On existence, optimality and asymptotic stability of the Kalman filter with partially observed inputs

Abstract

a b s t r a c tFor linear stochastic time-varying systems, we investigate the properties of the Kalman filter with partially observed inputs. We first establish the existence condition of a general linear filter when the unknown inputs are partially observed. Then we examine the optimality of the Kalman filter with partially observed inputs. Finally, on the basis of the established existence condition and optimality result, we investigate asymptotic stability of the filter for the corresponding time-invariant systems. It is shown that the results on existence and asymptotic stability obtained in this paper provide a unified approach to accommodating a variety of filtering scenarios as its special cases, including the classical Kalman filter and state estimation with unknown inputs.

Reduced-order disturbance observer design for discrete-time linear stochastic systems

“…Note that the latter assumption can be relaxed when some certain prior information on the disturbances d k is available; interested readers may refer to Su et al (2015a), Chang (2006) for different types of prior disturbance information. The process noise ω k ∈ R n and measurement noise υ k ∈ R p are assumed to be mutually independent, and each follows a Gaussian distribution with a zero-mean vector and known…”

“…Using the under-brace notations, (12) indicates that rank(P F ) = rank(P f ), where P f has a full columnrank for |z| ≥ 1 is part of the existence condition of FODO in Darouach and Zasadzinski (1997); Su et al (2015a). Now we choose…”

“…Hence, we only have to focus on condition part (i) with the constraint on K k+1 given by condition part (iii). Since CG has a full column-rank, there exists an invertible matrix M ∈ R p×p (Su et al (2015a)) such that…”

“…In addition, A, G and C are known matrices, where G is supposed to have a full column-rank (see, Darouach and Zasadzinski (1997); Li (2013); Su et al (2015a) for the rationality of this assumption). The initial state x 0 is independent of ω k and υ k with a known meanx 0 and covariance matrix P 0 > 0.…”

“…Later, Su et al (2015b) extends the results to the case where information on the disturbances is available at an aggregate level (Li (2013)). The assumption that the states are fully estimable inevitably restricts the applications of the FODOs (see, Su et al (2015a) for the existing condition of this kind of filter). An important earlier work on RODO can be traced back to Xiong and Saif (2003) where the concept of state function observer based on Lyapunov approach was investigated for continuous-time systems.…”