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record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd leader nam Ka controlfield tag 001 001795260 003 fts 005 20070702104857.0 006 med 007 cr mnuuuuuu 008 070702s2006 flu sbm 000 0 eng d datafield ind1 8 ind2 024 subfield code a E14SFE0001552 040 FHM c FHM 035 (OCoLC)150451213 049 FHMM 090 TP145 (ONLINE) 1 100 Garca Z., Yohn E. 0 245 Fuzzy logic in process control : b a new fuzzy logic controller and an improved fuzzyinternal model controller h [electronic resource] / by Yohn E. Garca Z. 260 [Tampa, Fla] : University of South Florida, 2006. 3 520 ABSTRACT: Two fuzzy controllers are presented. A fuzzy controller with intermediate variable designed for cascade control purposes is presented as the FCIV controller. An intermediate variable and a new set of fuzzy logic rules are added to a conventional Fuzzy Logic Controller (FLC) to build the Fuzzy Controller with Intermediate Variable (FCIV). The new controller was tested in the control of a nonlinear chemical process, and its performance was compared to several other controllers. The FCIV shows the best control performance regarding stability and robustness. The new controller also has an acceptable performance when noise is added to the sensor signal. An optimization program has been used to determine the optimum tuning parameters for all controllers to control a chemical process. This program allows obtaining the tuning parameters for a minimum IAE (Integral absolute of the error). The second controller presented uses fuzzy logic to improve the performance of the convention al internal model controller (IMC). This controller is called FAIMCr (Fuzzy Adaptive Internal Model Controller). Twofuzzy modules plus a filter tuning equation are added to the conventional IMC to achieve the objective. The first fuzzy module, the IMCFAM, determines the process parameters changes. The second fuzzy module, the IMCFF, provides stability to the control system, and a tuning equation is developed for the filter time constant based on the process parameters. The results show the FAIMCr providing a robust response and overcoming stability problems. Adding noise to the sensor signal does not affect the performance of the FAIMC.The contributions presented in this work include:The development of a fuzzy controller with intermediate variable for cascade control purposes. An adaptive model controller which uses fuzzy logic to predict the process parameters changes for the IMC controller. An IMC filter tuning equation to update the filter time constant based in the process paramete rs values. A variable fuzzy filter for the internal model controller (IMC) useful to provide stability to the control system. 502 Dissertation (Ph.D.)University of South Florida, 2006. 504 Includes bibliographical references. 516 Text (Electronic dissertation) in PDF format. 538 System requirements: World Wide Web browser and PDF reader. Mode of access: World Wide Web. 500 Title from PDF of title page. Document formatted into pages; contains 158 pages. Includes vita. 590 Adviser: Carlos A. Smith, Ph.D. 653 Artificial intelligence. Cascade control. Nonlinear chemical processes. Adaptive control. Mamdani and Sugeno inference systems. 690 Dissertations, Academic z USF x Chemical Engineering Doctoral. 773 t USF Electronic Theses and Dissertations. 4 856 u http://digital.lib.usf.edu/?e14.1552 PAGE 1 Fuzzy Logic in Process Control: A New Fuzzy Logic Controller and An Improved FuzzyInternal Model Controller by Yohn E. Garca Z. A dissertation submitted in partial fulfillment of the requirement s for the degree of Doctor of Philosophy Department of Chemical Engineering College of Engineering University of South Florida CoMajor Professor: Carlos A. Smith, Ph.D. CoMajor Professor: Marco E. Sanjuan, Ph.D. John Wolan, Ph.D. William Lee, Ph.D. John Llewlyn, Ph.D. Robert Carnahan, Ph.D. Date of Approval: February 7, 2006 Keywords: Artificial Intelligence, Cascade Control, Nonlinear Chemical Processes, Adaptive Control, Mam dani and Sugeno Inference Systems. Copyright 2006, Yohn E. Garca Z. PAGE 2 DEDICATION This work is dedicated with all my love to My Lord our God My parents, Francisco and Anita My sister Maribel and my brother