Distributed adaptive asymptotically consensus tracking control of nonlinear multi-agent systems with unknown parameters and uncertain disturbances

Automatica 77 (2017) 133–142

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Brief paper

Distributed adaptive asymptotically consensus tracking control of nonlinear multi-agent systems with unknown parameters and uncertain disturbances✩ Wei Wang a , Changyun Wen b , Jiangshuai Huang c,d,1 a

School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China

b

School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore

c

Key Laboratory of Dependable Service Computing in Cyber Physical Society, Ministry of Education, China

d

School of Automation, Chongqing University, Chongqing 400044, China

article

info

Article history: Received 20 September 2014 Received in revised form 28 June 2016 Accepted 1 October 2016

Keywords: Distributed coordination Multi-agent systems Consensus tracking Adaptive control Nonlinear systems

abstract Different from traditional centralized control, one major challenge of distributed consensus tracking control lies in the constraint that the desired reference trajectory is only accessible by part of the subsystems. Currently, most existing schemes require the availability of partial knowledge of the reference trajectories to all of the subsystems or information exchange of local control inputs. In this paper, we investigate distributed adaptive consensus tracking control without such requirements for nonlinear high-order multi-agent systems subjected to mismatched unknown parameters and uncertain external disturbances. By introducing compensating terms in a smooth function form of consensus errors and certain positive integrable functions in each step of virtual control design, a new backstepping based distributed adaptive control protocol is proposed. An extra estimator is designed in each subsystem to handle the parametric uncertainties involved in its neighbors’ dynamics, which avoids information exchange of local neighborhood consensus errors among connected subsystems. It is shown that global uniform boundedness of all the closed-loop signals and asymptotically output consensus tracking can be achieved. Simulation results are provided to verify the effectiveness of our scheme. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Consensus of multi-agent systems, due to its wide potential applications, has become a rapidly emerging topic in various research communities over the past decades. Distributed consensus control normally aims at achieving an agreement for the states or the outputs of network-connected systems, by designing controller for each subsystem based on only locally available information collected within its neighboring area. This control issue can

✩ This work was supported by the National Natural Science Foundation of China under Grant Nos. 61673035, 61203068 and 61290324. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Gang Tao under the direction of Editor Miroslav Krstic. E-mail addresses: [email protected], [email protected] (W. Wang), [email protected] (C. Wen), [email protected], [email protected] (J. Huang). 1 Corresponding author.

http://dx.doi.org/10.1016/j.automatica.2016.11.019 0005-1098/© 2016 Elsevier Ltd. All rights reserved.

be further classified into leaderless consensus control (Jadbabaie, Lin, & Morse, 2003; Moreau, 2005; Ren & Beard, 2005) and leaderfollowing consensus control (Arcak, 2007; Bai, Arcak, & Wen, 2008, 2009; Hong, Hu, & Gao, 2006; Ren, 2007) according to whether the final consensus values are predetermined. Note that in most of currently available results on the latter issue, the desired references are set by the behaviors of specific leaders with similar dynamics to the followers and zero/known inputs. With the consideration of more general cases of desired trajectories, such issue is also referred to as consensus tracking control in some references such as Li, Liu, Ren, and Xie (2013), Wang, Huang, Wen, and Fan (2014), Yoo (2013) and Zhang and Lewis (2012). As opposed to traditional tracking control of single system, the main challenge of distributed consensus tracking control of multiagent systems lies in the constraint that the common time-varying reference trajectory is only known by part of the subsystems. Some effective distributed control protocols have been proposed in this area. In Bai et al. (2008, 2009) and Hong et al. (2006), partial knowledge of the reference trajectories are assumed available to all of the subsystems. Distributed observers are then designed in

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W. Wang et al. / Automatica 77 (2017) 133–142

the subsystems which cannot fully access the reference trajectory to estimate its remaining uncertainties. In Ren (2007), to generate the local control law of each subsystem, the information of both states and control signals of its neighbors need be collected. Thus perfect consensus tracking can be achieved though the reference trajectory is totally unknown by some subsystems. However, such design of mutually dependent inputs may bring new challenges during implementation if they are generated without a prescribed priority (Wang, Wen, Huang, & Li, 2016). Alternative solutions to asymptotically consensus tracking are provided in Dong (2012) and Li et al. (2013), by introducing signum functions of local consensus errors in the proposed distributed control approaches. It is worth pointing out that most of the aforementioned results are developed based on system models with relatively simple structures such as pure integrators or linear systems. Besides, intrinsic subsystem uncertainties are not considered although they are often unavoidable in modeling and control of practical systems. As we know, adaptive control is a promising tool to handle parametric and structural uncertainties in the systems. However, the results on distributed adaptive consensus control of uncertain multi-agent systems are still limited due to the difficulty in constructing adaptive laws with only local interactions (Mei, Ren, & Ma, 2011). In Das and Lewis (2010), first-order nonlinear multi-agent systems with unknown nonlinear dynamics and disturbances are considered. By incorporating neural network and robust control techniques, semi-global uniform ultimate boundedness of consensus errors is ensured if the local control gains are chosen to be sufficiently large. The results are extended to systems with higher-order multi-agent systems in El-Ferik, Qureshi, and Lewis (2014) and Zhang and Lewis (2012). In Yu and Xia (2012), it is assumed that the reference trajectory is linearly parameterized and the basis function vectors are known by all the subsystems. Then distributed adaptive control algorithm is presented for achieving perfect consensus tracking of first-order multi-agent systems with unknown parameters. The results are extended in Yu, Shen, and Xia (2013) to solve adaptive finite-time consensus problem of uncertain higher-order distributed agents. Nevertheless, the proposed distributed adaptive laws can only be implementable with extra information transmission of local neighborhood consensus errors among the linked subsystems. Based on similar assumptions on linearly parameterized reference trajectories, two new distributed adaptive control schemes are proposed recently (Hu & Zheng, 2014; Wang et al., 2014) and the convergence of consensus tracking errors to zero is established. In Hu and Zheng (2014), second-order leader–follower system with unknown dynamics and relative position measurements is treated. In Wang et al. (2014), output consensus tracking of nonlinear multi-agent systems in parametric strict feedback form (Krstic, Kanellakopoulos, & Kokotovic, 1995) is investigated. Similar problem to Wang et al. (2014) is handled in Yoo (2013) by adopting dynamic surface design approach and semi-globally uniformly ultimately bounded consensus tracking errors are finally obtained. In this paper, a new distributed adaptive backstepping control scheme is developed to achieve output consensus tracking for nonlinear multi-agent systems. The main contributions of this paper can be summarized as follows.

