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Quasiconsensus of fractionalorder heterogeneous multiagent systems under eventtriggered impulsive control method
Advances in Continuous and Discrete Models volume 2022, Article number: 63 (2022)
Abstract
This paper investigates the quasiconsensus problem of fractionalorder heterogeneous multiagent systems, the distributed impulsive control protocol is designed for the multiagent system. In contrast to some existing results, the impulsive moments are determined by preset events, i.e., the eventtriggered mechanism is used. Based on the fractionalorder Lyapunov stability theory and fractionalorder differential inequality, the quasiconsensus criteria are derived; furthermore, the prescribed error bound is given. Then, Zeno behavior for the considered eventtriggered control method is excluded. Finally, numerical examples are given to shown the effectiveness of the proposed method.
1 Introduction
Multiagent systems have been a hot topic in the past decades, due to their wide applications in many different fields, such as unmanned aerial vehicles, multirobot formations, distributed optimization, etc. There are some results reflecting that modeling by fractionalorder differential equations would produce more accurate descriptions, for example, underwater robots that work on the ocean floor where microbes and sticky matter abound. Li has studied the consensus behavior of fractionalorder multiagent systems in [1] and [2]. Since then, many results have focused on consensus of fractionalorder multiagent systems (FOMASs), see for example [3–7] and references therein.
In a networked environment, communications among agents often block the channel under a continuoustransmission mechanism. Thus, discontinuous transmission mechanisms of information of agents have attracted much attention, in which, both timetriggered and eventtriggered methods have produced many significant results [8–10]. There are several kinds of timetriggered methods, such as impulsive control, intermittent control, sampleddata control, etc. All of them have been widely applied in the fractionalorder multiagent systems. See, for example [11–13] and references therein. The eventtriggered control method has been proposed by Tabuada in [14], and was first used for fractionalorder multiagent systems in [15]. Many outstanding results have been published recently [16–18].
Impulsive control makes the controlled systems convert their orbits just in some discrete instants and has extremely low cost. Noting that most of the existing results about impulsive control are time triggered, i.e., agents will change their states at some determined moments (periodic or aperiodic). A natural question is can we design the impulsive controllers based on an eventtriggered mechanism? In other words, agents will change their states at some moments when the preset events occur. The socalled “EventTriggered Impulsive Control (ETIC)” has aroused more and more attention in the last few years [19–21]. However, there is little research about ETIC for fractionalorder systems [22, 23].
In networked systems, the mismatched parameters for the subsystems are difficult to avoid. This phenomenon caused the heterogeneous multiagent systems to be widely investigated by researchers. For the fractionalorder multiagent systems, there are also a number of papers about heterogeneous models [24–26]. According to the discussion above, this paper will consider the quasiconsensus problem of fractionalorder heterogeneous multiagent systems via the ETIC method. The main contributions of this manuscript can be summarized as follows:
(1) This manuscript studies the consensus problem of fractionalorder heterogeneous multiagent systems using eventtriggered impulsive control, while most existing works about cooperative control for fractionalorder multiagent systems did not consider that the multiagent systems are heterogeneous.
(2) For the controllers given in this paper, the impulsive controllers based on an eventtriggered mechanism are provided, which can avoid the situation that impulsive instants for all agents should be always identical. Furthermore, Zeno behavior is successfully excluded.
(3) Distributed impulsive controllers are used, which can reduce channel blocking, under which, the bounded consensus criteria are given by some lowerdimensional matrix inequalities and scalar inequalities and a prescribed error bound is given.
The remainder of this paper is organized as follows. The preliminaries of fractionalorder calculus and problem formulation are introduced in Sect. 2. The quasiconsensus criteria for the considered fractionalorder multiagent systems are derived in Sect. 3. In Sect. 4, the effectiveness and feasibility of the developed methods are shown by two numerical examples. A concise discussion is given in Sect. 5.
