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Eventtriggered Hinfinity finitetime consensus control for nonlinear secondorder multiagent systems with disturbances
Advances in Difference Equations volume 2021, Article number: 315 (2021)
Abstract
The study considers the problem of finitetime eventtriggered Hinfinity consensus for secondorder multiagent systems (MASs) with intrinsic nonlinear dynamics and external bounded disturbances. Based on the designed triggering function, a distributed eventtriggered control strategy is presented on the basis of the designed triggering function to ensure consensus in the system, which effectively reduces the data transmission. Then, sufficient conditions for the finitetime consensus with Hinfinity performance level of the resulting eventtriggered MAS are derived by utilizing the Lyapunov function and finitetime stability theory. Furthermore, Zeno behavior is proven to be excluded under the proposed eventtriggered scheme. Finally, the validity of the proposed results is verified by numerical simulation.
Introduction
In recent years, multiagent systems (MASs) have been widely applied in many practical fields, such as unmanned aerial vehicles (UAVs) [1, 2], collective control [3], sensor network [4], and multiautonomous robot [5]. These systems have also attracted research interests from scholars in many fields. Consensus is an important and basic problem in MAS research. The consensus problem for MASs has been solved in many practical fields, such as UAV cruising and robotic arms. Therefore, achieving consensus among MASs is important and has become a major research area in this field in recent years [6–10].
At present, numerous advancements, such as asymptotic consensus, exponential asymptotic consensus, and finitetime consensus, have been made in MAS consensus control studies. In practical applications, MASs must achieve consensus within a limited amount of time. Therefore, the consensus convergence rate is a key performance index that should be considered by researchers. Finitetime consensus has many advantages over asymptotic consensus, such as higher accuracy, better robustness, and faster response times. Many researchers have been attracted to this field due to the good performances of finitetime implementations. Several interesting results have been obtained for different models. The finitetime consensus problem for a firstorder MAS with a continuous timevarying interaction topology was studied in [11]. Zhang et al. [12] studied the finitetime control of firstorder MASs. The abovementioned works focused on the finitetime consensus problem of firstorder MASs.
However, the model of the firstorder MASs has some limitations in practical application. Some highspeed MASs accrue large errors when the firstorder MAS model is used to describe the system for control. Thus, the description of the secondorder system could better represent the essence of object motion change. In recent years, many scholars have studied consensus control problems of secondorder MASs. For example, the author of [13] studied the finitetime consensus of secondorder MASs, and the robust finitetime consensus problem in secondorder nonlinear dynamic MASs was investigated in [14].
In the aforementioned studies, MAS consensus control mainly adopts continuoustime state feedback control. In our case, the agent controller is also in the field in some cases, that is, the agent must work in a limited energy environment, where the CPU frequency, memory capacity, and communication bandwidth in the control device are limited. These problems must be considered during the controller design to extend its service life and preserve communication capacity among agents. Therefore, changing the controller mode has become an important research topic. In response to the abovementioned issue, an eventtriggered control strategy has been proposed. Since its introduction, this method has become an important research area, has been widely applied in many fields, and research on MAS consensus based on eventtriggered control has obtained some achievements. For general linear firstorder MASs, the authors of [15–19] studied the eventtriggered finitetime control algorithm that can adjust the expected convergence time. The author of [20] provided a selftriggering algorithm to ensure consensus in the system.
However, the aforementioned research results apply mainly to eventtriggered firstorder MAS finitetime consensus control; thus, some scholars are currently studying the consensus control of secondorder MASs based on eventtriggered strategies. For example, Qian [21] emphasized that the eventtriggered control strategy is an effective way to reduce agent energy consumption that can significantly extend the operating life of MASs. Cao [22] proposed a distributed eventtriggered control strategy to ensure consensus in MASs within a certain period of time. The authors of [23] investigated the distributed finitetime consensus problem for a class of secondorder MASs under bounded perturbations and provided a continuous homogeneous finitetime consensus protocol based on nominal multiple agents. However, the multiagent models studied in these works ignore nonlinear dynamics and disturbance factors.
