RESEARCH ARTICLE

Robust consensus tracking control of multiple mechanical systems under fixed and switching interaction topologies Jianhui Liu1, Bin Zhang2* 1 School of Automation Science and Electrical Engineering, Beihang University (BUAA), Beijing 100876, P.R. China, 2 School of Automation, Beijing University of Posts and Telecommunications (BUPT), Beijing 100876, P.R. China * [email protected]

a1111111111 a1111111111 a1111111111 a1111111111 a1111111111

OPEN ACCESS Citation: Liu J, Zhang B (2017) Robust consensus tracking control of multiple mechanical systems under fixed and switching interaction topologies. PLoS ONE 12(5): e0178330. https://doi.org/ 10.1371/journal.pone.0178330

Abstract Consensus tracking problems for multiple mechanical systems are considered in this paper, where information communications are limited between individuals and the desired trajectory is available to only a subset of the mechanical systems. A distributed tracking algorithm based on computed torque approach is proposed in the fixed interaction topology case, in which a robust feedback term is developed for each agent to estimate the external disturbances and the unknown agent dynamics. Then the result is extended to address the case under switching interaction topologies by using Lyapunov approaches and sufficient conditions are given. Two examples and numerical simulations are presented to validate the effectiveness of the proposed robust tracking method.

Editor: Xiaosong Hu, Chongqing University, CHINA Received: December 8, 2016

Introduction

Accepted: May 11, 2017

Multi-agent system has emerged as an active area of research, and drawn attention of scholars from a varieties of disciplines in the past decades. This trend is triggered by the promising applications of multi-agent system in fields like disaster rescuing, industry assembly lines, surveillance, etc. Each agent in the multi-agent system has limited task abilities. However, through interactions with each other, they can work as a team and accomplish cooperative behaviors such as consensus [1, 2], flocking [3], formation [4, 5], and state estimation [6, 7]. Among the studies of these cooperative behaviors, consensus behavior is the most fundamental one [8]. The basic issue of consensus control in multi-agent system is to design a distributed consensus law such that all the agents could be driven to an agreement. In recent years, many consensus control approaches have been proposed for multi-agent systems with different interaction topologies and dynamic models. In the early literature [9], graph theory was used to represent the interaction topologies, and as a result, the relationship between system stability and Laplacian eigenvalues was precisely revealed. In [10], directed graphs were used to represent the interaction topology, and results under dynamically changing interaction topologies were derived. Other representative literatures are [11–13], to name a few. [11] studied finitetime consensus problems for first-order integrators. [12] extended the problems to systems

Published: May 25, 2017 Copyright: © 2017 Liu, Zhang. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All relevant data are within the paper. Funding: This work was supported by the National Natural Science Foundation of China (NSFC: 61603050) and the Fundamental Research Funds for the Central Universities. Competing interests: The authors have declared that no competing interests exist.

PLOS ONE | https://doi.org/10.1371/journal.pone.0178330 May 25, 2017

1 / 20

Robust consensus tracking control

with uncertain dynamics based on H1 control theory. [13] considered the constrained consensus problem with a global optimization function. In addition, some results extended the consensus problem to a more general consensus tracking problem where the agents track a time-varying trajectory instead of a static equilibrium. In [14], a robust adaptive control algorithm was proposed for uncertain nonlinear systems, where the reference trajectory was known to all the following systems. In [15], the problem was solved under the condition that the reference trajectory was available only to a few agents. In [16], finite-time tracking control of multi-agent systems was considered with a sliding-mode approach. In [17], distributed observers were established to estimate unavailable system states. The results in [18] addressed consensus problems with the assumption that the second-order derivatives of the reference signals conform to some given policy known to the system. Most literatures dealing with consensus tracking problem are presented under the assumption that the dynamics of the agents are linear and certain. However, almost all the physical plants exhibit some kinds of nonlinearities, and external disturbances are inevitable in their dynamical processes. Several works attempt to address the tracking problem with uncertainties, but assumptions of these results are too conservative to achieve in practical situations. Motivated by the desire to achieve practical results with only necessary constraints on mechanical systems and reference trajectory, we try to address the robust consensus tracking problem in this paper. We aim to design a controller such that a group of mechanical systems under both fixed and switching topologies could maintain a satisfactory collective performance in the presence of uncertainties or external disturbances. In the fixed topology case, a distributed robust control law is devised based on the computed torque approach and algebraic graph theory. We further extend the results to the switching topology case. Sufficient conditions are given, under which the states of the agents could converge to a neighbourhood of the origin. Two numerical simulations are conducted to validate the effectiveness of our results. The remainder of this paper is organized as follows. First, we introduce the problem formation and the relevant notations. Then, the robust tracking control under fixed and switching topologies is discussed. In the Simulation Section, two numerical simulations are conducted to validate the effectiveness of the proposed method. At last, some discussions are made to conclude this paper.

