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An Overview of Recent Advances in Fixed-Time Cooperative Control of Multi-Agent Systems Zongyu Zuo, Member, IEEE, Qing-Long Han, Senior Member, IEEE, Boda Ning, Member, IEEE, Xiaohua Ge, Member, IEEE, and Xian-Ming Zhang, Member, IEEE

Abstract—Fixed-time cooperative control is currently a hot research topic in multi-agent systems since it can provide a guaranteed settling time, which does not depend on initial conditions. Compared with asymptotic cooperative control algorithms, fixedtime cooperative control algorithms can achieve better closedloop performance and disturbance rejection properties. Different from finite-time control, fixed-time cooperative control produces the faster rate of convergence and provides an explicit estimation of the settling time independent of initial conditions, which is desirable for multi-agent systems. This paper aims at presenting an overview of recent advances in fixed-time cooperative control of multi-agent systems. Some fundamental concepts about finiteand fixed-time stability and stabilization are first recalled with insight understanding. Then recent results in finite- and fixedtime cooperative control are reviewed in detail and categorized according to different agent dynamics. Finally, this paper raises several challenging issues that need to be addressed in the near future. Index Terms—Multi-agent systems, consensus, finite-time cooperative control, fixed-time cooperative control

I. I NTRODUCTION OOPERATIVE control including consensus, flocking, coverage control, containment control, and formation control has received considerable research interest in multiagent systems due to its widespread applicable areas, such as wheeled robots, spacecraft flying, military surveillance, source seeking and exploration, search and rescue in disaster sites [1]–[7]. The key objective of cooperative control is to reach a desired global group behavior by using global/local information shared among neighboring agents in a distributed fashion. The cooperative control promises several advantages in implementing cooperative group tasks, such as strong adaptivity, high robustness, flexibility, scalability, and low operational costs. The settling time, which characterizes the rate of convergence rate of a closed-loop system, is well recognized as one of the performance specifications for control system design. Fast convergence is usually pursued in practice in order to achieve better performance and robustness. In the context

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This work was supported in part by the National Natural Science Foundation of China under Grant 61673034 and the Australian Research Council Discovery Project under Grant DP160103567. (Corresponding author: Q.-L. Han). Z. Zuo is with the Seventh Research Division, Beihang University (BUAA), Beijing 100191, China (e-mail: [email protected]). Q.-L. Han, B. Ning, X. Ge, and X.-M. Zhang are with the School of Software and Electrical Engineering, Swinburne University of Technology, Melbourne, VIC 3122, Australia. (emails: [email protected] (Q.-L. Han); [email protected] (B. Ning); [email protected] (X. Ge); [email protected] (X.-M. Zhang)).

of cooperative control, a pioneering result [8] shows that the second smallest eigenvalue of the Laplacian matrix of the interaction graph, i.e., algebraic connectivity, determines the consensus convergence rate. To increase the convergence rate, some investigations are carried out by proper design of interaction graph among agents [9]. Through working out the optimum vertex positional configuration in the presence of a proximity constraint, the consensus speed is accelerated. In practical multi-agent systems, it is of particular interest to realize cooperative control in a finite time to meet specific system requirements. Therefore, finite-time cooperative control problems, particularly finite-time consensus (one fundamental problem of cooperative control), have drawn considerable research attention and appreciation [10]–[19]. For example, in [12], a discontinuous controller together with a pinning control scheme is proposed, where only sign information of agents’ relative states is necessary to achieve finite-time consensus. In [15], to obtain a fast convergence rate, a switching strategy combining existing continuous and discontinuous finite-time consensus protocols is developed. Although convergence may be pursued in a finite time, an estimation on the settling time relies explicitly on initial conditions of systems in the group [10]–[19]. This may limit the application scope of those existing results in finite-time cooperative control when information on initial states of multiagent systems is unknown or unavailable a priori. Fortunately, a strategy called fixed-time cooperative control has been emerging since fixed-time stability was first investigated by Polyakov in 2012 [20]. It is shown that fixed-time stability provides guaranteed convergence (settling) time irrelevant to initial conditions. As a result, research in fixed-time cooperative control for multi-agent systems has come to the fore. In [21]–[23], an upper bound of finite settling time to reach an agreement is prescribed for networked multi-agent systems with simple dynamics. A distinctive feature of this approach is that the bound is merely dependent on design parameters and algebraic connectivity of graph Laplacian, and therefore, the closed-loop rate of convergence may be predefined or prescribed off-line. Based on the basic idea on fixed-time consensus, some related cooperative control problems, such as bipartite consensus [24], distributed optimization [25], average tracking [26], containment control [27], dispersion and flocking [28], are also studied. In this survey, we focus on fixed-time cooperative control problems of multi-agent systems with various agent dynamics, aiming at presenting an overview of recent advances. The remainder of this paper is organized as follows. The finite-

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and fixed-time stability and stabilization are briefly introduced and recalled in Section II, which serves as theoretic fundamentals for fixed-time cooperative control design. In Sections III and IV, finite- and fixed-time cooperative control results are reviewed and categorized according to different agent dynamics. Recent results in finite- and fixed-time consensus control of both simple and complex dynamic systems as well as related cooperative control problems are reviewed in detail in each category of finite- and fixed-time cooperative control. Some challenging issues and promising research directions are pointed out in Section V. II. F INITE -/ FIXED - TIME

STABILITY AND STABILIZATION

Before recalling the definition of the finite- and fixedtime stability, three simple examples are given to reveal the fundamental concepts and shed light on the motivation of developing fixed-time stability. Example 1: Consider a simple differential system 1

x(t) ˙ = −x 3 (t),

x(0) = x0 ,

(1)

where x(t) ∈ R represents the state. The system trajectory 4 3 satisfies x(t) = (x 3 (0)− 34 t) 4 , which reveals that x(t, x0 ) = 0 4

for t ≥ t∗1 = 34 x03 . Note that Example 1 is essentially a finite-time stable system, where the system’s solution reaches the origin at some finite moment. The key feature lies in the fractional exponent that enables the system to achieve the finite-time convergence to zero. Although the faster rate of convergence can be achieved in the neighborhood of the origin, the rate of convergence becomes slow when the system state is far away from 1. As an improvement of the rate of convergence, another example is given as follows. Example 2: Consider a scalar system 1

x(t) ˙ = −x(t) − x 3 (t),

x(0) = x0 .

(2)

4 3

The solution x(t, x0 ) = 0 for t ≥ t∗2 = 34 ln(x0 + 1). Due to the introduction of a linear term −x(t), it is expected that a faster speed of convergence than the one in system (1) can be obtained due to the fact that the linear term dominates the transient performance for any initial state x0 ≫ 1. This can also be verified by noting t∗1 > t∗2 . Then, a natural question one may ask is whether the system response can be even faster. The following example gives a positive answer. Example 3: Consider a polynomial feedback system 1 3

3

x(t) ˙ = −x (t) − x (t),

x(0) = x0 .

(3)

Integrating both sides of (3) yields that √ √ √ 2 4 √ 2 3 2 x 3 + 2x 3 + 1 2 + ln 4 √ 2 arctan( 2x 3 + 1) 2 8 4 x 3 − 2x 3 + 1 √ √ 2 2 + arctan( 2x 3 − 1) − C = −t. 4 where C denotes the integral constant about x0 . Thus, x(t, x0 ) = 0 for t ≥ t∗3 , where 4 √ √ 2 √ √ 2 2 3 2 x03 + 2x03 + 1 ∗ ln 4 √ 2 arctan( 2x03 + 1) + t3 = 2 8 4 x 3 − 2x 3 + 1 0

0

√ √ 2 2 + arctan( 2x03 − 1) . 4

(4) √

Additionally, it can be shown that limx0 →∞ t∗3 = 3 8 2 π. Example 3 indicates that the introduction of two polynomial terms with one fractional exponent and the other one larger than 1 may achieve high speed convergence and the upper bound of the reaching time is uniformly bounded by a constant which is not related to initial conditions. Generally, consider the following system x(t) ˙ = f (t, x(t)),

x(0) = x0 ,

(5)

where x ∈ Rℓ represents the state and f : R+ × D → Rℓ is an upper semi-continuous mapping in an open neighborhood D of the origin, such that the set f (t, x) is non-empty for any (t, x) ∈ R+ ×D and f (t, 0) = 0 for all t > 0. The solutions of system (5) are understood in the Filippov sense [29] if f (t, x) is discontinuous. Let x(t, x0 ) be an arbitrary solution of the Cauchy problem of system (5). Definition 1: [30] The origin is a finite-time stable equilibrium of (5), if and only if the origin is Lyapunov stable and there exist an open neighborhood N ⊆ D of the origin and a positive definite function T (x0 ) : N → R called the settling time function such that, for all x(0) ∈ N \{0}, limt→T (x0 ) x(t, x0 ) = 0 (6) x(t, x0 ) = 0, ∀t > T (x0 ). Furthermore, the origin is a globally finite-time stable equilibrium if the origin is finite-time stable with N = Rℓ . Also, finite-time stability of the origin implies the asymptotic stability of the origin. The systems given in Examples 1 and 2 are globally finite-time stable. In [30], a sufficient condition of finite-time stability is derived in the following theorem. Theorem 1: [30] Suppose that there exists a continuous positive definite function V (x) : D → R such that for any real numbers c > 0 and α ∈ (0, 1), the following inequality V˙ (x) + cV α (x) ≤ 0, x ∈ N \{0}

