Preface to the special issue on event-based analysis and synthesis for general systems

1. Introduction

The past decades have witnessed a rapid growth on the utilization of networked systems benefiting from the development of information technologies. The embedded communication network results in various advantages, such as simple installation and maintenance, high reliability, low power requirements as well as low cost. However, the limited bandwidth inevitably degrades the system performance due to the unwanted network phenomena. In order to establish a trade-off between system performance and network service quality, event-based schedules are usually adopted to intentionally decrease the unnecessary processing and communication. In the event-based designs, the information processing is triggered if some “interesting” event occurs. For instance, the change of the considered signal exceeds a certain threshold. To date, much research effort has been devoted to the applications of the event-based strategy to various systems such as energy-efficient information fusion over wireless networks, energy-efficient industrial control systems, complex dynamical networks and so on.

The event-based analysis and control problems for general systems have already become tremendously significant for control engineers, mathematicians and computer scientists. Nevertheless, the investigations on the general systems with event-based mechanism are still not sufficient. As such, this special issue would be a timely reflection of the latest progress, which aims to bring together the most recent approaches to understanding, estimating and controlling systems with event-based mechanism in a quantitative way. Topics include, but are not limited to, the event-based state estimation and systems control with various complexities including nonlinearities, uncertainties, time-delays, saturations, quantization, failures, channel fading and so on. In this case, it would be interesting to examine (1) how event-based schedules have substantial impacts on the dynamical behavior of general systems and (2) how can engineers design the desired controller/filter with an engineering-oriented purpose.

We have solicited submissions to this special issue from electrical engineers, control engineers, mathematicians and computer scientists. After a rigorous peer-review process, 10 papers have been selected that provide overviews, solutions or early promises to manage, analyze and interpret dynamical behaviors of various systems with event-based schedules. These papers have covered both the theoretical and practical aspects of control and filtering problems in the broad areas of networked control systems, sensor networks as well as complex networks.

2. The special issue papers

2.1. Event-based analysis and synthesis for industrial networked systems

As is well known, the nonlinearity is ubiquitous in almost all practical systems and may lead to the degradation of addressed system performance or even instability when it is not appropriately dealt with. The signal degradation is inevitable even with the latest wireless relay transmission technology. Accordingly, the filtering issue for nonlinear systems subject to channel fading has become an ongoing research topic. This special issue starts with a solution to the event-triggered nonlinear filtering subject to fading channels. In the paper entitled “Event-based recursive filtering for a class of nonlinear stochastic parameter systems over fading channels” by Shen et al., the filtering problem is investigated for a class of time-varying nonlinear systems with stochastic parameter matrices. A recursive filter is designed such that, in simultaneous presence of the stochastic parameter matrices and fading channels, the error covariance of the filtering error can be guaranteed to have an upper bound which is then minimized by appropriately selecting the filter parameters. The novelties of the paper are summarized: (1) the system model under consideration is comprehensive that covers stochastic parameter matrices and fading channels and (2) the recursive event-triggered filtering scheme is designed to minimize the derived upper bound on the error covariance.

In most available works, the implementation of filtering algorithms is usually dependent on an implicit assumption that the designed filter gains can be accurately realized. Such an assumption is, unfortunately, not always true in engineering practice due primarily to the finite resolution of instrumentation or the round-off errors in numerical computation. Therefore, it makes practical sense how to design a resilient or nonfragile filter capable of tolerating possible gain variations. In the paper entitled “Event-triggered resilient filtering with stochastic uncertainties and successive packet dropouts via variance-constrained approach” by Jia et al., the event-triggered resilient filtering issue is discussed for a class of time-varying systems in the simultaneous presence of stochastic uncertainties and successive packet dropouts. The considered nonlinear function is described by a predetermined upper bound of secondary moment. A compact form on actual measurements and target dynamics is proposed for the analysis purpose. In light of Kalman filtering idea, both one-step prediction error covariance and filtering error covariance are calculated, where the Hadamard product is skillfully utilized to deal with the covariance of nonlinear functions. In addition, a minimum upper bound of the filtering error covariance is given by designing the filter gain appropriately. The main novelty is that a sufficient condition is established to ensure the exponential boundedness of the filtering error in mean square sense.

In many industrial control systems, the proportional-integral-derivative (PID) control strategies have been widely employed owing to their merits of concise structure, convenient adjustment, strong adaptability and easy implementation. Traditionally, the process noises have been largely overlooked when tuning the parameters of the PID controller and, in this case, the desired disturbance attenuation capability cannot be guaranteed. As such, it makes practical sense to investigate the parameter tuning methods of PID controllers in the framework of the $H_{\mathrm{\infty }}$ control theory. In the paper entitled “$H_{\mathrm{\infty }}$ PID output-feedback control under event-triggered protocol” by Zhao et al., a design scheme on $H_{\mathrm{\infty }}$ PID output-feedback controller is provided for a class of linear discrete-time systems under event-triggered protocols. The controller and the actuators are connected through a bandwidth-limited communication network, and an event-triggered communication mechanism is adopted to decide when a certain control signal should be transmitted to the respective actuator. By means of the Lyapunov stability theory combined with the orthogonal decomposition, sufficient conditions are established under which the addressed PID controller design problem is recast into a linear convex optimization one. The advantages of the paper lie in that (1) a novel PID structure with a limited time-window is developed in the accumulative sum-loop and (2) some ingenious variable substitution is implemented to deal with the challenges from the coupling terms on both controller parameters and positive definite matrices in Lyapunov function.