Problem statement For a group of n mechanical systems, the dynamic model of the ith system is formulated by Euler-Lagrange equation [19, 20] Mi ðqi Þ€qi þ Ci ðqi ; q_i Þq_i þ Gi ðqi Þ þ fi ðq_i Þ þ ui ðtÞ ¼ ti

ð1Þ

where qi 2 Rm is the state of the ith system, Mi ðqi Þ 2 Rmm is the symmetric inertia matrix, Ci ðqi ; q_i Þ 2 Rmm is the matrix representing the centrifugal and Coriolis terms, Gi ðqi Þ 2 Rm is the vector of gravity terms, fi ðq_i Þ 2 Rm is the frictional term, ui ðtÞ 2 Rm denotes the bounded external disturbance and ti 2 Rm represents the control input vector. An assumption of the mechanical system equation described by Eq (1) is given as follows [20, 21]: Assumption 1. The symmetric inertial matrix Mi(qi), i 2 f1; . . . ; ng ≜ N , is positive definite, which satisfies:

PLOS ONE | https://doi.org/10.1371/journal.pone.0178330 May 25, 2017

=

=

lm k x k2 < xT Mi ðqi Þx < l k x k2 ; M

8qi ; x 2 Rm

ð2Þ

2 / 20

Robust consensus tracking control

where lm

≜ min minm lmin ðMi ðqi ÞÞ

lM

≜ max maxm lmax ðMi ðqi ÞÞ

8i2N 8qi 2R

8i2N 8qi 2R

In this paper, it is assumed that the information interchanges are bi-directional between the n agents through wireless networks or other sensors. Undirected graphs will be used throughout the paper to model the bi-directional interaction topologies among agents. Some basic knowledge and conventional notations in algebraic graph theory are given as follows. Let Gðn; εÞ be an undirected graph with n nodes ν = {ν1, ν2, . . ., νn} and the set of edges ε  ν × ν. The adjacency matrix A = [aij] is a symmetric matrix defined as aii = 0 and aij > 0 , (νi, νj) 2 ε. The Laplacian matrix of graph Gðn; εÞ is defined as L = D − A, where D = diag{d1, d2, . . ., dn} is Pn a diagonal matrix with diagonal entries di ¼ j¼1 aij for i = 1, 2, . . ., n. The set of neighbors of node νi is denoted by N i ¼ fnj 2 njðni ; nj Þ 2 εg. If there is a path between any two nodes of the graph Gðn; εÞ, then Gðn; εÞ is said to be connected. Suppose z is a nonempty subset of nodes ν, T then Gðn; ε ðz  zÞÞ is termed as an induced subgraph by z. A component of a graph Gðn; εÞ is defined as a maximal induced subgraph of Gðn; εÞ that is strongly connected. To characterize the variable interconnection topology, a piecewise-constant switching signal function sðtÞ : ½0; 1Þ ! f1; . . . ; Mg ≜ M is defined, where M 2 Zþ is the total count of possible interconnection graphs. The reference signals are denoted as qd ; q_d and €qd respectively. In this paper, qd ; q_d and €qd are only accessible to a subset of the n agents. The accesses of the agents to the trajectories are represented by a diagonal matrix C ¼ diagfc1 ; . . . ; cn g 2 Rnn , where ( ci ¼

1; if trajectory signals are available to agent i; ð3Þ 0; if trajectory signals not available to agent i:

The information exchange matrix [21] is defined as K ≜ L þ C, where L is the Laplacian matrix and C is defined in Eq (3). Lemma 1. [22] The Laplacian matrix of a component in graph Gðn; εÞ is a symmetric matrix with real eigenvalues that satisfy 0 ¼ l1  l2  l3 . . . lz  D where Δ = 2 × (max1iz di). Lemma 2. [18] If at least one agent in each component of graph G has access to the desired signals, then the information exchange matrix K = L + C is symmetric and positive definite. Definition 1. The robust tracking problem is said to be settled if for each ω > 0, there is T = T(ω) > 0 and a local distributed control law τi, i 2 {1, . . ., n}, such that k qi ðtÞ

qd ðtÞ k2  o

k q_i ðtÞ

q_d ðtÞ k2  o

ð4Þ

8t  t0 þ TðoÞ in the presence of frictional force and external disturbance.

