Second-order controllability of two-time-scale multi-agent systems

Applied Mathematics and Computation 343 (2019) 299–313

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Second-order controllability of two-time-scale multi-agent systems Mingkang Long a,b, Housheng Su b,∗, Bo Liu c a

School of Science, Hunan University of Technology, Zhuzhou 412008, PR China School of Automation, Image Processing and Intelligent Control Key Laboratory of Education Ministry of China, Huazhong University of Science and Technology, Luoyu Road 1037, Wuhan 430074, PR China c College of Science, North China University of Technology, Beijing 100144, PR China b

a r t i c l e

i n f o

Keywords: Second-order controllability Two-time-scale Multi-agent systems Coordination control Singularly perturbed systems

a b s t r a c t This paper addresses the controllability for an interconnected two-time-scale second-order multi-agent system. Firstly, to eliminate the singular perturbation parameter, we separate the multi-agent system into slow subsystem and fast subsystem by using singular perturbation methods. Then, based on matrix theory, some necessary and/or sufficient criteria are derived for second-order controllability of two-time-scale multi-agent systems with multiple leaders. Moreover, we propose some easy-to-use second-order controllability criteria determined only by eigenvalues of system matrices. Lastly, the effectiveness of the proposed theoretical results is illustrated by a simulation example. © 2018 Elsevier Inc. All rights reserved.

1. Introduction Recently, the investigation of multi-agent systems in control field has attracted a tremendous surge of interests and concerns, which mainly due to its wide applications in engineering projects [1–10]. As an important fundamental problem in modern control theory, the controllability problem of dynamic systems plays a key role in many research fields. A controllable multi-agent system enables each agent to achieve the desired state, thereby maximizing the system performance [11–19]. General speaking, controllability of multi-agent systems means that by controlling a part of agents, all the agents with arbitrary initial states can reach the expected states via an apposite controlled algorithm. Since many factors (intrinsic dynamics, control law, etc.) will affect the evolution behavior of multi-agent systems, the investigation of controllability for multi-agent systems is a challenging task. The controllability problem of multi-agent systems was first proposed by Tanner in 2004. The author proposed some necessary and/or sufficient criteria for controllability of multi-agent systems [20]. Afterwards, researchers proposed some theoretical results of controllability of multi-agent systems under different kinds of topologies or dynamic models, such as switching topology [21,22] and fixed topology [23], continuous-time model [21] and discrete-time model [24], first-order dynamic systems [21,22]. In fact, the investigation of high-order controllability of dynamic systems is of practical and theoretical significance. Then, Liu et al. investigated the controllability for second-order dynamic system [23]. In [25], Wang et al. discussed the high-order controllability of dynamic system, and proposed some necessary and/or sufficient criteria for high-order controllability. Recently, many researchers focused on investigating controllability of multi-agent systems from ∗

Corresponding author. E-mail address: [email protected] (H. Su).

https://doi.org/10.1016/j.amc.2018.09.033 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.

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M. Long et al. / Applied Mathematics and Computation 343 (2019) 299–313

graphical perspective and proposed some graphical criteria for controllability [26–28]. In [29–34], the authors characterized the relationship between controllability of systems and network topologies by algebraic criteria. Recently, some systems are divided into some subgroups to meet the needs of complex tasks, such as cluster consensus [35], group consensus [36]. In [37] and [38], Liu et al. gave the definition of group controllability and proposed some necessary and/or sufficient criteria for group controllability from the algebraic and graphical perspectives. During the most aforementioned works, researchers supposed that the dynamic systems just contain single time scale. Actually, most systems with multi-time scale are composed of slow and fast subsystems, which are evolved on different time scale. For example, the aircrafts of traffic network are much faster than the cars, then we can say that the network of aircrafts and the network of cars are evolved on different time scale. Since there exist important interactions between aircrafts and cars, we cannot investigate the aircrafts subnetwork or the cars subnetwork in isolation. Recently, the researches of multi-time-scale systems have attracted a tremendous surge of interests and concerns [39–41]. In [42], Kokotovic et al. proposed a mathematical model (singular perturbed system) to describe the systems with multiple time scale. However, due to the existence of singular perturbation parameter, the general system results in [42–44] cannot be applied in singular perturbed systems directly, and there will be ill-posedness problems, otherwise. At present, many researchers investigated the singular perturbed systems via singular perturbation method [42–47]. In [48], by using singular perturbation method, Long et al. investigated the group controllability of two-time-scale system with multiple single-order-integrator dynamic agents. The authors proposed several easy-to-use group controllability conditions based on PBH rank test. This paper focuses on second-order controllability of multi-agent systems with two-time-scale. The main contributions of this paper are as follows: (1) From a content perspective in this paper, we first investigate second-order controllability of two-time-scale multi-agent system, which is of practical and theoretical significance. (2) The controllable matrices rank test is the most common criterion for controllability. However, due to the increasing dimension of controllable matrices, the controllable matrices rank test is not easy to determine the second-order controllability for multi-agent system with twotime-scale feature. Then, we deduce some more effective second-order controllability conditions, where the second-order controllability can be determined only by the eigenvalues of system matrices. The rest of the paper is organized as follows. In Section 2, the problem formulation and mathematical preliminaries are stated. Section 3 gives some main results for second-order controllability of two-time-scale system. In Section 4, we present a simulation example. The conclusion is given in Section 5. Notation: Let I represent the identity matrix with compatible dimensions. R and C denote the sets of real numbers and complex numbers, respectively. Let 0 represent an all-zero vector or matrix with compatible dimensions. 2. Problem formulation and preliminaries 2.1. Mathematical preliminaries For convenience, the information flows among agents in systems are generally modeled by directed or undirected graphs. A weighted directed graph can be denoted by G = (V, E ), where V = {v1 , . . . , vN } is a nonempty finite set of agents, and E = {(vi , v j ) : vi , v j ∈ V } represents an edge set of ordered pairs of nodes. The weighted adjacency matrix is A = [ai j ], with aij > 0 if and only if (vi , v j ) ∈ E, and ai j = 0, otherwise. If (vi , v j ) ∈ E, then agent j is a neighbor of agent i. The set of neighbors of  agent i can be described as Ni = {v j ∈ V : (v j , vi ) ∈ E }. The Laplacian matrix of G is L = [li j ] ∈ RN×N , with lii = Nj=1 ai j and li j = −ai j , i = j. The Laplacian matrix L is generally asymmetric for a directed graph. 2.2. Problem formulation Consider a two-time-scale system ( G, x, z ) with multiple second-order integrator dynamic agents. The system ( G, x, z ) can be separated into two difference time scale subsystems (G1 , x ) and (G2 , z ), as shown in Fig. 1. There are M + Ml dynamic agents in slow subsystem and N + Nl dynamic agents in fast subsystem, respectively. In slow subsystem, the agents labeled as 1, 2, . . . , M are the followers and the agents labeled as M + N + 1, M + N + 2, . . . , M + N + Ml are the leaders. In fast subsystem, the agents labeled as M + 1, M + 2, . . . , M + N are the followers and the agents labeled as M + N + Ml + 1, M + N + Ml + 2, . . . , M + N + Ml + Nl are the leaders. The follower agents are modeled as



