A survey on distributed filtering, estimation and fusion for nonlinear systems with communication constraints: new advances and prospects

In this paper, some recent results on the distributed filtering, estimation and fusion algorithms for nonlinear systems with communication constraints are reviewed. First, some network-induced phenomena and communication protocols in the networked environment are recalled. Second, the recent advances of distributed fusion algorithms for nonlinear systems are discussed, which include distributed fusion extended Kalman filtering, distributed fusion unscented Kalman filtering, distributed fusion cubature Kalman filtering, distributed fusion particle filtering. Subsequently, some new distributed filtering, estimation and fusion algorithms for nonlinear systems subject to communication constraints are summarized and discussed. Finally, conclusion is given and several possible research topics on distributed filtering, estimation and fusion for nonlinear networked systems are pointed out.


Introduction
During the past decades, the nonlinear filtering problem has become one of the hottest research topics. Accordingly, the nonlinear filtering methods have been widely used in various domains, such as navigation and guidance system, radar tracking, sonar ranging, satellite orbit determination and so on (Crassidis et al., 2007). With the rapid developments of communication and computer technique, the rapid progress of the nonlinear filtering technologies has been made to handle the advantages and difficulties on the estimation for various complex dynamical systems. Compared with the linear filtering algorithm, the nonlinear filtering algorithm has wide application areas due to its feature on dealing with the nonlinear characteristic for complex dynamical systems. At present, some widely used nonlinear filtering algorithms include the extended Kalman filtering (EKF) (James & Petersen, 1998;Xiong et al., 2008), Gaussian-Hermite filtering (GKF) (Ito & Xiong, 2000), divided differences filtering (DDF) (NøRgaard et al., 2000), unscented Kalman filtering (UKF) (Julier & Uhlmann, 2004), cubature Kalman filtering (CKF) (Arasaratnam & Haykin, 2009), particle filtering (PF), Bayesian filtering (BF) (Doucet et al., 2000) and other nonlinear filtering algorithms.
In recent years, with the rapid developments of the science and technology, the problems of state estimation CONTACT Jun Hu jhu@hrbust.edu.cn, hujun2013@gmail.com or filtering for networked control systems (NCSs) have attracted increasing research attention (Hu, Zhang, Kao, et al., 2019;Hu, Zhang, Yu, et al., 2019;Q. Liu et al., 2020).
Here, the NCSs refer to the control systems, where the communications among the controllers, the sensors and the actuators are linked by the network channels (Tan & Liu, 2013;Tan et al., 2012Tan et al., , 2015Tan et al., , 2018. Recently, NCSs have gradually received the research interest because of their strong advantages, such as centralized resources, low cost, easy maintenance and so forth. However, due to the influence of communication network with limited transmission capacities, the incomplete information is commonly transferred to the remote estimator side, where the network-induced phenomena and communication protocols inevitably exist in the NCSs (Hu, Liu, et al., 2020;S. Liu et al., 2020;Y. Shen et al., 2020;Zou et al., 2017). Such network-induced phenomena mainly include signal quantization, missing/fading measurements, transmission delays and so on (see, e.g. Cui et al., 2019;Fei et al., 2018;Zhang et al., 2020;Zou, Wang, Hu, & Zhou, 2020). Besides, the communication protocols generally include the event-triggered protocol, stochastic communication protocol, round-robin protocol, try-once-discard protocol, etc.
As it is well known, the distributed estimation and filtering problems have been widely discussed due to its engineering importance in various fields such as target tracking, industrial equipment monitoring and environmental monitoring. According to different performance indices, a great number of results of distributed filtering have been developed, such as distributed varianceconstrained filtering (Z. , distributed H ∞ filtering (Dong et al., 2012a;Shen et al., 2010) and distributed consensus filtering . In addition, note that the Takagi-Sugeno (T-S) fuzzy model can approximate the nonlinear systems with admissible accuracy, and then many distributed filtering and fault detection algorithms have been developed in Su et al. (2012), D. Zhang, Cai, et al. (2014) for T-S fuzzy systems over sensor networks. In fact, with the rapid developments of the electronic technology, signal detection technology, network communication technology and control technology, increasing attention has been made on the utilization of the sensor networks, which can improve the tracking accuracy of moving objects (aircraft, satellite, vehicle, ship, etc.) or the state estimation accuracy of dynamic systems. Accordingly, a large number of multi-sensor methods with different application backgrounds have been presented. Generally, there are two commonly utilized multisensor fusion methods, i.e. centralized fusion algorithm (Z. Zhai et al., 2008) and distributed fusion algorithm (Sun, 2004;Zhu et al., 2001). Therefore, a great number of research results on estimation, filtering and fusion algorithms have been proposed for networked systems over sensor networks. For more information with respect to the challenging problems of estimation, filtering and fusion, the readers can refer to the literature (Hu et al., 2016). For the challenging problems regarding the multi-sensor distributed fusion estimation, we refer the readers to the survey .
In this paper, we aim to review the research progresses of the distributed estimation, filtering and fusion algorithms for nonlinear systems with communication constraints. The rest of this paper is organized as follows. In Section 2, some communication constraints are first reviewed. In Section 3, some distributed fusion estimation and filtering algorithms for nonlinear systems are discussed. In Section 4, some latest results on distributed estimation, filtering and fusion algorithms for nonlinear systems with communication constraints are summarized and discussed. Finally, conclusion and some future research topics are mentioned in Section 5.