Notations
Throughout this paper, \(I_{n}\) denotes an ndimensional identity matrix. \(\mathbb{R}^{n}\) denotes the ndimensional Euclidean space. \(\mathbb{R}^{m\times n}\) is the set of \(m\times n\) real matrices. ∗ stands for the symmetrical part in a matrix. \(\operatorname{diag}\{\ldots\}\) stands for a diagonal matrix. \( x  \) denotes the absolute value of x. \(\ \cdot \ \) denotes the Euclidean norm of the vector. \(\lambda _{\max}(P)\) and \(\lambda _{\min}(P)\) stand for the largest eigenvalue and smallest eigenvalue of matrix P, respectively. \(\sigma _{\max}(P)\) stands for the maximum singular value of matrix P.
2 Preliminaries and problem formulation
2.1 The Caputo fractional operator and Mittag–Leffler function
Definition 1
([27])
The \(\alpha >0\) order integral is defined as:
Definition 2
([27])
Caputo’s \(\alpha >0\) order derivative is defined as:
where \(n1<\alpha \leq n\), \(n\in \mathbb{N}\).
In the following, we will consider Caputo’s operation, by simply denoting:
We just consider the case that \(0<\alpha <1\), then, one has:
Noting that, for any constant C, one has \({}_{\varrho}\mathcal{D}^{\alpha}C=0\). The Mittag–Leffler function is the basis function of fractional calculus, as the exponential function is to the integerorder calculus, which is defined as follows.
Definition 3
([27])
The twoparameter Mittag–Leffler function is defined as:
where \(\alpha >0\), \(\beta >0\), \(\Gamma (.)\) is the Gamma function.
Definition 4
([27])
The oneparameter Mittag–Leffler function is defined as:
In the particular case when \(\alpha =1\), one has \(E_{1}(z)=\exp (z)\).
Lemma 1
([28])
Let \(x(t)\in R^{n}\) be a vector of differentiable functions. Then, for any time instant \(t\geq \varrho \), the following relationship holds
where \(P\in \mathbb{R}^{n\times n}\) is a constant, square, symmetric, and positivedefinite matrix.
Lemma 2
([29])
Suppose that \(V(t)\) is a continuous function satisfying \({}_{t_{k}}\mathcal{D}_{t}^{\alpha}V(t)\leqslant \theta V(t)\) for \(t>t_{k}\), then,
where \(0<\alpha <1\) and θ is a constant.
2.2 Model formulation
Consider the nonlinear FOMASs consisting of N followers (labeled by \(1,2,\ldots,N\)), which are described by
where \(x_{i}(t)=[x_{i1}(t),x_{i2}(t),\ldots,x_{in}(t)]^{T}\in \mathbb{R}^{n}\) denotes the state of the ith follower, \(A_{i}\) and \(B_{i}\) are constant matrices, \(g(x_{i}(t))=[g_{1}(x_{i}(t)),g_{2}(x_{i}(t)),\ldots,g_{n}(x_{i}(t))]^{T}\) is a vector value function with \(g_{k}(\cdot ):\mathbb{R}^{n}\rightarrow \mathbb{R}\), and \(u_{i}(t)\) is the communication protocol, which will be designed later. The dynamics of the leader (labeled by 0) is described by
where \(x_{0}(t)=[x_{01}(t),x_{02}(t),\ldots,x_{0n}(t)]^{T}\in \mathbb{R}^{n}\) denotes the state of the leader, \(x_{0}(t)\) may be an equilibrium point, a periodic orbit or event a chaotic orbit.
The distributed impulsive control protocol is designed as
where c is the coupling strength, \(d_{i}\geq 0\) are the gain between leader and the ith follower, \(i=1,2,\ldots,N\), when \(d_{i}=0\), there is no directed path from the leader to the ith follower. Consequently, it can be seen as a pinning control method. \(\delta (\cdot )\) is the Dirac delta function, and \(\delta (t)=\lim_{r\rightarrow 0}\chi (t)\) with \(\chi (t)=\frac {1}{r}\) when \(0\leq t< r\), and \(\chi (t)=0\) otherwise. \(\gamma _{k}\) is the impulsive gain in the kth impulsive moment; more information about the impulsive sequence \(\{t_{k}\}\) and impulsive gain \(\gamma _{k}\) will be given later.
