Multiple Shooting in a Microsecond

Abstract

Nonlinear Model Predictive Control (NMPC) is a feedback control technique that uses the most current state estimate of a nonlinear system to compute an optimal plan for the future system behavior. This plan is recomputed at every sampling time, creating feedback. Thus, NMPC needs to repeatedly solve a nonlinear optimal control problem (OCP). Direct multiple shooting is since long known as a reliable approach for discretization of OCPs. This is mainly due to the fact that the approach shows good contraction properties within the NMPC framework. Moreover, the procedure is easy to initialize and parallelize. In the context of real-time NMPC, the multiple shooting method was tailored to the Real-Time Iteration (RTI) scheme. This scheme uses a strategy known as Initial Value Embedding to deal efficiently with the transition from one optimization problem to the next. It performs two algorithmic steps in each sampling time, a long preparation phase and a short feedback phase to minimize the feedback time to the system. The two phases respectively prepare and solve a convex Quadratic Program (QP) that depends parametrically on the estimated system state. The solution of this QP delivers quickly a generalized tangential predictor to the solution of the nonlinear problem. Recent algorithmic progress makes the solution of NMPC optimization problems possible at sampling times in the milli- or even microsecond range on modern computational hardware. An essential part is the simulation of the nonlinear model together with the propagation of its derivative information. This article describes the developments and their efficient software implementations that made it possible to solve a classical NMPC benchmark problem within 1 μs sampling time.