Adaptive Intelligent Combustion Control Based on Data ...€¦ · Adaptive Intelligent Combustion Control Based on Data-Driven Low-Order Models Tongxun Yi Domenic Santavicca ... (Barbalat’s

Adaptive Intelligent Combustion Control Based on Data-Driven Low-Order Models

Tongxun YiDomenic Santavicca

Penn State University

Presented at the 2009 NASA Propulsion Control and Diagnostics WorkshopCleveland Airport Marriott Hotel, Cleveland, Ohio

December 8 – 10, 2009

*NASA Award No: NNX07C98A

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Content

4. Flame Transfer Functions and Control Design Perspective

2. Fuel Modulation Techniques

1. Why Data-Driven Model-Based Combustion Control

NASA Award No: NNX07C98A

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3. Combustion Sensing Techniques

5. Conclusions and Suggested Future Work

6. Reference

1. Why Data-Driven Model-Based Combustion Control


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Necessity of Advanced Combustion Control


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For gas turbine combustors, the design strategies favoring different performance indices are usually not compatible. Combustion control adds extra freedom to improve and optimize overall performance. A typical example is developing control systems to enhance lean combustion stability so that the engines can operate in clean, safe, and stable manner.

Mixed control performance has been reported, including insufficient suppression of slightly-damped modes. These are the intrinsic deficiencies of phase-shift control principles.

Three adaptive phase-shift controllers (Gatech, UTRC, and Yi&Gutmark). The last one is capable of identifying the dominant frequency within one pressure cycle and a half, with an estimation error within 5 Hz, and is free of stability concerns

Model-based control design is a standard routine for control engineers and theorists. Data-driven models are in particular attractive and practical.

Enough evidence suggests that model-based controllers can easily outperform the empirical ones, and for highly nonlinear systems, a simple nonlinear controller can well outperform linear ones.

No knowledge can be certain if it is not based on mathematicsNo knowledge can be certain if it is not based on mathematics.(Leonardo da Vinci)

Example 1. Empirical Vs. Model-based Control


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Mean Flow Control of the Goodrich Valve: PD Vs. LQG Control

Goodrich Magnetostrictive Valve

-8-6-4-202468

1012

0 1 2 3 4 5 6 7 8 9 10Time (s)

pressure (kPa) Mean Fuel Flow Rate (-2.5 g/s)

PD Control of Mean Flow

LQG Control of Mean Flow

9033.311744.27155.4728710431.66284.61389004198.0

)( 234

230

EsEsEssEsEss

sUM

++++−−−

−=&

XBu

GyXAXT

c

−=

+=&

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛−−

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−

=

4277.33638.8

3158.30633.7

1000;

0013.00027.0

005.03067.0

001.0 BG

System Identification Model

Example 2. Linear Vs. Nonlinear Control


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Adaptive control of Large-Vortex Shedding

No Control

N-S Equation:

POD-basedModel

⎩⎨⎧

=++=

⇒

⎪⎪⎭

⎪⎪⎬

⎫

+=

=∇+−∇=×××××

∞→

=∑ 0

1NNNNT

NN1NN

1iii0

XX(0)DXBXXAX

)S((t) Φxt),S(Ut),S(U

0U.;UΔRe1p

DtUD

&

vvvrv

vvv

Failure of the LQG Controller

Example 2. Linear Vs. Nonlinear Control (cont.)


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ueaxaxaxa 42

321 =++++&a1, a2 , a3 and a4: constant or slow-varying but unknown (a1>0); e: bounded unmodeled dynamics and external disturbance,

SISI Reduced-Order Model

Adaptive control Law

)aaaa(A;A~AA);1xxx(Y

kEΦ);

ΦsΦsat(ss;ΓYsA;xxsks;AYu

4321T2

dT

ΔΔdT

=+==

=−=−=−=−= &

212

1121

])([~)ˆ(

~ˆ)(;~~21

21

ΔΔ−

ΔΔ

−Δ

−Δ

−≤−Φ

Φ−Γ+−−=

Γ−−−=Γ+=

ksessatksAAYsks

AAeAYusVAAsaV

TT

TTT

&

&& is bounded, so global stability is guaranteed (Barbalat’s Lemma). V&&

Stability Proof

No Control

Adaptive Controller Performance

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

1 29 57 85 113

141

169

197

225

253

281

309

337

365

393

421

449

477

505

533

561

589

617

645

673

701

729

Tim e(x0.04)

