Stationary LQG control of singular systems

The stationary linear-quadratic-Gaussian control problem is formulated and solved for single-input single-output singular systems. The control system is required to be internally proper and stable in order to avoid both impulsive and unstable exponential behavior. The set of all controllers resulting in such a control system is specified in parametric form. All controllers that yield finite cost are identified, once again in parametric form, within this set. Necessary and sufficient conditions are then established for an optimal controller to exist. All optimal controllers are shown to possess the same transfer function. The problem is analyzed in the complex domain. The transfer functions are expressed as quotients of proper, strict-Hurwitz rational functions. By means of this maneuver, the powerful tools of algebra are made available. The synthesis of the optimal controller is reduced to the solution of two linear Diophantine equations whose coefficients are obtained by spectral factorization.