Improved Tobit Kalman filtering for systems with random parameters via conditional expectation

doi:10.1016/j.sigpro.2018.01.015

Signal Processing

Volume 147, June 2018, Pages 35-45

https://doi.org/10.1016/j.sigpro.2018.01.015 Get rights and content

Highlights

•
The Tobit Kalman filtering problem is, for the first time, investigated for random parameter systems via a novel conditional expectation approach.
•
The existing Tobit Kalman filtering algorithm is improved by making full use of the statistical information on measurement censoring.
•
The impacts from the censoring are clearly reflected on the covariance matrix of the filtering error.

Abstract

This paper is concerned with the Tobit Kalman filtering problem for a class of linear discrete-time system with random parameters. The elements of both the system matrix and the measurement matrix are allowed to be random variables in order to reflect the reality. The information matrix is employed to 1) derive the covariance between any two random variables; and 2) establish a novel weighting covariance formula to address the quadratic terms associated with the random matrices. A set of Bernoulli random variables is introduced to govern the censoring phenomenon on the measurement output. The conditional expectation, as a basic tool, is utilized to deal with the dependence among the random variables. Within the framework of the traditional Kalman filtering, the proposed filtering algorithm includes the information from both the random parameters and the censored measurements. A simulation example is presented to illustrate the effectiveness and applicability of the designed algorithm.

Introduction

For a few decades, the filtering or state estimation problem has long been a fundamental issue attracting an ever-increasing research attention from communities of signal processing and control engineering [5], [7], [10], [11], [18], [23], [24], [26], [27], [28], [30], [31], [34], [37], [39], [41], [42], [47], [48]. In today’s digital world, the measurement censoring [36], which results mainly from unavoidable data truncations, has now started to draw some attention due to its practical insights in many real applications such as economics [38], [40], chemistry [15] and information theory [1]. To be more specific, the sensor data available for the filters/estimators is often censored because of the limited communication capabilities, and such kind of censored measurements can typically lead to undesirable phenomena such as signal saturation, limit of detection and occlusion effects. Accordingly, it is practically significant and theoretically important to investigate the state estimation problem based on the censored measurements that have their distinctive features as summarized in [21], [40], and a well-noted challenge is how to make use of the statistical information of the censoring effect in the framework of Kalman filtering.

In the past few years, the Kalman filtering problem with Tobit-type censoring has been paid some initial attention, see e.g. [2], [3], [4], [21], [22], [29], [35], [36]. The description of Tobit-type I censoring dates back to [38] where a piecewise-linear transform has been used on the output variable with zero slope in the censored region. Based on such a description, the Tobit Kalman filter (TKF) has been originated in [35], which has proven to be a novel extension of the traditional Kalman filter to account for the censored measurements. The TKF algorithm with saturated data has been proposed in [2] where the measurements have known upper and lower limits. In [3], Monte Carlo methods have been employed to compare the performance of the TKF to that of the extended Kalman filter (EKF) and the unscented Kalman filter (UKF), and it has been shown that the TKF outperforms the EKF and the UKF in presence of censored data. In [4], a sequence of Bernoulli random variables has been employed to model the occurrence of the censored measurements and a new TKF has been designed. The TKF has been extended in [29] to linear discrete-time systems with time-correlated multiplicative measurement noises and in [17] for discrete time-varying system with fading measurements. It is worth pointing out that, despite the stirred research interests in developing TKFs, there is still much room for further improvements, for example, 1) the statistical information for the censoring could be better utilized in designing the TKF, and 2) the relationship between the censoring thresholds and the filter accuracy could be better reflected in the TKF algorithm. As such, this paper is mainly devoted to the identified improvements to be made on existing TKFs.

As is well known, a dynamical system in the real world could be subject to stochastic disturbances from multiple sources and a reasonable way of modeling such multiple stochastic effects is to formulate the system parameters as random matrices [16]. So far, the so-called random parameter system has drawn considerable research interest. For example, the unknown correspondence between echoes and targets has been represented in [32] by stochastic parameters in the time-varying measurement matrix. In [33], the KF and distributed KF fusion problems have been studied for random parameter systems. The distributed fusion estimation problem has been considered in [43] for a class of discrete-time Markovian jump linear systems with random parameter matrices and cross-correlated noises. In [44], the state estimation problem has been investigated for discrete-time Markovian jump linear systems with stochastic coefficient matrices. For some latest references in this regard, we refer the readers to [13], [20]. Nevertheless, the TKF design problem for random parameter systems has not been addressed yet and the main obstacle might be the complication in calculating the error covariance matrix due to the correlations among the large number of random parameters. Such an obstacle will be overcome in this paper.

