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Event‐triggered H ∞ Markovian switching pinning control for group consensus of large‐scale systems

2016; Institution of Engineering and Technology; Volume: 10; Issue: 11 Linguagem: Inglês

10.1049/iet-gtd.2015.0770

1751-8695

Yanliang Cui, Minrui Fei, Dajun Du,

Energy Efficient Wireless Sensor Networks

IET Generation, Transmission & DistributionVolume 10, Issue 11 p. 2565-2575 ArticleFree Access Event-triggered H∞ Markovian switching pinning control for group consensus of large-scale systems Yanliang Cui, Yanliang Cui Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronic Engineering and Automation, Shanghai University, Shanghai, 200072 People's Republic of China School of Mechatronic Engineering, Lanzhou Jiaotong University, Lanzhou, 730070 Gansu, People's Republic of ChinaSearch for more papers by this authorMinrui Fei, Corresponding Author Minrui Fei mrfei@staff.shu.edu.cn Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronic Engineering and Automation, Shanghai University, Shanghai, 200072 People's Republic of ChinaSearch for more papers by this authorDajun Du, Dajun Du Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronic Engineering and Automation, Shanghai University, Shanghai, 200072 People's Republic of ChinaSearch for more papers by this author Yanliang Cui, Yanliang Cui Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronic Engineering and Automation, Shanghai University, Shanghai, 200072 People's Republic of China School of Mechatronic Engineering, Lanzhou Jiaotong University, Lanzhou, 730070 Gansu, People's Republic of ChinaSearch for more papers by this authorMinrui Fei, Corresponding Author Minrui Fei mrfei@staff.shu.edu.cn Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronic Engineering and Automation, Shanghai University, Shanghai, 200072 People's Republic of ChinaSearch for more papers by this authorDajun Du, Dajun Du Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronic Engineering and Automation, Shanghai University, Shanghai, 200072 People's Republic of ChinaSearch for more papers by this author First published: 01 August 2016 https://doi.org/10.1049/iet-gtd.2015.0770Citations: 11AboutSectionsPDF 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 This study investigates the pinning control issue for group consensus of large-scale systems (LSSs) with communication topology changes and network disturbances. A node selection principle is firstly designed to choose appropriate control nodes, and a centralised pinning control protocol is proposed to drive LSSs achieving group consensus. For reducing control command updates, an event-trigger is employed to determine whether control signal need be changed. Since the node receives compound exogenous disturbances from its communication neighbours, an improved point-to-area H∞ index is advocated as a system robustness performance. Considering communication topology changes, an event-triggered H∞ Markovian switching pinning control law is finally established and a sufficient controller and event-trigger co-design method is presented. Under this strategy, the group consensus of LSSs with a pre-scheduled H∞ performance can be simultaneously achieved while the Markovian switching topologies are tolerated, moreover, only fewer nodes are controlled and seldom control commands are updated. Finally, illustrative examples are given to demonstrate the effectiveness of the proposed theoretical result. 1 Introduction Large-scale system (LSS) is a familiar concept nowadays for many typical real-world applications such as power grid, communication network, traffic transportation, and interpersonal circle and so on. From macro perspective, the overall LSS generally behaves complex dynamics that bring many obstacles both in theoretical research and practical implementation. Until the small-world phenomena are discovered in [1], which gives a clue that the original complicate LSS can be typically decomposed into some smaller simplified interconnected subsystems. Generally, an interconnected system is conceived as many organically coupled identical or similar nodes, which directly exchange their shared information via communication networks for achieving certain common interests [2–4]. To render such objective accessible, a fundamental problem is known as the consensus whose purpose is to probe effective control scheme for driving the nodes trend to certain synchronisation. In the past decades, many consensus strategies are gradually proposed such as decentralised fuzzy control [5], adaptive control [6], iterative learning control [7] and so on. The above-mentioned schemes mostly utilise all components of interconnected systems for fulfilling the