7/28/2019 06330126

1/5

Linear Approach to Performance assessment of

a 4-tank system

Yousef alipouri, Javad poshtanFaculty of Electrical Engineering

Iran University of Science and TechnologyTehran, Iran

E-mail: yalipouri@iust.ac.ir, jposhtan@iust.ac.ir

Majid PoshtanEECE Department

American University in DubaiDubai, UAE

E-mail: mposhtan@aud.edu

AbstractMonitoring performance of nonlinear system isan important task for many real world applications

especially for industrial environments; however its

realization is usually difficult. Many monitoring

performance methods have been introduced but they all

rely on loops with accurate linear models of the system.

The methods that are capable to identify MIMO nonlinearsystems are scarce and linear models are not so accurate in

modeling nonlinear systems. In this paper, a combined

series of two capable algorithms is used for identifying the

system. The method also identifies the existence of any

possible disturbance simultaneously in order to reduce the

complexity of the model. The logic behind the proposed

methods is to utilize and approximate the interaction

between the loops in the system. Moreover, the model

explicitly defines the relationship between the outputs and

the inputs of the system such that the performance indexes

can be easily calculated. A well designed control strategy is

introduced to determine the optimal values that are

required to calculate the performance index. A reasonable

decision on optimal situation is necessary to evaluate the

performance index of a system. A minimum variances

control strategy (MVC) is considered here as the most

suitable feedback controller because it achieves the

smallest possible closed-loop output variance. The above

method has been utilized for a 4-tank system and then a

minimum variance controller is designed for determining

the optimal output such that the minimum possible

variance, and consequently the performance index have

been calculated. The achieved Index can be then used for

monitor performance of systems in online and offline

applications.

Keywords- minimum variance controller, minimum

variance index, performance monitoring, Four-tank

benchmark system, maximum likelihood and CMA-ES

algorithms

I. INTRODUCTION

Control engineering deals with the theory, design andapplication of control systems. The primary objective ofcontrol systems is to maximize profits by transforming raw

materials into products while satisfying criteria such asproduct-quality specifications, operational constraints, safetyand environmental regulations [1]. The design, tuning andimplementation of control strategies and controllers areundertaken within the main phase in the solution of controlproblems.

The most widespread (stochastic) criterion considered forperformance assessment in the process control is variance (or,equivalently, the standard deviation), particularly for regulatorycontrol. The widespread use of variance as a performancecriterion is due to the fact that it typically represents theproduct-quality consistency. The reduction of variances ofmany quality variables not only implies improved productquality but also makes it possible to operate near theconstraints for increasing throughput, reducing energyconsumption and saving raw materials [2].

The main question in dealing with output variance is howmuch the variance can be decreased and how this aim can beperformed. The minimum possible variance is used forevaluating an index, namely minimum variance index, which

decides on the efficiency of loop performance. Minimumvariance index is one of the common forms of controlperformance index (CPI) used in industrial environments.Bialkowski [3] declared that almost 60 percent of controlperformance index used in industrial applications is minimumvariance index. The control performance index is a singlescalar which is usually scaled to lie within [0, 1], where valuesclose to 0 indicate poor performance and values close to 1mean better/tighter control [4]. This indeed holds when perfectcontrol is considered as benchmark. Perfect control determinesthe optimal value required for calculating control performanceindex. The minimum variances control (MVC) (also referred toas optimal H2 control and first derived in [5]) is the bestpossible feedback control for linear systems in the sense that itachieves the smallest possible closed-loop output variance.More specifically, the MVC task is formulated as minimizationof the variance of the error between the set point and the actualoutput. The output variance is defined as follows:

N

k

N

k

y kyN

yykyN 1 1

22 )(

1,)(

1

1V

MVC can be regarded as optimal solution to both questions[6]: how much and how output variance can be reduced.

Since 1998, a new approach (Interactor Matrix Method)was introduced for estimating minimum variance with high

(1)

___________________________________

978-1-4673-1332-2/12/$31.00 2012 IEEE

7/28/2019 06330126

2/5

7/28/2019 06330126

3/5

2

4

2

1

4

11

4

4

44

1

3

1

2

3

22

3

3

33

12

12

dA

k

A

kgh

A

a

dt

dh

dA

k

A

kgh

A

a

dt

dh

d

d

XJ

XJ

where kd1=1,kd1=2,d1 and d2 are disturbance inputs.

The manipulated valuables are input voltages of pumps.

Outputs of the model are the height of liquid in tanks 1 and 2.

