The Consensus of Different Fractional-Order Chaotic Multiagent Systems Using Adaptive Protocols

Firouzjahi, Masoumeh; Naderi, Bashir; Edrisi Tabriz, Yousef

doi:https://doi.org/10.1155/2022/5129072

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Fractional Operators in Modelling Chaotic and Real-World Problems

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Volume 2022 | Article ID 5129072 | https://doi.org/10.1155/2022/5129072

The Consensus of Different Fractional-Order Chaotic Multiagent Systems Using Adaptive Protocols

Masoumeh Firouzjahi,¹Bashir Naderi,¹and Yousef Edrisi Tabriz¹

Academic Editor: Ndolane Sene

Received02 Aug 2021

Revised12 Sept 2021

Accepted24 Dec 2021

Published11 Jan 2022

Abstract

This paper is concerned with the adaptive consensus problem of incommensurate chaotic fractional order multiagent systems. Firstly, we introduce fractional-order derivative in the sense of Caputo and the classical stability theorem of linear fractional order systems; also, algebraic graph theory and sufficient conditions are presented to ensure the consensus for fractional multiagent systems. Furthermore, adaptive protocols of each agent using local information are designed and a detailed analysis of the leader-following consensus is presented. Finally, some numerical simulation examples are also given to show the effectiveness of the proposed results.

1. Introduction

Study of multiagent systems over the past decades in various fields such as biology, mechanics, physics, and, more recently, control theories have been found (see [1]). As one of the most fundamental issues of multiagent systems, the consensus problem has attracted extensive research from various perspectives. Consensus of multiagent systems can be used to solve many complex problems in the control community and have been widely used in sensor network (see [2]), flocking (see [3]), formation control (see [4]), mobile robots (see [5]), and so on. In general, in multiagent systems, consensus means that agents with the arbitrary initial conditions converge to the desired target (position, attitude, speed, phase, etc.) by interacting information with their neighboring agents. In these systems, to achieve the conditions of consensus, graph theory, matrix theory, and the classical stability theorem of fractional-order system have been used, and in terms of classification, there are two general types, consensus with leader and consensus without leader; the latter is more challenging in terms of stability and connected topology than the former (see [6]). In a leader-following consensus, the control signals of the agents are appropriately selected such that their state trajectories follow the leader state, which can be achieved by local information exchanging from the leader and other agents. Prior to 2008, most articles related to multiagent systems worked on consensus of integer-order dynamics, such as consensus algorithms of first-order dynamic systems (see [7–9]) as well as second-order dynamics (see [10]) or even high order dynamic systems (see [11]). However, since many phenomena cannot be accurately explained by integer-order dynamics, such as macromolecule fluids and porous media, a clear example of the fractional order model is the relationship between heat flow and temperature in heat diffusion of a semi-infinite solid (see [12]); also, many systems in the nature can be described and more precisely modeled by a coordinated behavior of agents with fractional-order dynamics(see [13,14]); for instance, the group movement of bacteria in fatty and microbial environments (see [15]) and the coordination of submarines in stagnant water and the movement of land vehicles on sandy, muddy, or grassy roads (see [16]) amongst others, and moreover, compared to the integer-order model, fractional-order systems can provide an excellent method in description the characteristics of the system well; based on these facts, studying the consensus of fractional-order systems has become very important and was fully examined for the first time in [16–18]. Successively, the convergence speed of consensus for fractional-order multiagent systems was also discussed further in [19] and in [20] and presented the consensus of fractional-order multiagent systems with varying-order . The consensus problem of fractional-order systems with input delays and heterogeneous multiagent systems in [21, 22] was also examined.

An interesting issue in fractional-order multiagent systems is that agents follow a leader, where the leader is a special agent and his movement is independent of other agents. It has been reported that such models are an energy-saving mechanism (see [23]), which was found in many biological systems and can also strengthen group communication and orientation. Through the observer method, consensus of multiagent systems with the leader was given by the second-order model and the followers depicted differential order less than two, which was concerned in [24]. In [25], leader-following consensus of fractional-order multiagent systems under fixed topology has been studied.

