Stochastic multivariable control: A polynomial equation approach

A new technique to design optimal multivariable controllers is presented for stochastic plants described by rational transfer matrices. The objective is to minimize a weighted sum of the steady-state variances at the plant input and output subject to the asymptotic stability of the closed-loop system. The technique is based on polynomial matrices. Essentially, the design procedure is reduced to solving a linear matrix polynomial equation whose coefficient matrices are obtained by spectral factorization. The solution of this equation then directly yields the optimal controller transfer matrix as well as the associated minimum cost. The reported approach is relatively simple, computationally attractive, and lays bare the necessary and sufficient conditions for the optimal controller to exist. It is general enough to handle problems that cannot always be addressed by standard time-domain LQG techniques, such as problems involving plants with improper transfer matrices and/or singular noise intensities and weighting matrices.