- Email: [email protected]
Algebraic theory for multivariable linear systems
Automatica,Vol. 21, No.
1, pp. 109-112, 1985 Pergamon Press Ltd. Printed in Great Britain. International Federation of Automatic Control.
Book Reviews
Algebraic Theory for Muitivariable Linear Systems* H. Blomberg and R. Ylinen
The fourth and last part of the book does everything--once over lightly--for discrete time systems. The books ends with four quite nice appendices: fundamentals of abstract algebra; polynomials and polynomial matrices; polynomials and rational forms in an endomorphism; and finally an appendix on generalized functions. As a research monograph, this book is a success: it is highly original and deep and, as such, a very good source for researchers interested in the framework, the techniques, and the results available in this area. Surely, the book indulges a bit: some of the analysis seems overdone with detail and many topics are touched upon on many different occasions. At the end, the reader is left with a feeling of having been overwhelmed, in part because of the lack of examples, motivational discussion, and applications. However it seems to me that these shortcomings are the prerogative of a research monograph as distinguished from a textbook. The subject of the book is both timely and important. It is, of course, a matter above discussion that in many situations one starts with a model of the type Z and that for many applications it presents a very suitable starting point indeed. However, it is worthwhile to contrast this approach with other methods such as state space techniques and frequency domain methods. In their introduction, the authors argue that their approach falls within the frequency domain thinking which they view as making a come-back after the success of the time-domain, state space, approach, I personally have some difficulty with all this. It seems to me that the difference between starting with Z: A ( p ) y = B ( p ) u (assuming A - I B proper) or starting with = Ax + Bu; y = Cx + Du is relatively minor and mainly a matter of parametrization. The first description is more compact, more parsimoniously parametrized, while the second one has the advantage that it displays the important memory function x, the state, explicitly. However they are both descriptions of finitedimensional, linear, time-invariant systems and any method, algorithm, or synthesis technique available for the class of systems Z can in principle be translated into the state space framework, and vice versa. On the other hand, frequency domain techniques as one finds them in Nyquist stability theory, input/output (stability) methods, robustness criteria involving frequency bands, etc., transcend in a very significant way the finite-dimensional framework, do not require parametrization, and consequently allow a much more vague and imprecise model than the class of systems Z. This, it seems to me, is the main advantage of frequency domain techniques, but this advantage is unfortunately not shared by the system Z. The difference between A(p)y = B(p)u and ~ = Ax + Bu; y = Cx + Du is a question of input/output vs input/state/output, and not a matter of frequency vs time domain. Of course, this does not take away anything of the basic merits of the book: the development of polynomial methods is a very important aspect of finite-dimensional linear system theory and the book under review is definitely the most advanced treatment of it. The recent text by Barnett (1983) contains similar material, is less advanced, but in many senses more accessible, however. I have two points of criticism about the book which I would like to discuss briefly. The first one is the type of definitions which the authors choose to make. Undoubtedly motivated by the main theme of book they propose definitions which are very polynomial matrix oriented. For instance Z is defined to be stable if the roots of detA(p) lie in the left half plane, controllability is
Reviewer: J. C. WILLEMS
Department of Mathematics, Groningen, The Netherlands.
University of Groningen,
THiS BOOK is a research monograph on some foundational aspects of system theory, more specifically on the polynomial matrix approach to the description of linear time-invariant systems. This approach has proved to be a quite successful research area in the last decade. It is particularly the book by Rosenbrock (1970) which popularized this line of research. However, early work by the authors of the monograph under review and their co-workers (Blomberg et al., 1969) actually preceded Rosenbrock's seminal book. The basic description of the systems treated in this book is: Z: A(p)y = B(p)u with A and B polynomial matrices in the differential operator p (with coefficients in R or C) and with u the input and y the output. Most of the time it is assumed that A is square with detA(p) unequal to the zero polynomial. This is called the regular case. In part one of the book, a number of basic aspects of class of systems Z are discussed. A framework is developed in which to answer the following questions: In what sense does Z define a dynamical system? On what input space is it defined? What is the role of the initial conditions? What do we mean by an interconnection? What are the properties of an interconnection of a family of systems Z? The second part of the book is an in-depth study of the system Z through the theory of polynomial matrices. First a framework is set up in which polynomial matrices in a differential operator can be considered as bona fide operators on function spaces, in particular (for example) on the space of infinitely differentiable functions or on the space of distributions. The next topic discussed starts from the observation that two systems El: At(p)y = Bl(p)u and ~ 2 : A 2 ( p ) y = B2(p)u could define the same input/output pairs while [A1 ~ - B 1 ] # [A2 ! - B 2 ]. This leads to a natural notion of equivalence, of invariants, and of canonical forms. Also inclusion of behaviors is discussed and, in this context, controllability. Chapter 7, the last chapter of the second part, is the core of the book. It is entitled: "Analysis and Synthesis Problems" The analysis questions discussed are mainly decomposition and composition problems: decomposition of a system into its observable and non-observable parts, decomposition into its controllable and non-controllable parts, controllability of composition of controllable systems, series and parallel connection, decomposition into a state space system or into a Rosenbrock system. This leads--unavoidably--to a discussion of strict system equivalence. All this is followed up by the discussion of two synthesis problems: the synthesis of observers and of feedback controllers which achieve pole placement by (dynamic) output feedback. The third part of the book returns to more foundational questions associated with the class of systems Z and the interconnection of such systems. * Algebraic Theory for Multivariate Linear Systems, by H. Blomberg and R. Ylinen. Mathematics in Science and Engineering Series, Vol. 166. Published by Academic Press, London (1983). U.S. $48.00.
