Boundary Control of 2-D Burgers' PDE: An Adaptive Dynamic Programming Approach
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.
B. Talaei et al., "Boundary Control of 2-D Burgers' PDE: An Adaptive Dynamic Programming Approach," IEEE Transactions on Neural Networks and Learning Systems, vol. 29, no. 8, pp. 3669-3681, Institute of Electrical and Electronics Engineers (IEEE), Aug 2018.
The definitive version is available at https://doi.org/10.1109/TNNLS.2017.2736786
Electrical and Computer Engineering
Mathematics and Statistics
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)
Article - Journal
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