IET Control Theory & Applications Brief Paper
Leader-follower fixed-time consensus for multi-agent systems with unknown non-linear inherent dynamics
ISSN 1751-8644 Received on 4th December 2014 Revised on 13th May 2015 Accepted on 25th May 2015 doi: 10.1049/iet-cta.2014.1301 www.ietdl.org
Michael Defoort1 , Andrey Polyakov2 , Guillaume Demesure1 , Mohamed Djemai1,3 , Kalyana Veluvolu4 1
LAMIH, CNRS UMR 8201, University of Valenciennes and Hainaut-Cambresis, Le Mont Houy, 59313 Valenciennes Cedex 9, France NON-A team, Inria Lille-Nord Europe, 59300 Lille, France 3 Univ. Pierre et Marie CURIE, CNRS, UMR-7222, ISIR, 4 Place Jussieu, F-75005 Paris, France 4 School of Electronics Engineering, College of IT Engineering, Kyungpook National University, Daegu, South Korea E-mail:
[email protected] 2
Abstract: This study focuses on the design of fixed-time consensus for first-order multi-agent systems with unknown inherent non-linear dynamics. A distributed control protocol, based on local information, is proposed to ensure the convergence of the tracking errors in finite time. Some conditions are derived to select the controller gains in order to obtain a prescribed convergence time regardless of the initial conditions. Simulations are performed to validate the theoretical results.
1
Introduction
Multi-agent systems (MASs) have been studied with great attention during the last few decades [1] because multiple agents may perform a task more efficiently than a single one, reduce sensibility to possible agent fault and give high flexibility during the mission execution. Many recent works deal with cooperative scheme for MAS because of its broad range of applications in various areas, for example, flocking [2], rendezvous [3], formation control [4, 5], containment control [6], sensor network [7], coordination of mobile manipulators [8] and so on. Among them, the consensus problem, which objective is to design distributed control policies that enable agents to reach an agreement regarding a certain quantity of interest by relying on neighbours’ information [9], has received considerable attention. They can be categorised into two separated directions depending on whether there is a leader or not. Many consensus schemes have been developed recently [9–13]. However, most of the existing consensus protocols have been derived when there is no leader or when the leader is static. Nevertheless, in many missions, a dynamic leader is required. Many applications may require a dynamic leader, which could be virtual for its followers. Its behaviour is independent of the other agents. The problem of leader–follower consensus is also referred to the cooperative or distributed consensus tracking [14]. The coordination of a group of mobile autonomous agents following a leader has been firstly motivated in [15]. A multi-agent consensus problem with an active leader and a variable topology has been proposed in [16]. In [17], it has been highlighted the strong links between existing leader–follower formation control approaches and consensus-based strategies. The leader–following consensus problem for MASs with Lipschitz non-linear dynamics is discussed in [18]. Recently, a distributed output regulation approach has been proposed for the leader–follower consensus problem in [19]. Leader–follower consensus protocols for second-order non-linear MASs has been derived considering intermittent communication in [20]. In the study of consensus problem, the convergence rate has been an important topic. Indeed, this significant performance index is of high interest to study the effectiveness of a consensus protocol in the context of MAS. Most of the existing consensus algorithms focus on asymptotic convergence, where the settling
IET Control Theory Appl., 2015, Vol. 9, Iss. 14, pp. 2165–2170 © The Institution of Engineering and Technology 2015
time is infinite. However, many applications require a high speed convergence generally characterised by a finite-time control strategy [21]. Moreover, finite-time control allows some advantageous properties, such as good disturbance rejection and good robustness against uncertainties. A robust finite-time consensus tracking problem based on terminal sliding-mode control has been studied in [22], with input disturbances rejection. The finite-time consensus problem for MASs has been studied for single integrator [14, 23], double integrator [24] and inherent non-linear dynamics [25, 26]. A finite-time consensus protocol in directed networks with non-uniform time-varying delays has been proposed in [27]. Using finite time controller, an estimation of the settling time could be very useful when switching topology or networks of clusters are considered. It is worthy of noting that, for the