2211 , hyhy

The outputs-inputs model is as follows:

2

2

224

2

42

2

22

1

1

113

1

31

1

11

22

22

XJ

XJ

A

kgh

A

agy

A

a

dt

dy

A

kgh

A

agy

A

a

dt

dy

III. IDENTIFYING BY USING CMA-ES ALGORITHM

Equation (6) is a linear model is using for identifying SISO

systems. temtubtubntyatyaty mn )(1)()1( 11 ""where e(t) is white noise which is considered modelingmismatch.

This model has unknown parameters (7) which must beestimated.

> @Tmn bbaaa "" 121TBy replacing the unknown parameters ai in Eq. 2 with Ai

matrixes and replacing the scalary(t) with vectorY(t), (6) turnsto (8), which is a MIMO model which can model interactionamong loops.

,,

,,,

...1)(

)1()()()(

1

1111

1

1111

1

111

1

101

i

nm

i

n

i

m

i

i

n

nnn

n

n

i

nn

i

n

i

n

i

i

dp

d

ii

p

ii

bb

bb

B

a

a

a

tete

tete

t

ty

ty

tY

aa

aa

A

tadtUBtUBptYA

tYAtaitUBitYAtY

"

#%#

"

#

"

#%#

"

#

"

#%#

"

"

H

H

H

where p is model order, n is number of outputs, m is number of

inputs, a is average value of disturbance, Y(t) is vector ofoutput data and tH is model mismatch at sample time t.

ji BaA ,, are unknown constant parameters, which must be

estimated by a proper algorithm.For reaching higher accuracy, higher order (p) and capable

parameter estimation methods are needed. Consequently,higher order of model leads to higher number of unknownparameters. Estimating the value for high number ofparameters (high dimensional problems) is hard or evenimpossible for some methods. Most algorithms cannot be usedin such problems, as they are usually trapped in local

minimums. AI methods practically are used for finding bestparameters for this kind of models [20-22].

In this paper, CMA-ES which is state of art algorithm inevolutionary algorithms, introduced in [17], will be used. Forimplementing this method, at first Maximum likelihood (ML)algorithm begun to search possible values for modelparameters, but it is so likely that this algorithm trapped inlocal minima, and then CMA-ES algorithm is running and

searching for optimal values for parameters. Fig. 2 showsapproach of search.

Fig. 2. Shows approach of using CMA-ES and ML for estimating optimalmodel

Fig. 3 shows an example of raw data, primary model andoptimized model.

0 200 400 600 800 1000 1200-1

-0.5

0

0.5

1

1.5

2

samples

system output

0 200 400 600 800 1000 1200-1

-0.5

0

0.5

1

1.5

2

samples

outputs

modeloutput

system output

0 200 400 600 800 1000 1200-1

-0.5

0

0.5

1

1.5

2

Raw dataPrimiray Model outputOptimized model output

Fig. 3 shows example of reaching to optimal model by approachintroduced in Fig 2.

Interested readers can be referred to [20-22] for extensiveexplanation about procedure of model optimization by EAs.

IV. DESIGNING MINIMUM VARIANCE CONTROLLERIn Minimum Variance Controller (MVC), the main objective is

to decrease the control signal variance.

YYEJ TBy substituting (8) in (9) we have:

> @^ `211)(

2

)()(

)()1()1(min

)(min)(min

tztUqBtYqE

tYEtJ

tu

tutu

HI

Where

piAii dd 1I

zttUqBtYqtY

ttUqBzptYtYtY p

HI

HII

)1(1)(

)()()1()(

11

1

1 "

1111

1

1)(

d

d

p

p

qBBqB

qq

"

" III

Eq. (10) is minimized by Eq. (11):

> @ ^ `2)(

111 )()(min)()( tEtJztYqqBtUtu

HI u

Control signal in Eq. (38) minimizes the objective functiondefined in (36)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

7/28/2019 06330126

4/5

V. SIMULATION RESULTS

1) Empirical Tests-A Nonlinear Case

The most important part of designing MVC controller for anonlinear system is to estimate an accurate model, which wasthe subject of Section III. The sampling time for the four-tanksystem is 1 second and time constant is about 30 minutes [18].

The open loop response of system and estimated modelresponse are shown in Fig. 4.

