ABSTRACT
In the late 1960s, Basile and Marro (1969) (and later Wonham and Morse, 1970) discovered that the behavior of time-invariant linear control systems can be seen as a manifestation of the subspaces similar to the invariant subspaces characterized by the system matrices. As a result, the system behavior can be predicted and the solvability of many control problems can be tested by examining the properties of such subspaces. In many instances, one can understand essential issues intuitively in geometric terms. Moreover, thanks to good algorithms and software available in the literature (see Basile and Marro, 1992), the above subspaces can be generated and the properties can be readily examined by using personal computers. Thus, a large class of problems involving feedback control laws and observability of linear systems can be solved effectively by this geometric method, for example, problems of disturbance localization, decoupling, unknown input observability and system inversion, observer design, regulation
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and tracking, model following, robust control, and so on. Comprehensive treatments of the basic theory and many applications, including the ones mentioned above, can be found in the excellent books by Wonham (1985), Basile and Marro (1992), and Trentelman et al. (2002), with additional later results found in the newer books, for example, controlled and conditioned invariant subspaces and duality are treated in an organized fashion in the 2nd book and, in addition to the duality, an extension of the theory to include distributions as inputs and the relation between disturbance localization and H2 optimal control problem are also given in the 3rd book. The method is also useful in the analysis and design of decentralized control systems (Hamano and Furuta, 1975), perfect and near-perfect signal decoupling (and its applications to other control problems) for nonminimum-phase systems (Marro and Zattoni, 2006), and failure/fault detection (Massoumnia et al., 1989). This chapter serves as an introduction to the subject. Regrettably, citations in this chapter are rather limited due to editorial restrictions. Extensive references can be found in the above-mentioned books as well as in the chapter by Marro (2008). To prepare the present chapter, the book by Basile and Marro (1992) has been used as the primary reference, and the majority of the proofs omitted in this chapter can be found in this reference.
