Robust finite-time consensus control for multi-agent systems with disturbances and unknown velocities

ISA Transactions xxx (xxxx) xxx–xxx

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Research article

Robust finite-time consensus control for multi-agent systems with disturbances and unknown velocities Xuehong Tian, Huanlao Liu, Haitao Liu∗ School of Mechanical and Power Engineering, Guangdong Ocean University, Zhanjiang, 524088, PR China

A R T I C LE I N FO

A B S T R A C T

Keywords: Multi-agent systems Finite-time consensus Disturbances Backstepping method High-gain observer

In this paper, we investigated the finite-time consensus tracking problem for multi-agent systems with external bounded disturbances and input bounded disturbances and unknown velocities. Based on the Lyapunov finitetime theorem, a novel finite-time consensus control is constructed by using the backstepping method. For unknown velocities, the high-gain observer is used to estimate the velocity information. It is proved that the consensus can be achieved in finite time. The consensus shows fast response and strong robustness to various disturbances. Finally, the effectiveness of the results is illustrated by numerical simulations.

1. Introduction

manipulators by using backstepping method, which is proved by the finite-time Lyapunov stability theorem [28]. Li Shihua et al. designed a continuous distributed control algorithms for multi-agent systems described by double integrators based on the finite-time control technique [4]. Khoo et al. proposed a robust finite-time consensus tracking algorithm for multi-robot systems with input disturbances based on the terminal sliding mode control [22]. Similarly, Zhao et al. constructed a continuous consensus tracking control using a nonsingular terminal sliding mode scheme [23]. Liu et al. discussed the finite-time consensus problem for a class of time-varying nonlinear multi-agent systems, and proposed a finite-time controller based on the Lyapunov stability theorem [26]. He et al. constructed a finite-time consensus protocol by using the Lyapunov stability theorem [9]. Note that the abovementioned algorithms require velocity measurements to be available. In practice, the velocity is difficult to obtain or cannot be precisely measured [29], which makes it difficult to achieve consensus in a finite time using only the relative position information. There are some results about finite-time consensus algorithms without velocities [7,29–32]. For example, Zhang et al. proposed a finite time observerbased controllers for multi-agent systems to achieve finite-time consensus with unavailable velocities [32]. With the existence of disturbances, Zhao and Duan designed a finite-time containment protocol that uses only relative position measurements [31]. Hua et al. investigated a finite-time consensus control for second-order multi-agent systems without velocity measurements [7]. However, methods presented in those papers cannot consider input disturbances. Motivated by the abovementioned results, this paper discusses the finite-time tracking problem of second-order multi-agent systems with external

Recently, the consensus of problem of multi-agent systems has attracted increasing attention due to its applications in multi-vehicles formation, sensory networks, distributed computation, and so on [1–3]. The consensus means that all agents reach an agreement on a state under a designed protocol based only on local relative information between neighboring agents [4]. Currently, the consensus problem can be roughly categorized into two classes, namely leader-follower consensus (consensus with a leader) [5–7] and leaderless consensus (consensus without leader) [8,9]. There have been many consensus algorithms [5,10–15] developed by synthesizing algebraic graph theory and control theory. An important topic in the study of the consensus problem is the convergence rate. Furthermore, the abovementioned literature mainly focuses on the asymptotical convergence rate. The literature shows the best asymptotical convergence is exponential with infinite settling time, i.e., the states cannot reach a consensus in finite time. In practical applications, it may be more desirable to achieve consensus tracking in finite time. Therefore, it is very useful to investigate the finite-time consensus tracking control for multi-agent systems. Compared with asymptotic consensus, the finite-time consensus provides not only a faster convergence rate but also stronger robustness to uncertainty and disturbance rejection [16–20]. The finite-time consensus control has been studied in a number of recently published papers, see for instance [4,9,21–27]. For example, Hui et al. addressed some necessary and sufficient conditions for finitetime semi-stability of homogeneous multi-agent systems [21]. Zhao et al. proposed a robust finite-time stability control for robotic

∗

Corresponding author. E-mail address: [email protected] (H. Liu).

