Jun Hu, Chaoqing Jia, Hui Yu, and Hongjian Liu

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This letter investigates the recursive state estimation (RSE)problem for a class of coupled output complex networks via the dynamic event-triggered communication mechanism (DETCM) and innovation constraints (ICs). Firstly, a DETCM is employed to regulate the transmission sequences. Then, in order to improve the reliability of network communication, a saturation function is introduced to constrain the measurement outliers. A new RSE method is provided such that, for all output coupling, DETCM and ICs, an upper bound of state estimation error covariance (SEEC) is presented in a recursive form, whose trace can be minimized via parameterizing the state estimator gain matrix (SEGM). Moreover,the theoretical analysis is given to guarantee that the error dynamic is uniformly bounded. Finally, a simulation example is illustrated to show the effectiveness of the proposed RSE method.

Related work: Over the past few decades, complex networks(CNs) have been applied to plentiful domains such as biological networks and power grids. In recent years, the dynamical analyses regarding the CNs have attracted increasing research attention from many researchers, particularly the state estimation issue as mentioned in [1]?[3]. It should be noted that most of the published results have been devoted to proposing the state estimation algorithms for state coupled CNs. As is known to all, the CNs can be divided into state coupled CNs and coupled output CNs (COCNs) on the basis of the type of connection among different network units. As a matter of fact, the CNs could be coupled with the observation outputs due to the unknown system states, limited network resources or channel occupations. Up to now, some results on the dynamical analyses of COCNs have been presented, see e.g., [4]. Nevertheless, very few authors have proposed the state estimation strategies for COCNs [5],especially the RSE algorithm. Hence, it is essential to further develop novel RSE method to shorten such a gap.

Compared with the traditional systems, the networked systems have some potential advantages in simple installation, high flexibility and convenient maintenance [6], [7]. However, in the networked environment, a large number of data have rights to share an identical network channel, which results in the occupation of network resources. Nowadays, the research effort has been dedicated to employing the protocol-based scheduling strategies to reduce the transmission frequency thereby improving the communication quality [8]. The well-investigated protocol schedules mainly include event-triggered protocol, stochastic protocol, round-robin protocol,etc. Among them, the event-triggered protocol is a prevalent one to govern the transmission sequences in the process of developing the state estimation strategy [9], [10]. According to the triggered conditions, the event-triggered scheme can be categorized into the static event-triggered communication mechanism and the DETCM[11], [12]. As an extended version of static event-triggered situation,the DETCM has been widely applied to deal with the design issue of state estimation strategy for CNs. For example, [13] has developed a DETCM-basedH∞approach to handle the state estimation problem for CNs, where an auxiliary variable has been adopted to describe the scheduling criterion and the corresponding analysis has been provided to reveal the impact from communication constraint onto the estimation performance.

On another research frontier, the observation output may be contaminated due to sudden environmental changes or malicious cyber attacks. It should be pointed out that plenty of results have been reported on the design of state estimation strategy subject to malicious attacks [14], [15]. On the contrary, only a few references focus on the problem of state estimation with outliers. With the purpose of enhancing the reliability of the observation outputs, it is very essential to develop an outlier-resistant state estimation method for dynamical systems. Up to now, a saturation function has generally been employed to constrain the measurement outliers [16].For instance, [17] and [18] have presented the Kalman-type state estimation algorithms for networked systems with ICs, where the certain saturation level and the adaptive saturation threshold have been respectively adopted to attenuate the impacts of outliers. It is worth mentioning that the relevant research on the outlier-resistant RSE strategy in the context of CNs has not gained adequate attention,which is one of the current research motivations.

The main difficulty and challenge encountered in the design of RSE strategy can be listed as: 1) How to attenuate the effect caused by the measurement outliers properly in the framework of RSE algorithm; 2) How to handle the DETCM and ICs simultaneously in the design of RSE algorithm for a class of COCNs; and 3) How to provide a sufficient condition to ensure that the estimation error is bounded uniformly. The main contributions can be summarized as follows. a) We make the first attempt to develop an optimized RSE scheme for COCNs subject to DETCM and ICs. b) A minimized upper bound of SEEC is provided by parameterizing the SEGM. c) A sufficient condition is given to ensure that the error dynamic is uniformly bounded.

Problem statement: In this letter, we consider the following class of COCNs withNnodes:

Table 1.The Comparisons of log(MSE) With Different T

The simulation results are depicted in Fig. 1. The curves of actual state and estimation are presented in Figs. 1(a)?1(c). The sum of upper bounds under 200 iterations with and without saturation constraints are shown in Fig. 1(d). Similarly, the curves of log(MSE)under 200 iterations with and without saturation constraints are presented in Fig. 1(e). According to the comparison, it can be confirmed that the developed state estimation algorithm performs well than the strategy without saturation constraint scheme.Moreover, Fig. 1(f) draws the log(MSE) and the upper bound.

Fig. 1. The illustration of proposed RSE method.

Conclusions: This letter has investigated the DETCM-based RSE problem for a class of COCNs with ICs. Firstly, the DETCM has been adopted to adjust the mode of data transmission, thereby avoiding unnecessary energy consumption. Then, an attractive saturation constraint method has been introduced to attenuate the measurement outliers. An RSE approach to COCNs has been developed such that, for both DETCM and ICs, the trace of SEECUB is minimized via designing the SEGM properly. In addition, a theoretical result in regard to the mean-square uniform boundedness of error dynamic has been analyzed in detail.

Acknowledgments: This work was supported by the National Natural Science Foundation of China (12171124, 72001059), the Natural Science Foundation of Heilongjiang Province of China(YQ2020A004), the University Nursing Program for Young Scholars with Creative Talents in Heilongjiang Province of China(UNPYSCT-2020186), and the Alexander von Humboldt Foundation of Germany.