Boundary Control of Linear Uncertain 1-D Parabolic PDE using Approximate Dynamic Programming


This paper develops a near optimal boundary control method for distributed parameter systems governed by uncertain linear 1-D parabolic partial differential equations (PDE) by using approximate dynamic programming. A quadratic surface integral is proposed to express the optimal cost functional for the infinite-dimensional state space. Accordingly, the Hamilton-Jacobi-Bellman (HJB) equation is formulated in the infinite-dimensional domain without using any model reduction. Subsequently, a neural network identifier is developed to estimate the unknown spatially varying coefficient in PDE dynamics. Novel tuning law is proposed to guarantee the boundedness of identifier approximation error in the PDE domain. A radial basis network (RBN) is subsequently proposed to generate an approximate solution for the optimal surface kernel function online. The tuning law for near optimal RBN weights is created, such that the HJB equation error is minimized while the dynamics are identified and closed-loop system remains stable. Ultimate boundedness (UB) of the closed-loop system is verified by using the Lyapunov theory. The performance of the proposed controller is successfully confirmed by simulation on an unstable diffusion-reaction process.


Electrical and Computer Engineering

Second Department

Mathematics and Statistics

Research Center/Lab(s)

Intelligent Systems Center

Second Research Center/Lab

Center for High Performance Computing Research

Keywords and Phrases

Closed loop systems; Distributed computer systems; Distributed parameter control systems; Neural networks; Uncertainty analysis; Approximate dynamic programming; Diffusion-reaction process; Distributed parameter systems; Hamilton-Jacobi-Bellman equations; Neural network identifiers; Parabolic partial differential equations; Radial basis networks; Spatially varying coefficients; Dynamic programming; Partial differential equations; PDE Identifier

International Standard Serial Number (ISSN)

2162-237X; 2162-2388

Document Type

Article - Journal

Document Version


File Type





© 2018 Institute of Electrical and Electronics Engineers (IEEE), All rights reserved.

Publication Date

01 Apr 2018