Linear System Tools

Abstract

As will be evident in the coming chapters, the finite and infinite zero structures as well as the invertibility structure of the given system play dominant roles in the computation of the infima and the solutions to both continuous-time and discrete-time H _∞ optimization problems. Thus a good non-ambiguous understanding of linear system structure is essential for our study. In our opinion, the best way to display all the structural properties of linear systems is to transform them into a so-called special coordinate basis (SCB) developed by Sannuti and Saberi [93] and Saberi and Sannuti [89]. However, quite often it happens that the original special coordinate basis of Sannuti and Saberi is not fine enough to characterize all the details of the properties of linear systems. In order to see all the fine points of a given system, we would have to further decompose certain subsystems of its SCB using some well-known canonical forms such as the Jordan canonical form and the Brunovsky canonical form. Keeping this in mind, we recall in this chapter the following results: 1) the Jordan and real Jordan canonical forms for a square constant matrix; 2) the Brunovsky canonical form and the block diagonal controllability canonical form for a constant matrix pair; and 3) the special coordinate basis of a linear time invariant system characterized by either a matrix triple or a matrix quadruple. These canonical forms and the special coordinate basis will form a transformer for linear systems. Once a system is touched by this transformer, all its structural properties become clear and transparent. As such, we call it an X-transformer. We should note that the original work of [93,89] dealt only with the continuous time systems. In this chapter, we will unify the special coordinate basis for both continuous-time and discrete-time systems under a single framework. More importantly, we will provide rigorous proofs to all the properties of the special coordinate basis for the first time in the literature.