Polynomial optimization of multivariable control systems with both stochastic and deterministic inputs

The polynomial matrix method of optimization of multivariable control is formulated when the quadratic cost function contains both the stochastic steady-state response and the deterministic transient response. It is shown that the transient response to deterministic inputs may be included along with the steady-state response to stochastic inputs in the quadratic cost function as suggested by K. Park and J.J. Bongiorno (1989). This was incorporated in the polynomial method of optimization of multivariable control by a simple modification to the stochastic-only procedure given by M.M. Newmann and A.P. Roberts (1990). Two simple scalar examples are considered in order to illustrate the optimization technique in both discrete-time and continuous-time. The separate components of cost are calculated to show the resulting tradeoff between stochastic and deterministic costs