Distributed fault-tolerant control for a class of cooperative uncertain systems with actuator failures and switching topologies

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Distributed fault-tolerant control for a class of cooperative uncertain systems with actuator failures and switching topologies Xin Wang a, Guang-Hong Yang a,b,∗ a b

College of Information Science and Engineering, Northeastern University, Shenyang 110819, PR China Key Laboratory of Integrated Automation for Process Industry, Northeastern University, Shenyang 110819, PR China

a r t i c l e

i n f o

Article history: Received 13 January 2015 Revised 14 October 2015 Accepted 2 November 2015 Available online xxx Keywords: Fault-tolerant control Consensus tracking Multi-agent systems Switching undirected topologies

a b s t r a c t This paper studies the distributed fault-tolerant control (DFTC) problem for a class of highorder multi-agent systems (MASs) with switching undirected topologies, heterogeneous matching uncertainties, disturbances and actuator faults including loss of effectiveness, bias, outage and stuck. In this frame work, the communication network of dynamic agents may switch among several undirected connected graphs. It is assumed that the actuator efficiency factors, the lower and upper bounds of the time-varying failures and heterogeneous uncertainties, disturbances and the leader’s control input signal, are unknown. Based on the relative information of neighbors, a novel fault-tolerant consensus tracking protocol is designed for each follower node via distributed adaptive mechanism. By utilizing the multiple Lyapunov functions method and algebraic graph theory, a sufficient condition for consensus tracking is established. Furthermore, it is proved that if the topology dwell time is larger than a positive threshold, the state of each follower node synchronizes to that of the leader with a bounded residual error in the presence of actuator faults, heterogeneous matching uncertainties and disturbances. Finally, two numerical examples are given to show the effectiveness of the proposed control scheme. © 2015 Elsevier Inc. All rights reserved.

1. Introduction In recent year, designing a feasible and effective distributed protocol for each subsystem such that a networked of agents as a whole can perform complex tasks has attracted considerable attention in the integration of communication and control [5,22,26]. Due to its potential applications in many areas, (e.g., formation of flight of satellites, flocking of mobile vehicles, scheduling of automated highway systems [15,27,41,45]), the cooperative control of MASs has received tremendous attention. To date, a series of effective approaches have been made to study the emergence of consensus of high-order MASs, such as leader-following control [6,23,32,49], switching communication networks [25,33,34], cooperative optimal strategy [4,29,44], distributed adaptive protocol [9,16,17,24], coordination formation control [2], and so on. As well known, in order to achieve a special global common behavior, a large number of agents composed to form a cooperative system via communication networks. Unlike traditional networked control systems [13,14,42,47], the reliability demand of ∗ Corresponding author at: College of Information Science and Engineering, Northeastern University, Shenyang 110819, PR China. Tel.: +862483681939; fax: +862483681939. E-mail addresses: [email protected] (X. Wang), [email protected], [email protected] (G.-H. Yang).

http://dx.doi.org/10.1016/j.ins.2015.11.002 0020-0255/© 2015 Elsevier Inc. All rights reserved.

Please cite this article as: X. Wang, G.-H. Yang, Distributed fault-tolerant control for a class of cooperative uncertain systems with actuator failures and switching topologies, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.002

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MASs rapidly is enhanced due to an increasing number of actuators, sensors and other system components. However, in practical applications, the faulty components in some nodes may cause the system performance deterioration or lead to instability, even affect the global behavior of the multi-agent networks. A typical example is a power grid consisting of a large number of generators, a fault even in a single local power generator may have a severe undesirable impact on the synchronization of power supply. Unfortunately, the conventional fault-tolerant control methods for centralized systems are only useful to ensure the local stability of individual nodes. Motivated by this observation, researchers began to investigate distributed fault-tolerant control for MASs and tried to improve the safety and reliability of cooperative systems. Generally speaking, the distributed fault-tolerant control approaches can be broadly classified into two categories, that is, passive and active ones. Passive DFTC approaches mainly utilize fixed controllers via robust control theory with respect to a priori fixed set of failures, see for instance in [28,40]. In contrast to the passive strategy, the parameters and/or the structure of the controllers are adjustable with active DFTC approaches, see for instance in [8,10,18]. Recently, the very few active DFTC protocols for MASs subject to actuator failures have been proposed in the literature [30,49]. However, these works focus only on the distributed fault accommodation schemes against loss of effectiveness or additive failures but do not address the outage or stuck actuators in follower nodes. What’s more, different from centralized FTC problem [11,19–21,31,37,39,43], the task of designing DFTC controller based on neighborhood information is not only to compensate individual failures, but also to achieve consensus tracking objective via communication networks. Therefore, it is meaningful to further investigate the distributed faulttolerant tracking control for the agent networks. In addition, it is worth stressing that the existing DFTC methods in [30,32,49] are commonly required that the communication topology is fixed. But in practical scenario, the underlying network of all agents should be a time-varying graph as both communication links and nodes in the network may experience failures or reconstructions [8,33,34]. Essentially, due to the design of Lyapunov function by using the DFTC schemes in [30,32,49] is dependent on the Laplacian matrix of the underlying graph, the boundedness of closed-loop signals and the global tracking objective cannot be ensured when the network is switching. Therefore, the DFTC protocols in existing works cannot be directly applied any more. On the other hand, many network-connected systems may be controlled in complex operating environments, such as uncertainties, time delay, external disturbances and so on [1,3,12,17,38,42,46], which makes this problem even more difficult in practical applications. To the best of our knowledge, there are few effective techniques available for distributed fault accommodation of MASs under dynamic networks so far. In this paper, the distributed fault-tolerant tracking control problem for a class of uncertain networked agents with a dynamic leader under switching undirected topologies is investigated. Suppose that the network topology switches among several connected graphs and only part of agents can observe the information of the leader. Compared with the existing works, the results of this paper have three distinct features: 1. By using only relative state information of neighbors to estimate actuator efficiency factors and controller parameters online, a novel active DFTC protocol in individual node is designed, under which the time-varying actuator faults including loss of effectiveness, bias, outage and stuck, can be compensated in real time. Specifically, the distributed control allocation scheme is firstly used to achieve cooperative tracking performance, when some outage or stuck failures occur in the actuators. 2. For the non-identical matching uncertainties and external disturbances, the alternative distributed adaptive protocol with switching mechanism is employed, which can reduce effectively the global consensus tracking error. Moreover, designing of the proposed protocol does not require any prior knowledge, such as the bounds of the leader’s input, heterogeneous uncertainties and disturbances. 3. When the undirected network topology is fixed, which is relatively concise to design distributed adaptive fault-tolerant tracking protocol because of the compatibility with the single Lyapunov function in stability analysis; but for the switching undirected topologies case, the existing DFTC approaches given by [30,32,49] will not guarantee the convergence of the parameter estimate errors and global tracking error. To overcome this difficulty, motivated by the switching control approaches in [33–35,48], a constructive cooperative adaptive fault compensation methodology is established on the basis of the multiple Lyapunov functions method and algebraic graph theory. It is shown that with the proposed DFTC scheme, not only the boundedness of all closed-loop signals is ensured, but also global tracking of all followers’ states can be achieved. The rest of this paper is organized as follows. In Section 2, the preliminaries and problem statement are presented. The design approach of the DFTC protocol is presented in Sections 3. In Section 4, two numerical simulations are given to illustrate the proposed DFTC methodology. Finally, Section 5 draws the conclusion. Notation: Rn denotes the n-dimensional Euclidean space and · represents Euclidean norm of vectors or matrices. Let Ik stand for the identity matrix of dimension k. For matrix A ∈ Rn×n , λmax (A) and λmin (A) are its maximum eigenvalue and minimum eigenvalue, respectively. AT represents the transpose of matrix A. A ⊗ B denotes the Kronecker product of two matrices A ∈ Rm×n and B ∈ R p×q . Let diag(A1 , A2 , . . . , An ) denote a block-diagonal matrix with matrices Ai , i = 1, . . . , n, on its diagonal. Denote by P > 0 that P is a symmetric positive definite matrix with an appropriate dimension. 2. Preliminaries and problem statement 2.1. Basic graph theory Let Gσ (t ) = (V, Eσ (t ) , Aσ (t ) ) be a undirected graph with a set of N nodes V = {v1 , v2 , . . . , vN }, a set of edges Eσ (t ) = V × V, and an associated adjacency matrix Aσ (t ) = [ai j (t )] ∈ RN×N , where σ (t ) : [0, +∞) → P with P is an index set for all possible Please cite this article as: X. Wang, G.-H. Yang, Distributed fault-tolerant control for a class of cooperative uncertain systems with actuator failures and switching topologies, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.002