• The considered multi-agent system model is more general than those in most existing results on distributed consensus control including Arcak (2007), Bai et al. (2008, 2009), Das and Lewis (2010), Hong et al. (2006), Ren (2007) and Yu and Xia (2012) in the following terms. (i) The subsystems are nonlinear and allowed to have arbitrary relative degree and nonidentical dynamics; (ii) intrinsic mismatched unknown parameters and uncertain disturbances are simultaneously involved. • In contrast to Bai et al. (2008, 2009), Hu and Zheng (2014), Wang et al. (2014) and Yu and Xia (2012), the assumptions of

linearly parameterized reference signals and the corresponding basis function vectors being known by all subsystems are no longer needed. In this paper, it is assumed that the desired trajectory yr is known exactly for only part of the subsystems in the group, while it is only known that |˙yr | is upper bounded by an arbitrarily unknown constant for the rest subsystems. • It is worth noting that the non-differentiable signum function based distributed control approaches in Dong (2012) and Li et al. (2013) are not applicable to solve the output consensus tracking of the considered multi-agent system under the relaxed assumption on the reference signal. To address this issue, new compensating terms are introduced in the form of smooth functions with respect to consensus errors and certain positive integrable time-varying functions. Then the effects invoked by the uncertainties including bounded external disturbances and unknown y˙ r in part of subsystems can be counteracted completely. It is shown that global uniform boundedness of all the closed-loop signals and asymptotically consensus tracking for all subsystem outputs can be achieved in this paper, as opposed to El-Ferik et al. (2014), Yoo (2013) and Zhang and Lewis (2012). • In each subsystem i, extra estimates for the external uncertainties in its neighbors’ system dynamics are developed with only locally available information (i.e. xi , xj for j ∈ Ni ). Additional information transmission of local control inputs or local neighborhood consensus errors among connected subsystems required in Ren (2007), Yu et al. (2013) and Yu and Xia (2012) can be avoided. Simulation results on an application example with five onelink manipulators are provided to verify the effectiveness of the proposed distributed adaptive control scheme. 2. Problem formulation 2.1. System model In this paper, we consider a group of N nonlinear subsystems which can be transformable to the following parametric strictfeedback form. x˙ i,q = xi,q+1 + ϕi,q (xi,1 , . . . , xi,q )T θi + di,q (t ), q = 1, . . . , n − 1 x˙ i,n = bi βi (xi )ui + ϕi,n (xi )T θi + di,n (t ) y i = x i ,1 ,

for i = 1, . . . , N

(1)

where xi = [xi,1 , . . . , xi,n ]T ∈ ℜn , ui ∈ ℜ, yi ∈ ℜ are the states, the control input and the output of the ith subsystem, respectively. θi ∈ ℜpi is a vector of unknown constants and the control coefficient bi ∈ ℜ is an unknown non-zero constant. ϕi,j : ℜj → ℜpi for j = 1, . . . , n and βi : ℜn → ℜ1 are known smooth nonlinear functions. di,q (t ) ∈ ℜ for q = 1, . . . , n denotes uncertain external disturbances. 2.2. Information transmission condition among the N subsystems Suppose that the information transmission condition among the group of N subsystems can be represented by an undirected graph G , (V , E ), where V = {1, . . . , N } denotes the set of indexes corresponding to each subsystem, E ⊆ V × V is the set of edges between two distinct subsystems. Since G is undirected, if the edge (i, j) ∈ E , then (j, i) ∈ E . This indicates that subsystems i and j are directly connected with each other, i.e. they can obtain state and structure information from each other. In this case, subsystem j is called a neighbor of subsystem i, and vice versa. We denote the set of neighbors for subsystem i as Ni , {j ∈ V :

W. Wang et al. / Automatica 77 (2017) 133–142

(i, j) ∈ E }. Note that self edge (i, i) is not allowed, thus (i, i) ̸∈ E and i ̸∈ Ni . G is connected means that there is an undirected sequence of edges between every pair of distinct subsystems (Ren & Cao, 2010). The connectivity matrix A = [aij ] ∈ ℜN ×N is defined that aij = aji = 1 if (i, j) ∈ E , and aij = aji = 0 if (i, j) ̸∈ E . Clearly, A is symmetric and the diagonal elements aii = 0. We introduce an in-degree matrix ∆ such that ∆ = diag(∆i ) ∈ ℜN ×N with  ∆i = j∈Ni aij being the ith row sum of A. Then, the Laplacian matrix of G is defined as L = ∆ − A. The consensus tracking objective in this paper is to design distributed adaptive controllers ui (xi , xj ) for j ∈ Ni such that (i) all the signals in the closed-loop system are globally uniformly bounded; (ii) the outputs of all the N subsystems can track a common desired trajectory yr asymptotically though yr (t ) is allowed totally unknown by a subset of the N subsystems. Note that µi = 1 is adopted to indicate the case that the desired trajectory is accessible directly by subsystem i; otherwise, µi is set as µi = 0. To achieve the objective, the following assumptions are imposed. Assumption 1. The information transmission graph G is fixed and connected. The desired trajectory yr (t ) is available to at least one N subsystem in G, i.e. i=1 µi > 0. Assumption 2. The first nth-order derivatives of yr (t ) are bounded, piecewise continuous and only known by subsystem i with µi = 1. Besides, it is known by subsystem i with µi = 0 that |˙yr | ≤ Fr ,1 , where Fr ,1 is an unknown positive constant.

