Cooperative control of multi-agent systems with unknown control directions

Applied Mathematics and Computation 292 (2017) 240–252

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

Cooperative control of multi-agent systems with unknown control directions Qian Ma School of Automation, Nanjing University of Science and Technology, Nanjing 210094, PR China

a r t i c l e

i n f o

Keywords: Multi-agent systems Unknown control directions Adaptive backstepping High-order systems

a b s t r a c t This paper deals with the cooperative control problem of high-order multi-agent systems with unknown control directions. Adaptive backstepping technology is utilized to handle the difficulties caused by the unknown control direction in every order’s dynamics. By using a conditional inequality related to multiple terms of the Nussbaum-type function at each step, the consensus problem for systems with undirected and connected topology is solved. Then, it is shown that cooperative asymptotic regulation problem can be addressed by the same design procedure. The agents can achieve consensus and converge to the equilibrium asymptotically as well. Simulation examples are provided finally to demonstrate the effectiveness of the proposed design methods. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Cooperative control in networks of autonomous mobile agents is extensively studied over the past few years due to its extensive applications in sensor networks, mobile robots, spacecraft formation flying, and other areas. As one of the most fundamental research topics in the field of cooperative control of multi-agent systems, consensus plays an important role in achieving collective behavior through local interactions of agents. Consensus means that the states of all agents reach an agreement on a common value by using local information of each agent’s neighbors. Following some pioneering works [1–4], various consensus problems of multi-agent systems have been studied recently, such as consensus of systems with second-order dynamics [5–8], high-order dynamics [9] and linear dynamics [10,11], consensus of agents with time-delay [12–15], agreement over random networks [16–19], and distributed containment control [20–22], just to mention a few. Control directions, namely, the signs of the control coefficients, represent the motion directions of the system. In most existing works, a common assumption is that signs of the control coefficients are known and assumed to be positive without loss of generality. However, in many applications, control directions might not always be known in priori. For instance, when not all state variables are measurable, and when there exist large uncertainties in systems, it is difficult to detect the control directions directly [23]. The standard way to handle the unknown control directions is the Nussbaum gain approach, which was first proposed in [24]. Later, the Nussbaum gain approach has been extensively used to various adaptive control problems for single system in [25–27]. Recently, for a group of interacting systems involved an unknown control direction, a sub-Lyapunov function for each subsystem was constructed in [23,28], where only one Nussbaum-type function was employed. However, there are few results on the cooperative control problem for multi-agent systems with unknown control directions. Very recently, [29] discussed consensus problem of first-order multi-agent systems with unknown control directions by using the sub-Lyapunov function method. Note that the sign and amplitude of each agent’s control direction are E-mail address: [email protected] http://dx.doi.org/10.1016/j.amc.2016.07.016 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.

