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Discrete time systems. An introduction to the theory
Book reviews
L>. Filtration
377
problem
The last section contains one paper by R.M. nudaka and F. P. Vasil’ eva: Convergence and error estimation of the method of straignt lines for the solution of certain filtration problems. The paper discusses the application of the method of straight lines to the solution of non-stationary filtration problems in the linear approximation, when the required solution depends on two spatial variables and time. Similar problems are also discussed in the stationary case, for which an algorithm for the method of straight lines was earlier proposed by A.Ya. Alikhashkin.
FUEQiAN, H. Discrete-time John Wiley and Sons.,
systems.
Ltd.,
XIII
An introduction
+ 239 pp.,
75/,
to the
theory.
London,
196.5.
THE role of computers in investigating control systems increases in importance every year, and so, consequently, does the interest in the theory of discrete systems. The method of investigation of discrete systems differ in certain respects from the corresponding methods for continuous systems. There difference equations, are many books on quantized systems, finite numerical analysis and other topics, which rather form special chapters in the theory of discrete systems than present the theorv as an independent discipline with its own field of topics and methods. The author of the present book attempts a treatment of the origins of the theory of discrete systems. The book arose out of a course of lectures delivered by the author at New York university, and is aimed at engineers and post-graduate students. The book contains eight chapters and two appendices, a list of problems for each chapter.
and at the end,
Chapter 1 deals with basic concepts, the definition of a discrete of a system (input, output, state of the system), the phase description system, the equations of state connecting the input and the state, and the state and the output at adjacent instants, stationary systems, two forms of equivalent systems, and the concepts of controllability and observability. Chapter 2 is concerned with the analysis of numerical sequences. The topics considered are: the weight sequence of a system (the analogue of the impulsive weight function of continuous systems), transition matrices,
278
Book reviews
connecting the sequence of output signals with the sequence of input signals, the analogue of Cauchy’s formula for the solutions of the nonhomogeneous equations of state, the transition from the difference equation to the equations of state (two methods are described); general formulae for coordinate transformations, the canonical Jordan form of a matrix (briefly), and criteria for controllability and observability (incompletely). Chapter 3 on “Theory of transformations” describes the two-sided and one-sided Laplace transformations. direct and inverse theorems are given.
the analogues of The fundamental
The quantization of continuous functions is discussed in Chapter 4. The quantization theorem on the possibility of restoring a continuous function from its values at discrete instants is given. Interpolation formulae are obtained in the case of uniform, non-uniform and multidimensional quantization of functions having finite Fourier trrtnsforms. The finiteness of the Fourier transforms represents a restrictive condition, difficult to prove. It is removed in the next chapter, entitled “Polynomial interpolations and extrapolation”. Legendre polynomials, Newton - Gregory extrapolation, and the frequency responses of an extrapolation filter, are discussed here. Problems on continuous systems with time-quantized input are discussed in Chapter 6; the transition to the equations of state in the case of an impulsive and step-wise input is described, together with the problem of the approximate and exact inversion of the output transforms ; it is shown how to transform from a continuous system with quantized output to the equivalent discrete system; and finally, the method of modified z- transformation and the problem of determining the output at high rates of quantization are discussed. Chapter 7 deals with l’Discontinuously-acting control systems”, and covers the following: block diagrams, graphical analysis of the passage of signals, transport equations for discontinuously acting systems, Lyapunov stability, theorems of the second method for discrete systems, stability of stationary linear systems, an existence theorem for the Iivapunov function, stability in the case of external perturbations, the concept of reducible systems and stability with respect to the first approximation, Tsypkin Jury stability criteria, and certain other criterial. Chapter 8 covers certain aspects of discrete stochastic processes. The spectral theory of these systems is given, and the problem of the synthesis of an optimal filter is discussed. The concept of a Markov chain is introduced and a case of control by such a process described.
Book
reviews
279
The book ends with two appendices (numerical methods for inversion of and a table of t transforms), and z transforms, a table of z transforms, a list of exercises. It will be clear from what has been said that the book deals with a included in wide range of problems, some of which are traditionally books on the subject while others are familiar in the USSR only through journal articles. The writing is clear and easy to read. It is primarily of interest to theoretical workers; it contains no discussion of actual control systems, no method is described for obtaining the equations giving the behaviour of actual entities, and the examples are purely illustrative. In spite of some undoubted merits, translation of the book into Russian can hardly be recommended, since a better treatment of the traditional topics may be found in existing Russian monographs, while the treatment of the new topics is too cursory and incomplete.
