Arab J Sci Eng (2014) 39:20132019DOI 10.1007/s13369-013-0739-2

RESEARCH ARTICLE - ELECTRICAL ENGINEERING

Comparison of Fuzzy Identification Schemes for Robust ControlPerformance of an Adaptive Fuzzy Controller

Muhammad Bilal Kadri

Received: 14 May 2012 / Accepted: 11 July 2013 / Published online: 8 September 2013 King Fahd University of Petroleum and Minerals 2013

Abstract Fuzzy identification schemes play a vital role inthe control performance of an adaptive fuzzy controller. Non-linear uncertain systems are difficult to model and control. Agood fuzzy model of an uncertain non-linear system can beguaranteed if the adaptation mechanism is computationallyefficient as well as robust in the face of noisy data com-ing from the sensors. The prediction accuracy of the fuzzymodel depends on the quality of learning provided by theidentification algorithm. In adaptive fuzzy control problemsthe control performance is heavily dependent on the parame-ters estimates produced by the identification scheme. In thispaper the controller develops an inverse model of the plantonline. The problem of inverse model identification becomesmore challenging when external disturbances and plant de-lays are present in the control loop. In this research worktwo different computationally efficient fuzzy identificationschemes are discussed. They are used for estimating the ruleconfidences of Fuzzy Relational Models and are based on theprobabilistic learning approach. Both the classes of learningschemes are compared on the basis of robustness, rate ofconvergence and dependence on other controller parameterssuch as learning rate and forgetting factor. The control ob-jective is a tracking problem when the plant under control isnon-linear and controlled output is corrupted by sensor noise.

Keywords Fuzzy identification schemes Fuzzy relationalmodels Robust learning schemes RSK

M. B. Kadri (B)Electronics and Power Engineering Department, PN EngineeringCollege, National University of Sciences and Technology,Islamabad Pakistane-mail: bilal_kadri@yahoo.com; bilal.kadri@pnec.edu.pk

1 Introduction

Fuzzy logic controllers have shown exceptional performancewhen the plant under control is non-linear and little a prioriinformation is available about the system. Adaptive fuzzylogic controllers are able to update the controller parameterswhen the system dynamics changes over time. The adap-tive mechanism depends on the information available; thisinformation includes all the sensor outputs, e.g. controlledvariable, other states which are used in the control law, etc.Sensors no matter how they are designed can never be idealand they introduce noise into the measured variables. Thenoisy data produced by the sensors are hazardous for thelearning schemes. The adaptive mechanism might de-tunethe fuzzy controller based on the poor information available

123

2014 Arab J Sci Eng (2014) 39:20132019

from the sensors. The learning schemes used to adapt thefuzzy controllers play an essential part in the overall controlperformance of the system. Robustness of the fuzzy identi-fication schemes guarantees tight control performance. Theaim of the investigation was to compare the control perfor-mance of different fuzzy identification schemes which areused to estimate the parameters in a direct adaptive fuzzycontroller.

In order to test the performance of the identificationschemes the measured output is heavily corrupted with sen-sor noise. Two different fuzzy identification schemes havebeen used to identify an inverse plant model from the plantdata. The paper discusses fuzzy identification schemes withspecial focus on their performance in the presence of idealand non-ideal data. The paper is organized as follows: Sect. 2defines the fuzzy models with a special focus on fuzzy rela-tional models, adaptive fuzzy controller is discussed in Sects.3, and 4 includes a detailed explanation of the fuzzy identifi-cation schemes used in conjunction with the fuzzy controller.The simulation setup and the control performance are givenin Sects. 5, 6, respectively.

2 Fuzzy Models

Fuzzy Models can be broadly classified into linguistic fuzzymodels and rule-based fuzzy models [1]. The linguistic fuzzymodels are the Mamdani fuzzy models in which the rule baseis developed from expert knowledge. When input output dataare available then a fuzzy model can be developed by definingrules over the universe of discourse and by incorporatingsome adaptive mechanism. The rule-based fuzzy models canbe further categorized into TakagiSugeno Models (TS) andfuzzy relational models (FRM). FRM [2] have rules equal toall the possible different combinations of the input and outputfuzzy sets. A rule confidence [0, 1] is assigned to each of therule, which measures the amount of information a particularrule contains or to what extent a rule can contribute in forming

the output. The output of a multi input single output fuzzyrelational model can be described [3,4] by

Y = R X1 X2 . . . Xn (1)

y(x) =n

i=1 { f Ai (xi )[RAi ,B1U1 + RAi ,B2U2]}n

i=1 { f Ai (xi )[RAi ,B1 + RAi ,B2 ]}, (2)

where Y and y(x) are the fuzzified and defuzzified output ofthe fuzzy model, respectively. Y is a N1 array whose ele-ments are the membership grade of the output in the referencesets B1, B2,, BN. It is assumed without loss of generality[5] that N = 2 in this thesis. Ui is the position of the apexof the ith output set. x = [x1, x2, . . . , xn]T are the crisp in-puts to the fuzzy model. Xi = [Ai (xi )] is an array whoseentry is the membership grade of the input xi , in the multi-dimensional fuzzy set Ai which describe the ith input space.f Ai (xi ) = Ai (xi ). R is the fuzzy relational array contain-ing N n elements (rule confidences) and o is the fuzzycomposition operator. Choosing sum-product as the fuzzycomposition operator and using height defuzzification, theoutput of the FRM is calculated by Eq. (2).

3 Adaptive Fuzzy Control Structure

A block diagram of the fuzzy adaptive control is shown inFig. 1. The fuzzy control structure has shown robust con-trol performance when the plant is non-linear and little apriori information is available [3,6,7]. The fuzzy controlleris based on the model reference adaptive control (MRAC)strategy. MRAC has been widely reported in literature [8,9].The feedforward controller develops an inverse model of theplant to be controlled. The inverse is developed by with thehelp of the fuzzy identification scheme, which in turn incor-porates the feedback error learning [10,11] mechanism toestimate the correct control signal [12]. The estimated cor-rect control signal along with the plant output and referencesignal is fed into the fuzzy identification scheme. The fuzzy

Fig. 1 Block diagram of fuzzymodel reference adaptivecontroller

FuzzyIdentification

Algorithm

ProportionalController Plant

FeedforwardController

e(t)r(t) u(t)ub(t)

uf(t)

+

-

+

+

MeasurableDisturbances

UnmeasuredDisturbances

ReferenceModel

123

Arab J Sci Eng (2014) 39:20132019 2015

identification scheme computes an estimate of the controllerparameters which are updated at every instant. The propor-tional controller guarantees acceptable control performancewhen the rule base is empty, i.e. during the initial phase [3].The reference model is used to make the reference trajectoryachievable. The control signal produced from the feedfor-ward controller is uf (t) and the control signal generated bythe proportional controller is ub(t). In ideal conditions ub(t)should be zero when the controller is perfectly trained.

4 Fuzzy Identification Schemes

The fuzzy identification schemes discussed in this paper be-long to two different classes: the first class of identificationschemes is based on probabilistic model and is devised forfuzzy relational models (FRM). In the first case RSK learningscheme and modified RSK are discussed.

4.1 RSK Fuzzy Identification (RSK)

The scheme was proposed by [1315] and is one of thesimplest schemes for estimating the rule confidences in the

presence of noise. It can be used online effectively due toits computational simplicity. The algorithm is defined fora multi-input single-output system. A rule confidence ma-trix is formed which stores the mapping between the inputsx1(k), x2(k), . . . , xn(k) and the output y(k) where k is thesampling time. The entry RA1s1 ,...,Ansn ,B j (k) in the fuzzy re-lation array measures the possibility of obtaining an outputy(k) in set B j from inputs x1(k), x2(k), . . . , xn(k) in setsA1s1 , . . . , Ansn , respectively. The input space of each variableis divided into r referential sets using r fuzzy membershipfunctions. The fuzzy relational matrix are determined by

RA1s1 ,...,Ansn ,B j (k) =N

k=1 f A1s1 ,...,Ansn (x[k])B j (y[k])N

k=1 f A1s1 ,...,Ansn (x[k]),

(3)where, f A1s1 ,...,Ansn (x[k]) is the product A1s1 (x1[k]), . . . ,Ansn (xn[k]) and the summation runs over the relevant ob-servations. The RSK scheme can be modified to weight thedata exponentially [16]. This will enable the recent data tohave more impact on the rule confidences and to forget theold data.