Alfonso My daughter, Andrea Valenti na and her mommy Yayita My dear home country, Venezuela PAGE 3 ACKNOWLEDGMENTS I wish to express my entire gratitude To Dr. Carlos A. Smith, my major professor, his guidance and his wise advices taught me that everything is possible working hard. It has been an honor to be his student. Thanks Dr. Smith. To University of South Florida (USF), especially to the Chemical Engineering Department people. To Universidad de Los Andes (ULA), Mrida, Venezuela. To Jos Mara Andrez, my professor and one of my best friends. You have been a very important person in my hard and good times. Thank you, Jos. To Edinzo Iglesias for being my friend and my brother in all this time. To Dr. Marco Sanjuan, without his help and guidance this work could not be possible. PAGE 4 i TABLE OF CONTENTS LIST OF TABLES .... .v LIST OF FIGURES . ..x ABSTRACT . ... . ... .....xv CHAPTER 1 INTRODUCTION . ... . .1 1.1 Introduction . ..1 1.2 Contributions of This Research ......2 1.2.1 A Fuzzy Logic Controller With Intermediate Variab le...................2 1.2.2 Adaptive Internal Model Contro ller................ .................. .............2 1.2.3 IMC Filter Tuni ng Equation.............. ................ ................ ............3 1.2.4 Variable Fuzzy Filter for In ternal Model Control (IMC)....... ..........3 1.3 Survey and Discussion ....4 1.3.1 Recent Advanc es on Fuzzy Logic Control ...5 1.3.2 Recent Advances on Cascade Control . . .......... ..5 1.3.3 Advances on IM C ........... ...... . 6 1.4 Summary and Scope of the Thesis 6 CHAPTER 2 FUZZY CONTROLLER WITH INTERMEDIATE VARIABLE (FCIV) ... .... ... . .....8 2.1 Introduction . ..8 2.2 Fuzzy Controller With Intermediat e Variable (FCIV) ......... .10 PAGE 5 ii 2.2.1 Fuzzy Rules Set for FCIV ... .12 2.3 Testing the FCIV 15 2.4 Partial Results (FCIV Results) .. 19 2.5 Performance of the FCIV ..22 2.5.1 Zone 1...... ................. .................. .................. .................. ............23 2.5.2 Zone 2...... ................. .................. .................. .................. ............24 2.5.3 Zone 3...... ................. .................. .................. .................. ............24 2.6 FCIV Surfaces 25 2.7 Other Disturbances 27 2.8 Optimization Method .30 2.9 Summary .32 CHAPTER 3 A FUZZY ADAPTIVE INTERNAL MODEL CONTROLLER (FAIMCr) . . .. .... .... ..34 3.1 Introduction .....34 3.2 The Conventional Internal M odel Control (IMC) ... . 37 3.3 The FAIMCr Structure ... .38 3.3.1 The IMCFAM Unit (IMC Fuzzy Adaptive Module) 39 3.3.1.1 Module K ... ..41 3.3.1.1.1 Te sting the Module K ... ..44 3.3.1.2 Module TS . ..46 3.3.1.2.1 The FindingP1P2P3 Program . .. ..52 3.3.1.2.2 The Fmincon_TSIMC Program . . 52 3.3.1.2.3 The FuzzyTSIMC Program ... .53 PAGE 6 iii 3.3.1.2. 3.1 The TSIMC Fuzzy Inference System ..53 3.3.1.2. 3.2 TSIMC.fis Rules ... 56 3.3.1.2.4 The TSIMC_Con Program . 75 3. 3.1.3 Testing Module TS ..78 3.3.1.3.1 Te sting the IMCFAM Unit ..81 3.4 The IMC Filter T uning Equation .... 87 3.5 The IMCFF Unit (IMC With Variable Fuzzy Filter) .. ...90 3.5.1 Introducing the Fuzzy Filter into the IMC ... .95 3.5.2 Testing the IMCFF ... ...97 3.6 Testing the FAIMCr ...102 3.6.1 Testing the FAIMCr on a Nonlinear Process ... .109 3.7 Summary .114 CHAPTER 4 CONCLUSIONS AND FURTHER RESEARCH . .115 4.1 Conclusions 115 4.1.1 The FCIV Controlle r................. .................. .................. .............115 4.1.2 The IMCFAM Fu zzy Module................ ............... .............. ........115 4.1.3 The IMC Filter Tuning Equation .................. .................. ............116 4.1.4 The IMCFF Fuzzy Module................ ................ ................ ........116 4.2 Further Research................ ...................... ................ ................ ........116 REFERENCES .118 APPENDICES . ..120 Appendix 1 Process Model to Te st the FCIV . ... ... 121 PAGE 7 iv Appendix 2 Process Model to Te st the FAIMCr. .. ... 127 Appendix 3 Simulink Implementation for ) )( (* inv fF G ... 131 Appendix 4 Analyses of Variances for P1, P2, and P3 ... ..132 ABOUT THE AUTHOR . End Page PAGE 8 v LIST OF TABLES Table 1. Fuzzy Rules for the Firs t Unit, the FLC .. .12 Table 2. Basic Rules Used for the FCI Unit . .. 