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3

graphs {G p : p ∈ P }. An edge eij in Gσ (t ) is denoted by the ordered pair of vertices (v j , vi ) are called the parent and child vertices, respectively, and ei j ∈ Eσ (t ) if and only if aij (t) > 0, representing that node vi can observe the information of node v j . The graph is undirected, that is, the edges eij and eji in Eσ (t ) are considered to be the same. Two nodes vi and v j are neighbors to each other if ei j ∈ Eσ (t ) . The set of neighbors of node vi at time t is denoted by Ni (t ) = {vi ∈ V : ei j ∈ Eσ (t ) , j = i}. A path is a sequence of connected edges in a graph. A undirected graph is connected if there is a path between every pair of vertices. The degree  matrix Dσ (t ) is defined by Dσ (t ) = diag(d1 (t ), d2 (t ), . . . , dN (t )) ∈ RN×N with di (t ) = i∈N (t ) ai j (t ) and the Laplacian matrix as i Lσ (t ) = Dσ (t ) − Aσ (t ) . Furthermore, define the leader adjacency matrix Dσ (t ) = diag(d1 (t ), d2 (t ), . . . , dN (t )), where di (t) > 0 if and only if the node vi has access to the leader information. 2.2. Node dynamics Consider a group of multi-agent systems consisting of N follower nodes, indexed as 1, 2, . . . , N, and a leader with index as 0, which is described by

x˙ 0 (t ) = Ax0 (t ) + Br0 (t )

(1)

is the state of the leader, r0 (t ) ∈ is a smooth bounded reference input. It is assumed that the state of where x0 (t ) ∈ the leader is bounded, that is, x0 (t ) ≤ x¯0 , which is usually pre-assigned to the leader. Moreover, the ith follower node (i = 1, 2, . . . , N) has the following dynamics Rn

Rm

x˙ i (t ) = Axi (t ) + fi (xi (t ), t ) + Bui (t ) + E ωi (t )

(2)

where xi (t ) ∈ is the state of the ith follower, ui (t ) ∈ is the control input to be design, and ωi (t ) ∈ is the unknown bounded external disturbance for the agent i. Let A, B and E be given constant matrices, and (A, B) is assumed to be stabilizable. The nonlinear function fi (xi (t ), t ) ∈ Rn represents the unknown non-identical uncertainties, and fi (xi (t), t), E are assumed to satisfy the following matching conditions Rn

fi (xi (t ), t ) = B fˆi (xi (t ), t ),

Rm

Rl

E = BF

where F is a known matrix. Define gi (xi (t ), t ) = fˆi (xi (t ), t ) + F ωi (t ), then the dynamic of networked agent system (2) can be written as

x˙ i (t ) = Axi (t ) + B(ui (t ) + gi (xi (t ), t )),

i = 1, 2, . . . , N

(3)

Assumption 1. There exist two unknown constants α i > 0 and β i > 0 such that

gi (xi (t ), t ) ≤ αi + βi xi (t ), i = 1, 2, . . . , N

(4)

Remark 1. The subsystem model (2) considered is notable in regard to the MASs of consensus designs, which is widely used in literature [17,24]. The nonlinear function fi (xi (t), t) can be considered the weakly heterogeneous uncertainties in each subsystem. Moreover, Assumption 1 will be satisfied if fi (xi (t), t) and ωi (t) are both bounded. Remark 2. Note that the leader node (1) is a reference model, which generate the desired trajectory for the overall subsystems. It is assumed that only part of the followers can obtain the leader’s state information. 2.3. Agent actuator fault scenario The considered actuator faults in the follower nodes, which include loss of effectiveness, bias, stuck and outage are described (t ) represents the signal from the jth ( j = 1, 2, . . . , m) actuator that as follows. For the ith (i ∈ {1, 2, . . . , N}) follower agent, let uhF i, j has failed in the h-th fault mode, ui, j (t) represents the input signal of the j-th actuator. Then, the actuator fault model proposed in [21] is defined as h h uhF i, j (t ) = (1 − ρi, j )ui, j (t ) + ψi, j (t ),

t ≥ ti,h j ,

h = 1, 2, . . . , H

(5)

where ρi,h j is an unknown actuator efficiency factor of the jth actuator satisfying 0 ≤ ρ hi, j ≤ ρi,h j ≤ ρ hi, j ≤ 1, ψi,h j (t ) is an unknown time-varying bounded signal in the j-th actuator, the failure time instants ti,h j are unknown, and the index h represents the hth faulty mode and H is the number of all faulty modes. Note that, there is no fault for the jth actuator ui, j in the h-th fault mode when ρ hi, j = ρ hi, j = 0 and ψi,h j (t ) = 0. The case of 0 < ρ hi, j ≤ ρ hi, j < 1 and ψi,h j (t ) = 0 means that the jth actuator is loss of effectiveness in the hth fault mode. When ρ hi, j = ρ hi, j = 0 and ψi,h j (t ) = 0, the j-th actuator is bias in the hth fault mode. When

ρ hi, j = ρ hi, j = 1 and ψi,h j (t ) = 0, the j-th actuator is outage in the hth fault mode. When ρ hi, j = ρ hi, j = 1 and ψi,h j (t ) = 0 the stuck fault occurs on the jth actuator in the hth fault mode. Denote hF hF hF T h h uhF i (t ) = [ui,1 (t ), ui,2 (t ), . . . , ui,m (t )] = (I − ρi )ui (t ) + ψi (t )

Please cite this article as: X. Wang, G.-H. Yang, Distributed fault-tolerant control for a class of cooperative uncertain systems with actuator failures and switching topologies, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.002

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h , ρ h , . . . , ρ h ),ρ h ∈ [ρ h , ρ h ], and ψ h (t ) = [ψ h (t ), ψ h (t ), . . . , ψ h (t )]T , i = 1, 2, . . . , N, j = 1, 2, . . . , m, where ρih = diag(ρi,1 i, j i,2 i,m i, j i i,1 i,2 i,m i, j

h = 1, 2, . . . , H. For simplicity of presentation, the uniform actuator fault model is formulated by:

uFi (t ) = (Im − ρi )ui (t ) + ψi (t ) where ρi = diag(ρi,1 , ρi,2 , . . . , ρi,m ) ∈ 1, 2, . . . , H.

(6)

{ρi1 , ρi2 , . . . , ρiH },

ψi (t ) = [ψi,1 (t ), ψi,2 (t ), . . . , ψi,m

(t )]T ∈

{ψi1 (t ), ψi2 (t ), . . . , ψiH (t )},

h=

Assumption 2. There exists an unknown constant ψ¯ i, j > 0 such that

|ψi, j (t )| ≤ ψ¯ i, j , i = 1, 2, . . . , N,

j = 1, 2, . . . , m.

