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1
Fixed-Time-Synchronized Consensus Control of
Multi-Agent Systems
Dongyu Li, Shuzhi Sam Ge, and Tong Heng Lee
Abstract—In this paper, a unique multi-agent control problem
is defined — all the elements of all the agent states reaching
consensus at the same time, i.e., the multi-agent system achieves
time-synchronized consensus; and fixed-time-synchronized con-
sensus, where the upper bound of the synchronized settling
time is independent of the initial states of multi-agent systems.
To articulate this (fixed-) time-synchronized consensus problem,
we first propose Time-Synchronized Stability and Fixed-Time-
Synchronized Stability, a special kind of fixed/finite-time stabil-
ity, where all the elements of the system state synchronously
arrive at the equilibrium at the same time, with the upper
bound of the synchronized settling time dependent/independent
of initial conditions. Based on fixed-time-synchronized stability, a
singularity-free sliding mode control law is designed to solve the
fixed-time-synchronized consensus problem. Finally, numerical
simulations are conducted to showcase the effectiveness, and
further exploration of the merit of the proposed controller is
provided.
Index Terms—Fixed-Time-synchronized consensus, multi-
agent systems.
I. INTRODUCTION
NETWORKED system control has been an active area
with existing works focusing on interesting topics of
outspread applications, such as sensor networks [1], multi-
robot exploration [2], drone (or manned aircraft) cruise [3],
and spacecraft formation-flying [4], [5]. As one of the most
widely studied topics, the consensus problem is a kind of fun-
damental collective behaviors of multi-agent systems, where
each agent’s state aims to converge to a common value via
information flow exchanged with its neighbours [6]–[15].
To effectively enhance the convergence rate and to achieve
high-precision performance, finite-time consensus control is
introduced for multi-agent systems from different perspectives
in the works of [16]–[21]. Here, it can be noted that the
settling time of the classical finite-time consensus control
design invariably depends on initial conditions of the net-
worked systems. Next, fixed-time stability is introduced in
[22], where the uniform boundedness of the settling time is
independent of any initial states. Then in [23], the fixed-time
consensus problem is investigated for the case of second-order
multi-agent systems based on fixed-time stability. For higher-
order integrator dynamics, a fixed-time consensus controller
is further proposed in [24]. Additionally, some elegant fixed-
time consensus algorithms have also been designed with the
consideration of practical issues, e.g., the control of nonlinear
and disturbed systems in the work in [25], and the design for
Dongyu Li (corresponding author: levyli@nus.edu.sg), Shuzhi Sam Ge
and Tong Heng Lee are with the Department of Electrical and Computer
Engineering, National University of Singapore, Singapore 117576, Singapore.
control under discontinuous communications in the work in
[26], etc.
In summary, the fixed/finite-time consensus guarantees that
all the elements of the multi-agent states converge to the
equilibrium before a certain time instant. However, for a class
of practical missions, the fixed/finite-time consensus is not
enough during certain operations. For example, to achieve the
best defensive performance, a group of military drones are
required to form tactical formation synchronously. Otherwise,
the drones that arrive first will become vulnerable and be
easily exposed to the attack. Actually, not only the above case,
but also many other multi-agent missions depend greatly on
how and when the networked system states converge, e.g.,
considering spacecraft formation-flying (PROBA-3, EXO-S,
SWIFT, etc [4]), synthetic-aperture radar satellite lineup [27],
and cooperative missile attacks [28], the multi-agent systems
are normally ordered to achieve a specific consensus-based
configuration or to reach a target synchronously.
Motivated by the above observations, we further study a spe-
cial fixed/finite-time consensus problem — time-synchronized
consensus; focusing on how to design a controller which drives
all the elements of all the agent states to consensus at the
same time; and fixed-time-synchronized consensus, where the
upper bound of the synchronized settling time is independent
of the initial states of multi-agent systems. To better ad-
dress this (fixed-) time-synchronized consensus problem, time-
synchronized stability and fixed-time-synchronized stability are
stated, which describes the condition where all the elements
of a system are driven to the origin synchronously, with the
synchronized settling time dependent/independent of initial
conditions. Note that time-synchronized stability is first con-
ceptualized in [29], where control laws are properly designed
to drive all the state elements to the origin at the same time.
However, the result in [29] is confined to a single system which
cannot be trivially applied to multi-agent systems. Further, the
concept of fixed-time-synchronized stability and (fixed-) time-
synchronized consensus have not been touched on.
In this paper, we start from revisiting the properties of the
sign functions which are well-known in a broad range of
applications, such as signal processing, industrial electronics,
and control system design [30]–[40]. We are interested in
studying two kinds of multi-variable sign functions — the
classical sign function and the norm-normalized sign func-
tion. The definitions of these two sign functions had also
been previously described by the co-authors in earlier work
in [49], but the details of how differently they will affect
the convergence and stability of a control system are not
investigated. It is shown herein that both sign functions can
2
be integrated into a consensus controller to stabilize multi-
agent systems in fixed/finite time, while the norm-normalized
sign function-based controller guides the multi-agent systems
to achieve (fixed-) time-synchronized consensus. Actually,
the norm-normalized sign function is widely applied in the
existing results of sliding mode control and applications, e.g.,
in [37]–[40] to simplify the control design for various systems.
