SYSTEM ANALYSIS THROUGH BOND GRAPH MODELING by ...

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LIST OF FIGURES (continued) Figure 5.24. |Fin Power|: 5˚ Step Response, No Hinge Moment ................................ 193 Figure 5.25. Energy Input: 5˚ Step Response, No Hinge Moment ............................. 194 Figure 5.26. Energy Output: 5˚ Step Response, No Hinge Moment........................... 194 Figure 5.27. |Fin Energy|: 5˚ Step Response, No Hinge Moment............................... 196 Figure 5.28. Integral(|Fin Energy|): 5˚ Step Response, No Hinge Moment................ 196 Figure 5.29. Efficiency Worst Case ............................................................................ 197 Figure 5.30. 5˚ Step Response, Hinge Moment = -0.6 (N*m/deg.) ............................ 199 Figure 5.31. η: 5˚ Step Response, Hinge Moment = -0.6 (N*m/deg.) ........................ 199 Figure 5.32. 5˚ Step Response, Hinge Moment = -6 (N*m/deg.) ............................... 200 Figure 5.33. η: 5˚ Step Response, Hinge Moment = -6 (N*m/deg.) ........................... 200 Figure 5.34. 20˚ Step Response, Hinge Moment = 0 (N*m/deg.)............................... 201 Figure 5.35. η: 20˚ Step Response, Hinge Moment = 0 (N*m/deg.) .......................... 201 Figure 5.36. 20˚ Step Response, Hinge Moment = -0.6 (N*m/deg.) .......................... 202 Figure 5.37. η: 20˚ Step Response, Hinge Moment = -0.6 (N*m/deg.) ...................... 202 Figure 5.38. 20˚ Step Response, Hinge Moment = -6 (N*m/deg.) ............................. 203 Figure 5.39. η: 20˚ Step Response, Hinge Moment = -6 (N*m/deg.) ......................... 203 Figure 5.40. Two Non-Linear Actuators with Power Signal Analysis ....................... 205 Figure 5.41. 5˚ Step Response, Hinge Moment = 0 (N*m/deg.)................................. 205 Figure 5.42. Quantizer I/O for 5˚ Step Response, No Hinge Moment........................ 206 Figure 5.43. 5˚ Step Response (zoom), Hinge Moment = 0 (N*m/deg.) .................... 207 Figure 5.44. η: 5˚ Step Response, Hinge Moment = 0 (N*m/deg.) ............................ 207 Figure 5.45. 5˚ Step Response, Hinge Moment = -0.6 (N*m/deg.) ............................ 208 Figure 5.46. η: 5˚ Step Response, Hinge Moment = -0.6 (N*m/deg.) ........................ 208 Figure 5.47. 5˚ Step Response, Hinge Moment = -6 (N*m/deg.) ............................... 209 Figure 5.48. η: 5˚ Step Response, Hinge Moment = -6 (N*m/deg.) ........................... 209 Figure 5.49. 20˚ Step Response, Hinge Moment = 0 (N*m/deg.)............................... 210 Figure 5.50. 20˚ Step Response (zoom), Hinge Moment = 0 (N*m/deg.) .................. 211 Figure 5.51. η: 20˚ Step Response, Hinge Moment = 0 (N*m/deg.) .......................... 211 Figure 5.52. 20˚ Step Response (zoom), Hinge Moment = -0.6 (N*m/deg.).............. 212 Figure 5.53. η: 20˚ Step Response, Hinge Moment = -0.6 (N*m/deg.) ...................... 212 Figure 5.54. 20˚ Step Response (zoom), Hinge Moment = -6 (N*m/deg.)................. 213 Figure 5.55. η: 20˚ Step Response, Hinge Moment = -6 (N*m/deg.) ......................... 213 Figure 6.1. Autopilot Loop with the Autopilot Design Assumption......................... 215 Figure 6.2. Two Degree of Freedom Missile ............................................................ 218 Figure 6.3. Missile Distance Definitions................................................................... 220 Figure 6.4. Missile Pitch Plane Dynamics Bond Graph............................................ 221 Figure 6.5. Dymola Pitch Plane Dynamics: Diagram and Icon Windows................ 222 Figure 6.6. Dymola Pitch Plane Dynamics: Equation Window................................ 222 Figure 6.7. Dymola Pitch Plane Dynamics: Parameter Values................................. 223 Figure 6.8. Wing and Fin Chord Definitions............................................................. 224 Figure 6.9. Dymola Pitch Plane Instantiation............................................................ 224 Figure 6.10. Angle of Attack....................................................................................... 225 Figure 6.11. Missile Body Acceleration...................................................................... 225 Figure 6.12. Classic Three Loop Autopilot................................................................. 226 12
