A Time Domain Unknown Input Observer for a Class of Bilinear
Delayed System
WEJDENE ZARROUGUI, FATMA HAMZAOUI, MALEK KHADHRAOUI, HASSENI MESSAOUD
Department of Electrical Engineering,
National Engineering School of Monastir, (ENIM),
University of Monastir,
Ibn El Jazzar Str., 5019, Monastir,
TUNISIA
Abstract: - This paper deals with an unknown input observer design which estimates a linear function of the
dynamic and non-measured input vectors in the time domain for a bilinear model with the presence of an
unknown input and constant time delays cited in state and control vectors. The observer gain is based on
Lyapunov Krasovskii stability theory and results from a set of Linear Matrix Inequalities (LMI). The presented
approach proposes the decomposition of the unknown inputs and provides an estimation of a functional state
vector and a functional unknown input. Then, a numerical example is given to highlight the effectiveness of the
proposed approach.
Key-Words: - Bilinear, Functional observer, Constant delay, Unknown inputs system, Disturbance method,
LMI.
Received: May 2, 2024. Revised: October 4, 2024. Accepted: November 6, 2024. Published: December 13, 2024.
1 Introduction
Over the last decade, multiple studies have been
carried out on the development of functional
observers for nonlinear and linear systems
estimators with the presence of unknown inputs, [1].
Several studies have suggested methods that assume
prior knowledge of non-measurable inputs, which
may impose certain limitations, [2], [3].
A bilinear system can be written following
several mathematical models such as state
representation, [4]. So, an unknown input observer
of such a model is proposed using results developed
by [4], [5].
In addition, the estimation and the control
process of a bilinear system are still an open topic in
automation due to the nonlinear nature of these
systems, [6], [7] and the emergence of new
technologies. In this article, the authors present a
recent scheme design observer to reconstruct the
state vector for a bilinear system affected by an
unknown input in the time domain.
The exposed algorithm uses, principally,
stability notations [8] and an optimum gain solution
of an LMIs approach is given [9], [10], [11].
Hereby, the objective is to retrieve the dynamic
vector and a function of the non-measured input for
a bilinear system affected by a constant time delay.
The unknown inputs are injected in both
dynamic and output equations in order to represent
the effect of fault [11], on the estimation error
dynamic [12].
To attend to this objective the authors use the
de-composition of the unknown input vector and the
use of its parts fed by output measurement to design
the proposed observer, [13], [14], [15].
Considering that the delay is a physical
phenomenon that influences the stability of the
system dynamic, different linear matrix inequality
tools are deduced when the derivative a functional
of Lyapunov has been set.
The outlines of this paper are structured as
follows. Section 2 presents the related works. Then
the observer scheme is exposed in paragraph 3
verifying several hypotheses. Section 4 presents the
main contribution of this paper by detailing the
proposed design procedure in the time domain.
Then, we translate the problem into Linear Matrix
Inequalities to be solved. According to the
application of the LMI method, the observer gain Z
is optimized. Section 5 summarizes the observer
synthesis steps. The sixth paragraph confirms the
effectiveness of the given contribution by showing a
result of a numerical example and paragraph 7
concludes this paper.
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2024.19.38
Wejdene Zarrougui, Fatma Hamzaoui,
Malek Khadhraoui, Hasseni Messaoud
E-ISSN: 2224-2856
360
Volume 19, 2024
Several researchers have been interested in observer
development for a class of bilinear systems, [2]. The
main objective of these studies is to reconstruct the
state dynamics fed by a known input vector, [6]. As
done in [7], this paper proposes to synthesize a
software sensor for bilinear dynamics. In addition,
we consider time delays in state and input vectors to
take into account the stability effect of this
parameter. Compared to [8], this paper investigates
the problem of fault detection by reconstructing the
unknown input vector present in both dynamic and
measurement equations. To attain this goal [10],
transform the problem into a new configuration by
extending the state vector to a newer one containing
the unknown input vector. However, as rendered in
[12], we propose to isolate the measure part affected
by the fault vector from the output part independent
from the unknown input and reconstruct a functional
form of this kind of input besides the functional
state.
