3.1 Introduction

It is our belief that a good unambiguous understanding of linear system structures, i.e. the finite and infinite zero structures as well as the invertibility structures of linear systems, is essential for a meaningful control system design. As a matter of fact, the performance and limitation of an overall control system are primarily dependent on the structural properties of the given open-loop system. In our opinion, a control system engineer should thoroughly study the properties of a given plant before carrying out any meaningful design. Many of the difficulties one might face in the design stage may be avoided if the designer has fully understood the system properties or limitations. For example, it is well understood in the literature that a nonminimum phase zero would generally yield a poor overall performance no matter what design methodology is used. A good control engineer should try to avoid these kinds of problem at the initial stage by adding or adjusting sensors or actuators in the system. Sometimes, a simple rearrangement of existing sensors and/or actuators could totally change the system properties. We refer interested readers to the work by Liu et al. [70] and a recent monograph by Chen et al. [71] for details.

As such, we first recall in this chapter a structural decomposition technique of linear systems, namely the special coordinate basis of [72, 73], which has a unique feature of displaying the structural properties of linear systems. The detailed derivation and proof of such...