Linear parameter varying controller for a small turboshaft ...1310/... · Linear Parameter Varying...

Linear Parameter Varying Controller for a Small

Turboshaft Engine

A Thesis Presented

by

Jeffrey Michael Spack

to

The Department of Electrical & Computer Engineering

in partial fulfillment of the requirementsfor the degree of

Master of Science

in

Electrical Engineering

in the field of

Control Systems

Northeastern UniversityBoston, Massachusetts

September 2011

Linear Parameter Varying Controller for a Small Turboshaft

Engine

by

Jeffrey Michael Spack

Submitted to the Electrical & Computer Engineeringon May 12, 2011, in partial fulfillment of the requirements for the degree of

Master of Science

Abstract

Small turboshaft engine control systems are traditionally single input single output (SISO)and are designed utilizing classical control methods which require extensive gain schedulesresulting in considerable tuning effort. In addition, this method assumes that a system thatis stable at the linearized points will be stable in between the linearized points which may ormay not be true. With a linear parameter varying (LPV) control system, the gain scheduleis inherently built into the controller and the system is guaranteed to be stable betweenlinearizing points as long as the plant depends affinely on a set of time varying parametersthat are available as real time measurements and also vary along a fixed polytope. Anotherbenefit of using an LPV controller in place of a classic SISO controller is that modern robustcontrol techniques can be applied.

A model of a General Electric T700 turboshaft engine and baseline control systemwere developed in Simulink�using a NASA paper written by Mark G. Ballin [1]. Theengine plant was linearized at engine core speed points between 76% and 100% at intervalsof 2%. The LPV power turbine speed controller was developed using linearized plantsderived from the Simulink�model. The LPV controller was then compared to the baselinecontrol system showing performance greater than or equal to the baseline controller with asingle set of weighting functions. The referee transients used to determine a performancecomparison included steps and ramps in power turbine speed reference, helicopter collectiveload disturbances and uncompensated helicopter rotor load disturbances.

This research shows that an LPV controller can be applied to a small turboshaft engineapplication. It is also shown that while the LPV controller is more complex that is shouldtrade well against the traditional SISO methods based on tuning effort and performance.

Thesis Advisor: Professor Mario Sznaier

2

This thesis is dedicated to my wife and family, Constance, Penelope and Eleni.

Without their love, support and sacrifice my graduate coursework and research

would not have been achievable.

3

Acknowledgments

I would like to thank my advisor Professor Mario Sznaier whose guidance provided

me the necessary tools to complete this research.

I would also like to thank Dave Gutz, who provided great insight into making this

research applicable to a real world application, and my other colleagues at GE who

had to work around with my irregular work schedule and frequently fill in for me when

I was unavailable. The support from the GE Lynn Controls Department management

team was invaluable in allowing me to take the necessary time to complete my degree.

My degree was sponsored by the General Electric Company through the Advanced

Courses in Engineering, under the supervision of Tyler Hooper.

4

Contents

1 Introduction 21

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.2 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Engine Control System 25

3 Engine Model 27

3.1 Engine Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Engine Model Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Engine Model Architecture . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Engine Model Validation . . . . . . . . . . . . . . . . . . . . . . . . 40

3.5 Engine Model Linearization . . . . . . . . . . . . . . . . . . . . . . . 42

4 Load Model 48

4.1 Load Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 Load Model Architecture . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Engine Baseline Control System Model 52

5.1 Baseline Controller Assumptions . . . . . . . . . . . . . . . . . . . . . 53

5.2 Baseline Controller Tools . . . . . . . . . . . . . . . . . . . . . . . . 54

5.3 Baseline Controller Architecture . . . . . . . . . . . . . . . . . . . . 54

5.4 Baseline Controller Validation . . . . . . . . . . . . . . . . . . . . . . 61

6 Linear Parameter Varying (LPV) Control Theory 64

5

6.1 Classical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.2 LPV Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7 LPV Controller Design 71

7.1 LPV Controller Assumptions . . . . . . . . . . . . . . . . . . . . . . 71

7.2 LPV Controller Tools . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7.2.1 Changes to LMI toolset to deal with polytopic model . . . . . 72

7.3 LPV Controller Architecture . . . . . . . . . . . . . . . . . . . . . . . 72

7.4 LPV Controller Validation . . . . . . . . . . . . . . . . . . . . . . . . 80

7.4.1 Comparison of baseline controller versus LPV controller . . . . 81

8 Conclusions 90

8.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

A Engine Model Diagrams 92

B Engine Model Constants 97

C Engine Model Schedules 99

D Fuel Control Model Diagrams 118

E Fuel Control Constants 123

F Fuel Control Schedules 124

G Matlab�Code 136

G.1 LMI Toolbox Commands [2] . . . . . . . . . . . . . . . . . . . . . . . 136

G.2 Linearization Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

G.3 Changes to LMI toolset code . . . . . . . . . . . . . . . . . . . . . . . 145

G.4 LPV Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

G.4.1 Modified Plant Setup . . . . . . . . . . . . . . . . . . . . . . . 147

G.4.2 LPV Controller Design . . . . . . . . . . . . . . . . . . . . . . 148

G.4.3 LPV Controller Linear Simulation . . . . . . . . . . . . . . . . 149

6

G.4.4 Write Trajectory File . . . . . . . . . . . . . . . . . . . . . . . 149

G.4.5 Linear Simulation Input Function . . . . . . . . . . . . . . . . 150

G.4.6 LPV Controller s-function . . . . . . . . . . . . . . . . . . . . 150

H Results 152

H.1 Linear Plants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7

List of Figures

1-1 Pilot Controls [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1-2 Typical Helicopter Drive Train [3] . . . . . . . . . . . . . . . . . . . 23

2-1 Turboshaft Engine Schematic [4] . . . . . . . . . . . . . . . . . . . . . 26

3-1 Brayton Cycle [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3-2 Engine Stations [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3-3 Delta and Theta Simulink�Implementation . . . . . . . . . . . . . . . 30

3-4 Enthalpy at Station 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3-5 Static Pressure at Station 3 . . . . . . . . . . . . . . . . . . . . . . . 30

3-6√θ2 and Referenced Gas Generator Speed (NGR) . . . . . . . . . . . 31

3-7 Corrected Compressor Airflow . . . . . . . . . . . . . . . . . . . . . . 31

3-8 Compressor Airflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3-9 Compressor Outlet Temperature . . . . . . . . . . . . . . . . . . . . . 31

3-10 Bleed Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3-11 Fuel Air Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3-12 Combustor Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3-13 Enthalpy Drop Across the Gas Generator . . . . . . . . . . . . . . . . 34

3-14 Enthalpy Drop Across the Power Turbine . . . . . . . . . . . . . . . . 36

3-15 Corrected Power Turbine Mass Flow . . . . . . . . . . . . . . . . . . 36

3-16 Power Turbine Mass Flow . . . . . . . . . . . . . . . . . . . . . . . . 36

3-17 Power Turbine Outlet Pressure . . . . . . . . . . . . . . . . . . . . . 37

3-18 Compressor Outlet Pressure . . . . . . . . . . . . . . . . . . . . . . . 38

3-19 Combustor Outlet Pressure . . . . . . . . . . . . . . . . . . . . . . . 38

8

3-20 Power Turbine Inlet Pressure . . . . . . . . . . . . . . . . . . . . . . 39

3-21 Gas Generator Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3-22 Power Turbine Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3-23 Fuel Flow Step From 400 to 775 pph at 175 ft-lbs . . . . . . . . . . . 41

3-24 Fuel Flow Step From 400 to 125 pph at 175 ft-lbs . . . . . . . . . . . 42

3-25 Linear Model Configuration . . . . . . . . . . . . . . . . . . . . . . . 43

3-26 Model to Find Operating Points . . . . . . . . . . . . . . . . . . . . . 44

3-27 Compare Eigenvalues to Source Paper . . . . . . . . . . . . . . . . . 46

4-1 Load Shaft Horsepower . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4-2 Load Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5-1 Fuel Flow Governor Selection . . . . . . . . . . . . . . . . . . . . . . 54

5-2 NG Idle Fuel Flow Governor . . . . . . . . . . . . . . . . . . . . . . . 55

5-3 Percent Gas Generator Speed Lagged . . . . . . . . . . . . . . . . . . 55

5-4 NG Acceleration Topping Reference . . . . . . . . . . . . . . . . . . . 55

5-5 NG Deceleration Bottoming Reference . . . . . . . . . . . . . . . . . 56

5-6 Commanded Wf/Ps3 Signal Before Limiting Selector . . . . . . . . . 56

5-7 Fuel Flow Topping Reference . . . . . . . . . . . . . . . . . . . . . . 56

5-8 NG Topping Reference . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5-9 Collective Based Feedforward . . . . . . . . . . . . . . . . . . . . . . 57

5-10 Collective Based Feedforward Lagged . . . . . . . . . . . . . . . . . . 57

5-11 NP Governor Error Signal . . . . . . . . . . . . . . . . . . . . . . . . 58

5-12 NP Governor Error Signal Compensation . . . . . . . . . . . . . . . . 58

5-13 NP Governor Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 58

5-14 NP Governor Selection Logic . . . . . . . . . . . . . . . . . . . . . . . 59

5-15 NP Trim Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5-16 NP Trim Integrator Low Limit . . . . . . . . . . . . . . . . . . . . . . 60

5-17 NP Trim Integrator Low Limit Selection Logic . . . . . . . . . . . . . 60

5-18 Torque Motor Compensation . . . . . . . . . . . . . . . . . . . . . . . 60

5-19 Fuel Flow to the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 60

9

5-20 Baseline Controller Validation - Collective Pull . . . . . . . . . . . . . 62

5-21 Baseline Controller Validation - Cyclic Disturbance . . . . . . . . . . 63

6-1 Classic Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7-1 Modified Linear Plant Structure . . . . . . . . . . . . . . . . . . . . . 73

7-2 Modified Linear Plant Structure with Weighting Functions . . . . . . 76

7-3 100 RPM Step in NP reference . . . . . . . . . . . . . . . . . . . . . 77

7-4 40 ft-lb Step Load Disturbance . . . . . . . . . . . . . . . . . . . . . 78

7-5 Simulink�Changes to Figure 5-6 To Incorporate LPV Controller . . . 79

7-6 Bode Plot of the LPV Controller Weighting Functions Wc and We . . 82

7-7 100 RPM NP Step Response with Collective at 25 . . . . . . . . . . . 83

7-8 100 RPM NP Step Response with Collective at 77.2 . . . . . . . . . . 83

7-9 NP Reference Ramp at 50 RPM/sec with Collective at 25 . . . . . . 84

7-10 NP Reference Ramp at 50 RPM/sec with Collective at 77.2 . . . . . . 84

7-11 Rotor Load Disturbance of 40 ft-lbs with Collective at 25% . . . . . . 85

7-12 Rotor Load Disturbance of 40 ft-lbs with Collective at 77.2% . . . . . 86

7-13 Collective Instantaneous Pull from 25% to 35% . . . . . . . . . . . . 87

7-14 Collective Instantaneous Pull from 65.64% to 82.5% . . . . . . . . . . 87

7-15 Collective 1 Second Pull from 10% to 90% . . . . . . . . . . . . . . . 88



A-1 Simulink�Model Top Figure . . . . . . . . . . . . . . . . . . . . . . . 93

A-2 Engine Model Figure 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 94



C-1 F1 - Compressor Mass Flow . . . . . . . . . . . . . . . . . . . . . . . 100

C-2 F2 - Compressor Temperature . . . . . . . . . . . . . . . . . . . . . . 103

C-3 F3 - Seal Pressure Bleed Fraction . . . . . . . . . . . . . . . . . . . . 105

C-4 F4 - Power Turbine Balance Bleed Fraction . . . . . . . . . . . . . . . 108

10

C-5 F5 - Impeller Tip Leakage and Turbine Cooling Bleed Fraction . . . . 109

C-6 F6 - Combustor Efficiency . . . . . . . . . . . . . . . . . . . . . . . . 110

C-7 F7 - Gas Generator Turbine Energy . . . . . . . . . . . . . . . . . . . 111

C-8 F8 - Power Turbine Energy . . . . . . . . . . . . . . . . . . . . . . . 112

C-9 F9 - Power Turbine Mass Flow . . . . . . . . . . . . . . . . . . . . . 113

C-10 F10 - Exhaust Pressure Loss . . . . . . . . . . . . . . . . . . . . . . . 115

D-1 Engine Control Figure 1 . . . . . . . . . . . . . . . . . . . . . . . . . 119




F-1 Fec1 - ECU Thermocouple Sensor Time Constant . . . . . . . . . . . 125

F-2 Fhm1 - HMU Topping Line Schedule . . . . . . . . . . . . . . . . . . 127

F-3 Fhm2 - HMU Power Available Input Schedule . . . . . . . . . . . . . 128

F-4 Fhm3 - HMU Load Demand Compensation Schedule . . . . . . . . . 129

F-5 Fhm4 - HMU Load Demand Compensation Schedule Function 2 . . . 130

F-6 Fhm5 - HMU Idle Schedule Function 1 . . . . . . . . . . . . . . . . . 131

F-7 Fhm6 - HMU Idle Schedule Function 2 . . . . . . . . . . . . . . . . . 132

F-8 Fhm7 - HMU Maximum Fuel Parameter Limit During Acceleration . 133

11

List of Tables

1 Model Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1 Steady State Engine Model Results . . . . . . . . . . . . . . . . . . . 40

3.2 State Operating Point . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1 Rotor Inertia Values . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Collective to Shaft Horsepower Relationship . . . . . . . . . . . . . . 50

B.1 Engine Model Constants . . . . . . . . . . . . . . . . . . . . . . . . . 97

C.1 F1 - Compressor Mass Flow . . . . . . . . . . . . . . . . . . . . . . . 100

C.2 F2 - Compressor Temperature . . . . . . . . . . . . . . . . . . . . . . 103

C.3 F3 - Seal Pressure Bleed Fraction . . . . . . . . . . . . . . . . . . . . 105

C.4 F4 - Power Turbine Balance Bleed Fraction . . . . . . . . . . . . . . . 108

C.5 F5 - Impeller Tip Leakage and Turbine Cooling Bleed Fraction . . . . 109

C.6 F6 - Combustor Efficiency . . . . . . . . . . . . . . . . . . . . . . . . 110

C.7 F7 - Gas Generator Turbine Energy . . . . . . . . . . . . . . . . . . . 111

C.8 F8 - Power Turbine Energy . . . . . . . . . . . . . . . . . . . . . . . 112

C.9 F9 - Power Turbine Mass Flow . . . . . . . . . . . . . . . . . . . . . 113

C.10 F10 - Exhaust Pressure Loss . . . . . . . . . . . . . . . . . . . . . . . 115

F.1 Fec1 - ECU thermocouple sensor time constant . . . . . . . . . . . . 125

F.2 Fhm1 - HMU Topping Line Schedule . . . . . . . . . . . . . . . . . . 127

F.3 Fhm2 - HMU Power Available Input Schedule . . . . . . . . . . . . . 128

F.4 Fhm3 - HMU Load Demand Compensation Schedule . . . . . . . . . 129

12

F.5 Fhm4 - HMU Load Demand Compensation Schedule Function 2 . . . 130

F.6 Fhm5 - HMU Idle Schedule Function 1 . . . . . . . . . . . . . . . . . 131

F.7 Fhm6 - HMU Idle Schedule Function 2 . . . . . . . . . . . . . . . . . 132

F.8 Fhm7 - HMU Maximum Fuel Parameter Limit During Acceleration . 133

H.1 NG076 Linear Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

H.2 NG078 Linear Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

H.3 NG080 Linear Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

H.4 NG082 Linear Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

H.5 NG084 Linear Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

H.6 NG086 Linear Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

H.7 NG088 Linear Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

H.8 NG090 Linear Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

H.9 NG092 Linear Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

H.10 NG094 Linear Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

H.11 NG096 Linear Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

H.12 NG098 Linear Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

H.13 NG100 Linear Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

13

Nomenclature

Abbreviations

GE General Electric Company

FMU Fuel Metering Unit

HMU Hydro-Mechanical Unit

LTI Linear Time Invariant

LMI Linear Matrix Inequalities

LPV Linear Parameter Varying

LTI Linear Time Invariant

PID Proportional-Integral-Derivative Controller

PPS Pounds Per Second

PPH Pounds Per Hour

FADEC Full Authority Digital Electronic Controller

SISO Single Input-Single Output

14

Model Variables

Table 1: Model Variables

Variable Description Units

AWFP HMU deceleration schedule gain lbm∗in2

lbf∗hr∗%

B4 ECU nonlinear NP loop gain circuit (discrete switch)

B6 ECU speed error gain

BWFP HMU deceleration schedule bias lbm∗in2

lbf∗hr

CB ECU error threshold for additional NP governing gain % NP

CE ECU load share authority upper limit % NP

CH Hysteresis of gas-generator speed sensor % NG

CLMV metering valve lag time constant sec

CLLDS load demand spind1e lag time constant sec

CNTL lag time constant for gas generator speed sensor sec

CORR threshold for nonlinear NP loop gain circuit

CR engine torque level threshold for nonlinear NP loop gain circuit

CT2 lag time constant for ECU governor sec

CT7 lag time constant for ECU load-share circuit sec

CT9 ECU T4.5 compensation lag time constant sec

CT12 lag time constant for ECU governor sec

CT13 lag time constant for ECU governor rate compensation sec

CT14 lag time constant for ECU governor rate compensation sec

CT16 lag time constant for ECU proportional plus integral compensation sec

continued on next page

15

continued from previous page


CTPL lag time constant for power turbine speed sensor sec

CTPS3 lag time constant for compressor static discharge pressure sensor sec

DBIAS lower limit and bias for ECU load share authority % NP

DWFP fuel flow command for HMU load demand spindle lbm∗in2

lbf∗hr

DWFPL fuel flow command for HMU load-demand spindle including dynamics lbm∗in2

lbf∗hr

ET45 T4.5 error signal deg R

HMUSEL Wf/Ps3 demand after limiting by idle, acceleration, and deceleration

cams

lbm∗in2

lbf∗hr

KNDRP droop line slope delta(Wf/Ps3)/deltaPCNG lbm∗in2

lbf∗hr∗%

NG rotational speed of compressor and gas generator rpm

NGREF NG reference speed in percent of design speed % NG

P45 total pressure at power turbine inlet lbfin2

PAS angle of power available spindle deg

PCNG rotational speed of compressor and gas generator in percent of design

speed

% NG

PCNGI set point value of gas generator speed determined by idle schedule % NG

PCNGHL gas generator speed including sensor dynamics and hysteresis % NG

PCNP rotational speed of power turbine in percent of design speed % NP

PCPRF reference rotational speed of power turbine as set by cockpit control % NP

PNG set point value of gas generator speed determined by load demand-

compensation circuit

% NG

PS3 compressor static discharge pressure lbfin2

PS3HYS compressor discharge pressure sensor hysteresis lbfin2

PS3L compressor discharge pressure including sensor dynamics and hysteresis lbfin2


