THIS PAPER is concerned with the problem of persistent

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1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 9, SEPTEMBER Persistent Identification Adaptation: Stabilization of Slowly Varying Systems in Le Yi Wang Lin Lin, Member, IEEE Abstract In this paper, the problem of persistent identification adaptive stabilization of time-varying systems is studied within the framework of deterministic worst case identification slow H 1 adaptation. The plants under consideration are unstable time-varying cannot be stabilized by a fixed robust controller. Starting from an initial well-designed operating point, the controller must persistently adapt to the time-varying plant to maintain uniform stability over all future time. A key property which guarantees uniform stability is that the identification-adaptation iteration satisfies a certain invariance principle. We demonstrate that the adaptive design using periodic external inputs, least squares identification, slow H 1 adaptation possesses such an invariance property, leading to a successful adaptive stabilization methodology. Generic natures of our findings are discussed. Index Terms Adaptation, H 1 control, identification, stability, time-varying systems. I. INTRODUCTION THIS PAPER is concerned with the problem of persistent identification adaptive stabilization of singleinput/single-output (SISO) discrete-time time-varying systems. The plants under consideration are unstable time-varying cannot be stabilized by a fixed robust controller. As a result, adaptation is inevitable. Starting from an initial welldesigned operating point, the controller must persistently adapt to the time-varying plant to maintain uniform stability over all future time. Some challenging issues must be resolved: How can one design an identification algorithm which can persistently identify time-varying plants with uniformly small error bounds? How should the interaction between identification adaptation be resolved? In this paper, these issues are investigated for slowly time-varying systems which are, perhaps, the simplest most tractable time-varying systems. A design methodology is developed which employs the ideas of persistent probing signals, uncertainty-set identification, slow adaptation. Traditionally, research on adaptive control has been concentrated on parametric systems. Many useful adaptation algorithms have been developed [7]. Rigorous analysis of stability convergence of these algorithms have been pursued since Manuscript received January 23, 1995; revised October 1, Recommended by Associate Editor, M. Dahleh. This work was supported in part by the National Science Foundation under Grants ECS ECS L. Y. Wang is with the Department of Electrical Computer Engineering, Wayne State University, Detroit, MI USA. L. Lin is with MPD Technologies, Inc., Hauppauge, NY USA. Publisher Item Identifier S (98) the 1970 s [14]. Main results obtained in this direction address the issues of boundedness of signals convergence of parameter estimates when the underlying uncertain plants are time invariant. Extensions to time-varying systems have also been reported [18]. The work reported in this paper differs from the traditional problem formulation methodologies mainly in three aspects. First, plants are time-varying (versus time-invariant), contain norm-bounded unmodeled dynamics (versus finitedimensional systems), cannot be stabilized by a fixed controller (versus the assumption that there exists, albeit unknown, a time-invariant stabilizing controller for the system). Second, we focus on the issue of adaptation to the timevarying aspects of the plant, rather than the initial convergence to a satisfactory controller. This is clearly reflected in our assumption that at the starting time the designed frozentime controller can stabilize the frozen-time plant. The main challenge becomes: can the controller appropriately adapt itself when the plant varies beyond the robust regions of the initial controller? Finally, we impose a much stronger performance criterion, namely, uniform stability. This requires that controllers be adapted very carefully such that desired performance properties hold all the time after. Together, these aspects impose a great challenge on identification adaptation which must generate plant models sufficiently accurate for control design maintain performance levels persistently over all future time. We will use the term persistent identification adaptation for such problems. While these aspects are relatively unusual new to the traditional approaches of adaptive control, our approaches employ many ideas methods of classical adaptive control. For instance, our design can be easily characterized as an indirect adaptive control using certainty equivalence principles, although in the literature of time-varying systems our approaches have been called frozen-time design since the 1960 s. The rank conditions on input signals can be clearly recognized as a type of persistent excitation (PE) conditions, although the classical PE conditions often take very different forms depending on system parameterizations are used to guarantee asymptotic convergence of parameter estimates. While our estimates may not converge, they must generate sufficiently accurate uncertainty sets containing the plant, persistently over all time, so that robust control can perform. As a result, the conditions we impose take somehow different forms from classical PE conditions. The main assumptions we make on the time-varying plants are the following /98$ IEEE