13 Table 3. The 10 Inputs Variables for the TSIMC .... ..57 Table 4. Process Model Parameters Va lues for the 2025 Simulations .....58 Table 5. Results of the First 75 Simulations (NSS Matrix) . .60 Table 6. Analysis of Variance fo r P1 in the NSS Matrix ...61 Table 7. Set of Linear Equat ions to Determine P1* . .63 Table 8. Set of Linear Equat ions to Determine P2* . .63 Table 9. Set of Linear Equat ions to Determine P3* . .64 Table 10. 10 TSIMC.fis Fuzzy Rules . ..70 Table 11. Process Model Pa rameters Values ...... 88 Table 12. Process Parameters and Optimum f Values .. ...89 Table 13. Analysis of Variance for f ...90 Table 14. Fuzzy Rules for the Fuzzy F ilter Inference System .. ..93 Table 15. Constants and Steady Stat e Values for Preheating Tank Variables .... ..... ..... ................ ......... 124 Table 16. Steady State Values for the Reactor ... 125 Table 17. Steady State Values for Some Variables in the Process .....126 Table 18. Mixing Tank Operat ing Conditions . ..... 128 PAGE 9 vi Table 19. Analysis of Variance for P1 in the NSS Matrix .. ....132 Table 20. Analysis of Variance for P2 in the NSS Matrix .. .... ....132 Table 21. Analysis of Variance for P3 in the NSS Matrix .. ....132 Table 22. Analysis of Variance for P1 in the NSM Matrix ...... ...133 Table 23. Analysis of Variance for P2 in the NSM Matrix ... ....... ... ...133 Table 24. Analysis of Variance for P3 in the NSM Matrix ...... ...133 Table 25. Analysis of Variance for P1 in the NSB Matrix .. ....134 Table 26. Analysis of Variance for P2 in the NSB Matrix .. ....134 Table 27. Analysis of Variance for P3 in the NSB Matrix .. ....134 Table 28. Analysis of Variance for P1 in the NMS Matrix . ... .135 Table 29. Analysis of Variance for P2 in the NMS Matrix . ... .135 Table 30. Analysis of Variance for P3 in the NMS Matrix ........ .135 Table 31. Analysis of Variance for P1 in the NMM Matrix ..... 136 Table 32. Analysis of Variance for P2 in the NMM Matr ix ........ 136 Table 33. Analysis of Variance for P3 in the NMM Matrix ..... 136 Table 34. Analysis of Variance for P1 in the NMB Matrix . ... .137 Table 35. Analysis of Variance for P2 in the NMB Matrix ........ .137 Table 36. Analysis of Variance for P3 in the NMB Matrix . ... .137 Table 37. Analysis of Variance for P1 in the NBS Matrix .. .. . .138 Table 38. Analysis of Variance for P2 in the NBS Matrix . .. .. .138 Table 39. Analysis of Variance for P3 in the NBS Matrix .. .. . .138 Table 40. Analysis of Variance for P1 in the NBM Matrix . ... .139 Table 41. Analysis of Variance for P2 in the NBM Matrix ... . .139 PAGE 10 vii Table 42. Analysis of Variance for P3 in the NBM Matrix ... . .139 Table 43. Analysis of Variance for P1 in the NBB Matrix ......... .140 Table 44. Analysis of Variance fo r P2 in the NBB Matrix . .140 Table 45. Analysis of Variance for P3 in the NBB Matrix ... .. .140 Table 46. Analysis of Variance for P1 in the ZSS Matrix .. .141 Table 47. Analysis of Variance for P2 in the ZSS Matrix .. .141 Table 48. Analysis of Variance for P3 in the ZSS Matrix .. .141 Table 49. Analysis of Variance for P1 in the ZSM Matrix ... .. .142 Table 50. Analysis of Variance fo r P2 in the ZSM Matrix . .142 Table 51. Analysis of Variance for P3 in the ZSM Matrix ............ .142 Table 52. Analysis of Variance for P1 in the ZSB Matrix .. .143 Table 53. Analysis of Variance for P2 in the ZSB Matrix .. .143 Table 54. Analysis of Variance for P3 in the ZSB Matrix .. .143 Table 55. Analysis of Variance for P1 in the ZMS Matrix .. ... .144 Table 56. Analysis of Variance for P2 in the ZMS Matrix .. ... .144 Table 57. Analysis of Variance for P3 in the ZMS Matrix .. ... .144 Table 58. Analysis of Variance for P1 in the ZMM Matrix .. .. .145 Table 59. Analysis of Variance for P2 in the ZMM Matrix .... .145 Table 60. Analysis of Variance for P3 in the ZMM Matrix .... .145 Table 61 Analysis of Variance for P1 in the ZMB Matrix .. .146 Table 62. Analysis of Variance for P2 in the ZMB Matrix ... .. .146 Table 63. Analysis of Variance for P3 in the ZMB Matrix ... .. .146 Table 64. Analysis of Variance for P1 in the ZBS Matrix . ... ..147 PAGE 11 viii Table 65. Analysis of Variance for P2 in the ZBS Matrix . . .. ..147 Table 66. Analysis