Optimal controller synthesis for a class of LTI systems via switched feedback

We develop a switched feedback controller that optimizes the rate of convergence of the state trajectories to the origin for a class of second order LTI systems. Specifically, we derive an algorithm which optimizes the rate of convergence by employing a controller that switches between symmetric gains. As a byproduct of our investigation, we find that, in general, the controllers which optimize the rate of convergence switch between two linear subsystems, one of which is unstable. The algorithm we investigate will design optimal switching laws for the specific case of second order LTI plants of relative degree two.