Steady-state mmzmum-varzance controllers may be designed for stable or unstable plants with proper or tmproper ratmnal transfer functmns, and disturbances wtth rational spectra, using a stmple, computatzonally attractwe procedure for solwng two hnear polynomml equatmns whose coefficients are obtained by spectral factomzatton Key Word Index---Control system synthesis, hnear systems, polynomml equations, spectral factorlzation, stochastic control, computational methods Abstraet--A new technique to design optimal controllers ~s presented for plants described by rational transfer functions and additive disturbances with rational spectral densities The objective is to mmimlze a weighted sum of the plant input and output steady-state variances subject to asymptotic stabihty of the closed-loop system
The technique is based on polynomial algebra In fact, the design procedure is reduced to solving two linear polynomial equations whose unique solution directly yields the optimal controller transfer function as well as the tmmmized cost This approach is simple, computatlonally attractive, and can handle unstable and/or nonmmimum-phase plants with improper transfer functions An integral part of the paper are effective computational algorithms, which include the spectral factonzatlon, the so-luUon of polynomial equations, and the evaluation of minimum cost 1 INTRODUCTION THE DESIGN of optimum systems with random inputs ~s one of the most significant problems in optimal control In this paper we consider the problem of controlhng a scalar-input-output hnear plant m the presence of both background and measurement disturbances The plant is described in terms of its input-output properties and the objective ts to minimize a hnear combination of the plant input and output variances in steady state This problem was first attacked by the Wlener-Hopf technique in the frequency domain The classical treatment can be found in Newton, Gould and Kaiser (1957) and Chang (1961) Thts approach requires one first to find the optimal *