Adaptive iterative learning consensus control for second-order multi-agent systems with unknown control gains

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Adaptive iterative learning consensus control for second-order multi-agent systems with unknown control gainsR Guilu Li a, Chang-E Ren a,∗, C.L. Philip Chen b,c, Zhiping Shi a a

School of Information Engineering, Capital Normal University, Beijing, China School of Computer Science and Engineering, South China University of Technology, Guangzhou, China c Dalian Maritime University, Dalian, China b

a r t i c l e

i n f o

Article history: Received 19 May 2019 Revised 20 January 2020 Accepted 31 January 2020 Available online xxx Communicated by Jiahu Qin Keywords: Unknown control gains Multi-agent systems Adaptive consensus control Iterative learning Neural networks

a b s t r a c t We respond to the consensus control algorithms for second-order non-linear MAS with unknown control gains in the finite time interval by adopting iterative learning control (ILC) methods in this paper. Compared to the current findings, the control gains in the proposed method are unknown functions with unknown and non-identical signs. The topology graph between the follower agents is an undirected connected graph. By referencing the Nussbaum-type function and the neural networks, the adaptive control algorithms are intended to cope with the consensus control between the agents on the limited time interval. Simulation results illustrate the efficacy of the raised algorithms.

1. Introduction As multi-agent systems (MAS) have been extensively applied in various fields, more and more coordination problems of MAS have been brought to the researchers’ attention [1]. The consensus control problem is the primary research in the field of coordination control. The foremost intention of consensus control is that the status of all agents can be consensus under the control protocol. When there is no leader in the system, this control mode is called leaderless consensus control, as in [2,3]. However, when there is a leader existed, the states of each agent are identical to the leader. This control mode is referred to as the leader-following consensus control as in [4,5]. The practical cluster synchronization problem based on the leader-following method is solved in the universal linear MAS in [6]. However, some of the articles mentioned above don’t deal with uncertainty. In practical applications, the MAS inevitably is

R This work was supported in part by the National Natural Science Foundation of China under Grant 61803276, 61751202, 61751205, 61572540, U1813203 and U1801262, in part by the Macau Science and Technology Development Fund (FDCT) under Grants 079/2017/A2, 024/2015/AMJ and 0119/2018/A3, Beijing Municipal Education Commission Science Plan (General Research Project) No. KM201910 0280 04, CNU Young Yanjing Scholars Teacher Cultivation under Grant CNU0061955110, Beijing Municipal Excellent Talent Program (Youth Core Individual Project) No. 20170 0 0 020124G072 ∗ Corresponding author. E-mail address: [email protected] (C.-E. Ren).

© 2020 Elsevier B.V. All rights reserved.

influenced by uncertainty. Uncertainties can be divided into three types, the uncertainty of measurement, the uncertainty of model, and the uncertainty of interference. When dealing with uncertainties, different strategies are required to deal with various uncertainties [7]. In [8], for non-linear systems with unknown output uncertainties, adaptive output feedback control is studied by using neural networks. In [9], an observer is constructed to observe the mismatched and matched disturbance in the system, and the adaptive approach is designed to estimate the inexact information in the model [10]. In [11], an adaptive state observer is proposed for non-linear systems with unknown non-affine faults to observe unmeasured system states. The radial basis function neural networks (RBFNN) can estimate the unknown dynamics, which is designed as the approximator of the unknown dynamics as in [12]. The problem of output feedback control of uncertain systems is settled by RBFNN in [13]. In literature [14], the authors investigated the first-order non-affine non-linear MAS with unknown input dynamics by the implicit function theorem and Neural Network. In the above researches, only if time tends to infinity, the objective can be achieved. To obtain the leader-following consensus control of uncertain non-linear MAS within a finite time interval, we use the ILC method as in [15]. The ILC method can take the advantages of system duplication, which can be utilized to improve the performance of the entire iterative learning cycle. The tracking control of the uncertain non-linear systems is solved within a finite time by exploiting the

https://doi.org/10.1016/j.neucom.2020.01.108 0925-2312/© 2020 Elsevier B.V. All rights reserved.

Please cite this article as: G. Li, C.-E. Ren and C.L.P. Chen et al., Adaptive iterative learning consensus control for second-order multi-agent systems with unknown control gains, Neurocomputing, https://doi.org/10.1016/j.neucom.2020.01.108

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iterative Lyapunov functions in [16]. Many contributions have been made by using ILC in the consensus control and formation control of MAS, as [17,18]. For MAS with distributed parameter models, an ILC algorithm is constructed to obtain consensus control in [19]. A distributed ILC protocol is proposed only using itself and its directly available I/O data for the MAS in [20]. In [21], ILC is applied to the consensus and formation control of second-order MAS with unknown parameters. By using the backstepping design method of the composite energy function, a distributed adaptive ILC method for high-order non-linear MAS with parameter and non-parametric uncertainties is proposed in[18]. In [22], for the model-free non-linear MAS, a distributed adaptive ILC method is proposed to achieve consensus tracking of ideal trajectories. In [23], the ILC is used to reach the high-precision consensus of agents with switching topology and communication-delay in a limited time interval. However, when the control gain of the system is completely unknown, the control algorithms designed in the above articles are invalid. In most studies, the control gain is assumed to be known constants in order to facilitate the algorithm design. However, it is usually unknown in many real-world applications. In [24], the consensus problem for first-order MAS with unknown nonidentical and identical control signs is settled by the fuzzy neural network. The Nussbaum function is a convenient tool for dealing with unknown control gains and can be combined with the ILC algorithm design [25]. Furthermore, in [26–28], the Nussbaum function is used to achieve the consensus of MAS with incompletely known input gains. In [26], the uncertainties are not considered, and the model of the system is relatively simple. In [27], the researchers put forward to a new function to solve the completely unknown gains in the output regulation of non-linear systems. In [28], an ILC protocol based on the Nussbaum function is designed for first-order MAS to obtain consensus. Note that the systems in [28] is only the first-order MAS that non-linear dynamics satisfy the linearity-in-parameters conditions. We will utilise the Nussbaum function defined in [29] to solve the consensus problem of second-order MAS with completely unknown input gains. The objective of this paper is to come up with a distributed algorithms that achieve consensus in a finite time interval for second-order MAS with unknown dynamics and fully unknown control input gains. The control gains of the agents are unknown bounded smooth functions, and their signs are non-identical. Based on the Nussbaum function, the problem of unknown control gains is settled. By using ILC and RBFNN, the designed algorithms ensure that the agents achieve consensus in a limited time interval. Indeed, the main complexity of this paper is that there are multiple Nussbaum functions in the MAS. And in the repeated environment, the Nussbaum function is applied repeatedly. On the basis of the related literature, we summarise the contributions as follows. (a) In comparison with the existing results on unknown control gain, the model in our work is more general. First, the unknown control gain considered in this paper can be an unknown smooth function, while some studies assume that it is a constant [30]. Second, compared with the hypothesis that the sign of the gain is identical in [31], the control input sign considered in this paper can be non-identical. (b) To our best known, this is the first paper to use the ILC algorithm to achieve the consensus of second-order MAS with completely unknown input gains. Worthy of note is that followers can obtain consensus with the leader within a limited time interval, which indicates that the control algorithms designed in this paper have better control effect. (c) In this paper, both the unknown dynamics and disturbances of the systems are considered. The non-linear dynamics discussed in this paper are more general and do not need to satisfy the