(7)

Remark 3. Note that the various actuator failures may occur at any time in some subsystems, what’s more, the types of faults and uncertainties may be time-varying, non-identical and unknown, thus the considered overall MASs are essentially heterogeneous. Therefore, it is quite challenging to solve the DFTC problem using only the neighborhood information. 2.4. Switching communication networks Let Gσ (t ) be the interaction graph of MASs (1) and (2) at time t ≥ 0. For the switching topology, consider an infinite sequence of uniformly bounded non-overlapping time intervals [tk , tk+1 ), k ∈ N, with t1 = 0 and τ1 ≥ tk+1 − tk ≥ τ0 , k = 1, 2, . . . , across which the interaction graph is time-invariant. Here, the positive constant τ 0 is called the topology dwell time. The time sequence t1 , t2 , . . . , is called the switching sequence, at which the interaction graph changes. For each follower node, only relative state information of its neighbors can be used for the controller design. Assumption 3. For all p ∈ P, the undirected graph G p is fixed and connected across each interval [tk , tk+1 ), k = 1, 2, . . . , and at least one follower agent has access to the state of leader. Lemma 1 ([5]). Suppose Assumption 3 is satisfied, then H p = L p + D p is positive definite, for all p ∈ P. Remark 4. In the case when the communication network is time-varying, the influence of faulted nodes on global behavior may become much more serious than the case of fixed topology. In this case, how to design DFTC scheme for consensus tracking of MASs is certainly an interesting and important subject of its own right. 2.5. Distributed fault tolerant control objective Let x(t ) = [xT1 (t ), xT2 (t ), . . . , xTN (t )]T and u = [uT1 (t ), uT2 (t ), . . . , uTN (t )]T , then the dynamics of system (3) with actuator faults (5) can be written by the following form

x˙ (t ) = (IN ⊗ A)x(t ) + (IN ⊗ B)G(x, t ) + (IN ⊗ B)((Im − ρ)u(t ) + ψ(t ))

(8)

where G(x, t ) = [g1 (x1 (t ), t )T , g2 (x2 (t ), t )T , . . . , gN (xN (t ), t )T ]T , ρ = diag(ρ1 , ρ2 , . . . , ρN ), and ψ(t ) = [ψ1T (t ), ψ2T (t ), . . . , ψNT (t )]T . As only part of followers can receive the information of leader, the DFTC objective here is to design distributed

consensus tracking protocol for all follower nodes by using the neighborhood information under dynamic communication network such that the tracking error xi (t ) − x0 (t ) is as small as possible, even in the case that actuator faults, matching uncertainties and disturbances have effected on the follower dynamics simultaneously. Remark 5. Compared with the existing DFTC literature such as [30,32,49], this paper proposes a novel cooperative faults accommodation approach to relax the conservatism of fixed communication topology. Compared with the centralized FTC problem, only the state information of neighboring nodes can be used for the controller design, which makes the DFTC problem even more difficult. To achieve the DFTC objective, similar to conventional FTC results such as [19,21,36], we introduce the following standard assumptions. Assumption 4. For all ρi = diag(ρi,1 , ρi,2 , . . . , ρi,m ) ∈ {ρi1 , ρi2 , . . . , ρiH }, rank[B(Im − ρi )] = rank[B]. Assumption 5. In the presence of up to any m − q (1 ≤ q ≤ m − 1) actuators undergo stuck or outage fault, the remaining actuators can still achieve the DFTC objective. Moreover, all actuators are allowed to suffer from partial loss of effectiveness or bias. Remark 6. Assumption 4 introduces a condition of actuator redundancy in the system, and is also necessary for completely compensating the stuck-actuator faults, which have been given in [19,36]. As discussed in [19,21,36], Assumption 5 is a basic assumption to ensure the controllability of the plant and the existence of a nominal solution for the actuator failure compensation problem. In fact, many practical systems belong to this class of systems, and some studies have also been proposed based on the aforementioned assumptions [19,21,36]. Please cite this article as: X. Wang, G.-H. Yang, Distributed fault-tolerant control for a class of cooperative uncertain systems with actuator failures and switching topologies, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.002

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5 T

Lemma 2 ([9]). For any vector y = [y1 , y2 , . . . , yn ] ∈ Rn , there exists a vector v = [v1 , v2 , . . . , vn ] ∈ Rn , such that the following inequality

y  ≤ y T v

(9)

holds, where v satisfies vs = sgn(ys )|vs |, and

v2s

≥ 1, s = 1, 2, . . . , n.

Lemma 3 ([36]). For all ρi = diag(ρi,1 , ρi,2 , . . . , ρi,m ) ∈ {ρi1 , ρi2 , . . . , ρiH }, there exists a positive constant μi , such that the following inequality

ζ T PB(Im − ρi )BT Pζ ≥ μi ζ T PB2

(10)

holds. 3. Main results In this section, we first present an active DFTC scheme only based on state information shared by the neighboring agents. By adopting adaptive technique, the proposed DFTC protocol for individual nodes guarantees that actuator faults can be compensated in real time. Then, we can further conclude the consensus tracking objective can be achieved by exploiting the multiple Lyapunov functions method and algebraic graph theory. 3.1. Distributed fault-tolerant controller design Based on the relative states of neighboring agents, the following adaptive cooperative fault-tolerant controller is designed for each follower:

ui (t ) = c0 (Im + ρˆi )K0 eσi (t ) (t ) + Ki,c (t )

(11)

where c0 represents the coupling strength satisfying c0 > 1/(2λ0 ) with λ0 := min{λmin (H p ) : p ∈ P }, the feedback gain matrix K0 = −BT P with P is a positive definite matrix satisfying the following LMI:

PA + AT P − 2c0 λ0 PBBT P + σ (t )

ei

1

η0

In < 0

(12)

(t ) ∈ Rn represents the neighborhood tracking error, which is defined as follows eσi (t ) (t ) =



ai j (t )(xi (t ) − x j (t )) + di (t )(xi (t ) − x0 (t ))

(13)

j∈Ni (t )

ρˆi = diag(ρˆi,1 (t ), ρˆi,2 (t ), . . . , ρˆi,m (t )) is the estimate of ρ i , which is generated by the following projection algorithm

⎧ ⎨0,

dρˆi, j (t ) = Proj[min (ρ h ) max (ρ¯ h )] { i, j } = i, j i, j dt h h ⎩

i, j ,

if ρˆi, j > max (ρ¯ i,h j ) and i, j ≥ 0 or h

if ρˆi, j < min (ρ hi, j ) and i, j ≤ 0

(14)

h

otherwise

σ (t ) T

[PB] j 2 , φ i, j is a positive constant, [PB]j represents the jth column of matrix PB, j = where i, j = c0 φi, j (1 + ρˆi, j )ei 1, 2, . . . , m, Proj{ · } denotes the projection operator in [7], whose principle is to project the estimates ρˆi, j (t ) to the interval [min (ρ hi, j ), max (ρ¯ i,h j )], and ρˆi, j (t0 ) is finite. Ki, c (t) is an auxiliary control function addressed to compensate the effect of actuah

h

tor faults, matched uncertainties and disturbances, which will be given later. Then, the overall closed-loop dynamics with DFTC controller (11) can be written as

ˆ c0 (IN ⊗ K0 )eσ (t ) (t ) + K¯c (t )) + (IN ⊗ B)(G(x, t ) + ψ(t )) x˙ (t ) = (IN ⊗ A)x(t ) + (IN ⊗ B)(INm − ρ)(INm + ρ)(

(15)

σ (t ) σ (t ) σ (t ) where eσ (t ) (t ) = [e1 (t )T , e2 (t )T , . . . , eN (t )T ]T , and K¯c (t ) = [K1,c (t )T , . . . , KN,c (t )T ]T . Based on Assumption 1 and the projection algorithm (14), there also exists a positive constant ki, 1 satisfying

c0 ρˆi2  +

1 ηβ 2 ≤ μi ki,1 2 i

where β i is defined in (4), η is a given constant satisfying η >

(16)

η0 , and μi is chosen as in the inequality (10). In general, denote λ0

zi (t ) := (αi + βi x0 )vi + ψi (t ) − r0 (t ) = [zi1 (t ), zi2 (t ), . . . , zim (t )]T

