The concept of approximate dynamic programming and adaptive critic neural network based optimal controller is extended in this study to include systems governed by partial differential equations. An optimal controller is synthesized for a dispersion type tubular chemical reactor, which is governed by two coupled nonlinear partial differential equations. It consists of three steps: First, empirical basis functions are designed using the "Proper Orthogonal Decomposition" technique and a low-order lumped parameter system to represent the infinite-dimensional system is obtained by carrying out a Galerkin projection. Second, approximate dynamic programming technique is applied in a discrete time framework, followed by the use of a dual neural network structure called adaptive critics, to obtain optimal neurocontrollers for this system. In this structure, one set of neural networks captures the relationship between the state variables and the control, whereas the other set captures the relationship between the state and the costate variables. Third, the lumped parameter control is then mapped back to the spatial dimension using the same basis functions to result in a feedback control. Numerical results are presented that illustrate the potential of this approach. It should be noted that the procedure presented in this study can be used in synthesizing optimal controllers for a fairly general class of nonlinear distributed parameter systems.

Meeting Name

International Joint Conference on Neural Networks, 2003


Mechanical and Aerospace Engineering

Keywords and Phrases

Galerkin Method; Galerkin Projection; Adaptive Critic Neural Network; Approximate Dynamic Programming; Chemical Reactors; Closed Loop Systems; Control System Synthesis; Dual Neural Network Structure; Dynamic Programming; Feedback Control; Low Order Lumped Parameter System; Lumped Parameter Control; Lumped Parameter Networks; Neurocontrollers; Nonlinear Differential Equations; Nonlinear Partial Differential Equations; Optimal Control; Optimal Neurocontrollers; Orthogonal Decomposition; Partial Differential Equations; Process Control; Tubular Chemical Reactor

International Standard Serial Number (ISSN)


Document Type

Article - Conference proceedings

Document Version

Final Version

File Type





© 2003 Institute of Electrical and Electronics Engineers (IEEE), All rights reserved.

Publication Date

01 Jan 2003