INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
Online ISSN 1841-9844, ISSN-L 1841-9836, Volume: 15, Issue: 5, Month: october, Year: 2020
Article Number: 3898, https://doi.org/10.15837/ijccc.2020.5.3898

CCC Publications 

Lyapunov-based Methods for Maximizing the Domain of Attraction

H. Jerbi, F. Hamidi, S. Ben Aoun, S. C. Olteanu, D. Popescu

Houssem Jerbi*
University of Ha’il, College of Engineering
Department of Industrial Engineering
Hail 1234, KSA
*Corresponding author: h.jerbi@uoh.edu.sa

Faiçal Hamidi
Laboratoire de Recherche
Modélisation, Analyse et Commande des Systèmes
University of Gabes, ENIG, 6029 Gabes, Tunisie
faical.hmidi@isimg.tn

Sondess Ben Aoun
University of Ha’il, College of Computer Science and Engineering
Department of Computer Engineering
Hail 1234, KSA
s.benaoun@uoh.edu.sa

Severus Constantin Olteanu, Dumitru Popescu
University ‘Politehnica’ of Bucharest
Faculty of Automatic Control and Computer Science
Bucharest Romania
severus.olteanu@acse.pub.ro, popescu_upb@yahoo.com

Abstract

This paper investigates Lyapunov approaches to expand the domain of attraction (DA) of
nonlinear autonomous models. These techniques had been examined for creating generic numerical
procedures centred on the search of rational and quadratic Lyapunov functions. The outcomes are
derived from all investigated methods: the method of estimation via Threshold Accepted Algorithm
(TAA), the method of estimation via a Zubov technique and the method of estimation via a linear
matrix inequality (LMI) optimization and genetic algorithms (GA). These methods are effective for
a large group of nonlinear models, they have a significant ability of improvement of the attraction
domain area and they are distinguished by an apparent propriety of direct application for compact
and nonlinear models of high degree. The validity and the effectiveness of the examined techniques
are established based on a simulation case analysis. The effectiveness of the presented methods is
evaluated and discussed through the study of the renowned Van der Pol model.

Keywords: Lyapunov function, nonlinear model, asymptotic stability, equilibrium point, ge-
netic algorithm, threshold accepted algorithm, LMI.



https://doi.org/10.15837/ijccc.2020.5.3898 2

1 Introduction
During the previous decades, the complexity of approximating the attraction domain area has

remained the focus of several benchmarking works [25, 30, 35, 36, 41], the references contained in
them. The asymptotic stability area of physical systems is a significant property to be identified [18].
From a practical point of view, it is every so often insufficient to demonstrate the local asymptotic
stability for point of equilibrium [4, 20, 21, 34, 42], however one may also require discerning the size
of the stability region, as well [27, 33]. As a matter of fact, the stability area or attraction domain is
described as the set of original starting criteria where the system states meet at the points of equi-
librium [7, 8, 9]. Thus, it is vital to specify the state of this area. To achieve this, one can utilize
a Lyapunov function [11, 13, 20, 25, 27, 32]. In fact, for a specified energy function providing stable
local equilibrium, the biggest probable DA, the state of which is established by the Lyapunov energy
function theory, is referred to be as the prevalent smooth function set integrated in a bounded domain
where its derivative is negative [12, 22, 27, 40].

A significant number of approaches on estimating the area of attraction investigated in the liter-
ature are established exploiting the established statements of LaSalle and Lefschetz [26, 29, 31, 32].
These later are based upon appropriately selected Lyapunov functions in [23, 24, 27]. Considered
as one of the original outcomes as regard as the problematic of attraction domains estimation, the
Zubov’s method has designed a Lyapunov function restricted by an open ball over a closed interval
and consequently has sorted out an approximated DA [16, 43].

Similar complexity is detected when applying the algorithm synthesized by Knobloch and Kappel
[8]. An LMI method to design polynomial Lyapunov functions for non-polynomial class of systems is
provided in [10]. However, the problem of defining a truncation order for the Lyapunov function in
the Taylor series expansion is not well addressed.