(17)

T ki,2 (t ) := νi−1 (zi (t ) + i (t )) = [k1i,2 (t ), k2i,2 (t ), . . . , km i,2 (t )]

(18)

Please cite this article as: X. Wang, G.-H. Yang, Distributed fault-tolerant control for a class of cooperative uncertain systems with actuator failures and switching topologies, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.002

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where vi ∈ Rm is denoted in (9), i (t ) = [1i (t ), 2i (t ), . . . , m (t )]T is an unknown time-varying function, and νi := i diag(νi,1 , νi,2 , . . . , νi,m ) with ν i, j is defined by



νi, j =

1 − ρi, j , 1,

if ρi, j = 1 otherwise

(19)

Here, it should be emphasized that ki, 1 , and ki, 2 (t) are both unknown, because of α i , β i , ρ i, j , ψ i (t) and r0 (t) are unknown constants and signals, respectively. Then, we define the auxiliary control function Ki, c (t) as follows

Ki,c (t ) = −kˆ i,1 (t )BT Peσi (t ) (t ) − kˆ i,2 (t )

(20)

where kˆ i,1 (t ) is the estimate of ki, 1 , which is adjusted according to the adaptive law:

dkˆ i,1 (t ) = γi,1 BT Peσi (t ) 2 dt

(21)

where γ i, 1 is an any positive constant, and kˆ i,1 (t0 ) is finite. In addition, in order to accommodated outage or stuck actuator faults, a new distributed adaptive control allocation (DAAC) (t )]T is a control gain function, which is designed by algorithm is proposed. Let kˆ i,2 (t ) = [kˆ 1i,2 (t ), kˆ 2i,2 (t ), . . . ,kˆ m i,2 T j j j j kˆ i,2 (t ) = kˆ i,2 (t ) + (kˆ¯ i,2 (t ) − kˆ i,2 (t ))ϒ (eσi (t ) [PB] j )

(22)

where ϒ (·) is a function defined by



ϒ (z) =

1, 0,

if z ≥ 0 if z < 0

(23)

j j j kˆ i,2 (t ) and kˆ¯ i,2 (t ), j = 1, 2 . . . , m, are the estimates of lower and upper bounds of the unknown function ki,2 (t ), respectively, which are updated by the following adaptive laws: j

dkˆ i,2 (t ) j σ (t ) T = γi,2 ei [PB] j dt

(24a)

j

dkˆ¯ i,2 (t ) j σ (t ) T = γi,2 ei [PB] j dt

(24b) j

j

where γi,2 is an any positive constant, kˆ i,2 (t0 ) and kˆ¯ i,2 (t0 ) are finite. j

Remark 7. The reason of introducing DAAC approach in the consensus protocol is to handel the stuck or outage failures in some nodes, that is, the matching nonlinearities and unknown faults in stuck or outage actuators can be accommodated using the remaining actuators based on the DAAC strategy. 3.2. Error dynamic systems Define a tracking error δ(t ) = [δ1T (t ), δ2T (t ), . . . , δNT (t )]T with δi (t ) = xi (t ) − x0 (t ), then the dynamics of global tracking error can be written as

 ˙ t ) = IN ⊗ A + c0 (IN ⊗ B)(INm − ρ)(INm + ρ)( δ( ˆ Hσ (t ) ⊗ K0 ) δ(t )

+ (IN ⊗ B)(INm − ρ)K¯c (t ) + (IN ⊗ B)(G(x, t ) + ψ(t ) − r0 )

(25)

j j j j j j where r0 = 1N ⊗ r0 . Denote ρ˜i, j (t ) = ρˆi, j (t ) − ρi, j , k˜ i,1 (t ) = kˆ i,1 (t ) − ki,1 , k˜ i,2 (t ) = kˆ i,2 (t ) − ki,2 (t ), and k˜¯ i,2 (t ) = kˆ¯ i,2 (t ) − k¯ i,2 (t ), i = 1, 2, . . . , N, j = 1, 2, . . . , m, then, the error systems can be written in the following equations:

dρˆi, j (t ) dρ˜i, j (t ) = dt dt

(26a)

dk˜ i,1 (t ) = γi,1 BT Peσi (t ) 2 dt

(26b)

j

dk˜ i,2 (t ) j σ (t ) T ei [PB] j = γi,2 dt j dk˜¯ i,2 (t )

dt

T

j σ (t ) = γi,2 ei [PB] j

(26c)

(26d)

Please cite this article as: X. Wang, G.-H. Yang, Distributed fault-tolerant control for a class of cooperative uncertain systems with actuator failures and switching topologies, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.002

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7

3.3. Stability analysis Denote

ρ˜  (t ) = [ρ˜1,1 (t ), . . . , ρ˜1,m (t ), . . . , ρ˜N,1 (t ), . . . , ρ˜N,m (t )]T k˜ 1 (t ) = [k˜ 1,1 (t ), k˜ 2,1 (t ), . . . , k˜ N,1 (t )]T T T T k˜ 2 (t ) = [k˜ 1,2 (t ), k˜ 2,2 (t ), . . . , k˜ N,2 (t )]T

k˜¯ 2 (t ) = [k˜¯ T1,2 (t ), k˜¯ T2,2 (t ), . . . , k˜¯ TN,2 (t )]T ˜ t ) = [δ T , ρ˜ T , k˜ T , k˜ T , k˜¯ T ]T (t ), we denote a solution of the closed-loop error systems (25) and (26). Then the following theorem By δ( 1 2 2 can be derived. Theorem 1. Consider the closed-loop global tracking error system (25) and error systems (26) satisfying Assumptions 1, 2, 3, 4, 5, the DFTC protocol (11) with adaptive laws (14), (20) and (24). Suppose the topology dwell time satisfies τ 0 > (ln κ )/r0 , where κ > q¯ p˘ /q pˆ , ˜ t; t0 , δ( ˜ t0 )) to the (25) and (26) is uniformly bounded, and the global tracking ˆ p˘ ∈ P, and r0 = min (r p ). Then, the solution δ( for all p, p∈P

error vector δ (t) satisfies lim δ(t ) ≤ δ¯ , for some δ¯ ∈ R+ . t→∞

Proof. Consider the following multiple Lyapunov-Krasovskii functions candidate

Vσ (t ) (δ, ρ˜  , k˜ 1 , k˜ 2 , k˜¯ 2 ) =

m N  N    −1 δ T Hσ (t ) ⊗ P δ + φi,−1j ρ˜i,2 j + γi,1 μi k˜ 2i,1 i=1 j=1

+

m N  

i=1



−1 j j 2 (γi,2j ) (1 − ρi, j ) ij (k˜¯ i,2 ) + (1 − ij )(k˜ i,2 )2

(27)

i=1 j=1

σ (t ) T

[PB] j ). We first show that (1) during each subinterval [tk , tk+1 ), all signals in the closed-loop systems are where i := ϒ (ei bounded; and then (2) the consensus tracking error δ (t) in (25) is uniformly ultimately bounded for t > 0. j

(1) For each subinterval [tk , tk+1 ), i.e., σ (t ) = p, undirected graph G p is the fixed and connected. Then the time derivative of V (δ, ρ˜  , k˜ 1 , k˜ , k˜¯ 2 ) along (25) and (26) can obtained as 2

dVp (δ, ρ˜  , k˜ 1 , k˜ 2 , k˜¯ 2 )  dt = δ T Hp ⊗ (PA + AT P ) + 2c0 Hp2 ⊗ PBK0 δ + 2c0 δ T (Hp ⊗ PB)(ρ˜ + ρ˜ ρˆ − ρˆ 2 )(Hp ⊗ K0 )δ + 2δ T (Hp ⊗ PB)(INm − ρ)K¯c (t ) + 2δ T (Hp ⊗ PB)(G(x, t ) + ψ(t ) − r0 ) + 2

m N  

φi,−1j ρ˜i, j ρ˜˙ i, j + 2

i=1 j=1

+2

N  m 



j ˜¯ j ˜¯˙ j j ˜ j ˜˙ j (γ ) (1 − ρi, j ) i ki,2 ki,2 + (1 − i )ki,2 ki,2

N 

−1 γi,1 μi k˜ i,1 k˜˙ i,1

i=1

j −1 i,2

(28)

i=1 j=1

By noting that K0 = −BT P and (20), it is easy to get that

2δ T (Hp ⊗ PB)(ρ˜ + ρ˜ ρˆ − ρˆ 2 )(Hp ⊗ K0 )δ = −2c0

N 

T





eip PB ρ˜i (Im + ρˆi ) − ρˆi2 BT Peip

(29)

i=1

and

2δ T (Hp ⊗ PB)(INm − ρ)K¯c (t ) = −2

N 

T kˆ i,1 eip PB(Im − ρi )BT Peip

i=1

−2

N 

T eip PB(Im − ρi )(i kˆ¯ i,2 + (Im − i )kˆ i,2 )

(30)

i=1

where i = diag(i,1 , i,2 , . . . , i,m ). Please cite this article as: X. Wang, G.-H. Yang, Distributed fault-tolerant control for a class of cooperative uncertain systems with actuator failures and switching topologies, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.002

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Substituting (29) and (30) into (28) yields N   T δ T Hp ⊗ (PA + AT P) − 2c0 Hp2 ⊗ PBBT P δ − 2c0 eip PBρ˜i (Im + ρˆi )BT Peσi (t )

dVp (δ, ρ˜  , k˜ 1 , k˜ 2 , k˜¯ 2 ) = dt

i=1

+ 2c0

N 

T eip PB ˆi2 BT Peip

ρ

i=1 N 

−2



T

eip PB(Im − ρi )

−2

N 

T kˆ i,1 eip PB

(Im − ρi )BT Peip

i=1

 N  T i kˆ¯ i,2 + (Im − i )kˆ i,2 + 2 eip PBgi (xi , t )

i=1

+2

i=1

N 

T

eip PB(ψi (t ) − r0 (t )) + 2

m N  

i=1

+2

φi,−1j ρ˜i, j ρ˜˙ i, j + 2

N 

i=1 j=1

m N  

−1 γi,1 μi k˜ i,1 k˜˙ i,1

i=1

 j ˜˙ j −1 j j (γi,2j ) (1 − ρi, j ) ij k˜¯ i,2 k¯ i,2 + (1 − ij )k˜ i,2 k˜˙ i,2

(31)

i=1 j=1

1 2 m where kˆ¯ i,2 = [kˆ¯ 1i,2 (t ), kˆ¯ 2i,2 (t ), . . . , kˆ¯ m (t )]T and kˆ i,2 = [kˆ i,2 (t ),kˆ i,2 (t ),. . . , kˆ i,2 (t )]T . In light of Assumption 1, we have i,2

2

N 

T

eip PBgi (xi , t ) ≤ 2

i=1

N 

BT Peip (αi + βi xi )

i=1

≤ 2

N 

BT Peip (αi + βi x0 ) + 2

N 

i=1

ηβi2 BT Peip 2 + 2

i=1

N 

η−1 δi 2

(32)

i=1

where we have used the fact that

2

N 

βi BT Peip δi  ≤

i=1

N 

(ηβi2 BT Peip 2 + η−1 δi 2 )

i=1

Then according to Lemma 2, we can obtain

2

N 

T

eip PB(αi + βi x0 ) + 2

i=1

N 

T

eip PB(ψi (t ) − r0 (t ))

i=1

≤ 2

N  m 

T

(αi + βi x0 )eip [PB] j vi, j + 2

i=1 j=1

=2

m N  

N 

T

eip PB(ψi (t ) − r0 (t ))

i=1 T

eip [PB] j zij (t )

(33)

i=1 j=1

where zi (t ) is defined in (17). In what follows, we consider two cases. (i) All actuators of follower nodes cannot suffer from outage or stuck failures. j In this case, 0 ≤ ρ i, j < 1, we denote i (t ) = 0. In terms of (18) and (23), we can obtain j

2

m N  

T

eip [PB] j zij (t ) = 2

m N  

i=1 j=1

T

j eip [PB] j νi, j ki,2 (t ) ≤ 2

i=1 j=1

m N  

T

eip [PB] j νi, j



j j + ij k¯ i,2 (1 − ij )ki,2



(34)

i=1 j=1

Substituting (34) into (31) yields

dVp (δ, ρ˜  , k˜ 1 , k˜ 2 , k˜¯ 2 ) ≤ dt

 1 δ T Hp ⊗ (PA + AT P) − 2c0 Hp2 ⊗ PBBT P + INn δ η −2c0

N 

T

eip PBρ˜i (Im + ρˆi )BT Peip − 2

i=1

+2

m N  

T

eip [PB] j νi, j



N 

μi k˜ i,1 BT Peip 2

i=1 j j (1 − ij )ki,2 + ij k¯ i,2



i=1 j=1

−2

m N  

T



eip [PB] j (1 − ρi, j )

j

j (1 − ij )kˆ i,2 + ij kˆ¯ i,2



i=1 j=1

Please cite this article as: X. Wang, G.-H. Yang, Distributed fault-tolerant control for a class of cooperative uncertain systems with actuator failures and switching topologies, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.002

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+2

N  m  i=1 j=1

+2

m N  

φi,−1j ρ˜i, j ρ˜˙ i, j + 2

N 

9

−1 γi,1 μi k˜ i,1 k˜˙ i,1

i=1

j ˜¯ j ˜¯˙ j j ˜ j ˜˙ j (γ ) (1 − ρi, j ) i ki,2 ki,2 + (1 − i )ki,2 ki,2 j −1 i,2

(35)

i=1 j=1

Following the adaptive laws chosen in (14), (20) and (24), we can rewrite (35) as

 dVp (δ, ρ˜  , k˜ 1 , k˜ 2 , k˜¯ 2 ) 1 ≤ δ T Hp ⊗ (PA + AT P ) − 2c0 Hp2 ⊗ PBBT P + INn δ dt η

(36)

(ii) At least one outage or stuck actuator for some follower agents. Without loss of generality, in this case, we denote the index of follower nodes as i1 , i2 , . . . , il¯, 1 ≤ l¯ ≤ N, which have at least one outage or stuck actuator. According to Assumption 4, let rank[B(Im − ρi )] = rank[B] = r, which implies that at least r actuators are kept from outage or stuck failures for i-th follower agent. Here, it is assumed that the first r actuators cannot suffer from outage or stuck failures. Then, on the basis of linear algebra theory, there exist constants m1r+k , m2r+k , . . . , mrr+k such that

[B]r+k = m1r+k [B]1 + m2r+k [B]2 + · · · + mrr+k [B]r , 1 ≤ k ≤ m − r

(37)

For the faulty agent il , l = {1, 2, . . . , l¯}, denote the index of outage or stuck actuators as r + 1, r + 2, . . . , r + kl , 1 ≤ kl ≤ m − r. Then, it follows from (33) that m 

