Optimal Control for Rectangular Singular Systems

In this paper, the problem of linear quadratic regulator (LQR) optimal control, tracking optimal control and stochastic control for rectangular singular systems is considered. In this paper, for the LQR and tracking, the necessary and sufficient conditions for the controllability, observability, regularity, admissibility, and the existence of the solution for two types of rectangular singular systems have been obtained. In first case, the number of equations is greater than the number of unknowns and in the second case, the number of unknowns is greater than the number of equations, then, it is shown that the Hamiltonian form of a rectangular singular system is a square singular system. Therefore, the theorems concerning the uniqueness of the solution, stability and others which have been proved for the square singular systems can be extended to the rectangular singular systems. The only difference in the square singular systems is the necessary conditions for the existence of a solution. It is also shown that there are cases for which the matrix form system has never a unique solution. Simulation results demonstrate the good performance of the proposed methods.