Global practical conformable stabilization by output feedback
for a class of nonlinear fractional‐order systems
Hamdi Gassara1, Omar Naifar2, Abdellatif Ben Makhlouf3 and Lassaad Mchiri4
1Laboratory
of Sciences and Technology of Automatic Control and Computer Engineering, National School of Engineering of Sfax, University
of Sfax, P. O. Box 1173, 3038 Sfax, Tunisia
2Control
and Energy Management Laboratory, National School of Engineering, Sfax University, Sfax, Tunisia
3Department
4Department
of Mathematics, College of Science, Jouf University, Sakaka, Aljouf, Saudi Arabia
of Statistics and Operations Research, College of Sciences, King Saud University, Riyadh 11451, Kingdom of Saudi Arabia
Abstract : In this work, a global practical conformable stabilization by output feedback for a class of
nonlinear fractional‐order systems is shown. To begin, a method of stabilization is provided. Following that,
the proposed system's observer design is presented. Also, the principle of separation is described.
Furthermore, a new nonlinear condition is considered when developing the work. Finally, a numerical
example is offered to demonstrate the proposed methodology's validity.
Key words: General conformable derivative, stabilization, observer design, separation principle.
1. Introduction
Fractional calculus (FC) is a natural extension of regular calculus that covers derivatives and integrals of
non-integer order. FC has gotten a lot of attention over the last three decades since it is an effective and
widely utilised method for improving system modelling and control in a wide variety of fields of science and
engineering. Indeed, The book [1] presents major control systems and applications with either system
simulation results or genuine experimental data, or both. This book [2] proposes a method for implementing
and applying Fractional Order Systems (FOS). It is generally established that FOS may be used in control
applications and system modelling, and their efficacy has been demonstrated in a large number of theoretical
papers and simulation methods. [6] analyses the energy storage efficiency of a generic fractional-order
circuit element. Other applications such as: Fractional Dynamics [3], electric circuits [4], Adaptive
synchronization [5], Energy storage |6], constant phase element [7], mutual inductance [8] ,Nanotechnology
[9] and so on.
Fractional Derivatives (FDs) are nonlocal operators because they are constructed using integrals. As a
result, the FD includes information about the features at past places in time, producing a memory effect and
nonlocal spatial effects. In fact, they incorporate both the system's background and nonlocal scattering
effects, which are critical for a more precise and accurate description and analysis of complex and dynamic
control systems. It is already possible to define FDs and integrals in a number of methods [10]. The sources
cited in that section serve as illustrations of these notions. These facts are contained in the books and articles.
Indeed, In [11], The Adomian's decomposition approach has been devised to provide an approximation to
the resolution of a fractional order microbial cytolytic model in a tractor trailer environment. Furthermore,
the purpose of the work [12] is to make the solution of certain integral equations more visually appealing. It
provides content methodically and contains a list of sources relating to the subject's history from 1695 to the
present. Additionally, the volume [13] is devoted to the systematic and thorough presentation of classical
and contemporary achievements in the research and application of fractional integrals. Also, the book [14] is
a seminal work in the monitoring data of mathematics from integer to non-integer forms: from integer to real
numbers, from factorials to the gamma function, and from integer-order models to structures of any order.
[15] is a special book to combine an accessible introduction to fractional calculus with an examination of a
broad range of practical applications. The authors in [16] provide a novel definition of fractional derivative
with a smooth kernel that accommodates two distinct interpretations for the spatially and temporally
1
variables in the article. The authors in [17] suggested a novel fractional derivative with a non-local and nonsingular kernel in this publication.
One of the difficulties encountered in the discipline is selecting which FD should be utilized in instead of
the conventional derivative in a given case. The Caputo FDs and the Riemann-Liouville FDs are the most
often utilized. FC's potential has been proved in classical situations such as the tautochrone problem [18].
Additional applications include the fractional diffusion equation [20], memory-based models [19], and a new
linear capacitor theory [21]. Teka et al. [23] investigated numerous components of the fractional order
defective integrate and burn model that advanced multiple time scale brain dynamics provides. All FD
definitions satisfy the property of linearity. Almost all FDs, on the other hand, are devoid of mathematical
properties such as the product rule, chain rule, and so forth... These and other inconsistencies have created a
slew of problems in real-world applications, limiting our ability to examine fractional computations.
To address these issues, Khalil et al. [24] proposed a novel approach called conformable FD that extends
the traditional limit definitions of a function's derivatives. This concept permits alternative expansions of
some classical calculus theorems that are necessary for fractional differential models but are not permitted by
existing definitions. The conformable derivative is of interest to researchers because it appears to satisfy all
of the standard derivative's characteristics [25]. Additionally, computations with this new derivative are far
simpler than with previous FD formulations. As a result, this new concept has been incorporated into a lot of
stuff. In fact, In [27], a new precise solution to Burgers type equations with conformable derivatives was
proposed. Additionally, [28] investigates the time scales approach to conformable derivatives. Also, the
study [29] addressed parameter variation. Furthermore, the paper [30] discusses the Fractional Fourier series.
Moreover, the paper [31] discussed novel conformable derivative characteristics. By contrast, [33] examines
an extension of the classical conformable fractional derivative. Indeed, [33] pioneered a new class of
fractional derivatives known as the GCD. Additionally, the studies show multiple instances of the diffusion
equation solution being unique in [32]. Some further additional efforts to the conformable derivative are
recently done by researchers, for example the multi-agent systems with impulsive control protocols [37],
numerical methods [38], time power-based grey model [39], multivariate grey system model [40],
controllability and observability [41], the Barbalat lemma [42] and H∞ observers [43]. The conformable
derivative's broad application is exemplified by the large number of recent research publications, which
demonstrate the derivative's importance in solving diverse problems in science and engineering. Certain
notions remain unaddressed by the conformable derivative, and it represents an unexplored subject of study.
