Robust smoothing for continuous time uncertain nonlinear systems

This paper presents the derivation of a robust smoothing algorithm for a class of uncertain nonlinear systems. The uncertainties in the system are described in terms of an integral quadratic constraint which provides for a rich class of uncertainties. The smoothing problem is divided into two component filtering problems: a forward filtering problem and a reverse filtering problem. Each filtering problem is formulated in a set-valued state estimation framework and recast into an optimal control problem whose solution is described in terms of Hamilton-Jacobi-Bellman partial differential equations. An approximate solution is obtained for this optimal control problem by assuming a quadratic approximation for the value function. Linear approximations are used for various nonlinear functions in the system dynamics, as in the case of the linearized smoothing approach for nonlinear systems. The final recursion equations for the robust smoother consist of two sets of Riccati differential equations, filter state equations, and level shift scalar equations. One set corresponds to the forward filter while the other is associated with the reverse filter. Although the level shift scalar equations are required to complete the definition of the set-valued state estimator, they do not affect the recursion equations for the filters.