Robust Fault-tolerant Control for a Multi-tank System

Abstract

The paper deals with the problem of robust fault-tolerant control for non-linear discrete-time systems. The main part of this paper describes sensor a fault diagnosis scheme using virtual sensors, which recover the measurement of the fault sensor based on the fault-free ones. The virtual sensor is designed in such a way that a prescribed attenu-ation level is achieved with respect to the fault estimation error while guaranteeing the convergence of the robust observer underlying the vir-tual sensor. The subsequent part of the paper deals with the design of robust controller as well as the proposed fault-tolerant control scheme. The final part of the paper shows the experimental results regarding the multi-tank system, which confirm the effectiveness of proposed approach.

Full text

Robust Fault-tolerant Control for a Multi-tank
System
Mariusz Buciakowski, Marcin Witczak, and Marcel Luzar
Institute of Control and Computation Engineering, University of Zielona G´ora,
ul. Pog´orna 50, 65-246 Zielona G´ora, Poland
mariusz.bu@wp.pl
{m.luzar,m.witczak}@issi.uz.zgora.pl
Abstract. The paper deals with the problem of robust fault-tolerant
control for non-linear discrete-time systems. The main part of this paper
describes sensor a fault diagnosis scheme using virtual sensors, which
recover the measurement of the fault sensor based on the fault-free ones.
The virtual sensor is designed in such a way that a prescribed attenu-
ation level is achieved with respect to the fault estimation error while
guaranteeing the convergence of the robust observer underlying the vir-
tual sensor. The subsequent part of the paper deals with the design of
robust controller as well as the proposed fault-tolerant control scheme.
The final part of the paper shows the experimental results regarding the
multi-tank system, which confirm the effectiveness of proposed approach.
Keywords: Fault diagnosis, fault identication, robust estimation, non-
linear systems, virtual sensor
1 Introduction
The problem of fault diagnosis (FD) of non-linear industrial systems [1–4] has re-
ceived considerable attention during the last three decades. Indeed, it developed
from the art of designing a satisfactory performing systems into the modern the-
ory and practice that it is today. Within the usual framework, the system being
diagnosed is divided into three main components, i.e. plant (or system dynam-
ics [1, 2]), actuators and sensors. The paper deals with the problem of full fault
diagnosis of sensors, i.e. apart from the usual two steps consisting of fault detec-
tion and isolation (FDI), the fault identification is also performed. This last step
is especially important from the viewpoint of Fault-Tolerant Control (FTC) [5–
7], which is possible if and only if there is an information about the size of the
fault being a result of fault identification (or fault estimation). In this paper a
robust fault estimation approach is proposed, which can be efficiently applied
to realise the above-mentioned three-step procedure. The proposed approach is
based on the general idea of an Unknown Input Observer (UIO) [2,8], which was
initially designed to tolerate a degree of model uncertainty and hence increase
the reliability of fault diagnosis. The proposed approach can be perceived as a
combination of the linear-system strategies [9] and [10] for a class of non-linear

2 Mariusz Buciakowski, Marcin Witczak, and Marcel Luzar
systems [12]. The proposed approach is designed in such a way that a prescribed
disturbance attenuation level is achieved with respect to the fault estimation er-
ror while guaranteeing the convergence of the observer. While the fault-tolerant
control scheme is based on replacing the faulty sensor measurements and feed-
ing them into the robust controller. The paper is organised as follows. Section 2
describes the proposed virtual sensor. Whilst section 3 describes the robust con-
troller and an integration procedure with the virtual sensor strategy. The final
part of the paper presents a comprehensive case study regarding the multi-tank
system, which clearly indicate the performance of the proposed approach.
2 Virtual sensor design
The main objective of this section is to provide a detailed design procedure of
the virtual sensor, which can be used for sensor fault diagnosis. In other words,
the main role of this sensor is to provide the information about the sensor fault.
Indeed, apart from serving as a usual residual generator, the virtual sensor should
be designed in such a way that a prescribed disturbance attenuation level is
achieved with respect to the sensor fault estimation error while guaranteeing the
convergence of the observer.
Let us consider to following non-linear system:
xk+1 =Axk+Buk+g(xk) + W1wk,(1)
yk+1 =Cxk+1 +Lsfs,k +W2wk+1 ,(2)
where xk∈Rnis the state vector, uk∈Rrstands for the input, yk∈Rm
denotes the output, fs,k ∈Rmstands for the sensor fault. While , wk∈l2is
a an exogenous disturbance vector with W1∈Rn×n,W2∈Rm×nbeing its
distribution matrices while
l2={w∈Rn| ∥w∥l2<+∞},∥w∥l2=∞

k=0 ∥wk∥2
1
2
.(3)
Moreover, let us define the matrix Xbe partitioned in such a way that
X=


xT
1
.
.
.
xT
nx


(4)
where xjstands for the jth row of X. Let us also denote Xjas the matrix X
without the jth row and yjas a vector ywithout the jth element.
The sensor fault diagnosis will be realised by a set of mobservers of the form:
ˆ
xk+1 =Aˆ
xk+g(ˆ
xk) + Kj
oyj
k−Cjˆ
xk, j = 1, . . . , m, (5)
while the jth output (for Ls,k =I) is described by
yf,j,k =cT
jxf,k +wT
2,j wk+fs,j,k.(6)

Robust Fault-tolerant Control for a Multi-tank System 3
Thus:
fs,j,k =yf,j,k −cT
jxf,k −wT
2,j wk,(7)
and an jth fault estimate is
ˆ
fs,j,k =yf,j,k −cT
jˆ
xk.(8)
The fault estimation error εfj,k of the jth sensor is
εfj,k =fs,j,k −ˆ
fs,j,k =−cT
jxf,k +cT
jˆ
xk−wT
2,j wk=−cT
jek−wT
2,j wk,(9)
while the state estimation error is:
ek+1 =Aek+sk−Kj
oCjek−Kj
oW2wk+W1wk,(10)
ek+1 =A−Kj
oCjek+sk−¯
W wk,(11)
ek+1 =A1ek+sk−¯
W wk.(12)
where
sk=g(xk)−g(ˆ
xk).(13)
Note that both ek+1 and εfs,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) [11], it can be shown that
g(a)−g(b) = Mx(a−b),(14)
with
Mx=





∂g1
∂x (c1)
.
.
.
∂gn
∂x (cn)






,(15)
where c1, . . . , cn∈Co(a,b), ci̸=a,ci̸=b,i= 1, . . . , n. Assuming that
¯ai,j ≥∂gi
∂xj≥ai,j , i = 1, . . . , n, j = 1, . . . , n, (16)
it is clear that:
Mx=M∈Rn×n|¯ai,j ≥mx,i,j ≥ai,j, i, j = 1, . . . , n, (17)
Thus, using (14), the term A1ek+skin (12) can be written as
A1ek+sk= ( ¯
A+Mx,k −Kj
oCj)ek(18)
where Mx,k ∈Mx.
From (18), it can be deduced that the state estimation error can be converted
into an equivalent form
ek+1 =A2(α)ek−¯
W wk,(19)
A2(α) = ˜
A(α)−KoC,

4 Mariusz Buciakowski, Marcin Witczak, and Marcel Luzar
which defines an LPV polytopic system [13] with
˜
A=˜
A(α) : ˜
A(α) =
N

i=1
αi˜
Ai,
N

i=1
αi= 1, αi≥0,(20)
where N= 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, Nis given
by N= 2(n−c)2where cstands for the number of constant elements of Mx,k.
Thus, the state estimation error (12) can be described as
ek+1 =A2(α)ek−¯
W wk,(21)
A2(α) = ˜
A(α)−Kj
oCj,(22)
The problem of H∞observer design [14] is to determine the gain matrix Ko
such that
lim
k→∞
ek=0for wk=0,(23)
∥εf∥l2≤ω∥w∥l2for wk̸=0,e0=0.(24)
The general framework for designing robust observer is:
∆Vk+εT
fj,k εfj,k −µ2wT
kwk<0, k = 0, . . . ∞,(25)
with
∆Vk=Vk+1 −Vk, Vk=eT
kP(α)ek, ω =√µ. (26)
Consequently, it can be shown that:
∆Vk+εT
fj,k εfj,k −µ2wT
kwk=
eT
kA2(α)TP(α)A2(α)ek+
eT
kA2(α)TP(α)¯
Wwk+
wT
k¯
WTP(α)A2(α)ek+
wT
k¯
WTP(α)¯
Wwk<0
(27)
By defining
vk=eT
k,wT
kT,(28)
the inequality (27) becomes
∆Vk+εT
fj,kεfj,k −µ2wT
kwk=vT
kMVOvk<0,(29)
where
MVO=A2(α)TP(α)A2(α)−P(α) + cjcT
jA2(α)TP(α)¯
W+cjwT
2,j
¯
WTP(α)A2(α) + w2,j cT
j¯
WTP(α)¯
W+w2,j wT
2,j −µ2I.(30)
The following two lemmas can be perceived as the generalisation of those
presented in [13].

Robust Fault-tolerant Control for a Multi-tank System 5
Lemma 1. The following statements are equivalent
1. There exists X≻0such that
VTXV −W≺0 (31)
2. There exists X≻0such that
−W V TUT
UV X −U−UT≺0.(32)
Proof. Applying the Schur complement to (2) gives
VTUT(UT+U−X)−1UV −W≺0.(33)
Substituting U=UT=Xyields
VTXV −W≺0.(34)
Thus, (1) implies (2).
Multiplying (32) by T=I V Ton the left and by TTon the left of (32)
gives (31), which means that (2) implies (1) and hence the proof is completed.
Lemma 2. The following statements are equivalent
1. There exists X(α)≻0such that
V(α)TX(α)V(α)−W(α)≺0,(35)
2. There exists X(α)≻0such that
−W(α)V(α)TUT
UV (α)X(α)−U−UT≺0.(36)
Proof. The proof can be realised by following the same line of reasoning as the
one of Lemma 1.
It is easy to show that (36) is satisfied if there exist matrices Xi≻0such
that −WiVT
iUT
UV iXi−U−UT≺0, i = 1, . . . , N. (37)
Theorem 1. For a prescribed disturbance attenuation level µ > 0for the fault
estimation error (9), the H∞observer design problem for the system (1)–(2)
and the observer (5) is solvable if there exist matrices Pi≻0(i= 1, . . . , N), U
and Nsuch that the following LMIs are satisfied:



−Pi+cjcT
jcjwT
2,j AT
2,iUT
w2,j cT
jw2,j wT
2,j −µ2I¯
WTUT
UA2,i U¯
W P i−U−UT

≺0.(38)
where
UA2,i =U(ˆ
Ai−KoC) = Uˆ
Ai−NC.(39)

6 Mariusz Buciakowski, Marcin Witczak, and Marcel Luzar
Proof. Observing that the matrix (30) must be negative definite and writing it
as
A2(α)T
¯
WTP(α)A2(α)T¯
WT+−P(α) + cjcT
jcjwT
2,j
w2,j cT
jw2,j wT
2,j −µ2I≺0.(40)
and then applying Lemma 2 and (37) leads to (38), which completes the proof.
Finally, the gain matrix of the virtual sensor is given by
Ko=U−1N.(41)
3 Controler design
The main objective of this section is to present the design procedure of the robust
controller, for which a predefined disturbance attenuation level with respect to
the state of the system is achieved. Following the same line of reasoning as in the
preceding section, the state equation (1) can be written in an equivalent form:
xk+1 =A(α)xk+Buk+W1wk.(42)
Substituting
uk=−Kcxk(43)
into (42) yields
xk+1 =A3(α)xk+W1wk,(44)
where
A3(α) = (A(α)−BK c)xk,(45)
with
A=A(α) : A(α) =
N

i=1
αiAi,
N

i=1
αi= 1, αi≥0.(46)
Similarly as in Section 2, the general framework for designing robust controller
is:
∆Vk+xT
kxk−µ2wT
kwk<0, k = 0, . . . ∞,(47)
with
Vk=xT
kP(α)xk.(48)
Consequently, it can be shown that:
∆Vk+xT
kxk−µ2wT
kwk=
xT
kA3(α)TP(α)A3(α)xk+
xT
kA3(α)TP(α)W1wk+
wT
kWT
1P(α)A3(α)xk+
wT
kWT
1P(α)W1wk<0.(49)

Robust Fault-tolerant Control for a Multi-tank System 7
By defining
vk=xT
k,wT
kT,(50)
the inequality (49) becomes
∆Vk+xT
kxk−µ2wT
kwk=vT
kMVCvk<0,(51)
where
MVC=A3(α)TP(α)A3(α)−I A3(α)TP(α)W1
WT
1P(α)A3(α)WT
1P(α)W1−µ2I.(52)
Theorem 2. For a prescribed disturbance attenuation level µ > 0for the state
(42), the H∞controller design problem for the system (1)–(2) is solvable if there
exists matrices Pi≻0(i= 1, . . . , N), Uand Vsuch that the following LMIs
are satisfied:


I0A3,iU
0−µ2I W 1U
UTAT
3,i UTWT
1Pi−U−UT
≺0,(53)
with
A3,iU= (Ai−BKc)U=AiU−BV .(54)
Proof. Observing that the matrix (49) must be negative definite and writing it
as A3(α)T
WT
1P(α)A3(α)TWT
1+I0
0−µ2I≺0 (55)
and then applying transposed version of Lemma 2 leads to (53), which completes
the proof.
Thus, the final design procedure is: given a prescribed disturbance attenuation
level µ, obtain Pi≻0,U,Vby solving (53). Finally, the gain matrix of the FTC
controller is:
Kc=V U −1.(56)
Since the design procedure of the robust controller is provided, then the FTC
scheme can be described in details. The FTC scheme is portrayed in Fig. 1. The
main idea behind the proposed approach is that the faulty sensor measurement
is replaced by the fault-free one. The decision is realised by the FD part, which
aims at providing appropriate switching in case of a sensor fault. The decision
are based on thresholding the fault estimates provided by the virtual sensors [2].
4 Case study
To verify the proposed approach, it is implemented for the multi-tank system.
The considered multi-tank system (Fig. 2) is designed for simulating the real in-
dustrial multi-tank system in the laboratory conditions [15]. It can be efficiently

8 Mariusz Buciakowski, Marcin Witczak, and Marcel Luzar
SYSTEM
CONTROLLER FD
VIRTUAL
SENSOR
VIRTUAL
SENSOR
VIRTUAL
SENSOR
r
u
y
1
y2
n
y
_
y
1
y2
n
y
_
_
y
1
y2
n
y
^
^
^
Fig. 1. FTC scheme
used to practically verify both linear and non-linear control, identification and
diagnostics methods. The considered system 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 reser-
voir for the system. A variable speed water pump is used to fill the upper tank.
The water outflows the tanks due to 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.
The system matrices and non-linearities are
A=In,B=

0.014
0
0
,C=Im,
La=Im,
g(xf,k ) = 


1
β1(x1,k)C1xα1
1,k
1
β2(x2,k)C1xα1
1,k −1
β2(x2,k)C2xα2
2,k
1
β3(x3,k)C2xα2
2,k −1
β3(x3,k)C3xα3
3,k


.(57)
where xi,k, i ∈1, . . . , 3 is water level in the ith tank, βi(xi,k ) stand for cross
section area of the ith tank at the level xi,k and is, respectively, defined as:
β1(x1,k) = aw – constant cross-sectional area of the top tank,
β2(x2,k) = cw +x2,k
x2max bw – variable cross-sectional area of the middle tank,
β3(x3,k) = wR2−(R−x3,k )2– variable cross-sectional area of the bottom

Robust Fault-tolerant Control for a Multi-tank System 9
Fig. 2. Multi-tank system
tank.
The numerical values of above parameters are as follows: C1= 1.0057 ·10−4,
C2= 1.1963 ·10−4,C3= 9.8008 ·10−5,b= 0.34, c= 0.1, w= 0.035, R= 0.364,
x2max = 0.35, α1= 0.29, α2= 0.2256, α3= 0.2487, and h= 0.01s.
Due to the pump is not equipped with flow sensor, it is impossible to explicitly
identify the sensor fault in the upper tank. Thus, to present the results, the upper
tank liquid level is measured based on real sensor signal and the other levels in
the tanks are measured with the estimated and real values.
The assumed initial state and its estimate are equal and the system input is
generated using random signal 0.08 ≤uk≤2.3 while wk∼ N(0,0.01I).
The following fault scenario for second sensor was introduced:
fs2,k =y2,k −0.35,for 100 ≤k≤300,
0,otherwise.
Moreover, the strategy of switching between the virtual sensor output and system
output presented in Fig. 1 is based on following rule:
¯
y2,k =ˆ
y2,k,if |ˆ
fs,2,k|> ϵ,
y2,k,otherwise,
where ϵ > 0 is a given threshold and ˆ
fs,2,k is result of (8).
In Fig. 3 the second sensor output and its estimate is presented. It is clear, that

10 Mariusz Buciakowski, Marcin Witczak, and Marcel Luzar
the state observer designed in Section 2 estimates the second sensor output with
a satisfactory accuracy.
Fig. 4 shows the system performance with and without proposed FTC strat-
egy. Indeed, the reference signal (solid line) is the target that has to be achieved
by the controller. The fault of the second sensor appears for k= 100, . . . , 300,
which is clearly depicted on both figures. While in the right figure, it can be ob-
served that the faulty measurement is replaced by its estimate provided by the
virtual sensor. A straight comparison clearly indicates that the FTC controller
outperforms the usual robust controller. Finally it should be mentioned that
0 50 100 150 200
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Discrete time
y
y2
yf2
Fig. 3. Second output and its estimate
the similar results were obtained for the first and second sensor.
5 Conclusions
The paper deals with the problem of robust FTC for a class on non-linear sys-
tems. In particular, a combination of the celebrated generalised virtual sensor
scheme with the robust H∞approach is proposed to settle the problem of ro-
bust fault diagnosis. The proposed approach is designed in such a way that a
prescribed disturbance attenuation level is achieved with respect to the sensor
fault estimation error while guaranteeing the convergence of the observer. More-
over, the controller design, which realises the switching strategy between virtual
sensor and real sensor output, is carefully analysed. The final part of the paper
is concerned with a comprehensive case study regarding the multi-tank system.

Robust Fault-tolerant Control for a Multi-tank System 11
0 100 200 300 400
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Discrete time
y
y1
yf1
y2
0 100 200 300 400
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Discrete time
y
y1
yf1
y2
yf2
Fig. 4. Performance of the system with (right) and without FTC
The achieved results show the performance of the proposed approach, which
confirm its practical usefulness.
6 Acknowledgments
The work was financed as a research project with the science funds for years
2011-2014 with the kind support of the National Science Centre in Poland under
the grant NN514678440 Predictive fault-tolerant control for non-linear systems.
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