Enhancing the settling time estimation of a class of fixed-time stable systems 1
R. Aldana-L´opeza, David G´omez–Guti´erreza,b,∗, E. Jim´enez-Rodr´ıguezc, J. D. S´anchez-Torresd, M. Defoorte
aMulti-agent autonomous systems lab, Intel Labs, Intel Tecnolog´ıa de M´exico, Av. del Bosque 1001, Colonia El Baj´ıo, Zapopan, 45019, Jalisco,
M´exico.
bTecnologico de Monterrey, Escuela de Ingenier´ıa y Ciencias, Av. General Ram´on Corona 2514, Zapopan, 45201, Jalisco, M´exico.
cCINVESTAV, Unidad Guadalajara, Av. del Bosque 1145, colonia el Baj´ıo, Zapopan , 45019, Jalisco, M´exico.
dResearch Laboratory on Optimal Design, Devices and Advanced Materials -OPTIMA-, Department of Mathematics and Physics, ITESO,
Perif´erico Sur Manuel G´omez Mor´ın 8585 C.P. 45604, Tlaquepaque, Jalisco, M´exico.
eLAMIH, CNRS UMR 8201, Univ. Valenciennes, Valenciennes 59313, France.
Abstract
This paper deals with the convergence time analysis of a class of fixed-time stable systems with the aim to provide a
new non-conservative upper bound for its settling time. Our contribution is fourfold. First, we revisit the well-known
class of fixed-time stable systems, given in [1], while showing the conservatism of the classical upper estimate of
the settling time. Second, we provide the smallest constant that uniformly upper bounds the settling time of any
trajectory of the system under consideration. Third, introducing a slight modification of the previous class of fixed-
time systems, we propose a new predefined-time convergent algorithm where the least upper bound of the settling
time is set a priori as a parameter of the system. At last, predefined-time controllers for first order and second order
systems are introduced. Some simulation results highlight the performance of the proposed scheme in terms of settling
time estimation compared to existing methods.
Keywords: Predefined-time stability, Finite-time stability, Fixed-time stability, Lyapunov analysis.
1. Introduction
Convergence time is an important performance specification for a controlled system from a practical point of view
[2, 3]. Indeed, the design of controllers which guarantee predefined-time stability, instead of asymptotic stability, is
one of the desired objectives, which appear in many applications such as missile guidance [4], hybrid formation flying
[5], group consensus [6], online differentiators [7–9], state observers [2], etc. Furthermore, in the case of switching
systems, it is frequently required that the observer (or controller) achieves the stability of the observation error (or
tracking error) before the next switching [10, 11].
Lack of uniform boundedness of the settling-time function regardless of the initial conditions causes several re-
strictions to the practical application of finite time observer/controller [12–14]. These restrictions can be relaxed
using the fixed-time stability concept. It is an extension of global finite-time stability, and guarantees the convergence
(settling) time to be globally uniformly bounded, i.e., the bound does not depend on the initial state of the system
[1, 4, 15]. To this end, the class of systems
˙x = −(α|x|p + β|x|q)k signx, x(0) = x0,
(1)
1This is the preprint version of the accepted Manuscript: R. Aldana-L´opez, D. G´omez-Guti´errez, E. Jim´enez-Rodr´ıguez, J. D. S´anchez-Torres
and Michael Defoort, “Enhancing the settling time estimation of a class of fixed-time stable systems”, International Journal of Robust and Nonlinear
Control, 2019, ISSN: 1099-1239. DOI. 10.1002/rnc.4600. Please cite the publisher’s version. For the publisher’s version and full citation details
see: http://dx.doi.org/10.1002/rnc.4600
∗Corresponding Author.
Email addresses: rodrigo.aldana.lopez@gmail.com (R. Aldana-L´opez), David.Gomez.G@ieee.org (David G´omez–Guti´errez),
ejimenezr@gdl.cinvestav.mx (E. Jim´enez-Rodr´ıguez), dsanchez@iteso.mx (J. D. S´anchez-Torres),
michael.defoort@univ-valenciennes.fr (M. Defoort)
Preprint submitted to ArXiv
May 16, 2019
arXiv:1809.07012v2  [math.OC]  15 May 2019
where x is a scalar state variable and the real numbers α, β, p, q, k > 0 are system parameters which satisfy the
constraints kp < 1, and kq > 1, was proposed in [1, 15] and has been extensively used. Indeed, it represents a
wide class of systems which present the fixed-time stability property through homogeneity and Lyapunov analysis
frameworks. However, it is still difficult to derive a relatively simple relationship between the system parameters and
the upper bound of the settling time [16, 17]. This yields some difficulties in the tuning of the system parameters to
achieve a prescribed-time stabilization (see for instance [7]).
The computation of the least upper bound of the settling time is usually not an easy task. Therefore, it is common
to propose an upper bound of the settling time as an attempt to estimate the least upper bound. For instance, in [1], it
is shown that for system (1), the settling-time function T(x0) is bounded as
T(x0) ≤
1
αk(1 −pk) +
1
βk(qk −1) = Tmax,
∀x0 ∈R.
(2)
However, this bound significantly overestimates the least upper one. This overestimation can lead to restrictions
for the practical implementation of prescribed-time observer/controller. In this case, the gains will be over-tuned to
achieve a prescribed-time stabilization. It may lead to poor performances in terms of control magnitude or robustness
against measurement noise for instance.
Considering the extensive use of the class of systems represented by (1) and the overestimation exhibited by Tmax
in (2), this paper addresses the computation of the least upper bound of the settling-time function for this system
and the derivation of a new predefined-time convergent algorithm where the least upper bound of the settling time
is set a priori as a parameter of the system. Based on this result, new predefined-time controllers for first order and
second order systems are introduced to enhance the settling time estimation. In contrast to the common approaches of
homogeneity and Lyapunov analysis, the results in this paper are derived using the well-known geometric conditions
proposed in [12]. The overall contribution of this paper is divided into the following four main results:
1. The well-known class of fixed-time stable systems (1) is revisited. For this class of systems, the least upper
bound of the settling-time function,
γ =
Γ

