Finite-time consensus of multiple nonholonomic chained-form systems based on recursive distributed observer

Automatica 62 (2015) 236–242

Contents lists available at ScienceDirect

Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Finite-time consensus of multiple nonholonomic chained-form systems based on recursive distributed observer✩ Haibo Du a , Guanghui Wen b , Xinghuo Yu c,d , Shihua Li d , Michael Z.Q. Chen e a

School of Electrical Engineering and Automation, Hefei University of Technology, Hefei, Anhui 230009, China

b

Department of Mathematics, Southeast University, Nanjing, Jiangsu 210096, China

c

Platform Technologies Research Institute, RMIT University, Melbourne VIC 3001, Australia

d

School of Automation, Southeast University, Nanjing, Jiangsu 210096, China

e

Department of Mechanical Engineering, The University of Hong Kong, Hong Kong

article

info

Article history: Received 1 August 2014 Received in revised form 30 April 2015 Accepted 14 August 2015 Available online 11 November 2015 Keywords: Finite-time observer Nonholonomic systems Distributed control Chained-form

abstract The consensus tracking problem of multiple nonholonomic high-order chained-form systems is considered in this paper. A finite-time observer-based distributed control strategy is proposed. At the first step, a distributed finite-time convergent observer is proposed for each agent to estimate the leader’s state in a finite time. Then, a finite-time tracking controller is designed to track the estimated state and the leader’s attitude in a finite time. As an application of the proposed results, finite-time formation control of multiple wheeled mobile robots is studied and a finite-time formation control algorithm is proposed. To show effectiveness of the proposed approach, a simulation example is given. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Distributed control has attracted increasing interest in various research communities due to its many different applications, such as network synchronization (Li, Duan, Chen, & Huang, 2010; Yu, Chen, & Lü, 2009), formation control (Dong & Farrell, 2008; Fax & Murray, 2004; Ren & Atkins, 2007), attitude alignment (Ren, 2007), flocking (Olfati-Saber, 2006; Zhu, Lu, & Yu, 2013), etc. An active topic in the distributed control theory is the consensus problem, which requires that the states of all agents reach a common value (usually called consensus state) by using an appropriate algorithm. From the viewpoint of time optimization, the convergence rate of consensus is required to be as fast as possible. In Kim and Mesbahi (2006); Olfati-Saber and Murray (2004), it was shown that the convergence rate is related to the communication topology

✩ This work was supported in part by the Natural Science Foundation of China (61304007,61374053,61304168,61473080), and Science Foundation for Distinguished Young Scholars of Jiangsu Province (BK20130018). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Wei Ren under the direction of Editor Christos G. Cassandras. E-mail addresses: [email protected] (H. Du), [email protected] (G. Wen), [email protected] (X. Yu), [email protected] (S. Li), [email protected] (M.Z.Q. Chen).

http://dx.doi.org/10.1016/j.automatica.2015.09.026 0005-1098/© 2015 Elsevier Ltd. All rights reserved.

graph. Note that most of the existing consensus algorithms are required to satisfy the Lipschitz continuity condition, which means that the convergence rate is at best exponential with an infinite convergence time (Bhat & Bernstein, 1998). To enhance the convergence speed, the finite-time control technique (Bhat & Bernstein, 2000) has been employed, which guarantees the states consensus to be achieved in a finite time. Actually, besides a faster convergence rate, the finite-time control usually demonstrates better disturbance rejection properties, see some results about rigorous disturbance rejection analysis (Bhat & Bernstein, 2000; Li, Du, & Lin, 2011). In the literature, some finite-time consensus algorithms were proposed for first-order/second-order multiagent systems in Li et al. (2011), Xiao, Wang, Chen, and Gao (2009), Hui, Haddad, and Bhat (2008), Wang and Xiao (2010), Wang and Hong (2008), Lu, Lu, Chen, and Lu (2013) and Zhang, Yang, Zhao, and Wen (2013) and for multiple rigid spacecraft in Du, Li, and Qian (2011); Meng, Ren, and You (2010). As an important application of the consensus algorithm, the cooperative control of multiple nonholonomic mobile robots has attracted much interest. In Lin, Bruce, and Maggiore (2005), the formation control problem of multiple unicycles was discussed. In Dimarogonas and Kyriakopoulos (2007); Zhu et al. (2013), the rendezvous problems for multiple nonholonomic agents were studied. In Dong and Farrell (2008), a backstepping design scheme was employed to solve the problem of formation

H. Du et al. / Automatica 62 (2015) 236–242

control for multiple nonholonomic chained-form systems. Later, in Dong (2013), cascaded systems’ theory was employed to design distributed state and output feedback tracking controllers for multiple nonholonomic chained systems. In Liu and Jiang (2013), distributed nonlinear controllers without global position were proposed to solve the leader-following formation control of unicycle robots. Up to now, most results for cooperative control of multiple nonholonomic mobile robots have been about asymptotical stability. That is, the desired formation cannot be achieved in a finite time. The main results related to finite-time control are mainly about single nonholonomic mobile robot, for example the works in Hong, Wang, and Xi (2005); Wu, Wang, and Zong (2005). For multiple nonholonomic mobile robots, the works in Ou, Du, and Li (2014); Wang, Qiu, and Zhang (2012) proposed some finitetime cooperative formation control algorithms. However, these results are only for nonholonomic mobile robots with low-order models. Since many mechanical systems such as wheeled mobile robots, mobile vehicles with multiple trailers, the knife-edge, and the space robotics can be written in the form of high-order chained structure based on certain coordinate changes, see examples in Jiang and Nijmeijer (1999); Kolmanovsky and McClamroch (1995); Wu et al. (2005), it is interesting to investigate the distributed control for this kind of nonholonomic mobile robots. This paper will focus on solving the finite-time consensus problem for leader-following nonholonomic chained-form systems. Each nonholonomic system is described by a high-order chained structure. To this end, an observer-based distributed control strategy is employed. At the first step, design a distributed finite-time observer to estimate the leader’s state in a finite time. Then, the distributed finite-time consensus tracking problem for multiple nonholonomic chained-form systems is transformed to a finite-time tracking control problem for a single nonholonomic chained-form system. The technique of adding a power integrator (Qian & Lin, 2001) and a recursive design method are employed to explicitly construct a high-order finite-time controller. Finally, an observerbased distributed finite-time consensus algorithm is developed and a theoretic analysis is given. As an application of the proposed results, finite-time formation control of multiple wheeled mobile robots is studied.