109
110
Book Reviews
defined in terms of coprimeness or, even more technical, in terms of a certain minimality of the family of input/output pairs, etc. For someone who believes, as I do, that definitions should be made starting from an as high as possible intuitive basis, this is a bit disappointing. A second point is the fact that the authors choose to study models of systems in which the input and output basically play symmetric roles. True, when one adds regularity these roles are not completely symmetric anymore but even then one does not achieve non-anticipativeness of the resulting input/output behavior. It seems to me that certainly in the discrete time case it is quite objectionable to call a predictive map an input/output system. There is an appealing way to overcome these difficulties. Indeed, one can start, as we have done in Willems (1979, 1983), from a vantage point in which all the external variables play completely symmetric roles. This leads to the model:
matrix descriptions in an in-depth and authoritative way. As a research monograph it is simply a very good contribution to thc field.
Z': R(p)w = 0
References Barnett, S. (1983). Polynomials and Linear Control Systems. Dekker. Blomberg, H., J. Sinervo, A. Halme and R. Ylinen (1969). On algebraic methods in systems theory. Acta Polytechnica Scandinavica, 19. Rosenbrock, H. H. (1970). State Space and Muhivariable Theory. Nelson. Willems, J. C. (1979). System theoretic models for the analysis of physical systems. Ric. Automatica, 10, 71. Willems, J. C. (1983). Input-output and state space representations of finite-dimensional linear time-invariant systems. Linear Algebra and its Applications, Vol. 50, pp. 581-608.
It can now be shown that there exists a componentwise partition of w into w = (u, y) such that E' is equivalent (in the sense that it defines the same input/output pairs) to a system Z with A - IB proper. Actually it is possible to conceptualize E' even further. In the discrete time case one can in fact show that any linear, shift invariant, and closed (in the product topology) family of signals w: Z --, ~9 admits a representation as E'. In conclusion, this book presents the theory of linear, timeinvariant, finite-dimensional systems based on polynomial
About the reviewer Professor Jan C. Willems obtained his Ph.D. degree in electrical engineering from MIT in 1968. He is presently Professor of systems and control at the Mathematics Institute of the University of Groningen. He is on the editorial board of several journals in the field and a frequent contributor of research papers. His present interests lie principally in the geometric theory of linear systems and in problems of system representation.
Applied Control Theory* J. R. Leigh Reviewer: I. LEFKOWlTZ Department of Systems Engineering, Case Western Reserve University, Cleveland, Ohio, U.S.A. ONE OFTENhears concern expressed over the gap between theory and practice, particularly with respect to applications of control theory in industry. Indeed, the control literature abounds with references to state space, optimal control, stochastic control, adaptive control, etc., very little of which has found its way into the 'state-of-the-art' of industrial practice. This is not to say that industry has been lagging--in point of fact, industry has moved very aggressively in its application of modern computer and microprocessor-based control systems leading, as a result, to increasingly complex and sophisticated real-time systems for control and information processing. In this development, the focus has been more on factors related to economics, reliability, hardware and software implementations, than on the realizations of new theoretical concepts. One feels, nevertheless, that the enhanced capabilities of the new hardware and software being installed in plants will lead, in time, to increased consideration of modern and advanced control techniques for improving system performance. In this context, the book Applied Control Theory would appear to be particularly apropos. As implied by its title, the book is concerned with applications of control theory with emphasis, specifically, on the control of industrial sys~.ems. The author describes a variety of techniques which he considers useful or potentially useful in application to real problems in their industrial context. As the author states, "The book should be accessible to a wide variety of engineers. Preferably they should have an elementary knowledge of automatic control theory." The book is terse and compact (only 163 pp.), yet presents a broad mixture of theory and practice, hardware and applications, * Applied Control Engineering, by J. R. Leigh. IEE Control Engineering Series, Vol. 18. Published by Peter Peregrinus, London (1982). 163 pp., U.S. $56.00.
