The paper is devoted to the problem of the robust actuator fault diagnosis of the dynamic non-linear systems. In the proposed method, it is assumed that the diagnosed system can be modelled by the recurrent neural network, which can be transformed into the linear parameter varying form. Such a system description allows developing the designing scheme of the robust unknown input observer within H1 framework for a class of non-linear systems. The proposed approach is designed in such a way that a prescribed disturbance attenuation level is achieved with respect to the actuator fault estimation error, while guaranteeing the convergence of the observer. The application of the robust unknown input observer enables actuator fault estimation, which allows applying the developed approach to the fault tolerant control tasks.

1. Research Article
Neural network-based robust actuator fault diagnosis for a non-linear
multi-tank system
Marcin Mrugalski a
, Marcel Luzar a,n
, Marcin Pazera a
, Marcin Witczak a
,
Christophe Aubrun b
a
Institute of Control and Computation Engineering, University of Zielona Góra, ul. Podgórna 50, 65–246 Zielona Góra, Poland
b
Centre de Recherche en Automatique de Nancy, CRAN-UMR 7039, Nancy-Universite, CNRS, F-54506 Vandoeuvre-les-Nancy Cedex, France
a r t i c l e i n f o
Article history:
Received 14 November 2013
Received in revised form
3 November 2015
Accepted 10 January 2016
Available online 3 February 2016
This paper was recommended for publica-
tion by Dr. Didier Theilliol
Keywords:
Robust fault diagnosis
Fault estimation
Non-linear systems identification
Observers
Neural network
LPV systems
a b s t r a c t
The paper is devoted to the problem of the robust actuator fault diagnosis of the dynamic non-linear
systems. In the proposed method, it is assumed that the diagnosed system can be modelled by the
recurrent neural network, which can be transformed into the linear parameter varying form. Such a
system description allows developing the designing scheme of the robust unknown input observer
within H1 framework for a class of non-linear systems. The proposed approach is designed in such a way
that a prescribed disturbance attenuation level is achieved with respect to the actuator fault estimation
error, while guaranteeing the convergence of the observer. The application of the robust unknown input
observer enables actuator fault estimation, which allows applying the developed approach to the fault
tolerant control tasks.
& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
A growing complexity of industrial systems and requirement of
their reliabilities motivate the research in the field of the non-linear
systems Fault Detection and Isolation (FDI) [2,10,21,22,24,32,33] in
the last decade of the twentieth century. Among scientists, a com-
mon conviction dominated that fast detection of the small abrupt or
incipient faults could prevent systems larger failures. It was
assumed that early enough detection of the fault was able to stop
the process and in the consequence reduce the economical losses
resulting from the system malfunction. For such a reason, all efforts
were focused on the development of efficient and robust fault
detection methods and the problem of fault estimation and iden-
tification was marginalized.
However, with the years the expectations for the industrial
systems started to change. The efficient fault diagnosis was not
sufficient enough but it was expected that the systems and pro-
cesses can be operated efficiently despite existing faults. Such an
assumption caused that the intensive researches on developing
the so-called Fault Tolerant Control (FTC) [4,7,12,14,20,27,29]
approaches have appeared. In order to achieve such a goal, new
control methods should be developed which take into considera-
tion the existence of the system faults. Moreover, the efficient fault
estimation methods have to be proposed what is more difficult
task than developing fault detection schemes.
Industrial systems consist of the plant (or system dynamics
[40]), sensors and actuators. In the references, several efficient
fault estimation methods of the sensors can be found [17,18,24,36].
However from the FTC point of view much more important is the
problem of the actuators fault diagnosis, which is used during
control process. For such a reason actually intensive researches in
the field of the actuators fault detection and estimation are per-
formed. Appropriate actuator fault estimation allows for the
application of various FTC strategies enabling compensation of the
faulty actuator by increasing performance of the other actuator
existing in the control system.
The problem of the actuators fault estimation can be perceived
as the task of estimation of the system unknown inputs. Such a
challenging problem can be solved by the following approaches:
augmenting the state vector by an unknown input, two-stage
Kalman filter [11], minimum variance input and state estimator
[6], adaptive estimation [44], sliding mode high-gain observers
[34], proportional integral observer [28], an H1 approach [25],
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
http://dx.doi.org/10.1016/j.isatra.2016.01.002
0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
n
Corresponding author.
E-mail addresses: M.Mrugalski@issi.uz.zgora.pl (M. Mrugalski),
M.Luzar@issi.uz.zgora.pl (M. Luzar), marcin.pazera@o2.pl (M. Pazera),
M.Witczak@issi.uz.zgora.pl (M. Witczak),
christophe.aubrun@univ-lorraine.fr (C. Aubrun).