Thus, motivated by the above interpretations, our work presents the following contributions:
•
•
•
To our knowledge, there are no published publications that address the stabilization, observer design,
and separation principle of the proposed fractional order nonlinear system.
The nonlinear part's condition is generic, and hence cannot be a Lipschitz condition. In the example,
a function that is not Lipschitz is chosen.
The separation principle problem is unique in that it was developed and proved using the LMI
approach.
2. Preliminaries and system description
This section begins with a review of some theorems, definitions, and lemmas, it can be found in [22].
Definition 1. Assume a function 𝜙 which is defined on [𝑡0 𝑏); then, the general conformable derivative of 𝜙
is defined by
𝛼,𝜓𝑡0
𝑇𝑡0
𝜙(𝑡) = lim
𝜙 (𝑡 + 𝜀𝜓𝑡0 (𝑡, 𝛼 )) − 𝜙(𝑡)
,
𝜀
For all 𝑡 > 𝑡0 , where𝛼 ∈ (0,1) and 𝜓𝑡0 (𝑡, 𝛼 ) is a continuous nonnegative function that depends on 𝑡 and
satisfies:
𝜀⟶0
𝜓(𝑡, 1) = 1,
2
𝜓(. , 𝛼1 ) ≠ 𝜓(. , 𝛼2 ), 𝑤ℎ𝑒𝑟𝑒 𝛼1 ≠ 𝛼2 𝑎𝑛𝑑 𝛼1 , 𝛼2 ∈ (0,1].
𝛼,𝜓𝑡0
If 𝑇𝑡0
𝛼,𝜓𝑡0
𝜙(𝑡) exists, for every 𝑡 ∈ (𝑡0 , 𝑐) and for some 𝑐 > 𝑎, and lim+ 𝑇𝑡0
𝑡⟶𝑡0
𝛼,𝜓𝑡0
𝑇𝑡0
𝛼,𝜓𝑡0
𝜙(𝑡0 ): = lim+ 𝑇𝑡0
𝑡⟶𝑡0
𝜙(𝑡) exists, then
𝜙 (𝑡 ) ,
Remark 1. The general conformable derivative generalizes the classical derivative (𝛼 = 1) and the
conformable derivative 𝜓𝑡0 (𝑡, 𝛼) = (𝑡 − 𝑡0 )1−𝛼 (see [33]).
Remark 2. To further study the properties of the general conformable derivative, we suppose that
1
𝜓𝑡0 (𝑡, 𝛼) > 0 for all 𝑡 > 𝑡0 and
(. , 𝛼) is locally integrable.
𝜓𝑡0
Definition 2. Let 𝛼 ∈ (0.1]. The conformable fractional integral starting from a point 𝑡0 of a function ℎ ∶
[𝑡0 𝑏) ↦ ℝ of order 𝛼 is defined as
𝑡
𝛼,𝜓𝑡
𝐼𝑡0 0 ℎ(𝑡)
= ∫
𝑡0
ℎ (𝑠 )
𝑑𝑠.
𝜓𝑡0 (𝑠, 𝛼)
Property 1. Assume that ℎ is an absolutely continuous function on [𝑡0 𝑏).Then, for all 𝑡 ≥ 𝑡0 , we have
𝛼,𝜓𝑡0
𝐼𝑡0
𝛼,𝜓𝑡0
𝑇𝑡0
ℎ(𝑡) = ℎ(𝑡) − ℎ(𝑡0 ).
𝛼,𝜓𝑡0
Property 2. Let 𝛼 ∈ (0.1] and the function ℎ, 𝑔: [𝑡0 𝑏) → ℝ such that 𝑇𝑡0
𝛼,𝜓𝑡0
ℎ and 𝑇𝑡0
𝑔 exists on
(𝑡0 , 𝑏), then:
𝛼,𝜓𝑡0
𝛼,𝜓𝑡0
•
𝑇𝑡0
•
𝑇𝑡0
•
𝛼,𝜓𝑡
𝑇𝑡0 0 (ℎ𝑔)
•
𝑇𝑡0
𝛼,𝜓𝑡0
𝛼,𝜓𝑡0
(𝑐1 ℎ + 𝑐2 𝑔) = 𝑐1 𝑇𝑡
𝑐3 = 0,
ℎ
0
𝛼,𝜓𝑡0
ℎ + 𝑐2 𝑇𝑡0
𝑐3 ∈ ℝ ,
𝛼,𝜓𝑡0
𝛼,𝜓𝑡0
= ℎ𝑇𝑡0
( ) (𝑡 ) =
𝑔 + 𝑔𝑇𝑡0
𝛼,𝜓𝑡0
𝑔(𝑡)𝑇𝑡
0
ℎ,
𝛼,𝜓𝑡0
ℎ(𝑡)+ℎ(𝑡)𝑇𝑡
0
𝑔(𝑡)
𝑔2 (𝑡)
𝑔
𝜓
𝑡
Assume that 𝜐 ∈ ℝ∗ . If 𝑦(𝑡) ≔ 𝐸𝛼 0 (𝜐, 𝑡, 𝑡0 ) = 𝑒
1
𝜐
𝑔, 𝑐1 , 𝑐2 ∈ ℝ ,
, for each 𝑡 ∈ (𝑡0 , 𝑏) and 𝑔(𝑡) ≠ 0.