(7)

holds. Then, the origin is a finite-time stable equilibrium of (5) and the settling time is T (x0 ) ≤

1 V (1−α) (x0 ). c(1 − α)

(8)

Furthermore, if N = D = Rℓ , V is radially unbounded and V˙ < 0 on Rℓ \{0}, the origin is a globally finite-time stable equilibrium of system (5). Definition 2: [20] The origin is a fixed-time stable equilibrium of system (5) if it is globally finite-time stable and the settling time function T (x0 ) is bounded by a real number Tmax > 0, i.e. T (x0 ) ≤ Tmax , ∀x0 ∈ Rℓ . By Definition 2, it is easy to see that Example 3 demonstrates a fixed-time stable system. The following lemmas generalize Example 3 and present a first-order fixed-time stable system with the uniformly bounded settling time. Lemma 1: [22] Consider a scalar system p

p

x(t) ˙ = −αx2− q (t) − βx q (t),

x(0) = x0 ,

(9)




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where α, β > 0, and p, q satisfying q > p > 0 are odd integers. The equilibrium of system (9) is fixed-time stable and the settling time satisfies qπ T (x0 ) ≤ Tmax := √ . (10) 2 αβ(q − p) Lemma 2: [23] Consider a scalar system p

m

x(t) ˙ = −αx n (t) − βx q (t),

x(0) = x0 ,

(11)

where α, β > 0, and m, n, p, q satisfying m > n > 0 and q > p > 0 are odd integers. The equilibrium of system (11) is fixed-time stable and the settling time is obtained as T (x0 ) < Tmax :=

1 n 1 q + . αm−n β q−p

(12)

Furthermore, if ε , q(m−n) n(q−p) ≤ 1, a less conservative upperbound estimation of the settling time is given by r α q 1 1 √ tan−1 T (x0 ) < Tmax := . (13) + q−p β αε αβ It can be observed from Lemmas 1 and 2 that the settling time function eliminates the dependence of the initial condition x0 . As a result, the convergence time can be prescribed. In [20], a Lyapunov criterion for fixed-time stability is obtained in the following theorem. Theorem 2: [20] Suppose that there exits a continuous positive definite and radially unbounded function V (x) : Rℓ → R+ ∪ {0} such that D∗ V (x(t)) ≤ −(αV p (x(t)) + βV q (x(t)))k

(14)

for α, β, p, q, k > 0 satisfying pk < 1, qk > 1, the origin is fixed-time stable for system (5) with the settling time being estimated by T (x0 ) ≤ Tmax :=

1 1 + . αk (1 − pk) β k (qk − 1)

(15)

Then, in [21], Theorem 2 is refined as the following result. Theorem 3: [21] Suppose that there exits a continuous positive definite and radially unbounded function V (x) : Rℓ → R+ ∪ {0} such that D∗ V (x(t)) ≤ −αV p (x(t)) − βV q (x(t))

(16)

1 1 for α, β > 0, p = 1 − 2γ , q = 1 + 2γ , γ > 1, then the origin is fixed-time stable for system (5) with the settling time estimate πγ (17) T (x0 ) ≤ Tmax := √ . αβ

Note that asymptotic stability of the time-independent (autonomous) system always implies uniform asymptotic stability [31]. However, it is generally not the case for finite-time stable systems [32]. In [33], a uniformity of the settling time with respect to initial conditions is specified and another stability definition is presented. Definition 3: [33] The origin of system (5) is globally uniformly finite-time stable if it is globally uniformly asymptotically stable and there exists a locally bounded function T (x0 ) : Rℓ → R+ ∪ {0} such that x(t, x0 ) = 0 for all t ≥ T (x0 ).

By Definition 3, both systems (1) and (2) are globally uniformly finite-time stable since t∗1 and t∗2 are locally bounded. By a comparison with Definition 2, fixed-time stability requires the settling time T (x0 ) to be globally bounded, i.e., strong uniformity of finite-time stability [31]. It is worth mentioning that according to [34], different from Definition 3, a system is globally uniformly finite-time stable if there exists a constant Tmax such that x(t, x0 ) = 0 for all t > Tmax and any x0 ∈ Rℓ ; and a system is fixed-time stable if Tmax can be explicitly indicated. Notice that the autonomous system (5) with relative degree ℓ can be converted into an ℓ-th order integrator system by a proper selection of the output and a successive differentiation. A fixed-time stabilization control design is provided in [35] and recalled as follows. Theorem 4: [35] Consider an ℓth-order (ℓ ≥ 2) integrator system x(t) ˙ = Ax(t) + Bu(t), x(0) = x0 ,    0 0 1 ··· 0  ..  .. .. . . ..    . .  A= . .  and B =  .  0  0 0 ··· 1  0 0 ··· 0 1

(18) 

  , 

where x(t) = [x1 , x2 , . . . , xℓ ]T ∈ Rℓ denotes the state and u(t) ∈ R stands for the control input. Suppose that there exist constants ki and k¯i (i = 1, 2, . . . , ℓ) such that both the polynomial sℓ +kℓ sℓ−1 +· · ·+k2 s+k1 and sℓ + k¯ℓ sℓ−1 +· · ·+ k¯2 s+ k¯1 are Hurwitz. System (18) can be stabilized at the origin in fixed-time by using control input of u(t) = −

ℓ X i=1

ki |xi |γi sign(xi ) −

ℓ X i=1

k¯i |xi |βi sign(xi )

(19)

with parameters γi and βi satisfying, for i = 2, 3, . . . , ℓ, γi γi+1 βi βi+1 γi−1 = and βi−1 = , (20) 2γi+1 − γi 2βi+1 − βi

where γℓ+1 = βℓ+1 = 1, γℓ = γ0 ∈ (1 − ǫ, 1) and βℓ = β0 ∈ (1, 1 + ǫ) for a sufficiently small ǫ > 0. In addition, the convergence time is estimated as T ≤ Tmax =

1−γ0 γ0 λmax (P1 ) γ0 λmax (P1 ) (1 − γ0 )λmin (Q1 ) β0 −1 β0 λmax (P2 ) β0 (P2 ), + λmax (β0 − 1)λmin (Q2 )

(21)

where Pi = PiT > 0 is the solution of algebraic Lyapunov function Pi Ai + ATi Pi = −Qi for arbitrary Qi > 0, i = 1, 2, and both Ai are in the controllable canonical forms:   0 1 ··· 0  .. .. ..  ..  . . .  A1 =  . ,  0 0 ··· 1  −k1 −k2 · · · −kℓ   0 1 ··· 0  . .. ..  ..  .. . . .  .  A2 =   0 ··· 1   0 −k¯1 −k¯2 · · · −k¯ℓ




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A similar result is derived in [36] based on a bi-limit homogeneity technique [37] instead of the Lyapunov method. However, the settling time estimation cannot be explicitly provided by homogeneity formulation and only the existence of a globally bounded settling time can be guaranteed. In return, for the parameter ǫ used in Theorem 4, an explicit condition ǫ ∈ ( n−2 n−1 , 1) that depends upon the order of the system is derived in [36] to fulfill the homogeneity. To end this section, there is a need to clarify the difference between fixed-time stabilization and fixed-time control. Fixedtime stabilization requires the existence of an upper bound of convergence time, while the controller and observer gain based on the convergence time should be provided for fixed-time control. III. F INITE - TIME C OOPERATIVE C ONTROL

j∈Ni

The consensus problem is regarded as a fundamental issue for multi-agent cooperative control design, which motivates us to first outline the finite-time consensus problem for multiagent systems of simple dynamics so as to highlight basic ideas. Then, we review finite-time cooperative control for multi-agent systems with various system dynamics in practical systems. A. Finite-time Consensus Control of Simple Dynamical Systems Consider a group of N integrator-type agents of the form x˙ i (t) = ui (t),

i ∈ V,

(22)

where xi (t) ∈ R and ui (t) ∈ R denote, respectively, the state and the control input (or protocol) of agent i. Definition 4: System (22) is said to achieve the finitetime consensus (or agreement) if for any initial state x0 = [x1 (0), x2 (0), . . . , xN (0)]T ∈ RN and any i, j ∈ V, there exist a protocol ui and a locally bounded function T0 ∈ [0, +∞) such that limt→T0 |xi (t) − xj (t)| = 0 (23) xi (t) = xj (t), ∀t > T0 . Furthermore, if the final agreement state satisfies xi (t) = Pn k=1 xk (0) for all i ∈ V and all t > T0 , system (22) is said N to achieve the finite-time average consensus (or agreement). A typical finite-time consensus protocol is designed in the form of: α X X ui = sign aij (xj − xi ) aij (xj − xi ) , (24) j∈Ni

or

ui =

X

j∈Ni

j∈Ni

aij sign(xj − xi )|xj − xi |α ,

(25)

where α ∈ (0, 1). It is shown in [38] that if undirected G is connected, both protocols (24) and (25) ensure the finite-time agreement of the agents’ states, and in particular, the finite-time average agreement is achieved under (25). The work [10] further generalizes protocol (25) as X ui = aij (t)sign(xj − xi )|xj − xi |αij (t) , (26) j∈Ni

where αij (t) = αji (t) ∈ (0, 1) for all t > 0, and proves that finite-time average consensus is realized if the time-varying graph G(t) is undirected and the sum of time intervals, in which G(t) is connected, is sufficiently large or the timeinvariant graph G is directed and detail-balanced. In [39], a generalized vector form of protocol (24) is introduced and solves the finite-time consensus problem if the minimum dwell time is positive and the sum of time intervals, over which graph G(t) contains a spanning tree, is sufficiently large. Along the same line, in [11], by introducing a distributed linear consensus term, finite-time consensus is guaranteed with faster convergence (as inspired by Example 2) by the following protocol α X X ui = βsign aij (xj − xi ) aij (xj − xi ) +γ

X

j∈Ni

j∈Ni

aij (xj − xi ).

