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Computation of Sensitivity Measures Jack COMPUTATION OF SENSITIVITY MEASURES By JACK HAMMOND PRIDGEN /I Eachelor of Science Southe:rn Methodist University Dalle.s, Texas 196.3 Master of Science Oklahoma State University Stillwater, Oklahoma 1965 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY May, 1970 T~0\~ I c\~c_:, 3) ? q '-\ •'\c.. ,;. ti ~,,/' ,'j· ,l :-3"' I" / ,fl' }. :,!',/. ../ (.~·~ COMPUTATION OF SENSITIVITY MEASURES'"<, .. The~is Approved: ,;{; ~L,<JZ of: (?nn:,,,,;~. ~ t41<7M1"/ d~ T . ? ..,._....,...._ _.._D_e"'"an...._o_f_....,t"""h""'e"""G..:;:r~a"'"'a""'u-a-t-e-C-o-l-l-e-g"""e----Q, £4:1<4-- ..... ii ACKNOWLEDGMENTS First, I wish to take this opportunity to thank Dr. A. M. Breipohl and Dr. c. M. Bacon who served as thesis advisers for their interest, assistance, ~riticism, and encouragement during the course of this study. The time and effort expended by the other members of my graduate committee, Professors H. T. Fristoe, R. L. Cummins, and Jeanne L. Agnew, is also appreciated. Dr. W. A. Blackwell and Dr. L0 L. Grigsby assisted early in the study before leaving Oklahoma State University. Dr. J.M. Walden also graciously supplied his computer program listing and trial problems. I am very grateful for the financial assistance which I haver~ ceived during my graduate studies. Sandia Corporation has provided partial su~port for tqe research perfonned here, and the Departm~nt of Health, Education, and Welfare provided a three-year graduate fellowship. In addition, I offer my thanks to Miss Velda Davis (with the capa­ ble assistance of her colleague, Mrs. Margaret Estes) for the final tYJ)ing of this manuscript. Finally, I would like to express my deep appreciation to my wife, Mary, and my parents whose understanding, encouragement, and sacrifice were invaluable during my graduate studies. iii TABLE OF CONTENTS Chapter Page INTRO:QUCTION q • • • • e e • ~ ~ ~ o • e e o ~ o o • • • • • 1 1.1 Motivation •oooooQo~•••• 1 1.2 Scope of Study • 0 0000 o•••-i• 3 Ile SENSITIVITY MEASURES AND THEIR EVALUATION ~ . 7 Introduction. o .... o • o o o .. o o o • • •• 7 Frequency Domain Sensitivity Analysis •••• 8 Time Domain Sensitivity Operators o o o o •• 16 2o3o1 Linear Analysis. o o o • o o ••• 18 2o3.2 Npnlinear Analysis o o o a •••• 21 2o3o3 Example o • • .. .. .. .. .... , • 25 2.4 Time Domain Measures Based on Sensitivity Operators ••• o ........ n o c, e • • • • 32 2.5 Frequency Domain Measures Based on Sensitivity Operators ... 0 0 e e e e 39 2.6 Summary o ,. o • • • .. o .. 48 III. COMPUTER-AIDED S~N~ITIVlTY ANALYSIS OF LINEAR NETWORK.S e O O 8 8 e ~ 0 e 9, 0 0 0 & 0 0 n O O o o e • • • • 52 Introduction .... o o • o o • o .. o ........ 52 Extent Linear Ar:ialysis Programs With Sensitivity Capabilities.. .. o o ••••• 53 Terminology and Formulation of Networ~ Equations. 56 Formulation of the Sen~itivi~y Operators .. 60 Computer Progpam VARYIT O o o • o ... o ••••• 64 3o5o1 Tree Selection Algorithm • o o o . 68 3.5 .. 2 Cutset Matrix Formulation Algorithm. 70 3o5o3 Formulation Algorithm ..... o .. o •• • •• 71 3o5o4 Time Solution Technique ... o o o ...... 72 3a5o5 Impulse Response Solution Technique. 75 30506 Pole~Zero Sensitivity ..... o ..... , ••• 75 3 .. 5.7 Statistical Sensitivity Measure o . 76 30508 Time Domain Output of the Program VARYIT. ~ • a o • o o o .. a • • • • 77 J.6 Summary Q o e o a 0000.0•••• 77 !Va EXTENSION TO NONLINEAR NETWORKS .. • ...... o .. a o , • , • • 79 Introd4ction .. o •• o o ... 0 0 0 0 o . 79 Nonlinear Analysis Programs 0 0 Q O . ~ . 80 iv Chapter Page IV .. (CONTINUED) 4.J Inclusion of Nonlipear Elements in the State-Space Model o o • o •• o • o a O • • BJ 4.4 Formulation and Solution Techniques for Sensitiyity Operators for Nonlinear Networks 85 4.5 Computer Program VARNOL o ~ o o o • • 87 't. 6 Sununary • • e e 9 ., o e " o • o • ~ a o e • • • • .94 Vo EXAMPLES OF .SENSITIVITY ANALYSIS WITH VARYIT AND VARNOL • c, 0 0 e 4t O O n O O O e l!t O O O O O • • • • 95 Introduction. • •••• o • o . ' 95 Illustrations of the Use of VARYIT. o o •• 95 Illustrations of the Use of VARNOL. o ••• . ' . 110 Conclusions . 130 VIo SUMMARY AND CONCLUSIONS oeoeee 132 6.1 Summaqr OOftOOeeef)O . 132 6.2 Conclusions ••••• a e o 0 0 o . 134 0 0 6.J Suggestions for Further Study Q " • • • • 135 BIBLIOGRAPHY o o e e o e • • • & o a o e o o o o . 138 APPENDIX A - IMPULSE RESPONSE SOLUTION OF THE SYSTEM AND SENSITIVITY MODEL o ••••• o o o • a o o . 142 APPENDIX B - PROGRAM FOR FINDING THE POLE-ZERO SENSITIVITIES • • • 150 APPENDIX C ~ A THEOREM ON THE DEGREE OF PARAMETERS IN THE STATE MODEL MATRICES ••••• o o o o • o o o o • a •••• 16J ,APPENDIX D - DOCUMENTATION OF PROGRAM VARYlT O ft O O D . 171 APPENPIX E - DOCUMENTATION OF PROGRAM VARNOL e o o o e o • • • • • 188 v LIST OF '.L'ABLES Table Page I. Important References in Chapter JI 50 IL Sample Computer Output for Time Solution 100 III. Sample Computer Output for Pole-Zero Solution. 111 IVo Pole Sensitivities for Canonic Networ~s ••• 115 Vo Program Listing for the Impulse Re~ponse Solution 148 VIo Description of Input Data for the Pole-Zero Sensitivity. • • • • o •••••• Q o e ~ • • • • 152 VIIc Program Listing for the Pole-Zero Sensitivity Program ••• ,eooe.eoepeo e • ., • • 156 VIII. FORTRAN Variable Names and Definitions for VARYIT • • • 172 IX. Description of Input Data for Program VARYIT 0 0 0 e e e e 179 x. Input Data for Time Solution 8 0 0 0 Q • • 182 XL Input Data for Impulse Solution. 0 0 0 184 XIIo Input Data for Pole-Zero e e • G o o ~ o ~ o o ft • • • 185 XIII. Time Dependent Source Parameter Definitions. 186 XIV. Output Tape Format •••••••••••••••••••• 187 xv. Nonlinearity Representations Included in VARNOL. 189 J: .XVIo Descrtption of Input Data for Program VARNOL 0 0 0 •.. ~ • e 195 vi LIST OF FIGURES Figure Page 1. Diagram of Operations for Calculation of Sensitivity Oper~tors O 9 ~ 8 • 8 e • & e O O e D e e O ~ e O ., . 22 2. Example System • • • e e • e 0 • e O O O 8 0 0 9 0 0 0 e • e • 27 3. State Sensitivity Operators ~ Q O e O O O 8 ~ 0 0 0 0 • • e • 30 Estimated and Actual Solution for x(t) 0 0 0 0 0 0 0 ~ e e ~ e 30 5. Predicted and Actual x Increments. • • • • • • • • • • • • • 31 1 6. Predicted and Actual x Increments •••••• o • • • • • • • 31 2 Polarity Convention 0 0 0 8 ff O O ~ 0 0 0 ~ 0 0 0 0 0 e e e e 58 8. Defi~ition of Terms for E~uation (3.3.1) aoa:,ooe•••• 59 Operation~ Performed by VARYIT 0 & ~ 0 a O O O Q O O ~ • • e e 67 10. Operati9ns Performed by VARNOL 0 0 0 0 0 0 & 9 ~ 0 0 0 e • e • 89 Alternative Networks Realizing Same Impedance 0 0 0 • e • e • 97 12. Network A With Assigned Conventions 0 0 0 0 0 0 0 0 8 • e • e 98 Mean Square Error for Canonic Networks O a O O O 9 e • • e 106 Mean Square Error for Non-Canonic Networks e&ooeeo•••• 107 15. Mean Square Error for Canonic Networks •••• o o • o •••• 108 Mean Square Error for Non-Canonic Networks a ~ ~ 0 0 0 • e 109 17. Circuit for Example 5,3.1 o • o e • e e o o o e e o e • • • • 118 Output Voltage for Nominal Parameter Values for Exj:llllple 5.3.1 ... • • • • • • • • • • • . 119 19. Compa,rison of Actual and Preqicted Changes in Output Voltage for Five Percent Change in L ••••• o • e • • • • 120 20. Comparison of Actual and Predicted Changes in Output Voltag~ for Five Percent Change in RL •••• P • (t • • • • 121 vii Figure Page 21~ Filter Network (a) and Computer Model (b) 123 22. Step Response of Active Filter 125 23. Variance of Output Voltage Due to Parameter Variations • • o e 0 ., 0 0 126 2~. Comp~~s~tion Network of Example 5.3.3 " . 127 25. Rectifiers a.pd L-Section Filters 129 26. Circuit for Example 5,.3.5 • . 129 27. Circuit for Example 5.3.6 . 0 . 0 . 0 . .. 131 28. Flow C;hart for Computation of Impulse Response Solution 0 . n 0 . p . 0 . 0 . 0 . 0 .. • 146 Flow Chart for Pole-Zero Sensitivity Program 29. ~ • . 151 Chart 0 0 0 ~o. Input Data Preparation . ~ • . 178 viii CHAPTER I INTRODUCTION 1.1 Motivation As society has become more complex, it has become more difficult to fulfill its needs and desires with simple, unsophisticated machines and devices. Engineering systems have become so complex that in many cases engineers are unable to predict thei;r perf.ormcj.nce by "pencil and paper" methods. Indeed, in these cases the pro9ess of converting a physical system description to a mathematical model for analysis can easily occupy the total available time and capability of the practicing engineer. This mqdeling process usµally consists of: (a) selecting the major functions to be modeled; (b) isolating individual copstituent parts or components if possible; (c) constructing a mathematical model for each component which adequately describes it in terms of its system performance; anq ( d) fonnulating the system of equations comprising the ma the-, ·· matical model for the complete physical system from the component models and their interconnections.
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