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Distributed containment control of multi‐agent systems under asynchronous switching and stochastic disturbances

2019; Institution of Engineering and Technology; Volume: 13; Issue: 8 Linguagem: Inglês

10.1049/iet-cta.2018.6202

1751-8652

Quanli Liu, Tuo Zhou, Shixin Guo, Zehua Wang, Dong Wang, Wei Wang,

Mobile Agent-Based Network Management

IET Control Theory & ApplicationsVolume 13, Issue 8 p. 1105-1112 Research ArticleFree Access Distributed containment control of multi-agent systems under asynchronous switching and stochastic disturbances Quanli Liu, Quanli Liu School of Control Science and Engineering, Dalian University of Technology, Dalian, 116024 People's Republic of ChinaSearch for more papers by this authorTuo Zhou, Tuo Zhou School of Control Science and Engineering, Dalian University of Technology, Dalian, 116024 People's Republic of ChinaSearch for more papers by this authorShixin Guo, Shixin Guo School of Control Science and Engineering, Dalian University of Technology, Dalian, 116024 People's Republic of China Zhejiang Province Meteorological In formation Network Center, Hangzhou, 310017 People's Republic of ChinaSearch for more papers by this authorZehua Wang, Zehua Wang School of Control Science and Engineering, Dalian University of Technology, Dalian, 116024 People's Republic of ChinaSearch for more papers by this authorDong Wang, Corresponding Author Dong Wang dwang@dlut.edu.cn School of Control Science and Engineering, Dalian University of Technology, Dalian, 116024 People's Republic of ChinaSearch for more papers by this authorWei Wang, Wei Wang School of Control Science and Engineering, Dalian University of Technology, Dalian, 116024 People's Republic of ChinaSearch for more papers by this author Quanli Liu, Quanli Liu School of Control Science and Engineering, Dalian University of Technology, Dalian, 116024 People's Republic of ChinaSearch for more papers by this authorTuo Zhou, Tuo Zhou School of Control Science and Engineering, Dalian University of Technology, Dalian, 116024 People's Republic of ChinaSearch for more papers by this authorShixin Guo, Shixin Guo School of Control Science and Engineering, Dalian University of Technology, Dalian, 116024 People's Republic of China Zhejiang Province Meteorological In formation Network Center, Hangzhou, 310017 People's Republic of ChinaSearch for more papers by this authorZehua Wang, Zehua Wang School of Control Science and Engineering, Dalian University of Technology, Dalian, 116024 People's Republic of ChinaSearch for more papers by this authorDong Wang, Corresponding Author Dong Wang dwang@dlut.edu.cn School of Control Science and Engineering, Dalian University of Technology, Dalian, 116024 People's Republic of ChinaSearch for more papers by this authorWei Wang, Wei Wang School of Control Science and Engineering, Dalian University of Technology, Dalian, 116024 People's Republic of ChinaSearch for more papers by this author First published: 23 April 2019 https://doi.org/10.1049/iet-cta.2018.6202Citations: 3AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Distributed containment control of multi-agent systems with asynchronous switching and stochastic disturbances under a time-invariant directed graph is studied. Since both the process of identifying an active subsystem and that of applying a matched controller to the corresponding subsystem need to consume time, the controller switching usually lags behind the subsystem switching. In order to show that the controllers are still effective to tolerate some time delay, a multiple Lyapunov function is selected to present stability conditions. By employing stochastic stabilisation theory and matrix transformation, the problem of containment control is transformed to that of stability analysis of the closed-loop system and exponential stabilisation in mean square sense is achieved. Finally, two simulation examples are given to verify the effectiveness of the proposed results. 1 Introduction Over the past few years, the problem of distributed cooperative control of the multi-agent systems (MASs) [1] has received significant attention due to its wide applications, such as surveillance, food webs and economic markets. In the field of artificial intelligence and big data, researchers urgently need to store large amount of data into many subsystems, so, the applications of distributed MASs will be certainly broader. The research work of MASs are mainly carried out in the following aspects: consensus problem [2], which means to reach an agreement on a common value for a group by developing some distributed controllers based on the relative local information, flying formation problem [3] and containment control problem [4–11], which aims to design appropriate control protocols to drive the followers to a target area (convex hull formed by the leaders) asymptotically. As is well known, a variety of phenomena in nature occur randomly. The system with stochastic disturbance is difficult to achieve stability in practical applications. Owing to such a reason, stochastic problems receive more and more attention in science community. In [12], Kushner uses the Itô's stochastic