Optimal complementary matrices in systems with overlapping decomposition: A computational approach

The paper deals with linear quadratic (LQ) optimal control of linear time-invariant (LTI) systems which are decomposed into overlapped subsystems. A mathematical framework (inclusion principle) is available to formalize different structural properties and relations between the initial and the expanded systems, in which the so called complementary matrices play an important role. Up to now, only the structure and conditions on these matrices have been studied in the literature, but not the way to obtain their numerical values systematically. This paper presents a computational approach to select complementary matrices, which can be useful for a practical use of overlapping decompositions. The specific objective is to obtain the complementary matrices such that the quadratic performance for the expanded optimal control problem is minimum. An example is supplied to illustrate the use of the proposed algorithm