Contents 1 Introduction
Consider the following rectangular descriptor linear system: \eqalignno{ \tilde{E} \dot{\tilde{x}}(t) = & \tilde{A}\tilde{x}(t)+\tilde{B}u(t),\ \tilde{x}(t_{0})=\tilde{x}_{0}, &\hbox{(1a)} \cr &\quad y(t)=C\tilde{x}(t), &\hbox{(1b)}} where is the -dimensional state vector, is the q-dimensional control input vector, , is the -dimensional control output vector, and are constant matrices. Let rank , it is clear that . If , the system (1) is said to be square. And the system (1) is said to be regular if there exits such that . If , the system (1) is said to be rectangular. Since descriptor systems have comprehensive background, such as power systems [1], social economic systems [2], circuit systems [3], and so on, great progress has been made in the theory and its applications since 1970s for square descriptor systems [4]–[7]. In the last two decades, some results of square descriptor systems have been extended to rectangular descriptor systems (or named rectangular systems, non-square systems). Impulse solutions and impulse controllability have been discussed [8], [9]. Observer design has been tackled [10], [11]. Controllability, observability and elimination of impulsive mode have been presented [12]–[15]. The recursive estimation problem for general time-variant descriptor systems has been solved by using a game theory approach [16]. A new feedback structure, dynamic output feedback plus state feedback, has been utilized to stabilize rectangular descriptor systems [17]. An algebraic approach has been proposed to study the problems of regularization, impulse elimination and stabilization of rectangular descriptor systems by decentralized dynamic compensation [18].