Communication constraints
With the rapid developments of the science and technology, the networked control technology has been widely used in many fields. Compared with the traditional control systems, NCSs are a class of system, in which the sensors, controllers and actuators are connected through a shared network. NCSs can realize resource sharing, facilitate remote operation and improve the fault-tolerant diagnosis ability of the systems. However, due to the special characteristics of the communication network, the transmission efficiency will be affected by some networkinduced phenomena in the process of network transmission. In addition, there is a need to better utilize the limited network communication resources in order to avoid the data distortion, wrong sequence, collision and so on. These above communication constraints are common in the networked environment, which are one of the important factors leading to the deterioration of system performance. Hence, we decide to discuss the communication constraints from the following two aspects. First, during the process of the network transmissions, the transmission information will be affected by some networkinduced phenomena (e.g. communication delays, missing/fading measurements, randomly occurring incomplete information , etc.), which will cause the undesirable degradations (e.g. signal distortion/missing/fading, or instability and so on). Second, due to the limited abilities of the network communication, some communication protocols are introduced and utilized in the networked systems, which can avoid the distortion and other negative effects. In this section, we will introduce some communication constraints and summarize some recent methods used to handle those communication constraints.

Communication delays
During the process of data transmission within the networked environment, the time delays inevitably exist due to the limited communication capacity (Hu et al., 2014;J. H. Park et al., 2016;S. Zhang, Wang, et al., 2014). Generally speaking, the existence of the time-delay phenomenon makes the system characteristic more complicated and may cause the instability of the systems Tong et al., 2020;Zhang et al., 2018). In the past few decades, several methods have been proposed to attenuate the conservatism induced by the time-delays, such as the discretized Lyapunov functional method (L. Mozelli et al., 2011), the model transformation methods (the Park inequality method, the Moon inequality method and the descriptor model transformation) (Fridman & Shaked, 2002;Han, 2004), the free weighting matrix method (He et al., 2007;Wu et al., 2004), the integral inequality method (M. J. Park et al., 2015;Zhu & Yang, 2008), the augmented Lyapunov functional method , the bounded method (Moon et al., 2001), the slack matrix variables method (Souza et al., 2009), the delay-fractioning approach (Hu, Wang, Niu, et al., 2012) and so on. In summary, the study of delaydependent stability analysis mainly includes two core problems, one is to reduce the conservatism by proposing new delay-dependent conditions and the other is to simplify the delay-dependent conditions from the verification purpose. Regarding the conservatism issue, the maximum admissible upper bound of delay is usually seen as the performance index. On the other hand, with respect to the simplicity problem, a series of methods mainly introduce new matrix parameters to reduce the conservatism by increasing the computation complexity and hence cannot achieve initial research objective. Therefore, it is difficult to find a globally optimal method, which has less conservatism and low computational complexity. There indeed exists the tradeoff among the above delay analysis methods and proper methods can be adopted within admissible performance requirements.
With respect to the estimation and filtering problems, a series of methods has been given for linear/nonlinear systems with time delays in the literature. For instance, in terms of the Lyapunov method and free weighting matrix approach, the robust H ∞ filtering problem has been studied in Qiu et al. (2010) for linear systems with time-varying state delay. Based on the delay-fractioning approach, the H ∞ filtering approach has been proposed in Zhao et al. (2012) for linear systems with interval timevarying delays, where the stability and less conservatism have been discussed by presenting some sufficient conditions. In addition, an improved filtering algorithm has been designed in Fridman and Shaked (2004) for linear continuous time-invariant systems with time delays by using the descriptor model transformation and Park's inequality.
In contrast to the state estimation and filtering problems for linear systems, the corresponding study for nonlinear systems has received certain research attentions. For example, the distributed estimation and filtering algorithms have been studied based on the bounding method (Zhang et al., 2010;Zhong et al., 2015), the slack matrix variables method  and the delayfractioning method (Hu et al., 2015). In L. , the problem of the distributed filtering has been investigated for nonlinear systems with discrete time-delays and distributed time-delays over sensor networks, where the sufficient conditions have been obtained based on the average dwell time technique and stochastic analysis method. In Y. Sun et al. (2019), the problem of the distributed recursive filtering has been examined for a class of discrete-time stochastic nonlinear systems with timedelays, where the nonlinearities have been addressed by T-S fuzzy models and sufficient conditions have been given to ensure the optimized performance requirement by solving Riccati-like difference equations.
It should be noticed that many results have been given with respect to the deterministic delays, while the phenomenon of time delays caused by network transmissions may be random and time-varying. Generally, two approaches have been proposed to model the random communication/transmission delays, such as Bernoulli distributed random variables and Markov chain. Accordingly, the problems of the distributed filtering, estimation and fusion have been studied for linear/nonlinear systems with random communication delays (Dong et al., 2012a;Ge & Han, 2016;Ge et al., 2014;Liang et al., 2012;Sun, 2012a;Sun & Ma, 2014;Wang et al., 2015;Yang et al., 2019). For instance, based on the Markov chain, the issues of the filtering have been discussed in Ge et al. (2014), Ge and Han (2016) for linear systems with random delays. In Dong et al. (2012a), the problem of distributed H ∞ filtering has been investigated for nonlinear systems with random communication delays modelled by Markov chain. By using Bernoulli distributed random variables, the random communication delays have been modelled and the filtering problems have been investigated in Sun and Ma (2014), Sun (2012a) for linear systems with random delays. In Liang et al. (2012), the problem of distributed state estimation has been studied for a class of nonlinear systems with time delays over sensor networks solved by Lyapunov function and stochastic analysis technique. In Liang, Shen, et al. (2011), the problem of the robust distributed state estimation has been studied for nonlinear systems with multiple stochastic communication delays. The distributed fusion problems have been addressed in Wang et al. (2015), Yang et al. (2019) for nonlinear systems with random communication delays and some effective filtering algorithms have been given.

