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This paper is concerned with persistent identification of systems that involve deterministic unmodeled dynamics and stochastic observation disturbances, and whose unknown parameters switch values (possibly large jumps) that can be represented by a Markov chain. Two classes of problems are considered. In the first class, the switching parameters are stochastic processes modeled by irreducible and aperiodic Markov chains with transition rates sufficiently faster than adaptation rates of the identification algorithms. In this case, tracking real-time parameters by output observations becomes impossible and we show that an averaged behavior of the parameter process can be derived from the stationary measure of the Markov chain and can be estimated with periodic inputs and least-squares type algorithms. Upper and lower error bounds are established that explicitly show impact of unmodeled dynamics. In contrast, the second class of problems represents systems whose state transitions occur infrequently. An adaptive algorithm with variable step sizes is introduced for tracking the time-varying parameters. Convergence and error bounds are derived. Numerical results are presented to illustrate the performance of the algorithm. Index Terms--Error bounds, exogenous noise, parameter switching, parameter tracking, persistent identification, unmodeled dynamics.

Input design is of essential importance in system identification for providing sufficient probing capabilities to guarantee convergence of parameter estimates to their true values. This paper presents conditions on input signals that characterize their probing richness for strongly consistent parameter estimation of linear systems with binary-valued output observations. Necessary and sufficient conditions on periodic signals are derived for sufficient richness. These conditions are further studied under different system configurations including open-loop and feedback systems, and different scenarios of noises including actuator noise, input measurement noise, and output measurement noise. In addition to system parameter estimation, essential properties of identifiability and input conditions are also derived when sensor thresholds or noise distribution functions are unknown. The findings of this paper provide a foundation to study identification of systems that either use binary-valued or quantized sensors or involve communication channels, which mandate quantization of signals. Index Terms--Binary-valued observation, distribution function, identification, input design, parameter estimation, sensor threshold, sufficient excitation.

System identification is investigated for plants that are equipped with only binary-valued sensors. Optimal identification errors, time complexity, optimal input design, and impact of disturbances and unmodeled dynamics on identification accuracy and complexity are examined in both stochastic and deterministic information frameworks. It is revealed that binary sensors impose fundamental limitations on identification accuracy and time complexity, and carry distinct features beyond identification with regular sensors. Comparisons between the stochastic and deterministic frameworks indicate a complementary nature in their utility in binary-sensor identification. Index Terms--Binary sensors, estimation, system identification, time complexity.

The problem of fast identification of continuous-time systems is formulated in the metric complexity theory setting. It is shown that the two key steps to achieving fast identification, i.e., optimal input design and optimal model selection, can be carried out independently when the true system belongs to a general a priori set. These two optimization problems can be reduced to standard Gel'fand and Kolmogorov n-width problems in metric complexity theory. It is shown that although arbitrarily accurate identification can be achieved on a small time interval by reducing the noise-signal ratio and designing the input carefully, identification speed is limited by the metric complexity of the a priori uncertainty set when the noise/signal ratio is fixed. Index Terms: Adaptive control, H(super infinity), identification, model complexity, time complexity

In this paper, the problem of persistent identification and adaptive stabilization of time-varying systems is studied within the framework of deterministic worst case identification and slow [H.sup.[infinity]] adaptation. The plants under consideration are unstable and time-varying and 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, and slow [H.sup.[infinity]] adaptation possesses such an invariance property, leading to a successful adaptive stabilization methodology. Generic natures of our findings are discussed. Index Terms - Adaptation, [H.sup.[infinity]] control, identification, stability, time-varying systems.