2 1212 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 9, SEPTEMBER ) Time-varying plants are expressed in a factorization form, in which are frozen-time coprime in the rings of stable systems. 2) is known time invariant. 3) contains (possibly infinite-dimensional) unmodeled dynamics is slowly time-varying over a large range such that there exists no fixed robust controller to stabilize. 4) The external disturbances are uniformly bounded by a constant. Within these assumptions, the coprimeness is matory in our approach since our design relies on robust control to provide a nonzero robustness region for adaptation. It should be noted that coprimeness here is expressed in the rings of stable systems, rather than polynomials. Hence, common stable parts in are allowed. The second assumption is nonessential. The inclusion of this condition is for the simplicity clarity of our development. To demonstrate its ready extension to the case is unknown slowly time-varying, we cite the recent results on closedloop persistent identification [32]. The third condition ensures that our problems cannot be reduced trivially to the case of stable system identification by first applying a fixed stabilizing feedback to the plant. The fourth condition follows the paradigm of deterministic control-oriented identification in which a bounded uncertainty set is sought for robust control. The boundedness of the uncertainty set becomes impossible when is not bounded. A direct consequence is the potential conservativeness of our results: We do not allow occasional bursts of disturbances, no matter how rarely they may occur. The classical stochastic setting may offer advantages in this aspect. The pros cons of worst case identification versus averaging identification have been vigorously debated are far from being resolved. Our approach will carry this birthmark until a better framework is developed to incorporate worst case stochastic methodologies. It should be emphasized that this paper does not seek weakest conditions on the plant for the given problem. In fact, the assumptions we make on the plant are quite strong. Our main intention is to underst the nature essential features of persistent identification adaptation in a robust control framework. As it turns out, these features can be summarized in two invariance principles. 1) Feedback Invariance Property (FIP) for probing signals: The feedback system (see Fig. 1) which maps to is control-dependent alters signals significantly. As a result, closed-loop identification becomes very difficult. FIP characterizes the features of signals which are invariant, at least approximately, under the feedback mapping. Periodicity ranks are such properties. A pleasant surprise arises as these properties are shown to characterize desirable probing signals for persistent identification as well. 2) Invariance Principle (IP) for identification-adaptation iteration: To maintain uniform performance, identification errors performance levels should not increase Fig. 1. Feedback system. beyond the uniform limits in any steps of the iteration. This can be viewed as a condition of contraction or invariance on the identification-adaptation iteration. While this paper addresses these issues in a concrete problem, they seem to be generic in persistent adaptation problems. Despite a long history of research on adaptive control, some initial ( essential) objectives of adaptation are still not met with rigorous analysis. These objectives are: 1) adaptation is mainly useful for time-varying environment; 2) adaptation is useful only if robust (nonadaptive) control cannot achieve required performance; (3) adaptation is useful only if it can offer better performance. Our work here is a preliminary effort in establishing a framework in which these objectives can be directly rigorously addressed. The paper is organized as follows. After introducing necessary mathematics definitions in Section II, we describe the scenario key issues under study in Section III. The main problems are formulated in Section IV. General approaches are summarized. Features of probing signals which are desirable for closedloop persistent identification are sought in Section V. We argue that desirable probing signals should possess features which guarantee accurate identification, on one h, are feedback invariant, on the other. We show that full-rank periodic signals have such features hence are suitable for persistent identification adaptation of time-varying systems. Section VI studies the problem of system identification of slowly time-varying systems. Explicit error bounds on identification errors are established, which are related to metric complexities of prior uncertainty sets, variation rates of the plants, probing capability of input signals. The key property of these identification mappings, which is of essential importance for applications in adaptive systems, is expressed in the contraction property (Theorem 2) which establishes a sequence of vanishing invariant neighborhoods under the identification mappings. The main results on adaptation are presented in Section VII (Theorem 3) in which an adaptive control scheme is developed, employing the results of Sections V VI as well as the slow adaptation introduced in [41]. It is shown that when variation rates of the plants are small, the design procedure produces an adaptive control which achieves persistent bounds on posterior uncertainty sets stabilizes the time-varying uncertain plants. Since there exists no (nonadaptive) robust control for the time-varying plants, adaptive control is not only a superior choice, but inevitable indeed.