https://doi.org/10.1016/j.isatra.2018.07.032 Received 11 August 2017; Received in revised form 31 March 2018; Accepted 24 July 2018 0019-0578/ © 2018 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Tian, X., ISA Transactions (2018), https://doi.org/10.1016/j.isatra.2018.07.032

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Lemma 2. [33] Consider the nonlinear system x˙ = f (x , u) . Suppose that there exist continuous function V(x), scalars c > 0, 0 < α < 1 and 0 < ε < ∞ such that

bounded disturbances, input bounded disturbances and unknown velocities by using the backstepping method. The main contributions of this paper are as follows. First, a finitetime consensus for multi-agent systems is designed based on the finitetime Lyapunov stability theorem and the backstepping method motivated by paper [28,33]. Second, the external bounded disturbances and input bounded disturbances are considered in multi-agent systems. Finally, The high-gain observers are used to obtain velocity information, and the saturation input is introduced to eliminate the peaking phenomenon and make it more easily use in practice. In contrast to the previous works related to finite-time consensus, the proposed consensus control can enhance the robustness of multi-agent systems to various disturbances. The rest of the paper is organized as follows. Section 2 gives some preliminaries on graph theory. The main results are discussed in Section 3. In Section 4, two numerical examples are given to illustrate the theoretical results. Section 5 gives the conclusions.

V (x ) ≤ −cV α (x ) + ε

Then, the trajectory of system x˙ = f (x , u) is practical finite-time stable.

2.3. Description of the second order multi-agent systems In this paper, we consider a multi-agent system with leader and followers. The leader is active, and its behavior is independent of the followers. The dynamics of the leader are described as follows:

x˙ 0 = v0, x 0 ∈ Rm v˙ 0 = u 0 , v0 ∈ Rm

x˙ i = vi, x i ∈ Rm v˙ i = ui + δi, vi ∈ Rm , i = 1, ⋯, n

2.1. Graph theory notations For a multi-agent system consisting of one leader and n followers. Let G = {ν , κ } be a directed graph, where ν = {0,1,2, ⋯, n} is the set of nodes, node i represents the ith agent, κ is the set of edges, and an edge in G is denoted by an ordered pair (i, j )·(i, j ) ∈ κ if and only if the ith agent can send information to the jth agent directly, but not necessarily vice versa. In contrast to a directed graph, the pairs of nodes in an undirected graph are unordered, where the edge (i, j) denotes that agent i and j can obtain information from each other. Therefore, an undirected graph can be viewed as a special case of a directed graph. A directed tree is a directed graph, where every node has exactly one parent except for the root, and the root has a directed path to every other node. A directed spanning tree of G is a directed tree that contains all nodes of G. The matrix A = (aij ) ∈ R(n + 1) × (n + 1) where aij > 0, if (j, i) ∈ κ and aij = 0 otherwise, is called the weighted adjacency matrix of G with nonnegative elements. Let D = diag{d 0, d1, ⋯, dn} ∈ R(n + 1) × (n + 1) be a din agonal matrix, where di = ∑ j = 0 aij for i = 0, 1, …, n. Then, the Laplacian of the weighted graph can be defined as

Definition 1. The multi-agent system is said to achieve second-order finite-time consensus if for any initial conditions

lim x i (t ) − x 0 (t ) = 0, lim vi (t ) − v0 (t ) = 0

t→T

x i (t ) = x 0 (t ), vi (t ) = v0 (t ), ∀ t ≥ T , i = 1,2, ⋯, n. where T is a positive constant. The system consists of n+1 agents, where an agent indexed by 0 acts as the leader and the other agents indexed by 1, …, n, are referred to as the followers. The topological relationships between the leader and the followers are described by a directed graph G = {ν , κ } , with ν = {0,1, ⋯, n} and the adjacent matrix

0 ⎡ 0 a10 a11 ⎢ A= ⎢ ⋮ ⋮ ⎢ ⎣ an0 an1

(1)

⋯ 0 ⎤ ⋯ a1n ⎥ ∈ R(n + 1) × (n + 1) ⋱ ⋮ ⎥ ⋯ ann ⎥ ⎦

(6)

Denote G = {ν , κ } as the subgraph of G , which is formed by the n followers, where

a a ⋯ a1n ⎡ 11 12 ⎤ A = ⎢ ⋮ ⋮ ⋱ ⋮ ⎥ ∈ Rn × n ⎣ an1 an2 ⋯ ann ⎦

The Lyapunov finite-time stability theorem is discussed in Refs. [34,35].