𝑡
1
𝑑𝑥
0 𝜓𝑡0 (𝑥,𝜃)
𝜐 ∫𝑡
𝛼,𝜓𝑡0
, then 𝑇𝑡0
𝛼,𝜓𝑡0
𝑦(𝑡) = 𝜐𝑦(𝑡) and 𝐼𝑡0
𝑦 (𝑡 ) =
(𝑦(𝑠) − 1).
Consider the system
𝛼,𝜓𝑡0
𝑥(𝑡) = 𝐹 (𝑡, 𝑥 )
𝑥 (𝑡0 ) = 𝑥0 ,
where 𝐹: ℝ+ × ℝ𝑛 → ℝ𝑛 is a continuous function.
𝑇𝑡0
One presents the following definition related to the above nonlinear fractional order system.
Definition 3. The above system is said to be uniformly practically fractional exponentially stable if there
exist positive scalars 𝜎, λ and ρ such that
3
𝜓
‖𝑥(𝑡)‖ ≤ 𝜎 ‖𝑥0 ‖𝐸𝛼 𝑡0 (−𝜆, 𝑡, 𝑡0 ) +ρ, ∀𝑡 ≥ 𝑡0 .
Lemma 1. Let 𝑌 ∶ [𝑡0 , +∞) → ℝ that satisfies:
𝛼,𝜓𝑡0
𝑇𝑡0
𝑌(𝑡) ≤ −𝜆𝑌(𝑡) + ℎ(𝑡), ∀𝑡 ≥ 𝑡0
with ℎ(𝑡)satisfies:
𝑡
∫
𝑡0
1
𝜓𝑡
𝜓𝑡
𝐸 0 ( − 𝜆, 𝑡, 𝑡0 )𝐸𝛼 0 (𝜆, 𝑠, 𝑡0 )ℎ(𝑠)𝑑𝑠 ≤ 𝑀, ∀𝑡 ≥ 𝑡0 ≥ 0,
𝜓𝑡0 (𝑠, 𝛼) 𝛼
(1)
where 𝑀 > 0.
Then
𝜓
𝑡
𝑌(𝑡) ≤ 𝐸𝛼 0 (−𝜆, 𝑡, 𝑡0 )𝑌(𝑡0 ) + 𝑀,
∀𝑡 ≥ 𝑡0 .
Proof.
𝜓
𝛼,𝜓𝑡0
𝑡
Let 𝑥 (𝑡) = 𝐸𝛼 0 (𝜆, 𝑡, 𝑡0 ) 𝑌(𝑡) − 𝐼𝑡0
𝜓
𝑡
(𝐸𝛼 0 (𝜆, 𝑠, 𝑡0 )ℎ(𝑠)) (𝑡).
We have
𝛼,𝜓𝑡0
𝑇𝑡0
𝜓
𝜓
𝛼,𝜓𝑡0
𝑡
𝑡
𝑥(𝑡) = 𝜆𝐸𝛼 0 (𝜆, 𝑡, 𝑡0 )𝑌(𝑡) + 𝐸𝛼 0 (𝜆, 𝑡, 𝑡0 )𝑇𝑡0
𝜓
𝛼,𝜓𝑡0
𝑡
= 𝐸𝛼 0 (𝜆, 𝑡, 𝑡0 ) [𝜆𝑌(𝑡) + 𝑇𝑡0
𝜓
𝑡
𝑌(𝑡) − 𝐸𝛼 0 (𝜆, 𝑡, 𝑡0 )ℎ(𝑡)
𝑌(𝑡) − ℎ(𝑡)] ≤ 0.
Then,
𝑥 (𝑡) ≤ 𝑌(𝑡0 ), ∀𝑡 ≥ 𝑡0 .
So,
𝜓
𝛼,𝜓𝑡0
𝑡
𝐸𝛼 0 (𝜆, 𝑡, 𝑡0 )𝑌(𝑡) ≤ 𝑌(𝑡0 ) + 𝐼𝑡0
𝜓
𝑡
(𝐸𝛼 0 (𝜆, 𝑠, 𝑡0 )ℎ(𝑠)) (𝑡).
Thus,
𝑡
𝑌 (𝑡 ) ≤
𝜓𝑡
𝐸𝛼 0 (−𝜆, 𝑡, 𝑡0 )𝑌(𝑡0 )
1
𝜓𝑡
𝜓𝑡
𝐸 0 ( − 𝜆, 𝑡, 𝑡0 )𝐸𝛼 0 (𝜆, 𝑠, 𝑡0 )ℎ(𝑠)𝑑𝑠
𝜓𝑡0 (𝑠, 𝛼) 𝛼
+∫
𝑡0
𝜓
𝑡
≤ 𝐸𝛼 0 (−𝜆, 𝑡, 𝑡0 )𝑌(𝑡0 ) + 𝑀,
∀𝑡 ≥ 𝑡0 .
Remark 3. If ℎ(𝑡) = 𝑐, then (1) is satisfied. Indeed, one has:
𝜓𝑡
𝑡
1
0
𝜓𝑡0 (𝑠,𝛼)
∫𝑡
𝜓𝑡
𝜓𝑡
𝑡 𝐸𝛼 0 (−𝜆,𝑡,𝑡0 )𝐸𝛼 0 (𝜆,𝑠,𝑡0 )
𝜓𝑡
𝐸𝛼 0 ( − 𝜆, 𝑡, 𝑡0 )𝐸𝛼 0 (𝜆, 𝑠, 𝑡0 )ℎ(𝑠)𝑑𝑠 = 𝑐 ∫𝑡
𝑠
1
0
𝜓𝑡0 (𝑙,𝛼)
Using the change of variable 𝑣 = ∫𝑡
𝑡
𝜓
𝜓𝑡0 (𝑠,𝛼)
0
𝑑𝑙 = 𝑔(𝑠), one gets :
𝑔(𝑡)
𝜓
𝑡
𝑡
𝐸𝛼 0 (−𝜆, 𝑡, 𝑡0 )𝐸𝛼 0 (𝜆, 𝑠, 𝑡0 )
𝜓𝑡
𝑐∫
𝑑𝑠 = 𝑐𝐸𝛼 0 (−𝜆, 𝑡, 𝑡0 ) ∫ 𝑒 𝜆𝑣 𝑑𝑣
𝜓𝑡0 (𝑠, 𝛼 )
𝑡0
0
𝜓
𝑒 𝜆𝑔(𝑡) −1
𝑐
𝜆
𝜆
𝑡
= 𝑐𝐸𝛼 0 (−𝜆, 𝑡, 𝑡0 ) (
4
) ≤ .