For censored measurements, the random variable governing the censoring is dependent on the measurement output, and such a distinctive dependence leads to substantial difficulty in calculating the weighting covariance matrix associated with the random matrix. Note that the existing methods developed in [8], [19], [20] are no longer applicable when the dependence is involved. Fortunately, as an efficient tool in dealing with the dependence among random variables, the conditional expectation could be introduced to calculate the weighting covariance matrix regarding the dependent random matrices, and this provides us with a rigorous basis in designing the Tobit Kalman filter with random parameters. In response to the above discussions, in this paper, we aim to investigate the Tobit Kalman filtering problem for the discrete time-varying systems with random parameters. The main contributions of this paper are highlighted as follows: 1) the Tobit Kalman filtering problem is, for the first time, investigated for random parameter systems via a novel conditional expectation approach; 2) the existing Tobit Kalman filtering algorithm is improved by making full use of the statistical information on measurement censoring; and 3) the impacts from the censoring are clearly reflected on the covariance matrix of the filtering error.

The remaining of this paper is organized as follows. Section 1 lists the moments of the Truncated Normal distribution and the Censored Normal Variable. The Tobit Kalman filtering problem for the linear time-varying systems with random parameters is formulated in Section 2. The weighting covariance formulas associated with random matrix are provided in Section 3. The Tobit Kalman filter for the linear time-varying systems is developed in Section 4. An illustrative example is presented in Section 5 to demonstrate the effectiveness and the applicability of the proposed filtering algorithm. Finally, Section 6 draws conclusions and proposes a few future research topics.

Notation. In this paper, $R^{n}$ and $R^{n \times m}$ denote, respectively, the n-dimensional Euclidean space and the set of all n × m real matrices. M^T represents the transpose of matrix M. I denotes the identity matrix of compatible dimension. The notation X ≥ Y (respectively, X > Y), where X and Y are symmetric matrices, means that $X - Y$ is nonnegative-definite (respectively, positive definite). Symbol ⊗ stands for the Kronecker product. The shorthand $diag {N_{1}, N_{2}, \dots, N_{m}}$ denotes a block diagonal matrix whose elements are matrices $N_{1}, N_{2}, \dots, N_{m}$ . 1 denotes a special vector whose entries are all 1. Prob{E} is the probability of event E. $E {R}$ and Var(R) stand for, respectively, the expectation and variance of random matrix R. Cov(x, y) stands for the covariance of random vectors x and y. Matrices, if they are not explicitly stated, are assumed to have compatible dimensions.

Section snippets

Problem formulation

Consider the following discrete-time stochastic system: $\begin{matrix} x_{k} & = A_{k - 1} x_{k - 1} + B_{k - 1} w_{k - 1}, \\ y_{k}^{*} & = C_{k} x_{k} + v_{k} \end{matrix}$ where $x_{k} \in R^{n}$ is the state vector, $y_{k}^{*} \in R^{m}$ is the measurement output, $w_{k - 1} \in R^{n_{w}}$ and $v_{k} \in R^{m}$ are the Gaussian random vectors with zero mean and covariance $Q_{k} \in R^{n \times n}$ and $R = diag {σ_{1}^{2}, \dots, σ_{m}^{2}} \in R^{m \times m}$ $(σ_{i} > 0 \in R, \forall i = 1, \dots m),$ respectively. $B_{k - 1} \in R^{n \times n_{w}}$ is a known time-varying matrix with compatible dimension.

Without loss of generality, assume the random parameter matrices $A_{k - 1} \in R^{n \times n}$ and $C_{k - 1} \in R^{m \times n}$ to have the following structures: $\begin{matrix} A_{k - 1} = A \end{matrix}$

Formula for the weighting covariance matrix

Traditionally, the covariance matrix of a random vector $x_{k} \in R^{n}$ can be defined as $E {x_{k} x_{k}^{T}}$ and, more generally, the covariance matrix of a random matrix $p_{k} \in R^{m \times m}$ is of the form $E {p_{k} p_{k}^{T}}$ . Furthermore, if a weighting matrix W_k is considered, then one has $E {p_{k} W_{k} p_{k}^{T}},$ which is the so-called weighting covariance matrix (WCM) associated with random matrix p_k. In this section, we will present some preliminaries with respect to the WCM.

Lemma 1

[8, 19] Let the random matrices R, S and G be given by $\begin{matrix} \begin{matrix} R & : = diag {\begin{matrix} r_{1}, & r_{2}, & \dots \end{matrix} \end{matrix} \end{matrix}$

Main results

In this section, we derive the Tobit Kalman filter for the discrete-time linear systems with random parameters. The procedure is similar to the Tobit Kalman filter in [4] except extra efforts devoted to the complex computations of the gain matrix and the covariance matrices resulting from the random parameters. For this purpose, Lemma 2 and Corollary 2 will be exploited to deal with the random parameters so that the proposed filter can be executed in a recursive manner. According to [4], we

An illustrative example

In this section, a simulation example is employed to verify the effectiveness of the proposed algorithm through some necessary comparisons.

