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COMPARISON BETWEEN QR AND CHOLESKY METHOD

IN DYNAMICS SYSTEM

THRESEA DALAN

A report Submitted in partial fulfillment of the

requirements for the award of the degree of

Bachelor of Engineering (Electrical - Control and Instrumentation)

Faculty of Electrical Engineering

Universiti Teknologi Malaysia

MAY 2011


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To my beloved family especially mother. The love you gave me has been

tremendous and invaluable in my l i fe. I would not have succeeded without you as

my other half .


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ACNOWLEGMENT

Praise to God for His guidance and blessing, this project was finally

completed.First and foremost, I would like to express my heartily gratitude to my

supervisor, Dr. Norhaliza Binti Hj. Wahab for the guidance and enthusiasm given

throughout the progress of this project.

My appreciation also goes to my family who has been so tolerant and

supports me all these years. Thanks for their prayers, encouragement, love and

emotional supports that they had given to me. Without them, I feel very weak to face

every constraint in my life.

I would also like thanks for all of lectures and tutors that had given me so

much knowledge. May God give his blessing for all of their kindness. Nevertheless,

my great appreciation dedicated to all my best friends and SEI members. To friends

whom I have spent the year with - thanks for all the ideas and being supportive. Also

to those whom involve directly or indirectly with this project. There isno such

meaningful word than. Thank You So Much.


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v

ABSTRACT

In the complex system, often computation time is one of the main problems

in direct adaptive predictive controller. This is basically related to the identification

part to retrieve the model online. This thesis will tackle the problem of computation

time in the identification part using subspace identification algorithm (i.e N4SID).The main idea is to solve for linear predictor equation, f w p u fY L W L U using

least square solution. Both RQ and Cholesky methods will be used to solve

.These two algorithms basically used to reduces the effects noise in measured data

and use to extrapolating future plat input-output behavior from past input-output

data. Both methods of algorithms are compared in terms of computation time.

Y


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ABSTRAK

Dalam sistem yang kompleks, sering waktu pengkomputeran yang

merupakan salah satu masalah utama dalam pengawal aditif ramalan langsung. Hal

ini pada dasarnya berkaitan dengan bahagian pengenalan untuk memuatkan online

model. Tesis ini akan mengatasi masalah waktu pengkomputeran di bahagian

pengenalan menggunakan algoritma Subspace (N4SID). Idea utamanya adalah untuk

menyelesaikan persamaan linier prediktor, f w p u fY L W L U menggunakan

penyelesaian kuadrat terkecil. Kedua-dua RQ dan kaedah Cholesky akan digunakan

untuk menyelesaikan . Kedua-dua algoritma pada dasarnya digunakan untuk

mengurangkan kesan noise dalam data diukur dan digunakan untuk ekstrapolasi plat

perilaku input-output masa depan dari data input-output terkini. Kedua-dua kaedah

algoritma tersebut berbanding dalam hal masa pengkomputeran.

Y


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TABLE OF CONTENTS

CHAPTER TITLE PAGE

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENT iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENTS vii