x˙ i (t ) = wi (t ), w˙ i (t ) = ui (t ), ε z˙ j (t ) = θ j (t ), ε θ˙ j (t ) = ϑ j (t ),

i ∈ 1 , j ∈ 2 ,

(1)

where 0 < ε  1 is singular perturbation parameter; xi (t ) ∈ Rk and z j (t ) ∈ Rk represent the states of agent i in slow subsystem and agent j in fast subsystem, respectively; 1 , 2 , 3 and 4 represent the sets of integers {1, 2, . . . , M}, {M + 1, M + 2, . . . , M + N}, {M + N + 1, M + N + 2, . . . , M + N + Ml } and {M + N + Ml + 1, M + N + Ml + 2, . . . , M + N + Ml + Nl }, respectively. For simplicity, we suppose the agent state dimension k = 1. However, all the theoretical results in this paper can be generalized to any dimension k (k > 1 and k is an integer) according to Kronecker product operations. Let V 1 = {v1 , v2 , . . . , vM }, V2 = {vM+1 , vM+2 , . . . , vM+N }, V3 = {vM+N+1 , vM+N+2 , . . . , vM+N+Ml }, V4 =    {vM+N+Ml +1 , vM+N+Ml +2 , . . . , vM+N+Ml +Nl }, V = V1 V2 V3 V4 . Furthermore, the neighbor set of agent i is denoted by Ni , N1i = {v j ∈ V1 : (v j , vi ) ∈ E }, N2i = {v j ∈ V2 : (v j , vi ) ∈ E }, N3i = {v j ∈ V3 : (v j , vi ) ∈ E }, N4i = {v j ∈ V4 : (v j , vi ) ∈ E },    where Ni = N1i N2i N3i N4i .

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301

Fig. 1. The topology G of two-time-scale system.

In this paper, the protocol of the system (G, x, z ) is as follow:

ui (t ) =



j∈N1i

+





dip (w p (t ) − wi (t )),

ai j (z j (t ) − zi (t )) +

j∈N2i

+



ai j z j (t ) +

j∈N2i

p∈N3i

ϑi (t ) =



ai j (x j (t ) − xi (t )) +



bi j (w j (t ) − wi (t )) +

j∈N1i



bi j θ j (t ) +

j∈N2i



cip (x p (t ) − xi (t ))

p∈N3i

i ∈ 1 , ai j x j (t ) +

j∈N1i

dip (θ p (t ) − θi (t )),





bi j (θ j (t ) − θi (t )) +

j∈N2i

i ∈ 2 ,

 j∈N1i

bi j w j (t ) +



cip (z p (t ) − zi (t ))

p∈N4i

(2)

p∈N4i

where A = [ai j ] ∈ R(M+N )×(M+N ) and B = [bi j ] ∈ R(M+N )×(M+N ) represent position coupling matrix and speed coupling matrix among the followers, respectively; ∀ i, j ∈ 1 , aij ≥ 0, bij ≥ 0, ∀ i, j ∈ 2 , aij ≥ 0, bij ≥ 0 and ai j ∈ R, bi j ∈ R; C1 = [cip ] ∈ RM×Ml (i ∈ 1 ) with cip ≥ 0 and D1 = [dip ] ∈ RM×Ml (i ∈ 1 ) with dip ≥ 0 are the position and speed information flows from leaders to followers in slow subsystem; C2 = [cip ] ∈ RN×Nl (i ∈ 2 ) with cip ≥ 0 and D2 = [dip ] ∈ RN×Nl (i ∈ 2 ) with dip ≥ 0 are the position and speed information flows from the leaders to the followers in fast subsystem. Let y1 = (w1 , w2 , . . . , wM , x1 , x2 , . . . , xM )T and y2 = (θM+1 , θM+2 , . . . , θM+N , zM+1 , zM+2 , . . . , zM+N )T represent follower state vectors in slow subsystem and fast subsystem, respectively; and let u1 = (wM+N+1 , wM+N+2 , . . . , wM+N+Ml , xM+N+1 , xM+N+2 , . . . , xM+N+Ml )T and u2 = (θM+Ml +N+1 , θM+Ml +N+2 , . . . , θM+Ml +N+Nl , zM+Ml +N+1 , zM+Ml +N+2 , · · · , T zM+Ml +N+Nl ) represent leader state vectors in slow subsystem and fast subsystem, respectively. Then it follows that



y˙ 1 = A11 y1 + A12 y2 + B1 u1 ,

(3)

ε y˙ 2 = A21 y1 + A22 y2 + B2 u2 , where

 A11 

 A12 

 A21 

 A22 

 B1 

 B2 

−R11 − Qw I



−L11 − Qx , 0



−R12 0

−L12 , 0

−R21 0

−L21 , 0



−R22 − Qθ I



D1 0

C1 , 0

D2 0

C2 0



−L22 − Qz , 0



with A11 ∈ R2M×2M ,

A12 ∈ R2M×2N , A21 ∈ R2N×2M , A22 ∈ R2N×2N , B1 ∈ R2M×2Ml ,

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 B2 ∈ R

2N×2Nl

,

L11 L21



L12 = [li j ] ∈ R(M+N )×(M+N ) , L22



R11 R21



R12 = [ri j ] ∈ R(M+N )×(M+N ) , R22

w w x x x Qw = diag{qw 1 , q2 , . . . , qM }, Qx = diag{q1 , q2 , . . . , qM },

Qθ = diag{qθ(M+1) , qθ(M+2) , . . . , qθ(M+N ) }, Qz = diag{qz(M+1) , qz(M+2) , . . . , qz(M+N ) }, 

qw i =

cip , ∀i ∈ 1 ,

qxi =

p∈N2i

qθi =



dip , ∀i ∈ 1 ,

p∈N2i

cip , ∀i ∈ 2 ,

qzi =

p∈N4i

ri j =



⎧ −b , ⎪ ⎪ i j ⎪ ⎪ ⎨ bik ,



dip , ∀i ∈ 2 ,

p∈N4i

j = i, j = i, i ∈ 1 ,

k∈N1i

⎪  ⎪ ⎪ ⎪ bik , ⎩

j = i, i ∈ 2 ,

k∈N2i

li j =

⎧ −ai j , ⎪ ⎪  ⎪ ⎪ ⎨ aik ,

j = i, j = i, i ∈ 1 ,

k∈N1i

⎪  ⎪ ⎪ ⎪ aik , ⎩

j = i, i ∈ 2 .

