Solution of the stochastic optimal control problem in the s-domain for systems with time delay

The design of linear stochastic optimal tracking and regulating systems is considered for systems with time delay. A transfer-function solution for the optimal closed-loop controller is obtained in the s-domain. This solution can be realised physically, but involves the complex-domain delay operator. The state-space realisation of the controller is also obtained. The optimal controllers for both tracking and regulating systems include a Kalman predictor and state estimate feedback. This shows that a form of the separation principle holds for this class of problem. The design technique applies to multivariable systems which may be unstable, nonminimum phase and non-square. The process and measuring system noise terms may be correlated and be coloured or white. The linear quadratic performance criterion to be minimised includes a crossproduct weighting term. The solution to the usual stochastic regulator problem is obtained first and this result is then used to obtain the solution to the optimal tracking problem. Time delays in the control and in the measurement system are shown to have the same effect on the controller design, provided that the performance criterion is chosen appropriately. An expression for the minimum value of the performance criterion, that shows how the minimum cost is increased by the presence of the time delays, is obtained.