(27)

Note that motivated by Theorem 1, the fractional exponent α plays a vital role in achieving finite-time consensus. It can be seen that protocols (24) and (25) cover classical linear consensus protocols in some existing results in the published literature [8], [40]. More specifically, if α = 0, protocol (25) becomes discontinuous. In this case, the normalized and signed protocol in [41], and the set-valued Lyapunov functions and the nonsmooth analysis can be used to ensure finite-time stability. The work [6] introduces sign information of relative state into linear feedback control for integrator dynamics to achieve the decentralized finite-time formation tracking, i.e., let α = 0 in (27) with certain displacement drift according to specified formation pattern. It is worth noting that in spite of superiority of disturbance attenuation and robustness against uncertainties [42], the protocols involving discontinuous dynamics may produce undesirable chattering in the input, which could be avoided by applying continuous, non-Lipschitz type protocols like (24) and (25). The stability analysis of finite-time consensus for singleintegrator type multi-agent systems can be carried out based on Lyapunov functions and graph theory. In particular, applying Theorem 1, the regularity properties of the Lyapunov function and the settling time function are related by using the comparison principle of differential equations and the following well-known inequality used in Lyapunov function analysis. Lemma 3: [23], [38] Let ξ1 , ξ2 , . . ., ξN ≥ 0 and 0 < p ≤ 1. Then, N N p X X (28) ξi . ξip ≥ i=1

i=1

The rate of convergence of system (22) under the protocols (24)–(27) is closely related to initial conditions, interaction topology and protocol parameters. Especially, in the undirected interaction graph, larger algebraic connectivity of G will result in shorter settling time, which is similar to the linear counterpart of the protocol studied in [8]. In addition, the exponent α provides additional degree of freedom to prescribe the convergence rate.




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B. Finite-Time Cooperative Control of Complex Dynamical Systems Due to the existence of various dynamic behaviors in practical systems, it is natural to address the cooperative control problem for complex dynamical systems. Let’s begin with the following double-integrator type multiagent system: x˙ i (t) = vi (t) , i ∈ V, (29) v˙ i (t) = ui (t)

where xi ∈ R and vi ∈ R denote the position and the velocity for the i-th agent, respectively, and ui ∈ R denotes the control input. For a fixed undirected system of (29), some continuous distributed protocols [18], [43] are proposed and the finitetime stability is established with the aid of homogeneity with dilation (see the following lemma). Lemma 4: [30], [44] Suppose autonomous system (5) with f (t, x) = f (x) is homogeneous of degree σ with dilation (r1 , r2 , . . . , rn ), f is continuous and x = 0 is its asymptotically stable equilibrium. If homogeneity degree σ < 0, the equilibrium is finite-time stable. A typical homogeneous leaderless protocol takes the following form [43]: X ui = aij sign(xj − xi )|xj − xi |α1 j∈Ni

+

X

j∈Ni

aij sign(vj − vi )|vj − vi |α2 ,

The tools from homogeneity theory cannot provide an explicit estimation of the settling time. However, applying Theorem 1, one has to construct a proper Lyapunov function satisfying condition (7), which is usually nontrivial for complex dynamical systems including system (29). Based on Lyapunov functions, in [51], a leaderless consensus protocol is constructed for double-integrator networks under undirected and connected topology: X h i p2 −1 , (32) aij (xi − xj ) ui = −k1 vip + k2p j∈Ni

γ p21−1/p + β+N + k3 , k1 ≥ (2 − 1+p 1+p 1−1/p (β+N γ)21−1/p +(β+N γ)p 1−1/p 1+p 2 + + k3 ), k3 > 1/p)2 k2 ( P 1+p k2 0, β = max∀i∈V { j∈Ni aij }, γ = max∀i,j∈V {aij }, 1 < p = p1 p2 < 2 with p1 and p2 being positive odd integers. Note that

where k2

protocol (32) is not fully distributed due to the requirement of velocity measurement. From the Lyapunov stability analysis provided in [51], it can be seen that the construction and manipulation of Lyapunov functions are formidable to build the gap between the finite-time stability and the condition specified in (7). In spite of these difficulties, research efforts are made along this direction for systems with complex dynamics [17], [52]. In what follows, some typical complex dynamical systems are presented for illustration purpose. • Nonholonomic mobile robots [53]–[55]:

(30)

x˙ i = ui cos θi ,

and the corresponding leader-following protocol takes the form: X ui = aij sign(xj − xi )|xj − xi |α1 j∈Ni

− bi sign(xi − x0 )|xi − x0 |α1 X + aij sign(vj − vi )|vj − vi |α2 j∈Ni

− bi sign(vi − v0 )|vi − v0 |α2 ,

(31)

2α1 , x0 and v0 denote the leader where 0 < α1 < 1, α2 = 1+α 1 states. The work [18] introduces the pinning control into protocol (31) to solve the leader-following finite-time consensus problem for system (29) with fixed and switched topologies. Based on the homogeneous protocol (31) for nominal multiagent systems, the discontinuous signum function about an integral sliding mode is added to achieve the leader-following consensus with disturbance rejection [45]. Observer-based homogeneous protocols [46], [47], another extension of protocol (30) or (31), are also investigated for double-integrator systems to deal with finite-time consensus by considering relative outputs. To eliminate the dependence on the (relative) input information required in [46], [47], super-twisting algorithm based protocols are proposed in [14], [48] to solve the finitetime consensus tracking and the containment control problem without relying on velocity and acceleration measurements. In [49], [50], input saturated finite-time leaderless and leaderfollowing consensus issues for multiple double-integrators are addressed based on homogeneity.

≥

•

y˙ i = ui sin θi ,

θ˙i = ωi ,

(33)

where [xi , yi ] denotes the positions of the i-th agent, ui and ωi denote its translational and rotational velocity, respectively. The underactuated dynamics bring substantial difficulties in deriving consensus algorithms and analyzing finite- and fixed-time stability. The finite-time formation control for system (33) is achieved in [56], where a cooperative finite-time tracking protocol is developed using estimated leader states generated by a finite-time observer. Considering both uncertainties and external disturbances, an extended result is obtained in [57]. In [58], a distributed state observer is designed for each follower such that the leader state and input are estimated in finite-time, then a tracking protocol is established to guarantee that the estimated leader state is tracked in finite-time. Euler-Lagrange systems [59], [60]: Mi (qi )¨ qi + Ci (qi , q˙i )q˙i + gi (qi ) = τi ,

(34)

where qi ∈ Rp denotes generalized coordinates, Mi (qi ) ∈ Rp×p denotes a positive-definite inertia matrix, Ci (qi , q˙i )q˙i ∈ Rp denotes Coriolis and centrifugal torques, gi (qi ) ∈ Rp is gravitational torques, and τi ∈ Rp is the control torques to be designed. Dynamics (34) covers a large range of practical systems such as mechanical systems. Note that M˙ i (qi ) − 2Ci (qi , q˙i ) is of the skewsymmetry, which is very useful in Lyapunov stability analysis. In [61], the finite-time leader-following consensus problem is addressed for system (34) subject to external




•

6

disturbances. Protocol (32) is adapted in [62] to solve the attitude synchronization problem for spacecraft. By using the homogeneity with dilation, finite-time tracking is realized for agents with multiple Euler-Lagrange dynamics in [91], where a second-order sliding-mode observer based protocol is developed without using the velocity measurements. Higher-order linear systems [64], [65]: x˙ i (t) = Axi (t) + Bui (t),

(35)

where xi = [xi1 , xi2 , . . . , xiN ]T ∈ RN denotes the state for the i-th agent, ui ∈ R denotes the control input, A and B are system matrices defined as in (18). In [66], both leaderless and leader-following consensus are achieved through designing finite-time controllers such that proper Lyapunov functions are constructed in the form of (7). In the presence of unknown mismatched nonlinear dynamics, in [67], a neural network-based adaptive controller is proposed to achieve consensus tracking in finite time with sufficient accuracy. IV. F IXED - TIME C OOPERATIVE C ONTROL In this section, multi-agent systems with simple dynamics are first considered to demonstrate the definitions and the representative results in fixed-time consensus. Then, recent advances in fixed-time consensus and related cooperative control of complex dynamical systems are presented in detail. A. Fixed-time Consensus Control of Simple Dynamical Systems Definition 5: System (22) is said to achieve fixed-time consensus (or agreement) if for any initial state xi (0) and any i, j ∈ V, there exist a protocol ui and a globally bounded function T0 ∈ [0, +∞) such that limt→T0 |xi (t) − xj (t)| = 0 (36) |xi (t) − xj (t)| = 0, ∀t > T0 .