0 1 00 0 2 00 0 3 00 0 4 00 0 5 00 0 6 00 0 7 00 0 8 00 0 9 00 0 1 0 00 00

2

4

6

8

10

12

Samples

System Output1

Model Output 1

0 1 0 00 2 0 00 3 0 00 4 0 00 5 0 00 6 0 00 7 0 00 8 0 00 9 0 00 1 0 00 00

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Samples

System Output 2

Model output 2

Fig. 4 : The output of system Eq. 2 (blue curve) and identified model (redcurve)

The accumulated residuals for 10000 samples is 10106.1 uand the residual for each sample is in the order of 1410 ,

indicating the ability of model in identifying the property of thenonlinear four-tank system. The residuals between the sampleddata of model and system are shown in Fig. 5. Table 2 showsthe parameter of estimated model for the simulation data of thefour-tank system.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-8

-6

-4

-2

0

2

4

6x 10

-13

output1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-16

-14

-12

-10

-8

-6

-4

-2

0

2

4x 10

-14

output2

Fig. 5 : Residuals between the output of system (2) and optimal model(Table 1)

Table 1 : Estimated parameters of optimal model for system (2)

valuevaluevalue

6283

8362

EE

EE

Q

29.1

0

0

065.1

B

32.10

52.8

a

0289.000801.0

0077.00039.0A3

3649.00055.0

0084.0433.0A2

700.0005.0

004.0907.0A1

00541.00046.0

0099.00020.0A5

0054.00132.0

02164.00051.0A4

In Table 1, Q is covariance matrix of residuals and otherparameters were introduced in Eq. (8).

Table 2 lists simulation results after applying the designedMVC to the nonlinear four-tank system.

Table 2: The output and control signal variances for the nonlinear four-tank benchmark system

Var(y1)+

Var(y2)

Var(y2)Var(y1)Var(u1)+

Var (u2)

Var(u2)Var(u1)

1.49430.75410.74221.40920.68410.7251

Fig. 6 shows the disturbance samples which is applied onsystem.

0 5000 10000 15000-20

-15

-10

-5

0

5

10

15

20

25

samples

DisturbanceChanel2

0 5000 10000 15000-20

-15

-10

-5

0

5

10

15

20

sample

disturbancechanel1

Fig. 6: shows samples of disturbance

Fig. 7 shows the output of system by feedback (withoutcontroller). The system is going to be unstable (from samples10000) after increasing disturbance variance.

0 2 00 0 40 00 6 00 0 80 00 1 00 00 1 20 00 1 400 0 1 60 000

2

4

6

8

10

12

Samples

Data

Output1

0 2 00 0 4 00 0 6 00 0 8 00 0 1 00 00 1 20 00 1 40 00 1 60 000

1

2

3

4

5

6

7

8

9

10

Samples

Data

Output2

Fig. 7: shows output of feedback controller

Fig. 8 shows output of system after applying minimumvariance controller. It can be seen that system is stable and thevariance is decreased so considerably.

0 20 00 40 00 60 00 80 00 1 00 00 1 20 00 14 00 0 1 60 000

2

4

6

8

10

12

14

Samples

Data

Output1

0 2 00 0 4 00 0 6 00 0 8 00 0 1 00 00 1 20 00 1 40 00 1 60 00

0

2

4

6

8

10

12

Samples

Data

Output 2

Fig. 7: shows output of MV controller

2) Performance index

After designing optimal controller in (11), the minimumvariance index can be determined by (12).

act

opt

MVIndex 2

2

V

V (12)

7/28/2019 06330126

5/5

where opt2V is optimal variance which is output of minimum

variance controller and act2V is actual variance of system output

(without minimum variance controller). For example,performance index for feedback controller in Fig. can becalculated as follow:From Fig. 7 (variance calculated from samples 1 - 10000)

23.62 actV and from Table 2 4943.1

2 optV , Then:

23.023.6

4943.1 MVIndex

In the other word, the feedback controller is working in20% of optimal situation.

VI. CONCLUSION

Designing minimum variance controller (MVC) is anoption to reach the optimum variance in output which causes todetermine the minimum variance index. It needs an exact linearmodel of system and disturbance for determining the minimumpossible variance. Considering that there are no accuratenonlinear or linear models for most real-world systems,designing minimum variance controller is very challenging. Inthis paper, a new method was proposed for modeling nonlinear

four tank system by using heuristic algorithm CMA-ES. Aftermodeling the system minimum variance controller has beendesigned and applied on nonlinear system. It is shown that thecontroller can stabilize the system by decreasing the variance inoutput (feedback couldnt did it without MVC controller).Variance of output while controlled by minimum variancecontroller can be regarded as optimal variance, so actual outputvariance can be compared with optimal variance to determinethe performance of system. It is shown that the feedbackcontrol has just 20% accuracy of MVC in reaching to minimumvariance. This statement declares that performance of feedbackcontroller can be enhanced 80% by using suitable controller(such as MVC).