ISA Transactions 61 (2016) 318–328

2. and finally by the application of the Unknown Input Observer
(UIO) [5,36,38,39].
The UIO based approach seems to be especially attractive for
the actuator fault estimation. It follows from the fact that such a
method allows for the state and unknown inputs reconstruction
on the basis of the model and measurements of system inputs and
outputs. Moreover, UIOs have robustness properties because they
are designed to tolerate the model uncertainty and hence they
increase the reliability of fault diagnosis. Unfortunately, the main
weakness of UIOs and other approaches follows from the fact that
the analytical model of the diagnosed system is required, which in
the practice is often unavailable.
In order to solve such a problem a novel methodology of
designing the UIO on the basis of the Artificial Neural Network
(ANN) is developed in this paper. It was decided to use ANNs
because they have some interesting properties especially attractive
for modelling of complex non-linear dynamic systems for which
efficient analytic modelling methods do not exist. Among these
properties there are ability of approximation of any non-linear
functions, modelling of system dynamics, parallel processing,
generalization and adaptivity features [8,26,31]. However, the
main disadvantage of the ANNs is that the disturbances decou-
pling and convergence to the origin are not guaranteed. Thus, the
concept of this paper relies on the combination of the ANNs
modelling abilities with a Linear Parameter-Varying (LPV) techni-
que for designing the robust UIO in such a way that the influence
of disturbances is minimized in the H1 sense. Thus, the proposed
approach combines the positive features of the analytical and soft-
computing methods. In order to do it the state-space representa-
tion of the neural model is required. The above property is fulfilled
by the Recurrent Neural Network (RNN) [15] and such a neural
model was chosen for the UIO design in this paper. The RNN has
state-space description, which can be converted into a LPV form
[1]. Such a representation, especially attractive in the LPV gain-
scheduled control schemes [1,19,23], allows applying the observer
based methodology to design the robust actuators fault detection
and estimation schemes.
The paper aims at providing a novel observer synthesis pro-
cedure, which is based on the concept of the UIO for the actuators
fault detection and estimation, is developed. The proposed
approach is a combination of the linear-system strategies [6,25] for
a class of non-linear systems [30,42]. The UIO is designed in such a
way that a prescribed disturbance attenuation level is achieved
with respect to the actuator fault estimation error while guaran-
teeing the convergence of the observer. The resulting design pro-
cedures boil down to solving a set of linear matrix inequalities.
The paper is organized as follows. Section 2 presents the
structure of the RNN model and a method of its transforming into
a discrete-time polytopic LPV model. Section 3 describes the
design procedure of the robust UIOs using H1 framework for the
actuator fault estimation. Section 4 provides an illustrative
example of application of the proposed methodology to the robust
actuator fault diagnosis of the multi-tank system. Section 5 con-
cludes the paper.
2. LPV neural model
A dynamic non-linear system can be represented by the LPV
model in a relatively simple way. To design such a model, it is
necessary to linearize a non-linear system around a number of
operating points. The number of points determines the accuracy of
the LPV model. The local system behaviour around the operating
point is represented by each of these linear models. Let us consider
the following discrete-time non-linear model:
xk þ 1 ¼ hðxk; ukÞ; ð1Þ
yk ¼ Cxk: ð2Þ
where x ARn
is the state vector, y ARm
is the output, u ARr
is the
input vector and hðÁÞ is a non-linear function.
The goal of this section is to represent this model in the form of
a discrete-time polytopic LPV model:
xk þ 1 ¼ AðhkÞxk þBuk; ð3Þ
yk ¼ Cxk; ð4Þ
where AðhkÞ, B, C are state-space matrices and hk ARN
is a time-
varying parameter vector which ranges over a fixed polytope. Note
that the model (3)–(4) is input affine, which is motivated by its
furtherer application to the multitank system (cf. Section 4). The
dependence of A on hk represents a general discrete-time quasi-
LPV model. To obtain such a model, it is proposed to use the RNN
[15] with suitable modifications.
The general form of a discrete-time non-linear model repre-
sented by proposed RNN is given by:
xk þ 1 ¼ Axk þBuk þA1σðE1xkÞ; ð5Þ
ykþ 1 ¼ Cxk; ð6Þ
where
σ ¼ ½σðÁÞ; …; σðÁÞŠT
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
Nn
; ð7Þ
and Nn is the number of neurons and A, A1, B, C, E1 are real valued
matrices of appropriate dimensions and represent the weights,
which will be adjusted during the training stage of the RNN. The
non-linear activation function σðÁÞ, which is applied element-wise
in (5) is taken as a continuous, differentiable and bounded func-
tion. For that purpose let us write (5) as
xk þ 1 ¼ Axk þBuk þg xkð Þ; ð8Þ
where
g xkð Þ ¼ A1σðE1xkÞ: ð9Þ
This RNN leads to a general form of the neural state-space
model in the sense that if it is transformed into an LPV model in
the form (3)–(4), the matrix A will be parameter dependent.