Missing/fading measurements
When transferring the data through the network, it may lead to missing measurements (or packet dropouts/losses) due to the limited communication bandwidth (Hu, Wang, et al., 2019;. In general, the phenomenon of missing measurements has been described by the Bernoulli distributed random variables due to its simplicity and effectiveness. Recently, many results of the estimation, filtering and fusion for linear/nonlinear systems with missing measurements have been reported based on some methods. For example, the optimal H ∞ filtering method against the multiple packet dropouts has been given in Sahebsara et al. (2008) by employing the linear matrix inequality (LMI) method. In Sun, Xie, Xiao, and Soh (2008), , the optimal estimators have been constructed for linear systems with multiple packet dropouts based on the innovation analysis approach, where the optimal solutions have been given in the minimum mean-square error sense. It should be mentioned that the packet dropouts from sensors to estimator have been considered in above literature. Recently, some results have been given to handle the packet dropouts from both sides (e.g. from the sensors to the estimator and from the controller to the actuator). In Y. Liang et al. (2010), Sun (2012b) the optimal linear estimators have been designed to examine the effects from both-side packet losses based on the innovation analysis approach. In addition, the centralized fusion estimator has been proposed in Ma and Sun (2015) for linear systems with multiple packet losses.
The issues of the filtering, estimation and fusion for nonlinear systems with missing measurements have aroused many attentions and some results have been reported. For example, the H ∞ filtering issue has been studied in Shen et al. (2008) for nonlinear discrete-time stochastic systems with missing measurements, where new sufficient criterion has been given in terms of the Hamilton-Jacobi-Isaacs inequality. In Wang et al. (2013), the H ∞ filtering algorithm has been proposed for timevarying nonlinear systems with missing measurements, where the sub-optimal solutions have been given by utilizing the recursive Riccati difference equation approach. On the other hand, the distributed fusion estimation problem has been studied in Wang et al. (2015) for nonlinear systems with multi-step delays and packet losses over sensor networks, where some sufficient conditions have been given to ensure the convergence of the estimation error matrices. By using the Bernoulli distributed random variables, the phenomenon of the successive packet dropouts has been modelled in Dong et al. (2012b) and the distributed finite-horizon filtering algorithm has been presented based on LMI method for a class of nonlinear systems. In Han, Wei, et al. (2017), the distributed H ∞ consensus filtering approach has been proposed for nonlinear systems with multiple missing measurements modelled by Bernoulli distributed random variables.
On the other hand, the phenomenon of the fading measurements should also be addressed especially in the networked environment. To model the phenomenon of the fading measurements, a set of mutually independent random variables which satisfying certain probabilistic density distributions on the interval is commonly introduced. It is easy to see that the missing measurements are a special case of the fading measurements.
In the past few years, the phenomenon of fading measurements has received increasing research attention due to its practical application domains. Accordingly, the distributed filtering algorithms for nonlinear systems with fading measurements have been reported (Y. Chen et al., 2019;Q. Liu et al., 2015;Wen et al., 2016). For example, the distributed filtering problems have been studied in Wen et al. (2016), Q. Liu et al. (2015) for nonlinear systems with fading measurements over wireless sensor networks. In Y. Chen et al. (2019), the distributed nonfragile L 2 − L ∞ filtering scheme has been given for a class of discrete-time nonlinear systems with fading measurements, where the sufficient condition has been provided to guarantee the stability and disturbance attenuation performance of augmented filtering error systems by using stochastic analysis and Lyapunov function method.

