reconfigurable of active fault tolerant control for nonlinier actuator and sensor fault - sami...

8/10/2019 Reconfigurable of Active Fault Tolerant Control for Nonlinier Actuator and Sensor Fault - Sami Patton

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Reconfigurable Fault-Tolerant Control of Linear

System with Actuator and Sensor Faults

Katherin Indriawati, Trihastuti Agustinah, Achmad Jazidie

Department of Electrical Engineering

ITS

Surabaya, [email protected]

AbstractThis paper presents an active fault-tolerant control

for linear system in case of actuator and sensor faults where these

minor faults lead to degraded performance of the system. Three

steps are proposed to achieve fault tolerant control based on

simplified analytical redundancy. Firstly, a bank of linearobserver is proposed to estimate the actuator and sensor faults by

modeling a descriptor LTI system using the SVD technique.

Secondly, the estimated faults are used to design a fault decision

scheme to detect the faults correctly. Thirdly, a reconfigurable

fault-tolerant control scheme is designed by using the estimated

faults to compensate the fault effects on controller performance.

Simulation on the three tank system is given to illustrate the

performance of the proposed method.

Index TermsObserver, SVD, reconfigurable FTC.

INTRODUCTION

Control systems which have the ability to accommodate

component (actuator or/and sensor) failures automatically arecalled Fault Tolerant Control Systems. These systems are able

to maintain the stability and the desired performance of the

system in the presence of such failures [1]. FTCS is needed to

increase reliability and automation level in modern engineeringsystems. Generally, FTCS can be performed by passive

methods or by active methods. In passive methods, controller is

fixed and can be designed using robust control techniques to

ensure that a closed-loop system remains insensitive to certain

faults. This approach needs neither on-line fault information

nor controller reconfiguration, but it has limited fault-tolerantcapabilities [1]. On the other hand, in active methods, a new

control system is redesigned by using on line fault information

in order to maintain the stability and acceptable performance ofthe entire system, or in circumstances, to achieve accepted

degraded performance. Active FTCS are often referred to as

reconfigurable control. The design of an active FTCS requiresquick but effective fault detection and isolation (FDI) scheme

for adequate decision making that refers to the task of inferring

the occurrence of faults in a system.

A general approach of active FTCSis based on analytical

redundancy. Noura et al. in [2] has presented this approach for

discrete linear systems. They treat the sensor faults as the

actuator faults, then it used to estimate all faults by solving a

difference matrix equation. The estimated sensor faults are

used to modify the nominal control law to compensate for the

effects of the sensor faults. In [3], both sensor and actuator

faults are isolated and estimated by using of a unique structured

residual generator. The residual generator consist of a bank of

unknown input observer that each observer may be used to

detect a single fault. In [4] linear time invariant systems with

sensor faults are transformed into descriptor system and then

the proportional plus derivative observers are used to

simultaneously estimates the states of the system and the

sensor faults. However, all of those researches assume the

perfect condition of the plant regime and there is no

environmental noise in measurement system, which implies

that FDI algorithm detect faults instantaneously and always

correct [5].

To develop an active FTCS, it is required to examine

reconfigurable control and FDI to ensure that they can work inharmony. The kind of information needed from a FDI should

be examined to achieve a reasonable control strategy. An

imperfect FDI algorithm may not only result in loss of

performance, but also instability for the overall FTCS. This

paper focuses on active FTC based on analytical redundancy

which combines the functions of FDI and reconfigurablecontrol in noisy environment. Studies to this area is fewer than

other areas of fault-tolerant control research [6]. Furthermore,

many challenging issues still remain open for further research

and development for this area [1].

Generally reconfiguration scheme for the linear control

system that has been proposed in the scientific paper is only

detecting one sensor or actuator faults occur at a particulartime, as in [2] and [3]. This paper presents the results of a

simulation study for linear control system reconfiguration

scheme that tolerant of sensor and actuator faults that occur

sequentially. Furthermore, the measurement noise influences

are also considered in designing of FDI algorithm in order to

minimize the occurence of false alarm and missed alarm.

This paper is organized as follows. In section II, the

nominal linear control system and the reconfigurable control

probem dealing with actuator and sensor faults are presented.

In section III, the strategy of reconfigurable linear control

2013 IEEE International Conference on Control System, Computing and Engineering, 29 Nov. - 1 Dec. 2013, Penang, Malaysia

978-1-4799-1508-8/13/$31.00 2013 IEEE 22


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based on a bank of observers is propo

example of a three-tank system and its sim

given in section IV. Finally, concluding resection V.