of Variance for P3 in the ZBS Matrix ... . ..147 Table 67. Analysis of Variance for P1 in the ZBM Matrix ..... .148 Table 68. Analysis of Variance fo r P2 in the ZBM Matrix . .148 Table 69. Analysis of Variance for P3 in the ZBM Matrix .. ... .148 Table 70. Analysis of Variance for P1 in the ZBB Matrix . .... .149 Table 71. Analysis of Variance for P2 in the ZBB Matrix .. .149 Table 72. Analysis of Variance for P3 in the ZBB Matrix .. .149 Table 73. Analysis of Variance fo r P1 in the PSS Matrix .. .150 Table 74. Analysis of Variance fo r P2 in the PSS Matrix .. .150 Table 75. Analysis of Variance fo r P3 in the PSS Matrix .. .150 Table 76. Analysis of Variance for P1 in the PSM Matrix .. ... .151 Table 77. Analysis of Variance for P2 in the PSM Matrix .. ... .151 Table 78. Analysis of Variance for P3 in the PSM Matrix ..... .151 Table 79. Analysis of Variance fo r P1 in the PSB Matrix .. .152 Table 80. Analysis of Variance fo r P2 in the PSB Matrix .. .152 Table 81. Analysis of Variance fo r P3 in the PSB Matrix .. .152 Table 82. Analysis of Variance for P1 in the PMS Matrix .. ... .153 Table 83. Analysis of Variance for P2 in the PMS Matrix .. ... .153 Table 84. Analysis of Variance for P3 in the PMS Matrix .. ... .153 Table 85. Analysis of Variance for P1 in the PMM Matrix .. .. .154 Table 86. Analysis of Variance for P2 in the PMM Matrix .. .. .154 Table 87. Analysis of Variance fo r P3 in the PMM Matrix .154 PAGE 12 ix Table 88. Analysis of Variance for P1 in the PMB Matrix .. ... .155 Table 89. Analysis of Variance for P2 in the PMB Matrix .. ... .155 Table 90. Analysis of Variance for P3 in the PMB Matrix .. ... .155 Table 91. Analysis of Variance for P1 in the PBS Matrix .. ... .156 Table 92. Analysis of Variance fo r P2 in the PBS Matrix .. .156 Table 93. Analysis of Variance for P3 in the PBS Matrix ..... .156 Table 94. Analysis of Variance for P1 in the PBM Matrix .. ... .157 Table 95. Analysis of Variance for P2 in the PBM Matrix ..... .157 Table 96. Analysis of Variance for P3 in the PBM Matrix ..... .157 Table 97. Analysis of Variance for P1 in the PBB Matrix .. ... .158 Table 98. Analysis of Variance for P2 in the PBB Matrix .. ... .158 Table 99. Analysis of Variance for P3 in the PBB Matrix .. ... .158 PAGE 13 x LIST OF FIGURES Figure 1. Cascade Control Loop Using the FCIV .10 Figure 2. Scheme of the FCIV .11 Figure 3. Membership Functions for the Input s of the FCIV.......... .................. .13 Figure 4. Membership Functions for t he Outputs of the FCIV............ ...14 Figure 5. Responses from the Proc ess and from the Empirical Model When the Controller Output Signal is Increased by 10 %CO........ ........16 Figure 6. Responses from the Proc ess and from the Empirical Model (b) When the Controller Output Signal Increases (a) ...... .......... ......17 Figure 7. Responses from the Proc ess and from the Empirical Model (b) When the Controller Output Signa l Decreases (a) . ................ 18 Figure 8. Process Responses Under Di fferent Controllers ....... 20 Figure 9. Process Responses Un der Different Changes of +10 oF (+5.56 K),20 oF (11.11 K), +15 oF (+8.33 K), and 25 oF (13.89 K) in Ti(t).......... .................. ................ ................ .............21 Figure 10. Responses of Cascade Cont rol Strategies to Control the Output Concentration for the Ment ioned Disturbances ........ 22 Figure 11. Scheme for the FCIV Performance .. . ...23 Figure 12. Signal to the Valve fr om a PID, 2PIDs and the FCIV Controllers for Controlling the Output Concentration for a Disturbance of Te mperature by +10 oF (+5.56K) ......... ............ ....25 Figure 13. FLC Surface ...... ..26 Figure 14. FI Nonlinear Function .. . ..27 PAGE 14 xi Figure 15. Responses of PID, PIDs in Cascade and the FCIV to Control the Output Conc entration for the Mentioned Disturbances ... ............... .................. .................. ............... .......... 28 Figure 16. Signal to the Valve fr om PID, 2PIDs and the FCIV to Control the Output Concent ration for the Disturbances of Fig. 15. ...... .... ............... ................ ................ .............. ...........28 Figure 17. Signals of the Main Va riable (the Output Concentration) With and Without Noise for the FC IV Controller ....... .. . ..29 Figure 18. OptController Program Scheme 32 Figure 19. Scheme of the Conventional IMC Control Strategy . .....37 Figure 20. Scheme of the FAIMC 38 Figure 21. Scheme of the IMC Working With the IMCFAM Unit .. 39 Figure 22. IMCFAM Internal Structure ... .40 Figure 23. IMC Structure Used by Module K ... ..41 Figure 24. IMC Plant Internal Structure ... ...41 Figure 25. IMC Performances With and Wi thout Updating the Process Model Gain in the IMC Struct ure ....... .. ... .... 45 Figure 26. Process Model Gain Values () Calculated by the Module K Tracking the C hanges on the Process Gain Values (___).................. ................ .................... .............. .......................46 Figure 27. Normalized Modeling Error Response Showing P1, P2 and P3 ..................... ................ .................... .............. ................... 48 Figure 28. Scheme of the Module TS ... ..50 Figure 29. TSIMC.fis Scheme ... ... 54 Figure 30. A Sugeno Rule O peration Scheme . .....56 Figure 31. TSIMC.fis Internal Structure ... ... 65 Figure 32. Membership F unctions for the Input x1 ..... ..66 PAGE 15 xii Figure 33. Membership F unctions for the Input x2 ...... ..66 Figure 34. Membership F unctions for the Input x3 .. .... .. 67 Figure 35. Membership Function for the Inputs x4 and x7 ...... ..67 Figure 36. Membership Function for the Inputs x5 and x8 ... .. 68 Figure 37. Membership Function for the Input x6 .. 68 Figure 38. Membership Function for the Input x9 .. ... .69 Figure 39. Membership Function for the Input x10 ... . 69 Figure 40. Evaluation of the Membership Functions for x1 = 0.2 ....71 Figure 41. Evaluation of t he Membership Function for x2 = 3 . ... 71 Figure 42. Evaluation of t he Membership Function for x3 = 1.5 .. 71 Figure 43. Evaluation of t he Membership Function for x4 = 0.25 and x7 = 0.25 ..... .... ..72 Figure 44. Normalized Modeli ng Error Response Showing and P3 Signs, When the Process Dead Time Increases ( >0) ......... ..... ..77 Figure 45. Normalized Modeli ng Error Response Showing and P3 Signs, When the Process Dead Time Decreases ( <0).......... ........ 77 Figure 46. Performances of t he IMC and the IMC Working With Module K and Module TS for Severa l Set Point Changes.............79 Figure 47. Values of the Process Parameters (___), Values Obtained for the Process Model Pa rameters () and Values Calculated for the Filter Time Constant ( _ ) Testing Module TS................. .................... ................ .................... ..................80 Figure 48. Performances of the IMC and the IMCFAM Working for Several Disturbances and Set Point C hanges .............. ...... .....82 Figure 49. Values of the Process Parameters (___), Values Obtained for the Process Model Parameters () and Values Calculated for the Filter Time Constant ( _ ) Testing the IMC Working With the IMCFAM Un it ................. ............ .83 PAGE 16 xiii Figure 50. Performance of the IMC Working Wi th the IMCFAM Unit When Noise is Added to the Transmitter from the Controlled Variable .......... .................. ................ ............... ........... 85 Figure 51. Signal from the Controlle r Output from the IMC With the IMCFAM Unit in the Presenc e of Noise .... .... 86 Figure 52. IMC Responses for Different Values of f ... ... 87 Figure 53. Scheme of t he IMC Working With the IMCFF Unit. ... 91 Figure 54. Regions in the Response W here the Fuzzy Filter Module Acts................... .................. .................. ............... ................ ........... .92 Figure 55. Membership Functions fo r the Inputs of the IMCFF . 94 Figure 56. Membership Functions for the Output of t he IMCFF . .. .94 Figure 57. IMCFF Module Surface .. 95 Figure 58. Conventional IMC Briefly Modified 96 Figure 59. Performances of the C ontrollers IMC and the IMC Working With the IMCFF Unit Facing Proc ess Gain Changes .............. 98 Figure 60. Performances of the C ontrollers IMC and the IMC Working With the IMCFF Unit Faci ng Process Time Constant Changes.. ............ .................... ................ ................ .............. ...........99 Figure 61. Performances of the C ontrollers IMC and the IMC Working With the IMCFF Unit Facing Process Dead Time Changes .......... .............. .................. .................. ................ ........ 100 Figure 62. Performances of the C ontrollers IMC and the IMC Working With the IMCFF Unit Faci ng the Parameters Changes Shown in the Figures 59, 60, and 61 .................. .. 101 Figure 63. Performances of the IM C Working With the IMCFAM Unit and of the FAIMCr ..... ............. ....103 Figure 64. Process Parameter Changes (___) and the Process Parameters Changes Calculated by the FAIMCr () ..... ....104 Figure 65. Performances of the Conventional FAIMCr (___) and the IMC Working With the IMCFAM Unit () ..... ........... ...............105 PAGE 17 xiv Figure 66. Process Parameters C hanges Calculated by the IMC With IMCFAM Unit. ................ .... ............ .... ... ..106 Figure 67. Process Parameters Changes Calculated by the FAIMCr . ...107 Figure 68. FAIMCr Performances With and Without Noise ..108 Figure 69. Performance of t he FAIMCr for Set Point and Disturbances Changes ..... .... ................ ......... ...110 Figure 70. Process Model Parameters Changes and the Filter Time Constant Values Calculat ed by the FAIMCr for the Mixing Tank System ..... ....... 111 Figure 71. Responses from the PI D, IMC and FAIMCr Controllers, Facing the Set Points and Disturbances Introduced to the Mixing Tank Process ...... ................. .. ... 112 Figure 72. Performances of the IMC and FAIMCr Controllers .. ... ...113 Figure 73. Process Diagram........... .................. .................. .................. .............121 Figure 74. The Process: A Mixing Tank............ .................. .................. .............127 Figure 75. Implementation of the Transfer Function) )( (* inv fF G Using Simulink........... .................... .................. .................. ................131 PAGE 18 xv FUZZY LOGIC IN PROCESS CONTROL: A NEW FUZZY LOGIC CONTROLLER AND AN IMPR OVED FUZZYINTERNAL MODELCONTROLLER YOHN E. GARCA Z. ABSTRACT Two fuzzy controllers are presented. A fuzzy controller with intermediate variable designed for cascade control purposes is presented as the FCIV controller. An intermediate variable and a new set of fuzzy l ogic rules are added to a conventional Fuzzy Logic Controller (F LC) to build the Fuzzy Controller with Intermediate Variable (FCIV). The new c ontroller was tested in the control of a nonlinear chemical process, and its performance was co mpared to several other controllers. The FCIV shows the best cont rol performance regarding stability and robustness. The new controller also has an acceptable performance when noise is added to the sensor signal. An optimization program has been used to determine the optimum tuning parameters for all controllers to control a chemical process. This program allows obtaining the tuning parameters for a minimum IAE (Integral absolute of the error). The second controller presented uses fuzzy logic to improve the performance of the convent ional internal model controller (IMC). This controller is called FAIMCr (Fuzzy Adaptive Internal Model Controller). Two PAGE 19 xvi fuzzy modules plus a filter tuning equati on are added to the c onventional IMC to achieve the objective. The first fuzzy module, the IMCFAM, determines the process parameters changes. The second fuzzy module, the IMCFF, provides stability to the control system, and a tuning equation is develop ed for the filter time constant based on the process param eters. The results show the FAIMCr providing a robust response and overcoming stability problems. Adding noise to the sensor signal does not affect the performance of the FAIMC. The contributions presented in this work include The development of a fuzzy contro ller with intermediate variable for cascade control purposes. An adaptive model contro ller which uses fuzzy logi c to predict the process parameters changes for t he IMC controller. An IMC filter tuning equat ion to update the filter time constant based in the process parameters values. A variable fuzzy filter for the internal model controller (IMC) useful to provide stability to the control system. PAGE 20 1 CHAPTER 1 INTRODUCTION 1.1 Introduction Almost daily, control engineers fa ce the task