State-Equivalent Form and Minimum-Order Compensator Design for Rectangular Descriptor Systems

This technical note studies the analysis and control problem of rectangular descriptor systems (RDSs). First, we present the state-equivalent form of RDSs using the left (right) inverse of a matrix and demonstrate its equivalence to the original system in terms of admissibility. Subsequently, we introduce a minimum-order compensator (MOC) and establish a series of rank criteria for its application. The MOC is utilized to ensure that the closed-loop system is generalized regular, impulse-free, and admissible. In addition, we provide a necessary and sufficient condition in the form of bilinear matrix inequalities for designing the MOC, along with an iterative algorithm for computing the gains. Finally, the effectiveness of our methods are demonstrated through two numerical examples and a circuit example.