Quadratic polynomial solutions of the Hamilton-Jacobi inequality in reliable control design

Abstract

In the design of nonlinear reliable controllers, one major issue is to solve for the solutions of the Hamilton-Jacobi inequality. In general, it is hard to obtain a closed form solutions due to the nonlinear nature of the inequality. In this paper, we seek for the existence conditions of quadratic type positive semidefinite solutions of Hamilton-Jacobi inequality. This is achieved by taking Taylor's series expansion of system dynamics and investigating the negative definiteness of the associated Hamilton up to fourth order. An algorithm is proposed to seek for possible solutions. The candidate of solution is firstly determined from the associated algebraic Riccati inequality. The solution is then obtained from the candidate which makes the the truncated fourth order polynomial of the inequality to be locally negative definite. Existence conditions of the solution are explicitly attained for the cases of which system linearization possesses one uncontrollable zero eigenvalue and a pair of pure imaginary uncontrollable eigenvalues. An example is given to demonstrate the application to reliable control design problem.