mp

Γ

mq

αkΓ(k)(q −p)
 α
β
!mp
,
with mp = 1−kp
q−p and mq = kq−1
q−p , is found.
2. The following new predefined-time convergent system, where the least upper bound of the settling time Tc is
set a priori as a parameter of the system, is proposed:
˙x = −γ
Tc
(α|x|p + β|x|q)k signx, x(0) = x0,
(3)
where x is a scalar state variable, real numbers α, β, p, q, k > 0 are system parameters which satisfy the con-
straints kp < 1 and kq > 1 and Tc > 0. Notice that the only difference between (1) and the modified system
(3) is the constant gain γ/Tc. This slight change in the original system represents a considerable improvement
in its properties, since the tunable parameter Tc is directly the least upper bound of the convergence time. As a
consequence of this desirable feature, we say that the origin of system (3) is predefined-time stable with (strong)
predefined-time Tc, a notion formally defined in Section 2.
3. The bound given in the formula (2) for system (3) is shown to be a conservative estimation of the settling time
T f = supx0∈R T(x0) = Tc. Moreover, letting α = ϱ and β = 1
ϱ, it is shown that even if the least upper bound of
the convergence time is Tc, the upper estimate (2), given in [1] goes to infinity as ϱ →+∞and as ϱ →0.
4. New predefined-time controllers for first order and second order scalar systems with matched bounded pertur-
bations are introduced to enhance the settling time estimation.
The rest of the manuscript is organized as follows. In Section 2, we introduce the preliminaries on finite-time,
fixed-time and predefined-time stability. In Section 3, we present the main result on the least upper bound for the
settling time and propose a new strongly predefined-time convergent algorithm where the least upper bound of the
2
settling time is set a priori as a parameter of the system. We also present the analysis of how conservative the bound
provided in [1] may result and show some numerical results. In Section 4, we apply the previous result to derive
new predefined-time controllers for first and second order systems. Finally, in Section 5, we present some concluding
remarks.
2. Preliminaries and Definitions
Consider the nonlinear system
˙x = f(x; ρ), x(0) = x0,
(4)
where x ∈Rn is the system state, the vector ρ ∈Rb stands for the system (4) parameters which are assumed to be
constant, i.e., ˙ρ = 0. The function f : Rn →Rn is assumed to be nonlinear and continuous, and the origin is assumed
to be an equilibrium point of system (4), so f(0; ρ) = 0.
Let us first recall some useful definitions and lemma on finite-time, fixed-time and predefined-time stability.
Definition 1. (Lyapunov stability [18, Definition 4.1]) The origin of system (4) is said to be Lyapunov stable if for all
ϵ > 0, there is δ := δ(ϵ) > 0 such that for all ||x0|| < δ, any solution x(t, x0) of (4) exists for all t ≥0, and ||x(t, x0)|| < ϵ
for all t ≥0.
Definition 2. (Finite-time stability [13]) The origin of (4) is said to be globally finite-time stable if it is Lyapunov
stable, and for any x0 ∈Rn, there exists 0 ≤T < +∞such that the solution x(t, x0) = 0 for all t ≥T. The function
T(x0) = inf{T : x(t, x0) = 0, ∀t ≥T} is called the settling-time function.
