A Nonlinear Adaptive $H_{\infty }$ Optimal Control Method Without Solving HJIE: An Analytical Approach

This article presents an analytical approach to solve the infinite horizon
$H_{\infty }$
tracking control problem in nonlinear systems with unknown drift dynamics. A new quadratic cost function is presented that includes a feed-forward term and compensates the unknown nonlinearity effects in drift dynamics to improve the tracking performance. It is shown that the proposed cost function can be stated in another form. This enables us to extract the optimal solution without solving Hamilton–Jacobi–Isaacs equation (HJIE). To achieve the solution of
$H_{\infty }$
, a Riccati differential equation needs to be solved instead of solving the HJIE. The unknown drift dynamics are estimated by an adaptive neural network. The stability and disturbance attenuation of the proposed approach are studied. Since the
$H_{\infty }$
tracking control problem can be stated as a zero-sum game, it is shown that the obtained control and disturbance inputs provide a saddle point solution to the game.