Abstract
In this paper, we study the synchronization of a class of multiple neural networks (MNNs) with delay and directed disconnected switching topology based on state observer via impulsive coupling control. The coupling topology is connected sequentially, and the controller adjusts the state value through event-triggering strategies. Different from the related works on MNNs, its state in this paper is assumed to be unmeasurable, and the time delay is also unmeasurable. Therefore, the observer does not contain the time-delay term. The impulsive switching controller and observer controller adjust the system through the observed value. By constructing the corresponding augmented matrix, the system can finally achieve quasi-synchronization (synchronization). Through derivation, we give the sufficient conditions ensuring quasi-synchronization (synchronization) via the event-triggered impulse control mechanism. In addition, numerical simulation examples are given to test our results of the theorem.
1. Introduction
There has been rapid development of multi-agent systems (MASs). In practical application, the application of MAS mainly includes power engineering [1], bioengineering [2], robot formation control [3], vehicle formation control [4], and some other fields. Theoretically, the dynamic behavior of MAS, such as stability [5], robustness [6], synchronization [7], and so on, has become the basis of various theories and greatly promoted the development of MAS. So far, a number of achievements have been made in the study of MAS.
In addition, as a complex network, the topology of MAS plays an important role in the dynamic behavior. For example, in [8], second-order leaderless and leader-following consensus algorithms with communication and input delays in directed network topology are studied. In addition, this paper involves three different situations: leaderless consumption, consumption regulation, and consumption tracking. On the other hand, the network topology plays an important role in the asymptotical stability scheme. In the leader-following problem of multi-agent network, it is assumed that the network topology switches arbitrarily between limited topology sets and there is a time-varying delay in the coupling of agents [9]. Different from the general topology, the switching topology in this paper is of great significance to the sudden change or failure of the environment, so switching topology widely exists in MAS.
In recent years, MNNs have been widely used, especially in automatic control [10], signal processing [11], optimization [12], and so on. Such complex systems are extremely dependent on the synchronization and stability of MNNs. Therefore, synchronization problem is receiving more and more attention and has always been a very important research direction [13–16]. Especially, Chen et al. [16] considered synchronization for nonlinear neural complex networks by a switching topology. On the other hand, different from synchronous, quasi-synchronous is a special form of dynamical behavior, where all of the control systems in networks are almost synchronized with a given synchronization error, which could not tend to zero with time. Chen et al. [17] discussed the quasi-synchronization problem through a coupled memristor neural network with time-varying delay. The quasi-synchronization problem in fractional-order multi-layer networks with fractional mismatch is studied in [18].
With the development of industrial demand and information technology, some traditional control strategies have been replaced by other control schemes. The feasibility and advantages of event-triggered control (ETC) have been proposed for the first time since 1999. Different from traditional control scheme, ETC can ensure system performance while effectively reducing the execution of control tasks. In recent years, ETC has become a popular research subject [19–22]. Because we only need to adjust the state of the controller at the event-trigger instants via setting an appropriate event-trigger mechanism. Different from continuous control and ETC, controller status is updated only at the moment of event trigger, it is adjusted to meet the needs of the system. At present, existing work of ETC in the multiagent field (see [23, 24]). Especially, [25] the recurrent neural network triggered by finite-time event-triggered strategy is studied, and the stability of finite-time systems is proved by novel inequality methods such as, LyapunovCKrasovskii functional and Wirtinger single and double integral inequality. Compared with static trigger conditions, dynamic trigger conditions have more advantages. For example, a new fuzzy filter error system model under dynamic event-triggered control strategy is considered. In addition, there are different triggered thresholds for different fuzzy rules, which can save communication resources more effectively in [26].
In order to realize the synchronization of MNNs, we often add appropriate controllers to the system. According to Tang et al. [27], the leader following consistency problem for a class of nonlinear multiagent systems with mixed impulses and time-varying bounded delays is studied. The time-varying impulses in this paper is not only composed of synchronization impulses and desynchronization impulses but also placed in some nodes of the system. Based on Riemann Liouville derivative, Lyapunov functional method and comparison theorem, we can get the global synchronization problem of time-varying delay neural networks with impulsive fractional complex memristor [10]. The impulse controller is one of the most widely used controllers in recent years. Different from the traditional continuous control strategy, impulse control mechanism has the advantage of short action time, which makes it possible to use the impulse controller to occupy less communication resources for a system with a very large amount of information transmission (see [28, 29]). In order to reduce communication bandwidth and save communication costs, a new control strategy based on event-trigger impulse is given. For example, Yi et al. [30] proposed an impulsive control mechanism based on ETC. Except for above control strategy, the impulse coupling protocol is also studied. The coupling between neural networks only occurs at some discrete-time instants, that is, impulse instants. Consequently, the impulse coupling scheme is naturally proposed.