T eip [PB] j zij

(t ) =

j=1

r 

T eip [PB] j zij l l

l r+k 

(t ) +

l

j=1

=

r 

l r+k 

T

l

=

r 

r 

T

eip msj [PB]s zij (t ) + l





T

r 

T

eip [PB] j zij (t ) l

l

T

eip [PB] j zij (t )

l

l

l

j=r+kl +1 m 

T

eip [PB] j νil , j kij ,2 (t ) + l

r 

m 

eip [PB] j zij (t ) + il , j (t ) +

T

eip [PB] j νil , j kij ,2 (t )

l

j=1

≤

m  j=r+kl +1

j=r+1 s=1

l

l

l

j=1

=

l

j=r+kl +1

l

j=1

T

eip [PB] j zij (t )

l

j=r+1

eip [PB] j zij (t ) +

m 

T

eip [PB] j zij (t ) +

T

eip [PB] j νil , j



l

l

l

j=r+kl +1



 m  j j T (1 − ijl )kˆ il ,2 + ijl kˆ¯ ijl ,2 + eip [PB] j νil , j (1 − ij )kˆ il ,2 + ij kˆ¯ ij ,2 l l l l j=r+kl +1

j=1

(38) where zi (t ) and ki ,2 (t ) are denoted in (17) and (22), respectively, and il , j (t ) is defined as follows j

j

l

l

il , j (t ) =

⎧ r+kl ⎨ ⎩

msj zisl (t ),

if 1 ≤ j ≤ r

(39)

s=r+1

0,

otherwise

Define Il¯ = {i1 , i2 , . . . , il¯} as an index set, then, combining (31) with (38), we have

dVp (δ, ρ˜  , k˜ 1 , k˜ 2 , k˜¯ 2 ) dt



≤ δ T Hp ⊗ (PA + AT P ) − 2c0 Hp2 ⊗ PBBT P + −2

 i=1

+2

 i∈Il¯

μi k˜ i,1 BT Peip 2 + 2 

N  m 

1

η

T

 INn

eip [PB] j νi, j



δ − 2c0

N 

T

eip PBρ˜i (Im + ρˆi )BT Peip

i=1 j j (1 − ij )ki,2 + ij k¯ i,2



i∈I / l¯ j=1 r  j=1

T eip [PB] j i, j

ν



j

j ˆ¯ j k i i,2

(1 −  )kˆ i,2 +  j il



+

m 

T eip [PB] j i, j

ν



j

j ˆ¯ j k i i,2

(1 −  )kˆ i,2 +  j i

 

j=r+ki +1

Please cite this article as: X. Wang, G.-H. Yang, Distributed fault-tolerant control for a class of cooperative uncertain systems with actuator failures and switching topologies, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.002

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−2

N  m 



T

eip [PB] j (1 − ρi, j )

 N  m  j j (1 − ij )kˆ i,2 + ij kˆ¯ i,2 +2 φi,−1j ρ˜i, j ρ˜˙ i, j

i=1 j=1

+2

N 

i=1 j=1

−1 γi,1 μi k˜ i,1 k˜˙ i,1 + 2

i=1

m N  

j −1 j ˜¯ j ˜¯˙ j j ˜ j ˜˙ j (γi,2 ) (1 − ρi, j ) i ki,2 ki,2 + (1 − i )ki,2 ki,2

(40)

i=1 j=1

Considering the adaptive laws given in (14), (20) and (24), we have

 dVp (δ, ρ˜  , k˜ 1 , k˜ 2 , k˜¯ 2 ) 1 ≤ δ T Hp ⊗ (PA + AT P ) − 2c0 Hp2 ⊗ PBBT P + INn δ dt η

(41)

dVp (δ, ρ˜  , k˜ 1 , k˜ 2 , k˜¯ 2 ) satisfies (41). According to Lemma 1, dt we know that Hp is a positive definite matrix. Then, there exists an orthogonal matrix Tp ∈ RN×N , such that TpT H p Tp = p p diag(λ1 , . . . , λN ). By letting  = (Tp ⊗ In )δ, it follows from (41) that Therefore, based on the above two cases, we can get that

 N  dVp (δ, ρ˜  , k˜ 1 , k˜ 2 , k˜¯ 2 ) 1 ≤ λip iT PA + AT P − 2c0 λip PBBT P + In i dt η0

(42)

i=1

≤ −λmin (Hp ) min

i=1,...,N

(λ

(

p min Qi

))δ ≤ 0 2

where Qi = −(PA + AT P − 2c0 λi PBBT P + 1/η0 In ). Therefore, from the above equation, it is obvious that Vp (δ, ρ˜  , k˜ 1 , k˜ 2 , k˜¯ 2 ) is non-increasing. According to (27), one knows that all signals of the overall closed-loop systems are bounded for t ∈ [tk , tk+1 ).  m p −1 2 (2) In the following, motivated by [35], let λ p = λmin (H p ) mini=1,...,N (λmin (Qi )), M(t ) = N ˜i, j + i=1 j=1 φi, j ρ 

 N  j 2 j j j ˜ N m −1 −1 2 2 μi k˜ + i=1 γ (1 − ρi, j )  (k¯ ) + (1 −  )(k˜ i,2 ) , for t ∈ [tk , tk+1 ), and M¯ is the maxii=1 γ p

i,1

p

j=1

i,1

i,2

i

i,2

i

mum value of Mp (t). According the properties of Hp , there exist two positive constants q qp

δ2

≤

δ T (H

p

⊗ P )δ ≤ q¯ p

δ2 .

By choosing the parameters r p < λ p /q¯ p , when δ ≥



2r p M¯ p , −2r p q¯ p +λ p

p

and q¯ p such that

we have

˜ ≤ −λ δ2 ≤ −2r pVp (δ) ˜ V˙ p (δ) p

(43)

which implies

˜ ≤ −2r0Vp (δ) ˜ , V˙ p (δ)

for ∀ p ∈ P

 2r M¯ when δ ≥ δ¯1 with r0 = min (r p ), δ1 = max ( −2r pq¯ +p λ ). p p p∈P

p∈P

p

(44)



ˆ p˘ ∈ P, when δ ≥ δ2 = max ( On the other hand, for all p, ˆ p∈S ˘ p,

2M¯

p κ q pˆ −q¯ p˘ ) with κ > q¯ p˘ /q pˆ , it follows that

˜ ≤ κ Vp˘ (δ) ˜ Vpˆ (δ)

(45)

For an arbitrarily given T > 0, there exists a positive integer h ≥ 0 such that th < T ≤ thz+1 , and then, according to (43) and (45), when δ ≥ δ¯ = max (δ1 , δ2 ), one gets that

Vσ (T − ) (T − ) ≤ κ h e−2r0 T Vσ (t0 ) (t0 )

(46)

Since τ 0 > (ln κ )/2r0 , when T ∈ (th , th+1 ), it follows that

Vσ (T − ) (T − ) ≤ e

−

(h−1)τ0 h τ1

ςT

Vσ (t0 ) (t0 )

(47)

where ς = 2r0 − ( ln κ)/τ0 . And for the case of T = th+1 , one gets that

Vσ (T − ) (T − ) ≤ e

τ

− τ0 ς T 1

Vσ (t0 ) (t0 )

(48)

˜ t ) is continuous at each switching time tk , k = 1, 2, . . . , together with (47) and (48), one concludes that Since δ( ˜ t )|δ(t ) ≤ δ} ¯ is an invariant set, i.e., as time increases, all the tracking errors δ (t) will enter in the set 0 =  = {δ( ¯ , which implies lim δ(t ) ≤ δ¯ . The proof is completed.  {δ(t )|δ(t ) ≤ δ} t→∞

Remark 8. Note that the methodology of this work is different from the ones in [30,32,49]. For the fixed topology case, [30,32,49] present effective approaches for DFTC problem subject to actuator failures. For switching topologies case, however, the above methods cannot be applied directly. In this paper, by utilizing the multiple Lyapunov functions approach and adaptive technique, the proposed consensus tracking protocol for individual subsystems guarantees that the node actuator faults can be compensated as well as cooperative tracking objective can be achieved. Please cite this article as: X. Wang, G.-H. Yang, Distributed fault-tolerant control for a class of cooperative uncertain systems with actuator failures and switching topologies, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.002

