Event-TriggeredH∞ Filtering for Multiagent Systems with Markovian Switching Topologies

This paper is concerned with the problem of event-triggered H∞ filtering for multiagent systems with Markovian switching topologies and network-induced delay. An event-triggered mechanism is given to ease the information transmission. Consider that the network topology is directed in this paper, which represents the communication links among agents. Due to the existence of network-induced delay, the time-delay approach is adopted, which can effectively deal with filtering error system. By constructing a Lyapunov-Krasovskii functional and employing linear matrix inequality technique, sufficient conditions are established to ensure the filtering error system to achieve asymptotically stable withH∞ performance index. A simulation example is given to illustrate the effectiveness of the proposed method.


Introduction
Recently, multiagent systems (MASs) have received much interest due to their widespread applications in various fields such as formation control [1,2], sensor network [3], synchronization [4], and flocking [5,6].Due to the fact that communication networks consist of lots of agents, their states are usually not fully available in network outputs.Hence, filtering or state estimation problem is to estimate the states of the agent by the available output measurement, which is very important both in theory and in practice.In the past decades, the consensus-based filtering or estimation problem for a special multiagent system, i.e., the multi-sensor networked system, has received much attention.For example, Kalman filters are the filters primarily proposed in [7,8] for multisensor networked systems, while  ∞ filters are given in [9] for sensor networks with multiple missing measurements.In most existing literatures, the study of multiagent systems has been investigated in the ideal situation without considering external interference.However, in practical engineering, the existence of external disturbances cannot be ignored, which may have a great impact on agent dynamics.Furthermore, the research history of multiagent systems on  ∞ filtering or state estimation is very short; there are many problems which need to be studied extensively.Thus, it is necessary to study the  ∞ filtering problem of multiagent systems with many practical factors.
The exchange of information between agents on a communication network is usually completed on the basis of sampled packets.Many articles have presented more and more methods based on rapid development of the theory of sampling data systems.In the early results, timetriggered communication is a common way to transfer every sampled signal to the controller and update control signal periodically, which wastes a lot of network bandwidth resources and increases network load.Compared with the widely used time-triggered sampling scheme [10][11][12][13], eventtriggered communication schemes [14][15][16] can avoid the unnecessary transmission and reduce the release times of the sensor and the burden of communication network.Event-triggered filtering/estimation for different systems has received considerable attention in the past few years [17][18][19][20][21][22].Paper [17] addresses event-triggered state estimation problem for a class of complex networks with mixed time delays and [20] presents  ∞ filter design for a class of neural network systems with quantization.The filtering problem for discretetime networked control system (NCS) under event-triggered scheme is proposed in [18,19,21].Paper [22] designs the  ∞

Problem Formulation
2.1.Graph Theory.Let  = (, , ) denote a directed weighted graph of  order, where the set of nodes is denotes by  = {V 1 , V 2 , . . ., V  } and the set of edges is denoted by  ⊆  × .An edge in graph  is described by (V  , V  ), ,  ∈ 1, 2, . . .,  which represents that the ℎ agent transmit information to the ℎ agent.Matrix  = [  ] × is called weighted adjacency matrix.The set of neighbors of agent  is denoted as for  = 1, 2, . . ., , where   (),   (), and   () are the state, the measured output, and the signal to be estimated of agent .  () denotes the external disturbance which belongs to  2 [0, ∞).

Event-Triggered Scheme.
In order to reduce the communication network burden, event-triggered scheme is adopted in this paper.The sampler is assumed to be time-driven; the zero-order-hold (ZOH) is event-driven.Then, we construct the following event-triggered strategy.
Remark 1. From the event-triggered strategy (4), we can automatically exclude Zeno behavior.
Proof.Zeno behavior in the event-triggered communication framework is defined as the infinite number of communication in a finite time.Due to the fact that each agent's state is periodically sampled and the sampling time sequence is {0, ℎ, 2ℎ, . ..}, we can obtain that the event-triggered time sequence is a subsequence of the sampling time sequence, namely, {  0 ℎ,   1 ℎ,   2 ℎ, . ..} ⊆ {0, ℎ, 2ℎ, . ..}, which means that the minimum interevent time min  {  +1 ℎ −    ℎ}, ∀, is lower bounded by the sampling period ℎ.Since the number of agents is finite, the communication events in any finite time cannot be infinite.Therefore, the event-triggered strategy (4) can rule out Zeno behavior automatically.
Remark 2. Different from traditional filtering problem, due to the existence of network-induced delays, taking the property of ZOH into account, we get Substituting ( 5) into (3), we have Remark 3. Suppose that the network delay is bounded, i.e., 0 <   ≤    ≤ , where   and  denote the lower and upper bound of (), respectively.It is worth noting that when   = 0, all the sampled signals are transmitted, the event-triggered strategy (4) reduces to a periodic time-triggered scheme.[29], we assume that there is a integer  > 0, which satisfies   +1 =    +  + 1.Hence, we can obtain the following equality:

Time-Delay Modeling. Using the same technique as in
with For convenience, we define Then, we can know that To show the effect of the event-triggered scheme (4) in deriving system stability and stabilization criterion, we define a new variable   (), which satisfies the following equality: Substituting ( 8), ( 10) into (6), we have From ( 10) and ( 4), we can obtain where  ∈ [   ℎ + (   ),   +1 ℎ + (  +1 )).

