We consider a linear system affected by an additive stochastic disturbance and address the design of a finite horizon control policy that is optimal according to some cost criterion and accounts also for probabilistic constraints on both the input and state variables. The resulting policy can be implemented over a receding horizon according to the model predictive control strategy. Such a possibility, however, is hampered by the fact that a feasibility issue may arise when recomputing the policy. Infeasibility indeed can occur if the disturbance has unbounded support and the state is required to remain in a bounded set. In this paper, we propose a solution to this issue that is based on the introduction of a constraint relaxation that becomes effective only when the original problem turns out to be unfeasible. This is obtained via a cascade of two probabilistically-constrained optimization problems where, in the first one, performance is neglected and the policy is designed to fully recover feasibility or – if this is not possible – to determine the minimum level of relaxation which is needed to recover feasibility; in the second step, such a minimum relaxation level is imposed while optimally (re-)tuning the control policy parameters. Both problems are solved through a computationally tractable scenario-based scheme using a finite number of disturbance realizations and providing an approximate solution that satisfies with high confidence the original probabilistic constraints of the cascade.

我们考虑一个受累加随机扰动影响的线性系统，并提出了一种有限水平控制策略的设计，该策略根据某些成本标准是最优的，并且还考虑了输入变量和状态变量的概率约束。可以根据模型预测控制策略在较晚的范围内实施最终的策略。但是，这种可能性因重新计算策略时可能出现可行性问题而受到阻碍。如果干扰具有无限支持，并且要求状态保持在有界集中，则确实会发生不可行。在本文中，我们提出了一个针对此问题的解决方案，该解决方案是基于引入约束放松的，该约束放松仅在原始问题证明不可行时才有效。这是通过两个概率受限的优化问题的级联获得的，在第一个问题中，性能被忽略，该策略旨在完全恢复可行性，或者（如果不可能的话）确定所需的最小放松水平恢复可行性；在第二步中，在优化（重新）调整控制策略参数的同时施加了这样的最小松弛级别。这两个问题都是通过使用有限数量的扰动实现通过基于计算易处理的方案的方案来解决的，并提供了一种可满足置信度的原始概率约束的近似解决方案。绩效被忽略，该政策旨在完全恢复可行性，或者（如果不可能）确定恢复可行性所需的最小放松水平；在第二步中，在优化（重新）调整控制策略参数的同时施加了这样的最小松弛级别。这两个问题都是通过使用有限数量的扰动实现通过基于计算易处理的方案的方案来解决的，并提供了一种可满足置信度的原始概率约束的近似解决方案。绩效被忽略，该政策旨在完全恢复可行性，或者（如果不可能）确定恢复可行性所需的最小放松水平；在第二步中，在优化（重新）调整控制策略参数的同时施加了这样的最小松弛级别。这两个问题都是通过使用有限数量的扰动实现通过基于计算易处理的方案的方案来解决的，并提供了一种可满足置信度的原始概率约束的近似解决方案。