PLOS ONE | https://doi.org/10.1371/journal.pone.0178330 May 25, 2017

3 / 20

Robust consensus tracking control

To facilitate the subsequent analysis, we define x ¼ ½x1T

x2T Š

T

ð5Þ

x1 ¼ ½ðq1

qd Þ

T

;...;

ðqn

qd Þ Š

T T

ð6Þ

x2 ¼ ½ðq_ 1

T q_ d Þ

;...;

ðq_ n

T T q_ d Þ Š

ð7Þ

X ei ¼

aij ðqi

qj Þ þ bi ðqi

qd Þ

ð8Þ

aij ðq_ i

q_ j Þ þ bi ðq_ i

q_ d Þ:

ð9Þ

j2N i

X e_ i ¼ j2N i

Before proceeding, we now introduce an important lemma, which will be used in the system stability analysis. Lemma 3. [23] Let V(t)  0 be a continuously differentiable function such that _ V ðtÞ  gVðtÞ þ k, where γ and κ are positive constants. Then the following inequality is satisfied VðtÞ  Vð0Þe

gt

k þ ð1 g

e gt Þ:

Robust tracking control under fixed topology Distributed scheme design In this subsection, a distributed scheme based on computed-torque control will be given. Computed-torque control is an important approach to decouple complex robotic dynamics, which is shown as follows ti ¼ Mi ðqi Þvi þ Ci ðqi ; q_ i Þq_ i þ Gi ðqi Þ

vi ¼ bi €qd

ei

a_e i

Zi

ð10Þ

ð11Þ

where α is a positive constant and ηi is the robust control term. In this paper, ηi is used to estimate the uncertain terms based on information from neighboring agents. Combining Eqs (1), (10) and (11), we get x_ ¼ Fx þ HðZ þ gÞ

PLOS ONE | https://doi.org/10.1371/journal.pone.0178330 May 25, 2017

ð12Þ

4 / 20

Robust consensus tracking control

where  F

¼

H

¼

 In

Im aK

0 K



 0

Im In

ð13Þ T T n

T 1

;...;

Z Š

Z

¼

½Z

g

¼

½gT1

;...;

gTn Š

gi

¼

ðbi

1Þ€qd

Mi 1 ðqi Þðfi ðq_ i Þ þ ui ðtÞÞ:

T

For further analysis, we present the following assumptions. Assumption 2. The first order and second order derivatives of the desired trajectory are all bounded (i.e., q_ d ; €qd 2 L1 ). Assumption 3. The state velocity q_ i is bounded (i.e., q_ i 2 L1 ), and the frictional vector fi ðq_ i Þ and its first order and second order derivatives with respect to q_ i are bounded (i.e., 2

fi ðq_ i Þ; @f@i ðq_q_i i Þ ; @ @f2i ðq_q_ i Þ 2 L1 ). i

Under Assumption 2 and 3, it is easy to see k gi k1

¼ k ðbi  jbi

1Þ€qd

Mi 1 ðqi Þðfi ðq_ i Þ þ ui ðtÞÞk1

1j
k €qi k1 þ

ð14Þ

ðk fi ðq_ i Þk1 þ k ui ðtÞk1 Þ ≜ ri : lm T

To facilitate the analysis in the following subsection, we define r ¼ ½rT1 ; . . . ; rTn Š and χ = kρk2. Based on the previous preparations, we present the design of ηi as follows Zi ¼

w2 ðbei þ e_ i Þ 

ð15Þ

where  and β are positive parameters which have effects on the convergence precision. The definition of ηi can be utilized to eliminate the effect of frictional vector and external disturbance. In this section, we consider the fixed topology case where at least one agent in each component has access to the desired signals. By Lemma 2, we know that K = L + C is positive definite, and therefore we can define the smallest and largest eigenvalues of matrix K as λmin(K) > 0 and λmax(K) > 0. Before showing our main results, we present the following lemmas, which will be used in the prove of the theorems. " # " # K bI 2bK K Lemma 4. Let P ¼ and Q ¼ , where α and β are positive bI I K 2ðaK bIÞ parameters. If α and β satisfy ab ¼ 1