linearizable conditions as in [28]. In this work, RBFNN is used to assess the parametric uncertainties. 2. Preliminaries 2.1. Graph theory background An undirected graph is used to denote the topology connection between the agents by G = (V, E, A ), with V = {v1 , . . . , vn } denotes a finite nonempty set of agents, E ⊆ V × V denotes a set of arcs, and the relevant weighted matrix of the undirected graph A = (ai j ) ∈ Rn×n . The arc (vi , vj ) is presence when there is an undirected path between agent vi and agent vj , i.e. (vi , v j ) ∈ E. The element ai j = a ji = 1 when (vi , v j ) ∈ E, and when (vi , v j ) ∈ / E or i = j, ai j = 0. So all elements of weighted matrix A = (ai j ) ∈ Rn×n are nonnegative. L = (li j ) = H − A ∈ Rn×n represents the Laplacian matrix, where H = diag{h1 , . . . , hn } is a diagonal matrix and hi = n j=1 ai j . If there is a path between any two agents for information exchange, the topology is a connected graph. We will consider the MAS with a leader, and the topology between the followers is a connected undirected graph, denoted by G. In this work, not all followers can access the leader’s information directly, but at least one follower can receive it. We introduce a matrix B = diag{b1 , . . . , bn }, if the ith follower can directly access the leader’s state, bi = 1, otherwise, bi = 0. Lemma 1 [32]. The matrix P = L + B is symmetric positive definite, when G is connected and at least one bi > 0 exists. 2.2. Function approximation In this section, we show an approximation of the unknown dynamics of the agents. RBFNN has been opted for as the function approximator in the consensus algorithm design [33]. The approximation of the unknown non-linear dynamics of the agents can be demonstrated as

f (x ) = Φ T S (x ) + 

(1)

where Φ ∈ Rl denotes the ideal weight vector, l stands for the neurons’ number, S(x) ∈ Rl represents the NN activation function vector and  ∈ R denotes the approximation error bounded by a positive constant  ∗ , i.e., | | ≤  ∗ . In fact, it is hard to get the ideal neural network weight vector Φ . For compensating for the unknown dynamics, the approximation of the non-linear dynamics f(x) of a single agent can be indicated as

ˆ T S (x ) fˆ(x ) = Φ

(2)

ˆ is the estimation of Φ , it will be shown later how to where Φ select the estimations of the NN weight. Lemma 2 [29]. Let si (t) and V(t) are continuous scale functions over intervals [0, tf ), i = 1, . . . , n, besides, si (0 ) = 0 and V(t) ≥ 0. When the following inequality is satisfied

V (t ) ≤

n   i=1

+

t 0

gi (τ )N (si (τ ))s˙ i (τ )dτ

n   i=1

0

t

s˙ i (τ )dτ + c, ∀t ∈ [0, t f )

then, V(t), si (t) and

t i=1 0

n

(3)

gi (τ )N (si (τ ))s˙ i (τ )dτ are bounded on s2

[0, tf ), where function N (si ) is defined as N (si ) = − exp( 2i )(s2i + 2 ) sin(si ) and gi (t) is a variable that satisfies |gi (t )| ∈ [gmin , gmax ] with gmax  gmin > 0, and c is a constant.

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3

Remark 1. The Nussbaum-type function N (s ) is a function satisk fying the subsequent properties, limk→∞ sup 1k 0 N (s )ds = ∞ and  k limk→∞ inf 1k 0 N (s )ds = −∞.

Further, define the consensus error as

3. Problem statement

k (t ) = [ξ k (t ), . . . , ξ k (t )]T , we Let ekm = [ek1,m , . . . , ekn,m ]T , and ξm n,m 1,m

Consider the MAS with unknown input control gains and dynamics in the repeated environment. Suppose that the dynamics of the ith agent are represented as

x˙ ki,1 = xki,2 x˙ ki,2 = fi (xki,1 , xki,2 ) + gi (xki,1 , xki,2 )uki (t ) + dik (t )

(4)

where xki,1 ∈ R and xki,2 ∈ R are the states for agent i in the itera-

k ξi,m (t ) =

n 

ai j (xkj,m − xki,m ) + bi (x0,m − xki,m )

(9)

j=1

k (t ) = get ekm = xkm − 1n x0,m and the consensus tracking error ξm −(L + B )ekm . The result follows from Assumption 3 that eki,m (0 ) = −1 k (0 ) = ξ k−1 (T ), m = 1, 2. eki,m (T ) and ξi,m i,m To facilitate the design of the adaptive iterative learning input algorithm, we will introduce an auxiliary variable δik as the filtered error

δik (t ) = σ ξi,k1 (t ) + ξi,k2 (t ) (σ > 0 )

(10)

tion index k, uki ∈ R is the control input, fi (xki,1 , xki,2 ) denotes the

k (0 ) = ξ k−1 (T ), we can obtain δ k (0 ) = δ k−1 (T ). With the help of ξi,m i i,m i

parametric uncertainties, denotes the bounded external disturbance, and gi (xki,1 , xki,2 ) is the unknown smooth non-linear control gain with unknown non-identical sign. t ∈ [0, T], i = 1, 2, . . . , n. The overall models of the followers are represented as

δ k (t ) = σ ξ1k (t ) + ξ2k (t )

dik (t )

x˙ k1

xk2

=

x˙ k2

= f (xk ) + g(xk )uk (t ) + dk (t )

(5)

where xk1 = [xk1,1 , . . . , xkn,1 ]T , xk2 = [xk1,2 , . . . , xkn,2 ]T , uk (t ) = [uk1 (t ), . . . , ukn (t )]T , g(xk ) = diag{g1 (xk1,1 , xk1,2 ), . . . , gn (xkn,1 , xkn,2 )}, dk (t ) = [d1k (t ), . . . , dnk (t )]T , f (xk ) = [ f1 (xk1,1 , xk1,2 ), . . . , fn (xkn,1 , xkn,2 )]T . leader agent can be modeled by

The

x˙ 0,1 = x0,2 x˙ 0,2 = f0 (x0,1 , x0,2 )

(6)

where x0,1 and x0,2 are the position and velocity variable of the leader, respectively, and f0 (x0,1 , x0,2 ) is the unknown dynamic of the leader containing time-varying dynamics. The purpose is to design a input uki (t ), so that as the increasing of the iterative index k, all states xki,1 , xki,2 are consensus with the

leader, i.e., limk→∞ (xki,1 − x0,1 ) = 0 and limk→∞ (xki,2 − x0,2 ) = 0, i = 1, 2, . . . , n. Assumption 1. The leader’s dynamic f0 (x0,1 , x0,2 ) and the external disturbance di are bounded, i.e., there will be in existence unknown positive constants FM and di∗ satisfying ||f0 (x0,1 , x0,2 )|| ≤ FM   and dik (t ) ≤ di∗ , ∀t ≥ 0. Remark 2. Based on the above Assumption 1, we can know that the unknown external disturbance cannot be arbitrarily large. This assumption is very common and also necessary.

  Assumption 2. We suppose that we have gi ≤ gi (xki,1 , xki,2 ) ≤ gi , where gi and gi are unknown positive constants.

