Thesis - Beaver simulation
.pdfPreface |
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Preface.
This report is the result of my graduation research, which has been done in the period from may 1992 to the date of publishing, starting with some preliminary work which was published in ref.[25]. This report describes the construction of a model library in SIMULINKfor evaluating automatic control systems for the 'Beaver' aircraft and it presents a detailed analysis of the 'Beaver' autopilot as a case-study. The report has been divided in two separate parts:
Part I gives a detailed description of the SIMULINKmodels which were developed during this research. These models form a further improvement of the ones described in my preliminary thesis, ref.[25]. Part I of this report can be read independently of ref.[25], although there is of course some overlap (this version is better, however!).
Part I1 describes a case study of the 'Beaver' autopilot, which has been implemented in SIMULINK,using the models from part I. The autopilot structure is discussed and many simulation results are shown. Part I1 has been structured as a separate report, so the results of the autopilot analysis can be read independently, without having detailed knowledge of the SIMULINKimplementation.
Both parts of the report can (and should!) be used as references for future research on Automatic Aircraft Control Systems (AACSs), aimed at the new DUT/NLR laboratory aircraft, the Cessna 'Citation 11'. The process which leads from preliminary AACS design to in-flight testing of the system is quite complex and should be mastered in detail to ensure safe and satisfying AACS operation in flight. This can only be achieved if appropriate hardware and software tools for system analysis are developed.
The reader should be familiar with 'classical' control theory. Some basic knowledge about aircraft dynamics and MATLABis also required. A short introduction to SIMULINKhas been included in part I.
Acknowledgements |
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Acknowledgements.
I wish to express my thanks to all members of the 'Beaver Taskforce' for their support and for the pleasant working climate. The taskforce was founded by prof.dr.ir. Bob Mulder and lead by dr.ir. Hans van der Vaart of the disciplinary group for Stability and Control, for redesign, nonlinear evaluation, implementation, and in-flight testing of the 'Beaver' autopilot. The other taskforce members were Patrick Wever, Bardia Ghahramani, Marcel van Witzenburg, Samir Bennani, and Zainal Abidin. Later also Eric Kruijsen joined the group. Thank you all for the constant feedback of ideas!
Special thanks go to Samir Bennani for letting me (ab-?) use his personal computer. I really wouldn't have known what to do without it. I also would like to thank my father for his support during the preparation of this report, which really helped me out when time became a real issue.
Marc Rauw, Delft, september 1993.
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Summary of part I.
Since a couple of years, the Disciplinary Group for Stability and Control of Delft University of Technology, Faculty of Aerospace Engineering, has done research into automatic aircraft control technology with its laboratory aircraft, the De Havilland DHCd 'Beaver'. Recently, this aircraft has been replaced by a new Cessna Citation 11, which will be configured as National Fly-by-wire Testbed (NFT), to increase the research efforts in this field.
A prerequisite for successful fly-by-wire research is the availability of powerful tools for control system analysis and design. In this report, a flexible environment for the analysis of aircraft dynamics and control will be developed for the DHC-2 'Beaver' aircraft. This environment uses the power and flexibility of the simulation and system analysis programs SIMULINK and MATLAB.
The heart of the software package is formed by the SIMULINKimplementation of the nonlinear 'Beaver' simulation model. In part I1 of this report, this model will be used for the analysis of the Beaver' autopilot. Due to its modular structure, the model can easily be adapted for other aircraft. Aircraft trim and linearization tools have been included, to be able to do the whole linear and nonlinear control system design and analysis from within the same MATLABBIMULINKenvironment. The package also contains blocks to simulate the influence of atmospheric disturbances upon the motions of the aircraft, and to generate radio-navigation signals for the assessment of navigation and approach control laws.
The tools from this report can be used for a broad range of applications in the field of aircraft Stability and Control analysis. Although this report will emphasize their applicability to Automatic Aircraft Control Systems design, they can also be used for system identification and sophisticated open-loop analysis of the aircraft dynamics, due to the availability of the full range of MATLABtoolboxes. The current models can readily be used for educational purposes; application for the NFT project is possible after altering some subsystems for implementation of the Citation I1 model.
In the future, the tools need to be developed further into a standardized analytical tool which must be applicable to virtually any aircraft. If possible, easy links from this environment to the flightsimulator and flight control computers of the aircraft need to be made, in order to shorten the development cycle of automatic control systems. The tools do require sufficient computer capacity, but the resulting increase in productivity is absolutely necessary for the Disciplinary Group for Stability and Control to be able to set new standards for fly-by-wire research.
