Full-State Observer Synthesis for a Class of One-Sided Lipschitz
Nonlinear Fractional-Order Systems
MOHSEN MOHAMED HADJI1, SAMIR LADACI2
1EEA Department, GEPC Laboratory,
National Polytechnic School of Constantine,
Constantine,
ALGERIA
2Department of Automatic Control Engineering,
Ecole Nationale Polytechnique,
El Harrach,16200, Algiers,
ALGERIA
Abstract: - A full-order observer design for a large class of fractional-order continuous-time nonlinear systems
that satisfies the one-sided Lipschitz and quadratic inner boundedness conditions problem is addressed. By
employing the fractional-order extension of the Lyapunov direct approach, a sufficient condition for the
existence of the proposed observer is established in the form of a Nonlinear Matrix Inequality (NMI),
guaranteeing the asymptotic convergence of the observation error to the origin. The effectiveness and distinct
advantages of the proposed design are validated through numerical simulations on a representative fractional-
order system.
Key-Words: - Fractional-order observers, One-sided Lipschitz systems, Quadratic inner boundness, Riccati
equation, Lyapunov direct approach extension, Nonlinear systems, Nonlinear Matrix Inequality
(NMI).
Received: May 12, 2024. Revised: October 14, 2024. Accepted: November 16, 2024. Published: December 27. 2024.
1 Introduction
In the last few decades, state estimation of nonlinear
systems has emerged as a vital research focus for a
broad community of researchers, [1], [2], [3],
particularly in the control systems field, [4], [5], [6],
where state measurements are essential for
designing robust and effective control strategies.
Despite significant advancements, designing a state
observer for a general class of nonlinear systems
remains a challenging and open topic.
A review of the literature reveals that
established observer design techniques can be
categorized into two distinct approaches. The first
involves applying a coordinate transformation to the
observation error dynamical system, thus converting
it into a linear form, [7]. This transformation
facilitates the implementation of well-known
techniques applicable to linear systems; however, it
is often constrained by the complexities of
establishing such transformations. In contrast, the
second approach does not require any state
transformation, directly utilizing the system
dynamics in the observer design, [8]. The most
widely adopted strategy in this context is predicated
on solving a Riccati-like equation for specific
classes of nonlinear systems that satisfy the
Lipschitz continuity condition, [9], [10], [11], [12].
Notably, most existing results are grounded in the
principle that the linear component of the
observation error dynamics predominates over the
nonlinear component. This approach leverages the
Lipschitz property of nonlinearity, allowing for the
substitution of the nonlinear term with a linear
positive term, thereby simplifying the observer
design process, [13], [14]. However, for nonlinear
systems characterized by large Lipschitz constants,
this methodology may become ineffective, as the
associated Riccati-like equations may no longer be
solvable. To expand the class of nonlinear systems
that can be considered and surmounting the
drawbacks of the aforementioned techniques, a
more general condition for observer design is
introduced, referred to as the one-sided Lipschitz
continuity condition. Up to date, many interesting
observation schemes for this class of system have
been developed; readers can refer to [15], [16].
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Over the last three centuries, fractional-order
calculus has made substantial contributions to the
mathematical literature. Its fundamental concept lies
in generalizing the traditional definitions of
derivatives and integrals, [17]. With these powerful
tools, dynamical systems modeling, analysis, and
control have undergone considerable development
in various scientific fields, including viscoelastic
[18], electromagnetism [19], biology [20],
mechanics [21], robotics [22], aerodynamics [23],
renewable energy [24], and many others.
Nevertheless, the design of observers for fractional-
order systems, particularly those with one-sided
Lipschitz conditions, remains an underexplored
area, with few existing results in the current
literature [25], [26].
Motivated by the aforementioned discussion,
the primary contribution of this paper is the
development of a novel NMI-based observer for
fractional-order one-sided Lipschitz nonlinear
systems. Utilizing the one-sided Lipschitz property
and quadratic inner boundness, along with the
fractional-order extension of the Lyapunov direct
method, we derive sufficient conditions for the
observer’s existence and the asymptotic
convergence of the observation error, expressed in
the form of an NMI.
The rest of this paper is organized as follows:
Section 2 introduces the foundational concepts and
pertinent results related to fractional-order calculus.
In Section 3, the observation problem for fractional-
order one-sided Lipschitz systems is
comprehensively detailed. The primary
contributions and main results are presented in
Section 4. To validate the efficacy of the proposed
observation technique, simulation results for a
fractional-order nonlinear system are provided in
Section 5. Finally, conclusions are drawn in Section
6.
2 Preliminaries
Fractional calculus is concerned with the integrals
and derivatives of orders that might have real or
complex values. One of the fundamental notions of
fractional-order calculus is the Riemann-Liouville
fractional integral, which is stated in Definition 1.
Definition 1, [27]. The orderorder Riemann-
Liouville fractional integral of a continuous function
on the left half-axis of the real numbers is
defined by:
(1)
Where is the gamma function.
Definition 2, [27]. The Riemann-Liouville
fractional derivative of order of a function
is defined by:
(2)
Where
Definition 3, [27]. The Caputo fractional
Derivatives of order of a function is
defined as follows:
(3)
It is the one most frequently used in engineering
problems and the one used in this paper.