3 Problem Formulation
The bilinear systems are described by:
(1)
(2)
(3)
Where , , ,
, and are the state, the measured
output, the control inputs, the functional state and
the unknown input vectors respectively.
, , , , , , , and are evident
(known) and constant matrices of appropriate
dimensions.
: are the considered constant delays.
Assumption: [12]
Thus, we suggest that:
(4)
(5)
Then and are matrices used with
and is an orthogonal matrix [12]:
(6)
Multiplying in equation (1) by and using
(6) we have obtained:
(7)
(8)
With
(9)
Where
(10)
Replacing by its equation derived from
(7) in (1), the system (1-3) was transformed into the
following form:
(11)
(12)
(13)
(14)
Where
(15)
(16)
Where ,
(17)
(18)
(19)
The measure is decomposed into two parts
and , part is totally affected by the
unknown input and does not depend on
the fault design.
4 Time Domain Design
In this article, we aim at the development of a
functional observer for system described by
equation (1-3).
The proposed estimator is described by
equations (20-21):
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2 Related Works
(20)
(21)
The dynamic vector of the functional observer is
represented by and is the estimation
of.
, , , , , , , , and : will be
solved using LMI approach.
This observer defined by equations (20) and
(21) aims to reconstruct both functional state and
partial unknown input with a high accuracy to prove
its efficiency.
Remark
The objective is to determine the observer matrices
, , , , , , , , and such as
converges asymptotically to .
(22)
4.1 Observer Design Conditions
Having defined the functional error as follows:
(23)
Replacing and by their expressions given,
respectively, by (14) and (21). To get:
(24)
with:
(25)
(26)
Assuming that , then
(27)
becomes to the following form:
(28)
Theorem 1. The model (20-21) is an unknown input
functional observer for the system (1-3), if and only
if the next conditions are reached:
is asymptotically
stable
,
Proof 1. The derivative of (24), when considering
leads to:
(29)
According to the equations of and
represent by the equation (11) and (20),
becomes:
+
(30)
The unbiased estimation error defined by
condition as relative to the system (1-3) and
functional observer (20-21) and using the conditions
of Theorem 1.
The proposed observer performs both
minimizing the unknown input effect and the time
delay perturbation.
4.2 Computing of Observer Matrices
After inserting (25) in -from the theorem 1, we
have obtained:
(31)
(32)
(33)
(34)
with
(35)
(36)
(37)
They equations (31)-(34) can be written in the
following matrix form:
(38)
Where:
(39)
(40)
(41)
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There exists a solution of (38) if and only if:
(42)
The solution of equation (38) is:
(43)
With is the general inverse of the matrix
whereas is an observer gain of appropriate
dimension to be determined later using the LMI
approach.
In order to determine the matrix we have:
(44)
By inserting with it relation (43) in (44), we
obtain the following expression:
(45)
Knowing that:
,
(46)
Then:
(47)
The matrix, is given by:
(48)
Where:
,
(49)
Similarly, we can determine and .
At this stage, the gain is determined using the next
proposed theorem.
Theorem 2. System (20-21) is a functional observer
of the bilinear system (1-3), with the existence of
the two positive energy amplitudes ,
and , which are the solutions for the
following LMI:
(50)
With
(51)
Where the arbitrary matrix is determined by
equation (52):
(52)
Proof 2. Based on [8], we can consider this
functional:
(53)
Where
, (54)
Then, according to condition of theorem 1, the
time derivative of is obtained as follow:
(55)
Can be written as:
(56)
With
(57)
From (47), if and only if:
(58)
When is determined by (52), all observer matrices
will be calculated.
5 Observer Synthesis Steps in Time
Domain
1. Calculate matrices Ʃ and using (40) and
(41).
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2. Calculate matrices, , , and
from (46) and (49).
3. Checking if condition (42) is true.
4. Determine the gain using to expression
(52).
5. Computing , , , , and from
the equations (47) and (48).
6. Determine the matrices using expression
(27).