abovementioned works, the explicit expressions for the bound of the settling time depend on the initial states of agents. Therefore the knowledge of these initial conditions usually prevent us from the estimation of the settling time using distributed architectures. A new approach, called fixed-time stability has been recently proposed to define algorithms which guarantee that the settling time is upper bounded regardless to the initial conditions [28]. This concept is promising since one can design a controller such that some control performances are obtained in a given time and independently of initial conditions. In the case of leaderless strategies, the fixed-time average consensus problem has been solved for single integrator linear dynamics without disturbance in [29] and with bounded matched disturbances in [30] even in the case of switching communication topologies [31]. The fixed-time consensus tracking problem for second-order MASs without inherent non-linear dynamics has been solved in [32]. However, the agents sequentially update their control input according to their priority. This fact reduces the level of decentralisation. To the best of our knowledge, although the finite-time leader–follower consensus problem has been well studied, the estimation of the convergence time without the a priori knowledge of initial conditions of agents, has not yet been proposed for first-order MASs with unknown inherent non-linear dynamics. Motivated by the above works, in this paper, the fixed-time consensus tracking problem for MAS with inherent non-linear dynamics is considered. The contribution of this paper is twofold. First, a generalisation of the finite-time leader–follower consensus
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protocol, dealing with MAS with non-linear inherent dynamics, is proposed. Then, an explicit estimation of the settling time is provided regardless of the initial conditions. This paper is organised as follows. Section 2 is devoted to the formulation of the leader–follower consensus problem. Section 3 presents the distributed consensus protocol, based on local information, to ensure the convergence of the tracking errors in finite time. Some conditions are derived to select the controller gains in order to obtain a prescribed convergence time regardless of the initial conditions. In Section 4, some examples are discussed to show the effectiveness of the proposed scheme. Notations: We denote the transpose of a matrix M by M T and eig(M ) its eigenvalues. λmin (M ) [resp. λmax (M )] are the smallest and the largest eigenvalues of M . For a square symmetric matrix P ∈ RN ×N , P 0 (resp. P ≺ 0) indicates that P is positive (resp. negative) definite. Let 1N = (1, 1, . . . , 1)T ∈ RN and diag(a1 , a2 . . . aN −1 , aN ) the diagonal matrix ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
a1 0 .. . 0
0 ..
... .. . . .. .. . . ... 0
0 .. . 0 aN
⎞ ⎟ ⎟ ⎟ ∈ RN ×N ⎟ ⎠
T N xb = For x = (x1 , b. . . , xN ) ∈ R andb Tb ≥ 0, let us define sign (x1 ) |x1 | , . . . , sign (xN ) |xN | .
2 Formulation of the leader–follower consensus problem Consider a MAS consisting of a virtual leader and N followers. The dynamics of each follower and of the leader agent are given by x˙ i x˙ 0
= f (t, xi ) + ui , = f (t, x0 ) + u0
i ∈ {1, . . . , N }
(1)
where x0 ∈ Rm (resp. u0 ∈ Rm ) is the state (resp. control input) of the virtual leader, xi ∈ Rm is the state of agent i and ui ∈ Rm is the control input of agent i. Moreover, f : R+ × D → Rm is an uncertain non-linear dynamics of agent i, continuous in t, D ⊂ Rm a domain containing the origin. Furthermore, there exist positive constants l1 , l2 and l3 such that ∀x1 , x2 ∈ D, ∀t ≥ 0 f (t, x1 ) − f (t, x2 ) 2 ≤ l1 + l2 x1 − x2 2 + l3 x1 − x2 4
(2)
Remark 1: The presented class of systems includes Lipschitz (l1 = l3 = 0) and quasi-Lipschitz (l3 = 0) as well as some essentially non-linear (e.g. polynomial) models (l3 = 0). The polynomial term may appear, for example, in mechanical models with a viscous friction force proportional to square of velocity. √ Indeed, if xi is the velocity of some fast mobile vehicle, then l3 is proportional to a coefficient of viscous friction. It is worthy of noting that condition (2) is less restrictive than most of the existing works dealing on consensus for MAS with nonlinear inherent dynamics [25]. Since f (t, xi ) is uncertain, it is not possible to cancel it with the control, namely (1) is not feedback equivalent to an m-dimensional integrator. It is also assumed that the control u0 of the leader is bounded by a known positive constant l, that is u0 ≤ l
(3)
One should notify that the design of a saturated control input for one agent which solves the trajectory tracking problem has been solved in [33, 34].
