Hamilton-Jacobi-Isaacs Formulation for Constrained Input Systems: Neural Network Solution*

Abstract

In this paper we consider the H_∞ nonlinear state feedback control for constrained input systems. The constraints on the input to the system are encoded via a quasi-norm that will enable us to perform a quasi-L₂ gain analysis of the corresponding closed-loop nonlinear system. The quasi-norm allows using nonquadratic supply rates along with dissipativity theory to formulate the robust optimal control problem using Hamilton-Jacobi-Isaacs (HJI) equations. An iterative computationally efficient solution technique based on the game theoretic interpretation of the HJI equation is presented. The relation between attenuation gain and the region of asymptotic stability of the H_∞ controller is discussed from the game theoretic perspective. The solution is approximated at each iteration with a neural network over a predefined domain of the region of asymptotic stability of an initially stabilizing controller. The result is a closed-loop control based on a neural net that has been tuned a priori off-line.