Two-layer distributed formation-containment control of multiple Euler–Lagrange systems with unknown control directions

Two-layer distributed formation-containment control of multiple Euler-Lagrange systems with unknown control directions Communicated by Dr. Ma Lifeng Ma

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Two-layer distributed formation-containment control of multiple Euler-Lagrange systems with unknown control directions Luyan Xu, Chaoli Wang, Xuan Cai, Yujing Xu, Chonglin Jing PII: DOI: Reference:

S0925-2312(20)30077-1 https://doi.org/10.1016/j.neucom.2020.01.033 NEUCOM 21789

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Neurocomputing

Received date: Revised date: Accepted date:

8 October 2019 18 December 2019 6 January 2020

Please cite this article as: Luyan Xu, Chaoli Wang, Xuan Cai, Yujing Xu, Chonglin Jing, Two-layer distributed formation-containment control of multiple Euler-Lagrange systems with unknown control directions, Neurocomputing (2020), doi: https://doi.org/10.1016/j.neucom.2020.01.033

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Two-layer distributed formation-containment control of multiple Euler-Lagrange systems with unknown control directions ✩ Luyan Xua , Chaoli Wanga,∗, Xuan Caia , Yujing Xua , Chonglin Jinga a Department

of Control Science and Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China

Abstract This paper addresses the distributed formation containment control problem of Euler-Lagrangian systems(ELSs) with unknown control directions by two layers, namely, time-varying formation for the leaders and containment formation control for the followers. We first estimate the desired states and velocities for the leaders and the followers in finite time. Second, novel adaptive neutral networks are used to compensate the model uncertainties. Third, using the Nussbaum gain technique, the proposed control laws can guarantee the ultimate convergence of error signals to zero and that all of the signals in the closed-loop system are bounded. Finally, simulation examples are given to demonstrate the effectiveness of the proposed method. Keywords: distributed formation containment control, Nussbaum gain technique, neutral networks , , ✩ This paper was partially supported by The National Natural Science Foundation (61374040, 61673277, 61503262). ∗ Corresponding author Email address: [email protected] (Chaoli Wang)

Preprint submitted to Journal of LATEX Templates

January 13, 2020

1. Introduction In recent decades, distributed consensus control problem of multi-agent systems (MASs) has attracted the attention of numerous researchers in part due to 5

its great theoretical significance and extensive practical applications such as in unmanned air vehicles, multiple underactuated surface vessels [1], mobile robots [2–4], and sensor networks [5, 6]. Distributed control means that the control law of every agent can merely use the information exchanged between some agents called the neighbors of the agent, and this has been an interesting and chal-

10

lenging topic in the field of consensus control problems. The existing literature on distributed consensus control in the field of MASs has mostly focused on two fundamental problems [7, 8]. The first is the synchronization problem in which the agents eventually reach identical states by using appropriate control schemes, which is also called the leaderless tracking problem. The other is

15

named the leader-following problem in which there exist one leader and several followers, where the goal is to ensure that all of the followers track the position of the leader (see [9] and the references therein). The containment control problem – a more challenging problem with multiple leaders that is considered to be an extension of the leader-following problem

20

– is very different from the case with a single leader. The control objective of containment control requires all of the followers to reach the convex hull spanned by the leaders under the control laws. In [10], distributed attitude containment control problems of stationary and dynamic leaders are handled by using the neighbors information in finite time. In [11], using a neuroadaptive

25

control scheme, the problem of containment control with unknown non-affine dynamics and mismatched uncertainties was solved. Additionally, distributed containment control of ELSs was also studied due to the wide range of use of ELSs in practical applications (see [12–16] and [17–19]); for example, in [19], continuous distributed protocols were designed for ELSs to address the coor-

30

dinated tracking consensus issues under directed graphs, for which the control laws can avoid the undesirable chattering effect.

2

However, all of the results mentioned above were obtained for the case where control directions of the systems were known exactly ahead of time. In fact, very often, ELSs with unknown control directions occur in engineering. Nonlinear 35

systems with unknown high-frequency gains ([20]), adaptive control robotic systems ([21]), and multi-manipulator systems ([22]) are examples of such ELSs. Recently, a series of papers was published about the unknown control directions of systems (see [23–25]). The leader-following consensus problems of ELSs with unknown control directions were successfully addressed in [23] by employing

40

the Nussbaum function proposed in [25]. In [21], a novel Nussbaum gain was used to achieve adaptive control of robotic systems. Unfortunately, the studies regarding unknown control directions mentioned above only focused on the leader-following systems with a single leader, and the proposed control schemes cannot be directly applied to containment control, which remains an open ques-

45

tion. Therefore, Cheng first studied the containment control problem with unknown control directions in [26], for which the systems included one input and one output. To the best of our knowledge, the work described in this paper is the first to address the distributed formation containment control problems of multiple

50

ELSs with unknown control directions. Most previous works on containment control focused on the case where the state and velocity for every leader is known in advance (see [27] and the references therein), which means that we only need to design appropriate control laws for the followers. The goal of this paper is to not only require the followers to reach the convex hull spanned by

55

the leaders but also have information exchange between the leaders to form a certain formation and eventually maintain identical velocity (see [28]), which is more interesting. Compared with the existing results, the main contributions of this paper are as follows. 1) Using the control algorithms designed here, all of the error signals of

60

the agents can reach zero eventually, not only for the positions but also for the velocities. By contrast, in almost all of the reported studies of non-linear systems, such as [23, 24, 28], only the state errors can be controlled to reach 3

zero, while the velocities errors cannot be ensured to reach zero. While in this paper, by utilizing our control protocol, the velocities errors can also be ensured 65

to reach zero. 2) A new deformation of neural networks is proposed in this paper to compensate the model uncertainties of ELSs. It is well known that the conventional adaptation method for neural networks needs design adaptive laws to estimate the idealized weight matrix function, incurring a high computational cost (see

70

[28]). In this paper, we only need to estimate the bound of the idealized weight matrix, greatly simplifying the computations. 3) This work is the first to address the containment problem of ELSs with unknown control directions. First, although Cheng was the first to study the containment control problem with unknown control directions in [26] for systems

75

including one input and one output, our work includes multi-input and multioutput. Second, in [26], the trajectories of the leaders are indicated by a class of heterogeneous linear functions, while we extend the formation of the leaders to any time-varying circumstances in this paper, which is more rational and practical. The method proposed in [26] can not solve this problem.

80

The following notation will be used throughout this paper: k · k denotes the Euclidean norm of a vector; | · | is the absolute value of a real number; σ ¯ (·) and σ(·) denote the maximum and minimum singular value of a matrix, respectively; and A > 0 denotes that matrix A is positive definite.