REFERENCES

[1] Seborg DE, Edgar TF, Mellichamp DA (2004) Process Dynamics andControl. John Wiley & Sons.

[2] Shinskey FG (1996) Process-Control Systems: Application, Design, andTuning. McGraw Hill.

[3] W.L. Bialkowski, Dreams vs. reality: a view from both sides of thegap, Pulp & Paper Canada 94:1927, 1993.

[4] M, Jelali, (2010), Control System Performance Monitoring Assessment,Diagnosis and Improvement of Control Loop Performance in IndustrialAutomation, Springer.

[5] K J Astrom, (1970), Introduction to Stochastic Control Theory. NewYork: Academic Press.

[6] Martensson J., Rojas C. R. and Hjalmarsson H., 2011, Conditions whenminimum variance control is the optimal experiment for identifying aminimum variance controller, Automatica 47, 578583.

[7] W. DeVries, S Wu, Evaluation of process control effectiveness anddiagnosis of variation in paper basis weight via multivariate time-seriesanalysis, IEEE Trans Automat Control 23:702708, 1978.

[8] B. Huang, S.L. Shah, Practical issues in multivariable feedback controlperformance assessment, Proc IFAC ADCHEM, Banff, Canada, pp429434, 1997.

[9] B. Huang, S.L. Shah, Performance Assessment of Control Loops,Springer, 1999.

[10] Rogozinski M., Paplinski A. and Gibbard M., 1987 An algorithm forcalculation of nilpotent interactor matrix for linear multivariablesystems, IEEE Trans Automat Control 32:234237.

[11] Huang B., Shah S., Badmus L. and Vishnubhotla A., 1999, ControlPerformance Assessment: An Enterprise Asset Management Solution,www.matrikon.com/download/ products /lit /processdoctor _pa_eam.pdf.

[12] Huang B., 1997, Multivariate Statistical Methods for Control LoopPerformance Assessment, PHD thesis, University of Alberta, Canada.

[13] Huang B., Ding S. X. and Thornhill N., 2005, Practical solutions tomultivariate feedback control performance assessment problem: reduceda priori knowledge of interactor matrices, Journal of Process Control 15,573583.

[14] Kadali R. and Huang B., 2007, Multivariate controller performanceassessment without interactor matrixa subspace approach, IEEETrans on Control Systems Technology, Vol. 15, No. 1.

[15] Shah S. L., Mohtadi C. and Clarke D., 1987, Multivariable adaptivecontrol without a priori knowledge of the delay matrix, Systems &Control Letters 9:295306.

[16] Xia H., Majecki P., Ordys A. and Grimble M., 2004, Performanceassessment of MIMO systems under partial information, Proceeding ofthe 2004 American Control Conference Boston, Massachusetts.

[17] 10. Hansen, Nikolaus, The CMA Evolution Strategy: ATutorial.(www.bionik.tu-berlin.de/user/niko/cmatutorial.pdf), November24, 2010.

[18] Karl Henrik Johansson (2000), The Quadruple-Tank Process: AMultivariable Laboratory Process with an Adjustable Zero, IEEE Trans.on Control Systems Technology, Vol. 8, No. 3, May.

[19] E. P. Gatzke, E. S. Meadows, C. Wang, F. J. Doyle, (2000) Modelbased control of a four-tank system, Computers & ChemicalEngineering, Volume 24, Issues 2-7, 15 July, Pages 1503-1509.

[20] I. Rojas, F. Rojas, H. Pomares, L.J. Herrera, J. Gonzalez, O.Valenzuela,The synergy between classical and soft-computing techniques for timeseries prediction, in: Advances in Artificial Intelligence, Lecture Notesin Computer Science, Vol. 2972, 2004, pp. 3039

[21] C.-S. Ong, J.-J. Huang, G.-H. Tzeng, Model identification of ARIMAfamily using genetic algorithms, Appl. Math. Comput. 164 (3) (2005)885912.

[22] Valds, Julio; Barton, Alan, Multivariate Time Series Model Discoverywith Similarity Based Neuro-Fuzzy Networks and Genetic AlgorithmsApril 2003, published in Proceedings of the IEEE, INNS, IJCNN 2003International Joint Conference on Neural Networks. IEEE CatalogNumber: 03CH37464C,ISBN: 0-7803-7899-7. Oregon, Portand, USA.

(13)