𝑑𝑠.
Consider the system
𝛼,𝜓𝑡0
𝑥(𝑡) = 𝐴𝑥 + 𝐵𝑢 + 𝑓 (𝑡, 𝑥 )
(2)
𝑦(𝑡) = 𝐶𝑥,
where 𝐴 ∈ ℳ𝑛 (ℝ), 𝐵 ∈ ℳ𝑛,𝑝 (ℝ), 𝐶 ∈ ℳ𝑞,𝑛 (ℝ) and the nonlinear part 𝑓: ℝ+ × ℝ𝑛 → ℝ𝑛 which satisfies
the following assumption:
𝑇𝑡0
•
(𝐻) 𝑓 is a continuous function that satisfies:
‖𝑓(𝑡, 𝑥 ) − 𝑓(𝑡, 𝑦)‖ ≤ 𝑏‖𝑥 − 𝑦‖ + 𝜑(𝑡), ∀𝑡 ∈ ℝ+ , ∀(𝑥, 𝑦) ∈ ℝ𝑛 ,
𝐶 (ℝ+ , ℝ+ ).
where
𝑏>0
and
𝜑∈
Remark 4. We get from (𝐻) : ‖𝑓(𝑡, 𝑥 )‖ ≤ 𝑏‖𝑥‖ + 𝜑̃(𝑡), ∀𝑡 ∈ ℝ+ , ∀𝑥 ∈ ℝ𝑛 , where 𝜑̃(𝑡) = 𝜑(𝑡) +
‖𝑓(𝑡, 0)‖.
3. Stabilization
In this part, one presents the results of the system's stabilization. The following theorem proves that under the
suggested control law, the system (2) is practically stabilizable.
̃ ∈ ℳ𝑝,𝑛 (ℝ) and scalar 𝜀̃ > 0 such that:
Theorem 1. Suppose that (𝐻) holds and there exist 𝑋 ∈ 𝑆𝑛+ (ℝ), 𝐾
̃ ) + 𝜀̃𝐼
𝑠𝑦𝑚(𝐴𝑋 + 𝐵𝐾
(
∗
𝑋
𝜀̃ ) < 0,
− 2𝐼
2𝑏
the control 𝑢 = 𝐾𝑥 stabilizes practically the system (2) if :
𝜓𝑡
𝜓𝑡
𝑡 𝐸𝛼 0 (−𝜆,𝑡,𝑡0 )𝐸𝛼 0 (𝜆,𝑠,𝑡0 ) 2
𝜑̃ (𝑠)𝑑𝑠
𝜓𝑡0 (𝑠,𝛼)
0
∫𝑡
where 𝜆 = −
𝜆𝑚𝑎𝑥 (𝜃)
𝜆𝑚𝑎𝑥 (𝑃)
≤ 𝑀, ∀𝑡 ≥ 𝑡0 ≥ 0,
and 𝑀 > 0, in which
1
1
𝜀
𝜀
̃ 𝑋 −1 , 𝜀 = , 𝜃 = 𝑠𝑦𝑚(𝑃(𝐴 + 𝐵𝐾 )) + 2𝑏 2 𝜀𝐼 + 𝑃𝑃 .
𝑃 = 𝑋 −1 , 𝐾 = 𝐾
̃
Proof.
Let consider the Lyapunov function 𝑉 (𝑡, 𝑥 ) = 𝑥 𝑇 𝑃𝑥, one has:
𝛼,𝜓𝑡0
𝑇𝑡0
𝛼,𝜓𝑡0
𝑉(𝑡, 𝑥(𝑡)) = 2𝑥 𝑇 𝑃𝑇𝑡0
𝑥(𝑡)
= 2𝑥 𝑇 𝑃[(𝐴 + 𝐵𝐾 )𝑥 + 𝑓(𝑡, 𝑥)].
One has,
2𝑥 𝑇 𝑃𝑓 (𝑡, 𝑥 ) = 2𝑓 𝑇 (𝑡, 𝑥 )𝑃𝑥 ≤ 2‖𝑓(𝑡, 𝑥)‖‖𝑃𝑥‖
1
≤ 𝜀‖𝑓(𝑡, 𝑥 )‖2 + ‖𝑃𝑥‖2
𝜀
1
≤ 𝜀(2𝑏 2 ‖𝑥‖2 + 2𝜑̃ 2 (𝑡)) + ‖𝑃𝑥‖2
𝜀
1
≤ 𝑥 𝑇 (2𝑏 2 𝜀𝐼 + 𝑃𝑃) 𝑥 + 2 𝜀𝜑̃ 2 (𝑡).
𝜀
Thus,
5
(3)
𝛼,𝜓𝑡0
𝑇𝑡0
𝑉(𝑡, 𝑥 (𝑡)) ≤ 𝑥 𝑇 𝜃𝑥 + 2 𝜀𝜑̃ 2 (𝑡)
≤ −𝜆𝑉(𝑡, 𝑥 (𝑡)) + 2 𝜀𝜑̃ 2 (𝑡).