Unfortunately, the detailed properties of the norm-normalized
sign function, the (fixed-) time-synchronized stabilization of
system dynamics with the aid of the norm-normalized sign
function, and concept of (fixed-) time-synchronized consensus
have likewise not been explored there.
The present paper holds the following contributions.
First, we formally introduce the definition of (fixed-)
time-synchronized stability, as unique types of established
fixed/finite-time stability. Second, based on the discovery
in fixed-time-synchronized stability and the norm-normalized
sign function, we propose a fixed-time-synchronized con-
trol law for multi-agent systems to achieve fixed-time-
synchronized consensus. The merit of the proposed controller
compared with typical fixed-time consensus control laws has
also been revealed. Besides driving multi-agent systems to
achieve consensus at the same time, we further find that under
the proposed fixed-time-synchronized controller, the ratio of
each pair of the multi-agent system state elements is con-
stant during convergence, consequently yielding shorter and
smoother output trajectories. Last but not least, the proposed
fixed-time-synchronized consensus controller is singularity-
free. For the cases utilizing the classical sign function, the
singularity-avoidance problem has been well-studied, and
various major modifications have been developed, e.g., the
modified sliding model-based method [34], [41], [42], the
switching manifold-based method [43]–[45], the sine function-
based method [46], [47], the saturation function-based method
[48], etc. Despite the above mentioned effective non-singular
methods for fixed/finite-time control problems, this critical
singularity-avoidance issue still remains open for the fixed-
time-synchronized control design. The main challenge lies
in that these techniques are designed based on the classical
sign function. They are not applicable to the norm-normalized
sign function due to the inherent differences between the
two functions. This makes the present singularity-free result
certainly non-trivial.
In what follows, the rest of the developments in this paper
is expanded in suitable detail, where firstly in Section II,
properties of the sign functions, the graph theory, and the
problem formulation are described. Section III then covers the
main results of this study, where the fixed-time-synchronized
consensus control law is proposed. Next, in Section IV,
numerical simulations are furnished. Conclusions are stated
in Section V.
II. PRELIMINARIES AND PROBLEM FORMULATIONS
The abbreviations of this paper are listed in Table I.
A. Properties of Sign Functions
We introduce the following classical sign function sgncand
the norm-normalized sign function sgnn(while these functions
TABLE I
ABBREVIATION LIST
Abbreviations Meanings
TSC Time-Synchronized Consensus
FTSC Fixed-Time-Synchronized Consensus
TSS Time-Synchronized Stability/Stable
FTSS Fixed-Time-Synchronized Stability/Stable
and their properties have been provided previously in [29],
they are introduced here for completeness),
sgnc(x)∆
= [sgn (x1),sgn (x2),..., sgn (xn)]T,(1)
sgnn(x)∆
=x
kxk, x 6= 0,
0, x = 0,(2)
where x= [x1, x2, . . . , xn]T∈Rn,kxkis the L2-norm of
x, and
sgn (xi)∆
=
+1, xi>0,
0, xi= 0,
−1, xi<0,
(3)
with i= 1,2, . . . , n.
In this paper, signn(x)is called the norm-normalized sign
function, because essentially, as shown in (2), the stated
input vector and its specified norm are incorporated via the
normalization computation. Actually, signn(x)can be view
as a unit direction vector, which may not be typically of the
type usually associated with the classically-used concept of
sign, but as what follows, signn(x)holds many properties
similar to those of the classical sign function signc. Thus
here, we believe introducing these two functions in tandem
together will provide a more comprehensive comparison and
more acceptable logical applicability.
Excluding vectors on the axes, function sgnc(x)maps all
vectors from the same quadrant into one vector. Function
sgnn(x)maps each vector into the direction of x. Func-
tion sgnc(x)defines 3nvectors evenly distributed in an n-
dimensional space (including 0∈Rnfor x= 0), while
sgnn(x)defines an n-dimensional unit ball. 2-dimensional
and 3-dimensional examples are shown in Figs. 1(a) and 1(b),
where the blue arrows denote sgnc(x)and the red one denotes
sgnn(x). In brief, the classical sign function maps all vectors
from the same quadrant into one vector excluding vectors on
the axes, and the norm-normalized sign function maps each
vector into the direction of itself.