For stability and identifiability proofs of the proposed RNN the
reader is referred to [15]. The scheme of the proposed RNN is
depicted in Fig. 1.
Since the general modelling framework is given, it is possible to
proceed to robust observer design.
Fig. 1. State-space recurrent neural network.
M. Mrugalski et al. / ISA Transactions 61 (2016) 318–328 319

3. 3. UIO design for the actuator fault diagnosis
The main objective of this section is to provide a detailed
design procedure of the robust observer, which can be used for
actuator fault diagnosis. In other words, the main role of this
observer is to provide the information about the actuator fault.
Indeed, apart from serving as a usual residual generator [35], the
observer should be designed in such a way that a prescribed dis-
turbance attenuation level is achieved with respect to the actuator
fault estimation error while guaranteeing the convergence of the
observer.
Let us consider the following state-space model, which is an
alternative form of (3)–(4):
xkþ 1 ¼ Axk þBuk þg xkð ÞþLaf a;k þW1wk; ð10Þ
yk þ1 ¼ Cxk þ1 þLsf s;k þW2wk þ 1; ð11Þ
where xk AX & Rn
is the state vector, uk ARr
stands for the input,
yk ARm
denotes the output, f a;k ARs
, f s;k ARr
stand for the actuator
and sensor fault, respectively and La and Ls are their distribution
matrices. Moreover, wk Al2 is an exogenous disturbance vector
with W1 ARnÂn
, W2 ARmÂn
being its distribution matrices while
l2 ¼ wARn
j JwJl2
o þ1
È É
; JwJl2
¼
X1
k ¼ 0
Jwk J2
!1=2
: ð12Þ
Following [6,35], let us assume that the system is observable
and the following rank condition is satisfied:
rankðCLaÞ ¼ rankðLaÞ ¼ s: ð13Þ
Under the assumption (13), it is possible to calculate
H ¼ ðCLaÞþ
¼ ðCLaÞT
CLa
h iÀ1
ðCLaÞT
: ð14Þ
In the sequel, it is assumed that f s;k ¼ 0, which means that the
actuator faults are only considered. Moreover, following [13] the
system description (10)–(11) can be transformed in such a way
that the sensor faults can be treated as actuator ones. Thus, the
proposed actuator fault estimation scheme can be adapted for the
sensor fault estimation purposes. This is however beyond the
scope of this paper.
Substituting f s;k ¼ 0 into (11) as well as multiplying it by H, and
then substituting (10), it can be shown that
f a;k ¼ Hðyk þ1 ÀCAxk ÀCBuk ÀCg xkð ÞÀCW1wk ÀW2wk þ 1Þ: ð15Þ
Finally, by substituting (15) into (10) it can be shown that:
xkþ 1 ¼ Axk þBuk þGg xkð ÞþLykþ 1 þGW1wk ÀLW2wk þ1; ð16Þ
where G ¼ ðIn ÀLaHCÞ, A ¼ GA, B ¼ GB, L ¼ LaH. In order to esti-
mate (15), i.e., to obtain ^f k it is necessary to estimate the state of
the system, i.e., to obtain ^xk. Consequently, the fault estimate is
given as follows:
^f a;k ¼ Hðyk þ 1 ÀCA^xk ÀCBuk ÀCg ^xk
À Á
Þ: ð17Þ
The proposed observer structure is
^xk þ 1 ¼ A ^xk þBuk þGg ^xk
À Á
þLyk þ 1 þKaðyk ÀC ^xkÞ; ð18Þ
while the state estimation error is given by
ek þ 1 ¼ A ÀKaC
ek þGsk þðGW1 ÀKaW2Þwk ÀLW2wk þ 1
¼ A1ek þGsk þW 1wk þW 2wkþ 1; ð19Þ
where
sk ¼ g xkð ÞÀg ^xk
À Á
: ð20Þ
Similarly, the fault estimation error εf a;k can be defined as
εf a;k ¼ f a;k À ^f a;k ¼ ÀHC Aek þsk þW1wkð ÞÀHW2wk þ 1: ð21Þ
Note that both ek and εf a;k are non-linear with respect to ek.
To settle this problem within the framework of this paper, the
following solution is proposed.