k∈N2i

By using singular perturbation method in [43] and assuming A22 is nonsingular, we separate system (3) into slow subsystem and fast subsystem as follows

y˙ s (t ) = A0 ys (t ) + B0 us (t ),

(4a)

dy f = A22 y f (τ ) + B2 u f (τ ), dτ

(4b)

where ys (t) is state vector and us (t) represents control input in the slow subsystem (4a), y f = y2 + A−1 A y is the state 22 21 s vector of fast subsystem (4b), u f = u2 − us represents the control input of fast subsystem, A0 = A11 − A12 A−1 A , B0 = B1 − 22 21 B , τ = ε 0 represents the fast time-scale variable. A12 A−1 22 2 t−t

3. Main result 3.1. Controllability analysis of reduce-order subsystems Definition 1. (Second-order controllability of slow subsystem) A non-zero state ys0 of slow subsystem (4a) is controllable if it satisfies that there exist a finite time T > 0 and a piecewise input us , such that ys (0 ) = ys0 and ys (T ) = 0. The slow subsystem (4a) is said to be controllable when any non-zero state ys0 of slow subsystem (4a) is controllable. Lemma 1. Considering the slow subsystem (4a), the following propositions are equivalent: (1) Slow subsystem (4a) is controllable; (2) rank [B0 , A0 B0 , . . . , A0(2M−1 ) B0 ] = 2M; (3) rank [λi I − A0 , B0 ] = 2M, where λi ( ∀ i = 1, 2, . . . , 2M ) is the eigenvalues of matrix A0 ; (4) if there exist a vector v ∈ C2M satisfying vT A0 = λvT and vT B0 = 0, where λ ∈ C, then the vector must be v = 0. Proof. From Definition 1, obviously, propositions (1) and (2) are equivalent. The proof of proposition (3): Necessity: By contradiction, assume that there exists an eigenvalue λ1 belonging to A0 , such that

rank[λ1 I − A0 , B0 ] < 2M. So the rows of [λ1 I − A0 , B0 ] are linear dependent. Therefore, there exists a vector α = 0 such that α T [λ1 I − A0 , B0 ] = 0, then

α T A0 = λ1 α T , α T B0 = 0.

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303

Furthermore,

α T B0 = 0, α T A0 B0 = λ1 α T B0 = 0, . . . , α T A0(2M−1) B0 = 0. So, we have

α T [B0 , A0 B0 , . . . , A0(2M−1) B0 ] = 0. Since α = 0, then there must be



rank B0 , A0 B0 , . . . , A0(2M−1) B0 < 2M. According to proposition (2), the assumption does not hold. Thus, the necessity of proposition (3) is true. Sufficiency: By contradiction, we assume that the slow subsystem (4a) is uncontrollable, then we have



rank A0 , A0 B0 , . . . , A0(2M−1) B0 < 2M. According to the matrix theory, we have

β T [B0 , A0 B0 , . . . , A0(2M−1) B0 ] = 0, where β T ∈ C1×2M is a left eigenvector of A0 , whose associated eigenvalue is λ2 . Furthermore,

β T B0 = 0, β T A0 B0 = 0 = λ2 β T B0 , . . . , β T A0(2M−1) B0 = 0. This implies that β T [λ2 I − A0 , B0 ] = 0, where λ2 is an eigenvalue belonging to A0 . Moreover, rank [λ2 I − A0 , B0 ] < 2M. Obviously, the assumption does not hold. Thus, the sufficiency of proposition (3) is true, which completes the proof of proposition (3). The proof of proposition (4): Necessity: By contradiction, assume that there exists a vector v = 0, v ∈ C2M , such that vT A0 = λvT and vT B0 = 0, where λ ∈ C. Furthermore,

λvT B0 = 0, vT A20 B0 = λvT A0 B0 = 0, vT A0 B0 =

···

vT A0(2M−1) B0 = 0. So, we have

vT [B0 , A0 B0 , . . . , A0(2M−1) B0 ] = 0. Since v = 0, then there must be



rank B0 , A0 B0 , . . . , A0(2M−1) B0 < 2M. According to proposition (2), the assumption does not hold. Thus, the necessity of proposition (4) is true. Sufficiency: By contradiction, assume that the slow subsystem (4a) is uncontrollable, then we have



rank B0 , A0 B0 , . . . , A0(2M−1) B0 < 2M. So, there must exist a vector v = 0, v ∈ C2M , such that

vT [B0 , A0 B0 , . . . , A0(2M−1) B0 ] = 0. Furthermore,

vT B0 = vT A0 B0 = · · · = vT A0(2M−1) B0 = 0. This means that there must exist a vector v = 0 satisfying vT A0 = λvT and vT B0 = 0, where λ ∈ C. This contradicts to the fact that the vector v ∈ C2M must be v = 0. Thus, the sufficiency of proposition (4) is true, which completes the proof of proposition (4).  Remark 1. In Lemma 1, propositions (2)–(4) investigate the second-order controllability of slow subsystem (4a) from different perspectives. To be specific, proposition (2) is based on the system matrices; propositions (3) and (4) determine the second-order controllability via the eigenvalues and eigenvectors of system matrices, respectively. Theorem 1. Slow subsystem (4a) is controllable when it satisfies that: (1) The eigenvalues of A0 are all distinct; (2) All the row vectors of U −1 are not orthogonal to at least one column in B0 simultaneously, where U is composed of the eigenvectors of A0 .

