State and fault estimation based on fuzzy observer for a class of Takagi-Sugeno singular models

Indonesian Journal of Electrical Engineering and Computer Science
Vol. 25, No. 1, January 2022, pp. 172~182
ISSN: 2502-4752, DOI: 10.11591/ijeecs.v25.i1.pp172-182  172
Journal homepage: http://ijeecs.iaescore.com
State and fault estimation based on fuzzy observer for a class of
Takagi-Sugeno singular models
Kaoutar Ouarid1,2
, Mohamed Essabre3
, Abdellatif El Assoudi1,2
, El Hassane El Yaagoubi1,2
1
Laboratory of High Energy Physics and Condensed Matter, Faculty of Science Ain Chock, Hassan II University of Casablanca,
Casablanca, Morocco
2
ECPI, Department of Electrical Engineering, ENSEM, Hassan II University of Casablanca, Casablanca, Morocco
3
Laboratory of Condensed Matter Physics and Renewable Energy, Faculty of Sciences and Technologies Mohammedia,
Hassan II University of Casablanca, Casablanca, Morocco
Article Info ABSTRACT
Article history:
Received Jun 23, 2021
Revised Oct 23, 2021
Accepted Nov 30, 2021
Singular nonlinear systems have received wide attention in recent years, and
can be found in various applications of engineering practice. On the basis of
the Takagi-Sugeno (T-S) formalism, which represents a powerful tool
allowing the study and the treatment of nonlinear systems, many control and
diagnostic problems have been treated in the literature. In this work, we aim
to present a new approach making it possible to estimate simultaneously
both non-measurable states and unknown faults in the actuators and sensors
for a class of continuous-time Takagi-Sugeno singular model (CTSSM).
Firstly, the considered class of CTSSM is represented in the case of premise
variables which are non-measurable, and is subjected to actuator and sensor
faults. Secondly, the suggested observer is synthesized based on the
decomposition approach. Next, the observer’s gain matrices are determined
using the Lyapunov theory and the constraints are defined as linear matrix
inequalities (LMIs). Finally, a numerical simulation on an application
example is given to demonstrate the usefulness and the good performance of
the proposed dynamic system.
Keywords:
Fault diagnosis
Fuzzy observer
LMIs
Lyapunov theory
Takagi-Sugeno singular model
This is an open access article under the CC BY-SA license.
Corresponding Author:
Kaoutar Ouarid
Laboratory of High Energy Physics and Condensed Matter, Faculty of Science Ain Chock
Hassan II University of Casablanca
Km 8 Road El Jadida, B.P 5366 Maarif, 20100, Casablanca 20000, Morocco
Email: kaoutar.ouarid@gmail.com
1. INTRODUCTION
Over the last decades, the increase in the performance of equipment in terms of production quality
and gain in productivity was accompanied by the complexity of the equipment. However, the presence of
abnormal changes due to actuator or system or sensor faults can degrade system performances, hence the
need to integrate fault detection and diagnosis (FDD) tools [1], [2] to maintain, for a long time, the desired
performance of the whole system in various sectors. In particular, FDD has a very important role in
monitoring the behavior of system variables and revealing faults, and it is performed based on the relative
information to the system and its equipment. This information can be obtained by adding sensors to acquire
measured states or observers to estimate non-measured states requiring expensive or difficult sensors to
maintain. The state modeling of process dynamics is often obtained based on its state variables linked
together by mathematical equations. If these processes have constraints, then it is necessary to use static
equations to sufficiently characterize the studied process. Such systems composed of static and dynamic
equations are called singular, or descriptor or implicit systems [3]. Recently, the FDD problem for singular

Indonesian J Elec Eng & Comp Sci ISSN: 2502-4752 
State and fault estimation based on fuzzy observer for a class of Takagi-Sugeno singular… (Kaoutar Ouarid)
173
systems has attracted much attention in various fields such as mechanical engineering, computer science,
civil engineering, electrical engineering and automation.
Various techniques for detecting and estimating faults have been proposed for the class of linear
systems [4], [5], and for the class of nonlinear systems [6]-[8] allowing to provide a closer representation to
the real system, but which are difficult to exploit. Due to this complexity, it has become essential to work
with a precise class of nonlinear systems such as Lipschitz systems, uncertain systems, bilinear systems or
others. From these classes of nonlinear systems, we find the class of Takagi-Sugeno (T-S) [9] nonlinear
systems, in ordinary or singular form. It has been introduced to compromise between the good precision of
the nonlinear behavior of the studied system, and the use of techniques adapted to linear systems due to the
convex sum property of its activation functions [3], [10].
