Distributed Fixed-Time Coordinated Tracking for

Asian Journal of Control, Vol. 20, No. 4, pp. 1–13, July 2018 Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.1612

DISTRIBUTED FIXED-TIME COORDINATED TRACKING FOR NONLINEAR MULTI-AGENT SYSTEMS UNDER DIRECTED GRAPHS Boda Ning,

Zongyu Zuo, Jiong Jin, and Jinchuan Zheng ABSTRACT

This paper is concerned with the fixed-time coordinated tracking problem for a class of nonlinear multi-agent systems under detail-balanced directed communication graphs. Different from conventional finite-time coordinated tracking strategies, the fixed-time approach developed in this paper guarantees that a settling time bound is prescribed without dependence on initial states of agents. First, for the case of a single leader, a distributed protocol based on fixed-time stability techniques is proposed for each follower to accomplish the consensus tracking in a fixed time. Second, in the presence of multiple leaders, a new distributed protocol is proposed such that states of followers converge to the dynamic convex hull spanned by those of leaders in a fixed time. In addition, for a class of linear multi-agent systems, sufficient conditions that guarantee the fixed-time coordinated tracking are provided. Finally, numerical simulations are given to demonstrate the effectiveness of the theoretical results. Key Words: Fixed-time consensus, consensus tracking, containment control, multi-agent systems.

I. INTRODUCTION In recent years, cooperative control of networked multi-agent systems has attracted intensive research attention in both robotics and control communities [1–4]. One fundamental research topic of cooperative control is consensus, which aims at designing distributed control protocols to drive a group of agents to achieve agreement on states, such as position and velocity. According to the number of leaders in the network, existing explorations about the consensus problem are classified into three subareas, i.e., leaderless consensus without leaders, consensus tracking with a single leader, and containment control with multiple leaders. For leaderless consensus, control strategies are proposed for first-order [5], second-order [6], fractional-order [7], linear [8,9] multi-agent systems, and for synchronization [10]. For consensus tracking, the aim is to drive the states of followers to reach the state of the leader. This problem is investigated in [11] with integrator

Manuscript received March 03, 2017; revised May 06, 2017; accepted June 09, 2017. Boda Ning is with the School of Software and Electrical Engineering, Swinburne University of Technology, Melbourne, VIC 3122, Australia. Zongyu Zuo (corresponding author) is with The Seventh Research Division, Beihang University (BUAA), Beijing 100191, China (e-mail: [email protected]). Jiong Jin and Jinchuan Zheng are with the Department of Telecommunications, Electrical, Robotics and Biomedical Engineering, Swinburne University of Technology, Melbourne, VIC 3122, Australia. This work was supported in part by the National Natural Science Foundation of China (No. 61673034).

type dynamics. More recently, some control protocols have been developed in [12–16] for multi-agent systems with general linear dynamics or high-order dynamics. For containment control, the objective becomes to drive the states of followers to move into a convex hull spanned by those of multiple leaders. Indeed, in practical scenarios, sometimes it is favourable that followers travel into an area spanned by several leaders instead of tracking one specific trajectory [17]. For example, there may exist a scenario that a vehicle group moves from one place to a target one while only a small fraction of vehicles have the sensing capability to detect hazardous obstacles. In this case, the whole group will safely reach the destination as long as the followers (vehicles without sensing capability) remain inside the safety zone formed by those leaders (vehicles with sensing capability). Some results are reported on the containment control problem for integrator type [18], or nonlinear [19] multi-agent systems. Recently, the containment problem has been solved in [20] for networked Lagrangian systems in the presence of parametric uncertainties. Note that most of aforementioned work can only achieve exponential state agreement over an infinite time horizon. In practice, it is of particular interest to realize the consensus tracking/containment control in a finite time to meet specific system requirements. Therefore, the finite-time consensus tracking/containment control problem has received some research interest [21–24], e.g., a containment control protocol is proposed for nonlinear single-integrator multi-agent systems subject to external disturbances based on a sliding mode technique in [22],

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Asian Journal of Control, Vol. 20, No. 4, pp. 1–13, July 2018

and an observer-based approach is developed to achieve the finite-time containment control for double-integrator multi-agent systems in [23]. Although finite-time coordinated tracking has favorable properties, the estimation of convergence time depends on initial states of networked agents [21–23]. This restricts the applications in practice if the knowledge of initial conditions is unavailable in advance. Instead, a new strategy called fixed-time consensus is proposed in [25,26] where an upper bound of time to reach the consensus for single-integrator agents is obtained without dependence on initial states. The fixed-time strategy is further applied for agents with double-integrator dynamics [27,28]. For example, in [27] a guaranteed settling time independent of initial conditions is obtained with a proposed sliding surface. More recently, in wireless robotic networks, a fixed-time consensus based controller is proposed to guarantee an energy agreement is achieved before the next topology change [29]. In [30], collective behaviours of mobile robots are investigated by using the fixed-time consensus concept. Motivated by the above observations, in this paper, we aim to achieve the fixed-time coordinated tracking of nonlinear multi-agent systems under directed graph. Based on the fixed-time stability [31,32], the single leader case is first investigated. A distributed protocol is proposed for each follower to accomplish the fixed-time consensus tracking with the help of a properly designed Lyapunov function. Then, in the presence of multiple leaders, a new protocol is proposed such that the states of followers converge to the dynamic convex hull spanned by those of leaders in a fixed time. Note that the distinctive feature of our work is that an upper bound of the settling time for containment control is obtained, which is independent from initial states of agents. In addition, for a class of linear multi-agent systems, some sufficient conditions are provided to ensure the fixed-time containment control. Closely related to our work is [33], where coordinated tracking is investigated using observer based controllers together with the homogeneous theory. But the settling time of reaching a state agreement cannot be explicitly estimated. Another recent paper [22] does explicitly conclude a bound for the settling time, but such a bound varies if the initial conditions changes, while our approach guarantees that the coordinated tracking is achieved in a fixed time. In summary, the contributions of this paper are threefold. First, different from the existing consensus tracking work in [21,34] where first-order or second-order dynamics is investigated, in this paper the dynamics of each follower consists of two terms: one is the inherent nonlinear dynamics, the other is a communication protocol only relying on the information of its