Randomly occurring incomplete information
In recent years, with the rapid developments of the networked systems, the randomly occurring incomplete information (ROII) has become an important research issue that has received some special attentions. The ROII mainly includes, but are not limited to, randomly occurring uncertainties, randomly occurring nonlinearities, randomly occurring saturation, randomly occurring quantization and randomly occurring sensor delays. Up to now, many results of the distributed estimation and filtering for nonlinear systems with ROII have been reported in Z. Wang Liang, Wang, et al. (2011), Hu et al. (2015 for nonlinear systems with randomly occurring nonlinearities and missing measurements. In Dong et al. (2012a), the distributed filtering method has been given for nonlinear systems with randomly occurring quantization and packet dropouts. The distributed filtering problem has been examined in Dong et al. (2014) for nonlinear systems with randomly occurring saturation and successive packet dropouts modelled by the Bernoulli distributed random variables. In Hu et al. (2015), Dong, Wang, Bu, et al. (2016), the distributed filtering and estimation problems have been investigated for nonlinear systems with randomly occurring uncertainties and nonlinearities based on the LMI method. In Bu et al. (2018), the problem of the distributed non-fragile fault estimation has been studied for nonlinear systems with randomly occurring nonlinearities and gain variations. So far, some efficient analysis methods have been given to reveal and compensate the effects from the ROII to some extent.

Communication protocols
The communication protocols can alleviate the transmission pressure and regulate the order transmission of data packet so as to improve the communication frequency and prolong the service life of network components. Recently, some adopted communication protocols generally include the event triggering mechanisms/eventbased communication protocol, random communication protocol, round-robin protocol, and try-once-discard protocol and so on. In recent years, the issues of distributed state estimation, filtering and fusion algorithm for linear/ nonlinear systems based on communication protocols have received preliminary attention because of its practical significance and some interesting results have been found in terms of different methods. For example, the distributed set-membership filtering and estimation problems have been studied in Ge et al. (2017), Xia et al. (2016) for a class of discrete-time linear time-varying systems with event-triggered mechanism, where the sufficient conditions have been obtained based on convex optimization approach. In S. , the problem of distributed set-membership filtering has been tackled for a class of time-varying multi-rate systems with roundrobin communication protocol, where the sufficient condition of existence of the distributed filter has been presented by using the recursive linear matrix inequality method. The distributed Kalman consensus filtering problem has been investigated in Zhang and Jia (2017) for linear systems with event-triggered communication schedule over sensor networks, where the stability analysis of the estimation error has been derived based on the Lyapunov approach. By means of the vector dissipativity method, the distributed H ∞ consensus filtering problem has been studied in Han, Song, et al. (2017) for discrete time-varying linear systems with event-triggered protocol. In Ugrinovskii and Fridman (2014), the distributed H ∞ consensus estimation method has been proposed for linear systems with round-robin protocol by employing the LMI method.
Parallel to the issues of the distributed estimation and filtering for linear systems, the corresponding results under different indices have been given for nonlinear systems as in L.  Wen et al. (2018), the event-based distributed filtering scheme has been designed for a class of discrete-time state saturation systems with stochastic nonlinearities and missing measurement through the wireless sensor networks. In Hu et al. (2015), the problem of distributed estimation has been tackled for a class of nonlinear systems with ROII and event-triggered protocol. The distributed state estimation strategy has been proposed in L. Yan, Zhang, et al. (2014) for nonlinear discrete time-delay systems with event-triggered scheduling, where sufficient conditions have been obtained by solving some LMIs.
In addition, the distributed H ∞ estimation problem has been studied in Ding et al. (2015) for a class of discrete-time nonlinear systems with event-triggered mechanism. In H. Yan, Xu, et al. (2017), the eventtriggered distributed H ∞ state estimation method has been given for nonlinear systems modelled by T-S fuzzy models, where the H ∞ disturbance attenuation performance of the resulted error dynamics has been achieved by proposing the sufficient condition. In Z. , the distributed filtering method has been presented for T-S fuzzy nonlinear stochastic systems over wireless sensor networks with event-triggered mechanism, where the asymptotic stability has been ensured by using Lyapunov stability theory and an optimization solution of the algorithm has been obtained. In Q. , the problem of recursive distributed filtering has been studied for a class of nonlinear systems with dynamic event-triggered mechanism. The distributed fusion estimation problem has been discussed in L.  for nonlinear systems over sensor networks with random transmission delays and event-triggered protocols. In addition, the distributed state estimation issue has been discussed in Xu et al. (2016) for discrete-time nonlinear systems with round-robin protocol and fading channels, and the distributed resilient filtering scheme has been given in Sheng et al. (2019) for nonlinear time-varying systems over sensor networks with round-robin and stochastic communication protocol. Up to now, it should be pointed that few distributed estimation/filtering algorithms can be available for time-varying systems not to mention the optimized performance requirement Dong, Wang, Shen, et al., 2016;Dong et al., 2018Dong et al., , 2020. Thus additional effort should be devoted to solve this issue.

Distributed fusion filtering methods
In practical application, the dynamic process and measurement process of the systems are usually nonlinear. Hence, the traditional filtering algorithms cannot be used to provide satisfactory state estimation accuracy.
In recent years, some algorithms of distributed fusion filtering have been proposed for nonlinear systems, such as centralized fusion algorithm and distributed fusion algorithm. Compared with the centralized fusion algorithm, the distributed fusion algorithm has better robustness and reliability, which has been widely discussed and designed. In the following part, we introduce some literature and summarize several distributed fusion filtering algorithms for nonlinear systems.