Lemma 3 [34]. The following inequality is satisfied ∀ι(t) ∈ R and ∀θ (t ) ∈ R+

ι2 (t ) |ι(t )| −  ≤ θ (t ). ι2 (t ) + θ 2 (t )

(7)

Let δ k (t ) = [δ1k (t ), . . . , δnk (t )]T , we have

(11)

Combining (4), (5) and (6) with (9), we can get the following equations

ξ˙1k (t ) = ξ2k (t ) ξ˙2k (t ) = −(L + B )(x˙ k2 − 1n x˙ 0,2 )

(12)

[1, . . . , 1]T .

where 1n = Substituting the above equations into (11), we can obtain the derivative of (11) is

δ˙ k (t ) = σ ξ˙1k (t ) + ξ˙2k (t ) = σ ξ2k (t ) − (L + B )( f (xk ) + g(xk )uk (t ) + dk (t ) − 1n f0 (x0,1 , x0,2 ))

For the purposes of achieving consensus, construct the adaptive iterative learning consensus controller as follows

uki (t ) = −N (ski )u¯ ki (t )

(14)

u¯ ki (t ) = cik (t )δik (t ) + fˆi (xki,1 , xki,2 ) + φik (t )

(15)

s˙ ki (t ) = δik (t )u¯ ki (t ),

ski (0 ) = ski −1 (T ), s0i (0 ) = 0

where we choose the Nussbaum function as

(

(ski )2 2

)((ski )2

+ 2 ) sin(ski ),

and

φik (t )

= 

δik (ηik (t ))2

an auxiliary variable defined by (10).

, δik (t ) is

Si (xki,1 , xki,2 ) is the estimation of the unknown non-linear dynamic

fi (xki,1 , xki,2 ). θik (t ) > 0 is a positive integrable function that satis∞ fies 0 θik (t ) ≤ θi∗ with a positive constant θi∗ . The time-varying control gains cik (t ), ηik (t ) and the estimate of the weight Φik are updated as following



c˙ ik (t ) = αi (δik (t ))2

(17)

cik (0 ) = cik−1 (T )

  η˙ ik (t ) = βi δik (t ) ηik (0 ) = ηik−1 (T ), ηi (0 ) = 0

(18)

4. Controller design

Φˆ k (t ) = i

(8)

= − exp

ˆ k (t ))T fˆi (xki,1 , xki,2 ) = (Φ i

where α i and β i are positive design parameters. ˆ k can be designed as The updated law of Φ i

eki,m = xki,m − x0,m , m = 1, 2

(16) N (ski )

(δik ηik )2 +(θik (t ))2

Assumption 3. In the repeated environment, the following condi−1 tions are satisfied, i.e., the initial conditions xki,m (0 ) = xki,m ( T ), ∀k ∈ Z+ are satisfied, and the leader agent satisfies x0,m (0 ) = x0,m (T ), m = 1, 2.

In order to implement the control objectives, we introduce a new error variable

(13)

⎧ ⎨0

k = −1

−γi0 (t ) ( )δ (t ) ⎩ ˆ k−1 Φi (t ) − γi Si (xki,1 , xki,2 )δik (t ) Si x0i,1 , x0i,2

0 i

k=0

(19)

k>0

where γ i > 0 is the designed positive constant and γ i0 (t) is a strongly increasing and continuous function that satisfies γi0 (0 ) = 0 and γi0 (T ) = γi .

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Lemma 4 [28]. The controller function (14)–(16) and updating laws ˆ k (0 ) = Φ ˆ k−1 (T ) and s˙ k (0 ) = s˙ k−1 (T ). (17)–(19) guarantee that Φ i i i i Proof. Let us certificate the equation as below. Since Si (xki,1 , xki,2 ) is a function consisting of the system state xki,1 and xki,2 as the variable, and the system states satisfy Assumption 3, we can get Si (xki,1 (0 ), xki,2 (0 )) = Si (xki,−1 (T ), xki,−1 (T )). According to (19), 2 1 Φˆ 0 (0 ) = Φˆ −1 (T ) = 0 is obvious. Let us assume that for l = i

By substituting the designed controller (14) into the above formula (25), we can get

1 1 k (δ (0 ))T P−1 δ k (0 ) − (δ k−1 (t ))T P−1 δ k−1 (t ) 2 2  t −σ2 (δ k (τ ))T P−1 ξ1k (τ ) dτ

E1k (t ) =

i

+σ

1, . . . , k − 1, the following equation holds

Φˆ il (0 ) = Φˆ il−1 (T ), i = 1, . . . , n

−

Substituting (19) into (20), we can get

Φˆ ik (0 ) = Φˆ ik−1 (0 ) − γi Si (xki,1 (0 ), xki,2 (0 ))δik (0 )



(21)

−

and

ski (0 )

=

Theorem 1. Suppose that Assumption 1–3 hold. By using Lemma 1– 4, the consensus tracking control problem of second-order non-linear MAS (4) with a leader (6) is solved by the adaptive iterative learning controllers (14)–(16). Moreover, the updated laws for the control gains are chosen as (17) and (18), and the updated law for neural network weight is designed as (19). It can guarantee that all followers reach consensus with the leader in a finite interval [0, T], i.e., limk→∞ (xki,1 − x0,1 ) = 0 and limk→∞ (xki,2 − x0,2 ) = 0, i = 1, 2, . . . , n, t ∈ [0, T ].

E (t ) =

(t ) +

E2k

(t ) +

E3k

(t ) +

E4k

(t ) +

E5k

(t ) +

E6k

(t )

(23)

where each part is E1k (t ) = 12 (δ k (t ))T P −1 δ k (t ), E2k (t ) = 12 (ξ1k (t ))T   ξ1k (t ), E3k (t ) = 12 0t (Φ˜ k (τ ))T  −1 Φ˜ k (τ ) dτ , E4k (t ) = 12 tT (Φ˜ k−1 (τ ))T  n 1  −1 Φ˜ k−1 (τ ) dτ , E5k (t ) = i=1 2β (ηik (t ) − η0 )2 , and E6k (t ) = i n 1 k 2 i=1 2α (ci (t ) − c0 ) .  = diag{γ1 , . . . , γn } is a symmetric posi

itive definite matrix, P = L + B is the symmetric positive maˆ k (t ) is the parameter approximation error, ˜ k (t ) = Φ k − Φ trix, Φ i i i Φ˜ k (t ) = [Φ˜ k (t ), . . . , Φ˜ k (t )]T , and the positive constants η0 and c0 n

1

will determined later. We can calculate the difference of Ek (t) is

E k (t ) = E k (t ) − E k−1 (t ) = E1k (t ) + E2k (t ) + E3k (t ) + E4k (t ) + E5k (t ) + E6k (t )

(24)

According to the auxiliary error dynamic (13), we get



E1k

1 1 (t ) = (δ k (t ))T P−1 δ k (t ) − (δ k−1 (t ))T P−1 δ k−1 (t ) 2 2 1 1 k T −1 k = (δ (0 )) P δ (0 ) − (δ k−1 (t ))T P −1 δ k−1 (t ) 2 2  t k T −1 ˙ k + (δ (τ )) P δ (τ ) dτ