differential equations to prove stability of the stochastic systems. There are two main ways to design the protocol to cope with stochastic problems. The authors have presented a recursive control design for strict-feedback stochastic non-linear systems in [13]. Also, a controller which achieves global asymptotic Lyapunov stability in probability for a class of nonlinear continuous-time systems is designed in [14]. In the real world, the connection topology of the communication network may change very quickly. For general MASs, its topology is often switching because of communication failures or other complex and changeable environments. A switched system has a series of switching rules to make the whole system operate stably and meet some performance indicators. For example, many engineering systems [15–17] can be abstracted into a switched system. According to the switching time between the controller and the controlled subsystem, the switched system is divided into two parts: synchronous switching [18] and asynchronous switching [19]. Since time is required in identifying an active subsystem and applying the matched controller, the switching of controllers is generally later than that of the controlled subsystems and it is more meaningful to study asynchronous switching. In [20], Markovian switching topologies are taken into consideration and the dynamics of MASs can reach consensus. The switching signal being a designable variable is natural and important in practice. Moreover, the parameter model of the system may change throughout the time, and switching behaviours can also occur in the dynamic model of the system. Taking the power supply in the industrial park as an example, suppose that there are three power generating units in the power grid and these three power generating units supply power for the industrial park. Owing to the working habits of the workers, the factory usually has the electricity consumption peak and electricity stable period. When all the machines run at the high speed, the power generation of the three generating units usually increases. Similarly, the workers are off work, the power of the factory is reduced, thus, the power of the generator will also fall. The dynamic model of the three generating units changes with time, then, we can simplify them into a model when they supply power for the same region. Therefore, studying the problem of distributed MASs based on a switched system is meaningful. Up to now, many researchers have researched synchronisation under a switching topology in [21–25]. If one or more connection links are removed from a network permanently and thus synchronisation is broken. The study of modelling the MASs into an asynchronous switching system with the noise is relatively few although it has a lot of practical significance and engineering applications, like automatic high way system, chemical processes, smart grid etc. Furthermore, changes of the system model are also considered. The switching time of the controllers generally lags behind that of the subsystem, hence, this paper mainly studies exponential containment control of MASs under asynchronous switching and stochastic disturbances. The contribution of this paper mainly includes the following aspects. Firstly, in this paper, the switchings of the different subsystems and controllers are not synchronised which is different from [21], then, an asynchronous switching is applied to handle such problems. Secondly, stochastic disturbances are used to express the external noise which is common in nature, thus, we consider the impact of stochastic noises on distributed containment control. Thirdly, the problem of containment control is transformed to stability analysis and the multiple Lyapunov functions method is used to present the condition on exponential stability of the closed-loop system in mean square sense. The remaining part of this paper is as follows: In Section 2, the model of MASs under asynchronous switching and basic knowledge are given. In Section 3, the problem transformation of containment control and two important theorems are presented. Section 4 introduces two examples to verify effectiveness of the control method. Finally, Section 5 summarises the paper. 2 Preliminaries and problem formulation In this section, the model of MASs under asynchronous switching with stochastic disturbances and basic knowledge are given. 2.1 Notations The notations used in this paper are standard. The superscript T denotes the matrix transposition, and the symbol is the Kronecker product. is the Euclidean norm of a vector, the symbol * represents as an ellipsis in matrix expressions that are induced by symmetry, and denotes an identity matrix with dimension n. is the set of real matrix. In addition, an N dimension column vector with all the elements being is denoted as . Given real symmetric matrix X and Y, means that is positive definite. with a filtration is a complete probability space, which satisfies the accustomed conditions (i.e., the filtration contains all P -null sets and right continuous), and is the mathematical exception operator about the given probability measurement P. Moreover, , for , , is a diagonal matrix in which the i th diagonal element is . Without further clarification, the constants, parameters and coefficients defined in the paper are real numbers. 