Boundary Control of 2-D Burgers' PDE: An Adaptive Dynamic Programming Approach

Abstract

In this paper, an adaptive dynamic programming-based near optimal boundary controller is developed for partial differential equations (PDEs) modeled by the uncertain Burgers' equation under Neumann boundary condition in 2-D. Initially, Hamilton-Jacobi-Bellman equation is derived in infinite-dimensional space. Subsequently, a novel neural network (NN) identifier is introduced to approximate the nonlinear dynamics in the 2-D PDE. The optimal control input is derived by online estimation of the value function through an additional NN-based forward-in-time estimation and approximated dynamic model. Novel update laws are developed for estimation of the identifier and value function online. The designed control policy can be applied using a finite number of actuators at the boundaries. Local ultimate boundedness of the closed-loop system is studied in detail using Lyapunov theory. Simulation results confirm the optimizing performance of the proposed controller on an unstable 2-D Burgers' equation.

Department(s)

Electrical and Computer Engineering

Second Department

Mathematics and Statistics

Research Center/Lab(s)

Center for High Performance Computing Research

Comments

This work was supported in part by NSF under Grant ECCS1128281 and in part by the Intelligent Systems Center.

Keywords and Phrases

Actuators; Boundary conditions; Closed loop systems; Controllers; Dynamical systems; Estimation; Mathematical models; Neural networks; Nonlinear dynamical systems; Nonlinear equations; Optimal control systems; Partial differential equations; Approximate dynamic programming; Boundary controls; Burgers' equations; Optimal controls; Partial Differential Equations (PDEs); Reduced order systems; Stability analysis; Dynamic programming; 2-D partial differential equations (PDEs); Burgers' equation; PDE boundary control

International Standard Serial Number (ISSN)

2162-237X; 2162-2388

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

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

Publication Date

01 Aug 2018

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