## Linear stochastic systems.(English)Zbl 0658.93003

Wiley Series in Probability and Mathematical Statistics. New York etc.: Wiley. xiv, 874 p. £52.50 (1988).
A system is an aggregation of entities combined in a way as to form a whole and as such acting together and performing a certain objective. A system whose components have been assembled by nature is called natural while one whose components have been combined by man is called artificial. The word “system” should, therefore, be interpreted to imply physical, biological, economic, etc., systems. Roughly speaking, a system is a structural assemblage into which one puts at certain times something like energy, information or money, and from which one gets at certain times something of the same or of a different nature. Mathematical system theory is concerned with modelling, i.e. the mathematical description of systems, gathering of information of the system behaviour for the purpose of regulating or controlling its performance. Associated with these tasks there are others like estimation, prediction and realization. Systems are in general classified according to the type of equation used to describe them, Systems, for instance, for which the input, state and output processes are random are called stochastic systems. Linear systems are those for which the equation representing its dynamic is linear.
The book under review is concerned with stochastic systems that evolve in discrete time which are random and linear in their inputs-state-outputs behaviour. This extensive book covers in 12 chapters and three appendices the most standard topics on linear stochastic systems in discrete time as well as some new developments due mainly to the author and associates. The intention of the author has been to write a book for graduate students, teachers and research workers in the areas of systems and control theory and its applications, probability, statistics, time-series analysis, econometrics and related areas. In the appendices at the end of the book the author reviews basic definitions and results from probability theory, systems theory and harmonic analysis. However, the reader is assumed to have a working knowledge of linear algebra, analysis and probability at the first-year graduate level.
Chapter 1 and 2 present those concepts on which the rest of the book relies. Chapter 3 and 4 deal with various aspects of recursive estimation like filtering, prediction and smoothing but also with the inverse problem of the synthesis of a stochastic process known as the stochastic realization theory. Chapter 5 through 10 comprise topics ranging from system modelling to parameter estimation running under the label system identification on the shaping of which several strongly practice oriented disciplines have greatly contributed, i.e. applied mathematics, econometrics, statistics and systems theory itself. The interdisciplinary character of the problems dealt with, the consequent diversity of the methods used, the authoritative influence exerted by the established paradigms and nonrarely ideological views have given rise to intensive and long lasting controversies. In section 3.1 the author articulates the position in this regard that he has adopted.
The rest of the book deals with linear-quadratic and adaptive control. The underlying idea in most control systems reflects the fact that during the operation of a system various kinds of disturbances in the form of signals may arise tending to adversely affect the value of the system output. This gives rise to certain differences between the output or a desired state and certain reference input for the regulation of which one sets a device or feedback control mechanism aimed at avoiding this difference or at maintaining it at a certain level. However, the dynamic characteristics of most control systems are not constant or vary in an unpredictable way because of several reasons. Therefore, a satisfactory control system is expected to adapt in accordance with unpredictable changes in its environment or structure. Chapter 11 is dedicated to deriving feedback controls for the minimum-variance and linear quadratic stochastic control problem for time invariant and steady-state systems with complete and partial information. However, the case with time varying parameter state is briefly discussed and references given. Chapter 12 studies parameter-adaptive control algorithms by means of which to estimate the optimal control law for a given system, i.e. minimum-variance control case in which self-tuning techniques lead to asymptotically optimal performance. The problems of simultaneous control and parameter estimation as well as adaptive control with randomly varying parameter are also considered.
The book is certainly a welcome addition to the literature on systems theory for which the author should be congratulated. Concerning the remarks on Methodology on Economics and Econometrics, and Scientific Knowledge I would like to mention at least three references the inclusion of which, I believe, would have contributed to the interesting divagation of the author on the philosophical foundations of knowledge:
[1] E. Farjourn and M. Machover. Laws of chaos: A probabilistic approach to political economy (London 1983).
[2] K. G. Jöreskog and H. Wold. Systems under indirect observation, part I and II (Amsterdam 1982).
[3] J. Steindl, Random processes and the growth of firms: A study of Pareto law (New York 1965).
Reviewer: M.Gomez

### MSC:

 93-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory 93C05 Linear systems in control theory 93E20 Optimal stochastic control 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62P20 Applications of statistics to economics 93B15 Realizations from input-output data 93C40 Adaptive control/observation systems 93C55 Discrete-time control/observation systems 93E10 Estimation and detection in stochastic control theory 93E11 Filtering in stochastic control theory 93E12 Identification in stochastic control theory 62M20 Inference from stochastic processes and prediction 91B84 Economic time series analysis