Zero-sum two-player game theoretic formulation of affine nonlinear discrete-time systems using neural networks

Area of Science:

• Control Theory
• Nonlinear Systems
• Optimal Control

Background:

• Discrete-time (DT) affine nonlinear control systems often face challenges with partially unknown internal dynamics and external disturbances.
• Optimal control strategies are crucial for managing these complex systems effectively.

Purpose of the Study:

• To develop a nearly optimal control solution for discrete-time affine nonlinear systems with unknown dynamics and disturbances.
• To propose a method based on successive approximations of the Hamilton-Jacobi-Isaacs (HJI) equation.

Main Methods:

• Utilizing a successive approximation approach to update control and disturbance inputs for DT nonlinear affine systems.
• Deriving sufficient conditions for the convergence of the approximate HJI solution to a saddle point.
• Employing a neural network (NN) for iterative approximation of the HJI equation and a second NN for online approximation to relax the need for full system knowledge.

Main Results:

• A closed-loop optimal NN controller is achieved through offline learning.
• The approach effectively handles partially unknown internal system dynamics.
• A numerical example demonstrates the practical effectiveness of the proposed method.

Conclusions:

• The presented method provides a robust framework for optimal control of discrete-time nonlinear systems with uncertainties.
• The integration of neural networks offers a powerful tool for approximating complex control solutions.
• This research contributes to advancing control strategies for systems with partially unknown dynamics.