[IEEE 2013 IEEE International Conference on Control System, Computing and Engineering (ICCSCE) -...

Reconfigurable Fault-Tolerant Control of Linear System with Actuator and Sensor Faults

Katherin Indriawati, Trihastuti Agustinah, Achmad Jazidie Department of Electrical Engineering

ITS Surabaya, Indonesia [email protected]

Abstract—This 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 linear observer 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 Terms—Observer, SVD, reconfigurable FTC.

INTRODUCTION Control systems which have the ability to accommodate

component (actuator or/and sensor) failures automatically are called 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 engineering systems. 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-tolerant capabilities [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 of the 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 requires quick 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 in harmony. 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 reconfigurable control 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 particular time, 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

based on a bank of observers is proposexample of a three-tank system and its simgiven in section IV. Finally, concluding remsection V.

PROBLEM FORMULATION

Nominal Control Consider a discrete linear time invari

given by the following state space representa

⎩⎨⎧

=+=+

)()()()()1(

kkkkk

CxyBuAxx

where x∈Rn, u∈Rp and y∈Rm are the state vinput, the output vector, respectively. A∈Rnxn

Rmxn are the state, the control, and the respectively. The number of outputs m treference input vector yr do not exceed the inputs due to controlability requirement.

The nominal control system for that plastructure with integrator as shown in Figur(h≤p) represents the vector of the masuremerequired to follow the reference input vectorh) represents the vector of the unmeasuremnominal control system take into account th(U0, Y0). The state space representation of tshown in Figure 1 is

[ ]⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎥⎦

⎤⎢⎣

⎡=

⎥⎦

⎤⎢⎣

⎡+

⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

=⎥⎦

⎤⎢⎣

⎡++

)()(

0)(

)(0

0)()(0

)1()1(

,

,

,1

,

kzkx

Cky

kyIT

Bkzkx

ICTA

kzkx

pq

rps

pn

mpps

pn

where C1 is row of matrix C related to the cis the sample period to be chosen properlymatrix of dimension pxp, and 0n,p is dimension nxp. The nominal feedback cosystem is computed by

[ ] ⎥⎦

⎤⎢⎣

⎡−=−=

)()(

)()( 21 kzkx

KKkXKku

K = [K1 K2] is the feedback gain matrix obtatechniques such as a pole placement tquadratic optimization, and so on [8][9][10][

sed. A numerical mulation results are marks are given in

N

ant (LTI) system ation

(1)

vector, the control n, B∈Rnxp, and C∈

output matrices, that can track a number of control

ant uses feedback re 1,where y1∈Rh

ent outputs that are r yr while y2∈R(m-

ment output. The he operating point the control system

⎥⎦

⎤)(ku

(2)

controlled state, Ts y, Ip is an identity a null matrix of

ontrol law of this

(3)

ained using several technique, linear-[11].

Fig. 1 Nominal tracking contr

Reconfigurable Control The designed control syste

control signal automatically component faults so that the plaused algorithm in designing of the suitable model with the sim

To achieve a control systemsensor faults, the proposed mrecalculation of the control sigtype. The block diagram of theis shown in Figure 2. The nsystem is given by

)()()( ukukuku addsaddan ++=

where un (k) represents the norepresents the additive contractuator faults, and uadds (k) rsignal to compensate the sensor

The control signal reconfiprocess in order to detect acommonly known as fault Here, the FDI proposed methmathematical model (analyticobserver to generate residual.make the observer in a simplhandle both actuator and sensoorder to minimize the occurenalarm due to noise measuremodification algorithm of theto the FDI threshold values.

RECONFIGURABLE LI

Residual is the differencmeasurement value and the soperating condition. It is assumtaken by an actuator-sensor pobserver needed for the FDI syof actuator-sensor pairs conobserver then are used in the Fabout where may the faults be technique. Based on that deoccured fault is estimated and order to recalculate the control

rol with feedback structure [7]

em conducts reconfiguration of in order to accommodate the ant still operates as desired. The

f that control system is based on mple structure and techniques. m that tolerant from actuator and method of this paper consist of

gnal based on the occured fault e reconfigurable control system ew control law applied to the

)(ks (4)

ominal control signal, uadda (k) rol signal to compensate the represents the additive control r faults. iguration needs fault diagnosis and isolate the occured fault, detection and isolation (FDI). hod in this paper is based on cal redundancy), that is using . Thus the problem is how to le manner that can be used to r faults at once. Furthermore, in nce of false alarm and missed

ement, this paper propose the e Shewhart control chart related

INEAR CONTROL DESIGN e of the considered quantity

same quantity value in normal med that each controlled variable pair. Therefore, the number of ystem is equal with the number

nsidered. The results of each FDI system to make a decision occured by means of statistical

ecision, the magnitude of the the estimation result is used in signal.


23

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 11 0 , 0 ,0 , 00 ,

(5)

where is the magnitude of ith actuator 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 • 0 if 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 00 00 0 ;

00 ; 0 000 0 0 ; 1 ; 00 00 ; 1

The estimation of the actuator fault magnitude is

conducted by estimating the state vector , that is the last component of as shown in (6).

The next problem is obtaining an equation of . This problem can be solved by means of singular value decomposition (SVD) technique toward matrix on condition that it is of full column rank [12], with the result that

it can be declared in a product of three matrix: 0 (7)

Ti and Mi represent orthonormal matrices, and Si is a diagonal nonsingular matrix.

Substituting (7) to (6) and dividing matrix Ti to be two parts, Ti = [Ti1 Ti2] leads to

10 (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 vector estimate ̂ , and the ith actuator fault vector estimate , by using the free-fault control signal u = ufsf, and the measurement output signal y. Note that each is not sensitive 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 to produce alarm. Based on the behavior of , the statistical test is done by adopting Shewhart control chart of Statistical Process Control (SPC) method, that is evaluating each sample of to determine wheter it is in the in-control area or not. The in-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 of the successive sample data set of in a windowing. Length and overlap of the windowing determine false alarm rate and missed 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-control area, then the indicating signal Iai at that time is zero. On the other hand, if the sample value of at time instant k is outside the in-control area (out-of-control), then the


24

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 of Iai equal to one. 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:

)(ˆ)( 1 kfFBku aaadda−−= (12)

where matrix Fa is 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 , by using the free-fault control signal u = ufsf, and the measurement 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 anymore equal 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 getting worse, then the sensor fault magnitude may not decrease with increasing time. To minimize the possibility of false alarms caused by outliers, it is used j successive samples of y that are compared with on hold for three step sample times from tsf. The determination of can 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 is sensitive 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 then is used on satistical test in order to generate the sensor alarm. Based on the characteristic of , the statistical test is conducted by using deviation standard of the windowing sample 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 of Di, 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 k exceed the threshold Ti, the the indicating signal Isi at that time is equal to 1. Next, to each the sensor 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 sensor are combined into a vector of the sensor fault:

(16)

The additive control signal for compensating all sensor

faults is:

)(~

)(ˆ)( 21 kfKkfFKku sssadds += (17)

where sf~

is the integral of ss fF ˆ− 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]


25

where

(m, n = 1,2,3 ∀ m ≠ n)

The variables l1, l2, l3 denote the level in

respectively; qmn represents the flow rate fwhile q20 is the outflow rate at tank 2. The dnumerical values of the plant model paramTable 1. The controlled variables are l1 manipulated variables are q1and q2.The linplant can be derived in the equilibrium point0.325]T10-4 (m3/s);[0.4 0.2 0.3]T (m))

TABLE I. PARAMETER VALUES OF THE THREE

Parameter Symbol Tank cross sectional area Inter tank cross sectional area Inter tank outflow coefficient Outflow coefficient at tank 2 Maximum flow rate Maximum level

S Sp μ13=μ32

μ20 qmax lmax

0.05x10.50.61.20.6

TABLE II. ISOLATION TIME OF THE

Faults on Noise standard deviation of 10-4 m

Noisdeviat

Occurence Isolation OccurencePump 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 pumand level sensor faults as well, by reconfiguration scheme. Therefore, the simunot the component functional failure, but thewith small severity. The actuator faults areloss of effectiveness of the actuator pumpsdegradation of 0.3 on the output control sfaults are simulated as a constant offset or the piezoresistif level sensor so that the faused by the controller is equal to l + 0.03.