A comparison of sgnc(x)and sgnn(x)is provided:
i. sgnc(x) = sgnn(x)if and only if any one of following
conditions holds: 1) xis a scalar; 2) xcontains at least
two elements, where one element is non-zero and the rest
are all zero; and 3) all elements of xare zero. For the
other cases, sgnc(x)6= sgnn(x).
ii. ksgnc(x)k ≤ √n, while ksgnn(x)k= 1.
iii. xTsgnc(x) = Pn
i=1 |xi|=kxk1, while xTsgnn(x) =
kxk2/kxk=kxk.
iv. Consider a positive (negative) definite matrix A∈Rn×n.
xTAsgnn(x),x6= 0 is also positive (negative) definite
as xTAsgnn(x) = xTAx/kxkwhile the same property
3
𝑥
𝑥
1
1
‐1
‐1
𝑠
𝑠
𝑠
𝑠
𝑠𝑠
𝑠
𝑠
𝑥
‖𝑥‖
𝑎
(a) 2-D example
𝑥
𝑥
𝑥
1
1
1
𝑠
𝑠
𝑠
𝑠
𝑠𝑠
𝑠
𝑥
‖𝑥‖
𝑏
(b) 3-D example
Fig. 1. Functions sgnc(x)and sgnn(x)in 2- and 3-dimensions. (For
sgnc(x)in the 3-D case, only 7 vectors (s1, s2,...,s7) are shown for
simplicity.)
is not held by xTAsgnc(x).
The sign functions (1) and (2) are discontinuous, which are
hard to be implemented in practice and impede the control
performance due to the chattering phenomenon. We provide
the following modified continuous functions,
sigc(x)α= [sgn (x1)|x1|α,...,sgn (xn)|xn|α]T,(4)
sign(x)α=kxkαsgnn(x),(5)
where αis a positive constant.
Useful properties of sign(x)αare proposed.
Lemma 1: For z=zT
1, zT
2, . . . , zT
NT∈RNn ,zi∈Rn,
i∈ {1,2, ..., N }and 0< α < 1, we have the following
inequality holds element-wisely
hsign(z1)αT ,sign(z2)αT ,...,sign(zN)αT iT≥sign(z)α.(6)
Proof: In the case of zi6= 0, we can get
sign(zi)αT =zT
i
kzik1−α≥zT
i
pzT
1z1+zT
2z2+. . . +zT
NzN
1−α.(7)
In the case of zi= 0, it reads sign(zi)αT = 0.In this case,
sign(zi)αT is equal to the corresponding element in sign(z)α.
Based on the above, we have
hsign(z1)αT ,sign(z2)αT ,...,sign(zN)αT iT≥sign(z)α.(8)
Lemma 2: For any real vector x∈Rnand a positive
constant α,
∂
∂x kxkα+1 = (α+ 1) kxkα−1x, (9)
d
dtkxkα+1 = (α+ 1) kxkα−1xT˙x, (10)
∂
∂x sign(x)α+1 =αkxkα−2xxT+kxkαIn.(11)
Proof: The proof is omitted here.
For convenience, we refer sigcand signas the classical
sign function and the norm-normalized sign function in the
following sections.
B. Time-Synchronized Consensus Problem Formulation
We consider the following system,
˙x=f(x), f (0) = 0, x(0) = x0,(12)
where with respect to an open neighborhood D0⊆Rnof
the origin, f:D0→Rnis continuous. We assume that in
forward time, the system (12) has a unique solution for all
initial conditions.
Some well-known results on finite-time stability and fixed-
time stability are introduced.
Definition 1: [50] (Finite-Time Stability). The equilibrium of
system (12) is finite-time stable if for an open neighborhood
N0⊆ D0of the origin, the following statements hold:
(i) Finite-time convergence: For ∀x0∈ N0\{0},x(t)∈
N0\{0}, and limt→T(x0)x(t) = 0, where T(x0)is the
settling time.
(ii) Lyapunov stability: Given any open neighborhood Uεof
0, there exits an open subset Uδof N0containing 0 such
that, for ∀x∈ Uδ\{0},x(t)∈ Uε, for ∀t≥0.
The equilibrium is globally finite-time stable if it is finite-time
stable with N0=D0=Rn.
Lemma 3: [50] Considering the system (12), for any real
numbers c > 0and α∈(0,1), the equilibrium x= 0 of the
system (12) is finite-time stable if
˙
V(x) + cV α(x)≤0,(13)
where the settling time is estimated by
T(x0)≤V1−α(x0)
c(1 −α).(14)
Definition 2: [22] (Fixed-Time Stability). The equilibrium
of the system (12) is fixed-time stable if it is globally finite-
time stable with bounded settling-time T, where the bound is
independent of any initial system state, i.e., for ∀x0, we have
T < Tm, where Tmis a positive constant.
Lemma 4: [22] The equilibrium of system (12) is fixed-time
stable, if there exists a Lyapunov function V(x)and
˙
V(x)≤ −(αV p(x) + βV g(x))χ,(15)
where α, β, p, g and χare positive constants, satisfying pχ < 1
and gχ > 1. The settling time Tis bounded as
T≤1
αχ(1 −pχ)+1
βχ(gχ −1) .(16)
We here formally define time-synchronized stability and
fixed-time-synchronized stability.
Definition 3: (Time-Synchronized Stability). The equilibrium
x= 0 of the system (12) is time-synchronized stable if for
an open neighborhood N0⊆ D0of the origin, there exists
a function T:N0\{0} → (0,∞), called the synchronized
settling-time function, such that the following statements hold:
(i) Time-synchronized convergence: For ∀x0∈ N0\{0},
x(t)∈ N0\{0},xi(t)6= 0, and limt→T(x0)xi(t) = 0,
i∈ {1,2, . . . , n}, where T(x0)is the synchronized
settling time.