LIST OF TABLES x

LIST OF FIGURES xi

LIST OF APPENDICES xii

1 INTRODUCTION 1

1.1 Background 1

1.2 Objectives 2

1.3 Scope 3

1.4 Summary 3

2 LITERATURE REVIEW 5

2.1 Introduction 5

2.2

Least Square Subspace Predictor 5

2.3 QR method 8

2.4 Cholesky method 9

2.5

Fast updating and down dating ofw

L andu

L 10

2.6

Comparison between QR and Cholesky 11


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3 METHODOLOGY 13

3.1 Introduction 13

3.2 Create the MIMO Aircraft System 14

3.3

Create The Hankel matrices 15

3.4 Create QR algorithm 16

3.5 Create Cholesky algorithm 17

3.6

Updating and Down-dating data using QR method 17

3.7 Updating and down-date data using Cholesky method 18

3.8

Measure computation time for QR and Cholesky

method 18

4 RESULTS AND DISCUSSIONS 19

4.1 Discussions 19

4.2 Results 21

5 CONCLUSIONS AND FUTURE WORK 28

5.1 Conclusions 28

5.2

Future Works 29

REFERENCES 30

Appendices A-D 31-39


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LIST OF TABLES

TABLE NO. TITLE PAGE

1.1 Comparison between QR and

Cholesky method

12

4.1 Time taken by QR method to

determine and

21

4.2 Time taken by QR method to

updating data

22

4.3 Time taken by QR to down dating

data

23

4.4 Time taken by Cholesky method to

determine and

24

4.5 Time taken by Cholesky method to

updating data

25

4.6 Time taken by Choclesky method to

down dating data

26

wL

wLuL

uL


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x

LIST OF FIGURE

FIGURE NO. TITLE PAGE

1.1 Flow Chart 4

4.1 The performance of Aircraft system 27


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LIST OF APPENDICES

APPENDIX TITLE PAGE

A GANTTS CHART FOR FINAL

YEAR PROJECT 1

31

B GANTTS CHART FOR FINAL

YEAR PROJECT 1

32

C MATLAB PROGRAM FOR

CHOLESKY METHOD

33

D MATLAB PROGRAM FOR QR

METHOD

36


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CHAPTER 1

INTRODUCTION

1.1

Background

Nowadays, most of dynamics system use online tracking to monitor the

performance of the system. Online method make the engineer easier to monitor and

analysis the performance of the system especially dynamics system. The traditional

method requires more time to get the data from the plant and then do the analysis. In

traditional method the engineer need to calibrate every instrument in plant if

something wrong goes into the plant, so it take a long time. In online system the

engineer can detect the error of the system in second by monitoring the system and

control the system in control room. So in online system the engineer no needs to go

to the plant to do inspection.

However, online systems still have the problem in computation time in direct

adaptive predictive controller. So to tackle the problem, the subspace identification

algorithm was introduced in this paper. The subspace system identification methods

have recently gained much popularity for identification of linear time-invariant (LTI)

systems. The technique is typically employed in two step process: firstly, the best

least square subspace predictor is derived from available experimental data;

secondly, the predictor is used to derive a state space model of the dynamics system.

[1]


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A fast algorithm is very important to updating and computing a subspace

predictor data when new experiment data are available. For example in-flight modal

analysis of airplanes, a fast analysis of the test data is very important because test

flights are expensive. A batch of data samples is collected a used to estimate the

model parameters. While processing the data, the pilot has waited for the next

instructions. Online and in-flight exploration of the data would allow more direct

exploration of the flight domain, with improved confidence and reduced costs. [2] So

the fast algorithms used in this paper are RQ and Cholesky methods. Besides the

faster processing of the data, the algorithm also reduces the constraints on the

amount of data that can be processed so that more sensors can be used in dynamics

system. For example, is an airplane. In this project, the data from airplane will be

used.

1.2 Objective of Project

The main objective of this project is to tackle the problem of computation

time in identification part using subspace algorithms. Second is to solve for linear

predictor equationf w p u fY L W L U using RQ and Cholesky method. The

comparison between RQ and Cholesky in term mrthods of time is done. The last

objective is to extrapolate future plant input-output data behavior from past input-

output data.


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1.3 Scope of Project

In order to achieve objective of the project, there are several scopes have

been outlined. The scope of this project includes implement matrix algorithm for

subspace predictor, f w p u fY L W L U and interpret the data available to it, study on

RQ and Choclesky algorithm methods, to build / design matlab programming and

last but not least is compute the data using matlab Programming.

1.4 Summary of Work

Implementation and works of the project are summarized into the flow chart

as shown in figure 1.1. Gantt chart as shown in figure 1.2 and 1.3 shows the detail of

the work of the project that had been implemented in the first semester and the

planning of work for the second semester.


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Figure 1.1 Flow Chart


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CHAPTER 2

THEORY AND LITERATURE REVIEW

2.1 Introduction

This chapter includes the least square predictor, method use to solve least

square predictor in the adaptive control and use in-flight modal analysis of airplane.

So in this chapter the research about QR and Cholesky has done by read the journal

and book. Besides that, this chapter also discusses on floating point and the memory

require.

