reduced order modelling using evolutionary algorithms

8/10/2019 Reduced Order Modelling using Evolutionary Algorithms

1/32

1

Evoluonary lgorithm ssisted Reduced Order

Modelling

Keshav Sarraf

(10EE35020)

Under the guidance of

Professor J Pal

Department of Electrical Engineering

Indian Instute of Technology, Kharagpur

West Bengal, India


2/32

2

Declaraon

I cerfy that

The work contained in this report is original and has been done by me under the

guidance of my supervisor(s).

The work has not been submied to any other Instute for any degree or diploma.

I have followed the guidelines provided by the Instute in preparing the report.

I have conformed to the norms and guidelines given in the Ethical Code of Conduct of

the Instute.

Whenever I have used materials (data, theoretical analysis, figures, and text) from

other sources, I have given due credit to them by citing them in the text of the report

and giving their details in the references. Further, I have taken permission from thecopyright owners of the sources, whenever necessary.

.

Keshav Sarraf

10EE35020


3/32

3

CERTIFICATE

This is to cerfy that the report entled, Evoluonary Algorithm Assisted Reduced Order

Modelling submied to the Indian Instute of Technology Kharagpur, India, for the award ofthe degree of Bachelor of Technology by Mr. Keshav Sarrafis a record of bonade research

work carried out by him under my supervision and guidance. The thesis has reached the

standard fullling the requirements of the regulaons related to the degree. The results

embodied in the thesis have not been submied to any other University or Instute for the

award of any degree or diploma.

Dated:

Professor J Pal


Indian Instute of Technology


4/32

4

Acknowledgement

It is a maer of great privilege for me to be able to express my deep sense of gratude to my

project supervisor Professor J Pal for his unwavering support, insighul suggesons,encouragement and the fruiul discussions throughout the duraon of my project. He has

always made himself available for any sort of discussion and support. I have learned a great

deal in the areas of Evoluonary Algorithm based opmisaon techniques and Reduced Order

Modelling over the past year through his uninhibited help and movaon.

I am also grateful to all faculty members of the Department for their help, suggestions and

comments during the presentations and throughout the tenure of the work.

Finally, I would like to express my thanks and love to my parents for their support andencouragement.

Keshav Sarraf


Indian Instute of Technology, Kharagpur


5/32

5

Contents

Abstract: ..................................................................................................................................................6

Objecves:..............................................................................................................................................6

Genec Algorithms:................................................................................................................................7

Overview of opmizaon using GA.........................................................................................................7

Reduced Order Modelling of SISO System:.............................................................................................8

Results: ..................................................................................................................................................10

Reduced Order Modelling of SISO Systems (Contd.):...........................................................................12

Results: ..................................................................................................................................................14

Parallel Processing in Genec Algorithms: ............................................................................................16

Extension to MIMO systems:................................................................................................................16

Results: ..................................................................................................................................................19

Teaching Learning Based Opmisaon.................................................................................................26

Introducon..........................................................................................................................................26

Algorithm:.............................................................................................................................................26

Implementaon of TLBO in ROM: .........................................................................................................27

Results (ROM of SISO systems):............................................................................................................28

Results (ROM of MIMO): .......................................................................................................................29

Conclusions: ..........................................................................................................................................31

Future Work..........................................................................................................................................31

References .............................................................................................................................................32


6/32

6

Abstract:Industrial processes are very complex in nature. Accurate mathemacal analysis of these

processes yield models which are of very high order. With todays advanced computaon

technology, we are capable of handling fairly complex systems but there is always a need of

reduced order modelling to shrink the amount of me and storage required for studying andsimulang these complex processes without compromising the reliability of the results. This

project aims at developing a technique to reduce the order of complex plants using

Evoluonary Algorithms like GA (Genec Algorithms) and TLBO (Teaching Learning Based

Opmizaon).

Objecves:

Objecves of this project are as follows:-

To develop a technique for order reducon of high-order SISO linear systems using

Genec Algorithms.

To extend the developed technique to MIMO systems.

To check feasibility of TLBO for model order reducon


7/32

7

Genec Algorithms:Genec Algorithms is an opmizaon technique that involves probabilisc global search

methods which emulate the process of natural evoluon. Genec Algorithms have been found

to be capable of delivering good performance in complicated domains like opmizaon of

mul-variable systems. Thus it is expected that using Genec Algorithms for order reduconof systems should be praccally feasible and should result in opmum soluons which match

the desired specicaons.

Overview of opmizaon using GAWhile using GA for opmizaon, the following steps are needed to be followed.

