LQG CONTROLLERS FOR DISCRETE-TIME MULTIVARIABLE SYSTEMS WITH DIFFERENT TRANSPORT DELAYS IN SIGNAL CHANNELS

The design of stochastic LQG optimal tracking and regulating systems i s considered for discrete-time systems with different time delays in d ifferent signal channels. A Wiener frequency domain solution for the c losed-loop optimal controller is first obtained in the z-domain. This solution is physically realisable but involves the transport-delay ope rator. A state-space version of the controller is then derived from th e frequency domain results. It is shown that the state equation based controller includes a Kalman predictor and state-estimate feedback. Th is confirms that a form of the separation principle holds for linear s ystems containing different transport delays on input and output signa l channels. The Wiener solution applies to multivariable systems that may be unstable, nonminimum phase and nonsquare. The process and measu ring system noise terms may be correlated and be coloured or white. It is shown that for certain classes of system the optimal controller ca n be implemented using a combination of finite dimensional and pure tr ansport delay elements. The main advantage is that the estimator is of much lower order than the traditional solution. The gain computation involves a reduced state equal to that of the delay free system and is thereby independent of the length of the delay. The state-space form of the optimal controller may be implemented using either a finite imp ulse-response block, or alternatively in a Smith predictor form. In th is latter case it has the same limitation, namely the plant must be op en-loop stable. This restriction does not apply to either the Wiener o r finite impulse response state space solutions.