The system dynamics and the measurement output are modeled as follows: $\begin{matrix} {\bar{A}}_{k} & = [\begin{matrix} \cos (ω) & - \sin (ω) \\ \sin (ω) & \cos (ω) \end{matrix}] \end{matrix}$ where $ω = 0.005 * 2 π$ . $B_{k} = I, {\bar{C}}_{k} = I, {\tilde{A}}_{k} = 0, Υ_{k} = 0.05 I$ . All the elements of random matrix ${\tilde{C}}_{k}$ obey a normal distribution with zero means and standard deviation $σ = 0.5$ and are mutually independent. The conditional information matrix of ${\tilde{C}}_{k}$ conditioned on $p_{k} = I$ is given as

Conclusions

This paper has generalized the Tobit Kalman filtering algorithm to linear discrete-time systems with random parameters. Through the proposed weighting covariance matrix formula of the random matrix and statistical information from the truncation distribution as well as the censored random variable, a recursive filtering algorithm has been derived based on a rigorous calculation by utilizing the conditional expectation. The censored measurements and the random parameters have been addressed

References (48)

M. Basin et al.
Mean-square H_∞ filtering for stochastic systems: application to a 2DOF helicopter
Signal Process.
(2012)
B. Chen et al.
Distributed finite-horizon fusion kalman filtering for bandwidth and energy constrainded wireless sensor networks
IEEE Trans. Autom. Control
(2014)
H. Dong et al.
On H_∞ estimation of randomly occurring faults for a class of nonlinear time-varying systems with fading channels
IEEE Trans. Autom. Control
(2016)
D. Ding et al.
Event-based security control for discrete-time stochastic systems
IET Control Theory Appl.
(2016)
W.L. De Koning
Optimal estimation of linear discrete-time systems with stochastic parameters
Automatica
(1984)
H. Geng et al.
Tobit Kalman filter with fading measurements
Signal Process.
(2017)
H. Geng et al.
The linear minimum mean square error observer for multi-rate sensor fusion with missing measurements
IET Control Theory Appl.
(2014)
F. Han et al.
Local condition-based finite-horizon distributed H_∞-consensus filtering for random parameter system with event-triggering protocols
Neurocomputing
(2017)
B. Ibarz-Gabardos et al.
A kalman filter with censored data
2005 IEEE International Workshop On Intelligent Signal Processing
(2005)
S. Liu et al.
Extended kalman filtering for stochastic nonlinear systems with randomly occurring cyber attacks
Neurocomputing
(2016)

D. Liu et al.

A new framework for output feedback controller design for a class of discrete-time stochastic nonlinear system with quantization and missing measurement

Int. J. Gen. Syst.

(2016)

C. Miller et al.

Econometrica

R. Caballero-Aguila et al.

Fusion estimation using measured outputs with random parameter matrices subject to random delays and packet dropouts

Signal Process.

(2016)

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^☆: This work was supported in part by the National Natural Science Foundation of China under Grants 61329301, 61422301, 61374127 and 51575407, the Northeast Petroleum University Innovation Foundation for Postgraduate Students under Grant YJSCX2017-026NEPU, the Alexander von Humboldt Foundation of Germany, the Open Fund of the Key Laboratory for Metallurgical Equipment and Control of Ministry of Education in Wuhan University of Science and Technology under Grant 2017C01, and the China Postdoctoral Science Foundation under Grant 2017M621242.

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Improved Tobit Kalman filtering for systems with random parameters via conditional expectation☆

Highlights

Abstract

Introduction

Section snippets

Problem formulation

Formula for the weighting covariance matrix

Main results

An illustrative example

Conclusions

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IEEE Trans. Autom. Control

IET Control Theory Appl.

Automatica

Signal Process.

IET Control Theory Appl.

Neurocomputing

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Int. J. Gen. Syst.

Econometrica

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IET Control Theory Appl.

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Sci. Chin. Inf. Sci.

Decoding binary node labels from censored edge measurements: phase transition and efficient recovery

IEEE Trans. Network Sci. Eng.

Estimation of saturated data using the tobit kalman filter

2014 American Control Conference, Portland, Oregon, USA

Nonlinear estimators for censored data: a comparison of the EKF, the UKF and the tobit kalman filter

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The tobit kalman filter: an estimator for censored measurements

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Pinning controllability of autonomous boolean control networks

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Fusion estimation using measured outputs with random parameter matrices subject to random delays and packet dropouts

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