The dynamics of agents are complex. Consensus control for nonlinear MASs has also been studied given that many systems contain nonlinear dynamics. For example, an eventriggered sliding mode controller was proposed in [24] to achieve finitetime consensus for a firstorder nonlinear leaderfollowing MAS. Chen [25] proposed a distributed protocol based on relative position information for secondorder MASs with inherent nonlinear dynamics and communication delays. Accurately expressing the MAS model is difficult in many cases because the system may inherently be affected by uncertain factors, such as modeling errors and parameter fluctuations. Therefore, studying the MAS consensus control of systems with uncertain parameters is important. For example, Su and Huang [26] studied the consensus problem for leaderfollowing MASs by viewing it as an adaptive stability problem for an explicit error system to achieve consensus under unknown parameters. The author of [27] investigated the consensus control of MASs with uncertain model parameters.
MASs may also be subject to external disturbances in the field, such as while sending, transmitting, or receiving information. Therefore, nonlinear dynamics, parameter uncertainties, and external disturbances should be considered for practical MAS applications. However, the abovementioned works consider only nonlinear MAS and fail to consider external disturbances.
A few research achievements have been made on the finitetime consensus of secondorder MASs with disturbances. For example, Zhang [28] proposed a finitetime consensus problem for secondorder MASs with external bounded disturbances. The author assumed that a disturbance could be represented by a bounded constant. However, this assumption limits the negative influence produced by the disturbance and has difficulty fulfilling the purpose of precise control in some cases. Therefore, Hinfinity control methods that can effectively suppress the negative effects of disturbance have been proposed and have been successfully applied in some engineering fields [29]. So far, a lot of results have been published on this issue. For example, Jia and Huan [30, 31] studied firstorder and higherorder MASs with robustness for the Hinfinity consensus problem under external disturbances. Ban et al. [32] explored a firstorder MAS with leader for the finitetime Hinfinity consensus problem and introduced a nonlinear finitetime Hinfinity tracking control protocol. These works focused on firstorder systems, while the authors of [33] examined a class of secondorder MASs with a distributed Hinfinity composite spinning consensus problem. However, all these works ignored the finitetime consensus problem.
In recent years, a number of results on eventtriggered control have been derived for secondorder MASs with disturbance. However, to the best of our knowledge, the problem of eventtriggered Hinfinity finitetime consensus of nonlinear secondorder MASs with disturbance is rarely studied, which is the main motivation of the study. Compared with some previous relevant works, the main contributions of this study are summarized as follows:

1.
The problem of eventtriggered strategy control for a class of secondorder nonlinear MASs with external disturbances is addressed. To the best of the authors’ knowledge, few results on this topic for such systems are available;

2.
Under the proposed control protocol and distributed eventtriggered strategy, sufficient conditions are derived such that the MAS under study not only can achieve consensus but also can meet the requirements of suitable performance. Moreover, an Hinfinity optimal control algorithm that provides robust and dynamic characteristics for the secondorder MASs is proposed. Distributed eventtriggered control is proven to avoid Zeno behavior.
Problem description and preliminaries
This section presents the basic concepts of some algebraic graph theory and useful theorems.
Algebraic graph theory
The communication topology among agents in MAS can be modeled by graph theory, where each agent is a node and each communication path is an edge. Let \(G = \{ V,E,A\} \) be an undirected graph, where \(V = \{ 1,2,\ldots,n\} \) is a set of vertices, \(E \in V \times V\) is a set of edges, and \(A = {[{a_{ij}}]_{N \times N}}\) is a weighted adjacency matrix with weights \({a_{ij}} \ge 0\) for \(\forall i,j \in V\). If a path exists between node i and node j, then edge \((i,j) \in E\) and \({a_{ij}} = {a_{ji}} > 0\). The elements \({a_{ii}} = 0\) for all \(i \in V\) mean that no selfloops are present. If an edge exists between node j and node i, then node j is a neighbor of node i. The neighboring set of node i is \({N_{i}} = \{ j \vert {(i,j) \in E\} } \). The Laplacian matrix L of graph G is denoted as \(l_{ij} =  {a_{ij}}\) for \(j \ne i\) and \(l_{ii} = \sum_{j = 1,j \ne i}^{N} {{a_{ij}}} \). Simultaneously, a pair of angular matrices can be defined as \(D = \operatorname{diag}\{ d_{1},d_{2},\ldots,d_{N}\} \) with \(d_{i} = {\deg _{in}}({v_{i}})\). Then, the Laplacian matrix for the undirected graph G can be defined as \(L = D  A\).