2. Problem statement and preliminaries 85

2.1. Graph theory Consider a group of (N + M ) agents; the communication network topology among the N + M agents can be represented by a directed graph G = {V, E, A},

where V = {1, 2, ..., N + M } is the set of nodes, E ∈ V 2 represents the edge set, and A = [aij ] is the weighted adjacency matrix of graph G. An edge (j, i) ∈ E 90

means that node i can acquire information from node j, and node j is then called a neighbor of node i. We use Ni = {j ∈ V|(j, i) ∈ E} to represent the 4

set of all neighbors of agent i. In addition, a bidirectional graph is defined such that (i, j) ∈ E implies (j, i) ∈ E. A path from node i1 to node in means a sequence of ordered adjacent edges in the form of (ip , ip+1 ), p = 1, . . . , m − 1. 95

The weighted adjacent matrix A = [aij ] ∈ R(N +M )×(N +M ) of a directed graph G is defined as aij > 0 if (j, i) ∈ E and aij = 0 otherwise. A directed graph is

strongly connected if there exists a directed path from every node to every other node. A graph contains a spanning tree if there exists a root node that has a

100

path to every other node. The Laplacian matrix L = [lij ] ∈ R(N +M )×(N +M ) of P a directed graph G is defined as lii = j∈Ni aij and lij = −aij , i 6= j. 2.2. Neural networks Neural networks that are linear in the parameters, such as the radial basis function neural network (RBFNN), have been widely used in approximating some unknown nonlinearity over a compact set due to their simple and linear

105

parameter structure. An unknown and continuous nonlinear function f (Z) : Rl → Rm1 can be approximated by an RBFNN over a compact set ΩZ ∈ Rl as f (Z) = W T D(Z) + s(Z),

(1)

where Z ∈ ΩZ ⊂ Rl is the input vector, and W ∈ Rp1 ×m1 is the optimal weight matrix of the RBFNN, i.e., n o ˆ T D(Z)k . W = argminW∈R supZ∈ΩZ kf(Z) − W p1 ×m1 ˆ

(2)

Here, p1 is the number of neurons, s(Z) ∈ Rm1 is the approximation error 110

with the property that ks(Z)k ≤ sm ∀Z ∈ ΩZ , with m1 ∈ R≥0 , and D(Z) = [D1 (Z), . . . , Dp1 (Z)]T is a known smooth basis function vector with Dk (k = 1, . . . , p1 ), chosen as the commonly used Gaussian functions that have the form Dk (Z) = exp[−(Z − ιk )T (Z − ιk )/νk2 ],

(3)

where ιk = [ιk,1 , . . . , ιk,l ]T is the center of the receptive field and νk is the width of the Gaussian function. According to the universal approximation re5

115

sults in [29], if p1 is chosen to be sufficiently large, then W T D(Z) can approximate any continuous function of Z to any desired accuracy over a compact set. 2.3. Problem formulation In this paper, we consider the MASs of ELSs including N leaders, denoted by

120

1, 2, . . . , N , and M followers, labeled by N + 1, N + 2, . . . , N + M . The dynamic functions can be described as Mi (qi )¨ qi + Ci (qi , q˙i )q˙i + Gi (qi ) = τi + di ,

(4)

τi = Ki ui , where qi , q˙i , q¨i , di ∈ Rn denote the generalized position, velocity, and acceleration vectors and the environment disturbance of the ith agent, kdi (t)k ≤ d¯i , d¯i

is a positive constant, Mi (qi ) ∈ Rn×n is the inertial matrix, Ci (qi , q˙i ) ∈ Rn×n

125

is the Coriolis and centripetal matrix, Gi (qi ) ∈ Rn is the vector of gravitational force, τi ∈ Rn is the torque input vector, ui ∈ Rn is the voltage input vector provided to the amplifier, and Ki = diag{ki,1 , . . . , ki,n } ∈ Rn×n

with ki,j 6= 0, i, j = 1, . . . , N + M , is a constant diagonal transmission matrix that relates the voltage input ui to the torque input τi , i ∈ L ∪ F , with 130

L = {1, . . . , N } and F = {N + 1, . . . , N + M } representing the set of followers and leaders, respectively. In this study, Mi , Ci , Gi and Ki are all assumed to be unknown. Furthermore, assume that the sign of the control direction of the ith agent labeled by Ki,p is identical, i.e. sgn(Ki,d ) = sgn(Kj,p ) for 1 ≤ i 6= j ≤ N + M , d, p, = 1, . . . , n.

135

For the MASs (4) , we denote the communication topology by a directed graph G1+N +M = {V1+N +M , E1+N +M , A1+N +M }, with N and M representing the follower nodes and the leader nodes, respectively. Here, A = [aij ] ∈

R(1+N +M )×(1+N +M ) and L = [lij ] ∈ R(1+N +M )×(1+N +M ) denote the adjacency

and Laplacian matrices of graph G1+N +M , respectively. The communication 140

topology among all of the agents is bidirectional. Here, to form a desired formation, we introduce a virtual agent named agent 0 particular; ai0 = 1 if the

6

leader i knows the desired position and ai0 = 0 otherwise. L can be represented as 

0

01×N

  L =  LL2  0M ×1

LL1 LF 2



01×M

  01×M  ,  LF 1

(5)

where LL1 ∈ RN ×N , LL2 ∈ RN ×1 , LF 1 ∈ RM ×M , and LF 2 ∈ RM ×N . Other145

wise, we define 

  GL =  

a00 .. .

··· .. .

a0N .. .

aN 0

···

aN N



  . 

(6)

Assumption 1. GL is connected and there exists at least one leader that has a directed path to each leader. Assumption 2. The desired position qdi (t) and the desired velocity vdi (t) of the leaders are bounded, i ∈ L. 150

Property 1. The matrix (M˙ i (qi ) − 2Ci (qi , q˙i )) is skew-symmetric. Property 2. There exist unknown positive constants mi and mi such that Mi (qi ) satisfies the following inequalities for all i ∈ L ∪ F : mi kηk2 ≤ η T Mi (qi )η ≤ mi kηk2 , ∀η ∈ Rn . Property 3. There exist unknown positive constants M , C , and G such that Mi (qi ), Ci (qi , q˙i ) and Gi (qi ) satisfy the following inequalities for all i ∈ L ∪ F : kMi (qi )k ≤ M , kCi (qi , q˙i )k ≤ C kq˙i k, kGi (qi )k ≤ G .

155

Problem 1. Design distributed control laws for each leader belonging to L under Assumptions 1-2 with unknown control directions to form a desired timevarying formation, i.e., lim qi (t) = qdi (t), lim vi (t) = vdi (t), i ∈ L.

t→∞

t→∞

7

Problem 2. Design distributed containment control laws for each follower belonging to F under Assumptions 1-2 with unknown control directions to reach 160

the convex hull spanned by the leaders, i.e., −1 lim qi (t) = −L−1 F 1 LF 2 QL , lim vi (t) = −LF 1 LF 2 VL , i ∈ F,

t→∞

t→∞

T T T T where QL = [q1T , . . . , qN ] and VL = [q˙1T , . . . , q˙N ] .

Then, during the control design, we need the following lemmas. Lemma 1. [30] If x satisfies x˙ = f (x), f (0) = 0, x ∈ Rn and there exists a

positive definite continuous function V (x) : U → R such that V˙ (x) + cV (x)α ≤

165

0, where c ∈ R+ , α ∈ (0, 1) , x ∈ U0 , and U0 ∈ U is an open set excluding the origin 0, then V (x) is stable in finite time T , T (x0 ) ≤

Lemma 2. [25] Let M(ξi ) = exp(

ξi2 2 2 )(ξi

V 1−α (x(0)) c(1−α) .