(ii) Lyapunov stability: Given any open neighborhood Uεof
0, there exits an open subset Uδof N0containing 0 such
that, for ∀x0∈ Uδ\{0},x(t)∈ Uε, for ∀t≥0.
4
The equilibrium is globally time-synchronized stable if it is
time-synchronized stable with N0=D0=Rn.
Remark 1: According to Definition 3, time-synchronized
stability already requires the system state to converge to
the equilibrium in finite time. Therefore, time-synchronized
stability can be regarded as a stricter special kind of finite-
time stability, as it meets the requirements in Definition 1.
Comparing Definition 1and Definition 3, one can observe that
the main difference is that finite-time stability allows some
of the state elements to arrive at the origin before T(x0)
as long as the remaining state elements converge to zero
as t→T(x0); while the requirements of time-synchronized
stability are stricter and more than the above is required. To
be specific, all the system state elements xiare required to
arrived at the origin at the same time T(x0).
Definition 4: (Fixed-Time-Synchronized Stability). The equi-
librium x= 0 of the system (12) is fixed-time-synchronized
stable if it is globally time-synchronized stable with the upper
bound of the synchronized settling time Tindependent of any
initial system state, i.e., for ∀x(0), we have T≤Tm, where
Tmis a positive constant.
In the next subsection, the concept of time-synchronized
consensus is explored.
C. Time-Synchronized Consensus
Consider an N-agent network, where agent iis modeled as
˙xi=vi,
˙vi=fi(xi, vi) + bi(xi, vi)ui,(17)
where xi∈Rnand vi∈Rn,i∈ {1,2, . . . , N }, denote the
general position vector and velocity vector, respectively, ui∈
Rnis the control input, and fi(xi, vi)∈Rnand bi(xi, vi)∈
Rn×n(det (bi(xi, vi)) 6= 0) are the known parts.
We utilize a communication graph Gc= (Vc,Ec)to char-
acterize the underlying information flow among agents, where
Ec⊆ Vc× Vcand Vc={1,2, . . . , N }denote the arc set and
the vertex set, respectively. Denote Nias the in-neighbour set
of node i, where Ni={j: (j, i)∈ Ec}. The adjacency matrix
Ac= [aij ]∈RN×Nis defined as aij >0if (j, i)∈ Ec, and
aij = 0 otherwise, where aii = 0. The Laplacian matrix is
defined as Lc= [lij ]∈RN×N, where lii =Pk
j=1,j6=iaij and
lij =−aij ,i6=j.
Assumption 1: The undirected graph Gcis connected.
Assumption 1indicates that there always exists a path
between two nodes in Gc.
Definition 5: (Time-Synchronized Consensus). A group of
networked agent systems achieve time-synchronized consensus
if and only if all the agents reach consensus synchronously,
i.e., we have
lim
t→TX
i,j∈Vc,i6=jkxi(t)−xj(t)k= 0,(18)
with a positive time instant T, while for any time instants t1
and t2satisfying 0≤t1< t2< T and any i∈ Vc, we have
kxi,k (t)−χkk 6≡ 0,∀t1≤t≤t2,(19)
where χ=xi(T),i∈ Vcdenotes the final value of the
consensus, and xi,k (t)and χkdenote the kth element of xi(t)
and χ, respectively (k= 1,2, ..., n).
Different from the traditional consensus problem, the time-
synchronized consensus stated in Definition 5drives all the
elements of all the agent states to consensus at the same
time, i.e., the time-synchronized consensus is achieved in all
dimensions.
Definition 6: (Fixed-Time-Synchronized Consensus). A
group of networked agent systems achieve fixed-time-
synchronized consensus if and only if they achieve time-
synchronized consensus with bounded settling-time T, where
the bound is independent of any initial multi-agent system
states, i.e., for ∀xi(0),i∈ Vc, we have T < Tm, where Tm
is a positive constant.
III. FIX ED -TIME-SYNCHRONIZED CONS EN SU S CON TRO L
This section studies FTSC control design based on FTSS.
First, definitions on the consensus trajectory and the consensus
equilibrium persistence are introduced.
Definition 7: (Consensus Trajectory). Over a time interval
D ⊆ R+before the consensus is finally achieved, a consensus
trajectory is the trajectory of x(t)defined as
T(x, D) := nx|x(t) = xT
1, . . . , xT
NT∈RNn , t ∈ D ⊆ R+
o.
Definition 8: (Consensus Equilibrium Persistence). A group
of agents described by the framework Gc(x)is consensus
equilibrium persistent if there exist at most isolated time
instants when for an agent i∈ Vc, we have
X
j∈Ni
aij (xi−xj) = 0 and xi6=xj,(20)
which means that
Ωt∗:= {t∗|X
j∈Ni
aij (xi(t∗)−xj(t∗)) = 0,
xi(t∗)6=xj(t∗), i ∈ Vc}(21)
is a discrete set if not empty.
