Data-based Suboptimal Neuro-control Design with Reinforcement Learning for Dissipative Spatially Distributed Processes

Abstract

For many real complicated industrial processes, the accurate system model is often unavailable. In this paper, we consider the partially unknown spatially distributed processes (SDPs) which are described by general highly dissipative nonlinear partial differential equations (PDEs) and develop a data-based adaptive suboptimal neuro-control method by introducing the thought of reinforcement learning (RL). First, based on the empirical eigenfunctions computed with Karhunen−Loeve decomposition, singular perturbation theory is used to derive a reduced-order model of an ordinary differential equation that represents the dominant dynamics of the SDP. Second, the Hamilton−Jacobi−Bellman (HJB) approach is used for the suboptimal control design, and the thought of policy iteration (PI) is introduced for online learning of the solution of the HJB equation, and its convergence is established. Third, a neural network (NN) is employed to approximate the cost function in the PI procedure, and a NN weight tuning algorithm based on the gradient descent method is proposed. We prove that the developed online adaptive suboptimal neuro-controller can ensure that the original closed-loop PDE system is semiglobally uniformly ultimately bounded. Finally, the developed data-based control method is applied to a nonlinear diffusion-reaction process, and the achieved results demonstrate its effectiveness.

Data-Driven <inline-formula> <tex-math notation="LaTeX">$H_\infty$ </tex-math></inline-formula> Control for Nonlinear Distributed Parameter Systems

“…KLD has already been applied to compute orthogonal EEFs for PDE systems [8], [17], [45], [46]. Here, we omit its derivation and give the implementation procedure simply in Algorithm 1.…”

“…From Lemma 1 and Theorem 1, the H ∞ modal feedback control policy (18) relies on the solution of the HJI equation (17). However, the HJI equation (17) is a nonlinear PDE that is difficult to be solved analytically.…”

“…So far, many results [1]- [3], [14]- [17] have been reported to investigate the control problem of DPSs. For most of the real PDE systems, the existence of external disturbances is usually unavoidable.…”

Off-Policy Reinforcement Learning for <inline-formula> <tex-math notation="LaTeX">$ H_\infty $ </tex-math></inline-formula> Control Design

“…In the past few years, many RL approaches [5]- [23] have been introduced for solving the optimal control problems. Especially, some extremely important results were reported by using RL for solving the optimal control problem of discrete-time systems [7], [10], [14], [17], [18], [22].…”

“…In [19], an online neural network (NN) based decentralized control strategy was developed for stabilizing a class of continuous-time nonlinear interconnected large-scale systems. In addition, it worth mentioning that the thought of RL methods have also been introduced to solve the optimal control problem of partial differential equation systems [6], [12], [15], [16], [23]. However, for most of practical real systems, the existence of external disturbances is usually unavoidable.…”