Frequency Domain System Identification of Fixed-Wing Unmanned Aerial Vehicles

208 Pages·2014·3.67 MB·English

Frequency Domain System Identification

of Fixed-Wing Unmanned Aerial Vehicles

By
KAIWEN XU
8/14/2014

A Thesis
Submitted to the Faculty of Graduate Studies of the University of Manitoba
In Partial Fulfillment of the Requirements of the Degree of
MASTER OF SCIENCE

Department of Mechanical Engineering
The University of Manitoba
Winnipeg, Manitoba

Copyright 2014 © Kaiwen Xu
Abstract
The goal of this thesis is to identify airplanes’ reduced order transfer functions, and aerodynamic
derivatives in the longitudinal channel. The outcome of the research will benefit aircraft systems’
controller design, modeling and simulation. Aircraft under study are a conventional fixed wing
airplane called Cropcam, and a nonconventional delta-wing aircraft. To identify the system
transfer functions and aerodynamic derivatives, direct and indirect frequency domain
identification methods are applied. For the direct method, the Equation Error (EE) method is
adopted to process the Cropcam’s input-output data pairs and identify the aerodynamic
derivatives from the flight data directly. The indirect approach is called the Transfer Function
(TF) method. For this method, a commercially available system identification tool kit called
CIFER is utilized to identify the longitudinal transfer function of the aircraft first. Then the
aerodynamic derivatives are extracted from the identified transfer function. The derivatives
identified by the EE method and transfer function method are compared with the ones computed
from a Vortex Lattice based program called AVL. The identification results are further verified
by comparing computer simulation outputs with flight test responses. Issues such as input
excitation design, data gathering, data reliability analysis and result verification are also
investigated in this thesis.

i | P age
Table of Contents

Abstract ............................................................................................................................................ i
Acknowledgements ......................................................................................................................... v
List of Figures ................................................................................................................................ vi
List of Tables .................................................................................................................................. x
Nomenclature ................................................................................................................................. xi
1 Introduction ............................................................................................................................. 1
1.1 Motivation ........................................................................................................................ 1
1.2 Objectives ......................................................................................................................... 2
1.3 Methodology .................................................................................................................... 3
1.4 Thesis outline ................................................................................................................... 6
2 Background .............................................................................................................................. 9
2.1 Preliminary remark ........................................................................................................... 9
2.2 System identification ...................................................................................................... 10
3 Basic Aerodynamic Theory ................................................................................................... 14
3.1 Coordinate axes .............................................................................................................. 14
3.1.1 Earth axes ................................................................................................................ 14
3.1.2 Aircraft body-fixed axes ......................................................................................... 16
3.2 Definition of variables .................................................................................................... 17
3.3 Axes transformations...................................................................................................... 21
3.3.1 Linear acceleration, velocity and displacement transformation ............................. 21
3.3.2 Angular velocities transformation........................................................................... 26
3.4 Aircraft reference geometry and control surfaces .......................................................... 28
3.4.1 Reference geometry ................................................................................................ 28
3.4.2 Aircraft control surfaces ......................................................................................... 29
3.5 Summary ........................................................................................................................ 30
4 Aircraft Motion Equations ..................................................................................................... 31
4.1 General equations of motion .......................................................................................... 31
4.1.1 Force equations ....................................................................................................... 38
4.1.2 Moment equations ................................................................................................... 40
4.1.3 Kinematics equations .............................................................................................. 41
ii | P age
4.1.4 Navigation equations .............................................................................................. 41
4.2 Aerodynamic terms ........................................................................................................ 43
4.3 Equations of motion for small perturbations .................................................................. 45
4.4 Decoupled motion equations .......................................................................................... 47
4.4.1 Longitudinal motion equations ............................................................................... 47
4.4.2 Lateral motion equations......................................................................................... 51
4.5 Summary ........................................................................................................................ 53
5 Input Excitation ..................................................................................................................... 54
5.1 Input excitation requirements of system identification .................................................. 54
5.1.1 Frequency range ...................................................................................................... 54
5.1.2 Length of input excitation ....................................................................................... 56
5.1.3 Amplitude of input signal ....................................................................................... 57
5.1.4 Flight test conditions ............................................................................................... 59
5.2 Optimal input design ...................................................................................................... 61
5.3 Summary ........................................................................................................................ 69
6 Estimation of Aerodynamic Derivatives ............................................................................... 71
6.1 Transfer function method ............................................................................................... 72
6.1.1 Obtaining transfer function empirically .................................................................. 73
6.1.2 Obtaining transfer function analytically ................................................................. 77
6.1.3 Applying the transfer function method ................................................................... 82
6.2 Equation error method .................................................................................................... 85
6.2.1 Data transformation ................................................................................................ 86
6.2.2 Equation error method in frequency domain .......................................................... 87
6.3 Summary ........................................................................................................................ 91
7 Experimental Results ............................................................................................................. 93
7.1 Identification of Cropcam UAV ..................................................................................... 94
7.1.1 Transfer function method ...................................................................................... 103
7.1.2 Equation error method .......................................................................................... 115
7.2 Identification of a delta-wing UAV ............................................................................. 148
7.3 Summary ...................................................................................................................... 157
8 Conclusions ......................................................................................................................... 159
8.1 Contributions made by this thesis ................................................................................ 159
8.2 Future research ............................................................................................................. 161
iii | P age
References ................................................................................................................................... 163
Appendix 1 Derivatives Conversion ........................................................................................... 166
Appendix 2 Simulation Results .................................................................................................. 171
A2.1 Simulation responses .................................................................................................... 171
A2.2 Identification using transfer function based method .................................................... 180
A2.3 Identification using equation error method .................................................................. 189