Notation: The following notations will be used. Let R denote the real numbers set and \({R^{n}}\) denote the ndimensional real vector space. Given a vector \(x = [ {x_{1}},{x_{2}},\ldots,{x_{n}}{]^{T}} \in {R^{n}}\), denote \(\operatorname{sig}{(x)^{\alpha }} = {[\operatorname{sign}({x_{1}}){ \vert {{x_{1}}} \vert ^{\alpha }},\ldots, \operatorname{sign}({x_{n}}){ \vert {{x_{n}}} \vert ^{\alpha }}]^{T}}\), where \(\operatorname{sign}( \cdot )\) is the signum function, and \(\vert {{x_{i}}} \vert = {[ \vert {{x_{1}}} \vert \vert {{x_{ {2}}}} \vert ,\ldots, \vert {{x_{n}}} \vert ]^{T}}\).
Problem description
In a secondorder MAS, n dynamic agents in continuous time share a common state space R with all agents. \({x_{i}}\) represents the position state of agent i, and \({v_{i}}\) represents the degree state of agent i. The dynamic behavior of the agent i can be described as follows:
where \({x_{i}} \in {R^{n}}\), \({v_{i}} \in {R^{n}}\) denote the positive and velocity, respectively. \({u_{i}}(t) \in {R^{n}}\) is the control input of agent i. \(f(t,{x_{i}}(t),{v_{i}}(t))\) represents the nonlinear dynamic function of the ith agent. \({w_{i}}(t)\) is the exogenous disturbance input that satisfies \(\int _{0}^{\infty }{w_{i}^{T}(t)} {w_{i}}(t) < d\), \(d \ge 0\), \(i = 1,2,\ldots,n\).
Remark 1
In [16], secondorder multiagent consensus research was considered, but nonlinear functions and disturbance terms were ignored by the model. In [13], consensus under disturbance with MASs was studied, but the nonlinear dynamics of the system were ignored. In contrast, the model in the present study considers disturbance and nonlinear dynamics.
According to the relevant information, output control is defined as follows:
and \(z(t) = \frac{1}{{\sqrt{n} }}{ [ {({x_{i}}(t)  {x_{j}}(t) + \gamma ({x_{i}}(t)  {x_{j}}(t)))} ]^{\alpha }}\). The Hinfinity performance indicators refer to \(J(w) = \int _{0}^{T} {({z^{T}}(t)z(t)  {\delta ^{2}}{w^{T}}(t)} w(t))\,dt\), and δ is a positive number.
For system (1), we make the following two assumptions.
Assumption 1
The connection diagram of MAS (1) is undirected and connected.
Assumption 2
There exists a positive number μ, such that \(\Vert {f(t,{x_{i}}(t),{v_{i}}(t))} \Vert < \mu \), \(i = 1,2,\ldots,n\).
Definition 1
The finitetime consensus is achieved for secondorder systems (1) if for any initial conditions we have \({\kappa _{1}} > 0\), \({\kappa _{2}} > 0\) and a finite time T such that \(\vert {{x_{i}}(t)  {x_{j}}(t)} \vert < {\kappa _{1}}\) and \(\vert {{v_{i}}(t)  {v_{j}}(t)} \vert < {\kappa _{2}}\) if \(t \ge T\), where \(i,j \in V\).
Definition 2
Nonlinear MAS (1) with the eventtriggered state feedback controllers (4) is said to be FTB with a prescribed finitetime Hinfinity performance level \(\delta > 0\) if the following conditions hold:

1.
MAS (1) with eventtriggered state feedback controllers (4) is finitetime bounded.

2.
Under zeroinitial condition \(\forall t \in [0,T]\), the controlled output \(z(t)\) satisfies
$$\begin{aligned} \int _{0}^{T} {{z^{T}}(t)z(t)\,dt < { \delta ^{2}}} \int _{0}^{T} {{w^{T}}(t)w(t)\,dt} . \end{aligned}$$(3)
The following lemmas are used in this study.
Lemma 1
([15])
and \({x_{i}} \in R\), \(0 < p \le 1\).
