Linear and Nonlinear Multivariable Feedback Control

1Canonical representations and stability analysis of linear MIMO systems

1.1 INTRODUCTION

In the first section of this chapter, we consider in general the key ideas and concepts concerning canonical representations of linear multi-input multi-output (MIMO) control systems (also called multivariable control systems) with the help of the characteristic transfer functions (or characteristic gain functions) method (MacFarlane and Belletrutti 1970; MacFarlane et al. 1977; MacFarlane and Postlethwaite 1977; Postlethwaite and MacFarlane 1979). We shall see how, using simple mathematical tools of the theory of matrices and linear algebraic operators, one can associate a set of N so-called one-dimensional characteristic systems acting in the complex space of input and output vector-valued signals along N linearly independent directions (axes of the canonical basis) with an N-dimensional (i.e. having N inputs and N outputs) MIMO system. This enables us to reduce the stability analysis of an interconnected MIMO system to the stability analysis of N independent characteristic systems, and to formulate the generalized Nyquist criterion. We also consider some notions concerning the singular value decomposition (SVD) used in the next chapter for the performance analysis of MIMO systems. In the subsequent sections, we focus on the structural and geometrical features of important classes of MIMO systems – uniform and normal systems – and derive canonical representations for their transfer function ...