+σ

0

 +

t 0

0

(δ k (τ ))T (g(xk )N (sk ) + I )u¯ k (τ ) dτ (δ k (τ ))T [ε k + dk (τ ) −1n f0 (x0,1 , x0,2 ) − φ k (τ )] dτ

where S(xk ) = blockdiag{S1 (xk1,1 , xk1,2 ), . . . , Sn (xkn,1 , xkn,2 )} is a diagonal matrix composed of the known NN function vector Si (xki,1 , xki,2 ),

and φ k (t ) = [φ1k (t ), . . . , φnk (t )]T , N (sk ) = diag{N (sk1 ), . . . , N (skn )}, I is the identity matrix, and C k (t ) = diag{c1k (t ), . . . , cnk (t )}. And note that



t

0

(δ k (τ ))T [ε k + dk (τ ) − 1n f0 (x0,1 , x0,2 ) − φ k (τ )] dτ  t n 0

−

|δik (τ )|mi dτ

i=1

 t n 0



i=1

(δik (t ))2 (δ (t )ηik (t ))2 + (θik (t ))2 k i

(ηik (t ))2 dτ

(27)

where mi = FM + εik + di∗ . Similarly, we can get

1 1 k (ξ (t ))T ξ1k (t ) − (ξ1k−1 (t ))T ξ1k−1 (t ) 2 1 2  t 1 k T k = (ξ1 (0 )) ξ1 (0 ) + (ξ1k (τ ))T ξ2k (τ ) dτ 2 0 1 − (ξ1k−1 (t ))T ξ1k−1 (t ) 2  t 1 k = (ξ1 (0 ))T ξ1k (0 ) + (ξ1k (τ ))T δ k (τ ) dτ 2 0  t 1 −σ (ξ1k (τ ))T ξ1k (τ ) dτ − (ξ1k−1 (t ))T ξ1k−1 (t ) 2 0

E2k (t ) =

E3k (t ) =

1 2

(28)

(25)



t

(Φ˜ k (τ ))T  −1 Φ˜ k (τ )  ˜ k−1 (τ ))T  −1 Φ ˜ k−1 (τ ) dτ − (Φ 

≤

T −1 k

(δ k (τ ))T (x˙ k2 − 1n x˙ 0,2 ) dτ

t

(δ k (τ ))T S(xk )(Φ˜ k (τ ))T dτ

obtained

(δ (τ )) P δ (τ ) dτ k

0

(δ k (τ ))TC k (τ )δ k (τ ) dτ

(26)

0

0 t

t

(δ k (τ ))T P−1 δ k (τ ) dτ

˜ k (t ))T  −1 Φ ˜ k (t ) − (Φ ˜ k−1 (t ))T  −1 Φ ˜ k−1 (t ) = In the light of (Φ (Φˆ k−1 (t ) − Φˆ k (t ))T  −1 [2Φ˜ k (t ) + (Φˆ k (t ) − Φˆ k−1 (t ))], it can be

1 1 = (δ k (0 ))T P −1 δ k (0 ) − (δ k−1 (t ))T P −1 δ k−1 (t ) 2 2  t −σ2 (δ k (τ ))T P−1 ξ1k (τ ) dτ 

0

+

≤

Proof. We designed the composite energy function (CEF) as

E1k

t



(22)

ˆ k−1 (T ) = Φ ˆ k (0 ). With the help Therefore, it can be concluded Φ i i k−1 k−1 k−1 k k ˆ ˆ k ( 0 ), xk ( 0 ) = of ci (0 ) = ci (T ), ηi (0 ) = ηi (T ), Φi (T ) = Φ i i,m

k

0

+

ˆ k−1 (0 ) − γi Si (xk (0 ), xk (0 ))δ k (0 ) = Φ ˆ k (0 ) =Φ i,1 i,2 i i i

m = 1, 2, the equation

t



Φˆ ik−1 (T ) = Φˆ ik−2 (T ) − γi Si (xki,−1 (T ), xki,−1 (T ))δik−1 (T ) 1 2

−1 −1 xki,m (T ), and eki,m (0 ) = eki,m (T ) ski −1 (T ) is established. 

0



(20)

0 t



0

t 0

(Φˆ k−1 (τ ) − Φˆ k (τ ))T  −1 Φ˜ k (τ ) dτ

(29)

Substituting the designed adaptive law (19) into (29), we can get

Please cite this article as: G. Li, C.-E. Ren and C.L.P. Chen et al., Adaptive iterative learning consensus control for second-order multi-agent systems with unknown control gains, Neurocomputing, https://doi.org/10.1016/j.neucom.2020.01.108

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E3k (t ) ≤

 ≤

t 0 t 0

n  1 [(ck (0 ) − c0 )2 − (cik−1 (t ) − c0 )2 ] 2αi i i=1   1 T + (Φ˜ k−1 (τ ))T  −1 Φ˜ k−1 (τ ) − Φ˜ k−2 (τ )T  −1 Φ˜ k−2 (τ ) dτ 2 t (33)

( S(xk )δ k (τ ))T  −1 Φ˜ k (τ ) dτ (δ k (τ ))T S(xk )Φ˜ k (τ ) dτ

+ (30)

In addition, due to (17) and (18), we can obtain

E5k (t ) =

n  1 [(ηik (t ) − η0 )2 − (ηik−1 (t ) − η0 )2 ] 2βi

In order to simplify the above results, we carry out the following operations.

i=1

=

n  1 [(ηik (0 ) − η0 )2 − (ηik−1 (t ) − η0 )2 ] 2βi



i=1

+ =

0

i=1

t 0

 t n 1

βi

(ξ1k (τ ))T δ k (τ ) dτ + σ

(ηik (τ ) − η0 )η˙ ik (τ ) dτ

−

n  1 [(ηik (0 ) − η0 )2 − (ηik−1 (t ) − η0 )2 ] 2βi

+

0

( η ( τ ) − η0 ) | δ ( τ ) | d τ k i

k i

(31)

=−

0

(δ k (τ ))T P−1 δ k (τ ) dτ 

c0 (δik (τ ))2 dτ − σ 2

t 0

(δ k (τ ))T P−1 ξ1k (τ ) dτ

(ξ1k (τ ))T ξ1k (τ ) dτ

(Z k (τ ))TUZ k (τ ) dτ

(34)

where Z k (t ) = [(ξ1k (t ))T , (δ k (t ))T ]T , and

⎡ ⎢

U =⎣

i=1

i=1

 t n 1 0

i=1

αi

 t n

(cik (τ ) − c0 )c˙ ik (τ ) dτ

0

n  1 [(ck (0 ) − c0 )2 − (cik−1 (t ) − c0 )2 ] 2αi i

−

i=1

 t n 0

(cik (τ ) − c0 )(δik (τ ))2 dτ

σ 2 P−1 − I

σI

2

σ 2 P−1 − I

− σ P −1

⎤

⎥ ⎦.

c0 I 2 By utilizing Lemma 3, we have

n  1 = [(ck (0 ) − c0 )2 − (cik−1 (t ) − c0 )2 ] 2αi i

+

t 0

0

t

i=1

 t



i=1

n  1 E6k (t ) = [(ck (t ) − c0 )2 − (cik−1 (t ) − c0 )2 ] 2αi i

=

0



Or similarly

+

 t n

−σ

i=1

 t n

5

(32)

i=1

Hence, the inequality about the difference of CEF can be obtained as follow

E (t )

(ηik (τ ))|δik (τ )| dτ

i=1

 t n 0



i=1

(δik (t ))2 (ηik (t ))2 (δ (t )ηik (t ))2 + (θik (t ))2 k i

d τ ≤ nθ ∗

(35)

where θ ∗ = max{θ1∗ , . . . , θn∗ }. If we choose η0 and c0 large enough to satisfy the conditions η0 > max{m1 , . . . , mn }, and c0 > 2 ( λ σ (P ) −1 )2 min + λ σ (P ) , so that matrix U is positive definite without 4σ min

a doubt. Then, it follows that

k

1 k 1 (δ (0 ))T P−1 δ k (0 ) − (δ k−1 (t ))T P−1 δ k−1 (t ) 2 2  t  t −σ2 (δ k (τ ))T P−1 ξ1k (τ ) dτ + σ (δ k (τ ))T P−1 δ k (τ ) dτ ≤

