The stability of stochastic model-predictive control (MPC) subject to additive disturbances is often demonstrated in the literature by constructing Lyapunov-like inequalities that ensure closed-loop performance bounds and boundedness of the state, but tight ultimate bounds for the state and nonconservative performance bounds are typically not determined. In this article, we use an input-to-state stability property to find conditions that imply convergence with probability 1 of a disturbed nonlinear system to a minimal robust positively invariant set. We discuss implications for the convergence of the state and control laws of stochastic MPC formulations, and we prove convergence results for several existing stochastic MPC formulations for linear and nonlinear systems.

中文翻译：

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随机非线性系统的收敛性及其对随机模型预测控制的影响


文献中经常通过构造类李雅普诺夫不等式来证明随机模型预测控制 (MPC) 受加性扰动的稳定性，确保闭环性能边界和状态有界性，但状态和非保守性能的最终边界严格边界通常是不确定的。在本文中，我们使用输入到状态稳定性属性来查找意味着受扰非线性系统以概率 1 收敛到最小鲁棒正不变集的条件。我们讨论了随机 MPC 公式的状态收敛和控制律的含义，并证明了线性和非线性系统的几种现有随机 MPC 公式的收敛结果。