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Remark 9. From the derivation of Theorem 1, it can be seen that the topology dwell time condition is used such that the influence of actuator faults, matching uncertainties and disturbances can be compensated automatically via distributed adaptive consensus protocol. Furthermore, the relationship among the convergence domain of the consensus tracking error, the topology dwell time, and the connectivity of all possible graphs can be obtained. ¯ the DFTC For the case that the communication network is an undirected connected graph with fixed topology, i.e., Gσ (t ) = G, protocol for each follower is designed by (11). Then, the following corollary is established to achieve asymptotic synchronization for the MASs even in the presence of actuator faults, matching uncertainties and disturbances. Corollary 1. Consider the closed-loop global tracking error system (25) and error systems (26) satisfying Assumptions 1, 2, 3, 4, 5, the DFTC protocol (11) with adaptive laws (14), (20) and (24). Suppose the undirected communication graph G¯ is fixed and connected. Then, all closed-loop signals are bounded and the consensus tracking error converge asymptotically to zero, that is, lim xi (t ) − x0 (t ) = 0, t→∞

for i = 1, 2, . . . , N. Proof. Consider also the following Lyapunov–Krasovskii function candidate

V¯ (δ, ρ˜  , k˜ 1 , k˜ 2 , k˜¯ 2 ) =

m N  N    −1 δ T H¯ ⊗ P δ + φi,−1j ρ˜i,2 j + γi,1 μi k˜ 2i,1 i=1 j=1

+

N  m 

i=1

 j j 2 −1 γi,2 (1 − ρi, j ) ij (k˜¯ i,2 ) + (1 − ij )(k˜ i,2 )2

(49)

i=1 j=1

¯ Similarly to (1) in the proof of Theorem 1, the time derivative of V¯ (δ, ρ˜  , k˜ 1 ,k˜ 2 , k˜¯ 2 ) along (25) and (26) can be where H¯ = L¯ + D. written as

dV¯ (δ, ρ˜  , k˜ 1 , k˜ 2 , k˜¯ 2 ) ≤ −λmin (H¯ ) min (λmin (Q¯ i ))δ2 ≤ 0 dt i=1,...,N

(50)

¯ i PBBT P + 1/η0 In . Moreover, it is obvious that V¯˙ (t ) = 0 if and only if δi = 0, i = 1, 2, . . . , N, and a where −Q¯ i = PA + AT P − 2c0 λ T set can be found as  = {(δ, ρ˜  , kˆ 1 , kˆ , kˆ¯ 2 )|V¯˙ = 0} = {(δ, ρ˜  , kˆ 1 , kˆ , kˆ¯ 2 )|δ = 0}, where ρˆ  = [ρˆ1,1 , . . . , ρˆ1,m , . . . , ρˆN,1 , . . . , ρˆN,m ] , 2

T

2

T

T T T T kˆ 1 = [kˆ 1,1 , kˆ 2,1 , . . . , kˆ N,1 ] , kˆ 2 = [kˆ 1,2 , kˆ 2,2 , . . . , kˆ N,2 ] , and kˆ¯ 2 = [kˆ¯ T1,2 , kˆ¯ T2,2 , . . . , kˆ¯ TN,2 ] . Then, starting with arbitrary initial values j j j j j j δ i (0), ρˆi, j (0), kˆ i,1 (0), kˆ i,2 (0) and kˆ¯ i,2 (0), the orbits converge asymptotically to δi = 0, ρˆi, j = ρ˘i, j , kˆ 1 = k˘ i,1 , kˆ i,2 = k˘ i,2 and kˆ¯ i,2 = k˘¯ i,2 j j where ρ˘ , k˘ , k˘ and k˘¯ are some constants. According to the well-known LaSalle’s invariance principle, it can be seen that i, j

i,1

i,2

i,2

˙ ˙ the system trajectories converge to the largest positively invariant subset 0 = {(δ, ρˆ  , kˆ 1 , kˆ 2 , kˆ¯ 2 )|δ = 0, ρˆ˙  = 0, kˆ 1 = 0, kˆ 2 = 0, ˆk¯˙ = 0}. Therefore, all closed-loop signals are bounded and the global tracking error converges asymptotically to zero, that is, 2 limt→∞ xi (t ) − x0 (t ) = 0, i = 1, 2, . . . , N. The proof is completed.  Remark 10. It is worth noting that we simultaneously consider a general actuator fault model including loss of effectiveness, bias, outage and stuck, where the situation is turned out to be more complex than the existing works. What’s more, the novel distributed control allocation mechanism is to deal with the outage/stuck actuators out of operation. Remark 11. Different from the existing centralised design methods for networked control systems such as [13–15,42,43], not only the neighborhood information can be used for the controller design, but also multiple subsystems can be controlled to achieve a coordination objective. Compared with the reports of distributed consensus tracking scheme, for example, [6,16,17], where the time-varying topology and node fault scenarios have not been considered, in this work, the DFTC problem for MASs with switching networks and actuator failures is investigated. Algorithm 1. In terms of the above description, an algorithm is given to demonstrate the controller design procedure: 1. Choosing the coupling strength c0 > 1/(2λ0 ), and solving the LMI (12), we obtain the positive definite matrix P > 0 and the local control gain matrix K = −BT P. 2. According to the adaptive law (14), we obtain the estimate of fault effect factors ρˆi, j (t ). 3. Following (21) and (24), we obtain the estimate of the gains ki, 1 , i.e.,kˆ i,1 (t ), and the the estimates of upper and lower bounds for j

the parameter ki,2 (t ), i.e., kˆ¯ i,2 (t ) and kˆ i,2 (t ), respectively. j

j

Therefore, all controller parameters have been obtained by Algorithm 1. Remark 12. It should be pointed out that the design of the DFTC protocol in this paper only relies on relative state information, requiring neither the bounds of the time-varying failures, heterogeneous uncertainties, disturbances, and the leader’s input nor fault diagnosis mechanism. Please cite this article as: X. Wang, G.-H. Yang, Distributed fault-tolerant control for a class of cooperative uncertain systems with actuator failures and switching topologies, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.002

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Fig. 1. Communication topologies in Example 1.

4. Simulation examples In this section, two simulation examples are provided to validate the effectiveness of the theoretical results. 4.1. Example 1 Consider a network of four aircrafts and a virtual leader with the switching topologies G1 and G2 as shown in Fig. 1. Each agent is assumed to be a linearized B747-100/200 aircraft model where the technical data and the underlying differential equations have been obtained from [19]. In simulation, only the first four states are selected, which covered pitch rate, true airspeed, angle of attack and pitch angle, respectively. The inputs are elevator deflection, total thrust and horizontal stabiliser. The system and input distribution matrices are

⎡

−0.6803 ⎢ −0.1463 A=⎣ 1.0050 1

0.0002 −0.0062 −0.0006 0

−0.1049 −4.6726 −0.5717 0

⎤

0 −9.7942⎥ ⎦, 0 0

⎡

−1.5539 ⎢ 0 B=⎣ 0 0

0.0154 1.3287 0 0

⎤

−0.1556 0.2 ⎥ ⎦ 0 0

Here, the nonlinear function gi (xi (t), t) can be regraded as external disturbance, which is chosen as gi (xi (t ), t ) = [0.33 sin t, 0, 0]T . The input of leader node is designed as r0 (t ) = Kr xr + v(t ), where



Kr = and

18.5917 6.9189 2.9309

−8.3070 −5.0470 −1.6039

⎧ T ⎪ ⎨[0, 0, 0] ,

[1, 0.5, 0.5]T , v(t ) = T [0, , ⎪ ⎩ 0, −1] [0, 0, 0]T ,

2.7345 0.1822 0.3053



14.3555 3.4334 1.9757

0 < t < 10 10 ≤ t < 20 20 ≤ t < 35 35 ≤ t < 50

For all follower agents, the parameters are chosen as φi,1 = φi,2 = φi,3 = 0.01, γi,1 = 50, γi,2 = 20 and coupling strength is c0 = 30. The initial conditions for the five agents as x0 (0) = [0.1, −0.2, −0.1, 0]T , x1 (0) = [0, −0.5, −0.2, −0.1]T , x2 (0) = j j [0, −0.3, 0, 0.15]T , x3 (0) = [0.2, 0,−0.3, 0.1]T , x4 (0) = [−0.2, 0.3, −0.1, 0.05]T , ρˆi, j (0) = 0, kˆ i,1 (0) = 2, kˆ (0) = −2,kˆ¯ (0) = 2. i,2

i,2

In this simulation, the DFTC control objective will be achieved if the topology dwell time is larger than 3.9896 s. Suppose that the communication topology switches between G1 and G2 in every 4 s. Moreover, only the actuators in the 1st and 3rd agents become faulty at t = 13 s and t = 26 s, respectively. The faulty actuators are described as follows:



• Agent 1: • Agent 3:



ρ1,1 = 0.5, ρ1,2 = 0.3, ρ1,3 = 0, normal,

t ≥ 13 otherwise

ρ3,1 = 0, ρ3,2 = 0.5, ρ3,3 = 1, ψ3,3 (t ) = 20 + cos t, normal,

t ≥ 26 otherwise

The state trajectories of the closed-loop networks and global tracking errors are shown in Figs. 2 and 3, and the control efforts corresponding to all crafts are plotted in Fig. 4. In addition, compared with the DFTC protocol in [49] without considering switching topologies, actuator stuck faults and disturbances, Fig. 5 shows that the global tracking objective using the method in [49] cannot be achieved. These figures demonstrate the effectiveness of the proposed DFTC control strategy for multi-agent systems with switching undirected topologies. Please cite this article as: X. Wang, G.-H. Yang, Distributed fault-tolerant control for a class of cooperative uncertain systems with actuator failures and switching topologies, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.002

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Fig. 2. The state trajectories of Agent i.

4.2. Example 2 Consider a 6-node multi-agent systems with one leader node (Node 0) in which the nodes are the linearized reduced-order marine crafts model form [24]. The switching communication topologies are given in Fig. 6. The yaw dynamics of each marine craft is can be written in the form of (3), with xi (t ) = [xAi (t ), xRi (t )]T , and ui (t ) = ωi (t ), where xAi (t), xRi (t) and ωi (t) denote respectively the yaw angle, the yaw rate, and the yaw moment. The system and input distribution matrices are



A=

0 0



1 , 0



B=



0 0.0025

The unknown nonlinear function can be approximated by

gi (xi (t ), t ) = −0.1256xRi (t ) − 0.3576x3Ri (t ) − 0.0278x5Ri (t ) + · · · As done in [24], we chose a local feedback ωi (t ) = Kr xi + u¯ i (t ), where Kr = [−1, −40] is a feedback matrix. Let the input of leader node be r0 (t ) = Kr xr + r¯(t ), where

r¯(t ) =

⎧ 0, ⎪ ⎪ ⎨1.5,

−0.5,

⎪ ⎪ ⎩0.5, 0,

0 < t < 40 40 ≤ t < 80 80 ≤ t < 120 120 ≤ t < 160 160 ≤ t < 200

The following parameters are given in the designs: x0 (0) = [0, 0]T , x1 (0) = [0.1, −0.1]T , x2 (0) = [0.3, 0.08]T , x3 (0) = j [0.25, −0.03]T , x4 (0) = [−0.35, 0]T , x5 (0) = [−0.4, −0.15]T , x6 (0) = [−0.25, 0.05]T , ρˆi, j (0) = 0, kˆ i,1 (0) = 10, kˆ (0) = −2, i,2

j kˆ¯ i,2 (0) = 2, φi,1 = 0.01, γi,1 = 50, γi,2 = 20, and coupling strength is c0 = 50. Suppose the switching frequency is 5s. Here, the following fault scenarios including loss of effectiveness and bias failures are considered:



• Agent 1: • Agent 2:



ρ1,1 = 0.7, normal,

t ≥ 80 otherwise

ρ2,1 = 0, ψ2,1 (t ) = −20 + cos t, normal,

t ≥ 60 otherwise

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Fig. 3. The tracking errors of Agent i.

Fig. 4. Profiles of control efforts ui (t).

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Fig. 5. Response curves of the tracking error using the method in [49] without considering switching topologies, actuator stuck faults and disturbances.

Fig. 6. Communication topologies in Example 2.

 • Agent 4: • Agent 6:



ρ4,1 = 0, ψ4,1 (t ) = 20 + cos t, normal,

ρ6,1 = 0.45, normal,

t ≥ 150 otherwise

t ≥ 120 otherwise

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Fig. 7. The tracking errors of Agent i.

Fig. 7 shows that the states of each follower node synchronize to that of the leader under the condition that the communication topology is a switching undirected graph, and the information of actuator faults, matching uncertainties, and disturbances are unknown. 5. Conclusion In this paper, the distributed fault-tolerant tracking control problem for a class of networked non-identical uncertain systems with actuator faults and switching undirected topologies has been considered. Supposed that the communication topology switches among some possible connected graphs and only a group of follower nodes acquire the information of the leader. By making use of the neighborhood state information, a new adaptive DFTC protocol has been designed to achieve the consensus tracking objective. By using the multiple Lyapunov functions with topology dwell time approach, it is shown that all signals in the closed-loop systems are bounded and the global tracking performance can be ensured. The simulation results show the effectiveness of the proposed control approach. In the future, an interesting direction is to discuss the case with switching directed topologies and even in the presence of network disconnections. Acknowledgments This work was supported in part by the Funds of National Science of China (Grant nos. 61273148, 61420106016, 61403070, 61473067), the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Grant no. 201157), the Fundamental Research Funds for the Central Universities (Grant No. N120604006), and the Research Fund of State Key Laboratory of Synthetical Automation for Process Industries (Grant no. 2013ZCX01). References [1] W.W. Che, W. Guan, Y.L. Wang, Adaptive regulation synchronization for a class of delayed cohen-grossberg neural networks, Nonlinear Dynam. 74 (4) (2013) 929–942. [2] X.W. Dong, Z.Y. Shi, G. Lu, Y.S. Zhong, Time-varying output formation control for high-order linear time-invariant swarm systems, Inf. Sci. 295 (2015) 36–52. [3] X.H. Ge, Q.L. Han, Distributed event-triggered H∞ filtering over sensor networks with communication delays, Inf. Sci. 291 (2015) 128–142. [4] X.G. Guo, J.L. Wang, F. Liao, S. Suresh, S. Narasimalu, Quantized insensitive consensus of lipschitz nonlinear multi-agent systems using the incidence matrix, J. Franklin Inst. doi:10.1016/j.jfranklin.2015.07.015. [5] Y.G. Hong, L.X. Gao, D.Z. Cheng, J.P. Hu, Lyapunov-based approach to multiagent systems with switching jointly connected interconnection, IEEE Trans. Autom. Contr. 52 (5) (2007) 943–948. [6] W.L. He, G.R. Chen, Q.L. Han, F. Qian, Network-based leader-following consensus of nonlinear multi-agent systems via distributed impulsive control, Inf. Sci. doi:10.1016/j.ins.2015.06.005. [7] P.A. Ioannou, J. Sun, Robust Adaptive Control, Prentice-Hall, Englewood CliJs, NJ, 1996. [8] X.Z. Jin, G.H. Yang, Adaptive synchronization of a class of uncertain complex networks against network deterioration, IEEE Trans. Circ. Syst. I: Regular Pap. 58 (6) (2011) 1396–1409. [9] X.Z. Jin, G.H. Yang, W.W. Che, Adaptive pinning control of deteriorated nonlinear coupling networks with circuit realization, IEEE Trans. Neural Netw. Learn. Syst. 23 (9) (2012) 1345–1355. [10] X.Z. Jin, J.H. Park, Adaptive sliding-mode insensitive control of a class of non-ideal complex networked systems, Inf. Sci. 274 (2014) 273–285. [11] X.F. Jiang, On sampled-data fuzzy control design approach for t-s model-based fuzzy systems by using discretization approach, Inf. Sci. 296 (2015) 307–314.

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