Optimal Controller Synthesis for Second Order LTI Plants with Switched Output Feedback

We derive a switched output feedback control law for the class of second order linear systems of relative degree two which maximizes a quantity that measures the rate of convergence of the state trajectories to the origin. After providing a formal definition for the rate of convergence and formulating an infinite horizon optimization problem, we explore a corresponding finite horizon problem for the specific case where the plant is a double integrator (P(s)=1/s²) and use qualitative information about the behavior of the optimal state trajectories to derive a control law for the corresponding infinite horizon problem. We then derive an optimal controller for a general second order plant of relative degree two by relating it to the double integrator case study through appropriate transformations. We conclude with a design example