Event-Triggered
where Then, we define some new augment variables to compact system: Therefore, the filtering error system can be rewritten as follows: Definition 4. The filtering error system (16) with () = 0 is asymptotically stable in mean square, if for any initial conditions, such that lim Definition 5. Given a positive scalar , and for all nonzero () ∈  2 [0, ∞), the filtering error system ( 16) is asymptotically stable in mean square with a guaranteed  ∞ performance  if the filtering error () satisfies Before proceeding further, we introduce the following lemmas that will be helpful for deriving the following results.

𝐻 ∞ Performance Analysis
Theorem 9.For given scalars   > 0,   >   > 0, considering the existence of external disturbance, the filtering error system ( 16) can achieve being asymptotically stable with an  ∞ performance index  under Definition 5 if there exist appropriate dimensional matrices   > 0,  1 > 0,  2 > 0,  1 > 0,  2 > 0,   > 0 and  1 such that the following LMIs hold: where and the other parameters are defined as in Theorem 8. where By using the Schur complement, we can find easily that (32) guarantees Ω < 0. According to Definition 5, the filtering error system ( 16) is asymptotically stable with  ∞ norm bound  if ( 31), (32), and (33) are satisfied.The proof is completed.

8
} and define new variables: By the linear transformation above, we can find easily that (32) is equivalent to (38).Therefore, we can obtain (41).The proof is completed.
Remark 11.Comparing with the previous results [32], lower bounds theorem [31] is adopted in this paper, which not only achieves performance behavior identical to approaches based on the integral inequality lemma, but also decreases the number of decision variables dramatically.

Simulation
The parameters of MASs (1) are given as follows: In addition, the external disturbances of MASs (1) are defined as follows: () = 0.4 −0.2 cos (0.6) . ( All the possible information transmission relationships among agents are shown in Figure 1.Then, the corresponding Laplacian matrices are given as follows, respectively.
Figure 2 shows the switching of two modes in a Markov chain.Suppose that the probability transition matrix is defined as Given that ℎ = 0.1,  1 = 0.1,  2 = 0.2, minimum allowable delay   = 0.1 and the maximum allowable delay   = 0.5.Let the initial states In Figure 3, it shows the filtering error of agent  ( = 1, 2, 3, 4).In Figure 4, it depicts the state curves of   () and estimation signal   () ( = 1, 2, 3, 4).The event-triggered release instants and intervals of agent  ( = 1, 2, 3, 4) are shown in Figure 5. Letting the simulation time t=30, we can get that agent 1 triggers 84 times, agent 2 triggers 36 times, agent 3 triggers 40 times, and agent 4 triggers 92 times.If   = 0 (time-triggered), there are 300 times transmitted.We can find clearly that the event-triggered control strategy (12) efficiently saves the network resources.

Conclusion
In this paper, we study the problem of  ∞ filtering for MASs with Markovian switching topologies.Considering the effect of switching topologies and network-induced delay, we adopt a reasonable event-triggered mechanism to reduce the amount of network transmission.By employing Lyapunov stability theory and LMI technique, some sufficient conditions are obtained which can ensure filtering error system to achieve mean square stable with an  ∞ norm bound.Finally, a numerical example is provided to show that the method we proposed is feasible and effective.

( i )
The system parameters A, B, C1, C2, D used to support the findings of this study are included within the article.(ii) The external disturbance (t) used to support the findings of this study titled "Event-triggered H ∞ Filtering for Discrete-Time Neural Networks with Missing Measurements" is available from the corresponding author upon request.(iii) The initial states 1(0), 2(0), 3(0), 4(0) used to support the findings of this study are included within the article.(iv) The Laplacian matrices 1, 2 and probability transition matrix Π used to support the findings of this study are included within the article.(v) The filter parameters 1, 2, 1, 2, 1, 2 and event-triggered parameters Φ1, Φ2 used to support the findings of this study are derived by solving LMIs, which are included in the article.

Figure 5 :
Figure 5: The release instants and release interval of each agent.