There have been many methods of FDD [2], [11], [12] which can be classified into signal-based
approaches [13], knowledge-based approaches [12], [14], and process model-based approaches [12], [13]
which contain state observer-based method representing an analytical method having achieved several results
in this field, and which depends on the mathematical model of the studied system without needing other
components. Many publications have been interested in the design of observers for FDD [15]-[24] and have
presented fruitful results. A residual generator for detecting and isolating actuator faults for a class of T-S
fuzzy bilinear system is developed in [18]. Developing a novel fuzzy FD observer for FD of sensors faults of
T-S fuzzy systems is the aim of the work presented in [19]. In [20], depicted a T-S unknown input observer
to simultaneously estimate the interval of states and actuator faults for a class of T-S explicit systems.
Another technique based on a robust fault estimation observer has been introduced for estimating actuator
faults for a class of discrete-time singular systems [21]. In [22], a design of an adaptive observer is proposed
for detecting sensor faults of an industrial servo system. For the fault diagnosis and reconstruction of the
faults affecting the states of the system, in [23] suggested a new augmented linear parameter-varying (LPV)
observer for a class of LPV models. The design of a combination of reduced-order LPV and full-order LPV
unknown input observers, respectively, for FDD of actuator and sensor faults of industrial processes is
presented in [24].
Most of these observers are synthesized to estimate only actuator or sensor faults while guaranteeing
asymptotic convergence for various class of nonlinear system in continuous or discrete-time. The goal of our
work is not to compare our approach with those already carried out, but rather to extend our results from the
case of singular linear models [25] and T-S singular models with measurable premise variables [26] to the
case of T-S singular models with unmeasurable premise variables while ensuring an exponential
convergence, and simultaneously estimating the unmeasurable states and the faults at the level of actuators
and sensors. In this work, for simultaneous estimation of states and faults, the novel suggested technique
consists to associate for each local model a local observer. Then, the proposed fuzzy observer is obtained by
an aggregation of the local observers. Our contribution is based on the separation of the dynamic equations
from the static equations which makes it possible to facilitate and minimize the computation by obtaining the
static states just from the dynamic states already found. The design conditions are expressed in terms of
LMIs. This observer is applied for both actuators and sensors faults for a class of T-S singular model in the
case of unmeasurable premise variables. The paper is composed of five parts that are presented as follows:
Section 2 exposes the class of the studied system. Section 3 provides the synthesis of the proposed observer
and the stability conditions. The numerical results of the application example are given in section 4. Section 5
is devoted to a brief conclusion.
2. MATHEMATICAL FORMULATION OF THE CONSIDERED MODEL
In this paper, the following class of continuous-time Takagi-Sugeno singular model (CTSSM) with
unmeasurable premise variables in presence of actuator and sensor fault is considered (1),
{
𝑀𝑧̇ = ∑ 𝜌𝑖(𝛽)
𝑞
𝑖=1 (𝐴𝑖𝑧 + 𝐵𝑖𝜏 + 𝑀𝑎𝑖𝑣𝑎)
𝑦 = ∑ 𝜌𝑖(𝛽)
𝑞
𝑖=1 (𝐶𝑖𝑧 + 𝐷𝑖𝜏 + 𝐷𝑎𝑖𝑣𝑎 + 𝑀𝑠𝑖𝑣𝑠)
(1)
where 𝑧 = [𝑍1
𝑇
𝑍2
𝑇
]𝑇
∈ ℝ𝑛
is the state vector with 𝑍1 ∈ ℝ𝑟
is the vector of dynamic variables, 𝑍2 ∈ ℝ𝑛−𝑟
is
the vector of static variables, 𝜏 ∈ ℝ𝑝
is the control input, 𝑦 ∈ ℝ𝑚
is the measured output vector, 𝑣𝑎 ∈ ℝ𝑎
and 𝑣𝑠 ∈ ℝ𝑏
are the actuator and sensor fault vectors, respectively. The matrices 𝐴𝑖, 𝐵𝑖, 𝑀𝑎𝑖, 𝐶𝑖, 𝐷𝑖, 𝐷𝑎𝑖 and
𝑀𝑠𝑖 are real known constant matrices with adequate dimensions related with the 𝑖𝑡ℎ
local model with,
𝐴𝑖 = (
𝐴11𝑖 𝐴12𝑖
𝐴21𝑖 𝐴22𝑖
) ; 𝐵𝑖 = (𝐵1𝑖
𝐵2𝑖
) ; 𝑀𝑎𝑖 = (
𝑀𝑎𝑖
1
𝑀𝑎𝑖
2 ) ; 𝐶𝑖 = (𝐶1𝑖 𝐶2𝑖) (2)

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Indonesian J Elec Eng & Comp Sci, Vol. 25, No. 1, January 2022: 172-182
174
the rank of the matrices 𝐴22𝑖 are equal to 𝑛 − 𝑟 and it is supposed to be invertible. 𝑞 represents the number of
sub-models, and the premise variable 𝛽 is supposed to be real-time accessible.