X

No ControlAdaptive Control

71% Reduction in Turbulent Kinetic Energy

2. Fuel Modulation Techniques


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The NASA Combustion Rig

Static/Dynamic PressureTemperatureGas Fuel Bar

Quartz Tube

Liquid Fuel Tubing & Injector

Pressure & Ignitor

Preheated Air

NASA Venturi

NASA Swirler

NASA FuelInjector

Quartz Tube

• Visually-observed axisymmetric flame• Pressure drop within 4% up to 70 SCFM• Nice blue flame below Ф=0.38• Somewhat red/yellow flame above Ф=0.38

• Large fuel pressure drop, about 4 times larger

• Jet-like flow with vortex breakdown, not easy to ignite• Comparable LBO limits with the previous one• Comparable air pressure drop with the previous one

• Less efficient fuel/air mixing than the previous one which has no red/yellow flame up to Ф=0.60


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Experiment Setup


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To Exhaust Fan

0.36 m

0.15 m

Dynamic and Static Pressure Ports

ICCD Camera

1.25 m

Choking PlateTemperature

0.46 m

Radial Swirler

Fuel Injector

Choking PlateDynamic Pressure

0.13 m

Preheated Air

Liquid Fuel

Rotary Fuel Actuator

P

Fuel Pressure Measurement

ICCD Camera

Spectrometer

UV-Grade Long Optical Fiber

Spherical Lenses

Quartz Tube

0.56 m

Motor-Driven High-Frequency Fuel ValveFuel

OutletFuel Inlet

Motor Shaft

Fuel OutletFuel

Inlet

Motor Shaft

Rotor (20 teeth equally spaced)Rotor (20 teeth equally spaced)

Forcing at 300 Hz

0

10

20

30

40

50

60

70

80

0 0.01 0.02 0.03 0.04 0.05Time (s)

Pre

ssur

e (p

si)


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Fuel Transfer FunctionFlame Transfer Function

Pressurized Fuel Tank 0 1

Fuel Actuator

P1(t) P2(t)

Fuel Injector Combustor

Forcing Input u(t)

)(' tm& )(' tQ&P3(t)

Needle Valve

On/Off Valve

0

10

20

30

40

50

60

70

0 100 200 300 400 500 600Frequency(Hz)

Pre

ssur

e A

mpl

itude

(kP

a) MeasurementPrediction

-225

-175

-125

-75

-25

25

75

125

175

225

0 100 200 300 400 500 600

Frequency(Hz)

Pha

se(D

eg)

Measurement Prediction

Size 2.5’’(Ф)x2; Working up to 1 kHz

Motor-Driven High-Frequency Air Valve


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Shaft

Air In

Air Out

• Driven by a variable speed DC motor• Modulation frequency up to 900 Hz• Inlet velocity modulations above 50%

up to 800 Hz• Size 4’’x4’’x2.5’’

0.0

0.3

0.6

0.9

1.2

1.5

1.8

0 100 200 300 400 500 600 700 800Frequency (Hz)

Pre

ssur

e A

mp

(psi

)

0.0

0.1

0.2

0.3

0.4

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0.6

Nor

mal

ized

Vel

city

Rat

io

0.3

0.4

0.5

0.6

0.7

0.8

0.9

550 600 650 700 750 800 850Frequency (Hz)

Pre

ssur

e A

mpl

itude

(P3,

psi

) Prediction Measurement

Air Flow 60 SCFM

Flame Response to Air Modulations


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0

0.2

0.4

0.6

0.8

1

550 600 650 700 750 800 850 900 950Frequency (Hz)

Gai

n

Series1Series2Series3Series4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

550 600 650 700 750 800 850 900 950Frequency (Hz)

P0(p

si)


-90

-45

0

45

90

135

180

550 600 650 700 750 800 850 900 950

Frequency (Hz)

Pha

se(D

eg)


Air: 40 SCFM; Preheat Temperature: 373 K

)()()(

0

*

sPsCHsG =Flame Transfer Function:

Air modulations above 900 Hz

Nonlinearity in terms of the forcing amplitude, in particular around the one-wave resonant frequency

Quite some details need to be figured out

3. Combustion Sensing Techniques


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Background of Combustion Sensing

• The instantaneous heat release rate and equivalence ratios are two key parameters for combustion analysis and control. Chemiluminescence-based sensors are practical solutions.

