Review of Linear Systems Theory

Abstract

We review in this chapter some fundamental concepts of multivariable linear time-invariant systems. Many concepts, such as system responses and stability as well as controllability and observability, are widely discussed in the literature and can be found in most of the introductory text books on linear systems theory (see, e.g., Antsaldis and Michel [4], Cahier and Desoer [16], Chen [33], DeCarlo [46], Kailath [70], Rugh [114], and Zheng [158]). In particular, we will adopt some nice mathematical derivations of these results from Chen [33] and Zheng [158]. On the other hand, some issues, such as the invariant zero structure (also called finite zero structure) and infinite zero structure, invertibility structures as well as geometric subspaces of linear systems, might be somewhat abstract to general readers and new graduate students. These topics, which will be illustrated in detail in the coming sections, can be found in some advanced monographs and research theses in linear systems, such as Rosenbrock [112], Trentelman et al. [141], Verghese [146], Wonham [154], and some research articles (see, e.g., Commault and Dion [41], Kouvaritakis and MacFarlane [77], MacFarlane and Karcanias [96], Moylan [102], Owens [105], Pugh and Ratcliffe [110], and Saberi, Chen and Sannuti [115]). In particular, the concepts of finite and infinite zero structures as well as that of invertibility structures will be introduced using the well-known Kronecker canonical form, and the materials on geometric subspaces will be adopted mostly from Trentelman et al. [141]. These basic concepts and results will be essential to the development of this book and helpful in understanding the results presented in the coming chapters.