Additive Fault Detection in a Real Nonlinear System with Saturation

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ADDITIVE FAULT DETECTION IN A REAL NONLINEAR SYSTEM Wlm SATURATION L Felipe Blazquez*, Jose M. Foces*, L Javier de Miguelf *Dpto. Ingenierfa Electrica y Electranica. Univ. de Lean. Campus de Vegazana. sin. 24071 Lean (Spain). E-mail: {diejbq.diejfm}@Unileon.es tDpto. Ingenieria de Sistemos y Automatica. Univ. de Valladolid. Poseo del Cauce. sin. 47011 Valladolid (Spain). E-mail: [email protected]

Abstract: This paper describes an application of the parity equation method to a real nonlinear system with saturation. This nonlinear system is an industrial pilot plant, and the water level control of the pressurized reactor is the control process implemented in this pilot plant The saturation of the dynamic process is due to the inflow control valve. The parity equation method uses a nonIinear model of the plant to generate residuals, and only additive faults are considered. The goal of this application is to show the effects of saturation over the perfonnance of the model-based fault detection method. In this sense, the results obtained indicate the decrease of additive fault detectability due to presence of saturation in the dynamic process. Furthennore, these results also reveal the existence of a relation between the control strategy used in the process and additive fault detectability, in the sense that increases of fault detectability are obtained the due to use offaster control strategies. Copyright © 20031FAC Keywords: Fault detection, nonlinear systems, saturation. real systems.

1.

INfRODUCfION

of the various methods has been studied by several authors, see for example (Gert1er, 2000).

Model-based methods of fault detection and isolation rely on the idea of analytical redundancy. The essence of this idea is that measured plant outputs are compared to ones predicted with the model from the measured or actuated inputs. Discrepancies, expressed as residuals, are indications of faults in ideal conditions, though in reality they are also affected by disturbances, noise and modelling errors. There are various methods to generate residuals and to enhance them for fault isolation. These methods include diagnostic observers (Frank, 1990; Chen and Patton. 1999; Ding. et al., 2002), parity relations from the state-space model (Chow and WilIsky, 1984; Gertler and Luo, 1989), and parity equations from the input~utput model (Gertler, 1988; Gert1er and Singer, 1990; GertIer, 1998). The linear theory of these approaches is well developed and their relationship is also well understood. The equivalence

1017

For nonlinear systems, the fault diagnosis problem has traditionally been approached in two steps. Firstly, the model is linearized at an operating point, and then techniques are applied to generate residuals. To deal with systems with high nonlinearity and wide operation range, the fault diagnosis problem has to be tackled directly using nonlinear techniques. Results for nonIinear observers have been published by Frank and coworkers (Alcorta-Garcia and Frank, 1997) and others (Kinnaert, 1999). Krishnaswami and Rizzoni (1994), proposed a residual generator scheme integrated into a nonlinear parity equation that utilizes forward and inverse dynamic models of nonlinear dynamic systems. Blazquez and Miguel (2001), described by simulation the relation between the performance of a parity equation fault detection method and the degree of nonlinearity of the system due to the saturation effects (Blazquez, 2003).

This paper describes an application of the parity equation method to a real nonlinear system with saturation. This nonlincar system is an industrial pilot plant, and the water level control of the pressurized reactor is the control process implemented in this pilot plant. The saturation of the dynamic process is due to the inflow control valve. The parity equation method uses a nonlinear model of the plant to generate residuals, and only additive faults are considered. This application shows the effects of saturation on the performance of the nonlinear parity equation fault detection method. Furthermore, the results obtained also reveal the existence of a relation between the control strategy used in the process and additive fault detectability.

input/output control units that can be analogical or digital. The global control of the process is carried out with a controller connected to these analogical/digital units and also to a computer. The monitoring, control and acquisition software necessary for the process is installed in this computer. This software allows the carrying out of some tasks such as: the programming of control strategies, the downloading of these strategies to the controller, configuring the control loops, process monitoring and the generation of historical files. Furthermore, a web service and a data base managing SQL server are also installed.