ð16Þ

pffiffi 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 and λi − β2 > 0, which imply the roots of Eq (18) are both positive. Thus P is positive definite. Similarly, the eigenvalue ω of Q satisfies o2

2ðbli þ ali

bÞo þ 4abl2i

4b2 li

l2i ¼ 0:

ð19Þ

We have 2 bÞ ¼ ðb2 gi þ gi b

2ðbgi þ agi

2 b2 Þ > ðgi b 

4abg

2 i

2

4b gi

2 i

2

2 i

g ¼ 3g

4b gi ¼ 3gi gi

b2 Þ > 0

4 2 b 3

ð20Þ

 > 0:

ð21Þ

Thus the roots of Eq (19) are positive, i.e., Q is positive definite. Remark 1: The Assumptions used in this paper are idiomatically in the study of physical systems described by Euler-Lagrange equation. The elements of matrix Mi(qi) are rotary inertias of the joints and the readers can refer to [19, 20] for the precise algebraic expressions. An important property is that Mi(qi) is positive definite and bounded. Assumptions 2 and 3 mention that the physical parameters and desired trajectory are all bounded, which are naturally in the dynamic behaviour of physical systems [15]. Assumptions 2 and 3 are basically referring to the Lipschits condition. If k fi ðq_ i þ DÞ fi ðq_ i Þ k L k D k, then we can see that @f@i ðq_q_i i Þ 2 L1 .

Convergence analysis Theorem 1. Consider n Euler-Lagrange systems described as Eq (1). Let Assumptions 1, 2 and 3 be fulfilled. Suppose the interaction topology is fixed and at least one agent in each component has access to the desired signals. Then, under the control strategy Eqs (10), (11) and (15), rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for any sufficiently small constant  > 0 and  ¼ lmin1ðPÞ 1 þ 2lmin1ðKÞc , there is TðÞ ¼

lmax ðPÞ Vðt0 Þ ln lmin ðQÞ 

such that k xðtÞ k2  ;

8t  t0 þ TðÞ

ð22Þ

where V(t0) = xT(t0)(P Im)x(t0) and the control parameters satisfy ab ¼ 1

ð23Þ

pffiffi 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0, there is TðÞ ¼ c1 ln

Vðt0 Þ 

k xðtÞ k2  ;

such that

8t  t0 þ TðÞ

ð32Þ

which completes the proof. Remark 2: In contrast to the proof of Theorem 1, where algebraic approaches are conducted, we now give a geometric proof based on Fig 1. The Eq (28) shows that V_  xT ðQ Im Þx þ 2lmin ðKÞ which implies V_ < 0 for pffiffi k x k2 > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ≜ d. 2lmin ðQÞlmin ðKÞ

Let’s define Bd

¼

SðlÞ ¼ l

¼

fxj k x k2  dg fxjxT Px  lg

ð33Þ

minfljSðlÞ  Bd g

i.e., SðlÞ is the minimum ellipsoid containing Bδ. Next, we will show that for any initial value x (t0), the trajectory x(t) converges to SðlÞ in finite time. Obviously, xðtÞ 2 SðlÞ; 8t  t0 for any xðt0 Þ 2 SðlÞ. Thus in the following, we just need to consider the case when xðt0 Þ 2 = SðlÞ. Let k0 ¼ xT ðt0 ÞPxðt0 Þ and T c0 ¼ minfx ðQ Im Þx 2lmin ðKÞ jx 2 Sðk0 Þ SðlÞg. By Eq (28), we have Z

t

V_ dt 

c0 ðt

t0 Þ:

ð34Þ

t0

PLOS ONE | https://doi.org/10.1371/journal.pone.0178330 May 25, 2017

8 / 20

Robust consensus tracking control

Thus, the time point t1 when V(t) reaches the boundary of the ellipsoid SðlÞ satisfies t1  t0 þ

k0

l c0

:

ð35Þ

Then V(t) will be limited in the bounded ellipsoid SðlÞ for any t  t1. Thus, we have lmin ðPÞ k xðtÞ k22  VðtÞ  l;

8t  t1

ð36Þ

l

i.e., k xðtÞ k2  lmin ðPÞ ; 8t  t1 . It means that the norm of the trajectory error vector can be reduced to any prescribed positive value.