2.2 Least Square Subspace Predictor

The summary of the least square subspace predictor is in [1]. The

development of the predictor begins with experimental input- output data. Considerinput- output data of length n from plan m inputs uk R

mand loutputs yk Rl. The

designer then chooses a prediction horizon, i which should be chosen to be larger

than the expected order of the plant (if the plant is LTI). Typicallyi is chosen to be 2

and 3 times larger than the expected plant order. [1] The data set is then broken into j

prediction problems, wherej= n -2i+ 1. Usually there is a relatively large amount of

data, so that j>> i. The goal is to find a single predictor that simultaneously optimizes

(in the least squares sense) the j prediction problems.


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The j prediction problem can be formalized as follows. Define the block Hankel

matrices from the data as

(2.1)

The matrix in equation (2.1) represent past input.

(2.2)

The matrix in equation (2.2) represents future input.

(2.3)

The matrix in equation (2.3) represents past output.

(2.4)

pU

0 1 1

1 2

1 2

j

j

ii i j

u u u

u u u

u u u

fU

1 1

1 2

2 1 2 2 2

i i i j

i i i j

i i i j

u u u

u u u

u u u

pY

0 1 1

1 2

2 1 2 2 2

j

j

i i i j

y y y

y y y

y y y

fY

1 1

1 2

2 1 2 2 2

i i i j

i i i j

i i i j

y y y

y y y

y y y


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The matrix in equation (2.4) represents future output.

The predictor is determined by choosingw

L andu

L to simultaneously optimize

the prediction indicated by the matrix in equation (2.1), (2.2), (2.3) and (2.4).

The subscripts pand f essentially represent past and future time. Definepw ,

which contains all the past

(25)

To obtaining the best linear squares predictor of L , given and can be written as

a Fresenius norm minimization.

(2.6)

The least square represent below:

(2.7)

The and use to form an estimate of the future outputs

(2.8)

p

p

p

UW

Y

min

,w u

L L pf w u

f

WY L L

U

f w p u fY L W L U

1

k

k i

y

y

1

1

k i

k

w

k i

k

u

uL

y

y

1

k

u

k i

u

L

u

wu


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The equation (2.8) will be used to extrapolate future plant input-output behavior from

past input-output data.

2.3 QR method

The QR factorization of an m-by-n matrix A is given:-

A= QR (2.9)

The matrix A have full column rank then the first n column of Q form an

orthogonal basis for range(A).The RQ factorization is one way to compute an

orthogonal basis for set of vectors. They have five method in RQ factorization;

Householder, block householder, Givens and fast givens transformations.

So in this project the fast givens transformation will be to calculate the leastsquare predictor. The RQ decomposition forms the basis of a computationally

efficient and numerically reliable method to finding Lw and Lu. [1]

QR decomposition.

(2.10)

Then

(2.11)

p

f

f

W

U

Y

R Q

11

21 22

31 32 33

0 0

0

R

R R

R R R

T

Q

w uL L 31 32R R 11

21 22

0R

R R


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The indicated pseudo- inverse is usually calculated using the singular value

decomposition (SVD), so that

(2.12)

(2.13)

The QR decomposition has computational complexity and the SVD

algorithm has complexity o(i3), which produces an overall computational complexity

. The storage requirement for the RQ/SVD algorithm is large.

2.4 Cholesky Method

In order to enable an adaptive algorithm, rapid computation of wL and uL is

required. The following method is based on the cholesky factorization.

Let

(2.14)

(2.15)

11

21 22

0R

R R

T

svd svdU V

w uL L 31 32R R

1

1

1

2

1

0 0

0 0

0 0q

qV qU

2( )o i j

2 3( )o i j i

p

f

f

W

U

y

T T TA R Q QR


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Where R= chol(AAT) is the cholesky factorization of AAT. Since the

dimensions of AAT are much smaller than A when j>>i, the Cholesky factorization of

AAT can take significantly fewer computations to perform than the RQ

decomposition of A. The cholesky factorization is in fact O (i3) [1]. However, one

must also account for the computation of AAT.

The brute force method of computing AAT is O(i2j). Fortunately, the Hankel

structure of A can be exploited to drastically reduce this number.[1]

Let

(2.16)

By inspection, A is a permutation of B, thus AAT is a permutation of BBT, so

computing

(2.17)

is equivalent to computing AAT. Since BBTis symmetric, it is sufficient to compute

only the bloxk upper triangle of UUTand YYT, and to compute only UYT.

2.5 Fast updating and down dating of Lw and Lu

According to Woodley et al [1]. To update and downdate the data (sliding

window), the oldest experimental data are removed from R, and new data are added.