237

Assumption 2.1. There are two constants L1 , L2 such that |˙ud1 (t )| ≤ L1 , |˙ud2 (t )| ≤ L2 for ∀t ≥ 0. Graph theory. Each agent is a node and the information exchange is denoted by a directed graph G(A) = {V , E , A}. V = {vi , i = 1, . . . , n} is the set of nodes, E ⊆ V × V is the set of edges and A = [aij ] ∈ Rn×n is the weighted adjacency matrix of the graph G(A) with non-negative adjacency elements aij . If there is an edge from node j to node i, i.e., (vj , vi ) ∈ E, then aij > 0. Moreover, assume that aii = 0 for all i ∈ Γ . The set of neighbors of node i is denoted by Ni = {j : (vj , vi ) ∈ E }. The Laplacian matrix  of digraph n G is L = D − A, where D = diag {d1 , . . . , dn } with di = j=1 aij =  j∈Ni aij . A path in directed graph G from vi1 to vik is a sequence of vi1 , vi2 , . . . , vik of finite nodes starting with vi1 and ending with vik such that (vil , vil+1 ) ∈ E for l = 1, 2, . . . , k − 1. The directed graph G is strongly connected if there is a path between any two distinct vertices. As that in Zhang et al. (2013), a directed graph satisfies the detail-balanced condition in weights if there exist some scalars pi > 0, such that pi aij = pj aji , for all i, j ∈ Γ . Denote P = diag {p1 , . . . , pn }. Assume that the reference state is represented by a leader. The connection weight between the n robot and the leader is denoted by bi , i ∈ Γ . If the ith robot has access to the information of leader, then bi > 0, otherwise, bi = 0. Let B = diag {b1 , . . . , bn }. Assumption 2.2. For the considered multiple chained-form systems (1)–(2), the graph G for the n follower robots is strongly connected and detail-balanced, and there is at least one robot which can directly obtain the leader’s information, i.e., B ̸= 0. Definition 2.1. Define sigα (x) = sign(x)|x|α , where α > 0, x ∈ R and sign(·) is the standard sign function. Lemma 2.1 (Zhang et al. (2013)). For the considered multiple chained-form systems (1)–(2), if Assumption 2.2 holds, then matrix P (L + B) is positive definite. Lemma 2.2 (Bhat and Bernstein (2000)). Consider the nonlinear system x˙ = f (x), f (0) = 0, x ∈ Rn , where f (·) : Rn → Rn is a continuous function. Suppose there exists a positive definite continuous function V (x) such that V˙ (x) + c (V (x))α ≤ 0, where c > 0 and α ∈ (0, 1). Then V (x) approaches 0 in a finite time. In V (x(0))1−α c (1−α)

2. Problem formulation and preliminaries

addition, the finite convergence time T satisfies that T ≤

Consider the consensus tracking problem for n nonholonomic chained-form systems. Each chained-form system is represented by an agent. Let Γ = {1, 2, . . . , n}. As in Dong and Farrell (2008), each agent is described in the form of an extended chained structure, which is given as:

Lemma 2.3 (Qian and Lin (2001)). If 0 < p ≤ 1 and is a ratio of two odd integers, then |xp − yp | ≤ 21−p |x − y|p .

σ˙ i,1 = ui,1 , σ˙ i,2 = σi,3 ui,1 , . . . , σ˙ i,m−1 = σi,m ui,1 , σ˙ i,m = ui,2 i ∈ Γ ,

where σi = [σi,1 , . . . , σi,m ] is the system state of ith mobile agent and ui = [ui,1 , ui,2 ]T is the control input. The goal is to design a distributed control algorithm such that the states of all agents reach consensus in a finite time and the final consensus state is the reference state. Assume that the reference consensus state or trajectory is represented by a virtual leader, whose information is available only to a subset of the followers. Without loss of generality, the dynamics of the leader is described by:

σ˙ md = ud2 .

σ˙ 2d = σ3d ud1 ,

. . . , σ˙ md −1 = σmd ud1 ,

Lemma 2.4 (Qian and Lin (2001)). Let c , d > 0. For any γ > 0, the following inequality holds for ∀x, y ∈ R: |x|c |y|d ≤ c /(c + d)γ |x|c +d + d/(c + d)γ −c /d |y|c +d .

(1)

T

σ˙ 1d = ud1 ,

.