Lemma 1, [28]. Let be a derivable functions
vector. Then for any given time instant :
(4)
Theorem 1, [29]. Let be an equilibrium point
for the Caputo fractional non-autonomous system
(5)
where satisfies the Lipschitz condition with
as Lipschitz constant and .
Assume that there exists a Lyapunov
functionalsatisfying:
(6)
where are positive
constants and denotes an arbitrary norm. Then
the equilibrium point of the system (.) is Miattag-
Leffler stable.
Lemma 2, [30]. Consider a given matrix
knowing that
and
In
this case, the criteria set out below are equivalent:
1
2
3
3 Problem Statement
In this study, we are interested in the class of
fractional-order nonlinear systems, modeled by:
(7)
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Where is the state vector, is the
output, denotes the input,
, and represents a one-sided Lipschitz
nonlinear functions vector.
Before proceeding with our problem statement,
we require definitions of Lipschitz and one-sided
Lipschitz nonlinear functions.
Definition 4. The function is said to be
locally Lipschitz with respect to in a region if
there exists a constant such that the following
condition holds:
(8)
Where is the Lipschitz constant and is an
admissible control law. If ,then is
said to be globally Lipschitz.
Definition 5. The function is said to be one-
sided Lipschitz if , there exists a constant
satisfiying:
(9)
Where is called the one-sided Lipschitz constant
and it corresponds to the Jacobian’s logarithmic
matrix norm:
(10)
Definition 6. The function is quadratically
inner bounded in if there exsit such that
the following inequality holds .
(11)
Assuming that the defined system includes
certain states that cannot be directly measured, this
research aims to devise a fractional-order observer
that provides an accurate estimate of the full state
vector. To overcome this latter challenge, we
consider the observation scheme as follows:
(12)
Then the observation error system dynamics for
are provided by:
(13)
Where .
Here, the observer gain should be designed in
such a way as to guarantee the asymptotic
convergence of the error system trajectories toward
the origin.
4 Main Results
In this section, a sufficient condition for the
existence and asymptotic convergence of the
proposed observer is established. The following
theorem summarizes our main findings:
Theorem 2. Assuming that system (7) satisfies the
conditions (10) and (11) with constants and ,
and if there exist scalars such that the
following Riccati-like inequality has a symmetric
positive definite solution :
(14)
And the observer holds the form (13), with:
(15)
Then it can be assured that observer error
dynamicsare asymptotically stable.
Proof: Examining the following Lyapunov
functional candidate:
(16)
Applying the fractional derivative of orderto
expression (19) and referencing Lemma 1 yields:
(17)
By substituting Equation (14) into Equation (17), we
obtain:
(18)
The following property holds for all scalar values of
:
(19)
Since is quadratically inner bounded, it
follows from (-) that one can derive:
(20)
From the one sided Lipschitz definition, it is evident
that: (21)
On the basis of inequalities (20) and (21), we
can express equation (18) in the following
reformulated form:
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(22)
To achieve , the following
condition should be accomplished:
(23)
By referring to Lemma 2 the condition in (23) can
be expressed as :
(24)
Let
, thenconditions (24), (14) are
equivalent.
5 Simulation Results
This section presents a numerical simulation
example demonstrating the effectiveness of the
fractional-order observer design technique
introduced in Section 3. We consider the simulation
example provided in [31]. Consider a fractional-
order nonlinear dynamical system represented by Eq
(1), defined as follows:
(25)
(26)
By applying the mean value theorem one can derive:
(27)
(28)
Consequently, the system’s nonlinearity
satisfies the one-sided Lipschitz continuity
condition (.) and the quadratic inner boundness(.),
with and . In turn, Theorem
2 can be examined to design a full-state observer for
this system. By setting and utilizing the
YALMIP toolbox, the resolution of the NMI (−)
provides the following result:
(29)
Thus, the observer gain matrix:
(30)
The system’s initial conditions are selected as
, whereas the designed observer is
initialized with zero initial conditions.
Fig. 1: Actual state and its estimation time
history
Fig. 2: Actual state and its estimation time
evolution
Fig. 3: The observation error time evolution
The observation results of the system states
and are illustrated in Figure 1 and Figure 2,
respectively.The proposed observer demonstrates
significant effectiveness in accurately reconstructing
the system's full state vector, which. This
effectiveness is further corroborated by the
observation error time history Figure 3, which
exhibits asymptotic convergence to zero.
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6 Conclusion
An innovative NMI-based observer design for a
broader class of nonlinear systems is presented in
this study. Leveraging the one-sided Lipschitz
condition and the quadratic inner-boundedness
property, combined with the fractional-order
extension of Lyapunov's direct method, sufficient
conditions for the applicability of the proposed
observer are established. The stability analysis
ensures that the observation error converges
asymptotically to the origin. The effectiveness of the
proposed technique is validated through the state
estimation of a fractional-order nonlinear system.
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Contribution of Individual Authors to the Creation
of a Scientific Article (Ghostwriting Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
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