 + + − − +

0

0

t

0

ηik (τ )|δik (τ )| dτ

i=1



0

i=1

(δik (t ))2 (δik (t )ηik (t ))2 + (θik (t ))2

0



0

1 k 1 (δ (0 ))T P−1 δ k (0 ) − (δ k−1 (t ))T P−1 δ k−1 (t ) 2 2 1 − (ξ1k−1 (t ))T ξ1k−1 (t ) 2  t 1 + (ξ1k (0 ))T ξ1k (0 ) − (Z k (τ ))TUZ k (τ ) dτ + nθi∗ 2 0  t + (δ k (τ ))T (g(xk )N (sk ) + I )u¯ k (τ ) dτ n  1 + [(ηik (0 ) − η0 )2 − (ηik−1 (t ) − η0 )2 ] 2βi

i=1

c0 (δik (τ ))2 dτ

1 k (ξ (0 ))T ξ1k (0 ) + 2 1  t

−σ

 t n

n  1 + [(ck (0 ) − c0 )2 − (cik−1 (t ) − c0 )2 ] 2αi i i=1  1 T ˜ k−1 + {(Φ (τ ))T  −1 Φ˜ k−1 (τ ) 2 t ˜ k−2 (τ )} dτ ˜ k−2 (τ )T  −1 Φ −Φ

i=1

 t n 0

(mi − η0 )|δik (τ )| dτ +

i=1

 t n 0

0

(δ k (τ ))T (g(xk )N (sk ) + I )u¯ k (τ ) dτ

 t n 0

E k (t ) ≤

t

(ηik (t ))2 dτ

(ξ1k (τ ))T δ k (τ ) dτ

(ξ1k (τ ))T ξ1k (τ ) dτ

n  1 1 − (ξ1k−1 (t ))T ξ1k−1 (t ) + [(ηik (0 ) − η0 )2 − (ηik−1 (t ) − η0 )2 ] 2 2βi i=1

(36)

Furthermore, in accordance to Assumption 3 and Lemma 4, it can be easily obtained that δ k (0 ) = δ k−1 (T ). Then, along with ηik (0 ) = ηk−1 (T ), ck (0 ) = ck−1 (T ), and Φˆ k (0 ) = Φˆ k−1 (T ), let t = T in (36), i

one has

i

i

i

i

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Considering (38) and (39), we have

leader 0

E k (T ) ≤ E 0 (T ) − λmin (U ) +

kT

0

T 0

l=1

n   i=1

k  

(Z l (τ ))T Z l (τ ) dτ

(gi (xi,1 , xi,2 )N (si ) + 1 )s˙ i (τ ) dτ + nkθ ∗

(40)

From (23), we can get the differential of Ek (t)

follower4

follower1

E˙ k (t ) = (δ k (t ))T P −1 δ˙ k (t ) + (ξ1k (t ))T ξ˙1k (t )



1 (Φ˜ k (t ))T  −1 Φ˜ k (t ) − (Φ˜ k−1 (t ))T  −1 Φ˜ k−1 (t ) 2 n n   1 k 1 k + (ηi (t ) − η0 )η˙ ik (t ) + (ci (t ) − c0 )c˙ ik (t ) +

i=1

follower 3

E k ( T ) ≤ −

T 0

 +

T 0

(δ k (t ))T P−1 δ˙ k (t ) = σ (δ k (t ))T P−1 δ k (t ) − σ 2 (δ k (t ))T P−1 ξ1k (t ) − (δ k (t ))T φ k (t ) + (δ k (t ))T (ε k + dk (t ) + 1n f0 (x0,1 , x0,2 )) − (δ k (t ))TC k (t )δ k (t ) ˜ k (t ))T − (δ k (t ))T S(xk )(Φ

(Z k (τ ))TUZ k (τ ) dτ + nθi∗ (δ k (τ ))T (g(xk )N (sk ) + I )u¯ k (τ ) dτ

E k (T ) = E 0 (T ) +

k 

k   l=1

k  n  

T 0

l=1 i=1

T 0

(Z l (τ ))T Z l (τ ) dτ

0

l=1 i=1

=

( (

n   0

i=1



+

T

T

0

(gi (xli,1 , xli,2 ))N (sli ) + 1 )s˙ li (τ ) dτ +nkθ ∗ (38)



2T

T

=

i=1

0

kT

n 

(ηik (t ) − η0 )|δik (t )|

(45)

(cik (t ) − c0 )(δik (t ))2

(46)

i=1

Similar to (32), we derive n  1 i=1

αi

(cik (t ) − c0 )c˙ ik (t ) =

n  i=1

Substituting the above formulas (42)–(46) into (41), we have n 

E˙ k (t ) ≤ − (Z k (t ))TUZ k (t ) +

θik (t )

+ (δ k (t ))T (g(xk )N (sk ) + I )u¯ k (t ) ≤ (δ k (t ))T (g(xk )N (sk ) + I )u¯ k (t ) +

n 

θik (t )

(47)

According to (40) and (47), we can get the following results

( (

gi xki,1 , xki,2

)N ( ) + 1 ) ( τ ) d τ ski

s˙ ki



E k+1 (t ) = E k+1 (0 ) +

kT

(k−1 )T

(gi (xi,1 , xi,2 )N (si ) + 1 )s˙ i (τ ) dτ

(gi (xi,1 , xi,2 )N (si ) + 1 )s˙ i (τ ) dτ



t 0

E˙ k+1 (τ ) dτ

≤ E 0 (T ) − λmin (U ) +

(gi (xi,1 , xi,2 )N (si ) + 1 )s˙ i (τ ) dτ

+ ··· +

βi

(ηik (t ) − η0 )η˙ ik (t ) =

i=1

s˙ li

(gi (xi,1 , xi,2 )N (si ) + 1 )s˙ i (τ ) dτ



n  

T 0

T 0

(44)

i=1



i=1

+

)N ( ) + 1 ) ( τ ) d τ sli

(gi (x1i,1 , x1i,2 )N (s1i ) + 1 )s˙ 1i (τ ) dτ

+ ··· + =

n  1 i=1

(gi (x2i,1 , x2i,2 )N (s2i ) + 1 )s˙ 2i (τ ) dτ

n  

(43)

Similar to (31), we get

xki,m (t ), m = 1, 2. It’s easy to get that s˙ i (t ) and x˙ i,m (t ), ∀t ∈ [0, kT ] are continuous functions. Consequently,

gi xli,1 , xli,2

(ξ1k (t ))T ξ˙1k (t ) = (ξ1k (t ))T δ k (t ) − σ (ξ1k (t ))T ξ1k (t ) 1 ˜k {(Φ (t ))T  −1 Φ˜ k (t ) 2 ˜ k−1 (t ))T  −1 Φ ˜ k−1 (t )} ≤ (δ k (t ))T S(xk )Φ ˜ k (t ) − (Φ

Introduce an auxiliary function s˙ i (t + (k − 1 )T )  s˙ ki (t ) and si (t + (k − 1 )T )  ski (t ), t ∈ [0, kT ]. Similarly, we can define the function x˙ i,m (t + (k − 1 )T )  x˙ ki,m (t ) and xi,m (t + (k − 1 )T ) 

T

(42)

Similar to (28), we get

Similar to (30), we obtain

E l ( T )

≤ E 0 (T ) − λmin (U )

+ (δ k (t ))T (g(xk )N (sk ) + I )u¯ k (t )

(37)

l=1

k  n  

αi

After a series of simplifications similar to (26)–(31), the following results can be obtained.

