Fast Evaluation of Quadratic Control-Lyapunov Policy

摘要

The evaluation of a control-Lyapunov policy, with quadratic Lyapunov function, requires the solution of a quadratic program (QP) at each time step. For small problems this QP can be solved explicitly; for larger problems an online optimization method can be used. For this reason the control-Lyapunov control policy is considered a computationally intensive control law, as opposed to an #x201C;analytical#x201D; control law, such as conventional linear state feedback, linear quadratic Gaussian control, or ${bf H}_infty$, too complex or slow to be used in high speed control applications. In this note we show that by precomputing certain quantities, the control-Lyapunov policy can be evaluated extremely efficiently. We will show that when the number of inputs is on the order of the square-root of the state dimension, the cost of evaluating a control-Lyapunov policy is on the same order as the cost of evaluating a simple linear state feedback policy, and less (in order) than the cost of updating a Kalman filter state estimate. To give an idea of the speeds involved, for a problem with 100 states and 10 inputs, the control-Lyapunov policy can be evaluated in around 67 $mu$ s, on a 2 GHz AMD processor; the same processor requires 40 $mu$s to carry out a Kalman filter update.