Lemma 1 (Ren & Cao, 2010). If a graph G with connectivity matrix A ∈ ℜp×p is undirected and connected, then the matrix (L + diag{ξ1 , . . . , ξp }) is symmetric positive definite with at least one ξi > 0, for i = 1, . . . , p. Lemma 2 (Zuo & Wang, 2014). The following inequality holds 0 ≤ |z | − 

z2 z 2 + η2

Assumption 4. The disturbance di,q (t ) is bounded such that |di,q | ≤ Di,q , where Di,q is an unknown positive constant. Remark 1. As pointed in Section 1, the multi-agent system model described in (1) is more general than those in most of the currently available results on distributed consensus control including Arcak (2007), Bai et al. (2008, 2009), Das and Lewis (2010), Hong et al. (2006), Ren (2007) and Yu and Xia (2012) in the following terms. (i) The subsystems are nonlinear and allowed to have arbitrary relative degree and non-identical dynamics; (ii) intrinsic mismatched unknown parameters and uncertain disturbances are simultaneously involved. Moreover, (1) is in the parametric strict feedback form, which can be commonly encountered in many nonlinear control problems. The one-link manipulator with the inclusion of motor dynamics considered in the simulation section is a typical example. In Krstic et al. (1995), the conditions that a p class of general nonlinear systems χ˙ = f0 (χ ) + l=1 θl fl (x) + g (χ)u, y = h(χ ) are transformable into such form have been provided. Remark 2. Assumption 2 indicates that for the subsystems with µi = 0, the only information available about yr (t ) is |˙yr (t )| ≤ Fr ,1 with Fr ,1 being arbitrary positive constant. Such a condition is much weaker than the restrictions appeared in Bai et al. (2008, 2009), Hu and Zheng (2014), Wang et al. (2014) and Yu and Xia (2012) that the reference signals are linearly parameterized, such as yr (t ) = fr (t )T wr + cr in Wang et al. (2014), and the corresponding basis functions (i.e. fr (t )) are known by all the subsystems in the group. 3. Design of distributed adaptive controllers Before we proceed with the design procedure of distributed adaptive controllers, some useful lemmas are firstly presented.

≤η

(2)

for z ∈ ℜ and η > 0. Note that control design will be performed by following a stepby-step procedure known as backstepping technique (Krstic et al., 1995). Nevertheless, different from the standard backstepping technique, new actions need be made to handle the effects of uncertain disturbances and unknown desired trajectory in some subsystems. The detailed design procedure is presented as follows.

• Step 1. Introduce the first two error variables zi,1 =

N 

aij (yi − yj ) + µi (yi − yr )

(3)

j =1

zi,2 = xi,2 − αi,1

(4)

where αi,1 is a virtual control to be chosen. By defining z1 = [z1,1 , . . . , zN ,1 ]T , we have z1 = (L + B ) δ

(5)

where δ = y − y with y = [y1 , . . . , yN ]T ∈ ℜN and y r

r

=

yr [1, . . . , 1] = [yr , . . . , yr ] ∈ ℜ denotes the actual tracking error between each subsystem’s output and yr . B = diag{µi }. Obviously, the control objective is to ensure that limt →∞ δ(t ) = 0. T

Assumption 3. The sign of bi is known in each subsystem i and βi (xi ) ̸= 0.

135

T

N

From (1), (4) and (5), the derivative of z1 is computed as

 α1,1 + z1,2 + ϕ1T,1 θ1 + d1,1 − y˙ r   .. z˙1 = (L + B )  . . T αN ,1 + zN ,2 + ϕN ,1 θN + dN ,1 − y˙ r 

(6)

The virtual control αi,1 is chosen as

αi,1 = −ci,1 zi,1 − ϕiT,1 θˆi + µi y˙ r − 

zi,1 zi2,1 + ηi2 (t )

Fˆi,1

(7)

where ci,1 is a positive constant, θˆi is the estimate of θi , Fˆi,1 is the estimate of Fi,1 = Di,1 + (1 − µi )Fr ,1 , ηi (t ) is a function chosen to t satisfy that ηi (t ) > 0 and 0 ηi (τ )dτ ≤ η¯ i < ∞, ∀t ≥ 0. η¯ i is a positive constant. Based on Assumption 1 and Lemma 1, we can define a Lyapunov function candidate at this step as follows. V1 =

1 2

δ T (L + B )δ +

N 1

2 i=1

θ˜iT Γi−1 θ˜i +

N  1 2 F˜i,1 2 ε i,1 i=1

(8)

where θ˜i , F˜i denote the estimation errors that θ˜i = θi − θˆi and F˜i,1 = Fi,1 − Fˆi,1 , respectively. Γi is a positive definite matrix with appropriate dimension. From (5)–(7), the derivative of V1 is derived as V˙ 1 =

N 



zi,1 −ci,1 zi,1 + zi,2 + ϕiT,1 θ˜i + di,1

i =1

+ (µi − 1)˙yr − 

zi,1

zi2,1 + ηi2



Fˆi,1 +

N  i=1

  θ˜iT Γi−1 θˆ˙ i

136

W. Wang et al. / Automatica 77 (2017) 133–142

We select αi,2 as

N    1 ˙ F˜i,1 −Fˆ i,1 εi,1 i=1 N    ≤ −ci,1 zi2,1 + zi,1 zi,2 + θ˜iT ϕi,1 zi,1 + |zi,1 |Fi,1

+

∂αi,1 ∂αi,1 xi,2 + Γi τi,2 ∂ xi,1 ∂ θˆi T  ∂αi,1 ∂αi,1 ∂αi,1 ˙ˆ − ϕi,2 − ϕi,1 θˆi + η˙ i + F i ,1 ∂ x i ,1 ∂ηi ∂ Fˆi,1 N  ∂αi,1 ∂αi,1 ∂αi,1 + µi y˙ r + µi y¨ r + aij ∂ yr ∂ y˙ r ∂ x j ,1 j =1

αi,2 = −ci,2 zi,2 − zi,1 +

i=1

−

N 

zi2,1

 i=1



zi2,1 + ηi2



Fi,1 − F˜i,1 +

N 

  θ˜iT Γi−1 θ˙ˆ i

i=1

N    1 ˙ F˜i,1 −Fˆ i,1 εi,1 i=1    N 2  z i , 1 −ci,1 zi2,1 + zi,1 zi,2 + |zi,1 | −   ≤ i=1 zi2,1 + ηi2   1 ×Fi,1 + θ˜iT Γi−1 Γi ϕi,1 zi,1 − θ˙ˆ i + F˜i,1 εi,1  

+

× εi,1 

zi2,1

zi2,1 + ηi2

− F˙ˆ i,1  .

  (¯ai + 2)zi,2 φi2,1 × xj,2 + ϕjT,1 θˆij −  Fˆi,2 zi2,2 φi2,1 + ηi2 (t ) where ci,2 is a positive constant and a¯ i = (9)

φi,1

˙

zi2,1

zi2,1 + ηi2 (t )

.