(7)

LetD = diag{d1, d2, ⋯, dn} ∈ Rn × n be a diagonal matrix with n di = ∑ j = 1 aij for i = 1,2, ⋯, n . Then, it is clear that the Laplacian of the graph G can be defined as

Lemma 1. Consider the non-Lipschitz continuous nonlinear system x˙ = f (x ) with f(0) = 0. Suppose there are C1 positive definite function V(x) defined on a neighborhood of the origin and real numbers c > 0, and 0 < α < 1, such that (1) V(x)is positive definite;

L =D −A

(8)

The connection weight between agent i and the leader is denoted by (2a)

B = diag{b1, b2, ⋯, bn}

Then, the origin is a finite-time stable equilibrium, and the settling time, which depends on the initial state x (t0) = x 0 , satisfies

V 1 − α (x 0) c (1 − α )

t→T

and

2.2. Finite-time stability theory

T (x 0) ≤

(5)

where ui (i = 1, ⋯, n) represents the control inputs and δi represents the various disturbance, which is bounded, i.e., δ ∞ ≤ δ , δ > 0 .

The connection weight between the ith agent and the leader is denoted by bi with bi > 0 if there is an edge between the ith agent and the leader.

∂V V˙ (x ) ≤ −cV α (x ), where V˙ (x ) = f (x ). ∂x

(4)

where x0 is the position and v0 is the velocity of the leader. The dynamics of the ith follower agent are described by

2. Preliminaries and model description

L = D − A ∈ R(n + 1) × (n + 1)

(3)

(9)

In this paper, the following assumptions are considered. Assumption 1. The time-varying control input u0 is unknown to any follower but its upper bound u 0 is available to its neighbors.

(2b)

Assumption 2. The position of the leader x0 and its velocity v0 are available to its neighbors only.

for all x0 in some open neighborhood of the origin.

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3. Main result

According to Lemma 1, if we design the controller U as

U = (L + B )−1 [B 1u 0 + φ˙ (X1) − K2 Sig α (Z ) − X1 − (L + B ) δ ]

3.1. Finite-time consensus control with velocity measurements

(21)

whereK2 = diag(k21, k22, ⋯, k2n ). we have Define the error functions as

V˙ = V˙1 + Z TZ˙

n x͠ i = ∑ j = 1 aij (x i − x j ) + bi (x i − x 0)

= −

n v͠ i = ∑ j = 1 aij (vi − vj ) + bi (vi − v0)

(10)

X1T K1 Sig α (X1)

− φ˙ (X1)) = − X1T K1 Sig α (X1) − Z TK2 Sig α (Z )

1, if agent i is connected to the leader 1, if (j, i) ∈ ε aij = ⎧ , bi = ⎧ ⎨ ⎨ otherwise other 0, ⎩ ⎩ 0,

≤ − k1 V1μ − k2

X˙ 1 = X2 X˙ 2 = (L + B ) U + (L + B ) δ − B1u 0 ]T and

(11)

∑ aij (xi + Δi − xj − Δj) + bi (xi + Δi − x 0 − Δ0)

U = (L + B )−1 [φ˙ (X1) − K2 Sig α (Z ) − X1 − (diag(2nδ + B 1u 0

(12)

+ l 0)sgn(Z )]

(23)

where l0 > 0, then the closed-loop system is globally robust finite-time consensus stable.

x ⎡ Δ1 ⎤ ⎡ 1⎤ ⎢⋮ ⎥ = 1x 0 − ⎢⋮ ⎥ ⎢ Δn ⎥ ⎣ x2 ⎦ ⎣ ⎦

(13)

Proof. Substituting (23) into (20), we have

In the following, we will design the finite-time consensus control by the backstepping technique. Step 1. Introduce a new error variable

Z = X2 − φ (X1)