objective. However, as a complex large networked system, the LSS is usually unnecessary or even impossible for all of its nodes to be controlled. To overcome such defecation, a novel approach is to implement pinning control to a fraction of nodes while letting the rest nodes are propagated through their inner couplings. Due to the better features of saving costs and energy conservation, the pinning control strategies are intensively researched recently. For example, many pinning control principals and objectives are systematically introduced in [8]. Consider switching network topologies, a switching pinning control scheme is proposed in [9]. In [10], a pinning control method is advocated for achieving the consensus of second-order multi-agent systems. In [11], a random pinning control method is proposed to guarantee synchronisation of noise perturbed non-identical systems. In [12], a zero in-degree priority pinning control law is designed for consensus of first-order interconnected systems. The aforementioned pinning schemes provide a conception for controlling LSSs. However, there are many important problems still waiting to be answered. Specifically, the following issues will be addressed and analysed in this paper: The traditional consensus of LSSs requires its nodes converging to a unified target [13, 14]. However, due to instinct construct properties and intrinsic capabilities, it is more practical for nodes of LSS to be synchronised to different equilibrium values. Therefore, it is necessary to probe an effective pinning control scheme to drive the nodes of LSS for converging at a group of pre-scheduled equilibrium points. In the aforementioned literatures [8–12], the pinning control laws are mostly presented in continuous-time form. As far as an LSS is concerned, these control principles have a great chance of losing compatibility because the continuous-time control is an unrealistic scheme due to the excessive control burden and difficulties in implementation. In addition, with a continuous-time control scheme, the controllers and the actuators undergo heavy workloads which will lower life cycle of component and arise risk of fault. Therefore, to overcome the above defections meanwhile enable LSSs have better features of saving cost and conserving energy, a discrete-time command execution mechanism should be devised by which only necessary control signals are executed while the rest redundant commands are suspended. In spatially distributed interconnected systems, the nodes need to exchange necessary data via communication networks. Therefore, the data transmission quality of the networks plays a key role on the control performance. Although the network brings advantages in flexibility, effectiveness, easy implementation and so on, an undeniable negative effect is that some communication imperfections may deteriorate the control performance even endanger the system stability. Such network defections include exogenous disturbance [15], data transmission time-delay [16], data packet dropout [17], topology switching [18] and so on. Note that the topology temporarily disconnection phenomena is universal in LSSs, for encompassing situations of time-varying topologies so as to alleviate its influence on control performance, the designed controller should have the robustness to dynamic topologies. In addition, the exogenous disturbance is another challenging problem due to its impact upon control performances and unknown origination and transmutation mechanisms. Moreover, each node of LSSs will receive multi-source noises which are originated from different communication links. Therefore, the controller of LSSs should have the capability of suppressing the compound noises. Motivated by the above discussed factors, this paper further investigates group consensus of LSSs by event-triggered H∞ Markovian switching pinning control (EHMSPC). The contributions of this work are listed as follows: (i) A pinning node iterative selection method is proposed to determine the appropriate pinning node(s) for given topology of LSS. Based on the selected node(s), a centralised pinning control law is proposed for achieving group consensus of LSSs. (ii) An event-triggered control signal updating scheme is designed for reducing the command execution burden. In addition, an improved point-to-area H∞ performance is advocated for strengthening the noise suppression capability. The remaining of the paper is organised as follows. Section 2 formulates the problem. Section 3 presents the main result. Simulation examples are illustrated in Section 4. Conclusion remarks are outlined in Section 5. The following notations are given which will be used throughout the literature. Let and denote the real numbers and the integer numbers, respectively. is the n-dimensional Euclidean space and is the set of n1 × n2 real matrices. The superscript 'T' denotes the matrix transposition, the script '∗' denotes the corresponding transposed matrix item. The sign '⊗' and 'e' represent the matrix Kronecker product and Hadamard product, respectively. Given a vector x, the function denotes the Euclidean vector norm. Given any matrix A, sym(A) = A + AT. 2 Problem formulation The considered LSS is assumed to be consisted by m heterogeneous interconnected subsystems with a control framework as shown in Fig. 1. Fig. 1Open in figure viewerPowerPoint Overall structure of the LSS In Fig. 1, the information exchanging among the LSS can be conveniently captured by a group of non-weighted direction graphs of , l = 1, 2, …, m. For the lth interconnected system, its own a private communication graph as of order N(l), where and denote the sets of nodes and edges, respectively. The adjacency matrix is denoted as , where i, denote the ith and jth nodes of the lth interconnected system. An directed edge if the ith node has a direct link from the jth node, and then mark , otherwise, . Denote the adjacency matrix , the non-symmetrical Laplacian matrix associated with is defined as where , ; otherwise, . In Fig. 1, the lth subsystem is an interconnected system whose dynamic can be described as [8, 12] (1)where , and denote the system state, input vector and the state dimension of the lth subsystem, respectively. The matrix is the inner-coupling strength matrix of the lth interconnected system, , i, and l = 1, 2, …, m. The 'Pinning control centre' (PCC) dispatches the equilibrium points for the whole LSS. Denote x0, and , as the target equilibrium points of the overall LSS, the lth subsystem and the ith node of the lth subsystem, respectively. Apparently, their relationship can be described as Remark 1.In reality, there probably exists inter-graphs among the subsystems, an obviously question is that why the subsystems in Fig. 1 are independent with each other? It can be observed from the descriptions of x0 that, concerning with the different subsystems, their desired state equilibrium points are different. In other prospective, the concept of interact or cooperative is advocated for subjects to achieve their shared interests. Apparently, it is unnecessary for the subsystems exchanging their unconcerned state information to fulfil their own private mission. On the contrary, excessively insisting cooperative communication will definitely deteriorate control performance instead of improving it, because many unrelated information are introduced and the limited resource of interconnected systems is consumed. Therefore, for guaranteeing each interconnected system converges to its own equilibrium point, such unhelpful inter-graphs among the subsystems should be disconnected. Remark 2.From a practical standpoint, x0 is preferred to be a time-varying value as x0(t). However, to drive LSS to achieve the desired consensus, the PCC should give sufficient convergence time to the subsystems. Moreover, for saving cost and conserving energy, it will be better for the PCC to send out slowly time-varying commands. Therefore, x0(t) can be regarded as a time-invariant value as x0 in a relative longer time-scale. Remark 3.The equilibrium point x0 gives consensus target of whole LSS, it plays an important key in the system. Generally, by some optimal cost functions, x0 can be obtained thus enabling the LSSs to fulfil different objectives. For example (i) State rendezvous: for each of the subsystems, their state equilibrium points converge to a unified value. The cost functional is given as where xz denotes an acceptable common equilibrium point. (ii) Group consensus: it means that the state equilibrium points are arranged to some pre-given values. The cost functional is suggested as where denotes the desired state-steady value of the lth subsystem. (iii) Collision avoidance: the state equilibrium points are placed at the target locations with some pre-scheduled intervals. The cost functional is design as where denotes the specified collision radius. Since the above-mentioned cost functions generally have linear quadratic forms, it can be easily solved by dynamic programming or gradient descent methods. Therefore, the detailed optimal processing is omitted here, and it is assumed that the state equilibrium point x0 is the optimal value of LSS. In addition, note that the group consensus is a fundamental issue among the above-discussed tasks while the others can be regarded as its special cases, therefore, the group consensus is taken as the objective of this paper. Define error vector for the ith node of the lth subsystem as where and l = 1, 2, …, m; the corresponding state errors for the lth subsystem and the LSS can be described as Before the controller designing, the following definition is necessary. Definition 1.The control input solves the group consensus for lth interconnected system, if Furthermore, if (2)we say the group consensus of the LSS is achieved. Generally, a traditional group consensus means that the nodes in each subgroup reach their own private synchronisation [12, 19, 20], i.e. for all i, j ∈ V(l), such that holds. According to Definition 1, one can find that for the lth subsystem, it can own at most n(l) different equilibriums, such as , . In particular, if for , i, , it implies that the assigned equilibrium state of the lth subsystem is the same. In this sense, one discovers that the traditional group consensus is a special case of Definition 1. 2.1 Pinning controller For fulfilling the group consensus of a LSS, meanwhile reducing the control burden, a pinning controller is designed for the lth subsystem as (3)where and are the static control input and the dynamic pinning control input, respectively. The pinning scalar is designed as ; namely, if means the ith node of the lth subsystem is pinned, or if unnecessary; is the pinning gain and . In (3), is employed to specify different equilibrium points to each nodes, while is designed to implement real-time control on the pinned nodes for achieving group consensus for the lth subsystem. By (1)–(3), the close-looped system for the lth subsystem is written as (4) 2.2 H∞ performance During the above modelling process, the communication network is supposed to be trustworthy. However, due to exogenous disturbances, each node will inevitably receive deviated measurements instead of precise state values. Hence, the original neighbour state of can be perturbed as , where is the disturbance intensity of link of the ith and jth nodes of the lth subsystem; ω(l)(t) is the exogenous disturbances. Note that the pinned nodes also receive noises from pinning links, i.e. , where i is the index of the pinned node, is the noise intensity of the pinning link between i with the PCC. Due to the generation and transmutation mechanisms of exogenous disturbance still remain unrevealed, the exogenous disturbance can be modelled in many different prospectives, for example, the Gaussian noise [21, 22], impulsive noise [23, 24], energy-bounded noise [25, 26] and so on. Without loss generality, it is assumed that ω(l)(t) is an energy-bounded disturbance as ω(l)(t) ∈ L2[0, +∞); where L2[0, +∞) denotes the energy-bounded functions space, each function f(t) ∈ L2[0, +∞) satisfies . Accordingly, (4) is converted as (5)In the sense of cause-and-effect relationship, the controller has no influence upon the pinning link noise [i.e. not ], because the noise is originated from the pinning link which is happened after the controller action. As a traditional method, the H∞ control is ordinarily employed to reduce the effects of the energy-bounded exogenous disturbances. Its aim is to design an appropriate controller such that the closed-loop system is internally stable and its H∞ norm of the transfer function between the system state and the disturbances will not exceed a given H∞ performance level γ. Commonly, the H∞ performance index γ is defined as , where x and w denote the state vector and energy-bounded disturbance, respectively. By the well-known small-gain theorem, if γ < 1, the influence of disturbance to the state will be reduced down towards zero with time increasing. Concerning to the interconnected system, the disturbances, which are originated from different neighbours, are flocking towards the current node via communication network. Therefore, the proposed controller should have better capability of resisting the received compound disturbances. Accordingly, an improved point-to-area H∞ performance is preferred as where denotes the compound disturbance received by the ith node of the lth subsystem, ; it satisfies Here, the above H∞ performance is expressed by an equivalent linear quadratic index (LQI) as (6)For designing an appropriate pinning controller while guaranteeing the desired γ index, the following definition is necessary. Definition 2.For the lth interconnected system (5), the control input realises the group consensus with γ(l) performance if: the control input solves the group consensus when ω(l)(t) = 0; if ω(l)(t) ≠ 0 and ω(l)(t) ∈ L2[0, +∞), the group consensus is achieved, meanwhile, the γ performance of (6) is simultaneously guaranteed under the control input . Remark 4.It can be observed from (3) that and are two variables need to be determined. The authors of [12] say that, concerning with the interconnected systems, all of the zero in-degree nodes (i.e. ) should be pinned. Based on the thought of [12], we here give an adapted pinning node selection principle for the lth interconnected subsystem as follows: all of the zero in-degree nodes should be pinned, because these nodes should be informed what are their equilibrium points; if the group consensus with γ(l) performance cannot be achieved, create a pinning link with the maximal out-degree node for facilitating the consensus control; exclude all the pinned nodes and repeat step (ii) until the group consensus with γ performance for the lth subsystem can be realised. Since the subsystems are independent, the group consensus of LSS can be fulfilled by applying the principle on each subsystem. 2.3 Event-trigger design Due to the continuous-time control item of in (3), the amount of controller updates maintain at a relatively high level. To alleviate the defection, is firstly adapted to a discrete-time control law as , where T is a fixed sampling period, p(l) = {1, 2, …} denotes the index of a data sampling sequence. To further reduce the updating frequency, an event-trigger is introduced into the control scheme. With the event-triggered scheme, the control tasks are only triggered by the occurrence of an event instead at a fixed sampling period. Namely, only an event-triggering function is violated, the current control signal of is packed as and then updated by the controller. Otherwise, the current control action of remains unchanged; where denotes the task execution time instant, k = {1, 2, …}. Obviously, if , the controller updating frequency is reduced. By the event-triggered scheme, (5) is changed as a hybrid system as (7)Define , , , and Equation (7) can be rewritten as (8)where , , and . Note that if the ith node is pinned, ; otherwise, and i, . Generally, designing of an event-trigger is to determine the next update time instant of from the current time instant of by designing an appropriate event-triggering function. The event-triggering function is commonly related to the local measurement error and/or its neighbours information, it can be described as (9)Here, an event-triggering function is designed for the lth subsystem as (10)where denote the measurement deviations, denotes the kth measurement interval of the lth subsystem, it satisfies 0 < ζ(l) < 1 and are the thresholds of the lth event-trigger. Obviously, the key point of the event-trigger is to amplify as large as possible. Unfortunately, due to unpredictable and uncontrollable features of , the proposed event-trigger (9) is a trial strategy. Namely, if there exists at least one instant of k such that holds, (9) contributes to reduce the control execration amounts. In a worst-case scenario, if always holds, (9) becomes a failure strategy. The above shortcoming is still problematic in most event-triggered scheme. Up to now, many literatures show that when the concerned plant is not a highly oscillatory dynamic, the event-trigger is a successful strategy [27–30]. 2.4 Switching topologies From (10), one can easily find that the Laplacian matrix is assumed to be time-invariant. However, due to some practical reasons, for example unpredictable link failures, random data packet dropouts, unknown exogenous disturbances and so on, the communication topology changes actually [31–33]. Therefore, to confirm the reality, the topology dynamic phenomena are considered here. It is assumed that belongs to a finite set as , where S(l) = {1, 2, …, s(l)} denotes the switching index set with s(l) possible cases. By this description, is modelled as with a witching sequence of σ(l)(t), σ(l)(t) ∈ S(l) = {1, 2, …, s(l)}. As an effective method, the Markovian process is usually employed to describe the probability procedure of topology variants [34, 35]. Note that the link disconnection events are unpredictable and barely impossible to be detected in most communication networks, therefore, a practical description is to express the switching signal σ(l)(t) as a state transmission probability partly unknown homogeneous Markovian chain. Let r(l)(t) ∈ S(l) and denotes the Markovian chain and the state transform probability matrix, respectively; where (11)P{ · } is the conditional probability, ξ is the activation time of the ith Markovian state and o(ξ) is the corresponding higher order infinitesimal to ξ, it satisfies limξ→0(o(ξ)/ξ) = 0; denotes the state transmission probability from the ith state to the jth state, and the information of maybe unknown. By the Markovian switching topologies, the close-looped system (8) is accordingly changed as a Markovian switching system as (12)where It is worthy of note that to point out that under the influence of r(l)(t), the original pinning controller (3) has to be changed as a Markovian switching pinning controller as (13) Remark 5.Although the pinning lines play an important role in a pinning control, (13) shows that the pinning lines are still disconnectable (i.e. ). In this sense, the proposed control law (13) is more practical and has a better feature of robustness for tolerating the pinning line defections. Due to discrete-time item of , (12) is a hybrid system. For simplicity, the hybrid system usually needs to be transformed into a unified continuous- or discrete-time form. Note that the measurement time interval can be simply divided as Afterwards, a measurement deviation vector is defined as Since the sampling interval of [p(l)T, (p(l) + 1)T) is a fixed time sector, a substitutional time-delay can be defined as Apparently, one obtains τ(l)(t) ∈ [0, T) and , . By this description, one has hence the original hybrid system (12) is converted to an equivalent continuous time-delayed Markovian switching system as (14)Similarly, the original hybrid event-triggering functions (9) and (10) also need to be unified. Note that the event-trigger (9) is not violated when t ∈ [p(l)T, (p(l) + 1)T). Therefore, one has Accordingly, (10) has a form as (15)Synthesising all of the above mentioned factors, the objection of rest paper is to investigate a feasible design method for the pinning controller (13) and event-trigger (15) let the system (14) achieving group consensus (2) with pint-to-area H∞ performance (6). For ensuring a clear understanding of Theorem 1, the following necessary technique is given beforehand which will be used to derive the sufficient stability condition. Since the probability transmission probability information of Π(l) is partly unknown, denote and as the known and unknown information, respectively; . Assuming r(l)(t) = σ and σ ∈ S(l), the overall transmission probability set of the current σth Markovian state can be divided as , where and are the probability known and unknown sets, . Let the following lemma is obtained. Lemma 1.For a Markovian chain r(l)(t) who satisfies (11), when r(l)(t) = σ, the following inequality holds where is a positive matrix; , , otherwise, . Proof.According to Markovian property one has By the definition of , Lemma 1 is completed. □ 3 Main result For simplicity, the main result is given at first while the necessary proof is given afterwards. Theorem 1.Given a positive scalar γ(l)>0, if there exists positive scalar 0 < σ(l) < 1, positive symmetric matrices and Φ(l), σ ∈ S(l); semi-positive symmetric matrices and , i = 1, 2; and any appropriate dimension matrices and , j = 1, 2, 3; if (16)the group consensus with γ(l) performance of the lth subsystem (14) is achieved under pinning controller (13) and event-trigger (15), the controller gain is given as . Where Remark 6.Theorem 1 gives a concise collaborative design method for the EHMSPC for the lth subsystem. Since the subsystems are all independent, just simply applying Theorem 1 to the other subsystems, the desired group consensus for the overall LSS can be fulfilled. Remark 7.Due to the non-linear item of ζ(l)Φ(l), the inequality of (16) is still a non-linear matrix inequality (NLMI). To feasibly solve (16), we here suggest to set , where and Δζ(l) denote a small initial value and an increasing step, respectively. Giving some appropriate initial values, one can feasibly obtain the allowable maximum of ζ(l) by iteratively solving (16). Proof.Assuming r(l)(t) = σ, σ ∈ S(l), it implies that the σth switching subsystem is currently activated, hence (14) is rewritten as where When t ∈ [p(l)T, (p(l) + 1)T), a Lyapunov–Krasovskii functional is chosen for the σth subsystem as where denotes a positive symmetric matrix; denote semi-positive symmetric matrices, i = 1, 2.Defining an augment state vector as calculating the differential of V(l)(t) along the trajectory of system (14) yields where .For handling the last two integral items, the following zero items are appended into where and , denote any matrices, i = 1, 2, 3.According to Lemma 1, one obtains Merging the similar items of , gives where (17) Apparently, the last two integral items are naturally less than zero. Therefore, if Γ < 0, one obtains .By (15), gives Therefore, one has (18)Note that the LQI (6) is equivalent as Given zero initial condition of η(l)(t) = 0, one has V(l)(0) = 0. Assuming the close-looped system is asymptotically stable, gives .In addition, given any matrix and invertible matrix , the following inequality naturally holds Therefore, one has By the above inequality and the well-known Schur complement, if holds, system (14) is asymptotically stable with guaranteeing γ(l) performance.Due to non-linear of , (18) is a NLMI. Generally, it needs to be decoupled as an equivalent linear matrix inequal