Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1991 Feedback Linearization of Nonlinear Systems: Robustness and Adaptive Control. Weon Ho Kim Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Kim, Weon Ho, "Feedback Linearization of Nonlinear Systems: Robustness and Adaptive Control." (1991). LSU Historical Dissertations and Theses. 5130. https://digitalcommons.lsu.edu/gradschool_disstheses/5130 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. 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Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI University Microfilms International A Bell & Howell Information Company 300 Nortfi Zeeb Road, Ann Arbor, Ml 48106-1346 USA 313/761-4700 800/521-0600 Reproduced witfi permission of tfie copy rig fit owner. Furtfier reproduction profiibited witfiout permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Order Number 0200073 Feedback linearization of nonlinear systems: Robustness and adaptive control Kim, Weon Ho, Ph.D. The Louisiana State University and Agricultural and Mechanical Col., 1991 UMI 300N.ZecbRd. Ann Arbor, MI 48106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Feedback Linearization of Nonlinear Systems: Robustness and Adaptive Control A Dissertation Submitted to the Graduate Faculty o f the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department o f Chemical Engineering by Weon Ho Kim B.S. Hanyang University, Seoul, Korea, 1981 M.S. Seoul National University, Seoul, Korea, 1983 May 1991 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To my wife Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements I wish to thank Dr. Frank R. Groves for his guidance and patience in the preparation o f this work. Also, appreciation is extended to the members o f the advisory committee; Dr. Ralph W. Pike, Dr. Arthur M. Sterling, Dr. Armando B. Corripio, Dr. Jorge L. Aravena and Dr. Brian D. Marx. I thankfully recognize Dr. Hong G. Lee for his helpful comments on this work. I am certainly grateful for the support, financial and otherwise, o f the Department of Chemical Engineering and LSU Mineral Research Institute Fellowship. Finally, I especially extend appreciation to my family. Their help and encouragement has been essential throughout my college career. m Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents Feedback Linearization o f Nonlinear Systems: Robustness and Adaptive Control page Dedication ii Acknowledgements iii List o f Figures - vii List o f Tables xi Dissertation Abstract xii 1. Introduction ' 1 2. Feedback linearization 7 2.1 Introduction 7 2.2 Input-output linearization 10 2.3 Exact state-space linearization 23 3. Robustness analysis o f feedback linearization for parametric and structural uncertainties 28 3.1 Introduction 28 3.2 Literature review 30 3.3 Theoretical analysis 34 3.3.1 Feedback linearization o f an uncertain system 34 3.3.2 Robustness analysis 39 IV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4 Application; First order exothermic reaction in a CSTR 54 3.4.1 Introduction 54 3.4.2 Mathematical model o f a first order exothermic reaction in a CSTR 54 3.4.3 Feedback linearization o f the CSTR without model-plant mismatch 57 3.4.4 Robustness analysis 59 3.4.4.1 Parametric uncertainty in the reaction rate constant 59 3.4.4.2 Unmeasured disturbance in feed concentiation 72 3.4.4.3 Measurement error in concentration in the reactor 76 3.5 Conclusion 82 4. Robustness analysis o f the feedback linearization for parametric and structural uncertainties with unmodeled dynamics 83 4.1 Introduction 83 4.2 Literature review 85 4.3 Theoretical analysis 87 4.3.1 Introduction 87 4.3.2 Dimensional reduction 88 4.3.3 Feedback linearization based on the reduced dimensional model 90 4.4 Application: Multicomponent exothermic chemical reaction in a CSTR 103 