3 WANG AND LIN: PERSISTENT IDENTIFICATION AND ADAPTATION 1213 Finally, limitations of our approaches open research issues along the direction of this paper are summarized in Section VIII. Related Early Work: This paper employs ideas early findings in deterministic worst case identification, adaptation, classical adaptive control. Hence, it is no surprise that this paper is related to a vast array of early results in several fields. This paper is a continuation of some early work on identification frozen-time design of time-varying systems. The measures of persistent identification performance were introduced in [31] for linear time-invariant (LTI) slowly timevarying stable systems in impulse response models. Further back, these measures are essentially special cases of shiftinvariant identification -width introduced in [40]. Extensions to unstable systems were reported recently in [32]. A double algebra framework was introduced in [41] [35] for slow adaptation. The problem of worst case identification is now a very active research area. The concepts of -net -dimension in the Kolmogorov sense [11] were first introduced into the control field by Zames [37] in studies of model complexity system identification. Complexity issues were investigated by Tse et al. [28], [6], Poolla Tikku [25], [26], also [37] [34]. Milanese is one of the first researchers in recognizing the importance of worst case identification. He coworkers introduced the problem of set-membership identification produced many interesting results on the subject [20], [19], [21]. Within the framework of -frequency domain identification introduced in [10], many efficient algorithms have been developed, including Bai Raman [1], Chen et al. [3], Gu et al. [8], [9], Makila et al. [16], [17]. To enhance the feasibility of worst case identification in adaptive applications, major efforts have been made to develop algorithms using time-domain data. Numerous results have been reported, including [43], [22], [12], [4]. Furthermore, to relate identification to robust control in a direct manner, the problems of closed-loop identification interaction between identification control have also been pursued [2], [16]. The adaptation under study follows closely the fundamental philosophy of Zames in developing an information-based theory of identification adaptation [38], [39], [42]. There is a large treasury of literature on classical identification adaptive control. The reader is referred to [7] [14] the references therein for classical adaptation schemes for deterministic stochastic systems. An early version of this paper appeared in [33]. II. MATHEMATICS PRELIMINARIES A. Basic Notation The real numbers, complex numbers, integers are denoted by respectively. For is its complex conjugate its absolute value. Several vector matrix norms are used in this paper. For a vector, we define for For a matrix will denote its transpose, its largest smallest singular values. The following inequalities are stard. For We shall denote throughout this paper For (1) (2) denotes the normed spaces of sequences, for which is the space of sequences with Sequences in are upper bounded by an exponential decaying envelope. For becomes the Lebesgue space, is the usual norm. denotes the truncation operator: for zero otherwise. In this paper, we often require a system to be exponentially stable. It is well known that exponentially stable systems have transfer functions analytic on the disk of radius larger than one. Hence, we introduce the following notation: is the Hardy space of analytic functions on the open disk of radius, with norm For is reduced to the usual space. (3)

4 1214 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 9, SEPTEMBER 1998 B. Systems In this paper, stable systems will consist of singleinput/single-output (SISO) linear time-varying causal bounded discrete-time systems on with convolution representations the kernel of satisfies,. The norm of is defined by (4) The (time) variation rates of systems are defined by It is easy to verify that III. SCENARIOS AND ISSUES (6) Unstable systems will belong to, the extended space of. 1 The subspace of time invariant systems in will be denoted by. If, its kernel (impulse response). For with impulse response, we shall adopt the mathematics notation (rather than the engineering notation of form) of -transforms A. Feedback Configuration Consider the feedback system in Fig. 1. are the external input, plant input, plant output, disturbance, respectively. The interconnection of a feedback a plant in is well-posed if all elements of the closed-loop system (7) for in the region of convergence. denotes the right shift operator on are in stable if they are in. is said to robustly stabilize a subset of plants if is stable for all. The frozen-time system of at is defined as the (time-invariant) system with representation The kernel of will be denoted by,. Obviously,. Hence,. Several norms are frequently used in this paper. For with kernel, we denote, for, the time-domain norm the frequency-domain norm Since for we have 2 1 One may check, e.g., [36] for detailed discussions on extended spaces. However, we do not need further detail on IB e in this paper. 2 The last inequality in (5) is sometimes used without explicit citation. In particular, the estimation error bounds in the k1k norm established in Section VI are used in Section VII in the k1k () norm. (5) B. Main Scenarios Issues In adaptive control of time-varying systems, identification must be performed in closed loop. Since both identification adaptation rely on the input to fulfill their individual needs, they interact intimately often impose competing requirements on the input, leading to nonlinearity complications in system analysis design. Our problems will be formulated around Scenario 1. Scenario 1: Identification Adaptation of Time-Varying Systems: Suppose that the plant is time-varying. At each time, the frozen-time system of is identified by using input output observations near. The observations result in an uncertainty set which contains. A frozen-time controller is then designed which robustly stabilizes achieves certain desired performances. This identification controller adaptation procedure is repeated at each discrete time instant. This scenario, which is nothing but indirect adaptation with certainty equivalence principles, is characterized by several features: 1) The input to the plant is generated by the feedback loop cannot be manipulated directly. Only the external input can be selected at will. 2) Since the plant is time-varying, only a small window of observations near can provide useful information about the frozen-time plant, even if all past input output measurements are available. 3) One input must provide sufficient probing capability for all possible observation windows. 4) The controller must robustly stabilize for frozen-time performance, must change smoothly to ensure a tangible relationship between frozen-time performance that of the time-varying system, must not produce signals detrimental to identification.