+ 2) sin(ξi ) be a Nussbaum function,

where ξi denotes Bsmooth functions on [0, tf ) with ξi (0) being bounded, and V (t) ≥ 0. If ω is a suitable constant, χi are some constants with identical signs 170

that are not equal to 0, i, = 1, . . . , N + M , and the inequality V (t) ≤ ω + holds, then V (t), [0, tf ).

PN R t

N Z X i=1

0

t

(χi M(ξi (σ)) + 1)ξ˙i (σ)dσ

i=1 0 (χi M(ξi (σ))

(7)

+ 1)ξ˙i (σ)dσ and ξi (t) are bounded on

−1 Lemma 3. [10] Under Assumption 1, each entry of −L−1 L1 LL2 and −LF 1 LF 2

−1 is non-negative, and each row sum of −L−1 L1 LL2 and −LF 1 LF 2 is equal to 1, 175

LL1 ∈ RN ×N , LL2 ∈ RN ×1 , LF 1 ∈ RM ×M , and LF 2 ∈ RM ×N .

Lemma 4. [31] For any z(t) ∈ R, (t) ∈ R+ , the following inequality can be established: 0 ≤ |z(t)| − p

z 2 (t) z 2 (t) + 2 (t)

8

≤ (t).

(8)

3. Controller design 3.1. First-Layer – Distributed Consensus Formation Control of Leaders In this section, the distributed consensus formation control of leaders is considered, which means that the leaders need to form the desired time-varying 180

formation and eventually maintain the desired velocity under control algorithms. However, it is particularly difficult that only some leaders know the desired position and velocity. Therefore, in the first layer, we have two steps. First, distributed finite-time observers are given to trace the desired positions and velocities. Second, the distributed consensus formation control algorithms are

185

designed to eventually form the desired time-varying formation with desired velocity. Let qˆi = [ˆ qi1 , qˆi2 , . . . , qˆin ]T and vˆi = [ˆ vi1 , vˆi2 , . . . , vˆin ]T denote the estimates of the desired position qdi = [qdi1 , qdi2 , . . . , qdin ]T and the desired velocity vdi = [vdi1 , vdi2 , . . . , vdin ]T , respectively, i ∈ L. The distributed observers are inspired

190

by [28] as follows: qˆ˙i vˆ˙ i

PN = vˆi − β1 sgn j=0 aij (ˆ qi − qˆj − δij (t)), PN = −β2 sgn j=0 aij (ˆ vi − vˆj − δ˙ij (t)), i ∈ L,

(9)

where q0 and v0 denote the position and velocity of virtual agent 0, respectively, δij (t) = δi (t) − δj (t) and δi (t) define the relative positions of agent i and virtual agent 0, β1 and β2 are positive constants satisfying β2 > ($1 + $2 ) and β1 > ($3 + $4 ), vˆ0 = v0 , and qˆ0 = q0 . Here, $1 , $2 , $3 , and $4 are the bounds of 195

q¨0 , δ¨i , q˙0 , and δ˙i , i ∈ L ∪ F . Furthermore, we note that

T T ∆L (t) = [δ1T , δ2T , . . . , δN ] T T QL (t) = [q1T , q2T , . . . , qN ]

Q0 (t) = [q0T , q0T , . . . , q0T ]T . Theorem 1. Using the observers noted by (9), each leader can estimate the desired position and velocity individually in finite time T2 , i.e., the equalities 9

can be satisfied as follows when t > T2 : qˆi (t) =

q0 (t) + δi (t),

vˆi (t) =

q˙0 (t) + δ˙i (t), i ∈ L,

√ √ 1 2 λ (LL1 ) where T2 = T1 + (β1 −$3 −$max V2 2 (T1 ) and T1 = 4 )λmin (LL1 ) 200

(10) √

√

1 2 λmax (LL1 ) 2 (β2 −$1 −$2 )λmin (LL1 ) V1 (0).

Proof. Let v˜i (t) = vˆi (t) − q˙0 (t) − δ˙i (t) and q˜i (t) = qˆi − q0 (t) − δi (t), and note

T T T T that v˜L = [˜ v1T , v˜2T , . . . , v˜N ] and q˜L = [˜ q1T , q˜2T , . . . , q˜N ] . Define the following

Lyapunov candidate:

V1 (t) =

1 T v˜ (L1 ⊗ In )T v˜L . 2 L

(11)

Substituting (9) into the derivative of (11) yields V˙ 1 (t)

=

T v˜L (L1 ⊗ In )T v˜˙ L

=

T ¨0 − ∆ ¨ L )] v˜L (L1 ⊗ In )T [−β2 sgn((LL1 ⊗ In )˜ vL ) − Q

=

¨ ((L1 ⊗ In )˜ vL )T [−β2 sgn((LL1 ⊗ In )˜ vL ) + (L−1 L1 LL2 ⊗ In )Q0  ¨ +(L−1 L1 LL2 ⊗ In )∆L .

Then, since kqk2 ≤ kqk1 for any q ∈ Rn , we conclude that V˙ 1

  ≤ − β2 − ($1 + $2 )kL−1 vL k 1 L1 LL2 k k(L1 ⊗ In )˜ ≤

−(β2 − $1 − $2 )k(L1 ⊗ In )˜ vL k2 q T (L 2˜ ≤ −(β2 − $1 − $2 ) v˜L L1 ⊗ In ) v L

≤

205

−(β2 − $1 − $2 )λmin (LL1 )k˜ v L k2 √ 1 λmin (LL1 ) ≤ − 2(β2 − $1 − $2 ) p V12 . λmax (LL1 )

Thus, it is concluded from Lemma 1 that vˆi (t) → (q˙0 (t)+ δ˙i (t)), i = 1, . . . , N , in finite time:

T1 =

√ p 1 2 λmax (LL1 ) V1 2 (0). (β2 − $1 − $2 )λmin (LL1 ) 10

(12)

T Then, consider the Lyapunov function V2 = 21 q˜L (LL1 ⊗ In )T q˜L when t ≥ T1 ,

yielding

V˙ 2

T = q˜L (LL1 ⊗ In )T q˜˙L T ˙ i )] = q˜L (LL1 ⊗ In )T [−β1 sgn((LL1 ⊗ In )˜ qL ) − Q˙ 0 − ∆ T ˙ ˙ = q˜L (LL1 ⊗ In )T [−β1 sgn((LL1 ⊗ In )˜ qL ) + (L−1 L1 LL2 ⊗ In )(Q0 + ∆i )]

≤ −(β1 − $3 − $4 )k(LL1 ⊗ In )˜ vL k1 ≤ −(β1 − $3 − $4 )k(LL1 ⊗ In )˜ vL k2 q T (L 2˜ ≤ −(β1 − $3 − $4 ) q˜L L1 ⊗ In ) q L

≤ ≤

−(β1 − $3 − $4 )λmin (LL1 )k˜ qL k2 √ 1 λmin (LL1 ) V12 . − 2(β1 − $3 − $4 ) p λmax (LL1 )