neighbours. In other words, the system dynamics considered here is more generalized. Second, in contrast to the work in [22,23] where coordinated tracking is achieved in a finite time, in this paper, based on the fixed-time stability theory, we achieve the fixed-time coordinated tracking, of which an upper bound of the settling time is obtained using a class of newly proposed protocols. Such a settling time is independent of initial states, thus providing additional options for designers in practical scenarios where initial conditions are unavailable. Third, unlike the results obtained in [25,27,35,36] where fixed-time leaderless consensus or single leader consensus tracking problem is solved, we are dealing with the more complex multiple-leader case, i.e., containment control over directed topologies, which is of interest in scenarios such as safety-guaranteed vehicles moving. To the best of authors’ knowledge, it is the first to achieve the fixed-time containment control for single-integrator agents. Notation. The following notations are used throughout the paper. Let x = [xT1 , xT2 , … , xTN ]T , where xi = [xi1 , xi2 , … , xin ]T ∈ Rn , and denote sign(xi ) = [sign(xi1 ), sign(xi2 ), … , sign(xin )]T , where sign(⋅) is the signum function: sign(𝜐) equals to 1 if 𝜐 is positive; equals to −1 if 𝜐 is negative; and equals to 0 if 𝜐 = 0. The 1

p-norm is defined as ‖xi ‖p = (|xi1 |p +|xi2 |p +· · ·+|xin |p ) p , where p > 0. The fractional power of the vector xi is component-wise, i.e., [ ]T x𝜖i = x𝜖i1 , x𝜖i2 , … , x𝜖in ∈ Rn ,

(1)

where 𝜖 ∈ R is a constant. In addition, the notation ⊗ denotes the Kronecker product; for matrices A, B, C and D with appropriate dimensions, one has 1) (A ⊗ B)T = AT ⊗ BT ; 2) (A ⊗ B)(C ⊗ D) = AC ⊗ BD.

II. PRELIMINARIES AND PROBLEM FORMULATION 2.1 Graph theory The interaction among agents can be modeled by a directed graph f = (, ), where  = {v1 , v2 , ... , vN } is a vertex set indexed by an associated agent set  = {1, 2, … , N}, and  = {(vi , vj ) ∣ vi , vj ∈ } is an edge set consisting of communication links. In particular, an edge (vi , vj ) represents that agent i can directly access the state information of agent j, but not necessarily vice versa. The neighbour set for agent j is denoted as j = {vi ∈  ∣ (vi , vj ) ∈ }. An edge is undirected if and only if (vi , vj ) ∈  implies (vj , vi ) ∈ ; otherwise, the edge is directed. A directed path is a sequence of edges in a directed graph of the form (v1 , v2 ), (v2 , v3 ), … , where

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B. Ning et al.: Distributed Fixed-Time Coordinated Tracking for Nonlinear Multi-Agent Systems Under Directed Graphs

vi ∈ . A directed graph has a directed spinning tree if there exists at least one agent that has a directed path to any other agent. Mathematically, the interaction graph f can be represented by two matrices: the adjacency matrix f = [aij ]N×N with weights aij > 0 if (vi , vj ) ∈  and aij = 0 otherwise; and the Laplacian matrix f = [lij ]N×N ∑N with lij = −aij for i ≠ j, and lii = j=1 aij . It is common to assume no self-loops exist, i.e., (vi , vi ) ∉ , thus aii = 0. For an undirected graph, aij = aji .

in this article. In particular, the leader agents only send out information and receive no information from any other agent. Let  = [lij ](N+M)×(N+M) be the Laplacian matrix of . Then, it can be obtained that [ =

] Ωd f , 𝟎M×N 𝟎M×M

(5)

where f ∈ RN×N and Ωd ∈ RN×M . An assumption about the communication graph  is summarized as Assumption 3.

2.2 Problem formulation Suppose that there is a group consisting of N followers, labelled as agents 1 to N, and M leaders, labelled as agents N + 1 to N + M. The follower set and the leader set are denoted as  = {1, 2, … , N} and  = {N + 1, N + 2, … , N + M}, respectively. The dynamics of followers and leaders is described by {

ẋ i (t) = f (xi (t), t) + ui (t), ẋ i (t) = f (xi (t), t),

i∈ i ∈ ,

(2)

where xi (t) ∈ Rn is the state of the ith agent, f ∶ Rn × [0, ∞) → Rn is the inherent nonlinear dynamics, and ui (t) ∈ Rn denotes the control protocol to be designed for agent i. For notational convenience, let f (xi (t), t) = [f1 (xi (t), t)), f2 (xi (t), t), … , fn (xi (t), t)]T . Two separate assumptions about f (⋅) in the following are used in this paper. Assumption 1. The nonlinear function f (⋅) satisfies the condition of ‖f (𝜉1 (t), t) − f (𝜉2 (t), t)‖2 ≤ c1 ‖𝜉1 (t) − 𝜉2 (t)‖2

(3)

for all 𝜉1 (t), 𝜉2 (t) ∈ Rn , and c1 is a nonnegative constant. Assumption 2. Given nonnegative k1 , k2 , … , kM with ∑M k = 1, there exists a nonnegative constant c2 such i=1 i that the following inequality holds for all all 𝜉3 (t), xi (t) ∈ Rn ‖f (𝜉3 (t), t) −