Lemma 1. (Finite-time stability characterization for scalar systems [12, Fact 1]) Let n = 1 in system (4) (scalar
system). The origin is globally finite-time stable if and only if for all x ∈R \ {0}
(i) x f(x; ρ) < 0, and
(ii)
R 0
x
dz
f(z;ρ) < +∞.
Remark 1. A proof of Lemma 1 shall not be given here, but can be found in [19, Lemma 3.1]. Nevertheless, intuitively,
condition (i) implies Lyapunov stability. Moreover, under the conditions of Lemma 1, note that the settling time
function is T(x0) =
R T(x0)
0
dt. Since first-order systems do not oscillate, the solution x(·, x0) : [0, T(x0)) →[x0, 0) of
system (4) as a function of t defines a bijection. Using it as a variable change, the above integral equals (note that
1
f(x;ρ) is defined for all x ∈R \ {0} from condition (i)
T(x0) =
Z T(x0)
0
dt =
Z 0
x0
dx
f(x; ρ).
(5)
Thus, condition (ii) of Lemma 1 refers to the settling-time function being finite.
Definition 3. (Fixed-time stability [1]) The origin is said to be a fixed-time stable equilibrium of (4) if it is globally
finite-time-stable and the settling time function T(x0) is bounded on Rn, i.e. ∃Tmax > 0 : ∀x0 ∈Rn : T(x0) ≤Tmax.
Remark 2. Let the origin x = 0 of system (4) be fixed-time stable. Notice that there are multiple upper bounds of
the settling-time function Tmax; for instance, if T(x0) ≤Tmax, also T(x0) ≤λTmax with λ ≥1. However, from this
boundedness condition, the least upper bound of the settling-time function supx0∈Rn T(x0) exists.
Remark 3. It has been shown that fixed-time stability is guaranteed if the vector field of the system is homogeneous
in the bi-limit, a concept defined in [15, 20]. However, in these cases, the upper bound for the settling time is usually
not obtained. To differentiate this case to one where a settling time bound Tc is set in advance as a function of system
parameters ρ, i.e. Tc = Tc(ρ), we introduce the concept of predefined-time stability. A strong notion of this class of
stability is given when supx0∈Rn T(x0) = Tc, i.e., Tc is the least upper bound for the settling time.
3
Definition 4. (Predefined-time stability [21]) For the parameter vector ρ of system (4) and an arbitrarily selected
constant Tc := Tc(ρ) > 0, the origin of (4) 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) ≤Tc,
∀x0 ∈Rn.
If this is the case, Tc is called a predefined time. Moreover, if the settling-time function is such that supx0∈Rn T(x0) = Tc,
then Tc is called the strong predefined time.
Remark 4. The stability property, of any kind, refers to equilibrium points of a system. However, since this study only
focuses on the global stability of the origin of the system under consideration, it may be referred hereafter, without
ambiguity, to the stability of the system in the respective sense (asymptotic, fixed-time or predefined-time).
3. On the least upper bound for the settling time of a class of fixed-time stable systems
3.1. Least upper bound of the settling-time function of system (1)
Let us first revisit the well-known class of fixed-time stable systems. In the following theorem, based on an
appropriate use of the Gamma function, the least upper bound of the settling-time function of system (1) is provided.
Theorem 1. Let
γ =
Γ

