Use of semidefinite programming for solving the LQR problem subject to rectangular descriptor systems
This paper deals with the Linear Quadratic Regulator (LQR) problem subject to descriptor systems for which the semidefinite programming approach is used as a solution. We propose a new sufficient condition in terms of primal dual semidefinite programming for the existence of the optimal state-control pair of the problem considered. The results show that semidefinite programming is an elegant method to solve the problem under consideration. Numerical examples are given to illustrate the results.
Anderson, B. D. O. and Moore, J. B. (1990). Optimal Control: Linear Quadratic Methods, Prentice-Hall, Upper Saddle River, NJ.
Balakrishnan, V. and Vandenberghe, L. (2003). Semidefinite programming duality and linear time-invariant systems, IEEE Transactions on Automatic Control48(1): 30-41.
Bender, D. J. and Laub, A. J. (1987). The linear quadratic optimal regulator for descriptor systems, IEEE Transactions on Automatic Control32(8): 672-688.
Dai, L. (1989). Singular Control Systems, Lecture Notes in Control and Information Sciences, Vol. 118, Springer, Berlin.
Geerts, T. (1994). Linear quadratic control with and without stability subject to general implicit continuous time systems: Coordinate-free interpretations of the optimal cost in terms of dissipation inequality and linear matrix inequality, Linear Algebra and Its Applications203-204: 607-658.
Ishihara, J. Y. and Terra, M. H. (2001). Impulse controllability and observability of rectangular descriptor systems, IEEE Transactions on Automatic Control46: 991-994.
Jiandong, Z., Shuping, M. and Zhaolin, C. (2002). Singular LQ problem for nonregular descriptor system, IEEE Transactions on Automatic Control47(7): 1128-1133.
Katayama, T. and Minamino, K. (1992). Linear quadratic regulator and spectral factorization for continuous time descriptor system, Proceedings of the IEEE Conference on Decision and Control, Tucson, AZ, USA, pp. 967-972.
Klema, V. C. and Laub, A. J. (1980). The singular value decomposition: Its computation and some applications IEEE Transactions on Automatic Control25(2): 164-176.
Mehrmann, V. (1989). Existence, uniqueness, and stability of solutions to singular linear quadratic optimal control problems, Linear Algebra and Its Applications121: 291-331.
Rami, M. A. and Zhou, X. Y. (2000). Linear matrix inequalities, riccati equations, and indefinite stochastic linear quadratic controls, IEEE Transactions on Automatic Control45(6): 1131-1143.
Silva, M. S. and de Lima, T. P. (2003). Looking for nonnegative solutions of a leontif dynamic model, Linear Algebra and Its Applications364: 281-316.
Vandenberghe, L. and Boyd, S. (1999). Applications of semidefinite programming, Applied Numerical Mathematics29: 283-299.
Yao, D., Zhang, D. and Zhou, X. Y. (2001). A primal dual semidefinite programming approach to linear quadratic control, IEEE Transactions on Automatic Control46(9): 1442-1447.