Observer-based output feedback control is one of the traditional hot topics. It can be divided into two categories according to whether the variables are measurable or not. For the former, a relaxed stability condition based on state observer is proposed [31], and for the latter, a scheme based on fuzzy controller for a class of nonlinear systems is presented [32]. More recently, it is usually presumed that MAS state is measurable. Due to the limitations of measurement methods, many states cannot be measured. On the other hand, system states are unavailable for the state feedback control or too expensive to measure. Thus, it is imperative to research the observer for the system state is not measurable.
In general, we design an observer to estimate the value of different MNNs and then use the information to establish an observer based on feedback controller. However, the measured value is usually collected in discrete time. Therefore, an impulse observer is promoted. It was first proposed by Raf and Allgower in [33]. The observer is updated in the form of impulsive; hence, the measured output is discrete. So, use the impulse observer to estimate the error. Apart from this, designing a suitable control scheme based on impulsive observer is still a challenging problem.
Motivated by the previous research, this paper studies synchronization problem of MNNs with observer via an ETC mechanism. The main contributions of this paper are as follows:(1)An impulsive switching controller is designed via the event-triggered strategy of MNNs with disconnection switching topology. Considering the practical needs, the actual state may be unpredictable in reality. Thus, the system state in this paper is assumed to be unmeasurable.(2)A particular observer is constructed. Considering the unknown time delay in practical application, the observer does not exhibit time delay. Through the observation value of synchronization error and the tracking error of synchronization error, an augmented system is formed. The synchronization (quasi-synchronization) of the augmented system is also of the MNN system.(3)The MNNs with a switching topology is studied and the topology is disconnect. At the same time, impulse control, event-triggered strategies, and observers are used to study synchronization (quasisynchronization) issues. In the real system, the sufficient conditions of the synchronization (quasi-synchronization) are proved. We discuss this kind of question and give the relevant theorems. The Zeno behavior can be ruled out.
The remainder of this article is organized as follows. Section 2 describes problem formulation and some necessary preliminaries. In Section 3, a number of results are presented. In Section 4, a numerical simulation is presented to test the obtained theoretical analysis. Some conclusions are drawn in Section 5.
2. Preparation and Modeling
Notations. Throughout this study, sign( ) is the standard sign function. and represent a set of integer and positive integer. represents n-dimensional Euclid space. represents 1-norm. , and thus where , and is an identity matrix. Let denote a graph. is the set of nodes. is the set of edges. An arbitrary matrix is given, corresponding to , and thus from , the graph is indicated, where if and only if .
Also, a subset and a graph are given. Define the neighborhood of , such that }. If is a singleton set, represents the neighborhood of one single point.
When , the graph is undirected. has a directed path from node i to j if there is a sequence of edges in the form , , , and the , where, is called connected if there exists a directed path between each pair of nodes. The node is called a root of if has a directed path from to every other node contains a directed spanning tree if there exists at least one root.
For the graphs and , is the union of , . A sequence of graphs with common nodes is jointly connected if contains a spanning tree. A sequence of graphs with common nodes is sequentially connected if there exist node sets such that and , is a set of Singleton, .
Then, by the following dynamics:where , , , and are weight matrices; is transmission delay and satisfies ; is a controller; and indicates external input.
In the actual situation, considering that the system state value cannot be measured, we give the observer of the corresponding th node as follows:
is the estimated value of the corresponding th node. is the controller of the observer. Under (2), we can see that the observer does not contain time-delay term. Considering the fact that the time delay is unknown in practical application, observer (2) does not contain time delay.
