Hamiltonian techniques for the problem of set-membership state estimation



The problem of filtering under unknown input disturbances is addressed with set-membership bounds on the uncertain items. The possibility of solving this problem is considered using techniques of dynamic programming in continuous time via the related Hamilton–Jacobi–Bellman equations. The exact solutions to 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 which are given in two alternative versions. The suggested equations apply not only to the linear but also to the nonlinear case. However, in the nonlinear case the equations are especially difficult to calculate. This paper presents an alternative approach, based on a comparison principle 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 the comparison principle yields a parameterized array of ellipsoidal estimates, which ensure tight approximations of the convex information sets. It also indicates a deductive scheme for deriving these estimates in contrast with the earlier inductive schemes. Copyright © 2010 John Wiley & Sons, Ltd.