Introduction to mathematical systems theory. A behavioral approach.

*(English)*Zbl 0940.93002
Texts in Applied Mathematics. 26. New York, NY: Springer. xxix, 424 p. (1997).

This is an elegant textbook on Systems and Control and the authors have developed the subject in a nice historical setting with a novel approach.

Both the transfer function and the state space approach view a system as a signal processor that accepts inputs and transforms them into outputs. In the transfer function approach, this processor is described through the way in which exponential inputs are transformed into exponential outputs. In the state space approach, this processor involves the state as an intermediate variable, but the ultimate aim remains to describe how inputs lead to outputs. This input/output point of view plays an important role in this book at later stages. However, the starting point of this book is different, more general, and more adapted to modelling and more suitable for applications. The general model structure that has been developed in the first half of the book is referred to as the behavioural approach. A mathematical model is viewed as a subset of a universe of possibilities. It is presumed that all outcomes in the universe are in principle possible before a mathematical model as a description of reality is accepted.

It is claimed that only outcomes in a certain subset are possible. This subset is called the behaviour of the mathematical model. From this viewpoint, a dynamical system is viewed simply as a subset of time trajectories, as a family of time signals taking on values in a suitable signal space. The book is devoted to linear time-invariant differential systems. An accurate description of the behaviour of such a system means the description of the behaviour of the solution set of a system of differential equations.

In the first chapter, the mathematical definition of a dynamical system is presented. The basic ingredients of this definition are the behaviour of a dynamical system as the central object of study and the notions of manifest and latent variables. When one models an interconnected physical system from first principles, usually auxiliary variables will appear in the model apart from the variables modeled which are called manifest variables. The auxiliary variables appearing in a model are called latent variables. The interaction between manifest and latent variables is a recurring theme in this book.

The second chapter introduces linear time-invariant differential systems – the model class with which the book is concerned. A crucial concept discussed is the notion of a weak solution to give a mathematically sound basis to nondifferentiable solutions of a system of differential equations without the use of Schwartz’s Distribution Theory. It has been shown how systems of linear time-invariant differential equations are parameterized by polynomial matrices and also the authors have presented the interplay of polynomial matrices with differential equations.

The third chapter is devoted to a detailed discussion of the behaviour of linear differential systems. A dynamical system is determined by its behaviour. The behaviours studied are described by systems of behavioural differential equations. It is shown that systems of differential equations that can be transformed into each other by premultiplication by a unimodular matrix represent the same behaviour. Conversely, the relation between representations that define the same behaviour has been investigated and it is shown how the relation between inputs and outputs can be expressed as a convolution integral. A complete and explicit characterization of all weak solutions to \(R\left({d\over dt}\right)w= 0\) is given. This leads to the notion of input/output representation.

The fourth chapter deals with state models. The state of a dynamical system parameterizes its memory, the extent to which the past influences the future. That is, the state variables, an important class of latent variables, have the property that they parameterize the memory of the system. They split the past and future of the behaviour. It is shown that state equations which link the manifest variables to the state, turn out to be first-order differential equations and the output of a system is determined only after the input and the initial condition have been specified. The state space models for systems in input/output form have been discussed. Also linearization of nonlinear state space models has been dealt with.

The next chapter discusses controllability and observability which play a central role in control theory and which are two important features of dynamical systems from the behavioural point of view. Controllability refers to the question of whether or not one trajectory of a dynamical system can be steered towards another one and observability refers to the question of what one can deduce from the observation of one set of system variables about the behaviour of another set. By combining the concepts of controllability and observability applied to an input-output system, one obtains the Kalman decomposition.

The sixth chapter considers again the latent variable and state space systems. It is shown how a system of differential equations containing latent variables can be transformed to an equivalent system in which latent variables have been eliminated. The main result of this chapter shows that the elimination of latent variables in linear time-invariant differential systems is always possible by providing a systematic algorithm for doing this, which leads to a general theory of eliminating latent variables in linear systems.