Robust tracking control under switching topologies Distributed scheme design In this section, we extend the results in the above section to the switching topology case, where the switching signal is chosen as sðtÞ : ½0; 1Þ ! M. Consider an infinite sequence of nonempty, bounded, and contiguous time-intervals [tr, tr+1), r = 0, 1, . . . with t0 = 0, tr+1  tr+T for a constant T > 0. In each interval [tr, tr+1), there is a sequence of subintervals ½tr0 ; tr1 Þ; ½tr1 ; tr2 Þ; . . . ; ½trmr 1 ; trmr Þ;

tr ¼ tr0 ; trþ1 ¼ trmr

satisfying trjþ1 trj  t; 0  j  mr 1, for some integer mr  0 and a given constant τ > 0, such that the interaction graph G sðtÞ switches at trj and does not change during each subinterval ½trj ; trjþ1 Þ. Suppose the interaction graph G s in subinterval ½trj ; trjþ1 Þ has lσ  1 connected components with the corresponding node numbers denoted by y1s ; . . . ; ylss . For simplicity, we suppose that the first h(1  h  lσ) components have accesses to the desired signals. Then, by Lemmas 1 and 2, we know that matrix Kσ = Lσ + Bσ is semi-positive definite and there is matrix Ss 2 Rnn ; STs Ss ¼ I, such that STs Ks Ss ¼ diagfKs1 ; Ks2 ; . . . ; Ksls g ≜ Ls

ð37Þ

where ( Ki

s

¼

diagflis;1 ; lis;2 ; . . . ; lis;yis g;

Ksi

¼

diagf0; lis;2 ; . . . ; lis;yis g;

1  i  h; ð38Þ h < i  ls :

We define 8 i ^s K > > > > < ^ is K > > > > :^ Ls

PLOS ONE | https://doi.org/10.1371/journal.pone.0178330 May 25, 2017

¼ diagflis;1 ; lis;2 ; . . . ; lis;yis g; 1  i  h; ¼ diagflis;2 ; lis;2 ; . . . ; lis;yis g; h < i  ls ;

ð39Þ

^ 1s ; K ^ 2s ; . . . ; K ^ lss g: ¼ diagfK

9 / 20

Robust consensus tracking control

It is easy to see that Ks

¼

Ss Ls STs pffiffiffiffiffi qffiffiffiffiffiffi 1 qffiffiffiffiffiffipffiffiffiffiffi T ^ ^ ^ L L Ls Ss Ss Ls L s s s pffiffiffiffiffi pffiffiffiffiffi 1 pffiffiffiffiffi pffiffiffiffiffi T ^ Ls Ls Ss Ss Ls Ls L s

¼

^ 1 ST ÞðS L ST Þ ðSs Ls STs ÞðSs L s s s s s

¼

Ks Fs Ks

¼ ¼

ð40Þ

^ 1 ST is positive definite. where Fs ¼ Ss L s s In this case, the distributed robust tracking algorithm for Eq (1) is defined as ti ¼ Mi ðqi Þvi þ Ci ðqi ; q_ i Þq_ i þ Gi ðqi Þ vi ¼ bi €qd þ mbi q_ d

mq_ i

dei

de_ i

ð41Þ Zi

ð42Þ

where μ > 0 and d > 0 are positive constants; Zi is the robust input term used to eliminate the effect of uncertain terms, which will be shown in the following part. Combining Eqs (1), (41) and (42), we get x_ ¼ ZsðtÞ x þ EðZ þ gÞ

ð43Þ

where 2 ZsðtÞ

E

3

In

0

¼ 4 dKsðtÞ mI " # 0

Im ¼ In

dKsðtÞ

5 Im

ð44Þ T

Z

¼ ½ZT1

;...;

ZTn Š

g

¼ ½gT1

;...;

gTn Š

gi

¼ ðbi

1Þ€qd þ mðbi

T

1Þq_ d

Mi 1 ðqi Þðfi ðq_ i Þ þ ui ðtÞÞ:

Similar to Eq (14), we know gi is upper bounded and satisfies the following inequality k gi k1

¼

k ðbi



jb1

1Þ€qd

mðbi

1Þq_ d

Mi 1 ðqi Þðfi ðq_ i Þ þ ui ðtÞÞk1

1jðk €qd k1 þ m k q_ d k1 Þ þ

1 ðk fi ðq_ i Þ k1 þ k ui ðtÞk1 Þ ≜ ri : lm

ð45Þ

T

Thus g ¼ ½gT1 ; . . . ; gTn Š is upper bounded (i.e., g 2 L1 ). A necessary requirement in the investigation of multi-agent systems under switching topology is that there is a bounded piecewise continuous vector ϕσ(t) satisfying g ¼ ðKsðtÞ Im ÞsðtÞ

ð46Þ

where the upper bound of ϕσ(t) is denoted by φ (i.e., kϕσ(t)k1  φ). Based on the preparations, the robust input term is defined as Zi ¼

PLOS ONE | https://doi.org/10.1371/journal.pone.0178330 May 25, 2017

φ2 ðe þ e_ i Þ: 2 i

ð47Þ

10 / 20

Robust consensus tracking control

It is emphasized that φ is the upper bound of the uncertain terms and  is a design parameter which has an effect on the consensus precision. From Eq (47) we can see that there is significant positive correlation between the control energy and φ, and there is a significant negative correlation between the control energy and . The following lemma will be used in the robust convergence analysis. " # " # 2dKsðtÞ 2dKsðtÞ mI I Lemma 5. Let D ¼ and QsðtÞ ¼ , where μ and d 2dKsðtÞ 2ðmI þ dKsðtÞ IÞ I I are positive parameters. If μ > 1, then D is positive definite and Qσ(t) is positive semi-definite for 8t 2 [0, 1). Furthermore, Qσ(t) > 0 if and only if Kσ(t) > 0. Proof: The fact that D > 0 is obvious and the proof is omitted here. By Eq (37), we know Kσ can be transformed into a diagonal matrix as Ks ¼ Ss Ls STs . For simplicity, we redefine Λσ as Ls ¼ diagfl1s ; . . . ; lns g, where lis  0 is the ith eigenvalue of Kσ. Then Qσ can be written as " Qs

Ss

0nn

0nn

Ss

¼

#"

2dLs 2dLs

#"

2dLs 2ððm

1ÞI þ dLs Þ

STs

0nn

0nn

STs

# :

ð48Þ

For any eigenvalue ωσ of Qσ, we have that ðos

2dlis Þðos

2ðm

2

1 þ dlis ÞÞ

ð2dlis Þ ¼ 0

ð49Þ

i.e., o2s

2ðm

1 þ 2dlis Þos þ 4ðm

1Þdlis ¼ 0:

ð50Þ

For any μ > 1 and d > 0, we know 2ðm 1 þ 2dlis Þ > 0 and 4ðm 1Þdlis  0. It follows that ωσ  0, i.e., Qσ is positive semi-definite. Furthermore, ωσ > 0 if and only if 4ðm 1Þdlis > 0; i ¼ f1; 2; . . . ; ng, i.e., Kσ > 0.

Convergence analysis In this subsection, we will prove that the closed-loop system could maintain a satisfactory performance with switching topologies. Before giving the main result, we will present some preliminary definitions. Note that sðtÞ : ½0; 1Þ ! M is the finite switching signal and does not change during the time intervals no less than τ. We define x ¼ minsðtÞ2M lmin ðFsðtÞ Þ, % ¼ minsðtÞ2M lmin ðQsðtÞ Þ, and n ¼ lmax%ðDÞ, where D is defined in Lemma 5. Theorem 2. Consider n Euler-Lagrange systems described as Eq (1) with switching interaction topologies. Let Assumptions 1, 2 and 3 be fulfilled. Suppose that during each time interval [tr, tr+1), tr+1  tr + T, there is one subinterval ½trj ; trjþ1 Þ such that all the components have accesses to the desired signals. Then, under the control strategy Eqs (41), (42) and (47), for any ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi 1 sufficiently small constant  > 0 and  ¼ lmin ðDÞ 1 þ Tx þ ð1 1þnt , there is e nt Þnx   1 Vð0Þ wðÞ ¼ 1 þ ln T nt  such that k xðtÞ k2  ;