We assume that the networked N-agent system satisfies the
following assumption.
Assumption 2: The framework Gc(x)is consensus equilib-
rium persistent along the consensus trajectory.
Under Assumption 2, before achieving consensus, an agent
will not stay on fake equilibrium points which satisfy
Pj∈Niaij (xi−xj) = 0 while xi6=xj,i∈ Vc. In
accordance with a considerable amount of simulations with
different initial states and graphical conditions, we conjecture
that Assumption 2can always be met.
We propose the following auxiliary variables
qri =1
$iXj∈Ni
aij (xi−xj),(26)
zri =1
$iXj∈Ni
aij vj,(27)
where $i=Pj∈Niaij .
5
si= ˙qri +ssi,(22)
ssi =α1sign(qri)p1+β1sign(qr i)g1,if s∗
i= 0,or s∗
i6= 0,kqrik> ε,
l1qri +l2sign(qri )4,if s∗
i6= 0,kqrik ≤ ε, (23)
ui=−b−1
i(xi, vi) ( ˙ssi −˙zri +α2sign(si)p2+β2sign(si)g2+fi(xi, vi)) ,(24)
˙ssi =ρ1iqriqT
ri ˙qri +ρ2i˙qri,if s∗
i= 0 or s∗
i6= 0,kqrik> ε,
l1˙qri + 3l2kqri kqri qT
ri ˙qri +l2kqr ik3˙qr i,if s∗
i6= 0,kqrik ≤ ε, (25)
Based on the introduced norm-normalized sign function (5),
we design a switching terminal sliding mode (see equations
(22)-(23)), where the trigger sliding-mode variable is given as
s∗
i= ˙qri +α1sign(qri )p1+β1sign(qri )g1,(28)
α1,β1,g1and p1are positive constants satisfying g1>1and
p1=`1/`2∈(0.5,1) with positive odd integers `1and `2,l1
and l2are defined as
l1=α14
3−p1
3εp1−1+β14
3−g1
3εg1−1,(29)
l2=α1p1
3−1
3εp1−4+β1g1
3−1
3εg1−4,(30)
and εis a small constant.
We can verify that the design of l1and l2guarantee the
continuity of the switching law ssi (22) as well as its time
derivative ˙ssi (23).
Using the sliding mode (22), we propose the FTSC control
law for agent i(see equations (24) and (25)), where α2,β2,
p2and g2are positive constants satisfying p2<1and g2>1,
and auxiliary variables ρ1iand ρ2itake the forms
ρ1i=α1(p1−1) kqrikp1−3+β1(g1−1) kqr ikg1−3,(31)
ρ2i=α1kqrikp1−1+β1kqr ikg1−1.(32)
We can verify that no singularity problem occurs in the
following cases:
(i) In the case of s∗
i= 0 and qr i →0, we have
˙qri =−α1sign(qri )p1−β1sign(qri )g1,(33)
Taking (33) into the controller (24) yields,
˙ssi =−α2
1p1sign(qri)2p1−1−β2
1g1sign(qri)2g1−1
−α1β1(p1+g1) sign(qri)p1+g1−1.(34)
(ii) In the case of s∗
i6= 0 and qr i →0, we have
˙ssi =−l1˙qri −3l2kqri kqriqT
ri ˙qri −l2kqr ik3˙qr i.(35)
In both cases (i) and (ii), there is no singularity as p1>1/2
and g1>1.
The main theorem is formally proposed.
Theorem 1: Considering multi-agent systems governed by
(17), if Assumptions 1and 2hold, under the control law (24),
the closed-loop multi-agent systems achieve FTSC.
Proof: This proof is addressed by three steps: (i) the
system state reaches the sliding mode surface in fixed time;
(ii) the system state achieves consensus in fixed time; and (iii)
all the elements of all the agent states come to an agreement
synchronously.
Step 1: Consider a Lyapunov function Vm1=Pi∈VcsT
isi.
Substituting the control law (24) into the derivative of Vm1,
we have
˙
Vm1=−2sT
iα2X
i∈Vc
sign(si)p2+−2sT
iβ2X
i∈Vc
sign(si)g2
=−2α2X
i∈Vcksikp2+1 −2β2X
i∈Vcksikg2+1
=−2α2V
p2+1
2
m1−2β2V
g2+1
2
m1.(36)
Based on Lemma 4, we know that the agent states will reach
the surface si= 0 in a fixed-time instant bounded by
Tm1≤1
α2(1 −p2)+1
β2(g2−1).(37)
We can get s∗
i= 0 by choosing a sufficient small ε, which
directly yields
˙qri =−α1sign(qri )p1−β1sign(qri )g1.(38)
Step 2: Consider a second Lyapunov function Vm2=qT
rqr,
where qr=qT
r1, qT
r2, . . . , qT
rN T. The derivative of Vm2takes
the form
˙
Vm2=−2α1qT
r
sign(qr1)p1
· · ·
sign(qrN )p1
−2β1qT
r
sign(qr1)g1
· · ·
sign(qrN )g1
.
