Control of linear systems according to the minimal variance criterion--A new approach to the disturbance problem

The problem of minimizing the ensemble average of a performance index in the presence of control noise has received substantial attention in the literature. This work considers a generalization of viewpoint, in which the index variance is minimized while its expectation is constrained. Necessary and sufficient relations are derived for linear, time-invariant systems and disturbances having rational spectra. The open-loop, optimal-feedback solution is specified by its characteristic equation and boundary conditions for Gaussian noises and plants with distinct eigenvalues. The canonic structure of a noise-free plant incorporating covariance data from the disturbance process is shown to have fundamental significance in the optimal solution. Several examples are presented.