We conclude from Lemma 1, that the system (2) is practically fractional exponentially stable if the following
condition holds:
𝜃<0
Noting that the previous inequality is given in bilinear matrix inequality (BMI) form, which cannot be solved
using existing solvers such as LMI Toolbox or Yalmip in the MATLAB software. Accordingly, we need to
make further development to convert the bilinear condition in LMIs.
By multiplying the right part and the left part of 𝜃 by 𝑋 = 𝑃−1 , one gets:
̃ ) + 2𝜀𝑏 2 𝑋𝑋 + 1 𝐼 < 0.
𝑠𝑦𝑚(𝐴𝑋 + 𝐵𝐾
𝜀
So, using Schur complement, we get (3).
4. Observer design
The design of the system's observer is presented in this section. Consequently, one gets the following
outcome:
Theorem 2. Suppose that (𝐻) holds and there exist ∈ 𝑆𝑛+ (ℝ) , 𝐿̃ ∈ ℳ𝑛,𝑞 (ℝ), and scalar 𝜀 > 0 such that:
2
̃
(𝑠𝑦𝑚(𝑃𝐴 − 𝐿𝐶) + 2𝑏 𝜀𝐼
∗
𝑃 ) < 0.
−𝜀𝐼
(4)
The following system:
𝛼,𝜓𝑡0
𝑇𝑡0
𝑥̂ (𝑡) = 𝐴𝑥̂ + 𝐵𝑢 + 𝑓(𝑡, 𝑥̂ ) − 𝐿(𝐶𝑥̂ − 𝑦)
𝑦̂(𝑡) = 𝐶𝑥̂,
(5)
is a practical observer of the system (2) if:
𝑡
𝜓
𝜓
𝑡
𝑡
𝐸 0 (−𝜆, 𝑡, 𝑡0 )𝐸𝛼 0 (𝜆, 𝑠, 𝑡0 ) 2
∫ 𝛼
𝜑 (𝑠)𝑑𝑠 ≤ 𝑀,
𝜓𝑡0 (𝑠, 𝛼 )
∀𝑡 ≥ 𝑡0 ≥ 0,
𝑡0
where 𝜆 = −
𝜆𝑚𝑎𝑥 (𝜓)
𝜆𝑚𝑎𝑥 (𝑃)
1
and 𝑀 > 0, in which 𝐿 = 𝑃−1 𝐿̃,𝜓 = 𝑠𝑦𝑚(𝑃(𝐴 − 𝐿𝐶 )) + 2𝑏 2 𝜀𝐼 + 𝑃𝑃.
𝜀
Proof.
Let consider the error 𝑒 = 𝑥̂ − 𝑥, one has
𝛼,𝜓
𝑡
𝑇𝑡0 0 𝑒 = (𝐴 − 𝐿𝐶 )𝑒 + 𝑓(𝑡, 𝑥̂ ) − 𝑓(𝑡, 𝑥 ).
Consider now the Lyapunov function 𝑉 (𝑡, 𝑒) = 𝑒 𝑇 𝑃𝑒. One has :
𝛼,𝜓𝑡0
𝑇𝑡0
𝛼,𝜓𝑡0
𝑉(𝑡, 𝑒) = 2𝑒 𝑇 𝑃𝑇𝑡0
(6)
𝑒 = 2𝑒 𝑇 𝑃((𝐴 − 𝐿𝐶 )𝑒 + 𝑓(𝑡, 𝑥̂ ) − 𝑓(𝑡, 𝑥 )).
One has:
2𝑒 𝑇 𝑃(𝑓(𝑡, 𝑥̂ ) − 𝑓(𝑡, 𝑥 )) ≤ 2‖𝑓(𝑡, 𝑥̂ ) − 𝑓(𝑡, 𝑥 )‖‖𝑃𝑒‖
1
≤ 𝜀‖𝑓 (𝑡, 𝑥̂ ) − 𝑓(𝑡, 𝑥 )‖2 + ‖𝑃𝑒‖2
𝜀
6
1
≤ 𝜀 ( 2𝑏 2 𝜀‖𝑒‖ + 2𝜑 2 (𝑡)) + ‖𝑃𝑒‖2
𝜀
1
≤ 𝑒 𝑇 (2𝑏 2 𝜀𝐼 + 𝑃𝑃) 𝑒 + 2𝜀𝜑 2 (𝑡).
𝜀
Thus,
𝛼,𝜓𝑡0
𝑇𝑡0
𝑉(𝑡, 𝑒) ≤ 𝑒 𝑇 𝜓𝑒 + 2𝜀𝜑 2 (𝑡) ≤ −𝜆𝑉 (𝑡, 𝑒) + 2𝜀𝜑 2 (𝑡).
We conclude from Lemma 1 that system (5) is a practical observer of the system (2) if the following
condition holds:
𝜓 < 0.
(7)
By using Schur complement on (7), one gets (4)
5. Separation principle
In this part, it is assumed that the control law stabilized the system by using the estimated states
supplied by the observer. For that, one considers the control 𝑢(𝑥̂ ) = 𝐾𝑥̂. One has the following extended
system:
𝛼,𝜓𝑡0
𝑇𝑡0
𝐴 + 𝐵𝐾
𝑥̂
where 𝑥̃ = ( ) , 𝐺 = (
0
𝑒
𝑥̃ = 𝐺𝑥̃ + 𝜙(𝑡, 𝑥, 𝑥̂ ),
(8)
𝑓(𝑡, 𝑥̂)
−𝐿𝐶
), 𝜙(𝑡, 𝑥, 𝑥̂ ) = (
).