V˙ = V˙1 + Z TZ˙ = − X1T K1 Sig α (X1) − Z TK2 Sig α (Z ) + Z T ((L + B ) δ − B 1u 0 − (diag(2nδ + B 1u 0 + l 0)sgn(Z ))

(14)

X˙ 1 = Z + φ (X1) Z˙ = (L + B ) U + (L + B ) δ − B1u 0 − φ˙ (X1)

V˙ = V˙1 + Z TZ˙ = −

(15)

X1T K1 Sig α (X1)

≤ − X1T K1 Sig α (X1) − Z TK2 Sig α (Z ) ≤ − k1 V1μ − k2

(16)

If we design an auxiliary control j(X1) = −K1Sig α (X1) , 0 < α < 1, K1 = diag(k11, k12, ⋯, k1n ) , and whereSig α (X1) = [ x͠ 1 α sgn(x͠ 1), ⋯, x͠ n α sgn(x͠ n)]T , and sgn(·) is the sign function. Then we have

T (X ) ≤

Step 3. Consider the Lyapunov function

φ˙ i (X1i ) = (19)

Its derivative is

)

μ

(25)

1 V (X0 )1 − μ k (1 − μ)

(26)

− k1i α X1i α − 1 X˙ 1i , if X1i ≥ λ and X˙ 1i ≠ 0 ⎧ ⎪ − k1i α Δi α − 1 X˙ 1i , if X1i < λ and X˙ 1i ≠ 0 ⎨ ⎪ 0, if X˙ 1i = 0 ⎩

(27)

to avoid the singularity problem, where threshold value λ and Δi are both small positive constants [20], X1i is the ith element of vector X1, andφ˙ i (X1i ) is the ith element of vector φ˙ (X1) .

V˙ = V˙1 + Z TZ˙ = − X1T K1 Sig α (X1) + X1T Z + Z T ((L + B ) U + (L + B ) δ − B1u 0

Remark 2. The saturation function (28) is used to avoid the “chattering” phenomenon, which the variable structure term sgn(Z) in (23) may create, but this does not affect the finite-time stability of the closed-loop system. The saturation function is

− φ˙ (X1)) Z T (X1

n

∑i = 1 Zi2

Remark 1. The derivative of Sig α (X1) is infinite when X1i = 0 and X˙ 1i ≠ 0 in φ˙ (X1) . Therefore, the term φ˙ (X1) is modified as

(18)

1 T Z Z 2

1 2

Obviously, the closed-loop system is global finite-time consensus stable. The estimated setting time is

(17)

V˙1 = −X1T K1 Sig α (X1) + X1T Z

(

≤ − kV μ

The derivative of (16) is

V1 = X1T X˙ 1 = X1T (Z + φ (X1))

− Z TK2 Sig α (Z )

+ Z T ((L + B ) δ − B 1u 0 − (diag(2nδ + B 1u 0 + l 0)sgn(Z ))

Step 2. Select the following Lyapunov function

1 T X1 X1 2

(24)

If l 0 ≥ 0 , there comes

Substituting (14) into (11), we can obtain

+

μ

Theorem 1. For the uncertain multi-agents system (5), suppose that Assumptions 1 and 2 hold. If we design the consensus controller

and X1 = 0 , we have

= −

)

and k = min{k1, k2} . Therefore, the where k2 min = min{k2i},k2 = closed-loop system of (11) and (21) is finite-time consensus stable. According to Assumption 1, the control input u0 is unknown to any follower. In fact, the uncertaintiesδ are also unknown, so a variable structure term is introduced to improve the robustness to u0 and uncertaintiesδ . Therefore, we have the following theorem.

n

X1T K1 Sig α (X1)

n

∑i = 1 Zi2

(22)

whereU = [u1, ⋯, un δ = [δ1, ⋯, δn If we consider the formation control, the (10) can be defined as

V = V1 +

1 2

2 μk2 min

]T .

j=1

(

≤ − kV μ

Herein, we consider that the velocity of the leader is available to its neighbors only. Let X1 = [x͠ 1, ⋯, x͠ n]T , and X2 = [v͠ 1, ⋯, v͠ n]T . The error dynamics of the interconnection graph can be expressed as