16



SPDER ECU speed error signal % NP

SPDG ECU trim demand signal volts

SPDSI ECU compensated speed error signal volts

SPDSF ECU speed error signal with governor dynamics; input to ECU error

selector logic

volts

SPDSP ECU compensated trim demand signal volts

SPDSS ECU trim-demand signal before compensation volts

T2 inlet temperature deg R

T8 lead time constant for ECU T4.5 limiter compensation sec

T10 lag time constant for ECU T4.5 limiter compensation sec

T11 lead time constant for ECU governor sec

T17 lag time constant for ECU load share sec

T45 power turbine inlet temperature T4.5 deg R

T45COR ECU T4.5 thermocouple harness correlation bias deg R

T45E measured T4.5 deg R

T45EL T4.5 sensed by ECU thermocouple harness deg R

T45L measured T4.5 with hamess dynamics deg R

T45REF reference constant used as maximum T45 limit deg R

TAU45 T4.5 harness time-constant which varies based on station 4.5 flow pa-

rameter and T4.5

sec

TL1 lag-time constant 1 for ECU load-share torque sensor sec

TL2 lag-time constant 2 for ECU load share torque sensor sec

TLGE T4.5 harness time constant sec


17



TMDB HMU torque motor deadband ma

TMGN HMU torque motor sensitivity inma∗sec

TMLG HMU torque motor linkage gain lbm∗in2

lbf∗hr.in

TMLVG HMU torque motor LVDT feedback gain voltsin

TMRU Wf/Ps3 trim signal from torque motor based on ECU trim signal lbm∗in2

lbf∗hr

TORQ45 power turbine torque (identical to Q PT) ft-lbf

TRQER sensed torque error between two engines operating in parallel ft-lbf

TRQL sensed power turbine output torque ft-lbf

TSIG compensated T4.5 error signal; input to ECU error selector logic volts

W45 power turbine inlet mass flow rate lbmhr

W45R power turbine flow parameter

WF fuel flow lbmsec

WFIDM minimum value of Wf/Ps3 demand regardless of all other commands lbm∗in2

lbf∗hr

WFIRF set point value of Wf/Ps3 determined by idle schedule lbm∗in2

lbf∗hr

WFMAX maximum available fuel How; represents maximum aperture in metering

valve

lbmhr

WFMIN minimum available fuel flow; represents minimum flow stop in metering

valve

lbmhr

WFMV fuel How required by HMU; results from multiplication of Wf/Ps3and

Ps3 via mechanical linkage

lbmhr

WFPAC maximum value of Wf/Ps3 demand during acceleration transients; pre-

vents compressor stall

lbm∗in2

lbf∗hr

WFPDC minimum value of Wf/Ps3 demand during deceleration transients; pre-

vents engine flameout

lbm∗in2

lbf∗hr


18



WFPDCH maximum limit on Wf/Ps3 demand during deceleration lbm∗in2

lbf∗hr

WFPDCL minimum limit on Wf/Ps3 demand during deceleration lbm∗in2

lbf∗hr

WFPDM commanded Wf/Ps3 signal before limiting selector logic lbm∗in2

lbf∗hr

WFPRF maximum available Wf/Ps3 signal; set by power available spindle lbm∗in2

lbf∗hr

WFPTP Wf/Ps3 topping signal lbm∗in2

lbf∗hr

WFQPS3 set point value of Wf/Ps3 determined by load demand compensation

circuit

lbm∗in2

lbf∗hr

XCPC helicopter collective pitch position in percent of maximum %

XHILIM torque motor maximum limit insec

XKINTG ECU proportional plus integral compensation integral path gain

XKPROP ECU proportional plus integral compensation proportional path gain

XLDHYS load demand spindle hysteresis deg

XLDSA load demand spindle angle deg

XLDSH load demand spindle angle with hysteresis deg

XLOLIM torque motor minimum limit insec

XQLO load share error input % NP

YHILIM engine torque integrator maximum limit for nonlinear NP loop gain cir-

cuit

YLOLIM engine torque integrator minimum limit for nonlinear loop gain circuit

ZHILIM ECU proportional plus integral compensation integrator maximum limit

ZK1 NP loop additional proportional gain for ECU governor rate compensa-

tion used during high power operation

ZK3 ECU T4.5 compensationcircuit gain

ZK5 ECU governor rate compensation loop gain


19



ZK7 NP loop proportional gain for ECU governor rate compensation

ZK8 ECU load share path gain

ZK9 speed error trim gain for ECU governor rate compensation

ZK10 additional gain loop for ECU governor rate compensation for errors

greater than the absolute value of CB

ZLOLIM ECU proportional plus integral compensation integrator minimum limit

Greek Symbols

δ Ratio of the pressure over the standard pressure

θ Ratio of the temperature over the standard temperature

ρ Time varying parameter set

20

Chapter 1

Introduction

Classical linear single input-single output (SISO) controllers commonly used in helicopter

engine controls typically require extensive tuning and require additional margin between

linearized points for a reasonable assurance of stability. A linear parameter varying (LPV)

controller design provides gained scheduled controller that is stable in between plant lin-

earizing points as long as the plant satisfies a few assumptions:

1. the model depends affinely on a set of time varying parameters

2. the time varying parameters are available as real time measurements

3. the time-varying parameters range within a fixed polytope

Another benefit to an LPV controller is that the extensive tuning required for SISO

controls is significantly reduced. This research produced a controller that performed as well

or better than the baseline controller with a single set of weighting functions.

The downside to a LPV controller is that the complexity, when compared against the

baseline SISO controller[1], is increased. However, the reasonable assurance of stability

between linearized points and decreased tuning effort should trade well against the effort

and cost required to implement a more complex controller in a production environment.

1.1 Background

Small turboshaft engines are typically used for rotor craft, turbo propeller fixed wing aircraft

and large marine applications. In this research, the focus is on a model of a GE T700 turbo

21

shaft engine which is primarily used for rotor craft applications namely the Sikorsky UH-

60A Blackhawk. The Blackhawk is configured as a two engine system where both engines

feed a main gear box which drives the main and tail rotors. The main rotor of a helicopter

provides lift and directional control. The tail rotor counteracts the force applied by the

main rotor that pushes the helicopter body in the opposite direction of the main rotor. The

tail rotor also acts as a rudder to steer the helicopter left and right [3].

A helicopter pilot has three major controls [3]:

1. Collective: controls lift

2. Cyclic: controls left, right, forward and backward motions

3. Foot pedals: controls yaw rate in forward flight and left and right motion during

hover maneuvers

Figure 1-1 provides the general impact of the basic pilot controls on the helicopter.

Figure 1-1: Pilot Controls [3]

Typically knowledge of the collective lever is presented to the engine control system since

it represents the major portion of the load system. The other pilot controls, cyclic and pedal

movement, are typically unknown to the engine control system. Therefore, maneuvers with

22

significant cyclic and/or pedal movement without significant collective movement rely solely

on the power turbine speed governor while maneuvers with significant collective movement

are typically assisted by open loop anticipation logic.

Rotor craft in general present a complex controls problem for the engine control system

designer due to the integration with the helicopter. Since the engines are directly connected

to the main and tail rotors through the transmission the dynamics of the rotor system must

be included in the governor design. One of the most important requirements for the engine

control system is to keep the main rotor speed close to the datum under varying load con-

ditions while respecting several boundary conditions. Deviations below the datum is called

rotor “droop” and deviations above the datum are called rotor “upspeed” or “overshoot”.

Both are not desirable, too much rotor droop can lead to a unrecoverable loss of lift and

too much overshoot could lead to damage to the main gear box both of which can result in

loss of the aircraft.

The major boundary conditions that the engine control system must respect are: engine

acceleration limits, engine deceleration limits, maximum gas generator speed, minimum gas

generator speed, gas generator turbine temperature limits and engine fuel flow limits. In

addition, the engine control system must have good handling qualities at the aircraft level

such as avoiding torque reversals during engine acceleration.

Figure 1-2 shows a typical helicopter drive train.

Figure 1-2: Typical Helicopter Drive Train [3]

23

1.2 Scope

The purpose of this research is to replace the baseline controller [1] with an LPV controller

while respecting the boundary conditions stated in section 1.1. The engine model is based

from the paper written by Mark G. Ballin of the NASA Ames Research Center [1]. Since,

a rotor model is not specifically presented in the paper by Ballin the rotor model inertia is

taken from the thesis of William Pfiel [6]. This research does not require a perfect match

of engine performance as compared to the actual T700 engine as the control system can be

applied to any plant satisfying the assumptions stated in section 1.1. This research does not

attempt to re-produce the collective anticipation logic and instead assumes that the load is

known to the control system.

24

Chapter 2

Engine Control System

The modeled engine is General Electric T700-GE-700 [1]. Two T700 engines are coupled

to the main gear box of a Sikorsky model UH-60A Black Hawk helicopter. Each engine is

capable of producing approximately 1800 shaft horsepower. The engine compressor consists

of 5 axial compressor stages and 1 centrifugal compressor stage. The compressor utilizes

variable geometry on the engine stators for improved stall protection at off design points.

Air flows from the exit of the centrifugal compressor through an annular combustor where

fuel is introduced and ignited. The hot air flows through the gas generator turbine which

extracts a portion of the energy to drive the compressor. The air exiting the gas generator

flows through the power turbine module which extracts the remaining energy driving the

shaft connected to the helicopter main gear box. The gas generator or core of the engine

spins independently from the power turbine module allowing the helicopter gear box speed

to be modulated to the same reference under varying load conditions.

25

Figure 2-1: Turboshaft Engine Schematic [4]

26

Chapter 3

Engine Model

The T700 Engine model used as the plant for this research is based directly from the work

performed by Mark G. Ballin of the NASA Ames Research Center. The paper “A High

Fidelity Simulation of a Small Turboshaft Engine” [1] provides the equations to produce

a high fidelity non-linear simulation of the GE T700 engine. The equations are provided

in this thesis for reference and convenience however, this research is concentrated on the

development of an LPV controller provided in chapters 6 and 7. This engine is representative

of the current technology for gas turbine engines in this shaft horsepower class. Gas turbine

engines are described by the Brayton Cycle which is a thermodynamic process with four

major stages: compression, combustion or heat addition, expansion, and exhaust [6] [7]

[8]. Heat energy remaining in the gas generator turbine exhaust is expanded in the power

turbine to extract power to drive the helicopter rotor system.

Figure 3-1: Brayton Cycle [5]

27

3.1 Engine Model Assumptions

The engine model assumes the following for the purposes of simplification:

1. The inlet losses are approximately equal to the static ambient conditions.

2. The engine heat sink is not included.

3. The variable geometry and bleed valves are assumed to be on schedule and are not

included in the control system.

4. Other engine model assumptions are provided in Ballin’s paper [1].

3.2 Engine Model Tools

The engine model is implemented in native Matlab�Simulink�to take advantage of the

Matlab�toolset including the LMI toolbox [2] which contains the tools used to create the

LPV controller in chapter 7.

3.3 Engine Model Architecture

The equations to build the engine model are given in fragments pictorially as they appear

in the Simulink�model or in equation form depending on which method provided the best

clarity in communicating the implementation. The complete set of Simulink�diagrams is

given in Appendix A. The variables in the equations refer to engine stations associated with

different sections of the engine gas path. The numbers start with zero at the beginning of

the engine and end with nine at the exhaust of the engine. The engine stations are shown

in Figure 3-2.

28

Figure 3-2: Engine Stations [1]

In accordance with assumption 1, the pressures and temperatures at stations 1, 2 and

9 are approximately equal to the static ambient conditions.

P1 = P2 = Ps9 = P ambient (3.1)

T1 = T2 = T ambient (3.2)

The model normalizes temperature and pressure by creating delta and theta parameters

for stations 1 and 2.

δ =P

Pstd(3.3)

(where Pstd = 14.7 psia)

29

θ =T

Tstd(3.4)

(where Tstd = 518.67 R)

Figure 3-3 shows the implementation in the Simulink�model.

Figure 3-3: Delta and Theta Simulink�Implementation

Figure 3-4 shows the enthalpy at station 2. The constant Kh2 is defined in Table B.1.

Figure 3-4: Enthalpy at Station 2

Static pressure at station 3 (Figure 3-5) is a function of the total pressure. The constant

Kps3 is defined in Table B.1.

Figure 3-5: Static Pressure at Station 3

Figure 3-6 shows the calculation of referenced or corrected gas generator speed (core

speed).

30

Figure 3-6:√θ2 and Referenced Gas Generator Speed (NGR)

The corrected compressor airflow is a function of the pressures at stations 3 and 2 in

addition to the corrected core speed (Figure 3-7).

Figure 3-7: Corrected Compressor Airflow

The 2-D table utilized in Figure 3-7 was modified from its original form in order to ensure

that the linear interpolation would be performed properly. The plot and table definition

for the 2-D linear interpolation in Figure 3-7 is Figure C-1 and Table C.1 respectively.

Figure 3-8 implements the calculation for the uncorrected compressor airflow as a func-

tion of δ2 and θ2.

Figure 3-8: Compressor Airflow

The temperature at the compressor outlet (T3) is a function of the compressor pressure

ratio and T2 as shown in Figure 3-9. The plot and table definition for the 1-D linear

interpolation in Figure 3-9 is Figure C-2 and Table C.2 respectively.

Figure 3-9: Compressor Outlet Temperature

31

The enthalpy at station 3 is defined by equation 3.5.

H3 = KH31 ∗ T3 +KH32 (3.5)

(where KH31 and KH32 are defined in Table B.1)

There are three bleed fractions that are used to calculate the bleed for stations 2.4 and 3.

The bleed fractions B1, B2 and B3 are functions of corrected core speed and the corrected

airflow at station 2 as shown in Figure 3-10. The plot and table definition for the 1-D tables

in Figure 3-10 are Figure C-3 and Table C.3 for B1, Figure C-4 and Table C.4 for B2, and

Figure C-5 and Table C.5 for B3.

Figure 3-10: Bleed Fractions

The bleed flow extracted from station 2.4 is given by equation 3.6.

WA24 = WA2 ∗ (B1 +B2) (3.6)

The bleed flow extracted from station 3 is given by equation 3.7.

WA3bl = WA2 ∗ (B3 +Kb3) (3.7)

(where Kb3 is defined in Table B.1)

The flow at station 3 is a function of the station 2 flow minus the extracted bleed flow

from station 2 given by equation 3.8.

WA3 = WA2 −WA24 (3.8)

32

The flow at station 3.1, combustor inlet, is a function of combustor inlet pressure and

temperature as well as the combustor outlet pressure given by equation 3.9.

WA31 =

√P3 ∗ (P3 − P41)

Kdbp ∗ T3(3.9)

(where Kdbp is defined in Table B.1)

The fuel air ratio is the fuel flow over the airflow at the combustor inlet as shown by

Figure 3-11. A low limit was added on WA31 to avoid divide by zero issues.

Figure 3-11: Fuel Air Ratio

The combustor efficiency is a function of the fuel air ratio as shown by Figure 3-12. The

plot and table definition for the 1-D linear interpolation in Figure 3-12 is Figure C-6 and

Table C.6 respectively.

Figure 3-12: Combustor Efficiency

The no heat sink enthalpy at the combustor outlet, station 4.1, is a function of the

enthalpy at station 3, the efficient of the combustor and the fuel air ratio given by equa-

tion 3.10.

H41ns =H3 + (P3 ∗ FAR ∗HV F )

1 + FAR(3.10)

(where HVF is defined in Table B.1)

The temperature at the combustor outlet, station 4.1, is a function of enthalpy at that

station given by equation 3.11.

33

T41 = Kt411 ∗H41ns +Kt412 (3.11)

(where Kt411 and Kt412 is defined in Table B.1)

Note: a decision was made at this point not to implement the heat sink model described

in the NASA paper used to create the model[1]. Therefore, T41 was set equal to T41ns

where ns stands for “no sink”.

The heat sink enthalpy at station 4.1 is a function of T41 as given by equation 3.12.

H41 = Kh411 ∗ T41 +Kh412 (3.12)

(where Kh411 and Kh412 is defined in Table B.1)

The corrected temperature at station 4.1 is given by equation 3.13.

θ41 = Kth411 ∗ T41 +Kth412 (3.13)

(where Kth411 and Kth412 is defined in Table B.1)

The enthalpy drop across the gas generator is a function of pressure at station 4.5 and

4.1 as well as the corrected temperature at the combustor outlet, station 4.1, as shown in

Figure 3-13. The plot and table definition for the 1-D linear interpolation in Figure 3-13 is

Figure C-7 and Table C.7 respectively.

Figure 3-13: Enthalpy Drop Across the Gas Generator

The enthalpy at station 4.4 is the enthalpy at station 4.1 minus the enthalpy drop across

the gas generator, reference equation 3.14.

H44 = H41 −∆HGT (3.14)

34

The enthalpy at station 4.5 is a function of the enthalpy at station 4.5 as given be

equation 3.15.

H45 = H44 ∗Kh45 (3.15)

(where Kh45 is defined in Table B.1)

The airflow at the combustor outlet, station 4.1, is a function of the combustor outlet

pressure and corrected temperature as given by equation 3.16.

W41 =Kwgt ∗ P41√

θ41(3.16)

(where Kwgt is defined in Table B.1)

The temperature at the power turbine inlet, station 4.5, is a function of enthalpy at

station 4.5 as given by equation 3.17. A low limit was added on T41 to avoid divide by zero

issues.

T45 = Kt451 ∗H45 +Kt452 (3.17)

(where Kt451 and Kt452 is defined in Table B.1)

The corrected power turbine inlet temperature is a function of T45 as given by equa-

tion 3.18.

θ45 = Kth451 ∗ T45 +Kth452 (3.18)

(where Kth451 and Kth452 is defined in Table B.1)

The enthalpy drop across the power turbine is a function of pressure at station 4.9 and

4.5 as well as the corrected temperature at the power turbine inlet, station 4.5, as shown in

Figure 3-14. The plot and table definition for the 1-D linear interpolation in Figure 3-14 is

Figure C-8 and Table C.8 respectively.

35

Figure 3-14: Enthalpy Drop Across the Power Turbine

The corrected power turbine flow is a function of the pressure ratio between station 9 and

4.5 as shown in Figure 3-15. The plot and table definition for the 1-D linear interpolation

in Figure 3-15 is Figure C-9 and Table C.9 respectively.

Figure 3-15: Corrected Power Turbine Mass Flow

The uncorrected power turbine mass flow is a function of the corrected power turbine

mass flow, the power turbine inlet pressure and the corrected power turbine inlet tempera-

ture as shown in Figure 3-16.

Figure 3-16: Power Turbine Mass Flow

The enthalpy at the power turbine outlet is enthalpy at the power turbine inlet minus

the enthalpy drop across the power turbine, reference equation 3.19.

H49 = H45 −∆HPT (3.19)

The power turbine outlet pressure is a function of the corrected core speed and the

static pressure at station 9 as shown in Figure 3-17. The plot and table definition for the

1-D linear interpolation in Figure 3-17 is Figure C-10 and Table C.10 respectively.

36

Figure 3-17: Power Turbine Outlet Pressure

The compressor torque is a function of core speed, airflow at station 2 and 3, and

enthalpy at stations 2 and 3 as given by equation 3.20.

Qc = 778.12 ∗(

60

2π

)(1

NG

)(WA2 ∗ (KQC1 ∗H3 −H2) +WA3 ∗KQC2 ∗H3) (3.20)

(where KQC1 and KQC2 is defined in Table B.1)

The gas generator torque is a function of core speed, the flow at station 4.1 and enthalpy

drop across the gas generator as given by equation 3.21.

Qgt = 778.12 ∗(

60

2π

)(1

NG

)(W41 ∗∆HGT ) (3.21)

The power turbine torque is a function of power turbine speed, the power turbine flow

and enthalpy drop across the power turbine as given by equation 3.22.

Qpt = 778.12 ∗(

60

2π

)(1

NP

)(W45 ∗∆HPT −Kdamp ∗

(2π

60

)∗ (NP −NPdes)

)(3.22)

(where Kdamp is defined in Table B.1)

There are five states in the plant: the pressure at the compressor outlet (P3), the

pressure at the combustor outlet (P41), the pressure at the power turbine inlet (P45), core

speed (NG) and power turbine speed (NP).

The differential equation for the compressor outlet pressure(P3

)is a function of the

compressor outlet temperature and flow as given by equation 3.23.

P3 = Kv3 ∗ T3 ∗ (WA3 −WA3bl −WA31) (3.23)

(where Kv3 is defined in Table B.1)

37

Equation 3.23 is integrated in Simulink�as shown in Figure 3-18. A low limit is added

on P3 to avoid divide by zero issues.

Figure 3-18: Compressor Outlet Pressure

The differential equation for the combustor outlet pressure(

˙P41


combustor outlet temperature and flow as given by equation 3.24.

˙P41 = Kv41 ∗ T41 ∗ (WA31 −WF −W41) (3.24)

(where Kv41 is defined in Table B.1)



Figure 3-19: Combustor Outlet Pressure

The differential equation for the combustor outlet pressure(

˙P45


power turbine inlet temperature and power turbine flow including the cooling flow from the

compressor as given by equation 3.25.

˙P45 = Kv45 ∗ T45 ∗ (WA41 −W45 +B3 ∗Kbl ∗WA2) (3.25)

(where Kv45 and Kbl is defined in Table B.1)



38

Figure 3-20: Power Turbine Inlet Pressure

The differential equation for the core speed(

˙NG)

is a function of the compressor torque,

the gas generator torque and the combined inertia as given by equation 3.26.

˙NG =QGT −QC

JGT(3.26)

(where JGT is defined in Table B.1)


on NG to avoid divide by zero issues.

Figure 3-21: Gas Generator Speed

The differential equation for the core speed(

˙NP)

is a function of the power turbine

torque, the requested load and the combined inertia of the power turbine and the load as

given by equation 3.27.

˙NP =QPT −Qreq

JPT + Jload(3.27)

(where JPT is defined in Table B.1 and Jload is defined by equation 4.1 )


on NP to avoid divide by zero issues. This figure also includes the calculation for NP in

percent where NPdes is defined in Table B.1.

39

Figure 3-22: Power Turbine Speed

3.4 Engine Model Validation

The engine model is validated utilizing steady state points and transient cases provided

in reference [1]. In order to validate the model, the engine model with the rotor system

described in chapter 4 is connected but the fuel control system described in chapter 5 is

removed. Table 2 of reference [1] provides the steady state conditions including the load

demand and fuel flow. Table 3 of reference [1] provides the engine model results. The

Table 3.1 below provides the comparison between the data provided in reference [1] and

the model constructed for this research where “Reference” indicates it is from the reference

paper while “Simulink�” designates results from the model described in this thesis.

Table 3.1: Steady State Engine Model ResultsSteady State Conditions Model Results

Test Case Wf P2 T2 P49 Load Torque NG NP WA2 P3 T3 T45

Reference 1 140.1 14.37 516.7 14.4 30.1 70.0 83.8 3.6 63.4 855.9 1372.0

Simulink�1 140.1 14.37 516.7 15.9 29.2 72.6 77.1 3.8 64.3 893.7 1318.4

Reference 2 297.2 14.17 515.6 14.4 90.1 86.4 100.6 5.5 119.0 1043.0 1555.0

Simulink�2 297.2 14.17 515.6 14.6 87.4 86.7 106.4 6.1 123.8 1069.0 1460.3

Reference 3 372 14.16 508.3 14.5 148.3 88.1 97.8 6.3 141.7 1084.0 1629.0

Simulink�3 372 14.16 508.3 14.5 145.8 89.8 104.8 7.0 148.6 1107.5 1513.0

Reference 4 458.4 14.09 508 14.6 206.5 90.7 97.3 7.0 163.3 1130.0 1734.0

Simulink�4 458.4 14.09 508 14.5 202.1 92.5 105.6 7.8 170.9 1153.1 1609.2

Reference 5 560.6 14.02 507.2 14.6 274.3 92.8 98.2 7.7 186.4 1178.0 1855.0

Simulink�5 560.6 14.02 507.2 14.5 267.5 94.9 107.0 8.7 195.4 1202.1 1713.9

Reference 6 694.4 13.92 507.2 14.7 360.8 96.1 99.4 8.5 213.8 1233.0 2009.0

Simulink�6 694.4 13.92 507.2 14.7 349.4 97.5 106.6 9.4 220.9 1262.0 1874.6

The steady state results provided in Table 3.1 show relatively good agreement to the

results provided in reference [1] except for the NP parameter. The discrepancy between the

NP values can be explained based on the fact that the load model in the Simulink�model

is different than the one used in the reference paper.

Two transient case are analyzed: one, a step in fuel from 400 to 775 pph at an initial load

of 175 ft-lbs; two, a negative step in fuel from 400 to 125 pph at the same load conditions

40

as the first case. In both cases, the NP speed is maintained at 20900 RPM or 100% to

avoid a change horsepower due to the rotor speed. Figure 3-23 shows the transient case one

compared to the results provided in Figure 9 in reference [1].

−2 0 2 4 6400

500

600

700

800

sec

Wf (

pph)

−2 0 2 4 6150

200

250

sec

PS

3 (p

sia)

SimulinkPaper

−2 0 2 4 62000

2500

3000

3500

4000

sec

T41

(de

g R

)

SimulinkPaper

−2 0 2 4 61500

2000

2500

3000

sec

T45

(de

g R

)

SimulinkPaper

−2 0 2 4 690

95

100

sec

PC

NG

(%

)

SimulinkPaper

−2 0 2 4 6100

200

300

400

500

sec

TO

RQ

45 (

ft−lb

s)

SimulinkPaper

Figure 3-23: Fuel Flow Step From 400 to 775 pph at 175 ft-lbs

Figure 3-23 shows good agreement between the reference paper [1] results and the model

produced for this research. The critical parameters being core speed and torque since both

of these parameters will be outputs of the linear model.

Figure 3-24 shows the transient case two compared to the results provided in Figure 10

in reference [1].

41

−2 0 2 4 6100

200

300

400

sec

Wf (

pph)

−2 0 2 4 650

100

150

200

sec

PS

3 (p

sia)

−2 0 2 4 61400

1600

1800

2000

2200

sec

T41

(de

g R

)

−2 0 2 4 6800

1000

1200

1400

1600

sec

T45

(de

g R

)

−2 0 2 4 670

80

90

100

sec

PC

NG

(%

)

−2 0 2 4 6−100

0

100

200

sec

TO

RQ

45 (

ft−lb

s)

SimulinkPaper

SimulinkPaper

SimulinkPaper

SimulinkPaper

SimulinkPaper

Figure 3-24: Fuel Flow Step From 400 to 125 pph at 175 ft-lbs

Figure 3-24 shows fairly good agreement between the reference paper [1] results and

the model produced for this research. The Simulink�model decelerates at a different rate

than that of the reference model. However, the intent of this research is prove that an LPV

controller can be integrated into an engine control system. A perfect model match is not

required to achieve this end.

3.5 Engine Model Linearization

The purpose of this research was to replace the NP governor in the baseline fuel control

with a modern linear parameter varying H∞ controller. In order to find the new controller

a set of linear plants needs to be generated to use as input to the process. The engine

model was linearized by removing the baseline fuel control system and then providing the

inputs necessary to create the set of plants required. In this case, it was desired to produce

engine model plants from 76% to 100% NG with a linearization point every 2%. The

42

inputs required for the engine model are: fuel flow, load demand, ambient temperature and

ambient pressures (Wf, Qreq, Tamb, and Pamb respectively). The outputs of the plant are:

engine torque, power turbine speed, percent gas generator speed and gas generator rate

(Qpt, NP, PCNG, and NGDOT respectively). In addition to producing a plant every 2%

in core speed, the other requirement for the plant is that it is generated at an NP speed

of 100% or 20900 RPM. Figure 3-25 shows the top level Simulink�configuration used to

produce the linear plants where In1 and In2 are the inputs to the linearization process and

Out1 is the output of the linearization process.

Figure 3-25: Linear Model Configuration

In order to expedite the process required to determine the required values for WfInput

and QrefInput, an intermediate step is introduced. The model is modified as shown in

Figure 3-26.

43

Figure 3-26: Model to Find Operating Points

The fuel flow is proportional to core speed and therefore, an error signal between the

“NG goal” and the NG signal is created. The WfInput PID controller then produces a

WfInputBias that produces the desired core speed. The load demand correlates to the NP

speed and therefore, an error signal between the “NP goal” and the NP signal is created.

The QrefInput PID controller then produces a QrefInputBias that produces the desired

NP speed. These trimmers working together produce values for the WfInputBias and

QrefInputBias that simultaneously produce the desired NP and NG goals.

The process to find each operating point is as follows:

1. Set Tamb and Pamb to standard day. (For simplification, both are assumed to be

standard day)

2. Provide an “NG Goal”

3. Provide an “NP Goal”

4. Run the simulation until both NG and NP are with acceptable tolerances. In this

case, a 0.1% tolerance was used for both NP and NG.

The necessary inputs for WfInput and QrefInput were determined to be as shown in

Table 3-25. The first column correlates to WfInput and the second column correlates to

44

QrefInput. The table also shows the values of the states of the models NP , NG, P3, P41,

and P45.

Table 3.2: State Operating PointWf

(lbmhr

)Qreq (ft− lbf) NP (%) NG (%) P3 (psi) P41 (psi) P45 (psi)

832 460 99.15 99.26 251.1 238.2 51.06

732.1 401.4 99.92 98.12 240.5 228.1 48.83

617.8 328.4 99.89 96.16 217.4 205.9 44.08

526.3 263.3 99.81 94.23 195.6 185 39.71

445.2 205.4 99.86 92.18 175.3 165.6 35.69

377.8 157.5 99.93 90.09 156.6 147.8 32.14

336.1 127.3 99.82 88.22 143.5 135.4 29.74

287.6 90.46 99.92 86.08 126.2 119.1 26.73

251.9 60.7 99.84 84.16 111.3 105.1 24.22

220.5 38.74 99.93 82.09 99.35 93.83 22.44

187.7 18.7 99.9 80.13 87.44 82.55 20.82

172.7 8.906 99.98 78 81.5 76.96 20.07

161.5 1.585 99.98 76 76.7 72.44 19.49

150.3 -4.24 99.97 74 71.88 67.9 18.92

141 -3.949 100 72 67.24 63.55 18.39

133.4 -3.724 100 70 62.82 59.42 17.91

The following code was used to find the WfInputBias, QrefInputBias and state vector

to produce NG at 98% and NP at 100%:

1: WfInputBase = 832; %pph

2: QreqInputBase = 460; %Load ft-lbs

3: tol = 0.1;

4: %NG 98, NP 100

5: NgGoal = 98;

6: NpGoal = 100;

7: %Find the operating point

8: while not((xFinalFindOp(1)*100/NPdes>NpGoal-tol)&(xFinalFindOp(1)*100/NPdes<NpGoal+tol)...

9: &(xFinalFindOp(4)*100/NGdes>NgGoal-tol)&(xFinalFindOp(4)*100/NGdes<NgGoal+tol))

10: xInitialFindOp = xFinalFindOp;

11: sim(’NonLinearRTM FindOp’,50)

12: end

13: %Setup up inputs for linmod

14: WfInput = WfInputBase+WfInputBias.signals.values; %pph

15: QreqInput = QreqInputBase-QreqInputBias.signals.values; %Load ft-lbs

16: StateOpPoint = [ xFinalFindOp(1) xFinalFindOp(2) xFinalFindOp(3) xFinalFindOp(6) xFinalFindOp(4) ]

17: InputOpPoint = [0 0 ]

18: StateOpPointArchive(:,((100-NgGoal)/2)+1) = [StateOpPoint’;WfInput;QreqInput ];

19: %Linearize the operating point

20: NG098 = linmod(’NonLinearRTM Lin’,StateOpPoint,[0 0 ])

45

The complete m-file used to create all of the linearized plants can be found in ap-

pendix G.2.

The result from the code in appendix G.2 is a set of plants between 76% and 100%

NG with points every 2% NG. The state space matrices and eigenvalues for each of the

plants is in appendix H.1. At this point, the eigenvalues are reviewed to determine if the

plants are stable or unstable. In addition, the eigenvalues obtained are compared against

the eigenvalues from Ballin’s work the baseline paper “A High Fidelity Simulation of a

Small Turboshaft Engine” [1]. Figure 3-27 shows the eigenvalues obtained by linearizing

the model compared to the five degree of freedom model described in appendix B of the

model source paper [1].

Figure 3-27: Compare Eigenvalues to Source Paper

A review of Figure 3-27 reveals that there are three eigenvalues that are relatively low

frequency NP, NG and Ps3. The frequency of the eigenvalues of P45 and P41 are much

46

higher than bandwidth of the baseline controller and the LPV controller. Therefore, if

desired the model can be simplified by eliminating the three highest frequency eigenvalues.

47

Chapter 4

Load Model

The load demand model used for this research is based on the T700 engine model capabil-

ities. Two engines running at the maximum core speed producing the torque required to

maintain the rotor speed at 100% is defined as the 100% collective point. The zero percent

collective point is defined as zero shaft horsepower required. The curve between zero and

maximum load required follow a third order polynomial as given by equation 4.2. The load

model described in this chapter is compatible with the single pole rotor model described in

reference [1] that the engine model and baseline engine controller is derived from.

4.1 Load Model Assumptions

The following assumptions are applied to the construction of the load model:

1. The rotor can be represented the lumped inertia of the entire rotor system including

the main gear box. In practice, helicopter rotor models are normally much more

complex requiring attenuation for the main and tail rotor natural frequencies.

2. The collective to load application follows is a 3rd order polynomial.

3. The collective actuator response can be modeled by a simple lag.

4. The load remains constant across all airspeeds.

48

4.2 Load Model Architecture

The load model integrates the engine gas torque minus the load over the total inertia as

described by equation 3.27 and Figure 3-22. This architecture lumps the inertia of the

transmission, the main rotor and the tail rotor together. Table 4.1 shows the inertial values

used in this research based on the information provided by the work of W. Pfiel in his thesis

“Multivariate Control for the GE T700 engine using the LQR/LTR Design Methodology”

[6].

Table 4.1: Rotor Inertia Values

Description Signal Name Inertia (ft-lb-sec/RPM)

Transmission Inertia Jr 0.00538

Main Rotor Inertia Jr 0.1103

Tail Rotor Inertia Jet 0.01648

Since the power turbine inertia values are in units of ft ∗ lbf ∗ sec2 the inertial values

given in Table 4.1 must be converted. Equation 4.1 combines these parameters and performs

the conversion to the correct units.

Jload =(Jt+ Jmr + Jtr) ∗ 60

2 ∗ π(4.1)

The collective to shaft horsepower relationship used in this research as described in

the introduction of this chapter is based from a 3rd order polynomial which is given by

equation 4.2:

y = −0.0005 ∗ x3 + 0.47 ∗ x2 +−5 ∗ x+ 4 (4.2)

Table 4.2 resulting from this equation was modified slightly at the 10% point in Table 4.2

since the shaft horsepower required was negative at this point in the equation.

49

Table 4.2: Collective to Shaft Horsepower Relationship

Collective % Shaft Horsepower

0 4

10 10

20 88

30 263.5

40 524

50 866.5

60 1288

70 1785.5

80 2356

90 2996.5

100 3704

110 4475.5

The collective signal, labeled as “ACP”, is linearly interpolated to produce the load

shaft horsepower as shown in Figure 4-1. The figure shows that the relationship between

collective and shaft horsepower for a helicopter is a function of airspeed, however, for the

sake of simplicity this relationship is ignored for this research. Therefore, the values at 200

knots are set equal to the values at 0 knots in the model.

Figure 4-1: Load Shaft Horsepower

The engine requires that the load be in terms of torque in ft-lbs. Therefore, the shaft

horsepower is corrected for ambient temperature and pressure and then converted to torque

at 100% NP speed as shown in Figure 4-2. The output labeled “QrefLPV” is used in the

LPV controller implementation described in chapter 7. The time based input “QrefDist”

50

is used to input a load disturbance that is not compensated by the controller. The output

Qreq, used in the engine model as shown in Figure 3-22, is multiplied by half since the load

demand is for two engines.

Figure 4-2: Load Torque

51

Chapter 5

Engine Baseline Control System

Model

The T700 fuel control system model used as the baseline for this research was taken from

the paper written by Mark G. Ballin of the NASA Ames Research Center. The paper

“A High Fidelity Simulation of a Small Turbofan Engine” [1] provides the equations to

produce a high fidelity non-linear simulation of the GE T700 engine. The equations are

provided in this chapter for reference and convenience. The fuel control system provided

as the baseline is legacy technology involving a hydro-mechanical fuel unit (HMU) with

a digital trimming system. Current fuel control systems are based on a Full Authority

Digital Electronic Controller or FADEC which replace the most of the legacy controls with

electronic equivalents within the FADEC. However, the control system is still in use today

on several helicopter applications. In addition, the equations used to design HMU systems

are transferrable to a FADEC/FMU (fuel metering unit) system.

The control system on the T700 engine controls the fuel flow, the variable geometry, and

the operability bleed flow. The equations presented assume that the variable geometry and

operability bleed flow are controlled as scheduled. The equations for the plant make this

assumption as well. Therefore, the control system in this analysis considers only control of

the fuel flow into the engine. There are several boundary conditions that are considered

when designing a fuel control:

1. The following items are in order of decreasing importance:

52

(a) The core speed deceleration rate shall be greater than or equal to a minimum

scheduled reference.

(b) The core speed acceleration rate shall be less than or equal to a maximum

schedule reference.

(c) The core speed shall be greater than or equal to a minimum scheduled reference.

(d) The maximum fuel flow shall be less than or equal to a maximum scheduled

reference.

(e) The core speed shall be less than or equal to a maximum scheduled reference.

(f) The power available shall be less than or equal to the schedule set by the power

available spindle.

(g) The power turbine reference shall be kept at 100% or 20900 RPM.

2. It should be noted that the core speed acceleration is governed only byWf

Ps3and not

by ˙NG.

The purpose of this research is to determine if a modern linear parameter varying control

system can take the place of the baseline controller in controlling the NP speed.

5.1 Baseline Controller Assumptions

The following assumptions are applied to the construction of the baseline engine fuel control

model:

1. The model employs the following simplifications:

(a) The variable geometry and bleed valves are assumed to be on schedule and are

not included in the modeled control system.

(b) The T45 governor reference is set at a value such that it will not come into

regulation as this governor is not part of the primary analysis.

(c) The fuel flow actuator is modeled as a simple lag.

(d) The torque of the two engines is identical. Therefore, the load share governor

is ignored as it is not part of the primary analysis.

53

5.2 Baseline Controller Tools

The engine control system model is implemented in native Matlab�Simulink�since the

engine model and the LPV controller were also implemented in Simulink�facilitating the

back to back comparison.

5.3 Baseline Controller Architecture

The equations to build the baseline fuel control system model are given pictorially in frag-

ments as they appear in the Simulink�model. The complete set of Simulink�diagrams is

given in Appendix D. The variable names used in the diagrams and equations below are

defined in the Nomenclature section starting on page 14.

There are two inputs into the plant that determine where the engine will operate. The

fuel flow is provided by the fuel control system and the requested load is derived from the

rotor system model which is discussed in chapter 4.

As mentioned above, the fuel flow is determined by the fuel control system by selecting

between several governors utilizing a priority scheme. Part of this selection logic is shown

in Figure 5-1 where WFIDM is the NG idle fuel flow bottoming reference, WFPAC is the

NG acceleration topping reference and WFPDM is the deceleration bottoming reference.

WFPDM incorporates the fuel flow topping, NG toppping and the NP governor. WFPDM

also includes the open loop collective based feedforward (Note: The feedforward logic is not

within the scope of this research). The collective lever is part of the rotorcraft that the

pilot uses to control the lift of the helicopter. The purpose of this research is to replace the

baseline NP governor with a linear parameter varying H∞ controller.

Figure 5-1: Fuel Flow Governor Selection

(The units of HMUSEL are in ratio units or WfPs3

)

54

The NG idle fuel flow bottoming reference is defined in Figure 5-2. The plot and table

definitions for the 1-D linear interpolation labeled FHM5 HMUIdleSchedFunc1 in Figure 5-

2 is Figure F-6 and Table F.6 respectively. The plot and table definitions for the 1-D linear

interpolation labeled FHM6 HMUIdleSchedFunc2 in Figure 5-2 is Figure F-7 and Table F.7

respectively.

Figure 5-2: NG Idle Fuel Flow Governor

The signal PCNGHL is a lagged version of PCNG which is the gas generator speed in

percent as shown in Figure 5-3.

Figure 5-3: Percent Gas Generator Speed Lagged

(where NGdes is defined in Table B.1 and CNTL is defined in Table 1 )

The NG acceleration topping reference, WFPAC, is defined by Figure 5-4. The plot and

table definitions for the 2-D linear interpolation in Figure 5-4 is Figure F-8 and Table F.8

respectively.

Figure 5-4: NG Acceleration Topping Reference

The NG deceleration bottoming reference, WFPDC, is defined by Figure 5-5.

55

Figure 5-5: NG Deceleration Bottoming Reference

(where AWFP and BWFP is defined in Table 1 )

The WFPDM signal is a combination of the fuel flow topping schedule, the NG fuel

flow reference, the NG topping schedule, the load demand fuel flow reference and the NP

governor as shown in Figure 5-6.

Figure 5-6: Commanded Wf/Ps3 Signal Before Limiting Selector

(where NGREF and KNDRP is defined in Table 1 )

The fuel flow topping schedule is defined by Figure 5-7. The plot and table definitions

for the 1-D linear interpolation in Figure 5-7 is Figure F-2 and Table F.2 respectively.

Figure 5-7: Fuel Flow Topping Reference

The NG topping reference is based on the power available spindle which is another pilot

input as shown in Figure 5-8. The pilot uses this lever to set the NG speed between the

idle detent and the fly detent. This lever is also used to shutdown the engine. The plot and

table definitions for the 1-D linear interpolation in Figure 5-8 is Figure F-3 and Table F.3

respectively.

56

Figure 5-8: NG Topping Reference

The collective based feedforward logic is defined as shown in Figure 5-9. The plot and ta-

ble definitions for the 1-D linear interpolation labeled FHM3 HMULoadDmdCompSchedFunc1

in Figure 5-9 is Figure F-4 and Table F.4 respectively. The plot and table definitions for

the 1-D linear interpolation labeled FHM4 HMULoadDmdCompSchedFunc2 in Figure 5-2

is Figure F-5 and Table F.5 respectively.

Figure 5-9: Collective Based Feedforward

(Where ACP is the collective signal. Also, where NGREF and KNDRP are defined in

Table 1)

The feedforward logic is sent through a lag before entering the fuel flow command signal,

WFPDM, logic as shown in Figure 5-10.

Figure 5-10: Collective Based Feedforward Lagged

(Where CLLDS is defined in Table 1)

The NP governor has the most complex logic in the baseline fuel control system. The

NP error signal is calculated as shown in Figure 5-11. The XWLO Eng1 signal is the load

share bias. The torque of engine 1 has been set to the torque of engine 2 disabling the load

57

share logic. Therefore, the logic for load share is not provided here but it is provided in

appendix D.

Figure 5-11: NP Governor Error Signal

(Where CTPL is defined in Table 1)

The NP error signal is then compensated as shown in Figure 5-12.

Figure 5-12: NP Governor Error Signal Compensation

(Where CT13, CT14, CT14, ZK1,ZK5, ZK7, ZK9, ZK10, and B6 are defined in Table 1.

The input TRQL Eng1 is the torque of engine 1 from the engine model.)

The NP governor dynamics are then added to the compensated NP error signal as shown

in Figure 5-13.

Figure 5-13: NP Governor Dynamics

58

(Where T11, CT2, and CT12 are defined in Table 1)

The NP governor demand is then compared to the T45 governor signal and the maximum

between the two is chosen as shown in Figure 5-14. However, the T45 reference is set such

that the T45 is disabled as this is not in the primary area of this research. Therefore, the

logic for the T45 governor is not provided here but it is provided in appendix D.

Figure 5-14: NP Governor Selection Logic

The NP trim demand dynamics are shown in Figure 5-15 using the NP governor dy-

namics as the input.

Figure 5-15: NP Trim Dynamics

(Where XKPROP, XKING, and CT16 are defined in Table 1)

The low limit on the integrator is determined by the engine 2 torque level. This part

does not exist in native Simulink�and therefore, a variable low limit on the integrator was

implemented. Figure 5-16 shows the calculation for the low limit of the integrator and

Figure 5-17 shows the selection logic. While this logic was built into the model, the logic

was disabled when comparing the performance of the baseline NP governor to the LPV NP

Governor. The low limit on the integrator did not allow NP to return to the datum during

large load demand transients.

59

Figure 5-16: NP Trim Integrator Low Limit

Figure 5-17: NP Trim Integrator Low Limit Selection Logic

The torque motor dynamic compensation is added, as defined in the source paper [1] to

the NP trim demand signal as shown in Figure 5-18.

Figure 5-18: Torque Motor Compensation

(Where TMLG, TMGN, and TMLVG are defined in Table 1)

The selected fuel flow ratio unit signal, HMUSEL, is multiplied by Ps3 to obtain a fuel

flow demand in pounds per second, which is then applied to the fuel system dynamics

simulated as a simple lag, finally, the fuel flow from the fuel system is then converted

from pounds per second to pounds per hour for input into the engine model as shown in

Figure 5-19.

Figure 5-19: Fuel Flow to the Model

(Where CMLV is defined in Table 1)

60

5.4 Baseline Controller Validation

The baseline fuel control system is validated utilizing two transient cases provided in ref-

erence [1]. Figure 11 of reference [1] is a collective load disturbance while Figure 12 of the

reference paper provides a cyclic disturbance. The distinction between the two is important

to the engine control system. Currently, most helicopter engine control systems have knowl-

edge of the collective lever which is proportional to part of the load demand. However, there

are other factors that impact the rotor load such as cyclic movement, pedal movement, or

ambient conditions such as wind. Movement of the collective lever is called a compensated

maneuver due to the fact that engine control system reads the collective position and, in an

open loop fashion, adjusts the fuel flow in anticipation of the load application. Movement

of the cyclic or pedals is considered an uncompensated maneuver since the engine control

system does not have knowledge of the transient before the application of load. In this case,

the transient is driven solely from the error of the NP signal from the datum.

Figure 11 of reference [1] shows a collective pull producing a torque disturbance of about

40 ft-lbs with the engine model connected to the Sikorsky UH60A GENHEL rotor system

compared to flight test data. In this case, the aircraft is flying at approximately 90 knots.

However, the ambient pressures and temperatures are not provided. It should be noted that

the model prepared for this research does not include installed effects and therefore, does

not include the engine impacts due to ram air at the engine inlet. In addition, the rotor

system used in this research is much simpler than a typical rotor system model.

Since the ambient conditions were not disclosed, they were estimated by attempting to

match the core speed at the given fuel flow. It should also be noted that it is assumed

that there was an error in the labeling to the total torque parameter and that it should

have been “TOT.TORQUE, ft− lbsX10−1”. The T700 engines modeled for this research

are not capable of producing 12,000 ft-lbs or 1/2 of the 24,000 ft-lbs shown as the baseline

torque in Figure 11. The engine in the model is capable of providing approximately 460

ft-lbs at 100% NG and 100% NP speeds on a standard day at sea level. Finally, it should

be noted that the collective to load relationship is arbitrary, as discussed in chapter 4, for

this research while this relationship is modeled from the UH60A GENHEL model in the

reference paper.

Figure 5-20 shows a collective pull producing a load disturbance of approximately the

61

same magnitude as Figure 11 in the reference paper. The ambient pressure is set to 12.2

psia which corresponds to 5,000 feet on a standard day. The ambient temperature is set

to 518.67◦R. These parameters were adjusted in an attempt to match the transient in the

reference paper.

0 2 4 6 8280

300

320

340

360

380

400

420

sec

Wf (

pph)

0 2 4 6 899.5

100

100.5

101

secN

P (

%)

0 2 4 6 889

89.5

90

90.5

91

91.5

92

92.5

sec

PC

NG

(%

)

0 2 4 6 8200

220

240

260

280

300

sec

TO

RQ

45 (

ft−lb

s)

SimulinkPaper

SimulinkPaper

SimulinkPaper

SimulinkPaper

Figure 5-20: Baseline Controller Validation - Collective Pull

Figure 5-21 shows a cyclic movement producing a load disturbance of approximately

the same magnitude as Figure 12 in the reference paper. The ambient pressure is set to

12.2 psia which corresponds to 5,000 feet on a standard day. The ambient temperature is

set to 469.67◦R. These parameters were adjusted in an attempt to match the transient in

the reference paper.

62

−2 0 2 4 6 8280

300

320

340

360

380

400

sec

Wf (

pph)

0 2 4 6 898

98.2

98.4

98.6

98.8

sec

NP

(%

)

0 2 4 6 889.2

89.4

89.6

89.8

90

90.2

sec

PC

NG

(%

)

−2 0 2 4 6 8180

190

200

210

220

230

sec

TO

RQ

45 (

ft−lb

s)

SimulinkPaper

SimulinkPaper

SimulinkPaper

SimulinkPaper

Figure 5-21: Baseline Controller Validation - Cyclic Disturbance

While the two transients do not match absolutely, they do match fairly well dynamically.

In other words, the shape of the model response is very similar as well as the delta magnitude

for each of the parameters. Given the assumptions required to produce the transients and

the fact that the aircraft installation effects are not modeled, the match was considered

sufficient for the purposes of this research.

63

Chapter 6

Linear Parameter Varying (LPV)

Control Theory

6.1 Classical Approach

The classic approach to digital engine control design is to develop a non-linear model,

linearize that model at periodic core speed points, develop a controller for each linear point

and then interpolate or “gain schedule” between the LTI controllers as shown in Figure 6-1.

64

Figure 6-1: Classic Design Process

The major advantage to this process is that linear controller design methodologies can

be applied as long as the plant satisfies a few assumptions. While this is a proven method

with many fielded applications, there are inherent inefficiencies and potential de-stabilizing

results involved with this process. The most fundamental assumption that must be validated

when utilizing this approach is that the scheduling variable should vary slowly with respect

to the system dynamics [9]. This assumption is an indication the controller designs are

based from fixed time parameter plants and that those plants should represent the total time

varying plant in order to expect some reasonable guarantee of stability [10]. In engine control

design this is accomplished with extensive simulation and testing as well as conservative

gain and phase margin guidelines. This method can therefore sacrifice performance to cover

for un-modeled dynamics that are not compensated by the controller.

6.2 LPV Approach

A controller design that takes dynamics between the fixed operating points into account and

removes the slow varying parameter restriction could therefore, be desirable for an engine

control system. The controller design methodology utilized for this research is the linear

65

parameter varying (LPV) self scheduled H∞ control theory described in the work by Becker

and Packard [11]. This methodology utilizes a single quadratic Lyapunov function to find

a stabilizing linear finite dimensional controller, dependent on ρ, that performs well when

combined with a parameter varying plant also dependent upon ρ[11]. The symbol ρ stands

for a time varying parameter set. In the case of this research, ρ is the engine core speed

(NG).

This control methodology is limited to linear parameter varying plants where[12]:

� The model depends affinely on ρ

� The parameter set ρ is an available real time measurement(s)

� The time-varying parameter(s) ρ ranges within a fixed polytope

The state-space representation of the model is given by equation 6.1 [12]:

x = A(ρ(t))x+B(ρ(t))u

y = C(ρ(t))x+D(ρ(t))u(6.1)

The classical approach produces an LTI controller at specific values of ρ and then in-

terpolates between controllers to produce a “stable” system as ρ varies. However, this

approach does not guarantee stability or performance between linearized points. An LPV

controller takes advantage of the knowledge of the time varying parameter(s) resulting in a

controller structure given by equation 6.2 [12]:

xk = Ak(ρ(t))x+Bk(ρ(t))y

u = Ck(ρ(t))xk +Dk(ρ(t))y(6.2)

(where y and u designate the measurements and the controller inputs respectively)

The LPV control theory utilizes the Bounded Real Lemma, the definition of quadratic

H∞ performance and the Vertex property as the basis determining an LPV controller[12].

66

Lemma 6.2.1 (Bounded Real Lemma). Given a continuous transfer function G(s) = D+

C(sI −A)−1B, the following statements are equivalent:

(i) A is stable and ||G(s) = D + C(sI −A)−1B||∞ < γ

(ii) there exists a positive definite solution X to the matrix inequality:

M(X, γ) =

ATX +XA XB CT

BTX −γI DT

C D −γI

< 0 (6.3)

(Note: the Bounded Real Lemma is also applicable to discrete systems but the

equations presented here apply only to continuous systems)

The Bounded Real Lemma can be extended from purely LTI systems to LPV systems

utilizing the definition of Quadratic H∞ Performance.

Definition 6.2.2 (Quadratic H∞ Performance). The LPV system


y = C(ρ(t))x+D(ρ(t))u(6.4)

has quadratic H∞ performance γ if and only if there exists a single matrix X > 0 such

that

M(X, γ) =

A(ρ)TX +XA(ρ) XB(ρ) C(ρ)T

B(ρ)TX −γI D(ρ)T

C(ρ) D(ρ) −γI

< 0 (6.5)

for all admissible values of ρ.

The Quadratic H∞ Performance applies to systems with a fixed ρ, however, in the case

of polytopic LPV systems the matrix X in equation 6.5 can be found by solving a finite set

of LMIs. The Vertex property shows that 6.5 holds for all systems as a function of ρ if and

only if is holds at the vertices.

67

Theorem 6.2.3 (Vertex Property). Consider a polytopic LPV plant described by


y = C(ρ(t))x+D(ρ(t))u(6.6)

where all systems as a function of ρ exist within the collection of vertices

A(ρ) B(ρ)

C(ρ) D(ρ)

ε P = Co

Ai Bi

Ci Di

, i = 1, 2, ..., r

(6.7)

The following statements are equivalent:

(i) the LPV system is stable with quadratic H∞ performance γ

(ii) there exists a single matrix X > 0 such that, for all A(ρ) B(ρ)

C(ρ) D(ρ)

ε P,

M(X, γ) =

A(ρ)TX +XA(ρ) XB(ρ) C(ρ)T

B(ρ)TX −γI D(ρ)T

C(ρ) D(ρ) −γI

< 0 (6.8)

(iii) there exists X > 0 such that satisfying the system of LMIs

M(X, γ) =

AT

i X +XAi XBi CTi

BTi X −γI DT

i

Ci Di −γI

< 0, i = 1, 2, ..., r. (6.9)

Proof. Statements (i) and (ii) are equivalent by definition utilizing the Bounded Real

Lemma (6.2.1) and the definition of Quadratic H∞ Performance (6.2.2). The equivalence

of (ii) and (iii) is given by the fact that A(θ) B(θ)

B(θ) C(θ)

=

r∑i=0

αi

Ai Bi

Ci Di

(6.10)

with αi ≥ 0 andr∑

i=0αi = 1

68

Therefore, using the Bounded Real Lemma(6.2.1), the definition of Quadratic H∞ Per-

formance (6.2.2) and the Vertex property (6.2.3, a controller with quadratic H∞ perfor-

mance γ is found by finding X for the LMIs in equation 6.9[12]. In order to make this a

convex problem, the following restrictions are placed on the plant[12]:

Given the state space structure:

x = A(ρ(t))x+B1(ρ(t))w +B2(ρ(t))u

z = C1(ρ(t))x+D11(ρ(t))w +D12(ρ(t))u

y = C2(ρ(t))x+D21(ρ(t))w +D22(ρ(t))u

(6.11)

(where y is the measured output(s) and z is the controlled output(s))

(6.2a) D22(ρ(t)) = 0

(6.2b) B2(ρ(t)), C2(ρ(t)), D12(ρ(t)) and D21(ρ(t)) are parameter independent.

(6.2c) (A(ρ(t)), B2) and (A(ρ(t)), C2) are stabilizable and observable over the entire range

of polytope vertices.

Plant restriction (6.2a) can be worked around by using a loop shifting technique de-

scribed on page 77 of reference [13].

Plant restriction (6.2b) can be worked around by introducing new variables u and y

such that[12]

xu = Auxu +Buu

u = Cuxu

(6.12)

and

xy = Ayxy +Byy

y = Cyxy

(6.13)

(where Au and Ay are stable)

The resulting modified LPV is therefore,

69

x

xu

xy

=

A(ρ(t)) B2(ρ(t))Cu 0

0 Au 0

ByC2(ρ(t)) 0 Ay

x

xu

xy

+

B1(ρ(t))

0

ByD21(ρ(t))

w+

0

Bu

0

u

(6.14)

z =(C1(ρ(t)) D12(ρ(t))Cu 0

)x

xu

xy

+D11(ρ(t))w (6.15)

y =(

0 0 Cy

)x

xu

xy

(6.16)

It is clear from equations 6.14, 6.15 and 6.16 that the modified control and measurement

matrices are parameter independent satisfying Plant restriction (6.2b).

Plant restriction (6.2c) is necessary for the plant to be stabilizable via output feed-

back [13].

70

Chapter 7

LPV Controller Design

7.1 LPV Controller Assumptions

The following are the LPV controller assumptions:

� The plant must satisfy the assumptions detailed in section 6.2. Since the B matrix is

parameter dependent, the methods described in [12] must be utilized.

� It is assumed that the baseline controller provided in reference [1] was sufficiently

tuned for this application.

� It is assumed that there is no model uncertainty. In practice, this must be included

for robustness.

7.2 LPV Controller Tools

The LPV controller design described in this chapter is accomplished using the Matlab�LMI

control toolbox developed by Pascal Gahinet, Arkadi Nemirovski, Alan J. Laub and Mahout

Chilli [2]. There are several commands that were critical in the development of the LPV

controller for this research: “ltisys”, “psys”, “sconnect” and “hinfgs”. Descriptions of these

commands are provided in appendix G.1. The “ltisys” command packs the state-space

matrices A, B, C, D, and E into a single system matrix readable by the LMI toolbox. The

“psys” command takes the packed system matrices created by the “ltisys” command and

produces a polytopic model that can be referenced by “hinfgs”. The “sconnect” command

71

sets up the structure for the “hinfgs” command such that the inputs and outputs of the

controller are appropriately indicated. The “hinfgs” command takes the polytopic system

model and produces a gain scheduled H∞ controller.

In addition to the tools used to create the controller, the pdsimul command is used to

simulate the controller along parameter trajectories. The description for this command is

also provided in appendix G.1.

7.2.1 Changes to LMI toolset to deal with polytopic model

The pdsimul command was used to validate the controller against the linear plant. However,

as stated in the command description in appendix G.1 it was only designed for affine systems.

However, with a slight modification the code for pdsimul and the dependent function polsim,

it is possible to utilize these commands for the polytopic system in this research. The code

changes are provided in appendix G.3. To avoid confusion, the functions were renamed

from pdsimul and polsim to t700pdsimul and t700polsim respectively.

7.3 LPV Controller Architecture

The linearized plants obtained in section 3.5 have the following state space equation form:

x =

˙NP

˙P45

˙Ps3

˙P41

˙NG

= A

NP

P45

Ps3

P41

NG

︸︷︷︸

x

+B

Wf

Qref

︸︷︷︸

u

y =

Qpt

NP

NG%

˙NG

= C

NP

P45

Ps3

P41

NG

︸︷︷︸

x

+D

Wf

Qref

︸︷︷︸

u

(7.1)

72

However, the LPV controller for this research is based only on power turbine speed error

and therefore, the plant needs to be modified to support the architecture in Figure 7-1.

Figure 7-1: Modified Linear Plant Structure

As shown in Figure 7-1, in addition to the NP error output to the controller two other

outputs were added to the structure: NP error and Wf. These parameters serve as the

weighted outputs to drive the performance and ensure that the control input, Wf, remains

within the capability of the hardware. The state space equations are modified to take the

form:

x = Ax+(BQref 0

)︸︷︷︸

B

Qref

NPref

︸︷︷︸

w

+BWfWf︸︷︷︸u

z =

Wf

NPref

=

0

CNP

︸︷︷︸

C1

x+

0 0

0 −1

︸︷︷︸

D11

Qref

NPref

︸︷︷︸

w

+

1

0

Wf︸︷︷︸u

y =(NPref

)= CNPx+

(0 −1

)︸︷︷︸

D21

Qref

NPref

︸︷︷︸

w

+(0) Wf︸︷︷︸u

(7.2)

(where x remains the same as in equation 7.1)

The plant described by 7.2 provides the inputs Qref and NPref as external inputs, the

fuel flow input from the controller Wf , the outputs for weighting Wf and NPerror and

the final output NPerror as the input to the controller. This configuration was attempted

but unfortunately the B1 matrix was not parameter independent and the “hinfgs” script

returned an error per plant restriction (6.2b). Therefore, the plant must be further modified

73

as described in section 6.2. The state-space equations are further modified to produce a

pseudo-parameter independent B matrix as shown in equation 7.3. x

˙u

=

A BWf

0 −a

︸︷︷︸

A

x

u

+

BQref 0

0 0

︸︷︷︸

B

Qref

NPref

︸︷︷︸

w

+

0

1

Wf︸︷︷︸u

z =

Wf

NPref

=

0 0

CNP 0

︸︷︷︸

C1

x

u

+

0 0

0 −1

︸︷︷︸

D11

Qref

NPref

︸︷︷︸

w

+

1

0

Wf︸︷︷︸u

y =(NPref

)=(CNP 0

)x

u

+(

0 −1)

︸︷︷︸D21

Qref

NPref

︸︷︷︸

w

+(0) Wf︸︷︷︸u

(7.3)


The state space equation 7.3 parameter dependent vector BWfis now part of the mod-

ified A matrix satisfying plant restriction (6.2b). The consequence is that a new parameter

is introduced u with the dynamics of a low pass filter with a corner at a.

˙u = u− au (7.4)

or

us = u− au (7.5)

which when rearranged provides the transfer function from u to u:

u

u=

1

s+ a(7.6)

The input point for the linear plant obtained in section 3.5 is the output WF shown in

Figure 5-19. The dynamics of the fuel flow actuator is a low pass filter of the form:

u

u=

1

CLMV ∗ s+ 1(7.7)

(where CLMV is defined in Table E)

74

Therefore, to include the fuel flow actuator dynamics as part of the modified plant the

structure in equation 7.3 is changed to:

x

˙u

=

A BWf

0 −1

︸︷︷︸

A

x

u

+

BQref 0

0 0

︸︷︷︸

B

Qref

NPref

︸︷︷︸

w

+

0

CLMV

Wf︸︷︷︸u

z =

Wf

NPref

=

0 0

CNP 0

︸︷︷︸

C1

x

u

+

0 0

0 −1

︸︷︷︸

D11

Qref

NPref

︸︷︷︸

w

+

1

0

Wf︸︷︷︸u

y =(NPref

)=(CNP 0

)x

u

+(

0 −1)

︸︷︷︸D21

Qref

NPref

︸︷︷︸

w

+(0) Wf︸︷︷︸u

(7.8)


At this point, the modified plant is ready for the application of the weighting functions

which will determine the performance of the controller. In this case, the desired performance

is good disturbance rejection, fuel flow actuator commands appropriate for the hardware,

and zero steady state error. Therefore, the weighting function on the error signal should

be a low pass filter to achieve good steady state performance. The controller weighting

function on the other hand should be a high pass filter to penalize high frequencies and

keep the bandwidth of the control signal within the capabilities of the hardware.

We(s) =(10−3s+ 1)

(10−1s+ 1)(7.9)

(where We is the NP error weighting function)

Wc(s) =(10−2s+ 1)

(10−4s+ 1)(7.10)

(where Wc is the control input weighting function)

Once the weighting functions have been added to the modified plant, the LPV controller

must be setup using the sconnect command which indicates to the hinfgs command the plant

75

inputs and outputs as well as the controller inputs and outputs. The use of the sconnect

command is given in appendix G.4.2. At this point, the plant has been modified per the

plant restrictions given in section 6.2, the weighting functions on the control input and NP

error have been included and the inputs and outputs have been identified via the sconnect

command. Figure 7-2 shows this configuration.