𝑓(𝑡, 𝑥̂ ) − 𝑓 (𝑡, 𝑥 )
𝐴 − 𝐿𝐶
One supposes that 𝑟𝑎𝑛𝑘 (𝐶 ) = 𝑞,then there always exist two orthogonal matrices 𝑈 ∈ ℝ𝑞×𝑞 and 𝑊 ∈ ℝ𝑛×𝑛
such that :
𝑈 𝑇 𝐶𝑊 = [𝐶̃ 0],
(9)
where ̃𝐶 = 𝑑𝑖𝑎𝑔(𝑐1 ; … … ; 𝑐𝑞 ) and 𝑐𝑖 (𝑖 = 1; … … ; 𝑞) are nonzero singular values of 𝐶.
The following is the outcome:
+
̃ ∈
Theorem 3. Suppose that (𝐻) holds and there exist 𝑋1 ∈ 𝑆𝑛+ (ℝ), 𝑋211 ∈ 𝑆𝑞+ (ℝ) and 𝑋222 ∈ 𝑆𝑛−𝑞
(ℝ), 𝐾
ℳ𝑝,𝑛 (ℝ), 𝐿̃ ∈ ℳ𝑛,𝑞 (ℝ) and scalars 𝜀̃1 , 𝜀̃2 > 0 satisfying the following LMI:
̃ ) + 𝜀̃1 𝐼
𝑠𝑦𝑚(𝐴𝑋1 + 𝐵𝐾
(
∗
−𝐿̃𝐶
𝑠𝑦𝑚(𝐴𝑋2 − 𝐿̃𝐶) + 𝜀̃2 𝐼
∗
∗
∗
∗
𝑋11
where 𝑋2 = 𝑊 ( 2
0
−
𝑋1
0
0
𝑋2
𝜀̃1
2𝑏 2
𝐼
∗
0
) 𝑊𝑇.
𝑋222
The system (8) is practically fractional exponentially stable if
7
<0
0
−
𝜀̃1
2𝑏 2
𝐼)
(10)
𝑡
𝜓
𝜓
𝑡
𝑡
𝐸 0 (−𝜆, 𝑡, 𝑡0 )𝐸𝛼 0 (𝜆, 𝑠, 𝑡0 ) 2
∫ 𝛼
𝜑̃ (𝑠)𝑑𝑠 ≤ 𝑀,
𝜓𝑡0 (𝑠, 𝛼 )
∀𝑡 ≥ 𝑡0 ≥ 0,
𝑡0
where 𝜆 = −
𝜆𝑚𝑎𝑥 (Ω)
𝜆𝑚𝑎𝑥 (𝑃̃)
in which 𝑃1 =
𝑋1−1 ,
𝑃
and 𝑀 > 0, with 𝑃̃ = ( 1
0
𝑋11
𝑃2 = 𝑊 ( 2
0
−1
0
−1
𝑋222
0
)
𝑃2
−1
̃ 𝑋1−1 , 𝜀1 =
) 𝑊 𝑇 , 𝐿 = 𝐿̃𝑈𝐶̃ 𝑋211 𝐶̃ −1 𝑈 𝑇 , 𝐾 = 𝐾
1
,𝜀
𝜀̃1 2
=
1
𝜀̃2
and
Ω=(
𝑠𝑦𝑚(𝑃1 (𝐴 + 𝐵𝐾 )) + 2𝜀1 𝑏 2 𝐼 +
1
𝑃𝑃
𝜀1 1 1
−𝑃1 𝐿𝐶
𝑠𝑦𝑚(𝑃2 (𝐴 − 𝐿𝐶 )) + 2𝜀2 𝑏 2 𝐼 +
∗
1
𝜀2
𝑃2 𝑃2
).
Proof.
𝑃
0
) with 𝑃1 is a positive definite symmetric
Let consider the Lyapunov function 𝑉 (𝑡) = 𝑥̃ 𝑇 𝑃̃𝑥̃, 𝑃̃ = ( 1
0 𝑃2
𝑃11
0
+
matrix and 𝑃2 = 𝑊 ( 2
) 𝑊 𝑇 where 𝑃211 ∈ 𝑆𝑞+ (ℝ) and 𝑃222 ∈ 𝑆𝑛−𝑞
(ℝ).
0 𝑃222
The conformable derivative of the Lyapunov function 𝑉 (𝑡) gives:
𝛼,𝜓𝑡0
𝑇𝑡0
𝑉(𝑡) = 2𝑥̃ 𝑇 𝑃̃𝐺𝑥̃ + 2𝑥̃𝑃̃𝜙(𝑡, 𝑥, 𝑥̂).
One has
2𝑥̃ 𝑇 𝑃̃𝜙(𝑡, 𝑥, 𝑥̂ ) = 2𝑥̂ 𝑇 𝑃1 𝑓(𝑡, 𝑥̂ ) + 2𝑒 𝑇 𝑃2 (𝑓 (𝑡, 𝑥̂ ) − 𝑓(𝑡, 𝑥 )).
One has
2𝑥̂ 𝑇 𝑃1 𝑓(𝑡, 𝑥̂ ) ≤ 𝑥̂ 𝑇 (2𝜀1 𝑏 2 +
1
𝑃 𝑃 ) 𝑥̂ + 𝑆1 (𝑡),
𝜀1 1 1
where 𝑆1 (𝑡) = 2𝜀1 𝜑̃ 2 (𝑡).
