Homotopy Tracing of Static Output Feedback Gain for LQ Control

Abstract

In this paper, a method of finding a local minimum is examined for LQ optimal control problem of constant output feedback. A necessary condition for optimality is given as matrix algebraic equations for a modified LQ performance index, and a method of tracing a solution path from a state feedback gain to an output feedback gain is proposed. In order to trace the solution curve that may have branch, a new homotopy method based on series expansion of approximate algebra is applied. Initial gains are obtained from not only a positive definite solution but also the other solutions of Riccati equation, then it is examined numerically whether more than one local minimum can be obtained by tracing the solutions starting from the initial gains. A numerical example which has two local minima shows that one local minimum is connected to the optimal state feedback gain, but the other is not connected to any gains given by the solutions of Riccati equation.