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M. Long et al. / Applied Mathematics and Computation 343 (2019) 299–313

Proof. From condition (1), A0 can be decomposed as A0 = U DU −1 , where D = diag(λ1 , . . . , λ2M ), λi ( ∀ i = 1, 2, . . . , 2M ) is the eigenvalues of A0 , and U is composed of the eigenvectors of A0 . Then, we have

[B0 , A0 B0 , . . . , A0 (2M−1) B0 ] = [B0 , (U DU −1 )B0 , . . . , (U DU −1 )(2M−1) B0 ] = [B0 , U DU −1 B0 , . . . , U D(2M−1)U −1 B0 ] = U [U −1 B0 , DU −1 B0 , . . . , D(2M−1)U −1 B0 ]. Since U is a nonsingular matrix, the controllability can be determined only by the following matrix

[ U −1 B0 , DU −1 B0 , . . . , D(2M−1)U −1 B0 ]. Let ηi ( ∀ i = 1, 2, . . . , 2M ) and βi ( ∀ i = 1, 2, . . . , 2Ml ) represent the row vectors of U −1 and the column vectors of B0 , respectively, i.e.,

⎡ ⎤ η1 η2 ⎥ ⎢

U −1 = ⎢. ⎥ ∈ R2M×2M , B0 = β1 , β2 , . . . , β2Ml ∈ R2M×2Ml . ⎣.. ⎦ η2 M

Then, we have



U −1 B0 , DU −1 B0 , . . . , D(2M−1)U −1 B0

⎡⎡

⎤ ⎡ η1 λ1 ⎢⎢ η2 ⎥

⎢0 = ⎢⎢ . ⎥ β1 , β2 , · · · , β2Ml , ⎢ . ⎣⎣ .. ⎦ ⎣ .. η2 M 0 ⎡ 2M−1 λ1 0 ··· 0 λ22M−1 · · · 0 ⎢ 0 ··· , ⎢ . .. .. .. ⎣ .. . . . 0

λ2 .. . 0

⎤⎡

⎤ η1 ⎥⎢ η2 ⎥

⎥⎢ . ⎥ β1 , β2 , · · · , β2Ml , ⎦⎣ .. ⎦ λ2M η2 M ⎤ 0 0 .. .

⎤⎡

⎤ η1 ⎥⎢ η2 ⎥

⎥ ⎥⎢ . ⎥ β1 , β2 , · · · , β2Ml ⎥. ⎦⎣ .. ⎦ ⎦ η2 M λ22M−1 M

···

0

··· ··· .. . ···

0

By elementary column transformation of matrix,

⎧⎡ (η1 , β1 ) ⎪ ⎪ ⎨⎢ −1

U B0 , DU −1 B0 , · · · , D(2M−1)U −1 B0 → ⎢ ⎣ ⎪ ⎪ ⎩ ⎡ ⎢ ×⎢ ⎣

(η1 , β2 )

⎡ ⎢ ×⎢ ⎣

⎤ (η2 , β1 ) ..

(η2 , β2 )

⎥ ⎥M, · · · , ⎦ . (η2M , β2 ) ⎤ ⎫ ⎪ ⎪ (η2 , β2Ml ) ⎥ ⎬ ⎥ .. ⎦M⎪, . ⎪ ⎭ (η2M , β2Ml ) ..

(η1 , β2Ml )

where (ηi , β j ) represents vector inner product,

⎡

1 ⎢1 M = ⎢. ⎣ .. 1

λ1 λ2

λ21 λ22

λ2M

λ22M

.. .

.. .

⎤ λ21M−1 λ22M−1 ⎥ .. ⎥ ⎦. .

··· ··· .. . ···

λ22M−1 M

By condition (1), it is obvious that the rank of matrix M is full. Let

⎡ ⎢ ⎣

C=⎢

(η1 , β1 )

⎤

(η2 , β1 ) ..

.

⎥ ⎥. ⎦ (η2M , β1 )

.

⎥ ⎥M, ⎦ (η2M , β1 ) ⎤

M. Long et al. / Applied Mathematics and Computation 343 (2019) 299–313

305

Obviously, the rank of matrix C is full when all the diagonal elements of matrix C are nonzero, i.e., ηi are not orthogonal to β 1 simultaneously. By condition (2), we have the rank of matrix C is full. Thus,



rank U −1 B0 , DU −1 B0 , . . . , D(2M−1)U −1 B0 = 2M, i.e.,



rank B0 , A0 B0 , . . . , A0(2M−1) B0 = 2M. Then, slow subsystem (4a) is controllable.



Remark 2. During most general controllability criteria, controllability is determined by the rank criterion of controllable matrices. However, in Theorem 1, we investigate the second-order controllability of slow subsystem (4a) just by the eigenvalues and eigenvectors of system matrix. It is obvious that the Theorem 1 is more effective. Theorem 2. Slow subsystem (4a) is controllable when it satisfies that: (1) The eigenvalues of A0 are all distinct; (2) Every row of U −1 B0 has at least one nonzero element, where U is composed of the eigenvectors of A0 . Proof. According to Theorem 1, if



rank U −1 B0 , DU −1 B0 , . . . , D(2M−1)U −1 B0 = 2M, then,



rank B0 , A0 B0 , . . . , A0(2M−1) B0 = 2M, and slow subsystem (4a) is controllable. Furthermore, from Theorem 1, we know that



rank U −1 B0 , DU −1 B0 , . . . , D(2M−1)U −1 B0 = 2M, if and only if

rank



λi I − D, U −1 B0 = 2M.



Therefore, we can just analyze the rank of matrix λi I − D, U −1 B0 .

⎡



⎢ ⎢ λi I − D, U −1 B0 = ⎢ ⎣

λi − λ1 0 .. . 0

··· ··· .. .

0 λi − λ2 .. . 0

···

0 0 .. . λi − λ2 M

bˆ 1,1 bˆ 2,1 .. . bˆ 2M,1

bˆ 1,2 bˆ 2,2 .. . bˆ 2M,2

··· ··· .. . ···

⎤

bˆ 1,2Ml bˆ 2,2Ml ⎥ ⎥ .. ⎥, . ⎦ bˆ 2M,2M l

where λi ( ∀ i = 1, 2, . . . , 2M ) are the eigenvalues of A0 , and

⎡

λ1

0

λ2

⎢0 D=⎢ . ⎣ .. 0

.. . 0

⎡

bˆ 1,1 ⎢ bˆ 2,1 ⎢ U −1 B0 = ⎢ . ⎣ .. bˆ 2M,1

··· ··· .. . ···

0 0 .. .

⎤

⎥ ⎥, ⎦

λ2M

bˆ 1,2 bˆ 2,2 .. . bˆ 2M,2

··· ··· .. . ···

⎤

bˆ 1,2Ml bˆ 2,2Ml ⎥ ⎥ .. ⎥. . ⎦ bˆ 2M,2M l

By condition (1), λi ( ∀ i = 1, 2, . . . , 2M ) are all distinct, then only the ith element of λi I − D is equal to 0. From condition (2), then, we have

rank



λi I − D, U −1 B0 = 2M.