Figure 7-2: Modified Linear Plant Structure with Weighting Functions

The linear parameter varying gain scheduled controller is now be determined by the

hinfgs command as shown in appendix G.4.2. The controller and linear plant are combined

using the slft command shown in appendix G.4.3. The vertices of the model are defined by

the pvec command and then incorporated into the system matrix using the addpv command.

In order to use the t700pdsimul function, the trajectory of the varying parameter must be

defined as well as the input function. The varying parameter in this case is the core speed

of the engine. The parameter varying system matrix is built as a function of core speeds

between 92 through 98% at intervals of 2%. To specify, for example, a plant and controller

at a core speed of 93.5 the parameter matrix returned by the trajectory m-file is:

p =

P92

P94

P96

P98

=

(1−(

93.5− 92

2

)(

93.5− 92

2

)0

0

=

0.25

0.75

0

0

(7.11)

The linear system is simulated using the t700pdsimul command which requires a trajec-

tory and an input function. The Matlab�code to produce core speed trajectory files with a

sinusoid pattern with various frequencies and bias is provided in appendix G.4.4. The input

function is a step input for both the disturbance, Qref, and the reference, NPref. The plant

76

simulated is set back to the configuration shown in equation 7.3 removing the weighting

functions.

Figure 7-3 shows the response of the control system to a 100 RPM step in the NP

reference. The graphs to the right show the parameter varying matrix as it transitions from

92% to 98% core speed, see equation 7.11 for an example, at frequencies of 1, 5, 10, 15, 20,

and 25 hertz.

Figure 7-3: 100 RPM Step in NP reference

Figure 7-4 shows the response of the control system to a 40 ft-lb load disturbance. The

graphs to the right show the parameter varying matrix as it transitions from 92% to 98%

core speed, see equation 7.11 for an example, at frequencies of 1, 5, 10, 15, 20, and 25 hertz.

77

Figure 7-4: 40 ft-lb Step Load Disturbance

Figure 7-3 shows a good control system response to the NP reference step input for the

following core speed trajectories 95 + 2 ∗ sin(ω ∗ t) where ω is set equal to 1, 5, 10, 15, 20

and 25. Figure 7-4 also shows a good system response even given the change in steady state

NP speed. The reason for undamped response at higher frequencies is that the controller

is based from a delta plant model and the load disturbance moved the plant off of the

controller design point. In order to bring the NP speed back to the original NP reference,

the delta in steady state fuel flow must be added which is done in the non-linear simulation.

At this point, the linear model is validated and the focus is shifted to developing a controller

for the non-linear model.

The baseline non-linear model controller consists of five selectable controllers as shown

in Figure 5-1. The intent of this research is to replace the NP governor while leaving

the engine acceleration and deceleration governors in place. Therefore, the LPV controller

78

replaces the calculations for WFPDM parameter. However, since the linear plant model

was derived from the WF parameter shown in Figure 5-19, the units of the output of the

LPV controller must be changed from pph to pphpsi or wf

Ps3 . In addition, the steady state fuel

flow must be added to the output of the controller due to the fact that the linear model is

a delta model. The following modifications, shown in Figure 7-5, are made to the baseline

fuel control system to incorporate the new LPV controller.

1. The LPV controller is added as an s-function with core speed and power turbine speed

error as inputs and fuel flow as an output.

(a) The output of the controller is in PPS which needs to be converted to PPH.

(b) The steady state fuel flow is added to the controller output. Two options are

considered:

i. The torque parameter, Qpt, is used as the input to a table lookup producing

steady state fuel flow.

ii. The load reference, QreqLPV , is used as the input to a table lookup produc-

ing steady state fuel flow assuming that this parameter is known.

(c) The pressure at stage 3, Ps3, must be divided to produce the correct units for

downstream logic.

(d) A selector is added which allows the user to select between the baseline and the

LPV controller

Figure 7-5: Simulink�Changes to Figure 5-6 To Incorporate LPV Controller

79

The controller s-function is required due to the fact that there are no native Simulink�parts

that implement the LPV controller structure. However, the s-function is relatively straight

forward and is based from the Limited Integrator Level 1 M-File structure provided in the

basic Matlab�installation at "matlabroot\toolbox\Simulink\texttrademark\blocks\limintm.m".

The LPV controller, pdK, obtained via an m-file similar to the one provided in appendix G.4.2

except that the NG range is expanded to 76 to 100% is provided as an s-function parameter.

The following modifications are made to the base s-function:

1. The state space matrices are defined as global variables. This is due to the fact that

they are calculated during the derivative case.

2. The function “GetModel” is added which pulls the state space matrices from pdK.

3. The function “NGInputConversion” is added to produce the input necessary obtain

the model at the specified point.

4. The model is initialized with:

(a) The “GetModel” function is called at 92% NG.

(b) 2 inputs: NG and NP speed error

(c) 1 output: fuel flow

(d) The number of continuous states is set equal to the rows in the A matrix

(e) X0 is set to a vector of zeros the size of number of states

5. In the derivative case the “GetModel” function is called prior to calculating the

derivative.

6. The output case uses the state space matrices obtained in the derivative case to

calculate the output.

The LPV controller s-function is provided in appendix G.4.6.

7.4 LPV Controller Validation

The initial tuning of the LPV controller was performed during the linear controller design

analysis described in section 7.3. However, it was determined that additional tuning was

desired to achieve the following:

80

1. The performance of the LPV controller should be as good, if not better than the

baseline controller.

2. Good performance is defined as:

(a) Fast reaction to a step in the NP reference or a load disturbance

(b) The torque does not exhibit mid-transient torque reversals (especially those

large enough to be felt at the aircraft level).

(c) The NP speed should settle to within 0.5% of the reference in 3 cycles or less.

(d) The NP speed overshoot/undershoot should be better than or equal to the

baseline controller.

3. The LPV control action should not be faster than the torque motor actuator capability

which is modeled by a 30 msec lag.

4. The LPV control action should not drive the fuel flow commands out side the limits

of the wfPs3 acceleration and deceleration governors.

The next section shows the results LPV controller versus the baseline controller after

the tuning exercise was performed to achieve the goals stated above.

7.4.1 Comparison of baseline controller versus LPV controller

The LPV controller is validated by comparing the results from the baseline controller to the

LPV controller. The LPV controller was tuned to satisfy the goals described in section 7.4.

The tuning exercise resulted in the following controller weighting functions:

We(s) = 3 ∗ (10−2s+ 1)

(101s+ 1)(7.12)

(where We is the NP error weighting function)

Wc(s) = 0.5 ∗ (5 ∗ 103s+ 1)

(5 ∗ 10−3s+ 1)(7.13)

(where Wc is the control input weighting function)

81

The bode plot for both weighting functions 7.12 and 7.13 is shown in Figure 7-6

Figure 7-6: Bode Plot of the LPV Controller Weighting Functions Wc and We

Figures 7-7 and 7-8 show the baseline and LPV controller responses to a 100 RPM NP

reference step input at collective positions of 25% and 77.2% respectively. It is clear from

the figures that the LPV controller performance is better than the baseline controller due

to faster NP rise time, lower NP overshoot and faster settling time of both NP and torque.

82

Figure 7-7: 100 RPM NP Step Response with Collective at 25

Figure 7-8: 100 RPM NP Step Response with Collective at 77.2

Figures 7-9 and 7-10 show the baseline and LPV controller responses to a 50 RPM/sec

ramp of the NP reference RPM step input at collective positions of 25% and 77.2% respec-

tively. Again, it is clear from the figures that the LPV controller performance is better than

83

the baseline controller due to faster NP rise time, lower NP overshoot and faster settling

time of both NP and torque.

Figure 7-9: NP Reference Ramp at 50 RPM/sec with Collective at 25

Figure 7-10: NP Reference Ramp at 50 RPM/sec with Collective at 77.2

84

Figures 7-11 and 7-12 show the baseline and LPV controller responses to a 40 ft-lb

load disturbance in the rotor system at collective positions of 25% and 77.2% respectively.

The distinction between a transient involving a change in the NP reference and a load

disturbance is that during a load disturbance the controller does not have knowledge that a

transient occurred except for the feedback via the NP signal. The performance of the LPV

in this case is again superior to the baseline controller in terms of rise time, overshoot and

settling time.

Figure 7-11: Rotor Load Disturbance of 40 ft-lbs with Collective at 25%

85

Figure 7-12: Rotor Load Disturbance of 40 ft-lbs with Collective at 77.2%

Figures 7-13 and 7-14 show the baseline and LPV controller response to small instan-

taneous collective pulls. It should be noted that the LPV controller uses the load demand

generated from the rotor model. Therefore, it is provided exactly the correct amount of

steady state fuel flow where the baseline controller is using an estimate of the steady state

fuel flow. The purpose of including these figures, given that they cannot be used for com-

parison between the baseline and LPV controllers, is to show that the entitlement of the

LPV controller. The anticipation and estimate of steady state fuel flow as a function of

collective is outside the scope of this research. As expected, the performance of the LPV in

this case is significantly better than the baseline controller in terms of rise time, overshoot

and settling time.

86

Figure 7-13: Collective Instantaneous Pull from 25% to 35%

Figure 7-14: Collective Instantaneous Pull from 65.64% to 82.5%

Figures 7-15 through 7-17 show the baseline and LPV controller responses large col-

lective pulls at rates of 1 second, 2 seconds and 3 seconds respectively. Again, it should

be noted that the LPV controller uses the load demand generated from the rotor model.

87

Therefore, it is provided exactly the correct amount of steady state fuel flow where the

baseline controller is using an estimate of the steady state fuel flow. Due to the magnitude

of the 1 second collective pull, the fuel flow requested by the NP controllers in both cases

exceeds theWf

Ps3acceleration schedule. This is the reason why the “droop” or undershoot of

the NP signal is approximately equal to the baseline controller. This is shown by the differ-

ence between the “Fuel Flow Command” and the “HMUSEL” plots. The “HMUSEL” plot

shows that the selected fuel flow signal is lower than “Fuel Flow Command” from the NP

governors. In the case of the 2 second pull, the performance of the LPV controller is slightly

better because the commanded fuel flow drops below theWf

Ps3schedule before the baseline

controller producing less droop, lower overshoot and faster settling time. In the case of

the 3 second pull, the LPV controller performance is far better than the baseline controller

due to the fact that fuel flow command remains below theWf

Ps3acceleration schedule for the

entire transient.

Figure 7-15: Collective 1 Second Pull from 10% to 90%

88



89

Chapter 8

Conclusions

The results in chapter 7 show that a LPV control system can be effectively applied to an

engine control system with positive results. In the cases of the NP step, NP ramp and load

disturbances the performance of the LPV controller is significantly better than the baseline

controller. With further tuning of the baseline controller, it is most likely possible to obtain

the same performance as the LPV controller. However, it would most like require a more

significant tuning effort to cover the same range of engine torque conditions.

In retrospect, the linear tuning effort produces controller weighting functions that were

not very effective in the non-linear model. This was due mainly to the fact that the goals,

described in section 7.4 could not be tested using the linear model since it did not contain

the other fuel flow governors. However, the effort was not wasted since it did provide

validation of the LPV controller against the engine itself.

The LPV controller that produced the best results was the one where the linear plant

was simplified to contain only the rotor system pole. This pole is a lumped state that

includes the power turbine, gear box, main rotor and tail rotor. This pole is also the closest

to the jω axis. Without question, the P41 and P45 can be excluded since these poles are

very fast compared to the rotor system pole. There could be an argument to include the

NG and Ps3 parameters since they are close to the range of the rotor system pole and they

are measured parameters. The inclusion of these additional parameters would have to be

weighed against the increase complexity of the controller. The smaller the linear plant the

less complex the resulting controller which makes a production implementation easier. This

research did not show an improvement in controller performance with these states included

90

in the linear plant.

The fact that the rotor system is modeled by a single lumped state also impacts the

complexity of the controller. In a practical application, the rotor system should be modeled

to a much higher fidelity to include the natural frequencies and damping of both the main

and tail rotors. However, the methods used to create the LPV controller can easily be

applied to a more complex plant without changes to the algorithm.

8.1 Future Work

This research does not attempt to address the anticipation and estimate of the steady state

fuel flow based on helicopter load demands signals. In a practical application, the exact

load demand of the rotor system is not known and must be estimated using these signals

mainly the collective signal. However, in practice the linear analysis and tuning of the NP

governor is performed outside of the tuning of the anticipation logic and therefore, it was

valid to neglect this for the scope of this research.

This research assumes no model uncertainty and therefore, in a practical application

this would have to be addressed. In addition, the concept of stability for the baseline con-

troller is based on gain margin and phase margin which the aerospace industry is familiar

and comfortable with. It may be necessary to produce equivalent gain and phase margin

assessments to obtain acceptance from the certification authorities.

The LPV controller did not attempt to limit the output when the acceleration or de-

celeration governors are selected. Therefore, as shown in 7-15 there is the potential for

the controller to windup. The performance of the governor could be improved if additional

”anti-windup” logic is incorporated.

91

Appendix A

Engine Model Diagrams

92

Figure A-1: Simulink�Model Top Figure

93

Figure A-2: Engine Model Figure 1

94


95


96

Appendix B

Engine Model Constants

Table B.1: Engine Model Constants

Constant Value Units

HVF 18300.0 BTUlbm

Jgt 0.0445 ft ∗ lbf ∗ sec2

Jpt 0.062 ft ∗ lbf ∗ sec2

Kbl 0.7826 non-dimensional

Kb3 0.0025 non-dimensional

Kdamp 0.06854 (ft∗lbf∗sec)rad

Kdpb 0.03045 (lbf2∗sec2)(lbm2∗in4∗degR)

Kh2 0.239 BTU(lbm∗degR)

Kh31 0.2496 BTU(lbm∗degR)

Kh32 -8.4 BTUlbm

Kh411 0.3010 BTUlbm∗degR


97


Constant Value Units

Kh412 -86.905 BTUlbm

Kh45 0.9623 non-dimensional

Kps3 0.956 non-dimensional

Kqc1 0.71 non-dimensional

Kqc2 0.29 non-dimensional

Kt411 3.322 lbm∗degRBTU

Kt412 288.7 degR


Kt452 179.1 degR


Kt492 172.3 degR

Kth411 0.0018326 1degR

Kth412 0.0856 non-dimensional

Kth451 0.0018326 1degR

Kth452 0.0856 non-dimensional

Kv3 0.97 lbf(in2∗lbm∗degR)



Kwgt 0.0876 lbm∗in2

(lbf∗sec)

NGdes 44700 rpm

NPdes 20900 rpm

TCt41 0.29 lbm45 ∗sec

95

degR12

98

Appendix C

Engine Model Schedules

99

0 2 4 6 8 10 12 14 16 18 202

3

4

5

6

7

8

9

10

11

Compressor Static Pressure Ratio, Ps3/P2

Sta

tion

2 C

orre

cted

Mas

s F

low

WA

2R, l

b/se

c

F1 − Compressor Mass Flow

65808285878992949698100

Figure C-1: F1 - Compressor Mass Flow

Table C.1: F1 - Compressor Mass FlowPs3/P2 NGR-65 NGR-80 NGR-82 NGR-85 NGR-87 NGR-89 NGR-92 NGR-94 NGR-96 NGR-98 NGR-100

