A class of robust consensus algorithms with predefined-time convergence under
switching topologies 1
R. Aldana-Lópeza , David Gómez-Gutiérreza,b,∗, M. Defoortd , J. D. Sánchez-Torresc , A. J. Muñoz-Vázqueze
a Multi-agent autonomous systems lab, Intel Labs, Intel Tecnologı́a de México, Av. del Bosque 1001, Colonia El Bajı́o, Zapopan, 45019, Jalisco,

México.
b Tecnologico de Monterrey, Escuela de Ingenierı́a y Ciencias, Av. General Ramón Corona 2514, Zapopan, 45201, Jalisco, México.
c Research Laboratory on Optimal Design, Devices and Advanced Materials -OPTIMA-, Department of Mathematics and Physics, ITESO,

arXiv:1812.07545v3 [math.OC] 31 Jul 2019

Periférico Sur Manuel Gómez Morı́n 8585 C.P. 45604, Tlaquepaque, Jalisco, México.
d LAMIH, CNRS UMR 8201, Univ. Valenciennes, Valenciennes 59313, France.
e CONACYT – School of Engineering, Autonomous University of Chihuahua, Chihuahua, México.

Abstract
This paper addresses the robust consensus problem under switching topologies. Contrary to existing methods, the
proposed approach provides decentralized protocols that achieve consensus for networked multi-agent systems in a
predefined time. Namely, the protocol design provides a tuning parameter that allows setting the convergence time
of the agents to a consensus state. An appropriate Lyapunov analysis exposes the capability of the current proposal
to achieve predefined-time consensus over switching topologies despite the presence of bounded perturbations. Finally, the paper presents a comparison showing that the suggested approach subsumes existing fixed-time consensus
algorithms and provides extra degrees of freedom to obtain predefined-time consensus protocols that are less overengineered, i.e., the difference between the estimated convergence time and its actual value is lower in our approach.
Numerical results are given to illustrate the effectiveness and advantages of the proposed approach.
Keywords: Consensus, Multi-agent systems, Fixed-time consensus, Predefined-time consensus, Switching topology.

1. Introduction
The problem of cooperative control for multi-agent systems has received significant attention due to its wide range
of applications in different fields such as biology, power grid, robotics, etc. In cooperative control, a fundamental
challenge is the consensus problem, whose objective is to design decentralized protocols such that all the agents,
in the network, interact with their neighbors to reach a common value [1], called the consensus equilibrium. Many
decentralized consensus-based methods have been applied, for instance, to flocking [2], formation control [3–5] and
distributed resource allocation [6, 7]. Two consensus problems have been mainly addressed, the leader-follower
consensus problem [8, 9], where the agents along the network converge to the state of the leader (real or virtual) which
communicates its state only to a subset of agents; and the leaderless consensus problem [10–12], where the consensus
equilibrium is a function of the initial conditions of the agents, for instance achieving consensus to the average
value [2], the average min-max value [13, 14], the median value [15], and the maximum or minimum value [16]
of the agents’ initial conditions.
1 This is the preprint version of the accepted Manuscript: R. Aldana-López, D. Gómez-Gutiérrez, M. Defoort, J. D. Sánchez-Torres and
A. J. Muñoz-Vázquez, “A class of robust consensus algorithms with predefined-time convergence under switching topologies”, International
Journal of Robust and Nonlinear Control, 2019, ISSN: 1099-1239. DOI. 10.1002/rnc.4715. Please cite the publisher’s version. For the publisher’s
version and full citation details see: http://dx.doi.org/10.1002/rnc.4715
∗ Corresponding Author.
Email addresses: rodrigo.aldana.lopez@gmail.com (R. Aldana-López), David.Gomez.G@ieee.org (David Gómez-Gutiérrez),
michael.defoort@univ-valenciennes.fr (M. Defoort), dsanchez@iteso.mx (J. D. Sánchez-Torres),
aldo.munoz.vazquez@gmail.com (A. J. Muñoz-Vázquez)

Preprint submitted to ArXiv

December 7, 2016

Many existing consensus algorithms focus on asymptotic convergence (i.e. the settling time is infinite) [17] and
finite-time convergence (i.e. the settling time estimate is finite but depends on the initial conditions) [18, 19]. It is
clear that a finite settling time is useful for several applications (a network of clusters, manufacturing systems, etc.).
However, the initial conditions are usually unknown due to the decentralized architecture. Therefore, recently, there
has been a great deal of attention in the research community on algorithms which solve the decentralized consensus
problem in a finite-time with uniform convergence with respect to the initial value of the agents (see. e.g [20] and the
references therein).
Two main approaches have been proposed to address this problem. The first one is based on the fixed-time stability
theory of autonomous systems proposed in [21]. Based on this class of systems, fixed-time consensus algorithms have
been proposed either based on the homogeneity theory developed in [22, 23], as in [20], the settling-time estimation
provided in [21], as in [24, 25] or the subclass of fixed-time stable systems, with an upper bound of the settling
time that is less conservative than the one in [21], presented in [26], as in [8, 10, 11, 25, 27]. However, in spite of
the advantages in terms of settling time estimation, this approach does not enable to easily arbitrarily preassigned
the settling time. For this case, the consensus problem for agents with single integrator dynamics and affected by
disturbances has been addressed in [11, 12].
The second approach is based on time-varying consensus protocols and provides consensus protocols with predefinedtime convergence, where the convergence time is set a priori as a parameter of the protocol [28–33]. In these works,
non-conservative upper-bound estimates of the settling time are provided. However, this approach presents some
drawbacks. First, they are based on a common time-varying gain that is applied to each node on the network,
which requires a common time-reference along the network. Second, this time-varying gain becomes singular at
the predefined-time, such as methods based on the so-called time base generators [34] see for instance [28]. Third,
methods, such as [29, 33], are based on a time-varying gain that is a piecewise-constant function that produces Zeno
phenomenon [35], even in the unperturbed case. Additionally, the robustness, against external disturbances, of the
methods proposed in [28–31, 33] has not been analyzed.
In this paper, we address the robust predefined-time leaderless consensus problem for the cases where a common
time reference is not available. Thus, we focus on developing autonomous protocols. Our approach is based on recent
results on the upper bound estimation for the settling time of a class of fixed-time stable systems presented in [36] to
propose a broader class of consensus protocols for perturbed first-order agents with a fixed-time convergence whose
convergence upper bound is set a priori as a parameter of the system. In our opinion, [11] is the closest approach
in the literature, as it presents robust fixed-time consensus protocols that can be designed to satisfy time constraints.
We will show that the results in [11] and other fixed-time consensus protocols, such as [10, 24, 27], are subsumed
by our approach since our results allow more flexibility in the parameter selection to obtain, for instance, predefinedtime consensus protocols where the slack between the real-convergence time and the estimated upper bound of the
convergence time is lower. Contrary to previous fixed-time consensus protocols [10–12, 24, 25, 27] our approach
allows to straightforwardly specify the upper bound of the convergence time as a parameter of the consensus protocol.
We presents two robust consensus protocols. The first one is the one that is computationally simpler; we show that
it presents predefined-time convergence under static networks and fixed-time convergence under switched dynamic
networks. The second one, in the absence of disturbances, converges to a consensus state that is the average of the
agents’ initial conditions. This algorithm is shown to be a robust predefined-time consensus algorithm for static and
dynamic networks arbitrarily switching among connected graphs.
The rest of the manuscript is organized as follows. Section 2 introduces the preliminaries on graph theory, finitetime stability and predefined-time stability. Section 3 presents two approaches to address the robust consensus problem
for dynamic networks. Section 4 presents a comparison with [11] showing that having a broader class of predefinedtime consensus algorithms provides greater flexibility to improve the performance of the consensus algorithms. This
section also presents a qualitative comparison of both protocols with respect to other fixed-time consensus protocols
previously proposed in the literature highlighting the contribution of our approach. Finally, Section 5 presents some
concluding remarks.
2. Preliminaries
2.1. Graph Theory
The following preliminaries on graph theory are from [37]. In this paper, we will focus only on undirected graphs.
2

