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Stochastic modelling and control. (English) Zbl 0654.93001

Monographs on Statistics and Applied Probability. London-New York: Chapman and Hall. XII, 393 p. (Gö: 85A 14204) (1985).
One of the most sophisticated and active areas of statistical and probabilistic research is that of systems and control, a field that is often studied in departments of electrical engineering. Those working in systems theory frequently use a wide range of up-to-date probabilistic techniques and results, including recent work in stochastic calculus and its applications in filtering theory. The book by the authors deals only with discrete-time theory, but appearing in a statistical series it may bring its subject matter, as well as its motivating ideas and mathematics, to the attention of a wider audience of statisticians.
The basic concept considered by the authors is a “black box” analyzed through its input/output data. Futher, it is supposed that the data are measured at discrete-time points, that the relation between input and output is (approximately) linear and that additive random noise affects the output. The first chapter presents some concepts and results from probability theory and stochastic processes. Although the Stieltjes integral is defined the more subtle points of measure theory are avoided. The second half of Chapter One discusses basic concepts of linear systems, including results on controllability and observability. Linear systems with additive noise are discussed in chapter two and much attention is paid to ARMA models from time series, for which the noise process is the solution of a difference equation driven by white noise. Filtering is concerned with obtaining the best estimate of a signal process given noisy observations, and this is discussed for linear, discrete-time models in Chapter Three. The well-known Kalman filter is, therefore, derived in a natural way. An implicit assumption in, for example, the sections on time series and filtering is that the equations, (that is, the coefficients in the equations), are known.
The problem of determining (linear) equations, which adequately describe the system under investigation, from the input/output data is known as ‘system identification’ and it is studied in Chapter Four. Topics from point estimation theory, such as the Fisher information matrix and the Cramer-Rao lower bound, are discussed and particular attention is given to least-square and maximum likelihood estimates. Interesting recent results on order determination of ARMA models due to E. J. Hannan and J. Rissanen [Biometrika 69, 81-94 (1982; Zbl 0494.62083)] are also described. A variation of the idea of ‘consistency’ is the basis of Chapter Five. The question addressed is not whether the parameter (or coefficient) estimates will converge to their ‘true’ values as the amount of data increases, but whether the identification procedure will succeed in giving the optimum model, (within the class of models considered). The asymptotic analysis of identification methods given in Chapter Five largely is based on work of L. Ljung [see, e.g., IEEE Trans. Autom. Control AC-23, 770-783 (1978; Zbl 0392.93038)].
The optimal control of both deterministic and stochastic linear discrete- time systems with quadratic cost is discussed in Chapter Six. The role of the Riccati equation and the duality of the linear quadratic Gaussian problem with the Kalman filter are described. Finally, in Chapter Seven, as a synthesis of earlier ideas, the problem of concurrently identifying and controlling a system is discussed. This field is known as ‘adaptive control’ and fundamental results are due to K. Åström and his collaborators [see, e.g., K. Åström, U. Borisson, L. Ljung and B. W. Wittenmark, Automatica 13, 457-476 (1977; Zbl 0374.93024)]. Most significant are the results on self-tuning regulators.
In conclusion, the book provides an excellent introduction to topics from discrete-time, linear, stochastic system theory. Few misprints were noted, (for example, the statement of Proposition 5.1.2 requires some punctuation), and anxiety to avoid measure theory has led to the statement on page 81 that ‘almost every’ means ‘for all except a finite number of values of the variable in question’. The book is certainly suitable for graduate courses in stochastic system theory and is a valuable addition to the literature.
Reviewer: R.Elliott

MSC:

93-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory
93C55 Discrete-time control/observation systems
93E20 Optimal stochastic control
60G35 Signal detection and filtering (aspects of stochastic processes)
62F10 Point estimation
62G30 Order statistics; empirical distribution functions
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M20 Inference from stochastic processes and prediction
93C05 Linear systems in control theory
93C40 Adaptive control/observation systems
93C57 Sampled-data control/observation systems
93E03 Stochastic systems in control theory (general)
93E10 Estimation and detection in stochastic control theory
93E11 Filtering in stochastic control theory
93E12 Identification in stochastic control theory