Chapter 7 deals with the stability of a system. The classical stability conditions of systems of differential equations in terms of the roots of the associated polynomial or of the eigenvalue locations of the system matrix have been presented. The celebrated Routh-Hurwitz test, which specifies conditions for a polynomial to have only roots with negative real part, is also presented. Up to chapter 7, systems have been treated in their natural time-domain setting. There are several types of stability. In structural stability, one wants small parameter changes to have a similar small influence on the behaviour of a system. In dynamic stability, which is discussed in this chapter, it is the effect of disturbances in the form of an initial condition on the solution to the relevant dynamical equations. Intuitively, an equilibrium point is said to be stable if trajectories that start close to it remain close to it. Dynamic stability is thus not in the first instance a property of a system, but of an equilibrium point. However, in case of linear systems, stability is viewed as a property of the system. In input/output stability small input disturbances should produce small output disturbances.

Chapter 8 deals with the frequency domain description of systems and discusses some characteristic features and nomenclature for system responses related to the step response and the frequency domain properties. First, it is explained how a linear time-invariant system acts in the frequency domain. An important feature of such system is that it transforms sinusoidal or more generally exponential inputs into sinusoidal/exponential outputs. This leads to the transfer function and the frequency response as a convenient way of describing such systems. The transfer function specifies only the controllable part of a system, and uncontrollable modes are not represented by it. Also, properties of the time- and frequency-domain response are studied. Important characteristics of a system that can be deduced from its step-response or from its Bode and Nyquist plots have been described.

Chapters 9 and 10 deal with control theory. Control theory has two main roots: regulation and trajectory optimization. The regulation is the more important and engineering oriented. Whereas trajectory optimization is mathematics based. However, these roots have to a large extent merged in the second half of twentieth century. First, the authors discuss open-loop and feedback control and explain the difference and then they deal with an important result, the pole placement theorem – one of the central achievement of control theory. This theorem states that for a controllable system, there exists, for a suitable desired monic polynomial, a state feedback gain matrix such that the eigenvalues of the closed loop system are the roots of the desired polynomial. This chapter deals with an important control design question. More specifically, it deals with the choice of a control law such that the closed loop system is stable or, more generally, such that it has a certain degree of stability reflected – in a requirement on the location of the closed loop poles. The salient result obtained states that with this type of control, stabilization – in fact, pole placement – is always possible for controllable systems.

Then the tenth chapter deals with observers, that is, algorithms for deducing the system state from measured inputs and outputs. The design of an observer is very similar to the stabilization and pole placement procedures. However, it is not necessary to measure all the state variables for the design of a stabilizing feedback controller. By appropriate signal processing, it is often possible to obtain good estimates of all state variables from the measured outputs. The algorithm that performs this signal processing is called an observer. The observers discussed are actually nonoptimal, nonstochastic versions of the celebrated Kalman filter. The observers that have been obtained possess many appealing features, in particular, the recursivity of the resulting signal processing algorithm. By combining a state observer with a static control law, a feedback controller has been obtained, which is often called a compensator, that processes the measured outputs in order to compute the required control input.

This is an excellent textbook with many nice welcome features. First of all, the prface contains a historical introduction to systems and control and delineate thee philosophy of the approach to this subject, adopted by the authors, together with an outline of the topics covered. The behavioural approach originated in a series of six papers by J. C. Willems [the first one is “System theoretic models for the analysis of physical systems”, Ric. Autom. 10, 71-106 (1979)] over a period of ten years. This new approach to dynamical systems has many advantages, both from an engineering and from a mathematical perspective.

Each chapter contains sections called ‘Recapitulation’ and ‘Notes and References’ which gives the very useful summary of the discussion and sources used, respectively. Also the book contains two important appendices entitled ‘Simulation Exercises’ and ‘Background Material’, respectively. The exercises presented in the first appendix aim at giving the student insight into some simple modelling examples and control problems and require the use of a computer and numerical software packages. Also the text provides a good collection of problems as exercises after each chapter and some theory is also covered. The authors’ presentation is lucid and they have written the text with great care. The book is aimed at students in mathematics and control engineering and a certain background is needed to use it, since the material is presented in a mathematically rigorous way, while keeping physical modelling and applications aspects in mind. However, the text fully meets the objective of the TAM series. Finally, it is highly recommended for mathematics as well as mathematically oriented control engineering students.