iv | P age
Acknowledgements
First, I would like to thank my supervisor Dr. Nariman Sepehri and co-advisor Dr. Hamidreza
Bolandhemmat, whose invaluable guidance and support has helped steer my work in the right
direction over the past two years. I also thank the staffs from MicroPilot Inc., especially Howard
Loewen (president of the company), Asadilan Indrabudi and Adam Toews for providing me
technical support and fruitful suggestions. I thank my colleagues from the Fluid Power & Tele-
Robotics Research Laboratory, especially Morgan May and Divyarajsinh Raol, for their kind
help during this research. I also thank NSERC, MITACS and MicroPilot Inc., for sponsoring the
research. Finally, sincere thanks go to Drs. W Kinsner and S Balakrishnan, for the examination
of this thesis and their constructive comments.
v | P age
List of Figures

Figure 1.1 Frequency-response system identification procedure ................................................... 6
Figure 3.1 Earth axes (adapted from Cook, 2012) ........................................................................ 15
Figure 3.2 General aircraft coordinate system .............................................................................. 16
Figure 3.3 Aircraft motion variables ............................................................................................. 18
Figure 3.4 Longitudinal channel terms for typical aircraft climbing ............................................ 19
Figure 3.5 Quantities in body axes under symmetric flight (adapted from Cook, 2012) ............. 20
Figure 3.6 Axes transformation (adapted from Cook, 2012) ........................................................ 21
Figure 3.7 Transformation of velocities through incidence and sideslip angles (Cook, 2012) .... 24
Figure 3.8 Transformation of angular velocity (adapted from Cook, 2012) ................................ 26
Figure 3.9 Reference geometry ..................................................................................................... 28
Figure 3.10 Aircraft control surfaces (adapted from Cook, 2012) ............................................... 29
Figure 5.1 Elevator deflection....................................................................................................... 58
Figure 5.2 Body pitch rate ............................................................................................................ 58
Figure 5.3 Doublet input excitation .............................................................................................. 60
Figure 5.4 Body pitch rate ............................................................................................................ 60
Figure 5.5 Typical 3-2-1-1 signal ................................................................................................. 62
Figure 5.6 Frequency sweep input excitation ............................................................................... 64
Figure 5.7 Increasing trend of sweep frequency ........................................................................... 66
Figure 5.8 Typical Chirp input excitation ..................................................................................... 68
Figure 5.9 Finite Fourier transform of input signal ...................................................................... 69
Figure 6.1 Short-period mode flight (adapt from Cook, 2012)..................................................... 78
Figure 6.2 Phugoid mode flight (adapt from Cook, 2012) ........................................................... 78
Figure 6.3 Aircraft responses (w, q, α) to a 1° elevator step input ............................................... 79
Figure 6.4 Aircraft responses (u, θ, ) to a 1° elevator step input ................................................ 80
Figure 7.1 Cropcam aircraft ..........(cid:1)................................................................................................ 94
Figure 7.2 Manually generated Chirp input excitation ................................................................. 96
Figure 7.3 FFT of input signal ...................................................................................................... 97
Figure 7.4 Total axial velocity U .................................................................................................. 98
Figure 7.5 Total lateral velocity V ................................................................................................ 98
vi | P age
Figure 7.6 Total normal velocity W .............................................................................................. 99
Figure 7.7 Body roll rate p ............................................................................................................ 99
Figure 7.8 Body pitch rate q ....................................................................................................... 100
Figure 7.9 Body yaw rate r ......................................................................................................... 100
Figure 7.10 Roll angle φ ............................................................................................................. 101
Figure 7.11 Pitch angle θ ............................................................................................................ 101
Figure 7.12 Yaw angle ψ ............................................................................................................ 102