Using Lemma 1, we can get
˙
Vm2≤ − 2α1qT
rsign(qr)p1−2β1qT
rsign(qr)p1
=−2α1kqrk1+p1−2β1kqrk1+g1
=−2α1V
1+p1
2
m2−2β1V
1+g1
2
m2,(42)
where the settling time takes the form
Tm2≤1
α1(1 −p1)+1
β1(g1−1).(43)
For now, the fixed-time consensus has been achieved.
Step 3: Under the control law (24), to discuss the FTSC
property of systems (17), we first prove that the system of each
single agent i∈ Vcholds FTSS, which can be investigated by
comparing the convergence rate of elements of qri (t). From
the proof in Step 2, we know qri (t)will reach the equilibrium
6
d
dt q(k)
ri (t)
q(j)
ri (t)!=α1q(j)
ri (t)q(k)
ri (t)
kqri(t)k1−p1+β1q(j)
ri (t)q(k)
ri (t)
kqri(t)k1−g1−α1q(k)
ri (t)q(j)
ri (t)
kqri(t)k1−p1−β1q(k)
ri (t)q(j)
ri (t)
kqri(t)k1−g1
||q(j)
ri (t)||2= 0,(39)
¯ssi,k =
α1sigc(qri,k )p1+β1sigc(qri,k)g1,if ¯s∗
i= 0,
or ¯s∗
i6= 0,|qri|> ε,
l3qri,k +l4sigc(qri,k )2,if ¯s∗
i6= 0,|qri| ≤ ε,
(40)
˙
¯ssi,k =
α1p1|qri,k |p1−1˙qri,k +β1g1|qri,k |g1−1˙qri,k ,if ¯s∗= 0,
or¯s∗6= 0,|qr i,k|> ε,
l3˙qri,k +l4|qri,k |˙qri,k,if ¯s∗6= 0,|qr i,k| ≤ ε,
(41)
in fixed time. Then, FTSS can be proved if the proportion of
any two elements of qri (t)is time-invariant. Before the system
state converges to the equilibrium, under the control law (24),
the derivative of q(j)
ri (t)/q(k)
ri (t)(∀j, k ∈ {1,2, . . . , n},j6=
k) takes the following form
d
dt q(k)
ri (t)
q(j)
ri (t)!=˙q(k)
ri (t)q(j)
ri (t)−q(k)
ri (t) ˙q(j)
ri (t)
||q(j)
ri (t)||2.(44)
According to (38), we further have (see equation (39)),
which directly shows that ∀j, k ∈ {1,2, . . . , n},j6=k,
q(j)
ri (t)/q(k)
ri (t)is a constant, and finishes the proof of FTSS
of each single agent i∈ Vc.
We will next prove that the state elements for all the
agents reach the equilibrium synchronously. The proof is
given by contradiction. Assume that agent hreaches the
equilibrium first at the time instant tcand stays on it,
while agent kdoes not. According to Assumption 1, the
communication graph is connected, which correspondingly
means there exists a path from agent kto agent h. Without
loss of generality, suppose that there are lagents (labeled
as l1, l2, . . . , ll) among the path from agent kto agent h.
Based on Assumption 2, the position x+
l1(tc)of agent l1
which is directly connected to agent kmust not be at the
equilibrium point, since Pj∈Nl1al1j(xl1−xj)=0,xl16=xj
exists at most on isolated time instants. Accordingly, we
have the state of x+
l2(tc), . . . , x+
ll(tc)must not be at the
equilibrium point either. Again, based on Assumption 2, we
know Pj∈Nhahj (xh−xj) = 0,xh6=xjexists at most
on isolated time instants. Thus, x+
h(tc)will not stay at the
equilibrium as the state x+
ll(tc)of its neighbour agent llis not
at the equilibrium. This contradicts with the fact that agent h
reaches the equilibrium first at time instant tcand stay on it,
which completes the proof according to Definition 6.
Theorem 1is valid under an undirected graph. Interest-
ingly though, we are able to find several special cases of
directed graphs, under which (fixed-) time-synchronized can
be achieved. Unfortunately, the matter of how to extract
their mutual topographical features and propose a proper and
rigorous sufficient graphical condition is still challenging.
Remark 2: The proposed switching manifold (23) is actually
inspired by the finite-time control result integrated with the
switching law first proposed in [43]. Similar switching laws
have also been applied in [44], [45] for spacecraft attitude
tracking. However, the switching manifold in [43]–[45] is
designed for the classical sign function signc(·), which is
based on each element |xi|in the function sigc(x)α, and is thus
not applicable to sign(·)α. Compared with the result in [43]–
[45], the proposed switching law (23) is designed based on the
vector qri of the norm-normalized sign function signn(qri),
which completely changes the design procedure and the anal-
ysis. The new designed switching law (23), therefore, takes a
different form, and contributes not only to the avoidance of
the singularity but also to the achievement of FTSS.