In particular, there exists a constant Tmax such that T0 ≤ TPmax , for ∀x0 . If the final agreement state satisfies xi (t) = N k=1 xk (0) for all i ∈ V and all t > T (x0 ), it is said to achieve N the fixed-time average consensus (or agreement). In what follows, inspired by Lemmas 1 and 2, several typical fixed-time protocols for system (22) are summarized in a series of theorems: Theorem 5: [22] If undirected G of system (22) is connected, then the distributed consensus protocol X X pq 2− pq +β ui = α aij (xj − xi ) aij (xj − xi ) j∈Ni

j∈Ni

(37) solves the fixed-time consensus problem. In particular, the settling time function is bounded by q−p

πqN 4q , T (x0 ) ≤ Tmax := √ 2 αβλ2 (LA )(q − p)

where λ2 (LA ) denotes the algebraic connectivity of Laplacian matrix associated with graph G.

Theorem 6: [22] If undirected G of system (22) is connected, then the distributed consensus protocol X X p p aij (xj − xi ) q (38) aij (xj − xi )2− q + β ui = α j∈Ni

j∈Ni

solves the fixed-time average consensus problem. In particular, the settling time function is estimated as q−p

T (x0 ) ≤ Tmax

πqN 2q , := √ 2 αβλ(q − p)

where λ = min{λ2 (LB ), λ2 (LC )}, λ2 (LB ) and λ2 (LC ) denote, respectively, the algebraic connectivity of Laplacian 2q/(3q−p) ]N ×N and C = matrix associated with B = [aij 2q/(p+q) ]N ×N . [aij Theorem 7: [23] If undirected G of system (22) is connected, then the distributed consensus protocol X X m pq n ui = α +β (39) aij (xj −xi ) aij (xj −xi ) j∈Ni

j∈Ni

solves the fixed-time consensus problem. In particular, the settling time function is estimated as m−n 1 N 2n n 1 q T (x0 ) ≤ Tmax := , + λ2 (LA ) α m−n β q−p

or, if 0 < ε := q(m−n) n(q−p) ≤ 1, a less conservative bound for T (x0 ) can be derived: T (x0 ) ≤ Tmax := m−n

N 4n 1 √ tan−1 λ2 (LA )(q − p) αβ

s

n−m

1 αN 2n . + β εα

Letting m/n = 2 and p/q = 0, protocol (39) recovers the basic structure of the protocol developed in [68] which is adapted to solve the consensus tracking problem for a class of single-integrator nonlinear multi-agent systems with undirected topology. Theorem 8: [23] If undirected G of system (22) is connected, then the distributed consensus protocol X X p m ui = α aij (xj − xi ) n + β aij (xj − xi ) q (40) j∈Ni

j∈Ni

solves the fixed-time average consensus problem. In particular, the settling time function is estimated as m−n n 1N n 1 q , T (x0 ) ≤ Tmax := + λ α m−n β q−p

or, if 0 < ε := q(m−n) n(q−p) ≤ 1, a less conservative bound for T (x0 ) can be derived: T (x0 ) ≤ Tmax := m−n

N 2n 1 √ tan−1 λ(q − p) αβ

s

n−m

αN n β

+

1 , εα

where λ = min{λ2 (LB ), λ2 (LC )}, λ2 (LB ) and λ2 (LC ) denote, respectively, the algebraic connectivity of Laplacian 2n/(m+n) ]N ×N and C = matrix associated with B = [aij 2q/(p+q) ]N ×N . [aij




7 p

m

It is worth mentioning that, in real domain, x q (resp. x n ) is equivalent to sign(x)|x|α with 0 < α < 1 used in (24) (resp. with α > 1). To perform numerical simulation, one should p m use sign(x)|x| n and sign(x)|x| q instead to avoid complex solutions. By a closer inspection, it can be seen that protocols (39) and (40) are generalized versions of protocols (37) and (38), respectively. It should be also noted that protocols (38) and (40) are edge-based while protocols (37) and (39) are nodebased. Compared with the node-based ones, the edge-based protocols guarantee certain symmetry, which may facilitate the establishment of stability conditions for multi-agent systems with switching topology. The settling time estimates provided in Theorems 5–8 are derived based on comparison principle of differential equations and Lyapunov functions using a key inequality presented in the following lemma in addition to the inequality in (3). Lemma 5: [23], [38] Let ξ1 , ξ2 , . . ., ξN ≥ 0 and p > 1. Then, N N p X X (41) ξi . ξip ≥ N 1−p i=1

i=1

The work [69] further shows that if digraph G of system (22) is strongly connected, then the conclusions stated in Theorems 7 and 8 still hold, and if the sum of time-intervals, over which the switching digraph G(t) of system (22) is strongly connected, is larger than a finite time t∗ which is explicitly specified independent of initial conditions, protocol (40) also achieves the fixed-time average consensus. In [70], it is shown that the consensus can be reached in fixed-time by applying protocol (7) for systems (22) under directed topology containing a spanning tree and additionally, the fixed-time leader-following consensus and formation problem are solved by exploiting the same structure of (7). It is notable that the work [71] considers system (22) in the presence of a constant delay h in the input, and shows that with the Artstein model reduction transformation, either protocol (37) or (39) drives the states of all agents in an undirected network to reach agreement in a fixed time bounded by Tmax + h. In [72], fixed-time leader-following consensus is achieved taking discontinuous inherent dynamics into account by using tools from nonsmooth analysis. The problem is further investigated over switching topology in [73]. Containment problem is solved in [27], where a distributed protocol is designed such that all the followers’ states eventually converge to the convex hull spanned by leaders in fixed-time. In [74] and [28], energy based switching mechanisms are proposed using fixed-time consensus approach to investigate collective behavior of mobile robots or to optimize communication flows in a wireless robotic network. A uniform time interval between switching instants is accordingly set so that energy distribution is conducted in a fair manner. More recently, in [25], distributed optimization problems are solved by using an edge-based fixed-time consensus approach. In both cases of time-invariant and time-varying cost functions, distributed protocols are proposed such that the state agreement is reached in a fixed time while the sum of local convex functions known to individual agents is minimized.

B. Fixed-Time Cooperative Control of Complex Dynamical Systems The most obvious barrier of expanding the idea behind protocols (37)–(40) and the like to the consensus control of multiple high-order integrators or complex dynamical systems is the input singularity problem which is caused by successive differentiations of the term with fractional power. Several nonsingular fixed-time leader-following protocols have been reported for system (29) based on terminal sliding mode. Definition 6: System (29) is said to achieve the fixed-time consensus tracking (or leader-following consensus) if for any initial state ξi (0) := [xi (0), vi (0)]T , ∀i, j ∈ V, there exist a protocol ui and a constant Tmax > 0 such that the settling time function T0 < Tmax and limt→Tmax |ξi (t) − ξ0 (t)| = 0 (42) ξi (t) = ξ0 (t), ∀t > Tmax , where ξi (t) := [xi (t), vi (t)]T , i = 0, 1, 2, . . . , N , x0 and v0 are the position and the velocity of the leader which verifies the same dynamics as (29) with the input u0 . Let ep = [ep1 , ep2 , . . . , epN ]T and ev = [ev1 , ev2 , . . . , evN ]T be the vector for position and velocity disagreement, P respectively, p with their elements being defined by e = i j∈Ni aij (xi − P xj ) + bi (xi − x0 ) and evi = j∈Ni aij (vi − vj ) + bi (vi − v0 ). The work [75] studies, for the first time, the distributed fixed-time consensus tracking problem for double-integrator networks under a directed information flow. A nonsingular distributed protocol is proposed as X −1 h i X ui = udi + aij + bi aij uj − bi λsgn(si ) , (43) j∈Ni

j∈Ni

where λ > ρ, ρ is the known upper-bound of u0 , si denotes a sliding manifold, defined by q1

si = epi + [κi (epi ) · evi ] p1

(44)

with the virtual control signal udi being defined by p1 m1 1h m1 p1 udi = α1 (epi ) n1 − q1 −1 (κi evi )2 − κi n1 q1 p1 1− pq11 v 2− pq1 i p1 − pq11 τ v pq1 −1 1 − κi − κi µi ((ei ) 1 ) (ei ) q1 q1 m2 p2 q n2 v 1− p11 α2 si + β2 siq2 , · (ei )

where mk , nk , pk , qk are positive odd integers satisfying mk > nk , p1 < q1 < 2p1 , p2 < q2 and m1 /n1 − p1 /q1 > 1, and αk , βk > 0, k = 1, 2. µτi (·) is a C 1 function, defined by sin π2 · τx if |x| ≤ τ µτ (x) = (45) 1 otherwise with τ a positive constant, and κ(·) : R → R+ denotes a scalar positive function, defined by κ(x) =

1 . α1 xm1 /n1 −p1 /q1 + β1

(46)

The protocol (43) achieves fixed-time consensus tracking with the settling time T being uniformly bounded by T < Tmax := T1 + T2 + ǫ(τ ),