τi,2 (10)

N   −ci,1 zi2,1 + zi,1 zi,2 + Fi,1 ηi + θ˜iT Γi−1 i=1

  × Γi ϕi,1 zi,1 − θ˙ˆ i .

(11)

By defining a tuning function

τi,1 = ϕi,1 zi,1 , V˙ 1 in (11) can be rewritten as V˙ 1 ≤

(12)

      N  ∂αi,1 2  ∂αi,1 2 = 1 + + aij ∂ xi,1 ∂ x j ,1 j =1   ∂αi,1 = τi,1 + ϕi,2 − ϕi,1 zi,2 . ∂ xi,1

V2 = V1 + (13)

• Step 2. We firstly clarify the arguments of αi,1 . By examining (7) along with (3), we see that αi,1 for the ith subsystem is a function of xi,1 , θˆi , ηi , Fˆi,1 , yr , y˙ r if µi = 1 and its neighbors’ state xj,1 for aij = 1. Thus   αi,1 can be expressed as ˆ ˆ αi,1 xi,1 , θi , ηi , Fi,1 , aij xj,1 , µi yr , µi y˙ r . We introduce a new error variable as (14)

Taking the derivative of zi,2 in (4), it follows that z˙i,2 = x˙ i,2 − α˙ i,1

 T ∂αi,1 ∂αi,1 xi,2 + ϕi,2 − ϕi,1 θi ∂ xi,1 ∂ xi,1 ∂αi,1 ˙ ∂αi,1 ∂αi,1 ˙ˆ ∂αi,1 θˆ i − − η˙ i − F i ,1 − µi y˙ r ∂ηi ∂ yr ∂ θˆi ∂ Fˆi,1 N   ∂αi,1 ∂αi,1  − µi y¨ r − aij xj,2 + ϕjT,1 θj + di,2 ∂ y˙ r ∂ x j ,1 j=1

= zi,3 + αi,2 −

−

N  ∂αi,1 ∂αi,1 di,1 − aij d j ,1 . ∂ xi,1 ∂ xj,1 j=1

(18)

(19)

Remark 3. Similar to Wang et al. (2014), θˆij is introduced in subsystem i to estimate the unknown parameters involved in its neighbors’ dynamics (i.e. θj ). Based on this, additional information transmission of local consensus errors between two linked subsystems, which may involve state information of the subsystems beyond their neighboring areas, as required in Yu et al. (2013) and Yu and Xia (2012) can be avoided in this paper.

i=1

zi,3 = xi,3 − αi,2 .

(17)

Define a Lyapunov function candidate V2 at this step as

N   −ci,1 zi2,1 + zi,1 zi,2 + Fi,1 ηi + θ˜iT Γi−1

  × Γi τi,1 − θ˙ˆ i .

aij

Fˆi,2 , θˆij are estimates introduced in subsystem i to account for Fi,2 = max{Di,1 , Di,2 , Dj,1 } and θj , respectively, if aij = 1.

By applying Lemma 2 and substituting (10) into (9), we have V˙ 1 ≤

N  j=1

We choose the parameter update law for Fˆ i,1 as

˙ Fˆ i,1 = εi,1 

(16)

(15)

N 1

2 i=1

 zi2,2

N  (¯ai + 2) ˜ 2 Fi,2 + + aij θ˜ijT Γij−1 θ˜ij εi,2 j =1

 (20)

where θ˜ij , F˜i,2 are the estimation errors that θ˜ij = θj −θˆij , F˜i,2 = Fi,2 − Fˆi,2 and Γij is a positive definite matrix. From (11) and (15)–(19), the derivative of V2 can be computed to satisfy that V˙ 2 ≤

N 

  − ci,1 zi2,1 − ci,2 zi2,2 + zi,2 zi,3 + θ˜iT Γi−1

i =1

   ∂αi,1  ˙ Γi τi,2 − θˆ i × Γi τi,2 − θ˙ˆ i + Fi,1 ηi + zi,2 ∂ θˆi   N  ∂αi,1 − aij θ˜ijT Γij−1 Γij ϕj,1 zi,2 + θ˙ˆ ij ∂ xj,1 j =1   zi2,2 φi2,1  Fi,2 + (¯ai + 2) |zi,2 |φi,1 −  zi2,2 φi2,1 + ηi2   zi2,2 φi2,1 (¯ai + 2) ˜  + Fi,2 εi,2  − Fˆ˙ i,2  (21) εi ,2 zi2,2 φi2,1 + ηi2  where the inequality a1 + · · · + an ≤ n a21 + · · · + a2n for a1 ≥ 0, a2 ≥ 0, . . . , an ≥ 0 has been used.

W. Wang et al. / Automatica 77 (2017) 133–142

˙

We choose the parameter update law for Fˆ i,2 as

˙

Fˆ i,2 = εi,2 

zi2,2 φi2,1

Proof. The proof of Theorem 1 follows a step-by-step procedure starting with a Lyapunov function candidate V3 defined as below. (22)

zi2,2 φi2,1 + ηi2

∂αi,1 ϕj,1 zi,2 , ∂ x j ,1

if aij = 1,

(23)



2 i =1

zi2,3 +

2a¯ i + 3

εi,3

F˜i2,3

 (37)

φi,2 designed as in (27)–(32) and F˙ˆ i,3 chosen as (36) for Step 3, the derivative of V3 can be computed as

then V˙ 2 can be further derived based on Lemma 2 such that V˙ 3 ≤

N   V˙ 2 ≤ −ci,1 zi2,1 − ci,2 zi2,2 + zi,2 zi,3 + θ˜iT Γi−1

N 

 − ci,1 zi2,1 − ci,2 zi2,2 − ci,3 zi2,3 + zi,3 zi,4

   ∂αi,1  ˙ + θ˜iT Γi−1 Γi τi,3 − θ˙ˆ i + zi,2 Γi τi,2 − θˆ i ∂ θˆi   ∂αi,2  ∂αi,2  ˙ + zi,3 Γi τi,3 − θˆ i + zi,2 Γi τi,3 − τi,2 ∂ θˆi ∂ θˆi N    ˙ + aij θ˜ijT Γij−1 Γij τ¯ij,2 − θˆ ij

   ∂αi,1  ˙ × Γi τi,2 − θ˙ˆ i + zi,2 Γi τi,2 − θˆ i ∂ θˆi N    ˙ − aij θ˜ijT Γij−1 −Γij τ¯ij,1 + θˆ ij j =1