Remark 3: The acceleration information ˙zri is required in
the FTSC controller (24). In practice, in the case without
acceleration sensors, this acceleration can be indirectly cal-
culated by numerical differentiation of the velocities [52]. In
terms of the possible algebraic loops induced by the usage
of acceleration information, we can analyze the stability with
delayed acceleration information [53] to solve this problem.
To better showcase the performance of the FTSC controller,
another fixed-time consensus control law is proposed for
comparison studies.
Lemma 5: Considering multi-agent systems governed by
(17), if Assumptions 1and 2hold, under the following control
law
¯ui=−b−1
i(xi, vi) ( ˙
¯ssi −˙zr i +α2sigc(si)p2
+β2sigc(si)g2+fi(xi, vi)) ,(45)
with the sliding manifold ¯si= ˙qri + ¯ssi , the trigger sliding-
mode variable ¯s∗
i= ˙qri +α1sigc(qri )p1+β1sigc(qri )g1, and
the switching law ¯ssi (see (40) and (41)), where qri,k ,¯ssi,k
and ˙
¯ssi,k are respectively the kth element of qri,¯ssi and ˙
¯ssi,
k= 1,2, ..., n, and constant parameters l3and l4are designed
as follows to maintain the continuity of ¯ssi and ˙
¯ssi:
l3= (2 −p1)α1εp1−1+ (2 −g1)β1εg1−1,(46)
l4= (p1−1)α1εp1−2+ (g1−1)β1εg1−2,(47)
the closed-loop multi-agent systems achieve consensus in fixed
time.
Proof: The proof follows from that of Theorem 1.
Lemma 5implies that we can try to apply the norm-
normalized sign function to existing control laws using the
classical sign function, contributing an additional (F)TSS for a
7
single system (or (F)TSC property for multi-agent systems), as
according to Section II-A, the norm-normalized sign function
possesses most of the properties of the classical sign function.
However, as we have shown in this paper, certain efforts should
be taken to implement the norm-normalized sign function, as
there are substantial differences between the two sign functions
in terms of the control design and the stability analysis.
Remark 4: The switching law ¯ssi in Lemma 5is largely
inspired by the ones in [43]–[45], where the relevant proof
and more detailed analysis are also addressed. As shown
in Lemma 5, the switching law ¯ssi,k is designed element-
wisely, based on the state element qri,k and the classical
sign function signc(qri)(or sigc(qri)α), while the proposed
switching law (23) is designed based on the vector qri of the
norm-normalized sign function sign(qri)α, making the design
of ¯ssi and ssi (or ˙
¯ssi and ˙ssi), the forms of li,i= 1,2,3,4,
and the corresponding triggering conditions of the switching
laws differ substantially. This is in line with the discussion in
Remark 2that fundamental differences lie in the forms as well
as the performance of these two control algorithms.
Remark 5: Note that the controller (45) for the system (17)
is in Rn. Although in Lemma 5, the fixed-time stability can be
achieved, it is clear that the control law (45) with the classical
sign function generates ndecoupled scalar systems from each
single agent, where the achievement of (F)TSC cannot be
expected.
IV. SIMULATION
A. Numerical Examples of the FTSC Control Law
To demonstrate the full power of the proposed FTSC control
law, we consider a group of 4 networked agents labeled from
1to 4. The simulation parameters are given in Table II.
TABLE II
PARAMETERS FOR SIMULATION:CA SE 1
Parameters Values Parameters Values
α10.5 p10.7
α21p20.7
β10.1 g11.2
β20.05 g21.2
fi0biI3
qi−i(−1)i[4,−6,8]Tε0.001
Ac[0,1,0,0; 1,0,1,0; 0,1,0,1; 0,0,1,0]
Under the FTSC control law (24), the trajectories and the
closed-loop system states of the agents are given in Figs. 2
and 3. The control inputs are given in Fig. 4. It is shown
that the consensus is achieved and the states converge to the
equilibrium in fixed time. Unfortunately, we cannot clearly
observe the nature of the FTSC property from Fig. 3alone,
even though all the elements of all the agents appear to
converge synchronously in Fig. 3. Thus, compared with the
fixed-time consensus control law (45), additional simulations
are conducted to find out if all the elements of all the agents
reach the equilibrium synchronously. The simulation results in
Figs. 5and 6approve the declared FTSC property. Clearly, in
Fig. 5, under the fixed-time consensus control law, all the state
Fig. 2. Trajectories of 4 agents under the FTSC control law (24).
Fig. 3. The closed-loop system states under the FTSC control law (24).
Fig. 4. Control inputs of the FTSC control law (24).
elements of multi-agent systems converge to the equilibrium
separately, while in Fig. 6, under the FTSC controller, they
achieve consensus time-synchronously even when we zoom
in at the level of ×10−3. Moreover, as claimed in Step 3, the
proof of Theorem 1, after the sliding-mode surface is reached,
the proportion of the elements of the consensus error qri (t)
is time-invariant and preserved by the FTSC control law (24).
This property has also been shown in Fig. 7, which further
8
Fig. 5. Norms of the consensus errors under the fixed-time control law (45).
Fig. 6. Norms of the consensus errors under the FTSC control law (24).