PROBLEM FORMULATIO

Nominal Control

Consider a discrete linear time invarigiven by the following state space represent

=

+=+

)()(

)()()1(

kk

kkk

Cxy

BuAxx

where xRn, uR

pand yR

mare the state

input, the output vector, respectively. ARnx

Rmxn

are the state, the control, and the

respectively. The number of outputs m t

reference input vector yrdo not exceed the

inputs due to controlability requirement.

The nominal control system for that plstructure with integrator as shown in Figu(hp) represents the vector of the masureme

required to follow the reference input vectorh)

represents the vector of the unmeasure

nominal control system take into account t

(U0, Y0). The state space representation of tshown in Figure 1 is

[ ]

=

+

+

=

+

+

)(

)(0)(

)(0

0)(

)(0

)1(

)1(

,

,

,1

,

kz

kxCky

kyIT

B

kz

kx

ICT

A

kz

kx

pq

rps

pn

mpps

pn

where C1 is row of matrix C related to the

is the sample period to be chosen properly

matrix of dimension pxp, and 0n,p is

dimension nxp. The nominal feedback co

system is computed by

[ ]

==

)()()()( 21

kzkxKKkXKku

K = [K1 K2] is the feedback gain matrix obta

techniques such as a pole placement t

quadratic optimization, and so on [8][9][10][

ed. A numerical

ulation results are

arks are given in

ant (LTI) systemtion

(1)

ector, the control

, BRnxp

, and Coutput matrices,

hat can track a

number of control

ant uses feedbacke 1,where y1R

h

nt outputs that are

yrwhile y2R(m-

ent output. The

e operating point

he control system

)(ku

(2)

ontrolled state, Ts

, Ip is an identity

a null matrix of

ntrol law of this

(3)

ined using several

echnique, linear-

11].

Fig. 1 Nominal tracking cont

Reconfigurable ControlThe designed control syst

control signal automatically

component faults so that the pl

used algorithm in designing of

the suitable model with the sim

To achieve a control syste

sensor faults, the proposed m

recalculation of the control si

type. The block diagram of th

is shown in Figure 2. The n

system is given by

)()()( ukukuku addaddan ++=

where un (k) represents the n

represents the additive cont

actuator faults, and uadds (k)

signal to compensate the senso

The control signal reconfi

process in order to detect

commonly known as fault

Here, the FDI proposed met

mathematical model (analyti

observer to generate residual.

make the observer in a simplhandle both actuator and senso

order to minimize the occure

alarm due to noise measure

modification algorithm of the

to the FDI threshold values.

RECONFIGURABLE LI

Residual is the differenc

measurement value and the s

operating condition. It is assu

taken by an actuator-sensorobserver needed for the FDI s

of actuator-sensor pairs con

observer then are used in the

about where may the faults be

technique. Based on that de

occured fault is estimated and

order to recalculate the control

ol with feedback structure [7]

m conducts reconfiguration of

in order to accommodate the

ant still operates as desired. The

that control system is based on

ple structure and techniques.

that tolerant from actuator and

ethod of this paper consist of

gnal based on the occured fault

reconfigurable control system

ew control law applied to the

)(k (4)

ominal control signal, uadda (k)

ol signal to compensate the

epresents the additive control

faults.

guration needs fault diagnosis

nd isolate the occured fault,

detection and isolation (FDI).

od in this paper is based on

al redundancy), that is using

Thus the problem is how to

e manner that can be used tor faults at once. Furthermore, in

ce of false alarm and missed

ment, this paper propose the

Shewhart control chart related

NEAR CONTROL DESIGN

e of the considered quantity

ame quantity value in normal

ed that each controlled variable

air. Therefore, the number ofstem is equal with the number

sidered. The results of each

FDI system to make a decision

occured by means of statistical

cision, the magnitude of the

the estimation result is used in

signal.


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Fig. 2. Reconfigurable fault-tolerant control scheme [7]

Actuator Fault Compensation

The state-space representation of a system that may be

affected by ith

actuator fault is

1 1 0, 0, 0, 0 0,

(5)

where is the magnitude of ithactuator fault and is the related fault matrix.

Detection, isolation even estimation of the actuator fault

magnitude is conducted by designing an observer that is able

to generate fault signal estimate of ith

actuator,: 0 if there is no actuator fault 0if there is actuator fault

The observers in this paper are developed by modifying (5)

so that

to be component of state vector. The

modification results is [7]: 1 (6)

where

0 0 0 0 0 ; 00 ;

0 0 00 0 0; 1;

00 00 ;

1

The estimation of the actuator fault magnitude isconducted by estimating the state vector , that is the lastcomponent ofas shown in (6).

The next problem is obtaining an equation of . Thisproblem can be solved by means of singular value

decomposition (SVD) technique toward matrix oncondition that it is of full column rank [12], with the result that

it can be declared in a product of three matrix:

0 (7)

TiandMirepresent orthonormal matrices, and Siis a diagonal

nonsingular matrix.

Substituting (7) to (6) and dividing matrix Ti to be two

parts, Ti= [Ti1 Ti2] leads to

1 0 (8)where

; ; ; ; (9) ; ;

is pseudo-inverse of matrix

.

The ith

actuator fault compensation observer works using

the first equation in (8). Therefore, the each observer produce

results of the state vector estimate , the integral error vectorestimate , and the ith actuator fault vector estimate , byusing the free-fault control signal u = ufsf, and the

measurement output signal y. Note that each is notsensitive to setpoint changes as well as to faults of the other

couples of actuator-sensor

The result of then is used in statistical test in order toproduce alarm. Based on the behavior of , the statistical testis done by adopting Shewhart control chart of Statistical

Process Control (SPC) method, that is evaluating each sample

ofto determine wheter it is in the in-control area or not. Thein-control area is the region that has two boundary limits: the

upper control law (UCL) and the lower control law (LCL)

defined by

(10a) (10b)where is mean value and is deviation standard value ofthe successive sample data set of

in a windowing. Length

and overlap of the windowing determine false alarm rate andmissed alarm rate. In this case, those both parameters of the

windowing is determined by trial and error by reference to the

value of signal-to-noise ratio (SNR) of the measurement. is a constant determined by reference to the ratio value

between fault magnitude and noise measurement. If the

sample value of at time instant k is inside the in-controlarea, then the indicating signal Iaiat that time is zero. On the

other hand, if the sample value of at time instant k isoutside the in-control area (out-of-control), then the


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indicating signal Iai at that time is one. Effort to minimize the

occurrence of false alarms is done by way of the alarm signal

on if and only if there are four successive value ofIaiequal toone. Next, to each the actuator being considered, the alarm

signal displays 0 if there is no fault but 1 if there is a fault.

The estimated fault signals of the each actuator are

combined into an actuator fault vector:

(11)The additive control signal for compensating all actuator

faults is:

)()( 1 kfFBku aaadda

= (12)

where matrixFais the overall fault matrix (commonly equal to

matrix C1).

Sensor Fault Compensation

The output equation model with feedback control system

in (9) should be changed in case of sensor faults, i.e.:

0, (13)Thus, to detect and to estimate the magnitude of sensor

faults can be conducted by means of (13), using the

measurement output vector and the state vector estimate. As

mentioned above, the each developed observer produce results

of the state vector estimate

, the integral error vector

estimate , and the ith actuator fault vector estimate , byusing the free-fault control signal u = ufsf, and themeasurement output signal y. The state vector estimate

represent the free-fault condition state (the nominal condition

state). If there is an ith

sensor fault which is the couple of an ith

actuator, then the components value of is not anymoreequal with value of measurement results yi. The difference is

only occured at one time instant tsf, as a result of the nominal

control signal tries to bring the steady state error back to zero.

The difference turned out to be an estimate of the ith

sensor

fault magnitude,

. By using the assumption that the fault

sensor did not get better over time, but it may be gettingworse, then the sensor fault magnitude may not decrease with

increasing time. To minimize the possibility of false alarms

caused by outliers, it is usedjsuccessive samples ofythat are

compared with on hold for three step sample times from tsf.The determination ofcan be defined as: for 0 for (14)

where Ci is a null vector except ith

component equals to 1.

There is a trade off in determining the value of j, the delay

time detection and the false alarm. Note that each issensitive to setpoint changes, but is not senstivite to faults of

the other couples of actuator-sensor.

As with the actuator fault detection, the results in thenis used on satistical test in order to generate the sensor alarm.

Based on the characteristic of , the statistical test isconducted by using deviation standard of the windowingsample data set of . The two types of threshold used are: if

if (15)Di is detectability threshold that its value depends on the value

of SNR. The smaller the value ofDi, the higher occurrence of

false.

is a scalar that its value is influenced by the related

setpoint changes. If the sample value of at time instant kexceed the threshold Ti, the the indicating signalIsiat that timeis equal to 1. Next, to each the sensor being considered, thealarm signal displays 0 if there is no fault but 1 if there is a

fault.