5 WANG AND LIN: PERSISTENT IDENTIFICATION AND ADAPTATION 1215 Several fundamental issues arise in this scenario. 1) How can one characterize classes of plant inputs which can guarantee accurate identification for all frozen-time systems? 2) Can such plant inputs be generated by external inputs via the feedback loop? 3) How should controllers be designed? 4) How should the interaction conflicts between identification adaptation be resolved? but otherwise arbitrary. is assumed to be an exponentially decaying function of. 4) is slowly time-varying with rate 5) For any are coprime in. 6) The disturbance is bounded with IV. PROBLEM FORMULATION AND GENERAL APPROACHES A. Plants A pair is coprime in if for some A system has a left (or right) factorization representation in if (or ), is well defined. The factorization is coprime if the corresponding pair is coprime in. Plants controllers considered in this paper are represented in the feedback system by (8) (9) Examples of systems satisfying Assumption 1 include systems with unstructured additive or multiplicative uncertainties: or are LTI known, but the uncertainty part is unstructured slowly varying. It is noted that when a priori uncertainty on the plant is large, there exists no (nonadaptive) robust control for this class of systems. Consider, for instance, an uncertain timevarying plant with a priori information. Let be solutions to the Bezout equation Then, it is stard that stabilizable in if only if is robustly is the representation of the plant the controller; are the external input, disturbance, plant input output, respectively. Consequently, if B. Basic Assumptions Consider the feedback system in (9). The plant right factorization representation in admits a is not robustly stabilizable by a nonadaptive controller. In this case, adaptation becomes inevitable. Assumption 1: For some 1) is time invariant, known,. Without loss of generality, assume is inner in, namely,. Denote. 2) is exponentially stable,, is strictly causal. To be explicit, we sometimes write. Let. 3) A priori information about is that the frozen-time systems are decomposed into is the modeled part of order, is the unmodeled dynamics. We do not impose conditions on whose coefficients will be estimated by identification. The unmodeled dynamics is normbounded by C. Design Specifications Desired performance of the feedback system is characterized by the following specifications. Design Specifications: The property of the closedloop system (9), defined below, must be satisfied for all. 1) 2) 3) The estimate of must satisfy Here, 1) 2) are stability requirements (uniform boundedness of all signals). Item 3) dems uniform estimation error bounds. The norm is used since feedback design must generate exponentially stable systems so that time-varying effects can be compensated. The problem under study can be stated as follows.

6 1216 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 9, SEPTEMBER 1998 Problem 1: Given a time-varying plant satisfying Assumption 1, design an identification algorithm, an adaptation mapping, a probing signal such that Specification holds implies holds for all. Departing from traditional convergence analysis, Problem 1 concentrates on the ability of a feedback system in uniformly adapting to a time-varying environment (hence, the requirement holds for all ) after arriving at its initial equilibrium points (hence, the condition Specification holds ). Problem 1 will be solved for slowly varying plants, using the approaches of uncertainty-set identification slow adaptation. Although we select stability for concreteness in our development, the framework can certainly be extended to other performance measures. Here, the three main issues must be resolved: 1) selection of probing signals ; 2) construction of identification algorithms; 3) construction of adaptation mappings. In the sequel, the term an adaptive design procedure will mean a specification of these three items. D. General Approaches Adaptive control is an iteration process involving the interaction of identification adaptation at each time step. More precisely, one starts at some time with a certain estimation error For an identification algorithm is devised which provides estimates with estimation errors Then, an adaptation mapping is applied which produces robust controllers to achieve performance specifications based on the posterior uncertainty. This iteration process leads to a sequence of identification errors signal bounds Problem 1 can also be stated in the following invariance principle. Problem 2 (Invariance Principle): Define an adaptive design procedure such that the intervals Design Procedure: For a given real value (for estimation error bounds) an integer (for model complexity) we have the following. 1) The identification algorithm is the classical least squares estimation based on the most recent observations. 2) The adaptation mapping is the suboptimal design developed in [41]. 3) The probing signal is -periodic full rank, namely, the Toeplitz matrix of symbol is full rank. We will denote. While one might quote simplicity popularity for selecting least squares estimation, the justification here is more fundamental: for persistent identification, it is an optimal identification algorithm, under certain conditions, among all identification algorithms. This finding was first established for stable systems in [31]. In adaptive control of time-varying systems, a frozen-time designed controller must stabilize the frozen-time plant also vary smoothly. The design in [41] provides an explicit tradeoff between optimal robustness smoothness of controllers. Finally, periodicity rank conditions are feedback invariant, i.e., they are preserved after the feedback mapping from to. It turns out that these two properties also characterize desirable probing signals for persistent identification problems. The main result of the paper is Theorem 3. Theorem 3: Under Assumption 1, for any required, there exist such that an integer (model complexity) can be selected under which the design procedure guarantees are invariant, provided. The rest of this paper is devoted to establishing details properties of the design procedure this main result. V. PROBING SIGNALS We start with a search of probing signals suitable for persistent identification in a closed-loop setting. Consider a shift-invariant system with kernel. The input output relationship of is expressed as are invariant for under the procedure, namely. For a given model order, we assume that the unmodeled dynamics In this paper, an adaptive design procedure will be developed which consists of the following identification adaptation iterations. The details of the procedure will be postponed until later sections. is bounded by vectors. Define the truncated regression (10)