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Remark 2: In this paper, the bound l is assumed to be known for each follower. If it is not the case, an adaptive sliding mode controller could be used [35]. However, the design of adaptive scheme for fixed-time controller is still an open problem. To solve the coordination problem and model exchanged information between agents, graph theory is briefly recalled hereafter. The communication topology among all followers is fixed and is represented by an undirected graph G which consists of a nonempty set of nodes V = {1, 2, . . . , N } and a set of edges E ⊂ V × V. Here, each node in V corresponds to an agent i, and each edge (i, j) ∈ E in the undirected graph corresponds to an information link between agent i and agent j. The topology of graph G is represented by the weighted adjacency matrix A = (aij ) ∈ RN ×N given by aij > 0 if (j, i) ∈ E and aij = 0, otherwise. The Laplacian
matrix of G is defined as L = (lij ) ∈ RN ×N with lii = N j=1,j =i aij and lij = −aij for i = j. For an undirected graph, L is symmetric positive semi-definite. Graph G is connected if and only if its Laplacian matrix has a simple zero eigenvalue with associated eigenvector 1N . The graph G¯ is fixed and describes the communication topology of all followers and the virtual leader. The topology of G¯ is described by the weighted matrix H = L + D ∈ RN ×N where D = diag(a10 , . . . , aN 0 ) with ai0 > 0 if the leader state is available to follower i and with ai0 = 0 otherwise. The objective of this paper is to design a distributed control protocol ui , based on available local information, such that each follower tracks the virtual leader in a prescribed time Tmax , that is xi (t) = x0 (t),
∀t ≥ Tmax ,
∀i ∈ {1, . . . , N }
(4)
Remark 3: It is assumed that there is no communication delay. Hence, local information from the neighbours are available in continuous time without time delay. To reduce the network loads, a self-triggered controller has been proposed in [12] to solve the practical consensus for MASs under time-varying topology. The effect of communication delays has been recently accounted for in order to achieve the consensus exponentially [13, 20, 36]. A finitetime consensus protocol in directed networks with non-uniform time-varying delays has been proposed in [27]. Nevertheless, the estimation of the settling time regardless of the initial conditions (i.e. fixed-time stability) in the presence of time delay is still an open problem. Indeed, the communication delays clearly affect the convergence time. The estimation of the settling time regardless both of the initial conditions and communication delays still remains a challenging issue. Lyapunov–Krasovkii or Lyapunov– Razumikhin approaches should be used and may imply very conservative conditions in general cases [37]. Therefore the design of fixed-time consensus protocols considering delays is out the scope of this paper. Remark 4: For the sake of simplicity, in the following, it is assumed that m = 1. The analysis is valid for any arbitrary dimension m with the difference being that the expression should be written in terms of the Kronecker product.
3
Fixed-time consensus tracking protocol
In this section, a non-linear finite time distributed control protocol will be designed such that each follower tracks the virtual leader in a prescribed time Tmax . Before that let us introduce the concept of fixed-time stability and a useful lemma. 3.1
Preliminaries on fixed-time stability