{
𝑀𝑧̇ = 𝐴𝑖𝑧 + 𝐵𝑖𝜏 + 𝑀𝑎𝑖𝑣𝑎
𝑦 = 𝐶𝑖𝑧 + 𝐷𝑖𝜏 + 𝐷𝑎𝑖𝑣𝑎 + 𝑀𝑠𝑖𝑣𝑠
(3)
The transition between the contributions of each sub model (3) is ensured by the terms 𝜌𝑖(𝛽) which
represent the weighting functions, depending on the states of the system and verifying the property of the
convex sum,
∑ 𝜌𝑖(𝛽)
𝑞
𝑖=1 = 1 ; 0 ≤ 𝜌𝑖(𝛽) ≤ 1 ; 𝑖 = 1, … , 𝑞 (4)
the matrix 𝑀 whose 𝑟𝑎𝑛𝑘(𝑀) = 𝑟 < 𝑛 is assumed to have the following form,
𝑀 = (
𝐼 0
0 0
) (5)
Assumption 1: Assume that [3]:
− (𝑀, 𝐴𝑖) are regular, i.e. det(𝑠𝑀 − 𝐴𝑖) ≠ 0 ∀𝑠𝜖ℂ
− The sub-models (3) are impulse observable and detectable
The separation of the dynamic equations from the static equation in each sub-model (3) is the aim of
our approach, and then the aggregation of the resulting sub-models allows obtaining the global fuzzy model.
So, using the expression of the matrices (2) and (5), the sub-model (3) can be written in the following second
equivalent form [3],
{
𝑍̇1 = 𝐴11𝑖𝑍1 + 𝐴12𝑖𝑍2 + 𝐵1𝑖𝜏 + 𝑀𝑎𝑖
1
𝑣𝑎
0 = 𝐴21𝑖𝑍1 + 𝐴22𝑖𝑍2 + 𝐵2𝑖𝜏 + 𝑀𝑎𝑖
2
𝑣𝑎
𝑦 = 𝐶1𝑖𝑍1 + 𝐶2𝑖𝑍2 + 𝐷𝑖𝜏 + 𝐷𝑎𝑖𝑣𝑎 + 𝑀𝑠𝑖𝑣𝑠
(6)
by finding the expression of the static variable Z2, and replacing it in (6), we obtain,
{
𝑍̇1 = 𝐽𝑖𝑍1 + 𝐿𝑖𝜏 + 𝑁𝑎𝑖𝑣𝑎
𝑍2 = 𝑆𝑖𝑍1 + 𝑉𝑖𝜏 + 𝑅𝑎𝑖𝑣𝑎
𝑦 = 𝑃𝑖𝑍1 + 𝑇𝑖𝜏 + 𝐾𝑎𝑖𝑣𝑎 + 𝑀𝑠𝑖𝑣𝑠
(7)
where,
{
𝐽𝑖 = 𝐴11𝑖 + 𝐴12𝑖𝑆𝑖
𝐿𝑖 = 𝐵1𝑖 + 𝐴12𝑖𝑉𝑖
𝑁𝑎𝑖 = 𝑀𝑎𝑖
1
+ 𝐴12𝑖𝑅𝑎𝑖
𝑆𝑖 = −𝐴22𝑖
−1
𝐴21𝑖
𝑉𝑖 = −𝐴22𝑖
−1
𝐵2𝑖
𝑅𝑎𝑖 = −𝐴22𝑖
−1
𝑀𝑎𝑖
2
𝑃𝑖 = 𝐶1𝑖 + 𝐶2𝑖𝑆𝑖
𝑇𝑖 = 𝐷𝑖 + 𝐶2𝑖𝑉𝑖
𝐾𝑎𝑖 = 𝐷𝑎𝑖 + 𝐶2𝑖𝑅𝑎𝑖
(8)
let define,
𝑣 = (𝑣𝑎
𝑣𝑠
) (9)
which is equivalent to the following state representation,
{
𝑍̇1 = 𝐽𝑖𝑍1 + 𝐿𝑖𝜏 + 𝑁𝑖𝑣
𝑍2 = 𝑆𝑖𝑍1 + 𝑉𝑖𝜏 + 𝑅𝑖𝑣
𝑦 = 𝑃𝑖𝑍1 + 𝑇𝑖𝜏 + 𝐾𝑖𝑣
(10)

175
where,
{
𝑁𝑖 = (𝑁𝑎𝑖 0)
𝑅𝑖 = (𝑅𝑎𝑖 0)
𝐾𝑖 = (𝐾𝑎𝑖 𝑀𝑠𝑖)
(11)
then, from (10) ρi(β) can be rewritten as,