0 5 10 15 20 25 30
0
1
2
3
0 5 10 15 20 25 30
Time (s)
0
1
2
3
Fig. 7. Ratio of the elements of the consensus error under the FTSC control
law (24).
validates that FTSC is achieved.
Next, another simulation case is considered to illustrate
how the proposed controller performs with a larger number
of agents. Here, 20 agents are selected whose initial state
elements xi,k (0),i= 1,2, ..., 20,k= 1,2,3are random
values belong to [−40,40]. The control gains are α1= 0.5,
α2= 0.5,β1= 5,β1= 0.2, while the other parameters
remain the same as in Table II. The communication topology
is an arbitrary graph satisfying Assumption 1. Under the FTSC
control law (24), the agent trajectories and the norms of the
consensus errors are shown in Figs. 8and 9. From Fig. 9, it is
clear that the consensus is achieved time-synchronously, whose
Fig. 8. Trajectories of 20 agents under the FTSC control law (24).
Fig. 9. Norms of the consensus errors under the FTSC control law (24).
convergence is similar to that in the case of 4 agents (see
Fig. 6). This approves that the effectiveness of the proposed
algorithm is not influenced by the number of agents.
V. CONCLUSION
A multi-agent control problem called (fixed-) time-
synchronized consensus has been considered, which consists
in the control design that guarantees convergence of all the
elements of all the agent states to the equilibrium at the
same time. To articulate the described property, we have in-
troduced (fixed-) time-synchronized stability, based on which,
a singularity-free fixed-time-synchronized consensus control
law has been proposed.
Moreover, we have shown that it is feasible to apply this
norm-normalized-function-based approach to existing control
laws using the classical sign function, bring additional (fixed-)
time-synchronized stability to them, since most of the prop-
erties of the classical sign function are held by the norm-
normalized sign function but not vice versa. However, several
important directions remain open, e.g., how to apply this
technique to design distributed schemes under a directed graph
and to control more complex systems with practical issues.
9
ACKNOWLEDGMENT
The authors would like to extend their gratitude towards the
support and fruitful discussions of Prof. L. Xie and Dr. K. Cao
from Nanyang Technological University.
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Dongyu Li (S’16-M’19) received the B.S. and
Ph.D. degree from Control Science and Engineering,
Harbin Institute of Technology, China, in 2016 and
2020. He was a joint Ph.D. student supported by
China Scholarship Council with the Department of
Electrical and Computer Engineering at National
University of Singapore, where he is currently a
research fellow with the Department of Biomedi-
cal Engineering. His research interests include net-
worked systems, human-robot interaction, and intel-
ligent control systems.
Shuzhi Sam Ge (S’90-M’92-SM’00-F’06) received
the Ph.D. degree from the Imperial College London,
London, U.K., in 1993, and the B.Sc. degree from
the Beijing University of Aeronautics and Astronau-
tics, Beijing, China, in 1986. He is the Director with
the Social Robotics Laboratory of Interactive Digi-
tal Media Institute, Singapore, and the Centre for
Robotics, Chengdu, China, and a Professor with the
Department of Electrical and Computer Engineering,
National University of Singapore, Singapore, on
leave from the School of Computer Science and
Engineering, University of Electronic Science and Technology of China,
Chengdu. He has co-authored four books and over 300 international journal
and conference papers. His current research interests include social robotics,
adaptive control, intelligent systems, and artificial intelligence.
Dr. Ge is Editor-in-Chief of the International Journal of Social Robotics
(Springer). He has served/been serving as an Associate Editor for a number of
flagship journals, including IEEE Transactions on Automation Control,IEEE
Transactions on Control Systems Technology,IEEE Transactions on Neural
Networks and Automatica. He serves as a Book Editor for the Taylor and
Francis Automation and Control Engineering Series. He served as the Vice
President for Technical Activities, from 2009 to 2010, the Vice President of
Membership Activities, from 2011 to 2012, and a member of the Board of
Governors, from 2007 to 2009 at the IEEE Control Systems Society. He is a
Fellow of the International Federation of Automatic Control, the Institution
of Engineering and Technology, and the Society of Automotive Engineering.
Tong Heng Lee received the B.A. degree with First
Class Honours in the Engineering Tripos from Cam-
bridge University, England, in 1980; the M.Engrg.
degree from NUS in 1985; and the Ph.D. degree
from Yale University in 1987. He is a Professor
in the Department of Electrical and Computer En-
gineering at the National University of Singapore
(NUS); and also a Professor in the NUS Graduate
School, NUS NGS. He was a Past Vice-President
(Research) of NUS.
Dr. Lee’s research interests are in the areas of
adaptive systems, knowledge-based control, intelligent mechatronics and com-
putational intelligence. He currently holds Associate Editor appointments in
the IEEE Transactions in Systems, Man and Cybernetics; Control Engineering
Practice (an IFAC journal); and the International Journal of Systems Science
(Taylor and Francis, London). In addition, he is the Deputy Editor-in-Chief
of IFAC Mechatronics journal.