Parameters to be opmized are chosen.

These parameters are then encoded (Encoded parameters are called Genes)

Genes are combined to form a Chromosome (A chromosome represents an

individual person in a populaon).

Inial populaon of individual soluons is generated.

Fitness of each individual in the populaon is evaluated.

Based on the tness, parents are selected to determine the next generaon.

Next generaon of the populaon is then populated.

Again tness is evaluated for each individual and the cycle goes on ll the output

saturates or the number of iteraons are completed.

Initial Poputation

Evaluate fitness of each individual

Select parents based on fitness

Create new population


8/32

8

Reduced Order Modelling of SISO System:The preliminary approach involved direct determinaon of the reduced order model by opmizing

the value of coecients of the system transfer funcon using GA.

For example-

If the original system is represented by:

Transfer Funcon 1

Then a 3rdorder reduced system is taken as:

Transfer Funcon 2

Where a2, a1, b2, b1 and b0 represent the parameters that have to be opmised.

1. Creaon of inial Populaon:

o Parameter Selecon:Coecients of the reduced order transfer funcon are chosen

as parameters. To achieve similar steady state gain as the original system, the constant

terms in the numerator and denominator are taken in the same rao as the original

transfer funcon (can be observed in Transfer Funcon 1 and Transfer Funcon 2).

o Encoding: All the parameters (genes) are then encoded into binary strings of equal

length. For each soluon, these binary strings (genes) are concatenated to form a

chromosome. Each chromosome contains all the parameters of a single soluon.

o Populaon Generaon:A populaon of soluons (chromosomes) is created keeping

in consideraon that each parameter should lie within its viable range. This range is

xed inially and can be varied from o.

2. Fitness evaluaon of each individual:

Fitness of each individual is evaluated by calculang weighted sum of squared error at

various closely spaced instances of the step response. Error at a me instance is

calculated by taking the dierence of the step response of the original system and step

response of the reduced system at that me instant.

1 + = = 2 = =

Lower the value of J, beer is the tness of the soluon. The value of the variables w1,

w2, t1 and t2 are chosen by hit and trial to achieve best matching reduced order

model. To reduce complexity, sum of the weights is taken to be 1.

s5+ 1014 s4+ 14069 s3+ 69140 s2+ 140100 s + 100000

G(s) = ------------------------------------------------------------------------

s6+ 222 s5+ 14541 s4+ 248420 s3+ 1.454e06 s2+ 2.22e06 s + 1000000

(a2) s2+ (a1) s + (b0)/10

R(s) = -------------------------------

s3+ (b2) s2+ (b1) s + (b0)


9/32


10/32

10

Results:

10.38 s2+ 14.12 s + 59.69

-------------------------------s3+ 131.7 s2+ 619.2 s + 596.9

s5+ 1014 s4+ 14069 s3+ 69140 s2+ 140100 s + 100000

------------------------------------------------------------------------

s6+ 222 s5+ 14541 s4+ 248420 s3+ 14541006 s2+ 2.22e06 s + 1000000

Transfer Funcon 4: results obtained from GATransfer Funcon 3 G(s)

Figure 1: step response and bode plot of original system and reduced system

Populaon Size: 100

Bit Length per Variable: 16

Number of Variables: 5

Cross Over: random, Mutaon: 10%

GA parameters used to reduce transfer funcon 3


11/32

11

-4.148s3+ 13.01s2+ 20.2s + 6.538

--------------------------------------------------

s4+ 10.17s3+ 21.24s2+ 21.54s + 7.148

Transfer Funcon 6: results obtained from GA

Transfer Funcon 5 G(s)

Figure 2: step response and bode plot of original system and reduced system

Populaon Size: 100





-7.56 s^12 - 495.3 s^11 - 1.404e04 s^10 - 2.244e05 s^9 - 2.185e06 s^8 - 1.288e07 s^7 - 4.09e07 s^6 - 2.274e07 s^5 + 3.401e08 s^4 + 1.347e09 s^3 +

2.249e09 s^2 + 1.729e09 s + 4.682e08

-----------------------------------------------------------------------------------------------------------------

s^13 + 66.9 s^12 + 2079 s^11 + 4.246e04 s^10 + 6.087e05 s^9 + 6.152e06 s^8 + 4.373e07 s^7 + 2.184e08 s^6 + 7.613e08 s^5 + 1.824e09 s^4 +

2.937e09 s^3 + 3.053e09 s^2 + 1.872e09 s + 5.12e08


12/32

12

Reduced Order Modelling of SISO Systems (Contd.):It was observed that the reduced model obtained from opmizaon of transfer funcon coecients

did not perform very sasfactory and scope of beer results is present. Pal proposed an elegant

method to reduce the order of linear systems that is computaonally simple [1]. Inspired from that

work, an indirect approach was implemented. Instead of opmizing the coecients of the reducedorder model, new constant parameters -i (i = 1, 2, 3r) were introduced. Opmizaon of these

constants resulted in beer step response.