Let \(e_{i}(t)=x_{i}(t)x_{0}(t)\), \(e(t)=[e_{1}(t),e_{2}(t),\ldots,e_{N}(t)]^{T}\), then, the error dynamics can be described by
where \(f(e_{i}(t),x_{0}(t))=g(e_{i}(t)+x_{0}(t))g(x_{0}(t))\) and \(\varphi _{i}(x_{0}(t))=(A_{i}A)x_{0}(t)+(B_{i}B)g(x_{0}(t))\). Meanwhile, the control protocol can be rewritten as
According to [30], let \(\Delta e_{i}(t_{k})=e_{i}(t_{k}^{+})e_{i}(t_{k}^{})\), and \(e_{i}(t_{k})=e_{i}(t_{k}^{})=\lim_{h\rightarrow 0^{+}}e_{i}(t_{k}h)\), one can obtain the following error system:
Throughout this paper, the nonlinear FOMASs are assumed to satisfy the following assumptions.
Assumption 1
There are nonnegative constants \(q_{ij}\) (\(i,j=1,2,\ldots,n\)) such that, for any \(x=[x_{1},x_{2},\ldots,x_{n}]\in \mathbb{R}^{n}\) and \(y=[y_{1},y_{2},\ldots,y_{n}]\in \mathbb{R}^{n}\), \(g_{i}(x)g_{i}(y)\leq \sum_{j=1}^{n}q_{ij}x_{j}y_{j}\).
Assumption 2
\(x_{0}(t)\) is bounded, that is, for any initial value \(x_{0}(0)\), there is \(\hat{T}(x_{0}(0))\) such that for any \(t\geq \hat{T}(x_{0}(0))\), \(\ x_{0}(t)\ \leq \varrho \), where ϱ is a positive constant.
Assumption 3
There is a directed spanning tree with the leader as the root in the communication topology of the FOMAS, that is, the leader has a path to every follower.
Remark 1
Let \(Q=(q_{ij})_{n\times n}\). Then, for any diagonal matrices \(\Lambda _{g}>0\), Assumption 1 implies that \((xy)^{T}Q^{T}\Lambda _{g}Q(xy)\geq (g(x)g(y))^{T}\Lambda _{g}(g(x)g(y))\). Also, note that there are many systems that can be satisfied, such as Chua’s circuit, and some chaotic neural networks. In addition, according to Assumption 1 and Assumption 2, \(\varphi _{i}(x_{0}(t))\) is also bounded, that is, \(\max_{t\geq \hat{T}}\ \varphi _{i}(x_{0}(t)) \ =\varpi _{i}\), where \(\varpi _{i}\geq 0\), \(i=1,2,\ldots,N\), are constants.
2.3 The design of the eventtriggered impulsive controller (EIFC)
In this subsection, we will design the eventtriggered impulsive controller. In the impulsive control method, the states of the system will be jumped at some determined moment \(t_{k}\), however, when the states are converging at some impulsive moment, the states are unnecessary to jump. Therefore, the eventtriggered mechanism will be adopted in this paper, which is related with the states of the system.
Let \(T>0\) be the check period and \(0=t_{0}\), \(V(t)=\sum_{i=1}^{N}e_{i}^{T}(t)Pe_{i}(t)\) and \(P\in \mathbb{R}^{n\times n}\) is a positivedefinite matrix, \(\theta _{1}>1\) and \(\theta _{2}<1\). Then, the kth jumped moment and impulsive gain \(\gamma (k)\) are determined by the following algorithm (\(k= 1,2,\ldots\)):
Under the above EIFC, let \(D=\operatorname{diag}\{d_{1},d_{2},\ldots,d_{N}\}\) be the pinning control matrix, one can rewrite the error dynamics in a matrix form when \(t=t_{k}\):
3 Main results
In this section, we will prove that there is no Zeno behavior for the considered FOMAS with the EIFC. Then, some impulsive quasiconsensus criteria are established for FOMAS (1).
