A transfer function solution of the L.Q.G. control problem

Abstract

r o l . I n 1972 Weston and Bonqiorno [4J t r i e d t o ext.end Wiener's single input single output treatment of stochastic optimal control to the multivariable c a s e , b u t t h e i r s o l u t i o n was l i m i t e d t o s t a b l e systems and to the weighting of the system outputs, whose number should be less than or equal to the n W e r of inputs , i n the performance index. The main o b s t a c l e t o f u r t h e r development of the transfer function approach w a s t h e f a c t t h a t t h e s t a n da r d s p e c t r a l f a c t o r i s a t i o n method was unable t o deal W i t h unstable Plan*. The equivalence that was e s t a b l i s h e d i n MacFarlane [8] between the time domain and t h e frequency domain approaches i s Vali d i n the case of stable systems only. Recently, a comprehensive transfer function treatment of the problems of linear stationary filtering, deterministic quadratic optimal control and s t o c h a s t i c LQG optimal control has been introduced [9], kg. The present paper briefly summarises the results achieved there and discusses t h e d i f f e r e n t a s p e c t s o f t h e new theory.Abstract.Abstract A general transfer function approach t o t h e solution of the continuous stationary Kalman-Bucy f i l t e r and t h e time i n v a r i a n t l i n e a r d e t e m i n i s ti c and stochastic optimal control problems i s presented. A generalised spectral factor i s defined by which the equivalence between the time domain and the frequency domain approaches t o these problems is established. An e x p l i c i t expression for this generalised spectral factor i s given. I n the case where the number of system inputs and outputs is much smaller than the order of t h e system, the computation involved is much f a s t e r and requires less storage than is required to solve the Riccati equation of the time domain approach. The theory i s also extended to the case of coloured signals and l i n e a r dynamical ope r a t o r s i n the performance index and a simple soh ution to the singular aptimal control problem i s derived.