• For premixed gas-fueled combustion, linearity between chemiluminescence yield and heat release is valid for slightly turbulent or wrinkled flamelet region. But in the corrugated and broken flameletregion, nonlinearity cannot be ignored.

• In combustion instability analysis, it is usually assumed that chemilumienscence is proportional to the instantaneous heat release rate, which in fact, suffers from several major deficiencies.


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• Reported is an accurate correlation-function-based method for real-time combustion sensing, based on chemiluminescence measurements using PMTs. For the first time in combustion literature, the nonlinearity among heat release, chemiluminescence, equivalence ratios, and acoustics effects is taken into account

UV and VIS Flame Spectra

0123456789

1011

260 290 320 350 380 410 440 470Wavelength(nm)

Inte

nsity

(A.U

.) Phi=0.54 Phi=0.51Phi=0.48 Phi=0.45Phi=0.42 Phi=0.39Phi=0.36 Phi=0.33

05

1015202530354045

260 285 310 335 360 385 410 435 460 485Wavelength(nm)

Inte

nsity

(A.U

.)

Stable Combustion Unstable Combustion

10o

100o 190o


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0

2

4

6

8

10

12

0 100 200 300 400 500 600 700 800 900 1000 1100Experiment Index

OH

* Che

milu

min

esce

nce(

A.U

.)

Measurement Curve-Fitting

0

2

4

6

8

10

12


CO

2* C

hem

ilum

ines

cenc

e (A

.U.) Measurement Curve-Fitting

0123456789

1011


CH* Che

milu

min

esce

nce(

A.U.) Measurement Curve-Fitting

4045.04352.1

0735.36627.1430

2183.08217.0

2280.23637..1365

4522.01060.1

1369.27558.1307

~2000

~05.212

~2000

~85.115

~2000

~61.94

−

−

−

⎟⎠⎞

⎜⎝⎛ +=

⎟⎠⎞

⎜⎝⎛ +=

⎟⎠⎞

⎜⎝⎛ +=

pT

mI

pT

mI

pT

mI

ianm

ianm

ianm

φφ

φφ

φφ

&

&

&

Average Error: 2.4% for OH*; 2.0% for CO2*; 3.8% for CH*.

Combustion Sensing StrategyThe above correlation functions are mostly developed from stable combustion, thus they can be used to determine the mean heat release rate and the mean equivalence ratio (See JPP, Vol.129, No.5).

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) )(~7.281~00674.0~1.0

)(~)()()(1392.0)(

)(~)()()(1571.0)(~

2021.01437.0430

4942.1365

2966.1307

1974.05540.1430

7294.0365

4747.1307

kWttmttmHmm

ttmHtQ

tptItItIt

tptItItItm

R

trystoichiomea

fRR

nmnmnm

nmnmnm

φφφ

φ

&&&

&&&

&

=Δ=⎟⎟⎠

⎞⎜⎜⎝

⎛Δ=

=

=−−

−

They are obtained by eliminating the term of flame temperature from the three correlation functions.


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The estimated mean air consumption rate and the mean equivalence ratio. The air flow rate is 66.7g/s, the preheat temperature is 373 K, and the equivalence ratio is decreased from 0.41 to 0.31.

0.20

0.25

0.30

0.35

0.40

1 3 5 7 9 11 13 15 17Experiment Index

Equ

ival

ence

Rat

io

MeasurementPrediction

Error: 1.4%30

40

50

60

70

1 3 5 7 9 11 13 15 17Experiment Index

Air

Flow

Rat

e (g

/s)

MeasurementPrediction

Error: 1.9%

Combustion Sensing Strategy (Cont.)