The paper is organised as follows. In section 2 the industrial pilot plant and the control process are described. Section 3 shows the nonlinear model of the process and explains the fault detection scheme used in this paper. In section 4 the experiments carried out in the pilot plant are described and the results obtained are presented. Section 5 contains the conclusions of the paper and finally, some references are shown. Fig. 2. Control of level and pressure. 2.

SYSTEM AND PROCESS DESCRIPTION

2.1 Industrial pi/ot plant description. The real system used in this work to perform all experiments is a complex industrial pilot plant, figure I, existing in the Area de Automatica y Control of the Instituto de Automatica y Fabricaci6n of the University ofLe6n, Spain.

Fig. 3. Control oftemperature. The process that can be carried out with the pilot plant is the control of physical variables. These variables are level, pressure and reactor water temperature, and are controlled by six control loops, figures 2 and 3. These control loops have been designed to allow interactions between them. Thus, basic and multivariable control strategies can be studied.

Fig. 1. Industrial pilot plant. TillS pilot plant has been designed to test advanced control strategies in a 50 litre pressurized-water reactor. Specified conditions of pressure, temperature, flow and level can be kept in this reactor. This is due to six perfectly defined circuits existing in the pilot plant: process water, hot water, cool water, process air, instrumentation air and water drainage circuit. The field hardware is based on a distributed control system through which the available variables of the pilot plant are connected to input or output modules. These modules are grouped into distributed 1018

2.2 Control process description. The control of the water level in the pressurized reactor has been implemented in the industrial pilot plant as the process to study. Reactor inflow qj and reactor outflow 'lo are both through the bottom reactor; qj with a control valve and 'lo with a constant cross-sectional area pipe, noted as a . Two control strategies have been used in this process. One control strategy uses two PI digital controllers connected in cascade, as shown in figure 4. The other control

strategy uses only one PI digital controller, as shown in figure 5.

3. FAULT DETECTION SYSTEM DESIGN 3. J Nonlinear model ofthe plant.

h,.,

'le

I!t.

eq

"'c

h

qj

~p'-"99

In this paper, the fault detection system uses the parity equation approach based on a nonlinear model of the plant. This nonlinear model consists of equations (5), (6), (7) and (8); that gives the estimated water level h. In these equations, A = 962.11 cm2 is the cross sectional area of the reactor, g = 9.8 m/s" is the gravity constant and
Fig. 4. Control process with two PI controllers.

"'c

h..reh

~

h

~

Fig. 5. Control process with one PI controller. In the control strategy with two PI controllers, the control flow CIc is the manipulated variable. This signal CIc can be limited by PUn between 0 and 21 Vrnin (limited output of PUn), or not (unlimited output of PUn). These values calculated by experimentation are the real limits of the reactor inflow qj through the control valve, that have been. In the control strategy with one PI controller, the opening percentage of the control valve o/Ilc is the manipulated variable. In both control strategies, the water level in the reactor h is the controlled variable, and the nominal operating point is 200 mm. The water level setpoint hm. is a square signal with 190 mm and 210 mm as bottom limit and top limit respectively; taking into account that href begins and finishes at 200 mm.

qcn -qat-I = K[e bn -e hn _1+ <0

~ e hn ]

%cn - %cn-I =

K[

eqn -

eqn_1

%(2\ -%cn-I = K[ehn -e hn - I

d~=~-~~

00

qon = an ~2ghrern

(7)

a

(1)

(2)

3.2 Fault detection scheme.

+ ; eqn ]

(3)

+;

(4)

ehn ]

(5)

There are no available measurements of the outflow 'Lv so it must be estimated by equation (7). The parameter noted as a is not really constant, so it must also be estimated. This parameter is estimated as follows. By equation (8), the values of in the three steady points ofbmare known. that is to saya(190), 1(200) and a(21O). This is because <10 = qi in these three steady points of bm. If h > 200 mm then 3 is obtained by linear interpolation between l(200) and a(21O). On the other hand, if h < 200 mm then 3 is obtained by linear interpolation between 3(190) and 3(200).

then qc:n = 0

If qcn 0 < qc:n < 21 then qcn =qcn { > 21 then qat = 21

hn =hn-I + qin-I A - q""-I dt n

Figure 6 shows the fault detection scheme applied to the process with the control strategy of two PI controllers. Figure 7 shows the fault detection scheme applied to the process with the control strategy of one PI controller.