Both the transfer function and the state space approach view a system as a signal processor that accepts inputs and transforms them into outputs. In the transfer function approach, this processor is described through the way in which exponential inputs are transformed into exponential outputs. In the state space approach, this processor involves the state as an intermediate variable, but the ultimate aim remains to describe how inputs lead to outputs. This input/output point of view plays an important role in this book at later stages. However, the starting point of this book is different, more general, and more adapted to modelling and more suitable for applications. The general model structure that has been developed in the first half of the book is referred to as the behavioural approach. A mathematical model is viewed as a subset of a universe of possibilities. It is presumed that all outcomes in the universe are in principle possible before a mathematical model as a description of reality is accepted.

It is claimed that only outcomes in a certain subset are possible. This subset is called the behaviour of the mathematical model. From this viewpoint, a dynamical system is viewed simply as a subset of time trajectories, as a family of time signals taking on values in a suitable signal space. The book is devoted to linear time-invariant differential systems. An accurate description of the behaviour of such a system means the description of the behaviour of the solution set of a system of differential equations.

In the first chapter, the mathematical definition of a dynamical system is presented. The basic ingredients of this definition are the behaviour of a dynamical system as the central object of study and the notions of manifest and latent variables. When one models an interconnected physical system from first principles, usually auxiliary variables will appear in the model apart from the variables modeled which are called manifest variables. The auxiliary variables appearing in a model are called latent variables. The interaction between manifest and latent variables is a recurring theme in this book.

The second chapter introduces linear time-invariant differential systems – the model class with which the book is concerned. A crucial concept discussed is the notion of a weak solution to give a mathematically sound basis to nondifferentiable solutions of a system of differential equations without the use of Schwartz’s Distribution Theory. It has been shown how systems of linear time-invariant differential equations are parameterized by polynomial matrices and also the authors have presented the interplay of polynomial matrices with differential equations.

The third chapter is devoted to a detailed discussion of the behaviour of linear differential systems. A dynamical system is determined by its behaviour. The behaviours studied are described by systems of behavioural differential equations. It is shown that systems of differential equations that can be transformed into each other by premultiplication by a unimodular matrix represent the same behaviour. Conversely, the relation between representations that define the same behaviour has been investigated and it is shown how the relation between inputs and outputs can be expressed as a convolution integral. A complete and explicit characterization of all weak solutions to \(R\left({d\over dt}\right)w= 0\) is given. This leads to the notion of input/output representation.

The fourth chapter deals with state models. The state of a dynamical system parameterizes its memory, the extent to which the past influences the future. That is, the state variables, an important class of latent variables, have the property that they parameterize the memory of the system. They split the past and future of the behaviour. It is shown that state equations which link the manifest variables to the state, turn out to be first-order differential equations and the output of a system is determined only after the input and the initial condition have been specified. The state space models for systems in input/output form have been discussed. Also linearization of nonlinear state space models has been dealt with.

The next chapter discusses controllability and observability which play a central role in control theory and which are two important features of dynamical systems from the behavioural point of view. Controllability refers to the question of whether or not one trajectory of a dynamical system can be steered towards another one and observability refers to the question of what one can deduce from the observation of one set of system variables about the behaviour of another set. By combining the concepts of controllability and observability applied to an input-output system, one obtains the Kalman decomposition.

The sixth chapter considers again the latent variable and state space systems. It is shown how a system of differential equations containing latent variables can be transformed to an equivalent system in which latent variables have been eliminated. The main result of this chapter shows that the elimination of latent variables in linear time-invariant differential systems is always possible by providing a systematic algorithm for doing this, which leads to a general theory of eliminating latent variables in linear systems.