(2)

Lemma 2.5 (Hardy, Littlewood, and Polya (1952)). For ∀xi ∈ R, i = 1, . . . , n, and a real number p ∈ (0, 1], (|x1 | + · · · + |xn |)p ≤ |x1 |p + · · · + |xn |p . 3. Main results In this section, we will show that the finite-time consensus tracking problem for multiple chained-form systems (1)-(2) is solvable. The solution is given in two steps: • For each follower agent, a distributed finite-time observer is proposed to estimate the leader’s state in a finite time. • For each follower agent, an observer-based finite-time control law is proposed such that the desired state is tracked in a finite time.

238

H. Du et al. / Automatica 62 (2015) 236–242

3.1. Recursive design of distributed finite-time observer

By Lemma 2.5, we have

For ith agent, denote σˆ i = [ σi,1 , . . . ,  σi,m ] , uˆ i = [ˆui,1 , uˆ i,2 ] as the estimate values for desired state σ d = [σ1d , . . . , σmd ]T , ud = [ud1 , ud2 ]T . T

T



n 

V˙ m ≤ − (lm − |˙ud2 |)

p2i |εi,1 |2

i ∈ Γ, (3)

and



(9)







2 Vm (t ) ≤ 2 Vm (0) − (lm − L2 ) 2λmin (P (L + B))

>

 12

t.

( ) ≡ 0, ∀t ≥ Tm , where Tm = say that  ui,2 − ud2 ≡ 0, ∀t ≥ Tm ,

L2 , then Vm t 21/2 Vm (0)1/2 . That is to (lm −L2 )(λmin (P (L+B))1/2 )

∀i ∈ Γ .

j∈Ni

 σ˙ i,m

pi |εi,1 |.

As shown in Khalil (2002), a straightforward calculation leads to

Since lm

aij ( σi,s−1 −  σj,s−1 )

Second, consider the (m − 1)-th equation in (4) and the error variable ei,m =  σi,m − σmd , i ∈ Γ . It follows from (2) and (4) that

 ) + σi,s ui , 1 ,

+ bi ( σi,s−1 − σ s = 3, . . . , m,  = − lm−1 · sigη aij ( σi,m −  σj,m ) d s−1

i=1

 21

j∈Ni

 σ˙ i,s−1 = − ls−2 · sigη

n

  12 1 ≤ − (lm − L2 ) 2λmin (P (L + B)) Vm2 .

j∈Ni

 u˙ i,1 = − k2 · sign

≤

i=1



 aij ( ui,1 −  uj,1 ) + bi ( ui,1 − ud1 ) ,

 21

 21  = − (lm − |˙ud2 |) F2T (L + B)T PP (L + B)F2

aij ( σi,1 −  σj,1 ) + bi ( σi,1 − σ1d ) +  ui , 1 ,



2 2 i=1 pi |εi,1 |

Substituting this inequality into (8) yields

Theorem 1. For each agent under Assumptions 2.1 and 2.2, if the distributed observer is proposed as:

 σ˙ i,1 = − k1 · sigη

 n

e˙ i,m = − lm−1 · sigη

j∈Ni

aij ( σi,m −  σj,m ) + bi ( σi,m − σmd )



j∈Ni

 + bi ( σi,m − σmd ) +  ui,2 ,    aij ( ui , 2 −  uj,2 ) + bi ( ui,2 − ud2 ) , u˙ i,2 = − lm · sign

+ ui,2 −

ud2

,

i ∈ Γ.

(10)

Based on the results in the first step, when t ≥ Tm , the error dynamic system (10) is equivalent to

j∈Ni

i ∈ Γ,



(4)

then σˆ i → σ d , uˆ i → ud in a finite time, where k1 > 0, k2 > L1 , li > 0, i = 1, . . . , m − 1, lm > L2 , 0 < η < 1. Proof. Define

e˙ i,m = −lm−1 · sigη





aij ( σi,m −  σj,m ) + bi ( σi,m − σmd ) ,

j∈Ni

i ∈ Γ.

(11)

As that in the first step, define εi,2 = j∈Ni aij (ei,m − ej,m ) + bi ei,m , i ∈ Γ . System (11) can be rewritten as:



ei = [ei,1 , . . . , ei,m ]T = σˆ i − σ d

= [σˆ i,1 , . . . , σˆ i,m ]T − [σ1d , . . . , σmd ]T ,

i ∈ Γ,

(5)

e˙ i,m = − lm−1 · sigη εi,2 ,

(6)

Denote Em = [e1,m , . . . , en,m ]T and ε2 = [ε1,2 , . . . , εn,2 ]T , which means that (L + B)Em = ε2 . Use a similar calculation as that in the T first step and choose Lyapunov candidate Vm−1 = 12 Em (L + B)T PEm ,

and fi = [fi,1 , fi,2 ]T = uˆ i − ud = [ˆui,1 , uˆ i,2 ]T − [ud1 , ud2 ]T ,

i ∈ Γ,

as the observer error for ith agent. A bottom-up inductive argument will be employed to prove that ei , fi converge to zero in a finite time. Since the observer subsystem (3) can be regraded as a special case of subsystem (4), we only give the proof procedure for subsystem (4). First, consider the mth equation in (4) and the error variable fi,2 =  ui,2 − ud2 , i ∈ Γ . By (2) and (4), we have



f˙i,2 = − lm · sign

aij [( ui,2 − ud2 ) − ( uj,2 − ud2 )]

j∈Ni

 + bi ( ui,2 − ud2 ) − u˙ d2   = − lm · sign aij (fi,2 − fj,2 ) + bi fi,2 − u˙ d2 ,

i ∈ Γ.

(7)

j∈Ni

Define εi,1 = j∈Ni aij (fi,2 − fj,2 ) + bi fi,2 , i ∈ Γ . Let F2 = [f1,2 , . . . , fn,2 ]T and ε1 = [ε1,1 , . . . , εn,1 ]T . It follows from the definitions of L and B that (L + B)F2 = ε1 . Choose a Lyapunov candidate Vm = 21 F2T (L + B)T PF2 , where the matrix P is given in Section 2. Differentiating Vm along the dynamics (7) is



V˙ m =F2T (L + B)T P

  × lm · [−sign(ε1,1 ), . . . , −sign(εn,1 )]T − [˙ud2 , . . . , u˙ d2 ]T n

≤ − (lm − |˙ud2 |)

 i=1

pi |εi,1 |.

(8)

i ∈ Γ.

(12)

 1+η 2



which leads to V˙ m−1 ≤ −lm−1 2λmin (P (L + B))

1+η

Vm−2 1 . Since

lm−1 > 0 and 0 < 2 < 1, then it follows from Lemma 2.2 that Vm−1 converges to zero in a finite time. Combining the results of the first step, it can be concluded that  σi,m − σmd ≡ 0, ∀t ≥ Tm + Tm−1 , 1+η

∀i ∈ Γ , where Tm−1 =

1+η 2(1−η)/2 Vm−1 (0) 2

1+η lm−1 (1−η)(λmin (P (L+B))) 2

.

Note that the proof of previous two steps is available for observer subsystem (3) in the same way. That is to say that there is a finite time T ∗ such that σˆ i,1 ≡ σ1d , uˆ i,1 = ud1 , ∀t ≥ T ∗ , ∀i ∈ Γ . With this fact in mind, and based on the above recursive proof, it can be proven that there is a finite time T ∗ such that for j = 2, . . . , m − 1, σˆ i,j ≡ σjd , ∀t ≥ T ∗ , ∀i ∈ Γ , step by step. Therefore,

σˆ i ≡ σ d , uˆ i ≡ ud , ∀t ≥ T ∗ , ∀i ∈ Γ .



3.2. Design of finite-time tracking controller Based on the previous result, the consensus tracking control problem for multiple chained-form systems is transformed to the tracking control problem for single chained-form system. To this end, for each agent, define

σ¯ i = [σ¯ i,1 , . . . , σ¯ i,m ]T = σi − σˆ i ,

i ∈ Γ,

(13)

as the tracking error between the actual state σi and the estimated desired state σˆ i .

H. Du et al. / Automatica 62 (2015) 236–242

When t ≥ T ∗ , it follows from (1) and (3)–(4) that the error dynamics equation satisfies:

σ˙¯ i,1 = ui,1 − uˆ i,1 , σ¯˙ i,2 = σi,3 ui,1 − σˆ i,3 uˆ i,1 = σ¯ i,3 uˆ i,1 + σi,3 (ui,1 − uˆ i,1 ), .. . σ˙¯ i,m−1 = σi,m ui,1 − σˆ i,m uˆ i,1 = σ¯ i,m uˆ i,1 + σi,m (ui,1 − uˆ i,1 ), σ˙¯ i,m = ui,2 − uˆ i,2 , i ∈ Γ .

(14)

Inspired by Qian and Lin (2001), construct the Lyapunov function

σ¯ i∗,3

∗1/r2 2−r2

(s1/r2 − σ¯ i,3

)

ds.

(23)

Based on (21), the derivative of V2 along system (17) is 1+r2

V˙ 2 |(17) ≤ −(m − 1)|ˆui,1 |ξ1

+ uˆ i,1 σ¯ i,2 (σ¯ i,3 − σ¯ i∗,3 ) ∗1/r2

2−r2 2

+ξ  ×

(15)

such that σ¯ i,1 converges to zero in a finite time, where β1 > 0, 0 < α < 1. Then, subsystem (15) will reduce to

σ¯ i,3 σ¯ i∗,3

(16)

σ˙¯ i,2 = σ¯ i,3 uˆ i,1 , . . . , σ˙¯ i,m−1 = σ¯ i,m uˆ i,1 , σ¯˙ i,m = ui,2 − uˆ i,2 , i ∈ Γ .

σ¯ i,3

 V2 = V1 +

For subsystem (14), it is easy to design a first-order finite-time controller: ui,1 = uˆ i,1 − β1 sigα (σ¯ i,1 )

239

σ¯ i,4 uˆ i,1 +

]

dt ∗1/r2 1−r2

(s1/r2 − σ¯ i,3

)

ds.

(24)

First, it follows from Lemmas 2.3 and 2.4 that

 

 ) 

∗1/r2 r2 

1/r

uˆ i,1 σ¯ i,2 (σ¯ i,3 − σ¯ i∗,3 ) ≤ |ˆui,1 | · |σ¯ i,2 | · (σ¯ i,3 2 )r2 − (σ¯ i,3

1  ≤ |ˆui,1 | |ξ1 |1+r2 + c1 |ξ2 |1+r2

(25)

2

(17)

To solve the finite-time control problem for system (17), the work (Wu et al., 2005) employed the terminal sliding mode control (TSMC) method to design a finite-time controller. However, there is a singularity problem in the proposed TSM controller. To this end, here we will employ the technique of adding a power integrator (Qian & Lin, 2001) to design a finite-time controller.

d[−σ¯ i,3

for a positive constant c1 . Second, it follows (17) and (20) that

 r2    d[−σ¯ i∗,31/r2 ]      = β 1/r2 |ξ˙1 | = β 1/r2 |ˆui,1 | · ξ2 + σ¯ ∗1/r2  . i,3 2 2     dt

(26)

By Lemma 2.3, we further obtain Remark 3.1. For simplicity, in the following controller design, we define ri = 1 + (i − 1)τ , i = 1, . . . , m, and assume τ = −q/p with a positive even integer q and a positive odd integer p. It implies that ri , i = 1, . . . , m, are odd in both denominator and numerator, which simplifies the notation of the controller since sri = sigri (s). Theorem 2. Consider system (17) under the condition that uˆ i,1 (t ) does not pass through zero after a finite time T ∗∗ . For any τ ∈ 1 (− m− , 0) if u2,i is designed as (18), 1 u2,i

1  1 r r =ˆui,2 − βm |ˆui,1 | σ¯ i,mm−1 + βmm−−11 sign(ˆui,1 )  1 1 r r × σ¯ i,mm−−21 + · · · + β3 3 sign(ˆui,1 )  1  rm−1 +τ 1 r r × σ¯ i,32 + β2 2 sign(ˆui,1 )σ¯ i,2 · · · ,

σ¯ , whose derivative 2 i ,2

V˙ 1 |(17) = σ¯ i,2 σ¯ i,3 uˆ i,1 = σ¯ i,2 σ¯ i∗,3 uˆ i,1 + σ¯ i,2 (σ¯ i,3 − σ¯ i∗,3 )ˆui,1 .

(19)

(20)

1/r

∗1/r2

ξ2 = σ¯ i,3 2 − σ¯ i,3

.

−

≤ |ξ2 |1−r2 |σ¯ i,3 − σ¯ i∗,3 | ≤ 21−r2 |ξ2 |. (28)

It follows from (27) and (28), and Lemma 2.4 that

]

σ¯ i,3



σ¯ i∗,3

∗1/r2 1−r2

(s1/r2 − σ¯ i,3

)

ds

1  ≤ |ˆui,1 | |ξ1 |1+r2 + c2 |ξ2 |1+r2

(29)

2

for a positive constant c2 . Finally, it follows from (24), (25) and (29) that 1+r2

V˙ 2 |(17) ≤ − (m − 2)|ˆui,1 |ξ1 2−r2

+ ξ2

+ |ˆui,1 |(c1 + c2 )|ξ2 |1+r2

σ¯ i,4 uˆ i,1 .

(30)

Thus, a new virtual controller is chosen as r

r

σ¯ i∗,4 = −(m − 2 + c1 + c2 )sign(ˆui,1 )ξ23 := −β3 sign(ˆui,1 )ξ23

1+r2

V˙ 2 |(17) ≤ − (m − 2)|ˆui,1 |(|ξ1 2−r2

+ ξ2

Under this virtual control law, it follows from (19) that

Step 2: The error between the real state σ¯ i,3 and the virtual taken as a new state. Define

(s



 ∗1/r σ¯ i,3 2 )1−r2 

(31)

Under this virtual controller, by (30), we have

r

σ¯ i∗,3 = −β2 sign(ˆui,1 )ξ12 , ξ1 = σ¯ i,2 ,

1+r V˙ 1 |(17) ≤ − (m − 1)|ˆui,1 |ξ1 2 + uˆ i,1 σ¯ i,2 (σ¯ i,3 − σ¯ i∗,3 ).

σ¯ i∗,3

1/r2

with a constant β3 = m − 2 + c1 + c2

By backstepping design, design a virtual control σ¯ i∗,3 as: with a constant β2 = m − 1.

σ¯ i,3

dt

Proof. In the sequel, we first consider the case when uˆ i,1 (t ) ̸= 0. A recursive manner will be employed. Step 1: Choose a Lyapunov function: V1 = along system (17) is

   

∗1/r2

(18)

(27)

Meanwhile, by Lemma 2.3, we obtain

d[−σ¯ i,3

where βi , i = 2, . . . , m, are appropriate gains, the state of system (17) will be stabilized to zero in a finite time.

1 2

     d[−σ¯ i∗,31/r2 ]    ≤ β 1/r2 |ˆui,1 | · |ξ2 |r2 + β2 |ξ1 |r2 . 2   dt

(21)

σ¯ i∗,3

is

(22)

+ |ξ2 |1+r2 )

(σ¯ i,4 − σ¯ i∗,4 )ˆui,1 .

(32)

Following the same line shown in the first two steps, we can find appropriate gains β2 , β3 , . . . , βm such that under the controller (18) the following holds: 1+r2

V˙ m−1 |(17)–(18) ≤ − |ˆui,1 |(|ξ1

1+r2

+ · · · + |ξm−1 |

)

(33)

240

H. Du et al. / Automatica 62 (2015) 236–242

where 1/r

ξ1 = σ¯ i,2 ,

∗1/r

ξj = σ¯ i,j+j1 − σ¯ i,j+1j , r

σ¯ i∗,j+1 = −βj sign(ˆui,1 )ξj−j 1 , j = 2, . . . , m − 1.  σ¯i,3 1 ∗1/r Vm−1 = σ¯ i2,2 + (s1/r2 − σ¯ i,3 2 )2−r2 ds 2

σ¯ i∗,3



σ¯ i,m

+ ··· + σ¯ i∗,m

∗1/rm−1 2−rm−1

(s1/rm−1 − σ¯ i,m

)

ds.

In addition, it follows from Lemma 2.3 that ∗1/rm−1 2−rm−1

σ¯ i,m

)

 σ¯i,m σ¯ i∗,m

(s1/rm−1 − Fig. 1. The desired formation pattern (pentagon) and the information exchange among agents.

ds ≤ 21−rm−1 ξm2 −1 , which means that

Vm−1 ≤ ρ(ξ12 + ξ22 + · · · + ξm2 −1 ),

(34)

where ρ = max{1/2, 21−r2 , . . . , 21−rm−1 }. Since 1 + r2 = 2 + τ , and by Lemma 2.5, it follows from (33)–(34) that V˙ m−1 + |ˆui,1 |ρ

− 2+τ 2

2+τ 2

Vm−1 ≤ 0.

(35)

When uˆ i,1 (t ) = 0, the controller (18) becomes u2,i = uˆ i,2 and the closed-loop system (17) is

σ˙¯ i,2 = σ¯ i,3 uˆ i,1 , . . . , σ˙¯ i,m−1 = σ¯ i,m uˆ i,1 , σ˙¯ i,m = 0,

i ∈ Γ.

(36)

Hence, with this equation in mind, and based on the relation (35), it can be concluded that no matter uˆ i,1 (t ) pass through zero or not, the system states will be bounded before the time instant T ∗∗ . After T ∗∗ , by Lemma 2.2, it can be concluded from (35) that Vm−1 will converge to zero in a finite time. The proof is completed.  3.3. Design of distributed observer-based finite-time controller Theorem 3. Consider multiple nonholonomic chained-form systems (1)–(2) under the condition that the reference signal ud1 (t ) does not pass through zero after a finite time T ∗∗ . If Assumptions 2.1 and 2.2 are satisfied and the distributed observer-based controller is designed as (16) and (18) with observer (3)–(4), then the state consensus for systems (1)–(2) is achieved in a finite time. i.e., σi → σ d in a finite time for ∀i ∈ Γ . Proof. The proof is straightforward.



4. Finite-time formation control of wheeled mobile robots with a desired trajectory Consider a set of wheeled mobile robots which move on a plane as shown in Dong and Farrell (2008). The kinematic equations of each robot are described by x˙ i = vi cos θi ,

y˙ i = vi sin θi ,

θ˙i = ωi , i ∈ Γ ,

(37)

where (xi , yi ) denotes the position of the center of mass of the robot, θi is the heading angle of the robot, vi is the forward velocity, ωi is the angular velocity of the robot. A desired trajectory (xd , yd , θ d ) is generated by x˙ d = v d cos θ d , y˙ d = v d sin θ d , θ˙ d = ωd .

(38)

As shown in Dong and Farrell (2008), the formation control with a desired trajectory requires that the group of robots come into formation and move along the desired trajectory. As in Dong and Farrell (2008), assume that the desired geometric pattern in the plane is determined by vector △ij = [△ij,x , △ij,y ]T ∈ R2 , i, j ∈ Γ . That is to say that the desired position deviation between agents i and j is △ij = △i − △j = [△i,x , △i,y ]T − [△j,x , △j,y ]T . For simplicity of statement, we just consider the time-invariant formation pattern,

i.e., △i , ∀i ∈ Γ , are constant vectors. As in Dong and Farrell (2008), for each robot, define the following  coordinate change: σi,1 = −θi , σi,2 = −(xi − △i,x + 1n ni=1 △i,x ) sin θi + (yi −

△i,y + 1n

(yi −

)

n

 θ , σi,3 = (xi − △i,x + 1n ni=1 △i,x ) cos θi + ) sin θi , ui,1 = −ωi , ui,2 = vi + σi,2 ωi ,

i=1 △i,y cos i n △i,y 1n i=1 △i,y

+



which results in

σ˙ i,1 = ui,1 ,

σ˙ i,2 = σi,3 ui,1 ,

σ˙ i,3 = vi − σi,2 ui,1 = ui,2 . (39)

In the same way, for the desired trajectory (38), under the coordinate change: σ1d = −θ d , σ2d = −xd sin θ d + yd cos θ d , σ3d = xd cos θ d + yd sin θ d , ud1 = −ωd , u2 = v d + σ2d ωd , we obtain

σ˙ 1d = ud1 ,

σ˙ 2d = σ3d ud1 ,

σ˙ 3d = v d − σ2d ud1 = ud2 .

(40)

For systems (39) and (40), by Theorem 3, we can design the distributed observer-based finite-time controller in the form of (16) and (18) to achieve finite-time formation. Example 1. Consider a network of five wheeled mobile robots described by (37). The information exchange topology among agents is shown in Fig. 1. The weights of the undirected edges are: a12 = a21 = 0.5, a23 = a32 = 0.2, a25 = a52 = 0.3, a34 = a43 = 0.5, b3 = 1. The desired formation pattern in the plane is shown in Fig. 1, which is a pentagon. Then, the desired position deviations are: △ij = − △ji = △i − △j , △i = r [cos( 75π −

2iπ ), sin( 75π − 2i5π )]T , i, j = 1, . . . , 5. The initial conditions are 5 selected as follows: [x1 (0), y1 (0), θ1 (0)]T = [3, 2, 0.1]T , [x2 (0), y2 (0), θ2 (0)]T = [1, −3, 0.2]T , [x3 (0), y3 (0), θ3 (0)]T = [−2, 0, 0.5]T , [x4 (0), y4 (0), θ4 (0)]T = [3.5, 0.5, 0.4]T , [x5 (0), y5 (0), θ5 (0)]T = [2.5, −1, 1]T . In simulation, suppose the desired formation pattern is shown in Fig. 1 with r = 1, and the desired trajectory (xd , yd , θ d ) is generated by (38) with v d = 5, θ d = 0.5 where the initial conditions are chosen as (xd , yd , θ d ) = (0, 0, 0). Following the procedure detailed in the proof, the controller gains and observer gains can be calculated. In this specific example, we can simply choose relatively better control gains for a satisfactory performance after simulation tests by trial and error. The controller gains and observer gains are chosen as follows: η = 1/3, α = 2/3, β1 = 2, β2 = 4, β3 = 20, k1 = 2, k2 = 20, l1 = 2, l2 = 2, l3 = 20, τ = −2/9, r2 = 1 + τ = 7/9. The response curves of each agent are given in Figs. 2–4, from which it can be found that all the agents converge to a pentagon formation pattern in the plane and can track the desired formation trajectory.

5. Conclusion The finite-time consensus problem for a group of leaderfollowing nonholonomic chained-form systems has been solved. A high-order distributed observer-based finite-time controller

H. Du et al. / Automatica 62 (2015) 236–242

241

References

Fig. 2. The response curves of each agent’s position (xi , yi )T + △i and heading angle θi .

Fig. 3. The response curves of each agent’s forward velocity and angular velocity.

Fig. 4. State trajectories of each agent in the plane under the finite-time control law.

has been recursively constructed to guarantee the finite-time consensus of the states. The proposed consensus algorithm has also been applied to solve the finite-time formation control problem of multiple wheeled mobile robots. Future work includes how to relax the communication topology to directed graphs.

Bhat, S. P., & Bernstein, D. S. (1998). Continuous finite-time stabilization of the translational and rotational double integrators. IEEE Transactions on Automatic Control, 43(5), 678–682. Bhat, S. P., & Bernstein, D. S. (2000). Finite-time stability of continuous autonomous systems. SIAM Journal on Control and Optimization, 38(3), 751–766. Dimarogonas, D., & Kyriakopoulos, K. (2007). On the rendezvous problem for multiple nonholonomic agents. IEEE Transaction on Automatic Control, 52(5), 916–922. Dong, W. (2013). Distributed tracking control of networked chained systems. International Journal of Control, 86(12), 2159–2174. Dong, W., & Farrell, A. (2008). Cooperative control of multiple nonholonomic mobile agents. IEEE Transactions on Automatic Control, 53(6), 1434–1448. Du, H., Li, S., & Qian, C. (2011). Finite-time attitude tracking control of spacecraft with application to attitude synchronization. IEEE Transactions on Automatic Control, 56(11), 2711–2717. Fax, A., & Murray, R. (2004). Information flow and cooperative control of vehicle formations. IEEE Transactions on Automatic Control, 49(9), 1453–1464. Hardy, G., Littlewood, J., & Polya, G. (1952). Inequalities. Cambridge: Cambridge University Press. Hong, Y., Wang, J., & Xi, Z. (2005). Stabilization of uncertain chained form systems within finite settling time. IEEE Transactions on Automatic Control, 50(9), 1379–1384. Hui, Q., Haddad, W., & Bhat, S. (2008). Finite-time semistability and consensus for nonlinear dynamical networks. IEEE Transactions on Automatic Control, 53(8), 1887–1990. Jiang, Z., & Nijmeijer, H. (1999). A recursive technique for tracking control of nonholonomic systems in chained form. IEEE Transactions on Automatic Control, 44(2), 265–279. Khalil, H. K. (2002). Nonlinear systems (3rd edition) (pp. 552–553). Upper Saddle River, Nj: Prentice hall. Kim, Y., & Mesbahi, M. (2006). On maximizing the second smallest eigenvalue of a state-dependent graph Laplacian. IEEE Transactions on Automatic Control, 51(1), 116–120. Kolmanovsky, I., & McClamroch, N. H. (1995). Developments in nonholonomic control systems. IEEE Control Systems Magazines, 15(6), 20–36. Li, Z., Duan, Z., Chen, G., & Huang, L. (2010). Consensus of multiagent systems and synchronization of complex networks: a unified viewpoint. IEEE Transactions on Circuits and Systems I: Regular Papers, 57(1), 213–224. Li, S., Du, H., & Lin, X. (2011). Finite-time consensus algorithm for multi-agent systems with double-integrator dynamics. Automatica, 47(8), 1706–1712. Lin, Z., Bruce, F., & Maggiore, M. (2005). Necessary and sufficient graphical conditions for formation control of unicycles. IEEE Transactions on Automatic Control, 50(1), 121–127. Liu, T., & Jiang, Z. (2013). Distributed formation control of nonholonomic mobile robots without global position measurements. Automatica, 49(2), 592–600. Lu, X., Lu, R., Chen, S., & Lu, J. (2013). Finite-time distributed tracking control for multi-agent systems with a virtual leader. IEEE Transactions on Circuits and Systems I: Regular Papers, 60(2), 352–362. Meng, Z., Ren, W., & You, Z. (2010). Distributed finite-time attitude containment control for multiple rigid bodies. Automatica, 46(12), 2092–2099. Olfati-Saber, R. (2006). Flocking for multi-agent dynamic systems: Algorithms and theory. IEEE Transactions on Automatic Control, 51(3), 410–420. Olfati-Saber, R., & Murray, R. (2004). Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 49(9), 1520–1533. Ou, M., Du, H., & Li, S. (2014). Finite-time formation control of multiple nonholonomic mobile robots. International Journal of Robust and Nonlinear Control, 24(1), 140–165. Qian, C., & Lin, W. (2001). A continuous feedback approach to global strong stabilization of nonlinear systems. IEEE Transactions on Automatic Control, 46(7), 1061–1079. Ren, W. (2007). Distributed attitude alignment in spacecraft formation flying. International Journal of Adaptive Control and Signal Processing, 21(2-3), 95–113. Ren, W., & Atkins, E. (2007). Distributed multi-vehicle coordinated control via local information exchange. International Journal of Robust and Nonlinear Control, 17((10-11), 1002–1033. Wang, X., & Hong, Y. 2008. Finite-time consensus for multi-agent networks with second-order agent dynamics, in: Proceedings of IFAC World Congress, Korea, pp. 15185–15190. Wang, J., Qiu, Z., & Zhang, G. (2012). Finite-time consensus problem for multiple non-holonomic mobile agents. Kybernetika, 48(6), 1180–1193. Wang, L., & Xiao, F. (2010). Finite-time consensus problems for networks of dynamic agents. IEEE Transactions on Automatic Control, 55(4), 950–955. Wu, Y., Wang, B., & Zong, G. (2005). Finite-time tracking controller design for nonholonomic systems with extended chained form. IEEE Transactions on Circuits and Systems II: Express Briefs, 52(11), 798–802. Xiao, F., Wang, L., Chen, J., & Gao, Y. (2009). Finite-time formation control for multiagent systems. Automatica, 45(11), 2605–2611. Yu, W., Chen, G., & Lü, J. (2009). On pinning synchronization of complex dynamical networks. Automatica, 45(2), 429–435. Zhang, Y., Yang, Y., Zhao, Y., & Wen, G. (2013). Distributed finite-time tracking control for nonlinear multi-agent systems subject to external disturbances. International Journal of Control, 86(1), 29–40. Zhu, J., Lu, J., & Yu, X. (2013). Flocking of multi-agent non-holonomic systems with proximity graphs. IEEE Transactions on Circuits and Systems I: Regular Papers, 60(1), 199–210.

