main idea of this thesis is to introduce a new algorithm for adaptive controller convergence without using any constraint on the switching scheme (re- moving the constraints on the switching scheme, \e.g. dwell-time, average dwell- time, hysteresis switching"). The idea introduces in this thesis is investigated in the context of the unfalsied adaptive control algorithm. We believe that the unfalsied adaptive control algorithm is one of the best algorithms in adaptive control theory since it requires the minimum number of assumptions (i.e., at least one controller in the controller set has the ability to satisfy the adaptive control performance) about the plant to ensure convergence and stability. 41 Such a contribution could also be used under a dierent adaptive control algorithm (e.g., multiple model adaptive control) to enhance the performance, as we will show in the next chapter. 42 Chapter 6. Comparison Adaptive control using a continuum set of candidate controllers has recently re- ceived considerable attention, with several successful applications being reported (e.g., [HLM + 01, HLM03, SS08]). Some of these applications have been a source of inspiration for the idea introduced in this thesis. Our main goal in this thesis is to establish the conditions for performance criterion under which the convergence constraint on the switching schemes (i.e., strictly positive hysteresis constant) has been relaxed. The aim of this chapter is to show, by a literature review, how this new idea could be useful in relaxing some of the unnecessary assumptions. In [HLM + 01, HLM03], Hespanha and his coworkers introduced new modications to a hys- teresis switching technique that have the ability to deal with an innite set of candidate controllers (typically, a continuum of controllers) and ensure adaptive control convergence. The two switching logics are called hierarchical hystere- sis switching and local priority hysteresis switching logic, and were reported in [HLM03] and [HLM + 01], respectively. The primary idea of the rst switching 43 logic relies on a partition of the continuum set into a nite number of subsets. The switching strategy in this logic is based on two stages. The rst step is to choose controllers \system's parameters" that satisfy the minimum value for the performance criterion \monitoring signal" in each subset and then compare the signal values produced by these controllers to select the one that satises the overall minimum. For further details on hierarchical hysteresis switching, we refer the reader to [HLM03, LHM00]. The main idea of the second switching logic, local priority hysteresis switching logic, relies on giving priority to the neighborhood's parameters before switching to another one, as we will present in more detail in the sequel. The concepts of these switching logics are almost the same and the switching between controllers \system's parameters" occurs in a discrete switching form even though we use a continuum set of candidate controllers. Combining discrete switching with continuous dynamical systems will drive the designer to deal with hybrid dynamical systems instead of dealing with continuous systems. Since this will make the systems more complicated to deal with, we try to avoid this situ- ation in this thesis by introducing continuous adaption. In this chapter, we will choose the second switching logic, local priority hysteresis switching, for our case study. We perform a comparison and try to show how the idea presented in this thesis could contribute in this context and relax some assumptions. 44 We now present some concepts and equations about local priority hysteresis switching logic. For more details, the reader can refer to [HLM + 01]. The inputs of the local priority hysteresis switching logic are continuous signals, p , p2P, where p assumed to be strictly positive and monotone increasing int andP is a compact set. Dene a set D D (q) :=fp2P :jqpj g where is a proper positive constant andjj is a norm function inP. The output of the switching logic, at each instant of time, is a switching signal, (t). Pick a hysteresis constant h> 0 and set (0) = argmin p2P f p (0)g. Suppose that at time t i , has just switched to some q2P and kept xed until a time t i+1 > t i such that the following inequality is satised: (1 +h) min p2P f p (t i+1 )g min p2D (q) f p (t i+1 )g At this time, we set (t i+1 ) = argmin p2P f p (t i + 1)g. By repeating these steps we can generate a sequence of switching signal which will converge as time in- crease. In this study, the authors stated that the constant should be suciently small to ensure the tractability property for subset D . It is reasonable to ask about the kind of upper bound needed for the constant to ensure this prop- erty? Furthermore, if there is one, does this bound work for all possible per- formance criteria? The answers to these questions can be found in [SS08]. In 45 [SS08], Stefanovic succeeded to avoid these diculties by using the uniform con- tinuity property for the performance criterion. In this study, the constant is adjusted by choosing the hysteresis constant, h, using the continuity property of the function. The uniform continuity property of the performance criterion helps to ensure that the adaptive control system does not switch to another controller outside the neighborhood of radius , until all of the controllers in this neighbor- hood have been falsied. More details can be found in [SS08]. In all of above studies, the only way to ensure adaptive control convergence for the case of a continuum set of candidate controllers is by adding constraints to the switching logics through using strictly positive constants. These constraints may prevent the adaptive control system from reaching optimality, as we will show in Section 6.2. 6.1 Compactness Property In [HLM + 01], the authors used the \well-known" integral norms of the estima- tion errors performance criterion and the compactness property of the parameter set, P. In this part of the study we will show that the idea of this thesis can relax the compactness assumption by using the same performance criterion used in [HLM + 01]. 46 Suppose e p = (p T A +b)x E y and consider the integral norms of estimation errors performance criterion [HLM + 01] p () = Z 0 ke p (t)k 2 dt = Z 0 ((p T A +b)x E (t)y(t)) 2 dt (I) Then, r p ( p ()) = 2 Z 0 ((p T A +b)x E (t)y(t))x E (t) T A T dt and r 2 p ( p ()) = 2 Z 0 Ax E (t)x T E (t)A T dt Denition 6.1.1 We say that the system is persistently excitated if the hessian is strictly positive denite for all t suciently large. Under the persistent excitation assumption, the function p (t) is uniformly convex function in p for suciently large time t. Let ^ p(t) = arginf p p (t), t2R + . 47 Lemma 6.1.1 Let p :RR + !R + [f1g be a continuous function in p, and continuous monotonic increasing in t. Suppose that the system is persistently excited and that ^ p(t) (t) is bounded above. Then, there exists a time t C such that ^ p(t) lies in a compact subset L, LR, for all t>t C . Proof Since ^ p(t) minimizes p (t), we haver p ( ^ p(t) (t)) = 0 (1) Since p (t) is uniformly convex in p thenr 2 p ( p (t))> 0 (2) By Denition 2.1.5, equations (1) and (2), p (t) can be written as p (t) ^ p(t) (t) + 2 kp ^ p(t)k Hence, p (t) is equi-quasi-positive denite function (Def. 2.1.8) with a unique minimum and the proof proceeds like the proof of lemma 4.1.4. 6.2 Optimality The diculty of using the hysteresis switching algorithm [MMG92] and its mod- ications [HLM + 01, HLM03, SS08] is that when using the usual requirement that the hysteresis constant is strictly positive, this constraint may prevent the adaptive control system from achieving optimality. For this reason, our aim in this thesis is to reexamine the adaptive control convergence in the context of the 48 well-known hysteresis switching algorithm by setting the hysteresis constant to zero (relaxing the switching scheme constraint). Relaxing this constraint allows the adaptive controller to converge to a unique optimum in the case of an innite (continuum) candidate controller set as t!1. Another noticeable diculty with the local priority hysteresis switching logic is that other factors could prevent the adaptive control system from reaching optimality besides the hysteresis constant, including the choice of the other con- stant and the performance criterion \monitoring signal", p . Since there is no upper bound for choosing the constant and no clear rules for choosing the performance criterion, this drawback could be worse with bad choice for constant and the performance criterion. Lemma 6.2.1 Local priority hysteresis switching logic may stop switching (be- come prematurely stuck with one controller in the feedback loop) even though there are controllers in the controller set that satisfy the condition (1 +h) p other (t i )< p fl (t i ) where p fl is the monitoring signal associated with controller in the feedback loop and p other is the monitoring signal associated with other controllers in the controller set. 49 Proof Let's start with with (0) = argmin p2P f p (0)g and at certain time t i , has just switched to some q 2 P. Suppose that at time t i+1 > t i there exists a globally minimizing p m such that, p m = argmin p2P f p (t i+1 )g = argmin p2D (q) f p (t i+1 )g and p m 2D (q). Then equation (1 +h) min p2P f p (t i+1 )g min p2D (q) f p (t i+1 )g becomes h min p2P f p (t i+1 )g 0 Since (t) is positive function, this condition cannot be satised (i.e., if ppm is the monitoring signal associated with the controller p m and pq is the moni- toring signal associated with the controller q the system will not switch to the controller p m that satises the global minimum \p m = argmin p2P f p (t i+1 )g" what- ever the dierence between ppm and pq ). 