1.008 3.137 4.772 5.274 5.869 6.278 6.724 7.635 8.248 8.954 9.642 10.09

1.012 3.137 4.772 5.274 5.869 6.278 6.724 7.635 8.248 8.954 9.642 10.09

1.015 3.137 4.772 5.274 5.869 6.278 6.724 7.635 8.248 8.954 9.642 10.09

1.019 3.137 4.772 5.274 5.869 6.278 6.724 7.635 8.248 8.955 9.642 10.09

1.024 3.137 4.772 5.274 5.869 6.278 6.724 7.635 8.248 8.955 9.642 10.09

1.038 3.137 4.772 5.274 5.869 6.278 6.724 7.635 8.248 8.955 9.642 10.09

1.046 3.137 4.772 5.274 5.869 6.278 6.724 7.635 8.248 8.955 9.642 10.09

1.048 3.137 4.772 5.274 5.869 6.278 6.724 7.635 8.248 8.955 9.642 10.09

1.051 3.137 4.772 5.274 5.869 6.278 6.724 7.635 8.248 8.955 9.642 10.09

1.053 3.137 4.772 5.274 5.869 6.278 6.724 7.635 8.248 8.955 9.642 10.09

1.069 3.137 4.773 5.274 5.869 6.278 6.724 7.635 8.248 8.955 9.642 10.09

3.151 3.143 4.778 5.28 5.875 6.284 6.73 7.633 8.254 8.96 9.643 10.09

3.245 3.124 4.779 5.28 5.875 6.284 6.73 7.633 8.254 8.961 9.643 10.09

3.386 3.125 4.779 5.281 5.876 6.285 6.731 7.633 8.255 8.961 9.643 10.09

3.479 3.106 4.779 5.281 5.876 6.285 6.731 7.633 8.255 8.961 9.643 10.09

3.62 3.07 4.78 5.281 5.876 6.285 6.731 7.633 8.255 8.962 9.643 10.09


100


Ps3/P2 NGR-65 NGR-80 NGR-82 NGR-85 NGR-87 NGR-89 NGR-92 NGR-94 NGR-96 NGR-98 NGR-100

3.714 2.995 4.78 5.282 5.877 6.285 6.732 7.633 8.256 8.962 9.643 10.09

3.714 3.033 4.78 5.282 5.877 6.285 6.732 7.633 8.256 8.962 9.643 10.09

3.76 2.958 4.78 5.282 5.877 6.286 6.732 7.633 8.256 8.962 9.643 10.09

4.849 2.958 4.783 5.285 5.88 6.289 6.735 7.632 8.259 8.965 9.643 10.09

5.694 2.958 4.767 5.287 5.882 6.291 6.737 7.631 8.261 8.968 9.644 10.1

5.928 2.958 4.767 5.288 5.883 6.292 6.738 7.631 8.262 8.968 9.644 10.1

6.069 2.958 4.731 5.288 5.883 6.292 6.738 7.631 8.262 8.969 9.644 10.1

6.304 2.958 4.713 5.289 5.884 6.293 6.739 7.631 8.263 8.969 9.644 10.1

6.351 2.958 4.676 5.289 5.884 6.293 6.739 7.631 8.263 8.969 9.644 10.1

6.353 2.958 4.675 5.289 5.884 6.293 6.739 7.631 8.263 8.969 9.644 10.1

6.491 2.958 4.62 5.271 5.884 6.293 6.739 7.631 8.263 8.97 9.644 10.1

6.585 2.958 4.565 5.259 5.885 6.293 6.74 7.63 8.264 8.97 9.644 10.1

6.635 2.958 4.538 5.253 5.885 6.294 6.74 7.63 8.264 8.97 9.644 10.1

6.725 2.958 4.491 5.235 5.885 6.294 6.74 7.63 8.264 8.97 9.644 10.1

6.916 2.958 4.491 5.198 5.885 6.294 6.74 7.63 8.265 8.971 9.644 10.1

7.245 2.958 4.491 5.087 5.886 6.295 6.741 7.63 8.266 8.972 9.644 10.1

7.478 2.958 4.491 4.902 5.887 6.296 6.742 7.63 8.266 8.973 9.644 10.1

7.483 2.958 4.491 4.902 5.887 6.296 6.742 7.63 8.266 8.973 9.644 10.1

7.812 2.958 4.491 4.902 5.851 6.297 6.743 7.63 8.267 8.973 9.644 10.1

8.046 2.958 4.491 4.902 5.814 6.298 6.744 7.629 8.268 8.974 9.644 10.1

8.327 2.958 4.491 4.902 5.741 6.298 6.744 7.629 8.269 8.975 9.644 10.1

8.468 2.958 4.491 4.902 5.667 6.299 6.745 7.629 8.269 8.975 9.644 10.1

8.471 2.958 4.491 4.902 5.665 6.299 6.745 7.629 8.269 8.975 9.644 10.1

8.608 2.958 4.491 4.902 5.593 6.294 6.745 7.629 8.269 8.976 9.644 10.1

8.796 2.958 4.491 4.902 5.5 6.288 6.746 7.629 8.27 8.976 9.645 10.1

8.987 2.958 4.491 4.902 5.5 6.282 6.746 7.629 8.27 8.977 9.645 10.1

9.363 2.958 4.491 4.902 5.5 6.227 6.747 7.628 8.271 8.978 9.645 10.1

9.553 2.958 4.491 4.902 5.5 6.19 6.748 7.628 8.272 8.978 9.645 10.1

9.644 2.958 4.491 4.902 5.5 6.172 6.742 7.628 8.272 8.979 9.645 10.1

9.834 2.958 4.491 4.902 5.5 6.14 6.73 7.628 8.273 8.979 9.645 10.1

9.972 2.958 4.491 4.902 5.5 6.117 6.712 7.628 8.273 8.979 9.645 10.1

10.12 2.958 4.491 4.902 5.5 6.049 6.694 7.628 8.274 8.98 9.645 10.1

10.21 2.958 4.491 4.902 5.5 6.006 6.685 7.628 8.274 8.98 9.645 10.1

10.49 2.958 4.491 4.902 5.5 6.006 6.657 7.627 8.275 8.981 9.645 10.1

10.63 2.958 4.491 4.902 5.5 6.006 6.621 7.627 8.275 8.981 9.645 10.1

10.91 2.958 4.491 4.902 5.5 6.006 6.584 7.627 8.276 8.982 9.645 10.1

11.05 2.958 4.491 4.902 5.5 6.006 6.548 7.627 8.276 8.982 9.645 10.1

11.24 2.958 4.491 4.902 5.5 6.006 6.474 7.627 8.277 8.983 9.645 10.1

11.34 2.958 4.491 4.902 5.5 6.006 6.418 7.627 8.277 8.983 9.645 10.11

11.43 2.958 4.491 4.902 5.5 6.006 6.359 7.627 8.277 8.984 9.645 10.11

11.52 2.958 4.491 4.902 5.5 6.006 6.307 7.624 8.277 8.984 9.645 10.11

11.95 2.958 4.491 4.902 5.5 6.006 6.307 7.61 8.279 8.985 9.645 10.11

12.37 2.958 4.491 4.902 5.5 6.006 6.307 7.574 8.28 8.986 9.645 10.11

12.61 2.958 4.491 4.902 5.5 6.006 6.307 7.518 8.281 8.987 9.646 10.11

12.8 2.958 4.491 4.902 5.5 6.006 6.307 7.482 8.281 8.987 9.646 10.11

12.89 2.958 4.491 4.902 5.5 6.006 6.307 7.453 8.281 8.988 9.646 10.11


101


Ps3/P2 NGR-65 NGR-80 NGR-82 NGR-85 NGR-87 NGR-89 NGR-92 NGR-94 NGR-96 NGR-98 NGR-100

12.98 2.958 4.491 4.902 5.5 6.006 6.307 7.427 8.274 8.988 9.646 10.11

13.08 2.958 4.491 4.902 5.5 6.006 6.307 7.39 8.267 8.988 9.646 10.11

13.26 2.958 4.491 4.902 5.5 6.006 6.307 7.297 8.253 8.989 9.646 10.11

13.36 2.958 4.491 4.902 5.5 6.006 6.307 7.249 8.245 8.989 9.646 10.11

13.74 2.958 4.491 4.902 5.5 6.006 6.307 7.249 8.209 8.99 9.646 10.11

14.11 2.958 4.491 4.902 5.5 6.006 6.307 7.249 8.155 8.991 9.646 10.11

14.39 2.958 4.491 4.902 5.5 6.006 6.307 7.249 8.081 8.992 9.646 10.11

14.45 2.958 4.491 4.902 5.5 6.006 6.307 7.249 8.051 8.992 9.646 10.11

14.68 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.977 9.646 10.11

15.01 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.956 9.646 10.11

15.38 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.939 9.646 10.11

15.76 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.847 9.646 10.11

15.86 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.825 9.646 10.11

16.09 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.773 9.639 10.11

16.32 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.663 9.632 10.11

16.42 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.663 9.629 10.11

16.84 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.663 9.593 10.11

16.85 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.663 9.593 10.11

17.22 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.663 9.535 10.1

17.31 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.663 9.52 10.09

17.5 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.663 9.445 10.08

17.55 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.663 9.428 10.07

17.73 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.663 9.317 10.04

17.74 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.663 9.317 10.04

17.92 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.663 9.317 10.01

18.11 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.663 9.317 9.987

18.25 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.663 9.317 9.932

18.35 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.663 9.317 9.895

18.49 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.663 9.317 9.84

18.58 2.958 4.491 4.902 5.5 6.006 6.307 7.249 7.915 8.663 9.317 9.803

102

0 5 10 15 20 25

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

Compressor Static Pressure Ratio, Ps3/P2

Com

pres

sor

Tem

pera

ture

Rat

io, T

3/T

2

F2 − Compressor Temperature

Figure C-2: F2 - Compressor Temperature

Table C.2: F2 - Compressor TemperaturePs3/P2 T3/T2

1.033 1.306

2.018 1.448

3.003 1.571

4.034 1.685

5.018 1.784

6.001 1.872

6.985 1.955

8.015 2.025

8.998 2.091

10.03 2.152

11.01 2.209

11.99 2.263

13.02 2.317

14.01 2.374

15.03 2.434

16.02 2.498


103


Ps3/P2 T3/T2

17 2.561

18.03 2.637

19.06 2.72

20.04 2.808

104

65 70 75 80 85 90 95 1000

0.02

0.04

0.06

0.08

0.1

0.12

Corrected Gas Generator Speed, NGR%

Ble

ed F

ract

ion,

B1

F3 − Seal Pressure Bleed Fraction

Figure C-3: F3 - Seal Pressure Bleed Fraction

Table C.3: F3 - Seal Pressure Bleed FractionNGR Bleed Fraction-B1

65.08 0.1089

65.41 0.1089

66.23 0.1089

67.04 0.109

67.86 0.109

68.68 0.109

69.5 0.109

70.32 0.109

71.13 0.109

71.95 0.109

72.77 0.109

73.59 0.109

74.4 0.1091

75.22 0.1091

76.04 0.1091

76.86 0.1091


105


NGR Bleed Fraction-B1

77.59 0.1091

79.23 0.108

79.88 0.1066

80.21 0.1057

80.7 0.1038

81.19 0.102

81.6 0.09992

82.25 0.09624

82.5 0.09417

82.75 0.09186

83.24 0.0868

83.4 0.08449

83.56 0.08219

83.64 0.07988

84.14 0.07182

84.22 0.06951

84.3 0.06721

84.46 0.0649

84.54 0.0626

84.63 0.06029

84.71 0.05799

84.79 0.05591

84.95 0.05361

85.2 0.04808

85.36 0.04577

85.44 0.04347

85.85 0.03586

85.93 0.03355

86.1 0.03125

86.18 0.02964

86.34 0.02733

86.43 0.02503

86.59 0.02272

86.67 0.02042

86.83 0.01834

87.24 0.01143

87.41 0.009126

87.65 0.007053

87.82 0.004749

88.22 0.001986

88.88 0.0006112

89.29 0.0001554

90.11 0.0001661

90.92 0.0001769

91.74 0.0001877

92.56 0.0001985


106


NGR Bleed Fraction-B1

93.38 0.0002092

97.06 2.71e-005

97.87 3.79e-005

98.69 4.87e-005

100 6.59e-005

107

3 4 5 6 7 8 9 10 11 12 138.8

9

9.2

9.4

9.6

9.8

10

10.2

10.4

10.6x 10

−3

Station 2 Corrected Mass Flow, WA2R, lbm/sec

Ble

ed F

ract

ion,

B2

F4 − Power Turbine Balance Bleed Fraction

Figure C-4: F4 - Power Turbine Balance Bleed Fraction

Table C.4: F4 - Power Turbine Balance Bleed FractionWA2R (lbm/sec) Bleed Fraction-B2

3 0.01057

5.023 0.008996

12.02 0.009

108

2 4 6 8 10 12 140.077

0.078

0.079

0.08

0.081

0.082

0.083

0.084

0.085

0.086

Station 2 Corrected Mass Flow, WA2R, lbm/sec

Ble

ed F

ract

ion,

B3

F5 − Impeller tip leakage and turbine cooling bleed fraction

Figure C-5: F5 - Impeller Tip Leakage and Turbine Cooling Bleed Fraction

Table C.5: F5 - Impeller Tip Leakage and Turbine Cooling Bleed FractionWA2R (lbm/sec) Bleed Fraction-B3

2.981 0.08508

8.297 0.08463

12.02 0.07792

109

0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018 0.019 0.02−0.5

0

0.5

1

1.5

2

Fuel−to−air ratio, FAR

Com

bust

or E

ffici

ency

, ET

A

F6 − Combustor Efficiency

Figure C-6: F6 - Combustor Efficiency

Table C.6: F6 - Combustor EfficiencyFuel-to-air ratio-FAR Combustor Efficiency-ETA

0.01 0.985

0.02 0.985

110

0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.3620

25

30

35

40

45

50

Gas Generator Turbine Pressure Ratio, P45/P41

Gas

Gen

erat

or E

thal

py D

rop

Par

amet

er, B

TU

/LB

M

F7 − Gas Generator Turbine Energy

Figure C-7: F7 - Gas Generator Turbine Energy

Table C.7: F7 - Gas Generator Turbine EnergyGG Turbine Pressure Ratio-P45/P41 GG Ethalpy Drop-BTU/LBM

0.1996 48.23

0.2149 40.32

0.22 39.04

0.2249 38.29

0.2299 37.55

0.3502 24.66

111

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−5

0

5

10

15

20

25

30

35

Power Turbine Pressure Ratio, Ps49/P45

Pow

er T

urbi

ne E

thal

py D

rop

Par

amet

er, B

TU

/LB

M

F8 − Power Turbine Energy

Figure C-8: F8 - Power Turbine Energy

Table C.8: F8 - Power Turbine EnergyPT Pressure Ratio-Ps49/P45 PT Ethalpy Drop-BTU/LBM

0.2993 33.94

0.3503 30.33

0.4 26.58

0.4497 23.13

0.4982 20.13

0.5493 17.2

0.599 14.28

0.6488 11.35

0.6999 8.348

0.7484 5.272

0.7994 2.045

0.8491 -1.331

112

0.4 0.5 0.6 0.7 0.8 0.9 10.26

0.28

0.3

0.32

0.34

0.36

0.38

Power Turbine Pressure Ratio, P45/P41

Pow

er T

urbi

ne C

orre

cted

Mas

s F

low

, W45

R, l

bm/s

ec

F9 − Power Turbine Mass Flow

Figure C-9: F9 - Power Turbine Mass Flow

Table C.9: F9 - Power Turbine Mass FlowPT Pressure Ratio-P45/P41 PT Corrected Mass Flow-W45R (lbm/sec)

0.3013 0.372

0.3258 0.3718

0.3516 0.3717

0.3788 0.3708

0.3917 0.3706

0.4046 0.3699

0.4175 0.3694

0.4304 0.3683

0.442 0.3678

0.4549 0.3667

0.4665 0.366

0.4859 0.3642

0.4975 0.363

0.504 0.3623

0.5156 0.3612

0.5298 0.3594


113


PT Pressure Ratio-P45/P41 PT Corrected Mass Flow-W45R (lbm/sec)

0.5414 0.3582

0.5543 0.3566

0.566 0.3557

0.5789 0.3543

0.5879 0.3529

0.5982 0.3513

0.6034 0.3506

0.6124 0.349

0.6202 0.3471

0.6292 0.3453

0.637 0.3434

0.6447 0.3416

0.6538 0.3393

0.6602 0.3374

0.668 0.3353

0.6783 0.3323

0.6835 0.3302

0.7041 0.3237

0.7106 0.3216

0.7157 0.3195

0.7222 0.3177

0.7286 0.3154

0.7338 0.3133

0.739 0.3112

0.7441 0.3091

0.7674 0.2998

0.7725 0.2977

0.779 0.2949

0.7842 0.2928

0.7893 0.2907

0.8035 0.284

0.8087 0.2819

0.8138 0.2796

0.8177 0.2775

0.8293 0.2721

0.8345 0.27

0.8384 0.2677

0.841 0.2666

0.8461 0.2645

0.8513 0.2612

114

65 70 75 80 85 90 95 100 1051.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

Corrected Gas Generator Speed, NGR%

Exh

aust

Pre

ssur

e R

atio

, Ps9

/P49

F10 − Exhaust Pressure Loss

Figure C-10: F10 - Exhaust Pressure Loss

Table C.10: F10 - Exhaust Pressure LossNGR Exhaust Pressure Ratio-Ps9/P49

65.02 1.124

65.19 1.124

65.92 1.123

66.66 1.122

67.39 1.121

68.13 1.12

68.94 1.12

69.68 1.119

70.17 1.118

71.15 1.116

71.48 1.114

71.89 1.112

72.22 1.11

72.63 1.108

72.96 1.106

73.37 1.104


115


NGR Exhaust Pressure Ratio-Ps9/P49

73.78 1.102

74.11 1.1

74.52 1.098

74.85 1.096

75.26 1.095

75.59 1.093

76.16 1.09

76.57 1.088

76.98 1.086

77.39 1.084

77.81 1.082

78.22 1.081

78.71 1.079

78.96 1.077

79.28 1.074

79.61 1.073

79.94 1.071

80.27 1.069

80.52 1.066

80.85 1.065

81.18 1.063

81.75 1.059

82.08 1.057

82.41 1.055

82.74 1.053

83.07 1.051

83.48 1.049

83.81 1.047

84.14 1.045

84.47 1.043

84.8 1.041

85.13 1.039

85.7 1.036

86.03 1.034

86.61 1.03

87.02 1.028

87.59 1.026

88.16 1.025

88.66 1.023

89.31 1.022

89.88 1.021

90.94 1.021

91.68 1.022

92.08 1.022

92.65 1.024

93.14 1.025


116


NGR Exhaust Pressure Ratio-Ps9/P49

93.71 1.027

94.11 1.028

94.68 1.03

95.25 1.032

95.74 1.033

96.15 1.035

96.47 1.037

96.71 1.039

97.03 1.041

97.36 1.043

98 1.049

98.24 1.051

98.41 1.053

98.65 1.055

98.89 1.057

99.05 1.059

99.29 1.061

99.53 1.064

100 1.066

117

Appendix D

Fuel Control Model Diagrams

118

Figure D-1: Engine Control Figure 1

119


120


121


122

Appendix E

Fuel Control Constants

See the Model Variable Table 1 in the nomenclature section.

123

Appendix F

Fuel Control Schedules

124

0 2 4 6 8 10 12 14 161.5

2

2.5

3

3.5

4

4.5

5

5.5

Power Turbine Flow Parameter, W45R

Sen

sor

Tim

e C

onst

ant,

Tau

45, s

ec

Fec1 − ECU thermocouple sensor time constant

1260166020602460

Figure F-1: Fec1 - ECU Thermocouple Sensor Time Constant

Table F.1: Fec1 - ECU thermocouple sensor time constantPT Flow - W45R T45L=1260R T45L=1660R T45L=2060R T45L=2460R

1.009 5.001 4.214 3.642 3.406

1.021 4.992 4.206 3.636 3.4

1.062 4.963 4.18 3.617 3.383

1.069 4.958 4.175 3.614 3.38

2.525 3.934 3.224 2.938 2.768

2.528 3.932 3.224 2.936 2.766

2.529 3.931 3.223 2.936 2.766

2.564 3.907 3.216 2.929 2.76

6.486 2.52 2.35 2.168 2.081

6.533 2.516 2.339 2.158 2.073

6.535 2.516 2.339 2.158 2.073

6.536 2.516 2.339 2.158 2.073

9.989 2.199 1.991 1.922 1.858

9.992 2.199 1.991 1.922 1.858

9.992 2.199 1.991 1.922 1.858

10.04 2.197 1.986 1.92 1.857


125


PT Flow - W45R T45L=1260R T45L=1660R T45L=2060R T45L=2460R

14.95 1.989 1.82 1.726 1.714

14.99 1.987 1.818 1.724 1.713

15.04 1.985 1.817 1.722 1.712

15.04 1.985 1.817 1.722 1.712

126

350 400 450 500 550 600 6500

0.5

1

1.5

2

2.5

3

3.5

4

Inlet Temperature, T2, Deg R

Fue

l Flo

w T

oppi

ng P

aram

eter

, WF

PT

P

Fhm1 − HMU topping line schedule

Figure F-2: Fhm1 - HMU Topping Line Schedule

Table F.2: Fhm1 - HMU Topping Line ScheduleT2 (Deg R) Fuel Flow Topping - WFPTP

393.9 0.09091

501.2 3.509

515.3 3.309

534.3 3.136

554.9 3.118

575.6 3.218

594.8 3.291

615.4 3.336

634.6 3.373

127

20 30 40 50 60 70 80 90 100 110 1200

2

4

6

8

10

12

Power Available Spindle Angle, deg

Fue

l Flo

w P

ower

Ava

ilabl

e P

aram

eter

, WF

PR

F

Fhm2 − HMU Power Available input schedule

Figure F-3: Fhm2 - HMU Power Available Input Schedule

Table F.3: Fhm2 - HMU Power Available Input SchedulePower Available Spindle Fuel Flow Reference - WFPRF

28.62 10.4

30.07 9.836

39.91 7.022

50.21 5.736

59.92 4.844

70.21 3.812

79.93 2.808

89.65 1.945

99.92 1.224

109.9 0.7857

119.9 0.03619

128

0 10 20 30 40 50 60 70 80 90 10080

85

90

95

100

105

110

Load Demand Spindle Angle, XLDSA, deg

Gas

Gen

erat

or S

peed

Dem

and

Par

amet

er, P

NG

Fhm3 − HMU load demand compensation schedule

Figure F-4: Fhm3 - HMU Load Demand Compensation Schedule

Table F.4: Fhm3 - HMU Load Demand Compensation ScheduleLoad Demand Spindle Angle NG REF

0.2833 80.15

9.915 83.81

20.11 87.24

30.31 90.6

40.23 93.35

50.14 95.72

60.06 98.7

70.25 102.3

80.17 107.4

90.08 107.4

100 107.4

129

−20 0 20 40 60 80 100 1202.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

Load Demand Spindle Angle, XLDSA, deg

Fue

l Flo

w D

elta

Dem

and

Par

amet

er, W

FQ

PS

3

Fhm4 − HMU load demand compensation schedule function 2

Figure F-5: Fhm4 - HMU Load Demand Compensation Schedule Function 2

Table F.5: Fhm4 - HMU Load Demand Compensation Schedule Function 2Load Demand Spindle Angle Fuel Flow Delta - WFOPS3

-0.03453 2.294

19.9 2.361

29.84 2.531

40.07 2.69

50.29 2.879

60.22 3.119

70.14 3.396

80.36 3.571

90.34 3.576

100.6 3.577

130

350 400 450 500 550 600 6502.1

2.15

2.2

2.25

2.3

2.35

2.4

2.45

2.5

2.55

2.6


Fue

l Flo

w Id

le S

ched

ule

Lim

it P

aram

eter

, WF

IRF

Fhm5 − HMU idle schedule function 1

Figure F-6: Fhm5 - HMU Idle Schedule Function 1

Table F.6: Fhm5 - HMU Idle Schedule Function 1T2 (deg R) Fuel Flow Idle Reference

393.7 2.101

414.4 2.129

435 2.16

454.3 2.194

474.9 2.228

494.2 2.265

514.8 2.303

534.8 2.342

554.8 2.385

574.8 2.429

594.8 2.475

614.8 2.523

634.1 2.572

131

350 400 450 500 550 600 65062

64

66

68

70

72

74


NG

Dro

op L

ine

Idle

Ref

eren

ce P

aram

eter

, PC

NG

I

Fhm6 − HMU idle schedule function 2

Figure F-7: Fhm6 - HMU Idle Schedule Function 2

Table F.7: Fhm6 - HMU Idle Schedule Function 2T2 (deg R) NG Droop Idle Reference

394.4 63.89

441.5 65.89

505.8 68.47

560.7 70.68

609.3 72.59

635 73.63

132

50 60 70 80 90 100 1100.5

1

1.5

2

2.5

3

3.5

4

4.5

Sensed Gas Turbine Speed, PCNGHL%

Fue

l Flo

w A

ccel

erat

ion

Lim

it P

aram

eter

, WF

PA

C

Fhm7 − HMU maximum fuel parameter limit during acceleration

395430460480519555590

Figure F-8: Fhm7 - HMU Maximum Fuel Parameter Limit During Acceleration

Table F.8: Fhm7 - HMU Maximum Fuel Parameter Limit During AccelerationNG T2=395 R T2=430R T2=460R T2=480R T2=519R T2=555R T2=590R