2.2 Preliminaries The following graph related definitions are adopted from [26] and [27]. Assuming that there are n agents, we use to express it where is the node set. Elements of E are denoted as , which represents an edge from i to j and is represented by an arrow with tail at i and head at j. A directed path is a sequence of nodes such that , . Node i is connected to node j if there is a directed path from i to j. The adjacency matrix connected with the directed graph G is defined by , if , and otherwise. The Laplacian matrix is defined as where with and with . Moreover, if , j is an out-neighbour of i and if , j is an in-neighbour of i. A graph is an undirected graph if , . For undirected graphs, the qualifiers ‘in’ and ‘out’ in the aforementioned definitions are omitted. The adjacency matrix of the graph is a matrix in which the element on the i th row and j th column is . The weighted in-degree of node i is defined as . Define the diagonal in-degree matrix and the graph Laplacian matrix . An agent is called a leader if the agent has no neighbour. An agent is called a follower if the agent has at least one neighbour. 2.3 Problem formulation In this paper, we study MASs with m leaders and followers. In graph G, the leaders are the nodes whose in-degree is 0 and the follower is the node if there exists at least one leader can reach. Let and stand for the leader and the follower set, respectively. The dynamic of leaders is expressed as: (1)where and denote the state and switching signal, respectively. is a piecewise continuous (from the right) function and the switched system has l subsystems. is the noise intensity function matrix, is a one-dimensional (1D) Wiener process (Brownian motions) defined on , here, we can take a scalar, zeromean, Gaussian white noise process. Therefore, satisfies , . The followers have the following dynamics: (2)where represents the control input. Given , and represent the initial and k th switching instant, respectively. As the leader has only information output, the Laplacian matrix L connected with G can be divided into where and . It is noted that Assumption 1 and Assumption 2 refer to [28, 29] and [30], respectively, and Lemma 1 refers to [31]. Moreover, Definition 1, Definition 2, Definition 3 and Definition 4 refer to [31], [32], [19] and [33], respectively. Assumption 1.If there exists with and , then, the following Lipschitz expression is applied to the noise intensity function g, i.e., where is a non-negative known constant. Assumption 2.For each follower, there exists at least one leader that has a directed path to it. The objective is to design distributed controllers to solve the containment control problem, which is defined as follows. Definition 1.The containment control problem is solved for the agents in (1) and (2) if the states of the followers, under a certain distributed controller, asymptotically converge to the convex hull formed by those of the leaders. Lemma 1.Under Definition 1, every eigenvalue of is positive, all values of are non-negative, 1 is the same sum for all rows of . Definition 2.The definition of convex is that there exists a subset C of which satisfies where , and , the minimal convex set is denoted as , where and q is a positive integer. Definition 3.For any , denotes the switching numbers of over the period . If there exists , and the following condition holds: (3)then is the average dwell time and is the chatter bound. The following distributed control algorithm is proposed: (4)where denotes the switching signal of controllers. , where denotes the lag time satisfying and is a known value. Noted that both the process of identifying active subsystem and switching matched controller need time, so this paper studies the asynchronous switching which means that is generally equal or greater than 0. Therefore, the whole running time contains two parts: matched interval , where and unmatched interval . 3 Problem transformation and primary results Next, problem transformation and two important theorems are given in this section. Denote , and . Then, we can get (5)where , . Following from (1) and (2), (6) Suppose that the i th subsystem is activated at the moment and the j th subsystem is activated at the moment . Then, in order to write easily, define , . In the matched periods, (6) is denoted in the closed-loop form: (7)where In the unmatched periods, (6) is equivalent to (8)where Next, we give a definition and a lemma. Definition 4.Considering the closed-loop error system (7) and (8), if there exist positive values c and such that for any initial values , the following condition holds: (9)the system (7) and (8) can achieve exponentially stable in mean square sense and is the exponential decay rate. Remark 1.When , we obtain . Then, as . Owing to Lemma 1 and Definition 2, it is obvious that the followers converge to the convex hull. So, the control objectives is achieved. Under asynchronous switching and stochastic disturbances, specifically, our purpose is to design (4) to make the MASs (1) and (2) reach