模型预测控制的内在鲁棒性研究

Inherent robustness is the property of uncertainty tolerance of nominal stable systems, i.e., nominal stable systems can guarantee robustness with respect to specific classes of uncertainties or disturbances. This project will study the inherent robustness of nominal model predictive control (MPC) of nonlinear systems, including recursive feasibility of the related optimization problem and robust stability of systems under control..Inherent robustness of MPC has been studied by us for continuous time nonlinear systems with input constraints and terminal constraints. First, we will discuss the properties of the feasible set of MPC, in particular, the property of compactness and distance of the adjacent feasible sets. With the properties, the upper bound of the admissible disturbances will be estimated so as to guarantee recursive feasibility. Together with inherent robustness of MPC with input constraints and terminal constraint, inherent robustness of MPC with state constraints, input constraints and terminal constraints will be discussed. .Suppose that the involved optimization problem has a feasible solution at the initial time instant for discrete time systems. Assume the shifted control sequence with the terminal control law is a feasible solution with respect to disturbances or uncertainties at the next time instant. Then, the existence of a continuously upper bound on the optimal cost function of nominal model predictive control will be considered. Inherent robustness can be deduced directly since the considered systems has a candidate Lyapunov function which is continuous in system states. .Thirdly, for linear systems with stochastic disturbances, we will estimate the upper bound of the admissible covariance of the disturbances. Recursive feasibility can be guaranteed in probability if the covariance of the actual disturbance is less than the upper bound of the admissible covariance. For this case, robust stability is trivial since the optimal cost function of linear systems is continuous in system states. .This project is very important since the occurrence of the uncertainties or disturbances cannot be avoided and they might destroy the recursive feasibility in some scenarios. On one hand, the design of MPC with guaranteed robust stability and robust performance is still difficult at this time of day. On the other hand, the existing schemes already known are also very conservative. The study will permit us to design a nominal MPC if only minor disturbances will appear. It not only has the important academic value, but also will promote the popularization and application of model predictive control.

内在鲁棒性是指保证名义稳定性的系统 “自动” 具有的抑制扰动和不确定性的能力。本项目研究具有控制约束、状态约束和终端约束的保证名义稳定性的非线性预测控制算法的内在鲁棒性，包括优化问题的可行性和系统的鲁棒稳定性。对于连续时间系统，我们已经证明了具有控制约束和终端约束的预测控制算法的内在鲁棒性。首先研究预测控制算法的可行集的性质，包括可行集的紧性和两个相邻可行集的最小距离。利用两个相邻可行集之间的最小距离，以保证优化问题可行为条件估算允许扰动的数学特征。对于连续时间系统结合考虑具有控制约束和终端约束的预测控制算法内在鲁棒性结论，证明受控系统具有鲁棒稳定性。对于离散时间系统，证明存在关于系统状态连续的优化问题的值函数的上界，利用这个上界函数推断系统的鲁棒稳定性。证明了预测控制算法的内在鲁棒性，就可以忽略小扰动和不确定性的影响，直接设计保证稳定性的控制器。因而本项目具有重要的学术价值和应用价值。

本项目研究预测控制的内在鲁棒性。证明了保证名义稳定性的预测控制策略具有内在鲁棒性（优化问题中考虑控制约束、状态约束和终端约束）；提出了保证稳定性和满足控制约束、控制增量约束、状态约束和终端约束的非线性预测控制策略。提出了具有幅值有界扰动的线性系统最小鲁棒不变集的设计方法。研究预测控制的鲁棒性综合。提出了具有机会约束的系统的预测控制策略，算法可以保证机会约束满足和受控系统的均方稳定。针对具有慢变、幅值有界的扰动的系统，提出了基于扰动观测器的线性系统鲁棒预测控制策略，扰动观测器估计并且补偿扰动对系统动态的影响，预测控制保证约束满足并且优化系统的性能。提出具有扰动观测器的非线性预测控制策略，在设计预测控制器时用扰动观测值代替扰动值来预测系统的动态，并且用基于扰动观测器的非线性预测控制策略解决轮式移动机器人的路径跟踪控制和机械臂的位置控制。