Because it is in a repeated environment, (37) can be applied repeatedly, we can obtain

+

i=1

(41)

follower 2 Fig. 1. Communication topology.



βi

n   i=1



+ (39)

+

l=1 kT

0

 t n 0

T 0

(Z l (τ ))T Z l (τ ) dτ

(gi (xi,1 , xi,2 )N (si ) + 1 )s˙ i (τ ) dτ

θik (t ) dτ

i=1

n   i=1

k  

kT +t

kT

gi (xki,1 , xki,2 )N ((si ) + 1 )s˙ i dτ + nkθ ∗

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7

The position tracking of five robot manipulators

1.5

Position tracking of Link 1

1

0.5

0

-0.5

-1

-1.5 0

0.5

1

1.5

2

2.5

3

Time (s) Fig. 2. The angle tracking of link 1, k = 30.

The position tracking of five robot manipulators

1.5

Position tracking of Link 2

1

0.5

0

-0.5

-1

-1.5 0

0.5

1

1.5

2

2.5

3

Time (s) Fig. 3. The angle tracking of link 2, k = 30.

≤ E 0 (T ) − λmin (U ) +

n  kT +t  i=1

+

0

n   i=1

kT +t 0

k   l=1

0

T

(Z l (τ ))T Z l (τ ) dτ + nkθ ∗ + nθ ∗

gi (xi,1 , xi,2 )N (si )s˙ i (τ ) dτ

n  kT +t (gi (xi,1 , xi,2 )N (si ) + E k+1 (t ) and si (t) are bounded, i.e., i=1 0 1 )s˙ i (τ ) dτ ≤ N2 , N1 and N2 are limited positive constants. Furthermore, we obtain that

λmin (U )

k   0

l=1

s˙ i (τ ) dτ

(48)

Obviously, nθ ∗ + nkθ ∗ is bounded, i.e., nkθ ∗ + nθ ∗ ≤ N1 . Based n  kT +t on Lemma 2, we obtain gi (xi,1 , xi,2 )N (si )s˙ i (τ ) dτ , i=1 0

T

(Z l (τ ))T Z l (τ ) dτ

≤ E 0 (T ) − E k+1 (t ) + nkθ ∗ + nθ ∗ n  kT +t  + (gi (xi,1 , xi,2 )N (si ) + 1 )s˙ i (τ ) dτ i=1

0

≤ E (T ) − E k+1 (t ) + N1 + N2 0

(49)

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The angle tracking of five robot manipulators

4

3

Angle tracking of Link 1

2

1

0

-1

-2

-3

-4 0

0.5

1

1.5

2

2.5

3

Time (s) Fig. 4. The angular velocity tracking of link 1, k = 30.

The angle tracking of five robot manipulators

4

3

Angle tracking of Link 2

2

1

0

-1

-2

-3

-4 0

0.5

1

1.5

2

2.5

3

Time (s) Fig. 5. The angular velocity tracking of link 2, k = 30.

In summary, since the positiveness of E k+1 (t ), we conclude that if E0 (T) is bounded for any iteration k,   λmin (U ) kl=1 0T (Z l (τ ))T Z l (τ ) dτ is also limited. Therefore, we will provide the detailed proof of the limitation of E0 (T). According to the results of (47), we get the differential of E˙ 0 (t )

E (t ) ≤ (δ (t )) (g(x (t ))N (s ) + I )u¯ (t ) + ˙0

0

T

0

0

0

n 

θ (t ) 0 i

≤

i=1

(gi (x0i,1 , x0i,2 )N (s0i ) + 1 )s˙ 0i (t ) +

n  i=1

θi0 (t )

E 0 (t ) ≤

n   i=1

+

0

(50)

t

gi (x0i,1 , x0i,2 )N (s0i )s˙ 0i (τ ) dτ

n   i=1

i=1 n 

Integrating two sides of (50) gives

0

t

s˙ 0i (τ ) dτ + E 0 (0 ) + nθ ∗

Because E0 (0) and nθ ∗ conclusion that E0 (t), are bounded can be of E0 (t) means that

(51)

are bounded. On the basis of Lemma 2, the n  t 0 0 0 0 s0i (t ) and i=1 0 gi (xi,1 , xi,2 )N (si )s˙ i (τ ) d τ obtained. Furthermore, the boundedness E0 (T) is limited. In the light of (49),

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9

The angle error of the robot manipulators

0.3

Angle error of Link 1

0.25

0.2

0.15

0.1

0.05

0 0

5

10

15

20

25

30

Iteration number Fig. 6. Consensus error for the angle of link 1.

The angle error of the robot manipulators

0.14

0.12

Angle error of Link 2

0.1

0.08

0.06

0.04

0.02

0 0

5

10

15

20

25

30

Iteration number Fig. 7. Consensus error for the angle of link 2.

with the help of the boundedness of E 0 (t ) + N1 + N2 and the positiveness of E k+1 (t ), it is easy to acquire that the series   λmin (U ) kl=1 0T (Z l (τ ))T Z l (τ ) dτ are convergent. What’s more, T it is quite apparent that limk→∞ 0 (ξ1k (t ))T ξ1k (t ) dt = 0 and T k lim (δ (t ))T δ k (t ) dt = 0. It is easy to discover that ξ˙ k (t ) and k→∞ 0

1

1

1

δ˙ k (t ) are uniformly bounded on [0, T]. By using the Barbalar-like Lemma [35], we get that limk→∞ ξ1k (t ) = 0 and limk→∞ δ1k (t ) = 0. Then, we can obtain that limk→∞ ξ2k (t ) = 0. Obviously, because P is non-singular matrix, we can obtain that limk→∞ ek1 (t ) = 0 and limk→∞ ek2 (t ) = 0. Hence, it is obviously that each follower can

keep consensus with the leader as the iteration index k tend to infinity.  5. Simulation To illustrate the usefulness of the conclusions in this paper, the manipulator is used as an example model for simulation, more details to see [36]. The MAS are composed of a leader and four followers. The dynamic equation of 2-link manipulator ia considered as

Ji (qki )q¨ ki + Gi (qki ) = Bi uki − ωik

(52)

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The angular velocity error of the robot manipulators

0.6

Angular velocity error of Link 1

0.5

0.4

0.3

0.2

0.1

0 0

5

10

15

20

25

30

Iteration number Fig. 8. Consensus error for the angular velocity of link 1.