Definition 1. A graph X consists of a set of vertices V(X) and a set of edges E(X) where an edge is an unordered
pair of distinct vertices of X.
Writing i j denotes an edge and j ∼ i denotes that the vertex i and vertex j are adjacent or neighbors, i.e., there
exists an edge i j. The set of neighbors of i in the graph X is expressed by Ni (X) = { j : ji ∈ E(X)}.
Definition 2. A path from i to j in a graph is a sequence of distinct vertices starting with i and ending with j such
that consecutive vertices are adjacent. If there is a path between any two vertices of the graph X then X is said to be
connected. Otherwise, it is said to be disconnected.
Definition 3. A weighted graph is a graph together with a weight function W : E(X) → R+ .
Definition 4. Let X be a weighted graph such that i j ∈ E(X) has weight ai j and let n = |V(X)|. Then, the adjacency
matrix A is an n × n matrix where A = [ai j ].
Definition 5. Let X be a graph, the Laplacian of X is denoted by Q(X) (or simply Q when the graph is clear from the
P
context) and is defined as Q(X) = ∆(X) − A where ∆(X) = diag(d1 · · · , dn ) with di =
ai j .
j∈Ni (X)

Remark 1. For the Laplacian Q(X), there exists a factorization Q(X) = D(X)D(X)T (D(X) is known as the incidence
matrix of X [37]) where D(X) is an |V(X)| × |E(X)| matrix, such that if i j ∈ E(X) is an edge with weight ai j then
√
the column of D corresponding to the edge i j has only two nonzero elements: the i−th element is equal to ai j and
√
the j−th element is equal to − ai j . Clearly, the incidence matrix D(X), satisfies 1T D(X) = 0. The Laplacian matrix
Q(X) is a positive semidefinite and symmetric matrix. Thus, its eigenvalues are all real and non-negative.
When the graph X is clear from the context we omit X as an argument. For instance we write Q, D, etc to represent
the Laplacian, the incidence matrix, etc.
Lemma 1. [37] Let X be a connected graph and Q its Laplacian. The eigenvalue λ1 (Q) = 0 has algebraic multiplicity
one with eigenvector 1 = [1 · · · 1]T . The smallest nonzero eigenvalue of Q, denoted by λ2 (Q) satisfies λ2 (Q) =
xT Qx
min T .
x⊥1,x,0 x x
It follows from Lemma 1 that for every x⊥1, xT Qx ≥ λ2 (Q)kxk22 > 0. λ2 (Q) is known as the algebraic connectivity
of the graph X. The distance between two distinct nodes i and j is the shortest path length between them. The diameter
of X is the longest distance between two distinct vertices.
For a path graph and cycle graph of n nodes, Pn and Cn , λ2 can be computed as λ2 (Pn ) = 2 − 2 cos (π/n) and
λ2 (Cn ) = 2 − 2 cos (2π/n), respectively. For a star graph S n , λ2 (S n ) = 1. Graphs that are more connected have a larger
λ2 . A path graph Pn is the “most nearly disconnected” connected graph of n nodes.
Definition 6. A switched dynamic network Xσ(t) is described by the ordered pair Xσ(t) = hF , σi where F =
{X1 , . . . , Xm } is a collection of graphs having the same vertex set and σ : [t0 , ∞) → {1, . . . m} is a switching signal determining the topology of the dynamic network at each instant of time.
In this paper, we assume that σ(t) is generated exogenously and that there is a minimum dwell time between
consecutive switchings in such a way that Zeno behavior in network’s dynamic is excluded, i.e., there is a finite
number of switchings in any finite interval.
2.2. On finite-time, fixed-time and predefined-time stability
Consider the system
ẋ = f (x; ρ),

(1)

where x ∈ R is the system state, the vector ρ ∈ R stands for the parameters of system (1) which are assumed to
be constant, i.e., ρ̇ = 0. Furthermore, there is no limit for the number of parameters, so b can take any value in the
natural number set N. The function f : Rn → Rn is nonlinear, and the origin is assumed to be an equilibrium point of
system (1), so f (0; ρ) = 0. The initial condition of this system is x0 = x(0) ∈ Rn .
n

b

3

Definition 7. [38] The origin of (1) is globally finite-time stable if it is globally asymptotically stable and any solution
x(t, x0 ) of (1) reaches the equilibrium point at some finite time moment, i.e., ∀t ≥ T (x0 ) : x(t, x0 ) = 0, where T : Rn →
R+ ∪ {0} is called the settling-time function.
Definition 8. [21] The origin of (1) is fixed-time stable if it is globally finite-time stable and the settling-time function
is bounded, i.e. ∃T max > 0 : ∀x0 ∈ Rn : T (x0 ) ≤ T max .
To address the fixed-time stability analysis, one approach is based on the homogeneity theory.
Definition 9. [39] A function g : Rn → R is homogeneous of degree l with respect to the standard dilation if and only
if
g(λx) = λl g(x) for all λ > 0.
A vector field f (x), with x ∈ Rn , is homogeneous of degree d with respect to the the standard dilation if
f (x) = λ−(d+1) f (λx) for all λ > 0.
Definition 10. [22, 23] A function g : Rn → R, such that g(0) = 0, is homogeneous in the λ0 −limit with degree dλ0
with respect to the standard dilation if gλ0 : Rn → R, defined as
gλ0 (x) = lim λ−dλ0 g(λx),
λ→λ0

is homogeneous of degree dλ0 with respect to the standard dilation.
A vector field f : Rn → Rn is said to be homogeneous in the λ0 −limit with degree dλ0 with respect to the standard
dilation if the vector field fλ0 : Rn → Rn , defined as
fλ0 (x) = lim λ−(dλ0 +1) f (λx),
λ→λ0

(2)

is homogeneous of degree dλ0 with respect to the standard dilation.
A characterization of fixed-time stability, based on the homogeneity theory, is given in the following theorem.
Theorem 1. [22, 23] Let the vector field f : Rn → Rn be homogeneous in the 0−limit with degree d0 < 0 and
homogeneous in the +∞-limit with degree d∞ > 0 (if both conditions are satisfied it is said that the vector field f(x) is
homogeneous in the bi-limit). If for the dynamic systems ẋ = − f (x), ẋ = − f0 (x) and ẋ = − f∞ (x) the origin is globally
asymptotically stable (where f0 and f∞ are obtained from (2) with λ0 = 0 and λ0 = +∞, respectively), then the origin
of ẋ = − f (x) is a globally fixed-time stable equilibrium.
However, despite the advantages of having a finite settling time which allows uniform convergence with respect
to the initial conditions, the homogeneity based approach does not enable to easily arbitrarily preassigned the settling
time. For some applications such as state estimation, dynamic optimization, consensus of cluster networks, among
others, it would be convenient that the trajectories of system (1) reach the origin within a time T c ∈ T , which can be
defined in advance as a function of the system parameters, i.e., T c = T c (ρ). This motivates the following definitions.
Definition 11. [40] Let the origin be fixed-time stable for system (1). The set of all the bounds of the settling-time
function is defined as
T = {T max ∈ R+ : T (x0 ) ≤ T max , ∀ x0 ∈ Rn } .
Definition 12. [41, 42] For the parameter vector ρ of system (1) and a constant T c := T c (ρ) > 0, the origin of (1) is
said to be predefined-time stable if it is fixed-time stable and the settling-time function T : Rn → R is such that
T (x0 ) ≤ T c ,

∀x0 ∈ Rn .

T c is called a predefined time.
Remark 2. It would be desirable to choose T c = T c (ρ) not only as a bound of the settling-time function T c ∈ T , but
as the least upper bound, i.e., T c = min T = sup x0 ∈Rn T (x0 ). However, this selection requires complete knowledge
about the system, compromising its application to decentralized systems.
4

Let us recall some important results concerning predefined-time stability.
Theorem 2. [36, Theorem 1] Consider system
ẋ = −(α|x| p + β|x|q )k signx, x(0) = x0 ,

(3)

where x ∈ R, t ∈ [0, +∞). The parameters of the system are the real numbers α, β, p, q, k > 0 which satisfy the


constraints kp < 1 and kq > 1. Let ρ be the parameter vector ρ = α β p q k T ∈ R5 of (3). Then, the origin x = 0 of
system (3) is fixed-time stable and the settling time function satisfies T f = γ(ρ), where
  kq−1 

! 1−kp
Γ 1−kp
α q−p
q−p Γ q−p
,
(4)
γ(ρ) =
αk Γ(k)(q − p) β
R +∞
and Γ(·) is the Gamma function defined as Γ(z) = 0 e−t tz−1 dt [43, Chapter 1].
Using Theorem 2, one can obtain the following Lyapunov based characterization of predefined-time stable systems.
Theorem 3. [36, Theorem 3] Consider the nonlinear system
ẋ = f (x; ρ), x(0) = x0 ,

(5)

where x ∈ R is the system state, the vector ρ ∈ R stands for the system parameters which are assumed to be constant.
The function f : Rn → Rn is such that f (0; ρ) = 0. Assume there exists a continuous radially unbounded function
V : Rn → R such that:
n

b

V(0) = 0,
V(x) > 0,

∀x ∈ Rn \ {0},

and the derivative of V along the trajectories of (5) satisfies
D+ V(x) ≤ −

γ(ρ)
(αV(x) p + βV(x)q )k ,
Tc

∀x ∈ Rn \ {0}.

where α, β, p, q, k > 0, kp < 1, kq > 1 and γ is given in (4) and D+ V is the upper right-hand Dini derivative of V(x)
[44].
Then, the origin of (5) is predefined-time-stable with predefined time T c .
3. Main results
3.1. Problem statement
Consider a multi-agent system consisting of n agents with first-order dynamics given by
ẋi (t) = ui (t) + di (t),

i = 1, . . . , n,

where xi ∈ R and ui ∈ R are the system state and control input respectively. Term di represents external perturbations
and is assumed to be bounded by a known positive constant Li , i.e.
|di (t)| ≤ Li
The communication topology between agents is represented by the graph (which could be dynamic) Xσ(t) = hF , σi
where F = {X1 , . . . , Xm } is a collection of graphs having the same vertex set and σ : [0, ∞) → {1, . . . m} is the
switching signal determining the topology of the dynamic network at each instant of time.
The control objective is to design decentralized consensus protocols ui (i = 1, . . . , n), based on available local
information, such that the state of all the agents converges to an equilibrium, known as a consensus state x∗ , in a
predefined-time T c , regardless of the initial conditions xi (0) of the agents and the disturbance signals di (t) affecting
each agent, i.e.
x1 (t) = · · · = xn (t) = x∗ ,
∀t ≥ T c
5

Remark 3. If x∗ = n1

Pn

i=1 xi (0), the control objective is called as the predefined-time average consensus problem.

Remark 4. There exist methods for predefined-time consensus in the literature. However, they are based on timevarying gains [28–33], since the same gain must be applied to all agents, a common time-reference is needed along
the network. Moreover, this time-varying gain becomes singular at the predefined-time, see for instance [28] or as
in [29, 33] where it may produce a Zeno behavior (infinite switching in a finite interval). These drawbacks are not
present in our approach.
3.2. Robust predefined-time consensus algorithms
In this section, we propose and analyze two fixed-time consensus protocols, which have the following form
X
h
i
ai j (x j (t) − xi (t))
ui = κi (α|ei | p + β|ei |q )k + ζ signei , ei =

(6)

j∈Ni (Xσ(t) )

and
ui = κi

X

√

h
i
√
ai j (α|ei j | p + β|ei j |q )k + ζ signei j , ei j = ai j (x j (t) − xi (t))

(7)

j∈Ni (Xσ(t) )

where α, β, p, q, k > 0, kp < 1 and kq > 1. The constant ζ and κi will be designed later to guarantee the convergence
in a predefined-time T c , even in the presence of disturbances.
h
i
Remark 5. Let φ(·) = (α| · | p + β| · |q )k + ζ sign·. On the one hand, notice that the consensus protocol (6) is computationally simpler than consensus protocol (7) since at each time instance each agent only requires a single computation
of the nonlinear function φ(·) whereas for protocol (7) an agent requires to perform one computation of φ(·) for each
neighbor. On the other hand, we will show that (6) is a predefined-time consensus protocol for static networks and a
fixed-time consensus protocol for dynamic networks whereas control (7) is a predefined-time consensus protocol for
static and dynamic networks
3.2.1. Convergence analysis under consensus protocol (6)
Let e = [e1 · · · en ]T , where ei is given in (6), then e can be written as e = Q(Xσ(t) )x with x = [x1 · · · xn ]T . Notice
that, the dynamic of the network under the consensus algorithm (6) is given by
ẋ = −Φ(Q(Xσ(t) )x) + ∆(t),
where, for z = [z1 · · · zn ]T ∈ Rn , the function Φ : Rn → Rn is defined as
h
i


 κ1 (α|z1 | p + β|z1 |q )k + ζ signz1 


..
 ,
Φ(z) = 
.

h
i


κn (α|zn | p + β|zn |q )k + ζ signzn

(8)

(9)

and ∆(t) = [d1 (t) · · · dn (t)]T with k∆(t)k < L.
Let us now study the stability of the closed-loop system (8). First, using the homogeneity theory, let us derive
sufficient conditions for the design of control (6) such that the consensus is achieved in a fixed-time under switching
topologies.
Lemma 2. The vector field of (8) is homogeneous in the 0−limit with degree d0 = −1 < 0 and homogeneous in the
+∞-limit with degree d∞ = qk − 1 > 0 with respect to the standard dilation.
Proof. It follows straightforwardly from the definition of homogeneity in the bi-limit [22].
Theorem 4. Let F = {X1 , . . . , Xm } be a collection of connected graphs and let σ(t) : [0, ∞] → {1, . . . , m} be a
non-Zeno switching signal.
If
κζ ≥ L where κ = min κl
l∈{1,...,n}

then, protocol (6) guarantees the fixed-time consensus on switched dynamic network Xσ(t) under arbitrary switching
signals σ(t).
6

Proof. Since, according to Lemma 2 the vector field in (8) is homogeneous in the bi-limit then, according to Theorem 1, to show that fixed-time consensus is achieved it only remains to prove that (8) as well as
ẋ = −Φ0 (Q(Xσ(t) )x) + ∆(t)
and
ẋ = −Φ∞ (Q(Xσ(t) )x)
are asymptotically stable where, with d0 = −1 and d∞ = qk − 1,


 κ1 ζ signe1 



..
 + ∆(t)
−Φ0 (Q(Xσ(t) )x) + ∆(t) = lim λ−(d0 +1) (−Φ(Q(Xσ(t) )λx) + ∆(t)) = 
.
λ→0


κn ζ signen
and



 κ1 βk |e1 |qk signe1 


..
 .
−Φ∞ (Q(Xσ(t) )x) = lim λ−(d∞ +1) (−Φ(Q(Xσ(t) )λx) + ∆(t)) = 
.


λ→∞
k
qk
κn β |en | signen

To this aim, consider the (Lipschitz continuous) non-smooth Lyapunov function candidate
V(x) = max(x1 , · · · , xn ) − min(x1 , · · · , xn ),

(10)

which is differentiable almost everywhere and positive definite. Let xi = max(x1 , · · · , xn ) and x j = min(x1 , · · · , xn )
for a nonzero interval (τ, τ + ∆t ) then the time derivative of (10) along the trajectory of (8) is given by
D+ V(x) = κi (α|ei | p + β|ei |q )k signei − κ j (α|e j | p + β|e j |q )k signe j + κi ζ signei − κ j ζ signe j + di (t) − d j (t)
≤ κi (α|ei | p + β|ei |q )k signei − κ j (α|e j | p + β|e j |q )k signe j
Notice that, since xi = max(x1 , · · · , xn ) and x j = min(x1 , · · · , xn ) then ei ≤ 0 and e j ≥ 0, and unless consensus is
achieved, they cannot be both zero for a non zero interval, because since the graph is connected there is a path from
agent i to agent j such that there is a node i∗ satisfying xi = xi∗ but ei∗ < 0. Thus if ei (t) = 0 then ei (t+ ) < 0. A similar
argument follows to show that there is a node j∗ such that x j = x j∗ but e j∗ > 0 and thus if e j (t) = 0 then e j (t+ ) > 0.
Thus D+ V(x) < 0 almost everywhere and consensus is achieved.
Fixed-time convergence for dynamic networks, under arbitrary switching, follows from the stability theory of
switched systems [45, Theorem 2.1] by noticing that (10) is a common Lyapunov function for each subsystem evolving
with Xl connected.
Remark 6. Notice that Theorem 4 extends the results in [20], in which the fixed-time consensus problem was addressed using homogeneity for the case where k = 1 and no disturbances are present.
Using an appropriate Lyapunov function, let us derive sufficient conditions for the design of control (6) such that
the consensus is achieved in a predefined-time under a fixed topology.
Theorem 5. Let Xl be a connected graph and let
κi ≥

nγ(ρ)
and κζ ≥ L
λ2 (Q(Xl ))T c

where
κ = min κl
l∈{1,...,n}

and γ(ρ) is defined in Eq. (4), then, protocol (6) guarantees that consensus is achieved before a predefined-time T c .
7

Proof. Let δ = [δ1 , . . . , δn ]T be a disagreement variable such that x = α1 + δ where α is a consensus value, which is
unknown but constant. Consider the Lyapunov function candidate
V(δ) =