Contents |
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Contents (part I). |
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Preface ................................................... |
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Acknowledgements ......................................... |
vii |
Symbols and definitions (part I) ................................. |
1 |
0.1Symbols ........................................................ |
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0.2 Vectom and vector functions ......................................... |
5 |
0.3Matrices ........................................................ |
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0.4Function.s ....................................................... |
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0.5Indices ......................................................... |
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0.6Superscripts ..................................................... |
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0.7 Abbreviations .................................................... |
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0.8 Reference frames and sign conventions ................................. |
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0.8.1 Definitions ................................................. |
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0.8.2 Relationships between the reference frames ....................... |
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0.8.3 Sign conventions for deflections of control surfaces .................. |
11 |
Introduction |
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Chapter 1.Design and evaluation of Automatic Aircraft Control Systems |
19 |
1.1Introdudion .................................................... |
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1.2 The AACS design process .......................................... |
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1.3The place of this research within the AACS design process ................. |
22 |
1.4 Conclusions .................................................... |
. 24 |
Chapter 2. Dynamic models of the 'Beaver'. atmospheric turbulence. |
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navigation signals .......................................... |
27 |
2.1Inhduction .................................................... |
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2.2 Aircraft equations of motion ........................................ |
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2.2.1 General nonlinear equations of motion ........................... |
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2.2.2 Force and moment models ..................................... |
30 |
2.2.3 Writing the -equation explicitly ............................... |
36 |
2.2.4 Atmosphere model ........................................... |
37 |
2.2.5 Additional output equations ................................... |
40 |
2.3 Wind and atmospheric turbulence .................................... |
42 |
2.4 VOR navigation and ILS approach system signals ........................ |
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2.5Conclusions ..................................................... |
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Chapter 3.Analysis of nonlinear dynamical systems with SIMULINK.... |
49 |
3.lIntmdudion .................................................... |
49 |
3.2 The programs SIMULINKand WTLAB................................. |
49 |
3.3 Advantages and disadvantages of |
SIMULINK) compared with other |
simulation |
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environments |
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3.4 |
System representation in SIMULINK. . |
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3.5 The analytical functions of S ~ N |
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3.5.1 The integration routines ...................................... |
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54 |
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3.5.2 The linearization routines ..................................... |
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55 |
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3.5.3 The trim routines ........................................... |
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55 |
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3.6 SI- |
block-diagrams .......................................... |
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56 |
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3.7 Conclusions ..................................................... |
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56 |
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Chapter 4.Implementation of the nonlinear 'Beaver' model in SIMULINK. 59 |
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4.1Introduction .................................................... |
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59 |
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4.2 Modular structure of the aircraft models ............................... |
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4.3 |
Adding external disturbances. navigation and approach models) and control loops 66 |
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4.4 Conclusions ..................................................... |
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68 |
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Chapter 5. Assessment of the new SIMULINKimplementation of the 'Beaver' |
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model |
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71 |
5.1 |
Introduction .................................................... |
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71 |
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5.2 |
Results from the aerodynamic and engine ........................models |
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71 |
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5.3 Assessment of open-loop responses computed with the new SIMULINKmodels ... 72 |
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5.4 |
Open-loop responses to atmospheric turbulence .......................... |
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75 |
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5.5 |
Open-loop analysis of the linearized 'Beaver' ......................model |
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76 |
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5.6 |
Conclusions ..................................................... |
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77 |
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Chapter 6.Using the nonlinear SIWLIM~'Beaver' model in practice |
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6.1Introdudion ................................................... |
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101 |
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6.2 Creating nonlinear open-loop responses to deflections of |
control surfaces and engine |
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inputs ........................................................... |
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101 |
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6.3 Creating linear open-loop responses to deflections of |
control surfaces and engine |
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inputs ........................................................... |
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103 |
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6.4 |
Creating nonlinear open-loopresponses ..........to atmospheric turbulence |
104 |
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6.5 Linearization of the aircraft model and applications for the linear model |
..... 105 |
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6.6 |
Conclusions .................................................... |
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106 |
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Recommendations (part I) .................................... |
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113 |
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References ................................................ |
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115 |
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Appendix A.Nonlinear equations of motion ........for a rigid aircraft |
121 |
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A.l Introduction ................................................... |
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121 |
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A.2 |
Derivation of the equations for velocities and rotational velocities in the body-fixed |
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reference frame ................................................... |
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121 |
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A.2.1 General force equation for a rigid ..........................body |
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121 |
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A.2.2 General moment equation for a .......................rigid body |
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122 |
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A.2.3 Angular momentum about the centre of gravity ................... |
123 |
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A.2.4 General equations of motion for a rigid body ...................... |
124 |
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A.3 Using the angle of |