containment control, namely, the followers asymptotically converge to the convex hull of the leaders. Theorem 1.Given scalars , , , , , and , for , on each time period , the error in system (7) and (8) is exponentially stable in mean square sense if there exists such that the following inequalities hold: (10) (11) (12) (13)and the average dwell time satisfies (14) Proof.According to the above analysis of system (7) and (8), we know that the whole running time of the MASs contains two parts: matched periods and unmatched periods. So, the next proof process is also divided into two steps.Firstly, when , MASs runs in matched period. Select the Lyapunov function for (7) as (15)The infinitesimal operator of [34] is defined as According to Itô's differential formula, we can obtain the stochastic differential of along (7) (16)where can be obtained by (17)Denote . Owing to Lemma 1, we have . Then, it follows from Assumption 1 that (18)Owing to (13), one has (19)Combining (5) and (17)–(19), we can get (20)From (10), we can easily obtain (21)According to (16) and (21), we have (22)Integrating both sides of (22) from to t and taking expectation, we get (23)Secondly, when , MASs runs in unmatched period. Choose the Lyapunov function for (8) as (24)We can obtain (25)From (11), we have (26)Using (25), we get (27)Similar to (23), integrating from to t and taking its expectation, we have (28)The Lyapunov function in can be denoted as (29)From (29), when , the MASs runs in the matched period, . When , the MASs runs in the unmatched period, . At the same time, it's noted that a multiple Lyapunov function is selected according to the controller switching signal, the time period contains two parts: unmatched period and matched period . At the time instant , the controller switches, so, the Lyapunov functions chosen are different. Then, we combine (29) with (12), (23), and (28), when , to yield (30)where and represent the total time on the matched period and unmatched period during , respectively. When , the relation between and is (31)So, (30) can be converted into (32)By employing Definition 3, we get (33)Define then, we can get (34)i.e., (35)Owing to Definition 4 and condition (14), the error in system (7) and (8) is exponentially stable in mean square sense. The proof is complete. □ Remark 2.Compared with [21, 22], it is more convenient to choose the parameters and our method is easier to achieve performance requiremnets. In Theorem 1, the parameters are selected and the appropriate scalars according to Assumption 1 is known, then, there exist only two unknown parameter matrices and . By using the feasp function in the MATLAB LMI Toolbox, as long as two positive definite matrices are found, the conditions of Theorem 1 are satisfied. Next, we design the controller gains. Theorem 2.Given scalars , , , , , and , for , on each time period , for the MASs (1)) and (2) with controller (4) in the presence of stochastic disturbances, the distributed containment control problem can be solved if there exist matrices and matrices with appropriate dimensions such that the following inequalities hold: (36) (37) (38) (39)where then, MASs (1) and (2) reach containment control under the following controllers: (40) Proof.By Theorem 1, denote , and perform a multiplication on both sides of (10) by and . Then, (10) can be transformed into Assuming and applying Schur complement lemma [35], it can be turned into the following form: with Therefore, (36) is obtained. In the same way, (11) can be turned into (37) easily. According to and , then, (12) can be transformed into , so, (38) is achieved. Next, (13) can be transformed into by , then, (39) is derived. The proof is completed. □ Remark 3.Compared with [7], this paper studies containment control problem under asynchronous switching with stochastic disturbance. In (19), a coupling term caused by the multiplication of positive definite matrix and stochastic disturbances arises. To deal with this problem, we make the positive definite matrix satisfy the condition (13). Then, combined with Assumption 1, this problem has been successfully solved. After a congruence transformation by with (10)/(11), there is a non-linear matrix inequality. By using the Schur complement lemma, we can turn it into a linear matrix inequality. Remark 4.Compared with [21, 22, 29] where consensus problems or synchronous switching problems are discussed, however, this paper addresses the containment control problem under asynchronous switching. Meanwhile, we take into account the stochastic disturbances. In the process of proof, the multiple Lyapunov functions method is employed according to the switching signal of the controllers and decoupling problems are handled by Schur complement. If the switched system is simple, that is to say, the switching signal is constant. This means that no switching happens. We can obtain the following result. Corollary 1.Given scalars , , , , and , for the MASs (1) and (2) without any switching in the presence of stochastic disturbances, the following result is effective if there exists matrix such that the following inequalities hold: (41) (42)where the MASs (1) and (2) reach containment control and the controller