Exact Worst-Case Execution-Time Analysis for Implicit Model Predictive Control

We propose the first method that determines the exact worst-case execution time (WCET) for implicit linear model predictive control (MPC). Such WCET bounds are imperative when MPC is used in real time to control safety-critical systems. The proposed method applies when the quadratic programming solver in the MPC controller belongs to a family of well-established active-set solvers. For such solvers, we leverage a previously proposed complexity certification framework to generate a finite set of "archetypal" optimization problems; we prove that these archetypal problems form an execution-time equivalent cover of all possible problems; that is, that they capture the execution time for solving any possible optimization problem that can be encountered online. Hence, by solving just these archetypal problems on the hardware on which the MPC is to be deployed, and by recording the execution times, we obtain the exact WCET. In addition to providing formal proofs of the methods efficacy, we validate the method on an MPC example where an inverted pendulum on a cart is stabilized. The experiments highlight the following advantages compared with classical WCET methods: first, in contrast to classical static methods, our method gives the exact WCET; second, in contrast to classical measurement-based methods, our method guarantees a correct WCET estimate and requires fewer measurements on the hardware.