Distributed fusion EKF
The extended Kalman filtering (EKF) algorithm has been widely used as a classic nonlinear filtering method (Hu, Wang, Gao, et al., 2012;James & Petersen, 1998;Xiong et al., 2008). In recent years, the distributed fusion approaches based on the extended Kalman filtering have attracted some research interests and several approaches have been proposed for nonlinear systems (Battistelli & Chisci, 2016;Duan et al., 2019;Li et al., 2017;Rashedi et al., 2017a;Song et al., 2019;Z. Yu et al., 2019). For example, the distributed estimation algorithm has been studied in Battistelli and Chisci (2016) for nonlinear systems by designing the consensus extended Kalman filter, where the information has been fused by the consensus method and the stability properties of the algorithm have been analysed. In Li et al. (2017), the problem of distributed EKF with nonlinear consensus estimate has been addressed for nonlinear systems over sensor network by Riccati-like difference equation method. The robust distributed EKF has been proposed in Duan et al. (2019) for discrete nonlinear systems over sensor networks, where a sufficient condition has been proposed and the upper bound of the estimation error covariance has been obtained. In the power system, the distributed estimation algorithm has been developed in Z.  for nonlinear systems based on EKF approach. The problem of the distributed EKF has been investigated in Song et al. (2019) for nonlinear systems by utilizing the Raccati-like difference equation method to handle the variance constraint. In Rashedi et al. (2017a), the distributed adaptive highgain EKF algorithms have been developed for nonlinear systems, where the stability properties of the distributed estimation approach have been discussed.

Distributed fusion UKF
In 2000, Julier et al. proposed an unscented Kalman filtering (UKF) algorithm for nonlinear systems, which was based on the unscented transformation (UT) and adopted a nonlinear filter of linear Kalman filter structure under the criterion of minimum mean squared error estimation (Julier & Uhlmann, 2004;Julier et al., 2000). Compared with the EKF, the UKF has many advantages and has been widely used in the engineering field. For example, the UKF algorithm has the same accuracy as second-order Gaussian filter and does not need to calculate the Jacobian matrix (Li & Xia, 2015;K. Ma et al., 2019;Zheng et al., 2019). However, with respect to the distributed fusion UKF for nonlinear networked systems, few papers can be found. In fact, the distributed data fusion algorithm should only use local information and keep the flexibility and scalability. In , the problem of the distributed state estimation for nonlinear systems over sensor networks has been discussed based on the UKF. The problem of the distributed unscented Kalman filtering has been studied in Li and Jia (2011) for discrete-time jump Markov nonlinear systems based on the interacting multiple model approach. By using UKF, the problem of distributed fusion estimation has been proposed in L.  for nonlinear systems, and the distributed hybrid information fusion algorithm has been given in S.  for nonlinear systems. In Yao et al. (2019), the distributed unscented summation information weighted consensus filtering method has been designed for nonlinear systems based on the UT.

Distributed fusion CKF
In 2009, Arasaratnam et al. proposed a sampling CKF algorithm for nonlinear systems, which used the cubature numerical integration criterion to approximate the posterior probability distribution in the optimal framework through the recursive process (Arasaratnam & Haykin, 2009;Arasaratnam et al., 2010). Since the Jacobian matrix is not needed to be calculated and filtering parameters are not setting in advance, hence the CKF has attracted a lot of research attentions. The CKF can be regarded as a special case of UKF, which is used to address the issue of the filtering of high-dimensional nonlinear systems. Recently, the CKF has been introduced into the multi-sensor distributed fusion systems, and some results have been reported in Q. Chen et al. (2017), Hao and Sun (2019), Y. Liu et al. (2014), Z. Zhang, Li, et al. (2019). For example, the distributed cubature filtering algorithm has been developed in Q. Chen et al. (2017) for nonlinear systems based on weighted average consensus. In Hao and Sun (2019), the distributed fusion estimation problem for nonlinear systems over sensor networks has been studied based on cubature Kalman filter. In Y. Liu et al. (2014), the distributed squared-root cubature filtering algorithm for nonlinear systems over sensor networks has been proposed by employing an improved consensus method. Very recently, in Z. Zhang, Li, et al. (2019), the distributed consensus cubature Kalman filtering algorithm has been developed for nonlinear systems, where the boundedness analysis of the filtering algorithm has been discussed.