7. Calculate the matrices and which
correspond to (36)-(37).
8. Obtaining matrices and by using
from theorem 1.
6 Numerical Example
Assuming that system (1-3) is given by:
,
,
,
,
,
,
,
,
The constants delays are and
.
After verifying the condition (42), the matrices
observers are reached using the LMI approach (50).
We obtain as a result:
,
,
,
,
,
,
,
,
,
,
We use known software in the field of scientific
computing and the resolution of problems thanks to
algorithms, simulations and graphics, based on
MATLAB / SIMULINK toolbox.
Then, Figure 1 and Figure 2 represent the
evolution of the known and unknown input signals.
The unknown input describes the fault signals which
can affect the system dynamic.
Fig. 1: The known input signal
Fig. 2: Unknown inputs vectors
Figure 3 and Figure 4 present the difference
between unknown inputs and estimated vector, also
between the real functional state vectors and
estimated one.
0 2 4 6 8 10 12 14 16 18 20
-6
-4
-2
0
2
4
6
The known signal u1 (t)
The known signal u2 (t)
0 2 4 6 8 10 12 14 16 18 20
-6
-4
-2
0
2
4
6
Unknown input vector d1(t)
Unknown input vector d2(t)
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Fig. 3: The real and estimation unknown inputs
Note that the difference between the two
unknown inputs which is limited in a short time
.
The following figures illustrate a comparison
between estimated state vector and the real one.
A transitory phase will be shown using the
effect of disturbance.
We propose a zoom in at transitory phases due
to unknown input contribution and time delay effect
to highlight the efficiency of the proposed
estimation approach.
Fig. 4: The functional state vector
Figure 4 shows a comparison between the real
state and the estimated one using the (LMI)
technics, which are values with excellent precision.
Fig. 5: Zoom in the estimation functional state
Figure 5 gives the time delay effect on the
estimated functional state at during the
transitory phase.
Figure 6 illustrates the estimation error of the
functional state vectors.
Fig. 6: The estimation error
We can deduce the stability and the
convergence of the estimation error of the state
functional and the unknown input designed by the
observer (20-21) in the time domain. We can also
check the accuracy of the estimation error dynamic
through the comparison between the real and
estimated evolution.
This proves the efficiency of the proposed method.
0 2 4 6 8 10 12 14 16 18 20
-1
0
1
2
3
4
5
6
7
The unknown input vector d1
Estimation unknown input vector d1
0.4 0.6 0.8
-0.1
0
0.1
0.2
1.61.8 2
2.9
3
16.5 17 17.5
3
3
3
3
0 5 10 15 20 25 30 35 40
-20
-15
-10
-5
0
5
10
15
20
Real functional states vectors
Estimation functional states vectors
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-6
-4
-2
0
2
4
6
Real functional vector
Estimation functional vector
0 0.5 1 1.5 2 2.5
-6
-4
-2
0
2
4
6
Estimation error of unknown inputs vectors
Estmation error of functional state vectors
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DOI: 10.37394/23203.2024.19.38
Wejdene Zarrougui, Fatma Hamzaoui,
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A functional observer has been designed for a class
of bilinear delayed systems exited by unknown
inputs, to estimate a functional state and unknown
input vector in the time domain.
The performance of the proposed observer is based
on the LMI approach and the stability theory and
has demonstrated its effectiveness through a
numerical example which gives a very precise
measurement response.
As a future research work the proposed results can
be tested, checked, adapted, and applied to bilinear
systems when taking into consideration the effect of
the unknown input as a fault signal into the output
channel. Also, the filter problem for such systems
can be addressed.
Abbreviations and symbols:
LMI: Linear Matrix Inequalities
: Matrix of the appropriate dimension
The generalized inverse of
: Set of real numbers
: Size of the state vector
: Size of output vector
: Size of known vector
: Size of the functional state vector
: Size of unknown inputs vector
: Identity matrix of the appropriate dimension.
: Rank of matrix [.]
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7 Conclusion
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RAM47153.2019.9095793
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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DOI: 10.37394/23203.2024.19.38
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