If the original system is represented by a transfer funcon G(s) and the reduced order system is

represented by the transfer funcon R(s).|= |= i 1, 2, 3 rTaking the same test case as before (Transfer Funcon 3), 5 coecients of R(s) are needed to be

determined for which at least 5 equaons are needed. If suitable values of i (i ranges from 1 to 5)

can be determined then using the above equaon, all the coecients of the transfer funcon R(s)

could be determined. An approach similar to the method presented in the previous secon was used

to determine the value of these i (i = 1-5) using GA.

Results:

Figure 3: Step Response of Original (solid), GA Reduced (doed) and modred Reduced (dashed) systems

-0.01412 s3+ 5.415 s2+ 30.95 s + 58.57

------------------------------------------

s3+ 65.45 s2+ 793.2 s + 585.7

4.495 s2+ 25.56 s + 46.52

---------------------------------

s3+ 55.34 s2+ 645.1 s + 465.2

s5+ 1014 s4+ 14069 s3+ 69140 s2+ 140100 s + 100000

------------------------------------------------------------------------

s6+ 222 s5+ 14541 s4+ 248420 s3+ 14541006 s2+ 2.22e06 s + 1000000

Transfer Funcon 7: result obtained from modred () Transfer Funcon 8: Result obtained from GA

Transfer Funcon 9: G(s)


13/32

13

Figure 4: Enlarged Step response. Line (Original), dashed (Reference) and doed (GA obtained ROM)

Figure 5: Error in Step Response of the GA reduced system and modred Reduced systems, line (Reference), doed (GA ROM)

Figure 6: Bode Plot of Original, GA Reduced and modred Reduced Systems.

modred reduced system has non-minimum phase


14/32


15/32

15

Figure 9: Bode Plot of the Original (Solid), GA reduced (doed) and modred reduced (Dashed) system

Populaon Size: 100





Populaon Size: 100






16/32

16

Parallel Processing in Genec Algorithms:Matlab supports parallel processing on standalone systems with mulple cores. Parallel computaon

was implemented on a small secon of the algorithm - the tness evaluaon. Since the tness of an

individual is independent of the tness of other individuals, its evaluaon can be parallelised. In the

present implementaon, a pool of 4 local workers is created using parallel compung toolbox androughly 65% reducon in code runme has been observed.

Extension to MIMO systems:The original high order system is described by the following equaons in state space form. + The objecve is to nd a reduced order model represented by the following equaons +Yr(t) is a close approximaon of Y(t) for a step input. Ar, Br and Cr are constant matrices with

appropriate dimensions.


17/32

17

Procedure:

1.

Hankel Singular Values provide a measure of energy for each state in a system. Depending

upon the energy contribuon, order of the ROM is chosen. As shown below, out of 7 states

present in the system 3 states have major contribuon in the system energy so we can choose

the order of the reduced model to be at least 3.

Figure 10: Hankel Singular Values of the Original System

2. Characterisc polynomial of the original system is found by the relaon | |Using dominant pole retenon, a new characterisc polynomial may be found. The poles

which are closer to the origin are retained and they determine the value of the reducedpolynomial. This process requires human inspecon and various trials are needed before

arriving at an acceptable soluon.

+ + + + + 3. Ar may be formed from r(s) by

(

0 1 0 00 0 1 0. . . .0 0 0 1 )

4.

Alternavely, Ar may be determined from Balanced Reducon. Using this technique gives

beer results and requires no manual inspecon.

5. Numerical value of the original system transfer funcon is found at s = i (i varies from 1 to r)

and stored in a 3d array named T. These are found using the relaon [ ]|= 6. Similar to the SISO case, the value of T found for the original system is matched to the value

of T found for the reduced system.

[ ]|= [ ]|= 7.

For a 2 input - 2 output system, Br and Cr are assumed to be of the form


18/32

18

1 121 22. .1 2

11 12 . 111 12 . 1

8. Since the rst Column of Br matrix is taken to be all ones, the rst column of[ ]|=Is known. (Ar was previously determined)

9. Using the relaon menoned in step 5, the value of Cr can be determined, provided all the

values of i are known.