On the other hand, one has ;
2𝑒 𝑇 𝑃2 (𝑓(𝑡, 𝑥̂ ) − 𝑓(𝑡, 𝑥 )) ≤ 𝑒̂ 𝑇 (2𝜀2 𝑏 2 +
1
𝑃 𝑃 ) 𝑒̂ + 𝑆2 (𝑡),
𝜀2 2 2
where 𝑆2 (𝑡) = 2𝜀2 𝜑̃ 2 (𝑡).
Thus,
𝛼,𝜓𝑡0
𝑇𝑡0
𝑉(𝑡) ≤ ̃𝑥 𝑇 Ω𝑥̃ + 𝑆(𝑡) ≤
𝜆𝑚𝑎𝑥 (Ω)
𝑉 (𝑡 ) + 𝑆 (𝑡 ) .
𝜆𝑚𝑎𝑥 (𝑃̃)
(11)
where 𝑆(𝑡) = 𝑆1 (𝑡) + 𝑆2 (𝑡).
We conclude from Lemma 1, that the system (8) is practically fractional exponentially stable if the following
condition holds:
8
Ω < 0.
(12)
We are now in a position to transform the previous condition in terms of LMI, which can be solved
efficiently by using existing solvers such as LMI toolbox in the MATLAB software.
By multiplying the right part and the left part of Ω by 𝑋 where 𝑋 = (
with 𝑋211 = 𝑃211
−1
and 𝑋211 = 𝑃222
−1
0
) 𝑊𝑇
𝑋222
, one has the following inequality :
̃ ) + 2𝜀1 𝑏 2 𝑋1 𝑋1 +
𝑠𝑦𝑚(𝐴𝑋1 + 𝐵𝐾
(
𝑋11
0
), 𝑋2 = 𝑊 ( 2
𝑋2
0
𝑋1
0
1
𝐼
𝜀1
−𝐿𝐶𝑋2
1
𝑠𝑦𝑚((𝐴 − 𝐿𝐶 )𝑋2 ) + 2𝜀2 𝑏 𝑋2 𝑋2 + 𝐼
𝜀2 )
< 0.
(13)
2
∗
Considering (9), the term 𝐿𝐶𝑋2 can be expressed by:
𝑋11
𝐿𝐶𝑋2 = 𝐿𝑈[𝐶̃ 0]𝑊 𝑇 𝑊 ( 2
0
0
) 𝑊 𝑇 = 𝐿𝑈[𝐶̃ 𝑋211 0]𝑊 𝑇 = 𝐿𝑈𝐶̃ 𝑋211 𝐶̃ −1 𝑈 𝑇 𝑈[𝐶̃ 0]𝑊 𝑇 .
𝑋222
Let 𝐿̃ = 𝐿𝑈𝐶̃ 𝑋211 𝐶̃ −1 𝑈 𝑇 , we get
𝐿𝐶𝑋2 = 𝐿̃𝐶.
So, using Schur complement on (13), we get (10).
6. Numerical example:
0.1 −0.4
0.25
), 𝐵 = (
), 𝐶 = (1 0),
0.25 −0.1
0.3
1−𝛼
𝑡
𝑓(𝑡, 𝑥 ) = 𝑏(sin(𝑥1 ) , cos(𝑒 𝑥2 )), 𝛼 = 0.5, and 𝜓𝑡0 = (𝑡 − 𝑡0 ) . The LMI (10) was to be feasible with
0.2888
). The evolution of the errors
the following solutions: 𝐾 = 103 × (0.5403 −1.4636) and 𝐿 = (
−0.0098
are shown in figures 1 and 2 for 𝑡0 = 0.
One supposes a nonlinear system as in the form of (8) where 𝐴 = (
Figure 1. The evolution of the error 𝑒1 .
9
Figure 2. The evolution of the error 𝑒2 .
As illustrated in Figures 1 and 2, the achieved result is satisfactory, and we obtain the system's practical
stability.
7. Conclusion
A global practical conformable stabilisation by output feedback is demonstrated in this paper for a class of
nonlinear fractional order systems. To begin, a stabilisation method is described. Following that, the observer
design for the proposed system is presented. Additionally, the separation principle is discussed. When
developing the work, a novel nonlinear condition is examined. Finally, a numerical example is provided to
demonstrate the accuracy of the proposed methodology.
Referecnes
[1] Monje CA, Chen YQ, Vinagre BM, Xue D, FeliusV.Fractional‐Order Systems and Controls, Series:
Advances in Industrial Control. London: Springer; 2010.
[2] Caponetto R, Dongola G, Fortuna L, PetrásI.Fractional Order Systems: Modelling and Control
Applications. Singapore: World Scientific; 2010.
[3] GolmankhanehAlireza K, Lambert L.Investigations in Dynamics: With Focus on Fractional Dynamics.
Saarbrucken: Academic Publishing; 2012.
[4] Kaczorek T, Rogowski K. Positive fractional electric circuits. Fractional Linear Systems and Electrical
Circuits. Poland: Springer International Publishing; 2015:49‐80.
[5] Bao H, Park JH, Cao J. Adaptive synchronization of fractional‐order memristor‐based neural networks
with time delay. Nonlinear Dyn. 2015;82(3):1343‐1354.
[6] Hartley TT, Veillette RJ, Adams JL, Lorenzo CF. Energy storage and loss in fractional‐order circuit
elemnts. IET Circuits, Devices Syst. 2015;9(3):227‐235.
[7] Valsa J, Vlach J. RC models of a constant phase element. Int J Cicrcuit Theory Appl. 2013;41(1):59‐67.
[8] Soltan A, Radwan AG, Soliman AM. Fractional‐order mutual inductance: analysis and design. Int J
Cicrcuit Theory Appl. 2016;44(1):85‐97.