mp

Γ

mq

αkΓ(k)(q −p)
 α
β
!mp
,
(6)
where Γ(·) is the Gamma function defined as Γ(z) =
R +∞
0
e−ttz−1dt [22, Chapter 1], and mp = 1−kp
q−p and mq = kq−1
q−p
are positive parameters. The origin x = 0 of system (1) is fixed-time stable and the settling time function satisfies
supx0∈Rn T(x0) = γ.
Proof. Note that for system (1), the field is f(x; ρ) = −(α|x|p + β|x|q)k signx, where the parameter vector is ρ =
α β p q kT ∈R5. Furthermore, the product x f(x; ρ) = −(α|x|p + β|x|q)k |x| < 0 for all x ∈R \ {0}. Thus, V(x) = 1
2 x2
is a radially unbounded Lyapunov function for system (1), so its origin x = 0 is Lyapunov stable [18, Theorem 4.2].
Now, let x0 ∈R \ {0} (if x0 = 0, then x(t, 0) = 0 is the unique solution of (1) and T(0) = 0). From (5), the settling
time function is
T(x0) =
Z 0
x0
dx
f(x; ρ)
=
Z x0
0
signxdx
(α |x|p + β |x|q)k ,
z = |x|
=
Z |x0|
0
dz
(αzp + βzq)k .
Since the integrand
1
(αzp+βzq)k is positive for z ∈(0, |x0|), the settling time function is increasing with respect to |x0|.
Hence, the least upper bound of T(x0) (in the extended real numbers set, since we do not know yet if it is finite or not)
is obtained using Proposition 1, in the Appendix, as
sup
x0∈R
T(x0) =
lim
|x0|→+∞T(x0) =
Z +∞
0
dz
(αzp + βzq)k = γ,
with γ < +∞as in (6). Using Lemma 1 and by the definitions of finite-time and fixed-time stability, the origin x = 0
of system (1) is fixed-time stable and T f = γ, which completes the proof.
4
3.2. A class of predefined-time stable systems
Now, the result presented in Theorem 1 is used to derive a Lyapunov-like condition for characterizing predefined-
time stability of a system.
Theorem 2. (A Lyapunov characterization for predefined-time stable systems) If there exists a continuous positive
definite radially unbounded function V : Rn →R such that any solution x(t, x0) of (4) satisfies
˙V(x) ≤−γ
Tc
(αV(x)p + βV(x)q)k ,
∀x ∈Rn \ {0}.
(7)
where α, β, p, q, k > 0, kp < 1, kq > 1 and γ is given in (6).
Then, the origin of (4) is predefined-time stable and Tc is a predefined time. If in addition, the equality holds in (7),
then Tc is the strong predefined-time.
Proof. The negative definiteness of the time derivative of the function V implies Lyapunov stability of the origin
of (4). Now, suppose that there exists a function w(t) ≥0 that satisfies
˙w = −(ˆαwp + ˆβwq)k,
where ˆα = α
 γ(ρ)
Tc
 1
k and ˆβ = β
 γ(ρ)
Tc
 1
k , and V(x0) ≤w(0). Hence, by Theorem 1, w(t) will converge to the origin in a
strong predefined time
Γ

