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: (i) in contrast to classical static methods, our method gives the exact WCET; (ii) in contrast to classical measurement-based methods, our method guarantees a correct WCET estimate and requires fewer measurements on the hardware.

翻译：我们提出了第一个能够确定隐式线性模型预测控制（MPC）精确最差情况执行时间（WCET）的方法。当MPC用于实时控制安全关键系统时，此类WCET界限是必不可少的。所提出的方法适用于MPC控制器中的二次规划求解器属于一类成熟的有效集求解器的情况。针对此类求解器，我们利用先前提出的复杂度认证框架生成一组有限的原型优化问题；我们证明这些原型问题构成了所有可能问题的执行时间等价覆盖，即它们能够捕捉在线求解任何可能优化问题的执行时间。因此，只需在部署MPC的硬件上求解这些原型问题并记录执行时间，即可获得精确的WCET。除了提供该方法有效性的形式化证明外，我们还在一个倒立摆小车稳定控制的MPC实例上验证了该方法。实验凸显了与经典WCET方法相比的以下优势：（i）与经典静态方法不同，我们的方法能给出精确的WCET；（ii）与经典基于测量的方法不同，我们的方法能保证正确的WCET估计，且所需硬件测量次数更少。