In this paper, the problem of identification of time-varying systems is investigated in the framework of worst-case identification and information-based complexity. Measures of intrinsic errors, termed persistent identification errors, in such identification problems are introduced. For a selected model space of dimension n (finite impulse response models) and an observation window of length m, the persistent identification measures provide the worst-case posterior identification errors over all possible starting times of the observation windows when the input and identification algorithms are optimized. For linear time-invariant (LTI) plants with unmodeled dynamics belonging to certain types of prior unstructured uncertainty sets, upper and lower bounds of the persistent identification measures are explicitly computed. It is shown that when prior unmodeled dynamics are balls in the [l.sup.1] space, the lower and upper bounds coincide. In this case, any full-rank periodic probing signals are optimal, and the standard least-squares estimation is in fact an optimal identification algorithm. Motivated by closed-loop identification problems, the concept of nearly periodic signals is introduced. It is shown that such signals are asymptotically optimal for persistent identification and at the same time can be generated in a closed-loop configuration. For slowly varying systems, the persistent identification measures are shown to be continuous functions of the plant variation rates. Furthermore, periodic signals are asymptotically optimal in the sense that they achieve identification errors which approach the optimal persistent identification errors for LTI systems when the variation rates of the plants become small. This result verifies that the persistent identification measures are indeed benchmark values for the identification of time-varying systems.

The optimal worst-case uncertainty that can be achieved by identification depends on the observation time. In the first part of the paper, this dependence is evaluated for selected linear time invariant systems in the [l.sup.1] and [H.sup.[infinity]] norms and shown to be derivable from a monotonicity principle. The minimal time required is shown to depend on the metric complexity of the a priori information set. Two notions of n-width (or metric dimension) are introduced to characterize this complexity. In the second part of the paper, the results are applied to systems in which the law governing the evolution of the uncertain elements is not time invariant. Such systems cannot be identified accurately. The inherent uncertainty is bounded in the case of slow time variation. The n-widths and related optimal inputs provide benchmarks for the evaluation of actual inputs occurring in adaptive feedback systems.

In this paper, the problem of robust stabilization and robust disturbance attenuation is investigated for systems with linear nominal parts and norm-bounded nonlinear uncertainties on both states and control inputs. It is shown that the type of nonlinear uncertainty sets considered in this paper has an equivalent representation by linear uncertainty sets. Based on this key result and some standard Riccati inequality approaches for robust control of linear uncertain systems, a constructive design procedure is developed.

The fundamental issues of capability of robust and adaptive control in dealing with uncertainty are investigated in stochastic systems. It is revealed that to capture the intrinsic limitations of adaptive control, it is necessary to use [sup.sub.t] types of transient and persistent performance, rather than lim [sup.sub.t] types which reflect only asymptotic behavior of a system. For clarity and technical tractability, a simple first-order linear time-varying system is employed as a vehicle to explore performance lower bounds of robust and adaptive control. Optimal performances of nominal, robust and adaptive control are explicitly derived and their implications are discussed in an information framework. An adaptive strategy is scrutinized for its achievable performance bounds. The results indicate that intimate interaction and inherent conflict between identification and control result in a certain performance lower bound which does not approach the nominal performance even when the system varies very slowly. Explicit lower bounds are obtained when disturbances are either normally or uniformly distributed. Index Terms--Adaptive control, performance lower bounds, robust control, Stochastic system, time-varying parameter, uncertainty.

In this paper, a notion of information-based complexity is introduced to characterize complexities of plant uncertainty sets in feedback control settings, and to understand relationships between identification and feedback control in dealing with uncertainty. This new complexity measure extends the Kolmogorov entropy to problems involving information acquisition (identification) and processing (control), and provides a tangible measure of "difficulty" of an uncertainty set of plants. In the special cases of robust stabilization for systems with either gain uncertainty or unstructured additive uncertainty, the complexity measures are explicitly derived. Index Terms--Complexity, feedback, identification, information, metric robustness, uncertainty.

In this paper, a novel framework of system identification is introduced to capture the hybrid features of systems subject to both deterministic unmodeled dynamics and stochastic observation disturbances. Using the concepts of persistent identification, control-oriented system modeling, and stochastic analysis, we investigate the central issues of irreducible identification errors and time complexity in such identification problems. Upper and lower bounds on errors and speed of persistent identification are obtained. The error bounds are expressed as functions of observation lengths, sizes of unmodeled dynamics, and probability distributions of disturbances. Asymptotic normality and complexity lower bounds are investigated when periodic inputs and LS estimation are applied. Generic features of asymptotic normality are further explored to extend the asymptotic lower bounds to a wider range of signals and identification mappings. Index Terms--Error bounds, exogenous noises, identification, time complexity, uncertainties, unmodeled dynamics.