Equation (1) is the equation of the PUn controller of the control strategy with two PI controllers with unlimited output of PUn. Equations (1) and (2) are the equations of the PI in controller of control strategy with two PI controllers with limited output of the PI_in. Equation (3) is the equation of the PI_in controller of the control strategy with two PI controller. Equation (4) is the equation of the PI_in controller of the control strategy with one PI controller. The numeric values of the parameters of these equations are shown in table 1.

I

i. _~~-!~~~_ . _ . . :

Fig. 6. Residual generator with two PI controllers.

Table 1. Parameters of the PI digital controllers Strategy

2PI lPI

Controller Equation K Tj (s) T (s) (1) 7.8 0.333 O.S PUn (3) 1 0.02 O.S PUns 0.4 O.S PI in 9 ~4l

. I

. I

i . ~~~_g.~~._ . _ . i

Fig. 7. Residual generator with one PI controller.

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properties of the residual (mean JL and standard deviation er), CDF, 8 and CDF-B are presented. In experiment FNt, the fault occurs at 1,645.797 seconds. At this time, there is not saturation in the control valve, as shows the graph on the left of figure 10. On the other hand, in experiment FN+, the fault occurs at 2,231.068 seconds; at this time, there is saturation in control valve, as the graph on the right of figure 10 clearly shows. As it is observed in table 2, fault detectability in experiment FNt is greater than fault detectability in experiment FN+. So, in this case, the presence of saturation causes a decrease in the additive fault detectability.

An additive fault in level sensor fb can be caused by the software. This fault is tackled as abrupt change, so th is modeled as a step function of 5 mm. The residual equation (9), as shown in figures 6 and 7, is the difference between the measured level h given by the level sensor and the estimated level ii given by the nonlinear model. residual = h -

h

(9)

The fault detectability criterion CDF, is defined by equation (10) as a tool to ~ fault detectability. RF means the value of residual with fault considering a data window centering at the fault, while RNF means the value of residual without fault and SDRNF the standard deviation of residual without fault. CDF=

KRF)-(RNF) SDRNF

8;r-_~IHid= ·:::U=.I_ _- ,

5 4

1110

(10)

170 1800 5 10 15 20 25 30 35 40 -1 0 5 10 15 20 25 30 35 40 Tme (5) x10' Time (s) x10'

The fault is detected with the rule given by equation (11). Where B has a suitable value in order to detect incipient faults without detecting false alanns. In this work, B has been calculated applying CDF to a pair of residuals obtained from two different experiments without faults. This value is close to zero but not exactly zero due to the presence of noise in the measurement elements of the process and disturbances from the water network.

Fig. 8. Fault detection to FNt with 2PI and unlimited output ofPI_in. 8,...-._~noMi=::::u=.I_ _- ,

5 4

180

If CDF {~ B then no fault

170

(11)

> 8 then fault 4.

1800 5 10 15 20 25 30 35 40 Tome (s) x10'

TIme(s)

.10'

Fig. 9. Fault detection to FN+ with 2PI and non limited output of PUn.

