Transfer functions of distributed parameter systems:

Abstract In this tutorial article the rich variety of transfer functions for systems described by partial differential equations is illustrated by means of several examples under various boundary conditions. An important feature is the strong influence of the choice of boundary conditions on the dynamics and on system theoretic properties such as pole and zero locations, properness, relative degree and minimum phase. It is sometimes possible to design a controller using the irrational transfer function and several such techniques are outlined. More often, the irrational transfer function is approximated by a rational one for the purpose of controller design. Various approximation techniques and their underlying theory are briefly discussed. Key words: infinite-dimensional systems, partial differential equations, control theory, distributed parameter systems, relative degree, minimum phase Most of the analysis of control systems and the development of algorithms for controller design has been for systems modeled by ordinary differential equations, which provide adequate models for RLC circuits, rigid robots and many other systems. However, in many systems the physical quantity of interest depends on several independent variables. For instance, the temperature of an object depends on both position and time, as does the deflection produced by structural vibration. When there is more than one independent variable, the equation modeling the dynamics involves partial derivatives and is thus a partial-differential equation (PDE). The aim of this article is to acquaint the reader with a number of examples of systems with dynamics modeled by a partial-differential equation and in particular, to derive and analyze their transfer functions. Although the theory is generally applicable, for simplicity of exposition, all the examples are single-input-single-output and in one space dimension. Since the solution of the PDE reflects the distribution of a physical quantity such as the temperature of a rod or 1 Research supported by the National Science and Engineering Research Council of Canada the deflection of a beam, these systems are often called distributed-parameter systems (DPS). As is well known, the transfer functions of systems modeled by ordinary differential equations, often called lumped-parameter systems, are rational functions. In contrast, the transfer functions of distributed-parameter systems are irrational functions. Another difference is that the state space is infinite dimensional, usually a Hilbert space. Consequently, DPS are also called infinite-dimensional systems. In this article we do not discuss state-space realizations of these systems, but instead concentrate on the frequency-domain description in terms of the transfer function. The analysis of rational and irrational transfer functions differ in a number of important aspects. The most obvious differences between rational and irrational transfer functions are the poles and zeros. Irrational transfer functions often have infinitely many poles and zeros, as is the case fo