A numerical comparison of frozen-time and forward-propagating Riccati equations for stabilization of periodically time-varying systems

Feedback control of linear time-varying systems arises in numerous applications. In this paper we numerically investigate and compare the performance of two heuristic techniques. The first technique is the frozen-time Riccati equation, which is analogous to the state-dependent Riccati equation, where the instantaneous dynamics matrix is used within an algebraic Riccati equation solved at each time step. The second technique is the forward-propagating Riccati equation, which solves the differential algebraic Riccati equation forward in time rather than backward in time as in optimal control. Both techniques are heuristic and suboptimal in the sense that neither stability nor optimal performance is guaranteed. To assess the performance of these methods, we construct Pareto efficiency curves that illustrate the state and control cost tradeoffs. Three examples involving periodically time-varying dynamics are considered, including a second-order exponentially unstable Mathieu equation, a fourth-order rotating disk with rigid body unstable modes, and a 10th-order parametrically forced beam with exponentially unstable dynamics. The first two examples assume full-state feedback, while the last example uses a scalar displacement measurement with state estimation performed by a dual Riccati technique.