mp

Γ

mq

ˆαkΓ(k)(q −p)
 ˆα
ˆβ
!mp
= Tc
γ(ρ)
Γ

mp

Γ

mq

αkΓ(k)(q −p)
 α
β
!mp
= Tc
which is directly a tunable parameter of the system. Furthermore, by the comparison lemma [18, Lemma 3.4], it
follows that V(x(t)) ≤w(t), with equality only if (7) is an equality. Consequently, the origin of system (4) is predefined-
time stable with predefined time Tc. Moreover, if (7) is an equality, then supx0∈Rn T(x0) = Tc, i.e., Tc is the strong
predefined time.
Example 1. Consider system (3) and the continuous positive definite radially unbounded Lyapunov candidate function
V(x) = |x| for this system. The derivative of V(x) along the trajectories of system (3) is
˙V(x) = −signx γ
Tc
(α |x|p + β |x|q)k signx = −γ
Tc
(αV(x)p + βV(x)q)k .
Hence, by Theorem 2, the origin of system (3) is predefined-time stable with strong predefined time Tc.
To illustrate the above, some numerical simulations of system (3) are conducted, setting the parameters to α = 4,
β = 1
4, Tc = 1, p = 0.5, q = 3, k = 1.5. The simulations are conducted for several initial conditions x0, as presented
in Figure 1. It can be seen that supx0∈R T(x0) = Tc = 1 as stated above.
Remark 5. In [23], the least upper estimation of the settling time of (1) was addressed for the case where k = 1,
p = 1 −s, q = 1 + s, with 0 < s < 1, where γ(ρ) reduces to γ(ρ) =
Γ( 1
2 )2
2s √αβ =
π
2s √αβ. In [24], it was shown that,
in the case α = β =
π
2sTc , the least upper bound of the settling time is Tc. Thus, Theorem 2 is a generalization
of the results presented in [23] and [24]. Since only for the case where k = 1, p = 1 −s and q = 1 + s, with
0 < s < 1, a non-conservative upper bound estimate is provided, many applications, for instance fixed-time consensus
protocols [25–27], have been restricted to this case.
Given the relevance of fixed-time stability argumented in the introduction, inequality (7) is a result of paramount
importance, since as we show in the following, the upper estimate (2) is often too conservative. Thus, applications
based on the upper estimate of the settling time (2) presented in [1, Lemma 1] are often over-engineered.
5
time (t)
x(t, x0)
Tc = 1
Tmax(4) = 4.4331
x0 = 0.1
x0 = 1e20
x0 = 1
Figure 1: Trajectories of (3) for different initial conditions with α = 4, β = 1
4, Tc = 1, p = 0.5, q = 3, k = 1.5. The upper estimate in (9) is
Tmax(4) = 4.4331s. The least upper estimate is Tc = 1s.
6
3.3. Settling time bound analysis and comparison
Consider the predefined-time stable system (3), with strong predefined-time Tc. From (2), calculated in [1], an
upper bound for the settling time T(x0) is
T(x0) ≤Tc
γ(ρ)
 
1
αk(1 −pk) +
1
βk(qk −1)
!
,
∀x0 ∈R.
(8)
Let ϱ > 0, α = ϱ and β = 1
ϱ. Assuming that p, q and k remain constant, and noticing that γ is a function of ϱ, it
can be seen that varying ϱ the least upper bound of the settling time remains constant and equal to Tc. However, the
bound (8) becomes
T(x0) ≤Tmax(ϱ) := Tc
K
 