This sliding window procedure requires the storage of experimental data over thewhole window. In this case, the matrix R plays the role of maintaining a summary of

p

f

UU

U

p

f

YY

Y

p

f

p

f

UU U

BY Y

Y

T

T T

UU UY BB

YU YY


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all past information in order to update the predictor as new experimental data

installed. When the data are used to update R, then the Lw and Lu are computed

from R using SVD.

2.6 Comparison on RQ and Cholesky

Table 1.1Comparison between QR and Cholesky Method

RQ Cholesky

Floating point

operation

3.2 x 108 3.4 x 105

Memory(Bytes) 1.08x10 1.4x10

From B.R Woodley et.al [2], the result show that RQ method require more

floating point operation and memory byte in do the analysis. The Cholesky method is

the reduced storage requirements. The recursive method of computing BBT does not

require the complete formation of U or Y. Since any block i x j Hankel matrix can be

represented by a block vector of length i+j-1, the storage requirement. Based on this

result, we can expect outcome for this project is the cholesky method faster.


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CHAPTER 3

METHODOLOGY

3.1 Introduction

In this chapter discuss about the method and approach to implement the RQ

and Cholesky method in airplane analysis. Below the step will be taken to implement

this project.

a) Design matrix algorithm for subspace predictor, f w p u fY L W L U

b) Write or design programming matlab for RQ method to solve the Y .

c) Write or design programming matlab for Cholesky method to solve subspace

predictor.

d) Compute the data Y using RQ and Cholesky. Record the time might be taken

for RQ and cholesky method to solve the Y


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3.2 Create the MIMO Aircraft system

In this thesis the MIMO Aircraft system has been use to test the QR

algorithm and Cholesky algorithm. This system has two inputs and two outputs. That

mean this system has four states, beta (sideslip angle), phi (bank angle), yaw rate and

roll rate and also have two inputs, the rudder and aileron deflections. The units are

radians for beta (sideslip angle) and phi (bank angle) and radians/sec for yaw (yaw

rate) and roll (roll rate). The rudder and aileron are in degrees. Below show the

transfer function for this system:

Transfer function from input rudder to output

-0.475 s^3 0.2479 s^2 0.1187 s 0.05633

Yaw rate: ----------------------------------------------------------------- (3.1)

s^4 + 0.6358 s^3 + 0.9389 s^2 + 0.5116 s + 0.003674

Transfer function from input rudder to output

0.1148 s^2 0.2004 s 1.373

Bank angle: -------------------------------------------------------------------- (3.2)

s^4 + 0.6358 s^3 + 0.9389 s^2 + 0.5116 s + 0.003674

Transfer function from input aileron to output

0.0077 s^3 0.0005372 s^2 + 0.008688 s + 0.004523

Yaw rate: ---------------------------------------------------------------------- (3.3)

s^4 + 0.6358 s^3 + 0.9389 s^2 + 0.5116 s + 0.003674


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Transfer function from input aileron to output

0.1436 s^2 + 0.02737 s + 0.1104

Bank angle: -------------------------------------------------------------------- (3.4)

s^4 + 0.6358 s^3 + 0.9389 s^2 + 0.5116 s + 0.003674

The equation (3.1), (3.2), (3.3) and (3.4) are form of the transfer function of

airplane, the MATLAB programming or command was created and these systems

become are input and output for the Hankel matrix. In this programming the system

was representing in the A, B, C and D matrices. Then the step input was use in this

programming to test the response or performance of this MIMO Aircraft system.

3.3 Create the Hankel matrices

The block Hankel matrices were define from the Aircraft data. The input of

the the Hankel were represent as and . Then the output were represent as are

and . Then the MATLAB programming for block Hankel matrices was

created as an appendix C and D.

pU U

pYY


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3.4 Create QR algorithm

In this thesis the QR method is use to solve least square predictor,wLand

uL.

The MATLAB programming was use to create QR algorithm. In this programming,

term H was represent block Hankel matrices contained U , pU , p and . Then

the matrix H was factorizing using QR method. Matrix R and Q then transposed to

get lower triangular R. After that the matrix R divided in to two part, which is 1LR

and pseudo-inverse 2LR . The Singular Value Decomposition (SVD) was use to solve

the matrix pseudo-inverse 2LR .