𝜌𝑖(𝛽) = 𝜌𝑖(𝑍1, 𝑍2 = 𝑆𝑖𝑍1 + 𝑉𝑖𝜏 + 𝑅𝑖𝑣) = 𝜌𝑖(𝑍1, 𝜏, 𝑣) = 𝜌𝑖(⍵) (12)
with ⍵𝑇
= [𝑍1
𝑇
𝜏𝑇
𝑣𝑇
]. So, the system (1) can be rewritten under the following equivalent form,
{
𝑍̇1 = ∑ 𝜌𝑖(⍵)
𝑞
𝑖=1 (𝐽𝑖𝑍1 + 𝐿𝑖𝜏 + 𝑁𝑖𝑣)
𝑍2 = ∑ 𝜌𝑖(⍵)
𝑞
𝑖=1 (𝑆𝑖𝑍1 + 𝑉𝑖𝜏 + 𝑅𝑖𝑣)
𝑦 = ∑ 𝜌𝑖(⍵)
𝑞
𝑖=1 (𝑃𝑖𝑍1 + 𝑇𝑖𝜏 + 𝐾𝑖𝑣)
(13)
Assumption 2: Assume that 𝑣 is considered in the following form,
𝑣 = 𝑐0 + 𝑐1𝑡 + 𝑐2𝑡2
+ ⋯ + 𝑐𝑚𝑓
𝑡𝑚𝑓 (14)
where 𝑐𝑘 ; 𝑘 = 0,1, … , 𝑚𝑓 are real unknown constant parameters and the (𝑚𝑓 + 1)𝑡ℎ
time derivative of the
fault is null.
Let,
𝜑𝑘 = 𝑣𝑘−1
with 𝑘 = 1, … , 𝑚𝑓 + 1 (15)
then,
{
𝜑𝑘
̇ = 𝜑𝑘+1
𝜑̇𝑚𝑓+1 = 0 with 𝑘 = 1, … , 𝑚𝑓 (16)
thus, we rewrite the system (13) under the equivalent augmented state form as follows,
{
𝒳̇1 = ∑ 𝜌𝑖(𝜃)
𝑞
𝑖=1 (𝐽
̃𝑖𝒳1 + 𝐿
̃𝑖𝜏)
𝒳2 = ∑ 𝜌𝑖(𝜃)
𝑞
𝑖=1 (𝑆
̃𝑖𝒳1 + 𝑉𝑖𝜏)
𝑦 = ∑ 𝜌𝑖(𝜃)
𝑞
𝑖=1 (𝑃
̃𝑖𝒳1 + 𝑇𝑖𝜏)
(17)
where,
{
𝒳1
𝑇
= (𝑍1
𝑇
𝜑1
𝑇
⋯ 𝜑𝑚𝑓
𝑇
)
𝒳2 = 𝑍2
𝜃 = (
𝒳1
𝜏
)
𝐽
̃𝑖 =
(
𝐽𝑖 𝑁𝑖 0 ⋯ 0
0 0 𝐼 ⋯ 0
⋮ ⋱ ⋱ ⋱ ⋮
0 0 0 ⋯ 𝐼
0 0 ⋯ 0 0)
𝐿
̃𝑖 = (𝐿𝑖
𝑇
0 0 ⋯ 0)𝑇
𝑆
̃𝑖 = (𝑆𝑖 𝑅𝑖 0 ⋯ 0)
𝑃
̃𝑖 = (𝑃𝑖 𝐾𝑖 0 ⋯ 0)
(18)
3. RESEARCH METHOD
The following section shows the design of new structure of fuzzy observer allowing the
simultaneous estimation of the unmeasurable states and unknown faults of the equivalent structure (17) of the
CTSSM (1),

 ISSN: 2502-4752
176
{
𝒳
̂1
̇ = ∑ 𝜌𝑖(𝜃
̂)
𝑞
𝑖=1 (𝐽
̃𝑖𝒳
̂1 + 𝐿
̃𝑖𝜏 − 𝐺𝑖(𝑦
̂ − 𝑦))
𝒳
̂2 = ∑ 𝜌𝑖(𝜃
̂)
𝑞
𝑖=1 (𝑆
̃𝑖𝒳
̂1 + 𝑉𝑖𝜏)
𝑦
̂ = ∑ 𝜌𝑖(𝜃
̂)
𝑞
𝑖=1 (𝑃
̃𝑖𝒳
̂1 + 𝑇𝑖𝜏)
(19)
such that the estimated vectors of (𝒳1, 𝒳2) and y are denoted by (𝒳
̂1, 𝒳
̂2) and y
̂, respectively.
For 𝑖 = 1, ⋯ , 𝑞 the term 𝐺𝑖 expresses the observer gain for the 𝑖𝑡ℎ
submodel such as the estimated
of the augmented vector of the states and faults tends asymptotically towards the real vector.