Similarly, it is concluded from Lemma 1 that qˆi (t) → (q0 (t)+δi (t)), i = 1, . . . , N , 210

in finite time: √ p 1 2 λmax (LL1 ) V2 2 (T1 ). T2 = T1 + (β1 − $3 − $4 )λmin (LL1 )

(13)

Based on the above analysis, the observers (9) of the leaders have kept the desired position and the time-varying desired velocity when t > T2 , and we can then use qˆi and vˆi to replace the time-varying desired positions and the time-varying desired velocities of the leaders when t > T2 . 215

For convenience of the analysis below, we define the generalized tracking errors when i ∈ L as follows: z1i

= qi − qˆi ,

z2i

= q˙i − α1i ,

(14)

where z1i = [z1i1 , z1i2 , . . . , z1in ]T , z2i = [z2i1 , z2i2 , . . . , z2in ]T , α1i = vˆi (t) − a1i z1i , named the virtual controller, and a1i > 0 is the control gain. Differentiating (14), we have

11

z2i + α1i − vˆi ,

z˙1i

=

z˙2i

= Mi−1 (qi )[Ki ui + di − Ci (qi , q˙i )q˙i − Gi (qi )] − α˙ 1i .

(15)

220

The distributed controllers for the systems (4) of the leaders are designed as ui = M(ξi )(z1i + a2i z2i + pˆi Φi ), i ∈ L,

(16)

where M(ξi ) is the Nussbaum-type gain function given by lemma 2, pˆ˙i =

T γi z2i Φi , Φi =

z2i kDi k2 kz2i k2 kDi k2 +σ 2

+

z2i kz2i k2 +σ 2 ,

¯ a2i > 0 is pi = max{w ¯i , s¯i + d},

the control gain, d¯ is the boundary of the unknown environmental distur225

bances, w ¯i and s¯i are positive constants, Di = [Di1 (Z), . . . , Dil (Z)]T , Dil (Z) = T T T , α˙ 1i ] , ιi = [ιi1 , . . . , ιil ]T , and exp[−(Z − ιil )T (Z − ιil )/νil2 ], Zi = [qiT , q˙iT , α1i

νi = [νi1 , . . . , νil ]T , i ∈ L.

Remark 1. Traditional neural networks is in vector form, which means the dimension of dynamic equation is very high. While in this paper, one param230

eter scalar neural networks are used, which greatly simplifies the dimension of dynamic equation.

Theorem 2. Consider the Euler-Lagrange systems (4) under Assumptions 12 with unknown control directions. The leaders form the desired time-varying formation and eventually have the desired velocities if the controllers ui (t), i ∈ 235

L, are considered as (16). Proof. Consider the following Lyapunov function candidate: VL (t) =

N N N X 1X T 1X T 1 2 z1i z1i + z2i Mi z2i + p˜i , 2 i=1 2 i=1 2γ i i=1

where p˜i = pi − pˆi are the parameter estimation errors and γi ∈ R+ are positive constant gains, i ∈ L.

12

The derivative of V (t) along (15) satisfies V˙ L (t)

=

=

PN

+

240

P

N T T ˆi ) + i=1 z2i (Ki ui + di − Ci (qi , q˙i )q˙i i=1 z1i (z2i + α1i − v PN P N T T 1 ˙ − i=1 p˜i z2i Φi + i=1 z2i ( 2 Mi z2i − Mi (qi )α˙ 1i ) PN T 1 ˙ PN T ( 2 Mi z2i − Ci (qi , q˙i )z2i ) Φi + i=1 z2i − i=1 p˜i z2i PN T + i=1 z2i (−Ci (qi , q˙i )α1i − Gi (qi ) − Mi (qi )α˙ 1i )

PN

T i=1 (−a1i z1i z1i

− Gi (qi )) (17)

T T + z1i z2i + z2i Ki ui + di ).

Due to the uncertainties of the system parameters, we use the neural networks defined in (1) on a compact set Ωn , where the existence of the set can be confined as (2), which means that Mi (qi )α˙ 1i + Ci (qi , q˙i )α1i + Gi (qi ) = T T T T WiT Di (qiT , q˙iT , α1i , α˙ 1i ) + si (qiT , q˙iT , α1i , α˙ 1i ), Wi ∈ Rl×n , l > 1, ksi (Zi )k ≤ s¯i ,

and Z ∈ Ω4n ; then, (17) can be indicated as V˙ L (t)

245

PN

PN T T i=1 a1i z1i z1i + i=1 z2i (Ki ui PN T PN T + i=1 z1i z2i − i=1 p˜i z2i Φi .

= −

− WiT Di − si + di )

(18)

By using Lemma 4 and noting that kWi k ≤ w¯i , we obtain PN

T i=1 z2i di

PN

i=1

PN

≤

i=1 PN i=1

≤

T −z2i WiT Di (Zi ) ≤

≤ PN

i=1

T −z2i si

Then, we have

where Φ(t) =

T i=1 z2i di PN ≤ i=1

PN

i=1

T d¯i z2i z2i kz2i k2 +2 (t)

d¯i kz2i k PN ¯ + i=1 di (t), i=1

¯i kz2i kkDi k i=1 w T 2 PN w¯i z2i √ z2i2 kD2i k i=1 kz2i k + (t) i=1 PN i=1

≤

PN

√

PN

PN

PN

≤

kz2i kkdi k ≤

kz2i kksi k ≤ √

T s¯i z2i z2i

kz2i

√ z2i kD2 i k

2

kz2i k +(t)2

PN

i=1

√

13

i=1

w ¯i (t),

¯i kz2i k i=1 s PN + i=1 s¯i (t). PN

+

w ¯i kz2i kkDi k

PN

k2 +2 (t)

PN

i=1

PN

+

T T i=1 z2i Wi Di (Zi ) − PN T pi z2i Φ(t) + i=1 (d¯i + w ¯i

−

PN

≤

T i=1 z2i si

(19)

(20)

(21)

(22)

+ s¯i )(t),

z2i kz2i k2 +(t)2

and pi = max{w ¯i , s¯i +d¯i }.