(47)




where Tk = [N (mk −nk )/2nk nk ]/[αk (mk − nk )] + qk /[βk (qk − pk )], (k = 1, 2), and ǫ(τ ) denotes a small time margin. The key idea of the singularity avoidance behind protocol π (43) exploits the well-known limit limx→0 µτ (x)/x = 2τ . v In fact, the sliding mode (44) is a variant of si = ei + (evi )m1 /n1 + (epi )p1 /q1 , but the former facilitates the stability analysis. Noting that the small time margin, in spite of the fact limτ →0 ǫ(τ ) = 0, cannot be explicitly given, modified slightly from (45), the sliding mode si = (evi )m1 /n1 + (evi )m2 /n2 + (epi )m3 /n3 is proposed in the following form [76] ( 2 if |x| ≤ τ sin π2 · τx2 (48) µτ (x) = 1 otherwise, where 1 < m2 /n2 < m1 /n1 < m3 /n3 and m2 /n2 < 2. Then, limx→0 µτ (x)/x = 0 is used in the stability analysis and an 1 estimation ǫ(τ ) = 2τ m1 /n1 −1 /(λ − ρ − δ0 ) is provided, where λ0 is a small constant. The use of (45) or (48) plays a significant role in establishing the non-singular leader-following consensus protocols in [75], [76]. However, this artificially introduces a boundary layer activating the sinusoidal function which complicates the fixed-time stability analysis. To avoid such an implicit switching, one may use the following nonsingular second-order sliding mode firstly proposed in [20]: si = evi + sign sign(evi )|evi |2 + α1 epi + β1 sign(epi )|epi |3 1 · sign(evi )|evi |2 + α1 epi + β1 sign(epi )|epi |3 2 . (49)

In [77], a fixed-time distributed position-based consensus protocol is developed for double integrator multi-agent systems without velocity measurements to achieve the leader following within a bounded time, where the fixed-time stability is guaranteed by utilizing the bi-limit homogeneity technique without an explicit estimation for the settling time. Following the same lines as Subsection III-B, some representative results on this topic for more complex dynamical systems are presented. • Nonholonomic mobile robots. The sliding mode (49) is used in the consensus control of multiple nonholonomic systems (33) in [78]. In [79], robust fixed-time consensus tracking is investigated for double-integrator type multiagent systems with application for fixed-time formation tracking of unicycle-type robots. • Euler-Lagrange systems. The work [80] introduces a switching mechanism in the controller to avoid the singularity point and solves the fixed-time lag consensus problem with application to multiple single-link robotic manipulators described by (34). In [81], fixed-time coordination control problem for telerobotics systems with asymmetric time-varying delays is addressed with realtime experimental results. • Higher-order linear systems. In [65], [82], the leaderfollowing fixed-time consensus is realized for multi-agent systems with high-order dynamics and exogenous disturbances. Specifically, the work [65] proposes a distributed state observer such that each follower can estimate the leader state and therefore the communication loop problem encountered by [82] is eliminated. Bounded input

8

uncertainties are taken into account in [83] to achieve fixed-time consensus tracking by using a sliding mode based strategy. C. Some Insights and Remarks Generally speaking, existing results in finite- and fixed-time cooperative control have covered several interesting topics encountered in multi-agent systems with various dynamics, which reveals a number of advantages resulting from the fast rate of convergence with the guaranteed finite settling time. For example, in hybrid formation flying of unmanned aerial vehicles or satellites, the guaranteed rate of convergence to each formation pattern will eliminate the accumulated errors due to formation switching. However, the fast rate of convergence is usually achieved at the cost of large initial control force. In other words, an input constraint problem has not been considered in the existing results in fixed-time cooperative control. The main tools for analysis of finite- and fixed-time cooperative control are a Lyapunov function method and a homogeneity method. It is noticeable that the settling time estimation cannot be explicitly prescribed by using homogeneity instead of a Lyapunov tool. To derive an explicit estimation of the settling time, one has to construct a Lyapunov function, which is usually involved or even inhibitive for complex dynamical systems. It is therefore desirable to seek a systematic design tool or method for fixed-time cooperative control, e.g. iterative design or construction in a cascaded fashion. In addition, it should be pointed out that the existing results can provide an upper-bound estimation for the settling time instead of an accurate one, which may result in certain design conservatism in real applications. V. C ONCLUSIONS

AND SOME CHALLENGING ISSUES

The concepts about finite- and fixed-time stability and stabilization for single-agent-based systems have been first introduced and the relevant results have been sorted out as they act as theoretic fundamentals for finite- and fixed-time cooperative control. Then, an overview of some recent results in finite- and fixed-time cooperative control for multi-agent systems with both simple and complex dynamics has been conducted. In addition to the results that have been reported in the published literature, there still exist several issues that need to be addressed. Some important and yet challenging issues deserving research are highlighted as follows. • Fixed-time cooperative control for multi-agent systems with general linear or nonlinear dynamics and directed graphs. In [27], the directed graph needs to satisfy the detailed balance condition; in [70], directed graphs containing a spanning tree are investigated. However, in [27] and [70], results are obtained only for first-order multi-agent systems. Besides, note that in [101] switched system theory is used to solve the finite-time consensus problem for a multi-agent system under switching connection topology. The fixed-time cooperative control of switching multi-agent system also deserves more investigation. Obviously, it would be a challenge to develop




•

•

•

•

new controllers and to construct new Lyaponov functions to deal with general nonlinear dynamics or switched nonlinear systems; Fixed-time cooperative control with network-induced constraints. Most of existing fixed-time control results are obtained under an assumption that communication links between controllers and actuators (or, between sensors and controllers) are perfect. However, data packet dropouts, data packet disorders, network-induced delays, and/or quantization errors are inevitable in practical networked systems [84], [85]. Under these networkedinduced constraints, how to design proper networked fixed-time cooperative control protocols for multi-agent systems is an important and yet challenging issue; Event-triggered fixed-time cooperative control. Notice that the majority of existing finite- and/or fixed-time cooperative control algorithms are based on a periodic sampling or a consecutive transmission paradigm, which means that each agent’s state should be sampled regularly at equidistant sampling times irrespective of the current state of the system and resource usage or should be transmitted continually over a communication network at every instant of time. Event-triggered mechanisms have now shown their strengths in reducing the data sampling/transmission frequency while preserving desired system performance [86]–[89]. It should be also pointed out that in the context of finite- and/or fixedtime cooperative control, one should carefully design an event-triggered mechanism over the finite- and/or fixedtime interval, i.e. the time interval from the initial time to the exact settling time, such that the Zeno phenomenon can be effectively excluded after the computed exact settling time. Thus, how to well address the problem of fixed-time cooperative control under event-triggered sampling/transmission deserves a deep investigation; Prescribed-time or specified-time cooperative control. The fixed-time stabilization control design ensures a uniformly bounded settling time in terms of a constant, which certainly leads to a quite conservative estimation of the settling time due to the global stability. In [90], time-varying feedback control is proposed to stabilize normal-form nonlinear systems in a prescribed finite time, which is actually an optimal control with a terminal constraint by employing time-varying gains. In [91], based on Pontryagin’s maximum principle on Lie group, formation tracking control for vehicles is achieved with a prescribed-time. In [92]–[94], from the motion-planning perspective, a class of sample-data based protocols are developed to solve the prescribed-time consensus-related control problem for first-/second-order multi-agent systems as well as multiple harmonic oscillators over directed and periodically switching topologies. However, the prescribed-time or specified-time cooperative control for nonlinear multi-agent systems, especially with complex dynamics, is still an open problem; Fixed-time cooperative estimation and control under cyber attacks. Cyber-security has become an essential and practical issue with the fast development of real-world

9

•

network systems [95]–[98]. As a fundamental problem in control, filtering/estimation aims at developing an algorithm to provide an estimation of an unavailable state signal of the system through the disturbed plant and/or noisy measurement [99], [100]. However, how to protect networked multi-agent systems without sacrificing the state convergence performance under cyber attacks deserves deep investigation. In particularly, how to develop secure fixed-time cooperative filtering/estimation algorithms and how to achieve fixed-time cooperative control under cyber attacks are promising future directions; and Discretization of fixed-time stable cooperative systems. For implementation in networked environments, the finite- or fixed-time protocols have to be calculated in a digital controller with discrete measurements. Unfortunately, it has been shown in the existing literature [102], [103] that the discrete-time system with power rule has limit cycles rather than the origin or even exhibits unstable behaviors under improper discretization schemes, which is quite different from its continuous counterpart, although the design or analysis has been performed in continuous-time scenarios. In other words, the finite- or fixed-time convergence property in discretetime realization is impossible. However, the finite- or fixed-time stable system with a sampled continuous-time control input can still preserve some appealing properties like higher accuracy and faster convergence if the discretization scheme is carefully chosen or designed, such as the implicit Euler scheme proposed in [104] for homogeneous systems which guarantees asymptotic convergence of the approximating solutions. Hence, the discretization scheme for fixed-time stable cooperative control systems with an appropriate sampling mechanism that preserves the performance of the continuoustime counterpart except for the fixed-time convergence deserves a further attention and research. R EFERENCES

[1] X. Ge and Q.-L. Han, “Consensus of multiagent systems subject to partially accessible and overlapping Markovian network topologies,” IEEE Trans. Cybern., vol. 47, no. 8, pp. 1807-1819, Aug. 2017. [2] Y. Cao, W. Yu, W. Ren, and G. Chen, “An overview of recent progress in the study of distributed multi-agent coordination,” IEEE Trans. Ind. Informat., vol. 9, no. 1, pp. 427-438, Feb. 2013. [3] L. Ding, Q.-L. Han, and G. Guo, “Network-based leader-following consensus for distributed multi-agent systems,” Automatica, vol. 49, no. 7, pp. 2281-2286, Jul. 2013. [4] X. Ge, Q.-L. Han, and F. Yang, “Event-based set-membership leaderfollowing consensus of networked multi-agent systems subject to limited communication resources and unknown-but-bounded noise,” IEEE Trans. Ind. Electron., vol. 64, no. 6, pp. 5045-5054, Jun. 2017. [5] W. He, B. Zhang, Q.-L. Han, F. Qian, J. Kurths, and J. Cao, “Leaderfollowing consensus of nonlinear multiagent systems with stochastic sampling,” IEEE Trans. Cybern., vol. 47, no. 2, pp. 327-338, Feb. 2017. [6] Y. Cao, W. Ren, and Z. Meng, “Decentralized finite-time sliding mode estimators and their applications in decentralized finite-time formation tracking,” Syst. Control Lett., vol. 59, no. 9, pp. 522-529, Sept. 2010. [7] Y.-L. Wang and Q.-L. Han, “Network-based modelling and dynamic output feedback control for unmanned marine vehicles in network environments,” Automatica, vol. 91, pp. 43-53, May 2018. [8] R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time delays,” IEEE Trans. Autom. Control, vol. 49, no. 9, pp. 1520-1533, Sep. 2004.