PLOS ONE | https://doi.org/10.1371/journal.pone.0178330 May 25, 2017

8t  wðÞ

ð51Þ

11 / 20

Robust consensus tracking control

where V(0) = xT(0)(D Im)x(0) and the control parameters satisfy m>1

ð52Þ

d > 0:

ð53Þ

Proof: Take the Lyapunov function VðtÞ ¼ xT ðtÞDxðtÞ, where ð54Þ

D ¼ D Im :

We can see that V(t) is piecewise differentiable and the derivative of V(t) along the solutions of the closed-loop system during ½trj ; trjþ1 Þ is V_ ðtÞ ¼

T xT ðZsðtÞ D þ DZsðtÞ Þx þ 2xT DEðZ þ gÞ

¼

xT ðQsðtÞ Im Þx þ 2xT DEZ þ 2xT DEKsðtÞ φsðtÞ



xT ðQsðtÞ Im Þx þ 2xT DEZ þ 2φk ðKsðtÞ Im ÞET Dx k2 :

ð55Þ

From Eqs (40) and (47), we can get that xT DEZ

φ2 T x DEðe þ e_ Þ 2 φ2 T ¼ x DEðKsðtÞ Im ÞET Dx 2 φ2 T ¼ x DEðKsðtÞ Im ÞðFsðtÞ Im ÞðKsðtÞ Im ÞET Dx 2 xφ2  k ðKsðtÞ Im ÞET Dx k22 : 2

ð56Þ

pffiffi   xφ pffiffi k KsðtÞ Im ET Dxk2 x QsðtÞ Im x     xT QsðtÞ Im x þ x  nV ðt Þ þ : x

ð57Þ

¼

Thus, we have V_ ðtÞ



  

pffiffi 2   pffiffi þ x x



T

According to Lemma 3, we obtain jþ1

Vðtrjþ1 Þ  e

nðtr

 e

nðtr

jþ1

 ð1 nx j  tr Þ Vðtrj Þ þ : nx j

tr Þ

Vðtrj Þ þ

e

jþ1

nðtr

j

tr Þ

Þ ð58Þ

For any other subinterval ½tri ; triþ1 Þ; i 6¼ j, in which not all the components have accesses to the desired signals, we have V_ ðtÞ  x . It follows that  Vðtriþ1 Þ  Vðtri Þ þ ðtriþ1 x

PLOS ONE | https://doi.org/10.1371/journal.pone.0178330 May 25, 2017

tri Þ:

ð59Þ

12 / 20

Robust consensus tracking control

Thus, we get Vðtrþ1 Þ

  Vðtrjþ1 Þ þ ðtrþ1 x

trjþ1 Þ

  Vðtrj Þ þ þ ðtrþ1 trjþ1 Þ nx x   jþ1 j     e nðtr tr Þ Vðtr Þ þ ðtrj tr Þ þ þ ðtrþ1 x nx x   T 1  e nt Vðtr Þ þ þ : x nx

 e

jþ1

nðtr

j

tr Þ

ð60Þ trjþ1 Þ

It follows that Vðtrþ1 Þ  e

ðrþ1Þnt

 e

ðrþ1Þnt

Vð0Þ þ ðe

rnt

þe

ðr 1Þnt

þ . . . þ 1Þ

  T 1 þ  x nx

1 þ nT Vð0Þ þ : ð1 e nt Þnx

ð61Þ

Therefore, for any tr < t < tr+1 we have VðtÞ   

 Vðtr Þ þ T x



 T 1 þ nt e Vð0Þ þ þ  x ð1 e nt Þnx   T 1 þ nt t e nt½T Š Vð0Þ þ þ  x ð1 e nt Þnx rnt

 is the integer part of Tt which satisfies Tt > Tt 1.
Obviously, for any t  1 þ nt1 ln Vð0Þ T, we have    T 1 þ nt t VðtÞ  e nt½TŠ Vð0Þ þ þ  x ð1 e nt Þnx   T 1 þ nt ntðTt 1Þ Vð0Þ þ < e þ  x ð1 e nt Þnx   T 1 þ nt  1þ þ : x ð1 e nt Þnx

where

ð62Þ

t T

ð63Þ

It follows that  lmin ðDÞ k xðtÞ k22  VðtÞ