Identification of Reduced-Identification of Reduced-Oder Dynamic Models of Oder Dynamic Models of

Gas TurbinesGas Turbines

CSC Student SeminarsCSC Student Seminars(Spring/Summer, 2006)(Spring/Summer, 2006)

PhD Student: Xuewu Dai

Supervisor: Tim Breikin and Hong Wang

IntroductionIntroduction

1. Introduction1. Introduction 2. Reduced-order Model2. Reduced-order Model 3. Long-term Prediction3. Long-term Prediction 4. Dynamic Gradient Descent4. Dynamic Gradient Descent 5. Nonlinear Least-Squares Optimization5. Nonlinear Least-Squares Optimization 6. Future Works6. Future Works

1. Introduction1. Introduction

Modlling of Gas TurbinesModlling of Gas Turbines Fault Detection Fault Detection Condition MonitoringCondition Monitoring

AimsAims

Reducing Computational Complexity: Reducing Computational Complexity:

Real timeReal time Improving Prediction Accuracy:Improving Prediction Accuracy:

Long-term predictionLong-term prediction RobustnessRobustness

2. Reduced Order2. Reduced Order Thermodynamic models:Thermodynamic models: 1. High order : 261. High order : 26thth

2. Non-linear2. Non-linear

Linearisation Our ARX models :Our ARX models :

1. Reduced order: 11. Reduced order: 1stst, 2, 2ndnd … …

2. Linear: 2. Linear: )()(ˆ tzty T T

mn bbaa ] ... ... [ 11Tmtutuntytytz )]( ... )1( )( )...1([)(

3. Long-term Prediction3. Long-term Prediction

)(ty

)(ˆ ty

)(tu

Model

a. One-step Ahead Prediction Model

)(ty

)(ˆ ty

)(tu

Model

b. Long-term Prediction Model

Model EquationsModel Equations

)()(ˆ tzty T 1.One-step ahead prediction

2. Long-term prediction

)(ˆ)(ˆ tzty T Tmtutuntytytz )]( ... )1( )(ˆ ... )1(ˆ[)(ˆ

Tmtutuntytytz )]( ... )1( )( )...1([)(

ChallengesChallenges Computational BurdenComputational Burden How many iterations need to identify the parameters?How many iterations need to identify the parameters? Dependency of Prediction Errors (Non-Gaussian Noise) Dependency of Prediction Errors (Non-Gaussian Noise)

Ntyty /))(ˆ)(( 2

MSE=9.1318MSE=9.1318

Autocorrelation of prediction errorsAutocorrelation of prediction errors

4. Dynamic Gradient Descent4. Dynamic Gradient Descent

Objective FunctionObjective Function

Global Gradient and local gradientGlobal Gradient and local gradient

2

11

2 ))(ˆ)((2

1))((

2

1)(

N

t

TN

t

tztytE

N

i

N

t

tt

tE

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2 )()(

)(

2

1)(

)()(

tgt

Dynamic Gradient DescentDynamic Gradient Descent

(d) ]0...0 )()...1([)(ˆ)(

(c) )())(ˆ)(()(

(b) )(

(a) )(ˆ)(ˆ

1

1

ntgtgtztg

tgtytyE

E

tzty

N

t

kk

T

k

a

b

DGD SearchRoute

0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Results 1: deepest directionResults 1: deepest direction

a

b

DGD+BFGS SearchRoute

0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 50 60 70 80 90 1000

100

200

300

DGD+BFGS Iteration

Traing Error

BFGS directionBFGS direction

5. Nonlinear Least-squares 5. Nonlinear Least-squares Optimization (Gauss-Newton)Optimization (Gauss-Newton)

(e) )],( )......,2( ),1([

(d) 0...0] ),()...,1([)1(ˆ),(

(c) ),()](ˆ)([()(

(b)

(a) )(

)]([*

1

11

Tkkk

kkk

k

N

tk

k

T

k

kkk

NgggJ

ntgtgtztg

tgtytyE

JJR(k)

EkR

Search direction, step size and Search direction, step size and initial valueinitial value

Search direction:Search direction:

Deepest descent: inverse global gradientDeepest descent: inverse global gradient

Nonlinear Least Squares: Gauss-NewtonNonlinear Least Squares: Gauss-Newton Step size:Step size:

fixed, adjustable, line searchfixed, adjustable, line search Initial value:Initial value:

Blind guess: [0.5 0.5 0.5 0.5]Blind guess: [0.5 0.5 0.5 0.5]

LSE: LSE: [1.2805 -0.29191 0.10582 0.15903]

Result 3 Gauss-NewtonResult 3 Gauss-Newton

a

b

Gauss-Newton + Bisection SearchRoute

0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4 5 6 70

50

100

150

200

Gauss-Newton + Bisection Iteration

Traing Error

Prediction of 1Prediction of 1stst Order Model Order Model

0 500 1000 1500-50

0

50Real output(blue) vs Estimation output(red)

time

outp

ut

0 500 1000 1500-10

-5

0

5

10Error Curve The mean squared error is9.1318

time

err

or

Comparison of 1Comparison of 1stst Order Model Order ModelMethods MSE a b Iterations

LSE 23.49449 0.987395 0.032551 1

ANFIS 22.2925 N/A N/A 200

GD 11.0163 0.9809 0.0376 N/AExhausted

Search9.131926 0.977774 0.043542 10000

DGD1* 9.131816 0.977764 0.043568 1000

DGD2* 9.131786 0.777774 0.043544 101

DGD3* 9.131785 0.977776 0.043543 98•DGD1: Deepest descent direction and adjusting step size•DGD2: BFGS direction and adjusting step size•DGD3: Gauss-Newton and line search

High Order ModelHigh Order Model)2()1()2()1()(ˆ 2121 tubtubtyatyaty

initial (by LSE) : [1.2805 -0.29191 0.10582 0.15903]

final: [1.8604 -0.8641 0.07045 -0.007475]

0 500 1000 1500-50

0

50Engine output(dashed) vs Model Preditcion(solid)

sample Time

NH

P

0 500 1000 1500-5

0

5

10Long-term Prediction Error: The MSE is:3.31361166E+000

Sample Time

Err

or

6. Future Works6. Future Works

Initial value Problem: Initial value Problem: Robustness Problem: ???Robustness Problem: ??? Applying such learning algorithm to Neural Applying such learning algorithm to Neural

NetworksNetworks Model structure selection by autocorrelation Model structure selection by autocorrelation

of prediction errorsof prediction errors NARMX modelsNARMX models

ThanksThanks

CSC Student SeminarsCSC Student Seminars(Spring/Summer, 2006)(Spring/Summer, 2006)

AppendixAppendix

Initial value problemInitial value problem manual setting of initial value

[0.5 0.5 0.5 0.5]

[1.8604 -0.8641 0.07045 -0.007475]

Final MSE=8.60188

setting initial value by LSE[1.2805 -0.29191 0.10582 0.15903] [1.8604 -0.8641 0.07045 -0.007475]Final MSE=3.313612

appendixappendix

)(ˆ))(ˆ)(()( tytytyt

0 ... 0 ... 2 1),(ˆ

),(ˆ),(ˆ

)),(ˆ(

))(ˆ(

g(t-n))g(t-)g(ttz

tztz

tz

tyg(t)

T