Adaptive Horizon Model Predictive Control and Al’brekht’s Method

Abstract

A standard way of finding a feedback law that stabilizes a control system to an operating point is to recast the problem as an infinite horizon optimal control problem. If the optimal cost and the optimal feedback can be found on a large domain around the operating point, then a Lyapunov argument can be used to verify the asymptotic stability of the closed loop dynamics. The problem with this approach is that it is usually very difficult to find the optimal cost and the optimal feedback on a large domain for nonlinear problems with or even without constraints, hence the increasing interest in Model Predictive Control (MPC). In standard MPC a finite horizon optimal control problem is solved in real time, but just at the current state, the first control action is implemented, the system evolves one time step, and the process is repeated. A terminal cost and terminal feedback found by Al’brekht’s method and defined in a neighborhood of the operating point can be used to shorten the horizon and thereby make the nonlinear programs easier to solve because they have less decision variables. Adaptive Horizon Model Predictive Control (AHMPC) is a scheme for varying the horizon length of Model Predictive Control as needed. Its goal is to achieve stabilization with horizons as small as possible so that MPC methods can be used on faster and/or more complicated dynamic processes.