Defining,
𝑒 = (
𝑒1
𝑒2
) = (
𝒳
̂1 − 𝒳1
𝒳
̂2 − 𝒳2
) (20)
substituting (17) and (19) into (20) gives the following static and dynamic equations of the state estimation
error,
{
𝑒̇1 = ∑ 𝜌𝑖(𝜃
̂)
𝑞
𝑖=1 (𝐽
̃𝑖𝒳
̂1 + 𝐿
̃𝑖𝜏 − 𝐺𝑖(𝑦
̂ − 𝑦)) − ∑ 𝜌𝑖(𝜃)
𝑞
𝑖=1 (𝐽
̃𝑖𝒳1 + 𝐿
̃𝑖𝜏)
𝑒2 = ∑ 𝜌𝑖(𝜃
̂)
𝑞
𝑖=1 (𝑆
̃𝑖𝒳
̂1 + 𝑉𝑖𝜏) − ∑ 𝜌𝑖(𝜃)
𝑞
𝑖=1 (𝑆
(21)
equivalent to,
{
𝑒̇1 = ∑ 𝜌𝑖(𝜃
̂)
𝑞
𝑖=1 (𝐽
̃𝑖𝑒1 − 𝐺𝑖(𝑦
̂ − 𝑦)) − ∑ (𝜌𝑖(𝜃) − 𝜌𝑖(𝜃
̂))
𝑞
𝑖=1 (𝐽
̃𝑖𝒳1 + 𝐿
̃𝑖𝜏)
𝑒2 = ∑ 𝜌𝑖(𝜃
̂)
𝑞
𝑖=1 𝑆
̃𝑖𝑒1 − ∑ (𝜌𝑖(𝜃) − 𝜌𝑖(𝜃
̂))
𝑞
𝑖=1 (𝑆
(22)
let consider ℱ𝑖 = 𝐽
̃𝑖, 𝐿
̃𝑖, 𝑆
̃𝑖, 𝑉𝑖 and,
∑ (𝜌𝑖(𝜃) − 𝜌𝑖(𝜃
̂)) ℱ𝑖 =
𝑞
𝑖=1 ∑ 𝜌𝑖(𝜃)𝜌𝑗(𝜃
̂)
𝑞
𝑖,𝑗=1 ∆ ℱ𝑖𝑗 (23)
with ∆ ℱ𝑖𝑗 = ℱ𝑖 − ℱ𝑗.
Then, by using the expression (23) the system (22) becomes,
{
𝑒̇1 = ∑ 𝜌𝑖(𝜃
̂)
𝑞
𝑖=1 (𝐽
̃𝑖𝑒1 − 𝐺𝑖(𝑦
̂ − 𝑦)) − ∑ 𝜌𝑖(𝜃)𝜌𝑗(𝜃
̂)
𝑞
𝑖,𝑗=1 (∆𝐽
̃𝑖𝑗𝒳1 + ∆𝐿
̃𝑖𝑗𝜏)
𝑒2 = ∑ 𝜌𝑖(𝜃
̂)
𝑞
𝑖=1 𝑆
̃𝑖𝑒1 − ∑ 𝜌𝑖(𝜃)𝜌𝑗(𝜃
̂)
𝑞
𝑖,𝑗=1 (∆𝑆
̃𝑖𝑗𝒳1 + ∆𝑉𝑖𝑗𝜏)
(24)
as ∑ 𝜌𝑖(𝜃) = 1
𝑞
𝑖=1 , we obtain,
{
𝑒̇1 = ∑ 𝜌𝑖(𝜃)𝜌𝑗(𝜃
̂)
𝑞
𝑖,𝑗=1 (𝐽
̃𝑗𝑒1 − 𝐺𝑗(𝑦
̂ − 𝑦)) − ∑ 𝜌𝑖(𝜃)𝜌𝑗(𝜃
̂)
𝑞
𝑖,𝑗=1 (∆𝐽
̃𝑖𝑗𝒳1 + ∆𝐿
̃𝑖𝑗𝜏)
𝑒2 = ∑ 𝜌𝑖(𝜃)𝜌𝑗(𝜃
̂)
𝑞
𝑖,𝑗=1 (𝑆
̃𝑗𝑒1 − ∆𝑆
̃𝑖𝑗𝒳1 − ∆𝑉𝑖𝑗𝜏)
(25)
in the same way, we can get 𝑦 and 𝑦
̂ as follows,
{
𝑦 = ∑ 𝜌𝑖(𝜃)𝜌ℎ(𝜃
̂)
𝑞
𝑖,ℎ=1 ((𝑃
̃ℎ + ∆𝑃
̃𝑖ℎ)𝒳1 + (𝑇ℎ + ∆𝑇𝑖ℎ)𝜏)
𝑦
̂ = ∑ 𝜌𝑖(𝜃)𝜌ℎ(𝜃
̂)
𝑞
𝑖,ℎ=1 (𝑃
̃ℎ𝒳
̂1 + 𝑇ℎ 𝜏)
(26)
with ∆𝑃
̃𝑖ℎ = 𝑃
̃𝑖 − 𝑃
̃ℎ and ∆𝑇𝑖ℎ = 𝑇𝑖 − 𝑇ℎ. By the substitution of (26) in (25), we get,
{
𝑒̇1 = ∑ 𝜌𝑖(𝜃)𝜌𝑗(𝜃
̂)𝜌ℎ(𝜃
̂)
𝑞
𝑖,𝑗,ℎ=1 (𝛱𝑗ℎ𝑒1 + 𝛯𝑖𝑗ℎ𝒳1 + 𝛻𝑖𝑗ℎ𝜏)
𝑒2 = ∑ 𝜌𝑖(𝜃)𝜌𝑗(𝜃
̂)
𝑞
𝑖,𝑗=1 (𝑆