Let us consider the following system x˙ (t) ∈ F(t, x(t)) x(0) = x0
(5)
where x ∈ Rn is the state, F : R+ × Rn → Rn be an upper semicontinuous convex-valued mapping, such that the set F(t, x) is
IET Control Theory Appl., 2015, Vol. 9, Iss. 14, pp. 2165–2170 © The Institution of Engineering and Technology 2015
non-empty for any (t, x) ∈ R+ × Rn and F(t, 0) = 0 for t > 0. The solutions of (5) are understood in the Filippov sense [38]. Definition 1 [40]: The origin of system (5) is a globally finite time equilibrium if there is a function T : Rn → R+ such that for all x0 ∈ Rn , the solution x(t, x0 ) of system (5) is defined and x(t, x0 ) ∈ Rn for t ∈ [0, T (x0 )) and limt→T (x0 ) x(t, x0 ) = 0. T (x0 ) is called the settling time function. Definition 2 [28]: The origin of system (5) is a globally fixed-time equilibrium if it is globally finite-time stable and the settling time function T (x0 ) is bounded by some positive number Tmax > 0, that is, T (x0 ) ≤ Tmax for ∀x0 ∈ Rn . The fixed-time stabilisation at the origin can be demonstrated on the simplest scalar control system x˙ (t) = u(t), where x ∈ R is the state and u ∈ R is the control input. If the so-called negative relay feedback u = − sign (x) is applied, then the closed-loop system is finite-time stable with the settling time T (x0 ) = |x0 |. It is worthy of noting that the settling time depends on the initial conditions and is unbounded. The fixed-time control algorithm [28] has the form u = −(|x|2 + 1) sign (x). It guarantees finite convergence time independently of the initial conditions, namely, T (x0 ) ≤ (π/2), where π = 3.1415926 . . . is the well-know irregular number. More strong result is provided by the following lemma that refines the results of [28] and [39]. Lemma 1 [29]: Assume that there exists a continuously differentiable positive definite and radially unbounded function V : Rn → R+ such that ∂V (x) y ≤ −aV p (x) − bV q (x) sup t>0,y∈F(t,x) ∂x
for
Proof: Let us define the tracking errors as x˜ i = xi − x0 , ∀i ∈ {1, . . . , N }. Equations (1), (6) yield ⎢ ⎤2 ⎢ N ⎢N x˙˜ i = α ⎣ aij (xj − xi ) aij (xj − xi )⎥ ⎥ +β ⎥ j=0 j=0 ⎞ ⎛ N aij (xj − xi )⎠ + f (t, xi ) − f (t, x0 ) − u0 + γ sign ⎝
(9)
j=0
In a compact form, system (9) becomes x˙˜ = −α H x˜ 2 − βH x˜ − γ sign (H x˜ ) + f˜ − 1N u0 with x˜ = (˜x1 , . . . , x˜ N )T f˜ = (f (t, x1 ) − f (t, x0 ), . . . , f (t, xN ) − f (t, x0 ))T Considering the candidate Lyapunov function V =
x = 0
with a, b > 0, p = 1 − (1/μ), q = 1 + (1/μ) and μ ≥ 1. Then, the origin of the differential inclusion (5) is globally fixed-time stable with the settling time estimate
1 T x˜ H x˜ 2
H 0 since graph G is connected and there is at least one ai0 positive [14]. Its time derivative is given by V˙ = x˜ T H x˙˜ = −α x˜ T H H x˜ 2 − β H x˜ 22 − γ H x˜ 1 − x˜ T H 1N u0 + x˜ T H f˜ (10)
πμ T (x0 ) ≤ Tmax = √ . 2 ab In the next subsection, this lemma will be used to design a distributed non-linear fixed-time control protocol and to analyse its robustness with respect to uncertainties and disturbances. 3.2
with > 0, achieves the convergence of the tracking errors x˜ i = xi − x0 , ∀i ∈ {1, . . . , N } to zero in a finite time, which is bounded by π (8) Tmax =
The later considerations uses the classical result about equivalence of the norms · q
y q =
Fixed-time consensus protocol design
N
q1 |yi |q
,
y = (y1 , . . . , yN )T ∈ RN ,
q>0
i=1
Based on Lemma 1, a distributed control protocol is derived to guarantee that each follower tracks the leader in a prescribed time Tmax regardless the initial condition in the following theorem.
