SYSTEM ANALYSIS THROUGH BOND GRAPH MODELING by ...

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TABLE OF CONTENTS (continued) 6.5.2 Optimal Efficiency Comparisons............................................................ 258 6.6 Nonlinear Pitch Autopilot: An SDRE Approach........................................... 264 6.6.1 LQR Formulation and General Solution................................................. 264 6.6.2 LQR General Solution for Nonzero Feed-Through................................ 265 6.6.3 LQR Solution for Nonzero Feed-Through and Output Feedback .......... 268 6.6.4 LQR Tracking Solution for Nonzero Feed-Through, Output Feedback and Zero Steady State Error........................................................................................... 269 6.6.5 Dymola Implementation of the LQR Tracking Solution ........................ 271 6.6.6 Nonlinear Autopilot Results ................................................................... 279 6.7 Power Flow Analysis with Varying Mass Parameters................................... 286 6.7.1 Center of Gravity Shift............................................................................ 286 6.8 Conclusions.................................................................................................... 291 CHAPTER 7: Summary .......................................................................... 293 7.1 Contributions.................................................................................................. 293 7.1.1 Modeling................................................................................................. 293 7.1.2 Simulation............................................................................................... 293 7.1.3 System Analysis...................................................................................... 294 7.1.4 Controller Design.................................................................................... 295 7.2 Future Work................................................................................................... 296 7.2.1 Modeling................................................................................................. 296 7.2.2 Simulation............................................................................................... 296 7.2.3 System Analysis and Controller Design ................................................. 297 APPENDIX A1: Dymola Model, SDRE Code Listing .......................... 298 APPENDIX A2: Dymola Model, Riccati4 Code Listing....................... 302 APPENDIX A3: Dymola Model, Heig4 Code Listing........................... 306 APPENDIX A4: Dymola Model, Gen_Eigs Code Listing .................... 310 APPENDIX A5: Dymola Models, Misc. Functions, Code Listing ....... 313 A5.1 QuadRoots....................................................................................................... 313 A5.2 Complex_Mult ................................................................................................ 313 A5.3 Complex_Div.................................................................................................. 313 APPENDIX B1: Symmetry of Hamiltonian Eigenvalues ..................... 315 B1.1 Eigenvalue Symmetry about the Real Axis .................................................... 315 B1.2 Hamiltonian Eigenvalue Symmetry about the Imaginary Axis ...................... 316 APPENDIX B2: Vandermonde Representation of Controller Canonical Eigenvectors .............................................................................................. 317 APPENDIX C: Glossary of Terms.......................................................... 319 REFERENCES ......................................................................................... 320 8
9 LIST OF FIGURES Figure 3.1. Power Bond with the Flow of Power from A to B ................................... 29 Figure 3.2. Bond Graph Junctions and Conjugate Variable Relationships................. 31 Figure 3.3. Ideal Sources............................................................................................. 33 Figure 3.4. Basic Bond Graph 1-Port Elements .......................................................... 34 Figure 3.5. Basic 2-port Elements............................................................................... 34 Figure 3.6. Circuit Example ........................................................................................ 36 Figure 3.7. Simplified Circuit Power flow Diagram................................................... 37 Figure 3.8. Completed Circuit Power flow Diagram .................................................. 38 Figure 3.9. Causal Marks ............................................................................................ 39 Figure 3.10. Necessary Causality.................................................................................. 40 Figure 3.11. Possible Causal Assignments for 2-Port Elements ................................... 41 Figure 3.12. Integral Causal Assignments for 1-Port Elements.................................... 41 Figure 3.13. Differential Causal Assignments for 1-Port Elements.............................. 42 Figure 3.14. Possible Causal Assignments for a Resistive Element ............................. 43 Figure 3.15. Causal Assignments on Bond Graph Junctions ........................................ 44 Figure 3.16. Completed Circuit Bond Graph ................................................................ 47 Figure 3.17. General Bond Graph Structure Developed from the Lagrangian ............. 61 Figure 3.18. Single Degree of Freedom Pendulum....................................................... 62 Figure 3.19. Pendulum 1-Junction ................................................................................ 64 Figure 3.20. Complete Bond Graph of Pendulum......................................................... 65 Figure 3.21. Gyroscope Diagram .................................................................................. 67 Figure 3.22. Gyroscope Integral 1-Junctions ................................................................ 70 Figure 3.23. Gyroscope 1-Junctions: I-Element Connections....................................... 71 Figure 3.24. Gyroscope MGY Connections.................................................................. 74 Figure 3.25. Complete Gyroscope Bond Graph............................................................ 75 Figure 3.26. Ball Joint Table: Schematic ...................................................................... 80 Figure 3.27. Ball Joint Table: Brown’s Bond Graph .................................................... 81 Figure 3.28. Ball Joint Table: Bond Graph from Current Method................................ 81 Figure 4.1. Power Sensing Bond Model: Equation Window ...................................... 85 Figure 4.2. Power Sensing Bond Model: Diagram Window....................................... 86 Figure 4.3. Power Sensing Bond Model: Icon Window.............................................. 87 Figure 4.4. Spring Mass Damper: Example ................................................................ 88 Figure 4.5. Spring Mass Damper: Dymola Equation Window ................................... 89 Figure 4.6. Wheatstone Bridge Circuit Example: Dymola Diagram Window............ 92 Figure 4.7. Bond Graph of Wheatstone Bridge Circuit............................................... 93 Figure 4.8. Complete Bond Graph of Wheatstone Bridge Circuit .............................. 94 Figure 4.9. Gear Train Example................................................................................ 101 Figure 4.10. Gear Train Bond Graph .......................................................................... 103 Figure 4.11. Bond Graph Connector ........................................................................... 108