[9] Baleanu D, Günvenc ZB, Tenreiro Machado JA. New Trends in Nanotechnology and Fractional Calculus
Applications. Dordrecht: Springer; 2010.
[10] Capelas de Oliveira E, Tenreiro Machado JA. A review of definitions for fractional derivatives and
integrals. Math Prob Eng. 2014:Article ID:238459.
[11] Oldham KB, Spanier J. The Fractional Calculus. New York: Academic Press; 1974.
10
[12] Miller KS, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations.
NY: John Wiley; 1993.
[13] Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives: Theory and Applications.
New York: Gordon and Breach; 1993.
[14]. Podlubny I. Fractional Differential Equations. New York: Academic Press; 1999.
[15] Uchaikin V. Fractional Derivatives for Physicists and Engineers: Springer; 2013.
[16] Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Progr Fract
Differ Appl. 2015;1:73‐85.
[17] Atangana A, Baleanu D. New fractional derivatives with non‐local and non‐singular kernel, theory and
application to heat transfer model. Thermal Sci. 2016;20(2):763‐769.
[18] Abel NH. Résolution d'un Probléme de Mécanique. Œuvres Complétes (Tomo Premier, pp. 27‐30).
Chirstiana: Gr'́ondah; 1839a.
[19] Caputo M, Mainardi F. A new dissipation model based on memory mechanism. Pure Appl Geophys.
1971;91(8):134‐147.
[20] Wyss W. Fractional diffusion equation. J Math Phys. 1986;27:2782‐2785.
[21] Westerlund S. Capacitor theory. IEEE Transactions on Dielectrics and Electrical Insulation.
1994;1(5):826‐839.
[22] Hermann R. Fractional Calculus. New Jersey: World Scientific; 2011.
[23] Teka WW, Upadhyay RK, Mondal A. Fractional‐order leaky integrate‐and‐fire model with long‐term
memory and power law dynamics. Neural Networks. 2017;93:110‐125.
[24] Khalil R, MAlHorani, Yousef A, Sababheh M. A new definition of fractional derivative. J Comp Appl
Math. 2014; 264:65‐70.
[25] Katugampola UN. A new fractional derivative with classical properties. arXiv:1410.6535v1; 2014.
[26] Abdeljawad T. On conformable fractional calculus. J Comput Appl Math. 2015;279:57‐66.
[27] Cenesiz Y, Baleanu D, Kurt A, Tasbozan O. New exact solutions of Burger's type equations with
conformable derivative. Waves Random Complex Media. 2017;27(1):103‐116.
[28] Zhao D, Li T. On conformable delta fractional calculus on time scales. J Math Comput Sci.
2016;16:324‐335.
[29] Al Horani M, Abu Hammad M, Khalil R. Variation of parameters for local fractional non homogeneous
linear differential equations. J Math Comput Sci. 2016;16:147‐153.
[30] Hammad A, Khalil R. Fractional Fourier series with applications. Am J Comput Appl Math.
2014;4(6):187‐191.
[31] Atangana A, Dumitru D, Alsaedi A. New properties of conformable derivative. Open Math.
2015;13:889‐898.
[32] S. Li, S. Zhang, and R. Liu, “The existence of solution of diffusion equation with the general
conformable derivative,” Journal of Function Spaces, vol. 2020, Article ID 3965269, 10 pages, 2020.
[33] D. Zhao and M. Luo, “General conformable fractional derivative and its physical interpretation,”
Calcolo, vol. 54, pp. 903–917, 2015.
11
[34] Ellouze, I. and M. A. Hammami, “A separation principle of time varying dynamical systems: A
practical stability approach,” Math. Model. Anal., Vol. 12, No. 3, pp. 297–308 (2007).
[36] O Naifar, A Ben Makhlouf, MA Hammami, L Chen, Global practical MittagLeffler stabilization by
output feedback for a class of nonlinear fractional‐order systems, Asian journal of control 20 (1), 599-607,
2018.
[37] Jian Yang, Michal Fečkan, JinRong Wang, Consensus of linear conformable fractional order multiagent systems with impulsive control protocols, Asian journal of control, 2022, DOI : 10.1002/asjc.2775.
[38] Nadjwa Berredjem, Banan Maayah, Omar Abu Arqu, A numerical method for solving conformable
fractional integrodifferential systems of second-order, two-points periodic boundary conditions, Alexandria
Engineering Journal, 2022, DOI: 10.1016/j.aej.2021.11.025.
[39] Wen-Ze Wu,Liang Zeng, Chong Liu, Wanli Xie, Mark Goh, A time power-based grey model with
conformable fractional derivative and its applications, Chaos, Solitons & Fractals, 2022, DOI:
10.1016/j.chaos.2021.111657.
[40] Wenqing Wu, Xin Ma, Bo Zeng, Hui Zhang, Peng Zhang, A novel multivariate grey system model with
conformable fractional derivative and its applications, Computers & Industrial Engineering, 2022, DOI:
10.1016/j.cie.2021.107888.
[41] Zeyad Al-Zhour, Controllability and observability behaviors of a non-homogeneous conformable
fractional dynamical system compatible with some electrical applications, Alexandria Engineering Journal,
2022, DOI: 10.1016/j.aej.2021.07.018.
[42] A Ben Makhlouf, O Naifar, On the Barbalat lemma extension for the generalized conformable fractional
integrals: Application to adaptive observer design, Asian Journal of Control, 2022, DOI: 10.1002/asjc.2797.
[43] O Naifar, A Jmal, A Ben Makhlouf, Non‐fragile H∞ observer for Lipschitz conformable
fractional‐order systems, Asian Journal of Control, 2021, DOI: 10.1002/asjc.2626.
12