State estimation with partially observed inputs: A unified Kalman filtering approach

Abstract

For linear stochastic time-varying state space models with Gaussian noises, this paper investigates state estimation for the scenario where the input variables of the state equation are not fully observed but rather the input data is available only at an aggregate level. Unlike the existing filters for unknown inputs that are based on the approach of minimum-variance unbiased estimation, this paper does not impose the unbiasedness condition for state estimation; instead it incorporates a Bayesian approach to derive a modified Kalman filter by pooling the prior knowledge about the state vector at the aggregate level with the measurements on the output variables at the original level of interest. The estimated state vector is shown to be a minimum-mean-square-error estimator. The developed filter provides a unified approach to state estimation: it includes the existing filters obtained under two extreme scenarios as its special cases, i.e., the classical Kalman filter where all the inputs are observed and the filter for unknown inputs.

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

“…In addition, in comparison with the existing discrete-time RODOs, no assumption is made on disturbances and consequently it can obtain better disturbance estimation performance for generic disturbances. Hence it extends the applicability of the existing results in Gillijns and De Moor (2007); Darouach and Zasadzinski (1997); Li (2013); Su et al (2015a,b) to a much wider application area.…”

“…On the other hand, to relax the assumption on disturbances and incorporate noise information for stochastic systems, Gillijns and De Moor (2007) proposed a simultaneous state and disturbance observer on the basis of Darouach and Zasadzinski (1997) using the Minimum-VarianceUnbiased-Estimation (MVUE) method. 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).…”

“…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.…”

“…Consider a discrete-time linear stochastic system in the presence of disturbances (e.g., Gillijns and De Moor (2007); Darouach and Zasadzinski (1997);Li (2013)) as follows:…”

Point source estimation via finite element multiple-model Kalman filtering

“…Notice that, in equation (7), the intensity u k and the position p 0 of the point source input are unknown and hence must be estimated together with the state vector x k . As for the intensity u k , different models are possible: (a) u k is treated as an unknown input for which no information on the possible time evolution is available [15], [16]; (b) u k is unknown but a dynamic model for its time evolution is available, i.e., it is supposed that u k is generated as the output of an auxiliary linear system (called exosystem).…”