Using the Differential Mean Value Theorem (DMVT) [41], it can
be shown that
g að ÞÀg bð Þ ¼ MxðaÀbÞ; ð22Þ
with
Mx ¼
∂g1
∂x
ðc1Þ
⋮
∂gn
∂x
ðcnÞ
2
6
6
6
6
4
3
7
7
7
7
5
; ð23Þ
where c1; …; cn ACoða; bÞ, ci aa, ci ab, i ¼ 1; …; n. Assuming that
gi;j Z
∂gi
∂xj
Zgi;j
; i ¼ 1; …; n; j ¼ 1; …; n; ð24Þ
Eq. (9) can be rewritten as:
giðxkÞ ¼ ðai
1ÞT
σðE1xkÞ; ð25Þ
where ðai
1ÞT
stands for i-th row of A1 from (8). On the basis of the
(25) a gradient can be calculated:
∂giðxkÞ
∂xk
¼
∂giðxÞ
∂x1
; …;
∂giðxÞ
∂xn
!T
; ð26Þ
where each element of (26) can be calculated as:
∂giðxÞ
∂xj
¼
XNn
l ¼ 1
a1;i;lel;jσ0
lðE1xkÞ; ð27Þ
where a1;i;l and el;j are appropriate elements of matrices A1 and E1,
respectively.
On the basis of the above result, the boundary values of the
non-linear activation function derivatives (24) can be obtained as:
gi;j ¼ max
xk A X
XNn
l ¼ 1
a1;i;lel;jσ0
lðE1xkÞ
#
; ð28Þ
gi;j
¼ min
xk A X
XNn
l ¼ 1
a1;i;lel;jσ0
lðE1xkÞ
#
: ð29Þ
Thus, it is clear that:
Mx ¼ M ARnÂn
jgi;j Zmx;i;j Zgi;j
; i; j ¼ 1; …; n;
n o
: ð30Þ
Using (22), the term A1ek þGsk in (19) can be written as
A1ek þGsk ¼ ðA þGMx;k ÀKaCÞek; ð31Þ
where Mx;k AMx.
From (31), it can be deduced that the state estimation error (19)
can be converted into an equivalent form
ek þ1 ¼ A2ðhkÞek þW 1wk þW 2wk þ1;
A2ðhkÞ ¼ ~AðhkÞÀKaC; ð32Þ
which defines an LPV polytopic system [3] with
~A ¼ ~AðhkÞ : ~AðhkÞ ¼
XN
i ¼ 1
hki
~Ai;
XN
i ¼ 1
hki ¼ 1; hki Z0
( )
; ð33Þ
where N ¼ 2n2
. Note that this is a general description, which does
not take into account that some elements of Mx;k maybe constant.
In such cases, N is given by N ¼ 2ðnÀ cÞ2
where c stands for the
number of constant elements of Mx;k.
In a similar fashion, (21) can be converted into
εf a;k ¼ ÀHC A3ðhkÞek þW1wkð ÞÀHW2wk þ1; ð34Þ
M. Mrugalski et al. / ISA Transactions 61 (2016) 318–328320

4. with
A3 ¼ A3ðhkÞ : A3ðhkÞ ¼
XN
i ¼ 1
hkiA3;i;
XN
i ¼ 1
hki ¼ 1; hki Z0
( )
: ð35Þ
The objective of further deliberations is to design the observer
(18) in such a way that the state estimation error ek is asympto-
tically convergent and the following upper bound is guaranteed:
Jεf Jl2
rωJwJl2
; ð36Þ
where ω40 is a prescribed disturbance attenuation level. Thus,
on the contrary to the approaches presented in the literature, ω
should be achieved with respect to the fault estimation error but
not the state estimation error.
