Approximate Set-Valued Observers for Nonlinear Systems

A set-valued observer (SVO) produces a set of possible states based on output measurements and a priori models of exogenous disturbances and noises. Previous work considered linear time-varying systems and unknown-but-bounded exogenous signals. In this case, the sets of possible state vectors take the form of polytopes whose centers are optimal state estimates. These polytopic sets can be computed by solving several small linear programs. An SVO can be constructed conceptually for nonlinear systems; however, the set of possible state vectors no longer takes the form of polytopes, which in turn inhibits their explicit computation. This paper considers an “extended SVO.” As in the extended Kalman filter, the state equations are linearized about the state estimate, and a linear SVO is designed along the linearization trajectory. Under appropriate observability assumptions, it is shown that the extended SVO provides an exponentially convergent state estimate in the case of sufficiently small initial condition uncertainty and provides a nondivergent state estimate in the case of sufficiently small exogenous signals.