analysis and heuristics. The material is organized into 11 chapters. The first three chapters treat some introductory and general topics: definitions, interactions of control system design with process design and operation, considerations of human and economic factors, discussion of measurement problems and sensors. Another three chapters are concerned with single-input, single output control techniques and practices, including a description of simple control algorithms, tuning methods, techniques for handling dead-time and nonlinearities, and a potpourri of advanced process control approaches, e.g. cascade, feed forward, predictive and self-tuning control. A follow-up chapter extends the discussion to controller design for multiloop processes, including brief expositions of Bristol's relative gain approach, Rosenbrock's inverse Nyquist array technique, poleplacement algorithms, Richalet's method (based on a set of experimentally obtained impulse responses for the plant). The author stresses the growing importance of control implementation via digital computer and devotes two chapters to the theme. One chapter deals with direct digital control algorithms for single-input, single-output processes, including such topics as dead-beat control, design in the z-domain, comparison of DDC algorithms, time and memory requirements, quantisation and sampling interval problems, etc. The second chapter is entitled "Computer Control Methods," but is devoted almost entirely to descriptions of various manufacturers' microprocessor-based distributed control hardware. An extended chapter (some 2 0 ~ of the text) presents a selection of control applications. These are mostly drawn from the steel industry with a few miscellaneous examples drawn from other sources. A final chapter (which seems somewhat out of place) discusses D.C. motor drives and servomechanisms. I looked forward to the book making a valuable and much needed contribution to the control literature, particularly in view of the author's extensive experiences in industry and in academia. Indeed, there are many useful insights scattered throughout the book, based on his many years of practice. Nonetheless, I found myself very disappointed in the overall product--for many reasons.
1, pp. 109-112, 1985 Pergamon Press Ltd. Printed in Great Britain. International Federation of Automatic Control.
Book Reviews
Algebraic Theory for Muitivariable Linear Systems* H. Blomberg and R. Ylinen
The fourth and last part of the book does everything--once over lightly--for discrete time systems. The books ends with four quite nice appendices: fundamentals of abstract algebra; polynomials and polynomial matrices; polynomials and rational forms in an endomorphism; and finally an appendix on generalized functions. As a research monograph, this book is a success: it is highly original and deep and, as such, a very good source for researchers interested in the framework, the techniques, and the results available in this area. Surely, the book indulges a bit: some of the analysis seems overdone with detail and many topics are touched upon on many different occasions. At the end, the reader is left with a feeling of having been overwhelmed, in part because of the lack of examples, motivational discussion, and applications. However it seems to me that these shortcomings are the prerogative of a research monograph as distinguished from a textbook. The subject of the book is both timely and important. It is, of course, a matter above discussion that in many situations one starts with a model of the type Z and that for many applications it presents a very suitable starting point indeed. However, it is worthwhile to contrast this approach with other methods such as state space techniques and frequency domain methods. In their introduction, the authors argue that their approach falls within the frequency domain thinking which they view as making a come-back after the success of the time-domain, state space, approach, I personally have some difficulty with all this. It seems to me that the difference between starting with Z: A ( p ) y = B ( p ) u (assuming A - I B proper) or starting with = Ax + Bu; y = Cx + Du is relatively minor and mainly a matter of parametrization. The first description is more compact, more parsimoniously parametrized, while the second one has the advantage that it displays the important memory function x, the state, explicitly. However they are both descriptions of finitedimensional, linear, time-invariant systems and any method, algorithm, or synthesis technique available for the class of systems Z can in principle be translated into the state space framework, and vice versa. On the other hand, frequency domain techniques as one finds them in Nyquist stability theory, input/output (stability) methods, robustness criteria involving frequency bands, etc., transcend in a very significant way the finite-dimensional framework, do not require parametrization, and consequently allow a much more vague and imprecise model than the class of systems Z. This, it seems to me, is the main advantage of frequency domain techniques, but this advantage is unfortunately not shared by the system Z. The difference between A(p)y = B(p)u and ~ = Ax + Bu; y = Cx + Du is a question of input/output vs input/state/output, and not a matter of frequency vs time domain. Of course, this does not take away anything of the basic merits of the book: the development of polynomial methods is a very important aspect of finite-dimensional linear system theory and the book under review is definitely the most advanced treatment of it. The recent text by Barnett (1983) contains similar material, is less advanced, but in many senses more accessible, however. I have two points of criticism about the book which I would like to discuss briefly. The first one is the type of definitions which the authors choose to make. Undoubtedly motivated by the main theme of book they propose definitions which are very polynomial matrix oriented. For instance Z is defined to be stable if the roots of detA(p) lie in the left half plane, controllability is
Reviewer: J. C. WILLEMS
Department of Mathematics, Groningen, The Netherlands.