Lemma 2
([12])
Considering the system \(\dot{x} = f(x)\) with \(f(0) = 0\), \(x(0) = {x_{0}}\), \(x \in {R^{n}}\), if a positive definite continuous function \(V ( x ):U \to R\) exists, then we have real numbers \(c > 0\), \(\alpha \in (0,1)\), and \(d \ge 0\) such that
Then \(V(x)\) is finitetime bounded. If \(d = 0\), then \(V(x) \equiv 0\) for all \(t \ge T\), where the settling time T is determined as
Lemma 3
([16])
If the undirected graph of MAS (1) is connected, then the Laplacian matrix L is symmetric. \({\lambda _{1}} \le {\lambda _{2}} \le \cdots \le {\lambda _{n}}\) are defined as the eigenvalues of matrix L. Then \({\lambda _{1}} = 0\) and \({\lambda _{2}} > 0\). The algebraic connectivity is defined as follows: if \({1^{T}}r = 0\), \(r \ne 0\), then \(a = {\lambda _{2}} = \min \frac{{{r^{T}}Lr}}{{{r^{T}}r}}\), \({r^{T}}{L^{2}}r \ge a{r^{T}}Lr\).
Lemma 4
([24])
For \(x \in {R^{+} }\), \(y \in {R^{+} }\), and \(0 < h < 1\), the inequality \({(x + y)^{h}} \le {x^{h}} + {y^{h}}\) holds.
Lemma 5
([21])
For any positive numbers c, d, and any real numbers a, b,
Main results
This section presents the design of an appropriate eventtriggered protocol for a nonlinear secondorder MAS with external disturbances. To ensure consensus in a finite time in the system, we design a new finitetime control protocol based on the eventtriggered strategy:
where \(t \in [{t_{k}},{t_{k + 1}})\), \(k = 0,1,\ldots\) , and \(0 < \alpha < 1\), \(\beta > 0\), \(\gamma > 0\).
In the interval \(t \in [t_{k}^{i}, t_{k + 1}^{i})\), the state combination of agent i is given as follows:
The measurement error is
The combined measurement error is
MAS (1) can be rewritten as
Finitetime consensus protocol (4) can be converted to
The event triggering function of multiagent i is set as
where \(\sigma > 0\), and \(\Vert L \Vert \) denotes the 2norm of Laplacian matrix L. Then, the triggering condition is defined as
For system (1), the Lyapunov method is used to study the finitetime consensus under eventtriggered control when \({w_{i}}(t) = 0\).
Theorem 1
Let the assumption be that Assumption 1is satisfied and the undirected graph of MAS (1) is connected. With the eventtriggered control algorithm (4) and the triggering function (13), if suitable positive scalars β and γ exist, then the finitetime consensus problem can be solved when the following conditions are satisfied:
where \(0 < \sigma < 1\), \(\gamma > 0\), \(\beta > 0\), \(0 < \alpha < 1\), and \(\eta > 0\).
The finitetime T can be estimated using the following inequalities:
where \(V ( 0 ) = \sum_{i = 1}^{n} { \frac{\beta }{{1 + \alpha }}} { \vert {{q_{i}}(0) + \gamma {p_{i}}(0)} \vert ^{1 + \alpha }}\).
Proof
A Lyapunov function is established for MAS (1) as follows:
The derivative of \(V(t)\) is
Denote
From Lemma 4
Because of
we can obtain
For \(i \ne j\), \({l_{ij}} < 0\), which means
define \({\Phi _{i}}(t) = {q_{i}}(t) + {E_{xi}}(t) + \gamma ({p_{i}}(t) + {E_{vi}}(t)) = {q_{i}}(t_{k}^{i}) + \gamma {p_{i}}(t_{k}^{i})\), \({\Phi _{i}} = { [ {\Phi _{1}^{T},\Phi _{2}^{T},\ldots,\Phi _{n}^{T}} ]^{T}}\).