Distributed fusion PF
In 1999, an optimal estimation algorithm based on the Bayesian principle has been proposed in Carpenter et al. (1999), which could be applied to nonlinear and non-Gaussian state space model. The particle filtering (PF) plays an important role in the signal processing, automatic control, wireless communication and other fields.
In the past few years, the distributed PF problems have received the initial attention owing to its engineering significance. For example, in J. Y. , the distributed particle filters have been designed for nonlinear systems, which have low communication cost and robust performance. In Tang et al. (2014), the problem of distributed fusion PF based on information weighted consensus has been proposed for nonlinear systems. By using PF, the distributed sequential estimation problem has been studied and solved in Hlinka et al. (2015) for nonlinear systems over wireless sensor networks. In Hlinka et al. (2012), the distributed particle filter and the distributed Gaussian particle filter have been designed for nonlinear systems with wireless sensor networks based on the likelihood consensus method. In Mohammadi and Asif (2013), the distributed PF algorithm based on a multi-rate consensus/fusion framework has been given for nonlinear systems.
In addition, the problems of the detection and tracking of targets have attracted some attentions and many results of the distributed fusion algorithm via the PF have been proposed in Zhang and Ji (2012), Ghirmai (2016), Kang et al. (2018), Papa et al. (2018). For example, in Zhang and Ji (2012), the problem of the bearings-only tracking has been addressed by using the distributed fusion PF approach for nonlinear systems over sensor networks. In Ghirmai (2016), Kang et al. (2018), the distributed PF algorithms have been developed for nonlinear systems over the wireless sensor networks. In Papa et al. (2018), the distributed multi-sensor filtering algorithm based on particle filter has been proposed for nonlinear systems by using likelihood consensus approach.
Among the above distributed fusion algorithms, the distributed fusion approaches based on the EKF, which can deal with the filtering problems for nonlinear systems, have convergence shortcoming. In addition, the Taylor expansion technique is employed to linearize the nonlinear function during the design of EKF, and the calculation of Jacobian matrix leads to high computational burden. Compared with the distributed fusion EKF approaches, the distributed fusion UKF approaches neither need to calculate the Jacobian matrix nor require the linearization steps, which might result in low computational cost. The distributed fusion CKF algorithms can deal with the filtering and estimation problems for high-dimensional nonlinear systems. However, the fusion algorithms require the computation of integration over a high-dimensional spherical region, which lead to high computational burden. In addition, the distributed fusion PF algorithms can deal with the filtering problems for nonlinear systems and have low communication cost, which can be introduced to handle the tracking problem. To facilitate the readers, the advantages and disadvantages of the above algorithms are summarized as shown in Table 1.
On the other hand, the existing distributed fusion algorithms based on the EKF, the UKF, the CKF and the PF can deal with filtering and estimation problems with different estimation accuracies. In addition, those fusion algorithms have different fusion structures, which include series, parallel, layered, hybrid structure and so on. In the practical application, an appropriate fusion algorithm can be selected according to the actual needs.
Next, we will return to the research topics of distributed estimation, filtering and fusion for nonlinear systems, and the recent advances will be further introduced in the following section.

The latest progresses
In recent years, the problems of distributed estimation, filtering and fusion for nonlinear networked systems have received a growing number of research interests and a  great number of results have been proposed in the literature (Zou, Wang, Hu, & Han, 2020). Here, we introduce some new research contents. For example, the distributed H ∞ filtering and estimation problems for nonlinear networked systems with communication constraints are discussed and the multi-sensor fusion filtering problems for nonlinear systems subject to communication constraints are reviewed, respectively.

Distributed H ∞ filtering and estimation for nonlinear NSs
During the system analysis, there are many uncertain factors, such as parameter uncertainty, noise disturbance and system observation uncertainty. If those uncertainties are not properly handled, the performance of the filtering method might not be guaranteed. Because the H ∞ filtering method can provide the disturbance attenuation ability regarding the estimation error in the worst case, the issues of the distributed H ∞ estimation and filtering have been widely studied for nonlinear time-invariant/time-varying systems with communication constraints in the past few years. Therefore, several methods have been proposed, such as a stochastic sampled-data approach, LMI method, Riccati difference equation method and so on. For instance, by using a stochastic sampled-data approach, the problem of distributed H ∞ filtering problem has been investigated in Shen et al. (2011) for time-invariant nonlinear systems with event-triggered protocols over sensor networks, where a set of general nonlinearities has been modelled by the sector-bounded condition. In Shen et al. (2010), the problem of distributed H ∞ filtering has been addressed for a class of polynomial nonlinear stochastic systems over sensor networks in terms of the Lyapunov function and parameter-dependent LMI method. The distributed H ∞ filtering problem has been studied in Dong et al. (2011) for repeated scalar nonlinear systems over sensor networks with random packet dropouts, where sufficient conditions have been obtained to guarantee stochastic stability and the feasible solution of the filter parameters has been given by using the convex optimization method. In H. , the distributed H ∞ filtering algorithm has been proposed for switched repeated scalar nonlinear systems with randomly occurred sensor asynchronous switching. In Ding et al. (2015), the distributed H ∞ state estimation strategy has been given for a class of discrete-time nonlinear systems with packet dropouts and event-triggered mechanism, where sufficient conditions have been provided by using the stochastic analysis technique and convex optimization approach. The problem of the distributed H ∞ filtering has been investigated in Dong et al. (2012a) for a class of discrete-time Markovian jump nonlinear systems with time delays and randomly occurring incomplete information over sensor networks, where the nonlinearities have been represented by the Lipschitz-like manner and sufficient conditions have been provided to ensure the desirable performance. On the other hand, several methods on the distributed H ∞ filtering and estimation have been proposed for nonlinear time-varying networked systems. For example, in Ding et al. (2012), the problem of the distributed H ∞ state estimation has been examined for a class of discrete timevarying nonlinear systems with stochastic parameters over sensor networks, where a necessary and sufficient condition has been obtained to reach the H ∞ performance constraint, and the parameters of the distributed estimator have been given by solving the backward recursive Riccati difference equations. The distributed H ∞ filtering problem has been discussed in Ge and Han (2015) for nonlinear systems with event-triggered communication protocol and communication delays over sensor networks, where sufficient conditions have been derived to ensure the asymptotic stability of the filtering error system. In addition, the distributed H ∞ filtering algorithms have been proposed in Zhang et al. (2010) for a class of nonlinear time-varying delay systems, where the nonlinearities have been described by T-S fuzzy model method and sufficient condition of the algorithm has been given by the LMI method. In terms of the same model method in Zhang et al. (2010), the problems of decentralized piecewise H ∞ filtering have been studied in Zhong et al. (2015), Zhang et al. (2011) for a class of discrete-time nonlinear systems with time-varying delay. It can be shown that the stability and the prescribed H ∞ performance can be achieved based on the Lyapunov theory and the freeweighting matrix approach, where sufficient condition has been obtained.

