A comparison principle for equations of the Hamilton-Jacobi type in set-membership filtering

Abstract

This paper gives comparison principles for first-order PDEs of the Hamilton-Jacobi-Bellman type that arise in the problem of filtering under unknown disturbances with set-membership bounds on the uncertainty. The exact solutions of this problem, given in set-theoretic terms as “information sets,” are expressed as level sets to the solutions of some specific types of the HJB equation. But these solutions require complicated calculations. This paper presents an alternative approach that avoids exact solutions in favor of their upper and lower bounds, which in many cases may suffice for solving the required problems. For systems with linear structure ellipsoidal estimates are given, which ensure tight approximations of the convex information sets.