The angular velocity error of the robot manipulators

0.7

Angular velocity error of Link 2

0.6

0.5

0.4

0.3

0.2

0.1

0 0

5

10

15

20

25

30

Iteration number Fig. 9. Consensus error for the angular velocity of link 2.



where

Ji (· )=

Ji11

Ji12

Ji21

Ji22





is the inertia matrix,

G i ( · )=

ci11

ci12

ci21

ci22

 q˙ ki +

i = 1, . . . , 4, is the matrix including gravitational torque, centripetal and Coriolis term, uki denotes the input of joint torques, Bi is the unknown gains of the input and Bi = bi [(cos qki1 + li1 /li2 )2 , (cos qki2 + li1 /li2 )2 ]T , ωik is the external disturbance, qki = [qki1 , qki2 ]T . We assume that di11 = (mi1 + mi2 )li21 + mi2 li22 + 2mi2 li1 li2 cos qki2 . di12 = di21 = mi2 (li22 + li1 li2 cos qki2 ), di22 = mi2 li22 , ci11 = −mi2 li1 li2 q˙ ki2 sin qki2 , ci12 = −mi2 li1 li2 (q˙ ki1 + q˙ ki2 ) sin qki2 , ci21 = mi2 li1 li2 q˙ ki1 sin qki2 , ci22 = 0. gi1 = [gi1 ,gi2 ]T ,

g(mi1 + mi2 )li1 sin qki1 + gmi2 li2 sin(qki1 + qki2 ), gi2 = gmi2 li2 sin(qki1 + qki2 ), ωik (t ) = (0.5i )[0.3 sin(t ), 0.1(1 − e−t )]T , parameters are set as b1 = b3 = 0.4 and b2 = b4 = −0.4, the gravity acceleration g = 9.8m/s2 , the link lengths of agent are set as li1 = 1.5m, li2 = 1m, the masses are set as mi1 = 1kg and mi2 = 2kg. The ideal angle of the leader manipulator is set as q0 (t ) = [sin 3t , cos 3t ]T . The Nussbaum function is designed in Lemma 2. The state initials of the manipulators are chosen as q0i (0 ) = [0, 1]T . We set the iteration number as k = 30. Select γ10 = 0.3t, γ20 = 0.6t, γ30 = 0.3t, γ40 = 0.6t; for T = 3, γ1 = 0.9, γ2 = 1.8, γ3 = 0.9,

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11

10

5

0 0

5

10

15

20

25

30

0

5

10

15

20

25

30

0

5

10

15

20

25

30

0.6 0.4 0.2 0 0.15 0.1 0.05 0

Iteration number Fig. 10. The responses of parameters.

γ4 = 1.8; α1 = 1, α2 = 0.5, α3 = 0.5, α4 = 0.5; β1 = 1, β2 = 1, β3 = 0.2, β4 = 0.2. The initial values of the parameters are Φi (0 ) = [0.5; 0.5; 1; 1]T ; s1 (0 ) = 0.2, s2 (0 ) = 0.3, s3 (0 ) = −0.1, s4 (0 ) = 0.4; c1 (0 ) = 0.3, c2 (0 ) = 0.1, c3 (0 ) = 0.2, c2 (0 ) = 0.15; ηi (0 ) = 0; θik (t ) = e−0.05t , for i = 1, . . . , 4. The basis function of the RBFNN are Gaussian functions, as set in [37]. The topology is described as Fig. 1. By observing Figs. 2–5, we can see from Fig. 2 that the angles of link 1 of all agents keep consensus at the 30th iteration, and Fig. 3 shows that the angles of link 2 of all agents can achieve agreement at the 30th iteration. Figs. 4 and 5 show that the angular velocity of link 1 and link 2 of all agents reach consensus within a finite time interval [0,3] at the 30th iteration. We can get a conclusion that in the thirtieth iteration, not only the angle of the two-joint manipulator can reach consensus with the state of the leader, but also the angular velocity of the manipulator can be agreed with the leader. By observing the images, we obtain that Figs. 6 and 7 show that the angular errors of link 1 and link 2 between the follower agents and the leader tend to zero, respectively. As shown in Figs. 8 and 9, the angular velocity errors of link 1 and link 2 between the followers and the leader tend to zero. We obtain from Figs. 6 to 9 that the change of the tracking error of the manipulator is evident in 30 iterations. With the growth of the number of iterations k, the tracking error becomes smaller and smaller. As k approaches infinity, the tracking error of the system eventually tends to zero. Fig. 10 shows the trajectories of all adaptive update parameters, from which we can observe that all parameter variables are bounded. 6. Conclusion A novel consensus control algorithm is proposed for MAS with unknown control gains by using ILC in this work. The control gains are unknown and bounded smooth functions, and the unknown signs of different agents can be non-identical. Besides, we firstly use the ILC algorithms to achieve consensus control for secondorder MAS with unknown input gains. The considered model is more general so that we can use the proposed algorithms in more

widespread areas. The designed algorithm can guarantee the followers reach consensus within a finite time interval. The efficacy of the proposed algorithms is illustrated by simulation results. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement Guilu Li: Software, Validation, Formal analysis, Writing - original draft, Investigation, Methodology. Chang-E Ren: Conceptualization, Visualization, Resources, Writing - review & editing. C.L. Philip Chen: Supervision. Zhiping Shi: Supervision. References [1] F.L. Lewis, H. Zhang, K. Hengster-Movric, A. Das, Cooperative Control of Multi-agent Systems: Optimal and Adaptive Design Approaches, Springer, London, 2014. [2] H. Du, G. Wen, G. Chen, J. Cao, F.E. Alsaadi, A distributed finite-time consensus algorithm for higher-order leaderless and leader-following multiagent systems, IEEE Trans. Syst., Man, Cybern.: Syst. 47 (7) (2017) 1625–1634. [3] R. Sakthivel, R. Sakthivel, B. Kaviarasan, H. Lee, Y. Lim, Finite-time leaderless consensus of uncertain multi-agent systems against time-varying actuator faults, Neurocomputing 325 (2019) 159–171. [4] C.-E. Ren, C.L.P. Chen, Sliding mode leader-following consensus controllers for second-order non-linear multi-agent systems, IET Control Theory Appl. 9 (10) (2015) 1544–1552. [5] C.-E. Ren, L. Chen, C.P. Chen, Adaptive fuzzy leader-following consensus control for stochastic multiagent systems with heterogeneous nonlinear dynamics, IEEE Trans. Fuzzy Syst. 25 (1) (2017) 181–190. [6] J. Qin, W. Fu, Y. Shi, H. Gao, Y. Kang, Leader-following practical cluster synchronization for networks of generic linear systems: an event-based approach, IEEE Trans. Neural Netw. Learn. Syst. 30 (1) (2019) 215–224. [7] C.-E. Ren, L. Chen, C.L.P. Chen, T. Du, Quantized consensus control for second-order multi-agent systems with nonlinear dynamics, Neurocomputing 175 (A) (2016) 529–537. [8] Z. Liu, G. Lai, Y. Zhang, C.L.P. Chen, Adaptive neural output feedback control of output-constrained nonlinear systems with unknown output nonlinearity, IEEE Trans. Neural Netw. Learn. Syst. 26 (8) (2015) 1789–1802. [9] X. Wang, S. Li, J. Lam, Distributed active anti-disturbance output consensus algorithms for higher-order multi-agent systems with mismatched disturbances, Automatica 74 (2016) 30–37.