The estimated fault signals of the each sensor are

combined into a vector of the sensor fault:

(16)The additive control signal for compensating all sensor

faults is:

)(~

)()( 21 kfKkfFKku sssadds += (17)

where sf~

is the integral of ssfF

Therefore, the free fault control signal is

ufsf(k) = un(k) + uadds(k) (18)

Note that vector ufsf is the control signal which is used by

the observer in the FDI system..

APPLICATION EXAMPLE

To illustrate the proposed method, a three tank benchmark

system is considered. The dynamical model of the system is

given by [7]


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where

(m, n= 1,2,3 mn)

The variables l1, l2, l3 denote the level i

respectively; qmn represents the flow rate

while q20is the outflow rate at tank 2.The d

numerical values of the plant model para

Table 1. The controlled variables are l1

manipulated variables are q1and q2.The liplant can be derived in the equilibrium poin

0.325]T10

-4(m

3/s);[0.4 0.2 0.3]

T(m))

TABLEI.PARAMETER VALUES OF THE THRE

Parameter Symbol

Tank cross sectional areaInter tank cross sectional area

Inter tank outflow coefficient

Outflow coefficient at tank 2Maximum flow rate

Maximum level

SSp

13=3220qmax

lmax

0.05x1

0.5

0.61.2

0.6

TABLEII.

ISOLATION TIME OF TH

Faults on Noise standard

deviation of 10-4m

Nois

deviat

Occurence Isolation Occurenc

Pump 1 100 s 107 s 100 s

Sensor 1 50 s 51 s 50 s

Pump 2 400 s 407 s 400 s

Sensor 2 500 s 501 s 500 s

The simulation is intended to determine

developed control system to overcome pu

and level sensor faults as well, by

reconfiguration scheme. Therefore, the simu

not the component functional failure, but the

with small severity. The actuator faults ar

loss of effectiveness of the actuator pumps

degradation of 0.3 on the output control s

faults are simulated as a constant offset or

the piezoresistif level sensor so that the fa

used by the controller is equal to l+ 0.03.

(19)

tank 1, 2, and 3

rom tank m to n

escription and the

eters are listed in

and l2 while the

ear model of thes (U0;Y0) = ([0.35

TANK SYSTEM

Value

154 m2

0-5m2

75x10-4m3/s

2 m

FDI

e standard

ion of 10-3m

Isolation

119 s

51 s

427 s

501 s

the ability of the

p actuator faults

means of the

lated fault types is

component faults

simulated as the

, by using a gain

ignal. The sensor

biasof 0.03 m on

ulty measurement

The measurement noise

Gaussian distribution. The st

distribution that used in the si

The simulation results are

concluded that the proposed F

environment. Note that the lar

the larger the delay time of th

fault detection time is largertime. This is because the actua

on the dynamics of the syste

fault detection process is don

direct measurements and the s

the controller from reacting.

Figure 3 shows the syste

change in the reference value

faults, i.e the pump 1 at t = 10

It is noticed that the respons

system (with FTC) is better t

system (without FTC). Altho

still able to track the referenc

and the overshoot of its outp

system outputs with FTC. Th

additional control signal uaddasystem capables of compensate

In the second simulation

faults appears in the tank 1 at i

tank 2 at instant 500 s. The

illustrated in Fig. 4. It can be

the real level follows the set

classical control law. It is ca

used to generate the control s

the real situation. The fault is i

for sensor 1 and sensor 2reconfiguration approach pres

the system in the presence of s

To know the ability of t

dealing with more than one

faults occurrence is simulated

occured before the actuator fa

the tank 2, the actuator fault is

fault. Figure 5 illustrates the

that the developed control

compensate more than one fau

first occured. The analysis of t

also emphasizes the better perfto the classical control, i.e. 4

FTC for the level of tank 1; as

without FTC for the level of ta

is assumed to be zero-mean

andard deviation of the noise

ulations is 10-3

m and 10-4

m.

illustrated in Table 2. It is

I is able to work well in noisy

ger the noise standar deviation,

e FDI. In addition, the actuator

than the sensor fault detectionor fault detection process relies

m response. While, the sensor

e by comparing the results of

ate estimate thus be preventing

m responses when there is a

and followed with the actuator

0 s and the pump 2 at t = 400 s.

of the reconfigurable control

an that of the classical control

gh the system without FTC is

e value, but the time response

uts are larger than that of the

s, it has been proved that the

rom the reconfiguration control

the actuator faults properly.