1
ϱ2mp(1 −pk) + ϱ2(k−2mp)
(qk −1)
!
,
(9)
where K =
Γ(mp)Γ(mq)
Γ(k)(q−p) . It is easy to see that
lim
ϱ→0 Tmax(ϱ) = lim
ϱ→∞Tmax(ϱ) = +∞,
i.e., Tmax(ϱ) in (9) has no upper bound as ϱ increases or is close to zero.
Moreover, the best upper estimate of the bound (9) is achieved at arg minϱ>0 Tmax(ϱ) = 1, and its value is
min
ϱ>0 Tmax(ϱ) = Tmax(1) = Tc
K
 
1
(1 −pk) +
1
(qk −1)
!
> Tc.
An illustration of this argument, showing Tmax(ϱ) as a function of ϱ, with Tc = 1s, is presented in Figure 2.
Although, by Theorem 2 the least upper bound of the settling time is Tc = 1s, it can be seen that in the best case
the bound (8) provides an overestimation of εTc with ε =
1
K

1
(1−pk) +
1
(qk−1)
p=0.5,q=3,k=1.5 = 1.1249. Moreover, in
Figure 1, it can be seen that although the least upper bound of the settling-time function is Tc = 1s, the upper bound
estimation provided by (9) is Tmax(ϱ)|ϱ=4 = 4.4331s.
4. Application to robust predefined-time stabilization for first and second order systems
4.1. First-order predefined-time controllers
Consider the following perturbed first-order system
˙x(t) = u(t) + ∆(t),
(10)
where x ∈R is the state variable, u ∈R is the control input and ∆(t) ∈R is an unknown but bounded perturbation
term of the form |∆(t)| ≤δ, with 0 ≤δ < ∞a known constant.
The objective is to to design the control input u such that the origin x = 0 of system (10) is predefined-time stable,
in spite of the unknown perturbation term ∆(t).
Theorem 3. Let α, β, p, q, k > 0, kp < 1, kq > 1, Tc > 0, ζ ≥δ and γ be as in (6). If the control input u is selected as
u = −
" γ
Tc
(α |x|p + β |x|q)k + ζ
#
signx
(11)
then, the origin x = 0 of system (10) is predefined-time stable with predefined time Tc.
7
Tmax(ϱ)
ϱ
Tc
Figure 2: Function Tmax(ϱ) for p = 0.5, q = 3, k = 1.5. The least upper bound of the settling time is Tc = 1s. The smallest value of Tmax(ϱ) is
minϱ>0 Tmax(ϱ) = Tmax(1) = 1.1249s.
8
x0 = 0.1
time (t)
x0 = 1
x0 = 1e20
Tc = 1
Tmax = 4.4331
x(t, x0)
∆(t)
Figure 3: State trajectory of the closed-loop system (10)-(11) with Tc = 1s and the considered perturbation.
Proof. Consider the continuous radially unbounded Lyapunov function candidate V(x) = |x|. Its derivative along the
trajectories of the closed system (10)-(11) yields
˙V(x) = −signx
" γ
Tc
(α |x|p + β |x|q)k signx + ζ signx −∆
#
= −γ
Tc
(αV(x)p + βV(x)q)k −ζ + ∆signx
≤−γ
Tc
(αV(x)p + βV(x)q)k −ζ +
∆signx