1p
λ2 (Q(Xl ))δT Q(Xl )δ,
n

By noticing that δ̇ = ẋ, then it follows that
p
p
λ2 (Q(Xl )) T
λ2 (Q(Xl )) T
+
δ Q(Xl )δ̇ = p
δ Q(Xl )(−Φ(Q(Xl )δ) + ∆(t)),
D V(δ) = p
T
n δ Q(Xl )δ
n δT Q(Xl )δ
Let v = Q(Xl )δ = (v1 , . . . , vn )T , therefore:


p

λ2 (Q(Xl )) 
1
δT Q(Xl )
+
T

D V(δ) =
v Φ(v) + p
∆(t)
− p
n
δT Q(Xl )δ
δT Q(Xl )δ


p
n
n
T
X
X

λ2 (Q(Xl )) 
1
ζ
δ
Q(X
)
l
p
q
k
− p
=
κi |vi |(α|vi | + β|vi | ) − p
κi |vi | + √
∆(t) (11)
n
vT δ
δT Q(Xl )δ i=1
δT Q(Xl )δ i=1
Using Lemma 7, the first term can be rewritten as:
n
n
X
X
1
1
n
κn
κi |vi |(α|vi | p + β|vi |q )k ≥ p
|vi |(α|vi | p + β|vi |q )k
p
δT Q(Xl )δ i=1 n
δT Q(Xl )δ i=1 n
!
!p
!q !k
1
κn
1
1
≥ p
kvk1 α kvk1 + β kvk1
n
n
δT Q(Xl )δ n
P
where κ = min{κ1 , . . . , κn } and kvk1 = ni=1 |vi | Moreover, using Lemma 8, one can obtain
p
kvk1 ≥ kvk2 = vT v = δT Q(Xl )2 δ

Expressing the disagreement variable δ as a linear combination of the eigenvectors of Q(Xl ), the term δT Q(Xl )2 δ can
be bounded as
δT Q(Xl )2 δ ≥ λ2 (Q(Xl ))δT Q(Xl )δ
Therefore, by Lemma 5:
n
p

n
X
1

δT Q(Xl )δ i=1 n

κi |vi |(α|vi | p + β|vi |q )k ≥ p

κn
δT Q(Xl )δ

p
V(αV p + βV q )k = κ λ2 (Q(Xl ))(αV p + βV q )k

(12)

Furthermore, from the last two terms of (11) the following is obtained:


n
T
 ζ X

v
−p
κi |vi | + p
∆(t) ≤ λ2 (Q(Xl )) −
κi |vi | +
∆(t)
kvk i=1
kvk
δT Q(Xl )δ i=1
δT Q(Xl )δ
ζ

n
X

vT

≤ λ2 (Q(Xl ))(−ζκ + L) ≤ 0
Therefore, the following inequality is obtained from (11), by combining (12) and (13):
D+ V(δ) ≤ −

κλ2 (Q(Xl ))
γ(ρ)
(αV p + βV q )k ≤ −
(αV p + βV q )k
n
Tc

Then, according to Theorem 3, protocol (6) guarantees that consensus is achieved before a predefined-time T c .

8

(13)

Figure 1: Static network X used for Example 1.

Example 1. Consider the multi-agent system (3.1) composed of n = 10 agents with external perturbation di (t) =
sin(40t + 0.1i). The communication topology, given in Figure 1 is undirected and static. The corresponding algebraic
connectivity is λ2 (Q(X)) = 0.27935. The initial conditions of the agents are randomly generated and are as follows:
x(0) = [134.51, 40.72, 214.15, 40.04, −241.50, −189.57, 181.35, −7.8517, 172.42, −145.29]T .
According to Theorem 5, protocol (6) with T c = 1, p = 1.5, q = 3.0, k = 0.5, α = 1, β = 2, κ = 178.88, ζ = 0.0177
guarantees that the consensus is achieved before a predefined-time T c under the graph topology X. Figure 2 shows
the corresponding result. For this experiment the settling time is of 0.095s.
Example 2. Consider the multi-agent system (3.1) composed of n = 10 agents with external perturbation di (t) =
sin(40t + 0.1i). The collection of communication topologies F = {X1 , . . . , X4 }, given in Figure 3a-3d are undirected.
The switched dynamic network evolves according to the switching signal σ(t) given in Figure 4 which satisfies the
minimum dwell time condition. The corresponding algebraic connectivity is
λ2 (Q(Xl )) ∈ {0.16548, 0.73648, 0.15776, 0.57104}.
The initial conditions of the agents are randomly generated and are as follows:
x(0) = [−210.02, 117.66, 161.32, −78.30, −181.93, 82.97, 165.22, 86.81, −180.27, −60.58]T .
According to Theorem 4, protocol (6) with p = 1.5, q = 3.0, k = 0.5, α = 1, β = 2,
{κ1 , . . . , κ4 } = {301.9585, 67.8472, 316.7348, 87.5037},
κ = 67.8472 and ζ = 0.0466 guarantees that the consensus is achieved in a fixed-time under the switched dynamic
network Xσ(t) . Figure 4 shows the corresponding result.
3.2.2. Convergence analysis under consensus protocol (7)
Similarly to the previous subsection, let us define x = [x1 · · · xn ]T . Notice that the dynamic of the network under
the consensus algorithm (7) is given by
ẋ = −D(Xσ(t) )Φ(D(Xσ(t) )T x) + ∆(t).

(14)

where, for z = [z1 · · · zn ]T ∈ Rn , the function Φ : Rn → Rn is defined as (9) and ∆(t) = [d1 (t) · · · dn (t)]T with
k∆(t)k < L.
9

200

150

100

50

x(t)
0

-50

-100

-150

-200

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

time (s)
Figure 2: Convergence of the consensus algorithm for Example 1.