̃𝑗𝑒1 − ∆𝑆
̃𝑖𝑗𝒳1 − ∆𝑉𝑖𝑗𝜏)
(27)
with,
{
𝛱𝑗ℎ = 𝐽
̃𝑗 − 𝐺𝑗𝑃
̃ℎ
𝛯𝑖𝑗ℎ = 𝐺𝑗∆𝑃
̃𝑖ℎ − ∆𝐽
̃𝑖𝑗
𝛻𝑖𝑗ℎ = 𝐺𝑗∆𝑇𝑖ℎ − ∆𝐿
̃𝑖𝑗
𝑖, 𝑗, ℎ 𝜖 (1, ⋯ , 𝑞)
(28)

177
therefore, to demonstrate the convergence of 𝑒 towards zero, it suffices to demonstrate that 𝑒1 converges to
zero. Considering 𝑒̃1 = (𝑒1
𝑇
𝒳1
𝑇
)𝑇
, we get,
{
𝑒̃1
̇ = ∑ 𝜌𝑖(𝜃)𝜌𝑗(𝜃
̂)𝜌ℎ(𝜃
̂)
𝑞
𝑖,𝑗,ℎ=1 ( 𝛴𝑖𝑗ℎ 𝑒̃1 + 𝛹𝑖𝑗ℎ𝜏)
𝑒1 = 𝑄𝑒̃1
(29)
with,
{
𝛴𝑖𝑗ℎ = (
𝛱𝑗ℎ 𝛯𝑖𝑗ℎ
0 𝐽
̃𝑖
)
𝛹𝑖𝑗ℎ = (
𝛻𝑖𝑗ℎ
𝐿
̃𝑖
)
𝑄 = (𝐼 0)
(30)
guaranteeing the stability of (29) while attenuationg the effect of 𝜏 on 𝑒1 is linked to the determination of the
observer gains 𝐺𝑖 for 𝑖 = 1, ⋯ , 𝑞.
Theorem: Under assumptions 1 and 2, if for the CTSSM (1) there are matrices 𝑃1,𝑃2,𝑊𝑖 for
𝑖 = 1, ⋯ , 𝑞, and a positive scalar ξ for a given 𝜕 > 0 which satisfy the LMIs (31), then it will be possible to
determine the observer gains, that ensure the exponential convergence to zero of the estimation error,
(
𝛿𝑗ℎ 𝛾𝑖𝑗ℎ ɸ𝑖𝑗ℎ
𝛾𝑖𝑗ℎ
𝑇
ʎ𝑖 𝑃2𝐿
̃𝑖
ɸ𝑖𝑗ℎ
𝑇
𝐿
̃𝑖
𝑇
𝑃2 −ξI
) < 0 ∀(𝑖, 𝑗, ℎ) ∈ (1, ⋯ , 𝑞)3
(31)
with,
{
𝛿𝑗ℎ = 𝐽
̃𝑗
𝑇
𝑃1 + 𝑃1𝐽
̃𝑗 − 𝑃
̃ℎ
𝑇
𝑊
𝑗
𝑇
− 𝑊
𝑗𝑃
̃ℎ + 2𝜕𝑃1 + 𝐼
𝛾𝑖𝑗ℎ = 𝑊
𝑗(𝑃
̃𝑖 − 𝑃
̃ℎ) − 𝑃1(𝐽
̃𝑖 − 𝐽
̃𝑗)
ɸ𝑖𝑗ℎ = 𝑊
𝑗(𝑇𝑖 − 𝑇ℎ) − 𝑃1(𝐿
̃𝑖 − 𝐿
̃𝑗)
ʎ𝑖 = 𝐽
̃𝑖
𝑇
𝑃2 + 𝑃2𝐽
̃𝑖 + 2𝜕𝑃2
(32)
the gains of the observer 𝐺𝑖, 𝑖 = 1, ⋯ , 𝑞 are obtained by,
𝐺𝑖 = 𝑃1
−1
𝑊𝑖 (33)
the attenuation level is,
𝛼 = √ ξ (34)
Proof of Theorem: Let us consider the following quadratic Lyapunov function as follows,
𝑉(𝑒̃1) = 𝑒̃1
𝑇