As we know, the behavior of some fractional dynamical systems in nature can be often chaotic. Chaotic systems are a class of nonlinear deterministic systems and describe numerous complex and unusual behaviors. At first, it has been found that a large number of fractional-order differential systems exhibit chaotic behavior, such as the fractional-order Lorenz system (see [26]), the fractional-order Rossler system (see [27]), the fractional-order Chua system (see [28]), the fractional-order Lü system (see [29]), and the fractional-order Chen system (see [30]). Therefore, control and consensus of fractional-order chaotic systems have attracted great attention. According to one type of classification in terms of the derivative order, consensus in leader-following chaotic fractional systems is divided into two categories. The first group of two dynamic systems have the same order, which is introduced to the commensurate order system, and in the second group, the fractional order of the two systems is different, which is called incommensurate order system, and it is desirable that, by exercising control, the follower system follows the behavior of the leader system. In recent years, many successful attempts have been made to consensus of chaotic commensurate fractional-order systems, but in applied and practical systems, two chaotic systems cannot necessarily be assumed to be the same orders. On the contrary, the methods used to consensus of the same orders’ chaotic systems are not easily applicable to consensus of different orders chaotic systems, and also, the incommensurate order system has advantages compared to the previous case. One of them is that the fraction derivative order of the state variables in the follower system, which are to be consensus, is freed from the derivative order component in the leader system, and this can create more flexibility in the choice of leader and follower system. As a result, due to the many applications of chaotic systems consensus in data security in fuzzy systems (see [31]), secure communication (see [32]), the effect of market trust on the financial system (see [33]), and the study and treatment of some diseases, such as the study of tumor cell chaos in the tumor immunity fractional model (see [34]) and its application in neural networks to solve fractional differential equations (see [35]), motivated us in this paper to deal with the consensus of the different fractional-order chaotic systems using adaptive control.

Adaptive control is a technique of applying some systems’ identification techniques to obtain a model and using this model to design a controller (see [36]). The parameters of the controller are adjusted automatically during the operation.

The rest of this paper is organized as follows. In Section 2, the graph theory notations, Caputo fractional operator and one kind of it, the commensurate and incommensurate fractional order system, and some necessary lemmas and theorems are introduced. In Section 3, the main results on adaptive consensus for chaotic fractional multiagent systems with are presented. In Section 4, the corresponding simulation results are provided in this section to demonstrate the effectiveness of the proposed method. Finally, the concluding remarks are given in Section 5.

2. Preliminaries

In this section, first, some basic definitions of algebraic graph theory and Caputo fractional operator will be mentioned. Then, fractional systems with different orders and some necessary lemmas and theorems for the stability of such systems were discussed.

2.1. Graph Theory

Using graphs is the simplest and most effective way to model the exchange of information between factors in multiagent systems with a leader and agents. Each graph is a pair of that which is a set of empty and finite nodes and is a set of edges of a graph that connect nodes to each other. Each edge of the graph is shown as an ordered pair which means that the agent can transfer its information to the agent , but not necessarily the opposite. An undirected graph has the property that implies . The set of neighbors of node is denoted by . Edges of graph can be weighted or weightless. Weight can indicate cost, time, relocation, or any other factor. The weighted adjacency matrix of with nonnegative entries is defined as if and , otherwise. The degree matrix of is , where diagonal elements if the agent is a neighbor of the leader and , otherwise. The Laplacian matrix of the weighted graph G is defined as and for .

Lemma 1 (see [25]). For any undirected graph , matrix is positive definite if there is at least one directed path from (the leader) to all the other nodes. Also, it is obvious that all the elements on the main diagonal of the matrix are all positive.

