TABLE OF CONTENTS This chapter describes the modelling and analysis tools that will be used throughout the book. They all concern Linear Time Invariant systems or transfer functions and can be classified in three categories.
Closed-loop system representation and analysis: We briefly define two possible representations of a system G 0 in closed loop with some stabilising controller K. The first one is a general representation based on a standard block-diagram description of the closed loop (Section 2.2) and on which we shall base the notions of closed-loop stability and generalised stability margin. The second one is based on a linear fractional transformation (LFT) (Section 2.3), which is sometimes more convenient for the manipulation of multivariable closed-loop systems or for robust control design and analysis. We show, in Section 2.3, that any closed-loop system represented by means of an LFT can be put in the general form defined in Section 2.2, allowing use of standard stability analysis tools.
Linear systems analysis: Two important tools are presented. The first one is coprime factorisation (Section 2.4). A coprime factorisation is a way of representing a possibly unstable transfer function or matrix by two stable ones (a numerator and a denominator) with particular properties related, e.g., to closed-loop stability. Procedures are given to build such (possibly normalised) factorisations. It is also shown how the generalised closed-loop transfer matrix of a system can be expressed in function of the plant and controller coprime factors. The second tool is the ?
