A Game Theoretic Algorithm to Solve Riccati and Hamilton—Jacobi—Bellman—Isaacs (HJBI) Equations in H ∞ Control

Abstract

In this chapter, we propose a new algorithm to solve Riccati equations and certain Hamilton—Jacobi—Bellman—Isaacs (HJBI) equations arising in $$H_{\infty}$$ control. The need for the algorithm is motivated by the existence of $$H_{\infty}$$ problems for which standard Riccati solvers break down, but which can be handled by the algorithm. By using our algorithm, we replace the problem of solving $$H_{\infty}$$ Riccati equations or HJBI equations by the problem of solving a sequence of H 2 Riccati equations or Hamilton—Jacobi—Bellman (HJB) equations. The algorithms have some advantages such as a simple initialization, local quadratic rate of convergence, and a natural game theoretic interpretation. Some numerical examples are given to demonstrate advantages of our algorithm.