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Article

Min–Max Dynamic Programming Control for Systems with Uncertain Mathematical Models via Differential Neural Network Bellman’s Function Approximation

by
Alexander Poznyak
1,
Sebastian Noriega-Marquez
1,
Alejandra Hernandez-Sanchez
2,
Mariana Ballesteros-Escamilla
3 and
Isaac Chairez
4,*
1
Automatic Control Department, Centro de Investigacion y Estudios Avanzados del Instituto Politecnico Nacional, Ciudad de Mexico 07360, Mexico
2
Institute of Advanced Materials for the Sustainable Manufacturing, Tecnologico de Monterrey, Ciudad de Mexico 14380, Mexico
3
Centro De Innovacion Y Desarrollo Tecnologico En Computo, Instituto Politecnico Nacional, Ciudad de Mexico 07700, Mexico
4
Institute of Advanced Materials for the Sustainable Manufacturing, Tecnologico de Monterrey, Jalisco 45201, Mexico
*
Author to whom correspondence should be addressed.
Mathematics 2023, 11(5), 1211; https://doi.org/10.3390/math11051211
Submission received: 20 January 2023 / Revised: 16 February 2023 / Accepted: 24 February 2023 / Published: 1 March 2023
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
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Abstract

This research focuses on designing a min–max robust control based on a neural dynamic programming approach using a class of continuous differential neural networks (DNNs). The proposed controller solves the robust optimization of a proposed cost function that depends on the trajectories of a system with an uncertain mathematical model satisfying a class of non-linear perturbed systems. The dynamic programming min–max formulation enables robust control concerning bounded modelling uncertainties and disturbances. The Hamilton–Jacobi–Bellman (HJB) equation’s value function, approximated by a DNN, permits to estimate the closed-loop formulation of the controller. The controller design is based on an estimated state trajectory with the worst possible uncertainties/perturbations that provide the degree of robustness using the proposed controller. The class of learning laws for the time-varying weights in the DNN is produced by studying the HJB partial differential equation. The controller uses the solution of the obtained learning laws and a time-varying Riccati equation. A recurrent algorithm based on the Kiefer–Wolfowitz method leads to adjusting the initial conditions for the weights to satisfy the final condition of the given cost function. The robust control suggested in this work is evaluated using a numerical example confirming the optimizing solution based on the DNN approximate for Bellman’s value function.
Keywords: robust optimal control; artificial neural networks; approximate models; Kiefer–Wolfowitz method robust optimal control; artificial neural networks; approximate models; Kiefer–Wolfowitz method

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MDPI and ACS Style

Poznyak, A.; Noriega-Marquez, S.; Hernandez-Sanchez, A.; Ballesteros-Escamilla, M.; Chairez, I. Min–Max Dynamic Programming Control for Systems with Uncertain Mathematical Models via Differential Neural Network Bellman’s Function Approximation. Mathematics 2023, 11, 1211. https://doi.org/10.3390/math11051211

AMA Style

Poznyak A, Noriega-Marquez S, Hernandez-Sanchez A, Ballesteros-Escamilla M, Chairez I. Min–Max Dynamic Programming Control for Systems with Uncertain Mathematical Models via Differential Neural Network Bellman’s Function Approximation. Mathematics. 2023; 11(5):1211. https://doi.org/10.3390/math11051211

Chicago/Turabian Style

Poznyak, Alexander, Sebastian Noriega-Marquez, Alejandra Hernandez-Sanchez, Mariana Ballesteros-Escamilla, and Isaac Chairez. 2023. "Min–Max Dynamic Programming Control for Systems with Uncertain Mathematical Models via Differential Neural Network Bellman’s Function Approximation" Mathematics 11, no. 5: 1211. https://doi.org/10.3390/math11051211

APA Style

Poznyak, A., Noriega-Marquez, S., Hernandez-Sanchez, A., Ballesteros-Escamilla, M., & Chairez, I. (2023). Min–Max Dynamic Programming Control for Systems with Uncertain Mathematical Models via Differential Neural Network Bellman’s Function Approximation. Mathematics, 11(5), 1211. https://doi.org/10.3390/math11051211

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