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In this chapter, some of the important issues concerning centralized linear time-invariant (LTI) finite-dimensional systems are studied. The notion of a centralized fixed mode (CFM) is introduced, and its relationship with the notions of controllability and observability is spelled out. In particular, a procedure is given to determine the controllability and/or observability of different modes of the system using the concept of CFMs. This procedure enables one to determine the controllability and/or observability of the modes of a system, which can be computationally more efficient compared to existing analytical techniques such as the PBH test (Brogan in Modern control theory. Prentice Hall, New Jersey, 1991 [4]). It is also shown that a LTI system can be stabilized using a LTI controller if and only if the system does not have any unstable CFMs. A highly effective algorithm is then presented to find the minimal realization of a LTI system using the concept of CFMs. The problem of model reduction for LTI systems is then studied, and different techniques are discussed. The idea behind these techniques is to retain the dominant modes of the system and delete the less dominant ones. The dominant modes may be shifted slightly in the reduced-order model in order to compensate for the neglected ones (and to maintain certain features of the system such as the DC gain). In particular, the balanced realization technique relies on the Gramians of controllability and observability and provides a procedure to find a similarity transformation under which the Gramians of the controllability and observability matrices of the transformed system are diagonal and equal. An upper bound for the infinity norm of the approximation error in terms of the transfer functions of the original system and the reduced-order model is also provided, which can be used as a quantitative measure for the effectiveness of the technique in approximating a high-order model. |