 + Fi,1 ηi + (¯ai + 2)Fi,2 ηi .



i=1

i=1

j =1

(24)

+ • Step q (q = 3, . . . , n). By following the ideas in the above two

(q)

Fˆi,q , ηi , . . . , ηi , µi yr , . . . , µi yr . ϱˆ i and Fˆi,q are respectively the estimates of ϱi = 1/bi and Fi,q = max{Di,1 , . . . , Di,q , Dj,1 , . . . , Dj,q−1 } if aij = 1. ci,q and γi are positive constants.

compensating terms of Fˆi,2 in (16), (27) are introduced to account for the effects of di,1 , . . . , di,q and dj,1 , . . . , dj,q−1 if aij = 1, i.e. uncertain disturbances in its local dynamics as well as its neighbors’ dynamics. 4. Stability analysis

aij zi,3

j =1

a function of xi,1 , . . . , xi,q , aij xj,1 , . . . , aij xj,q , θˆi , aij θˆij , Fˆi,1 , . . . ,

Remark 4. In Dong (2012), Li et al. (2013) and Mei et al. (2011), distributed consensus control problems of relatively simple multiagent system models such as pure-integrator dynamics and linear systems were considered. Signum functions of consensus errors were utilized to treat uncertain disturbances existing in system dynamics. However, the non-differentiability of these functions restricts such scheme being applicable to solve asymptotically consensus tracking problem for high-order nonlinear multi-agent systems in this paper. To address this issue, a new compensating term of Fˆi,1 associated with a smooth function of distributed consensus error variable (zi,1 ) and a positive integrable timevarying function ηi (t ), is introduced in designing αi,1 to account for the effects of di,1 and unknown y˙ r for µi = 0 (see (7)). Furthermore, in designing αi,q for 2 ≤ q ≤ n, similar forms of

N 

 ∂αi,2  ˙ Γij τ¯ij,2 − θˆ ij ∂ θˆij

+ Fi,1 ηi + (¯ai + 2)Fi,2 ηi + (2a¯ 1 + 3)   zi2,3 φi2,2 ¯  Fi,3 + 2ai + 3 F˜i,3 × |zi,3 |φi,2 −  εi,3 zi2,3 φi2,2 + ηi2   εi,3 zi2,3 φi2,2 ˙ ×  − Fˆ i,3  zi2,3 φi2,2 + ηi2  N 3     ˙ ≤ − ci,q zi2,q + zi,3 zi,4 + θ˜iT Γi−1 Γi τi,3 − θˆ i

steps, the final adaptive controllers are obtained recursively as summarized in Table 1, where αi,q for q = 2, . . . , n is (q−1)

N 1

V3 = V2 +

where F˜i,3 = Fi,3 − Fˆi,3 . Based on V˙ 2 derived in (24), αi,3 together with ωi,3 , ω ¯ ij,2 , τi,3 , τ¯ij,2 ,

and introduce the following tuning function

τ¯ij,1 = −

137

i =1

q=1

  ∂αi,2  ∂αi,1  ˙ ˙ Γi τi,3 − θˆ i + zi,3 Γi τi,3 − θˆ i + zi,2 ∂ θˆi ∂ θˆi N    ˙ + aij θ˜ijT Γij−1 Γij τ¯ij,2 − θˆ ij j =1

+

N 

aij zi,3

 ∂αi,2  ˙ Γij τ¯ij,2 − θˆ ij ∂ θˆij

j =1  + Fi,1 ηi + (¯ai + 2)Fi,2 ηi + (2a¯ 1 + 3)Fi,3 ηi .

(38)

Then for Step q, q = 4, . . . , n−1, we define the following Lyapunov function candidate. Vq = Vq −1 +

N 1



2 i =1

zi2,q +

(q − 1)¯ai + q ˜ 2 Fi,q εi,q

 (39)

where F˜i,q = Fi,q − Fˆi,q . Based on (38), αi,q together with ωi,q , ωij,q−1 , τi,q , τ¯ij,q−1 , φi,q−1

˙

The main results of our distributed adaptive control design scheme in Section 3 can be formally stated in the following theorem. Theorem 1. Consider the closed-loop adaptive system consisting of N uncertain nonlinear subsystems (1) satisfying Assumptions 1–4, the local controllers (26) and the parameter estimators (33)–(36). All the signals in the closed-loop system are globally uniformly bounded and asymptotic consensus tracking of all the subsystems’ outputs to yr (t ) is achieved, i.e. limt →∞ δi (t ) = 0 for i = 1, . . . , N.

designed as in (27)–(32) and Fˆ i,q chosen as (36) for Step q, the derivative of Vq is computed as V˙ q ≤

N 

 −

i=1

 +



l =1

N  j =1

˙

ci,l zi2,l + zi,q zi,q+1 + θ˜iT Γi−1 Γi τi,q − θˆ i

q  ∂αi,l−1 l =2

+

q 

∂ θˆi

 zi,l





˙ Γi τi,q − θˆ i ˙

aij θ˜ijT Γij−1 Γij τ¯ij,q−1 − θˆ ij







138

W. Wang et al. / Automatica 77 (2017) 133–142

Table 1 The design of distributed adaptive controllers for Step q (q = 3, . . . , n). Introducing error variables: zi,q+1 = xi,q+1 − αi,q , q = 3, . . . , n − 1

(25)

Control Laws: ui =

ϱˆ i αi,n βi (xi )