M ∑

ki f (xi (t), t)‖2

i=1

Assumption 3. For each follower in the multi-agent system (2), there exists at least one leader that has a directed path to the follower and there exist some scalars pi > 0, i ∈  , such that pj aij = pi aji for all i, j ∈  , where aij is the (i, j)th entry of the adjacency matrix associated with topology f . In what follows, for notational simplicity, we occasionally drop the dependence on t for states without causing confusion. Before presenting our main results, we provide some definitions and lemmas that will be used in Section III. Definition 1. Fixed-time consensus tracking problem. Given a multi-agent system (2), the consensus tracking problem is to design distributed ui (t) for the followers such that xi → x0 (t) as t → T, ∀i ∈  . In particular, the settling time T is upper bounded by Tmax that is independent of initial states. Note that x0 (t) denotes the single leader’s state. Definition 2. Fixed-time containment control problem. Given a multi-agent system (2), the fixed-time containment control problem is to design distributed ui (t) for the followers such that xi (t) → Co(XL ) as t → T, ∀i ∈  . In particular, the settling time T is upper bounded by Tmax that is independent of initial states. Note that Co(XL ) denotes the convex hull of XL , where XL is a set of leaders’ state. Lemma 1. [20] Under Assumption 3, each entry of −−1 Ωd is non-negative and each row sum of −−1 Ωd is f f equal to one.

(4)

Lemma 2. [22] Under Assumption 3, and denote P = diag{p1 , p2 , … , pN }, then the matrix f P is positive definite.

The communication topology among the N + M agents, denoted as , is assumed to be fixed and directed

Lemma 3. [37] For any vector x ∈ Rn , and if p > r > 0, where p and r are scalar constants, then the following

≤ c2 ‖𝜉3 (t) −

M ∑

ki xi (t)‖2 .

i=1

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Asian Journal of Control, Vol. 20, No. 4, pp. 1–13, July 2018

where a < b, a and b are a positive even integer and a positive odd integer, respectively. And

inequality holds 1

‖x‖p ≤ ‖x‖r ≤ n r

− 1p

‖x‖p .

(6)

Lemma 4. [37] Given a differential equation 1

1

ẏ = −𝛼y1− 𝜅 − 𝛽y1+ 𝜅 , y(0) = y0 ,

(7)

where y ∈ R+ ∪ {0}, 𝛼, 𝛽 > 0, 𝜅 > 1. Then, the equilibrium is fixed-time stable for (7) and the following upper bound of the settling time T holds 𝜋𝜅 T ≤ Tmax = √ . 2 𝛼𝛽

(8)

Remark 1. The detailed proof of Lemma 4 is presented in [37], and it is a generalization of Lemma 4 in [32]. Note that, the settling time Tmax in (8) is independent of initial states. Compared to the results in [25,27,38], Lemma 4 extends the parameter range. Instead of restricting the power of y to be the quotient of two odd integers, i.e., odd p and q in Lemma 4.1 of [25], odd m, n, p and q in Lemma 4.1 of [38], it is only required that 𝜅 > 1. The advantage of such an extension is that, under the same 𝛼 and 𝛽, a less conservative settling time can be obtained through adjusting 𝜅.

c p ⎧𝛼 ≥ √ 1 max( ) ⎪ 1 pmin 𝜆min Tf f ⎪ ( ) a −1 ⎨ 𝛽 = 𝛽p 2𝜆min (f P) 2b 1 max ⎪ ) a a ( ⎪ 𝛾1 = 𝛾pmax (nN) 2b 2𝜆min (f P) − 2b −1 , ⎩

(10)

where pmax and pmin denote the largest and the smallest diagonal entry of P, respectively, and 𝜆min (⋅) denotes the smallest eigenvalue of the matrix. Note that the first term in (9) is to cancel the effect caused by the inherent nonlinear dynamics, and the last two terms is to guarantee the fixed-time consensus tracking. Theorem 1. Given a multi-agent system (2) over a directed graph , and Assumptions 1 and 3 hold. The control protocol (9) enables the system to achieve fixed-time consensus tracking, i.e, limt→T ‖xi (t) − xN+1 (t)‖2 = 0, ∀i ∈  . In particular, the 𝜋b settling time T ≤ √ . a 𝛽𝛾

Proof. Substituting (9) into (2), one obtains ∑

N+1

ẋ i = −𝛼1

aij (xi − xj ) + f (xi (t), t)

j=1

− 𝛽1

III. MAIN RESULTS

3.1 Fixed-time consensus tracking over directed graph In this subsection, the single leader case (M=1) for (2) is investigated. Under directed graphs, we propose a fixed-time distributed protocol for each follower ∑

ui = −𝛼1

aij (xi − xj ) − 𝛽1

(N+1 ∑

j=1

− 𝛾1

(N+1 ∑

)1− a b

aij (xi − xj )

j=1

)1+ a b

aij (xi − xj )

,

)1− a b

(11)

aij (xi − xj )

j=1

In this section, under directed graphs, the single leader case for (2), i.e., the fixed-time consensus tracking problem, is first investigated, then the multiple leader case, i.e., the fixed-time containment control problem, is explored.

