This paper considers linear discrete-time systems with additive disturbances, and designs a Model Predictive Control (MPC) law incorporating a dynamic feedback gain to minimise a quadratic cost function subject to a single chance constraint. The feedback gain is selected from a set of candidates generated by solutions of multiobjective optimisation problems solved by Dynamic Programming (DP). We provide two methods for gain selection based on minimising upper bounds on predicted costs. The chance constraint is defined as a discounted sum of violation probabilities on an infinite horizon. By penalising violation probabilities close to the initial time and ignoring violation probabilities in the far future, this form of constraint allows for an MPC law with guarantees of recursive feasibility without an assumption of boundedness of the disturbance. A computationally convenient MPC optimisation problem is formulated using Chebyshev's inequality and we introduce an online constraint-tightening technique to ensure recursive feasibility. The closed loop system is guaranteed to satisfy the chance constraint and a quadratic stability condition. With dynamic feedback gain selection, the conservativeness of Chebyshev's inequality is mitigated and closed loop cost is reduced with a larger set of feasible initial conditions. A numerical example is given to show these properties.

中文翻译：

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具有动态反馈增益选择和折扣概率约束的随机 MPC

本文考虑了具有加性扰动的线性离散时间系统，并设计了一个模型预测控制 (MPC) 法则，该法则结合了动态反馈增益，以最小化受单一机会约束的二次成本函数。反馈增益是从由动态规划 (DP) 解决的多目标优化问题的解生成的一组候选中选择的。我们提供了两种基于最小化预测成本上限的增益选择方法。机会约束被定义为无限范围内违规概率的折扣总和。通过惩罚接近初始时间的违规概率并忽略遥远未来的违规概率，这种形式的约束允许 MPC 法则保证递归可行性，而无需假设干扰有界。使用切比雪夫不等式制定了一个计算方便的 MPC 优化问题，我们引入了一种在线约束紧缩技术来确保递归的可行性。保证闭环系统满足机会约束和二次稳定性条件。通过动态反馈增益选择，切比雪夫不等式的保守性得到缓解，并且闭环成本通过更大的可行初始条件集降低。给出了一个数值例子来显示这些特性。通过动态反馈增益选择，切比雪夫不等式的保守性得到缓解，并且闭环成本通过更大的可行初始条件集降低。给出了一个数值例子来显示这些特性。通过动态反馈增益选择，切比雪夫不等式的保守性得到缓解，并且闭环成本通过更大的可行初始条件集降低。给出了一个数值例子来显示这些特性。