Optimization and invariance of linear stationary control systems

The survey discusses the relationships between optimization of linear stationary systems subjected to stochastic exogenous signals and robustness or consistent inimplementability of optimal controllers. The relationships between the optimization theory and the invariance theory is determined. The difficulties facing the invariance theory (instability or non-robustness when absolute invariance in attempted), are shown to be attributable to the same causes to which the recently discovered similar difficulties in the stochastic control theory are. A functional is introduced which determines the measure of invariance of a specified output with a specified (deterministic) input of the system and the minimization of this functional is shown to be equivalent to some stochastic optimization. In this way problems of system optimization are found to be solvable by using the invariance property. The relationship with analytical controller design is shown and the role of singular problems demonstrated. Methods of obtaining minimizing sequences are considered for cases where an optimal controller is inexistent. Examples are given where statistical optimization problems are solved with a coarse and structurally and consistently realizable stabilizing controller.