2.2. Caputo Fractional Derivative

At present, there are several different definitions regarding the fractional derivative of order ; Caputo and Riemann–Liouville (R–L) fractional operators are the two most commonly used in different fields of fractional dynamic systems. The main advantage of the Caputo fraction derivative over the R-L fraction derivative is that the initial conditions for fractional differential equations with the Caputo derivatives are the same as the integer order for the differential equations. Therefore, in this paper, we will adopt the Caputo fractional derivative to model the multiagent systems’ dynamics. The Caputo fractional derivative of with order can be written as follows (see [13]):where , and is the Gamma function:

For simplicity, the Caputo derivative is replaced by notation in this paper.

One of the important concepts in fractional systems’ theory is stability, where necessary and sufficient conditions for the stability of these systems have been thoroughly studied in (see [37, 38]). For this object, we consider the following linear system of fractional differential equations:where , the matrix indicates the fractional orders, and , for , and is the least common multiple of the denominators .

Theorem 1 (see [37]). If , then system (3) is called a commensurate fractional-order system. In this case, system is asymptotically stable if and only if is satisfied for all eigenvalues matrix .

Theorem 2 (see [38]). If are not identically equal to each other, then system (3) is called an incommensurate fractional-order system. In this case, the system is asymptotically stable if all roots of the equation satisfy .

3. Main Results

In this section, the leader-following consensus problem of incommensurate fractional-order chaotic multiagent systems is discussed, and a distributed adaptive protocol is designed to achieve consensus under an undirected interaction fixed graph. We first consider the different fractional-order multiagent system consisting of agents and a leader. The dynamics of each agent is given byand the dynamics of the leader(labeled as ) is depicted bywhere means the Caputo fractional derivative of order , , and , , and represent the state of ith agent, the state of the leader, and the control input, respectively. are the system matrices and denote the nonlinear part of the leader-following system.

Remark 1. The leader’s dynamic is independent of others. We take the different nonlinear dynamical functions and for leader and all the agents, respectively.

Definition 1. The leader-following consensus of systems (4) and (5) will be achieved if, for each agent , there is the appropriate control of such that the closed-loop system satisfies

Theorem 3. Consider the leader-following multiagent systems (4) and (5), where , but are equal to each other if we define the distributed adaptive control law as follows:such thatwhere is feedback control gain and is the largest eigenvalue of ; then, all the agents follow the leader under any initial conditions.

Proof. Consider the following system (4), and now, by substituting from (7) in (4), we haveBy taking the fractional derivative of order from the above relation, we will haveLet us define the state error between the agents and the leader as . Then, the dynamics of isThus, according to Theorem 1, system (11) is asymptotically stable ifWe select so that the following relation is established:and on the contrary, from Lemma 1, all eigenvalues of matrix are nonnegative which implies thatBy definition, , all the agents follow the leader from any initial conditions; it is enough .
The main purpose of this controller is to convert the following system to a fractional-order system equivalent to the leader system so that the derivative order is equal in the corresponding state variables in the two systems.

Corollary 1. Consider Theorem 3 under the control law (7). Ifwhere is the smallest eigenvalue of , the agents never follow the leader.

Theorem 4. Consider the leader-following multiagent systems (4) and (5), where , but are not necessarily equal to each other if we define the distributed adaptive control law as follows:such thatwhere is feedback control gain and and matrix are shown as the largest elements and the elements on the main diagonal of the matrix , respectively. Matrices and are the Laplacian matrix and degree matrix of graph and Jacobian matrix , respectively. Then, all the agents follow the leader under any initial conditions.

Proof. Consider system (4), and assuming , now, by substituting from (16) in (4), we haveNow, if we define the state error between the agents and the leader as , then the dynamics of isBy using Taylor expansion around , the following statement is obtained:By combining equations (19) and (20), we haveSince, are not assumed to be equal, so according to Theorem 2, all roots of the equation satisfy in which are obtained as follows:So, holds in at least one of equations . Suppose to get n’th roots of a complex number; we haveOn the contrary, all the elements on the main diagonal of the matrix are all positive, and as can be seen, if the relation is , then the condition is in all cases, and therefore, the stability condition is established.

Corollary 2. Consider Theorem 4 under the control law (16). Ifthen the agents never follow the leader.