In [11, 15] authors designate an LMI based technique for approximating the so-titled robust DA
for a polynomial class of systems with model polytope uncertainties.

In [9], authors formulate a fundamental analytical background in which particular classes of min-
imum based distance problems are resolved via LMI calculations [14]. The previous specified ap-
proaches, nevertheless, are appropriate for the class of smooth non-linear ODE systems.

In general, there is no generic techniques for determining Lyapunov functions as a group of non-
linear models [6, 8] are established. Nonetheless, Lyapunov theory is still considered as the most
efficient method to analyze the nonlinear models stability, although the Lyapunov theorem does not
need the class of algebraic functions that must be retained [1, 2, 3, 37]. In [11, 13] the maximum
uncertain domain to preserve the stability and nominal performances (robustness) of the nonlinear
optimal control systems, is estimated.

The goal of this paper is to recommend analytical methods motivating systematic approximating
approaches of the DA. With the aim of getting a definite expression of the estimated DA, an investi-
gation of the Lyapunov theory exploiting parameterized Lyapunov function is conducted as described
in [6]. As the DA is related with a specified function of Lyapunov, the proposal entails selecting the
optimal parameters to attain the most significant DA area. These later are designed as results to an
optimization problem [5]. Output response designs with quantifiable premise parameters have also
been employed in [22].

The paper structure is as follows. Section 2 provides a representation of the Carleman Linearization
and the methodology of obtaining the Lyapunov functions based upon Lyapunov stability techniques
and TAA. In Section 3, a recursive technique to calculate the DA through rational Lyapunov function
is discussed. The fundamental principles of the GA are evaluated in the last part of this section.
The assessment of the global accomplishment for the different techniques is performed by means of
a simulation study carried out on the Van der Pol model. Section 4 focuses on the conclusions and



https://doi.org/10.15837/ijccc.2020.5.3898 3

future works.

2 Searching Lyapunov Function via Threshold Accepting Algorithms

2.1 The Carleman Linearization

The development of f (x) into a Taylor series expansion provides [17, 19]:

f (x) =
∑

l

Alx[l] (1)

wherein:
Al =

{
1
l!

∂lf (0)
∂xi1 . . .∂xil

}
(2)

and:
x[l] = x⊗x⊗x.. .⊗x︸ ︷︷ ︸

l−T imes

(3)

denotes the l power Kronocker product of vector x Performing algebraic operations one can rewrite
the vector derivative ẋ[l] as follows:

ẋ([l]) =
∑l

j=1
x⊗ ..⊗ .. ẋ︸︷︷︸

j

⊗..⊗x (4)

This means
ẋ([l]) =

∑l
j=1

x⊗ ..⊗ ..f(x) ⊗ ..⊗x (5)

By substituting f (x) with (1), one obtains:

ẋ[l] =
∑l

j=1
x⊗ ..⊗ ..

∑
k
Akx[k] ⊗ ..⊗x (6)

Thus, it becomes
ẋ[k] =

∑
l

Alkx
[k+l−1] (7)

where
Alk = A

l
1 ⊗ I

[k−1] + I ⊗Alk−1 (8)

Let the change of variables given below be considered now:

Ψ =
(
x,x[2],x[3], . . . ,x[k], . . .

)T
(9)

an infinite dimension linear system given as:

Ψ̇ = H Ψ (10)

is attained with:

H =



H11 . . . H

i
1 . . . . . .

0 H12 H22 . . . . . .
0 0 H13 H23 . . .
...

...
...

...
...


 (11)



https://doi.org/10.15837/ijccc.2020.5.3898 4

2.2 A Lyapunov theory case investigation

In this paragraph, an overall Lyapunov hypothesis for stabilizing a linear infinite dimension dy-
namical model is given. Consider the nonlinear system (1) where it is assumed that the origin is an
equilibrium state (i.e.f(0) = 0). It is shown that system (1) can be expressed under a linear operator
form with an infinite dimension as shown in (11) where the changed state vector is given by (10).
Consider the nonlinear function given by [28]:

V =
〈(
x,x[2],x[3], ...