Let us now study the stability of the closed-loop system (14). First, let us derive a useful lemma concerning average
consensus in the absence of disturbance. Then, using an appropriate Lyapunov function, let us derive sufficient
conditions for the design of control (7) such that the consensus is achieved in a predefined-time under switching
topologies
Lemma 3. Let Xσ(t) be a switching dynamic network composed of connected graphs and assume that under the
protocol (7) and in the absence of disturbance consensus is achieved at a time T c . Then, the consensus state is
x∗ = n1 1T x(t0 ), i.e. consensus to the average is achieved.
Proof. Let s x = 1T x be the sum of the agent states. In the absence of disturbances, i.e. if ∆(t) = 0, since 1T D(Xσ(t) ) =
0, then ṡ x = −1T D(Xσ(t) )Φ(D(Xσ(t) )T x) = 0. Thus, s x is constant during the evolution of the system, i.e. ∀t ≥ 0,
s x (t) = 1T x(t0 ) = s x (t0 ).
Therefore, s x (t) − s x (t0 ) = 1T x − 1T x(t0 ) = 0. Thus, if t ≥ T c , x1 = · · · = xn = x∗ . Thus, 1T x = nx∗ and
x∗ = n1 1T x(t0 ), i.e. consensus to the average is achieved.
Theorem 6. Let F = {X1 , . . . , Xm } be a collection of connected graphs and let σ(t) : [0, ∞] → {1, . . . , m} be a
non-Zeno switching signal. If
Mγ(ρ)
L
κi ≥ ∗
and ζ ≥ p ∗
λ2 T c
κ λ2
with
M = min |E(Xl )|, κ = max κl and λ∗2 = min λ2 (Q(Xl )),
l∈{1,...,m}

l∈{1,...,n}

l∈{1,...,m}

and γ(ρ) is defined in Eq. (4), then, protocol (7) guarantees that consensus is achieved before a predefined-time T c
on switched dynamic networks Xσ(t) under arbitrary switching signals σ(t). Moreover, in the absence of disturbances
consensus to the average is obtained.

10

(a) X1

(b) X2

(c) X3

(d) X4

Figure 3: Collection of communication topologies F = {X1 , . . . , X4 } used for the switched dynamic network in Example 2.

11

x(t)

σ(t)

time (s)
Figure 4: Convergence of the consensus algorithm for Example 2 and the corresponding switching signal.

Proof. Let δ = [δ1 , . . . , δn ]T be a disagreement variable such that x = α1 + δ where α is a consensus value, which is
unknown but constant. Consider the Lyapunov function candidate
1 q ∗ T
λ2 δ δ.
(15)
V=
M
To show that consensus is achieved on dynamic networks under arbitrary switchings, we will prove that (15) is a
common Lyapunov function for each subsystem of the switched nonlinear system (14) [45, Theorem 2.1]. To this
aim, assume that σ(t) = l for t ∈ [0, T c ]. By noticing that δ̇ = ẋ, then it follows that
p ∗
p ∗
λ2 T
λ
+
D V(x) = √
δ δ̇ = − √ 2 δT (D(Xl )Φ(D(Xl )T δ) + ∆(t)),
M δT δ
M δT δ
Let v = D(Xl )T δ = (v1 , . . . , vm )T , therefore:
p ∗
!
λ2
1 T
δT
+
D V(x) =
−
v Φ p (v) +
∆(t)
M
kδk
kδk

p ∗
m
m
T
X

λ2  1 X
ζ
δ
p
q
k
−
=
κi |vi |(α|vi | + β|vi | ) −
κi |vi | +
∆(t) .
M
kδk
kδk
kδk
i=1

i=1

Using Lemma 7, the first term can be rewritten as:
m
m
κm X 1
m X1
κi |vi |(α|vi | p + β|vi |q )k ≥
|vi |(α|vi | p + β|vi |q )k
kδk i=1 m
kδk i=1 m
!
!p
!q !k
κm 1
1
1
≥
kvk1 α
kvk1 + β
kvk1
,
kδk m
m
m

12

(16)

where κ = min{κ1 , . . . , κn } and kvk1 =

Pm

i=1 |vi |. Moreover, it follows from Lemma 8 that

kvk1 ≥ kvk2 = vT v =

p

δT Q(Xl )δ ≥

q
λ∗2 kδk.

Therefore, by Lemma 5:
m

q
m X1
κm
κi |vi |(α|vi | p + β|vi |q )k ≥
V(αV p + βV q )k ≥ κ λ∗2 (αV p + βV q )k .
kδk i=1 m
kδk

(17)

Furthermore, from the last two terms of (16), the following is obtained:
m

−

q
δT
κζ
ζ X
κi |vi | +
∆(t) ≤ −
kvk1 + k∆(t)k ≤ −ζκ λ∗2 + L ≤ 0
kδk i=1
kδk
kδk

(18)

Therefore, the following inequality is obtained from (16), by combining (17) and (18):
D+ V(x) ≤ −

κλ∗2
γ(ρ)
(αV p + βV q )k ≤ −
(αV p + βV q )k
M
Tc

Then, according to Theorem 3, protocol (7) guarantees that the consensus is achieved before a predefined-time T c .
Moreover, since the above argument holds for any Xl ∈ F , then protocol (7) guarantees that the consensus is achieved
before a predefined-time T c , on switching dynamic networks under arbitrary switching. Furthermore, it follows from
Lemma 3 that if ∆(t) = 0 then the consensus state x∗ is the average of the initial values of the agents.
Remark 7. Note that existing consensus protocols such as [10, 11, 24, 27] are subsumed in our approach. In fact,
the consensus protocols in [10, 11, 27] are derived from (6) and (7) by taking k = 1, p = 1 − s and q = 1 + s,
s ∈ (0, 1). Moreover, we show that our analysis provides less conservative estimate than the ones proposed, for
instance in [24]. Thus, our paper contributes to a broader class of consensus protocols with fixed-time convergence.
Moreover, unlike [10, 11, 24], in our method the convergence time is determined a priori.
Remark 8. Notice that, the Lyapunov functions in our proofs are different than in previous methods, such as [10,
11, 18, 24]. Consequently, the proposed predefined-time consensus results do not follow trivially from the existing
literature. An essential part of our approach is the convexity result provided in Lemma 6.
Example 3. Consider the multi-agent system (3.1) composed of n = 10 agents with external perturbation di (t) =
sin(40t + 0.1i). The collection of communication topologies, given in Figure 3a-3d are undirected. The switched
dynamic network evolves according to the switching signal σ(t) given in Figure 5 which satisfies the minimum dwell
time condition. The corresponding algebraic connectivity is λ2 (Q(Xl )) ∈ {0.16548, 0.73648, 0.15776, 0.57104}. The
initial conditions of the agents are randomly generated and are as follows:
x(t0 ) = [72.31167.49 − 226.30, 45.68, 246.20, −121.78, −196.90, −128.59, −88.57, 29.05]T .
According to Theorem 6, protocol (7) with T c = 1, p = 1.5, q = 3.0, k = 0.5, α = 1, β = 2, {κ1 , . . . , κ4 } =
{241.5668, 54.2777, 253.3879, 70.0029}, κ = 54.2777 and ζ = 0.3693 guarantees that the consensus is achieved
in a predefined-time T c under the switched dynamic network Xσ(t) . Figure 4 shows the corresponding result. For this
example, the settling time is 0.187s.
4. Comparison and Discussion
In this section, we present a discussion on the contribution with respect to the state of the art in the literature. Our
main argument is that our approach subsumes existing methods, as it allows a broader range of parameter selection,
which provides extra degrees of freedom to design more efficient consensus protocols (for instance, allowing to reduce
the slack between the true convergence time and the predefined-convergence time).
13

x(t)
σ(t)

time (s)
Figure 5: Convergence of the consensus algorithm for Example 3 and the corresponding switching signal.