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0

1

2

3

4

5

6

7

8

0 0.005 0.01 0.015 0.02 0.025Time(s)

CO2*(A.U.) CH*(A.U.)OH*(A.U.) Combustor Pressure(x0.1, kPa)

(a)0

1

2

3

4

5

6

7

0 0.005 0.01 0.015 0.02 0.025Time(s)

CH*(A.U.) Combustor Pressure(x0.1, kPa)Estimated Air Consumptiion Rate(x0.05,g/s) Estimated Equivalence Ratio(x10)Estimated Heat Release(x0.05, kW)

(b)

(a) Time traces of chemiluminescence and combustor pressure; (b) The estimated instantaneous air consumption rate, the estimated instantaneous equivalence ratio, and the estimated instantaneous heat release rate.

Self-Excited Combustion Oscillations

-0.32

-0.24

-0.16

-0.08

0.00

0.08

0.16

0.24

0.32

0.40

0 0.005 0.01 0.015 0.02

Time(s)

Normalized CH* Chemiluminescence Normalized Air Consumption RateNormalized Equivalence Ratio Normalized Heat Release RateNormalized Fuel Pressure

Forcing at 296 Hz -0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0 0.001 0.002 0.003 0.004 0.005 0.006

Time(s)

Normalized CH* Chemiluminescence Normalized Air Consumption RateNormalized Equivalence Ratio Normalized Heat Release RateNormalized Fuel Pressure

Forcing at 874 Hz

Fuel Forcing Induced Combustion Oscillations

4. Flame Transfer Functions and Control Design Perspective


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Background

• Combustion instability and lean blowout are major technical challenges for liquid-fueled DLE combustion. Both phenomena can be attributed to the increased sensitivity in heat release to external disturbances or intrinsic acoustic oscillations at very lean conditions.

• Active control of both phenomena can be achieved using small-amplitude fuel modulations, employing the same control hardware and fuel actuators. However, major differences exist.

• First-principle low-order modeling is challenging. The measured flame transfer functions (FTFs), i.e. heat release responses to inlet air and/or fuel modulations, provides an accurate description of combustion dynamics around the working conditions where they are derived.

• Acoustic responses are system- and geometry-dependent, but heat-release-based open-loop FTFs can be used for different types of engines employing the same type of burners.


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Flame Transfer Functions

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0 200 400 600 800 1000Frequency(Hz)

Gai

n(A

.U.)

-200

-150

-100

-50

0

50

100

150

200

Pha

se(D

eg)

Gain(A.U.)Phase(Deg)

)(/)(* sPsCH f

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0 200 400 600 800 1000Frequency(Hz)

Gai

n(A

.U.)

-200

-150

-100

-50

0

50

100

150

200

Pha

se(D

eg)

Gain(A.U.)Phase(Deg)

)(/)( sPs fφ

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 200 400 600 800 1000Frequency(Hz)

Gai

n(A

.U.)

-200

-150

-100

-50

0

50

100

150

200

Pha

se(D

eg)Gain(A.U.)

Phase(Deg)

)()( sPsQ fR&

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0 200 400 600 800 1000Frequency(Hz)

Gai

n(A

.U.)

-120

-100

-80

-60

-40

-20

0

20

40

Pha

se(D

eg)

Gain(A.U.) Phase(Deg)

)()( * sCHsQR&


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Control-Oriented Low-Order ModelingThe measured flame transfer function provides an accurate description of combustion dynamics.

1.5

1.7

1.9

2.1

2.3

2.5

2.7

2.9

3.1

3.3

200 300 400 500 600Frequency(Hz)

Am

plitu

de

MeasurementModel

-180

-150

-120

-90

-60

-30

0200 300 400 500 600

Frequency(Hz)

Pha

se(D

eg) Measurement

Model

321

432

10

1*1

5222.0525.1952.11147.1604.14213.0

)()()( −−−

−−−

−

−−

−+−−+−

==zzzzzz

zPzCHzW

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 100 200 300 400 500 600Frequency(Hz)

Gai

n

75.1 g/s66.7 g/s58.4 g/s

-300

-250

-200

-150

-100

-50

00 100 200 300 400 500 600

Frequency(Hz)

Pha

se(D

eg)

75.1 g/s66.7 g/s58.4 g/s

Equivalence Ratio

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 100 200 300 400 500 600Frequency(Hz)

Gai

n

473 K423 K373 K

-270

-225

-180

-135

-90

-45

00 100 200 300 400 500 600

Frequency(Hz)

Pha

se(D

eg)

473 K423 K373 K

Preheat Temperature

Examination of the gain and phases of the measured FTFs at different working conditions sheds insight on adaptive robust control design.


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Fast Control of LBOLBO limits can be extended by increasing the amount of pilot fuel. But this approach is too slow, not suitable for transient LBO. In addition, locally hot regions form and exacerbate emissions.

Small-amplitude fuel modulations, based on a feedback controller, are capable of quickly attenuating small deviations from the equilibrium points within a small fraction of a second. Also detection of incipient LBO is not needed. The spatial fuel distribution is not modified, thus favoring low emissions.

Near-LBO combustion dynamics is rather slow, typically below 200 Hz. Thus the requirements of the actuator bandwidth and challenges associated with time delay are no longer major technical challenges.

0.7

0.8

0.9

1.0

1.1

1.2

0 50 100 150 200 250Frequency(Hz)

Am

plitu

de

Measurement Model

-100

-80

-60

-40

-20

00 50 100 150 200 250

Frequency(Hz)

Pha

se(D

eg)

MeasurementModel

2.119e6) 6.6s (s )7.26(s)1262)(9793(2618.37)(

2 +++++

=sssW

-0.05

-0.03

-0.01

0.01

0.03

0.05

0.00 0.02 0.04 0.06 0.08 0.10

Time(s)

Closed-loop output Control signal Open-loop output


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FTFs around Resonant Frequencies

0

1

2

3

4

5

0 150 300 450 600 750 900 1050Frequency(Hz)

Gai

n

-180

-135

-90

-45

0

45

90

135

180

Phas

e(D

eg)

Gain(No Baffle)Gain(Baffle)Phase(No Baffle)Phase(Baffle)

Ф=0.40 , the air flow rate of 44.5 g/s, and Ti=373 K. Stable combustion is achieved by inserting three baffle plates inside the combustion chamber.

The FTFs around the acoustic resonant frequencies are no longer open-loop and linear. In this figure, off the resonant frequencies, i.e. around 340 Hz and 670 Hz, differences in both the gain and phases are rather small. Considerable differences exit around the acoustic resonant frequencies.


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Nonlinear Responses of Combustion InstabilityShown here are the quenching and entrainment of self-excited combustion instability with fuel modulation approaching the unstable frequency.

Unstable combustion occurs at Ф=0.40 , the air flow rate of 44.5 g/s, and Ti=373 K. The fundamental resonant frequency is 672 Hz, corresponding to one-wave mode of the combustion chamber, 1.05-m-long.

0.0

0.4

0.8

1.2

1.6

2.0

650 660 670 680 690 700Frequency (Hz)

Ampl

itude

CH* (A.U.) Pressure (psi)Fuel Pressure (x0.2, psi)

Forcing at 655 Hz

0.0

0.4

0.8

1.2

1.6

2.0

650 660 670 680 690 700Frequency (Hz)

Ampl

itude

CH* (A.U.)

Pressure (psi)

Fuel Pressure (x0.2, psi)

Forcing at 661 Hz

0.0

0.8

1.6

2.4

3.2

4.0

650 660 670 680 690 700Frequency (Hz)

Ampl

itude

CH* (A.U.)

Pressure (psi)

Fuel Pressure (x0.2, psi)

Forcing at 666 Hz

0.0

0.5

1.0

1.5

2.0

2.5

100 400 700 1000 1300 1600Frequency(Hz)

Am

plitu

de

CH* (A.U.)Pressure(psi)

Baseline Spectrum


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Characterization of Combustion Instability


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Characterization of Combustion Instability (Cont.)


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Phase Portrait of Self-Excited Combustion Instability

Probability Density Function of Pressure Amplitude and Period

5. Conclusions and Suggested Future Work


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Conclusions


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• Performed systematic investigations of flame response to fuel modulations up to 1 kHz and to air modulations up to 900 Hz.

• Developed strategies for accurate determination of the instantaneous heat release rate and equivalence ratios, which take into account of the nonlinearity among heat release, chemiluminescence, equivalence ratios, and acoustics-induced chemiluminescence oscillations.

• Proposed that a single adaptive robust controller be used for simultaneously control of both combustion instability and lean blowout.

Suggested Future Work• Development of high-frequency fuel-modulation technologies

• Quantification of flame response within a large range of working conditions

• Implementation of combustion control experiments

Reference


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1. T. Yi and D. A. Santavicca, “Flame Spectra for Turbulent Liquid-Fueled Swirl-Stabilized LDI Combustion,” Journal of Propulsion and Power, Vol.25, No.5, pp.1058-1067, 2009.

2. T. Yi and D. A. Santavicca, “Forced Flame Response of Turbulent Liquid-Fueled Swirl-Stabilized LDI Combustion to Fuel Modulations,” Journal of Propulsion and Power, Vol.25, No.6, pp.1259-1271, 2009.

3. T. Yi and D. A. Santavicca, “Combustion Instability in a Turbulent Liquid-Fueled Swirl-Stabilized LDI Combustion,”under review at Journal of Propulsion and Power (similar to AIAA2009-5014).

4. T. Yi and D. A. Santavicca, “Determination of Instantaneous Fuel Flow Rates out of a Fuel Injector,” ASME Journal of Engineering for Gas Turbines and Power, Vol.132, No.2, 2010.

5. T. Yi and D. A. Santavicca, “Flame Transfer Functions for Turbulent Liquid-Fueled Swirl-Stabilized LDI Combustion,”ASME Journal of Engineering for Gas Turbines and Power, Vol.132, No.2, 2010.

6. T. Yi and E. J. Gutmark, “Stability and Control of Lean Blowout in Chemical-Kinetics-Controlled Combustion Systems,” Combust. Sci. and Technol., Vol.181, No.2, pp.226-244, 2009.

7. T. Yi and E. J. Gutmark, “Adaptive Control of Combustion Instability Based on Dominant Acoustic Modes Reconstruction,” Combust. Sci. and Technol., Vol.180, No.2, pp.249-263, 2008.

8. T. Yi and E. J. Gutmark, “Online Prediction of the Onset of Combustion Instability based on the Computation of Damping Ratios,” Journal of Sound and Vibration, Vol.310, No.1-2, pp.442-447, 2008.

9. T. Yi and E. J. Gutmark, “Real-Time Prediction of Incipient Lean Blowout in Gas Turbine Combustors,” AIAA Journal, Vol.45, No.7, pp.1734-1739, 2008.

10. T. Yi and E. J. Gutmark, “Dynamics of a High Frequency Fuel Actuator and its Applications for Combustion Instability Control,” ASME J. Eng. Gas Turbines Power, Vol.129, pp. 648-654, 2007.

11. D. Wee, T. Yi, A. M. Annaswamy, and A. F. Ghoniem, “Self-Sustained Oscillations and Vortex Shedding in Backward-Facing Step Flows: Simulation and Linear Instability Analysis,” Physics of Fluids, Vol. 16, No. 9, pp. 3361-3373, 2004.

12. T. Yi, D. Wee, A. M. Annaswamy, and A. F. Ghoniem, “Self-Sustained Oscillations in Separating Flows I: Stability Analysis and Reduced-Order Modeling,” Proceedings of the International Symposium on Combustion Noise and Control, pp. 214-220, Cranfield University, 2003.

13. T. Yi, A. M. Annaswamy, and A. F. Ghoniem, “Self-Sustained Oscillations in Separating Flows II: Reduced-Order Modeling and Control,” Proceedings of the International Symposium on Combustion Noise and Control, pp. 221-227, Cranfield University, 2003.

Thanks !


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Adaptive Intelligent Combustion Control Based on Data ...€¦ · Adaptive Intelligent Combustion Control Based on Data-Driven Low-Order Models Tongxun Yi Domenic Santavicca ... (Barbalat’s