University of Groningen,
THiS BOOK is a research monograph on some foundational aspects of system theory, more specifically on the polynomial matrix approach to the description of linear time-invariant systems. This approach has proved to be a quite successful research area in the last decade. It is particularly the book by Rosenbrock (1970) which popularized this line of research. However, early work by the authors of the monograph under review and their co-workers (Blomberg et al., 1969) actually preceded Rosenbrock's seminal book. The basic description of the systems treated in this book is: Z: A(p)y = B(p)u with A and B polynomial matrices in the differential operator p (with coefficients in R or C) and with u the input and y the output. Most of the time it is assumed that A is square with detA(p) unequal to the zero polynomial. This is called the regular case. In part one of the book, a number of basic aspects of class of systems Z are discussed. A framework is developed in which to answer the following questions: In what sense does Z define a dynamical system? On what input space is it defined? What is the role of the initial conditions? What do we mean by an interconnection? What are the properties of an interconnection of a family of systems Z? The second part of the book is an in-depth study of the system Z through the theory of polynomial matrices. First a framework is set up in which polynomial matrices in a differential operator can be considered as bona fide operators on function spaces, in particular (for example) on the space of infinitely differentiable functions or on the space of distributions. The next topic discussed starts from the observation that two systems El: At(p)y = Bl(p)u and ~ 2 : A 2 ( p ) y = B2(p)u could define the same input/output pairs while [A1 ~ - B 1 ] # [A2 ! - B 2 ]. This leads to a natural notion of equivalence, of invariants, and of canonical forms. Also inclusion of behaviors is discussed and, in this context, controllability. Chapter 7, the last chapter of the second part, is the core of the book. It is entitled: "Analysis and Synthesis Problems" The analysis questions discussed are mainly decomposition and composition problems: decomposition of a system into its observable and non-observable parts, decomposition into its controllable and non-controllable parts, controllability of composition of controllable systems, series and parallel connection, decomposition into a state space system or into a Rosenbrock system. This leads--unavoidably--to a discussion of strict system equivalence. All this is followed up by the discussion of two synthesis problems: the synthesis of observers and of feedback controllers which achieve pole placement by (dynamic) output feedback. The third part of the book returns to more foundational questions associated with the class of systems Z and the interconnection of such systems. * Algebraic Theory for Multivariate Linear Systems, by H. Blomberg and R. Ylinen. Mathematics in Science and Engineering Series, Vol. 166. Published by Academic Press, London (1983). U.S. $48.00.
109
110
Book Reviews
defined in terms of coprimeness or, even more technical, in terms of a certain minimality of the family of input/output pairs, etc. For someone who believes, as I do, that definitions should be made starting from an as high as possible intuitive basis, this is a bit disappointing. A second point is the fact that the authors choose to study models of systems in which the input and output basically play symmetric roles. True, when one adds regularity these roles are not completely symmetric anymore but even then one does not achieve non-anticipativeness of the resulting input/output behavior. It seems to me that certainly in the discrete time case it is quite objectionable to call a predictive map an input/output system. There is an appealing way to overcome these difficulties. Indeed, one can start, as we have done in Willems (1979, 1983), from a vantage point in which all the external variables play completely symmetric roles. This leads to the model:
matrix descriptions in an in-depth and authoritative way. As a research monograph it is simply a very good contribution to thc field.