gain in (4) is (43) 4 Simulation results In this section, we present two different examples to verify the given results. Example 1.A mechanical revolving model is considered in [36], which shows the schematic diagram of a physical system in Fig. 1. In Fig. 1, J represents the inertia of the revolving cylinder, R is the radius, H denotes the viscous friction coefficient, is the angular rotation and F denotes the force added by a motor whose power is a constant value. Based on the above analysis, we build the state space description as follows: where is the angular velocity, , a is a constant. Let three unforced systems be the leaders, which generate a convex hull, while the other four systems act as the followers. The aim is to use the protocol (4) for follower i such that they can move into the convex hull. The communication topology with seven subsystems is as shown in Fig. 2, where nodes 1, 2, 3 are leaders and 4, 5, 6, 7 are followers.Considering the switching phenomenon of the system model, we suppose that the MASs with (1) and (2) have two subsystems. Fig. 3 shows the switching signals where dashed line and solid line denote the system signal and controller signal , respectively. Selecting the appropriate parameters of switchings and taking account of the impact with the disturbances on the performance of the system, we can obtain that It can be easily determined that . The values of are . , , , and . According to (14), the average dwell time is 1.749. Substituting the above parameters into (36)–(39), we can calculate Owing to (40), the control gains are Fig. 4 shows the state trajectories of the agents with 3D state vector. The initial values are chosen as , , , , , , . The leaders are labelled as and the followers are labelled as , respectively at the instants . From Fig. 4, we can see that the followers move into the convex hull as . Fig. 5 shows the state trajectories of the agents with 2D state vector. The initial values are chosen as , , , , , , . The figures of all the agents' state trajectories at the same instant T =0, 8, 20s are individually shown in Fig. 5 with the same marker. It is observed from these two figures that the followers enter into and stay in the convex hull formed by all leaders with time going to infinite.Also, Fig. 6 displays the trajectories of the containment control errors. The initial values are chosen as , , , , , , . It can be seen that the errors go to zero in the mean square sense, i.e., the containment control is achieved. Example 2.Consider a 3D state vector. This situation is more complex in comparison with Example 1, and it is shown to enrich simulation results.Considering the switching phenomenon of the system model, we suppose that the MASs with (1) and (2) have two subsystems. With the same communication topology in Fig. 2, selecting the appropriate parameters in which the state vector is 3D and taking into account the impact of the disturbances on the performance of the system, we can obtain that Also, adopting the same parameters as in Example 1 and substituting the above parameters into (36)–(39), we can calculate Owing to (40), the control gains are Fig. 7 displays the trajectories of the containment control errors. The initial values are chosen as , , , , , , . It can be seen that the errors go to zero in the mean square sense, i.e., the containment control is achieved. Fig. 1Open in figure viewerPowerPoint Mechanical revolving model Fig. 2Open in figure viewerPowerPoint Communication topology Fig. 3Open in figure viewerPowerPoint Switching signal Fig. 4Open in figure viewerPowerPoint State trajectories of the agents in 3D Fig. 5Open in figure viewerPowerPoint State trajectories of the agents in 2D (a) Trajectory at instant T =0s, (b) Trajectory at instant T =8s, (c) Trajectory at instant T =15s Fig. 6Open in figure viewerPowerPoint Containment error of Example 1 (a) Containment error of , (b) Containment error of Fig. 7Open in figure viewerPowerPoint Containment error of Example 2 (a) Containment error of , (b) Containment error of , (c) Containment error of 5 Conclusion This paper has studied containment control problems of MASs under asynchronous switching and stochastic disturbances. Based on the controller switching signal, the multiple Lyapunov functions have been designed and the closed-loop system has been proved to be exponentially stable in mean square sense. Finally, two simulation results have demonstrated the effectiveness of the proposed controller protocol. In addition, asynchronous switching time for each individual agent may be different in practical applications, which is important and interesting. Therefore, our future work will be devoted to relax this limitation. 6 Acknowledgments This work was supported in part by the National Natural Science Foundation of China under grants 61773085, 61473055 and 61773089, by Fundamental Research Funds for the Central Universities under grant DUT17ZD227, and by Youth Star of Dalian Science and Technology under grants 2015R052 and 2016RQ014. 7 References 1Mannor S., and Shamma J.S.: ‘Multi-agent learning for engineers’, Artif. Intell., 2007, 171, (7), pp. 417– 422 2Ma Q. 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