Dissipativity-Based Consensus Tracking Control of Nonlinear Multiagent Systems With Generally Uncertain Markovian Switching Topologies and Event-Triggered Strategy

This article focuses on the dissipativity-based consensus tracking control (DBCTC) problems of time-varying delayed leader-following nonlinear multiagent systems (LFNMASs) with the event-triggered transmission strategy. The switching topologies of the LFNMASs are subject to the uncertain and partially unknown generally Markovian jumping process. The control inputs of the following agents are updated according to the proposed event-triggered transmission strategy, which could reduce the communication burden. Based on the event-triggered transmission condition and distributed consensus protocol, some dissipativity-based criteria obtained by adopting the delay-product-term Lyapunov–Krasovskii functional (DPTLKF) and higher order polynomial-based relaxed inequality (HOPRII) are proposed to guarantee the LFNMAS consensus. The validity of the main results is verified by two simulation examples.


I. INTRODUCTION
D URING the past decade, a great many problems of multiagent systems (MASs), such as tracking control, containment control, consensus analysis, formation control, and deep reinforcement learning, were extensively investigated by researchers due to their widespread applications in unmanned aerial vehicle formation control, autonomous underwater vehicles cooperative control, spacecraft attitude coordination, and sensor networks information fusion (see [1]- [22] and the references therein). By utilizing the shared communication network topology and cooperative control theory, a group of agents could carry out difficult tasks with high quality and efficiency [23], [24]. Hence, the cooperative control issue of MASs becomes a research hotspot. The neuro-adaptive cooperative tracking control with prescribed performance of unknown higher-order nonlinear MASs was investigated in [25]. Considering switching directed topologies, the hierarchical cooperative control was investigated for a two-layer networked MASs in [26]. After that, based on the sampled-data setting, the research progresses of distributed cooperative control of MASs were addressed in [27]. Based on [27], the investigation of fault-tolerant cooperative control in MASs was presented in [28]. For the cooperative control of leader-following MASs, the leader usually has an impact on the movement of the followers. Compared with leaderless MASs, leader-following formation could help the communication and orientation of the team. Hence, the consensus problems of leader-following MASs were developed in recent years (see [3], [24], [29]- [35] and the references therein).
The communication resources in the network environment are usually limited due to channel bandwidth and capacity. Hence, much effort has been made to design energy-saving control strategies to reduce the communication and calculation burden caused by the continuous communication [8], which could save network energy consumption. Compared with the time-triggered method, the key advantage of the event-triggered strategy (ETS) is that the measurement transmission and control tasks are executed only at the violation of event-triggered condition. Hence, event-triggered data transmission has become popular for its great benefits in saving communication resources [36]. The distributed event-triggered control problem of linear MASs was investigated in [37]. The novel event-triggered control schemes were addressed for the MASs with general linear dynamics in [38]. Based on [38], the leader-following event-triggered consensus control of MASs with different topology structures was investigated in [39]. The distributed event-based consensus of linear MASs with jointly connected switching topologies was investigated in [40]. However, due to the speed of data transmission and the limited bandwidth resources of channels, the communication delay commonly exists in the networks and is a non-negligible element in the consensus problem of MASs [41]. The eventtriggered consensus problem of linear MASs with time-varying communication delays was investigated in [41]. Some novel event-triggered control strategies of linear MASs with timevarying delays were investigated in [3] and [41]. To the best of our knowledge, the event-triggered consensus tracking control issues of NMASs with time-varying delay and Markovain switching topologies have not been studied extensively, which is the first motivation of this article.