(26)

with

  q−1 ∂αi,q−1 ∂αi,q−1 q−1 ∂αi,l−1 αi,q = −zi,q−1 − ci,q zi,q − ωiT,q θˆi + Γi τi,q + xi,l+1 + zi,l Γi ωi,q l=1 l= 2 ∂ xi , l ∂ θˆi ∂ θˆi   q−1 ∂αi,l−1 N q−1 ∂αi,q−1 ∂αi,q−1 T ˆ θ + xj,l+1 + ω ¯ Γ τ ¯ − z Γ ω ¯ + aij ij ij ij , q − 1 i , l ij ij , q − 1 ij,q−1 j=1 l=3 l=1 ∂ xj,l ∂ θˆij ∂ θˆij q ∂αi,q−1 q−1 ∂αi,q−1 (l) q−1 ∂αi,q−1 ˙ [¯ a ( q − 1) + q]zi,q φi2,q−1 i  + µ η y(rl) − Fˆi,q Fˆ i,l + + i i ( l − 1 ) ( l − 1 ) l=1 l=1 l= 1 ∂ Fˆi,l ∂η ∂ yr z2 φ2 + η 2 (t ) i

i,q

q−1 ∂αi,q−1 ωi,q = ϕi,q − ϕi,l l= 1 ∂ xi,l q−1 ∂αi,q−1 ω¯ ij,q−1 = ϕj,l l=1 ∂ xj,l τi,q = τi,q−1 + ωi,q zi,q τ¯ij,q−1 = τ¯ij,q−2 − ω¯ ij,q−1 zi,q  q−1  ∂αi,q−1 2 N q−1  ∂αi,q−1 2 + φi,q−1 = 1 + aij l= 1 l=1 j = 1 ∂ xi,l ∂ xj,l

i,q−1

(27)

i

(28) (29) (30) (31) (32)

Parameter estimators:

ϱˆ˙ i = −γi sgn(bi )αi,n zi,n θ˙ˆ i = Γi τi,n θ˙ˆ ij = Γij τ¯i,n−1 zi2,q φi2,q−1 ˙ Fˆ i,q = εi,q  zi2,q φi2,q−1 + ηi2

(33) (34) (35) (36)

  q    ∂αi,l−1 ˙ zi,l Γij τ¯ij,q−1 − θˆ ij + aij ∂ θˆij j =1 l =3  q  + Fi,1 ηi + (¯ai (q − 1) + q)Fi,l ηi . N 

−

q=1

(40)

+

l =2

≤ N 1

2 i =1



zi2,n

(¯ai (n − 1) + n) ˜ 2 |bi | 2 Fi,n + ϱ˜ i + εi ,n 2γi



−

i=1

Based on V˙ n−1 in (40)–(32), the derivative of Vn is computed as follows.

n 

zi,n bi βi ui + ϕ θ + di,n − α˙ i,n−1

N   a¯ i (n − 1) + n

εi,n

i =1





+



˙



|bi |  ˙  ϱ˜ i −ϱˆ i γi

−

zi,n bi (ϱi − ϱ˜ i )αi,n + ϕiT,n θi + di,n



j =1

∂ xi,q

aij



N  |bi |

γi ˙



ci,q zi2,q + θ˜iT Γi−1 Γi τi,n − θˆ i

∂αi,q−1 zi,q ∂ θˆi q=1

N 



  ϱ˜ i −ϱ˙ˆ i



  ˙ Γi τi,n − θˆ i ˙



aij θ˜ijT Γij−1 Γij τ¯ij,n−1 − θˆ ij

+

N 

 aij

∂αi,q−1 zi,q ∂ θˆij q =1

n 







˙ Γij τ¯ij,n−1 − θˆ ij



 + Fi,1 ηi +

n−1 

(¯ai (q − 1) + q)Fi,q ηi + [¯ai (n − 1) + n]



xi,q+1 + ϕiT,q θi + di,q

n −1  ∂αi,n−1  q =1

∂ xj,q

 × |zi,n |φi,n−1 − 

 ∂αi,n−1  N 

˙

q =2

i=1

q =1



F˜i,n −Fˆ i,n +

y(rq)

i=1

n 

j =1



F˜i,n −Fˆ i,n +

n −1

−

∂ y(rq−1)

q =1

N

= V˙ n−1 +

q=1

j =1 T i ,n i



i=1

+

εi,n



n  ∂αi,n−1

(41)

where F˜i,n = Fi,n − Fˆi,n , ϱ˜ i = ϱi − ϱˆ i .

N 

∂ηi

− µi

N  a¯ i (n − 1) + n

N 

+

V˙ n = V˙ n−1 +

η(q) (q−1) i

i=1

Finally, we introduce the following Lyapunov function at Step n as

Vn = Vn −1 +

n−1  ∂αi,n−1



xj,q+1 + ϕjT,q θi + dj,q



N n −1   ∂αi,n−1 ˙ ∂αn−1 ˙ ∂αi,n−1 ˙ˆ − θˆ i − aij θˆ ij − F i ,q ˆ ˆ ∂ θi ∂ θij ∂ Fˆi,q j =1 q =1



zi2,n φi2,n−1 zi2,n φi2,n−1 + ηi2

 Fi,n

 zi2,n φi2,n−1 [¯ai (n − 1) + n] ˜  + Fi,n εi,n  εi ,n z2 φ2 + η2 i,n

 − F˙ˆ i,n  +

i,n−1

i

  |bi |  ˙ ϱ˜ i −ϱˆ i − γi sgn(bi )αi,n  . γi

(42)

W. Wang et al. / Automatica 77 (2017) 133–142

139

By choosing the parameter update laws as (33)–(36), V˙ n can be rendered negative definite such that V˙ n ≤

N 

 −

n 

i=1

ci,q zi2,q + Fi,1 ηi

q =1

n  + (¯ai (q − 1) + q)Fi,q ηi



q =2

≤−

n 

2

cq ∥zq ∥ +

N 

q=1



 n  Fi,1 + (¯ai (q − 1) + q)Fi,q ηi (43)

i =1

q=2

where cq = min{ci,q }. Integrating both sizes of (43) yields that Vn (t ) +

n 

∥zq (τ )∥2 dτ 0

q =1

≤ Vn (0) +

t

 cq N 

Fig. 1. Communication topology for a group of the 5 subsystems.

 Fi,1 +

i =1

n 

 (¯ai (q − 1) + q)Fi,q η¯ i .

