Particle observers for state estimation and adaptation in deterministic systems with random piecewise constant inputs

In this paper we develop a methodology for state estimation of partially contracting systems inspired by the particle filter (PF). If a system is partially contracting, in the sense that once some state variables are assumed known and viewed as inputs, the remaining reduced system is contracting, then only those variables need to be estimated. Indeed, once those variables are known, the remaining variables are asymptotically recovered owing to the contraction properties. To estimate the state of such a system, we thus suggest to use a PF approach to estimate only the non-contracting part, whereas the other part is naturally recovered. The methodology is applied to a nontrivial tracking problem. The trajectories of objects one seeks to track, such as aircraft and marine vehicles, typically consist of smooth sections with large, but infrequent, unpredictable changes. As a result, their motion is well modeled by nonlinear deterministic ordinary differential equations driven by piecewise constant random control inputs. Following our methodology, we combine a type of PF called the variable rate particle filter (VRPF) with a bank of observers assumed to have convergence properties. Using the notion of virtual system, we devise a hybrid variable rate particle observer, that uses the particles to detect changes, and the observers for state estimation in between changes.