𝑃 𝑒̃1 , 𝑃 = 𝑃𝑇
> 0 (35)
with,
𝑃 = (
𝑃1 0
0 𝑃2
) (36)
the derivative of 𝑉 with respect to time is,
𝑉̇ = ∑ 𝜌𝑖(𝜃)𝜌𝑗(𝜃
̂)𝜌ℎ(𝜃
̂)
𝑞
𝑖,𝑗,ℎ=1 (𝑒̃1
𝑇
(𝛴𝑖𝑗ℎ
𝑇
𝑃 + 𝑃𝛴𝑖𝑗ℎ) 𝑒̃1 + 𝑒̃1
𝑇
𝑃𝛹𝑖𝑗ℎ𝜏 + 𝜏𝑇
𝛹𝑖𝑗ℎ
𝑇
𝑃 𝑒̃1) (37)
to guarantee the stability of (29) and the boundedness of the transfer from the input 𝜏 to 𝑒1,
||𝑒1||2
||𝜏||2
< 𝛼, ||𝜏||2 ≠ 0 (38)
we consider,

 ISSN: 2502-4752
178
𝑉̇ + 𝑒1
𝑇
𝑒1 − 𝛼2
𝜏𝑇
𝜏 < 0 (39)
so, the exponential convergence of the estimation error is guaranteed if,
𝑉̇ + 𝑒1
𝑇
𝑒1 − 𝛼2
𝜏𝑇
𝜏 < −2𝜕𝑉 𝜕 > 0 (40)
inserting (29) and (37) into (40) leads to the following inequality,
𝑉̇ = ∑ 𝜌𝑖(𝜃)𝜌𝑗(𝜃
̂)𝜌ℎ(𝜃
̂)
𝑞
𝑖,𝑗,ℎ=1 (𝑒̃1
𝑇
𝜏𝑇)Г𝑖𝑗ℎ(𝑒̃1 𝜏)𝑇
< 0 (41)
with,
Г𝑖𝑗ℎ = (
𝛴𝑖𝑗ℎ
𝑇
𝑃 + 𝑃𝛴𝑖𝑗ℎ + 𝑄𝑇
𝑄 + 2𝜕𝑃 𝑃𝛹𝑖𝑗ℎ
𝛹𝑖𝑗ℎ
𝑇
𝑃 −𝛼2
𝐼
) (42)
then, the inequality (41) is contented if,
Г𝑖𝑗ℎ < 0 ∀𝑖, 𝑗, ℎ ∈ (1, ⋯ , 𝑞) (43)
taking account (28), (30), (36), and the following change of variables,
{
𝑊𝑖 = 𝑃1𝐺𝑖
ξ = 𝛼2 (44)
we can deduce the LMIs (31) presented in the Theorem that complete the proof.
4. RESULTS AND DISCUSSION
To display the benefits of the suggested observer, we consider the following CTSSM which is
affected by faults, at the level of actuator and sensor, and subjected to unmeasurable premise variable,
{
𝑀𝑧̇ = ∑ 𝜌𝑖(𝛽)
2
𝑖=1 (𝐴𝑖𝑧 + 𝐵𝜏 + 𝑀𝑎𝑣𝑎)
𝑦 = 𝐶𝑧 + 𝑀𝑠𝑣𝑠
(45)
where 𝑧 ∈ ℝ4
, 𝜏 ∈ ℝ, 𝑦 ∈ ℝ2
, 𝑣𝑎 ∈ ℝ, and 𝑣𝑠 ∈ ℝ are the vectors of states, input, output, actuator fault
and sensor fault, respectively.
