The book deals with mathematical theory of automatic regulation; it consists of 8 chapters, 5 appendices and bibliography.
Chapter 1 (nearly 25% of the text) outlines the theory of stability; the author mentions that in most of his theorems both the necessary and sufficient conditions are given (instead of sufficient conditions only); moreover it is shown that restrictions to be imposed on Lyapunov’s function can be weakened and that it is possible to calculate limits of the asymptotic stability region (the “region of attraction”) in an appropriate parameter space. In chapter 2 linear differential systems with variable coefficients are investigated; it is shown that in some special cases the problem is considerably simplified; the question of stability plays again an important role. The contents of this chapter bear mostly on the transient behavior of control systems. It is shown how one can determine the region of stability in the parameter space and also the influence of the parameter variation on the corresponding variation of this region. More applied questions are investigated in chapter 3; among these questions are: the estimate of a transient deviation from the stationary state; calculation of damping minimum time; applicability of equations of the first approximation; manifestations of stability during finite time intervals; behavior of linear systems with retarded actions.
Chapter 4 is devoted to the classical problem of the determination of the form of a solution in neighbourhood of a singular point in terms of infinite series. The author generalizes the earlier work (of Briot-Bouquet, Poincaré, Picard and others); the results are substantially the same as those used in the control theory, namely: the analysis of real parts of the roots of the characteristic equation with a corresponding construction of a region of stability. Effect of perturbations is investigated in chapter 5; in addition to the stability calculations, a procedure for calculating deviations caused by perturbations is given. When these perturbations are of a random character, the classical analysis in terms of the infinite series still holds but in such a case it is necessary to introduce appropriate correlation functions in the infinite series representing the solution; if the series is broken off at a certain number of terms an approximation of a stochastic process is thus obtained. Difficult problems of stability in “critical” cases are analysed in chapter 6. In addition to the classical results (of Lyapunov, Malkin and others), the author treats some special cases, for instance when the series solution begins with higher order terms, and shows how the construction of an appropriate function of Lyapunov can be obtained.
Self-excited oscillations are treated in chapter 7 by classical analytical methods (of Poincaré and Krylov-Bogolyubov). Here the possibilities are more restricted than in problems of stability to which the preceding 6 chapters are mostly devoted. In fact, a purely analytic approach to the problem of oscillations in the theory of automatic regulation seems to be less promising than that based on an essentially “piecewise linear” idealization which constitutes the existing (non-analytic) approach to these problems. In view of this, connections between the theory and experimental facts here are less definite than in other chapters; this in no respect reacts on the intrinsic mathematical value of conclusions reached in this chapter.
The last chapter 8 (written by Chernetskiĭ) does not belong to the text properly speaking and concerns applications of electronic computing machines to analysis and synthesis of systems of automatic regulation.
There are several appendices on methods of approximations, on linear difference-differential equations, on correlation functions of stochastic solutions, on programming of computing machines and on auto-oscillations.