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.
B. Talaei et al., "Boundary Control of Linear Uncertain 1-D Parabolic PDE using Approximate Dynamic Programming," IEEE Transactions on Neural Networks and Learning Systems, vol. 29, no. 4, pp. 1213-1225, Institute of Electrical and Electronics Engineers (IEEE), Apr 2018.
The definitive version is available at https://doi.org/10.1109/TNNLS.2017.2669944
Electrical and Computer Engineering
Mathematics and Statistics
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)
Article - Journal
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