[9] Y. Kim and M. Mesbahi, “On maximizing the second smallest eigenvalue of a state-dependent graph laplacian,” IEEE Trans. Autom. Control, vol. 51, no. 1, pp. 116-120, Jan. 2006. [10] L. Wang and F. Xiao, “Finite-time consensus problems for networks of dynamic agents,” IEEE Trans. Autom. Control, vol. 55, no. 4, pp. 950-955, Apr. 2010. [11] F. Xiao, L. Wang, J. Chen, and Y. Gao, “Finite-time formation control for multi-agent systems,” Automatica, vol. 45, no. 11, pp. 2605-2611, 2009. [12] G. Chen, F. L. Lewis, and L. Xie, “Finite-time distributed consensus via binary control protocols,” Automatica, vol. 47, no. 9, pp. 1962-1968, 2011. [13] H. Liu, L. Cheng, M. Tan, Z. Hou, and Y. Wang, “Distributed exponential finite-time coordination of multi-agent systems: containment control and consensus,” Int. J. Control, vol. 88, no. 2, pp. 237-247, 2015. [14] Y. Zhao, Z. Duan, G. Wen, and G. Chen, “Distributed finite-time tracking for a multi-agent system under a leader with bounded unknown acceleration,” Syst. Control Lett., vol. 81, pp. 8-13, Jul. 2015. [15] X. Liu, J. Lam, W. Yu, and G. Chen, “Finite-time consensus of multiagent systems with a switching protocol,” IEEE Trans. Neural Netw. Learn. Syst., vol. 27, no. 4, pp. 853-862, Apr. 2016. [16] Y. Cao and W. Ren, “Distributed coordinated tracking with reduced interaction via a variable structure approach,” IEEE Trans. Autom. Control, vol. 57, no. 1, pp. 33-48, Jan. 2012. [17] S. Khoo, L. Xie, and Z. Man, “Robust finite-time consensus tracking algorithm for multirobot systems,” IEEE/ASME Trans. Mechatronics, vol. 14, no. 2, pp. 219-228, Apr. 2009. [18] Z.-H. Guan, F.-L. Sun, Y.-W. Wang, and T. Li, “Finite-time consensus for leader-following second-order multi-agent networks,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 59, no. 11, pp. 2646-2654, Nov. 2012. [19] Y. Zhang, Y. Yang, Y. Zhao, and G. Wen, “Distributed finite-time tracking control for nonlinear multi-agent systems subject to external disturbances,” Int. J. Control, vol. 86, no. 1, pp. 29-40, 2013. [20] A. Polyakov, “Nonlinear feedback design for fixed-time stabilization of linear control systems,” IEEE Trans. Autom. Control, vol. 57, no. 8, pp. 2106-2110, Aug. 2012. [21] S. Parsegov, A. Polyakov, and P. Shcherbakov, “Nonlinear fixed-time control protocol for uniform allocation of agents on a segment,” in Proc. IEEE Conf. Decision Control, Hawaii, USA, Dec. 2013, pp. 7732-7737. [22] Z. Zuo and L. Tie, “A new class of finite-time nonlinear consensus protocols for multi-agent systems,” Int. J. Control, vol. 87, no. 2, pp. 363-370, Jun. 2014. [23] Z. Zuo and L. Tie, “Distributed robust finite-time nonlinear consensus protocols for multi-agent systems,” Int. J. Syst. Sci., vol. 47, no. 6, pp. 1366-1375, 2016. [24] D. Meng, Y. Jia, and J. Du, “Finite-time consensus for multiagent systems with cooperative and antagonistic interactions,” IEEE Trans. Neural Netw. Learn. Syst., vol. 27, no. 4, pp. 762-770, Apr. 2016. [25] B. Ning, Q.-L. Han, and Z. Zuo, “Distributed optimization for multiagent systems: an edge-based fixed-time consensus approach,” IEEE Trans. Cybern., to be published, doi: 10.1109/TCYB.2017.2766762. [26] H. Hong, W. Yu, X. Yu, G. Wen, and A. Alsaedi, “Fixed-time connectivity-preserving distributed average tracking for multi-agent systems,” IEEE Trans. Circuits Syst. II, Exp. Brief, vol. 64, no. 10, pp. 1192-1196, Oct. 2017. [27] B. Ning, Z. Zuo, J. Jin, and J. Zheng, “ Distributed fixed-time coordinated tracking for nonlinear multi-agent systems under directed graphs,” Asian J. Control, to be published, doi: 10.1002/asjc.1612. [28] B. Ning, Q.-L. Han, Z. Zuo, J. Jin, and J. Zheng, “Collective behaviors of mobile robots beyond the nearest neighbor rules with switching topology,” IEEE Trans. Cybern., to be published, doi: 10.1109/TCYB.2017.2708321. [29] A. Filippov, Differential equations with discontinuous right-hand side. Kluwer Academic, Dordrecht, 1988. [30] S. Bhat and D. Bernstein, “Finite time stability of continuous autonomous systems,” SIAM J. Control Optim., vol. 38, no. 3, pp. 751766, 2000. [31] A. Polyakov, D. Efimov, and W. Perruquetti, “Finite-time and fixedtime stabilization: Implicit lyapunov function approach,” Automatica, vol. 51, no. 2, pp. 332-340, 2015. [32] A. Polyakov and L. Fridman, “Stability notions and lyapunov functions for sliding mode control systems,” J. Franklin Inst., vol. 351, no. 4, pp. 1831-1865, 2014.

10

[33] Y. Orlov, “Finite time and robust control synthesis of uncertain switched systems,” SIAM J. Control Optim., vol. 43, no. 4, pp. 12531271, 2005. [34] M. Basin, C. Panathula, and Y. Shtessel, “Adaptive uniform finite/fixed-time convergent second-order sliding-mode control,” Int. J. Control, vol. 89, no. 9, pp. 1777-1787, 2016. [35] M. Basin, Y. Shtessel, and F. Aldukali, “Continuous finite- and fixedtime high-order regulators,” J. Franklin Inst., vol. 253, no. 18, pp. 5001-5012, 2016. [36] B. Tian, Z. Zuo, and H. Wang, “Uniform finite-time stabilization of a class of high-order systems with mismatched disturbances,” Automatica, submitted. [37] V. Andrieu, L. Praly, and A. Astolfi, “Homogeneous approximation, recursive observer design and output feedback,” SIAM J. Control Optim., vol. 47, no. 4, pp. 1814-1850, 2008. [38] F. Xiao and L. Wang, “Reaching agreement in finite time via continuous local state feedback,” in 26th Chinese Control Conference, Zhangjiajie, Hunan, China, Jul. 2007, pp. 711-715. [39] F. Xiao and L. Wang, “Finite-time consensus in networks of integratorlike dynamics agents with directional link failure,” IEEE Trans. Autom. Control, vol. 56, no. 3, pp. 756-762, 2014. [40] W. Ren and R. Beard, “Consensus seeking in multiagent systems under dynamically changing interaction topologies,” IEEE Trans. Autom. Control, vol. 50, no. 5, pp. 655-661, May 2005. [41] J. Cortés, “Finite-time convergent gradient flows with applications to network consensus,” Automatica, vol. 42, no. 11, pp. 1993-2000, 2006. [42] M. Franceschelli, A. Pisano, A. Giua, and E. Usai, “Finite-time consensus with disturbance rejection by discontinuous local interactions in directed graphs,” Automatica, vol. 60, no. 4, pp. 1133-1138, 2015. [43] X. Wang and Y. Hong, “Finite-time consensus for multi-agent networks with second-order agent dynamics,” in Proc. the 17th IFAC World Congress, Seoul, Korea, Jul. 2008, pp. 15185-15190. [44] Y. Hong, “Finite-time stabilization and stabilizability of a class of controllable systems,” Syst. Control Lett., vol. 46, no. 4, pp. 231-236, 2002. [45] S. Yu and X. Long, “Finite-time consensus for second-order multiagent systems with disturbances by integral sliding mode,” Automatica, vol. 54, pp. 158-165, 2015. [46] Y. Zhao, Z. Duan, G. Wen, and Y. Zhang, “Distributed finite-time tracking control for multi-agent systems: An observer-based approach,” Syst. Control Lett., vol. 62, no. 1, pp. 22-28, 2013. [47] H. Du, Y. He, and Y. Cheng, “Finite-time synchronization of a class of second-order nonlinear multi-agent systems using output feedback control,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 61, no. 6, pp. 1778-1788, Jun. 2014. [48] Y. Zhao and Z. Duan, “Finite-time containment control without velocity and acceleration measurements,” Nonlinear Dyn., vol. 82, no. 1-2,pp. 259-268, Oct. 2015. [49] X. Lu, S. Chen, and J. Lü, “Finite-time tracking for double-integrator multi-agent systems with bounded control input,” IET Control Theory Appl., vol. 7, no. 11, pp. 1562-1573, 2013. [50] Y. Zhao, Z. Duan, and G. Wen, “Finite-time consensus for second-order multi-agent systems with saturated control protocols,” IET Control Theory Appl., vol. 9, no. 3, pp. 312-319, 2014. [51] S. Li, H. Du, and X. Lin, “Finite-time consensus algorithm for multiagent systems with double-integrator dynamics,” Automatica, vol. 47, no. 8, pp. 1706-1712, 2011. [52] Y. Zhao, Z. Duan, G. Wen, and G. Chen, “Distributed finite-time tracking for multiple non-identical second-order nonlinear systems with settling time estimation,” Automatica, vol. 64, pp. 86-93, 2016. [53] X. Ge and Q.-L. Han, “Distributed formation control of networked multi-agent systems using a dynamic event-triggered communication mechanism,” IEEE Trans. Ind. Electron., vol. 64, no. 10, pp. 81188127, Oct. 2017. [54] W. Dong, “Tracking control of multiple-wheeled mobile robots with limited information of a desired trajectory,” IEEE Trans. Autom. Control, vol. 53, no. 6, pp. 1434-1448, Jul. 2008. [55] X. Ge, Q.-L. Han, and X.-M. Zhang, “Achieving cluster formation of multi-agent systems under aperiodic sampling and communication delays,” IEEE Trans. Ind. Electron., vol. 65, no. 4, pp. 3417-3426, Apr. 2018. [56] M. Ou, H. Du, and S. Li, “Finite-time formation control of multiple nonholonomic mobile robots,” Int. J. Robust Nonlin. Control, vol. 24, no. 1, pp. 140-165, 2014. [57] H. Du, G. Wen, Y. Cheng, Y. He, and R. Jia, “Distributed finitetime cooperative control of multiple high-order nonholonomic mobile




