LQG controllers for discrete-time multivariable systems with different transport delays in signal channels

The design of stochastic LQG optimal tracking and regulating systems is considered for discrete-time systems with different time delays in different signal channels. A Wiener frequency domain solution for the closed-loop optimal controller is first obtained in the z-domain. This solution is physically realisable but involves the transport-delay operator. A state-space version of the controller is then derived from the frequency domain results. It is shown that the state equation based controller includes a Kalman predictor and state-estimate feedback. This confirms that a form of the separation principle holds for linear systems containing different transport delays on input and output signal channels. The Wiener solution applies to multivariable systems that may be unstable, nonminimum phase and nonsquare. The process and measuring system noise terms may be correlated and be coloured or white. It is shown that for certain classes of system the optimal controller can be implemented using a combination of finite dimensional and pure transport 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 impulse-response block, or alternatively in a Smith predictor form. In this latter case it has the same limitation, namely the plant must be open-loop stable. This restriction does not apply to either the Wiener or finite impulse response state space solutions