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241

all unknown and the sign of each agent can be different. However, the method proposed in [29] is not valid for secondorder or high-order system. For second-order linearly parameterized multi-agent systems with unknown identical control directions, consensus problem was solved in [30] by constructing a new Nussbaum-type function to estimate the unknown control direction of each agent and establishing a novel lemma to deal with the multiple terms of the proposed Nussbaumtype function. It is worth noting that only the control direction in the second-order dynamics is unknown, whereas the control direction in the first-order dynamics is known and the control gain is assumed to be 1. If the control directions at every order are unknown or even high-order system are considered, the controller design and consensus analysis will become more difficult. It is worth noting that some interesting results on cooperative control of nonlinear system with unknown control directions can be found in [31–33]. In [31] and [32], adaptive neural networks-based control approach is used, where switching topology and fixed topology are considered, respectively. In [33], output consensus of heterogenous agents is studied with two distributed Nussbaum-type control laws. In this paper, we consider the cooperative control problem for high-order multi-agent systems with unknown control directions. Since that each order’s control direction is unknown, adaptive backstepping technology is utilized. By using a conditional inequality related to multiple terms of the Nussbaum-type function at each order, the consensus problem for systems with undirected and connected topology is solved. Then, it is shown that cooperative asymptotic regulation problem can be addressed by the same design procedure. The agents can achieve consensus and converge to the equilibrium asymptotically as well. The main contributions of this paper are summarized as follows: 1) This paper makes one of the first few attempts to deal with the cooperative control problem of high-order multi-agent systems with unknown control directions; 2) Each order’s control direction is unknown; 3) To accomplish the consensus analysis with backstepping technology, a conditional inequality related to multiple terms of the Nussbaum-type function is used at each step; 4) The cooperative asymptotic regulation problem can be solved by the same approach. Although nonlinear system are not considered in our paper, the approach used in the paper is more simpler than that in [31–33]. Furthermore, both cases of consensus and decentralized asymptotic regulation are addressed in the paper. Notations. Throughout this paper, for real symmetric matrices X and Y, the notation X ≥ Y (respectively, X > Y) means that the matrix X − Y is positive semi-definite (respectively, positive definite). Rn denotes the set of n × 1 real vectors. Rn×n denotes the set of n × n real matrices. I denotes an identity matrix of appropriate dimension. 1N ∈ RN be the vector with all entries being 1. The notation ‘∗’ is used as an ellipsis for terms that are induced by symmetry. 2. Preliminaries In this section, some basic concepts and definitions about graph theory and model formulations are briefly introduced. Let G = {V, E, A} be an undirected graph with the set of nodes V = {v1 , v2 , . . . , vN }, the set of edges E ⊆ V × V, and an adjacency matrix A = (ai j )N×N . A path is sequence of edges with the form (v1 , v2 ), (v2 , v3 ), . . .. An undirected graph is connected if there is path between every pair of distinct nodes. The elements of the adjacency matrix A are defined as aij > 0 if and only if there is an edge (v j , vi ) in G; otherwise, ai j = 0, and ai j = a ji . Define the degree matrix as D = diag{di }N×N  with di = Nj=1 ai j and the Laplacian matrix as L = D − A. For the undirected and connected graph G, L is symmetric. Consider a group of N agents with the following dynamics:

x˙ i,k = bi,k xi,k+1 x˙ i,n = bi,n ui

(1) ]T ,

Rn

where i = 1, . . . , N, k = 1, . . . , n − 1; xi = [xi,1 , . . . , xi,n xi ∈ and ui ∈ R are the state and the control input, and bi,q , i = 1, . . . , N, q = 1, . . . , n (bi,q = 0) are unknown constant parameters. Assumption 1. bi,q , q = 1, . . . , n from different agents have the identical unknown sign. That is, the sign of bi,1 , i = 1, . . . , N are the same, the sign of bi,2 , i = 1, . . . , N are the same, . . . , the sign of bi,n , i = 1, . . . , N are the same. Definition 1. Consensus in multi-agent system (1) is achieved if for any initial conditions,

lim |xi,q − x j,q | = 0,

t→∞

∀i, j = 1, . . . , N, q = 1, . . . , n

Remark 1. A Nussbaum-type function N (· ) is the one with the following properties [24]:

lim sup

k→∞

lim in f

k→∞

1 k 1 k

 

k 0 k 0

N ( τ )d τ = ∞ N (τ )dτ = −∞

To overcome the obstacle of the multiple Nussbaum-type function terms coexisting in the same condition inequality, the author in [30] constructed a new Nussbaum-type function for solving the consensus problem of first-order and secondorder multi-agent systems:

N0 (k ) = cosh(λk )sin(k )

(2)

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However, just second-order system with unknown control direction in the second order dynamics is considered in [30]. Note that the n-order multi-agent systems with n unknown control directions are considered in this paper, some novel approaches will be presented on the basis of the backstepping technique and the following Lemma. Lemma 1 ([30]). Let V(t) and ki (t ), i = 1, 2, . . . , N, be smooth functions defined on [0, tf ) with V(t) ≥ 0 and ki (0 ) = 0. Also let N0 (· ) be defined by (2) with λ > max{ π1 ln ηmaxη (N−1 ) , 0}, where ηmax > ηmin > 0. If the following inequality holds: min

V (t ) ≤

N 

ηi

i=1



t 0

N0 (ki (τ ))k˙ i (τ )dτ +

N 

 ai

i=1

t 0

k˙ i (τ )dτ + c,

∀t ∈ [0, t f )

(3)

where ai and ηi are constants with ai > 0, ηi has the same sign, and |ηi | ∈ [ηmin , ηmax ], i = 1, 2, . . . , N, then, V(t), ki (t) t  ˙ and N i=1 ηi 0 N0 (ki (τ ))ki (τ )d τ are bounded on [0, tf ). 3. Design procedure Now we are in a position to present consensus controller on the basis of adaptive backstepping technique. The design procedure consists of n steps. At the jth step for every agent i, 1 ≤ j ≤ n − 1, 1 < i ≤ N, the state variable xi, j+1 is viewed as the fictitious control, for which a reference signal α i,j is designed. At the nth step, the fictitious control coincides with the actual control ui . Assume that G is undirected and connected. Step 1. Define

zi,1 =

N 

ai j (xi,1 − x j,1 )

(4)

j=1

zi,2 = xi,2 − αi,1

(5)

αi,1 = N (ki,1 )zi,1

(6)

k˙ i,1 = zi,21

(7)

Let

where N (ki,1 ) is an Nussbaum-type function with the form (2), namely, N (ki,1 ) = cosh(λ1 ki,1 )sin(ki,1 ), λ1 > max{ π1 ln

b1 max(N−1 ) , 0} b1 min

and |bi,1 | ∈ [b1 min,b1 max], i = 1, 2, . . . , N. b1 min and b1 max represent the minimum and maximum of

|bi,1 |, respectively. Define

V1 =

1 T x Lx 2

(8)

where x = [x1,1 , x2,1 , . . . , xN,1 ]T . Then, the time derivative of V1 is given by

V˙ 1 =

N 

zi,1 bi,1 xi,2

i=1

=

N 

zi,1 bi,1 zi,2 +

i=1

=

N 

zi,1 bi,1 zi,2 −

N 

zi,21 +

i=1

N 

zi,21 +

i=1

≤−

zi,1 bi,1 αi,1

i=1

i=1

=−

N 

N 

N  i=1

zi,21 +

N 

(bi,1 N (ki,1 ) + 1 )k˙ i,1 +

i=1

zi,1 bi,1 αi,1

i=1 N 

bi,1 zi,1 zi,2

i=1

N N N   3 2 zi,1 + (bi,1 N (ki,1 ) + 1 )k˙ i,1 + b2i,1 zi,22 4 i=1

i=1

(9)

i=1

Step j (2 ≤ j ≤ n − 1 ). Define

zi, j = xi, j − αi, j−1 zi, j+1 = xi, j+1 − αi, j

(10)

αi, j = N (ki, j )(zi, j − α˙ i, j−1 )

(11)

Let

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243

k˙ i, j = zi,2 j − α˙ i, j−1 zi, j N (ki, j )

where

is

an

b max(N−1 ) max{ π ln j b min , 0} j 1

(12)

Nussbaum-type

function

with

the

form

(2),

namely,

N (ki, j ) = cosh(λ j ki, j )sin(ki, j ),

λj >

and |bi,j | ∈ [bj min, bj max], i = 1, 2, . . . , N.

Define

Vj =

1 T z zj 2 j

(13)

where z j = [z1, j , z2, j , . . . , zN, j ]T Then, one has

V˙ j =

N 

zi, j (bi, j xi, j+1 − α˙ i, j−1 )

i=1

=

N 

zi, j bi, j zi, j+1 +

i=1

=−

bi, j zi, j αi, j −

N 

i=1

N 

zi,2 j +

i=1

≤−

N 

N 

zi, j α˙ i, j−1 −

i=1

(bi, j N (ki, j ) + 1 )k˙ i, j +

i=1

N 

zi,2 j +

i=1 N 

zi,2 j

i=1

bi, j zi, j zi, j+1

i=1

N N N   3 2 zi, j + (bi, j N (ki, j ) + 1 )k˙ i, j + b2i, j zi,2 j+1 4 i=1

N 

i=1

(14)

i=1

Step n. Define

zi,n = xi,n − αi,n−1

(15)

ui = N (ki,n )(zi,n − α˙ i,n−1 )

(16)

2 k˙ i,n = zi,n − α˙ i,n−1 zi,n

(17)

Let

where N (ki,n ) is an Nussbaum-type function with the form (2), namely, N (ki,n ) = cosh(λn ki,n )sin(ki,n ), λn > (N−1 ) max{ π1 ln bn max , 0} and |bi,n | ∈ [bn min, bn max], i = 1, 2, . . . , N. bn min Set

Vn =

1 T z zn 2 n

(18)

where zn = [z1,n , z2,n , . . . , zN,n ]T . Then, we have

V˙ n =

N 

zi,n (bi,n ui − α˙ i,n−1 )

i=1

=

N 

zi,n bi,n ui −

N 

i=1

=−

zi,n α˙ i,n−1 −

i=1

N 

2 zi,n +

i=1

N 

N 

2 zi,n +

i=1

N 

2 zi,n

i=1

(bi,n N (ki,n ) + 1 )k˙ i,n

(19)

i=1

Integrating both sides of (19) from 0 to t, we get

Vn ≤

N   i=1

t 0

bi,n N (ki,n )k˙ i,n dτ +

N   i=1

t 0

k˙ i,n dτ + c1

(20)

N  t ˙ where c1 is a constant. From Lemma 1, it is easy to obtain that Vn , ki,n and i=1 0 bi,n N (ki,n )ki,n d τ are bounded on [0, 2 is integrable on [0, t ). Thus, on the tf ). Thus, no finite-time escape phenomenon may happen and t f = ∞. Furthermore, zi,n f basis of Barbalat’s Lemma, it can be concluded that limt→∞ zi,n = 0, 1 ≤ i ≤ N. For step n − 1, using Lemma 1 again, one has limt→∞ zi,n−1 = 0, 1 ≤ i ≤ N. Then using an induction argument and applying Lemma 1 (n − 2 ) times, it can be shown from the above design procedure that limt→∞ zi, j = 0, 1 ≤ i ≤ N, 1 ≤ j ≤ n − 2. Thus, one has limt→∞ zi, j = 0, 1 ≤ i ≤ N, 1 ≤ j ≤ n, and limt→∞ xi, j = 0, 1 ≤ i ≤ N, 2 ≤ j ≤ n. Since G is connected, it is implied that limt→∞ |xi,q − x j,q | = 0, i, j = 1, . . . , N, q = 1, . . . , n. Then, consensus in system (1) is achieved.

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Theorem 1. Suppose that the undirected graph G is connected and Assumption 1 holds. Under the virtual control signal (6), (11) and control law (16), and adaptive laws (7), (12) and (17), system (1) can achieve consensus. Proof. The proof can be obtained by following the above design procedure form Step 1 to n.



Remark 2. By using adaptive backstepping technique, consensus problem of high-order multi-agent system with unknown control directions is solved. It is worth noting that each order’s dynamics of system has unknown control directions. It is assumed that the amplitude of control directions is known. By using Lemma 1 and Barbalat’s Lemma at each order respectively, it can be verified that consensus is achieved. Peng and Ye [29] and Chen et al. [30] discuss the similar problem and focus on first-order and second-order systems respectively. However, if the control direction in every order’s dynamics is unknown or even high-order system is considered, the methods proposed in [29] and [30] may not be applicable directly. In [29], system x˙ i = bi ui is considered, which can be seen as a special case of system (1). It is clear that the con troller ui = k2i sin(ki ) ai j (xi − x j ) fails to tackle the consensus problem of second-order and high-order systems. In [30], second-order system x˙ i = vi , v˙ i = bi ui is studied, which also can be covered by system (1) in this paper. Differ from the single unknown control direction bi in [29] and [30], there are unknown control directions at every order in high-order system in this paper. Even if the second-order case of x˙ i1 = bi1 xi2 , x˙ i2 = bi2 ui be considered, it is quite different from that in [30]. There exists unknown control directions in the first order dynamics and second order dynamics. In this case, adaptive backstepping technique is crucial to deal with the difficulty caused by multiple unknown control directions. It is necessary to use Nussbaum-type function N (ki j ) = cosh(λ j ki j )sin(ki j ) with λ j > max{ π1 ln

b j max(N−1 ) , 0} b j min

and adaptive law kij at every

order to constitute the virtual control signal α ij and the real control law ui . And a conditional inequality (3) related to multiple terms of the Nussbaum-type function is used at each step. Without this design procedure, the problem of multiple unknown control direction cannot be addressed. Remark 3. Backstepping technique has been used in distributed control problem of nonlinear multi-agent systems recently; see, e.g., [34–37] and the reference therein. However, the existing methods cannot be used for the case considered in this paper because they cannot deal with the consensus problem of system with unknown control direction. To tackle the difficulty caused by unknown control direction, the Nussbaum-type function is necessary to used in virtual control signals α i,q and controller ui (i = 1, . . . , N, q = 1, . . . , n ). As pointed out in [29], in some cases, we need not only xi → xj but also to drive all xi to some desired values such as the equilibrium. This case is called decentralized asymptotic regulation. Let node xN+1 = 0 representing the equilibrium and it can be seen as the leader. Assume that there exits at least one node ir can observe information from the leader directly, i.e., ar,N+1 > 0. The Laplace matrix L¯ can be expressed as



L¯ =

L1 ∗1×N

L2 0



(21)

In step 1, we set

zi,1 =

N+1 

ai j (xi,1 − x j,1 ),

1≤i≤N

(22)

j=1

Then

αi,1 = N (ki,1 )zi,1 = N (ki,1 )

N+1 

ai j (xi,1 − x j,1 )

(23)

j=1

k˙ i,1 = zi,21

(24)

And

V1 =

1 T x L¯ x 2

(25)

where x = [x1,1 , x2,1 , . . . , xN+1,1 ]T . Based on the same line of design procedure in Theorem 1, we then can obtain that limt→∞ zi, j = 0, 1 ≤ i ≤ N, 1 ≤ j ≤ n, and limt→∞ xi, j = 0, 1 ≤ i ≤ N, 2 ≤ j ≤ n. Let xl = xN+1,1 , x f = [x1,1 , x2,1 , . . . , xN,1 ]T and z1 = [z1,1 , z2,1 , . . . , zN,1 ]T , then for 1 ≤ i ≤ N, one has

z1 = L1 x f + L2 xl

(26)

Recall that we have got limt→∞ z1 = 0, one has

lim x f = −L−1 1 L2 xl

t→∞

(27)

It is followed that L2 = −L1 1N since L¯ 1N+1 = 0. Then we know that −L−1 L2 = 1N . Then, it is easy to obtain that 1 limt→∞ xi,1 = limt→∞ x j,1 = xN+1,1 = 0, 1 ≤ i ≤ N, 1 ≤ j ≤ N, which implies that limt→∞ xi = limt→∞ x j = 0, 1 ≤ i ≤ N, 1 ≤ j ≤ N.

Q. Ma / Applied Mathematics and Computation 292 (2017) 240–252

245

Fig. 1. Topology structure of network in Example 1.

Fig. 2. Topology structure of network in Example 2.

0.3 x1 x2 x3

0.2 0.1

x

0 −0.1 −0.2 −0.3 −0.4

0

5

10

15

t Fig. 3. Trajectories of xi in Example 1, i = 1, 2, 3.

Theorem 2. Suppose that the undirected graph G is connected and Assumption 1 holds. Let the equilibrium to be the leader. Assume that there exits at least one node ir can observe information from the leader directly. Under the virtual control signal (23), (11) and control law (16), and adaptive laws (24), (12) and (17), system (1) can achieve consensus and be regulated to the equilibrium. Proof. Refer to the above discussion, the proof can be obtained easily and is omitted here.



Remark 4. Compared to the leaderless consensus problem, a particularly interesting topic is consensus of a group of agents with a leader. Such a problem is named the leader-following consensus or distributed tracking. We can see that decentralized asymptotic regulation problem discussed above can be seen as a case of the leader-following consensus. It has been shown that the obtained consensus results can be extended to the leader-following case. Note that Theorems 1 and 2 can also be applied to high-order linearly parameterized multi-agent systems with unknown control directions. system (1) can be changed as follows:

x˙ i,k = bi,k xi,k+1 x˙ i,n = bi,n ui + φi (xi,1 , . . . , xi,n )T θi

(28) ]T ,

Rn

where i = 1, . . . , N, k = 1, . . . , n − 1; xi = [xi,1 , . . . , xi,n xi ∈ and ui ∈ R are the state and the control input, and bi,q , i = 1, . . . , N, q = 1, . . . , n are unknown constant parameters. θ i is an unknown constant vector, φ i is a known continuous vectorvalued function. To deal with linearly parameterized uncertainties, let

ui = N (ki,n )(zi,n − α˙ i,n−1 + φiT θˆi )

(29)

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Q. Ma / Applied Mathematics and Computation 292 (2017) 240–252

1 v1 v2 v3

0.8 0.6

v

0.4 0.2 0 −0.2 −0.4

0

5

10

15

t Fig. 4. Trajectories of vi in Example 1, i = 1, 2, 3.

5 w1 w2 w3

4 3 2

w

1 0 −1 −2 −3 −4

0

5

10

15

t Fig. 5. Trajectories of wi in Example 1, i = 1, 2, 3.

2 k˙ i,n = zi,n − α˙ i,n−1 zi,n + φiT θˆi zi,n

ˆθ˙ i =

φi zi,n

(30)

(31)

Corollary 1. Suppose that the undirected graph G is connected and Assumption 1 holds. Under the virtual control signal (6), (11) and control law (29), and adaptive laws (7), (12), (30) and (31), system (28) can achieve consensus. Proof. For steps 1, 2, . . . , n − 1, apply the same design procedure in Theorem 1. For step n, apply (15), (29)–(31). Let

Vn =

N 1 T 1 ˆ zn zn + (θi − θi )T (θˆi − θi ) 2 2 i=1

where zn = [z1,n , z2,n , . . . , zN,n ]T .

(32)

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247

1.1 k11 k21 k31

1 0.9 0.8 0.7 k

0.6 0.5 0.4 0.3 0.2 0.1

0

5

10

15

t Fig. 6. Trajectories of ki,1 in Example 1, i = 1, 2, 3.

0.9 0.8 0.7 0.6

k12 k22 k32

k

0.5 0.4 0.3 0.2 0.1 0

0

5

10

15

t Fig. 7. Trajectories of ki,2 in Example 1, i = 1, 2, 3.

Then, we have

V˙ n =

N 

zi,n (bi,n ui + φiT θi − α˙ i,n−1 ) +

N 

i=1

=

N 

zi,n bi,n ui −

N 

i=1

=−

(θˆi − θi )T φi zi,n

i=1

N  i=1

i=1 2 zi,n +

N 

zi,n α˙ i,n−1 +

N 

zi,n φiT θˆi −

i=1

N  i=1

2 zi,n +

N 

2 zi,n

i=1

(bi,n N (ki,n ) + 1 )k˙ i,n

i=1

Therefore, based on the same proof procedure of Theorem 1, it can be concluded that system (28) achieves consensus.



Corollary 2. Suppose that the undirected graph G is connected and Assumption 1 holds. Let the equilibrium to be the leader. Assume that there exist at least one node ir can observe information from the leader directly. Under the virtual control signal

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2.5

k13 k23 k33

2

k

1.5

1

0.5

0

0

5

10

15

t Fig. 8. Trajectories of ki,3 in Example 1, i = 1, 2, 3.

20 u1 u2 u3

15

u

10

5

0

−5

−10

0

5

10

15

t Fig. 9. Trajectories of ui in Example 1, i = 1, 2, 3.

(23), (11) and control law (29), and adaptive laws (24), (12), (30) and (31), system (28) can achieve consensus and be regulated to the equilibrium. Proof. Refer to the proof in Theorem 2 and Corollary 1, the proof can be obtained easily and is omitted here.



Remark 5. In this paper, the case of undirected and connected topology is considered. On the other hand, if the network topology is directed or switching (jointly-connected), we cannot get L = LT . Therefore, the all results cannot hold. To address the consensus problem of high-order system with unknown control directions and directed or switching topology is a very meaningful and challenging problem, which is still an open problem deserve the further research. 4. Numerical examples In this section, we provide two examples to demonstrate the effectiveness of the proposed methods.

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249

1.2 x1 x2 x3

1

0.8

x

0.6

0.4

0.2

0

−0.2

0

5

10

15

t Fig. 10. Trajectories of xi in Example 2, i = 1, 2, 3.

20 v1 v2 v3

15

10

v

5

0

−5

−10

−15

0

5

10

15

t Fig. 11. Trajectories of vi in Example 2, i = 1, 2, 3.

Example 1. Consider a third-order multi-agent system (1) with the topology shown in Fig. 1. The system is set as follows:

x˙ 1 = −2v1 , x˙ 2 = −v2 , x˙ 3 = −v3 , v˙ 1 = 1.5w1 , v˙ 2 = w2 , v˙ 3 = 2w3 , w˙ 1 = 0.5u1 ,

w˙ 2 = 2u2 ,

w˙ 3 = 1.5u3 .

According to Theorem 1, choose λ1 = 0.5 > max{ π1 ln 21∗2 , 0} = 0.4413 in N (ki,1 ), λ2 = 0.5 > max{ π1 ln 21∗2 , 0} = 0.4413 in N (ki,2 ), and λ3 = 1 > max{ π1 ln 20∗2 .5 , 0} = 0.6619 in N (ki,3 ), i = 1, 2, 3. The simulation results are shown in Figs. 3–9. It can be seen that the agents achieve consensus. Note that both the results in [29] and [30] are not valid in this system. Example 2. Consider the decentralized regulation problem. Assume the agent 1 can obtain the leader’ information directly. The topology is shown in Fig. 2. The second-order system is set as follows:

x˙ 1 = −2v1 , x˙ 2 = −1.2v2 , x˙ 3 = −v3 ,

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1.6

1.4 k11 k21 k31

1.2

k

1

0.8

0.6

0.4

0.2

0

5

10

15

t Fig. 12. Trajectories of ki,1 in Example 2, i = 1, 2, 3.

4 3.5 3 2.5

k

2 k21 k22 k32

1.5 1 0.5 0 −0.5

0

5

10

15

t Fig. 13. Trajectories of ki,2 in Example 2, i = 1, 2, 3.

v˙ 1 = 1.5u1 , v˙ 2 = u2 , v˙ 3 = 2.5u3 . According to Theorem 2, choose λ1 = 0.8 and λ2 = 1, the simulation results are shown in Figs. 10–14. It can be seen that the agents achieve consensus and are regulated to the equilibrium eventually. Only Chen et al. [30] discuss the second-order system with unknown control direction. But the system in [30] is x˙ i = vi , v˙ i = bi ui , where just the second order’s dynamics have unknown control direction. In this example, both the first order’s dynamics and the second order’s dynamics have unknown control directions. Therefore, the result in [30] does not work in this system. 5. Conclusions In this paper, cooperative control problem of high-order multi-agent systems with unknown control directions is considered. It is assumed that the control direction in every order’s dynamics is unknown. Adaptive backstepping technique is utilized. By using a conditional inequality related to multiple terms of the Nussbaum-type function at each step, the con-

Q. Ma / Applied Mathematics and Computation 292 (2017) 240–252

251

80 u1 u2 u3

60

u

40

20

0

−20

−40

0

5

10

15

t Fig. 14. Trajectories of ui in Example 2, i = 1, 2, 3.

sensus problem for system with undirected and connected topology is solved. Later, it is shown that the agents not only achieve consensus but also can converge to the equilibrium asymptotically by the same design procedure. Finally, simulation examples are provided to demonstrate the effectiveness of the proposed design methods. There are still a number of related interesting cooperative control problems of multi-agent systems with unknown control directions. In our future work, we may study the case of system with directed or switching network topologies. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants 61403199, 61503189, U1509217, the Program for Changjiang Scholars and Innovative Research Team in University under Grant IRT13072, the Fundamental Research Funds for the Central Universities 30916015105, and the Natural Science Foundation of Jiangsu Province under Grant BK20140770. References [1] A. Jadbabaie, J. Lin, A. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Autom. Control 48 (6) (20 03) 988–10 01. [2] T. Vicsek, A. Czirok, E.B. Jacob, I. Cohen, O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. E 75 (6) (1995) 1226–1229. [3] R.O. Saber, R.M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control 49 (9) (2004) 1520–1533. [4] W. Ren, R.W. Beard, Consensus seeking in multiagent systems under dynamically changing interaction topologies, IEEE Trans. Autom. Control 50 (5) (2005) 655–661. [5] W. Yu, G. Chen, M. Cao, Some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems, Automatica 46 (6) (2010) 1089–1095. [6] H. Hu, A. Liu, Q. Xuan, G. Xie, Second-order consensus of multi-agent systems in the cooperation-competition network with switching topologies: A time-delayed impulsive control approach, Syst. Control Lett. 62 (12) (2013) 1125–1135. [7] W. Ren, On consensus algorithms for double-integrator dynamics, IEEE Trans. Autom. Control 58 (6) (2008) 1503–1509. [8] Q. Ma, S. Xu, F.L. Lewis, Second-order consensus for directed multi-agent systems with sampled data, Int. J. Robust Nonlin. Control 24 (16) (2014) 2560–2573. [9] P. He, Y. Li, J.H. Park, Noise tolerance leader-following of high-order nonlinear dynamical multi-agent systems with switching topology and communication delay, J. Frankl. Inst. 353 (1) (2016) 108–143. [10] Z. Zhao, Z. Lin, Global leader-following consensus of a group of general linear systems using bounded controls, Automatica 68 (2016) 294–304. [11] M. Davoodi, N. Meskin, K. Khorasani, Simultaneous fault detection and consensus control design for a network of multi-agent systems, Automatica 66 (2016) 185–194. [12] T. Ma, F.L. Lewis, Y. Song, Exponential synchronization of nonlinear multi-agent systems with time delays and impulsive disturbances, Int. J. Robust Nonlin. Control 26 (8) (2016) 1615–1631. [13] T. Ma, Synchronization of multi-agent stochastic impulsive perturbed chaotic delayed neural networks with switching topology, Neurocomputing 151 (2015) 1392–1406. [14] Q. Xiao, Z. Huang, Consensus of multi-agent systems with distributed control on time scales, Appl. Math. Comput. 277 (2016) 54–71. [15] Y. Shang, Continuous-time average consensus under dynamically changing topologies and multiple time-varying delays, Appl. Math. Comput. 244 (2014) 457–466.

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