N+1

(N+1 ∑

i ∈ ,

j=1

(9)

− 𝛾1

(N+1 ∑

)1+ a b

aij (xi − xj )

,

i ∈ .

j=1

Then the dynamics of N followers can be written in a compact vector form ẋ F = −𝛼1 ((f ⊗ In )xF + (Ωd ⊗ In )xN+1 ) [ ]1− a − 𝛽1 ((f ⊗ In )xF + (Ωd ⊗ In )xN+1 ) b [ ]1+ a (12) − 𝛾1 ((f ⊗ In )xF + (Ωd ⊗ In )xN+1 ) b + F(xF , t), [ ]T [ where xF = xT1 , xT2 , … , xTN , F(xF , t) = f (x1 (t), t)T , ]T f (x2 (t), t)T , … , f (xN (t), t)T and xN+1 is the state of the single leader. Define the error vector as e = [ T T ]T e1 , e2 , … , eTN , where ei = xi − xN+1 , i ∈  . It can be further calculated that (f ⊗ In )xF + (Ωd ⊗ In )xN+1 = (f ⊗ In )e, where e = xF − 𝟏N ⊗ xN+1 . Note that here the

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B. Ning et al.: Distributed Fixed-Time Coordinated Tracking for Nonlinear Multi-Agent Systems Under Directed Graphs

fact that f 𝟏N + Ωd = 𝟎N has been used. Thus, one gets ]1− a [ ė = −𝛼1 (f ⊗ In )e − 𝛽1 (f ⊗ In )e b [ ]1+ a − 𝛾1 (f ⊗ In )e b + F(xF , t)

By Lemma 3, it can be obtained that ⎧ ‖(f ⊗ In )e‖2− a ≥ ‖(f ⊗ In )e‖2 b ⎪ ( )−1 1 − 1 ⎨ 2 2+ a b a ‖(f ⊗ In )e‖2 . ⎪ ‖(f ⊗In )e‖2+ b ≥ (nN) ⎩ (17)

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− 𝟏N ⊗ f (xN+1 (t), t). Now, construct a Lyapunov candidate )( )T ( ) 1( (f ⊗ In )e (f P)−1 ⊗ In (f ⊗ In )e 2 ) 1 T ( T −1 = e (f P ) ⊗ In e. 2 (14)

V1 =

Since f P is positive definite according to Lemma 2, it is invertible and the inverse is also positive definite. Therefore, V1 is well defined. Differentiating V1 with respect to t, one gets ( ) V̇ 1 = eT (Tf P−1 ) ⊗ In ė ( )T = (P−1 f ⊗ In )e ė [ ( −1 )T = (P ⊗ In )(f ⊗ In )e − 𝛼1 (f ⊗ In )e

−

𝛾1 p−1 max

2𝜆min (f P)V1

2

=

⊗ In )e‖2

≥

. Therefore,

( )1− a 2− ab ⎧ 2b ⊗ I )e‖ ≥ 2𝜆 ( P)V ‖( a f n min f 1 ⎪ 2− b a ⎨ )1+ a 2+ −a ( ⎪ ‖(f ⊗ In )e‖2+ ba ≥ (nN) 2b 2𝜆min (f P)V1 2b . ⎩ b (18) ( ) a −1 Inserting 𝛽1 = 𝛽pmax 2𝜆min (f P)V1 2b and 𝛾1 = )−a2b a ( 𝛾pmax (nN) 2b 2𝜆min (f P)V1 −1, and combining (16) and (18), one finally obtains a

(19)

For V̇ 1 = −𝛽V 1− 2b − 𝛾V 1+ 2b = −𝛽V 1− 𝜅 − 𝛾V 1+ 𝜅 , where the parameter 𝜅 = 2b > 1. By Lemma 4, the equia a

a

1

librium of V1 = 0 is fixed-time stable, and T =

1

𝜋b √ a 𝛽𝛾

is obtained, which is obviously independent of initial states. Then using the comparison principle, for V̇ 1 ≤ a a −𝛽V 1− 2b − 𝛾V 1+ 2b , the equilibrium is still fixed-time stable and an upper bound of the settling time is estimated 𝜋b as T = √ . Note that V1 = 0 indicates e = 𝟎nN . There-

2− ab

‖(f ⊗ In )e‖2− a b

2+ ab

‖(f ⊗ In )e‖2+ a b

a 𝛽𝛾

c1 p−1 min

fore, xi (t) = xN+1 (t), i ∈  for t ≥ T, which completes the proof.

+√ ‖(f ⊗ In )e‖22 . T 𝜆min (f f ) (15) Note that the fact that ‖F(xF , t) − 𝟏N ⊗ c √ 1 T ‖(f ⊗ In )e‖2 has f (xN+1 (t), t)‖2 ≤ 𝜆min (f f )

been inserted to get the inequality of (15). Since c p √1 max T , one gets 𝛼1 ≥ pmin

⊗ )1

a

2 ≤ −𝛼1 p−1 max ‖(f ⊗ In )e‖2

−

(

⊗ In )e‖22

1− 1+ V̇ 1 ≤ −𝛽V1 2b − 𝛾V1 2b .

[ ]1− a [ ]1+ a − 𝛽1 (f ⊗ In )e b − 𝛾1 (f ⊗ In )e b ] + F(xF , t) − 𝟏N ⊗ f (xN+1 (t), t)

𝛽1 p−1 max

1 ‖(f 2𝜆min (f P)

( ) 1 𝜆 (f P)−1 ‖(f 2 max In )e‖22 , one gets ‖(f

≤

Since V1

𝜆min (f f )

a

2− V̇ 1 ≤ −𝛽1 p−1 ‖(f ⊗ In )e‖2− ba max b

2+ ab

− 𝛾1 p−1 ‖(f ⊗ In )e‖2+ a . max b

(16)

Theorem 2. Set f to zero, i.e., the nonlinear multi-agent system (2) is modified to a linear form of {

ẋ i (t) = ui (t), ẋ i (t) = gi (t),

i∈ i ∈ ,

(20)

where gi (t) ∈ Rn is responsible for controlling trajectory of the ith leader, and we assume ‖gi (t)‖2 ≤ 𝜂, where 𝜂 is a known positive constant. Given a multi-agent system (20) over a directed graph , and Assumption 3 holds. The fixed-time consensus tracking is achieved by using the following control protocol for each follower

© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd

Asian Journal of Control, Vol. 20, No. 4, pp. 1–13, July 2018

ui = −𝛼1 sign

(N+1 ∑

( )T ( [ where the fact that (P−1 ⊗In )(f ⊗In )e −𝛼1 sign (f ⊗ ]) ‖(f ⊗ In )e‖1 due to P−1 is a positive In )e ≤ −𝛼1 p−1 max definite diagonal matrix has√been used to obtain the first

) aij (xi − xj )

j=1

− 𝛽1

(N+1 ∑

)1− a b

aij (xi − xj )

(21)

j=1

− 𝛾1

(N+1 ∑

)1+ a b

,

aij (xi − xj )

i ∈ ,

where 𝛼1 ≥

(

) a −1 2𝜆min (f P) 2b and

𝜂pmax N , pmin

𝛾1 = 𝛾pmax (nN)

a 2b

𝛽1 = 𝛽pmax )− a 2𝜆min (f P) 2b − 1.

(

Proof. Since the proof is similar to that of Theorem 1, only key steps are presented. Note that all notations are the same as those in Theorem 1. First, the time derivative of e becomes [ ] [ ]1− a ė = −𝛼1 sign (f ⊗ In )e − 𝛽1 (f ⊗ In )e b [ ]1+ a − 𝛾1 (f ⊗ In )e b − 𝟏N ⊗ gN+1 (t).

N

a 𝛽𝛾

j=1

√

𝜂p

inequality, and 𝛼1 ≥ max has been used to obtain pmin the second inequality. Then, following the same lines in Theorem 1, the fixed-time consensus is achieved within 𝜋b T≤ √ , which completes the proof.

(22)

Differentiating V1 with respect to t, one gets [ ( ) [ ] ̇V1 = (P−1 ⊗ In )(f ⊗ In )e T − 𝛼1 sign (f ⊗ In )e [ ]1− a [ ]1+ a − 𝛽1 (f ⊗ In )e b − 𝛾1 (f ⊗ In )e b ] − 𝟏N ⊗ gN+1 (t)

Remark 2. In [22,23], the containment control problem has been achieved within a finite time, which depends on initial states. However, in practical scenarios, the knowledge of initial conditions is sometimes unavailable in advance, which may limit the applications of the existing finite-time approaches. Instead, in this paper, the upper 𝜋b bound of the settling time, i.e., √ in Theorems 1 and a 𝛽𝛾

2, does not depend on the initial conditions. In fact, once the system parameters are determined, the upper bound is directly estimated. Remark 3. Compared to some finite-time controllers [39–43], the fixed-time controller in this paper has an additional fractional power term with an index larger than 1, i.e., 1 + ab in (9) and (21). This power term ensures the containment control is achieved in a fixed time. On the other hand, the initial control effort, i.e., ui in (9) and (21), could be relatively large due to the possibly large initial state difference among agents.

3.2 Fixed-time containment control over directed graph In this subsection, the containment control, i.e., M > 1, for (2) is investigated. The aim is to guarantee the states of followers to converge to a convex hull formed by those of leaders in a fixed time regardless of initial states. Under directed graphs, the fixed-time distributed protocol for each follower is proposed as

≤ −𝛼1 p−1 ‖(f ⊗ In )e‖1 max √ + 𝜂p−1 N‖(f ⊗ In )e‖2 min 2− a

− 𝛽1 p−1 ‖(f ⊗ In )e‖2− ba max b

−

𝛾1 p−1 max

( = − 𝛼1 p−1 max

2+ ab

‖(f ⊗ In )e‖2+ a b √ ) −1 − 𝜂pmin N ‖(f ⊗ In )e‖2

∑

N+M

ui = −𝛼2

2− a

b − 𝛽1 p−1 max ‖(f ⊗ In )e‖2− a

−

𝛾1 p−1 max

b 2+ ab 2+ ab 2− ab 2− ab 2+ ab 2+ ab

− 𝛽2

(N+M ∑

)1− a b

aij (xi − xj )

j=1

‖(f ⊗ In )e‖

− 𝛾2

≤ −𝛽1 p−1 ‖(f ⊗ In )e‖ max

− 𝛾1 p−1 ‖(f ⊗ In )e‖ max

aij (xi − xj )

j=1

(N+M ∑

(24)

)1+ a b

aij (xi − xj )

,

i ∈ ,

j=1

, (23)

where a and b are the same as those defined in Section 3.1, and 𝛼2 , 𝛽2 , 𝛾2 are feedback gains to be determined.

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B. Ning et al.: Distributed Fixed-Time Coordinated Tracking for Nonlinear Multi-Agent Systems Under Directed Graphs

Theorem 3. Given a multi-agent system (2) over a directed graph , and Assumptions 2 and 3 hold. If ( ) a −1 c p √2 max T , 𝛽2 = 𝛽pmax 2𝜆min (f P) 2b and 𝛼2 ≥ pmin

𝜆min (f f )

)− a −1 a ( 𝛾2 = 𝛾pmax (nN) 2b 2𝜆min (f P) 2b , then the control protocol (24) enables the system to converge to a convex hull formed by those of leaders in a fixed time, i.e, xi → Co(XL ) in a finite time T, ∀i ∈  . In particular, the 𝜋b . settling time T ≤ √ a 𝛽𝛾

