Identifying the input-output relationship of a system or discovering the evolutionary law of a signal on the basis of observation data, and applying the constructed mathematical model to predicting, controlling or extracting other useful information constitute a problem that has been drawing a lot of attention from engineering and gaining more and more importance in econo­ metrics, biology, environmental science and other related areas. Over the last 30-odd years, research on this problem has rapidly developed in various areas under different terms, such as time series analysis, signal processing and system identification. Since the randomness almost always exists in real systems and in observation data, and since the random process is sometimes used to model the uncertainty in systems, it is reasonable to consider the object as a stochastic system. In some applications identification can be carried out off line, but in other cases this is impossible, for example, when the structure or the parameter of the system depends on the sample, or when the system is time-varying. In these cases we have to identify the system on line and to adjust the control in accordance with the model which is supposed to be approaching the true system during the process of identification. This is why there has been an increasing interest in identification and adaptive control for stochastic systems from both theorists and practitioners.

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1 Probability Theory Preliminaries.- 1.1 Random Variables.- 1.2 Expectation.- 1.3 Conditional Expectation.- 1.4 Independence, Characteristic Functions.- 1.5 Random Processes.- 1.6 Stochastic Integral.- 1.7 Stochastic Differential Equations.- 2 Limit Theorems on Martingales.- 2.1 Martingale Convergence Theorems.- 2.2 Local Convergence Theorems.- 2.3 Estimation for Weighted Sums of a Martingale Difference Sequence.- 2.4 Estimation for Double Array Martingales.- 3 Filtering and Control for Linear Systems.- 3.1 Controllability and Observability.- 3.2 Kalman Filtering for Systems with Random Coefficients.- 3.3 Discrete-Time Riccati Equations.- 3.4 Optimal Control for Quadratic Costs.- 3.5 Optimal Tracking.- 3.6 Model Reference Control.- 3.7 Control for CARIMA Models.- 4 Coefficient Estimation for ARMAX Models.- 4.1 Estimation Algorithms.- 4.2 Convergence of ELS Without the PE Condition.- 4.3 Local Convergence of SG.- 4.4 Convergence of SG Without the PE Condition.- 4.5 Convergence Rate of SG.- 4.6 Removing the SPR Condition By An Overparameterization Technique.- 4.7 Removing the SPR Condition By Using Increasing Lag Least Squares.- 5 Stochastic Adaptive Tracking.- 5.1 SG-Based Adaptive Tracker With d = 1.- 5.2 SG-Based Adaptive Tracker With d ?1.- 5.3 Stability and Optimality of Åström-Wittenmark Self-Tuning Tracker.- 5.4 Stability and Optimality of ELS-Based Adaptive Trackers.- 5.5 Model Reference Adaptive Control.- 6 Coefficient Estimation in Adaptive Control Systems.- 6.1 Necessity of Excitation for Consistency of Estimates.- 6.2 Reference Signal With Decaying Richness.- 6.3 Diminishingly Excited Control.- 7 Order Estimation.- 7.1 Order Estimation by Use of a Priori Information.- 7.2 Order Estimation by not Using Upper Bounds for Orders.- 7.3 Time-Delay Estimation.- 7.4 Connections of CIC and BIC.- 8 Optimal Adaptive Control with Consistent Parameter Estimate.- 8.1 Simultaneously Gaining Optimality and Consistency in Tracking Systems.- 8.2 Adaptive Control for Quadratic Cost.- 8.3 Connection Between Adaptive Controls for Tracking and Quadratic Cost.- 8.4 Model Reference Adaptive Control With Consistent Estimate.- 8.5 Adaptive Control With Unknown Orders, Time-Delay and Coefficients.- 9 ARX(?) Model Approximation.- 9.1 Statement of Problem.- 9.2 Transfer Function Approximation.- 9.3 Estimation of Noise Process.- 10 Estimation for Time-Varying Parameters.- 10.1 Stability of Random Time-Varying Equations.- 10.2 Conditional Richness Condition.- 10.3 Analysis of Kalman Filter Based Algorithms.- 10.4 Analysis of LMS-Like Algorithms.- 11 Adaptive Control of Time-Varying Stochastic Systems.- 11.1 Preliminary Results.- 11.2 Systems with Random Parameters.- 11.3 Systems with Deterministic Parameters.- 12 Continuous-Time Stochastic Systems.- 12.1 The Model.- 12.2 Parameter Estimation.- 12.3 Adaptive Control.- References.