54.12 3.639 3.741 3.798 3.778 3.752 3.681 3.675

54.12 3.638 3.741 3.798 3.778 3.752 3.681 3.675

54.13 3.638 3.741 3.798 3.778 3.752 3.681 3.676

54.29 3.628 3.734 3.795 3.777 3.754 3.686 3.681

54.29 3.628 3.734 3.794 3.777 3.754 3.686 3.681

54.29 3.628 3.734 3.794 3.777 3.754 3.686 3.681

54.46 3.618 3.728 3.791 3.776 3.756 3.691 3.687

60.24 3.26 3.505 3.664 3.728 3.824 3.862 3.877

60.24 3.26 3.505 3.664 3.728 3.824 3.862 3.877

60.24 3.26 3.504 3.664 3.728 3.824 3.862 3.877

60.24 3.26 3.504 3.664 3.728 3.824 3.862 3.877

60.25 3.259 3.504 3.664 3.728 3.824 3.862 3.878

60.26 3.259 3.504 3.664 3.727 3.824 3.862 3.878

60.41 3.252 3.497 3.658 3.723 3.822 3.867 3.882

64.14 3.092 3.324 3.524 3.609 3.786 3.9 3.976

64.17 3.091 3.323 3.523 3.608 3.786 3.9 3.976


133


NG T2=395 R T2=430R T2=460R T2=480R T2=519R T2=555R T2=590R

64.31 3.088 3.316 3.518 3.603 3.784 3.901 3.978

64.32 3.088 3.316 3.518 3.603 3.784 3.901 3.978

64.33 3.088 3.316 3.518 3.603 3.784 3.901 3.978

64.33 3.088 3.315 3.518 3.603 3.784 3.901 3.978

64.33 3.088 3.315 3.517 3.603 3.784 3.901 3.978

68.06 3.013 3.216 3.378 3.518 3.726 3.883 4.02

68.22 3.009 3.212 3.372 3.514 3.723 3.88 4.021

68.23 3.009 3.212 3.372 3.514 3.723 3.88 4.021

68.24 3.009 3.211 3.371 3.514 3.723 3.88 4.021

68.25 3.009 3.211 3.371 3.514 3.723 3.88 4.021

68.25 3.009 3.211 3.371 3.513 3.722 3.88 4.021

68.41 3.008 3.209 3.368 3.51 3.719 3.877 4.021

72.14 2.981 3.15 3.296 3.401 3.63 3.822 4.004

72.14 2.981 3.15 3.296 3.401 3.63 3.822 4.004

72.16 2.981 3.15 3.295 3.4 3.629 3.821 4.004

72.3 2.98 3.15 3.293 3.396 3.626 3.818 4.003

72.32 2.98 3.15 3.292 3.395 3.626 3.817 4.003

72.33 2.98 3.15 3.292 3.395 3.626 3.817 4.002

72.49 2.985 3.149 3.289 3.393 3.622 3.813 3.998

76.04 3.094 3.143 3.26 3.349 3.545 3.727 3.91

76.06 3.094 3.143 3.259 3.349 3.544 3.726 3.909

76.24 3.1 3.142 3.261 3.347 3.54 3.722 3.905

76.24 3.1 3.142 3.261 3.347 3.54 3.722 3.905

76.39 3.105 3.147 3.263 3.345 3.537 3.718 3.902

76.4 3.105 3.147 3.263 3.345 3.537 3.718 3.902

76.4 3.106 3.148 3.263 3.345 3.537 3.718 3.902

80.13 3.24 3.267 3.305 3.348 3.491 3.646 3.817

80.13 3.241 3.267 3.305 3.348 3.491 3.646 3.817

80.14 3.241 3.268 3.305 3.348 3.491 3.646 3.816

80.14 3.241 3.268 3.305 3.348 3.491 3.646 3.816

80.31 3.248 3.273 3.311 3.348 3.488 3.644 3.814

80.48 3.255 3.28 3.316 3.348 3.486 3.643 3.811

80.48 3.255 3.28 3.316 3.348 3.486 3.643 3.811

84.05 3.404 3.436 3.436 3.452 3.499 3.608 3.758

84.05 3.404 3.436 3.436 3.452 3.499 3.608 3.758

84.21 3.404 3.442 3.442 3.457 3.499 3.606 3.756

84.21 3.404 3.442 3.442 3.457 3.499 3.606 3.756

84.22 3.404 3.442 3.443 3.457 3.499 3.606 3.756

84.39 3.404 3.448 3.449 3.464 3.5 3.608 3.755

86.77 3.406 3.538 3.546 3.558 3.576 3.627 3.748

86.93 3.364 3.544 3.552 3.565 3.582 3.628 3.748

87.1 3.324 3.546 3.559 3.571 3.587 3.63 3.747

87.1 3.323 3.546 3.559 3.571 3.587 3.63 3.747

87.1 3.322 3.546 3.559 3.572 3.587 3.63 3.747

87.78 3.153 3.554 3.586 3.598 3.618 3.662 3.748

87.95 3.11 3.556 3.593 3.605 3.626 3.666 3.748


134


NG T2=395 R T2=430R T2=460R T2=480R T2=519R T2=555R T2=590R

88.12 3.069 3.556 3.6 3.612 3.633 3.67 3.748

90.16 2.559 3.557 3.682 3.693 3.726 3.717 3.688

90.5 2.474 3.558 3.696 3.706 3.742 3.708 3.678

90.5 2.473 3.558 3.696 3.706 3.742 3.707 3.677

90.67 2.432 3.519 3.696 3.713 3.75 3.703 3.673

92.6 1.949 3.068 3.698 3.789 3.704 3.648 3.616

92.7 1.899 3.043 3.698 3.794 3.702 3.645 3.613

92.71 1.898 3.042 3.698 3.794 3.702 3.644 3.613

92.71 1.897 3.042 3.698 3.794 3.702 3.644 3.612

92.71 1.896 3.041 3.698 3.794 3.702 3.644 3.612

92.73 1.887 3.037 3.698 3.794 3.701 3.644 3.612

94.67 0.9806 2.546 3.699 3.795 3.656 3.605 3.561

94.75 0.9806 2.526 3.699 3.795 3.654 3.603 3.559

94.92 0.9808 2.484 3.699 3.795 3.65 3.6 3.555

94.96 0.9808 2.473 3.689 3.795 3.649 3.6 3.554

95.09 0.9809 2.442 3.655 3.795 3.646 3.597 3.55

96.95 0.9822 2.009 3.184 3.797 3.603 3.565 3.504

96.97 0.9822 2.004 3.178 3.791 3.602 3.565 3.503

97.01 0.9822 1.995 3.168 3.781 3.601 3.564 3.502

97.13 0.9823 1.969 3.139 3.753 3.598 3.562 3.499

101.5 0.9854 0.9854 2.064 2.677 3.497 3.461 3.391

101.6 0.9855 0.9855 2.042 2.655 3.495 3.459 3.389

101.7 0.9856 0.9856 2 2.614 3.491 3.415 3.384

101.7 0.9856 0.9856 2 2.613 3.49 3.414 3.384

101.8 0.9856 0.9856 1.993 2.606 3.484 3.407 3.377

101.8 0.9856 0.9856 1.988 2.601 3.478 3.402 3.371

104 0.9872 0.9872 0.9979 2.044 2.931 2.817 2.722

104.1 0.9873 0.9873 0.9497 2.017 2.905 2.789 2.69

104.2 0.9873 0.9873 0.9227 2.002 2.89 2.773 2.673

104.3 0.9874 0.9874 0.8759 1.976 2.864 2.745 2.642

104.3 0.9874 0.9874 0.8725 1.974 2.862 2.743 2.64

104.3 0.9874 0.9874 0.8624 1.968 2.856 2.737 2.633

104.4 0.9875 0.9875 0.8477 1.96 2.848 2.729 2.624

135

Appendix G

Matlab�Code

G.1 LMI Toolbox Commands [2]

LTISYS

sys = ltisys(a,b,c,d,e) sys = ltisys(’tf’,n,d)

Stores the state-space realization (A,B,C,D,E) of an LTI system as the SYSTEM matrix

SY S =

A+ j(E − I) 0 N/A

C D 0

0 0 −∞

(G.1)

where na = size(A,1). The matrices A through E should be real. When omitted, D and E

are set to the default values D=0 and E=I. The values 0 and 1 for E are interpreted as

E=0 and E=I.

SYS = LTISYS(A) and SYS = LTISYS(A,E) specify the autonomous systems

x = Ax

Ex = Ax(G.2)

SISO systems can also be specified via their transfer function N(s)/D(s). The syntax is

then SYS = LTISYS(’tf’,N,D) where N and D are the vector representation of the

polynomials N(s) and D(s).

136

PSYS

PSYS Specify a parameter-dependent system (P-system)

S = PSYS([S1,...,SN]) takes N LTI models S1,...,SN (created with LTISYS) and constructs

a polytopic uncertain model S with equations Specification of uncertain state-space

models

Ex = Ax+Bu

y = Cx+Du(G.3)

whose matrices A, B, C, D, and E can take any value in the convex hull of the matrices

(Aj, Bj, Cj, Dj, Ej) of the models S1,...,SN.

S = PSYS(PV,[S0,S1,...,SN]) creates an affine parameter-dependent uncertain model

S = S0 + p1*S1 + ... + pN*SN

The state-space matrices (A(p),...,E(p)) of S depend affinely on those of S0,...,SN, eg A(p)

= A0 + P1*A1 + ... + pN*AN. The first input PV describes the ranges and rates of the

parameters p1,...,pN (see PVEC).

SCONNECT

[P,r] = sconnect(inputs,outputs,K in,G1 in,g1,G2 in,g2,...)

SCONNECT computes the SYSTEM matrix P of the standard H-infinity plant P(s)

associated with a given control structure. It also returns the 1x2 vector R such that

R(1) = nbr of measurements (controller inputs)

R(2) = nbr of controls (controller outputs)

The control structure is described by the relations between the various signals flowing in

the control loop:

� the string INPUTS lists the exogenous input signals. For instance, INPUTS = ’r(2)

; d’ specifies two inputs, a vector r of size 2 and a scalar input d

� the string OUTPUTS lists the output signals. The output of a system G is denoted

simply by G. For instance, OUTPUTS = ’e=r-G ; G’ specifies two output signals

e=r-y and y where y is the output of the system G

137

� the string K IN gives the inputs of the controller in terms of the exogenous inputs

and the system outputs. For instance, K IN = ’K:e’ specifies a controller named K

and with input e

� each LTI system G in the loop but the controller is specified by its input list G IN

and its SYSTEM matrix G. For instance, G1 IN = ’S:K’ inserts a system named S

whose input is the output of the controller K

The syntax G([2,4]) selects the 2nd and 4th output of a system G. Note that the names

given to the various systems are immaterial provided that they are consistent throughout.

If one of the systems is polytopic or parameter-dependent, SCONNECT returns a

polytopic model of the interconnection.

HINFGS

HINFGS Design gain-scheduled H-infinity controllers

[GOPT,PDK,R,S]=HINFGS(PDS,R,GMIN,TOL,TOLRED) performs gain-scheduled

H-infinity control for the parameter-dependent systems with an affine dependence on the

time-varying parameters. These parameters are assumed to be measured in real-time.

This-function implements the quadratic H-infinity performance approach.

Input:

� PDS: parameter-dependent plant (see PSYS).

� R: 1x2 vector specifying the dimensions of D22:

– R(1) = nbr of measurements

– R(1) = nbr of measurements

� GMIN: target value for GOPT. Set GMIN = 0 to compute the optimum GOPT,

and set GMIN = GAMMA to test whether the performance GAMMA is achievable

(Default = 0)

� TOL: desired relative accuracy on the optimal performance GOPT (Default = 1e-2)

� TOLRED: optional (default value = 1e-4). Reduced-order synthesis is performed

whenever

138

– rho(X ∗ Y ) >= (1− TOLRED) ∗GAMA2

Output

� PDK: polytopic representation PDK = [K1 , ... , Kn] of the gain-scheduled

controller. The vertex controller Kj is associated with the j-th corner of the

parameter box as given by POLYDEC

� GOPT: optimal performance if GMIN=0, and some achievable performance <

GMIN otherwise

� R,S: solutions of the characteristic LMI system

PDSIMUL

PDSIMUL Simulate P-systems along parameter trajectories

[T,X,Y] = PDSIMUL(PDS,’TRAJ’,TF,’UT’,XI,OPTIONS) simulates the (affine)

parameter-dependent system PDS along the parameter trajectory specified by the

function ’TRAJ’. The function PDSIMUL calls ODE15s to integrate the time response of

this linear time-varying system.

Without output arguments, PDSIMUL plots the output

trajectories.

Input:

� PDS: affine parameter-dependent system (see PSYS)

� ’TRAJ’: name of the function p=TRAJ(t) specifying the parameter trajectory.

This-function takes a time t and returns a value p of the parameter vector.

� TF: final time for the integration (initial=0)

� ’UT’: name of the input function u=UT(t). The default is a step input.

� XI: initial state of the system (Default = 0)

OPTIONS control parameters for the ODE integration (see ODESET)

Output:

139

� T: integration time points

� X: state trajectories (X(:,1) = first state, etc.)

� Y: output trajectories (Y(:,1) = first output, etc)

G.2 Linearization Scripts

1: %Linearized Cases

2: clear StateOpPointArchive

3:4: WfInputBase = 832; %pph

5: QreqInputBase = 460; %Load ft-lbs

6:7: tol = 0.1;

8:9: %NG 100, NP 100

10: NgGoal = 100;

11: NpGoal = 100;

12:13: xInitialFindOp = 1.0e+004*[ 2.0900 0.0053 0.0258 4.4698 0.0002 0.0245 -0.0010 ];

14:15: sim(’NonLinearRTM FindOp’,50)

16: disp ’NG100 FindOp Complete’

17:18: WfInput = WfInputBase+WfInputBias.signals.values; %pph



21: InputOpPoint = [ 0 0 ]

22: StateOpPointArchive(:,((100-NgGoal)/2)+1) = [ StateOpPoint’;WfInput;QreqInput ];

23:24: NG100 = linmod(’NonLinearRTM Lin’,StateOpPoint,InputOpPoint)

25:26: disp ’NG100 Complete’

27:28: %NG 98, NP 100

29: NgGoal = 98;

30: NpGoal = 100;

31:32: %Find the operating point





37: end

38:39: disp ’NG098 FindOp Complete’

40:41: %Setup up inputs for linmod

42: WfInput = WfInputBase+WfInputBias.signals.values; %pph





140

47:48: %Linearize the operating point

49: NG098 = linmod(’NonLinearRTM Lin’,StateOpPoint,[ 0 0 ])


52:53: %NG 96, NP 100

54: NgGoal = 96;

55: NpGoal = 100;

56:57: SimCounter = 0;





62: SimCounter = SimCounter +1;

63: end







72:73: NG096 = linmod(’NonLinearRTM Lin’,StateOpPoint,[ 0 0 ])

74:75: %NG 94, NP 100

76: NgGoal = 94;

77: NpGoal = 100;







85: end









98:99: %NG 92, NP 100

100: NgGoal = 92;

101: NpGoal = 100;






141


109: end









122:123: %NG 90, NP 100

124: NgGoal = 90;

125: NpGoal = 100;







133: end









146:147: %NG 88, NP 100

148: NgGoal = 88;

149: NpGoal = 100;







157: end








168:

142

169: disp ’NG088 Complete’

170:171: %NG 86, NP 100

172: NgGoal = 86;

173: NpGoal = 100;







181: end









194:195: %NG 84, NP 100

196: NgGoal = 84;

197: NpGoal = 100;







205: end









218:219: %NG 82, NP 100

220: NgGoal = 82;

221: NpGoal = 100;







229: end

143









242:243: %NG 80, NP 100

244: NgGoal = 80;

245: NpGoal = 100;







253: end









266:267: %NG 78, NP 100

268: NgGoal = 78;

269: NpGoal = 100;







277: end









290:

144

291: %NG 76, NP 100

292: NgGoal = 76;

293: NpGoal = 100;







301: end









G.3 Changes to LMI toolset code

1 % dx=polsim(t,x,flag,ufun,ptraj,pds,pv,range)2 %3 % Called by PDSIMUL: integrate a polytopic PDS with ODE15s45 % Author: P. Gahinet 6/946 % Copyright 1995−2004 The MathWorks, Inc.7 % $Revision: 1.1.6.1 $89 function dx=t700polsim(t,x,flag,ufun,ptraj,pds,pv,range)

101112 % current parameter and input values13 pt = feval(ptraj,t);14 ut = feval(ufun,t);1516 % check position in box17 if any(pt < range(:,1) | pt > range(:,2)),18 error(sprintf('The parameter trajectory leaves the parameter box at t = %6.3e',t));19 end2021 % get A(p),B(p),C(p),D(p)22 %The following line is modified to deal with a polytopic model23 %[ap,bp,cp,dp,ep]=ltiss(psinfo(pds,'eval',polydec(pv,pt))); %Old Code24 [ap,bp,cp,dp,ep]=ltiss(psinfo(pds,'eval',pt)); %New Code252627 if isempty(flag)28 % Return state derivative29 if ¬isequal(ep,eye(size(ap)))30 ap=ep\ap;31 bp=ep\bp;32 end33 dx=ap*x + bp*ut;34 else35 % Return output36 dx=(cp*x + dp*ut).';37 end

145

1 function [tout,x,y, p] = t700pdsimul(pds,ptraj,tf,ut,xi,options)2 % PDSIMUL Simulate P−systems along parameter trajectories3 %4 % [T,X,Y] = PDSIMUL(PDS,'TRAJ',TF,'UT',XI,OPTIONS) simulates the (affine)5 % parameter−dependent system PDS along the parameter trajectory specified6 % by the function 'TRAJ'. The function PDSIMUL calls ODE15s to integrate7 % the time response of this linear time−varying system.8 %9 % Without output arguments, PDSIMUL plots the output

10 % trajectories.11 %12 % Input:13 % PDS affine parameter−dependent system (see PSYS)14 % 'TRAJ' name of the function p=TRAJ(t) specifying the15 % parameter trajectory. This−function takes a time16 % t and returns a value p of the parameter vector.17 % TF final time for the integration (initial=0)18 % 'UT' name of the input function u=UT(t). The default19 % is a step input.20 % XI initial state of the system (Default = 0)21 % OPTIONS control parameters for the ODE integration22 % (see ODESET)23 %24 % Output:25 % T integration time points26 % X state trajectories (X(:,1) = first state, etc.)27 % Y output trajectories (Y(:,1) = first output, etc)28 %29 % See also GEAR, PSYS, PVEC.3031 % Author: P. Gahinet 6/94, Revised 11/13/9632 % Copyright 1995−2004 The MathWorks, Inc.33 % $Revision: 1.1.6.1 $34 %35 % J.M. Spack36 % Modified code such that pvec is polytopic instead of affine3738 if nargin<2,39 error('usage: pdsimul(pds,ptraj,tf)');40 elseif ¬ispsys(pds),41 error('PDS must be a parameter−dependent system');42 else43 if nargin≤2, tf=5; end44 if nargin≤3, ut='stepsml'; end45 if nargin≤4, xi=[]; end46 if nargin≤5, options=[]; end47 end48 %49 %50 if exist(ptraj) < 2,51 error(sprintf(['Undefined function ' ptraj]));52 else53 p0 = feval(ptraj,0);54 end5556 if exist(ut) < 2,57 error(sprintf(['Undefined function ' ut]));58 else59 u0 = feval(ut,0);60 end61 %62 %63 [pdtyp,nv,ns,ni,no]=psinfo(pds);64 pv=psinfo(pds,'par');65 if isempty(pv),66 error(sprintf([...67 'The parameter vector description is missing! \n' ...68 ' Execute the command pds = addpv(pds,pv) \n' ...69 ' and rerun PDSIMUL']));70 end71 [pvtyp,np]=pvinfo(pv);7273 % check consistency74 %if ¬strcmp(pvtyp,'box'), %Commented Out75 % error('PV must be of type ''box'' '); %Commented Out76 if ¬isempty(xi) & length(xi)6=ns,77 error(sprintf('Initial condition XI should have %d states',ns));78 elseif np6=length(p0),79 error(sprintf(['The function ' ptraj ...80 ' should return a parameter vector of length %d'],np));81 elseif ni6=length(u0),82 error(sprintf(['The input function UT ' ...83 'should return a parameter vector of length %d'],ni));84 elseif isempty(xi),85 xi = zeros(ns,1);86 end8788 %range=pvinfo(pv,'par'); %Old Code

146

89 range=pvinfo(pv,'par',[1 2]); %New Code90 pds=pds(:,1:1+nv*(ns+ni+2));9192 if strcmp(pdtyp,'aff'),93 [t,x] = ode15s('affsim',[0 tf],xi,options,ut,ptraj,pds,pv,range);94 y = zeros(length(t),no);95 for i=1:length(t),96 y(i,:) = affsim(t(i),x(i,:).','output',ut,ptraj,pds,pv,range);97 end9899 else