R. Gabasou
L>. Filtration
377
problem
The last section contains one paper by R.M. nudaka and F. P. Vasil’ eva: Convergence and error estimation of the method of straignt lines for the solution of certain filtration problems. The paper discusses the application of the method of straight lines to the solution of non-stationary filtration problems in the linear approximation, when the required solution depends on two spatial variables and time. Similar problems are also discussed in the stationary case, for which an algorithm for the method of straight lines was earlier proposed by A.Ya. Alikhashkin.
FUEQiAN, H. Discrete-time John Wiley and Sons.,
systems.
Ltd.,
XIII
An introduction
+ 239 pp.,
75/,
to the
theory.
London,
196.5.
THE role of computers in investigating control systems increases in importance every year, and so, consequently, does the interest in the theory of discrete systems. The method of investigation of discrete systems differ in certain respects from the corresponding methods for continuous systems. There difference equations, are many books on quantized systems, finite numerical analysis and other topics, which rather form special chapters in the theory of discrete systems than present the theorv as an independent discipline with its own field of topics and methods. The author of the present book attempts a treatment of the origins of the theory of discrete systems. The book arose out of a course of lectures delivered by the author at New York university, and is aimed at engineers and post-graduate students. The book contains eight chapters and two appendices, a list of problems for each chapter.
and at the end,
Chapter 1 deals with basic concepts, the definition of a discrete of a system (input, output, state of the system), the phase description system, the equations of state connecting the input and the state, and the state and the output at adjacent instants, stationary systems, two forms of equivalent systems, and the concepts of controllability and observability. Chapter 2 is concerned with the analysis of numerical sequences. The topics considered are: the weight sequence of a system (the analogue of the impulsive weight function of continuous systems), transition matrices,
278
Book reviews
connecting the sequence of output signals with the sequence of input signals, the analogue of Cauchy’s formula for the solutions of the nonhomogeneous equations of state, the transition from the difference equation to the equations of state (two methods are described); general formulae for coordinate transformations, the canonical Jordan form of a matrix (briefly), and criteria for controllability and observability (incompletely). Chapter 3 on “Theory of transformations” describes the two-sided and one-sided Laplace transformations. direct and inverse theorems are given.
the analogues of The fundamental
The quantization of continuous functions is discussed in Chapter 4. The quantization theorem on the possibility of restoring a continuous function from its values at discrete instants is given. Interpolation formulae are obtained in the case of uniform, non-uniform and multidimensional quantization of functions having finite Fourier trrtnsforms. The finiteness of the Fourier transforms represents a restrictive condition, difficult to prove. It is removed in the next chapter, entitled “Polynomial interpolations and extrapolation”. Legendre polynomials, Newton - Gregory extrapolation, and the frequency responses of an extrapolation filter, are discussed here. Problems on continuous systems with time-quantized input are discussed in Chapter 6; the transition to the equations of state in the case of an impulsive and step-wise input is described, together with the problem of the approximate and exact inversion of the output transforms ; it is shown how to transform from a continuous system with quantized output to the equivalent discrete system; and finally, the method of modified z- transformation and the problem of determining the output at high rates of quantization are discussed. Chapter 7 deals with l’Discontinuously-acting control systems”, and covers the following: block diagrams, graphical analysis of the passage of signals, transport equations for discontinuously acting systems, Lyapunov stability, theorems of the second method for discrete systems, stability of stationary linear systems, an existence theorem for the Iivapunov function, stability in the case of external perturbations, the concept of reducible systems and stability with respect to the first approximation, Tsypkin Jury stability criteria, and certain other criterial. Chapter 8 covers certain aspects of discrete stochastic processes. The spectral theory of these systems is given, and the problem of the synthesis of an optimal filter is discussed. The concept of a Markov chain is introduced and a case of control by such a process described.
Book
reviews
279
The book ends with two appendices (numerical methods for inversion of and a table of t transforms), and z transforms, a table of z transforms, a list of exercises. It will be clear from what has been said that the book deals with a included in wide range of problems, some of which are traditionally books on the subject while others are familiar in the USSR only through journal articles. The writing is clear and easy to read. It is primarily of interest to theoretical workers; it contains no discussion of actual control systems, no method is described for obtaining the equations giving the behaviour of actual entities, and the examples are purely illustrative. In spite of some undoubted merits, translation of the book into Russian can hardly be recommended, since a better treatment of the traditional topics may be found in existing Russian monographs, while the treatment of the new topics is too cursory and incomplete.
R. Gabasou