Education ArticleRobust extended fractional Kalman filter for nonlinear fractional system with missing measurements
Introduction
The idea of fractional calculus was firstly proposed by Leibniz and L’Hospital in 1695 [1], then in the later of 19th century, the definition of fractional derivative was expressed for the first time by Liouville and Riemann. Due to some practical systems can be modeled more accurately by using the fractional system, therefore, in recent years, much attention has been paid to investigate the dynamic properties of fractional system, such as stability analysis [2], [3], bifurcations analysis [4], and synchronization control [5], [6], [7]. Furthermore, it has been widely and successfully used in many engineering aspects [8], [9], [10], [11], [12]. For example, in [8], based on a fractional order impedance spectra model, the state of charge (SOC) of Lithium-ion battery was estimated by the fractional order Kalman filter with a high accuracy. In [9], by using a fractional calculus element, the frequency dependence was modeled successfully. In [10], by using the fractional feedback Kalman filter, the problem of vehicle tracking in video was solved, in [11], the fractional order systems in industrial automation were analyzed, and in [12], the extended fractional Kalman filter was applied to predict the air pollutant emission.
It is well known that the fractional Kalman filter (FKF) has played an important role in the application of fractional calculus, since it can accurately identify or estimate the unknown model parameters and state variables [13], [14]. In the applications of conventional FKF, the measurements from sensors are assumed complete and without packet losses [15]. However, in the practical applications, these assumptions are usually not satisfied, especially in network computing systems and navigation systems [16], [17], [18]. As expressed in [19], data dropouts inevitably occur in the transmission process from meters to the control centers, which severely affect the estimation performance and further influence the monitoring of system running status. In recent years, the missing data problem has attracted much attention, due to sensor networks have been widely deployed in various applications, and the missing data phenomena has become a very severe and common problem [20], [21], [22], [23].
In fact, in the 1960s, the estimation problem with missing measurements has already been discussed, and the model of missing data was proposed by using a binary switching sequence in [24], [25], [26]. After that, in order to deal with this important problem, lots of meaningful works have been developed. For instance, by utilizing the jump Riccati equation method, several filtering approaches have been proposed in [27], [28]. In [29], in order to estimate the state variables of discrete-time systems with missing measurements, a variance-constrained filtering was proposed. In addition, a robust finite-horizon estimator was developed in [30], and the statistical convergence of Kalman filter with missing measurements was investigated in [31]. However, it should be noted that these results can be only used for the estimation of linear integer order system with missing measurements. For the estimation problem of fractional linear system with missing measurements, an improved fractional Kalman filter was designed in [15], which can be used to estimate the states accurately even with intermittent missing measurements.
On the other hand, since most of systems confronted in the practical application are nonlinear, therefore, it is more important and meaningful to investigate the state estimation method for nonlinear system with missing measurements. During the past few years, several useful methods have been proposed to deal with this problem, in [32], by using a sequence of independent Bernoulli random variables, missing measurements in the transmission channel were modeled, then a recursive state estimation method for nonlinear system with missing measurements was proposed. In [33], based on the conventional extended Kalman filtering (EKF), a finite-horizon EKF was designed, which can deal with the stochastic nonlinearities and missing measurements of time-varying systems. In addition, several filtering methods considering the missing measurements have also been developed in [34], [35], [36], [37], [38]. Following these researches, recently, a novel stochastically resilient extended Kalman filtering addressing the sensor failures problem in nonlinear discrete systems has been developed in [20], furthermore, in [39], this method was successfully used to estimate voltage synchronization of smart power grid with noisy missing measurements, which can provide more accurate results than EKF. It is known that the state estimation of nonlinear integer order system with missing measurements has attract much attention and fully studied. However, to the best of the authors’ knowledge, until now, state estimation of nonlinear fractional system with missing measurements has not been fully studied, which is very important and challenging. In addition, the additive uncertainty in the feedback gain caused by computational or tuning errors was also neglected in the design of conventional EFKF, which maybe degrade the performance of estimation results [20], this topic is also interesting and there are few results considering this issue.
Based on the above discussions, in this work, the missing measurements phenomenon in each channel is assumed to be independent of each other with different missing probabilities, then by using a sequence of independent Bernoulli random variables, the nonlinear fractional system with missing measurements is modeled. Furthermore, in order to degrade the impact of missing measurements, a robust extended fractional Kalman filter (REFKF) is designed and proved, in which not only the missing measurements are considered, but also the additive uncertainty in the gain is taken into account, therefore, the designed robust filter is expected to be more general and practical.
The rest of this paper is organized as follows. In Section 2, some related definitions are introduced in advance, then the nonlinear fractional system model with missing measurements is presented. In Section 3, the robust extended fractional Kalman filter is developed and proved in detail. In Section 4, two numerical illustrative examples are utilized to demonstrate the effectiveness and usefulness of the proposed method. Finally, the conclusions are drawn in Section 5.
Notation : The notations used in this work are fairly standard except where otherwise stated. x ∈ RN indicates dimensional real state variables with norm where ( · )T indicates matrix transpose. D ≥ 0 denotes D is a positive semi-definite symmetric matrix, stands for the expectation of the stochastic variable x. denotes the Kronecker delta function, while and if k ≠ j. diag{·} stands for a block diagonal matrix, Prob( · ) indicates the probability of an event.
Section snippets
Problem formulation and preliminaries
In this section, the Grünwald–Letnikov definition to be used in the following derivation is introduced in advance [1]. Then, the nonlinear fractional order system with missing measurements will be modeled.
Definition 1 The fractional order difference of Grünwald–Letnikov can be expressed by the following equation
[1]
where Δ is the operator of fractional order system, n indicates the fractional order, h is the sampling interval, and k is the sampling number for which the derivative is
Main results
In this section, firstly, some related lemmas are introduced. Then, based on model (3), the locally unbiased, resilient, and minimum variance state estimator for the nonlinear fractional system is developed, where not only the missing measurements phenomenon is considered, but also the stochastic additive uncertainty in the gain are taken into account.
Lemma 1 Let A and B be n × m matrices, then, the Hadamard product of A and B can be given by
Lemma 2 Given a symmetric matrix [40]
[41]
Simulation studies
In this section, two numerical examples are provided to demonstrate the usefulness and effectiveness of the proposed robust extended fractional Kalman filter, respectively. In order to fairly evaluate the average estimation performance of all states by different methods, the notation average mean square error (AMSE) is adopted, which is defined as where k indicates the time instant, N is the number of the state variables, xj denotes the jth state.
The algorithms
Conclusions
In this paper, in consideration of missing measurements problem, by using a sequence of independent Bernoulli distributed random variables, the model of nonlinear fractional system with missing measurements has been modeled. Then, based on the nonlinear fractional Luenberger observer, a robust extended fractional Kalman filter for state estimation of nonlinear fractional system has been designed and proved, in which not only the missing measurements, but also the stochastic additive uncertainty
Acknowledgment
This work was supported in part by the National Natural Science Foundation of China under Grants 61673161 and 61603122, in part by the Natural Science Foundation of Jiangsu Province of China under Grants BK20161510 and BK20151500, in part by the 111 Project under Grant B14022, in part by the Fundamental Research Funds for the Central Universities of China under Grants 2017B13914 and 2017B655X14, in part by Postgraduate Research & Practice Innovation Program of Jiangsu Province under Grant
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