The inverse problem of the optimal regulator

The inverse problem of the optimal regulator is considered for a general class of multi-input systems with integral-type performance indices. A new phase variable canonical form is shown to be convenient for this analysis. The advantage of the canonical form is to separate the state variables into subvectors of directly controlled, indirectly controlled, and uncontrollable components. Necessary and sufficient conditions for optimized performance indices are given. With the nonlinearities of the system restricted to functions of the directly controlled state variables, additional results are developed about the nonnegative property of optimized, loss functions. Answers are given to the following two questions. What performance indices can be optimized by an identically zero feedback control law? What performance indices can be optimized in a linearly synthesized feedback control system?