)
, Υ
(
x,x[2],x[3], ...

)〉
(12)

then
V̇ =

〈(
HT Υ + ΥH

)(
x,x[2], ..

) (
x,x[2], ..

)〉
= −

∥∥∥(x,x[2],x[3], ...)∥∥∥2
= −

(
e ‖x‖

2
− 1

) (13)
whereΥ is a symmetric positive definite operator and ‖.‖ denotes the norm on the subset = ={(
x,x[2], ....

)
,x ∈<n

}
considered on the tensor algebra space <n over <n. Consequently the defined

function is a Lyapunov function and the studied system is globally asymptotically stable.

2.3 Main outcomes

Here, a finalized algorithm yielding a Lyapunov function for the class of investigated systems
is presented. To achieve this aim, proposed functions of Lyapunov characterized by a quadratic
formulation [39]:

V (x) = xT Υx, Υ = ΥT (14)
are taken into account. A standard TAA [38] has been adjusted for sake of establishing the matrices
and which satisfies the Lyapunov equation:HT Υ + ΥH = −Θ. The steps given below represent the
synthesized technique for stability assessment.

– Initial step: Set zero matrixes Υ and Θ
– First step: Randomize matrix elements and validate the definite positive characteristic. IF Υ

isn´ t positive definite, REDO first step.
– Second step: Calculate matrix Θ established in relation with Υ. Validate the definite negative

propriety of Θ. If not satisfied, REDO first step.
– Final step: Approve final solutions.

2.4 Illustrative example

Consider the nonlinear system given as [43]:{
ẋ1 = −x2
ẋ2 = x1 +

(
x21 − 1

)
x2

(15)

The Carleman Linearization is used while the polynomial state equation is truncated to the third
order. The equivalent linear system (11) is acquired as follows:

Ψ =
[
x1,x2,x

2
1, . . .x

3
2

]
(16)

and

H =




0 −1 0 0 0 0 0 0 0
1 −1 0 0 0 0 1 0 0
0 0 0 −2 0 0 0 0 0
0 0 1 −1 −1 0 0 0 0
0 0 0 2 −2 0 0 0 0
0 0 0 0 0 0 −3 0 0
0 0 0 0 0 1 −1 −2 0
0 0 0 0 0 0 2 −2 −1
0 0 0 0 0 0 0 3 −3




(17)



https://doi.org/10.15837/ijccc.2020.5.3898 5

Applying the TAA, yields:
Υ =

[
Υ1 Υ2 Υ3

]
(18)

where:

Υ1 =




1.7894 0.6500 −0.4242
0.6500 1.2914 0.0615
−0.4242 0.0615 1.3590
0.0693 −0.0081 0.3250
0.0146 0.2092 0.3114
0.0515 −0.4560 0.2696
0.0520 0.0914 0.0695
0.0540 0.0325 −0.1123
0.1635 0.0717 −0.1578




Υ2 =




0.0693 0.0146 0.0515
−0.0081 0.2092 −0.4560
0.3250 0.3114 0.2696
0.6614 0.3636 −0.0943
0.3636 0.6136 −0.1231
−0.0943 −0.1231 2.1146
0.2623 0.0292 0.2667
0.1107 0.0571 0.1462
0.0732 0.0815 0.0435




Υ3 =




0.0520 0.0540 0.1635
0.0914 0.0325 0.0717
0.0695 −0.1123 −0.1570
0.2623 0.1107 0.0732
0.0292 0.0571 0.0815
0.2667 0.1462 0.0435
0.5518 0.2324 0.1160
0.2324 0.3183 0.1784
0.1160 0.1784 0.3950




and:
Θ =

[
Θ1 Θ2 Θ3

]

Θ1 =




−1.3000 −0.1000 −0.2001
−0.1000 −1.1000 −0.4000
−0.2001 −0.4000 −1.3000
−0.5000 0.2001 −0.6002
−0.0998 −0.6000 −0.7000
0.3000 0.5000 −0.0199
−0.1999 −0.1000 −0.0999
−0.2000 0.3000 0.3000
−0.4002 0.0902 −0.0099