[58]

[59]

[60]

[61]

[62]

[63]

[64]

[65]

[66]

[67]

[68]

[69]

[70]

[71]

[72]

[73]

[74]

[75] [76]

[77]

[78]

[79]

robots,” IEEE Trans. Neural Netw. Learn. Syst., vol. 28, no. 12, pp. 2998-3006, Dec. 2017. H. Du, G. Wen, X. Yu, S. Li, and M. Z. Chen, “Finite-time consensus of multiple nonholonomic chained-form systems based on recursive distributed observer,” Automatica, vol. 62, pp. 236-242, Dec. 2015. J. Mei, W. Ren, and G. Ma, “Distributed coordinated tracking with a dynamic leader for multiple euler-lagrange systems,” IEEE Trans. Autom. Control, vol. 56, no. 6, pp. 1415-1421, Jun. 2011. W. Dong, “On consensus algorithms of multiple uncertain mechanical systems with a reference trajectory,” Automatica, vol. 47, no. 9, pp. 2023-2029, 2011. W. He, C. Xu, Q.-L. Han, F. Qian, and Z. Lang, “Finite-time L2 leaderfollower consensus of networked Euler-Lagrange systems with external disturbances,” IEEE Trans. Syst., Man, Cybern., Syst., to be published, doi: 10.1109/TSMC.2017.2774251. H. Du, S. Li, and C. Qian, “Finite-time attitude tracking control of spacecraft with application to attitude synchronization,” IEEE Trans. Autom. Control, vol. 56, no. 11, pp. 2711-2717, Nov. 2011. Y. Zhao, Z. Duan, and G. Wen, “Distributed finite-time tracking of multiple Euler-Lagrange systems without velocity measurements,” Int. J. Robust Nonlinear Control, vol. 25, pp. 1688-1703, 2015. S. Mondal and R. Su, “Finite time tracking control of higher order nonlinear multi agent systems with actuator saturation,” in Proc. the 14th IFAC Symp. Control in Transportation Syst., Taksim, Istanbul, Turkey, May 2016, pp. 165-170. Z. Zuo, B. Tian, M. Defoort, and Z. Ding, “Fixed-time consensus tracking for multi-agent systems with high-order integrator dynamics,” IEEE Trans. Autom. Control, vol. 63, no. 2, pp. 563-570, Feb. 2018. H. Du, G. Wen, G. Chen, J. Cao, and F. E. Alsaadi, “A distributed finite-time consensus algorithm for higher-order leaderless and leaderfollowing multiagent systems,” IEEE Trans. Syst., Man, Cybern., Syst., vol. 47, no. 7, pp. 1625-1634, Jul. 2017. L. Zhao, J. Yu, C. Lin, and Y. Ma, “Adaptive neural consensus tracking for nonlinear multiagent systems using finite-time command filtered backstepping,” IEEE Trans. Syst., Man, Cybern., Syst., to be published, doi: 10.1109/TSMC.2017.2743696. M. Defoort, A. Polyakov, G. Demesure, M. Djemai, and K. Veluvolu, “Leader-follower fixed-time consensus for multi-agent systems with unkonwn non-linear inherent dynamics,” IET Control Theory Appl., vol. 9, no. 14, pp. 2165-2170, 2015. Z. Zuo, W. Yang, L. Tie, and D. Meng, “Fixed-time consensus for multiagent systems under directed and switching interaction topology,” in Proc. the 2014 American Control Conf., Portland, Oregon, USA, Jun 2014, pp. 5133-5138. J. Fu and J.-Z. Wang, “Finite-time consensus for multi-agent systems with globally bounded convergence time under directed communication graphs,” Int. J. Control, vol. 90, no. 9, pp. 1807-1817, 2017. C. Wang, Z. Zuo, Y. Wu, and Z. Ding, “Fixed-time nonlinear consensus algorithms for multi-agent systems with input delay,” in Proc. the 13th IEEE Int. Conf. Control & Automation, Ohrid, Macedonica, Jul. 2017, pp. 52-57. B. Ning, J. Jin, J. Zheng, and Z. Man, “Finite-time and fixedtime leader-following consensus for multi-agent systems with discontinuous inherent dynamics,” Int. J. Control, to be published, doi: 10.1080/00207179.2017.1313453. B. Ning, J. Jin, and J. Zheng, “Fixed-time consensus for multiagent systems with discontinuous inherent dynamics over switching topology,” Int. J. Syst. Sci., vol. 48, no. 10, pp. 2023-2032, 2017. B. Ning, J. Jin, B. Krishnamachari, J. Zheng, and Z. Man, “Twostage deployment strategy for wireless robotic networks via a class of interaction models,” IEEE Trans. Syst., Man, Cybern., Syst., vol. 47, no. 7, pp. 1510-1521, Jul. 2017. Z. Zuo, “Non-singular fixed-time consensus tracking for second-order multi-agent networks,” Automatica, vol. 54, no. 4, pp. 305-309, 2015. J. Fu and J. Wang, “Fixed-time coordinated tracking for second-order multi-agent systems with bounded input uncertainties,” Syst. Control Lett., vol. 93, pp. 1-12, 2016. B. Tian, H. Lu, Z. Zuo, and W. Yang, “Fixed-time leader-follower output feedback consensus for second-order multiagent systems,” IEEE Trans. Cybern., to be published, doi: 10.1109/TCYB.2018.2794759. M. Defoort, G. Demesure, Z. Zuo, A. Polyakov, and M. Djemai, “Fixed-time stabilization and consensus of non-holonomic systems,” IET Control Theory Appl., vol. 10, no. 18, pp. 2497-2505, 2016. X. Chu, Z. Peng, G. Wen, and A. Rahmani, “Robust fixed-time consensus tracking with application to formation control of unicycles,” IET Control Theory Appl., vol. 12, no. 1, pp. 53-59, 2018.