attack. sideslip angle. and total aimpeed in stead of |
the velocity |
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components in Fs .................................................. |
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127 |
A.3.1 Why use flight-path axes for the state equations? .................. |
127 |
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A.3.2 Transformations of fomes and velocities from body-axes to flight-path axes 129 |
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A.3.3 Derivation of the V-equation ................................. |
130 |
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A.3.4 Derivation of the a-equation ................................. |
131 |
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A.3.5 Derivation of the P-equation .................................. |
132 |
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A.4 Equations of motion in a non-steady atmosphere ....................... |
133 |
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A.5 Kinematic relations |
and transformation of gravity forces h m earth-axes to body- |
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axes ............................................................ |
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136 |
A.5.1 The Euler angles ......................................... |
. 136 |
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A.5.2 Gravity forces in the body-fixed reference frame ................... |
136 |
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A.5.3 The position of the aircraft ................................... |
136 |
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A.6 Summary of the state equations .................................... |
137 |
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A.7 Conclusions ................................................... |
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Appendix B.Mathematical models of wind and atmospheric turbulence ..141 |
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B.1 Intmduction ................................................... |
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141 |
B.2 Wind profiles and wind shear ...................................... |
141 |
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B.3 Modelling atmospheric turbulence .................................. |
142 |
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B.4 Power spectra of atmospheric turbulence ............................. |
145 |
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B.4.1 The von K |
h |
k spectra ..................................... |
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B.4.2 The Dryden spectra ........................................ |
145 |
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B.5 Filter design for atmospheric turbulence .............................. |
146 |
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B.5.1 Modelling atmospheric turbulence as filtered white noise ............ |
146 |
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B.5.2 Filter design for the Dryden spectra ............................ |
147 |
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B.5.3 Filter design for the von K h h spectra ........................ |
148 |
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B.6 Conclusions ................................................... |
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148 |
Appendix C. The Instrument Landing System and the VOR navigation |
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system ................................................... |
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151 |
C.l Intmduction ................................................... |
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151 |
C.2 The Instrument Landing System ................................... |
151 |
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C.2.1 Nominal I I S signals ........................................ |
151 |
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C.2.2 Steady-state ILS offset errors ................................. |
158 |
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C.2.3 ILS noise characteristics ..................................... |
159 |
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C.3 The VOR navigation system ....................................... |
162 |
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C.3.1 Nominal VOR signals ....................................... |
162 |
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C.3.2 VOR coverage and cone of silence .............................. |
163 |
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C.3.3 VOR steady-state errors ..................................... |
165 |
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C.4 Conclusions ................................................... |
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165 |
Appendix D. Simulation of |
dynamic systems in SIMULINK. theoretical |
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backgrounds .............................................. |
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167 |
D.l Introduction . . . . . . . . . . |
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167 |
D.2 The integrators ................................................ |
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D.2.1 Some fundamental concepts .................................. |
167 |
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D.2.2 A review of several numerical integration methods ................. |
171 |
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a.Taylor series methods ...................................... |
171 |
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b.Runge-Kutta methods ...................................... |
171 |
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c.Multistep methods |
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176 |
d.Extrapolation methods ..................................... |
180 |
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D.2.3 Stiff differential equations ................................... |
180 |
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D.2.4 Differential algebraic equations ............................... |
182 |
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D.3 Simulation of digital controllers .................................... |
183 |
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D.4 Obtaining state models h |
m transfer functions ........................ |
184 |
D.5 Algebraic loops ................................................. |
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186 |
D.6 Conclusions ................................................... |
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189 |
Appendix E.Steady-state trimmed flight ......................... |
191 |
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E.lIntroduction ................................................... |
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191 |
E.2 Definition of steady-state flight .................................... |
191 |
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E.3 Specification of the flight condition .................................. |
192 |
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E.4 The rate-of-climb and turncoordination constraints ...................... |
193 |
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E.5The steady-state aircraft trim algorithm .............................. |
194 |
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E.6 Implementation in M A ~ S W N K : basic principles ................... |
196 |
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E.7 Conclusions ................................................... |
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198 |
Appendix F.The SIMULINKsimulation models..................... |
201 |
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F.l Introduction ................................................... |
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201 |
F.2 Nonlinear model of the aircraft dynamics: state and output equations ....... |
202 |
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F.2.1 Introduction .............................................. |
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202 |
F.2.2 List of variables. used for the aircraft state and output models ........ |
202 |
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F.2.3 The a k r a f t model library and the complete aircraft model ........... |
206 |
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F.2.4 First level of the simulation model BEAVER ...................... |
206 |
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F.2.5 Beaver dynamics and output equations (second level of BEAVER) ...... |
209 |
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a.AirdataGmup ........................................... |
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210 |
b.Aerodynamics Group ....................................... |
213 |
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217 |
c Engine Group ............................................ |
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d.Gravity forces ............................................ |
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218 |
e.Windforces ............................................. |
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218 |
f.Total forces and moments ................................... |
218 |
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g.Equations of motion ....................................... |
220 |
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h.Otheroutpu ts ............................................ |
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226 |