(19)

n tank 1, 2, and 3 from tank m to n description and the meters are listed in

and l2 while the near model of the ts (U0;Y0) = ([0.35

E TANK SYSTEM

Value 0154 m2

10-5 m2

675 2x10-4 m3/s 62 m

E FDI

se standard tion of 10-3 m e Isolation

119 s 51 s 427 s 501 s

e the ability of the mp actuator faults

means of the lated fault types is

e component faults e simulated as the s, by using a gain signal. The sensor bias of 0.03 m on

aulty measurement

The measurement noise Gaussian distribution. The stdistribution that used in the simThe simulation results are concluded that the proposed FDenvironment. Note that the larthe larger the delay time of thfault detection time is larger time. This is because the actuaton the dynamics of the systefault detection process is dondirect measurements and the stthe controller from reacting.

Figure 3 shows the systechange in the reference value faults, i.e the pump 1 at t = 100It is noticed that the responsesystem (with FTC) is better thsystem (without FTC). Althoustill able to track the referencand the overshoot of its outpsystem outputs with FTC. Thuadditional control signal uadda fsystem capables of compensate

In the second simulation faults appears in the tank 1 at itank 2 at instant 500 s. Theillustrated in Fig. 4. It can be the real level follows the set pclassical control law. It is cauused to generate the control sithe real situation. The fault is ifor sensor 1 and sensor 2reconfiguration approach presethe system in the presence of se

To know the ability of thdealing with more than one faults occurrence is simulated occured before the actuator fathe tank 2, the actuator fault isfault. Figure 5 illustrates the that the developed control compensate more than one faufirst occured. The analysis of talso emphasizes the better perfoto the classical control, i.e. 4FTC for the level of tank 1; aswithout FTC for the level of tan

is assumed to be zero-mean tandard deviation of the noise mulations is 10-3 m and 10-4 m. illustrated in Table 2. It is

DI is able to work well in noisy rger the noise standar deviation, e FDI. In addition, the actuator than the sensor fault detection tor fault detection process relies m response. While, the sensor

ne by comparing the results of tate estimate thus be preventing

em responses when there is a and followed with the actuator 0 s and the pump 2 at t = 400 s. e of the reconfigurable control han that of the classical control ugh the system without FTC is ce value, but the time response puts are larger than that of the us, it has been proved that the from the reconfiguration control e the actuator faults properly.

experiment, an abrupt sensor instant 50 s, and followed in the e results of this experiment is

noticed that with FTC method point; it is not the case for the used the tank level information ignal is not in accordance with solated at instant 51 s and 501 s

2 respectively. Therefore, the erves the dynamical behavior of ensor faults. he proposed control system in fault, the sensor and actuator sequentially. The sensor fault is ult in the tank 1. In contrast to

s occured first before the sensor simulation results which prove

system has the ability to lts, no matter what type of fault the integral absolute error (IAE) formance of the FTCS compared 01 with FTC and 428 without

s well as 200 with FTC and 214 nk 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 then sensor 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”, IEEE Control Syst. Mag.,vol. 20, pp. 33-49, 2000.

D. Theilliol, H. Noura, J.C. Ponsart, "Fault diagnosis and accommodation of a three-tank system based on analytical redundancy", ISA Transactions, vol. 41, no. 3, pp.365–382, 2002.

Z. Gao, H. Wang, “Descriptor observer approaches for multivariable system with measurement noises and application in fault detection and diagnosis,” Systems & Control Letters, vol. 55, pp. 304–313, 2006.

M. Mahmoud, J. Jiang, Y.M. Zhang, "Active fault tolerant control systems: Stochastic analysis and synthesis", Lecture notes in control 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-tolerant Control Systems: Design and Practical Applications, Springer-Verlag London, 2009.

K. Ogata, Modern Control Engineering - 4th ed., Prentice Hall, 2006. R.L. Williams-II and D.A. Lawrence, Linear state-space control

systems, John Wiley & Sons, Inc., 2007. K.J. Astrom, R.M. Murray, Feedback systems: an indtroduction for

scientists and engineers, Princenton University Press, 2008. D. Xue, Y. Chen, D.P. Atherton, “Linear feedback control: analysis

and design with MATLAB” (Advances ind design and control), Society for Industrial Mathematics, first ed., 2008.

A. Bassong-Onana, M. Darouach, G. Krzakala, "Optimal estimation of state and inputs for stochastic dynamical systems with unknown inputs", Proceedings of International Conference on Fault Diagnosis, pages 267–275, 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)

leve

l (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)

leve

l (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)

leve

l (m

)

measured

measured

real

Tank 2

Tank 1

Tank 3

real


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