of upgrading control strategies and controllers to handle proc ess nonlinearities. Chemical processes are notoriously nonlinear. In addition, most chemical processes have slow dynamics. These characteristics of proc ess nonlinearities and slow dynamics make chemical processes control quite challenging. Due to this challenge, most of the recent studies address the impr ovement of process control regarding stability, and robustness [Chiu, 1998], [Gormandy and Edgar, 2000], [Gormandy and Postlethwait, 2001]. In 1965 the theory of fuzzy logic wa s developed, and in 1974 its first application to industrial processes wa s presented [Mamdani, 1974]. Fuzzy logic provides means to deal with nonlinear syst ems, and its flexibility and simplicity makes fuzzy logic controllers suitable for many industrial applications. This research uses fuzzy logic to des ign a controller to improve the control performance of nonlinear pr ocesses with slow dynami cs using a variation of cascade architecture, and to improve t he performance of an already existing internal model controller. PAGE 21 2 1.2 Contributions of This Research This research provides the followi ng contributions to process control: 1.2.1.1 A Fuzzy Logic Controller With Intermediate Variable (Chapter 2) This controller substitutes the two conventional PIDs used in a process cascade control strategy by a single fuzzy controller. The controller consists of a three inputs two outputs Fuzzy Inference System (FIS) with the rules of a PItype FLC (Fuzzy Logic Controller) coupled with seven rules to deal with an intermediate variable change. The first par t of the FIS is a regular fuzzy logic controller (FLC) with the error of the primary controlled variable ( e ) and its change (e ) as inputs. The second part of the FIS, called Fuzzy Intermediate Rules ( FI ), handles an intermediate variable changes and adjusts the controller output anticipating the controlled variab le change. The output to the valve, m, depends on the contribut ion from the FLC, mFB, and the contribution from the FI mINT. This controller is referred to as Fuzzy Controller with Intermediate Variable, FCIV, 1.2.2 Adaptive Internal Mode l Controller (Chapter 3) Based on the response from the modeling error ( em) in the IMC (Internal Model Control) controller, and using fuzzy logic, it is possible to estimate the amount of change of each model param eter due to the nonlinear process behavior. The new values are then up dated into the IMC controller structure to PAGE 22 3 improve its performance in nonlinear processes. The ma ximum value, an inverse peak to the maximum value, and steadystate value of the modeling error ( em) are key to predict the parameters change. Da ta analysis and regression models are used to relate modeling error response an d process characteristics. Finally, a set of fuzzy rules are used to form the TakagiSugeno model to obtain the new parameters values. 1.2.3 IMC Filter Tuning Equation (Chapter 3) Based on a First Order Plus Dead Time (FOPDT) process model, the optimum filter value for a Single Input Single Output (SISO) IMC controller is tuned as a function of the model parameters: Process Gain ( KP), Time Constant () and Dead Time ( t0). The tuning equation for a wide range of process parameter values is developed for this purpose. 1.2.4 Variable Fuzzy Filter for Internal Model Control (IMC) (Chapter 3) Using fuzzy logic, it is possible to substitute the constant filter in the conventional IMC, by a variable filter. This new filter helps the IMC to overcome the inconvenience of instability when exce ssive oscillations appear. A Mamdanitype FIS with 49 rules is used to set a f ilter value that pur sues control loop stability. The error ( e ), and its change (e ) are the inputs to the FIS, and the change on the value of the filter constant, (F ) is the output. This improved IMC controller is called Inter nal Model Control with Fuzzy Filter, (IMCFF). PAGE 23 4 1.3 Survey and Discussion Much has been written over the pas t 30 years about advanced control with respect to the underlying theory, implem entation studies, benefits that its applications will bring, and projections of future trends. During the 1960s, advanced control was understoo d to be to