LIST OF FIGURES (continued) Figure 6.13. Classic Three Loop Autopilot: Dymola Model ...................................... 227 Figure 6.14. Three Loop AP: Closed Loop System .................................................... 227 Figure 6.15. Three Loop AP: Angle of Attack............................................................ 228 Figure 6.16. Three Loop AP: Achieved Acceleration................................................. 229 Figure 6.17. Three Loop AP: Necessary Fin Deflection............................................. 229 Figure 6.18. Three Loop AP: Closed Loop System with Actuator ............................. 231 Figure 6.19. Angle of Attack with Actuator Dynamics .............................................. 232 Figure 6.20. Achieved Acceleration with Actuator Dynamics ................................... 232 Figure 6.21. Necessary Fin Deflection with Actuator Dynamics................................ 233 Figure 6.22. Angle of Attack with Actuator Dynamics: Zoom................................... 233 Figure 6.23. Achieved Acceleration with Actuator Dynamics: Zoom........................ 234 Figure 6.24. Necessary Fin Deflection with Actuator Dynamics: Zoom.................... 234 Figure 6.25. Linearized Pitch Plane Block Diagram................................................... 239 Figure 6.26. Open Loop Linear and Nonlinear Angle of Attack ................................ 240 Figure 6.27. Open Loop Linear and Nonlinear Missile Body Acceleration ............... 241 Figure 6.28. Linear Pitch Plane and Autopilot Block Diagram .................................. 242 Figure 6.29. Closed Loop Linear and Nonlinear Angle of Attack.............................. 243 Figure 6.30. Closed Loop Linear and Nonlinear Missile Body Acceleration............. 243 Figure 6.31. Closed Loop Linear and Nonlinear Necessary Fin Deflection ............... 244 Figure 6.32. Optimal Gain Selection Angle of Attack................................................ 248 Figure 6.33. Optimal Gain Selection Missile Body Acceleration............................... 248 Figure 6.34. Optimal Gain Selection Necessary Fin Deflection ................................. 249 Figure 6.35. Optimal Gain Selection: (Command – AZ) 2 ........................................... 250 Figure 6.36. Optimal Gain Selection: Performance Index .......................................... 250 Figure 6.37. Nonlinear Missile with Optimal Gains: Angle of Attack ....................... 252 Figure 6.38. Nonlinear Missile with Optimal Gains: Body Acceleration................... 252 Figure 6.39. Nonlinear Missile with Optimal Gains: Fin Deflection.......................... 253 Figure 6.40. Nonlinear Missile with Optimal Gains: Body Acc. (zoom) ................... 253 Figure 6.41. Nonlinear Missile with Optimal Gains: Fin Deflection (zoom) ............. 254 Figure 6.42. Autopilot η AP : Autopilot Efficiency Signals for Gain Sets 1-3 .............. 255 Figure 6.43. Optimal Gain Set 4: Body Acceleration ................................................. 257 Figure 6.44. Optimal Gain Set 4: Body Acceleration (zoom)..................................... 257 Figure 6.45. Optimal Gain Set 4: Autopilot Efficiency η AP ........................................ 258 Figure 6.46. Body Acceleration: Gain Sets 4-6........................................................... 260 Figure 6.47. Autopilot η AP Gain Sets 4-6.................................................................... 261 Figure 6.48 Nonlinear PI for Gain Sets 4-6................................................................ 