4.4.1 Introduction 103 4.4.2 Mathematical model 104 4.4.3 Reduced dimensional model 111 4.4.4 Feedback linearization based on the reduced dimensional model 113 4.4.5 Robusmess analysis 116 4.5 Conclusion 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5. Adaptive control o f feedback linearizable systems 125 5.1 Introduction 125 5.2 Literature review 127 5.3 Adaptive regulation o f a feedback linearizable process 129 5.4 Adaptive output tracking of a feedback linearizable process 140 5.5 Applications o f adaptive regulation 146 5.5.1 First order exothermic reaction in a CSTR 146 5.5.2 Biological reaction in a CSTBR 163 5.5.3 First order exothermic reaction in a CSTR with uncertainty in the activation energy 184 5.6 Application for adaptive output tracking 197 5.7 Conclusion 202 6. Conclusion and Future work 203 Notation 206 References 209 Appendices 214 Appendix I 214 Appendix II 216 Appendix IB 217 Appendix IV 218 Appendix V 220 Appendix V I 221 Appendix VII 222 Vita 242 VI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures Page Figure 2.1 Feedback linearization 14 Figure 3.1 Equilibrium point for each parametric value o f D where b^ = -2.1 and b2 = -2.0 62 Figure 3.2 Estimated bounded and converging region o f the solution trajectories for the parametric uncertainty in kg using Theorem 2, where b^ =-2.1 and b2 = -2.0 and line 1: initial condition, C = 2.944 x 10'3 gmole/cc, T = 404.51 K and D = 31.8188, line 2; initial condition, C = 5.004 X 10-3 gmole/cc, T = 391.60 K and D = 31.7788 66 Figure 3.3 Estimated bounded and converging region o f the solution trajectories for the parametric uncertainty in kg using Theorem 3, where b j = -2.1 and b2 = -2.0 and line 1: initial condition, C = 4.867 x 10"3 gmole/cc, T = 392.30 K and D = 31.8188, line 2: initial condition, C = 3.136 x 10-3 gmole/cc, T =403.58 K and D = 31.7788 71 Figure 3.4 Estimated bounded and converging region o f the solution trajectories for the unmeasured disturbance in feed concentration, where b^ = - 2.1 and b2 =- 2.0 and line 1: initial condition, C = 3.008 x 10-3 gmole/cc, T = 404.51 K, line 2: initial condition, C = 4.956 x 10-3 gmole/cc, T = 391.48 K 75 Figure 3.5 Estimated bounded and converging region o f the solution trajectories for measurement error in concentration in the reactor, where bj = - 2.1 and b2 = -2.0 and line 1: initial condition, C = 3.273 x 10*3 gmole/cc, T = 402.68 K, line 2: initial condition, C = 4.748 x 10-3 gmole/cc, T = 392.83 K 80 Figure 3.6 Estimated bounded and converging region o f the solution trajectories for measurement error in concentration in the reactor, where b j = -4.0 and b2 = -2.7 and line 1 : initial condition, C = 3.393 x 10-3 gmole/cc, T = 401.84 K, line 2: initial condition, C = 4.619 x 10-3 gmole/cc, T = 393.69 K 81 vu Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.1 Control system o f a multicomponent exothermic chemical reaction in CSTR 104 Figure 4.2 Three steady state solutions o f energy equation w he.. Ql = ( i H.T* - ( a To V pCpvj V pCpV V QR=^-^S^(klCj-k 2CB) 107 pCp Figure 4.3 The set U where the reduced dimensional model has relative degree 2 114 Figure 4.4 Estimated bounded and converging region o f system trajectories with unmodeled dynamics, where b^ = - 2.1,62 = -2.0 and line 1: Xi(0) = -0.116, X2(0) = 0.546, w(0) = 2.5 x 10"3 , D i = 31.789 line 2: x i(0 ) = 0.116, XgCO) = - 0.451, w(0) = 2.5 x 10'^ , D i = 31.809 121 Figure 4.5 System response o f the state variable o f the unmodeled dynamics, w, where bj = -2.1, b2 = -2.0 and Xj(0) = - 0.116, X2(0) = 0.546, w(0) = 2.5 X 10-3, D i = 31.789 122 Figiu-e 4.6 System response when the reaction coefficient k^Q has the nominal value, where b j = - 2.1, 62 = -2.0 and xi(0 ) = - 0.116, X2(0) = 0.546, w(0) = 2.5 x 10 ^ 123 Figure 5.1 The set Qc and 0% for output tracking.