RESULTS

Several experiments have been carried out for each of three operation schemes: two PI controllers with unlimited output of PUn. two PI controllers with limited output of PUn and one PI controller with unlimited output of PUn. All experiments present the same system reference input h,..r, as a square signal with limit values of 190 mm and 200 mm. This signal has been chosen so that there is saturation in the control valve. The experiments that have been carried out are noted as follows:

25i~_ql:...-_and----:qc::....-_oI_F_N_f~

25,~_,\:...-_an_d_q::...c-_oI_FNJ._~

20 15

20 15

1 ~ 5,1-I-\--,1.....,.,-,~.-4-+ ~

~

~O

~O

~5

~5

-201!:-0-=5,.......,.,10,-1=-=5-=2O:-:--::25~3O,......,35,....,14O -201!:-0-=5,.....,.,10~'5,..,2O=--=25~3O~35,....,14O Time (s)

x10'

Time (5)

x10'

Fig. 10. Inflow and control flow with 2PI and unlimited output of PUn.

• SFl, SF2. Two experiments without faults to obtain 15. • FNt. With fault in level sensor at rise step of hrcr. • FN+. With fault in level sensor at dropped step

Table 2. Fault detection with 2PI and unlimited output of PI in

ofhref.

Experiment

SFl SFl

4.1 Control strategy with two PI digital controllers . with unlimited output ofPI_in.

FNt FN.j.

Figure 8 shows the real and estimated reactor level and the corresponding residual, obtained for experiment FNt. Figure 9 shows the same variables obtained for experiment FN+. Figure 10 shows inflow and control flow for both experiments, FNt and FN.l. respectively. In table 2, the statistical

Residual ~~nun~ 0.21387 -0.42708 2.00549 1.21741

(J~mm~

CDF

6

CDF-6

0.49833 0 0.29885 0.64309 0.64309 1.15452 1.92518 1.79761 1.78768 1.0069 0.36381

4.2 Control strategy with two PI digital controllers with limited output ofPI_in. Figure 11 shows the real and estimated reactor level and the corresponding residual, obtained from

1020

experiment FNt. Figure 12 shows the same variables obtained for experiment FN.J.. Figure 13 shows the inflow and control flow for both experiments, FNt and FN.J. respectively. In table 3, the statistical properties of the residual (mean ~ and standard deviation a), CDF, l) and CDF-l> are presented.

4.3 Control strategy with one PI digital controller with unlimited output ofPI_in. Figure 14 shows the real and estimated reactor level and the corresponding residual, the obtained for experiment FNt. Figure 15 shows the same variables obtained for experiment FN.J.. Figure 16 shows opening percentage and control opening percentage of the control valve for both experiments, FNt and FN.J. respectively. In table 4, the statistical properties of the residual (mean ~ and standard deviation a), CDF, l) and CDF-l> are presented.

h- ;h-

2~r---~~~----,

215

185 1800 5 10 15 ~ 25 30 35 <10 Time (s) 110'

detectability due to the presence of saturation in the control valve is confirmed once more.

Time(s)

Fig. 11. Fault detection to FNt with 2PI and limited output of PI_in.

h-;h-

midu.'

10 15 ~ 25 30 35 <10 Tim. (5) 11 0'

185 180 0 5 10 15 ~ 25 30 35 <10 Tome (5) 110'

Tim.(s)

Time (5)

x10'

Fig. 14. Fault detection to FNt with IPI and unlimited output of PUn.

• 10'

Fig. 12. Fault detection to FN.J. with 2PI and limited output of PUn. 25 ~

C

15

'~10 ~

10 15

5

o11

~

~

I

~

25 30 35 <10 .10'

Time (s)

TIme (s)

-5 0 5 10 15 ~ 25 30 35 <10 -5 0 5 Tim. (5) 110'

Fig. 15. Fault detection to FN.J. with IPI and unlimited output of PI_in.

10 15 ~ 25 30 35 <10 T..... (s) .10'

Fig. 13. Inflow and control flow with 2PI and limited output of PUn.

,.. -

and

"0 -.

of FN1

Table 3. Fault detection with 2PI and limited output of PI in Experiment SFl SF2

FNt FNJ.

Residual

~~mm! 0.03341 0.52733 2.3934 2.05142

a~mm!