Multi-sensor distributed fusion filtering for nonlinear NSs
Multi-sensor information fusion algorithms utilize the information of each sensor synthetically, overcome the uncertainty and limitation of a single sensor, improve the effective performance of the whole sensor systems, and more accurately describe the tested object. In recent years, some review papers with respect to distributed fusion filtering for linear networked systems have been proposed in Hu et al. (2016), Sun et al. (2017). Unlike the one in Hu et al. (2016), Sun et al. (2017), the multisensor distributed fusion filtering results for nonlinear networked systems can be discussed and summarized as follows.
Compared with the centralized fusion filtering approaches, the distributed fusion filtering methods are commonly suboptimal. It is difficult to get the optimal fusion estimation algorithm due to the existence of many uncertainties in NSs. Note that the distributed fusion H ∞ filtering approach has the characteristics of robustness, hence it has been extensively applied in various fields. For example, in M. , the problem of the multi-sensor optimal H ∞ fusion has been studied for a class of nonlinear intelligent systems with time-delays, where a unified model including the linear systems and nonlinear systems has been described by neural networks and T-S fuzzy model. Moreover, the H ∞ performance analysis of the unified model has been discussed by using the LMI method, and the centralized as well as distributed fusion filtering algorithms have been developed. Besides, the asymptotic stability of the fusion error systems has been discussed and the parameters of the fusion filters have been proposed by conducting the eigenvalue problem. In M. Liu, Qi, et al. (2011), according to the method in M. , both the centralized and distributed fusion filtering algorithms have been designed for a class of discrete nonlinear stochastic systems with missing measurements and time delays by employing the LMI method, where the missing measurements have been modelled by a binary switching sequence.
In practical applications, in order to effectively identify unknown information and improve the accuracy of state estimation, the adaptive/self-tuning fusion algorithms have gained much attention. For instance, based on the adaptive fading UKF, the multi-sensor optimal information fusion algorithm has been proposed in Gao et al. (2018a) for nonlinear stochastic systems. It has been shown that this method has a two-level fusion structure, where the adaptability and robustness of local state estimations have been promoted at the bottom, and the globally optimal state estimation has been achieved in terms of the principle of linear minimum variance for the fusion of local state estimations at the top level. In Gao et al. (2018b), the multi-sensor optimal fusion approach for nonlinear systems has been developed for the same fusion structure as in Gao et al. (2018a), where the local optimal state estimations and the global optimal state estimation have been obtained based on UKF and the principle of linear minimum variance. Moreover, the multi-rate sampling sensors can get more information than single rate sensor. Therefore, the problem of the asynchronous fusion estimation has received some interest. For example, the distributed fusion estimation algorithm for a single nonlinear model with asynchronous multiple sensor has been developed in Jeon and Eun (2014) by employing the information matrix and information state contribution, which have been reconstructed based on the UT.
For multi-sensor systems, it is very important to identify unknown information and provide adaptive/selftuning fusion approach, where the convergence analysis of the fusion approach is a key point. During the last few years, many adaptive/self-tuning fusion algorithms have been developed for linear systems. For example, the problem of self-tuning information fusion estimation has been studied in Sun (2007) for discrete linear stochastic systems with correlated noises by using the weighting average method. In Ran et al. (2009), the self-tuning decoupled fusion Kalman predictor has been designed for multi-sensor systems with unknown noise variances, where the convergence and asymptotic optimality of the fusion algorithm have been proved. Up to now, the problems of adaptive/self-tuning fusion approach for nonlinear systems are relatively few. On the other hand, the problems of the distributed fusion estimation for linear systems with multi-rate sensors have been investigated. For instance, the problem of the distributed fusion estimation has been discussed in Lin and Sun (2018) for multi-sensor multi-rate systems with correlated noises, where the stability of the estimation algorithm has been discussed. In , the multi-rate distributed fusion estimation methods have been developed by employing the fusion approach in terms of weighted matrices. Furthermore, in L. Yan, Jiang, et al. (2016), the centralized, sequential and distributed fusion linear estimation algorithms for multi-rate sensor systems have been proposed. Besides, there are also some fusion results for non-uniform sampling systems. For example, the distributed fusion estimation algorithms for multi-sensor asynchronous sample systems with correlated noises and missing measurements have been proposed in Lin and Sun (2017), Lin and Sun (2016). However, the study on the distributed asynchronous fusion for nonlinear systems with communication constrains has not been thoroughly made in the existing literature.
By employing the EKF, the distributed fusion filtering problem has been investigated in Song et al. (2019) for nonlinear systems with multiple fading measurements and event-triggered communication protocol, where the estimation error covariance upper bound has been given by using variance-constrained approach and sufficient condition has been obtained to ensure the stability. The distributed adaptive high-gain EKF problem has been discussed in Rashedi et al. (2016) for nonlinear systems with data losses and time delays, where sufficient conditions have been given to guarantee the convergence of the algorithm. Based on the results in Rashedi et al. (2016), the issue of the distributed adaptive high-gain EKF for nonlinear systems with triggering communication protocol has been studied in Rashedi et al. (2017b), where both the convergence and boundedness of the estimation error have been discussed and related conditions have been given.
Based on the CKF, the distributed cubature information filtering method has been designed in Q. Tan et al. (2017) for nonlinear systems with event-triggered protocol addressed by using weighted average consensus method, where the amount of data transmission has been reduced by employing the communication scheme and consensus method. In the light of the results in Q. Tan et al. (2017), an improved distributed cubature information filtering approach has been proposed in J.  for nonlinear systems with randomly occurring incomplete measurements based on the consensus method, where a new fusion weight method has been designed and the convergence of the fusion algorithm has been achieved. In Q. Ge et al. (2016), a series of multisensor information fusion algorithms has been given for a class of nonlinear systems based on cubature squareroot information filter approach, which include the augmented measurements fusion, measurements weighted fusion, sequential filtering fusion and distributed fusion. Moreover, the estimation accuracies of the above mentioned nonlinear fusion methods have been discussed and the fusion algorithm based on EKF for complex nonlinear systems has higher estimation accuracy.
Overall, the distributed H ∞ filtering algorithms mentioned in Section 4.1 are generally applicable for the time-invariant systems subject to energy bounded noises and communication constraints, where the linear matrix inequality technique is commonly adopted to tackle related issues. Accordingly, the robust estimation methods can be available. When the time-varying system become the concern, some multi-sensor distributed fusion algorithms have been discussed for nonlinear systems as mentioned in Section 4.2. It should be noted that most of the algorithms of the distributed adaptive/selftuning fusion and distributed asynchronous fusion are suitable for linear systems, which are seldom applicable for nonlinear systems with degraded estimation performance. Therefore, there is a need to further develop efficient fusion methods for nonlinear networked systems catering the time-varying situations.

