IEEE Trans Cybern. 2013 Dec;43(6):1641-55. doi: 10.1109/TSMCB.2012.2227253.
In this paper, the nearly optimal solution for discrete-time (DT) affine nonlinear control systems in the presence of partially unknown internal system dynamics and disturbances is considered. The approach is based on successive approximate solution of the Hamilton-Jacobi-Isaacs (HJI) equation, which appears in optimal control. Successive approximation approach for updating control and disturbance inputs for DT nonlinear affine systems are proposed. Moreover, sufficient conditions for the convergence of the approximate HJI solution to the saddle point are derived, and an iterative approach to approximate the HJI equation using a neural network (NN) is presented. Then, the requirement of full knowledge of the internal dynamics of the nonlinear DT system is relaxed by using a second NN online approximator. The result is a closed-loop optimal NN controller via offline learning. A numerical example is provided illustrating the effectiveness of the approach.
本文针对存在部分未知内部系统动态和干扰的离散时间（DT）仿射非线性控制系统，考虑了几乎最优解。该方法基于最优控制中出现的 Hamilton-Jacobi-Isaacs（HJI）方程的连续近似解。针对 DT 非线性仿射系统的控制和干扰输入的更新，提出了连续近似方法。此外，还推导出了近似 HJI 解收敛到鞍点的充分条件，并提出了一种使用神经网络（NN）近似 HJI 方程的迭代方法。然后，通过使用第二个在线 NN 逼近器，放宽了对非线性 DT 系统内部动态的完全了解的要求。结果是通过离线学习获得的闭环最优 NN 控制器。通过数值示例说明了该方法的有效性。