In the past few decades, a rich body of results has been reported in the literature with respect to the $H_{\mathrm{\infty }}$ control problem of various engineering systems. It should be pointed out that, in some applications, it is of particular engineering significance to guarantee the hard constraint with respect to the controlled outputs. Therefore, the ${\ell }_2$–${\ell }_{\mathrm{\infty }}$ performance, called the guaranteed energy-to-peak performance, has received some preliminary research attention. In the paper entitled “Nonfragile  ${\ell }_2$ –${\ell }_{\mathrm{\infty }}$ control for discrete-time stochastic nonlinear systems under event-triggered protocols” by Sun et al., the nonfragile ${\ell }_2$–${\ell }_{\mathrm{\infty }}$ control problem is investigated for a class of discrete-time stochastic nonlinear systems under event-triggered communication protocols. In virtue of the Lyapunov function and S-procedure method, a sufficient condition is established to guarantee both the exponential mean-square stability and the ${\ell }_2$–${\ell }_{\mathrm{\infty }}$ performance for the closed-loop system. In addition, the orthogonal decomposition is utilized to obtain the desired nonfragile controller parameter. The novelties of the paper are summarized: (1) the addressed control issue is very interesting which includes the event-triggered protocol, the nonfragility and the stochastic nonlinearities into a uniform framework and (2) an easy-to-execute scheme is proposed to get the desired controller parameter by using the orthogonal decomposition approach.

Recent years have witnessed a constant research interest in the filtering/estimation problem for the nonlinear systems. Actually, in reality, the nonlinearities are pervasive and the linearity assumption on the studied system is over conservative and cannot reflect the reality closely. In the paper entitled “Discrete-time state estimation for stochastic polynomial systems over polynomial observations” by Hernandez-Gonzalez et al., the the mean-square state estimation problem is studied for stochastic nonlinear polynomial systems over polynomial observations driven by additive white Gaussian noises. A two-step procedure is used to obtain the desired filter parameters, that is, to compute the time-update equations and compute the measurement-update equations for the state estimate and error covariance matrix. In particular, the mean-square filtering equations are derived for a third degree polynomial system with second degree polynomial measurements. The contributions of this paper can be highlighted: (1) a solution is provided to the mean-square state estimation problem for a class of discrete-time stochastic nonlinear systems over nonlinear polynomial observations confused with white Gaussian noises; (2) the solution is obtained computing the time-update and measurement-update equations for the state estimate and error covariance and (3) finite dimensional filtering equations are derived for a third degree polynomial term in the state equation and a second degree polynomial term in the observation one.

2.2. Event-based analysis and synthesis for complex networks

In reality, a number of complex systems can be modeled by certain complex networks, such as the power networks, aviation networks, transportation networks, computer networks, social networks and so on. In order to understand and monitor the dynamical behavior of complex networks, the state estimation plays an important and irreplaceable role. Different from distributed filtering, it is by no means trivial to design an appropriate filter with consideration of the coupling structure. In addition, it is worth pointing out that only a partial network node can be accessed by sensors in practice due to the cost constraint. In a case where the measurement cannot be obtained from a particular node, it is very interesting and critical to estimate the dynamical behavior of that node via estimates (or measurements) from neighboring nodes. As such, researchers should make great effect to discuss the state estimation issues with partial node information. In the paper entitled “Event-triggered state estimation for time-delayed complex networks with gain variations based on partial nodes” by Hou et al., the event-triggered nonfragile state estimation is investigated for a class of time-delayed complex networks with randomly occurring sensor saturations (ROSSs) and gain variations on the basis of measurements from partial nodes. Some random sequences with predetermined probability distributions are employed to govern the phenomena of ROSSs and gain variations. A weight-based event-triggered rule is defined to govern the authorization of information transmission. A set of local nonfragile state estimators is constructed with purpose of distributed calculation. With the help of Lyapunov–Krasovskii functional and stochastic analysis techniques, sufficient conditions are established for the existence of the desired state estimator which ensures that the estimation error dynamics is exponentially ultimately bounded in the mean square sense. The main novelties of this paper lie in: (1) the event-triggered mechanism, measurements of partial nodes and randomly occurring gain variations are considered in the target network, which makes the investigation more practical and (2) by adopting intensive stochastic analysis techniques, sufficient conditions are obtained to guarantee the existence of the desired exponentially ultimately bounded state estimator in the mean square sense for the addressed complex networks.