in a finite-dimensional linear space. Namely, for p > r > 0 we always have 1
Theorem 1: Suppose that graph G is connected and at least one ai0 > 0. The control protocol ⎢ ⎤2 ⎢ ⎢N ui = α ⎣ aij (xj − xi )⎥ ⎥ ⎥ j=0 ⎛ ⎞ N N +β aij (xj − xi ) + γ sign ⎝ aij (xj − xi )⎠ (6) j=0
j=0
where the gains are defined as √ √ N l3 (λmax (H ))3/2 + α= 3 7/2 λmin (H ) (2λmin (H )) 2 √ l2 β= λmin (H ) λmax (H ) γ = l + l1 N + 2λmin (H )
IET Control Theory Appl., 2015, Vol. 9, Iss. 14, pp. 2165–2170 © The Institution of Engineering and Technology 2015
y p ≤ y r ≤ N r
− 1p
y p
The first term of (10) can be estimated as follows 3 N N 2 T −α x˜ H H x˜ = −α aij (xj − xi ) = −α H x˜ 33 i=1 j=0 ≤ −αN − 2 (2λmin (H )) 2 V 2 1
3
3
The last term of (10) admits the following representation
(7)
1/2 x˜ T H f˜ ≤ H x˜ 2 f˜ 2 ≤ H x˜ 2 l1 N + l2 ˜x 22 + l3 ˜x 44 ≤ H x˜ 2 l1 N + l2 ˜x 2 + l3 ˜x 24 √ l2 H x˜ 22 ≤ l1 N H x˜ 1 + λmin (H ) √ 1 l3 N − 2 (2λmax (H ))3/2 3/2 V + λ2min (H )
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4
0
1
Fig. 1
2
3
6
4
5
Let us consider the MAS (1) with N = 6 followers and one virtual leader. The communication topology, given in Fig. 1 is undirected and fixed. It is connected and the corresponding matrix H is given by ⎡ ⎤ 2 −1 0 0 0 0 ⎢−1 3 −1 0 0 0⎥ ⎢ ⎥ ⎢ 0 −1 3 −1 −1 0 ⎥ H =⎢ 0 −1 2 −1 0 ⎥ ⎢0 ⎥ ⎣0 0 −1 −1 3 −1⎦ 0 0 0 0 −1 2
Communication topology
It follows that V˙ ≤ −αN − 2 (2λmin (H )) 2 + 1
− β−
3
√
Numerical simulations
√ 1 3 l3 N − 2 (2λmax (H ))3/2 V2 λ2min (H )
l2 H x˜ 22 − γ H x˜ 1 + (l + l1 N ) H x˜ 1 λmin (H ) (11)
Let the gains be designed by (7). Therefore, one obtains V˙ ≤ −V q − V p with μ=2 1 μ 1 q=1+ μ p=1−
Hence, Lemma 1 guarantees that the origin of the closed-loop system (9) is globally fixed-time stable with the settling time bounded by Tmax = π/. Remark 5: It is worthy of noting that the settling time does not depend on the initial conditions. If a prescribed convergence time is required, one can easily tune the controller gains according to (7) to solve the leader–follower consensus problem. Such a requirement occurs when considering switching topologies. An extension of these results to the case of switching network is possible. Indeed, in this case, the sum of time intervals over which the graph is strongly connected must be larger that the settling time to guarantee that each follower tracks the virtual leader.
One can see that agents 3, 4 and 5 do not have direct information from the leader. Hereafter, two scenarios are discussed to highlight the effectiveness of the proposed fixed-time consensus protocol for MAS with inherent non-linear dynamics. It will be shown that the control law ensures the convergence of the tracking errors in a prescribed convergence time regardless of the initial conditions (i.e. fixed-time stability). Although the finite-time leader–follower consensus problem has been well studied, the estimation of the convergence time regardless of the initial conditions for first-order MASs with unknown inherent non-linear dynamics has not yet been proposed. Scenario 1: We consider the case m = 1. The non-linear dynamics, assumed uncertain, are f (t, xi ) = 0.1 sin (xi ), ∀i = {0, . . . , N }. The Lipschitz condition is verified. Indeed, in this case, l1 = l3 = 0 and l2 = 0.1. The control input of the virtual leader is chosen as u0 = 2 cos (0.1πt). It is bounded by l = 2. The virtual leader starts from x0 (0) = 0. From Theorem 1, the control protocol (6) achieves the convergence of the tracking errors x˜ i = xi − x0 , ∀i ∈ {1, . . . , N } to zero in finite time. Contrary to existing finite-time controllers, here an explicit estimation of the settling time is provided without the a priori knowledge of initial conditions of agents. The estimated upper bounds of the settling time for the proposed distributed control scheme (6) is Tmax = 1.5 s. The settling time remains bounded even for large initial conditions, testifying to the fixed-time nature of the developed control protocol. To illustrate this fact, two initial conditions are considered. For the first case, the six followers start from the following initial conditions: x1 (0) = −10, x2 (0) = −5, x3 (0) = −20, x4 (0) = −15, x5 (0) = 10 and x6 (0) = 5. Large initial conditions of each follower has been chosen for the second case, that is, x1 (0) = −150, x2 (0) = −75, x3 (0) = −300, x4 (0) = −225, x5 (0) = 150 and x6 (0) = 75. Fig. 2a (resp. Fig. 2b) depicts the trajectories of the virtual leader and the followers for the first case (resp. for the second case) when the control protocol (6)
