Mathematical Systems Theory I

In this chapter we present a range of dynamical systems from different areas of application and use them as examples to illustrate some typical problems from systems and control theory. Several of the mathematical models we introduce and discuss in the following sections will be taken up as examples in later chapters. The development of mathematical systems theory starts in the next chapter. The readers who prefer to go directly to Chapter 2 can do so without any difficulty as the mathematical exposition in that chapter is self-contained and independent of following material. On encountering an example based on a dynamic model from Chapter 1, they may wish to look back to its origin here to find more details and get additional background information.

State space theory deals with dynamical models describing both the internal dynamics of a given physical process and the interaction of the process with the outside world. In this chapter we introduce the general notion of a dynamical system and set the basis for the study of various important system classes. We emphasize that for us a dynamical system is a mathematical model and hence should be carefully distinguished from the physical process for which it is a model. Dynamical systems of different types may be used as models of one and the same physical process. Nevertheless it will sometimes be convenient to use the word “system” for the real physical process described by the dynamical model and in this case we shall add the epithet “real” or “physical” whenever this is necessary for a clear distinction.

The Oxford English Dictionary's definition of stable is not easily moved, changed or destroyed”. Most of us have an intuitive notion of stability which corresponds more or less with this definition. However, in order to build a theory of stability it is necessary to be more precise about terms like “not easily moved or changed”. We need to define the basic class of objects to which the notion of stability is applied and also specify the type of perturbations which are considered. In this chapter we study the stability of state trajectories under the influence of perturbations in the initial state and in the next two chapters we consider perturbations in the system parameters. The stability of output trajectories under the influence of perturbations in the input signal will be discussed in Volume II. The development of modern stability theories was initiated by Maxwell (1868) [364] and Vyshnegradskiy (1876) [511] in their work on governors, but the importance of the concept of stability in many other scientific fields was soon recognized and now it is a cornerstone of applied mathematics. For example, the prediction of instabilities from a mathematical model has in many instances led to a confirmation that the model adequately represents the corresponding physical process. In 1923 G.I. Taylor [492], using the Navier-Stokes equations, showed that the flow of a viscous fluid between rotating cylinders would become unstable at a particular value of a parameter, now known as the Taylor number. He confirmed this experimentally and so increased confidence in modelling viscous fluid flows by the Navier-Stokes equations. Perhaps more relevant to this text is the fact that almost all control system designs are founded on a stability requirement and our treatment of the subject will be slanted in this direction.

The aim of this chapter is to study how the root and eigenvalue locations of polynomials and matrices change under perturbations. The chapter is quite a substantial one since we address a number of different issues. First and foremost we consider a variety of perturbation classes, ranging from highly structured perturbations which are determined via a single parameter to unstructured perturbations where all the entries of the matrix or coefficients of the polynomial are subject to independent variation. The size of the perturbations will, in the main, be measured by arbitrary operator norms. Moreover we will develop the theory for both complex and real perturbations which often require quite different approaches. The first section is concerned with polynomials. We establish some continuity and analyticity results for the roots, then describe the sets of all Hurwitz and Schur polynomials in coefficient space. We also consider the problem of determining conditions under which all polynomials with real coefficients belonging to prescribed intervals are stable and prove Kharitonov's Theorem. The effect of perturbations on the eigenvalues of matrices is considered in Section 4.2. We first state some simple continuity and analyticity results which follow directly from the results of Section 4.1. Then we assume that the matrix depends analytically on a single parameter and examine the smoothness of eigenvalues, eigenprojections and eigenvectors. Section 4.3 deals with singular values and singular value decompositions which are important tools in the quantitative perturbation analysis of linear systems. Section 4.4 is dedicated to structured perturbations and presents some elements of μ-analysis, both for complex and for real parameter perturbations. We finish the chapter in Section 4.5 with a brief introduction to some numerical issues which are important for Systems Theory, focussing on those aspects which have a relationship with the material of this and the previous chapters.

The first step in most applications of mathematics is to determine a mathematical model for the system under investigation. The model may be used in a number of different ways. For example, a mathematical and computational analysis of the model often leads to a better understanding of the real physical system it represents. From a more practical viewpoint the model can be used to make predictions about the future behaviour of the system, or to design algorithms of automatic control which ensure that the system behaves in some desirable fashion. However, in each of these applications it is of fundamental importance to keep in mind that the model is only a model, its behaviour and that of the real system might be quite different. The origins and causes of this possible discrepancy are many and in the systems theory literature are collectively referred to as model uncertainties: