Dynamic Modelling of Gas Turbine Engines
23
Identification of Identification of Reduced-Oder Dynamic Reduced-Oder Dynamic Models of Gas Models of Gas Turbines Turbines CSC Student Seminars CSC Student Seminars (Spring/Summer, 2006) (Spring/Summer, 2006) PhD Student: Xuewu Dai Supervisor: Tim Breikin and Hong Wang
description
Dynamic Modelling of Gas Turbine Engines
Transcript of Dynamic Modelling of Gas Turbine Engines
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
11
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