Z': R(p)w = 0
References Barnett, S. (1983). Polynomials and Linear Control Systems. Dekker. Blomberg, H., J. Sinervo, A. Halme and R. Ylinen (1969). On algebraic methods in systems theory. Acta Polytechnica Scandinavica, 19. Rosenbrock, H. H. (1970). State Space and Muhivariable Theory. Nelson. Willems, J. C. (1979). System theoretic models for the analysis of physical systems. Ric. Automatica, 10, 71. Willems, J. C. (1983). Input-output and state space representations of finite-dimensional linear time-invariant systems. Linear Algebra and its Applications, Vol. 50, pp. 581-608.
It can now be shown that there exists a componentwise partition of w into w = (u, y) such that E' is equivalent (in the sense that it defines the same input/output pairs) to a system Z with A - IB proper. Actually it is possible to conceptualize E' even further. In the discrete time case one can in fact show that any linear, shift invariant, and closed (in the product topology) family of signals w: Z --, ~9 admits a representation as E'. In conclusion, this book presents the theory of linear, timeinvariant, finite-dimensional systems based on polynomial
About the reviewer Professor Jan C. Willems obtained his Ph.D. degree in electrical engineering from MIT in 1968. He is presently Professor of systems and control at the Mathematics Institute of the University of Groningen. He is on the editorial board of several journals in the field and a frequent contributor of research papers. His present interests lie principally in the geometric theory of linear systems and in problems of system representation.
Applied Control Theory* J. R. Leigh Reviewer: I. LEFKOWlTZ Department of Systems Engineering, Case Western Reserve University, Cleveland, Ohio, U.S.A. ONE OFTENhears concern expressed over the gap between theory and practice, particularly with respect to applications of control theory in industry. Indeed, the control literature abounds with references to state space, optimal control, stochastic control, adaptive control, etc., very little of which has found its way into the 'state-of-the-art' of industrial practice. This is not to say that industry has been lagging--in point of fact, industry has moved very aggressively in its application of modern computer and microprocessor-based control systems leading, as a result, to increasingly complex and sophisticated real-time systems for control and information processing. In this development, the focus has been more on factors related to economics, reliability, hardware and software implementations, than on the realizations of new theoretical concepts. One feels, nevertheless, that the enhanced capabilities of the new hardware and software being installed in plants will lead, in time, to increased consideration of modern and advanced control techniques for improving system performance. In this context, the book Applied Control Theory would appear to be particularly apropos. As implied by its title, the book is concerned with applications of control theory with emphasis, specifically, on the control of industrial sys~.ems. The author describes a variety of techniques which he considers useful or potentially useful in application to real problems in their industrial context. As the author states, "The book should be accessible to a wide variety of engineers. Preferably they should have an elementary knowledge of automatic control theory." The book is terse and compact (only 163 pp.), yet presents a broad mixture of theory and practice, hardware and applications, * Applied Control Engineering, by J. R. Leigh. IEE Control Engineering Series, Vol. 18. Published by Peter Peregrinus, London (1982). 163 pp., U.S. $56.00.
analysis and heuristics. The material is organized into 11 chapters. The first three chapters treat some introductory and general topics: definitions, interactions of control system design with process design and operation, considerations of human and economic factors, discussion of measurement problems and sensors. Another three chapters are concerned with single-input, single output control techniques and practices, including a description of simple control algorithms, tuning methods, techniques for handling dead-time and nonlinearities, and a potpourri of advanced process control approaches, e.g. cascade, feed forward, predictive and self-tuning control. A follow-up chapter extends the discussion to controller design for multiloop processes, including brief expositions of Bristol's relative gain approach, Rosenbrock's inverse Nyquist array technique, poleplacement algorithms, Richalet's method (based on a set of experimentally obtained impulse responses for the plant). The author stresses the growing importance of control implementation via digital computer and devotes two chapters to the theme. One chapter deals with direct digital control algorithms for single-input, single-output processes, including such topics as dead-beat control, design in the z-domain, comparison of DDC algorithms, time and memory requirements, quantisation and sampling interval problems, etc. The second chapter is entitled "Computer Control Methods," but is devoted almost entirely to descriptions of various manufacturers' microprocessor-based distributed control hardware. An extended chapter (some 2 0 ~ of the text) presents a selection of control applications. These are mostly drawn from the steel industry with a few miscellaneous examples drawn from other sources. A final chapter (which seems somewhat out of place) discusses D.C. motor drives and servomechanisms. I looked forward to the book making a valuable and much needed contribution to the control literature, particularly in view of the author's extensive experiences in industry and in academia. Indeed, there are many useful insights scattered throughout the book, based on his many years of practice. Nonetheless, I found myself very disappointed in the overall product--for many reasons.