Figure 7.13 Altitude z ................................................................................................................. 102
Figure 7.14 -q data pair ........................................................................................................... 104
Figure 7.15 (cid:2)(cid:3)-w data pair .......................................................................................................... 104
Figure 7.16 B(cid:2)(cid:3)ode plot of -w .................................................................................................... 105
Figure 7.17 Bode plot of (cid:2)(cid:3)-q..................................................................................................... 105
Figure 7.18 Verification o(cid:2)f(cid:3) magnitude of -w .......................................................................... 107
Figure 7.19 Verification of phase of -w(cid:2) ..(cid:3)................................................................................ 107
Figure 7.20 Verification of magnitud(cid:2)e(cid:3) of -q ........................................................................... 108
Figure 7.21 Verification of phase of -q (cid:2)..(cid:3)................................................................................ 108
Figure 7.22 Comparison of simulated(cid:2) (cid:3)w and measured w .......................................................... 109
Figure 7.23 Comparison of simulated q and measured q............................................................ 110
Figure 7.24 Unseen Doublet signal ............................................................................................. 111
Figure 7.25 Comparison of simulated w and measured w with the unseen Doublet signal ........ 112
Figure 7.26 Comparison of simulated q and measured q with the unseen Doublet signal ......... 112
Figure 7.27 Delay removed -q data pair ................................................................................. 116
Figure 7.28 Delay removed (cid:2)(cid:3)-w data pair ................................................................................ 117
Figure 7.29 Delay removed (cid:2)(cid:3)-u data pair ................................................................................. 117
Figure 7.30 Original data sam(cid:2)(cid:3)pling time .................................................................................... 118
Figure 7.31 Verification of EE identified pitching moment related derivatives ......................... 122
Figure 7.32 Verification of AVL estimated pitching moment related derivatives ..................... 123
Figure 7.33 Unseen random signal used for verification ............................................................ 124
Figure 7.34 Verification of EE identified pitching moment related derivatives with unseen
random signal .............................................................................................................................. 125
Figure 7.35 Verification of AVL computed pitching moment related derivatives with unseen
random signal .............................................................................................................................. 126
Figure 7.36 Unseen Doublet signal ............................................................................................. 127
vii | Page
Figure 7.37 Verification of EE identified pitching moment related derivatives with unseen
Doublet signal ............................................................................................................................. 128
Figure 7.38 Verification of AVL computed pitching moment related derivatives with unseen
Doublet signal ............................................................................................................................. 129
Figure 7.39 Verification of EE identified lift force related derivatives with original signal ...... 131
Figure 7.40 Verification of AVL computed lift force related derivatives with original signal .. 132
Figure 7.41 Verification of EE identified lift force related derivatives with unseen random signal
..................................................................................................................................................... 133
Figure 7.42 Verification of AVL computed lift force related derivatives with unseen random
signal ........................................................................................................................................... 134
Figure 7.43 Verification of EE identified lift force related derivatives with unseen Doublet signal
..................................................................................................................................................... 135
Figure 7.44 Verification of AVL computed lift force related derivatives with unseen Doublet
signal ........................................................................................................................................... 136
Figure 7.45 Verification of EE identified axial force related derivatives with original signal ... 138
Figure 7.46 Verification of AVL computed axial force related derivatives with original signal 139
Figure 7.47 Verification of EE identified axial force related derivatives with unseen random
