Boundary Control of 2-D Burgers' PDE: An Adaptive Dynamic Programming Approach: An adaptive dynamic programming approach

Behzad Talaei; Jagannathan Sarangapani; John R. Singler

doi:10.1109/TNNLS.2017.2736786

Boundary Control of 2-D Burgers' PDE: An Adaptive Dynamic Programming Approach: An adaptive dynamic programming approach

Research output: Contribution to journal › Article › peer-review

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.

Original language	American English
Pages (from-to)	3669-3681
Number of pages	13
Journal	IEEE Transactions on Neural Networks and Learning Systems
Volume	29
Issue number	8
DOIs	https://doi.org/10.1109/TNNLS.2017.2736786
State	Published - Aug 1 2018

Keywords

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

Disciplines

Electrical and Computer Engineering
Mathematics
Statistics and Probability

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10.1109/TNNLS.2017.2736786License: CC BY

https://doi.org/10.1109/TNNLS.2017.2736786

Cite this

@article{dd27c6fd561a474c816c8018b3c8ba2f,

title = "Boundary Control of 2-D Burgers' PDE: An Adaptive Dynamic Programming Approach: 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.",

keywords = "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",

author = "Behzad Talaei and Jagannathan Sarangapani and Singler, \{John R.\}",

note = "Publisher Copyright: {\textcopyright} 2012 IEEE.",

year = "2018",

month = aug,

day = "1",

doi = "10.1109/TNNLS.2017.2736786",

language = "American English",

volume = "29",

pages = "3669--3681",

journal = "IEEE Transactions on Neural Networks and Learning Systems",

issn = "2162-2388",

number = "8",

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TY - JOUR

T1 - Boundary Control of 2-D Burgers' PDE: An Adaptive Dynamic Programming Approach

T2 - An adaptive dynamic programming approach

AU - Talaei, Behzad

AU - Sarangapani, Jagannathan

AU - Singler, John R.

PY - 2018/8/1

Y1 - 2018/8/1

N2 - 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.

AB - 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.

KW - Actuators

KW - Boundary conditions

KW - Closed loop systems

KW - Controllers

KW - Dynamical systems

KW - Estimation

KW - Mathematical models

KW - Neural networks

KW - Nonlinear dynamical systems

KW - Nonlinear equations

KW - Optimal control systems

KW - Partial differential equations

KW - Approximate dynamic programming

KW - Boundary controls

KW - Burgers' equations

KW - Optimal controls

KW - Partial Differential Equations (PDEs)

KW - Reduced order systems

KW - Stability analysis

KW - Dynamic programming

KW - 2-D partial differential equations (PDEs)

KW - Burgers' equation

KW - PDE boundary control

UR - https://scholarsmine.mst.edu/ele_comeng_facwork/3629

UR - https://doi.org/10.1109/TNNLS.2017.2736786

U2 - 10.1109/TNNLS.2017.2736786

DO - 10.1109/TNNLS.2017.2736786

M3 - Article

C2 - 28866603

SN - 2162-2388

VL - 29

SP - 3669

EP - 3681

JO - IEEE Transactions on Neural Networks and Learning Systems

JF - IEEE Transactions on Neural Networks and Learning Systems

IS - 8

ER -

Boundary Control of 2-D Burgers' PDE: An Adaptive Dynamic Programming Approach: An adaptive dynamic programming approach

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