Distributed event‐triggered output feedback H∞ control for multi‐agent systems with transmission delays

Correspondence Hui Yu, College of Science, China Three Gorges University, Yichang, China. Email: yuhui@ctgu.edu.cn Abstract The output feedback H∞ consensus control problem of multi-agent systems is studied using an event-triggered control strategy. Two types of transmission delays, one from the system output to the output feedback controller (OFC) and the other from the OFC to the zero-order holder, are considered. This causes the OFC and the system not to be updated in the same time intervals. An interval dividing approach is applied to such that the whole system can be updated in the same time intervals. An event-triggered OFC with H∞ performance is proposed for multi-agent systems to achieve consensus. By constructing an appropriate Lyapunov–Krasovskii functional, sufficient conditions based on linear matrix inequality are derived to guarantee the consensus achievement. Finally, the theoretical results are validated using computer simulation.


Introduction
Multi-agent systems have aroused extensive attention due to their autonomy, fault tolerance, flexibility, extensibility and collaboration. In recent decades, coordination of MASs has been extensively applied in different fields such as formation control, flocking, software development, multi-robot coordination and smart grids [1][2][3][4]. Consensus means that all the agents can reach a common value by only local information exchange. Many scholars have carried out a series of researches on the related issues of consensus from different aspects, such as the problem of finite-time consensus [5][6][7][8], consensus with time-varying delays [9][10][11], and consensus with different topologies [12][13][14], to name just a few. The main idea of ETC strategy is to use the opportunistic aperiodic sampling instead of the classic periodic sampling to improve the efficiency. The ETC method uses a trigger function to replace the time constant in classic periodic sampling. When system is still running under the ideal state, the event will not be triggered. Otherwise, it will be triggered. As a result, ETC method can reduce the frequency of information transmission between agents to save energy. Therefore, how to accurately determine the updating time instants of control signals is the key to study this kind of problems. In 1999, [15] and [16] first proposed the ETC method. In 2012, [17] adopted centralized and distributed ETC method to analyze the consensus problem of MASs, respectively. Since then, more and more scholars have applied event-triggered strategies to MASs with different topologies [18][19][20], such as output feedback control [20][21][22][23][24], H ∞ control [25][26][27][28][29][30] etc, and have achieved fruitful research results in this field.
Event-triggered H ∞ consensus control is an important aspect for MASs, which has been deeply studied by a large number of literatures so far. In [25], the consensus control of MASs with switched topologies is investigated. Considering the uncertainty of communication networks in practical application, an event-triggered H ∞ consensus controller is proposed in switching networks subject to Markov chains using local information exchange via state-feedback. A sufficient condition based LMI for H ∞ consensus is given. In [26], aperiodic and periodic ETC methods are proposed for MASs to achieve H ∞ consensus. The event-triggered method is combined with the time-triggered method, and a fixed lower limit of sampling time interval is given to guarantee the avoidance of the Zeno behaviour. In [27], H ∞ control of MASs is investigated in directed networks via ETC method. In the case with external disturbances, a new distributed sampling method is proposed, and the Zeno-behaviour is completely excluded. In [28], the H ∞ consensus problem of MASs with missing measurements and external disturbance is considered, in which the considered system is in discrete-time and time-varying. Redundant channels are introduced to enhance the reliability of information transmission. An observer-based ETC method is proposed to reach consensus with H ∞ performance in a limited range. In [29], the H ∞ consensus control for discrete-time MASs with Markov switching topology is studied. An ETC strategy is proposed, which takes into account the influence of information exchange between neighbors and the channel noise due to environmental uncertainty. In [30], the consensus problem for MASs with external disturbance is investigated based on event-triggered scheme. A control algorithm is presented to achieve the control object by defining a control output to turn the consensus problem into H ∞ one. Time-delay is also a key factor in information transmission in practical applications. In the literatures mentioned above, only part of them consider the information transmission delay and the others do not. The above literature analysis inspires us to do the work in this paper, in which two kind of transmission delay are considered.
The output feedback H ∞ consensus problem of MASs is considered in this paper based on ETC strategy. Using the ETC method, the output signal is sampled and transmitted to the OFC side, and then sampled and transmitted to the ZOH. There are two kinds of transmission delays in this process, one from the output of system to the OFC and the other from the OFC to the ZOH. This causes the output feedback controller and system to be updated in different time intervals. By using interval decomposition method, the output feedback controller and system are unified into identical time intervals, and then the closed-loop system (CLS) of whole system is obtained.
Since the system states are not measurable, an observer-based event-triggered OFC is presented for the followers to follow the leader. By constructing a Lyapunov-Krasovsky functional, sufficient conditions for consensus convergence and H ∞ performance are obtained in the form of LMI. The contributions of this work are summarized as follows. First, a novel eventtriggered distributed output feedback controller is proposed for MAS to achieve leader-following consensus. In the proposed algorithm, both the controller and the trigger function are distributed only depending on the local information of the neighboring agents. Second, sufficient conditions based on LMI are derived to guarantee asymptotic stability and H ∞ performance of the considered system. The algorithms based on LMI to solve ETC problem were also proposed in [31][32][33][34], however, only one kind of transmission delay is considered in these literatures. Third, compared with [25-31, 35, 36], two kinds of transmission delay are considered in this paper. As far as we know, the work in this paper has rarely appeared in the literature except for [32]. In [32], the ETC problem via output feedback is applied to network control systems to achieve H ∞ performance. Two kinds of transmission delay are also considered and then a kind of interval decomposition method is applied to acquire a unified closedloop system. However, due to the distributed requirement of MASs for controller and trigger function, the method proposed in [32] cannot be applied directly and the interval decomposition for MASs is more challenging.
The structure of this work is given below. In Section 2, we introduce some needed lemmas and concepts on algebraic graphic theory. The system model and problem are specified in Section 3. In Section 4, we propose the output feedback controller and analyze its stability. In Section 5, two instances of simulations are given to verify the feasibility of the results. We conclude this article in Section 6.

