Article
Globally Optimal Distributed Kalman Filtering for
Multisensor Systems with Unknown Inputs
Yali Ruan, Yingting Luo * and Yunmin Zhu
College of Mathematics, Sichuan University, Chengdu 610064, Sichuan, China; ruanyali2018@163.com (Y.R.);
ymzhu@scu.edu.cn (Y.Z.)
* Correspondence: ytluo@scu.edu.cn; Tel.: +86-138-8227-9920
Received: 10 July 2018; Accepted: 3 September 2018; Published: 6 September 2018
Abstract:
In this paper, the state estimation for dynamic system with unknown inputs modeled as an
autoregressive AR (1) process is considered. We propose an optimal algorithm in mean square error
sense by using difference method to eliminate the unknown inputs. Moreover, we consider the state
estimation for multisensor dynamic systems with unknown inputs. It is proved that the distributed
fused state estimate is equivalent to the centralized Kalman filtering using all sensor measurement;
therefore, it achieves the best performance. The computation complexity of the traditional augmented
state algorithm increases with the augmented state dimension. While, the new algorithm shows
good performance with much less computations compared to that of the traditional augmented state
algorithms. Moreover, numerical examples show that the performances of the traditional algorithms
greatly depend on the initial value of the unknown inputs, if the estimation of initial value of the
unknown input is largely biased, the performances of the traditional algorithms become quite worse.
However, the new algorithm still works well because it is independent of the initial value of the
unknown input.
Keywords:
optimal estimate; unknown inputs; distributed fusion; augmented state Kalman filtering
(ASKF)
1. Introduction
The classic Kalman filtering (KF) [
1
] requires the model of the dynamic system is accurate.
However, in many realistic situations, the model may contain unknown inputs in process or
measurement equations. The issue concerning estimating the state of a linear time-varying discrete
time system with unknown inputs is widely studied by researchers. One widely adopted approach is
to consider the unknown inputs as part of the system state and then estimate both of them. This leads
to an augmented state Kalman filtering (ASKF). Its computational cost increases due to the augmented
state dimension. It is proposed by Friedland [
2
] in 1969 a two-stage Kalman filtering (TSKF) to reduce
the computation complexity of the ASKF, which is optimal for the situation of a constant unknown
input. On the basis of the work in [
2
], it is proposed by Hsieh et al. an optimal two-stage algorithm
(OTSKF) for the dynamic system with random bias and a robust two-stage algorithm for the dynamic
system with unknown inputs in 1999 [
3
] and 2000 [
4
] respectively. It is assumed in [
3
–
5
] that the
unknown inputs were an autoregressive AR (1) process, with the two-stage algorithms being optimal
in the mean square error (MSE) sense. However, the optimality of the ASKF and OTSKF depends on
the premise that the initial value of the unknown measurement can be estimated correctly. Under the
condition of incorrect initial value of the unknown measurement, the ASKF and OTSKF will have
poor performance (see Examples 1 and 2 in Section 5), especially, when the unknown measurement
is not stationary as regarded in [
4
,
5
]. Due to the difficulty of knowing the exact initial value of the
Sensors 2018, 18, 2976; doi:10.3390/s18092976 www.mdpi.com/journal/sensors