6.3 Convergence The primary goal of this thesis is to establish conditions for the performance criterion under which the convergence constraint on the switching schemes (i.e., strictly positive hysteresis constant) may be relaxed. In this section, we shall get to the main point of this chapter by showing that using the same perfor- mance criterion that was used in our case study [HLM + 01], it is possible to 50 prove adaptive control convergence without a strictly positive hysteresis constant (e.g., withouth> 0 and > 0), which allows the adaptive controller to converge to a unique optimal solution where the \optimal performance has been satised". To make this comparison we need to recall some required and necessary no- tations and denitions from [HLM + 01]. The switching process in the local priority hysteresis switching logic is orches- trated by a supervisory unit, which is responsible for switching into the feedback loop, at each instant of time, the best controller from the controller set= based on the measured data and performance criterion. This supervisor consists of three subsections, as shown in Fig. 6.1. 1. Multi-estimator E | a dynamical system whose inputs are the output y and the input u of the processP and whose outputs are the signals y p , p2P. 2. Monitoring signal generator M | a dynamical system whose inputs are the estimation errors e p = y p y, p2P and whose outputs p , p2P are suitably dened integral norms of the estimation errors, called monitoring signals. 3. Switching logic S | a switched system whose inputs are the monitoring signals p , p2P and whose output is a switching signal taking values in P, which is used to dene the control law u. 51 y u d + - Supervisor n Figure 6.1. Supervisory control block diagram State-space equations for the supervisory system is described in detail in [Mor96], recall the state-space equations for the three subsystems. As p ranges overP, let realizations of the transfer functions of the candidate controllers be: _ x C =A p x C +b p y u =k p x C +r p y where x C is controller state andC q is one controller parameter in the candidate controller set= (i.e. fC q : q2=g). It have been assumed that there is a con- troller in the candidate controller set that able to solve the tracking error and regulation problems for each unknown processP. 52 The state space realization of multi-controllerC can be dened as: _ x C =A x C +b y u =k x C +r y and multi-estimator E has the following realization: _ x E =A E x E +b E y +d E u y p =c p x E , p2P where x E is estimated state and its assumed to be available for the controller in all time, A E is a stable matrix and d is a process disturbance. The matrices c p , p2 P is design in such way for each p2 P, c p exists and unique (See Reference [Mor96] Section IV). Moreover, for the case of P to be continuum c p , p2P assumed to depend linearly on p to ensure the tractability property (See Reference [Mor96] Section XI). So the matrixc p can be represented in the form: c p =p T A +b For SISO system, A is nn nonzero matrix, p is n 1 unknown process parameters and b is 1n vector. 53 In our case study [HLM + 01], authors used the \well-known" performance criterion integral norms of estimation errors: p () = Z 0 ke p (t)k 2 dt where e p =y p y and y p =c p x E so, p can be written as p () = Z 0 k(p T A +b)x E (t)y(t)k 2 dt or p () = Z 0 ((p T A +b)x E (t)y(t)) T ((p T A +b)x E (t)y(t))dt Then, r p ( p ()) = 2 Z 0 ((p T A +b)x E (t)y(t))x E (t) T A T dt and r 2 p ( p ()) = 2 Z 0 Ax E (t)x T E (t)A T dt Then p is uniformly convex inp when the hessian satisesr 2 p ( p (t))>> 0, which holds since the systems is persistently excited. For this example, we can derive explicit conditions on e p (t) that guarantee parameter convergence by consideringkx E k 2 does not tend to zero. Therefore, whenever the systems is persistently excited, this performance criterion has the 54 the uniform convexity property. The persistent excitation (PE) property de- ned by us is crucial in many adaptive schemes where parameter convergence is one of the objectives and is closely related to the persistent excitation of [NA87, Eyk74, BS86, Bit84, AB66, And77]. Under the persistent excitation assumption, the function p (t) has a unique minimum parameter ^ p(t) \controller" for suciently large timet, let the optimal parameter ^ p(t) at each t is dened as ^ p(t) = argmin p2P p (t) In the sequel, we will use the same assumptions that have been used in [HLM + 01], except: 1. Compactness property: we showed that by using the same performance criterion that has been used in [HLM + 01] we were able to relax the com- pactness property for the parameter setP (See Section 6.1). 2. Strictly positive constants (i.e.,h> 0 and > 0): these two constants have been used in local priority hysteresis switching logic to ensure adaptive con- trol convergence for the case of a continuum set of candidate controllers. In this section, we provide the main result for this chapter, which relies on proving that the adaptive controller convergence for the case of a continuum set of candidate controllers without using any constraints on the switching logic (i.e., h = 0, = 0). Relaxing these constraints allows the adap- tive control system to overcome these limitations and ensure the optimal performance. 