Many existing results with respect to event-triggered consensus of MASs usually focused on fixed topologies [15], [39], switching topologies [15], [39], [40], directed topologies [42], and undirected topologies [43]. The new leader-follower relation-invariable persistent formation with various switching topologies was proposed in [21]. Due to environment factors and communication interference, the topology structures of MASs may vary according to time [3], [11], [44]. In order to represent the switching structure, it is always characterized as Markovian jumping topologies. Hence, the consensus issues of MASs with Markovian switching topologies were researched in [3] and [44]- [47]. Under most circumstances, it is not easy to obtain the exact values of the transition rates because of the equipment performance influence and external environment uncertainties [48], [49]. However, there are few papers for the consensus control of MASs with generally uncertain transition rates. Due to the internal and external environment factors in practical application, it is essential to consider more general consensus tracking control of delayed leader-following nonlinear multiagent systems (LFNMASs) with generally uncertain Markovian switching topologies, which is the second motivation of this article.
For input-output energy-related feature of control systems, the dissipativity problems have attracted considerable attention in static neural networks [50], singular MASs [51], complex networks [52], and fuzzy systems [53]. In [54], based on the sliding-mode control strategy, the dissipative consensus control issue was researched for multiagent networks. After that, the dissipativity consensus issues were investigated for singular MASs [51] and fuzzy MASs with switching directed topologies [55]. It is also noted that the common Lyapunov function was constructed to deal with the dissipativity-based consensus in [51] and [54], [55]. However, the dissipative performance has not been introduced into the consensus tracking control of LFNMASs with generally uncertain Markovian switching topologies and ETS, which is the third motivation of this article.
Motivated by the aforementioned research and based on ETS, the dissipativity-based consensus tracking issue of delayed NMASs with generally uncertain Markovian switching topologies is addressed in this article, and each agent is generally nonlinear dynamics. In the light of the graph theory, augmented Markovian delay-product-term Lyapunov-Krasovskii functional (DPTLKF), and dissipativity analysis, the consensus tracking problems are proposed in this article. In addition, the corresponding controllers are obtained from dissipativity-based consensus conditions. The primary contributions are summarized as follows.
1) Compared with the Markovian switching topologies in [3], [44], [45], and [51], the generally uncertain Markovian switching topologies of NMASs are investigated in this article, which is more practical because it is hard to precisely estimate the transition rates due to the costs and the stochastic factors.
2) The DPTLKF and higher order polynomial-based relaxed inequality (HOPRII) with a tighter lower bound are adopted in this article to tackle the consensus tracking control issues of delayed NMASs, and more information about time-varying delay and its derivative could be contained in the dissipativity-based consensus conditions. 3) Each agent of MASs is nonlinear and time-varying system, which is capable of providing more nature representations than the normal linear MASs. In addition, ETS is only required to be computed at sampling instants, which could reduce the communication loads. The structure of this article is as follows. Some preliminaries and LFNMASs are presented in Section II. In Section III, the novel dissipativity-based consensus conditions of NMASs are proposed based on DPTLKF and ETS. The validity of the proposed results is demonstrated by the simulation examples in Section IV. The summaries are proposed in Section V.

