An Algorithm for Solving Zero-Sum Differential Game Related to the Nonlinear <inline-formula><math display="inline"><semantics><msub><mi mathvariant="script">H</mi><mo>∞</mo></msub></semantics></math></inline-formula> Control Problem

Abstract

This paper presents an approach for the solution of a zero-sum differential game associated with a nonlinear state-feedback H∞ control problem. Instead of using the approximation methods for solving the corresponding Hamilton–Jacobi–Isaacs (HJI) partial differential equation, we propose an algorithm that calculates the explicit inputs to the dynamic system by directly performing minimization with simultaneous maximization of the same objective function. In order to achieve numerical robustness and stability, the proposed algorithm uses: quasi-Newton method, conjugate gradient method, line search method with Wolfe conditions, Adams approximation method for time discretization and complex-step calculation of derivatives. The algorithm is evaluated in computer simulations on examples of first- and second-order nonlinear systems with analytical solutions of H∞ control problem.

Keywords

• H ∞ control%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm22222222"> <mml:semantics> <mml:msub> <mml:mi mathvariant="script">H</mml:mi> <mml:mo>∞</mml:mo> </mml:msub> </mml:semantics> </mml:math> </inline-formula></named-content> control
• zero-sum differential game
• conjugate gradient method
• quasi-Newton method
• complex-step method