RA1s1 ,...,Ansn ,B j (k)

=N

k=1 Nk f A1s1 ,...,Ansn (x[k])B j (y[k])N

k=1 Nk f A1s1 ,...,Ansn (x[k]), (4)

where 0 < 1 is the forgetting factor and N is thenumber of times a particular combination of input sets is fired.The RSK scheme can be converted into a recursive form tosave memory space and processing power. By defining anF array of size

ni=1 si the denominator of Eq. (4) can be

written in recursive form:

FA1s1 ,...,Ansn (k)

=

f A1s1 ,...,Ansn (x[k]) + FA1s1 ,...,Ansn (k 1)if f A1s1 ,...,Ansn (x[k]) = 0

FA1s1 ,...,Ansn (k 1)otherwise

(5)

The F array indicates the frequency with which a partic-ular combination of input has been fired. The recursive formof the RSK scheme will then be of the following form:

RA1s1 ,...,Ansn ,B j (k) =

f A1s1 ,...,Ansn (x[k])B j (y[k])+RA1s1 ,...,Ansn ,B j (k1)FA1s1 ,...,Ansn (k1)FA1s1 ,...,Ansn (k)

if f A1s1 ,...,Ansn (x[k]) = 0RA1s1 ,...,Ansn ,B j (k 1) otherwise

(6)

4.2 Modified RSK Fuzzy Identification (MRSK)

The MRSK algorithm was proposed by [17,18] in which theauthor introduced a second f A1s1 ,...,Ansn (x[k]) matrix givenin Eq. (7), which makes the algorithm more robust to thenon-ideal data. When data are encountered the strength withwhich it fires the rules, i.e. f A1s1 ,...,Ansn (x[k]) is calculated.If the value is above the current activation level f A1s1 ,...,Ansn(x[k 1]) then the rule is fired; otherwise, the rule confi-dence value is not updated. This methodology helps in re-ducing the effect of non-ideal data, which may corrupt theprevious learning. f A1s1 ,...,Ansn (x[k]) matrix is calculated asfollows:

f A1s1 ,...,Ansn (x[k])= max( f A1s1 ,...,Ansn (x[k 1), f A1s1 ,...,Ansn (x[k]) (7)

The recursive form of the MRSK learning scheme utilizes

the F(k) array which is represented by

123

2016 Arab J Sci Eng (2014) 39:20132019

FA1s1 ,...,Ansn (k)

=

f A1s1 ,...,Ansn (x[k]) + FA1s1 ,...,Ansn (k 1)if f A1s1 ,...,Ansn (x[k]) f A1s1 ,...,Ansn (k)

FA1s1 ,...,Ansn (k 1)otherwise

(8)The recursive form of the MRSK scheme will then be of thefollowing form:

RA1s1 ,...,Ansn ,B j (k) =

f A1s1 ,...,Ansn (x[k])B j (y[k])+RA1s1 ,...,Ansn ,B j (k1)FA1s1 ,...,Ansn (k1)FA1s1 ,...,Ansn

(k) if f A1s1 ,...,Ansn (x[k]) f A1s1 ,....,Ansn (k)RA1s1 ,....,Ansn ,B j (k 1) otherwise

(9)

5 Simulation Results

In order to thoroughly test the control performance two dif-ferent plants are considered. The first plant is a Hammer-stein model of a cooling coil whereas the second plant is anon-linear model of a three tank system. The fuzzy adaptivecontroller is modeled as a FRM.

5.1 Cooling Coil Modeled as a Hammerstein Model

The first example is a cooling coil of an air-conditioningunit. Many industrial processes can be approximated by aHammerstein Model (a combination of a static non-linearityand linear dynamics). A cooling coil in an air-conditioningsystem can be represented by such a model [19,20]. Thetransfer function in Laplace form is given by

y(s) = L { f (u[t])} . esTd

s + 1 , (10)where Td is the dead time of the plant and is the timeconstant of the cooling coil. u(t) determines the positionof the control valve and y(s) is the Laplace transform ofthe temperature difference across the coil. The static non-linearity f (u) is defined as

f (u) = 13.433

ln(30u + 1) (11)Td=10 s and = 120 s in all of the simulations.