PRELIMINARIES
In multi-agent systems, a directed graph denoted by (, , ) is used to represent the communication relationship between agents, where vertex set  = { 1 , 2 , … , N } represents N agents, and  ⊂  ×  is the edge set. A directed edge ( j , i ) ∈  means that agent i can sense information from agent j , in other words, agent i can receive information from agent j . For the weighted adjacency matrix to be a diagonal matrix, and b i > 0, if agent i can sense the leader, otherwise b i = 0.
The following lemmas are useful in our theoretical analysis.
Lemma 1 [32]. For any positive definite matrix Q, if constant > 0, then in the interval [0, ], the following inequality holds for the integrable vector function (s): .
if and only if or equivalently where T 11 , T 12 , and T 22 are matrices with appropriate dimensions.

PROBLEM STATEMENT
Consider a class of MASs with N followers and a leader. The ith, i = 1, 2, … , N , follower's dynamic is where A ∈  n×n , B ∈  n×p , B ∈  n×q , C 1 ∈  r×n , and C 2 ∈  n×n are matrices, x i ∈  n , u i ∈  p , y i 2 ∈  n , y i 1 ∈  r , i ⊂  q are the state vector, controller, measured output, controlled output, and disturbance input, respectively. The dynamic of the leader labelled by 0 is We denote the release times of agent i by The output y i 2 (t ) takes h as the sampling period and samples at time instant kh, where h > 0. Two types of transmission delays are considered. One is the transmission delay from system output to output feedback controller, denoted by i k . The other is from the OFC to the ZOH, denoted by i k . We make the hypothesis that i k ∈ [0,̄) and i k ∈ [0,̄), wherēand̄are upper bounds of i k and i k , respectively. Without loss of generality, let̄= m 1 h and̄= m 2 h, where m 1 , m 2 > 0. Motivated by the works in [37], a novel event-triggered condition requiring only local information: is constructed to judge whether the output signal is being transferred to the OFC or not, Remark 1. When the inequality (3) holds, the sample output y i 2 (t i k h + p i h) of agent i will not be transferred to the OFC. Only when the inequality (3) fails to hold, it will be transmitted to output feedback controller. It can be seen from the information transmission mechanism that event-triggered design can save network bandwidth and energy. Obviously, when in (3) is equal to 0, it becomes time-triggered scheme as the special case of ETC scheme.

OUTPUT FEEDBACK H ∞ CONTROL VIA ETC STRATEGY
From the event-triggered condition (3), the (k Letx i andx 0 be the estimates of x i and x 0 , respectively, and construct observers aṡx An observer-based dynamical OFC is presented as the following: , and thenx i (t ) on [t i 0 , +∞) are continuous. For the same reason,x i (t ) is continuous on [t i 0 , +∞) as well.
Remark 3. Note that the event-triggered condition (3) and controller (8) are distributed depending only on local information of neighboring agents. The event-triggered control method is applied in this paper, which can reduce unnecessary energy consumption.
Because there are two type of time-delays i k and i k , the dynamic output feedback controller (7) is updated based on In other words, systems (7) and (9) are updated in different time intervals, so the CLS cannot be obtained from the two equations directly. In the following, the closed-loop system is derived by using an interval partition method. We divide the time interval of (9) using the updating time instants of (7).
Then we have the following interval decomposition: and From (11) and (12), the system (4), (9) and the dynamic OFC (7) can be rewritten aṡx and respectively.
We denoted the stack column vectors of In the following lemma, CLS is derived according to (10). (13) and (14), the following CLS can be obtained:

Lemma 3. Based on systems
where the functions (t ) and i (t ) will be determined later.
Proof. Similar to [37], we decompose the time interval  i . For From the definition of i , one has are considered. We can find some constant N 0, i such that (3). Then,  0, i can be divided into Define and Just like in Case 1, we can also obtain that Similarly,  i s and  1, i can be divided into and respectively, where and i = t i m k 2 . To facilitate the understanding of interval decomposition methods, an illustrative example is given in Figure 1. Let Define and By a similar analysis, we can obtain that From the definition of i (t ) and (3), for t ∈  i , we have that is, On the basis of the above analysis, we can easily derive that From (4) to (13), the error dynamics is given bẏ By (16) and (25), the CLS (17) can be obtained. □ Remark 4. The updating interval of (7) and (9) is different due to the transmission delays i k and i k . It is challenging for stability analysis. An interval decomposition method is used such that system (7) and (9) are updated in the same time interval.
Remark 5. In [31, 34 37], the interval decomposition method has also been used in the event-triggered control problem. The main difference is that we need to obtain a unified closed-loop system due to existing two kinds of transmission delay. In Lemma 2, a CLS is obtained. (17) and given > 0, if:
In the following, sufficient conditions based on LMI are given to ensure the existence of the H ∞ consensus OFC.

Lemma 4.
There exists an H ∞ consensus OFC (7) for system (1) and (2), if there exist matrices L, K , Ω > 0 and W > 0, and constants h > 0 and > 0 such that where The time derivatives of U 1 (t ) and U 2 (t ) along trajectories of CLS (17) arė From Lemma 1, we obtain that Let we havė anḋ Note that col col and col Then, (32) and (33) can be rewritten aṡ , , Moreover, by (38) and (39), we havė Define U 3 (t ) as: that is, From (38) to (41), we havė , From Lemma 2, the condition (27) is equivalent to In the case of t ∈ [t i 0 h, t i 0 h +̄+̄), the derivative of U 1 (t ) is given bẏ From (38) and (44), we havė The condition (27) implies that the matrix is negative definite. Therefore, from (45) and the above matrix, one haṡ , e(t ) are continuous on [t 0 , +∞), thus, U (t ) is continuous on [t 0 , +∞). If (t ) = 0, we geṫ U (t ) +ȳ 1 (t ) Tȳ 1 (t ) < 0, Therefore, when the disturbance vanishes, the CLS is asymptotically stable. Furthermore, lim t →∞x i (t ) = 0. Since U (t ) is continuous on [t 0 , +∞), integrating the inequality (43) from t 0 to t yields Using the 0 initial condition and when t → ∞, one has Thus, ‖ȳ 1 (t ) Tȳ The matrix inequality (27) with respect to W , L and K is not solvable. In the following theorem, we transform the matrix inequality (27) into an LMI-based feasible problem. Theorem 1. There exists an H ∞ consensus OFC (7) for system (1) and (2), if there exist matricesL,K , W > 0 and Ω > 0, and given constants h > 0 and > 0 such that wherē ) , Under this setting, the control gain and the observer gain are K = (B T B) Proof. LetK = WBK andL = WL. Then, we can obtain (46) from (27). □ Remark 6. The sufficient condition proposed in Theorem 1 for H ∞ consensus achieving is based on LMI. For LMI based algorithm, how to reduce its conservatism is an interesting topic worthy of further investigation in the future.

NUMERICAL SIMULATIONS
In this section, we give two examples to show the validity of results. Consider a MAS consisting of one leader and four followers shown in Figure 2. Choose the parametric matrices of the MAS as: Remark 7. For multi-agent systems, computational complexity is an important problem we face when the number of agents is large. However, the LMI (46) in theorem 1 can be solved offline, and the event-triggered condition (3) and controller (8) are distributed only depending on local information exchange, which greatly reduce the computational burden when the number of agents is large.

CONCLUSION
The consensus control of leader-following MASs is studied in this paper via event-triggered H ∞ consensus OFC. Due to taking two class of time-delay into account, the system and the output feedback controller have different update time intervals. By interval dividing, we obtain the CLS updated in the same time intervals. The event-triggered condition is adopted to reduce times of sampling and improve efficiency. Output feedback H ∞ control method is applied such that leader-following consensus is reached. In the future, it is important to reduce the conservatism of the sufficient conditions.