signal ........................................................................................................................................... 140
Figure 7.48 Verification of AVL computed axial force related derivatives with unseen random
signal ........................................................................................................................................... 141
Figure 7.49 Verification of EE identified axial force related derivatives with unseen Doublet
signal ........................................................................................................................................... 142
Figure 7.50 Verification of AVL computed axial force related derivatives with unseen Doublet
signal ........................................................................................................................................... 143
Figure 7.51 A delta-wing UAV (From uas.trimble.com) ........................................................... 148
Figure 7.52 Input excitation for delta-wing aircraft.................................................................... 149
Figure 7.53 Measured output of the delta-wing aircraft ............................................................. 149
Figure 7.54 Finite Fourier transform plot of input excitation ..................................................... 150
Figure 7.55 Sampling time .......................................................................................................... 151
Figure 7.56 Bode plot of δ -q ...................................................................................................... 152
e
Figure 7.57 Comparison of magnitudes ...................................................................................... 153
Figure 7.58 Comparison of phases.............................................................................................. 153
Figure 7.59 Comparison of simulated output and measured output ........................................... 154
Figure 7.60 Unseen input excitation used for transfer function verification .............................. 155
Figure 7.61 Comparison of simulated and measured outputs with unseen input signal ............. 155
Figure A 1 Input excitation ......................................................................................................... 172
viii | Page
Figure A 2 Total velocity V ........................................................................................................ 173
Figure A 3 Angle of attack α ...................................................................................................... 173
Figure A 4 Sideslip angle β ......................................................................................................... 174
Figure A 5 Roll rate p ................................................................................................................. 174
Figure A 6 Pitch rate q ................................................................................................................ 175
Figure A 7 Yaw rate r ................................................................................................................. 175
Figure A 8 Roll angle φ .............................................................................................................. 176
Figure A 9 Pitch angle θ ............................................................................................................. 176
Figure A 10 Yaw angle ψ ........................................................................................................... 177
Figure A 11 Axial position x ....................................................................................................... 177
Figure A 12 Lateral position y .................................................................................................... 178
Figure A 13 Altitude z................................................................................................................. 178
Figure A 14 Chirp input excitation ............................................................................................. 180
Figure A 15 FFT of input excitation ........................................................................................... 181
Figure A 16 Bode plot of -w ................................................................................................... 182
Figure A 17 Bode plot of (cid:2)(cid:3)-q .................................................................................................... 183
Figure A 18 Magnitude of(cid:2) (cid:3) -w ................................................................................................. 184
Figure A 19 Phase of -w(cid:2) ..(cid:3)....................................................................................................... 185
Figure A 20 Magnitud(cid:2)e(cid:3) of -q .................................................................................................. 185
Figure A 21 Phase of -q .(cid:2)..(cid:3)....................................................................................................... 186
Figure A 22 Verificati(cid:2)o(cid:3)n of vertical velocity w .......................................................................... 187
Figure A 23 Verification of pitch rate q ...................................................................................... 187

ix | P age