5.2 Identification of a Fuzzy Relational Model (FRM)

RSK and MRSK are used for training purpose. In both theschemes, the R and F vectors are initialized to 0.01. The inputto the controller is x(t) = [r(t), y(t td)]. The apexes of themembership functions for r (t), u(t) and y(t td) areat 0:0.25:1.0. The value of the forgetting factor () is 0.999.Learning rate ( ) of 0.5 is used for feedback error learning.

The simulation is run for 1.0105samples. The samplingtime of the controller is 10 s.

5.3 Control Objective and Disturbance Modeling

The control objective is setpoint tracking which varies be-tween 0.25 and 0.75 with a time period of 4,000 samples.The controller along with the plant and sensor is shown inFig. 2. The sensor dynamics is modeled as a first-order system

having a time constant of 1,000 s. The sensor noise is gen-erated by a random number generator. The random numberis normally distributed with a zero mean () and a standarddeviation ().

6 Control Performance of the Adaptive FuzzyController

6.1 Parameter Convergence of the Learning Schemes

In the following discussion two terminologies ideal dataand non-ideal data are used. Ideal data signifies the factthat the rule is completely fired, whereas non-ideal datarefers to those data points which do not fire a rule completely.Figures 3 and 4 show that with ideal training data the ruleconfidences converge, whereas with non-ideal data they donot converge and change with the setpoint. Ideal training datacan be assumed to be the data at the steady state, whereas thenon-ideal data are related to the plants transient behavior.Monotonic behavior for some of the rule confidences can beobserved in case of ideal data, but with non-ideal data thereis no such behavior. Whenever the rule is fired completelythe rule confidence moves towards its final value. Even withideal data the rule confidences do not reach the final valuein the first epoch. When the same data set is presented tothe algorithm then the rule converges after certain numberof iterations. With ideal data the initial rule confidence valuedoes not play an important role, but the initial F values (Eq.[5]) have a great impact on the rule confidences. If the Fvalues are initialized to some high value this implies that theinitial rule confidences are close to their final values. If thedata coincide with the apexes of the membership functionsthen the rule is fired completely, updating the rule confidenceto the maximum extent. However, if the data do not coincidewith the apexes, then the estimate of the rule confidence os-cillates as shown in Figs. 3 and 4. Figures 3 and 4 showthat when ideal data are presented to the system, most of

123

Arab J Sci Eng (2014) 39:20132019 2017

Fig. 2 Block diagram of thecontrol loop

PlantAFCr(t)

u(t) y(t)

e(t)+

-

ReferenceSignal

Sensorym(t)

0 5000 100000

0.51

time

R 221

0 5000 100000

0.51

timeR 2

22

0 5000 100000

0.51

time

R 241

0 5000 100000

0.51

time

R 242

0 5000 100000

0.20.4

time

R 321

0 5000 100000

0.51

time

R 322

0 5000 100000

0.5

time

R 331

0 5000 100000

0.51

time

R 332

Fig. 3 Rule confidences with RSK learning scheme

the rule confidences either grow up or down monotonicallyand converge to some final values after prolonged training.When the same rule is fired repeatedly then the update in therule confidences becomes smaller the reason being that theF value which is updated by a value of 1 (in case of idealdata with f A1s1 ,...,Ansn (x[k]) = 1) at every sample instantmaking the denominator in Eq. (9) larger. This also indicatesthat the weights are converging. If at the current operatingpoint the rules are fired fully and then the reference signalslightly moves to another operating point such that the pre-vious rules with some additional new rules are fired then thisalgorithm will not learn the new behavior, which is a majordrawback. The pre-filter, i.e. Eq. (7), which is applied to thedata, can cause a constant steady-state error in the responseof the system. The convergence rate of the rule confidences isalso dependent on the forgetting factor () which is discussedin the next section.

6.2 Impact of Sensor Noise on the Learning Schemes

In order to investigate the behavior of the learning schemeswhen the controlled output is heavily corrupted by noise, thestandard deviation of the sensor noise was increased from0.1 to 0.9 while maintaining all the other parameters con-stant. The RMSE of the control error is shown in Fig. 5. TheRMSE of the control error increases monotonically with boththe learning schemes. This is exactly what is expected: as thelevel of the sensor noise increases the measurement will befar less accurate and more noise is fed into the control loop.

0 5000 100000

0.51

time

R 221

0 5000 100000

0.51

time

R 222

0 5000 100000

0.51

time

R 241

0 5000 100000

0.51

time

R 242

0 5000 100000

0.20.4

time

R 321

0 5000 100000

0.51

time

R 322

0 5000 100000

0.5

timeR 3

31

0 5000 100000

0.51

time

R 332

Fig. 4 Rule confidences with MRSK learning scheme

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.081

0.082

0.083

0.084

0.085

0.086

0.087

0.088

0.089Comparison of RMSE of RSK and MRSK

Standard deviation of sensor noise

RM

SE o

f the

con

trol e

rror

RSKMRSK

Fig. 5 Impact of sensor noise on the RMSE of the control error

The learning schemes are unable to estimate the correct con-troller parameters which will drive the system towards thesetpoint. It can be observed that the RMSE of the control er-ror for MRSK is always higher than the RMSE with the RSKlearning scheme. RSK scheme instantaneously updates thecontroller parameter as soon as a rule is fired, whereas MRSKupdates the rule confidences only when the firing strength isgreater than certain threshold.

The MAE of the control error is shown in Fig. 6. The MAEof the control error for RSK is always greater than the MAEof the control error for MRSK. This clearly indicates that the

123

2018 Arab J Sci Eng (2014) 39:20132019

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91.9

2

2.1

2.2

2.3

2.4

2.5

2.6x 10-3 Comparison of MAE of RSK and MRSK

Standard deviation of sensor noise

MAE

of t

he c

ontro

l erro

r

RSKMRSK

Fig. 6 Impact of sensor noise on the MAE of the control error

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10.082

0.084

0.086

0.088

0.09

0.092

0.094

0.096

0.098

0.1Comparison of RMSE of RSK and MRSK

Forgetting factor

RM

SE o

f the

con

trol e

rror

RSKMRSK

Fig. 7 Impact of forgetting factor () on the RMSE of the control error

control signal is more active when RSK learning scheme isused with the fuzzy controller. Since RSK quickly updatesthe controller parameters, the defuzzified control signal isalways different from the previously calculated control sig-nal resulting in large control activity. It can be concludedfrom the results presented in Figs. 5 and 6 that RSK learn-ing scheme is more suitable for the plants where excessiveactuator movement (and hence the wear tear of the actuator)does not degrade the actuator performance with time. Thelarge actuator movement results in tight control performancewhich can be verified from the RMSE plots.

6.3 Impact of Forgetting Factor () on the Learning Scheme

Forgetting factor can be related to the window size of theincoming data which has an impact on the control perfor-

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 11.5

2

2.5

3

3.5

4

4.5

5x 10-3 Comparison of MAE of RSK and MRSK

Forgetting factor

MAE

of t

he c

ontro

l erro

r

RSKMRSK

Fig. 8 Impact of forgetting factor () on the MAE of the control error

mance [17]. The larger the forgetting factor greater will bethe window size and hence more will be the dependence ofthe rule confidences on the pat data. The standard deviationof the sensor noise is zero during this experiment. It can beobserved from Fig. 7 that the RMSE of the control error in-creases for both RSK and MRSK learning schemes, whenthe forgetting factor is increased from 0.6 to 0.9. As soonas the forgetting factor is increased above 0.9 the RMSE ofthe control error decreases continuously and drastically. Theminimum RMSE is achieved when the value of the forgettingfactor is 1. A value of 1 indicates that all the past data are usedin calculating the rule confidences. Coincidentally, RMSE ofthe control error is same for both the learning schemes when = 1. The impact of forgetting factor on the MAE of thecontrol activity is shown in Fig. 8; MAE reduces monotoni-cally when the forgetting factor is increased from 0.6 to 1.0.The control scheme becomes more robust with the increasingforgetting factor; hence there is lesser control activity.

7 Conclusions

Fuzzy adaptive controllers based on the inverse model philos-ophy were investigated. The fuzzy controllers were modeledas fuzzy relational models. Robustness of the fuzzy controlleris dependent on the fuzzy identification schemes. Two differ-ent fuzzy identification schemes, namely RSK and MRSK,which are based on the fuzzy relational model have beencompared. Convergence of the rule confidences as well asimpact of the forgetting factor in the presence of heavily cor-rupted controlled output is discussed. Both learning schemeswere able to provide satisfactory control which is evidentfrom the RMSE and MAE of the control error and control

123

Arab J Sci Eng (2014) 39:20132019 2019

activity, respectively. In most of the cases, RSK outperformsModified RSK but with a larger control activity. Future workis directed towards the comparison of fuzzy identificationschemes when the adaptive fuzzy controller is modeled as aTakagiSugeno (TS) fuzzy model.

Acknowledgments The research work was conducted in the ControlLaboratory, Department of Engineering Science, University of Oxford,UK, under the supervision of Professor Arthur Dexter. The researchwas funded by the National University of Sciences and Technology,Islamabad, Pakistan.

References

1. Babuska, R.; Verbruggen, H.B.: An overview of fuzzy modelingfor control. Control Eng. Pract. vol. 4, 15931606 (1996)

2. Czogala, E.; Pedrycz, W.: On identification in fuzzy systems and itsapplications in control problems. Fuzzy Sets Syst. 6, 7383 (1981)

3. Tan, W.W.; Dexter, A.L.: A self-learning fuzzy controller for em-bedded applications Automatica. 36, 11891198 (2000)

4. Kadri, M.B.; Hussain, S.: Model free adaptive control based onFRM with an approach to reduce the control activity. Presented at2010 IEEE International Conference on Systems Man and Cyber-netics (SMC), Istanbul, Turkey, 2010

5. Tan, W.W.; Dexter, A.L.: Properties of a fuzzy relational model.Technical Report OUEL 2119/96. University of Oxford, Depart-ment of Engineering Science, 1996

6. Tan, W.W.; Dexter, A.L.: Self-learning neurofuzzy control of aliquid helium cryostat. Control Eng. Pract. 7, 12091220 (1999)

7. Kadri, M.B.: Robust model free fuzzy adaptive controller withfuzzy and crisp feedback error learning schemes. Presented at 12thInternational Conference on Control, Automation and Systems (IC-CAS), 2012, Jeju, South Korea, 2012

8. Butler, H.: Model Reference Adaptive Control, from Theory toPractice. Prentice Hall, USA (1992)

9. Blazic, S.; Skrjanc, I.; Matko, D.: Globally stable direct fuzzymodel reference adaptive control. Fuzzy Sets Syst. 139, 333(2003)

10. Nakanishi, J.; Schaal, S.: Feedback error learning and nonlinearadaptive control. Neural Netw. 17, 14531465 (2004)

11. Tan, W.W.; Lo, C.H.: Development of feedback error learningstrategies for training neurofuzzy controllers on-line. IEEE Int.Fuzzy Syst. Conf. 10161021 (2001)

12. Kadri, M.B.: Disturbance rejection in model free adaptive con-trol using feedback. Presented at Modelling, Identification, andControl-2010, Innsbruck, Austria, 2010

13. Ridley, J.N.; Shaw, I.S.; Kruger, J.J.: Probabilistic fuzzy model fordynamic systems. Electron. Lett. 24(14), 890892 (1988)

14. Kadri, M.B.: Disturbance rejection in information poor systemsusing model free neurofuzzy control. D.Phil Thesis, University ofOxford, 2009

15. Kadri, M.B.; Dexter, A.L.: Disturbance rejection in information-poor systems using an adaptive model-free fuzzy controller. Pre-sented at Annual Meeting of the North American Fuzzy Informa-tion Processing Society, NAFIPS, 2009

16. Postlethwaite, B.E.: Empirical comparisons of methods of a fuzzyrelational identification. IEEE Proc. 138, 199206 (1991)

17. Tan, W.W.: Self-learning neurofuzzy control of non-linear systems.D.Phil Thesis, University of Oxford, 1997

18. Kadri, M.B.: Disturbance rejection in nonlinear uncertain systemsusing feedforward control. Arab. J. Sci. Eng. (2012)

19. Wang, S.W.: Dynamic simulation of building VAV air-conditioningsystem and evaluation of EMCS on-line control strategies. Build.Environ. 34, 681705 (1999)

20. Kadri, M.B.: System identification of a cooling coil using recurrentneural networks . Arab. J. Sci. Eng. 37, 21932203 (2012)

123