Decompositions of Large-Scale Systems

Abstract

From the standpoint of control theory, it is usually convenient to represent a large-scale system as a collection of interconnected subsystems. In certain cases such a decomposition can be derived directly from the physical description of the problem, which suggests a “natural” grouping of the state variables. More often than not, however, the only information that we have about the system dynamics comes from a mathematical model whose properties provide little or no insight into how the subsystems should be chosen. In order to deal with such problems in a systematic manner, one obviously needs to develop decomposition algorithms that are based exclusively on the structure of the underlying equations.

In this chapter we will consider three decompositions, all of which are designed to exploit the sparsity of large-scale state space models. The algorithms are based on graph theoretic representations, which provide the necessary mathematical framework for partitioning the system. We should also point out that each of the three decompositions corresponds to a different gain matrix structure. With that in mind, it is fair to say that the purpose of these decompositions is not only to simplify the computation, but also to help us identify the most appropriate type of control law for a given large-scale system. We will take a closer look at this aspect of the problem in Chaps. 2 and 3, which are devoted to control design with information structure constraints.