Thus, the problem of H1 observer design [16,42] is to deter-
mine the gain matrix Ka such that
lim
k-1
ek ¼ 0 for wk ¼ 0; ð37Þ
Jεf Jl2
rωJwJl2
for wk a0; e0 ¼ 0: ð38Þ
In order to settle the above problem, it is sufficient to find a Lya-
punov function Vk such that:
ΔVk þεT
f a;kεf a;k Àμ2
wT
k wk Àμ2
wT
k þ 1wk þ1 o0; k ¼ 0; …; 1; ð39Þ
where ΔVk ¼ Vk þ1 ÀVk, μ40. Note that the structure of (39) is
uncommon in the literature. Indeed, the novelty is that the term
Àμ2
wT
k þ1wk þ 1 is introduced. This is caused by the fault decou-
pling procedure (cf. (15)). Indeed, if wk ¼ 0 ðk ¼ 0; …; 1Þ, then (39)
boils down to
ΔVk þεT
f a;kεf a;k o0; k ¼ 0; …; 1; ð40Þ
and hence ΔVk o0, which leads to (37). If wk a0 ðk ¼ 0; …; 1Þ
then (39) yields
J ¼
X1
k ¼ 0
ΔVk þεT
f a;kεf a;k Àμ2
wT
k wk Àμ2
wT
kþ 1wk þ 1
o0; ð41Þ
which can be written as
J ¼ ÀV0 þ
X1
k ¼ 0
εT
f a;kεf a;k Àμ2
X1
k ¼ 0
wT
k wk Àμ2
X1
k ¼ 0
wT
kþ 1wk þ 1 o0:
ð42Þ
Bearing in mind that
μ2
X1
k ¼ 0
wT
k þ 1wk þ1 ¼ μ2
X1
k ¼ 0
wT
k wk Àμ2
wT
0w0; ð43Þ
inequality (42) can be written as
J ¼ ÀV0 þ
X1
k ¼ 0
εT
f a;kεf a;k À2μ2
X1
k ¼ 0
wT
k wk þμ2
wT
0w0 o0: ð44Þ
Knowing that V0 ¼ 0 for e0 ¼ 0, (44) leads to (38) with ω ¼
ffiffiffi
2
p
μ.
Since the general framework for designing the robust observer
is given, then the following form of the Lyapunov function is
proposed [41]:
Vk ¼ eT
k PðhkÞek; ð45Þ
where PðhkÞ40. On the contrary to the design approach presented
in the literature [42] it is not assumed that PðhkÞ ¼ P is a constant.
Indeed, PðhkÞ can be perceived as a parameter-depended matrix of
the form (cf. [3])
PðhkÞ ¼
XN
i ¼ 1
hkiPi; Pi 40: ð46Þ
As a consequence:
ΔVk þεT
f a;kεf a;k Àμ2
wT
k wk Àμ2
wT
k þ 1wk þ1
¼ eT
k A2ðhkÞT
Pðhkþ 1ÞA2ðhkÞþA3ðhkÞT
H1A3ðhkÞÀPðhkÞ
ek
þeT
k A2ðhkÞT
Pðhk þ 1ÞW 1 þA3ðhkÞT
H1W1
wk
þeT
k A2ðhkÞT
Pðhk þ 1ÞW 2 þA3ðhkÞT
H2
wk þ1
þwT
k W
T
1Pðhk þ1ÞA2ðhkÞþWT
1H1A3ðhkÞ
ek
þwT
k W
T
1Pðhk þ1ÞW 1 þWT
1H1W1 Àμ2
I
wk
þwT
k W
T
1Pðhk þ1ÞW2 þWT
1H2
wkþ 1 þwT
kþ 1 W
T
2Pðhk þ 1ÞA2;k
þHT
2A3ðhkÞ
ek þwT
k þ1 W
T
2Pðhk þ1ÞW1 þHT
2W1
wk
þwT
k þ 1 W
T
2Pðhk þ 1rÞW 2 þWT
2HT
HW2 Àμ2
I
wk þ 1 o0; ð47Þ
where ΔVk ¼ Vk þ 1 ÀVk, H1 ¼ CT
HT
HC and H2 ¼ CT
HT
HW2.
By defining the following vector
vk ¼ eT
k ; wT
k ; wT
k þ 1
 ÃT
; ð48Þ
inequality (47) receives the following form:
ΔVk þεT
f a;kεf a;k Àμ2
wT
k wk Àμ2
wT
k þ1wk þ 1 ¼ vT
k MV vk o0; ð49Þ
where MV is given by the equation:
MV ¼
A2ðhkÞT
Pðhk þ1ÞA2ðhkÞþA3ðhkÞT
H1A3ðhkÞÀPðhkÞ
W
T
1Pðhk þ 1ÞA2ðhkÞþWT
1H1A3ðhkÞ
W
T
2Pðhk þ 1ÞA2ðhkÞþHT
2A3ðhkÞ
2
6
6
6
4
A2ðhkÞT
Pðhk þ1ÞW 1 þA3ðhkÞT
H1W1
W
T
1Pðhk þ 1ÞW 1 þWT
1H1W1 Àμ2
I
W
T
2Pðhkþ 1ÞW1 þHT
2W1
A2ðhkÞT
Pðhk þ1ÞW 2 þA3ðhkÞT
H2
W
T
1Pðhkþ 1ÞW2 þWT
1H2
W
T
2Pðhkþ 1ÞW 2 þWT
2HT
HW2 Àμ2
I
3
7
7
7
7
7
5
:
ð50Þ
The following theorem constitutes the main result of this section:
Theorem 1. For a prescribed disturbance attenuation level μ40 for
the fault estimation error (21), the H1 observer design problem for
the system (10)–(11) and the observer (18) is solvable if there exist
matrices Pi g0 ði ¼ 1; …; NÞ, U and N such that the following LMIs
are satisfied:
AT
3;iH1A3;j ÀPi AT
3;iH1W1
WT
1H1A3;i WT
1H1W1 Àμ2
I
HT
2A3;i HT
2W1
UA2;i UW 1
2
6
6
6
6
6
4
AT
3;iH2 A2;iUT
WT
1H2 W
T
1UT
WT
2HT
HW2 Àμ2
I W
T
2UT
UW 2 Pj ÀU ÀUT
3
7
7
7
7
7
7
5
!0; ð51Þ
for i ¼ 1; …; N and j ¼ 1; …; N where (cf. (19) and (32)):
UA2;i ¼ Uð ~Ai ÀKaCÞ ¼ U ~Ai ÀNC; ð52Þ
M. Mrugalski et al. / ISA Transactions 61 (2016) 318–328 321

5. UW 1 ¼ UðGW1 ÀKaW2Þ ¼ UGW1 ÀNW2: ð53Þ
Proof. For the purpose of subsequent deliberations, let us remind
the following lemma [3].
Lemma 1. The following statements are equivalent:
(i) There exists X g0 such that
VT
XV ÀW !0: ð54Þ
(ii) There exists X g0 such that
ÀW VT
UT
UV X ÀU ÀUT
#
!0: ð55Þ
Subsequently, observing that the matrix (50) must be negative
definite and writing it as
A2ðhkÞT
W
T
1
W
T
2
2
6
6
6
4
3
7
7
7
5
Pðhk þ 1Þ A2ðhkÞ W 1 W 2
 Ã
þ ð56Þ
A3ðhkÞT
H1A3ðhkÞÀPðhkÞ A3ðhkÞT
H1W1 A3ðhkÞT
H3
WT
1H1A3ðhkÞ WT
1H1W1 Àμ2
I WT
1H2
HT
2A3ðhkÞ HT
2W1 WT
2HT
HW2 Àμ2
I
2
6
6
4
3
7
7
5!0;
ð57Þ
and then applying Lemma 1 leads to (51), which completes the
proof.□
Finally, the design procedure boils down to solving LMIs (51)
and then (cf. (52)–(53)) Ka ¼ U À1
N.
It can also be observed that the observer design problem can be
treated as a minimization task, i.e.
μn
¼ min
μ40;P1;…;PN 40;U;N
μ; ð58Þ
under (51).
To summarize the fault and state estimator design procedure of
the proposed scheme is:
1. Collect the input–output data from the system.
2. Select the structure of RNN (5)–(6) and obtain its parameters e.
q., with the software provided in MATLAB.
3. Compute the bounds (28)–(29).
4. Obtain the LPV description underlying (32).
5. Solve (58) under (51) and obtain Ka ¼ U À 1
N.
4. Fault diagnosis of the multi-tank system
In order to show the effectiveness of the developed approach in
the actuator fault estimation task, the multi-tank system pre-
sented in Fig. 2 is chosen. Such a system is designed for simulating
the real industrial multi-tank systems in the laboratory conditions
[9]. It consists of three separate tanks placed each above other and
equipped with drain valves and level sensors based on a hydraulic
pressure measurement. Each of them has a different cross-section
in order to reflect system nonlinearities. The lower bottom tank is
a water reservoir for the system. A variable speed water pump is
used to fill the upper tank. The water outflows the tanks due to
Fig. 2. Multi-tank system.
Fig. 3. Distribution of the disturbances for the top tank level sensor.
M. Mrugalski et al. / ISA Transactions 61 (2016) 318–328322

6. gravity. The considered multi-tank system has been designed to
operate with an external, PC-based digital controller. The control
computer communicates with the level sensors, valves and a
pump by a dedicated I/O board and the power interface. The I/O
board is controlled by the real-time software, which operates in a
Matlab/Simulink environment. For further information the reader
is referred to the INTECO manufacturer documentation.
The distribution matrices W1 and W2 should express the
influence and magnitude of wk onto the state and output (10)–
(11), respectively. To obtain appropriate proportion between the
elements of W1 and W2, series of constant liquid level measure-
ments were performed for the top tank. Subsequently, the mean
was removed, which represents the constant liquid level, and then
the disturbances were analysed. Fig. 3 depicts the histogram of the
estimated disturbances. The standard deviation of the disturbance
is equal to 1:75 Â 10À4
(obtained for 1000 measurements). Almost
identical results were obtained for the sensors in the middle and
bottom tanks. This is not surprising since all sensors are identical.
It should be underlined that the term W1wk in (10) represents the
inaccuracy of the pump with respect to a desired control action.