Substituting 22 into the derivative of Vi , we obtain V˙ L (t)

=

PN T PN T T i=1 z2i Ki ui i=1 z1i z2i + i=1 a1i z1i z1i + PN ¯ PN T Φi , + i=1 (di + w¯i + s¯i )(t) − i=1 pˆi z2i

−

PN

(23)

and then, by considering that ξ˙i = diag(z2i,1 , . . . , z2i,n )(z1i + a2i z2i + pˆi Φi ), 250

we further obtain that

V˙ L (t) ≤

PN Pn T i=1 a1i z1i z1i + i=1 q=1 (Ki,q M(ξi,q ) PN PN T ¯i + s¯i + d¯i )(t) − i=1 a2i z2i z2i . + i=1 (w

−

PN

+ 1)ξ˙i,q

(24)

Thus, the following inequality can be further obtained by integrating both sides of (24): PN R t PN R t T T ≤ VL (t) + i=1 0 a1i z1i z1i dt + i=1 0 a2i z2i z2i dt (25) PN Pn R t ≤  + i=1 q=1 0 (ki,q M(ξi,q (ν)) + 1)ξ˙i,q (ν)dν, PN R t where  = VL (0) + i=1 0 (w ¯i + s¯i + d¯i )(t)dt. PN Pn R t By lemma 2, we conclude that VL (t), ξi,q (t), i=1 q=1 0 (ki,q M(ξi,q (ν))+ 0

255

1)ξ˙i,q (ν)dν ∈ L∞ for ∀t ∈ [0, tf ), i ∈ L, q = 1, . . . , n. Noting that based on

T Mi (qi )z2i , we conclude that z1i (t), z2i (t), p˜i ∈ L∞ . property 2, mi kz2i (t)k2 ≤ z2i

From assumption 2, qdi , vdi ∈ L∞ , and by the definition of z1i (t), z2i (t), and ui , we further have qi (t), vi (t), z˙1i (t), ui ∈ L∞ ∀t ∈ [0, tf ), i = 1, . . . , N . Since ui

is bounded according to Property 3, the boundedness of q¨i (t) can be obtained, 260

and accordingly, z˙2i ∈ L∞ ∀t ∈ [0, tf ), i = 1, . . . , N . Therefore, we conclude that the solution of the systems remains bounded under the control scheme of (16) without a finite-time escape phenomenon, so when time approaches ∞, the solution can be extended to infinity by Theorem 3.3 in [32]. Furthermore, (25) clearly shows z1 (t) z2 (t) ∈ L2 ∀t ∈ [0, ∞), as z1i (t), z˙1i (t), z2i (t), z˙2i (t) ∈ L∞

265

∀t ∈ [0, ∞); it can be obtained than limt→∞ z1i = 0 and lim limt→∞ z2i = 0 as a result of Barbalat’s lemma in [33].

14

3.2. Second-Layer – Distributed Containment Control of the Followers In this section, two steps are given to compel the followers to eventually reach the convex hull spanned by the leaders. First, distributed finite-time observers 270

similar to (9) are proposed to trace the desired position and velocity for each follower. Second, the distributed consensus formation control algorithms are designed to achieve containment. In this section, F = {N + 1, . . . , N + M }.

Let qˆi = [ˆ qi1 , qˆi2 , . . . , qˆin ]T and vˆi = [ˆ vi1 , vˆi2 , . . . , vˆin ]T , i ∈ F , denote the

estimates of the desired position qdi = [qdi1 , qdi2 , . . . , qdin ]T and desired velocity 275

vdi = [vdi1 , vdi2 , . . . , vdin ]T . Inspired by [28], the distributed observer is given as qˆ˙i vˆ˙ i

PN +M = vˆi − β3 sgn j=1 aij (ˆ qi − qˆj ), PN +M = −β4 sgn j=1 aij (ˆ vi − vˆj ), i ∈ F,

(26)

where β3 and β4 are positive constants satisfying β3 > ($1 + $2 ) and β4 > ($3 + $4 ). Here, $1 , $2 , $3 , and $4 are defined as before. Define = [cTN +1 , cTN +2 , . . . , cTN +M ]T ,

CF

(27)

= −(L−1 F 1 LF 2 ⊗ In )QL .

Theorem 3. Using the observers noted by (26), each follower can acquire their 280

desired position and velocity in finite time T4 ; in other words, when t > T4 , the following equalities can be satisfied for i ∈ F : qˆi (t)

√

vˆi (t) √

2

λ

=

ci ,

(28)

= c˙i , i ∈ F, (L

)

1

F1 where T4 = T3 + (β3 −$3 −$max V4 2 (T3 ), T3 = 4 )λmin (LF 1 )

√

√

1 2 λmax (LF 1 ) 2 (β4 −$1 −$2 )λmin (LL1 ) V3 (0),

V3 = 12 v˜FT (LF 1 ⊗ In )T v˜FT , V4 = 21 q˜FT (LF 1 ⊗ In )T v˜FT , v˜i = vˆi − c˙i , q˜i = qˆi − ci , and ci is defined in 28, i ∈ F . 285

The proof of Theorem 3 is similar to that of Theorem 1. For convenience of the following illustration, we define the tracking errors for i ∈ F as z1i

= qi − qˆi ,

z2i

= q˙i − α1i , 15

(29)

where z1i = [z1i1 , z1i2 , . . . , z1in ]T , z2i = [z2i1 , z2i2 , . . . , z2in ]T , α1i = vˆi (t) − a3i z1i , named the virtual controller, and a3i > 0 is the control gain. Differenti290

ating (29), we obtain

z˙1i z˙2i

= z2i + α1i − vˆi ,

= Mi−1 (qi )[Ki ui + di − Ci (qi , q˙i )q˙i − Gi (qi )] − α˙ 1i .

(30)

The distributed controllers for the system (4) are designed as ui = M(ξi )(z1i + a4i z2i + pˆi Φi ), i ∈ F,

(31)

T where M(ξi ) is the Nussbaum-type gain function given by (2), pˆ˙i = γi z2i Φi ,

Φ = 295

z2i kDi k2 kz2 ik2 kDi k2 +σ 2

+

z2i kz2ik2 +σ2

, pi = max{w ¯i , s¯i + d¯i }, a4i > 0 is the control

gain, w ¯i and s¯i are positive constants, Di = [Di1 (Z), . . . , Dil (Z)]T , Di (Z) = T T T exp[−(Z − ιi )T (Z − ιi )/νi2 ], Zi = [qiT , q˙iT , α1i , α˙ 1i ] , ιi = [ιi1 , . . . , ιil ]T , and

νi = [νi1 , . . . , νil ]T , i ∈ F .

Theorem 4. Consider the systems (4) under Assumptions 1-2 with unknown control directions; if the distributed controllers ui (t) are given by (31), then the 300

followers can eventually reach the convex hull spanned by the leaders, which means that di → ci and vi → c˙i , i ∈ F . The candidate Lyapunov function can be defined as

Vf (t) =

PN

1 T i=1 2 z1i z1i

+

PN

1 T i=1 2 z2i Mi z2i

+

PN

1 2 ˜i , i=1 2γi p

(32)

where p˜i = pi − pˆi are the parameter estimation errors and γi ∈ R are positive constant gains, i ∈ F . 305

To compensate for the uncertainties of the unknown system parameters, we utilize the RBFNN defined in 1 on a compact set Ωzf ; the existence of the set can be confined as in 2. Then, Mi (qi )α˙ 1i + Ci (qi , q˙i )α1i + Gi (qi ) = T T WiT Di (qiT , q˙iT , α1i , α˙ 1i ) + si , Wi ∈ Rl×n , l > 1, ksi (Zi )k ≤ s¯i , and Zi =

16

T T T [qiT , q˙iT , α1i , α˙ 1i ] , Z ∈ Ωzf . Similar to Theorem 2, the derivative of V (t) along 310

(30) satisfies V˙ f (t) ≤

PN Pn T i=1 t1i z1i z1i + i=1 q=1 (ki,q M(ξi,q ) PN PN T + i=1 (w ¯i + s¯i + d¯i )(t) − i=1 a4i z2i z2i ,