10.Once Cr is known, using relaon 5, the value of Br can be determined.

11.To determine the opmum values of i, GA is used. Opmizaon is done to minimize the error

in sum of all the step responses of the MIMO system.


19/32

19

Results:The model of a synchronous machine connected to an innite busbar [2] is considered in this

example. The system has 2 inputs, two outputs and seven states.

Original Model:

Hankel Singular Values of the system are:

Figure 11: Hankel Singular Values of the original system


20/32

20

On observaon of the relave magnitudes of the singular values, it is decided that the

reduced model should be of 3rdorder.

The eigenvalues of A matrix are:

o -0.20

o

-0.47 j9.35

o -37.48

o -46.34

o -13.55 j376.33

Retaining the rst 3 dominant eigenvalues (ones which are closer to the origin), isdetermined to be: +1.1346 + 87.9027 + 17.9080

This is transformed into Ar matrix and subsequently Cr and Br are determined.

Reduced Model Reference Model

Reference Model is determined using balanced reducon. Matlab command used are:

sysOld = ss(A,B,C,D);

[hb,g] = balreal(sysOld);

hmdc = modred(hb,4:7,'MatchDC');

hmdc is a structure that contains the Ar, Br and Cr Matrices


21/32

21

Figure 12: Step Response of the MIMO system, Original (Solid), GA Reduced (Doed) and modred Reduced (Dashed)

Figure 13: Grouped Step Response of the Original system (Solid), GA Reduced system (Doed) and modred Reduced system

(Dashed)


22/32

22

Figure 14: Bode Plot of the Original System (Solid), GA reduced system (Doed) and modred reduced system (Dashed)

A linearized model of a nuclear reactor [3] is considered in this example. The system has 4

inputs, 4 outputs and 11 states.


23/32

23

Figure 15: Hankel Singular Values of the original system

-


For a 3rdorder reduced model, eigenvalues

chosen are:

-0.0006

-0.0082 - j0.0079

-0.0082 +j 0.0079

Eigenvalues of A:

-0.0006

-0.0020

-0.0082 j0.0079

-0.0127

-0.1923

-1.0030

-2.8313-3.6275

-6.9349


24/32

24

Figure 16: Step Response of the Original System (Solid), GA reduced system (Doed) and modred reduced system (Dashed)

Figure 17: Combined step response of the Original System (Solid), GA reduced system (Doed) and modred reduced system

(Dashed)


25/32

25

Figure 18: Bode Plot of the Original System (Solid), GA reduced system (Doed) and modred reduced system (Dashed)


26/32

26

Teaching Learning Based Opmisaon

Introducon:

TLBO is a teaching-learning process inspired algorithmbased on the eect of inuence of a teacher

on the output of learners in a class. Teacher and learners are the two vital components of thealgorithm. Two basic modes of the learning are:

Through the teacher (known as teacher phase). A teacher is considered as a highly learned

person who trains learners so that they can have beer results.

Through interacng with the other learners (known as learner phase).

TLBO is populaon based opmizaon algorithm in which a group of learners is considered as a

populaon and dierent design variables are considered as dierent subjects oered to the learners.

Result of the learners is analogous to the tness value used in GA.

Algorithm:The working of TLBO is divided into two parts, Teacher phase and Learner phase.

Teacher phase

This phase of the algorithm simulates the learning of the students through a teacher. During this

phase a teacher conveys knowledge among the learners and puts eorts to increase the mean result

of the class.

At any sequenal teaching-learning cycle, letMbe the mean result of the learners. The best

soluon among the pool will act as a Teacher for the running iteraon and aempt to increases the

knowledge level of the whole class.

During teaching phase each soluon is updated by a factor called dierence mean (DM). If the

updated soluon has a beer tness then only it is accepted, otherwise the update is rejected. r is a random number between [0,1] whose value changes at every iteraon.

TF can assume only one of the 2 values {1, 2}. Its value is chosen randomly at every iteraon.

+ X denotes the updated value of the soluon X. Each learner is updated by the same amount and theupdate is kept only if it gives beer results.Learner Phase:

This phase of the algorithm simulates the learning of the students through interacon among

themselves. The students can also gain knowledge by discussing and interacng with the other

students. A learner will learn new informaon if the other learner has more knowledge than him or

her. The learning phenomenon of this phase is expressed below.

Randomly two disnct learners P and Q are selectedand the learner P is updated as below

XP XP + rXP XQ, IfitnessXP > itnessXQXP XP + rXQ XP, IfitnessXP < itnessXQ


27/32

27

r is again a random number between [0,1]. Similar to the teaching phase, if the XP has beer

tness than XP then only it is accepted otherwise it is rejected.

