Model Predictive Control (MPC) is a well-established approach to solve infinite horizon optimal control problems. Since optimization over an infinite time horizon is generally infeasible, MPC determines a suboptimal feedback control by repeatedly solving finite time optimal control problems. Although MPC has been successfully used in many applications, applying MPC to large-scale systems -- arising, e.g., through discretization of partial differential equations -- requires the solution of high-dimensional optimal control problems and thus poses immense computational effort. We consider systems governed by parametrized parabolic partial differential equations and employ the reduced basis (RB) method as a low-dimensional surrogate model for the finite time optimal control problem. The reduced order optimal control serves as feedback control for the original large-scale system. We analyze the proposed RB-MPC approach by first developing a posteriori error bounds for the errors in the optimal control and associated cost functional. These bounds can be evaluated efficiently in an offline-online computational procedure and allow us to guarantee asymptotic stability of the closed-loop system using the RB-MPC approach. We also propose an adaptive strategy to choose the prediction horizon of the finite time optimal control problem. Numerical results are presented to illustrate the theoretical properties of our approach.
翻译:模型预测控制(MPC)是解决无限地平线最佳控制问题的既定方法。由于在无限时间平线上优化一般不可行,MPC通过反复解决有限时间的最佳控制问题确定亚最佳反馈控制。尽管MPC在许多应用中被成功使用,但将MPC应用于大型系统 -- -- 例如,通过部分差异方程式的离散处理 -- -- 需要解决高维最佳控制问题,从而带来巨大的计算努力。我们认为,由平衡的参数偏差部分差异方程式所管理的系统,并采用缩小基数(RB)方法作为有限时间最佳控制问题的低维度替代模型。减少的订单最佳控制是原始大型系统的反馈控制。我们分析拟议的RB-MPC方法,首先开发出最佳控制错误的后继误圈,并随之产生成本功能。这些界限可以在离线计算程序中得到高效评估,并使我们能够保证使用固定地平线部分偏差部分偏差的偏差方程式作为有限基数控制问题的低维基(RBM-MPC)方法。我们还提议了对最佳时空预测结果的调整战略。