Optimizing Static Linear Feedback: Gradient Method

The linear quadratic regulator is the fundamental problem of optimal control. Its state feedback version was set and solved in the early 1960s. However static output feedback problem has no explicit-form solution. It is suggested to look at both of them from another point of view as a matrix optimization problem, where the variable is a feedback matrix gain. The properties of such a function are investigated, it turns out to be smooth, but not convex, with possible non-connected domain. Nevertheless, the gradient method for it converges to the optimal solution in state feedback case and to a stationary point in output feedback case. The results can be extended for the general framework of reduced gradient method for optimization with equality-type constraints.