7 WANG AND LIN: PERSISTENT IDENTIFICATION AND ADAPTATION 1217 the vectors of system parameters A. Toeplitz Systems with Periodic Symbols Let be an -periodic signal (i.e., a periodic signal of period ) the corresponding Toeplitz matrices defined in (11). Define the discrete Fourier transform of as It follows that As shown in [34] [40], for typical classes of uncertainty sets, is the optimal choice of models of order. A stard procedure for estimating proceeds as follows. First, observations on in are performed the input output relationship is rewritten in the form of In particular, for verify the following shift property: for any, it is easy to The following results are fairly stard, especially (1) (2). The proofs are hence omitted. Lemma 1: 1) 2) (11) Note that are Toeplitz matrices. If the input is selected such that is invertible, can be represented as 3) Suppose is an -periodic signal, are the Toeplitz matrices with symbols, respectively. Then have the same set of singular values. It follows immediately from Lemma 1 that is invertible if only if its symbol satisfies is an estimate of, (12) Now, let us revisit the identification errors in (13). Proposition 1: If is -periodic, then (14) (13) is a term of identification error, its dependence on the current time of observation windows is expressed explicitly. In persistent identification problems, estimation errors must be uniformly small over all. Hence, an appropriate measure of estimation errors is is defined in (2). Proof: See the proof in Appendix 1. Remark: Express, with its dependence on the symbol explicitly denoted. Let be the normalized input. Obviously,. Denote, the noise/signal ratio. By linearity As a result, (14) can be expressed as To obtain bounds on the identification error which depends on the probing signal, some basic properties of periodic signals must be established, which will be discussed in the next subsection..

8 1218 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 9, SEPTEMBER 1998 B. Feedback Invariance of Periodic Signals Plant input signals appearing in feedback systems are not directly accessible for experiment design in identification. For example, the external probing signal the plant input are related by a feedback system. is LTI (or slowly time-varying) when both are LTI (or slowly timevarying). We will show that periodicity ranks of a signal are invariant, at least approximately, under such feedback mappings. LTI Systems: Consider signals which are generated by - periodic signals passing through an LTI stable system is the kernel of. Let. Denote. Lemma 2 is a stard result [23]. Lemma 2: If is -periodic, then is -periodic Proof: Since is slowly time-varying with rate,by (6) we have It follows that since is -periodic Next, we study rank conditions. The frozen-time system of at is denoted by. Without loss of generality, assume define. are the Toeplitz matrices of symbols, respectively. That is (15) An immediate consequence of Lemmas 1 2 is that the Toeplitz matrix is invertible if It is noted that the condition is always satisfied if is a stable feedback system in which the plant controller do not have boundary poles zeros. As a result, periodicity ranks of are preserved in. Slowly Varying Systems: However, when is slowly timevarying, in general its output is not -periodic even when the input is -periodic. Suppose is slowly varying with rate First, we will show that if is -periodic is slowly varying, then is approximately -periodic. Observe that is -periodic if only if Definition: A signal is said to be nearly -periodic of deviation if Proposition 3: If, then is approximated by. Proof: See the proof in Appendix 1. Since the frozen-time system is shift invariant, the extreme singular values are characterized by Lemmas 1 2. It follows from Proposition 3 that if is -periodic full rank, is slowly time-varying, then Consequently, implies whenever is sufficiently small. VI. PERSISTENT IDENTIFICATION OF SLOWLY VARYING SYSTEMS The problem here is to identify a slowly varying system with rate via observations from (16). Initially, a priori information on about its dynamics at some target time is that. The prior uncertainty set will consist of systems with exponentially decaying memory, we denote Proposition 2: If is -periodic, then is nearly -periodic of deviation For systems with exponential decaying memory, is an exponentially decaying function of. Take, for instance, the uncertainty set (17)