≤−γ
Tc
(αV(x)p + βV(x)q)k −(ζ −δ)
≤−γ
Tc
(αV(x)p + βV(x)q)k .
Hence, using Theorem 2, the origin x = 0 of system (10) is predefined-time stable with predefined time Tc.
Example 2. An example of this approach is shown in Figure 3, where the control input u given in (11), with ζ = 1,
p = 0.5, q = 3, k = 1.5 and α = 1/β = ϱ = 4, is applied to the perturbed system (10) with disturbance ∆(t) =
sin(2πt/5). Note that although the upper bound of the settling-time function is Tc = 1s using the proposed scheme,
the upper bound estimation provided by [1] is Tmax(ϱ)|ϱ=4 = 4.4331s.
4.2. Second-order predefined-time controllers
Consider the following perturbed second-order system
˙x1 = x2
˙x2 = u + ∆(t),
(12)
where x1, x2 ∈R are the state variables, u ∈R is the control input and ∆(t) ∈R is an unknown but bounded
perturbation term of the form |∆(t)| ≤δ, with 0 ≤δ < ∞a known constant.
The objective is to design the control input u such that the origin (x1, x2) = (0, 0) of system (12) is predefined-time
stable, in spite of the unknown perturbation term ∆(t).
The following definition will be useful for stating Theorem 4.
9
Definition 5. For any real number r, the function x →⌊x⌉r is defined as ⌊x⌉= |x|r signx for any x ∈R if r > 0, and
for any x ∈R \ {0} if r ≤0.
Theorem 4. Let α1, α2, β1, β2, p, q, k > 0, kp < 1, kq > 1, Tc1, Tc2 > 0, ζ ≥δ, and
γ1 =
Γ
 1
4
2
2α1/2
1 Γ
 1
2

 α1
β1
!1/4
, and γ2 =
Γ

mp

Γ

mq

αk
2Γ(k)(q −p)
 α2
β2
!mp
,
with mp = 1−kp
q−p and mq = kq−1
q−p . If the control input is selected as
u = −

γ2
Tc2
(α2 |σ|p + β2 |σ|q)k + γ2
1
2T 2c1

α1 + 3β1x2
1

+ ζ
signσ,
(13)
where the sliding variable σ is defined as
σ = x2 +
⌊x2⌉2 + γ2
1
T 2c1

α1 ⌊x1⌉1 + β1 ⌊x1⌉3
1/2
,
(14)
then the origin (x1, x2) = (0, 0) of system (12) is predefined-time stable with predefined time Tc = Tc1 + Tc2.
Proof. The time-derivative of the sliding variable σ in (14) is
˙σ = u + ∆+
|x2| (u + ∆) +
γ2
1
2T 2c1

α1 + 3β1x2
1

x2
⌊x2⌉2 +
γ2
1
T 2c1

α1 ⌊x1⌉1 + β1 ⌊x1⌉3
1/2
= −γ2
Tc2
(α2 |σ|p + β2 |σ|q)k signσ −ζ signσ + ∆−γ2
1
2T 2c1

α1 + 3β1x2
1

signσ
−
γ2
Tc2 (α2 |σ|p + β2 |σ|q)k signσ + ζ signσ −∆
⌊x2⌉2 +
γ2
1
T 2c1

α1 ⌊x1⌉1 + β1 ⌊x1⌉3
1/2
−
γ2
1
2T 2c1

α1 + 3β1x2
1

⌊x2⌉2 +
γ2
1
T 2c1

α1 ⌊x1⌉1 + β1 ⌊x1⌉3
1/2
|x2| signσ −x2

 .
Now, considering V2(σ) = |σ| as a continuous radially unbounded positive definite Lyapunov candidate function
it can be easily checked using the above that
˙V(σ) ≤−γ2
Tc2
(α2V2(σ)p + β2V2(σ)q)k ,
and using Theorem 2, the origin σ = 0 of the sliding variable dynamics is predefined-time stable with predefined time
Tc2.
Once the system trajectories are constrained to the manifold σ = 0, i.e. for t ≥Tc2, the solutions of system (12)
satisfy the following reduced-order dynamics (see (14)):
˙x1 = x2 = −γ1
Tc1