Theorem 1
Consider the FOMAS (1) with the checked period \(T>0\), impulsive instants \(t_{k}\) for \(k=1,2,\ldots \) determined by the Algorithm 1. If Assumptions 1–3hold, and there are positive matrices P, \(\Psi _{1i}\), \(\Psi _{2i}\), \(\Xi _{1i}\), \(\Xi _{2i}\) and constants \(a_{i}\), positive constants \(\xi _{i}\), \(i=1,2,\ldots,N\), such that
Then, there is no Zeno behavior for the concerned FOMAS, that is, there is a constant \(\tau >0\) such that \(\inf \{t_{k}t_{k1}\}\geq \tau >0\), where
and \(a=\max_{1\leq i\leq N}\{a_{i}\}\), Q and \(\Lambda _{g}\) are defined in Remark 1.
Proof
Choose a Lyapunov function as \(V(t)=\sum_{i=1}^{N}e_{i}^{T}(t)Pe_{i}(t)\), according to Lemma 1 and Remark 1 for any \(t\in (t_{k1},t_{k}]\), \(k=1,2,\ldots \) , one has
where \(\eta _{i}(t)=[e_{i}^{T}(t), f^{T}(e_{i}(t)), \varphi _{i}^{T}(x_{0}(t))]^{T}\), according to (7) and (8), one has
Noting that, \({}_{t_{k}}\mathcal{D}^{\alpha}C=0\) for any constant C, then, we have
According to Lemma 2, one has
Taking any \((t_{k1},t_{k}]\), let us consider the event at \(t=t_{k}\). Based on Algorithm 1, if the first condition “\(\exists t\in (t_{k1},t_{k1}+T]\) such that \(V(t)\geq \theta _{1}V(t_{k1}^{+})\)” is not met, then \(t_{k}t_{k1}=T>0\), it is obvious that there is no Zeno behavior. Consequently, we should investigate the case that “\(\exists t\in (t_{k1},t_{k1}+T]\) such that \(V(t)\geq \theta _{1}V(t_{k1}^{+})\)”, if this event occurs at \(t_{k}\), we have \(V(t_{k})=\theta _{1}V(t_{k1}^{+})\), combined with \(\theta _{1}>1\) and (11), we have
Thus, we have \(E_{\alpha}(a(t_{k}t_{k1})^{\alpha})>1\), then, we obtain \(t_{k}t_{k1}>0\). That is, Zeno behavior is excluded for the system. The proof is completed. □
Theorem 2
Consider the FOMAS (1) with the checked period \(T>0\), impulsive instants \(t_{k}\) for \(k=1,2,\ldots \) determined by Algorithm 1. If Assumptions 1–3, (7), (8) hold, and parameters of the FOMAS are satisfied by
then, the trajectory of the error system (5) can exponentially converge into a ball \(\mathbb{M}\) with a convergence rate \(\frac {\ln (\theta _{2})}{2T}\), where \(\mathbb{M}= \{e(t) \ e(t)\ \leq \sqrt{ \frac {(\eta 1)\sum_{i=1}^{N}\xi _{i}\varpi _{i}^{2}}{a\lambda _{\min}(P)}} \}\), in which,
Proof
Choose a Lyapunov function as \(V(t)=\sum_{i=1}^{N}e_{i}^{T}(t)Pe_{i}(t)\). If “\(\exists t \in (t_{k1},t_{k1}+T]\) such that \(V(t)\geq \theta _{1}V(t_{k1}^{+})\)”, according to (12) and definition of \(t_{k}\), we have
If “\(\exists t\in (t_{k1},t_{k1}+T]\) such that \(V(t)\geq \theta _{1}V(t_{k1}^{+})\)” is not met, but “\(\exists t \in (t_{k1},t_{k1}+T]\) such that \(V(t)\geq \theta _{2}V(t_{k1}^{+})\)”, similarly, we have
If “\(\exists t\in (t_{k1},t_{k1}+T]\) such that \(V(t)\geq \theta _{1}V(t_{k1}^{+})\)” is not met, and “\(\exists t \in (t_{k1},t_{k1}+T]\) such that \(V(t)\geq \theta _{2}V(t_{k1}^{+})\)” is also not met, one can conclude that
According to (11), one has
where \(\zeta =\varepsilon (\eta 1)\), \(\varepsilon =\frac {\sum_{i=1}^{N}\xi _{i}\varpi _{i}^{2}}{a}\). By mathematical induction, we can derive that
Noting that \(\tau \leq t_{k}t_{k1}\leq T\) and for any t, there must be k such that \(t\in (t_{k1},t_{k}]\), one has \(\frac {t}{T}\leq k\leq \frac {t}{\tau}\), which implies that
Therefore, one can conclude that