100 %Old Code101 %[t,x] = ode15s('polsim',[0 tf],xi,options,ut,ptraj,pds,pv,range);102 %New Code103 [t,x] = ode15s('t700polsim',[0 tf],xi,options,ut,ptraj,pds,pv,range);104 y = zeros(length(t),no);105 for i=1:length(t),106 %Old Code107 %y(i,:) = polsim(t(i),x(i,:).','output',ut,ptraj,pds,pv,range);108 %New Code109 y(i,:) = t700polsim(t(i),x(i,:).','output',ut,ptraj,pds,pv,range);110 p(i,:) = feval(ptraj,t(i)); %New Line111 end112113 end114115 if nargout>0,116 tout = t;117 else118 plot(t,y)119 title('Output trajectory')120 end

G.4 LPV Controller

G.4.1 Modified Plant Setup

1 function PaugReturn = SetupGSAugPlantDistContLinear(A,B,C,D,WithWeighting)23 %Jeffrey M Spack4 %Create Augmented Plant56 %Xdot = Ax + Bu7 %z = C1x+D1u −> error8 %y = C2x+D2u −> NP9

10 Aaug = A1112 %Include parameter dependent B matrix in Aaug1314 B1 = B(:,2)15 B2 = B(:,1)1617 % [xdot] = [Aaug(rho) B2(rho)] [x] + [0][uprime] + [B1][w]18 % [udot] [ 0 −a ] [u] [1] [0]1920 % udot = uprime − a*u21 % a = pole of actuator dynamics22 CLMV = 0.03;2324 [m,n] = size(Aaug);25 Aaug = [Aaug B2; zeros(1,n) −1]2627 [m,n] = size(B1)28 %Baug = [B1 zeros(m,n) B2]29 Baug = [[B1;0] zeros(m+1,n) [zeros(m,n);CLMV]]3031 C1 = C(2,:)32 C2 = C(2,:)3334 %Add control signal to output35 if WithWeighting ==136 Cwf = [zeros(size(C2)) 0] %utilda37 else38 Cwf = [zeros(size(C2)) 0] %u39 end

147

40 Cwe = C24142 %Create augmented plant with only weighted error and control input43 Caug = abv(Cwe,C2)4445 D2 = [0 −1 D(2,1)]4647 %Add control signal to output48 if WithWeighting ==149 Dwf = [0 0 1] %u Tilda50 else51 Dwf = [0 0 1] %u52 end53 Dwe = [0 −1 0]5455 %Add D12, D22 to augmented C56 Caug = [Caug [D(2,1); D(2,1)]]5758 %Add control input to output59 Caug2 = abv(Cwf,Caug)6061 %Add parameter dependent D matrix to Caug6263 %D matrix is already setup for augmented C since D(2,1) = 064 [m,n] = size(Dwe);65 Daug = abv(Dwf,Dwe,D2);6667 Paug = pck(Aaug,Baug,Caug2,Daug)6869 if WithWeighting ==170 w e = nd2sys([10ˆ−3 1],[10ˆ−1 1],1);71 else72 w e = 1;73 end7475 %Weighting function for control action76 if WithWeighting ==177 WC = nd2sys([10ˆ−2 1],[10ˆ−4 1],10); %Baseline78 else79 WC = 1;80 end81 Waug = daug(WC,w e,1);8283 %Baseline Control without weighting84 %Paug = sysbal(Paug)8586 %Weight only error term87 PaugW = mmult(Waug,Paug);8889 [AW,BW,CW,DW] = unpck(PaugW);90 PaugWlti = ltisys(AW,BW,CW,DW,1);9192 PaugReturn = PaugWlti;

G.4.2 LPV Controller Design

12 clear all3 close all4 cd 'D:\Users\a276svt\My Documents\MATLAB\Thesis\GainSchController'56 load '../LinearModelsNew101124'78 A92 = NG092.a;9 B92 = NG092.b;

10 C92 = NG092.c;11 D92 = NG092.d;1213 A94 = NG094.a;14 B94 = NG094.b;15 C94 = NG094.c;16 D94 = NG094.d;1718 A96 = NG096.a;19 B96 = NG096.b;20 C96 = NG096.c;21 D96 = NG096.d;2223 A98 = NG098.a;24 B98 = NG098.b;25 C98 = NG098.c;

148

26 D98 = NG098.d;2728 PaugW92lti=SetupGSAugPlantDistContLinear(A92,B92,C92,D92,1)29 PaugW94lti=SetupGSAugPlantDistContLinear(A94,B94,C94,D94,1)30 PaugW96lti=SetupGSAugPlantDistContLinear(A96,B96,C96,D96,1)31 PaugW98lti=SetupGSAugPlantDistContLinear(A98,B98,C98,D98,1)3233 pdG = psys([PaugW92lti PaugW94lti PaugW96lti PaugW98lti])3435 [pdP,r] = sconnect('dist; ref','G(1);G(2)','K:G(3)','G:dist;ref;K',pdG);3637 [gopt,pdK] = hinfgs(pdP,r);

G.4.3 LPV Controller Linear Simulation

12 pcl = slft(pdPorig,pdK)34 pvbox = pvec('pol',[0 1; 0 1; 0 1; 0 1])56 pclpv = addpv(pcl,pvbox);78 %plot step response at multiple frequencies9 bias = 95; ampl = 2;

10 tf = 50;11 MyUt = 'GSInputFunction2';1213 freq1 = 1; freq=freq1; WriteTrajFile92 9814 [t1,x,y1,pt1]=t700pdsimul(pclpv,['trajectory',num2str(freq)],tf,MyUt);1516 freq2 = 5; freq=freq2; WriteTrajFile92 9817 [t2,x,y2,pt2]=t700pdsimul(pclpv,['trajectory',num2str(freq)],tf,MyUt);1819 freq3 = 10; freq=freq3; WriteTrajFile92 9820 [t3,x,y3,pt3]=t700pdsimul(pclpv,['trajectory',num2str(freq)],tf,MyUt);2122 freq4 = 15; freq=freq4; WriteTrajFile92 9823 [t4,x,y4,pt4]=t700pdsimul(pclpv,['trajectory',num2str(freq)],tf,MyUt);2425 freq5 = 20; freq=freq5; WriteTrajFile92 9826 [t5,x,y5,pt5]=t700pdsimul(pclpv,['trajectory',num2str(freq)],tf,MyUt);2728 freq6 = 25; freq=freq6; WriteTrajFile92 9829 [t6,x,y6,pt6]=t700pdsimul(pclpv,['trajectory',num2str(freq)],tf,MyUt);

G.4.4 Write Trajectory File

1 fid = fopen(strcat('trajectory',num2str(freq),'.m'), 'w');2 fprintf(fid, strcat('function p = trajectory',num2str(freq),'(t)\n'));3 fprintf(fid, strcat('NGtrajectory = ',num2str(bias),'+',num2str(ampl),'*sin(',num2str(freq),'*t);\n'));4 fprintf(fid, 'flag92 94=0;\n');5 fprintf(fid, 'flag94 96=0;\n');6 fprintf(fid, 'flag96 98=0;\n');7 fprintf(fid, 'if ((NGtrajectory≥92)&&(NGtrajectory<94))\n');8 fprintf(fid, ' flag92 94=1;\n');9 fprintf(fid, 'elseif ((NGtrajectory≥94)&&(NGtrajectory<96))\n');

10 fprintf(fid, ' flag94 96=1;\n');11 fprintf(fid, 'elseif ((NGtrajectory≥96)&&(NGtrajectory≤98))\n');12 fprintf(fid, ' flag96 98=1;\n');13 fprintf(fid, 'end\n');14

15 fprintf(fid, 'p92=flag92 94*(1−(NGtrajectory−92)/2);\n');16 fprintf(fid, 'p94=flag92 94*((NGtrajectory−92)/2)+flag94 96*(1−(NGtrajectory−94)/2);\n');17 fprintf(fid, 'p96=flag94 96*((NGtrajectory−94)/2)+flag96 98*(1−(NGtrajectory−96)/2);\n');18 fprintf(fid, 'p98=flag96 98*((NGtrajectory−96)/2);\n');19 fprintf(fid, 'p = [p92;p94;p96;p98];\n');20 fclose(fid);

149

G.4.5 Linear Simulation Input Function

1 function u = GSInputFunction2(t)2 if t>0.13 if t<124 temp1=40;5 else6 temp1=0;7 end8 else9 temp1=0;

10 end1112 if (t>25)13 if t<4014 temp2=100;15 else16 temp2=0;17 end18 else19 temp2=0;20 end2122 u=[temp1;temp2];

G.4.6 LPV Controller s-function

1 function [sys,x0,str,ts] = LPVControllerSFunc2(t,x,u,flag,pdK)23 warning off all45 global ak bk ck dk pdK67 switch flag,8 %%%%%%%%%%%%%%%%%%9 % Initialization %

10 %%%%%%%%%%%%%%%%%%11 % Initialize the states, sample times, and state ordering strings.12 case 01314 [sys,x0,str,ts]=mdlInitializeSizes;1516 case 1,17 sys=mdlDerivatives(t,x,u);1819 %%%%%%%%%%%20 % Outputs %21 %%%%%%%%%%%22 % Return the outputs of the S−function block.23 case 324 sys=mdlOutputs(t,x,u);2526 %%%%%%%%%%%%%%%%%%%27 % Unhandled flags %28 %%%%%%%%%%%%%%%%%%%29 % There are no termination tasks (flag=9) to be handled.30 % Also, there are no continuous or discrete states,31 % so flags 1,2, and 4 are not used, so return an empty u32 % matrix33 case { 1, 2, 4, 9 }34 sys=[];3536 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%37 % Unexpected flags (error handling)%38 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%39 % Return an error message for unhandled flag values.40 otherwise41 error(['Unhandled flag = ',num2str(flag)]);4243 end4445 %46 %=============================================================================47 % mdlInitializeSizes48 % Return the sizes, initial conditions, and sample times for the S−function.49 %=============================================================================

150

50 %51 function [sys,x0,str,ts] = mdlInitializeSizes()52 global ak bk ck dk pdK5354 [ak,bk,ck,dk] = GetModel(pdK,92);5556 sizes = simsizes;57 sizes.NumContStates = size(ak,1); %dynamically sized58 sizes.NumDiscStates = 0;59 sizes.NumOutputs = 1; % 1 outputs: fuel flow (pps)60 sizes.NumInputs = 2; % 1 inputs: NP error61 sizes.DirFeedthrough = 1; % has direct feedthrough62 sizes.NumSampleTimes = 1;6364 sys = simsizes(sizes);65 str = [];66 x0 = zeros(sizes.NumContStates,1);67 ts = [0 0]; % inherited sample time6869 % end mdlInitializeSizes7071 %72 %=============================================================================73 % mdlDerivatives74 % Return the derivatives for the continuous states.75 %=============================================================================76 %77 function sys = mdlDerivatives(t,x,u)7879 global ak bk ck dk pdK8081 ak,bk,ck,dk] = GetModel(pdK,u(1));8283 sys=ak*x + bk*u(2);8485 % end mdlDerivatives8687 %88 %=============================================================================89 % mdlOutputs90 % Return the output vector for the S−function91 %=============================================================================92 %93 function sys = mdlOutputs(t,x,u)94 global ak bk ck dk pdK9596 sys=(ck*x + dk*u(2)).';9798 % end mdlOutputs99

100101 function [ak,bk,ck,dk] = GetModel(pdK,NGInput)102 [ak,bk,ck,dk]=ltiss(psinfo(pdK,'eval',NGInputConversion(NGInput)));103104 %end GetModel105106 function p = NGInputConversion(NGinput)107 x = 1;108 NGUL=100;109 NGLL=76;110 for i=NGLL:2:NGUL111112 if ((i==NGLL)&&(NGinput ≥i && NGinput<(i+2)))113 temp(x)= (1−(NGinput−i)/2);114 elseif ((i==NGUL)&&(NGinput ≥i && NGinput≤(i+2)))115 temp(x)= ((NGinput−(i−2))/2);116 elseif (NGinput >(i−2) && NGinput<(i))117 temp(x)= ((NGinput−(i−2))/2);118 elseif (NGinput ≥i && NGinput<(i+2))119 temp(x)= (1−(NGinput−i)/2);120 else121 temp(x)=0;122 end123 x=x+1;124125 end126 p = temp';127128 %End NGInputConversion

151

Appendix H

Results

H.1 Linear Plants

Table H.1: NG076 Linear PlantA B

-0.05483 74.11 -0.3139 0.2915 0.01312 59.11 -7.559

0 -6516 -267.2 1018 0.3269 5.012e+004 0

0 0 -373.3 358.3 0.1941 0 0

0 0 3674 -4033 0 1.866e+005 0

0 -809.2 -604.3 910.3 -1.157 8.374e+004 0

-0.007253 9.803 -0.04153 0.03856 0.001736 7.82 0

1 0 0 0 0 0 0

0 0 0 0 0.002237 0 0

0 -809.2 -604.3 910.3 -1.157 8.374e+004 0

C D

λNP = −0.05483;λP45 = −6516;λNG − 0.5975;λPs3 = −43.99;λP41 = −4363;


-0.05747 74.26 -1.667 1.546 0.01172 312.2 -7.559

0 -6576 -266.3 1019 0.3274 4.964e+004 0

0 0 -376.1 360.6 0.2032 0 0

0 0 3684 -4044 0 1.867e+005 0

0 -793.1 -594.5 902.3 -1.19 8.374e+004 0

-0.007602 9.824 -0.2205 0.2045 0.00155 41.3 0

1 0 0 0 0 0 0

0 0 0 0 0.002237 0 0

0 -793.1 -594.5 902.3 -1.19 8.374e+004 0

C D

152

A B

C D

λNP = −0.05747;λP45 = −6576;λNG − 0.5903;λPs3 = −44.71;λP41 = −4376;


-0.06099 78.8 -3.255 3.032 0.01709 604.1 -7.559

0 -5934 -265.7 1024 0.7109 4.904e+004 0

0 0 -379.3 363.7 0.4608 0 0

0 0 3705 -4070 0 1.869e+005 0

0 -778.5 -602.5 898.4 -2.653 8.364e+004 0

-0.008069 10.42 -0.4306 0.4011 0.00226 79.92 0

1 0 0 0 0 0 0

0 0 0 0 0.002237 0 0

0 -778.5 -602.5 898.4 -2.653 8.364e+004 0

C D

λNP = −0.06099;λP45 = −5933;λNG − 1.475;λPs3 = −45.78;λP41 = −4404;


-0.06826 82 -6.064 5.672 0.01698 1083 -7.559

0 -2869 -138.7 519.8 0.2854 2.385e+004 0

0 0 -392.4 371.3 0.4164 0 0

0 0 3776 -4150 0 1.87e+005 0

0 -774.2 -594.9 920.8 -2.326 8.493e+004 0

-0.00903 10.85 -0.8021 0.7503 0.002247 143.2 0

1 0 0 0 0 0 0

0 0 0 0 0.002237 0 0

0 -774.2 -594.9 920.8 -2.326 8.493e+004 0

C D

λNP = −0.06826;λP45 = −2869;λNG − 1.204;λPs3 = −51.57;λP41 = −4492;


-0.07621 84.47 -8.508 7.927 0.01765 1489 -7.559

0 -2496 -135.1 521.7 0.3817 2.297e+004 0

0 0 -397 376.3 0.8461 0 0

0 0 3810 -4185 0 1.874e+005 0

0 -765.3 -588.8 921.8 -4.04 8.481e+004 0

-0.01008 11.17 -1.125 1.049 0.002334 196.9 0

1 0 0 0 0 0 0

0 0 0 0 0.002237 0 0

0 -765.3 -588.8 921.8 -4.04 8.481e+004 0

C D

λNP = −0.07621;λP45 = −2496;λNG − 1.798;λPs3 = −52.73;λP41 = −4531;

153


-0.08699 87.31 -10.99 10.06 0.01959 1936 -7.559

0 -2234 -130.4 500.4 0.383 2.208e+004 0

0 0 -403.1 380.6 0.7216 0 0

0 0 3816 -4197 0 1.877e+005 0

0 -1054 -581.8 983.8 -3.714 8.476e+004 0

-0.01151 11.55 -1.454 1.33 0.002592 256.1 0

1 0 0 0 0 0 0

0 0 0 0 0.002237 0 0

0 -1054 -581.8 983.8 -3.714 8.476e+004 0

C D

λNP = −0.08699;λP45 = −2234;λNG − 1.805;λPs3 = −54.81;λP41 = −4547;


-0.1004 92.2 -13.71 11.82 0.008076 2347 -7.559

0 -1966 -126.5 456.7 0.3872 2.124e+004 0

0 0 -406.7 387.1 0.6577 0 0

0 0 3867 -4254 0 1.881e+005 0

0 -1793 -601.3 1153 -3.574 8.414e+004 0

-0.01328 12.2 -1.813 1.563 0.001068 310.5 0

1 0 0 0 0 0 0

0 0 0 0 0.002237 0 0

0 -1793 -601.3 1153 -3.574 8.414e+004 0

C D

λNP = −0.1004;λP45 = −1966;λNG − 1.654;λPs3 = −52.89;λP41 = −4610;


-0.1112 97.11 -15.79 13.69 0.00132 2614 -7.559

0 -3752 -258 928.3 0.966 4.18e+004 0

0 0 -412.9 393 0.6894 0 0

0 0 3931 -4326 0 1.884e+005 0

0 -1781 -617.7 1163 -4.074 8.36e+004 0

-0.01471 12.85 -2.089 1.811 0.0001746 345.8 0

1 0 0 0 0 0 0

0 0 0 0 0.002237 0 0

0 -1781 -617.7 1163 -4.074 8.36e+004 0

C D

λNP = −0.1112;λP45 = −3752;λNG − 2.442;λPs3 = −53.67;λP41 = −4687;


-0.1286 106.1 -19.1 16.66 -0.0107 2961 -7.559

0 -3722 -262.4 955.8 1.047 4.068e+004 0

C D

154

A B

0 0 -418 402 0.7379 0 0

0 0 4046 -4453 0 1.889e+005 0

0 -1780 -669.3 1190 -4.38 8.253e+004 0

-0.01701 14.04 -2.526 2.204 -0.001415 391.7 0

1 0 0 0 0 0 0

0 0 0 0 0.002237 0 0

0 -1780 -669.3 1190 -4.38 8.253e+004 0

C D

λNP = −0.1286;λP45 = −3722;λNG − 3.063;λPs3 = −50.6;λP41 = −4822;


-0.1495 126.8 -23.06 17.7 -0.01199 3295 -7.559

0 -2665 -278.2 745.3 0.9913 3.98e+004 0

0 0 -428.9 413.3 0.8415 0 0

0 0 4202 -4624 0 1.891e+005 0

0 -3597 -709.9 1618 -4.935 8.137e+004 0

-0.01977 16.77 -3.05 2.341 -0.001586 435.9 0

1 0 0 0 0 0 0

0 0 0 0 0.002237 0 0

0 -3597 -709.9 1618 -4.935 8.137e+004 0

C D

λNP = −0.1495;λP45 = −2664;λNG − 2.582;λPs3 = −52.87;λP41 = −5003;


-0.1731 128 -26.63 20.82 -0.02577 3583 -7.559

0 -2624 -283.9 768.6 0.9343 3.823e+004 0

0 0 -437.8 423.3 0.8101 0 0

0 0 4328 -4768 0 1.9e+005 0

0 -3614 -759.8 1651 -4.765 7.988e+004 0

-0.0229 16.94 -3.523 2.754 -0.003409 474 0

1 0 0 0 0 0 0

0 0 0 0 0.002237 0 0

0 -3614 -759.8 1651 -4.765 7.988e+004 0

C D

λNP = −0.1731;λP45 = −2624;λNG − 3.317;λPs3 = −52.21;λP41 = −5156;


-0.1995 133.8 -31.02 24.82 -0.0397 3809 -7.559

0 -2695 -300.2 807.4 0.6872 3.687e+004 0

0 0 -450.9 437.6 0.5982 0 0

0 0 4531 -4996 0 1.905e+005 0

0 -3655 -827.4 1710 -3.543 7.816e+004 0

C D

155

A B

-0.02639 17.7 -4.104 3.284 -0.005251 503.9 0

1 0 0 0 0 0 0

0 0 0 0 0.002237 0 0

0 -3655 -827.4 1710 -3.543 7.816e+004 0

C D

λNP = −0.1995;λP45 = −2694;λNG − 2.863;λPs3 = −51.57;λP41 = −5396;


-0.2168 139.5 -34.68 28.07 -0.04104 3858 -7.559

0 -2782 -324.3 848.2 0.7437 3.598e+004 0

0 0 -458.2 444.2 0.6295 0 0

0 0 4689 -5170 0 1.938e+005 0

0 -3716 -869.8 1771 -3.746 7.572e+004 0

-0.02868 18.46 -4.587 3.713 -0.005429 510.4 0

1 0 0 0 0 0 0

0 0 0 0 0.002237 0 0

0 -3716 -869.8 1771 -3.746 7.572e+004 0

C D

λNP = −0.2168;λP45 = −2781;λNG − 3.021;λPs3 = −52.87;λP41 = −5576;

156

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