Table 1 and Table 2 present a comparison between the current proposal and some of the existing fixed-time
consensus protocols for the leaderless consensus problem when using autonomous protocols. Notice that contrary
to the results existing in the literature, our approach does not present any restriction on the parameters p, k and q
(additional to those given in Theorem 2 for (3)). Regarding consensus protocols in the form (6), it is illustrated in
Table 1 that for switched dynamic networks under arbitrary switching among connected graphs, our approach provides
fixed-time convergence (no upper bound convergence is provided in this case); whereas for static networks, it provides
predefined-time convergence, even if the agents are affected by disturbances. Regarding consensus protocols in the
form (7), see Table 2, for switched dynamic networks among connected graphs, our approach provides predefinedtime convergence for agents with single integrator dynamics affected by disturbances.
Regarding the estimation of the upper bound of the convergence time, the results in [24] are based on the estimate
provided in [21], which has been shown in [36] to be very conservative. Moreover, to address the robust consensus
problem in [12] an estimate improving the result in [21] was presented for the case where k = 1, q = m̂n̂ , p = q̂p̂
satisfying
q̂(m̂ − n̂)
0<
≤ 1.
n̂(q̂ − p̂)
However, such estimation is also too conservative. The proposed approach is based on the predefined-time stability
results provided in [36] which extend the results in [26] (in [26], they were limited to k = 1, p = 1 − s, q = 1 + s,
0 < s < 1) by showing that (3) is predefined-time stable, with T c as the least upper bound of the convergence-time, for
every p, q, k > 0 satisfying kp < 1 and kq > 1. Thus, our approach has the same upper estimate as [11] for the case
when k = 1, p = 1− s, q = 1+ s, 0 < s < 1 but provides a broader class of consensus protocols than [10, 11, 20, 24, 27]
by having extra degrees of freedom on p, q, k as long as the conditions p, q, k > 0, kp < 1 and kq > 1 are met. As we
show below, such broader parameter selection allows to design consensus protocols with improved estimates of the
convergence time bound.
Regarding consensus protocols in the form (6), in addition to the above mentioned advantages, in this paper we
show, using an homogeneity approach, that (6) is a fixed-time consensus algorithm for switched dynamic networks under arbitrarily switching among connected graphs, which extends the algorithm in [20] to allow k , 1 and disturbances
14

Reference
[20]

Additional restrictions on k,
p, q
k=1

[24]

k = 1, p = 1 − s, q = 1 + s,
0<s<1
k=1

[12]

k=1

[12]

k = 1, q = m̂n̂ , p = q̂p̂ satisfy-

[10]

m̂−n̂)
ing 0 < q̂(
n̂(q̂− p̂) ≤ 1.

Ours

k = 1, p = 1 − s, q = 1 + s,
0<s<1
None

Ours

None

[11]

Network
Type
Jointly
Connected
Static Connected
Static Connected
Static Connected
Static Connected

Robustness

Static Connected
Dynamic
Connected
Static Connected

Yes

from [26], Same

Yes

None provided. Based
on homogeneity.
from [36]

No
No
No
Yes
Yes

Yes

Upper bound Estimate/
Comparison to ours
None provided. Based
on homogeneity.
from [26], Same
from [21], More conservative
from [21], More Conservative
from [12], More Conservative

Table 1: Comparison of papers presenting autonomous protocols of the form (6) for the consensus problem.

affecting the agents.
4.1. Comparison
Notice from Table 1 and Table 2 that previous fixed-time consensus protocols were limited by a parameter selection
with k = 1 and most of them allow only p = 1 − s, q = 1 + s, 0 < s < 1. In this subsection, we present numerical
simulations and comparisons to illustrate that by having greater flexibility in the protocol design, one can obtain
protocols with improved estimates of the convergence time bound.
In our opinion, relevant robust consensus algorithms for agents with single integrator dynamics are given in [11]
for the leaderless consensus problem. We show that, by selecting parameters different from the ones proposed in [11],
we obtain robust consensus protocols where the slack between the real convergence time and the estimated upper
bound is lower.
This simulation results are presented in Example 4 for protocol (6) and in Example 5 for protocol (7). The first
case, p = 0.1, q = 0.9, k = 1 corresponds to a selection allowed by [11] for which the gains are computed from [11] to
have an upper bound for the convergence time T c = 1s. For the second case, our aim is to show that by varying k , 1,
the slack between the true convergence time and the upper bound of the convergence time T c = 1 can be reduced.
Moreover, we show that such slack can be further reduced by selecting values for p and q that are not symmetrical
w.r.t. one, i.e. values that are outside the parameter selection allowed in [11]. Thus, our approach subsumes existing
fixed-time consensus protocols and provides more flexibility in the parameter selection to improve, for instance, the
convergence.
Example 4. Consider the multi-agent system (3.1) composed of n = 10 agents with external perturbation di (t) =
sin(40t + 0.1i).. The collection of communication topologies, given in Figure 3a-3d are undirected. The switched
dynamic network evolves according to the switching signal σ(t) given in Figure 6 which satisfies the minimum dwell
time condition. The initial conditions of the agents are randomly generated and are as follows:
x(t0 ) = [−210.02, 117.66, 161.32, −78.30, −181.93, 82.97, 165.22, 86.81, −180.27, −60.58]T .
Protocol (6) with, α = 1, β = 2, {κ1 , . . . , κ4 } = {301.9585, 67.8472, 316.7348, 87.5037}, κ = 67.8472 and ζ = 0.0466
and the parameter selection described in Table 3 achieves consensus in fixed-time. Figure 6 and Table 3 show the
corresponding result.
15