COF

8

COF-8

o

o

-~O 5 10 15 ~ 25 30 35 40 -~O 5 10 15 ~ 25 30 35 40

0.35926 0.49255 0.68742 0.68742 0 1.95775 3.28455 2.59713 1.79107 2.80858 2.12117

Tome (si

.10'

x10'

Time (s)

Fig. 16. Opening % and control opening % of the valve with IPI and unlimited output of PUn.

In experiment FNt, the fault occurs at 1,634.06 seconds. In experiment FN.J., the fault occurs at 2,240.081 seconds. At these times, there is no saturation in the control valve, as the graphs offigure 13 show, due to the limited output of PUn. As can be seen in table 3, fault detectability in experiment FNt is similar to, although greater than fault detectability in experiment FN.J.. However, both values of detectability are very clearly greater than those values of detectability obtained in table 2, that is to say, in the case of the control valve with saturation. So, the idea of the decrease in

Table 4. Fault detection with IPI and unlimited output of PI in Experiment

Residual

I!!mm!

a~mm!

SF!

-C.266 13 0.475435

SF2

0.01802 1.474303 1.90081

FNt FN'!'

0.30744 1.64825 1.87082

COF

I)

0.29883 0.29883 1.83036 2.27891

COF-I) 0 1.S3153

1.98007

In experiment FNt, the fault occurs at 1,618.948 seconds. In experiment FN.J., the fault occurs at 2,236.977 seconds. At these times, there is saturation 1021

in the control valve, although this saturation is greater in the experiment FNt. as the graphs offigure 16 show. As can be seen in table 4, fault delectability in experiment FNt is less than tabt in experiment FN+. So. in this case too, the idea of decrease in detectability due to the presence of saturation in the control valve is confinned again.

4.4 Relation between control strategy and fault detectability.

detection method are shown. In this sense, the results obtained indicate a decrease in additive fault detectability due to the presence of saturation in the dynamic process. Furthennore, these results also reveal the existence of a relation between the control strategy used in the process and additive fault detectability, in the sense that increases of fault delectability are obtained due to the use of faster control strategies. REFERENCES

Table 5 shows the statistical properties (mean J1 and standard deviation 0') of the inflow qj and Aqj. equation (12), for the control strategy with one PI controller and the control strategy with two PI controllers. Both control strategies have unlimited output of PI_in; taking into account the experiments without fault SF 1 and SF2. ~i = (qi2 -qil,qi3 -qi2 ... ·'qiN -qi(N-I»

(12)

It is shown that the mean values of inflow qj are not the same. This is due to the total time being different for each experiment. The values of the standard deviation of qj indicate that there is a lesser number of changes in the case of one PI controller than with two PI controllers. Furthermore. the values of the mean of Aqj indicate that these changes are greater in the case of one PI controller than with two PI controllers. So, in this case, the control strategy with one PI controller with unlimited output of PI_in is faster than the control strategy with two PI controllers with unlimited output of PI_in. Table 5. Statistical properties of qj and Aqj 'I; &!;

Noolimitcd output of PI in IPI 2PI

SFl SF2 SFI SF2

I' (Urnin)

(J

S.S4797 S.SOI49 S.43324 HS398

2.1S63 2.07021 2.404S7 2.44893

(Vmin)

I' (Vrnin) -S.019e-OOS -3.S44e-OOS - 3.40Se-OOS l.S2e-00.5

(J

(Urnin)

0.31166 0.3S498 0.3674 0.31204

Tables 2 and 4 show that fault detectability, in the experiment FNt, is less in the case of the control strategy with two PI controllers with unlimited output of PUn than in the case of control strategy with one PI controller with unlimited output ofPI)n; even despite the presence of saturation in the control valve in the case of the control strategy with one PI controller with unlimited output of PUn. So, there is a relation between the control strategy and additive fault detectability in the sense that increases of fault detectability are achieved due to the use faster control strategies. 5.

CONCLUSIONS

In this paper, the parity equation method has been applied to a level control process, implemented in an industrial pilot plant, using a nonlinear model of the plant. The effects of saturation in the control valve on the perfonnance of the model-based fault 1022

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