V1 =

+ Z T ((L + B ) U + (L + B ) δ − B 1u 0

= − X1T K1 Sig α (X1) + Z T (X1 + (L + B ) U + (L + B ) δ − B 1u 0 − φ˙ (X1))

where

x͠ i =

+

X1T Z

+ (L + B ) U + (L + B ) δ − B1u 0 − φ˙ (X1)) (20)

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()

⎧ sgn Z , ⎪ ζ sat(Z ) = ⎨ Z ⎪ ζ, ⎩

Z ζ

Select the candidate Lyapunov function

≥1

others

1 T 1 X1 X1 + Z2T Z2 2 2

V2 = (28)

The derivative of (38) along the trajectories of (37) is given by

whereζ is a smaller positive constant called boundary layer thickness.

V˙2 = X1T X˙ 1 + Z2T Z˙ 2

Remark 3. The consensus controller (24) can be expressed as −1

n

ui =

= −

n

≤ − k1

1 2

n

∑i = 1 Zi2

)

μ

+ω

+ω

K (1 − μ)

t →∞

be made arbitrarily small. This completes the Proof. Remark 4. In practice, the peaking phenomenon made by the high-gain observer has a various impact on the performance of the system. To eliminate the peaking phenomenon, the bounded control is designed as follows

U (X , Z ) = S1 sag ⎛ ⎝

⎜

U1 (X ) ⎞ U (X ) ⎞ + S2 sag ⎛ 2 S1 ⎠ ⎝ S2 ⎠ ⎟

⎜

⎟

(40)

whereU1 (X1) = (L + B )−1 [φ˙ (X1) − X1], U2 (Z )

n x͠ i = ∑ j = 1 aij (x i − x j ) + bi (x i − x 0)

, and

= (L + B )−1 [−K2 Sig α (Z2) − (diag(2nδ + B 1u 0 + l1)sgn(Z2)] sag(·) is the saturation function, which is different from (28), defined by

(33)

Xˆ2 = [η͠ 1, ⋯, η͠ n]T , and v͠ i = vˆi − vi, i = 1, ⋯, n . Let X1 = [x͠ 1, ⋯, x͠ n Then, the error dynamics of the interconnection graph can be expressed as ]T ,

ui (·) > Si ⎧ 1 ⎪ u (·) u (·) ui (·) ≤ Si sag ⎛ i ⎞ = iS i ⎝ Si ⎠ ⎨ ⎪ − 1 ui (·) < − Si ⎩ ⎜

⎟

and Si ≥ max ui (·) , where Ωc = {x ∈

(34)

X , Z ∈ Ωc

(41)

Rn

X1 < c, V2 (X1 , Z2) ≤ p} ,

c > 0. Inside Ωc , we have

whereU = [u1, ⋯, un]T , δ = [δ1, ⋯, δn]T , E = [θ1, ⋯, θn]T and

U (X , Z ) = U1 (X ) + U2 (Z )

n

θi =

kV2μ

(

is practical finite-time stable. V˙2 ≤ −ηV2μ ≤ 0 means that V2 (t ) is eventually bounded. Thus, X1, Z2 are also uniformly and ultimately bounded. Furthermore, since lim ω = 0 , the ultimate error bound can

ˆ

X˙ 1 = Xˆ2 − E Xˆˆ2 = (L + B ) U + (L + B ) δ + E˙ − B 1u 0

− k2

t →∞

(32)

ˆ − ˆvj ) + bi (ˆvi − v0)

μ

defined by V2 (X1 , Z2) = p , lettingk > η + σ / p μ , there is V˙2 ≤ −ηV2μ on V2 = p . Thus, V2 ≤ p is an invariant set, i.e., if V2 (0) ≤ p, V2 (t ) ≤ p for all 1 T (s ) ≤ ∼ Vs (s0)1 − μ . According to Lemma 2, the closed-loop system

where x i andˆ vi are the estimates of x i and vi , respectively. The values h1 and h2 are chosen such that the roots of s 2 + h1 s + h2 = 0 have negative real parts. Redefine the error functions as

η͠ i =

)

where ω = X1 E , k2 min = min{k2i}, k2 = and k = min{k1, k2} . From Ref. [36], we know that the observation errors are bounded, and lim ω = 0 . Herein, we define ω = X1 E ≤ σ , σ > 0 . On the level set

i

n ∑ j = 1 aij ( vi

n

∑i = 1 Xi2

(39)

In practice, it is difficult to obtain the velocity information. In this section, a high-gain observer is used to obtain the velocity. We design an observer for the ith agent as

i

− X1 E

2 μk2 min

3.2. Finite-time consensus control without velocity measurements

vˆˆi = ui −

1 2

≤ −

(31)

ˆ−x) ˆ−x)

(

(30)

I − [D + B ]−1A = [D + B ]−1 [[D + B ] − A ] = [D + B ]−1 [L + B ]

h1 (xi ε h2 (xi ε2

−

Z2T K2 Sig α (Z2)

≤ − X1T K1 Sig α (X1) − Z2T K2 Sig α (Z2) + X1 E

which can be transformed into (23) by the following equation:

xˆˆi = ˆ vi −

X1T K1 Sig α (X1)

+ Z2T (E˙ + (L + B ) δ − B 1u 0 − (diag(2nδ + B 1u 0 + l1)sgn(Z2))

⎞ ⎡ ⎛ a + bi ⎢ ∑ aij uj + φ˙ i (x͠ i ) − x͠ i ⎟ ⎜ ∑ ij = ≠ j j i 1, ⎠ ⎣ j = 1, j ≠ i ⎝ ⎤ − (diag (2nδ + bi u 0 + l 0)sgn(z i ) ⎥ ⎦

(38)

(42)

∑ aij (v͠ i − v͠ j) + bi v͠ i j=1

(35)

From the paper [36], θi is bounded. Therefore, we have the following consensus control without velocity measurements.

4. Simulation A simulation is presented to illustrate the performance of the proposed FTC algorithms. We consider one leader indexed by 0 and four followers indexed by 1,2,3 and 4, respectively. Suppose that the leader's dynamics are

Theorem 2. For the uncertain multi-agents system (5), suppose that Assumptions 1 and 2 hold. If the velocity observers are designed as (32) and the consensus controller is designed as

U = (L + B )−1 [φ˙ (X1) − K2 Sig α (Z2) − X1 − (diag(2nδ + B 1u 0 + l1)sgn(Z2)]

x˙ 0 = v0 v˙ 0 = u 0

(36)

whereZ2 = Xˆ2 − φ (X1) , l1 > E˙ > 0 , then the closed-loop system is practical finite-time stable.

and the dynamics of ith followers are expressed as follows

x˙ i = vi v˙ i = ui + 0.05 sin(x i ), i = 1,2,3,4

Proof. The φ (X1) is defined the same as above. Combining that expression with (34), we can obtain

X˙ 1 = Z2 + φ (X1) − E Z˙ 2 = (L + B ) U + (L + B ) δ + E˙ − B 1u 0 − φ˙ (X1)

(43)

(44)

The initial condition of the four followers are x1(0) = 1, x2(0) = 1.2, x3(0) = 2, x4(0) = −1.2, and v1(0) = v2(0) = v3(0) = v4(0) = 0. The directed graph is shown in Fig. 1, which is used to model

(37)

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Fig. 1. Directed graph.

Fig. 3. Position tracking errors of four followers with velocity measurements.

(1) Finite-time consensus control with velocity measurements The control parameters are α = 0.8, k1 = 5, k2 = 10, l0 = 10u 0 = 1, ζ = 0.1, and S1 = S2 = 50. The simulation results are shown in Figs. 2 and 3. It can be seen that the followers can track the leader within 2.5 s in the presence of disturbances. To illustrate the formation control strategy, the error functions for formation control (12) are considered, we set the values Δ1 = −2, Δ2 = −3, Δ3 = −4, Δ 4 = −5. As shown in Figs. 4 and 5, x1 → x 0 − 2 x2 = x 0 − 3, x3 → x 0 − 4 , and x 4 → x 0 − 5within 2.5 s. (2) Finite-time consensus control without velocity measurements The high-gain observers are designed as Fig. 2. Position tracking of four followers with velocity measurements.

