Functional methods for the study of nonlinear systems have developed steadily for the 70 years since the publication of the seminal paper by Volterra. For engineers, the most important contributions were made 40 years ago, by a group at the Massachusetts Institute of Technology. In particular, Brilliant showed that the Volterra series representation of a nonlinear system converges when the system is composed entirely of subsystems that are either linear with memory or nonlinear without memory. This allowed functional methods to be applied to systems described by block-oriented models which are not only favoured by many engineers but also form the basis of much computer software for dynamic system simulation.

In this chapter we have shown how symbolic computation is more than simply the use of symbolic algebra computer packages. The essential feature of modelling with hierarchical bond graphs is that information is kept in a symbolic format which represents the structure of the system and is transformed and translated into the desired representations and languages when required. Many different programs are used to achieve the manipulation of hierarchical bond graphs but in most cases the calculations and manipulations are performed symbolically rather than numerically, thus ensuring that the loss of information at any stage is minimised. The use of hierarchical bond graphs allows for the reuse of many components and the creation of libraries of bond graph models of subsystems. This aids the translation from initial conceptual modelling into a detailed simulation or design model and would not be possible without the use of significant symbolic computation.

The application of symbolic mathematical computing to control problems has begun to be recognised as a significant benefit of modern technological achieve ments, the results from which now appear in high ranking journals and international conference proceedings with increasing regularity. They are testimony to the growing profile of research that embraces algebraic computation, and they affirm the acceptability of such work as a worthwhile addition to the control profession throughout industry and academia. The uptake of computer algebra across the broad spectrum of control, however, has been relatively slow compared with some scientific disciplines, and there is evidently much scope within control engineering to find new uses for this potent resource. Control of a physical system is provided for by a dynamic representation of it. This chapter gives a concise account of the consistent spread of customised symbolic computer code over the last quarter of a century or so as a means to construct mathematically faithful dynamic models of multibody systems.

The following sections are included: introduction; notation and preliminaries; real parameter stability margin; extremal results in RPRCT; frequency-domain results in RPRCT; concluding remarks; acknowledgments; and references.

In the analysis of linear systems algebraic methods have been important since the early work of Routh and Hurwitz. Much of the work has been concerned with transfer functions or transfer matrices. For nonlinear systems the use of algebraic methods was for a long time made difficult by the complexity of the calculations involved. In recent years the availability of symbolic computer based methods has made it possible to explore algebraic methods in the nonlinear context. Here we concentrate on two approaches: differential algebra, which can be used to eliminate variables in systems of differential equations, and real algebra, which gives computational tools to handle systems of polynomial equations and inequalities. We give very brief overviews of the computational algorithms and then present some control related applications. To begin with we also note that the restriction to polynomial systems is not very severe. It can be shown that systems where the nonlinearities are not originally polynomial may be rewritten in polynomial form if the nonlinearities themselves are solutions to algebraic differential equations.

This chapter deals with the computation of certain types and values of invariants, the presence of which on a family of linear models is nongeneric. The computation of such invariants on models with numerical inaccuracies requires special methods, which may lead to approximate meaningful results to the computation problem. A classification of the algebraic computations according to their behaviour on numerically uncertain models is given, and then two of the key problems underlying the computation of a number of system invariants are considered; these are the problems of approximate computations of the greatest common divisor (GCD) and least common multiple (LCM) of polynomials. Some fundamental issues in the transformation of the GCD and LCM algebraic computations in an analytic, 'approximate' sense are considered, and methodologies yielding approximate solutions to GCD, LCM problems are examined.

This chapter examines the robust stability of multi-input/multi-output (MIMO) systems with parametric uncertainty, based on Rosenbrock's Direct Nyquist Array stability theorem. Two classes of structured uncertainty are considered, namely the interval and the affine linear types. Various measures of the diagonal dominance of a transfer function matrix are reviewed, and the original matrix row/column dominance concept, the subsequent generalised dominance concept, and more recent fundamental diagonal dominance measure are all extended to incorporate uncertainty in the system parameters. The fundamental dominance condition is also used as a nominal system approximation measure, generalising the diagonal system approximation assumption inherent in all diagonal dominance measures. Illustrative examples are presented.

Chapter 4 presented some fundamental results that are aids in analysing the behaviour of control systems subject to real parameter uncertainty. This chapter illustrates the application of these results to design by: (a) showing how classical design techniques can be robustified, by using these results and (b) developing a new linear programming approach to controller design that exploits these results. These examples should suggest to the reader how to formulate other design questions that take advantage of the fundamental results in real parametric robust control theory (RPRCT). The design of control systems is an art that should combine practical experience, analytical results and computational capabilities blended with imagination and control expertise to produce efficient systems. In this process the role of the analytical computations is to supply ready answers to various questions that the designer may pose. A typical design procedure may involve iterative loops within which the analytical calculations are embedded. In the following sections it shows how the theory described in Chapter 4 can be employed in this fashion.

This chapter gives a brief description of the dynamic sliding mode control methodology and shows how it can be implemented in a straightforward manner using symbolic algebra. A particular Mathematica implementation is described and some examples of design sessions presented.

In this chapter we have presented the possible use of symbolic algebra in the context of robust pole assignment for parametric uncertain systems. In particular, we have shown that symbolic algebra has a wide area of application when general forms of pole assignment compensators are sought. It has also been shown that in some cases it is possible to convert the dynamic output feedback problem to a general static output-feedback pole assignment problem. Then, following the methods proposed in Section 10.3.1 it may often be possible to find general forms of the pole assignment compensators that assign l + q poles of the closed-loop system. The remaining closed-loop system poles can then be readily calculated using the same formulas. This is particularly useful in reducing the search space, when robust compensators are sought.

This chapter presents algebraic synthesis and algebrogeometric approaches in the study of control problems rely on the theory of invariants and canonical forms of polynomial matrices and use the theory of Groebner bases for the computation of solutions of polynomial equations in many variables. This chapter provides first a classification of the different types of algebraic computations, according to whether they can be performed symbolically and/or numerically, and then considers three representative families of algebraic control theory problems, which require the use of symbolic computation tools.

This chapter describes the role of symbolic computation in several phases of the control system design cycle, namely modelling, analysis and synthesis. It provides some mathematical background of nonlinear system theory and presents a guided tour through those phases using typical examples, both contrived and real life. To name them, the symbolic computation of (i) the zero dynamics, used for different purposes, (ii) the input-output exact linearising state feedback, and (iii) the state-space exact linearising state feedback, all for nonlinear dynamic systems, are discussed. The examples come from a diverse range of applications and provide a picturesque landscape of this area of research. The possibilities and limitations of the so-called NON^CON package, based on the symbolic computation program Maple, are outlined with these examples. This package was developed to aid with analytical computations in the area of nonlinear control systems.

The use of formal optimisation techniques for the determination of optimal designs and operating strategies is increasing rapidly in all branches of engineering. Many of the problems of interest require the solution of nonlinear programming (NLP) problems involving the minimisation or maximisation of nonlinear objective functions Φ() subject to nonlinear equality and inequality constraints g(-) and Λ(-), respectively.

The following sections are included: introduction; quantifier elimination; Bernstein expansion; approximation of the solution set; algorithm; examples; conclusions; acknowledgment; and references.

Integrated design of systems and their control require tools for symbolic modelling and manipulation. Our approach involves the integration of symbolic and numerical computing. Successful modelling of complex vehicle dynamics and design of adaptive tracking controls for a detailed model of a magnetic bearing demonstrate that this methodology can solve realistic problems. The commercial packages TSiControls and TSiDynamics are now in use at several hundred installations.