Nonuniform sampling Kalman filter for networked systems with Markovian packets dropout

Abstract

This paper focuses on a state estimation problem on networked systems with Markovian packets dropout. An event-based nonuniform sampling scheme is applied in intelligent samplers to save resources of the samplers and networks. Another sampling scheme combined with time-trigger and event-trigger is applied in a Kalman filter to detect the packets dropout. A delta operator Kalman filter is designed for the nonuniform sampling networked system. Two sufficient conditions of peak covariance stability and usual covariance stability are given to guarantee convergence of the delta operator Kalman filter. Numerical examples are shown to illustrate effectiveness of the developed techniques.

Introduction

Kalman filter has played a significant role in state estimation problems since it was proposed in 1960 [1]. Filtering for networked systems is an important research area which has attracted considerable interests owing to rapid development of networks [2]. For the filtering of networked systems, packets dropout is usually happened in the process of networked transmission, which influences the performance of networked Kalman filters [3]. It means that packets dropout should be considered in the process of networked filters designing. A Kalman filter has been designed and analyzed with stochastic stability for a linear system subject to Bernoulli packets dropout in [4]. A notion of peak covariance stability has been proposed to analyze stability of Kalman filters subject to Markovian packets dropout in [5]. The idea has been extended to unscented Kalman filtering problems with Markovian packets dropout in [6], [7]. A composite disturbance-observer-based control problem has been studied for a Markovian jump nonlinear system in [8]. In [9], a disturbances observer has been designed for nonlinear Markovian jump systems with multiple disturbances. Saving communication energy and bandwidth of networks are also interesting problems for the filtering of networked systems. Event-trigger communication mechanisms are widely adopted in networked systems for saving limited communication energy and network bandwidth [10], [11]. In [12], [13], a send-on-delta scheme based on output has been applied on networked Kalman filters. However, the send-on-delta scheme is not triggered in a long time for some low amplitude shaking signals, which leads to poor sampling effects [14]. To modify the scheme, a send-on-area scheme has been proposed based on difference accumulation of signal [15]. All the schemes have a defect that resources of samplers are wasted in the process of event-trigger communication. Therefore, it is an interesting idea to obtain an event-based sampling scheme which saves resources of networks and samplers. In addition, the problem of packets dropout is not considered among the above mentioned literatures of event-trigger. However, the packets dropout is an inevitable problem in practical networked systems. Therefore, it is a challenge to design an event-based networked Kalman filter on networked systems with packets dropout.

It is well known that discrete systems are obtained by sampling continuous systems and the shorter the sampling period is, the better the system performances are [16]. However, parameters in traditional shift operator systems not tend to the ones in corresponding continuous systems when sampling frequencies are gradually increased, the problem usually leads to poor performances and instability of control systems [17]. To solve this problem, Goodwin used delta operator to sample continuous systems with small periods [18]. Delta operator systems are widely studied due to numerical advantages in fast sampling [19], [20]. A delta operator Kalman filter has been proposed and the convergence has also been analyzed for the filter in [21]. Delta operator systems also have another advantage that sampling periods and sampling times are explicit parameters [22]. Therefore, delta operator is a powerful tool to analyze systems with nonuniform sampling periods. In networked systems, all signals are sampled by samplers, and sampling rates for each signals may be varying according to actual situations [23]. The kind of systems are called nonuniform sampling networked systems that have been widely studied in past years [24], [25], [26]. The delta operator method has been extensively studied in control systems especially for networked systems. However, to the best of our knowledge, the delta operator Kalman filter for nonuniform sampling networked systems has not been investigated yet.

Motivated by the above observations, in this paper, a state estimation problem is studied for networked systems subject to Markovian packets dropout. An event-based nonuniform sampling scheme is proposed to save the network bandwidth. Another nonuniform sampling scheme combined with event-trigger and time-trigger is applied in Kalman filters to detect packets dropout. Moreover, a nonuniform sampling delta operator Kalman filtering algorithm is introduced subject to the packets dropout. Convergence of the filter is proved by analyzed boundness of estimation error covariance. At last, two numerical examples are given to illustrate the feasibility and effectiveness of the developed technique.

The rest of the paper is organized in the following. Section 2 formulates the delta operator networked systems and the nonuniform sampling schemes. Section 3 gives main results for the delta operator networked Kalman filter and its convergence analysis. In Section 4, two numerical examples are studied to show effectiveness of proposed results. This paper is concluded by Section 5.

Notation: In the sequel, if not explicitly stated, matrices are assumed to have compatible dimensions. Rⁿ denotes the n-dimensional Euclidean space. The set of natural numbers and the set of positive integers are denoted by N and $Z^{+}$. The notation X > Y (X ≥ Y) means that the matrix $X-Y$ is positive definite ($X-Y$ is semi-positive definite, respectively). I is the identity matrix of appropriate dimension. For any matrix A, A^T denotes the transpose of matrix A, $A^{-1}$ denotes the inverse of matrix A. ‖ · ‖ denotes the norm for vectors or the spectral norms of matrices. $E\{·\}$ stands for the mathematical expectation.