Hence, slow subsystem (4a) is controllable.



Similar to the controllability analysis of slow subsystem (4a), we will propose some controllability criteria for fast subsystem (4b). Definition 2. (Second-order controllability of fast subsystem) A non-zero state yf0 of fast subsystem (4b) is controllable if it satisfies that there exist a finite time T > 0 and a piecewise input uf , such that y f (0 ) = y f 0 and y f (T ) = 0. The fast subsystem (4b) is said to be controllable when any non-zero state yf0 of fast subsystem (4b) is controllable. Lemma 2. Considering the fast subsystem (4b), the following propositions are equivalent:

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(1) (2) (3) (4)

Fast subsystem (4b) is controllable; (2N−1 ) rank [B2 , A22 B2 , . . . , A22 B2 ] = 2N; rank [μi I − A22 , B2 ] = 2N, where μi ( ∀ i = 1, 2, . . . , 2N ) are the eigenvalues of matrix A22 ; if there exist a vector v ∈ C2N satisfying vT A22 = λvT and vT B2 = 0, where λ ∈ C, then the vector must be v = 0.

Proof. The proofs of propositions (1)–(4) are similar to Lemma 1.



Theorem 3. Fast subsystem (4b) is controllable when it satisfies that: (1) The eigenvalues of A22 are all distinct; (2) All the row vectors of U0−1 are not orthogonal to at least one column in B2 simultaneously, where U0 is composed of the eigenvectors of A22 . Proof. The proof is similar to Theorem 1.



Theorem 4. Fast subsystem (4b) is controllable when it satisfies that: (1) The eigenvalues of A22 are all distinct; (2) Every row of U0−1 B2 has at least one nonzero element, where U0 is composed of the eigenvectors of A22 . Proof. The proof is similar to Theorem 2.



Theorem 5. Fast

subsystem (4b) is controllable if it satisfies one of conditions as follows: (1) rank C2 = N;



(3) rank C2 , D2 = N; 2

(4) rank μi IN + μi (R22 + Qθ ) + (L22 + Qz ), C2 = N; 2

(5) rank μi IN + μi (R22 + Qθ ) + (L22 + Qz ), D2 = N; 2

(6) rank μi IN + μi (R22 + Qθ ) + (L22 + Qz ), D2 , C2 = N; where μi ( ∀ i = 1, 2, . . . , 2N ) are the eigenvalues of A22 . (2) rank D2 = N;

Proof. By elementary row and column transformation of matrix,



μi IN + (R22 + Qθ ) rank μi I2N − A22 , B2 = rank

C2



μi IN 0 0 μ2i IN + μi (R22 + Qθ ) + (L22 + Qz )

D2

C2

−IN

0

0

0

 = rank

D2

 μi IN + (R22 + Qθ )

−IN

= rank

(L22 + Qz )

0

μ

2 I i N

+ μi (R22 + Qθ ) + (L22 + Qz )

−IN

0

D2

C2

0

0





,

where μi ( ∀ i = 1, 2, . . . , 2N ) are the eigenvalues of A22 . Obviously, if fast subsystem (4b) satisfies one of the conditions (1)–(6), from Lemma 2, fast subsystem (4b) is controllable.  Specially, if the position topology is the same as the speed topology, i.e., A = B, C1 = D1 , C2 = D2 , then we can derive a special result on second-order controllability for fast subsystem (4b). Theorem 6. If A = B, C2 = D2 in two-time-scale system ( G, x, z ), then fast subsystem (4b) is controllable if and only if (L22 + Qz , C2 ) is controllable. Proof. Since A = B, C2 = D2 , A22 and B2 can be written as



A22 =

−L22 − Qz I





−L22 − Qz C , B2 = 2 0 0



C2 . 0

Sufficiency: By contradiction, assume that fast subsystem (4b) is uncontrollable, from Lemma 2, then there exists an eigenvalue λ of A22 , whose corresponding left eigenvector is [β1T , β2T ] with β1 ∈ RN , β2 ∈ RN , such that







β1T , β2T A22 = λ β1T , β2T , β1T , β2T B2 = 0.

Furthermore,

⎧ T T ⎪ ⎨−β1 (L22 + Qz ) = λβ2 , β2 − β1T (L22 + Qz ) = λβ1T , ⎪ ⎩β T C = 0. 1 2

(5)

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307

Consequently,



−(λ + 1 )β1T (L22 + Qz ) = λ2 β1T ,

(6)

β1T C2 = 0.

We can know that f (λ )  λ + 1 = 0. Otherwise, if f (λ ) = 0, from Eqs. (5) and (6), β1T = β2T = 0. This contradicts the fact that [β1T , β2T ] is a left eigenvector of matrix A22 . Therefore, from Eq. (6), we have − λλ+1 is one of eigenvalues of matrix 2

(L22 + Qz ), and β1T is a corresponding left eigenvector, such that

β1T (L22 + Qz ) = −

λ2

λ+1

β1T , β1T C2 = 0.

(7)

Eq. (7) implies that (L22 + Qz , C2 ) is uncontrollable. Obviously, the assumption does not hold. Thus, the sufficiency of proposition (3) is true. Necessity: By contradiction, assume that (L22 + Qz , C2 ) is uncontrollable, then we have

β T (L22 + Qz ) = μβ T , β T C2 = 0, where μ is an eigenvalue of (L22 + Qz ), and β ∈ RN is an associated left eigenvector of (L22 + Qz ).