Terminaon:

The algorithm terminates aer xed number of iteraons, set by the user.

Flow:

Implementaon of TLBO in ROM:Implementaon of TLBO for Reduced Order Modelling is similar to the case of GA where in both SISO

and MIMO case the parameter being opmised are i

For SISO case,

|= |= i 1, 2, 3 rAnd for MIMO case,[ ]|= [ ]|=


28/32

28

Results (ROM of SISO systems):

Figure 19: Step Response of the original system (Solid), Reduced System(Doed) and modred reduced system (Dashed) 2

Figure 20: Bode Plot of the original system (Solid), Reduced System (Doed) and modred reduced system (Dashed)

-0.01412 s3+ 5.415 s2+ 30.95 s + 58.57

------------------------------------------

s3+ 65.45 s2+ 793.2 s + 585.7

4.922 s2+ 25.97 s + 48.07

---------------------------------

s3+ 62.54 s2+ 678.8 s + 480.7

s5+ 1014 s4+ 14069 s3+ 69140 s2+ 140100 s + 100000

------------------------------------------------------------------------

s6+ 222 s5+ 14541 s4+ 248420 s3+ 14541006 s2+ 2.22e06 s + 1000000

Transfer Funcon 13: result obtained from

modred ()Transfer Funcon 14: Result obtained

from GA

Transfer Funcon 15: G(s)


29/32

29

Results (ROM of MIMO):The model of a synchronous machine connected to an innite busbar [] is considered in this

example. The system has 2 inputs, two outputs and seven states.

Original Model:



30/32

30

Figure 21: Step Response of the original system (Solid), Reduced System (Doed) and modred reduced system (Dashed)

Figure 223: Grouped Step Response of the original system, Reduced System and modred reduced system

Figure 234: Bode Plot of the original system (Solid), Reduced System (Doed) and modred reduced system (Dashed)


31/32

31

Conclusions: First Method (direct opmizaon of coecients):

o This technique performs poorly with respect to the balanced reducon technique.

Second Method (opmizaon of i and using it to nd the coecients)

o

This technique performs almost similar to the balanced reducon funcon in medomain, but on close observaon it can be seen that it doesnt outperform the

reference (balanced reducon)

Reference technique (Balanced Reducon) outputs a non-minimum phase system

TLBO has much less complexity as compared to GA but provides equally good results.

GA Parallel - GA TLBO

Runme (s) 42.218 17.390 36.1563

Future Work GA and TLBO are few of the many opmisaon techniques available in the

literature. Analysis of other opmisaon techniques could lead to faster and

ecient implementaons.

The technique developed for model order reducon for MIMO systems is

computaonally expensive and further research could lead to simple and ecient

procedures.

TLBO is a fairly new opmisaon algorithm that was introduced in recently (2012)

and further analysis of the algorithm could lead to even beer results.


32/32

32

References[1]

Pal J (1986), An algorithmic method for simplicaon of linear dynamic scalar systems,Int.

J. Control, Vol 43, No. 1, 257-269

[2]

Pal J, Ghosh M.K, Sarvesh B (1995), A new method for model order reducon, J. Instn.

Electronics and Telecom. Engrs., Vol 41, No. 5&6[3]

Hickin, J., & Sinha, N. K. (1980). Model reducon for linear mulvariable systems.

Automac Control, IEEE Transacons on, 25(6), 1121-1127.

[4] Al-Baiyat, S. A., & Bettayeb, M. (1994). Appliation of recent model reduction techniques

to nuclear reactors. International journal of modelling & simulation, 14(2), 65-69.

[5] Rao, R. V., Savsani, V. J., & Balic, J. (2012). Teachinglearning-based optimization

algorithm for unconstrained and constrained real-parameter optimization problems.

Engineering Optimization, 44(12), 1447-1462.

[6] Rao, R. V., & Patel, V. (2013). An improved teaching-learning-based optimization

algorithm for solving unconstrained optimization problems. Scientia Iranica, 20(3), 710-

720.

[7]

Sinha, A. K., & Pal, J. (1990). Simulation based reduced order modelling using a clusteringtechnique. Computers & Electrical Engineering, 16(3), 159-169.

[8] Rommes, J., & Martins, N. (2006). Efficient computation of multivariable transfer

function dominant poles using subspace acceleration. Power Systems, IEEE Transactions

on, 21(4), 1471-1483.

reduced order modelling using evolutionary algorithms

Documents