Li Jie, Nagamune Ryozo, Zhang Yuhang, Li Shengbo Eben
IEEE Trans Neural Netw Learn Syst. 2025 Jan;36(1):896-910. doi: 10.1109/TNNLS.2023.3335138. Epub 2025 Jan 7.
Model error and external disturbance have been separately addressed by optimizing the definite performance in standard linear control problems. However, the concurrent handling of both introduces uncertainty and nonconvexity into the performance, posing a huge challenge for solving nonlinear problems. This article introduces an additional cost function in the augmented Hamilton-Jacobi-Isaacs (HJI) equation of zero-sum games to simultaneously manage the model error and external disturbance in nonlinear robust performance problems. For satisfying the Hamilton-Jacobi inequality in nonlinear robust control theory under all considered model errors, the relationship between the additional cost function and model uncertainty is revealed. A critic online learning algorithm, applying Lyapunov stabilizing terms and historical states to reinforce training stability and achieve persistent learning, is proposed to approximate the solution of the augmented HJI equation. By constructing a joint Lyapunov candidate about the critic weight and system state, both stability and convergence are proved by the second method of Lyapunov. Theoretical results also show that introducing historical data reduces the ultimate bounds of system state and critic error. Three numerical examples are conducted to demonstrate the effectiveness of the proposed method.
在标准线性控制问题中，通过优化确定性能已分别解决了模型误差和外部干扰问题。然而，同时处理这两者会给性能引入不确定性和非凸性，这对求解非线性问题构成了巨大挑战。本文在零和博弈的增广哈密顿 - 雅可比 - 伊萨克斯（HJI）方程中引入了一个额外的成本函数，以同时处理非线性鲁棒性能问题中的模型误差和外部干扰。为了在所有考虑的模型误差下满足非线性鲁棒控制理论中的哈密顿 - 雅可比不等式，揭示了额外成本函数与模型不确定性之间的关系。提出了一种批评家在线学习算法，该算法应用李雅普诺夫稳定项和历史状态来增强训练稳定性并实现持续学习，以逼近增广HJI方程的解。通过构造关于批评家权重和系统状态的联合李雅普诺夫候选函数，利用李雅普诺夫第二方法证明了稳定性和收敛性。理论结果还表明，引入历史数据会降低系统状态和批评家误差的最终界限。进行了三个数值例子以证明所提方法的有效性。
