Stochastic stability of a modified unscented Kalman filter with stochastic nonlinearities and multiple fading measurements

doi:10.1016/j.jfranklin.2016.10.028

Journal of the Franklin Institute

Volume 354, Issue 2, January 2017, Pages 650-667

https://doi.org/10.1016/j.jfranklin.2016.10.028 Get rights and content

Abstract

Stochastic stability of a modified unscented Kalman filter is investigated for a class of nonlinear systems with stochastic nonlinearities and multiple fading measurements. Note that the stochastic nonlinearities are represented by statistical means which indicate multiplicative stochastic disturbances. The fading probability of fading measurements is individual for each sensor. Convergence on the modified unscented Kalman filter is established if there exists a lower bound for the mean of fading probability. Sufficient conditions are obtained to ensure stochastic stability of the modified unscented Kalman filter. The effectiveness of the filtering algorithm is verified by an illustrative numerical example.

Introduction

Filtering for nonlinear dynamic systems is an important research area which has attracted considerable interest, for the reason that almost all practical systems involve nonlinearities of one kind or another. During the past several decades, extended Kalman filter (EKF) is widely used to solve problems on nonlinear state estimation [10]. However, there exist some well-known limitations due to linearization technique, such as the difficulties in implementing and tuning [17]. Unscented transformation (UT) has been used to substitute linearization in unscented Kalman filter (UKF) which is an improvement for EKF [3]. The second-order accuracy of Taylor expansions is obtained by the UKF, but only the first-order accuracy can be got by the linearization technique. Due to that both explicit Jacobian and Hession calculations are not necessary, the computational complexity of UKF is comparable to EKF. Performance of UKF for nonlinear stochastic discrete-time systems with linear measurement equation has been analyzed in [8]. Afterwards, stochastic stability of UKF with intermittent observations is investigated [11]. In [18], [20], adaptive UKF based on some rules is studied. In addition, there exist a multitude of promising application potentials for UKF in various practical engineering problems, such as mobile target tracking [3], [4], position determination [7], training of neural networks and so on.

In engineering practices, nonlinearity is a common feature that increases the complexity of system modeling greatly [26], [27]. Due to the fact that nonlinearity may cause oscillation or even instability, works for nonlinear systems are the main trend of research topics. In [22], a robust EKF for nonlinear systems is derived by guaranteeing an optimized upper bound on the state estimation error covariance. In [17], an error behavior of a discrete-time EKF is analyzed for general nonlinear systems, and stochastic stability of EKF is also discussed. A fault detection method using the residuals generated by UKF is proposed for nonlinear systems [21]. Note that the nonlinearities are often described by a deterministic way in aforementioned literatures, but there also exist another kind of nonlinearities, namely stochastic nonlinearities described by statistical means. The stochastic nonlinearities occur randomly due to high moving speed of tracked targets, intermittent network congestion and modification of linearized model for nonlinear systems. A lot of results on stochastic nonlinearities have been reported, refer to [19], [23], [24], [25]. A finite-horizon EKF problem is investigated for nonlinear systems with stochastic nonlinearities in [6], where the considered stochastic nonlinearities are quantified by known matrices.

In networked systems, perfect communication is not always available for the extensive application of wireless sensors and actuators [28], [29]. Fading measurements as the most continually occurred phenomena are widely investigated [5]. Missing measurements occur when the fading measurements reach an extreme case. Thereby, many results on fading measurements and missing measurements are obtained in recent years. An envelope-constrained filtering problem is investigated for time-varying stochastic systems with fading measurements [2]. Kalman filter is designed in terms of handling noisy packets occurring in wireless fading channels [13]. Furthermore, fading channels are partially compensated through control of bit-rates and power levels used by radio amplifiers, see, e.g., [14], [15], [16]. For nonlinear filtering, EKF is investigated for nonlinear systems with multiple missing measurements, where missing measurements are governed by certain probability distribution [6], but convergence on the proposed filter is not analyzed. Considering the limitations of EKF, [12] shows a modified UKF for wireless sensor networks with fading channel, and statistical convergence of the estimation error covariance is studied. However, to the best of our knowledge, UKF with stochastic nonlinearities and multiple fading measurements has not been fully investigated, which motivates us to make an effort in this paper.

In this paper, we focus on the problem of UKF-based state estimation for a class of nonlinear systems over wireless sensor network. Two phenomena of stochastic nonlinearities and multiple fading measurements are considered for the systems. Both deterministic and stochastic nonlinearities described by statistical means are taken into account. The fading probability governing the fading measurements is allowed to be any discrete distribution taking values over the interval $[β_{k}, γ_{k}]$ . A modified unscented Kalman filter (MUKF) is designed to deal with the nonlinear filtering problem. Sufficient conditions for convergence on the MUKF are derived, which indicates that the estimation error covariance is bounded when the mean of fading probability has a lower bound. Stochastic stability of the MUKF is analyzed if noise covariance and initial estimation error are small enough. Finally, a numerical example is provided to illustrate the effectiveness of the proposed nonlinear filter.

The paper is organized as follows: Section 2 is a brief introduction of current discussion on the issue. The MUKF with stochastic nonlinearities and multiple fading measurements is proposed. In Section 3, convergence and stochastic stability on the MUKF are demonstrated. In Section 4, an illustrative example is given to show the effect of the improved filtering algorithm. Some conclusions are given in Section 5. The main contributions of this paper are summarized as below:

(i)
A modified unscented Kalman filter is designed for a class of nonlinear systems with stochastic nonlinearities and multiple fading measurements.
(ii)
Sufficient conditions are obtained to ensure convergence and stochastic stability of the modified unscented Kalman filter.
(iii)
A critical value for mean of fading probability exists to ensure convergence of the proposed filter.

Notation: The notations are standard in the paper. $R^{n}$ represents the n-dimensional Euclidean space. For a matrix P, P^T and P⁻¹ denote the transpose and inverse of matrix P, respectively. $P > 0$ means that the matrix P is real symmetric and positive definite. $tr (\cdot)$ stands for the trace of a matrix and $∥ \cdot ∥$ denotes the Euclidian norm of corresponding vectors or the spectral norm of corresponding matrices. $ρ (P)$ represents its spectral radius, $E {x}$ stands for the expectation of random variable x and $E {x | y}$ presents the expectation value of x conditional on y. I and 0 represent the identity matrix and the zero matrix with appropriate dimensions, respectively. $diag {X_{1}, X_{2}, \dots, X_{n}}$ stands for a block-diagonal matrix with matrices ${X_{1}, X_{2}, \dots, X_{n}}$ on the diagonal.

Section snippets

Problem formulation

Consider a discrete time-varying system with stochastic nonlinearities, multiple fading measurements and random noises, where the structure diagram of the system is shown in Fig. 1.

The system model is represented by: $x_{k + 1} = f (x_{k}) + g (x_{k}, η_{k}) + D_{k} ω_{k}$ $y_{k} = Ξ_{k} h (x_{k}) + s (x_{k}, ζ_{k}) + ν_{k}$ where $k \in N$ is the discrete time, $N = {0, 1, \dots}$ . $x_{k} \in R^{n}$ represents the state vector to be estimated, and $y_{k} \in R^{m}$ represents the measurement output. Both η_k and ζ_k are zero-mean Gaussian disturbances. D_k is a known matrix with appropriate

Error bounds for the modified unscented Kalman filter

Theorem 1

Suppose that the linearized forms of the nonlinear system (1), (2) satisfy uniform observability condition. There exist real constants $\underset{̲}{c}$ , $\underset{̲}{e}$ , ${\underset{̲}{n}}_{2}$ , $\underset{̲}{l}$ , $\bar{r}$ , $\underset{̲}{ξ}$ , $\hat{p} > 0$ , such that the following bounds on various matrices are satisfied for every $k \geq 0$ : ${\underset{̲}{c}}^{2} I \leq C_{k} C_{k}^{T}, {\underset{̲}{e}}^{2} I \leq E_{k} E_{k}^{T}, {\underset{̲}{n}}_{2}^{2} I \leq Δ_{2, k} Δ_{2, k}^{T}$ ${\underset{̲}{l}}^{2} I \leq L_{k} L_{k}^{T}, R_{k} \leq \bar{r} I$

If $\frac{\sqrt{(1 + ε_{1}) n ∥ Π_{k + 1}^{2} ∥ ∥ Γ_{k + 1}^{2} ∥}}{\underset{̲}{c} + \underset{̲}{e} {\underset{̲}{n}}_{2} \underset{̲}{l}} I < \underset{̲}{ξ} I \leq {\bar{Ξ}}_{k}$ , then ${\hat{P}}_{k + 1 | k + 1} \leq \hat{p} I$

Proof

Combining Eqs. (16), (22), it can be seen that ${\hat{P}}_{k + 1 | k + 1} = {(A_{k} + B_{k} Δ_{1, k} L_{k}) {\hat{P}}_{k | k} {(A_{k} + B_{k} Δ_{1, k} L_{k})}^{T} + E [g (x_{k}, η_{k}) g^{T} (x_{k}, η_{k})] + D_{k} Q_{k} D_{k}^{T}} - {(A_{k} + B_{k} Δ_{1, k} L_{k}) {\hat{P}}_{k}$

Numerical simulation

In this section, a simulation example is presented to illustrate the effectiveness and applicability of the MUKF. Consider a maneuvering target tracking system with stochastic nonlinearities and multiple fading measurements as follows [6]: $x_{k + 1} = f (x_{k}) + g (x_{k}, η_{k}) + D_{k} ω_{k} y_{k} = Ξ_{k} h (x_{k}) + s (x_{k}, ζ_{k}) + ν_{k}$ where $f (x_{k}) = [\begin{matrix} 0.8 x_{k}^{1} + x_{k}^{1} x_{k}^{2} \\ 1.5 x_{k}^{2} - x_{k}^{1} x_{k}^{2} \end{matrix}], D_{k} = [\begin{matrix} 0.01 \\ 0.03 \end{matrix}] h (x_{k}) = 7.5 \sin (x_{k}^{2})$ where $x_{k}^{1}$ and $x_{k}^{2}$ represent the position and velocity of the target, respectively. $ω_{k} \in R$ and $ν_{k} \in R$ stand for Gaussian white noises with means 0

Conclusions

In this paper, the UKF-based nonlinear filtering problem has been investigated for general nonlinear systems with stochastic nonlinearities and multiple fading measurements. Sufficient conditions for the convergence on the MUKF have been established based on the existence of an upper bound for estimation error covariance when the mean of the fading probability has a lower bound. Stochastic stability of the filter has been proved by analyzing the boundedness of estimation error. Moreover, it has

Acknowledgements

The authors would like to thank the anonymous reviewers for their detailed comments which helped to improve the quality of the paper. The work was supported by the National Natural Science Foundation of China (61573301, 61473161, 61403330, 61403125), the Postdoctoral Science Foundation of China (2014M551052), the Science Fund for Distinguished Young Scholars of Hebei Province (F2016203148), the Natural Science Fund of Hebei Provincial (F2015203163), the Natural Science Foundation of Hebei

References (29)

D. Ding et al.
Envelope-constrained $H_{\infty}$ filtering with fading measurements and randomly occurring nonlinearitiesthe finite horizon case
Automatica
(2015)
G. Hu et al.
A derivative UKF for tightly coupled INS/GPS integrated navigation
ISA Trans.
(2015)
J. Hu et al.
Recursive filtering with random parameter matrices, multiple fading measurements and correlated noises
Automatica
(2013)
J. Hu et al.
Extended Kalman filtering with stochastic nonlinearities and multiple missing measurements
Automatica
(2012)
L. Li et al.
Stochastic stability of unscented Kalman filter with intermittent observations
Automatica
(2012)
L. Li et al.
UKF-based nonlinear filtering over sensor networks with wireless fading channel
Inf. Sci.
(2015)
D. Quevedo et al.
On Kalman filtering over fading wireless channels with controlled transmission powers
Automatica
(2012)
G. Wei et al.
Robust filtering with stochastic nonlinearities and multiple missing measurements
Automatica
(2009)
E.E. Yaz et al.
State estimation of uncertain nonlinear stochastic systems with general criteria
Appl. Math. Lett.
(2001)
N. Hou et al.
Non-fragile state estimation for discrete Markovian jumping neural networks
Neurocomputing
(2016)

Y. Yu et al.

Design of non-fragile state estimators for discrete time-delayed neural networks with parameter uncertainties

Neurocomputing

(2016)

R. Bhatia, Matrix Analysis. Graduate Texts in Mathematics,...

S. Julier, J. Uhlmann, H.F. Durrant-Whyte, A new approach for filtering nonlinear system, in: Proceedings of the...

S. Julier, The scaled unscented transformation, in: Proceedings of the American Control Conference, Anchorage, AK,...

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Stochastic stability of a modified unscented Kalman filter with stochastic nonlinearities and multiple fading measurements

Abstract

Introduction

Section snippets

Problem formulation

Error bounds for the modified unscented Kalman filter

Numerical simulation

Conclusions

Acknowledgements

Automatica

ISA Trans.

Automatica

Automatica

Automatica

Inf. Sci.

Automatica

Automatica

Appl. Math. Lett.

Neurocomputing

Neurocomputing