For complex networks, the inherent node coupling leads to various physical phenomenon such as chaos, furcation as well as synchronization. Coupling strengths could be random due mainly to the random nature of the working conditions. As such, stochastic coupling model is capable of describing the above network case in a realistic way, whereas it is unavoidable to bring certain mathematical complexities and challenges in the dynamical analysis. In the paper entitled “Resilient filtering for time-varying stochastic coupling networks under the event-triggering scheduling” by Wang et al., the resilient filtering is considered for a class of time-varying networks with stochastic coupling strengths. A set of random variables with known statistical characteristics are exploited to describe both the uncertainty of coupling strengths and the nonfragility of filter gains. An event-triggered strategy with state-independent threshold is adopted to save the network resources by scheduling the signal transmission from the sensors to the filters. For the adopted communication scheme, a novel distributed filter is designed to effectively fuse the node information. Under the framework of distributed Kalman filtering, an upper bound on the estimation error variance is established for each node according to the stochastic analysis. Subsequently, the recursive filter is designed by locally minimizing the derived upper bound. It is easy to see that the developed filtering scheme is of a recursive form applicable for the online computations. The novelties of the paper are summarized: (1) the considered network covers several commonly occurred network-induced phenomena, especially stochastic coupling, and (2) rigorous analysis reveals the monotonicity of the minimal upper bound regarding the triggering threshold.

Uncertain coupling in complex networks usually has various behavior patterns, such as the randomness of coupling strengths, switching characteristics of coupling connection, time-varying characteristic and so forth. As such, considerable research attention should be paid to the performance analysis of various complex networks with stochastic coupling. In the paper entitled “Event-based state estimation for time-varying stochastic coupling networks with missing measurements under uncertain occurrence probabilities” by Zhang et al., the event-triggered state estimation problem is investigated for a class of time-varying stochastic coupling complex networks in the simultaneous presence of both time-delays and missing measurements with uncertain occurrence probabilities. A set of random variables are used to characterize the phenomenon of stochastic coupling behavior between nodes in an on–off fashion. In addition, the statistical characteristics of both stochastic coupling and missing measurements are uncertain. The innovation only from node itself is utilized to update the estimated states, and thus the developed filtering scheme can be executed in a distributed form. The main novelties of this paper are as follows: (1) a new event-triggered state estimation algorithm is presented by optimizing the upper bound of the estimation error covariance and (2) the relationship between the occurrence probability of missing measurements and the trace of the upper bound is discussed in theory.

The wireless sensor networks, regarded as a special case of complex networks, have become one of the hottest research fields due to the revolutionary development of sensor technologies. Wireless sensor networks usually consist of plenty of distributed miniature sensor nodes in some certain regions. Due to the vulnerability of sensors, measurement data could suffer from malicious damage. This kind of damage cannot be reduced generally while utilizing the event-triggered scheduling. In the paper entitled “Event-triggered distributed filtering over sensor networks with deception attacks and partial measurements” by Bu et al., the distributed filtering is discussed for a class of time-varying systems over sensor networks with deception attacks on the measurement outputs. The new measurement model is proposed to describe randomly occurring deception attacks and an innovation-based weight event-triggered rule is adopted to manage the information exchange among neighbors. Considering the partial node measurements, a novel distributed filtering scheme, named as partial-nodes-based one, is first developed with the help of recursive linear matrix inequalities. A simulation example is presented to demonstrate the effectiveness of the proposed filtering method. The main contributions of this paper involve the following: (1) a distributed filtering framework is constructed for sensor networks subject to deception attacks and (2) a new partial-nodes-based scheme is introduced for sensor networks where the measurements are only from a fraction of sensor nodes. In the paper entitled “DECHADE: DEtecting slight Changes with HArd DEcisions in Wireless Sensor Networks” by Ciuonzo et al., the problem of change detection is investigated through a wireless sensor network whose nodes report only binary decisions (on the presence/absence of a certain event to be monitored) due to bandwidth/energy constraints. The design objective is to provide a systematic framework collecting a plethora of viable detectors which can be used for each instance of the problem. The main contributions of this paper are summarized as follows: (1) a framework is provided for viable rules for DCD, which is denoted as DECHADE, and (2) simulation results pertaining to two relevant DCD problems in WSNs are provided so as to compare the considered BH-originated rules under both cases.

Acknowledgements

This special issue is a timely reflection of the research progress in the area of analysis and synthesis for general systems under event-triggered communication scheduling. The investigated systems include networked control systems, complex networks as well as sensor networks. We would like to acknowledge all authors for their efforts in submitting high-quality papers. We are also very grateful to the reviewers for their thorough and on-time reviews of the papers. Last but not least, our deepest gratitude goes to Professor Radim Belohlavek, Editor-in-Chief of the International Journal of General Systems, for his consideration, encouragement and advice to publish this special issue.