9 WANG AND LIN: PERSISTENT IDENTIFICATION AND ADAPTATION 1219 For Remark 1): The first term in is a function of the metric complexity (optimal modeling error) of the a priori information. The second term depends on the variation rate probing capability of. Note that the second term is a function of. As a result, the reduction of the second term cannot be achieved by increasing the magnitude of the probing signal. The third term is a function of noise/signal ratio, as expected in filtering problems. 2): Asymptotically is a constant. As a result, approaches zero exponentially as. Note that the identification problem for the systems described in (9) can be easily transformed into a case of (16) with, noting that is known time invariant. The identification of the frozen-time system (with kernel ) is to be performed on consecutive observations on in the time interval. Without loss of generality, assume (or any fixed ). Define which, according to [31], is the optimal radius of the a posteriori uncertainties for some typical classes of LTI systems. B. Closed-Loop Identification In an adaptive control setting in which both are slowly varying, the probing signal to the plant is the output of a slowly varying system, 3 is -periodic full rank, is slowly time-varying with rate. Theorem 1: Under Assumption 2 1) as the Toeplitz matrices of symbol as in (11). Also, define, the normalized probing signal with,, the noise/signal ratio. Assumption 2: 1) has variation rate. 2) is the estimate of (18) provided (20) 3) is an exponentially decaying function of for some (19) It is observed that for the uncertainty set (17), condition (19) is satisfied. A. Open-Loop Identification with Periodic Probing Signals Proposition 4: Under Assumption 2, if is -periodic full rank with, then,. 2) Suppose, in addition,. Denote Then a) (21) Proof: See the proof in Appendix 2. 3 To make the notation consistent for this section, we use x u in place of u r in Fig. 1.

10 1220 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 9, SEPTEMBER 1998 b) For any given, there exist an integer (model order) such that one can construct an -periodic full rank probing signal, for which provided. Proof: See the proof in Appendix 2. Conceptually, it is quite straightforward to see that to reduce below, one can reduce by increasing (Assumption 2), reduce the second third terms by allowing smaller, the fourth term by increasing. However, when is actually a feedback system, its variation rate is a function of the plant controller variation rates, as well as estimation error. To avoid circular arguments, we must demonstrate that is uniformly bounded. This will be sought in the next subsection. C. Contraction Properties When one applies the identification mapping (18) in an adaptation scheme, the feedback mapping is a time-varying system. Conceptually, it is obvious that due to identificationadaptation interaction, the variation rates of will depend on the variation rates of the plant controller, as well as estimation errors. It will be shown in Section VII that for the adaptation schemes employed in this paper, the variation rates of are bounded by (22) are constants, is the identification error bound at. As a result, according to (21) (22) the error bound in (21) evolves dynamically in an iteration inequality (23) To establish uniform boundedness of identification errors in adaptation, we must show that there exists a neighborhood of at zero which is invariant under the iteration mapping (23). Theorem 2 (Contraction Properties): There exists a sequence of neighborhoods (24) with as, such that for each, there exist for which is invariant under the iteration mapping (23), provided Proof: First, let. Then by (22). By (21) Define Then for To have exists such that, we have A sufficient condition for (25) is However, by Assumption 2, since with which, for sufficiently large, we only need to show that there (25) (26) is bounded by, is monotone decreasing in As a result, there exists for which the strict inequality (25) is valid for. Define as (27) Now, by the continuity of with respect to, as well as the strict inequality of (25), there exist such that for any provided. Therefore, is invariant under the iteration mapping (23). Finally, since defined in (27) satisfies,as, the invariant neighborhoods are vanishing indeed. This completes the proof. VII. PERSISTENT ADAPTATION A. Adaptation Mapping The feedback controller is updated at every step via the frozen-time approach. At time, the posterior information of the plant is obtained via identification is designed which robustly stabilizes. 4 For the uncertainty sets considered in this paper, (to be constructed later) takes the form of is an estimate of the identification error in. By the theory of robust stabilization, 4 It should be pointed out here that in general, Y 01 t X t 6= (Y 01 X) t.to avoid confusion, the symbol F(t) is used to represent Y 01 t X t.

11 WANG AND LIN: PERSISTENT IDENTIFICATION AND ADAPTATION 1221 robustly stabilizes in if one solves the Bezout equation Let (28) (29) (30) the infimum is taken over all satisfying (28) (with some ). More precisely, if solve (28), then by Youla s parameterization, all stabilizing controllers for are parameterized by B. Identification Precision Robust stabilizability requirements mate that the identification error must satisfy (35) For (35) to hold for all must be uniformly bounded. Observe that is a function of, which will be explicitly denoted by. Since is a function of, it depends on the a priori information as well as identification. A uniform upper bound on can be computed. Observe that which implies with well-posed. It follows that Define (31) (36) In particular, for let be the -suboptimal central solution to (31) (see, e.g., [35] for detail). Then is an upper bound on when the underlying identification error is bounded by. Note that is a monotone increasing function of. Since Consequently, we can compute robustly stabilizes in if Define the adaptation mapping as (32) (33) a sufficient condition for (35) is (37) Inequality (37) is always satisfied for sufficiently small, since. C. Design Specifications Procedures The closed-loop (time-varying) system is given by It has been shown [35] that the mapping is Lipschitz continuous, namely By defining,we get (38) (34) the Lipschitz constant. 5 The Lipschitz continuity (34) is required to guarantee that are slowly time-varying when is slowly varying. 5 The H 2 norm is G G = s G G dz : The design objective is expressed by design specifications given in Section IV. Namely, we would like to establish holds for all Now, we specify our design parameters procedures. Let be the monotone decreasing invariant neighborhoods defined in Theorem 2, the corresponding parameters. Also, define by (36).