After a similar experiments like for the sensors, it was derived that
the maximum magnitude of W1wk is approximately 5 times larger
than that of W2wk. As a result, the following settings of the dis-
tribution matrices were established
W1 ¼ diagð0:05; 0; 0Þ; W2 ¼ 0:01Im: ð59Þ
At the beginning of the development, the neural LPV model
of the multi-tank system according to the proposed methodology
(cf. Section 2) have to be obtained. It should be mentioned that
the neural network was trained using Levenberg–Marquardt
backpropagation algorithm. 70% of the data set gathered from the
system was taken as a training set, 15% as validation set and 15% as
testing set. Fig. 4 presents the performance of the neural network.
The training process stops after 12 iterations which confirms that
prescribed mean squared error level is reached. Fig. 5 shows the
system and model outputs representing measurement and esti-
mate of the liquid level in the upper tank of the multi-tank system
for the validation data set. Moreover, in this figure the scaled input
voltage representing the control signal is depicted. As it can be
seen, the proposed neural model has an appropriate approxima-
tion properties and with relatively high accuracy reflects the real
system.
The obtained neural model of the multi-tank system can be
used to validate the effectiveness of the fault diagnosis method
developed in Section 3. For that purpose, it is assumed that matrix
C has the following form:
C ¼
1 0 0
0 0 1
!
; ð60Þ
which means that the state x2, representing the level of the second
tank is unavailable.
Let us consider the following different types of fault scenarios
which may be perceived i.e. as a permanent or temporary decrease
of the pomp efficiency:
(a) Stack in place fault:
f a;k ¼
À5 Á 10À5
þuk; for 10 000rkr15 000;
0; otherwise:
(
ð61Þ
(b) 20% abrupt actuator loss of effectiveness fault:
f a;k ¼
À0:2uk; for 5000rkr10 000;
0; otherwise:
(
ð62Þ
(c) Incipient fault:
f a;k ¼
À0:2ðkþ5000Þuk for 5000rkr10 000;
0; otherwise:
(
ð63Þ
According to the methodology of the UIO design for the
actuators fault identification described in Section 3 as a result ofFig. 4. Neural network performance.
Fig. 5. System and model outputs for the upper tank. Fig. 6. Evolution of ΔVk þεT
f a ;kεf a ;k Àμ2
wT
k wk Àμ2
wT
kþ 1wk þ 1.
M. Mrugalski et al. / ISA Transactions 61 (2016) 318–328 323

7. solving the problem (51), the following values were obtained:
μ ¼ 0:45; Ka ¼
0 0
À0:0720 0
0 0:0678
2
6
4
3
7
5: ð64Þ
Let us assume that the initial condition for the system and the
observer is x0 ¼ ½0:001; 0:001; 0:001ŠT
and ^x0 ¼ ½0:003; 0:002; 0:001
ŠT
while the input is uk ¼ 0:00009. First, let us consider the case
when ^x0 ¼ x0 for e0 ¼ 0. Fig. 6 clearly indicates that condition (38)
is satisfied, which means that an attenuation level μ¼0.45 is
achieved. Now let us assume that wk ¼ 0 and ^x0 ax0. Note that, to
check the disturbance-free behaviour, the MATLAB simulator
provided by INTECO was employed while the rest of the experi-
ments were performed with real system exclusively. Fig. 7 clearly
shows that (37) is satisfied as well.
It should be pointed out that for the purpose of comparison, an
adaptive fault estimator [43] was used with the system matrices
provided in the INTECO documentation [9].
Fig. 8 shows the stack in place fault introduced into the system
and its estimate for the nominal case (^x0 ax0 and wk a0). In order
to show the performance of the proposed approach, the fault
identification results obtained with the linear observer are also
presented in Fig. 8. As it can be seen, the robust UIO estimates the
Fig. 7. Evolution of Jek J (for k ¼ 0; …; 20).
Fig. 8. Stack in place fault estimated with linear and robust UIO.
Fig. 9. Estimation error of the stack in place fault.
Fig. 10. State estimation error – first tank.
Fig. 11. State estimation error – second tank.
Fig. 12. State estimation error – third tank.
M. Mrugalski et al. / ISA Transactions 61 (2016) 318–328324

8. real stack in place fault with better accuracy than linear observer.
Fig. 9 shows the fault estimation error obtained with the proposed
robust UIO, which oscillates around zero.
Figs. 10, 11 and 12 show the estimation errors in the first,
second and third tank, respectively. Taking into account the fact
that the level in each tank varies from 0 to 0.35 m, these results
should be perceived as very satisfactory ones.