Reference

Additional restrictions on k,
p, q

Network
Type

Robustness

[25]

k=1

No

[25]

[24]

k = 1, p = 1 − s, q = 1 + s,
0<s<1
k = 1, p = 1 − s, q = 1 + s,
0<s<1
k=1

[12]

k=1

[12]

k = 1, q = m̂n̂ , p = q̂p̂ satisfy-

[46]

m̂−n̂)
ing 0 < q̂(
n̂(q̂− p̂) ≤ 1.
k=1

Static Connected
Static Connected
Static Connected
Dynamic
Connected
Static Connected
Static Connected
Static Connected
Dynamic
Connected
Dynamic
Connected
Dynamic
Connected
Dynamic
Connected

Yes

[10]

[27]
[47]
[11]
Ours

k = 1, p = 1 − s, q = 1 + s,
0<s<1
k = 1, p = 1 − s, q = 1 + s,
0<s<1
k = 1, p = 1 − s, q = 1 + s,
0<s<1
None

No

Upper bound Estimate/ Comparison
to ours
from [21], More
Conservative
from [26], Same

No

from [26], Same

No

from [21], More
Conservative
from [21], More
Conservative
from [12], More
Conservative

Yes
Yes

No

from [21], More
Conservative
from [26], Same

No

from [26], Same

Yes

from [26], Same

Yes

from [36]

Table 2: Comparison of papers presenting autonomous protocols of the form (7) for the consensus problem.

Parameters
p = 0.1, q = 0.9, k = 1.0
p = 0.1, q = 1.9, k = 0.75
p = 1.5, q = 12, k = 0.1

Convergence time for protocol (6)
0.138s
0.185s
0.258s

Convergence time for protocol (7)
0.105s
0.127s
0.212s

Table 3: Comparison of convergence time for protocol (6) and protocol (6) under different parameter selection.

16

σ(t)

δT δ

δT δ

p = 1.5, q = 12.0, k = 0.1
p = 0.1, q = 1.9, k = 0.75
p = 0.1, q = 1.9, k = 1.0

time (s)
Figure 6: Convergence of the consensus algorithm for Example 4 and the corresponding switching signal.

Example 5. Consider the multi-agent system (3.1) composed of n = 10 agents with external perturbation di (t) =
sin(40t + 0.1i).. The collection of communication topologies, given in Figure 3a-3d are undirected. The switched
dynamic network evolves according to the switching signal σ(t) given in Figure 6 which satisfies the minimum dwell
time condition. The initial conditions of the agents are randomly generated and are as follows:
x(t0 ) = [72.31167.49 − 226.30, 45.68, 246.20, −121.78, −196.90, −128.59, −88.57, 29.05]T .
Protocol (7) with T c = 1, α = 1, β = 2, {κ1 , . . . , κ4 } = {241.5668, 54.2777, 253.3879, 70.0029}, κ = 54.2777 and
ζ = 0.3693 and the parameter selection described in Table 3 achieves consensus in fixed-time. Figure 7 and Table 3
show the corresponding result.
5. Conclusion
In this paper, we presented two robust consensus protocols for perturbed agents with single integrator dynamics.
The first one is the one that is computationally simpler; we show that it presents predefined-time convergence under
static networks and fixed-time convergence under switched dynamic networks. The second one, in the absence of disturbances, converges to a consensus state that is the average of the agents’ initial conditions. This algorithm is shown
to be a robust predefined-time consensus algorithm for static and dynamic networks arbitrarily switching among connected graphs. The effectiveness of our approach is shown in simulations where we show that our approach subsumes
existing fixed-time consensus protocols and provides more flexibility in the parameter selection, for instance, to improve the convergence. Numerical results were given to illustrate the effectiveness and advantages of the proposed
approach and qualitative comparisons were presented to expose the contribution.

17

σ(t)

δT δ

δT δ

p = 1.5, q = 12.0, k = 0.1
p = 0.1, q = 1.9, k = 0.75
p = 0.1, q = 1.9, k = 1.0

time (s)
Figure 7: Convergence of the consensus algorithm for Example 5 and switching signal

Appendix A. Some useful inequalities
Lemma 4. [48, Formula 5] Let f (x) be a real-valued function and λ1 , . . . , λn ∈ [0, 1] satisfy
f (x) is a convex function if and only if it satisfies Jensen’s inequality:
 n

n
X
 X
f  λi xi  ≤
λi f (xi )
i=1

i=1 λi = 1. Therefore,

Pn

(A.1)

i=1

Lemma 5. Let
f (x) = x(αx p + βxq )k
for α, β, p, q, k > 0, pk < 1 and qk > 1. Then, f (x) is monotonically increasing for x > 0.
Proof. The direct differentiation of f (x) results in:
d
f (x) = (αx p + βxq )k + kx(αx p + βxq )k−1 (αpx p−1 + βxq−1 ) ≥ 0
dx
for x > 0. Therefore, f (x) is monotonically increasing.
Lemma 6. Let the function f (x; k) defined as in (A.2). Then, f (x) is a convex function for x > 0.
Proof. Taking the second derivative of f (x) with respect to x leads to:


d2
k
f (x) = (αx p + βxq )k−2 α2 p(kp + 1)x2p + ab(2kpq + p + q + (q − p)2 )x p+q + b2 q(kq + 1)x2q .
2
x
dx
2

d
Since dx
2 f (x) > 0 ∀x > 0, f (x) is convex for x > 0.

18

(A.2)

Lemma 7. Let n ∈ N. If a = (a1 , . . . , an ) is a sequence of positive numbers, then the following inequality is satisfied
 n    n p
 n q k
n
 1 X  
1X
 1 X    1 X 
q k
p
ai (αai + βai ) ≥ 
ai  α 
ai  + β 
ai  
n i=1
n i=1
n i=1
n i=1  

(A.3)

for α, β, p, q, k > 0, pk < 1 and qk > 1.
P
Proof. First, recognize n1 ni=1 ai as a convex combination of the elements of the sequence a. Moreover, since f (x; k)
 P

defined in (A.2) is convex due to Lemma 6, then it satisfies Jensen’s inequality (A.1). Evaluating f 1n ni=1 ai in
Jensen’s inequality results in expression (A.3).
Lemma 8. [49, Property A.6.1] Let z = [z1 · · · zn ]T ∈ Rn and
1
 n
X p  p

kzk p =  |zi | 
i=1

then,
kzkl ≤ kzkr
for l ≥ r.
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