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100 80
5
60 40
0
−10 −15 −20 0
Leader Follower Follower Follower Follower Follower Follower 5
x (m )
x(m )
20 −5
0
−20
1 2 3 4 5 6
−40 −60 −80 10
t (s )
15
20
−100 0
Leader Follower Follower Follower Follower Follower Follower
1 2 3 4 5 6
5
15
20
t (s )
a
Fig. 2
10 b
Leader–follower consensus for MASs with unknown Lipschitz non-linearities: trajectories of each agent
a Case 1 b Case 2
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IET Control Theory Appl., 2015, Vol. 9, Iss. 14, pp. 2165–2170 © The Institution of Engineering and Technology 2015
a
Fig. 3
b
Leader–follower consensus for MASs with unknown Lipschitz non-linearities: tracking errors
a Case 1 b Case 2
a
Fig. 4
b
Leader–follower consensus for MASs with unknown non-Lipschitz non-linearities
a Trajectories of each agent b Norm of the tracking errors
is used. Fig. 3 shows the tracking errors x˜ i for i = 1, . . . , 6. It is clear that from Figs. 3a and b that the settling time is about 0.95 s for case 1 and 1.25 s for case 2. One can see that the estimate of the settling time is of moderate conservatism. These two cases confirm the theoretical results obtained from Theorem 1. Scenario 2: Now, we consider the case m = 2. The state of leader and the xi are written as x0 = thex virtual followers y T y T η0 , η0 ∈ D and xi = ηix , ηi ∈ D where D = [−10, 10] × [−10, 10]. The non-linear dynamics, assumed uncertain, are y T f (t, xi ) = 0.01 0.01(ηix )2 , 0.01 sign (ηi ) . This function does not satisfy the Lipschitz condition. Nevertheless, in this case, condition (2) is fulfilled since l1 = 0.02 and l3 = 0.1 on D. It is worthy of noting that condition (2) is less restrictive than most of the existing works dealing on consensus for MAS with non-linear inherent dynamics [25]. The control input of the virtual leader is chosen as u0 = (0.1, 2 cos(π/2t)). It is bounded by l = 0.64. From Theorem 1, the control protocol (6) achieves the convergence of the tracking errors x˜ i = xi − x0 , ∀i ∈ {1, . . . , N } to zero in finite time. The estimated upper bounds of the settling time for the proposed distributed control scheme (6) is Tmax = 1.5 s. The six followers start from the following initial conditions: x1 (0) = (−2, −2)T , x2 (0) = (−1, −3)T , x3 (0) = (−4, 2)T , x4 (0) = (−3, 4)T , x5 (0) = (2, 2.4)T and x6 (0) = (1, 3)T . The virtual leader starts from x0 (0) = (0, 0)T . Fig. 4a depicts the trajectories of the
IET Control Theory Appl., 2015, Vol. 9, Iss. 14, pp. 2165–2170 © The Institution of Engineering and Technology 2015
virtual leader and the followers when the control protocol (6) is used. Fig. 4b shows that the consensus is achieved after 0.5 s. It confirms the theoretical results obtained from Theorem 1.
5
Conclusion
In this paper, the fixed-time consensus tracking problem for MAS with inherent non-linear dynamics has been considered. The contribution of this paper is twofold. First, a generalisation of the finite-time leader–follower consensus protocol, dealing with MAS with non-linear inherent dynamics, has been proposed. Then, an explicit estimation of the settling time has been provided regardless of the initial conditions. Some conditions have been derived to select the controller gains in order to obtain a prescribed convergence time regardless of the initial conditions. As future works, we will consider communication delays and adaptive scheme to avoid the knowledge of the upper bound of the leader control input.
6
Acknowledgments
This work was supported by the International Campus on Safety and Intermodality in Transportation, the European Community, the Regional Delegation for research and Technology, the Nord-Pasde-Calais Region, the Ministry of Higher Education and Research
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and the National Center for Scientific Research under the FUI EQUITAS and the ARCIR SUCRe projects.
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References Cao, Y., Yu, W., Ren, W., Chen, G.: ‘An overview of recent progress in the study of distributed multi-agent coordination’, IEEE Trans. Ind. Inf., 2013, 9, (1), pp. 427–438 Wen, G., Duan, Z., Su, H., Chen, G., Yu, W.: ‘A connectivity-preserving flocking algorithm for multi-agent dynamical systems with bounded potential function’, IET Control Theory Appl., 2012, 6, (6), pp. 813–821 Song, C., Feng, G., Wang, Y., Fan, Y.: ‘Rendezvous of mobile agents with constrained energy and intermittent communication’, IET Control Theory Appl., 2012, 6, (10), pp. 1557–1563 Fax, A., Murray, R.: ‘Information flow and cooperative control of vehicle formations’, IEEE Trans. Autom. Control, 2004, 49, (9), pp. 1453–1464 Defoort, M., Floquet, T., Kokosy, K., Perruquetti, W.: ‘Sliding mode formation control for cooperative autonomous mobile robots’, IEEE Trans. Ind. Electron., 2008, 55, (11), pp. 3944–3953 Wang, P., Jia, Y.: ‘Distributed containment control of second-order multi-agent systems with inherent non-linear dynamics’, IET Control Theory Appl., 2014, 8, (4), pp. 277–287 Zhang, H., Yan, H., Yang, F., Chen, Q.: ‘Distributed average filtering for sensor networks with sensor saturation’, IET Control Theory Appl., 2013, 7, (6), pp. 887–893 Li, Z., Yang, C., Tang, Y.: ‘Decentralised adaptive fuzzy control of coordinated multiple mobile manipulators interacting with non-rigid environments’, IET Control Theory Appl., 2013, 7, (3), pp. 397–410 Ren, W., Beard, R.: ‘Consensus seeking in multi-agent systems under dynamically changing interaction topologies’, IEEE Trans. Autom. Control, 2005, 50, pp. 655–661 He, W., Cao, J.: ‘Consensus control for high-order multi-agent systems’, IET Control Theory Appl., 2011, 5, (1), pp. 231–238 Seyboth, G., Dimarogonas, D., Johansson, K.: ‘Event-based broadcasting for multi-agent average consensus’, Automatica, 2013, 49, (1), pp. 245–252 Defoort, M., Di Gennaro, S., Djemai, M.: ‘Self-triggered control for multi-agent systems with unknown nonlinear inherent dynamics’, IET Control Theory Appl., 2015, 8, (18), pp. 2266–2275 Huang, N., Duan, Z., Zhao, Y.: ‘Consensus of multi-agent systems via delayed and intermittent communications’, IET Control Theory Appl., 2014, 9, (1), 62–73. Cao, Y., Ren, W.: ‘Distributed coordinated tracking with reduced interaction via a variable structure approach’, IEEE Trans. Autom. Control, 2012, 57, (1), pp. 33–48 Jadbabaie, A., Lin, J., Morse, A.: ‘Coordination of groups of mobile autonomous agents using nearest neighbor rules’, IEEE Trans. Autom. Control, 2003, 48, (6), pp. 988–1001 Hong, Y., Hu, J., Gao, L.: ‘Tracking control for multi-agent consensus with an active leader and variable topology’, Automatica, 2006, 42, (7), pp. 1177–1182 Ren, W.: ‘Consensus strategies for cooperative control of vehicle formations’, IET Control Theory Appl., 2007, 1, (2), pp. 505–512 Xu, W., Cao, J., Yu, W., Lu, J.: ‘Leader-following consensus of non-linear multiagent systems with jointly connected topology’, IET Control Theory Appl., 2014, 8, (6), pp. 432–440
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Hong, Y., Wang, X., Jiang, Z.P.: ‘Distributed output regulation of leader-follower multi-agent systems’, Int. J. Robust Nonlinear Control, 2013, 23, (1), pp. 48–66 Huang, N., Duan, Z., Zhao, Y.: ‘Leader-following consensus of second-order non-linear multi-agent systems with directed intermittent communication’, IET Control Theory Appl., 2014, 8, (10), pp. 782–795 Liu, X.W., Chen, T.P., Lu, W.L.: ‘Consensus problem in directed networks of multi-agents via nonlinear protocols’, Phys. Lett. A, 2009, 373, (35), pp. 3122–3127 Khoo, S., Xie, L., Man, Z.: ‘Robust finite-time consensus tracking algorithm for multirobot systems’, IEEE/ASME Trans. Mechatron., 14, (2), 2009, pp. 219–228 Wang, L., Xiao, F.: ‘Finite-time consensus problems for networks of dynamic agents’, IEEE Trans. Autom. Control, 2010, 55, (4), pp. 950–955 Li, S., Du, H., Lin. X.: ‘Finite-time consensus algorithm for multi-agent systems with double-integrator dynamics’, Automatica, 2011, 47, (8), pp. 1706–1712 Wen, G., Yu, Y., Peng, Z., Rahmani, A.: ‘Distributed finite-time consensus tracking for nonlinear multi-agent systems with a time-varying reference state’, Int. J. Syst. Sci., 2015, In Press, DOI:10.1080/00207721.2014.955279 Cao, Y., Ren, W.: ‘Finite-time consensus for multi-agent networks with unknown inherent nonlinear dynamics’, Automatica, 2014, 50, (10), pp. 2648–2656 Sun, Y.G., Wang, L.: ‘Consensus of multi-agent systems in directed networks with nonuniform time-varying delays IEEE Trans. Autom. Control, 2009, 54, (7), pp. 1607–1613 Polyakov, A.: ‘Nonlinear feedback design for fixed-time stabilization of linear control systems’, IEEE Trans. Autom. Control, 2012, 57, (8), pp. 2106–2110 Parsegov, S., Polyakov, A., Shcherbakov, P.: ‘Fixed-time consensus algorithm for multi-agent systems with integrator dynamics’. Fourth IFAC Workshop on Distributed Estimation and Control in Networked Systems, 2013, pp. 110–115 Zuo, Z., Tie, L.: ‘Distributed robust finite-time nonlinear consensus protocols for multi-agent systems’, Int. J. Syst. Sci., 2015, In press, DOI: 10.1080/00207721. 2014.925608 Zuo, Z., Yang, W., Tie, L., Meng, D.: ‘Fixed-time consensus for multi-agent systems under directed and switching interaction topology’. IEEE Proc. American Control Conf., 2014, pp. 5133–5138 Zuo, Z.: ‘Nonsingular fixed-time consensus tracking for second-order multi-agent networks’, Automatica, 2015, 54, pp. 305–309 Teel, A.R.: ‘Global stabilization and restricted tracking for multiple integrators with bounded controls’, Syst. Control Lett., 1992, 18, (3), pp. 165–171 Defoort, M., Palos, J., Kokosy, A., Floquet, T., Perruquetti, W.: ‘Performancebased reactive navigation for nonholonomic mobile robots’, Robotica, 2009, 27, (2), pp. 281–290 Plestan, F., Shtessel, Y., Bregeault, V., Poznyak, A.: ‘Sliding mode control with gain adaptation – Application to an electropneumatic actuator’, Control Eng. Pract., 2013, 21, (5), pp. 679–688 Munz, U., Papachristodoulou, A., Allgower, F.: ‘Delay robustness in consensus problems’, Automatica, 2010, 46, (8), pp. 1252–1265 Efimov, D., Polyakov, A., Fridman, E., Perruquetti, W., Richard, J.P.: ‘Comments on finite-time stability of time-delay systems’, Automatica, 2014, 50, (7), pp. 1944–1947 Filippov, A.: ‘Differential equations with discontinuous right-hand side’ (Kluwer Academic, Dordrecht, 1988) Parsegov, S., Polyakov, A., Shcherbakov, P.: ‘Nonlinear fixed-time control protocol for uniform allocation of agents on a segment’. 51st Conf. on Decision and Control, 2012, pp. 7732–7737 Bhat, S., Bernstein, D.: ‘Geometric homogeneity with applications to finite time stability’, Math. Control Signals Syst., 2005, 17, pp. 101–127
IET Control Theory Appl., 2015, Vol. 9, Iss. 14, pp. 2165–2170 © The Institution of Engineering and Technology 2015