Θ2 =




−0.5000 −0.0998 0.3000
0.2001 −0.6000 0.5
−0.6002 −0.7000 −0.0199
−1.4000 −0.0702 −0.2999
−0.0702 −1.0000 −0.0300
−0.2999 −0.0300 −1.6002
−0.8000 0.1997 −0.0999
−0.0501 −0.0301 −0.4997
−0.2000 −0.0898 −0.0399






https://doi.org/10.15837/ijccc.2020.5.3898 6

Θ3 =




−0.1999 −0.2000 −0.4002
−0.1000 0.3000 0.0902
−0.0999 0.2992 −0.0123
−0.8000 −0.0501 −0.1992
0.1997 −0.0301 −0.0898
−0.0999 −0.4997 −0.0399
−1.4998 −0.2000 −0.0801
−0.2000 −0.7004 −0.1001
−0.0801 −0.1001 −1.2996




Therefore, the Lyapunov function V (Ψ (x)) and its first derivative V̇ (Ψ (x)) are expressed under in
the forms V (Ψ) = ΨT ΥΨ and V̇ (Ψ) = ΨT ΘΨ. The area plots respectively for the function V (Ψ) and
V̇ (Ψ) are demonstrated in Figure 1 and Figure 2. Subsequently system (15) is globally asymptotically
stable at origin.

0

2

500

2.51 2

V
(
x

)

1000

1.5

x
2

10 0.5

x
1

1500

0-1 -0.5-1-1.5-2 -2-2.5

Figure 1: Area plots for the Lyapunov function V (Ψ)

2.5 Results and discussion

In this section, the investigation of the Carleman linearization letting the computation of an infi-
nite dimensional linear model is provided. An analytical step procedure which defines a function of
Lyapunov and ensures nonlinear models stability via the outlining of a Lyapunov equation is consid-
ered. This technique is based on the generic TAA [38]. The most important conclusion that is derived
as a result of the implementation of this method on a second order system is given as follows:

Raising the truncation degree of the Lyapunov equation creates better accuracy on the typical DA.
However, it is noteworthy that there is a quick increase in the linear matrix dimension. This causes
difficulties in the calculation of the Lyapunov function with no appreciable enhancement in the DA



https://doi.org/10.15837/ijccc.2020.5.3898 7

-2500

-2000

2

-1500

2.51 2

d
V

(
x

)
/
d

t -1000

1.5

x
2

10

-500

0.5

x
1

0-1 -0.5-1-1.5-2 -2-2.5

Figure 2: Area plots for the function V̇ (Ψ)

dimension. It would be an innovative technique to examine beforehand the developed technique to
define the best possible degree of truncation for the highest accuracy in defining the DA.

3 Estimation of the DA using a Zubov approach

3.1 Rational Lyapunov function

This paragraph aims to explore an analytical approach for approximating the DA of nonlinear
models. This later is established based on the calculation of a Lyapunov function providing a solution
for the subsequent equation :

V̇ (x) = (
dV (x)
dx

)T f (x) = −φ (x) (19)

With φ (x) a positive definite function. To solve the equation (19), a structured procedure using the
conversion of f(x) into Taylor series expansion was given in [43]. This procedure started with writing
f(x) under the following form:

f (x) =
∞∑

i=1
Fi (x) (20)

As the Lyapunov candidate function should be restricted when evolving towards the border (that’s
mean when x →∞), therefore, this attribute can be stated by considering a rational defined by [43] :

V (x) =
N (x)
D (x)

(21)



https://doi.org/10.15837/ijccc.2020.5.3898 8

where N(x) and D(x) are polynomial functions. Subsequently, V (x) should be represented as follows:

V (x) =

∞∑
Ni (x)
i=2

1 +
∞∑
Di (x)
i=1

(22)

Equation (20) and equation (22) yield:

(κ(Di,Ni))
∞∑

i=1
Fi = −xT Inx(1 +

∞∑
Ni

i=1
)2 (23)

wherein:
κ(Di,Ni) =

(
1 +

∞∑
i=1

Di

) ∞∑
i=2
∇NTi −

( ∞∑
i=1
∇DTi

) ∞∑
Ni

i=2
(24)

From this, finally one obtains:
∞∑

i=2

∞∑
j=2
∇NTi Fj +

∞∑
i=1

∞∑
j=2

∞∑
k=1

Di∇NTj Fk −
∞∑

i=1

∞∑
j=2

∞∑
k=1
∇DTi NjFk

= −xT Inx
(

1 + 2
∞∑

k=1
Dk +

∞∑
i=1

∞∑
j=1

DiDj

) (25)
Identifying the coefficient of equal degrees from both sides of the second order equation gives:

∇NT2 F1 = −x
T I2x (26)

and:
F1 (x) = H1x =

[
h11 h12
h21 h22

][
x1
x2

]
(27)

When k ≥ 3 the generic solution can be written as:
k∑

i=2
∇NTi Fk+1−i

+
j−2∑
i=1

∞∑
k=2

(
Di∇NTk −∇D

T
i Nk

)
Fj+1−i−k

= −xT Qnx
(

1 + 2Dk−2 +
k−3∑
i=1

DiDk−2−i

) (28)

Hence, in every step of the synthesized algorithm, the below linearity under a set of established
equations as the same form for equations (27) and (28) is obtained:

Hnq = bn (29)

where Hn is a matrix of an appropriate dimension given by:

Hn =
[
Hn1 Hn2 Bn

]
(30)

with:

Hn1 =



h11n h21 0
h21n (h22 + h11 (n− 1)) 2h21

0 0 h12 (n− 1)
...

...
...

0 0 · · ·


 (31)

Hn2 =




. . . . . . 0
0 . . . 0

[h22 + h11 (n− 2)] 3h21 0
. . . 0 0
0 h12 nh22


 (32)



https://doi.org/10.15837/ijccc.2020.5.3898 9

Bn is obtained by expanding the quantity
(
Di∇NT2 −∇D

T
i N2

)
F1, bn contains the numerical values

of the equation (28).
q = [ h1, ..., hn+1 g1 , ...,gn−1]T (33)

Now the derivation of the Lyapunov function is as follows:

V̇n = −xT Qnx +
e (x)(

1 +
n−2∑
i=1

Qi (x)
)2 (34)

Where e(x) is a sum of monomials of degree greater than n. To enlarge the attraction area, in the
proximity of the origin, one needs to analyze the condition V̇ (x) < 0 minimization process of the
quantity e (x). The problem thus implies the next reducing criterion:{

Min en (q)
Hnq = bn

(35)

with:

en (q) =
∥∥∥∥∥ coefficients of the numerator termsof (34)

∥∥∥∥∥
2

2
(36)

3.2 Study of an illustrative example

The earlier example of Van der Pol model is being investigated here. To identify the boarder of
the DA, use a Lyapunov function defined by:

V4 (x) =

4∑
i=2

Ri (x)

1 +
2∑

i=1
Qi (x)

(37)

with
Ri (x) = h1xi1 + h2x

i−1
1 x2 + · · · + hix

i
i

and
Qi (x) = g1xi1 + g2x

i−1
1 x2 + · · · + gix

i
i

From equation (26) for degree 2, one can easily establish the equation of R2(x) described by:

R2 (x) = 1.5x21 −x1x2 + x
2
2 (38)

For k ≥ 3 the generic solution can be obtained based on the optimization method described by (35).
The coefficients and are suggested to be coded into a 7 bits digital word as follows:

hi = ( h00, h01,h02,h03,h04,h05,h06)
gi = (g00,g01,g02,g03,g04,g05,g06)

(39)

The GA is employed where the genetic properties are summarized as follows: the size of population is
50, the rate of mutation is 0.1, the crossover rate is 0.65 and the maximum number for the generations
is taken equal to 100 [24]. When implementing the recommended method based on the GA, the
following outcomes are obtained:

V4 (x) =
Vn4
VD4

= 3.18 (40)

wherein:
Vn4 = 1.5x21 −x1x2 + x22−0.2449x41+0.4082x31x2

+0.1582x21x22−0.1293x1x32−0.0459x42

VD4 = 1 + 0.2041x21+0.0560x1x2−0.0252x
2
2



https://doi.org/10.15837/ijccc.2020.5.3898 10

-3 -2 -1 0 1 2 3
x

1

-3

-2

-1

0

1

2

3

x
2

Limit Cycle
Estimated DA

Figure 3: Stability region of Van der Pol system

3.3 Results and discussion

In this section, an algorithm leading to construct a rational Lyapunov function to approximate
the DA of autonomous nonlinear systems is provided. Using this algorithm is advantageous compared
to other one since there is no major complexity in selecting initial starting values. In fact, this
technique is a linear programming based method which can be validated by an equivalence constrained
minimization criterion. Moreover, the described technique can be of a practical use for a large set of
nonlinear models. Indeed, it is not restricted for polynomial nonlinear systems which the case for the
majority of existing algorithms. Equally important, the approximated DA is satisfactory because it
is adequately large in comparison to the theoretical bound. However, the main disadvantages of the
presented method can be mentioned as follows:

– The obtained DA has a non-definite geometric shape which means that it does not support a
control strategy requiring the mathematical expression of the domain bound.

– The convergence of the developed algorithm is not demonstrated since it depends on a heuristic
method (that is, the GA).

– The initial condition of the given algorithm impacts heavily the final outcome.

4 Conclusion
In this paper, the Lyapunov theory for approximating and the expanding of the domain of asymptotic
stability of nonlinear models is examined. The investigated techniques have been performed with
reference to numerical algorithms that were established using the integration of a rational Lyapunov
function and a quadratic function. The primary motivation has been to study the various analytical
approaches for the synthesis of DA in the proximity of an equilibrium point. The outcomes of the
developments which are performed using the technique of threshold algorithms, the approximation



https://doi.org/10.15837/ijccc.2020.5.3898 11

method via a rational Lyapunov function and the approximation method based on LMI and GA
demonstrated that this variety of approaches has the following attributes:

– They are effective for large classes of nonlinear models.
– They have the capability to enhance the domain of asymptotic stability.
– They can be performed for both high and low order nonlinear models.
– All the investigated techniques are developed in order to convert the DA estimation problem

into a standard generic one.
The primary method known as the Carleman method demonstrated some difficulties which can

be summed up as follows: when increasing the truncation order which enables the achievement of a
Lyapunov function, an apparent complexity illustrates the digital operations of the dependent matrices
describing the established Lyapunov function. The remaining paper discussed couple of approaches:
the first is based on the result of the Zubov equation and the other is based on a statistical approach.
To discover a representation of the Lyapunov function, it is demonstrated that solving the given
problem corresponds to an optimization problem. A merging of the GA, LMI and RTM is employed
in this respect. To analyse the performance of the various studied methods, a simulation analysis is
accomplished based on the oscillator model of Van der Pol. This later is described by an asymptotic
stability domain limited by a closed contour. The achieved outcomes have confirmed the effectiveness
of the presented methods and approaches to ensure asymptotic stability in case of controlled systems.
With the recommendations in this paper, conservativeness in the context of solutions from earlier
literature was reduced: Solutions based on Lyapunov equations can be utilised as a seed for the
techniques developed here.

Funding

The authors would like to acknowledge the support of the Deanship of Scientific Research at Hail
University - KSA under the research project (RG-0191315).

Author contributions

The authors contributed equally to this work.

Conflict of interest

The authors declare no conflict of interest.

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Cite this paper as:
Jerbi, H.; Hamidi, F.; Ben Aoun, S.; Olteanu, S. C. ; Popescu, D. (2020). Lyapunov-based Methods
for Maximizing the Domain of Attraction, International Journal of Computers Communications &
Control, 15(5), 3898, 2020. https://doi.org/10.15837/ijccc.2020.5.3898