Then, as \(t\rightarrow +\infty \), the error \(e(t)\) converges exponentially into the ball \(\mathbb{M}= \{e(t) \ e(t)\ \leq \sqrt{ \frac {(\eta 1)\sum_{i=1}^{N}\xi _{i}\varpi _{i}^{2}}{a\lambda _{\min}(P)}} \}\) at a convergence rate \(\frac {\ln (\theta _{2})}{2T}\). The proof is completed. □
Remark 2
Note that conditions in Theorem 1 are independent of the order α; however, α effects the value of \(E_{\alpha}(a(t_{k}t_{k1})^{\alpha})\), which implies that α will impact the time interval of two successive triggers. In addition, \(E_{\alpha}(a\tau ^{\alpha})\) is also related with α, which is significant in Theorem 2. Consequently, the consensus results in this paper are closely related to the order α.
Remark 3
In the above, the topology structure of the network is considered as a directed graph. When the topology is undirected, one has a symmetric Laplacian matrix L, then, the condition (12) can be replaced as \(\lambda _{\max}^{2}(I_{N}\frac {c}{\Gamma (\alpha +1)}(L+D)^{T}) \leq \rho \). In addition, if the FOMAS is homogeneous, which means that all nodes are identical, then it is easy to obtain \(\varpi _{i}=0\), \(i=1,2,\ldots,N\), according to the above, one can obtain the complete exponential consensus.
Remark 4
More detailed results about error estimation, optimization for quasiconsensus of heterogeneous dynamic networks via distributed impulsive control have been discussed in [31], in which, the pinning strategy also has been investigated. Some similar results also can be derived in this paper, therefore, we omit them here.
Remark 5
Compared with some existing results about impulsive control or the distributed impulsive control method, this paper has considered the eventtriggered mechanism. Conditions in this manuscript are unrelated to the checked period T, which is important, the checked period T just effects the converge rate. Furthermore, due to the eventtriggered mechanism, some unnecessary impulsive jumping can be avoided, which would be verified in the simulation part.
Remark 6
There are some results about impulsive control with an eventtriggered mechanism. In [32–35], the eventbased impulsive control method has been investigated, in which, the impulsive instants are determined by a certain event. However, the feedback controllers are also used in the systems, which is different from this paper. Distributed impulsive control for heterogeneous multiagent systems based on an eventtriggered scheme has been studied in [36], compared with which, events and impulsive controllers are simpler. Furthermore, this paper has discussed a FOMAS with fractionalorder dynamics. Of course, letting \(\alpha =1\), the corresponding results about consensus of integerorder multiagent system can be obtained.
Remark 7
The consensus problem has been analyzed in this paper, results about synchronization of a coupled dynamical network or master–slave system can be derived easily. For example, if there is only one follower, then the consensus problem converts to the synchronization problem of a master–slave system directly, an example will be given in the simulation part.
4 Numerical simulations
In this section, three examples will be given to show the effectiveness of the above theoretical results. A master–slave system with mismatched parameters and a heterogeneous FOMAS will be studied in two examples. The predictor–corrector algorithm has been used to simulate the fractionalorder dynamical networks in this paper [37] with step 0.001.