Chapter 7 deals with the stability of a system. The classical stability conditions of systems of differential equations in terms of the roots of the associated polynomial or of the eigenvalue locations of the system matrix have been presented. The celebrated Routh-Hurwitz test, which specifies conditions for a polynomial to have only roots with negative real part, is also presented. Up to chapter 7, systems have been treated in their natural time-domain setting. There are several types of stability. In structural stability, one wants small parameter changes to have a similar small influence on the behaviour of a system. In dynamic stability, which is discussed in this chapter, it is the effect of disturbances in the form of an initial condition on the solution to the relevant dynamical equations. Intuitively, an equilibrium point is said to be stable if trajectories that start close to it remain close to it. Dynamic stability is thus not in the first instance a property of a system, but of an equilibrium point. However, in case of linear systems, stability is viewed as a property of the system. In input/output stability small input disturbances should produce small output disturbances.

Chapter 8 deals with the frequency domain description of systems and discusses some characteristic features and nomenclature for system responses related to the step response and the frequency domain properties. First, it is explained how a linear time-invariant system acts in the frequency domain. An important feature of such system is that it transforms sinusoidal or more generally exponential inputs into sinusoidal/exponential outputs. This leads to the transfer function and the frequency response as a convenient way of describing such systems. The transfer function specifies only the controllable part of a system, and uncontrollable modes are not represented by it. Also, properties of the time- and frequency-domain response are studied. Important characteristics of a system that can be deduced from its step-response or from its Bode and Nyquist plots have been described.

Chapters 9 and 10 deal with control theory. Control theory has two main roots: regulation and trajectory optimization. The regulation is the more important and engineering oriented. Whereas trajectory optimization is mathematics based. However, these roots have to a large extent merged in the second half of twentieth century. First, the authors discuss open-loop and feedback control and explain the difference and then they deal with an important result, the pole placement theorem – one of the central achievement of control theory. This theorem states that for a controllable system, there exists, for a suitable desired monic polynomial, a state feedback gain matrix such that the eigenvalues of the closed loop system are the roots of the desired polynomial. This chapter deals with an important control design question. More specifically, it deals with the choice of a control law such that the closed loop system is stable or, more generally, such that it has a certain degree of stability reflected – in a requirement on the location of the closed loop poles. The salient result obtained states that with this type of control, stabilization – in fact, pole placement – is always possible for controllable systems.

Then the tenth chapter deals with observers, that is, algorithms for deducing the system state from measured inputs and outputs. The design of an observer is very similar to the stabilization and pole placement procedures. However, it is not necessary to measure all the state variables for the design of a stabilizing feedback controller. By appropriate signal processing, it is often possible to obtain good estimates of all state variables from the measured outputs. The algorithm that performs this signal processing is called an observer. The observers discussed are actually nonoptimal, nonstochastic versions of the celebrated Kalman filter. The observers that have been obtained possess many appealing features, in particular, the recursivity of the resulting signal processing algorithm. By combining a state observer with a static control law, a feedback controller has been obtained, which is often called a compensator, that processes the measured outputs in order to compute the required control input.

This is an excellent textbook with many nice welcome features. First of all, the prface contains a historical introduction to systems and control and delineate thee philosophy of the approach to this subject, adopted by the authors, together with an outline of the topics covered. The behavioural approach originated in a series of six papers by J. C. Willems [the first one is “System theoretic models for the analysis of physical systems”, Ric. Autom. 10, 71-106 (1979)] over a period of ten years. This new approach to dynamical systems has many advantages, both from an engineering and from a mathematical perspective.

Each chapter contains sections called ‘Recapitulation’ and ‘Notes and References’ which gives the very useful summary of the discussion and sources used, respectively. Also the book contains two important appendices entitled ‘Simulation Exercises’ and ‘Background Material’, respectively. The exercises presented in the first appendix aim at giving the student insight into some simple modelling examples and control problems and require the use of a computer and numerical software packages. Also the text provides a good collection of problems as exercises after each chapter and some theory is also covered. The authors’ presentation is lucid and they have written the text with great care. The book is aimed at students in mathematics and control engineering and a certain background is needed to use it, since the material is presented in a mathematically rigorous way, while keeping physical modelling and applications aspects in mind. However, the text fully meets the objective of the TAM series. Finally, it is highly recommended for mathematics as well as mathematically oriented control engineering students.

Reviewer: R.K.Bose (New Delhi)

##### MSC:

93-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory |

34-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations |

93A10 | General systems |

93D25 | Input-output approaches in control theory |

93B25 | Algebraic methods |