Singular LQ problem for nonregular descriptor systems

Use of semidefinite programming for solving the LQR problem subject to rectangular descriptor systems

“…A great number of results on solving the LQR problem subject to descriptor systems have appeared in the literature (Bender and Laub, 1987;Geerts, 1994;Jiandong et al, 2002;Katayama and Minamino, 1992;Mehrmann, 1989). However, almost all of these results consider the regularity assumption of descriptor systems and the positive definiteness assumption of the control weighting matrix in the quadratic cost functional.…”

“…Thus no answer is provided to a basic question such as when the LQR problem subject to descriptor systems possesses an optimal solution in the form of a conventional control, in particular, one that does not involve an impulse distribution. However, this issue was discussed by Jiandong et al (2002),…”

“…Here, since we do not assume that the descriptor system is regular, our work is more general than some previous results (Bender and Laub, 1987;Geerts, 1994;Katayama and Minamino, 1992;Mehrmann, 1989). The method of Jiandong et al (2002) is still maintained to transform the original problem into the equivalent singular LQR one.…”

“…Nonetheless, there still remains an open problem, that is, the new standard LQR problem may be singular and this is not answered by Jiandong et al (2002).…”

“…In this paper, we reconsider the problem stated by Jiandong et al (2002) and, in particular, the open problem, i.e., the singular version of the new standard LQR problem, is solved using the semidefinite programming approach. A new sufficient condition in terms of primal dual semidefinite programming for the existence of the optimal state-control pair of such a problem is proposed.…”