Paper Info
Title
Approximate optimal control design for nonlinear one-dimensional parabolic PDE systems using empirical eigenfunctions and neural network.
Abstract
This paper addresses the approximate optimal control problem for a class of parabolic partial differential equation (PDE) systems with nonlinear spatial differential operators. An approximate optimal control design method is proposed on the basis of the empirical eigenfunctions (EEFs) and neural network (NN). First, based on the data collected from the PDE system, the Karhunen-Loève decomposition is used to compute the EEFs. With those EEFs, the PDE system is formulated as a high-order ordinary differential equation (ODE) system. To further reduce its dimension, the singular perturbation (SP) technique is employed to derive a reduced-order model (ROM), which can accurately describe the dominant dynamics of the PDE system. Second, the Hamilton-Jacobi-Bellman (HJB) method is applied to synthesize an optimal controller based on the ROM, where the closed-loop asymptotic stability of the high-order ODE system can be guaranteed by the SP theory. By dividing the optimal control law into two parts, the linear part is obtained by solving an algebraic Riccati equation, and a new type of HJB-like equation is derived for designing the nonlinear part. Third, a control update strategy based on successive approximation is proposed to solve the HJB-like equation, and its convergence is proved. Furthermore, an NN approach is used to approximate the cost function. Finally, we apply the developed approximate optimal control method to a diffusion-reaction process with a nonlinear spatial operator, and the simulation results illustrate its effectiveness.
Year
DOI
Venue
2012
10.1109/TSMCB.2012.2194781
IEEE Transactions on Systems, Man, and Cybernetics, Part B
Keywords
Field
DocType
karhunen-loève decomposition (kld),optimal control,neural network (nn),hamilton-jacobi-bellman (hjb) equation,singular perturbation (sp),nonlinear parabolic partial differential equation (pde) systems,function approximation,parabolic equations,asymptotic stability,partial differential equations,approximation theory
Hamilton–Jacobi–Bellman equation,Parabolic partial differential equation,Mathematical optimization,Nonlinear system,Optimal control,Ordinary differential equation,Control theory,Mathematical analysis,Algebraic Riccati equation,Partial differential equation,Mathematics,Ode
Journal
Volume
Issue
ISSN
42
6
1941-0492
Citations 
PageRank 
References 
30
1.16
23
Authors
2
Name
Order
Citations
PageRank
Biao Luo155423.80
Huai-Ning Wu2210498.52