(44)

q =2

From the definition of Vn in (41) along with (8), (20), we establish that z, θˆi , θˆij when aij ̸= 0 and ϱˆ i , Fˆi,q for q = 1, . . . , n are bounded. Besides, zi,q ∈ L2 . From (3) and Lemma 1, yi , i.e. xi,1 for i = 1, . . . , N are bounded. From (7) and the smoothness of ϕi , αi,1 for i = 1, . . . , N are bounded. From the definition of zi,2 in (4), it follows that xi,2 is bounded. By following similar procedure, the boundedness of αi,q for q = 2, . . . , n, xi,q for q = 3, . . . , n is ensured. From (26), we can conclude that the control signal ui is bounded. Therefore the boundedness of all the signals in the closed-loop adaptive systems is guaranteed. Thus z˙i,q is bounded. By applying Barbalat’s lemma, we have limt →∞ zi,q (t ) = 0 for i = 1, . . . , N and q = 1, . . . , n. From (5) and (L + B ) being positive definite, it follows that asymptotic consensus tracking of all the N subsystems’ outputs to a desired trajectory yr (t ) is also achieved, i.e. limt →∞ δi (t ) = 0 for i = 1, . . . , N.  Remark 5. From Theorem 1, it can be seen that global uniform boundedness of all the closed-loop system and asymptotically output consensus tracking can be achieved in this paper. This is different from many existing results on consensus control of uncertain multi-agent systems including El-Ferik et al. (2014), Yoo (2013) and Zhang and Lewis (2012), where only semi-global uniform ultimate boundedness of tracking errors can be shown. Nevertheless, since the derivatives of virtual controls need be computed in each recursive step, the design procedure will be more involved if the system order is higher. This constitutes the main deficiency of our proposed distributed adaptive backstepping control scheme.

x˙ i,1 = xi,2 x˙ i,2 = xi,3 − [sin(xi,1 ), xi,2 ]θi,1 +

τd,i Di

x˙ i,3 = bi ui − [xi,2 , xi,3 ]θi,2 with θi,1 =



Ni Di

, DBii

T

, bi =

(46) 1 , Mi D i

θi,2 =

In this section, an application example with five one-link manipulators is considered to illustrate the effectiveness of proposed design scheme. As described in Yang, Feng, and Ren (2004), each manipulator with the inclusion of motor dynamics can be modeled as follows, Di q¨ i + Bi q˙ i + Ni sin(qi ) = τi + τd,i 1≤i≤5

(45)

where qi , q˙ i and q¨ i denote the position, velocity and acceleration of the ith link, respectively. τi and τ˙i are the motor shaft angle and velocity. ui is the control input which represents the motor torque. τd,i is the torque disturbance. Parameters Di , Bi , Ni , Mi , Hi and Km,i are unknown constants.



Km,i Mi Di

, MHii

T

.

The control objective is to design distributed adaptive controllers such that the outputs yi (t ) = xi,1 (t ) = qi (t ) of all the five subsystems can asymptotically follow a desired trajectory yr (t ). In simulation, the desired trajectory is selected as yr (t ) = 1 + 0.1 sin(0.05t ) + 0.1 cos(0.1t ). The system parameters are set as Di = 1, Ni = 2, Bi = 1, Km,i = 10, Mi = 10, Hi = 0.5, τd,i = 0.1 cos(10t ) for 1 ≤ i ≤ 5. The communication topology for the 5 subsystems is given in Fig. 1. It is clear that µ1 = 1, whereas µ2 = µ3 = 0. In the recursive design procedure, the error variables are defined as zi,1 =

N 

aij (xi,1 + ∆i − xj−1 − ∆j ) + µi (xi,1 + ∆i − yr )

j =1



zi,2 = xi,2 − αi,1 xi,1 , aij xj,1 , µi yr , µi y˙ r , (1 − µi )ηi , (1 − µi )Fˆi,1





zi,3 = xi,3 − αi,2 xi,1 , xi,2 , aij xj,1 , aij xj,2 , µi yr , µi y˙ r ,

µi y(r2) , θˆi,1 , (1 − µi )ηi , (1 − µi )η˙ i , (1 − µi )Fˆi,1 , Fˆi,2



(47)

where αi,1 and αi,2 are the virtual controls chosen as

αi,1 = −ci,1 zi,1 + µi y˙ r − (1 − µi ) 

5. Simulation results

Mi τ˙i + Hi τi = ui − Km,i q˙ i ,

By defining xi,1 = qi , xi,2 = q˙ i and xi,3 = τi /Di , (45) can be transformed to the following parametric strict-feedback form

zi,1 zi2,1

+ η (t ) 2 i

Fˆi,1

∂αi,1 x i ,2 ∂ x i ,1 N  ∂αi,1 ∂αi,1 (2) ∂αi,1 + aij xj,2 + µi y˙ r + µi y ∂ x ∂ y ∂ y˙ r r j ,1 r j =1 ∂αi,1 ∂αi,1 ˙ˆ + (1 − µi ) η˙ i + (1 − µi ) F i,1 ∂ηi ∂ Fˆi,1

αi,2 = −ci,2 zi,2 − zi,1 + [sin(xi,1 ), xi,2 ]θˆi,1 +

−

zi,2

zi2,2 + ηi2 (t )

Fˆi,2 .

(48)

The control input ui is designed as ui = ϱˆ i αi,3

(49)

140

W. Wang et al. / Automatica 77 (2017) 133–142

with

∂αi,2 x i ,2 ∂ x i ,1 N    ∂αi,2 aij xi,3 − [sin xi,1 , xi,2 ]θˆi,1 + x j ,2 ∂ xj,1 j=1

αi,3 = −ci,3 zi,3 − zi,2 + [xi,2 , xi,3 ]θˆi,2 + +

+

∂αi,2 ∂ xi,2 N 

 ∂αi,2  xj,3 − [sin xj,1 , xj,2 ]θˆij,1 ∂ x j ,2

aij

j =1

∂αi,2 ∂αi,2 (2) ∂αi,2 y˙ r + µi y + µi (2) y(r3) ∂ yr ∂ y˙ r r ∂ yr ∂αi,2 ˙ ∂αi,2 (2) ∂αi,2 + η˙ i + (1 − µi ) η θˆ i,1 + ∂ηi ∂ η˙ i i ∂ θˆi,1    ∂αi,2 ˙ˆ ∂αi,2 ˙ˆ + (1 − µi ) F i,1 + F i ,2 − 1 + aij ∂ Fˆi,1 ∂ Fˆi,2 j =1       N  ∂αi,2 2 ∂αi,2 2 zi,3 + aij ∂ xj,2 ∂ xi,2 j=1 ×  Fˆi,3 .    2  2  N  ∂αi,2 ∂α z 2 + aij ∂ xji,,22 + ηi2 (t ) i,3 ∂ xi,2 + µi

Fig. 2. Tracking performance of the outputs yi (t ) for 1 ≤ i ≤ 5 to the desired trajectory yr (t ).