A. Notation
R p and R p×q are p real vector space and p × q real matrices set, respectively. · is regarded as the Euclidean norm. ⊗ represents the Kronecker product. Z T is regarded as the transpose of matrix Z. Z > 0 is regarded as that Z is a real symmetric positive-definite matrix.
For any matrices Z 1 , Z 2 , and Z 3 , B. Algebraic Graph Theory G = (V, E, A) is a directed graph between the N followers. Here, V = {v 1 , v 2 , . . . , v N } denotes a set of followers.
This article primarily focuses on a directed graphG, including a leader v 0 and N followers. In addition,G comprises G, v 0 , and directed edges from leader v 0 to followers. The leader could not acquire the information from the followers, and the leader is a neighbor of part of the followers.
= diag{γ 1 , γ 2 , . . . , γ N } is the leader adjacency matrix to describe the graphG, where γ k > 0 if and only if the node v k could acquire the information from v 0 ; otherwise, γ k = 0.

C. Assumptions
Assumption 1 [56]: Each graphG, including the leader and followers, has a directed spanning tree with a root of the leader.
Assumption 2 [3]: All sampling periods of agents are synchronized by the same clock.
where "?" denotes the completely unknown transition rate. π ij and ij denote the estimate value and estimate error of the uncertain transition rate π ij , respectively. ij ≤ ij and ij ≥ 0.π ij and ij are known values. For ∀ i ∈ S, we denote The estimate value of π ij is known for j ∈ S} and S i uk = {j : The estimate value of π ij is unknown for j ∈ S}.
On the basis of the characteristics of the transition rates, the following cases are assumed.
Remark 1: Different from most existing TRM, the generally uncertain TRM (1), including both bounded uncertain and partially unknown elements, is considered for NMASs in this article, which is more general and applicable. When ij = 0, generally uncertain TRM (1) is simplified as a special case of partially unknown TRM.
The following NMASs with N following agents are considered in this article. The dynamics of the kth agent are given as where x k (t) ∈ R n , u k (t) ∈ R n , and ψ k (t) ∈ R n are the state, control input, and output signal of the kth agent, respectively.
The dynamics of the leader are described as where x 0 (t) ∈ R n and ψ 0 (t) ∈ R n represent the state and output of the leader.
To save network communication resources and make the NMASs reach consensus, a distributed ETS is equipped with each agent. If the ETS is satisfied, the agent could receive the sampled data from its neighbors where θ k > 0 means the threshold parameter. means the event-triggered matrix, which satisfies ∈ R n×n and > 0. h denotes the sampling period, and t k p denotes the pth eventtriggered instant of the agent k. t k p +βh represents the currently sampled instant and y k (t k for ∀ k is greater than or equal to the sampling period h. Thus, Zeno behavior does not happen.
Consider the LFMASs consensus protocol Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
where t ∈ [t k p , t k p+1 ) [qh, (q+1)h), and q denotes an integral. K(r t ) is the feedback gain matrix. a kl (r t ) is the element of the weighted adjacency A(r t ). γ k (r t ) is the element of leader adjacency matrix. γ k (r t ) > 0 if the kth agent could receive information from the leader; otherwise, γ k (r t ) = 0.
Remark 3: The ETS (4) is adopted to decrease needless communication. In addition, the generally uncertain Markovian switching topologies in this article, including the uncertain and unknown transition process, are more general than the Markovian switching topologies with completely known transition rates.
Define the next transmission instant t k p+1 for the kth agent According to (2), (3), and (7), one obtains the following error dynamics system: where ). The structure of the control system for error system (8) is given in Fig. 1.
1) Under the condition w k (t) = 0, the nonlinear multiagent error system (9) is the leader-following consensus. 2) With the zero-initial condition, scalar ρ > 0, the real symmetric matrices P ≤ 0, R, and real matrix T , the nonlinear multiagent error system (9) is strictly (P, T , R)-ρ-dissipative if the following inequality holds for any σ ≥ 0: , v(κ 1 ) = 0, and any matrix Z > 0, the following inequality holds: Lemma 2 (HOPRII, [59]): x(t) is a differentiable function in [a 1 , a 2 ] → R n for a time-varying scalar a(t) ∈ [a 1 , a 2 ]. For symmetric matrices Z l > 0 (l = 1, 2), any matrices N l > 0 (l = 1, 2), the following inequality holds: Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. Remark 4: According to [59], Lemma 3 provides a tighter lower bound, which is close to the integral term in (12). In addition, there is no positive requirement for matrix , which could provide with extra freedom.

III. MAIN RESULTS
In this part, with w k (t) = 0, the consensus issue of error system (9) is introduced first. After that, based on the strict (P, T , R)-ρ-dissipativity theory, the dissipativity-based consensus tracking problem is investigated by protocol (5).

A. Consensus Analysis
In this section, the tracking consensus of NMASs (2) and (3) with w k (t) = 0 is obtained based on DPTLKF and the consensus protocol.
If i ∈ S i k , according to π ii = − j∈S,j =i π ij , there is P j ≤ P ε (j, ε ∈ S i uk ), and one has Then, one obtains Thus, one obtains Hence, one obtainŝ Hence, from (21)- (27), one obtains E{φ γ } < 0 (γ = 1, 2, 3, 4). From (60), we obtain the feedback gain matrices as follows: Remark 5: Compared with [3], [51], and [55], the generally uncertain Markovian switching topologies of MASs are investigated in this article, which is more practical because it is hard to precisely estimate the transition rates due to the influence of device performances and the uncertainties of the external environment. Compared with the authors' past paper with generally uncertain Markovian jumping [49], [62], without adopting the Schur complement and matrix inequality lemma to deal with generally uncertain Markovian, only one set of relaxation variables is utilized in this article, which could reduce the computational complexity of consensus conditions.