Page 1 and 2: SYSTEM ANALYSIS THROUGH BOND GRAPH
Page 3 and 4: 3 STATEMENT BY AUTHOR This disserta
Page 5 and 6: 5 DEDICATION To Shannon, for her un
Page 7: TABLE OF CONTENTS (continued) 4.3.3
Page 11 and 12: LIST OF FIGURES (continued) Figure
Page 13 and 14: LIST OF FIGURES (continued) Figure
Page 15 and 16: 15 LIST OF TABLES Table 3.1. Effort
Page 17 and 18: 17 CHAPTER 1: Introduction 1.1 Prob
Page 19 and 20: 19 Chapter 5 provides a means of me
Page 21 and 22: 21 CHAPTER 2: Related Work 2.1 Intr
Page 23 and 24: 23 dissipated energy in the form of
Page 25 and 26: 25 of a thermo-fluid bond graph can
Page 27 and 28: 27 between the describing parameter
Page 29 and 30: 29 variables associated with it. Na
Page 31 and 32: 31 3.2.2 Bond Graph Junctions Power
Page 33 and 34: 33 power. They are called 1-port el
Page 35 and 36: 35 the conjugate variables. As seen
Page 37 and 38: 37 As seen in figure 3.6 the power
Page 39 and 40: 39 between all of the variables are
Page 41 and 42: 41 Figure 3.11. Possible Causal Ass
Page 43 and 44: 43 Figure 3.14. Possible Causal Ass
Page 45 and 46: 45 3.2.7 Bond Graph Causal Mark Ass
Page 47 and 48: 47 information is obvious from the
Page 49 and 50: 49 3. Use the conjugate variables a
Page 51 and 52: 51 3.2.9 Conversion of Bond Graph V
Page 53 and 54: 53 obtaining dynamic equations, and
Page 55 and 56: 55 3.3.1 Lagrangian to Hamiltonian
Page 57 and 58: 57 dL ∂L ∂L ∂L & (3.22) ∂q
Page 59 and 60:
59 An underlying assumption in the
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61 7. Develop the Hamiltonian for t
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63 The partial of the Lagrangian wi
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65 bond graph shows a single degree
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67 of a Lagrangian/Hamiltonian appr
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69 d dt through 3.48 and the simple
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71 since flows sum around a 0-junct
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73 2 2 ( + A′ )& θ − ( A + B
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75 of the bond graph, i.e., they ar
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77 The simplest bond graph equation
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79 2 ( A + B′ )&& φ sin θ + ( A
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81 ω & 3 = θ (3.102) The method d
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83 Also, the bond graph maps the fl
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85 4.2 Dymola The Dymola framework
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87 such that the user can assign a
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89 A possible set of equations for
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91 Vδ1 = V 2 −V1 Vδ 2 = Vi −V
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93 The icon window has been left bl
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95 derive the equations for a bond
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97 ⎡ 1 ⎛ q ⎤ B4 ⎞ 1 ⎛ qB4
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99 f f f 2 3 1 : : : x 2 1 1 1 x 1
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101 4.2.3.1 Structural Singularitie
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103 Figure 4.10. Gear Train Bond Gr
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105 This section shows that bond gr
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107 [Pan88]. Also, note that the co
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109 The iconic representation of th
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111 Figure 4.14. A-Causal Bond As s
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113 The bond models are complete an
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115 Figure 4.19. Three-Port One The
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117 Figure 4.22 shows a bond graph
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119 Figure 4.25. C-Element Model Fi
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121 The transformer model defines t
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123 Figure 4.32. Modulated Effort S
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125 from the bond graph 3-tuple, an
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127 Figure 4.38. Q Sensor Naturally
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129 These models can now be used to
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131 Figure 4.43. Gyroscope Model: E
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133 4.4.2 Inertial Rate Sensor Mode
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135 The gyroscope in figure 4.45 is
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137 4.4.2.3 Roll Gyro Figure 4.50 s
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139 Figure 4.53. Sensor Delays The
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141 As seen in figure 4.55, the pla
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143 The icon labeled Platform_Cntrl
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145 pointed at a fixed inertial poi
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147 in the camera, a pitch command
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149 The icon of the completed model
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151 inertial axes. These inertial c
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153 The yaw achieved response is sh
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155 80 Camera: Pitch Pitch Angle (d
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157 bond graph and drop it into a h
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159 can be used to monitor the syst
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161 5.2.1 Servo Positioning System:
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163 diode is seen in figure 5.3 as
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165 Figure 5.5. Gear Train and Fin
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167 The modulated effort source is
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169 Figure 5.7. Linear Fin Dynamics
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171 Here the electrical dynamics ar
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173 Similarly, the state space equa
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175 Magnitude (dB) 50 0 -50 -100 -1
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177 showing up on terms (3, 4) and
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179 outputs are fin position and se
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181 5.4 Servo Controllers Separate
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183 Figure 5.14. Linear Controller/
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185 5.4.3 Non-Linear Control Scheme
Page 187 and 188:
187 Figure 5.17. Content of the Y3
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189 The power signal vector is pass
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191 6 5 (deg) Step: Hinge Moment =
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193 1.5 x 106 5 (deg) Step: Hinge M
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195 PID1 delivers less energy to th
Page 197 and 198:
197 Figure 5.28 shows a clear and c
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199 The step responses and efficien
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201 Figures 5.34 and 5.35 correspon
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203 Figures 5.38 and 5.39 correspon
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205 Figure 5.40. Two Non-Linear Act
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207 Figure 5.43 through 5.47 show t
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209 5.3 5 (deg) Step: Hinge Moment
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211 Figures 5.49 through 5.54 show
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213 20.3 20.2 20 (deg) Step: Hinge
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215 CHAPTER 6: Optimal Gain Compari
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217 need to be optimized further, o
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219 S ref C N 2 πd = (6.3) 4 2 1.5
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221 Figure 6.5 shows a Dymola model
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223 The equations used to execute t
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225 Angle of attack and body accele
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227 A Dymola model of the above aut
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229 1 3-Loop AP: -12.92 G step resp
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231 Figure 6.18. Three Loop AP: Clo
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233 6 3-Loop AP: -12.92 G step resp
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235 than the missile response. Prop
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237 Equation 6.15 shows that C N is
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239 M δ Q * S * d * C ref Mδ I yy
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241 5 0 5 Deg. Fin Deflection, at S
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243 0 -0.5 3 Loop AP: -12.92 G step
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245 can be exploited to place the c
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247 Although equations 6.44 through
Page 249 and 250:
249 Figures 6.32 through 6.34, show
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251 Figures 6.35 and 6.36, show how
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253 6 Complete System: -12.92 G ste
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255 view. Gain set 1 rises higher i
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257 Figures 6.43 and 6.44 show the
Page 259 and 260:
259 compare controller efficiencies
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261 1.2 x 10-6 Complete System: -12
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263 It is interesting to note, that
Page 265 and 266:
265 where −1 T K = −R B P (6.52
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267 ∂J − ∂t * = H 1 ⎛ ∂J
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269 u * = KC −1 * [ y − Du ] (6
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271 Where −1 ⎡1⎤ [ − K D] K
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−1 T ( Q − SR S ) 273 Ch = (6.1
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275 real matrix, and an imaginary m
Page 277 and 278:
277 ~ the characteristic polynomial
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279 6.6.6 Nonlinear Autopilot Resul
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281 15 10 Complete System: -12.92 G
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283 0.07 Complete System: 1 G step
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Page 287 and 288:
287 Shifting the cg towards the nos
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291 much increase in efficiency red
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293 CHAPTER 7: Summary 7.1 Contribu
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295 Various control schemes were pr
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297 Naturally this expansion is not
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299 Real Hric[1, 2]; Real Lric[1, 1
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Text( extent=[-16, -62; 18, -80], s
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Modelica.Blocks.Interfaces.OutPort
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Text(extent=[36, 38; 78, 24], strin
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307 AB2 = A*AB1; AB3 = A*AB2; for j
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309 for j in 1:n loop Reigvec_outpu
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311 eig3[2] = eig_input.signal[7];
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313 APPENDIX A5: Dymola Models, Mis
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315 APPENDIX B1: Symmetry of Hamilt
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317 APPENDIX B2: Vandermonde Repres
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319 APPENDIX C: Glossary of Terms 2
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321 Cur84 Dym Elm94 Fah99 Fah94 Fav
Page 323 and 324:
323 Mas91 Mat McB05a Maschke, B.,
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325 Zar02 Zei95 Zho96 Zarchan, P.,
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