(50)

j =1

Parameter update laws are chosen as

  ˙θˆ = −Γ [sin x , x ]T z − ∂αi,2 z i,1 i,1 i,1 i ,2 i,2 i ,3 ∂ xi,2 θˆ˙ = −Γ [x , x ]T z i,2

i,2

i ,2

i,3

Fig. 3. The states xi,2 (t ) for 1 ≤ i ≤ 5.

i ,3

θ˙ˆ ij,1 = Γij,1 [sin xj,1 , xj,1 ]

∂αi,2 zi,3 , ∂ xj,2

if aij = 1

ϱ˙ˆ i = −γi αi,3 zi,3 zi2,1

˙

Fˆ i,1 = εi,1 

zi2,1 + ηi2

if µi = 0

zi2,2

˙

Fˆ i,2 = εi,2 

,

zi2,2 + ηi2

 

∂αi,2 ∂ xi,2

N 

2



∂αi,2 ∂ xj,2

2



ε + aij j =1 ˙Fˆ =  . i,3    2   2 N  ∂α ∂α i,2 i,2 z 2 + aij + η2 2 i,3 zi,3

i ,3

∂ xi,2

j =1

∂ xj,2

(51)

Fig. 4. The states xi,3 (t ) for 1 ≤ i ≤ 5.

i

All the state initials are set as zero, except that x1,1 (0) = −1, x2,1 (0) = −0.5, x3,1 (0) = 0, x4,1 (0) = 0.5, x5,1 (0) = 1. The controller coefficients are chosen as ci,1 = 3, ci,2 = ci,3 = 1, Γi,1 = Γi,2 = Γi,3 = 3I, γi = 1, εi,1 = εi,2 = εi,3 = 0.01, ηi (t ) = e−0.01t for 1 ≤ i ≤ 5. The tracking performance of all the subsystems’ outputs yi (t ) = xi,1 (t ) for 1 ≤ i ≤ 5 with comparison to the desired trajectory yr (t ) are shown in Fig. 2. It can be seen that asymptotically consensus tracking can be achieved with the proposed distributed adaptive control scheme. Moreover, the boundedness of the states xi,2 , xi,3 and the parameter estimates including θˆi,1 , θˆi,2 , ϱˆ i , θˆij,1 , Fˆi,1 , Fˆi,2 , Fˆi,3 can be observed in Figs. 3–9, respectively. 6. Conclusion In this paper, a new smooth function based distributed adaptive tracking control scheme is proposed for nonlinear

Fig. 5. Parameter estimates θˆi,1 for 1 ≤ i ≤ 5.

multi-agent systems with arbitrary system order, mismatched unknown parameters and uncertain external disturbances. Based on the scheme, asymptotically consensus output tracking can be achieved without the assumptions on parameterized reference

W. Wang et al. / Automatica 77 (2017) 133–142

141

Note that if the information transmission condition is represented by a directed graph, the existence of asymmetric matrix (L + B ) will make it rather difficult to design distributed adaptive laws. Thus the extension of current results to the case of directed graph will be an interesting topic for further investigation. Acknowledgments The authors wish to thank the Associate Editor and the anonymous reviewers for their helpful comments to improve the paper. Fig. 6. Parameter estimates θˆi,2 for 1 ≤ i ≤ 5.

Fig. 7. Parameter estimates ϱˆ i for 1 ≤ i ≤ 5.

Fig. 8. Parameter estimates θˆij,1 for 1 ≤ i ≤ 5 if aij = 1.

Fig. 9. Parameter estimates Fˆi,1 , Fˆi,2 and Fˆi,3 for 1 ≤ i ≤ 5.

trajectory and known basis functions. Moreover, except for subsystem states, extra information transmission of local control inputs and local neighborhood consensus errors among linked subsystems can also be avoided.

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Wei Wang received her B.Eng. degree in Electrical Engineering and Automation from Beihang University (BUAA) in 2005, M.Sc. degree in Radio Frequency Communication Systems with Distinction from University of Southampton (UK) in 2006 and Ph.D. degree from Nanyang Technological University (Singapore) in 2011. From January 2012 to June 2015, she was a Lecturer with the Department of Automation at Tsinghua University, China. Currently, she is an Associate Professor with the School of Automation Science and Electrical Engineering at Beihang University and supported by BUAA Young Talent Recruitment Program. Her research interests include adaptive control of uncertain systems, distributed cooperative control of multi-agent systems, fault tolerant control and security control of cyber–physical systems. Wei Wang is the receipient of Zhang Si-Ying Outstanding Youth Paper Award in 2013 25th Chinese Control and Decision Conference.

networks, model based online learning and system identification, signal and image processing. Dr. Wen is an Associate Editor of a number of journals including Automatica, IEEE Transactions on Industrial Electronics a and IEEE Control Systems Magazine. He is the Executive Editor-in-Chief of Journal of Control and Decision. He served the IEEE Transactions on Automatic Control as an Associate Editor from January 2000 to December 2002. He has been actively involved in organizing international conferences playing the roles of General Chair, General Co-Chair, Technical Program Committee Chair, Program Committee Member, General Advisor, Publicity Chair and so on. He received the IES Prestigious Engineering Achievement Award 2005 from the Institution of Engineers, Singapore (IES) in 2005. He is a Fellow of IEEE, was a member of IEEE Fellow Committee from January 2011 to December 2013 and a Distinguished Lecturer of IEEE Control Systems Society from February 2010 to February 2013.

Changyun Wen received the B.Eng. degree from Xi’an Jiaotong University, Xi’an, China, in 1983 and the Ph.D. degree from the University of Newcastle, Australia in 1990. From August 1989 to August 1991, he was a Postdoctoral Fellow at University of Adelaide, Adelaide, Australia. Since August 1991, he has been with School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, where he is currently a Full Professor. His main research activities are in the areas of control systems and applications, intelligent power management system, smart grids, cyber–physicalsystems, complex systems and

Jiangshuai Huang received his B.Eng. and M.Sc. degrees in School of Automation from Huazhong University of Science & Technology, Wuhan, China in July 2007 and August 2009 respectively, and Ph.D. from Nanyang Technological University in 2015. He is currently with the Key Laboratory of Dependable Service Computing in Cyber Physical Society, Ministry of Education, China and with School of Automation, Chongqing University, Chongqing 400044, China. His research interests include adaptive control, nonlinear systems control, underactuated mechanical system control and multi-agent system control.