Chapter 5 - Linear State-Space Models and Solutions of the State Equations

Abstract

This chapter presents linear state-space models and describes some simple familiar physical systems in state-space forms. Very often, the mathematical model of a system is not obtained in first-order form; it may be a system of nonlinear equations, a system of second-order differential equations, or partial differential equations. It is shown how such systems can be reduced to the standard first-order state-space forms. The computational methods for the state equations are then considered both in time and frequency domains. The major computational component of the time-domain solution of a continuous-time system is the matrix exponential eAt. Some results on the sensitivity of this matrix and various well-known methods for its computation: the Taylor series method, the Padé approximation method, the ordinary-differential equation methods, and the methods based on decompositions of the matrix are described in the chapter in details. A comparative study of these methods is also included. The Padé method (with sealing and squaring) and the method based on the Real Schur decomposition of a matrix are recommended for practical use. The problem of computing the frequency response matrix for many different values of the frequencies is considered in the chapter. The computation of the frequency response matrix is necessary to study various system responses in frequency domain. A widely used method based on the one-time reduction of a state matrix to a Hessenberg matrix is described in the chapter in detail, and the references to the other recent methods are given.