We can obtain that the roots of the polynomial s2 + μs + μ = 0 with respect to s are the eigenvalues of A22 and β T , β0T is their corresponding eigenvectors with

β0 = (s + μ )β . However,





β , β B2 = β , β T

T 0

T

T 0



C2 0





C2 = β T C2 , β T C2 = 0. 0

From Lemma 2, we have fast subsystem (4b) is uncontrollable. Thus, the assumption does not hold, which means that the necessity of Theorem 6 is true.  Remark 3. By using controllable matrix rank test to determine the controllability, the rank of controllable matrix of fast subsystem (4b) with large dimension is so difficult to be computed. However, the second-order controllability of fast subsystem (4b) can be determined by some matrices with smaller dimension in Theorems 5 and 6, which are simpler and more effective. 3.2. Controllability analysis of two-time-scale multi-agent systems

Definition 3. (Second-order controllability of two-time-scale system) Non-zero states ys0 and yf0 of system (4) are controllable when ys0 and yf0 satisfy that: (1) There exist a finite time T1 > 0 and a piecewise input us , such that ys (0 ) = ys0 and ys (T1 ) = 0; (2) There exist a finite time T2 > 0 and a piecewise input uf , such that y f (0 ) = y f 0 and y f (T2 ) = 0, where T2  T1 . The system (4) is said to be controllable when any non-zero states ys0 and yf0 of system (4) are controllable. Lemma 3. Two-time-scale system (4) is controllable, if and only if rank Q1 = 2M, rank Q2 = 2N, where

Q1 = [B0 , A0 B0 , . . . , A0(2M−1) B0 ], (2N−1 ) Q2 = [B2 , A22 B2 , . . . , A22 B2 ].

Here, Q1 and Q2 are called as the controllability matrices of two-time-scale system (4). Proof. From Definition 3, the result is obvious. Theorem 7. Two-time-scale system (4) is controllable if A0 and H have no common eigenvalue, A22 and H have no common eigenvalue, where

⎡

A0 ⎢BT0 H=⎣ 0 0

B0 I 0 0

0 0 A22 BT2

⎤

0 0⎥ . B2 ⎦ I

Proof. By contradiction, we assume that the slow subsystem (4a) is uncontrollable. From Lemma 1, we can know that there must exist a vector ζ = 0, ζ ∈ C2M , such that

ζ T A0 = γ ζ T , ζ T B0 = 0,

308

M. Long et al. / Applied Mathematics and Computation 343 (2019) 299–313

Fig. 2. The initial states of follower agents.

Fig. 3. The paths and final states of follower agents.

where γ ∈ C is an eigenvalue belonging to A0 . Furthermore,

⎡



ζT 0 0 0 H = ζT =



A0

⎢BT 0 0 0 ⎣ 0 0 0

ζ T A0 ζ T B0 0 0

B0 I 0 0

0 0 A22 BT2

⎤

0 0 ⎥ B2 ⎦ I

M. Long et al. / Applied Mathematics and Computation 343 (2019) 299–313

309

Fig. 4. The final states of follower agents.

γ ζT 0 0 0

= γ ζT 0 0 0 , =



which implies that matrix A0 and matrix H have the same eigenvalue γ . Thus, the assumption does not hold, slow subsystem (4a) is controllable. Similarly, fast subsystem (4b) is controllable. By Lemma 3, we have two-time-scale system (4) is controllable. This completes the proof of Theorem 7.  Remark 4. Lemma 3 determines second-order controllability of system (4) via the controllable matrices rank test. However, the rank of controllable matrices of system (4) is too complex to be computed. Thus, it is difficult to investigate the secondorder controllability by the rank criterion. Then, we propose a more effective criterion (Theorem 7), which determines the second-order controllability of system (4) only by eigenvalues of system matrices.

4. A simulation example Consider a two-time-scale system with multiple second-order-integrator agents, which is composed of eleven follower agents and four leader agents, where five followers and two leaders are in slow subsystem, six followers and two leaders are in fast subsystem (i.e., M = 5, N = 6). The follower agents are shown in Fig. 2. We suppose that the system matrices of system (3) are as follows

⎡

A11

−4 ⎢0 ⎢1 ⎢ ⎢0 ⎢ ⎢0 =⎢ ⎢1 ⎢0 ⎢ ⎢0 ⎣0 0

1 −2 0 0 0 0 1 0 0 0

0 0 −4 1 0 0 0 1 0 0

1 1 1 −2 1 0 0 0 1 0

1 0 0 1 −2 0 0 0 0 1

−4 0 1 0 0 0 0 0 0 0

1 −3 0 0 0 0 0 0 0 0

0 0 −3 1 0 0 0 0 0 0

1 1 1 −3 1 0 0 0 0 0

⎤

1 0⎥ 0⎥ ⎥ 1⎥ ⎥ −2⎥ ⎥, 0⎥ 0⎥ ⎥ 0⎥ ⎦ 0 0

310

M. Long et al. / Applied Mathematics and Computation 343 (2019) 299–313

Fig. 5. Velocity convergence.

⎡

A12

0 ⎢1 ⎢0 ⎢ ⎢0 ⎢ ⎢0 =⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎣0 0

1 0 1 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0

1 0 1 1 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0

1 0 1 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0

⎤

1 0⎥ 1⎥ ⎥ 1⎥ ⎥ 0⎥ ⎥, 0⎥ 0⎥ ⎥ 0⎥ ⎦ 0 0

M. Long et al. / Applied Mathematics and Computation 343 (2019) 299–313

Fig. 6. Position errors of follower agents.

⎡0 ⎢1 ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢0 A21 = ⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎣ 0 0

⎡−3 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ A22 = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

1 0 0 1 1 1 0 0 0 0 0

1 0 0 1 0 1 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0

1 −3 0 0 1 0 0 1 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0

0 0 −2 1 0 1 0 0 1 0 0 0

0 1 0 0 0 1 0 0 0 0 0 0

0 0 1 −5 1 0 0 0 0 1 0 0

0 1 0 0 0 0 0 0 0 0 0 0

1 0 0 1 0 1 0 0 0 0 0 0

1 0 0 1 −4 0 0 0 0 0 1 0

0 0 0 0 1 0 0 0 0 0 0 0

0 1 0 1 0 −3 0 0 0 0 0 1

0 0 1 0 0 0 0 0 0 0 0 0

⎤

0 1⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ 1⎥ ⎥, 0⎥ ⎥ 0⎥ 0⎥ ⎥ 0⎥ ⎦ 0 0

−3 1 0 0 1 1 0 0 0 0 0 0

1 −4 0 0 1 0 0 0 0 0 0 0

0 0 −2 1 0 1 0 0 0 0 0 0

⎤

0 1 0 0 0 1⎥ 1 0 0⎥ ⎥ −4 1 1⎥ ⎥ 1 −3 0⎥ 0 0 − 3⎥ ⎥, 0 0 0⎥ ⎥ 0 0 0⎥ 0 0 0⎥ ⎥ 0 0 0⎥ ⎦ 0 0 0 0 0 0

