Optimal linear-quadratic control: From matrix equations to linear matrix inequalities

The results of the present authors' studies on the design of optimal linear-quadratic control based on the linear matrix inequalities were summarized. Uniform consideration was given both to the nondegenerate and degenerate problems of optimal control by state and measured output, the so-called orthogonality condition being either satisfied or not. In the problem of control by the measured output, the matrix of parameters of the optimal feedback depends substantially on the initial state of the plant, which makes this control law practically impossible. The need for passing from the integral functional as the system performance index to the maximal value of the ratio of this functional to the square of the initial state norm over all initial states and the corresponding passage from the optimal control law to the minimax linear-quadratic law were substantiated for this case.