262 Figure 6.49. Nonlinear PI for Gain Sets 4-6 (zoom)................................................... 262 Figure 6.50. Autopilot Efficiency η AP: Gain Sets 4-6 (zoom)..................................... 263 Figure 6.51. SDRE Autopilot: Icon and Diagram Window ........................................ 272 Figure 6.52. Algebraic Riccati Equation Solver Riccati4: Diagram Window ............ 273 Figure 6.53. Hamiltonian Eigenvalue Solver Heig4: Diagram Window .................... 275 Figure 6.54. Pitch Plane Dynamics with SDRE Autopilot.......................................... 279 Figure 6.55. Pitch Plane Dynamics with SDRE Autopilot and Actuator.................... 280 13
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 and 8: TABLE OF CONTENTS (continued) 4.3.3
Page 9 and 10: 9 LIST OF FIGURES Figure 3.1. Power
Page 11: 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
Page 61 and 62: 61 7. Develop the Hamiltonian for t
Page 63 and 64:
63 The partial of the Lagrangian wi
Page 65 and 66:
65 bond graph shows a single degree
Page 67 and 68:
67 of a Lagrangian/Hamiltonian appr
Page 69 and 70:
69 d dt through 3.48 and the simple
Page 71 and 72:
71 since flows sum around a 0-junct
Page 73 and 74:
73 2 2 ( + A′ )& θ − ( A + B
Page 75 and 76:
75 of the bond graph, i.e., they ar
Page 77 and 78:
77 The simplest bond graph equation
Page 79 and 80:
79 2 ( A + B′ )&& φ sin θ + ( A
Page 81 and 82:
81 ω & 3 = θ (3.102) The method d
Page 83 and 84:
83 Also, the bond graph maps the fl
Page 85 and 86:
85 4.2 Dymola The Dymola framework
Page 87 and 88:
87 such that the user can assign a
Page 89 and 90:
89 A possible set of equations for
Page 91 and 92:
91 Vδ1 = V 2 −V1 Vδ 2 = Vi −V
Page 93 and 94:
93 The icon window has been left bl
Page 95 and 96:
95 derive the equations for a bond
Page 97 and 98:
97 ⎡ 1 ⎛ q ⎤ B4 ⎞ 1 ⎛ qB4
Page 99 and 100:
99 f f f 2 3 1 : : : x 2 1 1 1 x 1
Page 101 and 102:
101 4.2.3.1 Structural Singularitie
Page 103 and 104:
103 Figure 4.10. Gear Train Bond Gr
Page 105 and 106:
105 This section shows that bond gr
Page 107 and 108:
107 [Pan88]. Also, note that the co
Page 109 and 110:
109 The iconic representation of th
Page 111 and 112:
111 Figure 4.14. A-Causal Bond As s
Page 113 and 114:
113 The bond models are complete an
Page 115 and 116:
115 Figure 4.19. Three-Port One The
Page 117 and 118:
117 Figure 4.22 shows a bond graph
Page 119 and 120:
119 Figure 4.25. C-Element Model Fi
Page 121 and 122:
121 The transformer model defines t
Page 123 and 124:
123 Figure 4.32. Modulated Effort S
Page 125 and 126:
125 from the bond graph 3-tuple, an
Page 127 and 128:
127 Figure 4.38. Q Sensor Naturally
Page 129 and 130:
129 These models can now be used to
Page 131 and 132:
131 Figure 4.43. Gyroscope Model: E
Page 133 and 134:
133 4.4.2 Inertial Rate Sensor Mode
Page 135 and 136:
135 The gyroscope in figure 4.45 is
Page 137 and 138:
137 4.4.2.3 Roll Gyro Figure 4.50 s
Page 139 and 140:
139 Figure 4.53. Sensor Delays The
Page 141 and 142:
141 As seen in figure 4.55, the pla
Page 143 and 144:
143 The icon labeled Platform_Cntrl
Page 145 and 146:
145 pointed at a fixed inertial poi
Page 147 and 148:
147 in the camera, a pitch command
Page 149 and 150:
149 The icon of the completed model
Page 151 and 152:
151 inertial axes. These inertial c
Page 153 and 154:
153 The yaw achieved response is sh
Page 155 and 156:
155 80 Camera: Pitch Pitch Angle (d
Page 157 and 158:
157 bond graph and drop it into a h
Page 159 and 160:
159 can be used to monitor the syst
Page 161 and 162:
161 5.2.1 Servo Positioning System:
Page 163 and 164:
163 diode is seen in figure 5.3 as
Page 165 and 166:
165 Figure 5.5. Gear Train and Fin
Page 167 and 168:
167 The modulated effort source is
Page 169 and 170:
169 Figure 5.7. Linear Fin Dynamics
Page 171 and 172:
171 Here the electrical dynamics ar
Page 173 and 174:
173 Similarly, the state space equa
Page 175 and 176:
175 Magnitude (dB) 50 0 -50 -100 -1
Page 177 and 178:
177 showing up on terms (3, 4) and
Page 179 and 180:
179 outputs are fin position and se
Page 181 and 182:
181 5.4 Servo Controllers Separate
Page 183 and 184:
183 Figure 5.14. Linear Controller/
Page 185 and 186:
185 5.4.3 Non-Linear Control Scheme
Page 187 and 188:
187 Figure 5.17. Content of the Y3
Page 189 and 190:
189 The power signal vector is pass
Page 191 and 192:
191 6 5 (deg) Step: Hinge Moment =
Page 193 and 194:
193 1.5 x 106 5 (deg) Step: Hinge M
Page 195 and 196:
195 PID1 delivers less energy to th
Page 197 and 198:
197 Figure 5.28 shows a clear and c
Page 199 and 200:
199 The step responses and efficien
Page 201 and 202:
201 Figures 5.34 and 5.35 correspon
Page 203 and 204:
203 Figures 5.38 and 5.39 correspon
Page 205 and 206:
205 Figure 5.40. Two Non-Linear Act
Page 207 and 208:
207 Figure 5.43 through 5.47 show t
Page 209 and 210:
209 5.3 5 (deg) Step: Hinge Moment
Page 211 and 212:
211 Figures 5.49 through 5.54 show
Page 213 and 214:
213 20.3 20.2 20 (deg) Step: Hinge
Page 215 and 216:
215 CHAPTER 6: Optimal Gain Compari
Page 217 and 218:
217 need to be optimized further, o
Page 219 and 220:
219 S ref C N 2 πd = (6.3) 4 2 1.5
Page 221 and 222:
221 Figure 6.5 shows a Dymola model
Page 223 and 224:
223 The equations used to execute t
Page 225 and 226:
225 Angle of attack and body accele
Page 227 and 228:
227 A Dymola model of the above aut
Page 229 and 230:
229 1 3-Loop AP: -12.92 G step resp
Page 231 and 232:
231 Figure 6.18. Three Loop AP: Clo
Page 233 and 234:
233 6 3-Loop AP: -12.92 G step resp
Page 235 and 236:
235 than the missile response. Prop
Page 237 and 238:
237 Equation 6.15 shows that C N is
Page 239 and 240:
239 M δ Q * S * d * C ref Mδ I yy
Page 241 and 242:
241 5 0 5 Deg. Fin Deflection, at S
Page 243 and 244:
243 0 -0.5 3 Loop AP: -12.92 G step
Page 245 and 246:
245 can be exploited to place the c
Page 247 and 248:
247 Although equations 6.44 through
Page 249 and 250:
249 Figures 6.32 through 6.34, show
Page 251 and 252:
251 Figures 6.35 and 6.36, show how
Page 253 and 254:
253 6 Complete System: -12.92 G ste
Page 255 and 256:
255 view. Gain set 1 rises higher i
Page 257 and 258:
257 Figures 6.43 and 6.44 show the
Page 259 and 260:
259 compare controller efficiencies
Page 261 and 262:
261 1.2 x 10-6 Complete System: -12
Page 263 and 264:
263 It is interesting to note, that
Page 265 and 266:
265 where −1 T K = −R B P (6.52
Page 267 and 268:
267 ∂J − ∂t * = H 1 ⎛ ∂J
Page 269 and 270:
269 u * = KC −1 * [ y − Du ] (6
Page 271 and 272:
271 Where −1 ⎡1⎤ [ − K D] K
Page 273 and 274:
−1 T ( Q − SR S ) 273 Ch = (6.1
Page 275 and 276:
275 real matrix, and an imaginary m
Page 277 and 278:
277 ~ the characteristic polynomial
Page 279 and 280:
279 6.6.6 Nonlinear Autopilot Resul
Page 281 and 282:
281 15 10 Complete System: -12.92 G
Page 283 and 284:
283 0.07 Complete System: 1 G step
Page 285 and 286:
Page 287 and 288:
287 Shifting the cg towards the nos
Page 289 and 290:
Page 291 and 292:
291 much increase in efficiency red
Page 293 and 294:
293 CHAPTER 7: Summary 7.1 Contribu
Page 295 and 296:
295 Various control schemes were pr
Page 297 and 298:
297 Naturally this expansion is not
Page 299 and 300:
299 Real Hric[1, 2]; Real Lric[1, 1
Page 301 and 302:
Text( extent=[-16, -62; 18, -80], s
Page 303 and 304:
Modelica.Blocks.Interfaces.OutPort
Page 305 and 306:
Text(extent=[36, 38; 78, 24], strin
Page 307 and 308:
307 AB2 = A*AB1; AB3 = A*AB2; for j
Page 309 and 310:
309 for j in 1:n loop Reigvec_outpu
Page 311 and 312:
311 eig3[2] = eig_input.signal[7];
Page 313 and 314:
313 APPENDIX A5: Dymola Models, Mis
Page 315 and 316:
315 APPENDIX B1: Symmetry of Hamilt
Page 317 and 318:
317 APPENDIX B2: Vandermonde Repres
Page 319 and 320:
319 APPENDIX C: Glossary of Terms 2
Page 321 and 322:
321 Cur84 Dym Elm94 Fah99 Fah94 Fav
Page 323 and 324:
323 Mas91 Mat McB05a Maschke, B.,
Page 325:
325 Zar02 Zei95 Zho96 Zarchan, P.,
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SYSTEM ANALYSIS THROUGH BOND GRAPH MODELING by ...

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