B. Leader-Following Dissipativity Consensus Control
Under the condition w k (t) = 0, the strictly (P, T , R) − ρ dissipative capability is addressed to reach the leader-following consensus of NMASs (2) and (3).
Letting ς(t) = [ξ T (t), w T (t)] T , from (29)-(62), one obtains From (82)-(96), the Schur complement lemma, and the proof of Theorem 1, it implies that For (100), integrating [0, σ ], one obtains According to Dynkin's formula, one has For any σ > 0, we have Thus, the strictly (P, T , R) − ρ-dissipative capacity of the leader-following error system (9) is achieved. From Definitions 1 and 2, the leader-following dissipativity consensus issue of NMASs (2) and (3) is solved by consensus protocol (5). According to Theorem 1, we obtain the following feedback gain matrices: Remark 6: Based on DPTLKF and ETS, the dissipativitybased consensus issue of delayed NMASs with generally uncertain Markovian switching topologies is investigated for the first time in this article. In addition, more information about time-varying delay and its derivative is contained in the dissipativity-based consensus conditions by utilizing DPTLKF and HOPRII.
Remark 7: In view of [63] and [51], H ∞ criteria could be obtained by setting P = −I, T = 0, and R − ρ = γ 2 I, and the passivity problem could be obtained by setting P = 0, T = I, and R − ρ = γ I.
Remark 8: Because of the uncertainty disturbance and the weak communication of the current and wave on the sea, the Markovian switching topologies and event-triggered transmission strategy in this article could be used in remotely operated underwater vehicles (ROVs). As a kind of NMS, the tracking control of ROVs was investigated [64] based on the motion modelM kηk +C k (η k )η k +D k (η k )η k +ḡ k (η k ) = τ k (k =  1, 2, . . . , N). k = 1 is selected as the leader agent, whose aim is to track the target point. Meanwhile, other ROVs are
selected as the following agents, which keeps the desired relative state from the leader agent. Based on the method in this article and the following ROVs, the distributed controller based on the Markovian switching topologies could be designed N). To save the network communication resources and achieve tracking control of ROVs efficiently, the distributed ETS for each following ROV could be designed as . When we establish the appropriate Lyapunov-Krasovskii functional, the corresponding position tracking control conditions of the ROVs could be obtained based on the Markovian switching topologies and event-triggered transmission strategy.
From Theorem 2, the dissipativity-based consensus tracking design algorithm is given in Algorithm 1.

IV. SIMULATION
In order to verify the proposed results, two simulation examples based on the generally uncertain Markovian switching topologies and ETS are presented in this section.
Example 1: Consider the consensus tracking control of the following time-varying NMASs: where  With w k (t) = 0, this example focuses on the consensus tracking control problem. The communication network between the agents is expressed as a set of Markovian switching topologies in Fig. 2, which is dominated by a generally uncertain Markovian switching process with three modes. The leader agent is denoted as 0. The follower agents are denoted as 1, 2, . . . , 7.
Example 2: With w k (t) = 0, the dissipative capacity control issue of the leader-following NMASs (3) and (105)    When w k (t) = 0, the error states and event-triggered instants are given in Figs. 5 and 6. From Figs. 5 and 6 and Definitions 1 and 2, the dissipative consensus control is achieved by ETS (4) and consensus protocol (5).

V. CONCLUSION
In this article, the dissipativity-based consensus tracking control (DBCTC) problems of LFNMASs with time-varying delays have been investigated. Different from previous topologies, the Markovian switching topologies with uncertain and partially unknown transition rates were considered, which is more general. The ETS protocol has been established to reduce unnecessary communication when obtaining more information from the agent network. Based on DPTLKF and HOPRII, some consensus tracking control criteria have been proposed to guarantee the leader-following consensus. Moreover, the consensus gain matrices have been obtained by solving the LMIs. Finally, two simulation examples have been presented to illustrate the validity of the theoretical results. In the future, we would like to focus on the distributed fault-tolerant containment control of the delayed NMASs under the adaptive event-triggered transmission strategy, as well as the consensus control of generally uncertain Markovian NMASs against denial-of-service attacks and actuator faults.