Dual Set Membership Filter with Minimizing Nonlinear Transformation of Ellipsoid

In this paper, we propose a dual set membership filter for nonlinear dynamic systems with unknown but bounded noises, and it has three distinctive properties. Firstly, the nonlinear system is translated into the linear system by leveraging a semi-infinite programming, rather than linearizing the nonlinear function. In fact, the semi-infinite programming is to find an ellipsoid bounding the nonlinear transformation of an ellipsoid, which aims to compute a tight ellipsoid to cover the state. Secondly, the duality result of the semi-infinite programming is derived by a rigorous analysis, then a first order Frank-Wolfe method is developed to efficiently solve it with a lower computation complexity. Thirdly, the proposed filter can take advantage of the linear set membership filter framework and can work on-line without solving the semidefinite programming problem. Furthermore, we apply the dual set membership filter to a typical scenario of mobile robot localization. Finally, two illustrative examples in the simulations show the advantages and effectiveness of the dual set membership filter.Comment: 26 pages, 9 figure

Rethinking the Mathematical Framework and Optimality of Set-Membership Filtering

Set-Membership Filter (SMF) has been extensively studied for state estimation in the presence of bounded noises with unknown statistics. Since it was first introduced in the late 1960s, the studies on SMF have used the set-based description as its mathematical framework. One important issue that has been overlooked is the optimality of SMF. In fact, the optimality has never been rigorously established. In this work, we put forward a new mathematical framework for SMF using concepts of uncertain variables. We first establish two basic properties of uncertain variables, namely, the law of total range (a non-stochastic version of the law of total probability) and the equivalent Bayes' rule. This enables us to put forward, for the first time, an optimal SMFing framework. Furthermore, we obtain the optimal SMF under a non-stochastic Markovness condition, which is shown to be fundamentally equivalent to the Bayes filter. Note that the classical SMF in the literature is only equivalent to the optimal SMF we obtained under the non-stochastic Markovness condition. When this condition is violated, we show that the classical SMF is not optimal and it only gives an outer bound on the optimal estimation.Comment: 8 pages, 3 figure