Conclusion and new research directions
In this paper, we have received some distributed estimation, filtering and fusion for nonlinear systems. First, some communication constraints in the NSs have been mentioned including a series of network-induced phenomena and communication protocols. Second, some distributed fusion estimation algorithms have been introduced and summarized, where the distributed fusion EKF, distributed fusion UKF, distributed fusion CKF and distributed fusion PF have been listed. Finally, some recent advances on distributed estimation, filtering and fusion algorithms for nonlinear systems with communication constraints have been reviewed, which generally include: (i) distributed H ∞ estimation, filtering and fusion; (ii) multi-sensor distributed fusion filtering, distributed adaptive/self-tuning fusion and distributed asynchronous fusion. Based on the above literature review, some topics can be further studied.
• The problems of estimation, filtering and fusion with network-induced phenomena and communication protocols are of engineering significance. It may be an interesting yet important research topic to analyse the dynamical behaviours of different networked systems (e.g. complex dynamical networks and sensor networks). • The distributed estimation, filtering and fusion problems for nonlinear systems are an interesting yet challenge topic, especially when multiple networkinduced phenomena and communication mechanisms exist at the same time. Although there have been a few methods to handle the problem of the distributed fusion filtering for nonlinear systems, the available results are seldom considering the design of fusion algorithm for time-varying case. Hence, more effective approaches and strategies should be given to deal with this problem. • It would be meaningful to address the problems of robust distributed estimation, filtering and fusion for nonlinear networked systems with multiple communication constraints. In particular, additional effort should be devoted to deal with the performance indices (e.g. variance constraint, finite time index, probability constraint). • It is noteworthy that the adaptive/self-tuning fusion algorithms have become an effective method to solve the case that the model parameters/noise information of the system may be (partially) known. However, most of the algorithms of adaptive/self-tuning algorithms are applicable for linear systems, the corresponding problem of distributed adaptive/self-tuning fusion estimation for nonlinear systems with communication constraints might be a challenging research topic. • Note that different types of sensors might have different sampling rates, thus it can be seen that the study of asynchronous fusion is one of the important problems of multi-rate multi-sensor systems. At present, some results can be available for linear systems, but few for nonlinear systems. However, the problem of the distributed asynchronous estimation, filtering and fusion for nonlinear systems would be a challenging research direction. • When the communication constraints occur, the existing fusion schemes face the issues of computational burden and communication burden, which are mainly caused by the calculations of covariance and the augmented matrix of systems. Therefore, it is necessary to develop new methods to reduce the computational complexity. Furthermore, the performance analysis of multi-sensor fusion estimation algorithm is an interesting research topic, such as the convergence, monotonicity and sensitivity of the newly proposed fusion estimation schemes.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This work was supported in part by the National Natural Sci-