−

PN

+ 1)ξ˙i,q

(33)

where ξi = ξ˙i = diag(z2i,1 , . . . , z2i,n )(z1i + a4i z2i + pˆi Φi ). Integrating both sides of 33, we have 0

PN R t PN R t T T ≤ Vf (t) + i=1 0 t1i z1i z1i dt + i=1 0 t2i z2i z2i dt PN Pn R t ≤ σ + i=1 q=1 0 (ki,q Mξi,q (ν) + 1)ξ˙i,q (ν)dν,

(34)

Rt ¯i q=1 0 (w

PN Pn

+ s¯i + d¯i )(t)dt. PN Pn R t Then, by utilizing 2, we obtain that Vf (t), ξi,q (t), i=1 q=1 0 (ki,q Mξi,q (ν)+

where σ = V (0) +

i=1

T Mi (qi )z2i , 1)ξ˙i,q (ν)dν ∈ L∞ for ∀t ∈ [0, tf ), i ∈ F . Because mi kz2i (t)k2 ≤ z2i

we obtain z1i (t), z2i (t), p˜i ∈ L∞ . From the analysis of Theorem 2, ci , c˙i ∈ L∞ , 315

and thus, by the definition of z1i (t), z1i (t), and ui , we further obtain qi (t), vi (t), z˙1i (t), ui ∈ L∞ ∀t ∈ [0, tf ), i ∈ F . Since ui is bounded, the boundedness of q¨i (t) can be obtained, and accordingly, z˙2i ∈ L∞ ∀t ∈ [0, tf ), i ∈ F . Therefore, we conclude that the solution of the systems remains bounded under the control scheme of (31) and that finite-time escape will never arise, so the solution can

320

be extended to infinity utilizing Theorem 3.3 in [32] or another time as time approaches ∞. Furthermore, (34) also indicates that z1 (t), z2 (t) ∈ L2 when t ∈ [0, ∞); based on Barbalat’s lemma as in the proof of Theorem 2, we conclude that limt→∞ z1i = 0 and limt→∞ z2i = 0 ∀t ∈ [0, ∞). This finishes the proof.

325

4. Illustrative example In this section, a sample simulation has been presented to verify the effectiveness of the proposed method. The leader-following system of this example consists of eight satellites, where they are four leaders labeled 1, 2, 3, and 4 and four followers labeled 5, 6, 7, and 8. The communication topology of the eight

330

agents is shown in Fig. 1. 17

0

1

2

3

4

5

6

7

8

Fig. 1. Network topology among 8 satellites.

Here we consider a team of eight networked agents with the dynamics of the ith agent, for i = 1, 2, 3, 4, given by µ(R0 +xi ) − Rµ3 Ri3 0 µyi i − = 2ω x˙ i − ω 2 xi + µy 3 3 Ri Ri τiz i z¨i + µz = K iz mi , Ri3

x ¨i + 2ω y˙ i − ω 2 xi + y¨i −

= Kix τmixi , τ

Kiy miyi ,

(35)

where mi is the weight of the ith satellite, µ is the Earth gravitational con-

335

stant, R0 is the orbit radius, Ri is the distance between the geocenter and the p ith satellite, and ω = µ/R03 is the orbital angular velocity. τi = [τix , τiy , τiz ]

is the control input. Ki = [Kix , Kiy , Kiz ] represents the unknown control directions of the ith agent. Here, let qi = [xi , yi , zi ] is the relative coordinate of the ith satellite in the local-vertical-local-horizontal rotating frame, Mi = mi I3 , mi = 35kg, 

0

  Ci = mi  2ω  0 

  Gi = mi  

−2ω 0 0

0



  0 ,  0

µ(R0 +xi ) − Rµ3 Ri3 0 µyi i −ω 2 xi + µy − 3 3 Ri Ri µzi Ri3

−ω 2 xi +

18

(36)



  . 

(37)

Table 1 Parameters of the robots.

Parameters

Values

Parameters

Values

mi

35 kg

β1

1.5

β2

1.5

β3

1.5

Ki

I3

a1l

1

a2l

1

a1f

1

a2f

1

Rl

1

R0

1 km

w

30◦

δ1

[1; 0; 0]T km

δ2

[−1; 0; 0]T km

δ3

[0; 1; 0]T km

δ4

[0; −1; 0]T km

Rf

1

q0

[cos(t); sin(t); sin(t)]T km

q10

[0.3; −0.2; 0.8]T km

q20

[−0.2; 0.1; 0.5]T km

q40

[−0.2; −0.2; 0.6]T km

q30 q50

340

[−0.1; 0.2; −0.8]T km

[1.15; 0.15; −0.5]T km

q60

[0.3; −0.4; −0.6]T km

q70

[−0.1; 0.4; 0]T km

q80

[0.2; 0; 0.6]T km

q˙10

[0; 0; 0]T km/s

q˙20

[0; 0; 0]T km/s

q˙30

[0; 0; 0]T km/s

q˙40

[0; 0; 0]T km/s

q˙50

[0.7; 0; 0]T km/s

q˙60

[0.3; 0; 0]T km/s

q˙70

[−0.2; 0.4; 0]T km/s

q˙80

[0.5; 0; 0]T km/s

Then, it can be easily verified that the system 35 is a form of ELSs as in 2.3. Furthermore, the other parameters for simulation are given in Table 1. In the first layer, suppose that the group of the leaders is constructed by the four satellites labeled 1, 2, 3, and 4, for which the initial states and desired circles are given in Table 1 . Figs. 2, 3 and 4 present the trajectory errors and

345

Figs. 5, 6 and 7 present the velocity errors of the four satellites, which show that the four leaders are able to achieve the desired trajectories with the desired velocities. In the second layer, the followers are indicated by the four satellites labeled

19

1 leader1 leader2 leader3 leader4

0.5

0

-0.5

-1

-1.5

0

1

2

3

4

5

6

7

8

Fig. 2. Trajectory errors ei1 , i=1,. . . ,4.

0.5 leader1 leader2 leader3 leader4

0

-0.5

-1

-1.5

-2

0

1

2

3

4

5

6

7

8

9

Fig. 3. Trajectory errors ei2 , i=1,. . . ,4.

1 leader1 leader2 leader3 leader4

0.5

0

-0.5

-1

-1.5

0

1

2

3

4

5

6

7

8

Fig. 4. Trajectory errors ei3 , i=1,. . . ,4.

20

9

9

8 leader1 leader2 leader3 leader4

7 6 5 4 3 2 1 0 -1 -2

0

1

2

3

4

5

6

7

8

9

Fig. 5. Velocity errors ci1 , i=1,. . . ,4.

4 leader1 leader2 leader3 leader4

3

2

1

0

-1

-2

-3

0

1

2

3

4

5

6

7

8

9

Fig. 6. Velocity errors ci2 , i=1,. . . ,4.

2 leader1 leader2 leader3 leader4

1

0

-1

-2

-3

-4

-5

0

1

2

3

4

5

6

7

8

Fig. 7. Velocity errors ci3 , i=1,. . . ,4.

