Zero-Sum Two-Player Game Theoretic Formulation of Affine Nonlinear Discrete-Time Systems Using Neural Networks

• Published in: IEEE transactions on cybernetics Vol. 43; no. 6; pp. 1641 - 1655
• Main Authors: Mehraeen, Shahab, Dierks, Travis, Jagannathan, S., Crow, Mariesa L.
• Format: Journal Article
• Language: English
• Published: United States IEEE 01.12.2013; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Approximation methods; Computer Simulation; Equations; Feedback; Game Theory; Games; Hamilton-Jacobi-Isaacs (HJI); Humans; Models, Statistical; Neural networks; Neural Networks (Computer); neural networks (NNs); nonlinear discrete-time (DT) systems; Nonlinear Dynamics; Nonlinear systems; Operations research; Optimal control; Taylor series
• ISSN: 2168-2267; 2168-2275
• DOI: 10.1109/TSMCB.2012.2227253

In this paper, the nearly optimal solution for discrete-time (DT) affine nonlinear control systems in the presence of partially unknown internal system dynamics and disturbances is considered. The approach is based on successive approximate solution of the Hamilton-Jacobi-Isaacs (HJI) equation, which appears in optimal control. Successive approximation approach for updating control and disturbance inputs for DT nonlinear affine systems are proposed. Moreover, sufficient conditions for the convergence of the approximate HJI solution to the saddle point are derived, and an iterative approach to approximate the HJI equation using a neural network (NN) is presented. Then, the requirement of full knowledge of the internal dynamics of the nonlinear DT system is relaxed by using a second NN online approximator. The result is a closed-loop optimal NN controller via offline learning. A numerical example is provided illustrating the effectiveness of the approach.