Adv Syst Sci Appl 2023; 02:70–78
Published online at https://ijassa.ipu.ru.

A New Approach on the Modelling and Analysis Stability
of a Class of Fractional-Order Quasi-Polynomial Systems

Mohamed Reda Lemnaouar1, Mohamed Khalfaoui1,
Rabie Zine2*, Younes Louartassi1,3,

1Mohammed V University in Rabat, EST Salé, LASTIMI, Salé, Morocco
3School of Science and Engineering, Al Akhawayn University in Ifrane, Morocco

3 Mohammed V University in Rabat, Faculty of Sciences,
Laboratory of Mathematics, Computing and Applications, Rabat, Morocco

Abstract: Stabilization and observation for nonlinear fractional derivative systems remain open
problems in automatic due to the fractional nature and nonlinearity of these systems. The present
paper studies global stability by the return output for fractional systems. First, we give some
definitions of fractional calculus and the quasi-polynomial (QP) and Lotka-volterra (LV) systems.
Then, we analyze their stabilities as well as linear (LMI) and bilinear (BMI) matrix inequalities.
In order to solve the controller design problem. The goal of this paper is to investigate the
global and local stability of a dynamic fractional order system using the quasi-polynomial and
LV representation. Then, we use the LMI to study the stabilization of this fractional system.

Keywords: quasi-polynomial systems, fractional derivative, LMI, BMI, stability

1. INTRODUCTION

In the last decades, the analysis problem of control and stability of dynamic systems has
attracted the attention of researchers [10, 20]. These studies are applied in several fields in
reality such as chemistry, technology and physical systems including electrical networks,
economic systems and electrical systems [2].

Stabilization and observation for nonlinear fractional derivative systems remain open
problems in automatic due to the fractional nature and nonlinearity of these systems. Indeed,
stabilization is one of the major concerns of both researchers and engineers. There are many
methods of stabilizing linear systems; they are generally based on pole placement techniques
or the minimization of a quadratic criterion and lead to feedback, the implementation of
which requires the use of observers when the condition is partially measured. Lyapunov’s
first method makes it possible to use these results for the control of nonlinear systems from
their linearizations and the use observers.

Many dynamical systems are represented by a non-integer order dynamic model, usually
based on the notion of differentiation or integration of the non-integer order. Studying the
stability of fractional order systems [12–17] is more difficult than for their counterparts
(integer order systems). On the one hand, fractional systems are regarded as memory
systems, in particular for the consideration of initial conditions, and on the other hand they

∗Corresponding author: r.zine@aui.ma



A NEW APPROACH ON THE MODELLING AND ANALYSIS STABILITY. . . 71

have a much more complex dynamic.

Lotka-Volterra and quasi-polynomial are models proven to be well adapted to canonical
forms generally applicable to models of ordinary non-linear ODE differential equations, since
a large class of these models can be written in these types [5]. Furthermore, the approach of
global and local stability analysis of Generalized LV models are abundant [3, 4, 11].

LV systems are generally used in some scientific fields, such as population biology. The
form and properties of the generalized and classical LV algebraic forms were treated by
Hernandez-Bermejo and Fairen in [19].

The goal of this paper is to investigate the global and local stability of a dynamic fractional
order system using the quasi-polynomial and LV representation. Then, we use the LMI to
study the stabilization of this fractional system.

2. BACKGROUND MATERIAL AND PRELIMINARY RESULTS

Throughout this paper, Mp,m(R) denotes the set of real matrices with p rows and m columns,
Mm(R) the set of square real matrices of size m, diag(ω1, . . . , ωm) represent the diagonal
matrix which contains the elements ω1, . . . , ωm and Bjk is the element of the j-th row and
the k-th column of a matrix B.

In this section, we use the new easy definition of the fractional derivative (see [7]). This
new definition is a natural extension of the usual derivative, and it verifies the first four
properties of the fractional derivative in the sense of Riemann-Liouville and in the sense
of Caputo (see [7]). Also, this definition coincides with the known fractional derivatives on
the polynomials.

Definition 2.1:
Let h : [0,∞) −→ R, u > 0 and α ∈ (0, 1). We define the conformable fractional derivative
of h of order α by :

T α(h)(u) = lim
ϵ→0

h(u+ ϵu1−α)− h(u)

ϵ
. (2.1)

Furthermore, if h is α-differentiable (the conformable fractional derivative T α(h)(u) exist)
in some (0, u1), u1 > 0 and lim

u→0+
T α(u) exists, then clearly T α(u)(0) = lim

u→0+
T αh(u).

Theorem 2.1:
Let α ∈ [0, 1[ and the functions y1, . . . , yp are α-differentiable at a point u > 0. Then

(1) T α(k1y1 + k2y2) = k1T α(y1) + k2T α(y2) for all k1, k2 ∈ R.
(2) T α(u) = 0, for all u ∈ R.
(3) T α(y1y2) = y1T α(y2) + y2T α(y1).

(4) T α
(

y1
y2

)
= y2T α(y1)−y1T α(y2)

y22
.

(5) If y is differentiable, then T α(y) = u1−α dy(u)
du

.
(6) T α(ym) = mym−1Tα(y) for all m ∈ N∗.

(7) T α(
p∏

k=1

yk) =
p∑

k=1

T α(yk)
p∏

i=1
i ̸=k

yi for all m ∈ N∗.

Proof
Properties (1) through (5) have already been approved by Khalil et al. in [7].

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



72 M. R. LEMNAOUAR, M. KHALFAOUI, R. ZINE AND Y. LOUARTASSI

To prove property (6). Let u > 0 and m ∈ N∗, we have

T α(ym)(u) = lim
ϵ→0

ym(u+ϵu1−α)−ym(u)
ϵ

,

= lim
ϵ→0

(
y(u+ϵu1−α)−y(u)

ϵ

)lim
ϵ→0

m−1∑
k=0

yk(u+ ϵu1−α)ym−k−1(u)

 ,

= T α(y)(u)
m−1∑
k=0

yk(u)ym−k−1(u),

= T α(y)(u)
m−1∑
k=0

ym−1(u),

= mym−1(u)T α(y)(u).

Now, for the last property, we use recurrence reasoning. For all u > 0, we suppose that yp+1

is α-differentiable and T α(
p∏

k=1

yk) =
p∑

k=1

T α(yk)
p∏

i=1
i̸=k

yi.

We have

T α(
p+1∏
k=1

yk) = T α((
p∏

k=1

yk)yn+1),

= yn+1T α(
p∏

k=1

yk) + (
p∏

k=1

yk)T α(yn+1),

= (
p∑

k=1

T α(yk)
p∏

i=1
i̸=k

yi)yn+1 + (
p∏

k=1

yk)T α(yn+1),

=
p∑

k=1

T α(yk)
p+1∏
i=1
i ̸=k

yi + (
p∏

k=1

yk)T α(yn+1),

=
p+1∑
k=1

T α(yk)
p+1∏
i=1
i ̸=k

yi

According to the reasoning by recurrence we proved the relation (7).

3. LOTKA-VOLTERRA AND QUASI-POLYNOMIAL MODELS

We will define some notions concerning the QP and LV fractional order systems as well as
their stability analysis and that of LMI and BMI.

QP models are systems of FDEs in this form :

T α(yi) = yi

Li +
m∑
j=1

Mij

p∏
k=1

y
Bjk

k

 , i = 1, . . . , p. (3.2)

where T α is defined in Definition 2.1, y = [y1, . . . , yp]
T ∈ Rp

+, L = [L1, . . . , Lp]
T ∈ Rp,

M ∈Mp,m(R) and B ∈Mm,p(R). without the loss of generality, we can assume that
rank(B) = p and m ≥ p (see [5]).

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



A NEW APPROACH ON THE MODELLING AND ANALYSIS STABILITY. . . 73

Considering zj =
p∏

k=1

y
Bjk

k and using the 6th and 7th properties of Theorem 2.1. If rank(B) =

p, then the fractional differential system FDEs (3.2) becomes in the LV form :

T α(zj) = zj

Nj +
m∑
i=1

Ajizi

 , j = 1, . . . ,m (3.3)

where A = BM and N = BL.

3.1. Input-affine QP system models
We consider the input-affine nonlinear system fractional order model following :

T α(y) = ϕ(y) +
r∑

l=1

φl(y)ϑl,

θ = ψ(y).

(3.4)

where y, ϑ and θ are the state vector, the input vector and the output vector, respectively. The
function ϕ, φl for l = 1, . . . , r and ψ are in QP -form. Then the input-affine QP -system with
p-inputs is :

T α(yi) = yi

L0 +
m∑
j=1

M0ij

p∏
k=1

y
Bjk

k

+
r∑

l=1

yi

Lli +
m∑
j=1

Mlij

p∏
k=1

y
Bjk

k

ϑl, i = 1, . . . , p.

(3.5)
where M0,Ml ∈Mp,m(R), B ∈Mm,p(R) and L0, Ll ∈Mp,1(R) for l = 1, . . . , r. The related
input-affine LV model in this form :

T α(zj) = zj

N0j +
m∑
k=1

A0jkzk

+
r∑

l=1

zj

Nlj +
m∑
k=1

Aljkzk

ϑl, j = 1, . . . ,m (3.6)

where N0 = BL0 ∈Mm,1(R), Nl = BLl ∈Mm,1(R), A0 = BM0 ∈Mm(R) and
Al = BMl ∈Mm(R) for l = 1, . . . , r.

3.2. Transforming non-QP FDEs models into QP -form
The non-linear FDEs is writing in this form :

T α(yi) =
∑

is1,...,isn,js

ais1,...,isn,jsy
is1
1 . . . yisnn g(ȳ)js ,

yi(0) = yi0, i = 1, . . . , p

(3.7)

where g(ȳ) is not written in this form :
p∏

k=1

y
Cjk

k , j = 1, . . . ,m with C ∈Mm,p(R).

In addition,
T αg

∂yi
=

∑
es1,...,esn,es

bes1,...,esn,esy
es1
1 . . . yesnn g(ȳ)es . (3.8)

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



74 M. R. LEMNAOUAR, M. KHALFAOUI, R. ZINE AND Y. LOUARTASSI

We introduce a new auxiliary variable

η = f q

p∏
k=1

ypkk , q ̸= 1. (3.9)

Then, equations (3.7) can write in QP -form :

T α(yi) = yi

 ∑
is1,...,isn,js

ais1,...,is1,jsη
js/q

p∏
k=1

y
isk−δsk−jspk/q
k

 , i = 1, . . . , n (3.10)

where δsk = 1 if s = k and 0 otherwise. Additionally, a new QP FDEs appears for the new
variable η :

T αη = η

p∑
s=1

psT α(ys)y
−1
s +

∑
isβ

esβ,es

,js

aisα,js
besβ,esqη

(es+js−1)/q

p∏
k=1

y
isk+esk+(1−es−js)pk/q
k

 .

(3.11)
The choose of p and q can be in (3.9) in different approach in order to facilitate the
calculations. The simplest choice is ps = 0, s = 1, . . . , n and q = 1. So, if the initial values
of the variables introduced according to (3.9) is fixed, at that moment the dynamics of the
integrated system will be equivalent to the original non-QP system shown in (3.9). Hence,
it is evident that the original system (3.9) is stable whenever the systems (3.10)-(3.11) are
stable.

3.3. QP models of process systems
Nonlinear process system models with localized parameters are divided into two forms in
terms of their representation in the form of QP . The systems operate ϕ from the input
affine spatial state model (3.4) and they are not always in the form of QP . Consequently,
the integration of these models in the described QP -form crucial practice.
The characteristics of the input function φi of the input-affine space-state model (3.4).
Frequently, φi is a simple homogeneous linear function of the corresponding state variable yi
: φi(y) = const.yi implies Al = 0 in (3.2) and Ml = 0 in (3.6).
If φi = const, this form is not accepted when an equation of state in QP -form comes from
the incorporation of variables.

3.4. Application
New, we consider the following non-QP model (see [9]) :

T α(X) = r1X
(
1− X

K

)
− σ1X + σ2Y − uX2 − βXS

α′+X
− q1E1X − n1XY − γXI,

T α(Y ) = (r2 − σ2)Y + σ1X − vY 2 − n2XY,
T α(S) = σβXS

α′+X
− δSI − µS − q2E2S,

T α(I) = δSI + σγXI − q3E3I − ηI.

(3.12)
Where X , Y , I and S represent biomass densities of the unreserved area, reserved area,
infected and susceptible predator, respectively. For the harvesting in the unreserved area,
E1, E2 and E3 denote the effort applied the susceptible and infected predator populations,

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



A NEW APPROACH ON THE MODELLING AND ANALYSIS STABILITY. . . 75

respectively. r1, r2 the fish population growth rates in the reserved and unserved areas,
respectively. q1 and q2 represents the coefficient of catchability in the unreserved area and
the predator species. σ1 and σ2 represent the migration rates between the two zones. µ and η
represent the rate of death of susceptible and infected predators. n1 and n2 are the coefficients
of competition. γ the strength of intra-specific between X and I . δ the coefficient of disease
transmission. β the rate of search of the prey toward susceptible predators. α′ constant of
saturation while susceptible predators S attack the prey X . σ is the rate of conversion of
susceptible predator I due to prey X . uX2 and vY 2 represents the reduction terms, in the
unreserved area and reserved area respectively.
To transform the model (3.12) into the QP form, we introduce a new variable Z = 1

α′+X
, we

get the following QP-model :
T α(X) = X

(
r1 − σ1 − q1E1 + σ2V2 − (u+ r1

K
)X − βV1 − n1Y − γI

)
,

T α(Y ) = Y (r2 − σ2 + σ1V3 − vY − n2X) ,
T α(S) = S (σβV4 − δI − µ− q2E2) ,
T α(I) = I (δS + σγX − q3E3 − η) ,
T α(Z) = Z

(
−V4(r1 − σ1 − q1E1) + ( r1

K
+ u)V5 + n1V6 + βV7 − σ2V8 + γV9

)
.

(3.13)
The 14 quasi-monomials of the QP system model given are : X , Y , Z, I , S, V1 = SZ,
V2 = X−1Y , V3 = XY −1, V4 = XZ, V5 = X2Z, V6 = XY Z, V7 = XZ2S, V8 = Y Z and
V9 = XZI . Then, the QP-model (3.13) is transfomed in the Lotka-Volterra form :

T α(X) = X
(
r1 − σ1 − q1E1 + σ2V2 − (u+ r1

K )X − βV1 − n1Y − γI
)
,

T α(Y ) = Y (r2 − σ2 + σ1V3 − vY − n2X) ,
T α(S) = S (σβV4 − δI − µ− q2E2) ,
T α(I) = I (δS + σγX − q3E3 − η) ,
T α(Z) = Z

(
−V4(r1 − σ1 − q1E1) + ( r1K + u)V5 + n1V6 + βV7 − σ2V8 + γV9

)
,

T α(V1) = V1
(
(σβ + r1 − σ1 − q1E1)V4 + ( r1K + u)V5 + n1V6 + βV7 − σ2V8 + γV9 + σγX − η − q3E3

)
,

T α(V2) = V2
(
r2 − σ2 − (r1 − σ1 − q1E1) + ( r1K + u− n2)X + (n1 − v)Y + βV1 − σ2V2 − γI

)
,

T α(V3) = −V3
(
r2 − σ2 − (r1 − σ1 − q1E1) + ( r1K + u− n2)X + (n1 − v)Y + βV1 − σ2V2 − γI

)
,

T α(V4) = V4

(
T α(X)

X +
T α(Z)

Z

)
,

T α(V5) = V5

(
T α(X)

X + 2
T α(Z)

Z

)
,

T α(V6) = V6

(
T α(X)

X +
T α(Z)

Z +
T α(Y )

Y

)
,

T α(V7) = V7

(
T α(X)

X + 2
T α(Z)

Z +
T α(S)

S

)
,

T α(V8) = V8

(
T α(Z)

Z +
T α(Y )

Y

)
,

T α(V9) = V9

(
T α(X)

X +
T α(Z)

Z +
T α(I)

I

)
.

(3.14)

4. STABILITY OF QP SYSTEMS FRACTIONAL ORDER

According on the basic concepts of LV and QP systems gave in section 3, we will analysis of
the stability of QP systems fractional order and a proposed numerical algorithm to solve the
case of non-strict LMI.
The analysis of stability is carried out around an equilibrium point y∗ = (y∗1, y

∗
2, . . . , y

∗
p). The

solution of the equilibrium point is determined from (3.2) as follows :

0 = yi

Li +
m∑
j=1

Mij

p∏
k=1

y
Bjk

k

 , i = 1, . . . p, and m ≥ p (4.15)

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



76 M. R. LEMNAOUAR, M. KHALFAOUI, R. ZINE AND Y. LOUARTASSI

Suppose that y∗ is a positive equilibrium point of (3.2), so z∗ = (z∗1 , z
∗
2 , . . . , z

∗
m) is a positive

equilibrium in the case LV systems (3.6). The Lyapunov function associated with LV-system
in the following:

F (z) =
m∑
i=1

ωi

(
zi − z∗i − z∗i ln(

zi
z∗i
)
)

ωi > 0, i = 1, . . . ,m

(4.16)

where z∗ is correspondent point of equilibrium y∗ of system (3.2). The fractional derivate of
Lyapunov function is defined by :

T αF (z) = F ′(z).T α(z) =
1

2
(z − z∗)(WA+ ATW)(z − z∗), (4.17)

where W = diag(ω1, . . . , ωm) and A is the invariant characterizing the LV-form (3.6).
Consequently, the decreasing nature of Lyapunov’s function is equivalent to a feasibility
problem of the following LMI :

WA+ ATW ≤ 0
W > 0,

(4.18)

The possibilities of finding a function of Lyapunov that shows the global asymptotic stability
of a QP system can be raised with the help of temporal reparametrization (See [3]). If the
model (3.6) are ordered so that the first p lines of B are linearly independent, then ωi > 0 for
i = 1, . . . , p and cj = 0 for j = p+ 1, . . . ,m ensure the global stability.
The global stability of (3.6) is studied and demonstrated in [3,4] with the Lyapunov function
(4.16), which gives the boundedness of the solutions and the global stability of (3.2). It can
be noted that the global stability is limited to the positive orthant only for LV and QP models,
which is compatible with the nature of the variables.
The global stability of the equilibrium points (3.2) as a function of Lyapunov (4.16) are not
related on the value of the vector L as will as the equilibrium points are in the positive orthant.
This makes it possible to reach the equilibrium point of the closed-loop system (CLS) when
designing the stabilization controller [18].
By using temporal reparametrization [18], we increase the possibilities to find a function of
Lyapunov which demonstrates the global asymptotic stability of a QP system.

5. THE CONTROLLER DESIGN PROBLEM

Now, we will study the global feedback stabilization of (3.4) where the state feedback control
is on the QP form of the CLS. Also, we will study its global stability using LMI if the return
parameters are known and fixed. Otherwise, a feedback design problem will be defined that
globally stabilizes the CLS.

The global stabilization of the feedback design problem for QP systems is modeled on
(5.19). Let QP entries:

φl =
s∑

i=1

kilqi, l = 1, . . . , r (5.19)

where the quasi-monomial functions qi = qi(y1, . . . , yp) for i = 1, . . . , s of the state variables
of (3.7) and kil are constants gain. Moreover, the CLS will also be a QP system with the

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



A NEW APPROACH ON THE MODELLING AND ANALYSIS STABILITY. . . 77

following matrices :

M̄ = M0 +
r∑

l=1

s∑
i=1

kilMil, B̄

L̄ = L0 +
r∑

l=1

s∑
i=1

kilLil.
(5.20)

The qi in the CLS, as well as the matrix B̄ can vary considerably depending on the choice
of the feedback structure. Hence, in the closed-loop LV, the matrix of the coefficients Ā is
expressed in this form :

Ā = B̄.M̄ = A0 +
r∑

l=1

s∑
i=1

kilAil. (5.21)

Thereafter, the analysis of the global stability of the CLS with an unspecified gain feedback
kil gives the next BMI :

ĀTW +WĀ = AT
0W +WA0 +

r∑
l=1

s∑
i=1

kil(A
T
ilW +WAil) ≤ 0. (5.22)

where the feedback gain parameters is kil and the positive coefficients of the Lyapunov
function is cj , j = 1, . . . ,m. Then, there exists globally stabilizing feedback whenever the
BMI above is attainable.

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Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)


	Introduction
	Background Material and Preliminary Results
	Lotka-Volterra and Quasi-Polynomial Models
	Input-affine QP system models
	Transforming non-QP FDEs models into QP-form
	QP models of process systems
	Application

	Stability of QP systems fractional order
	The controller design problem