This article presents a method for designing stabilizing terminal-costs for linear model predictive controllers whose stage-costs are a mixture of weighted 1-norms and ∞-norms of the system states and inputs. This problem is abstracted by replacing the 1/∞-norm stage-costs with convex Minkowski functions. We show that a convex Minkowski terminal-cost is a control Lyapunov function if and only if the dual of its defining set satisfies a set-inclusion condition. Furthermore, we show that this set-inclusion is satisfied by all robust positive invariant sets of a certain dual system. In particular, we show that the minimal convex set that satisfies the set-inclusion produces the optimal terminal-cost which is the cost-to-go of the corresponding unconstrained infinite-horizon optimal control problem with convex Minkowski stage-costs. To reduce the number of constraints in the MPC, we show that the defining set of the Minkowski function can be used as an invariant terminal-constraint. Finally, we demonstrate our terminal-cost design procedure for three case studies; a double-integrator, a galvanometer laser scanner, and a nanosatellite.

中文翻译：

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具有线性阶段成本的模型预测控制的终端成本设计：一种集合论方法

本文提出了一种为线性模型预测控制器设计稳定终端成本的方法，其阶段成本是系统状态和输入的加权 1 范数和∞范数的混合。这个问题通过替换 1/ ∞- 具有凸 Minkowski 函数的范数阶段成本。我们证明，当且仅当其定义集的对偶满足集包含条件时，凸 Minkowski 终端成本是控制李雅普诺夫函数。此外，我们表明某个对偶系统的所有稳健正不变集都满足这种集合包含。特别是，我们表明满足集合包含的最小凸集产生最优终端成本，这是相应的无约束无限范围最优控制问题的成本，具有凸 Minkowski 阶段成本。为了减少 MPC 中的约束数量，我们展示了 Minkowski 函数的定义集可以用作不变的终端约束。最后，我们展示了三个案例研究的终端成本设计程序；双积分器，