Thus, according to the eventtriggered function
we have
According to Assumption 2, we can find that
It is given by Lemma 3
and \({c_{i}} = \vert {\sum_{j \in {N_{i}}}^{n} {{l_{ij}}} } \vert \), \({c_{\max }} = \mathop{\max }_{i \in V} {c_{i}}\), \({c_{\min }} = \mathop{\min }_{i \in V} {c_{i}}\)
where \(\eta = \gamma {2^{\alpha }}{(\frac{1}{{1  \sigma }})^{\alpha }}\rho {(1 + \alpha )^{\frac{{2\alpha }}{{1 + \alpha }}}}{\beta ^{ \frac{2}{{1 + \alpha }}}} \), and \(\eta > 0 \). Then we have
and
According to Definition 1, the secondorder MAS (1) with control protocol (4) and eventtriggered condition (13), the system can achieve finitetime consensus, and \({w_{i}}(t) = 0\).
When \({w_{i}}(t) \ne 0\), we will prove that MAS (1) has an Hinfinity performance. □
Theorem 2
Let Assumption 1be satisfied and the undirected graph of MAS (1) be connected. With the eventtriggered control algorithm (4) and the triggering function (13), if suitable positive scalars β and γ exist, then the finitetime Hinfinity tacking consensus problem can be solved when the following conditions are satisfied:
Proof
In view of the proof of Theorem 1, we have
And
Then we can obtain
According to condition (27), we have
By \(V(t) \ge 0\), \(V(0) = 0\), we have \(\int _{0}^{T} {{z^{T}}(t)z(t)  {\delta ^{2}}{w^{T}}(t)} w(t)\,dt < 0\), \(\Vert {z(t)} \Vert _{2}^{2} < {\delta ^{2}} \Vert {w(t)} \Vert _{2}^{2}\). Thus, from Definition 2, the multiagent system (1) has an Hinfinity performance level δ, and the proof of this theorem is completed. □
Remark 2
Theorem 2 shows that, when the design parameter δ is closer to the optimal value \({\delta _{\mathrm{opt}}}\) [34] of the Hinfinity norm, the Hinfinity control performance is better and the antiinterference is stronger. Our subsequent simulation results also verify this conclusion. The conclusion of Theorem 2 also indicates that, when the controller ensures consensus in the MAS, the selected control gains β and γ are larger and the performance of Hinfinite control is better. However, when the control gain is greater, the cost of control is higher.
Corollary 1
The following MAS is considered:
with the eventtriggered control algorithm (4) and the triggering function (13), if there exist suitable positive scalars β and γ, the finitetime consensus problem can be solved when the following conditions are satisfied:
where \(\gamma > 0\), \(\beta > 0\), \(0 < \alpha < 1\), \(0 < \sigma < 1\).
In addition, the finite time T can be estimated using the following inequality:
where \(V ( 0 ) = \sum_{i = 1}^{n} { \frac{\beta }{{1 + \alpha }}} { \vert {{q_{i}}(0) + \gamma {p_{i}}(0)} \vert ^{1 + \alpha }}\).
Remark 3
In [24], the consensus control problem was studied for a nonlinear secondorder MAS based on the eventtriggered mechanism. The current study additionally considers external disturbances. The author of [28] investigated the finitetime consensus of secondorder MASs with disturbances. However, the disturbances were limited to positive numbers, which brings difficulty in achieving accurate control. In [31],a firstorder MAS was studied with disturbance for finitetime Hinfinity consensus, but this study ignored eventtriggered strategy control.
Theorem 3
MAS (1) with an eventtriggered function (13) and control strategy (4) is considered. A positive lower bound \({T_{\min }}\) of the eventtriggered execution interval is given as follows:
Thus, each agent i can avoid Zeno behavior before consensus is achieved.