12 1222 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 9, SEPTEMBER 1998 Design Procedure : For the selected, we have the following. 1) The adaptation mapping is defined by the central - suboptimal design, given by (33). 2) The estimate of is constructed from observations on in via the least squares estimation is the Toeplitz matrix of symbol,, 3) The probing signal is -periodic full rank. D. Main Results Theorem 3: Under Assumption 1, there exists such that for any a corresponding can be found for which the design procedure guarantees that holds for all with in (39), provided that holds the variation rate of the plant satisfies. Remark: Theorem 3 claims that for any disturbance bounds any a priori information satisfying Assumption 1, an adaptive design procedure can be devised to achieve uniform stabilization uniform identification error bounds, provided that the variation rate of the plant is sufficiently small. The allowable variation rate as well as the signal bounds, however, are functions of the metric complexity of, disturbance bound, the required identification precision. The remaining part of this section is devoted to the proof of Theorem 3. First, Property will be exped. Design Specifications :, consisting of the following components, must hold for all. a) Uniform Identification Error Bounds: b) Slowly Varying Estimates: (39) (40) Therefore implies with. Hence, we only need to demonstrate here. Theorem 3 will be proved by induction. Namely, we will establish the implication Then the conclusion follows by induction. Recall that the closed-loop system is expressed by (38) with satisfying Assumption 1 as well as designed by the design procedure. To facilitate the analysis, we define the following artificial time-varying systems: At time, define for In other words, assumes the same frozen-time systems as before is time invariant after with dynamics, similarly for. Then, the following artificial system can be constructed: By defining By causality,we get Define the local global resolvents of the artificial system by c) (41) It is easy to show that the local product of two time-varying systems is defined by Indeed, since we have, by the causality of Two lemmas will be established first. Lemma 3: Under Assumption 1, there exist a design procedure (with the corresponding ), being independent of, such that is slowly time-varying with are constants, provided holds with. Proof: See the proof in Appendix 3.

13 WANG AND LIN: PERSISTENT IDENTIFICATION AND ADAPTATION 1223 Lemma 4: Under Assumption 1, there exist a design procedure (with the corresponding ), being independent of, such that 1) 2) of are satisfied whenever holds with. Proof: See the proof in Appendix 3. Now, we are ready to prove Theorem 3. Proof of Theorem 3: Suppose holds. Let By Lemma 4, 1) 2) of are satisfied, when the design procedure is employed. To prove 3) of, we define the artificial systems by the suboptimal design. Since we get. This verifies 3) of. By induction, holds implies holds for all. for More precisely, the artificial system is given by By defining,we get Denote the corresponding global resolvent as As a result Following the same arguments leading to (53), we conclude (42). Therefore VIII. CONCLUDING REMARKS Historically, adaptation was introduced as a means of achieving superior performance for time-varying plants environment. While some practical adaptive systems often demonstrate these expected benefits, rigorous analysis of them remains a daunting problem. This paper is a preliminary effort in establishing a framework in which the ability of a controller to adapt to a time-varying environment to achieve performance beyond robust control can be rigorously studied. Conceptually, the notion of persistent identification adaptation is introduced to capture these aspects of adaptive systems. It is shown that the fundamental features of adaptive schemes in such problems are feedback invariance properties for probing signals invariance principles for identification-adaptation iteration. A stabilization problem is solved in detail. There remain numerous unresolved issues in persistent adaptation problems. It is highly desirable that system performance, beyond stability, can be analyzed. The main obstacle is the lack of tangible robust performance synthesis methods (available design methods like robust performance, synthesis, etc., still encounter difficulties in characterization numerical solutions). Also, in trading for a concise introduction of the framework, we used some academic arguments in the design procedure. For instance, we assume that noise/signal ratios can be freely changed by increasing the signal magnitudes. When this is not the case (as always in practice), signals must be more carefully selected to minimize the effect of disturbances on identification errors. Furthermore, to fit into the robust control framework, identification is performed in a worst case paradigm. Inevitably, results may become very conservative. The conservativeness becomes especially severe in persistent adaptation problems since allowable variation rates of the plant are functions of time complexity of the prior uncertainty sets. It is now well understood that time complexity in worst case identification can be extremely high. the last inequality follows from APPENDIX 1 PROOFS FOR SECTION V Proof of Proposition 1: It follows from Lemma 1 that

14 1224 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 9, SEPTEMBER 1998 As a result We have Furthermore, by (1) Since the observation is performed on, not, we express as (45) Therefore (43) When is -periodic Proof of Proposition 3: by Lemma 1. Consequently is an estimate of. The error term is bounded in the norm by Since is a Toeplitz matrix of symbol APPENDIX 2 PROOFS FOR SECTION VI Proof of Proposition 4: By the definition of frozen-time systems by Assumption 2 (43). Moreover, by (1) Observe that is approximated by in the interval of observation by (44). Therefore (44) Finally Define