Example 1
Consider \(N=1\), then, the consensus problem of a leaderfollowing FOMAS (1) becomes a synchronization problem between \(x_{1}(t)\) and \(x_{0}(t)\). Let \(n=3\), for any \(z\in \mathbb{R}^{3}\),
Without any control, the chaotic behavior of the leader \(x_{0}(t)\) and the error response are shown in Fig. 1 and Fig. 2, respectively. In which, the initial values are selected as \(x_{0}(0)=[0.1,0.2,0.3]^{T}\) and \(x_{1}(0)=[1,3,4]^{T}\).
Consider the eventtriggered impulsive controllers that have been designed in this paper, one can let \(T=1\), \(\theta _{1}=25\), \(\theta _{2}=0.9\), \(P=I_{3}\), \(\mu _{1}=0.8\), \(\mu _{2}=0.5\), then, the consensus states are shown in Fig. 3. Furthermore, the errors are shown in Fig. 4, and the eventtriggered instants and the interval between this triggered moment and the next triggered moment is shown in Fig. 5.
Example 2
Let us consider \(N=4\), \(n=3\) in this example, for any
Also,
Without any control, the chaotic behavior of the leader \(x_{0}(t)\) is shown in Fig. 6, and the phase spaces of the followers can be seen in Fig. 7. One can see that the chaotic, stable, unstable or periodic behaviors have been shown for the followers. Obviously, without any control, the consensus can not be achieved, the error response is shown in Fig. 8.
The topology of the multiagent system is shown in Fig. 9, obviously, just the 1st and 2nd agents have been selected to be controlled. Let \(d_{1}=d_{2}=1\), \(d_{3}=d_{4}=0\), \(T=1\), \(\theta _{1}=1.2\), \(\theta _{2}=0.9\), \(P=I_{3}\), \(\mu _{1}=0.95\), \(\mu _{2}=0.8\), then, the consensus states are shown in Fig. 10. Furthermore, the errors are shown in Fig. 11, and the eventtriggered instants and the interval between this triggered moment and the next triggered moment is shown in Fig. 12.
5 Conclusion
The quasiconsensus problem of a fractionalorder multiagent system has been studied in this paper, the heterogeneous case is considered for the multiagent system. By using the designed eventtriggered impulsive control protocol, the quasiconsensus can be reached under some conditions that are formulated by a number of lowerdimensional matrix inequalities and scalar inequalities. The upper bound of the consensus error was estimated precisely. Furthermore, Zeno behavior was excluded successfully. Numerical simulation examples have been given to check the validity of the theoretical results. Noting that the centralized control method has been used in this paper, however, the distributed strategy will be more robust, thus, we will pay more attention to the distributed control methods in our future works. As is known, time delays are difficult to avoid in realworld networked systems, thus, the fractionalorder multiagent system with time delays based on the control method in this manuscript will be researched in our future works.
Availability of data and materials
All data generated or analyzed during this study are included in this article.
Abbreviations
 ETIC:

EventTriggered Impulsive Control
 FOMAs:

fractionalorder multiagent systems
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Acknowledgements
The authors are highly grateful to the anonymous reviewers for their careful reading of this paper and their insightful comments and suggestions.
Funding
This work was jointly supported by the highend research and training project of professional leaders of teachers in vocational colleges in Jiangsu Province (Sugao Peihan [2022] No. 11), the China Postdoctoral Science Foundation No. 2020M672027, the Natural Science Foundation of Shandong Province of China under Grant No. ZR2022QF075, ZR2019MA034, the Youth Creative Team SciTech Program of Shandong Universities (grant no. 2019KJI007), and the National Natural Science Foundation of China under Grant 61973183.
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Huang, C., Wang, F. & Zheng, Z. Quasiconsensus of fractionalorder heterogeneous multiagent systems under eventtriggered impulsive control method. Adv Cont Discr Mod 2022, 63 (2022). https://doi.org/10.1186/s1366202203739z
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DOI: https://doi.org/10.1186/s1366202203739z