Paper Info
Title
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 "infor- mation 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. 1. Introduction. The solution to many problems of state estimation and control synthesis for systems described by ODEs may be reduced to the investigation of first order PDEs of the Hamilton-Jacobi-Bellman (HJB) type and their modifications. HJB equations may also be used to calculate forward and backward reachability sets for control systems without disturbances, and the HJBI (HJB-Isaacs) equation may be used for systems with unknown but bounded disturbances (1, 2, 3, 4, 5, 6, 7). Another application for Hamiltonian techniques is the solution of the "guaran- teed" or "set-membership" filtering problem, which is to estimate the state of a dynamic system, based on observations corrupted by unknown and bounded noise with no statistical description. The solution to such a filtering problem is given by "information sets" of states consistent with the system dynamics and the available measurements. In turn, the information sets may be expressed as level sets of certain functions, called"information states," which are the solutions to special types of the HJB equation. The problem of calculating the information sets turns into one of finding reachability sets under state constraints that arrive on-line, in contrast with problems in which state constraints that are given in advance (8, 9, 10, 11, 12). Solutions to equations of the HJB type are rather difficult to calculate, and the design of computational algorithms is still under development (14, 15). However, for many applied problems one may often be satisfied with approximate solutions that �We dedicate this paper to Professor Tyrone E. Duncan on the occasion of his birthday, in
Year
DOI
Venue
2006
10.4310/cis.2006.v6.n3.a2
Commun. Inf. Syst.
Keywords
DocType
Volume
comparison principle,ellipsoidal calculus.,guaranteed estimation,set-membership uncertainty,information state,hjb equation,dynamic programming,information set,reach- ability,first order,level set,exact solution,upper and lower bounds,reachability
Journal
6
Issue
Citations 
PageRank 
3
1
0.40
References 
Authors
3
2
Name
Order
Citations
PageRank
Alexander B. Kurzhanski120425.02
Pravin Varaiya22543298.93