55 Theorem (Main Result) Consider the feedback adaptive control system in Fig. 6.1. Assume that the adaptive control problem is feasible (Def. 3.1.2), and that the associated performance criterion \ p (t)" is suitably dened integral norms of the estimation errors. Assume further that p (t) is monotone increasing in t and continuous in t and p. Then, the adaptive control system converges to a unique optimal controller as time proceeds. Proof By using Taylor's theorem the performance criterion p (t) can be written as: p (t) = ^ p(t) (t) + (p ^ p(t)) T r p ( ^ p(t) (t)) + 1 2 (p ^ p(t)) T r 2 p ( (t) (t)) (p ^ p(t)) Where (t) can be written as p + (1)^ p(t) ; 2 [0; 1] Since ^ p(t) minimizes p (t), we haver p ( ^ p(t) (t)) = 0 (1) Since p (t) is uniformly convex in p thenr 2 p ( (t) (t)c> 0 (2) By denition 2.1.5, equations (1) and (2), p (t) can be written as p (t) - ^ p(t) (t) c 2 kp ^ p(t)k 2 56 or, equivalently, ^ p(tm) (t n ) - ^ p(tn) (t n ) c 2 k^ p(t m ) ^ p(t n )k 2 (?) From monotonicity) ^ p(tm) (t m ) ^ p(tm) (t n ) 8t m t n Then (?) can be written as ^ p(tm) (t m ) - ^ p(tn) (t n ) c 2 k^ p(t m ) ^ p(t n )k 2 8t m t n (??) (from lemma 4.1.3) for each > 0 there exists t N such that ^ p(tm) (t m ) - ^ p(tn) (t n ) 8t m ;t n t N Then, (??) can be written as c 2 k^ p(t m ) ^ p(t n )k 2 ) k^ p(t m ) ^ p(t n )jj r 2 c |{z} 8 t m ;t n t N It is clear that as ! 0 implies that ! 0 57 By Cauchy criterion (lemma 4.1.7) ^ p(t n ) converges. Since ^ p(t n ) is unique by uniformly convexity property and since for each parameter p there is a unique controller that satisfy the adaptive control performance then, the adaptive con- trol system converges to a unique optimal controller. 6.4 Performance Improvement The main reason for introducing the supervisory control approach [Mor96, Mor97] is to ensure a satisfactory performance (e.g., regulation and tracking problem) of a closed-loop system by switching among a given set of candidate controllers. The basic idea behind the controller selection strategy is to determine which nominal process model is associated with the smallest monitoring signals, and then select the corresponding candidate controller. According to the certainty equivalence concept [Mor92]: The nominal process model with the smallest performance criterion signal \best" approximates the actual process, and therefore the candidate controller associated with that model can be expected to do the best job of controlling the process. The contribution of the local priority hysteresis switching logic in the context of supervisory control is to introduce a new switching logic that has the ability to deal with the case where the unknown parameters belong to a continuum set. 58 Now, suppose the unknown processP shown in Fig. 6.1 whose input and output signals u and y are the input of multi-estimator E where the output of E is y p , p2 P. Each y p would converge to y if the transfer function ofP was equal to the nominal process model transfer function # p in the absence of disturbances, unmodeled dynamics and noises. Disturbance input and noise sig- nal are represented by d and n respectively. Assumed that the transfer function ofP fromu toy belongs to a family of admissible process model transfer functions F = S p2P F (p) (???) for each p, F (p) denotes a family of transfer functions `centered' around some known nominal process model transfer function# p wherep is a parameter taking values in some index set P. In the absence of noises, unmodeled dynamics and disturbances equation (???) will be equivalent to V = S p2P # p In our case study [HLM + 01], the authors assumed that a candidate controllers set= =fC p :p2Pg is chosen in such a way that for each p2P;C p will satisfy the adaptive control performance, whereP is any element of F. In the sequel we assume that all assumption in [HLM + 01] are hold except the compactness property ofP that has been relaxed in Section 6.1. 59 6.4.1 Performance In the Presence of Disturbances In the Multiple model adaptive control, the adaptive control problem is placed in a setting of a standard optimization problem and the nominal process model with the smallest performance criterion signal is the model that best ts the available data ('certainty equivalence'). Therefore the candidate controller associated with this model can be expected to do the best job of controlling the process. As shown in Section 6.2, local priority hysteresis switching logic may fail to optimize the performance criterion (I) while the new idea introduced in this thesis (which relies on relaxing the local priority hysteresis switching logic constraints) ensure the optimal signal for the performance criterion (I) as shown in Section 6.3. By certainty equivalence concept [Mor92], our idea improves the adaptive control performance. 6.4.2 Performance In the Absence of Disturbances For the case of free of disturbance, unmodeled dynamics and noise the transfer functionF (p) is equal to # p ,p2P. In Section 6.3, we showed that our idea suc- ceeded to ensure convergence to optimal solution \y p ! y" therefore the exact match between the actual processP and nominal process model # p is achieved, since there is a controller parameter that has the ability to achieve adaptive con- trol performance for each # p , p2 P, then targeting performance is satised as t!1. 