311

312

M. Long et al. / Applied Mathematics and Computation 343 (2019) 299–313

⎡

1

⎢1 ⎢0 ⎢ ⎢1 ⎢ ⎢0 B1 = ⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎣0 0

0 1 1 0 1 0 0 0 0 0

0 1 1 0 1 0 0 0 0 0

⎤

⎡1

1 ⎢1 ⎢0 0⎥ ⎢ ⎢1 1⎥ ⎥ ⎢0 0⎥ ⎢ ⎥ ⎢0 0⎥ ⎥, B = ⎢ 0⎥ 2 ⎢ 0 ⎢ 0⎥ ⎢0 ⎥ ⎢0 0⎥ ⎢ ⎦ ⎢0 0 ⎣ 0 0 0

0 1 1 0 0 1 0 0 0 0 0 0

1 0 1 1 0 0 0 0 0 0 0 0

⎤

0 1⎥ 0⎥ ⎥ 1⎥ ⎥ 1⎥ 1⎥ ⎥, 0⎥ ⎥ 0⎥ 0⎥ ⎥ 0⎥ ⎦ 0 0

By calculation, the eigenvalues of A0 , A22 and H are {−0.65, −0.92 ± 1.30i, −1.00, −1.15 ± 1.25i, −1.63, −2.83 ± 0.53i, 2.39}, {−0.63 ± 0.86i, −0.91 ± 1.04i, −1.00, −1.51 ± 0.53i, −2.17 ± 1.26i, −2.70, −4.86}, {0.03, 1.01, 1.09, 1.19, 1.28, 1.30 ± 0.03i, 2.72, 8.60, −0.14, −0.81, −0.88, −1.03 ± 0.79i, −1.03 ± 1.08i, −1.06 ± 1.30i, −1.11, −1.26, −1.63, −2.24 ± 1.00i, −2.50, −3.14 ± 0.71i, −3.37, −5.28, −6.34}, respectively. Obviously, according to Theorem 7, system (4) is controllable. The initial positions and velocities of follower agents are chosen randomly from the boxes [0, 20] × [0, 20] and [0, 2] × [0, 2], respectively. The follower agents in fast subsystem and slow subsystem are depicted by black star dots and red circle dots, respectively, in Fig. 2. The trajectories and final states of follower agents are given in Fig. 3. From random states, the follower agents of fast subsystem are finally steered to be a regular-pentagon configuration, and those in slow subsystem are steered to be a regular-triangle configuration, as shown in Fig. 4. To keep the objective configuration, all the follower agents in the same subsystem reach the same speed, ultimately. In Figs. 5 and 6, the dotted lines show the velocity convergence and position errors of follower agents, respectively. Obviously, the follower agents of fast subsystem achieve the desired goal more quickly. 5. Conclusion In this paper, we have investigated the second-order controllability of two-time-scale multi-agent systems. Inspired by the singular perturbed systems, we have established the second-order dynamic equation of two-time-scale multi-agent system. Then, we have given the definition of second-order controllability for two-time-scale multi-agent systems. Consequently, according to the PBH rank test, we have derived several necessary and/or sufficient criteria for the second-order controllability of two-time-scale multi-agent systems. Future efforts will be concentrated on investigating the second-order controllability of discrete-time multi-agent systems with two-time-scale feature. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 61873318, 61473129, and 61773023, the Natural Science Foundation of Hubei Province of China under Grant No. 2018CFA058, the Wuhan Morning Light Plan of Youth Science and Technology under Grant No. 2017050304010288, the Fundamental Research Funds for the Central Universities [grant number HUST: 2017KFYXJJ178]. References [1] B. Liu, D.J. Hill, Z. Sun, Input-to-state-KL-stability with criteria for a class of hybrid dynamical systems, Appl. Math. Comput. 326 (2018) 124–140. [2] Z. Yu, H. Jiang, X. Mei, C. Hu, Guaranteed cost consensus for second-order multi-agent systems with heterogeneous inertias, Appl. Math. Comput. 338 (2018) 739–757. [3] X. He, Q. Wang, Distributed finite-time leaderless consensus control for double-integrator multi-agent systems with external disturbances, Appl. Math. Comput. 295 (2017) 65–76. [4] X. Dong, G. Hu, Time-varying formation tracking for linear multi-agent systems with multiple leaders, IEEE Trans. Autom. Control 62 (2017) 3658–3664. [5] H. Su, H. Wu, J. Lam, Positive edge-consensus for nodal networks via output feedback, IEEE Trans. Autom. Control. doi:10.1109/TAC.2018.2845694. [6] X. Dong, G. Hu, Time-varying formation control for general linear multi-agent systems with switching directed topologies, Automatica 73 (2016) 47–55. [7] H. Su, H. Wu, X. Chen, M.Z.Q. Chen, Positive edge consensus of complex networks, IEEE Trans. Syst. Man Cybern. Syst. (2017), doi:10.1109/TSMC.2017. 2765678. [8] X. Dong, Y. Zhou, Z. Ren, Y. Zhong, Time-varying formation tracking for second-order multi-agent systems subjected to switching topologies with application to quadrotor formation flying, IEEE Trans. Ind. Electr. 64 (2017) 5014–5024. [9] H. Su, Y. Ye, Y. Qiu, Y. Cao, M.Z.Q. Chen, Semi-global output consensus for discrete-time switching networked systems subject to input saturation and external disturbances, IEEE Trans. Cybern. doi:10.1109/TCYB.2018.2859436. [10] H. Wu, H. Su, Observer-based consensus for positive multiagent systems with directed topology and nonlinear control input, IEEE Trans. Syst. Man Cybern. Syst. (2018), doi:10.1109/TSMC.2018.2852704. [11] X. Xu, Z. Rong, Z. Wu, T. Zhou, C.K. Tse, Extortion provides alternative routes to the evolution of cooperation in structured populations, Phys. Rev. E 95 (2017) 052302. [12] Y. Ye, H. Su, Y. Sun, Event-triggered consensus tracking for fractional-order multi-agent systems with general linear models, Neurocomputing 315 (2018) 292–298. [13] Y. Mao, X. Xu, Z. Rong, Z. Wu, The emergence of cooperation-extortion alliance on scale-free networks with normalized payoff, EPL 122 (2018) 50 0 05.