𝐴1 = (
0 1 0 0
−2.5 −0.75 0 0.025
0 1 −0.4 0
−2.5 −0.75 0 0.075
) ; 𝐴2 = (
0 1 0 0
−2.696 −0.75 0 0.025
0 1 −0.4 0
−2.696 −0.75 0 0.075
) (46)
𝐵 = (
0
0
0
−0.125
) ; 𝑀𝑎 = 𝐵; 𝐶 = (
1 0 1 0
0 1 0 1
) ; 𝑀𝑠 = (
1
0
) (47)
ρ1(β) and ρ2(β) represent the weighting functions,
{
𝜌1(𝛽) =
𝛽−𝛽𝑚𝑖𝑛
𝛽𝑚𝑎𝑥−𝛽𝑚𝑖𝑛
𝜌2(𝛽) =
𝛽𝑚𝑎𝑥−𝛽
𝛽𝑚𝑎𝑥−𝛽𝑚𝑖𝑛
(48)
where the expression of the premise variable 𝛽 is,
𝛽 =
−5−5𝑧1
2
2
∈ [𝛽𝑚𝑖𝑛, 𝛽𝑚𝑎𝑥] (49)
in order to apply the suggested fuzzy observer (19) on our application example (45), it suffices to represent it
in its equivalent form (17). Thus, by using the Theorem with 𝜕 = 0.1, we obtain the following observer gains
𝐺1 and 𝐺2,

179
𝐺1 =
(
−16.2542 0.4992
60.4899 19.3705
443.5258 136.5402
77.1069 23.0732
35.2472 −5.6930
12.2730 −3.6819 )
; 𝐺2 =
(
−16.3254 0.5089
59.6945 28.7403
437.8298 202.7100
76.1530 34.2843
35.5444 −7.7807
12.4475 −5.1612 )
(50)
the simulation results are given in Figures 1 to 4 where the input signal is given by,
𝜏(𝑡) = {
𝑡 − 2 𝑤ℎ𝑒𝑛 𝑡 ≤ 2
0 𝑒𝑙𝑠𝑒
(51)
Under Assumption 2, the trajectories of actuator and sensor fault signals, which are applied respectively
during the intervals [40, 160s] and [200, 320s], their first order derivatives, and their estimates are shown in
Figures 3 and 4.
These results demonstrate that the suggested fuzzy observer gives good performances in estimating
unmeasurable states and unknown faults while catching up with unwanted variations. This approach has the
benefit of being applied at the level of a large class of nonlinear systems. This is due to the fact that it is not
required to know the value of the Lipschitz constant that can influence the resolution of LMIs [27], as well as
without being limited by the condition of the rank between the matrices such as in [28].
Figure 1. z1 and z2 with their estimates
Figure 2. z3 and z4 with their estimates

 ISSN: 2502-4752
180
Figure 3. 𝑣𝑎 and 𝑣̇𝑎 with their estimates
Figure 4. 𝑣𝑠 and 𝑣̇𝑠 with their estimates
5. CONCLUSION
This work is addressed to the design of fuzzy observer for simultaneous estimation of unmeasurable
states and unknown faults, for Takagi-Sugeno singular models in continuous time. The main idea of this
paper is to extend the results developed in the case of measurable premise variables. The diagnostic
procedure is based on the separation of static equations from dynamic ones. Using this, the determination of
the static variables will be deduced from the computation of the dynamic variables. At last, an example of
application is presented in order to highlight and confirm the effectiveness of the proposed approach in the
estimation of the states, and the faults of actuators and sensors.
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BIOGRAPHIES OF AUTHORS
Kaoutar Ouarid received the M.Sc. degree in Electrical Engineering from
Faculty of Science and Technology Marrakech, Morocco, in 2016. Currently, she is working
toward the PhD degree at Faculty of Science Ain Chock (FSAC), in Hassan II University of
Casablanca, Morocco. Her research interests include observer design, nonlinear systems,
Takagi-Sugeno fuzzy systems, fuzzy control, and fault diagnosis. She can be contacted at
email: kaoutar.ouarid@gmail.com.

 ISSN: 2502-4752
182
Mohamed Essabre received the PhD in Engineering Science-Electrical
Engineering in 2015 from the National Higher School of Electricity and Mechanics (ENSEM),
Hassan II University of Casablanca, Morocco. His research interests include nonlinear control
and state observer theory. Currently, he is an Associate Professor at the Department of
Electrical Engineering, Faculty of Science and Technology Mohammedia, Hassan II University
of Casablanca, Morocco. He can be contacted at email: mohamed.essabre@fstm.ac.ma.
Abdellatif El Assoudi is a professor at the department of Electrical Engineering
in National Higher School of Electricity and Mechanics (ENSEM), in Hassan II University of
Casablanca (Morocco). His research interests focus in nonlinear implicit model, nonlinear
observer design, unknown input observer design, fault detection, Takagi-Sugeno fuzzy
control. He can be contacted at email: a.elassoudi@ensem.ac.ma.
El Hassane El Yaagoubi is a professor at the department of Electrical
Engineering in National Higher School of Electricity and Mechanics (ENSEM), in Hassan II
University of Casablanca (Morocco). His areas of interest include Nonlinear implicit model,
Nonlinear observer design, Unknown Input Observer Design, fault detection, Takagi-Sugeno
fuzzy control. He can be contacted at email: h.elyaagoubi@ensem.ac.ma.

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