α1 |x1| + β1 |x1|31/2 signx1.
Thus, considering V1(x1) = |x1| as a continuous radially unbounded positive definite Lyapunov candidate function and
using Theorem 2, it is concluded that the origin x1 = 0 of the reduced order system is predefined-time stable with
predefined time Tc1. Moreover, from (14), if σ = 0 and x1 = 0, then x2 = 0. Hence, it is concluded that the origin
(x1, x2) = (0, 0) of the closed-loop system (12)-(13) is predefined-time stable with predefined time Tc1 + Tc2.
Example 3. An example of this approach is shown in Figure 4, where the control input u given in (13), with ζ = 1,
p = 0.5, q = 3, k = 1.5 and α1 = α2 = 1/β1 = 1/β2 = 4 is applied to the perturbed system (12) with disturbance
∆(t) = sin(2πt/5). It follows from Theorem 4 that with Tc1 = Tc2 = 0.5s, the origin (x1, x2) = (0, 0) of system (12) is
predefined-time stable with predefined time Tc = 1s. Note that although the upper bound of the settling-time function
is Tc = 1s using the proposed scheme, the upper bound estimation provided by [1] is Tmax =
Tc2
γ2

1
αk
2(1−pk) +
1
βk
2(qk−1)

+
2Tc1
γ1

1
√α1 +
1
√β1

= 5.1073s.
10
0            0.5            1.0           1.5           2.0            2.5           3.0            3.5           4.0           4.5           5.0
0            0.5            1.0           1.5           2.0            2.5           3.0            3.5           4.0           4.5           5.0
1.0
0.5
0
-0.5
-1.0
1.0
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1.0
time (t)
∆(t)
x2(t, x0)
x1(t, x0)
x2(0) = 100
x2(0) = 1
x2(0) = 0.1
x1(0) = 100
x1(0) = 1
x1(0) = 0.1
Tc = 1
Tmax = 5.1073
Figure 4: State trajectory of the closed-loop system (12)-(13) with Tc = 1s and the considered perturbation.
5. Conclusion
In this paper, we studied the convergence time of a class of fixed-time stable systems with the aim to provide a
new non-conservative upper bound for its settling time. We showed that the well-known upper bound condition for
the settling time of this class of systems is often too conservative. To illustrate our claim, we showed how by changing
one parameter the upper estimate of the settling time tends to infinity even though the actual settling time is always
bounded by a constant Tc. To address this problem, we proposed a modification to the classical fixed-time algorithm
to transform it into a strongly predefined-time (in which the upper bound for the settling time is set in advance as a
parameter of the system and is the lowest upper estimate of the settling time) with strong predefined-time Tc. With
this result, the Lyapunov inequality, which is a sufficient condition for fixed-time stability, was modified in a way
that it becomes predefined-time parametrized by Tc. When such inequality becomes equality, Tc becomes the lowest
upper estimate of the settling time of the system. At last, predefined-time controllers for first order and second order
systems were introduced. Some simulation results have shown the performance of the proposed scheme in terms of
settling time estimation compared to existing methods. This is an important contribution toward online differentiators,
observers and controllers satisfying prescribed-time objectives.
Appendix A. Auxiliary Results
Definition 6. [22, Pg. 87] Let a, b > 0. The Beta function, denoted by B(a, b), is defined as
B(a, b) =
Z 1
0
za−1(1 −z)b−1dz = Γ(a)Γ(b)
Γ(a + b) .
Proposition 1. Let α, β, p, q, k > 0, with pk < 1, qk > 1. Hence,
Z +∞
0
dx
(αxp + βxq)k =
Γ

mp

Γ

mq
  α
β
mp
αkΓ(k)(q −p)
.
(A.1)
11
Proof. The left-hand side of (A.1) can be rewritten as
Z +∞
0
dx
(αxp + βxq)k =
Z +∞
0
β−kx−qkdx
 α
β xp−q + 1
k .
(A.2)
Note that the term z =
 α
β xp−q + 1
−1 goes to 0 when x →0 and to 1 when x →+∞if p −q < 0. Furthermore, using
z as a variable change and by Definition 6, the integral (A.2) becomes
Z +∞
0
dx
(αxp + βxq)k =
 α
β
mp
αk(q −p)
Z 1
0
zmp−1(1 −z)mq−1dz
=
B

mp, mq

αk(q −p)
 α
β
!mp
=
Γ

mp

Γ

mq

αkΓ(k)(q −p)
 α
β
!mp
,
concluding the proof.
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