Proof. Substituting (24) into (2), one obtains

f (xN+M (t), t)T ]T . Then, constructing a Lyapunov candi( )T ( )( ) (f P)−1 ⊗ In (f ⊗ In )̂e , and date V2 = 12 (f ⊗ In )̂e differentiating it with respect to t, one gets [ ( ( ) ) ̇V2 = (P−1 ⊗ In ) f ⊗ In ê T − 𝛼2 (f ⊗ In )̂e [ ]1− a [ ]1+ a − 𝛽2 (f ⊗ In )̂e b − 𝛾2 (f ⊗ In )̂e b ] −1 + F(xF , t) + (f Ωd ⊗ In )F(xL , t) ≤ − 𝛼2 p−1 ‖(f ⊗ In )̂e‖22 max

∑

2− a

N+M

ẋ i = −𝛼2

− 𝛽2 p−1 ‖(f ⊗ In )̂e‖2− ba max

aij (xi − xj ) + f (xi (t), t)

b

j=1

− 𝛽2

(N+M ∑

)1− a

−

− 𝛾2

‖(f ⊗ In )̂e‖2+ a

+

p−1 ‖(f ⊗ In )̂e‖2 ‖F(xF , t) min −1 (f Ωd ⊗ In )F(xL , t)‖2 .

b

b

aij (xi − xj )

(25)

j=1

(N+M ∑

2+ ab

𝛾2 p−1 max

+

)1+ a b

aij (xi − xj )

,

i ∈ .

(28)

j=1

Then the dynamics of N followers can be written in a compact vector form ẋ F = −𝛼2 ((f ⊗ In )xF + (Ωd ⊗ In )xL ) ]1− a [ − 𝛽2 ((f ⊗ In )xF + (Ωd ⊗ In )xL ) b [ ]1+ a − 𝛾2 ((f ⊗ In )xF + (Ωd ⊗ In )xL ) b

(26)

+ F(xF , t), where xL = [xTN+1 , xTN+2 , … , xTN+M ]T , xF and F(xF , t) are the same as those defined in Section 3.1. Define x∗ = −(−1 Ωd ⊗ In )xL , and invoke Lemma 1, it can be conf cluded that the states of followers will converge to the convex hull Co(XL ) if xF → x∗ . Then, an error vector is defined as ê = xF − x∗ , and ê = [̂eT1 , ê T2 , … , ê TN ]T . It can be further calculated that (f ⊗ In )xF + (Ωd ⊗ In )xL = (f ⊗ In )̂e. Therefore, the following equality holds ]1− a [ ê̇ = −𝛼2 (f ⊗ In )̂e − 𝛽2 (f ⊗ In )̂e b [ ]1+ a − 𝛾2 (f ⊗ In )̂e b + F(xF , t)

(27)

To deal with the last term in (28), let hij be the (i, j)th entry of −−1 Ωd . By Lemma 1, hij is nonnegative and f ∑M h = 1, i = 1, 2, … , N. Then, under Assumption 2, j=1 ij it can be obtained that ( ) ‖F(xF , t) + −1 Ω ⊗ I F(xL , t)‖2 d n f ( M ∑ ‖ ( )T =‖ f (x (t), t) − h1j f (xN+j (t), t) , 1 ‖ ‖ j=1 M ∑ ( )T + … , f (xN (t), t) − hNj f (xN+j (t), t) j=1

‖ ‖ ‖ ‖2

( M ∑ ‖ ‖ =‖ ‖ f (x (t), t) − h1j f (xN+j (t), t)‖ ‖ 1 ‖2 , ‖ j=1 + …,‖ ‖f (xN (t), t) −

M ∑ j=1

hNj f (xN+j (t), t)‖ ‖2

)T

‖ ‖ ‖ ‖2

‖( )T ‖ ‖ ≤ c2 ‖ ‖ ‖̂e1 ‖2 , ‖̂e2 ‖2 … , ‖̂eN ‖2 ‖ ‖ ‖2 c2 = c2 ‖̂e‖2 ≤ √ ‖(f ⊗ In )̂e‖2 . 𝜆min (Tf f ) (29)

+ (−1 Ωd ⊗ In )F(xL , t), f where F(xL , t) = [f (xN+1 (t), t)T , f (xN+2 (t), t)T , … ,

)T

Inserting (29) into (28), the following inequality holds

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Asian Journal of Control, Vol. 20, No. 4, pp. 1–13, July 2018

( ) c p−1 ̇V2 ≤ − 𝛼2 p−1 − √ 2 min ‖(f ⊗ In )̂e‖22 max T 𝜆min (f f )

[ ] [ ]1− a ê̇ = −𝛼2 sign (f ⊗ In )̂e − 𝛽2 (f ⊗ In )̂e b [ ]1+ a − 𝛾2 (f ⊗ In )̂e b + (−1 Ωd ⊗ In )g(t), f

2− a

− 𝛽2 p−1 ‖(f ⊗ In )̂e‖2− ba max b

− ≤

𝛾2 p−1 max

−𝛽2 p−1 max

2+ ab

‖(f ⊗ In )̂e‖2+ a ‖(f ⊗ In )̂e‖2− a

b 2+ ab 2+ ab

where g(t) = [gN+1 (t)T , gN+2 (t)T , … , gN+M (t)T ]T . Differentiating V2 with respect to t, one gets

, (30)

c pmax

pmin

(33)

b

2− ab

− 𝛾2 p−1 ‖(f ⊗ In )̂e‖ max

where 𝛼2 ≥

Proof. Note that all notations are the same as those in Theorem 2. Under (32), the time derivative of ê becomes

√2

𝜆min (Tf f )

[ ( ) [ ] ̇V2 = (P−1 ⊗ In )(f ⊗ In )̂e T − 𝛼2 sign (f ⊗ In )̂e ]1− a ]1+ a [ [ − 𝛽2 (f ⊗ In )̂e b − 𝛾2 (f ⊗ In )̂e b ] −1 + (f Ωd ⊗ In )g(t)

has been used. Then follow

a similar procedure in Section 3.1 and insert ) a −1 ( a 𝛽2 = 𝛽pmax 2𝜆min (f P) 2b and 𝛾2 = 𝛾pmax (nN) 2b a ( )− −1 2𝜆min (f P) 2b , it can be finally obtained that

≤ −𝛼2 p−1 e‖1 max ‖(f ⊗ In )̂ 2− a

a

− 𝛽2 p−1 ‖(f ⊗ In )̂e‖2− ba max

a

b

1− 1+ V̇ 2 ≤ −𝛽V2 2b − 𝛾V2 2b .