the information exchange among agents. The figure shows that the information of leader is available only to followers 3 and 4, and that follower 4 has no directed path to all other followers, but there exists a directed path from the leader to all followers. The adjacent matrix of the graph is

⎡0 ⎢0 A = ⎢0 ⎢1 ⎢ ⎣1

0 0 0 0 0

0 1 0 1 0

0 1 1 0 0

xˆˆi = vˆi −

h1 (xˆi ε

− xi)

vˆˆi = ui −

h2 (xˆi ε2

− x i ) i = 1,2,3,4

(46)

where h1 = 2, h2 = 1, ε = 0.01. The disturbances are selected in the same manner as in (1). The control parameters are also the same as those in (1). The results of position and velocity tracking errors are shown in Figs. 6 and 7, respectively. It can be seen that the state errors

0⎤ 1⎥ 0⎥ 0⎥ 0⎥ ⎦

The Laplacian of the follower system is

⎡ 3 − 1 − 1 − 1⎤ −1 0 ⎥ L = ⎢0 1 0 ⎥ ⎢0 − 1 1 ⎢ 0 0 ⎥ ⎦ ⎣0 0 and the diagonal matrices for the interconnection relationship between the leader and the followers is

B = diag(0 0 1 1) The time-varying control input to the leader is designed as

u0 = −

sin(x 0) 1 + e−t

(45) Fig. 4. Formation control of four followers with velocity measurements.

where x 0 = π/2 and v0 = 0 . 5

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Fig. 8. Control input of four followers without velocity measurements.

Fig. 5. Formation errors of four followers with velocity measurements.

converge to zero with 2.5 s, which presents the fast response. From Fig. 8, there no exists chattering phenomenon. In order to demonstrate advantages of the proposed finite-time consensus (Theorem 2), the comparison with the following state feedback finite-time consensus control [37] is presented here.

⎧ N +1 ui = −λ1 sgn ∑ j = 1 aij (vj − vi ) ⎨ ⎩

i = 1, ⋯, N

N +1 ⎫ + λ2 Sig1/2 ⎡∑ j = 1 aij (x j − x i ) ⎤ , ⎢ ⎥⎬ ⎣ ⎦⎭

(47)

where λ1 = 5, λ2 = 1, the leader is labeled by N+1 in this controller. Simulation results are shown in Fig. 9, Fig. 10 and Fig. 11. The proposed consensus controller has comparable performances with the finite-time consensus in Ref. [37], but the control in Ref. [37] suffers from chattering (see Fig. 12). From the two numerical simulation examples associated with Theorems 1 and 2, we can see that the high-gain observer and the finitetime consensus control presented in this paper are highly efficient and perform well in multi-agent systems.

Fig. 6. Position tracking errors of four followers without velocity measurements.

Fig. 7. Velocity tracking errors of four followers without velocity measurements.

Fig. 9. Position tracking of four followers (Paper [37]). 6

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5. Conclusion In this paper, the robust finite-time tracking control problem for a class of second-order multi-agent systems with various disturbances and unknown velocities has been investigated via the backstepping method and by using a high-gain observer. It has been theoretically proved that the position states of the followers can converge to that of the leader in a finite-time, which satisfies our performance goals. Future work will focus on experimental research of this work, and distributed finite-time consensus problem of multi-agent systems with various disturbances and communication delay. Acknowledgements This work was supported by the Training Plan of Outstanding Young Teachers in Higher Education Institutions of Guangdong Province [grant number YQ2015087], the Natural Science Foundation of Guangdong Province in China [grant number 2015A030310307], the National Natural Science Foundation of China [grant number 51375100], the Science and Technology Planning Project of Zhanjiang City [grant number 2017A02025, 2016A02018], and the Science and Technology Planning Project of Guangdong Province [grant number 2014A020208118].

Fig. 10. Position tracking errors of four followers (Paper [37]).

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Fig. 11. Velocity tracking errors of four followers (Paper [37]).

Fig. 12. Control input of four followers (Paper [37]).

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