Matrix2L

(3.5)

Matrix1L

(3.6)

Pseudo-inverse = (3.7)

p

f

f

W

U

Y

11

21 22

31 32 33

0 0

0

R

R R

R R R

Q

1 31 32LR R R

11

21 22

0R

R R

2L


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3.5 Create Cholesky algorithm

In this project another method that was use to calculate least square predictor

was cholesky method. In this section, the MATLAB programming for Cholesky

algorithm are created. The block hankel for this method was similar with QR

method, which is contained U , pU , p and . For Cholesky method, the Matrix

R was equal to cholesky factorization of matrix hankel H multiply by Matrix hankel

H transpose. To get R lower triangular, the command [R,p]=chol(B,lower);

was used. Matrix B is equal to Matrix H multiply with transpose H. After that the

matrix R was divided in to two part, which is 1LR and pseudo-inverse 2LR . Then the

same method (SVD) in QR was use in this section in order to solve Pseudo-inverse

2LR . Last but not least, the command for wL and uL was created.

R= chol(AAT) (3.8)

3.7 Updating and Down dating Data using QR method

In dynamic system updating and down dating the new and old data is very

important in order to engineer analysis data in short time. So in this section the QR

method has been use to update the data. The MATLAB programming command that

used to update the data is [Q1,R1] = qrupdate(Q,H,f,v); and the command

used to down- date the data is [Q2,R2] = qrdelete(Q,H,j,row);. The

programming for upadating and downdating for QR method showed at appendix D.


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3.7 Updating and down-date data using Cholesky method

In this thesis, other method that was used to update data and down-date data

is cholesky. The command that was used are Anew= cholupdate(H,a,+);

andAnews = cholupdate(H,b,-);. The programming for upadating and

downdating for Cholesky method showed at appendix C.

3.8 Measure computation Time for QR and Cholesky

In this thesis, the command that was used to measure the computation time

for QR and cholesky method is tic-toc command. This command was measure the

Computer Processing Unit (CPU) time. So time measured was display at command

window in unit second(s). This command applies at the bottom and top of the

MATLAB programming statement. This command functioning like a stop watch, the

command tic means the watch start counting and the command toc mean the

watch stop to counting the time. Below show how the tic toc command was used:

Statement

The watch stop counting time

The watch start countin


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CHAPTER 4

RESULT AND DISCUSSION

4.1 Discussion

The time taken for QR and Cholesky method to compute the data have been

measured and recorded. The measured showed at Table 4.1, Table 4.2, Table 4.3,

Table 4.4, Table 4.5 and Table 4.6. The average time taken by QR to compute data to

solve the least square predictor is 0.492350 seconds and the time average for

Cholesky method to compute data was 0.466970 seconds. So the Cholesky method is

faster than QR method to compute the data. The time different between QR and

Cholesky was 25.38 mili second.

Table 4.1 shows the time taken for QR and Table 4.4 show the time taken for

Cholesky to compute the data. Next, Table 4.2 show the time taken for QR method to

updating the new data added to the plant. From the table, the average time for QR

updating data is 8.617 mili seconds. Table 4.5 shows that time taken for Cholesky to

update data. The average time is 8.436 mili second. Then time different between this

two method to update data is 0.181 mili second (t (QR) t(chol)). So the time taken for

Cholesky method to update data is shorter than QR method.


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Table 4.3 and Table 4.6 show that the time taken for QR and Cholesky to

down-dating the data. From the Table 4.3, the result shows that the average time for

QR to down-date the data is 2.291 mili second. Then from table 4.6, average time for

Cholesky to down-dating the data is 0.031 mili second. So, ime different between

this two method is 2.26ms. The time different for down-dating is more short than

time different for compute the data to find least square predictor and update data.

The Figure 4.1 shows that the performance of MIMO Aircraft system. From

the figure, the results show that the Aircraft system is stable. There have four graphs,

which show the performance of four transfer function of Aircraft system with two

inputs and two outputs. The Graph 1, shown the performance of the input rudder to

output Yaw rate. The second graph showed the performance of the input rudder to

bank angle. The third graph showed the performance of input aileron to output yaw

rate. The fourth graph showed the performance of input aileron to output bank angle.

