IEEE Trans Neural Netw Learn Syst. 2015 Apr;26(4):684-96. doi: 10.1109/TNNLS.2014.2320744.
Highly dissipative nonlinear partial differential equations (PDEs) are widely employed to describe the system dynamics of industrial spatially distributed processes (SDPs). In this paper, we consider the optimal control problem of the general highly dissipative SDPs, and propose an adaptive optimal control approach based on neuro-dynamic programming (NDP). Initially, Karhunen-Loève decomposition is employed to compute empirical eigenfunctions (EEFs) of the SDP based on the method of snapshots. These EEFs together with singular perturbation technique are then used to obtain a finite-dimensional slow subsystem of ordinary differential equations that accurately describes the dominant dynamics of the PDE system. Subsequently, the optimal control problem is reformulated on the basis of the slow subsystem, which is further converted to solve a Hamilton-Jacobi-Bellman (HJB) equation. HJB equation is a nonlinear PDE that has proven to be impossible to solve analytically. Thus, an adaptive optimal control method is developed via NDP that solves the HJB equation online using neural network (NN) for approximating the value function; and an online NN weight tuning law is proposed without requiring an initial stabilizing control policy. Moreover, by involving the NN estimation error, we prove that the original closed-loop PDE system with the adaptive optimal control policy is semiglobally uniformly ultimately bounded. Finally, the developed method is tested on a nonlinear diffusion-convection-reaction process and applied to a temperature cooling fin of high-speed aerospace vehicle, and the achieved results show its effectiveness.
高度耗散非线性偏微分方程（PDE）广泛用于描述工业空间分布过程（SDP）的系统动态。本文研究了广义高耗散 SDP 的最优控制问题，并提出了一种基于神经动态规划（NDP）的自适应最优控制方法。首先，基于快照法，采用 Karhunen-Loève 分解计算 SDP 的经验本征函数（EEF）。然后，将这些 EEF 与奇异摄动技术一起用于获得一个准确描述 PDE 系统主导动态的有限维常微分方程慢子系统。随后，基于慢子系统重新制定最优控制问题，并进一步转换为求解 Hamilton-Jacobi-Bellman（HJB）方程。HJB 方程是一个非线性 PDE，已经证明无法进行解析求解。因此，通过 NDP 开发了一种自适应最优控制方法，该方法使用神经网络（NN）在线求解 HJB 方程，以逼近值函数；并提出了一种在线 NN 权重调整律，无需初始稳定控制策略。此外，通过引入 NN 估计误差，我们证明了带有自适应最优控制策略的原始闭环 PDE 系统是半全局一致最终有界的。最后，将所提出的方法应用于非线性扩散-对流-反应过程，并将其应用于高速航天飞行器的冷却散热片，结果表明其有效性。