The next results are regarding the 20% abrupt actuator loss of
effectiveness fault. Similarly, as in the previous fault scenario,
Fig. 13 presents the results of the fault estimation with the appli-
cation of the robust UIO and linear observer.
Fig. 14 presents the fault estimation error obtained with the
application of the developed approach. Figs. 15, 16 and 17 present
the estimation errors in the first, second and third tank, respec-
tively. From the above results it is clear that the estimate of the
fault obtained with the UIO has good accuracy.
In the real systems the fault value often increases with the
time. To simulate such kind of fault, the incipient fault scenario is
introduced. Fig. 18 presents the results of the incipient fault esti-
mation obtained with the robust UIO and linear observer. It is easy
to observe that the fault is estimated with relatively small error,
which is depicted in Fig. 19. Figs. 20, 21 and 22 present the esti-
mation errors in the first, second and third tank, respectively.
Fig. 13. 20% abrupt actuator loss of effectiveness fault estimated with linear and
robust UIO.
Fig. 15. State estimation error – first tank.
Fig. 16. State estimation error – second tank.
Fig. 17. State estimation error – third tank.
Fig. 18. Incipient fault estimated with linear and robust UIO.
Fig. 14. Estimation error of the 20% abrupt actuator loss of effectiveness fault.
M. Mrugalski et al. / ISA Transactions 61 (2016) 318–328 325

9. Similarly, as in the case of stack in place and 20% abrupt
actuator loss of effectiveness faults the robust UIO estimates the
incipient fault much better than the linear one.
The next goal of this section is to show the capability of the
developed actuators fault estimation method. To achieve this the
comparison of the approaches, which are based on the LPV models
obtained with the RNN and analytical model in the state estima-
tion task, is required. For this reason, the result of work [37] is
utilized, in which the analytical model, which is given in the multi-
tanks system documentation [9], is transformed into LPV model
using the DMVT according to the methodology presented in [37].
Note that such a method is based on the physics laws governing
the behaviour of the multi-tank system whereas the method
developed in this paper is designed entirely on the basis of the
input–output measurement data, without any prior knowledge
about the system.
Fig. 23 presents the state estimates obtained with the Robust
Fault Estimator (RFE) designed according to the technique pre-
sented in [37] and obtained with a proposed Robust Neural
Network-based Fault Estimator (RNNFE). It is clear that both
techniques estimate the first tank state x1 with similar, satisfactory
accuracy.
In Figs. 24 and 25 the second x2 and the third x3 tank state
estimates are presented, respectively. Similarly as in the case of the
first tank, both RFE and RNNFE estimate the second tank state x2
with similar quality. In the case of the third tank, some deviations
Fig. 19. Estimation error of the incipient fault.
Fig. 20. State estimation error – first tank.
Fig. 21. State estimation error – second tank.
Fig. 22. State estimation error – third tank.
Fig. 23. Comparison of the real system state x1 with state estimated by RFE
and RNNFE.
Fig. 24. Comparison of the real system state x2 with state estimated by RFE
and RNNFE.
M. Mrugalski et al. / ISA Transactions 61 (2016) 318–328326

10. can be seen, however, their values are close to the measurement
errors.
5. Conclusions
The main objective of this paper was to propose a novel
structure of the RNN-based robust UIO and its design procedure
for a fault estimation purpose for a class of non-linear discrete-
time systems. First, a procedure for transforming neural state-
space model into a discrete-time polytopic LPV model is proposed.
Such an approach allows to combine positive features of analytical
and soft-computing methods. Moreover, a combination of the
celebrated generalized observer scheme with the robust H1
approach is developed to settle the problem of robust fault diag-
nosis. The proposed approach is designed in such a way that a
prescribed disturbance attenuation level is achieved with respect
to the actuators fault estimation error while guaranteeing the
convergence of the observer. The final part of the paper is con-
cerned with a comprehensive case study regarding the multi-tank
system. The application of the multi-tank-system is motivated by
the fact that its analytical model is well-known and it can be used
to obtain the LPV model. Such model is applied to the evaluation of
the RNNFE approach developed in this paper. The obtained results
show that both approaches have similar quality. However, it
should be clearly underlined that the RNNFE can be easily applied
in several practical cases in contrast to the RFE approach which
requires the analytical model of the diagnosed system which is not
always available. The natural extension of this paper, based on the
achieved fault identification results, may be to design the FTC
strategy.
Acknowledgements
The authors would like to express their sincere gratitude to the
referees for their valuable comments, which contributed sig-
nificantly to the current shape of the paper.
The work was supported by the National Science Centre of
Poland under Grants: UMO-2013/11/B/ST7/01110 and UMO-2014/
15/N/ST7/00749.
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