21

9

2.5 Follower1 Follower2 Follower3 Follower4

2

1.5

1

0.5

0

-0.5

0

1

2

3

4

5

6

7

8

9

Fig. 8. Trajectory errors ei1 , i=5,. . . ,8.

1.4 Follower1 Follower2 Follower3 Follower4

1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6

0

1

2

3

4

5

6

7

8

9

Fig. 9. Trajectory errors ei2 , i=5,. . . ,8.

5, 6, 7, and 8, for which the initial states are defined in TABLE I. Figs. 8, 9 and 350

10 present the trajectory errors and Figs. 11, 12 and 13 present the velocity errors of the four satellites, which show that the four followers reach the convex hull spanned by the leaders. Furthermore, Fig. 14 show the trajectories of the eight agents in the space plane, on which all the trajectories of the leaders are depicted by green and all the trajectories of the followers are depicted by red.

355

It can be observed that the followers achieved convergence. The above simulation results demonstrate that the control algorithms for the ELSs with unknown control directions given in this paper are effective and

22

0.4 Follower1 Follower2 Follower3 Follower4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

0

1

2

3

4

5

6

7

8

9

Fig. 10. Trajectory errors ei3 , i=5,. . . ,8.

8 Follower1 Follower2 Follower3 Follower4

6

4

2

0

-2

-4

0

1

2

3

4

5

6

7

8

9

Fig. 11. Velocity errors ci1 , i=5,. . . ,8.

7 Follower1 Follower2 Follower3 Follower4

6 5 4 3 2 1 0 -1 -2 -3

0

1

2

3

4

5

6

7

8

Fig. 12. Velocity errors ci2 , i=5,. . . ,8.

23

9

2 Follower1 Follower2 Follower3 Follower4

1

0

-1

-2

-3

-4

0

1

2

3

4

5

6

7

8

9

Fig. 13. Velocity errors ci3 , i=5,. . . ,8.

2

Follower1 Follower2 Follower3 Follower14 leader1 leader2 leader3 leader4

1 0

3

-1

2 1 0 -1 -2 -3 -2

-1

0

2

1

3

Fig. 14. trajectories of the leaders and followers.

24

practical.

5. Conclusion 360

This paper is the first work to address the containment problem of ELSs with unknown control directions. Two steps are given to develop the control laws based on the use of Nussbaum gain techniques and neural networks to enforce that all of the leaders form a desired time-varying formation with the desired velocities and ultimately drive all of the followers into the convex hull

365

spanned by the leaders. In addition, simulations are performed to demonstrate the effectiveness of the proposed control laws. Future research efforts will be devoted to discuss time-delay ELSs systems to further solve practical problems in engineering.

References 370

References [1] W. Xie, B. Ma, T. Fernando, H. H.-C. Iu, A new formation control of multiple underactuated surface vessels, International Journal of Control 91 (5) (2018) 1011–1022. doi:10.1080/00207179.2017.1303849. [2] B. Ning, Q. Han, Prescribed finite-time consensus tracking for multi-agent

375

systems with Nonholonomic chained-form dynamics, IEEE Transactions on Automatic Control (2018) 1–1doi:10.1109/TAC.2018.2852605. [3] D. Panagou, D. M. Stipanovi, P. G. Voulgaris, Distributed coordination control for multi-robot networks using Lyapunov-like barrier functions, IEEE Transactions on Automatic Control 61 (3) (2016) 617–632.

380

doi:10.1109/TAC.2015.2444131. [4] W. Dong, J. A. Farrell, Cooperative control of multiple Nonholonomic mobile agents, IEEE Transactions on Automatic Control 53 (6) (2008) 1434– 1448. doi:10.1109/TAC.2008.925852. 25

[5] Y. Luo, Z. Wang, G. Wei, F. E. Alsaadi, Nonfragile l2 − l∞ fault es385

timation for markovian jump 2-d systems with specified power bounds, IEEE Transactions on Systems, Man, and Cybernetics: Systems (2018) 1–12doi:10.1109/TSMC.2018.2794414. [6] Y. Luo, Z. Wang, J. Liang, G. Wei, F. E. Alsaadi, h∞ control for 2-d fuzzy systems with interval time-varying delays and missing measurements, IEEE

390

Transactions on Cybernetics 47 (2) (2017) 365–377. doi:10.1109/TCYB. 2016.2514846. [7] L. Ma, Z. Wang, Q.-L. Han, Y. Liu, Consensus control of stochastic multiagent systems: a survey, Science China Information Sciences 60 (12) (2017) 120201. doi:10.1007/s11432-017-9169-4.

395

[8] L. Ma, Z. Wang, H. Lam, Mean-square h∞ consensus control for a class of nonlinear time-varying stochastic multiagent systems: The finite-horizon case, IEEE Transactions on Systems, Man, and Cybernetics: Systems 47 (7) (2017) 1050–1060. doi:10.1109/TSMC.2016.2531657. [9] A. Jadbabaie, J. Lin, A. S. Morse, Coordination of groups of mobile au-

400

tonomous agents using nearest neighbor rules, IEEE Transactions on Automatic Control 48 (6) (2003) 988–1001. doi:10.1109/TAC.2003.812781. [10] Z. Meng, W. Ren, Z. You, Distributed finite-time containment control for multiple Lagrangian systems, in: Proceedings of the 2010 American Control Conference, 2010, pp. 2885–2890. doi:10.1109/ACC.2010.5531499.

405

[11] Y. Wang, Y. Song, Fraction dynamic-surface-based neuroadaptive finitetime containment control of multiagent systems in nonaffine pure-feedback form, IEEE Transactions on Neural Networks and Learning Systems 28 (3) (2017) 678–689. doi:10.1109/TNNLS.2015.2511005. [12] M. Lu, L. Liu, Leader-following consensus of multiple uncertain Euler-

410

Lagrange systems subject to communication delays and switching net-

26

works, IEEE Transactions on Automatic Control 63 (8) (2018) 2604–2611. doi:10.1109/TAC.2017.2771318. [13] Q. Yang, H. Fang, J. Chen, Z. Jiang, M. Cao, Distributed global outputfeedback control for a class of EulerLagrange systems, IEEE Transactions 415

on Automatic Control 62 (9) (2017) 4855–4861. doi:10.1109/TAC.2017. 2696705. [14] H. Cai, J. Huang, The leader-following consensus for multiple uncertain Euler-Lagrange systems with an adaptive distributed observer, IEEE Transactions on Automatic Control 61 (10) (2016) 3152–3157.

420

doi:

10.1109/TAC.2015.2504728. [15] C. Ma, P. Shi, X. Zhao, Q. Zeng, Consensus of EulerLagrange systems networked by sampled-data information with probabilistic time delays, IEEE Transactions on Cybernetics 45 (6) (2015) 1126–1133. doi: 10.1109/TCYB.2014.2345735.