242

H. Du et al. / Automatica 62 (2015) 236–242 Haibo Du received his B.S. degree in Mathematics from Anhui Normal University, China, in 2004, and the Ph.D. degree in Automatic Control from the Southeast University, China, in 2012. He is currently an Associate Professor in the School of Electrical Engineering and Automation, Hefei University of Technology. His research interests include nonlinear system control, cooperative control of multi-agent systems, and spacecraft attitude control.

Paper Award. He is serving/served as Associate Editor of IEEE Transactions on Automatic Control, IEEE Transactions on Circuits and Systems — Part I, IEEE Transactions on Industrial Electronics, IEEE Transactions on Industrial Informatics and several other scholarly journals. He holds fellowship of IEEE, Institution of Engineering and Technology (UK), Engineers Australia, Australian Computer Society, Australian Institute of Company Directors, and International Energy Foundation. He is an IEEE Distinguished Lecturer and Vice-President (Publications) of IEEE Industrial Electronics Society.

Guanghui Wen received the Ph.D. degree in Mechanical Systems and Control from Peking University, China, in 2012. From September 2012 to January 2013, he was a Research Associate and Post-doctoral Fellow in the University of New South Wales at Australian Defence Force Academy, Australia. Currently, he is a Lecturer in the Department of Mathematics, Southeast University, China. His research focuses on cooperative control of multi-agent systems and cyber–physical systems. He was the recipient of the Best Student Paper Award in the 6th Chinese Conference on Complex Networks in 2010.

Shihua Li was born in Pingxiang, Jiangxi Province, China in 1975. He received his bachelor, master, Ph.D. degrees all in Automatic Control from the Southeast university, Nanjing, China in 1995, 1998 and 2001, respectively. Since 2001, he has been with the School of Automation, Southeast University, where he is currently a Professor. His main research interests lie in modeling, analysis and nonlinear control theory (nonsmooth control, disturbance rejection control, adaptive control, etc.) with applications to mechatronic systems, including manipulator, robot, AC motor, power electronic systems and others. He serves as Associate Editors or Editor of IET Power Electronics, International Journal of Electronics, Journal of Power Electronic, Mathematical Problems in Engineering and guest editors of IEEE Transactions on Industrial Electronics, International Journal of Robust & Nonlinear Control and IET Control Theory & Applications.

Xinghuo Yu received BEng and MEng degrees from the University of Science and Technology of China, Hefei, China, in 1982 and 1984, respectively, and PhD degree from the Southeast University, Nanjing, China in 1988. He is currently with RMIT University (Royal Melbourne Institute of Technology), Melbourne, Australia, where he is the Founding Director of RMIT Platform Technologies Research Institute. His research interests include variable structure and nonlinear control systems, complex and intelligent systems and applications. He has received a number of awards and honors for his contributions, including the 2013 Dr.-Ing. Eugene Mittelmann Achievement Award of IEEE Industrial Electronics Society and 2012 IEEE Industrial Electronics Magazine Best

Michael Z.Q. Chen received the B.Eng. degree in Electrical and Electronic Engineering from Nanyang Technological University, Singapore, and the Ph.D. degree in Control Engineering from the Cambridge University, Cambridge, UK. He is currently an Assistant Professor at the Department of Mechanical Engineering, The University of Hong Kong. He is a Life Fellow of the Cambridge Philosophical Society. He is now a Guest Associate Editor for the International Journal of Bifurcation and Chaos. His research interests include passive network synthesis, complex networks, mechanical control, and robotic control.