A numerical computational approach of Hamilton-Jacobi-Isaacs equation in nonlinear H∞ control problems

We propose a new numerical approach to computing an approximate solution of the Hamilton-Jacobi-Isaacs Equation (HJE) developed in nonlinear H_∞ control problems. The new method is motivated by a method proposed by Lu and Doyle (1995) that derived a state dependent Riccati equation (sRE) from the HJE under some kinds of integratability constraint. They formulated a NLMI (nonlinear matrix inequality)-problem to solve a numerical solution of the sRE. However, the resulting sRE solution from their method cannot lead us to a promising HJE solution since its integratability assumption is not considered at all. Here, we propose a method to treat the constraints by means of the least-square-error approximation method. Specifically, we realize the problem by choosing the to-be-minimized square-error function in approximating a HJE solution from the sRE solution. Then, minimization of the square-error function provides us with a meaningful approximation method to determine an optimal HJE solution from the sRE solution in the sense of least square-error. We discuss the advantages of the new method in comparison with another approximation approach-Taylor series approximation. Finally, we give numerical examples of a nonlinear H_∞, control problem, and it is verified that a nonlinear H_∞ controller derived from the new approach could result more satisfactory system performance than the one from Taylor series approximation method or linear H_∞ theory