Remark 2. The case and are not necessarily equal to each other; initially, using controller (7), the fractional order of the follower system is equal to the order of the leader system; then, to consensus of the systems, we use the stability of Theorem 4 and select from equation (15).

4. Numerical Example

In this section, some illustrative examples are presented to verify the efficiency of the proposed leader-following consensus approach. The first example considers the case when fractional orders in the leader-following system are not equal, but are equal and the second one considers the case when fractional orders in the leader-following system are equal, but are not necessarily equal to each other. The third example is considered and are not necessarily equal to each other, which are all considered under the undirected graph.

Example 1. Consider a multiagent of chaotic different fractional-order consisting of a leader and three agents, and in order to facilitate the solution of our examples, without loss of generality, we assume that and , for all examples in this section. They are the state variables of the leader and state variables of the agents satisfyingwhere and .
Assume the Lorenz system (25) is the leader system (see [26]) and the Chua system (26) is the agent systems (see [28]). Also, suppose the topology is described as in Figure 1. For convenience, let if and , otherwise.Thus, the Laplacian and matrix are as follows:A straightforward calculation shows the largest eigenvalue of is . In simulation, we choose , the initial conditions of leader system as and the initial conditions of agents system as . Under the control law (7), the fractional order of the follower system is equal to the order of the leader system; now, according to (8), by choosing , we can see that three agents follow the leader. The consensus errors are shown in Figure 2.
Now, if we increase the feedback control gain from −0.1 to 0.75 in the above example, it is expected that the consensus speed will increase; in other words, the time to reach the zero error will decrease, which we see in Figure 3. Time to zero error is reduced from above 5 seconds to below 2 seconds. Table 1 also shows the average time to zero of the variables error with increasing .
Also, according eigenvalues of are in the above example; if we suppose , for example , we will see that the followers do not follow the leader, which is shown in Figure 4. And this example is presented as a proof of Corollary 1.

(a)

(b)

(c)

(a)

(b)

(c)

(a)

(b)

(c)

Example 2. Consider a multiagent of chaotic fractional-order consisting of a leader and three agents. They are the state variables of the leader and state variables of the agents satisfyingwhere and .
Assume the Lü system (28) is the leader system (see [29]) and the Chen system (29) is the agent system (see [30]). Also, suppose the topology is described as in Figure 5. Hence, the Laplacian and matrix are as follows:In simulation, we choose and the initial conditions of leader system as and the initial conditions of agents syttem as . Under the control law (16) and according to equation (17), by choosing , consensus occurs in the leader-following system. The consensus errors are shown in Figure 6.
We can see in Table 2 the average time to zero of the variables error with increasing in Example 2.

(a)

(b)

(c)

Example 3. Consider Example 1, except that the orders of the leader and follower systems are , respectively. Now, using controller (7), we equalize the orders of the two systems, and then, according to equation (17), we choose . The consensus errors are provided in Figure 7.
Table 3 shows the average time to zero of the variables’ error with increasing in Example 3.

(a)

(b)

(c)

5. Conclusion

In the present paper, we study the consensus of different fractional-order systems with adaptive protocols via an undirected fixed interaction graph. Therefore, using the stability theorems of the fractional order system as well as the concepts of information exchange between graph vertices and corresponding matrices, several types of nonlinear adaptive controllers have been designed to suit the problem conditions. As mentioned in the study, these controllers were designed in such a way that, after applying the controller, the fractional order of all agents of the incommensurate leader-following multiagent system is equal to the fractional order of the leader system so that the two systems can be matched. It should be noted that, in the design of the controllers, the feedback rate coefficient was used and the effective range for the desired coefficient was considered. Future research includes the study of adaptive consensus for fractional time-delayed multiagent systems and or time-varying communication constraints. We can also expand on the condition of consensus in a limited time. Finally, examples and numerical results show that using the appropriate interest rate, the consensus of multiagent systems is guaranteed.

Data Availability

The data used to support the findings of the study are provided within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2022 Masoumeh Firouzjahi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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