425

[16] H. Cai, J. Huang, Leader-following consensus of multiple uncertain EulerLagrange systems under switching network topology, International Journal of General Systems 43 (3-4) (2014) 294–304. doi:10.1080/03081079. 2014.883714. [17] J. R. Klotz, T.-H. Cheng, W. E. Dixon, Robust containment control in a

430

leaderfollower network of uncertain EulerLagrange systems, International Journal of Robust and Nonlinear Control 26 (17) (2016) 3791–3805. doi: 10.1002/rnc.3535. [18] L. Chen, C. Li, J. Mei, G. Ma, Adaptive cooperative formation-containment control for networked EulerLagrange systems without using relative veloc-

435

ity information, IET Control Theory Applications 11 (9) (2017) 1450–1458. doi:10.1049/iet-cta.2016.1185. [19] B. Cheng, Z. Li, L. Shen, Distributed containment control of EulerLagrange systems over directed graphs via distributed continuous controllers, IET 27

Control Theory Applications 11 (11) (2017) 1786–1795. doi:10.1049/ 440

iet-cta.2016.0699. [20] Z. Ding, Global adaptive output feedback stabilization of nonlinear systems of any relative degree with unknown high-frequency gains, IEEE Transactions on Automatic Control 43 (10) (1998) 1442–1446.

doi:

10.1109/9.720504. 445

[21] C. Chen, Z. Liu, Y. Zhang, C. L. P. Chen, Adaptive control of robotic systems with unknown actuator nonlinearities and control directions, Nonlinear Dynamics 81 (3) (2015) 1289–1300. doi:10.1007/ s11071-015-2068-3. [22] L. Cheng, Z.-G. Hou, M. Tan, Adaptive neural network tracking control

450

for manipulators with uncertain kinematics, dynamics and actuator model, Automatica 45 (10) (2009) 2312 – 2318. doi:https://doi.org/10.1016/ j.automatica.2009.06.007. [23] G. Wang, C. Wang, Y. Shen, Distributed adaptive leader-following tracking control of networked Lagrangian systems with unknown control directions

455

under undirected/directed graphs, International Journal of Control (2018) 1–13doi:10.1080/00207179.2018.1465204. [24] X. Cai, C. Wang, G. Wang, D. Liang, Distributed consensus control for second-order nonlinear multi-agent systems with unknown control directions and position constraints, Neurocomputing 306 (2018) 61 – 67.

460

doi:https://doi.org/10.1016/j.neucom.2018.03.063. [25] C. Chen, C. Wen, Z. Liu, K. Xie, Y. Zhang, C. L. P. Chen, Adaptive consensus of nonlinear multi-agent systems with non-identical partially unknown control directions and bounded modelling errors, IEEE Transactions on Automatic Control 62 (9) (2017) 4654–4659. doi:10.1109/TAC.2016.

465

2628204.

28

[26] H. Cheng, Y. Dong, Distributed containment control for a class of uncertain nonlinear multi-agent systems with unknown control direction, in: 2017 36th Chinese Control Conference (CCC), 2017, pp. 1174–1179. doi:10. 23919/ChiCC.2017.8027507. 470

[27] X. Wang, S. Li, P. Shi, Distributed finite-time containment control for double-integrator multiagent systems, IEEE Transactions on Cybernetics 44 (9) (2014) 1518–1528. doi:10.1109/TCYB.2013.2288980. [28] D. Li, W. Zhang, W. He, C. Li, S. S. Ge, Two-layer distributed formationcontainment control of multiple Euler-Lagrange systems by output feed-

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back, IEEE Transactions on Cyberneticsdoi:10.110NLS.2015.2511005. [29] M. J. D. Powell, Approximation Theory and Methods, Cambridge Univ. Press, 1981. [30] S. Bhat, D. Bernstein, Finite-time stability of continuous autonomous systems, SIAM Journal on Control and Optimization 38 (3) (2000) 751–766.

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doi:10.1137/S0363012997321358. [31] Z. Zuo, C. Wang, Adaptive trajectory tracking control of output constrained multi-rotors systems, IET Control Theory Applications 8 (13) (2014) 1163–1174. doi:10.1049/iet-cta.2013.0949. [32] H. K. Khalil, Nonlinear systems, New Jersey: Prentice hall, 1996.

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[33] W. L. Jean-Jacques Slotine, Applied nonlinear control, Prentice-hall Englewood Cliffs, 1991.

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Luyan Xu was born in Xinyang, China, in 1994. She received the B.Sc. degree in Henan Normal University in 2016. She is currently pursuing the Ph.D. degree in control science and engineering at the University of Shang490

hai for Science and Technology, Shanghai, China. Her research interests include distributed control of nonlinear systems, adaptive control, and multi-agent systems.

Chaoli Wang received the B.S. and M.Sc. degrees from Mathematics Department, Lanzhou University, 495

Lanzhou, China, in 1986 and 1992, respectively, and the Ph.D. degree in control theory and engineering from the Beijing University of Aeronautics and Astronautics, Beijing, China, in 1999. He is a Professor with the School of OpticalElectrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai, China. From 1999 to 2000, he was a Post-Doctoral Re-

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500

search Fellow with the Robotics Laboratory of Chinese Academy of Sciences, Shenyang, China. From 2001 to 2002, he was a Research Associate with the Department of Automation and Computer-Aided Engineering, the Chinese University of Hong Kong, Hong Kong. Since 2003, he has been with the Department of Electrical Engineering, University of Shanghai for Science and Technology,

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Shanghai, China. His current research interests include nonlinear control, robust control, robot dynamic and control, visual servoing feedback control, and pattern identification.

Xuan Cai received the B.Sc. degree in Automation from Shanghai Dianji University, Shanghai, China in 2012 and M.E. degree in 510

Control Engineering from University of Shanghai for Science and Technology, Shanghai, China in 2015. He is currently pursuing the Ph.D. degree in control science and engineering at University of Shanghai for Science and Technology, Shanghai, China. His current research interests include nonlinear control theory distributed control of nonlinear systems and adaptive control.

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Yujing Xu was born in Shangqiu, Henan Province,

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China, in 1990. She received the M.Sc. degree in Shandong University of Technology in 2018. She is currently pursuing the Ph.D. degree in control science and engineering at the University of Shanghai for Science and Technology, Shanghai, China. Her research interests include nonlinear adaptive control, formation 520

control, multi-agent systems, and the control of Nonholonomic mobile robot.

Chonglin Jing was born in Henan, China, in 1992. He received the B.Sc. degree in Information and Computing Science from Henan University of Science and Technology. He is currently pursuing the Ph.D. degree in control science and 525

engineering at the University of Shanghai for Science and Technology, Shanghai, China. His research interests include adaptive control and adaptive dynamic programming.

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Conflict of Interest The authors declare that they have no known competing financial interests or 530

personal relationships that could have appeared to influence the work reported in this paper.

Author Statement Luyan Xu: Writing - Original Draft, Investigation, Methodology, Data curation , Software, Writing- Reviewing and Editing. Chaoli Wang: Supervi535

sion, Writing- Reviewing and Editing, Project administration, Funding acquisition. Cai Xuan: Methodology, Software, Investigation, Writing- Reviewing and Editing. Yujing Xu:Resources, Validation. Chonglin Jing: Resources, Validation.

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