Let be a limited set of index and be a directed graph set. represent function of switching in , and (where ) represent instants of switching impulsive time. Let indicate the graph of directed at , where . Thus, obviously is the switch time unchanged for . We use to express of adjacent matrix, where when the system sends information from node to node and otherwise. In addition, we give , namely, there is no self-loop in where . In order to convenient calculation, we assume that and
Consider the following assumptions:(A1)The set of discrete graphs is sequentially connected, if there exists a positive integer , where .(A2)The set of discrete graphs is jointly connected, if there exists a positive integer , where .
denotes a sequence of triggering time. Hence, we will give the event-trigger protocol (ETP). In order to make the system achieve synchronization (quasi-synchronization), we design the impulsive switching controller with ETP and the controller of the corresponding observer. In order to make the system achieve synchronization (quasi-synchronization), we design the impulsive switching controller and the corresponding observer controller of the th node as follows:where .
The state is not measurable.
Therefore, (3) and (4) are only related to the observed values. For, the measurement error is defined as follows:
Meanwhile, by Figure 1, we can get the block diagram for ETC and the ETP:where , ; moreover, and . They are both threshold parameters.
From (1)–(4), we havewhere , , , , and .
Let be a Banach space and . Let represent all continuity functions, and thus the initial value of (7) and (8) can be given:for and . Let denote the synchronization error for , where and denotes the observed value of .
Then,
Let denote tracking error of synchronization error of th node and th node.
Then,
Let ; from (10) and (11), there are the following augmentation systems:
In addition, assume that , , and where and.
Thus, from (13), it follows that
Definition 1. (14) is called to achieve final synchronization, if .
Definition 2. (14) is called to achieve final quasi-synchronization, if , where .
Remark 1. From augmented matrix (14), we can find that there are if for any . The synchronization (quasi-synchronization) of augmented system (13) is also of (1). Moreover, by observer of error systems and of tracking error of error system. We do not directly quote the state of the error system, and thus it has certain significance for the system. Its state is not measurable in practical application. For notational convenience, we denote at any time instant for , and denotes .
3. Main Results
where .where .
Lemma 1. A sequence of graphs is jointly connected if a sequence of graphs is sequentially connected.
Lemma 2. There exists such thatfor any and where denotes the largest eigenvalue of , and we obtain that
Proof. Let for . From (14), we can deduceHere let .
We haveThen, one has , and so , i.e., . The proof is completed.
From Assumption (A2), there exists a sequence of graphs which is sequentially connected where . Hereafter, represents . A sequence of graphs is sequentially connected for any ; meanwhile, and is a set of singleton.
Letwhere and .
Lemma 3. By Assumption (A2), if, thenwhere,,,, and.
Proof. First, we review the state equation of (14) at the impulse instant.for and .
Considering and , where , we give the following three cases.
Case 1. For , under (21), we getFrom (6), ETP, and , we deducefor . Hence, due to (22), we haveFrom and , we can deduceTherefore, (20) is established.
Case 2. Let , and if a, s .
Under (21), one hasFrom (26), we can deduceHence, (20) is established.
Case 3. , , and if .
Becausefrom (21), we haveBy (29),Obviously, (20) holds. Now the proof is completed.
It is always important to assure that Zeno behavior can not be occurred under ETC (3) and (4) in order to prove the synchronization or quasi-synchronization that can be reached by (14).
For error observation (11), it follows that
Theorem 1. Assume that all the conditions of Lemma 2, hold, and satisfy following conditions: C1: and C2: whereis the maximum eigenvalue of matrix, and thus (14) does not exhibit Zeno behavior. Event-triggered time sequenceis generated under event-triggered strategy (6), which satisfieswhen.
Proof. Let be a bounded set and also be a bounded set. It is assumed that .
For any , , there areLet such that . By (11), where .
Similar to Lemma 2, there is , where and is maximum eigenvalue of . Thus, we can deduce , where and .
Then, we havewhere . Obviously, there are for .
Similar to Lemma 3, at impulse instants, (11) hasif exists , where, if, thusLet . Combine (32)–(35), and we haveFrom (6), event-triggered strategy is triggered to update the controller when ; in other words, . Hence, from (36), we obtain thatwhich meansand thensuch that the event-triggered sequence has time interval. This is contrary to the assumption that the sequence is bounded. This ends the proof.