Proof
At \(t = t_{k}^{i}\), the controller of agent i updates its control output. Thus, the measurement error is set to 0, that is, \(\vert {{e_{ix}}(t_{k}^{i})} \vert = 0\), \(\vert {{e_{iv}}(t_{k}^{i})} \vert = 0\). During the interval \([t_{k}^{i},t_{k + 1}^{i})\), we have
When the event is triggered, we have
Theorem 1 indicates that \(\vert {{q_{i}}(t_{k}^{i}) + \gamma {p_{i}}(t_{k}^{i})} \vert > 0\) before the system trajectory reaches consensus. Thus, we have
We can conclude that \(t_{k + 1}^{i}  t_{k}^{i} > 0\) before consensus is achieved. In turn \(t_{k + 2}^{i}  t_{k + 1}^{i} > 0\). Thus, the Zeno behavior can be excluded. □
Remark 4
At the current moment \(t_{k}^{i}\), given that the agents have not yet achieved consensus, \({q_{i}}(t_{k}^{i})\) and \({p_{i}}(t_{k}^{i})\) are not equal to 0. Obviously, they are certain constants, and the next triggering time \(t_{k}^{i}\) is determined by constants such as \({q_{i}}(t_{k}^{i})\) and \({p_{i}}(t_{k}^{i})\). Theorem 3 proves that the size of \((t_{k + 1}^{i}t_{k}^{i})\) satisfies formula (35). Obviously, the righthand side of formula (35) is a certain constant. Therefore, Zeno behavior can be avoided.
Numerical simulation
This section presents a numerical example to verify the theoretical results. Figure 1 shows the undirected connection topology of a MAS with five nodes.
The Laplacian matrix L is
We set \(\alpha = 0.1\), \(\beta = 0.5\), \(\gamma = 0.3\), \(\sigma = 0.5\), and \(D = \operatorname{diag}([ 3\ 3\ 2\ 2\ 4])\). To satisfy the conditions of Theorem 1, we set the initial state of system (1) to \({x_{1}}(0) = {[1,  1]^{T}}\), \({x_{2}}(0) = {[0.5,  0.5]^{T}}\), \({x_{3}}(0) = {[0.2,  0.8]^{T}}\), \({x_{4}}(0) = {[0.7,  0.4]^{T}}\), and \({x_{5}}(0) = {[0.2,  0.8]^{T}}\). The initial value of velocity for the five agents is set to 0. The simulation results of the system are given below.
Figures 2 and 3 represent the state and velocity trajectory diagrams of all the agents under controller (4). The figures show that all agents can achieve consensus under the eventtriggered control strategy. As shown in Fig. 2, each agent’s controller updates at its own event time only and remains unchanged during the triggering interval. Figure 4 shows the input to the distributed controller. Figures 5–9 show the measurement error of agents \({x_{1}}\), \({x_{2}}\), \({x_{3}}\), \({x_{4}}\), and \({x_{5}}\) under the eventtriggered control strategy in finite time. This figure indicates that the system error converges to 0 in finite time. Thus, the system can achieve consensus in finite time. Figure 10 shows the event triggering interval of each agent under the eventtriggered strategy (13). Figure 11 shows that the energy of the output signal \(z(t)\) is smaller than that of the external disturbance \(w(t)\). The results of the numerical simulation verify the validity of the conclusion. The designed controller and algorithm can ensure consensus in a MAS in finite time.
Conclusion
Eventtriggered finitetime Hinfinity consensus has been studied for secondorder multiagent nonlinear systems with external disturbances. An eventtriggered strategy has been introduced to save communication resources. The data can be sampled by the system only when the eventtriggered condition is satisfied. A sufficient condition on finitetime consensus has been obtained by employing the Lyapunov method and analysis technology. A theoretical analysis proves that the designed finitetime controller can suppress the influence of disturbances on the system and satisfy the robust Hinfinity performance. The analysis also proves that the system has good antiinterference performance under external disturbance. Moreover, the Zeno behavior can be avoided given that a positive lower bound of the eventtriggered execution interval is ensured. The validity of the proposed method has also been verified by numerical simulation. In future research, we will guarantee cost finitetime consensus of secondorder uncertain MAS based on distributed eventtriggered strategy.
Availability of data and materials
The data used to support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
The authors are thankful to Nature Research Editing Service for providing professional language editing services in the improvement of the present work.
Funding
This work was jointly supported by the National Natural Science Foundation of China (Grant No. 11972156) and Natural Science Foundation of Hunan Province (Grant No. 2017JJ4004).
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Luo, Y., Zhu, W. Eventtriggered Hinfinity finitetime consensus control for nonlinear secondorder multiagent systems with disturbances. Adv Differ Equ 2021, 315 (2021). https://doi.org/10.1186/s1366202103467w
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DOI: https://doi.org/10.1186/s1366202103467w
Keywords
 Multiagent systems
 Finitetime consensus
 Eventtriggered control
 Hinfinity consensus
 Disturbances