11

[80] J. Ni, L. Liu, C. Liu, and J. Liu, “Fixed-time leader-following consensus for second-order multi-agent systems with input delay,” IEEE Trans. Ind. Electron., vol. 64, no. 11, pp. 8635 - 8646, Nov. 2017. [81] Y. Yang, C. Hua, J. Li, and X. Guan. “Fixed-time coordination control for bilateral telerobotics system with asymmetric time-varying delays,” J. Intell. Robot. Syst., vol. 86, no. 3-4, pp. 447-466, Jun. 2017. [82] B. Tian, Z. Zuo, and H. Wang, “Leader-follower fixed-time consensus of multi-agent systems with high-order integrator dynamics,” Int. J. Control, vol. 90, no. 7, pp. 1420-1427, 2017. [83] Y. Huang and Y. Jia, “Fixed-time consensus tracking control for second-order multi-agent systems with bounded input uncertainties via nfftsm,” IET Control Theory Appl., vol. 11, no. 16, pp. 2900-2909, 2017. [84] X.-M. Zhang, Q.-L. Han, and X. Yu, “Survey on recent advances in networked control systems,” IEEE Trans. Ind. Informat., vol. 12, no. 5, pp. 1740-1752, Oct. 2016. [85] X. Ge, F. Yang, and Q.-L. Han, “Distributed networked control systems: A brief overview,” Inform. Sci., vol. 380, pp. 117-131, Feb. 2017. [86] X.-M. Zhang, Q.-L. Han, and B.-L. Zhang, “An overview and deep investigation on sampled-data-based event-triggered control and filtering for networked systems,” IEEE Trans. Ind. Informat., vol. 13, no. 1, pp. 4-16, Feb. 2017. [87] L. Ding, Q.-L. Han, X. Ge, and X.-M. Zhang, “An overview of recent advances in event-triggered consensus of multiagent systems,” IEEE Trans. Cybern., vol. 48, no. 4, pp. 1110-1123, Apr. 2018. [88] X. Ge, Q.-L. Han, and Z. Wang, “A dynamic event-triggered transmission scheme for distributed set-membership estimation over wireless sensor networks,” IEEE Trans. Cybern., to be published, doi: 10.1109/TCYB.2017.2769722. [89] X.-M. Zhang and Q.-L. Han, “A decentralized event-triggered dissipative control scheme for systems with multiple sensors to sample the system outputs,” IEEE Trans. Cybern., vol. 46, no. 12, pp. 2745-2757, Dec. 2016. [90] Y. Song, Y. Wang, J. Holloway, and M. Krstic, “Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time,” Automatica, vol. 83, pp. 243-251, Sep. 2017. [91] Y. Liu, Y. Zhao, and G. Chen, “Finite-time formation tracking control for multiple vehicles: A motion planning approach,” Int. J. Robust Nonlinear Control, vol. 26, no. 14, pp. 3130-3149, 2016. [92] Y. Liu, Y. Zhao, W. Ren, and G. Chen, “Appointed-time consensus: Accurate and practical designs,” Automatica, 2018, to be published, doi:10.1016/j.automatica.2017.12.030. [93] Y. Liu and Y. Zhao, “Specified-time coordination control algorithms of multiple harmonic oscillators over directed graphs,” Nonlinear Dynamics, vol. 91, no. 1, pp. 343-358, Jan. 2018. [94] Y. Liu, Y. Zhao, Z. Shi, and D. Wei, “Specified-time containment control of multi-agent systems over directed topologies,” IET Control Theory Appl., vol. 11, no. 4, pp. 576-585, 2017. [95] L. Ma, Z. Wang, Q.-L. Han, and H.-K. Lam, “Variance-constrained distributed filtering for time-varying systems with multiplicative noises and deception attacks over sensor networks,” IEEE Sensors Journal, vol. 17, no. 7, pp. 2279–2288, Apr. 2017. [96] L. Hu, Z. Wang, Q.-L. Han, and X. Liu, “State estimation under false data injection attacks: Security analysis and system protection,” Automatica, vol. 87, pp. 176-183, Jan. 2018. [97] D. Ding, Z. Wang, Q.-L. Han, and G. Wei, “Security control for discrete-time stochastic nonlinear systems subject to deception attacks,” IEEE Trans. Syst., Man, Cybern., Syst., to be published, doi: 10.1109/TSMC.2016.2616544. [98] D. Ding, Q.-L. Han, Y. Xiang, X. Ge, and X.-M. Zhang, “A survey on security control and attack detection for industrial cyber-physical systems,” Neurocomputing, vol. 275, pp. 1674-1683, 2018. [99] X. Ge and Q.-L. Han, “Distributed event-triggered H∞ filtering over sensor networks with communication delays,” Inf. Sci., vol. 291, pp. 128-142, Jan. 2015. [100] S. Zhu, Q.-L. Han, and C. Zhang, “l1 -gain performance analysis and positive filter design for positive discrete-time Markov jump linear systems: A linear programming approach,” Automatica, vol. 50, no. 8, pp. 2098-2107, Aug. 2014. [101] H. Yang, B. Jiang, and J. Zhao, “On finite-time stability of cyclic switched nonlinear systems,” IEEE Trans. Autom. Control, vol. 60, no. 8, pp. 2201-2206, Aug. 2015. [102] X. Yu, J. Xu, Y. Hong, and S. Yu, “Analysis of a class of discrete-time systems with power rule,” Automatica, vol. 43, no. 3, pp. 562-566, Mar. 2007. [103] H. Du, X. Yu, M. Chen, and S. Li “Chattering-free discrete-time sliding mode control,” Automatica, vol. 68, pp. 87-91, Jun. 2016.




[104] D. Efimov, A. Polyakov, A. Levant, and W. Perruquetti, “Realization and discretization of asymptotically stable homogeneous systems,” IEEE Trans. Autom. Control, vol. 62, no. 11, pp. 5962-5969, Nov. 2017.

Zongyu Zuo (M’14) 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 with the School of Electrical and Electronic Engineering, University of Manchester from September 2014 to September 2015 and held an inviting associate professorship with 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.

Qing-Long Han (M’09-SM’13) received the B.Sc. degree in Mathematics from Shandong Normal University, Jinan, China, in 1983, and the M.Sc. and Ph.D. degrees in Control Engineering and Electrical Engineering from East China University of Science and Technology, Shanghai, China, in 1992 and 1997, respectively. From September 1997 to December 1998, he was a Post-doctoral Researcher Fellow with the Laboratoire d’Automatique et d’Informatique Industielle (currently, Laboratoire d’Informatique et ´ d’Automatique pour les Systémes), Ecole Supérieure ´ d’Ing’enieurs de Poitiers (currently, Ecole Nationale Supérieure d’Ingé nieurs de Poitiers), Université de Poitiers, France. From January 1999 to August 2001, he was a Research Assistant Professor with the Department of Mechanical and Industrial Engineering at Southern Illinois University at Edwardsville, USA. From September 2001 to December 2014, he was Laureate Professor, an Associate Dean (Research and Innovation) with the Higher Education Division, and the Founding Director of the Centre for Intelligent and Networked Systems at Central Queensland University, Australia. From December 2014 to May 2016, he was Deputy Dean (Research), with the Griffith Sciences, and a Professor with the Griffith School of Engineering, Griffith University, Australia. In May 2016, he joined Swinburne University of Technology, Australia, where he is currently Pro Vice-Chancellor (Research Quality) and a Distinguished Professor. In March 2010, he was appointed Chang Jiang (Yangtze River) Scholar Chair Professor by Ministry of Education, China. His research interests include networked control systems, neural networks, time-delay systems, multi-agent systems and complex dynamical systems. Prof. Han was a recipient of the World’s Most Influential Scientific Minds from 2014 to 2016, and the Highly Cited Researcher Award in Engineering by Thomson Reuters. He is an Associate Editor of a number of international journals, including the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, the IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, the IEEE TRANSACTIONS ON CYBERNETICS, and Information Sciences.

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Boda Ning (S’14-M’17) received the B.Eng. degree in automatic control from East China University of Science and Technology, Shanghai, China, in 2011, and the M.Sc. degree with Distinction in control systems from University of Manchester, Manchester, U.K., in 2012. As a Vice-Chancellor’s Research Scholarship (2013-2017) recipient, he received the Ph.D. degree in 2017 from Swinburne University of Technology, Melbourne, Australia, where he is currently a Research Fellow. From 2010 to 2011, he was an exchange student at University of Dundee, Dundee, U.K. His current research interests include consensus and formation control of multi-agent systems. He was a recipient of the Chinese Government Award for Outstanding Students Abroad in 2017.

Xiaohua Ge (M’18) received the B.Eng. degree in electronic and information engineering from Nanchang Hangkong University, Nanchang, China, in 2008, the M.Eng. degree in control theory and control engineering from Hangzhou Dianzi University, Hangzhou, China, in 2011, and the Ph.D. degree in computer engineering from Central Queensland University, Rockhampton, QLD, Australia, in 2014. He was a Research Assistant with the Centre for Intelligent and Networked Systems, Central Queensland University, from 2011 to 2013, where he was a Research Fellow with the Centre for Intelligent and Networked Systems, in 2014. From 2015 to 2016, he was a Research Fellow with the Griffith School of Engineering, Griffith University, Gold Coast, QLD, Australia. In 2017, he joined the Swinburne University of Technology, Melbourne, VIC, Australia, where he is currently a Lecturer with the School of Software and Electrical Engineering. His current research interests include distributed networked control systems, multiagent systems, and sensor networks.

Xian-Ming Zhang (M’16) received the M.Sc. degree in applied mathematics and the Ph.D. degree in control theory and engineering from Central South University, Changsha, China, in 1992 and 2006, respectively. In 1992, he joined Central South University, where he was an Associate Professor with the School of Mathematics and Statistics. From 2007 to 2014, he was a Post-Doctoral Research Fellow and a Lecturer with the School of Engineering and Technology, Central Queensland University, Rockhampton, QLD, Australia. From 2014 to 2016, he was a Lecturer with the Griffith School of Engineering, Griffith University, Gold Coast, QLD, Australia. In 2016, he joined the Swinburne University of Technology, Melbourne, VIC, Australia, where he is currently a Senior Lecturer with the School of Software and Electrical Engineering. His current research interests include H-infinity filtering, event-triggered control systems, networked control systems, neural networks, distributed systems, and time-delay systems. Dr. Zhang was a recipient of first Hunan Provincial Natural Science Award in Hunan Province in China in 2011, second National Natural Science Award in China in 2013, both jointly with Prof. M. Wu and Prof. Y. He, and the IET Premium Award in 2016, jointly with Prof. Q.-L. Han. He is an Associate Editor of the Journal of the Franklin Institute and the Neurocomputing, and he is an Editorial Board Member of Neural Computing and Applications.