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[10] Z. Ding, Consensus disturbance rejection with disturbance observers, IEEE Trans. Ind. Electron. 62 (9) (2015) 5829–5837. [11] Y. Li, K. Sun, S. Tong, Observer-based adaptive fuzzy fault-tolerant optimal control for SISO nonlinear systems, IEEE Trans. Cybern. 49 (2) (2019) 649–661. [12] C.P. Chen, G. Wen, Y. Liu, F. Wang, Adaptive consensus control for a class of nonlinear multiagent time-delay systems using neural networks, IEEE Trans. Neural Netw. Learn. Syst. 25 (6) (2014) 1217–1226. [13] Y. Liu, S. Tong, D. Wang, T.-S. Li, C.P. Chen, Adaptive neural output feedback controller design with reduced-order observer for a class of uncertain nonlinear SISO systems, IEEE Trans. Neural Netw. 22 (8) (2011) 1328–1334. [14] J. Qin, G. Zhang, W.X. Zheng, Y. Kang, Neural network-based adaptive consensus control for a class of nonaffine nonlinear multiagent systems with actuator faults, IEEE Trans. Neural Netw. Learn. Syst. 30 (12) (2019) 3633–3644. [15] D. Meng, Y. Jia, Iterative learning approaches to design finite-time consensus protocols for multi-agent systems, Syst. Control Lett. 61 (1) (2012) 187–194. [16] Y. Li, K. Li, S. Tong, Adaptive neural network finite-time control for multi-input and multi-output nonlinear systems with positive powers of odd rational numbers, IEEE Trans. Neural Netw. Learn. Syst. (2019) 1–12. [17] X. Bu, L. Cui, Z. Hou, W. Qian, Formation control for a class of nonlinear multiagent systems using model-free adaptive iterative learning, Int. J. Robust Nonlinear Control 28 (4) (2018) 1402–1412. [18] X. Jin, Adaptive iterative learning control for high-order nonlinear multi-agent systems consensus tracking, Syst. Control Lett. 89 (2016) 16–23. [19] Q. Fu, L. Du, G. Xu, J. Wu, Consensus control for multi-agent systems with distributed parameter models via iterative learning algorithm, J. Frankl. Inst. 355 (10) (2018) 4453–4472. [20] X. Bu, P. Zhu, Z. Hou, J. Liang, Finite-time consensus for linear multi-agent systems using data-driven terminal ILC, IEEE Trans. Circuits Syst. II Express Briefs (2019) 1. [21] J. Li, J. Li, Adaptive iterative learning control for coordination of second-order multi-agent systems, Int. J. Robust Nonlinear Control 24 (18) (2014) 3282–3299. [22] X. Bu, Q. Yu, Z. Hou, W. Qian, Model free adaptive iterative learning consensus tracking control for a class of nonlinear multiagent systems, IEEE Trans. Syst., Man, Cybern.: Syst. 49 (4) (2019) 677–686. [23] D. Meng, Y. Jia, J. Du, Consensus seeking via iterative learning for multi-agent systems with switching topologies and communication time-delays, Int. J. Robust Nonlinear Control 26 (17) (2016) 3772–3790. [24] C.-E. Ren, C.L.P. Chen, T. Du, Y. Guan, Fuzzy adaptive leader-Following consensus control for nonlinear multi-Agent systems with unknown control directions, Int. J. Fuzzy Syst. 21 (7) (2019) 2066–2076. [25] J. Xu, R. Yan, Iterative learning control design without a priori knowledge of the control direction, Automatica 40 (10) (2004) 1803–1809. [26] W. Chen, X. Li, W. Ren, C. Wen, Adaptive consensus of multi-agent systems with unknown identical control directions based on a novel nussbaum-type function, IEEE Trans. Autom. Control 59 (7) (2014) 1887–1892. [27] Z. Ding, Adaptive consensus output regulation of a class of nonlinear systems with unknown high-frequency gain, Automatica 51 (2015) 348–355. [28] J. Li, S. Liu, J. Li, Adaptive iterative learning protocol design for nonlinear multi-agent systems with unknown control direction, J. Frankl. Inst. 355 (10) (2018) 4298–4314. [29] C. Chen, C. Wen, Z. Liu, K. Xie, Y. Zhang, C.P. Chen, Adaptive consensus of nonlinear multi-agent systems with non-identical partially unknown control directions and bounded modelling errors, IEEE Trans. Autom. Control 62 (9) (2017) 4654–4659. [30] D. Wang, J. Huang, Adaptive neural network control for a class of uncertain nonlinear systems in pure-feedback form, Automatica 38 (8) (2002) 1365–1372. [31] G. Wang, C. Wang, Y. Shen, Distributed adaptive leader-following tracking control of networked lagrangian systems with unknown control directions under undirected/directed graphs, Int. J. Control 92 (12) (2018) 2886–2898. [32] W. Ren, R.W. Beard, Distributed Consensus in Multi-Vehicle Cooperative Control, Springer, London, 2008. [33] S.S. Ge, C. Wang, Adaptive neural control of uncertain mimo nonlinear systems, IEEE Trans. Neural Netw. 15 (3) (2004) 674–692. [34] W. Dong, On consensus algorithms of multiple uncertain mechanical systems with a reference trajectory, Automatica 47 (9) (2011) 2023–2028. [35] M. Sun, A barbalat-like lemma with its application to learning control, IEEE Trans. Autom. Control 54 (9) (2009) 2222–2225. [36] J. Liu, Intelligent Control Design and MATLAB Simulation, Springer, Singapore, 2018. [37] Z. Hou, L. Cheng, M. Tan, Decentralized robust adaptive control for the multiagent system consensus problem using neural networks, IEEE Trans. Syst., Man, Cybern., Part B 39 (3) (2009) 636–647.

Guilu Li received the B.S. degree in automation from Qufu Normal University, Rizhao, in 2017. She is currently a postgraduate in the College of Information Engineering, Capital Normal University, Beijing, China. Her research interests include nonlinear control and consensus control of multi-agent systems.

Chang-E Ren received the Ph.D. degree from University of Macau, Macau, China. She is currently a lecture in Capital Normal University. Her research interests include adaptive fuzzy control, nonlinear control, and consensus control of multi-agent systems.

C.L. Philip Chen (S’88-M’88-SM’94-F’07) is the Chair Professor and Dean of the College of Computer Science and Engineering, South China University of Technology. Being a Program Evaluator of the Accreditation Board of Engineering and Technology Education (ABET) in the U.S., for computer engineering, electrical engineering, and software engineering programs, he successfully architects the University of Macau’s Engineering and Computer Science programs receiving accreditations from Washington/Seoul Accord through Hong Kong Institute of Engineers (HKIE), of which is considered as his utmost contribution in engineering/computer science education for Macau as the former Dean of the Faculty of Science and Technology. He is a Fellow of IEEE, AAAS, IAPR, CAA, and HKIE; a member of Academia Europaea (AE), European Academy of Sciences and Arts (EASA), and International Academy of Systems and Cybernetics Science (IASCYS). He received IEEE Norbert Wiener Award in 2018 for his contribution in systems and cybernetics, and machine learnings. He is also a 2018 highly cited researcher in Computer Science by Clarivate Analytics. His current research interests include systems, cybernetics, and computational intelligence. Dr. Chen was a recipient of the 2016 Outstanding Electrical and Computer Engineers Award from his alma mater, Purdue University, after he graduated from the University of Michigan at Ann Arbor, Ann Arbor, MI, USA in 1985. He was the IEEE Systems, Man, and Cybernetics Society President from 2012 to 2013, and currently, he is the Editor-in-Chief of the IEEE Transactions on Systems, Man, and Cybernetics: Systems, and an Associate Editor of the IEEE Transactions on Fuzzy Systems, and IEEE Transactions on Cybernetics. He was the Chair of TC 9.1 Economic and Business Systems of International Federation of Automatic Control from 2015 to 2017 and currently is a Vice President of Chinese Association of Automation (CAA).

Zhiping Shi is currently a professor in the College of Information Engineering at the Capital Normal University, Beijing, China. From 2005 to 2010, he was on the faculty at the Institute of Computing Technology, Chinese Academy of Sciences where he received his Ph.D. degree in Computer Software and Theory in 2005. His research interests include formal verification and visual information analysis. He is the (co-)author of more than 100 research papers.

Please cite this article as: G. Li, C.-E. Ren and C.L.P. Chen et al., Adaptive iterative learning consensus control for second-order multi-agent systems with unknown control gains, Neurocomputing, https://doi.org/10.1016/j.neucom.2020.01.108