Theorem 2. By Assumption (A1), ifTheorem 1,Lemma 2, andLemma 3hold, then the quasi-synchronization of (7) can be obtained based on observer ifsatisfies (15) besideswhere.
Proof. According to Lemma 2, we can see thatwhere . By Theorem 1, we can know that Zeno behavior cannot occur.
Meanwhile, according to Lemma 3, it follows thatCombining (41) and (43), we can obtain thatThen, from (41)–(44), through the iterative method, we can obtain thatandand thusFrom the introduction of the previous preparation part, we have and . Meanwhile, from which is a single point, we can obtain . By using (47), one hasBy applying (47) and (48), we can prove the following result:where , , and . From (43), it follows thatTherefore, we can obtain(14) can achieve quasi-synchronization. In other words, (7) can also quasi-synchronization-based observer.
In addition, if . From Theorem 2, (7) can reach synchronization-based observer if the sufficient conditions are satisfied.
Remark 2. Under all the conditions of Theorem 2, (14) can be quasi-synchronized. According to the definition of (14), we can find that the system is composed of an observation error system and a tracking error system . Therefore, when (14) can be quasi-synchronized by means of and, we can have the quasi-synchronization of (1). In this process, we do not directly use the state value of the original system (1). It is consistent with the situation that the state value of the system is unknown in practical application.
Theorem 3. Under Assumption (A3), usingLemma 2,Lemma 3, andTheorem 1, system (7) can obtain quasi-synchronization-based observer if
Proof. From Lemma 1, when the sequence of graphs satisfies , the sequence of graphs satisfies . Therefore, under Lemma 2, system (7) can have quasi-synchronization-based observer and can have synchronization if .
4. Numerical Simulations
In this section, a numerical example is given to verify the validity of theory analyses.
Review original system (7) with controller and observer system (8):
Let , and the parameters in the system are designed as follows: and , and the weight matrix is designed as
Meanwhile, the initial function is given as follows: , , , , , , , , , , , , , , , , and , and we set , . Figures 2 and 3 show the switching topology.
From Figure 2 () and Figure 3 (), we can find that is sequential connection. In the switching period , it is also a joint connection and . The change of node set is as follows: , , and . Satisfy Assumptions (A1) and (A2), and the coupling matrix is as follows:
Thus, we know , .
From the above data, augmented matrix (14), we have
For Lemma 2, by solution of the integral equation reach . Let , and the condition of Lemma 2 can be satisfied.
For Lemma 3, we can obtain , and Lemma 3 is also satisfied. According to the conditions in Theorem 1, we have and . Let and satisfy the condition of Theorem 1. For C2, we have . Thus, according to Theorem 1, there is no Zeno behavior.
For Theorem 2, obtain . According to , the condition of Theorem 2 is also established where . Then, (14) can reach the quasi-synchronization, namely, (1) achieves quasi-synchronization based on observers.
Take , , and .
Under quasi-synchronization, the state diagram of 8 nodes of (1) and (2) when is shown in Figure 4. Under synchronization, the state diagram of 8 nodes of (1) and (2) when is shown in Figure 5.
Figure 6 shows the state diagram of 8 nodes of system (14) when .
Figure 7 shows the state diagram of 8 nodes of system (14) when .
In Figure 8, the state of sampling value is given when . In Figure 9, the state of sampling value is given when . When the system meets the conditions given in this paper, all nodes reach synchronization. Thus, we can find that the theorem given in this article is valid.
5. Conclusion
The question of quasi-synchronization (synchronization) in MNNs with observers in impulsive coupling controller via event-trigger strategy is discussed. An event-triggering mechanism is designed by using the combination measurement method. The real system state in this paper is assumed to be unmeasurable, and the system time delay is also unmeasurable. The state of (1) is measured by observer. The time delay is unknown, so the observer does not have time delay too. The augmented system is composed of the observer and the tracking error system of the error system. In the real system, the sufficient conditions of the quasi-synchronization and synchronization are proved. Compared with existing works, this paper considers the real state and unmeasurable time delay, and the controller used in this paper is a impulsive controller with event-triggered mechanism, so it plays a significant role in saving communication resources. In addition, we consider trying to spread it to the more general system and more complex systems.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This study was supported by the Natural Science Foundation of China (62072164 and 11704109).