60 6.5 Summary The idea introduces in this thesis is investigated in the context of the unfalsied adaptive control algorithm. We believe that the unfalsied adaptive control al- gorithm is one of the best algorithms in adaptive control theory since it requires the minimum number of assumptions \feasibility" about the plant to ensure con- vergence and stability. The aim of this chapter is to show that such contribution could also be used under dierent adaptive control algorithm like multiple model adaptive control in order to enhance the system performance. Common goal for the dierent adaptive control algorithms is to satisfy the best performance for the system under minimum assumptions about the plant and its structure. Therefore combining this contribution with the unfalsied adaptive control may help to achieve this goal. 61 Chapter 7. Conclusion and Future Direction 7.1 Conclusion In this thesis we discussed recent progress in the design and analysis of the hys- teresis switching algorithm for the case of a continuum set of candidate con- trollers. The main contribution of this thesis is to study the Morse-Mayne- Goodwin hysteresis switching algorithm for continuous adaptive control and es- tablish condition on performance criterion under which the hysteresis constant may be set to zero. It has been shown that using an equi-quasi-positive denite performance criterion is sucient to ensure adaptive control convergence. The primary focus of this dissertation is to relax the usual requirement that the hys- teresis constant is strictly positive. Relaxing this constraint allows the adaptive controller to converge to a unique optimum, yielding an improved performance (regulation and tracking), as shown in Chapter 5.1. 62 7.2 Future Directions The method of controlling a system using adaptive control is not new. The idea was discovered more than a half century ago. It seemed natural to switch be- tween dierent controllers when no single controller was capable of achieving the performance goal. At that time, the stability and convergence proofs of adaptive control were based on several plant assumptions, which could cause limited prac- tical applications of this method. Since then, a fair amount of research has been done to relax these assumptions. It has been found that some of these assump- tions are not crucial and can be relaxed. In this thesis, we examined continuously switched adaptive control systems in the context of unfalsied adaptive control, without using any of the usual con- straints on the switching process (e.g., hysteresis switching, dwell-time, average dwell-time), and were able to theoretically prove the system convergence to a unique \optimum" controller based on the feasibility and some assumptions on the performance criterion used for controller selection. The main contribution of the unfalsied adaptive control algorithm is that it does not require any assumption about the plant (i.e., plant-assumption-free method) in order to ensure the stability of the system, given the feasibility of the adaptive control problem and a cost detectable performance criterion. 63 The cost-detectability property is a condition of the performance criterion that ensures closed-loop stability for the switched multi-controller adaptive control (MCAC) system whenever stabilization is feasible. For this reason, an adaptive control system that employs cost-detectability has been called a \safe adaptive control system" [WPSS05]. Unfortunately, the performance criterion introduced in this thesis does not have this property. The possibility of achieving cost- detectable safe adaptive control with continuous switching is a topic for future research. 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Abstract (if available)
Abstract
Adaptive control convergence has been proved for long time by using slow switching schemes through separating the two successive switching events by a positive time interval (e.g., dwell-time, average dwell-time, hysteresis switching technique). This thesis addresses the inherent limitations of some logic-based switching among infinite (i.e. continuum) set of candidate controllers. In this thesis, we examine adaptive control convergence in the context of well-known hysteresis switching algorithm by relaxing the usual requirement that the hysteresis constant is strictly positive. Relaxing this constraint allows the adaptive controller to converge to a unique optimum in the case of an infinite (continuum) candidate controller set, provided that at least one controller in the controller set has the ability to satisfy adaptive control performance.
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Alharashani, Mubarak
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Relaxing convergence assumptions for continuous adaptive control
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06/22/2010
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