The graphs show that all the performance of the system is stable.


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4.2 Result

Table 4.1Time taken by QR method to determine Lu and Lw

No. of measure Time (s)

1 0.503669

2 0.460108

3 0.486851

4 0.571558

5 0.495226

6 0.462641

7 0.592830

8 0.442809

9 0.517873

10 0.390042

AVERAGE 0.492350


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Table 4.2Time taken by QR method to updating data

No. of measure Time(ms)

1 8.889

2 7.659

3 12.09

4 8.036

5 8.274

6 8.188

7 8.960

8 8.594

9 7.428

10 8.048

AVERAGE 8.617


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Table 4.3Time taken by QR method to down-dating data


1 2.081

2 1.916

3 2.177

4 2.793

5 2.827

6 2.184

7 2.665

8 2.174

9 1.958

10 2.137

AVERAGE 2.291


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Table 4.4Time taken by Cholesky method to determine Lu and Lw

No. of measure Time(s)

1 0.500651

2 0.412089

3 0.453118

4 0.498101

5 0.457423

6 0.444088

7 0.435379

8 0.482314

9 0.473129

10 0.513413

AVERAGE 0.466970


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Table 4.5Time taken by Cholesky method to updating data


1 8.815

2 8.888

3 9.113

4 7.396

5 7.792

6 8.505

7 8.929

8 8.466

9 7.894

10 8.570

AVERAGE 8.436


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Table 4.6Time taken by Cholesky method to down-dating data


1 0.190

2 0.091

3 0.209

4 0.087

5 0.167

6 0.194

7 0.191

8 0.239

9 0.206

10 0.784

AVERAGE 0.031


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Figure 4.1The Performance of Aircraft system


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CHAPTER 5

CONCLUSIONS AND FUTURE WORK

5.1 Conclusions

The main objective of this thesis is to compare the computation time between

Cholesky method and QR method. Chapter 4 have shows the result that Cholesky is

faster than QR method in order to compute data during solves the least square

predictor, updating and down-date the data. However, the time different between

these two methods in this thesis is quite small because the data that have been used is

in small amount. Beside that, the CPU of RAM speed may be can influence the time

computation; if the CPU has high speed so the time different between these two

methods is small. Then if the CPU has low speed so the time between these two

methods is large.

The Cholesky method faster than QR because the QR method is requires

more storage location than Cholesky. The QR method has produced more zeroelements in matrix than Cholesky method. In Cholesky method zero elements in the

matrix has been eliminated. So the size of matrix R after instruction chol is small

but for the QR method the size of R matrix become large because zero element

become more.

As a conclusion, for this thesis is the main objective of the thesis is

successful. As an expected outcome, the analysis of this thesis proves that the

Cholesky method is faster than QR method.


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5.2 Future Work

As a recommendation, use more data input to the system, so that the time

different between QR and Cholesky method is larger. The matrix size that has been

use in this thesis is 32 times 32 so in the future work use larger matrix sizes like 64

times 64. So the CPU will execute the program in long time. Then the different time

between these two methods can measure in large value of time.


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REFERENCES

[1] G.H GOLUB AND C.F. VAN LOAN, matrix computations, John Hopkins

University Press, Baltimore MD, third edition (1996).

[2] Bruce R. Woodley et.al, Subspace based direct adaptive control, Stanford

University, 2001.

[3] Katrien De Cock et.al, Recursive subspace identification for in-flight modal

analysis of airplane, K.U.Leuven Department of Electrical Engineering,

Belgium, 2006.

[4] Che Rahim Che The, MATLAB Programming Mathematical Approach, Jabatan

Matematik Fakulti Sains, Universiti Teknologi Malaysia,2008.

[5] Matlab, State space Jet Transport Aircraft,2008

[6] I.Geothals et.al, Recursive output only subspace identification for in-flight flutter

monitoring, Dearborn Michigan,2004.


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APPENDIX A

GANTTS CHART FOR FINAL YEAR PROJECT 1


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APPENDIX B

GANTTS CHART FOR FINAL YEAR PROJECT 1


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APPENDIX C

MATLAB PROGRAMMING FOR CHOLESKY METHOD


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APPENDIX D

MATLAB PROGRAMMING FOR QR METHOD


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Comparison Between Qr and Cholesky Method2

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