On convergence of the unscented Kalman–Bucy filter using contraction theory

J.P. Maree, L. Imsland, J. Jouffroy
2014 International Journal of Systems Science  
Contraction theory entails a theoretical framework in which convergence of a nonlinear system can be analysed differentially in an appropriate contraction metric. This paper is concerned with utilizing stochastic contraction theory to conclude on exponential convergence of the Unscented Kalman-Bucy Filter. The underlying process and measurement models of interest are Itô-type stochastic differential equations. In particular, statistical linearisation techniques are employed in a virtual-actual
more » ... ystems framework to establish deterministic contraction of the estimated expected mean of process values. Under mild conditions of bounded process noise, we extend the results on deterministic contraction to stochastic contraction of the estimated expected mean of process state. It follows that for regions of contraction, a result on convergence, and thereby incremental stability, is concluded for the Unscented Kalman-Bucy Filter. The theoretical concepts are illustrated in two case studies. Proof. Simple algebraic manipulation for the left-hand side of relation (26) reveals where we have appliedṀ (t) = −M (t)Ṗ (t)M (t). From Lemmata 2.1 and 3.1 we have that (27) evaluates where from Lemma 2.1, we have applied the relationP T xf T (x) = f (χ(t), t)Wχ T (t). Relation (28) can further be simplified by incorporating the UKBF variance dynamics (7c). Hence, (28) evaluates Next, for an intermediate calculation, multiply the UKBF Kalman gain (7a), from right, with By applying the relationP T xh T (x) = h(χ(t), t)Wχ T (t) in the context of Lemma 2.1, it follows that (30) can be expressed as Substituting relation (31), and conjugate transpose thereof, into (29), and applying Assumption 4, implies that (27) evaluates
doi:10.1080/00207721.2014.953799 fatcat:wxwth4eybzdlreyqb5zgo43wl4