For batch processes with partial actuator failures and unknown system dynamics, an innovative two-dimensional (2D) model-free Q-learning algorithm is proposed to obtain the optimal controller's gains, achieving output feedback fault-tolerant control. First, a 2D linear model is constructed to describe batch processes with partial actuator failures. Then, the state increments in the batch direction and the output errors in the time direction are used as novel state variables to construct a multi-degree-of-freedom model. Second, a 2D Bellman equation is proposed through a connection between a 2D value and a 2D Q functions. Next, a 2D off-policy model-free Q-learning algorithm is highlighted, which incorporates target policies into a multi-degree-of-freedom model and focuses on using policy iteration to solve the fault-tolerant tracking control problem. The robustness analysis rigorously proves the stability of the closed-loop system. Lastly, the simulation results of the holding stage prove the feasibility and effectiveness of the presented algorithm.

对于具有部分执行器故障和未知系统动力学的批处理，提出了一种创新的二维（2D）无模型Q学习算法来获得最优控制器增益，实现输出反馈容错控制。首先，构建二维线性模型来描述具有部分执行器故障的批处理过程。然后，将批量方向上的状态增量和时间方向上的输出误差用作新的状态变量来构造多自由度模型。其次，通过二维值和二维Q函数之间的联系提出了二维贝尔曼方程。接下来，重点介绍一种2D离策略无模型Q学习算法，该算法将目标策略纳入多自由度模型中，并专注于使用策略迭代来解决容错跟踪控制问题。鲁棒性分析严格证明了闭环系统的稳定性。最后，保持阶段的仿真结果证明了该算法的可行性和有效性。