M. Long et al. / Applied Mathematics and Computation 343 (2019) 299–313

313

[14] H. Su, H. Wu, X. Chen, Observer-based discrete-time nonnegative edge synchronization of networked systems, IEEE Trans. Neural Netw. Learn. Syst. 28 (2017) 2446–2455. [15] H. Wu, H. Su, Discrete-time positive edge-consensus for undirected and directed nodal networks, IEEE Trans. Circ. Syst. II Expr. Briefs 65 (2018) 221–225. [16] B. Liu, D.J. Hill, Z. Sun, Input-to-state exponents and related ISS for delayed discrete-time systems with application to impulsive effects, Int. J. Robust Nonlinear Control 28 (2018) 640–660. [17] Y. Ye, H. Su, Leader-following consensus of general linear fractional-order multi-agent systems with input delay via event-triggered control, Int. J. Robust Nonlinear Control. (2018), doi:10.1002/rnc.4339. [18] H. Su, Y. Qiu, L. Wang, Semi-global output consensus of discrete-time multi-agent systems with input saturation and external disturbances, ISA Trans. 67 (2017) 131–139. [19] X. Dong, Y. Hua, Y. Zhou, Z. Ren, Y. Zhong, Theory and experiment on formation-containment control of multiple multirotor unmanned aerial vehicle systems, IEEE Trans. Autom. Sci. Eng. doi:10.1109/TASE.2018.2792327. [20] H. Tanner, On the controllability of nearest neighbor interconnections, IEEE Conf. Decis. Control 3 (3) (2004) 2467–2472. [21] B. Liu, T. Chu, L. Wang, Z. Zuo, G. Chen, H. Su, Controllability of switching networks of multi-agent systems, Int. J. Robust Nonlinear Control 22 (6) (2012) 630–644. [22] B. Liu, T. Chu, L. Wang, G. Xie, Controllability of a leader-follower dynamic network with switching topology, IEEE Trans. Autom. Control 53 (4) (2008) 1009–1013. [23] B. Liu, H. Feng, L. Wang, R. Li, J. Yu, H. Su, G. Xie, Controllability of second-order multiagent systems with multiple leaders and general dynamics, Math. Probl. Eng. 2013 (4) (2013) 14–26. [24] B. Liu, H. Su, R. Li, D. Sun, W. Hu, Switching controllability of discrete-time multi-agent systems with multiple leaders and time-delays, Appl. Math. Comput. 228 (9) (2014) 571–588. [25] L. Wang, F. Jiang, G. Xie, Z. Ji, Controllability of multi-agent systems based on agreement protocols, Sci. China 52 (11) (2009) 2074–2088. [26] A. Rahmani, M. Ji, M. Mesbahi, M. Egerstedt, Controllability of multi-agent systems from a graph theoretic perspective, SIAM J. Control Optim. 48 (1) (2009) 162–186. [27] Z. Ji, H. Yu, A new perspective to graphical characterization of multi-agent controllability, IEEE Trans. Cybern. 47 (6) (2017) 1471–1483. [28] N. O’Clery, Y. Yuan, G. Stan, M. Barahona, Observability and coarse graining of consensus dynamics through the external equitable partition, Phys. Rev. E 88 (1) (2013) 042805. [29] M.A. Rahimian, A.G. Aghdam, Structural controllability of multi-agent networks: robustness against simultaneous failures, Automatica 49 (11) (2013) 3149–3157. [30] G. Parlangeli, G. Notarstefano, On the reachability and observability of path and cycle graphs, IEEE Trans. Autom. Control 57 (3) (2012) 743–748. [31] M. Egerstedt, S. Martini, M. Cao, K. Camlibel, A. Bicchi, Interacting with networks: how does structure relate to controllability in single-leader consensus networks? IEEE Control Syst. Mag. 32 (4) (2012) 66–73. [32] M. Cao, S. Zhang, M.K. Camlibel, A class of uncontrollable diffusively coupled multiagent systems with multichain topologies, IEEE Trans. Autom. Control 58 (2) (2013) 465–469. [33] Z. Ji, H. Lin, H. Yu, Leaders in multi-agent controllability under consensus algorithm and tree topology, Syst. Control Lett. 61 (9) (2012) 918–925. [34] G. Notarstefano, G. Parlangeli, Controllability and observability of grid graphs via reduction and symmetries, IEEE Trans. Autom. Control 58 (7) (2013) 1719–1731. [35] J. Zhan, X. Li, Cluster consensus in networks of agents with weighted cooperative-competitive interactions, IEEE Trans. Circ. Syst. II Expr. Briefs 65 (2) (2017) 241–245. [36] J. Yu, L. Wang, Group consensus of multi-agent systems with undirected communication graphs, in: Proceedings of the 7th Asian Control Conference, 2009, pp. 105–110. [37] B. Liu, Y. Han, F. Jiang, H. Su, J. Zou, Group controllability of discrete-time multi-agent systems, J. Frankl. Inst. 353 (14) (2016) 3524–3559. [38] B. Liu, H. Su, F. Jiang, Y. Gao, L. Liu, J. Qian, Group controllability of continuous-time multi-agent systems, IET Control Theory Appl. 12 (11) (2018) 1665–1671. [39] R. Al-Ashoor, K. Khorasani, A decentralized indirect adaptive control for a class of two-time-scale nonlinear systems with application to flexible-joint manipulators, IEEE Trans. Ind. Electon. 46 (5) (2002) 1019–1029. [40] A. Meyer-Bäse, F. Ohl, H. Scheich, Singular perturbation analysis of competitive neural networks with different time scales, Neural Comput. 8 (8) (1996) 1731–1742. [41] M. Balldes, P. Daoutidis, Control of integrated process networks-a multi-time scale perspective, Comput. Chem. Eng. 31 (5) (2005) 426–444. [42] P. Kokotovic, H. Khalil, J. O’Reilly, Singular Perturbation Methods in Control Analysis and Design, SIAM, Philadephia, 1999. [43] E. Robert, J. O’Malley, Singular Perturbation Methods for Ordinary Differential Equations, Springer Science and Business Media, 2012. [44] C. Cai, Z. Wang, J. Xu, Finite Frequency Analysis and Synthesis for Singular Perturbation Systems, Springer, Swirzerland, 2017. [45] J. Yu, L. Wang, Group consensus in multi-agent systems with switching topologies and communication delays, Syst. Control Lett. 59 (6) (2010) 340–348. [46] B. Jihene, M. Irinel-Constantin, D. Jamal, Synchronization in networks of linear singularly perturbed systems, in: American Control Conference, 2016, pp. 4293–4298. [47] M. Posfai, J. Gao, S.P. Cornelius, A. Barabsi, R.M. D’Souza, Controllability of multiplex, multi-time-scale networks, Phys. Rev. E 94 (3) (2016) 032316. [48] M. Long, H. Su, B. Liu, Group controllability of two-time-scale multi-agent networks, J. Frankl. Inst. 355 (13) (2018) 6045–6061.