experiment, an abrupt sensor

nstant 50 s, and followed in the

results of this experiment is

noticed that with FTC method

oint; it is not the case for the

sed the tank level information

gnal is not in accordance with

solated at instant 51 s and 501 s

respectively. Therefore, therves the dynamical behavior of

nsor faults.

e proposed control system in

fault, the sensor and actuator

sequentially. The sensor fault is

ult in the tank 1. In contrast to

occured first before the sensor

simulation results which prove

system has the ability to

lts, no matter what type of fault

he integral absolute error (IAE)

ormance of the FTCS compared01 with FTC and 428 without

well as 200 with FTC and 214

k 2.


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Fig. 3. The output measurement responses when the actuator faults occurred

in the tank 1 and the tank 2

Fig. 4. The real output responses when the sensor faults occurred in the tank 1

and the tank 2

Fig. 5. The system responses when the actuator and sensor faults occured

sequential (sensor fault then actuator fault for tank 1; actuator fault thensensor fault for tank 2)

CONCLUSION

After The simulation example of the fault tolerant control

of linear system with more than one fault has been conducted.

The three tank system is used to illustrate the abilities of the

proposed method to compensate for both sensor and actuator

faults. A bank of observers has been developed to detect,

isolate, and estimate faults of the sensor-actuator pairs. Based

on the simulation results, it is concluded that the control system

with FTC has the output responses which are closer to the

nominal outputs rather than that of the system with the classical

control law.

REFERENCES

Y. Zhang, J. Jiang, "Bibliographical review on reconfigurable fault-tolerant control systems", Annual Reviews in Control, vol. 32,issue 2, pp. 229-252, December 2008.

H. Noura, D. Sauter, F. Hamelin, D. Theilliol, Fault-tolerant control

in dynamic systems: Application to a winding machine, IEEEControl Syst. Mag.,vol.20, pp. 33-49, 2000.

D. Theilliol, H. Noura, J.C. Ponsart, "Fault diagnosis andaccommodation of a three-tank system based on analyticalredundancy", ISA Transactions, vol. 41, no. 3, pp.365382,2002.

Z. Gao, H. Wang, Descriptor observer approaches for multivariablesystem with measurement noises and application in faultdetection and diagnosis, Systems & Control Letters, vol. 55, pp.304313, 2006.

M. Mahmoud, J. Jiang, Y.M. Zhang, "Active fault tolerant controlsystems: Stochastic analysis and synthesis", Lecture notes incontrol and information sciences, vol. 287, Berlin, Germany:Springer, 2003.

R.J. Patton, Fault-tolerant control: The 1997 situation (survey),Proseding IFAC SAFEPROCESS'97, Hull, U.K., vol.2, 1033-1055, 1997

H. Noura, D. Theilliol, J.C. Ponsart, A. Chamseddine,Fault-tolerantControl Systems: Design and Practical Applications, Springer-Verlag London, 2009.

K. Ogata,Modern Control Engineering- 4thed., Prentice Hall, 2006.

R.L. Williams-II and D.A. Lawrence, Linear state-space controlsystems, John Wiley & Sons, Inc., 2007.

K.J. Astrom, R.M. Murray, Feedback systems: an indtroduction forscientists and engineers, Princenton University Press, 2008.

D. Xue, Y. Chen, D.P. Atherton, Linear feedback control: analysisand design with MATLAB (Advances ind design and control),Society for Industrial Mathematics, first ed., 2008.

A. Bassong-Onana, M. Darouach, G. Krzakala, "Optimal estimationof state and inputs for stochastic dynamical systems withunknown inputs", Proceedings of International Conference onFault Diagnosis, pages 267275, Toulouse, France, 1993.

0 100 200 300 400 500 600 700 800 900 1000

0.2

0.25

0.3

0.35

0.4

0.45

0.5

time (s)

level(m)

Tank 1

Tank 3

Tank 2

without FTC

with FTC

without FTC

with FTC

0 100 200 300 400 500 600 700 800 900 10000.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

time (s)

level(m)

Tank 1

Tank 2

Tank 3

with FTC

without FTC

with FTC

without FTC

0 100 200 300 400 500 600 700 800 900 1000

0.2

0.25

0.3

0.35

0.4

0.45

0.5

time (s)

level(m)

measured

measured

real

Tank 2

Tank 1

Tank 3

real


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reconfigurable of active fault tolerant control for nonlinier actuator and sensor fault - sami...

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