15 WANG AND LIN: PERSISTENT IDENTIFICATION AND ADAPTATION 1225 Proof of Theorem 1 1): By Proposition 3 by (46). Suppose. Then for is invertible by (43) (19). Therefore (46) 2)-a): It follows from Lemmas 1 2 that It follows that the error in (45) can be expressed as Let be the normalized probing signal. By linearity,. Now, by (20) b): Let. Then, for, we have It is easy to verify that Since by Assumption 2, with, for any, there exists for which As a result Moreover, by selecting to satisfy we have The proof is completed by defining

16 1226 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 9, SEPTEMBER 1998 APPENDIX 3 PROOFS FOR SECTION VII Proof of Lemma 3: By the Lipschitz continuity of the adaptation mapping The right-h side of (47) is bounded by (47) As a result Now Furthermore, by -suboptimality (48) Also, by [41] By (5), we have Now, select, appears in (24) satisfying Then, there exists such that Also (52) Note that by Assumption 1 (a) of whenever. In this case. Moreover (53) It follows that Furthermore, is slowly varying. Indeed are constants. Now the local resolvent can be expressed as (49) (50) (51) by (51), Lemma 3 follows with. Proof of Lemma 4: Let us consider the identification problem at. Here, is to be estimated based on input output observations Since is strictly causal,, the identification of requires output observations input. Denote Define the difference operator by Since is time-invariant, by [41] we have Since

17 WANG AND LIN: PERSISTENT IDENTIFICATION AND ADAPTATION 1227 Observe that when in (52), Let By linearity Now If sufficiently small (which is always true for sufficiently large ), we have. By Lemma 3 Proposition 4, the identification error is bounded by Select to satisfy (54) is the required rate in the design specifications. Then for, by Theorem 2 there exists for which the neighborhood is invariant under the iteration mapping (54) when. As a result provided. This implies 1) of. Finally, by (48) Since, there exists such that whenever, we have Therefore, 2) of by choosing is satisfied. Now, Lemma 4 follows REFERENCES [1] E. Bai S. Raman, A linear robustly convergent interpolatory algorithm for system identification, in Proc. ACC Conf., 1992, pp [2] R. Bitmead M. Gevers, An iterative identification control strategy, in Proc. European Control Conf., Grenoble, France, 1991, pp [3] J. Chen C. N. 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18 1228 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 9, SEPTEMBER 1998 [38], Information-based theory of adaptation, in Proc. IEEE CDC Conf., Plenary Speech, Kobe, Japan, [39], Information, complexity adaptation, Control Using Logic- Based Switching, A. S. Morse, Ed. New York: Springer-Verlag, [40] G. Zames, L. Lin, L. Y. Wang, Fast identification n-widths uncertainty principles for LTI slowly varying systems, IEEE Trans. Automat. Contr., vol. 39, pp , [41] G. Zames L. Y. Wang, Local-global double algebras for slow H1 adaptation Part I: Inversion stability, IEEE Trans. Automat. Contr., vol. 36, pp , Feb [42], Adaptive vs. robust control: Information based concepts, Adaptive Syst. Contr. Signal Processing, vol. 8, [43] T. Zhou H. Kimura, Time domain identification for robust control, Syst. Contr. Lett., vol. 20, pp , Lin Lin (S 91 M 94) received the B.E. M.E. degrees from Tsinghua University, Beijing, China, in , respectively, the Ph.D. degree from McGill University, Montreal, Canada, in 1993, all in electrical engineering. From 1985 to 1987, he participated in a research project on power system stabilization at Je Jiang Hydro, China. Since 1993, he has worked on various control signal processing problems at several places including Nortel CAE Electronics in Montreal, Canada, MPD Technologies, New York, NY, he is currently an Engineering Manager. He has been an Adjunct Professor in the Department of Electrical Engineering, McGill University since His research interests include wireless communications industrial process control. Le Yi Wang received the M.E. degree in computer control from the Shanghai Institute of Mechanical Engineering, China, in 1982, the Ph.D. degree in electrical engineering from McGill University, Montreal, Canada, in Since 1990 he has been with Wayne State University, Detroit, MI, he is currently an Associate Professor in the Department of Electrical Computer Engineering. His research interests include H1 optimization, robust control, time-varying systems, system identification, adaptive systems, as well as hybrid nonlinear systems with automotive applications. Dr. Wang was awarded the Research Initiation Award in 1992 from the National Science Foundation. He also received the Faculty Research Award from Wayne State University in 1992 the College Outsting Teaching Award from the College of Engineering, Wayne State University, in 1995.