2+ ab

(31)

− 𝛾2 p−1 ‖(f ⊗ In )̂e‖2+ a max b √ −1 + 𝜂pmin N‖(f ⊗ In )̂e‖2

Following the same lines in Section 3.1 and using Lemmas 4 and the comparison principle, it can be concluded that the equilibrium of V2 = 0, is fixed-time stable and an upper bound of the settling time is estimated as 𝜋b T= √ . Note that V2 = 0 indicates e = 𝟎nN . Therefore,

2− a

≤ −𝛽2 p−1 ‖(f ⊗ In )̂e‖2− ba max b

−

a 𝛽𝛾

Ωd ⊗ In )xL (t) for t ≥ T, which completes xF (t) = −(−1 f the proof. Theorem 4. If set f to zero, i.e., the nonlinear multi-agent system (2) is modified to the form of (20), then the fixed-time containment control problem over a directed graph  under Assumption 3 is solved by the following control protocol for each follower

𝛾2 p−1 max

2+ ab

‖(f ⊗ In )̂e‖2+ a , b

(34) √ where ‖−1 Ω ⊗ I )g(t)‖ ≤ 𝜂 N (based on Lemmas 1 d n 2 f 𝜂p

√

N

and 3) and 𝛼2 ≥ max has been used to get the first pmin and last inequality, respectively. Then, following the same lines in Theorem 2, the fixed-time containment control 𝜋b , which completes the for (20) is achieved within T ≤ √ a 𝛽𝛾

ui = −𝛼2 sign

(N+M ∑

− 𝛽2

(N+M ∑

) aij (xi − xj )

j=1

− 𝛾2

IV. SIMULATIONS

)1− a b

aij (xi − xj )

j=1

(N+M ∑

proof.

(32)

)1+ a b

aij (xi − xj )

,

i ∈ ,

j=1

where 𝛼2 ≥

√ 𝜂pmax N , pmin

𝛾2 = 𝛾pmax (nN)

a 2b

(

) a −1 2𝜆min (f P) 2b and

𝛽2 = 𝛽pmax )− a −1 2𝜆min (f P) 2b .

(

In this section, two numerical examples are provided to illustrate the effectiveness of the theoretical analysis. Without loss of generality, it is assumed that n = 2 for both examples. Example 1 (Fixed-time consensus tracking). Consider a multi-agent system consists of five followers (N = 5) and one leader. A directed graph shown in Fig. 1 is used as the interaction topology, and its associated matrix  and

© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd

B. Ning et al.: Distributed Fixed-Time Coordinated Tracking for Nonlinear Multi-Agent Systems Under Directed Graphs

Fig. 1. A directed interaction topology for the consensus tracking.

Fig. 3. Fixed-time consensus tracking under protocol (9) - the second dimension. [Color figure can be viewed at wileyonlinelibrary.com]

where xi = (xi1 , xi2 , xi3 )T , ∀i ∈  ∪ . In this case, it has been shown in [22] that c1 = 4.3871. Set the initial states of followers as xF (0) = [(2, 1, 1), (−3, −3, −2), (1, −2, −1), (−2, 3, 1), (−1, 2, 1)]T . The leader evolves from xN+1 (0) = (1, 1.5, −2)T . Select 𝛽 = 1, 𝛾 = 0.3,√a = 2, b = 5 in (9). Therefore, Tmax = 𝜋b∕a 𝛽𝛾 = 14.34 s. Further, c p √1 max T = 17.86, it can be calculated 𝛼1 = Fig. 2. Fixed-time consensus tracking under protocol (9) - the first dimension. [Color figure can be viewed at wileyonlinelibrary.com]

(

) a −1

pmin

𝜆min (f f )

𝛽1 = 𝛽pmax 2𝜆min (f P) = 2.00, and 𝛾1 = 𝛾pmax )− a −1 a ( 2b (nN) 2b 2𝜆min (f P) = 0.95. Under protocol (9), in Figs. 2, 3 and 4, the settling time is about t = 8.5 s, which demonstrates the effectiveness of Theorem 1. 2b

P are denoted as Example 2 (Fixed-time containment control). Consider a multi-agent system consists of five followers (N = 5) and two leaders (M = 2). A directed graph shown in Fig. 5 is used as the interaction topology, and its associated matrix  and matrix P are denoted as

⎡ 4.4 −0.4 0 0 0 −4 ⎤ ⎢ −0.2 0.6 −0.4 0 0 0 ⎥ ⎢ ⎥ 0 −0.8 4.8 −1 0 −3 ⎥ =⎢ ; 0 −0.75 2.25 −1.5 0 ⎥ ⎢ 0 ⎢ 0 0 0 −1 4 −3 ⎥ ⎢ ⎥ 0 0 0 0 0 ⎦ ⎣ 0 P = diag{2, 1, 2, 1.5, 1}. The inherent nonlinear dynamics for each agent is described by the Chua’s circuit ⎡ 𝜁(−xi1 + xi2 − l(xi1 )) ⎤ ⎥, xi1 − xi2 + xi3 f (xi (t), t) = ⎢ ⎢ ⎥ −𝜚xi2 ⎣ ⎦

(35)

0 0 −2 ⎡ 2.4 −0.4 0 ⎢ −0.2 0.6 −0.4 0 0 0 ⎢ 0 −0.8 1.8 −1 0 0 ⎢ 0 −0.75 2.25 −1.5 0 =⎢ 0 ⎢ 0 0 0 −1 2 0 ⎢ 0 0 0 0 0 0 ⎢ 0 0 0 0 0 ⎣ 0 P = diag{2, 1, 2, 1.5, 1}.

© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd

0 ⎤ 0 ⎥ 0 ⎥ ⎥ 0 ⎥; −1 ⎥ 0 ⎥ ⎥ 0 ⎦

Asian Journal of Control, Vol. 20, No. 4, pp. 1–13, July 2018

Fig. 4. Fixed-time consensus tracking under protocol (9) - the third dimension. [Color figure can be viewed at wileyonlinelibrary.com]

Fig. 6. Fixed-time containment control under protocol (24) the first dimension. [Color figure can be viewed at wileyonlinelibrary.com]

Fig. 5. A directed interaction topology for containment control.

The inherent nonlinear dynamics is set as [

] x1 cos (t) f (xi (t), t) = , x2 sin (2t2 )

(36)

where xi = (xi1 , xi2 )T , ∀i ∈  ∪ . In this case, c2 = 1 according to Assumption 2. Set the initial states of followers as xF (0) = [(5, 1), (−3, −3), (−15, −15), (−2, 5), (−1, 2)]T , and the leader xL (0) = [(1.2, 1.5), (−1.2, −1)]T . The parameters 𝛽, 𝛾, a, b keep unchanged. Therefore, Tmax = 14.34 s. Moreover, it can be calculated 𝛼1 = ) a −1 ( c p √2 max T = 9.24, 𝛽1 = 𝛽pmax 2𝜆min (f P) 2b = pmin

𝜆min (f f )

)− a −1 a ( 3.37, and 𝛾1 = 𝛾pmax (nN) 2b 2𝜆min (f P) 2b = 2.08. In Figs. 6 and 7, the settling time is about t = 0.7 s under protocol (24), which is smaller than Tmax and thereby demonstrating the effectiveness of Theorem 3.

Fig. 7. Fixed-time containment control under protocol (24) the second dimension. [Color figure can be viewed at wileyonlinelibrary.com]

V. CONCLUSION This paper has investigated the fixed-time coordinated tracking problem of nonlinear multi-agent systems with single-integrator dynamics under directed graphs. A new class of nonlinear protocols are proposed, under which an upper bound of the settling time is directly estimated. Such an approach is able to provide system information in advance, i.e., the estimated settling time, when designing controllers in practical scenarios

© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd

B. Ning et al.: Distributed Fixed-Time Coordinated Tracking for Nonlinear Multi-Agent Systems Under Directed Graphs

where the knowledge of initial conditions is unavailable. The effectiveness of the proposed controllers is illustrated in simulations for both consensus tracking and containment control. Future work will concentrate on containment control with more general directed graphs, double-integrator dynamics, or switching topology.

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37.

38.

39.

40.

41.

42.

43.

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Boda Ning was born in Henan, China, in 1989. He received the B.Eng. degree in automatic control from East China University of Science and Technology, Shanghai, China, in 2011, the M.Sc. degree with Distinction in control systems from University of Manchester, Manchester, U.K., in 2012, and the Ph.D. degree from Swinburne University of Technology, Melbourne, Australia, in 2017. From 2010 to 2011, he was an exchange student at University of Dundee, Dundee, U.K. He was a recipient of the Vice-Chancellor’s Research Scholarship (2013-2017). His current research interests include fixed-time consensus and cooperative control of multi-agent systems.

© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd

B. Ning et al.: Distributed Fixed-Time Coordinated Tracking for Nonlinear Multi-Agent Systems Under Directed Graphs

Zongyu Zuo received his B.Eng. degree in automatic control from Central South University, Hunan, China, in 2005, and Ph.D. degree in control theory and applications from Beihang University (BUAA), Beijing, China, in 2011. He was an academic visitor at the School of Electrical and Electronic Engineering, University of Manchester from September 2014 to September 2015 and held an inviting associate professorship at Mechanical Engineering and Computer Science, UMR CNRS 8201, Université de Valenciennes et du Hainaut-Cambrésis in October 2015 and May 2017. He is currently an associate professor at the School of Automation Science and Electrical Engineering, Beihang University. His research interests are in the fields of nonlinear system control, control of UAVs, and coordination of multi-agent system.

Jinchuan Zheng received his B.Eng. and M.Eng. degrees in mechatronics engineering from Shanghai Jiao Tong University, Shanghai, China, in 1999 and 2002, respectively, and the Ph.D. degree in electrical and electronic engineering from Nanyang Technological University, Singapore, in 2006. In 2005, he joined the Australian Research Council (ARC), University of Newcastle, Callaghan, Australia, as a Research Academic. Currently, he is serving as a Senior Lecturer at Swinburne University of Technology, Melbourne, Australia. His research interests include mechanism design and control of high-precision mechatronic systems, sensing and vibration analysis, dual-stage actuation, and vision-based control.

Jiong Jin received his B.E. degree with First Class Honours in Computer Engineering from Nanyang Technological University, Singapore, in 2006, and Ph.D. degree from The University of Melbourne, Australia, in 2011. He is currently a Senior Lecturer in Swinburne University of Technology, Australia. Prior to it, he was a Research Fellow in The University of Melbourne from 2011 to 2013. His research interests include network design and optimization, nonlinear systems and sliding mode control, networked robotics, Internet of things, cyber-physical systems and applications in smart grids and smart cities.

© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd