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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

oyj

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

oCjek+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

kA2(α)TP(α)A2(α)ek+

eT

kA2(α)TP(α)¯

Wwk+

wT

k¯

WTP(α)A2(α)ek+

wT

k¯

WTP(α)¯

Wwk<0

(27)

By deﬁning

vk=eT

k,wT

kT,(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 Ton 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

¯

WTP(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

kA3(α)TP(α)A3(α)xk+

xT

kA3(α)TP(α)W1wk+

wT

kWT

1P(α)A3(α)xk+

wT

kWT

1P(α)W1wk<0.(49)

Robust Fault-tolerant Control for a Multi-tank System 7

By deﬁning

vk=xT

k,wT

kT,(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

1P(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) = wR2−(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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