an algorithm or strategy that deviated from the classical Pro portionalIntegralDerivative (PID) controller. Depending on anyones point of view, the concept of advanced control can be defined according to the specific proc ess where it is used: implementation of feedforward control; cascade control sc hemes; dead time compensators as IMC; tuning or adaptive algorithms of optimization strategies; and even combinations of some of them with Ar tificial Intelligence. Today, process plants must handle th e quality and the required production due to market demands, environmental concerns, and of course, keeping in mind energy and material costs. On the other hand currently ther e is no one technique that will solve all the cont rol problems that can mani fest in modern plants. Indeed, different plants have different requirements. The present work focuses on the use of Artificial Intelligence (Fuzzy Logic) for developing a simpler and more robus t controller as an alternative to the conventional cascade control, and for im proving Internal Model Control. PAGE 24 5 1.3.1 Recent Advances on Fuzzy Logic Control Fuzzy Logic is much closer in spir it to human thin king and natural language than traditional logi cal systems. In recent y ears, fuzzy logic has been successfully applied in the area of nonlin ear process control [Burden, Tantalean and Deshpande, 2003], [Foulloy and Galichet, 2003], [Govender and Bajic, 2003]. Fuzzy Logic is a practical alternative to a variety of challenging control applications since it provides a conv enient method for constructing nonlinear controllers via the use of heuristic information [Passino, 2001]. A Fuzzy Logic Controller (FLC) is ess entially a set of linguistic control rules with the objective to analyze vague input variables (fuzzification); to make a logic decision (inference mechanism); and to convert the conclusions reached into the output from the controller ( defuzzification) [Passino and Yurkovich, 1998]. In general, the FLC provides an algor ithm which can convert the linguistic control strategy based on expert knowledge into an automatic control strategy. 1.3.2 Recent Advances on Cascade Control Cascade control is one of the most popular control structures and significantly improves the performance provided by feedback control in some applications. Recently, Fuzzy Logic has been applied on cascade control of mechanical systems [Lepetic, 2003]. Using fuzzy identification, a TakagiSugeno fuzzy model was established for predictive purposes. PAGE 25 6 The present research develops a new fuzzy controller unit that uses the secondary variable information, similar to a cascade control strategy, to minimize the negative effect of distur bances on the main variable. 1.3.3 Advances on IMC Internal Model Control (IMC), in troduced by Garcia and Morari in 1982 has been considered one of the best strategies based on disturbance rejection and robustness analysis [Morari and Zafiri ou, 1989]. Recent developments have shown that combining Fuzzy Logic and IMC significantly improves the control performance on a variety of linear system s [Gormandy and Post lethwaite, 2002]. Due to the relevance of usi ng IMC over the last years, this research also focuses on the improvement of the performance of the IMC cont rol structure to overcome its limitations when deali ng with highly nonlinear chem ical processes. A self tuning filter, which becomes a variable filt er for excessive oscillations, and an adaptive process model are designed, using Fuzzy Logic, to improve the conventional IMC structure. 1.4 Summary and Scope of the Thesis This chapter summarizes the principal objectives of this dissertation to improve the performance of the conv entional control system facing the nonlinearities of the processes. Chapter 2 presents the design of a new fuzzy logi c controller, the FCIV, useful for cascade control purposes. PAGE 26 7 Chapter 3 presents an impr oved conventional IMC (I nternal model control) controller. Three modules have been designed and attached to the IMC controller: an adaptive internal model controller, an IMC filter tuning equation, and a variable fuzzy filter. Chapter 4 presents the conclu sions of the dissertation. PAGE 27 8 CHAPTER 2 FUZZY CONTROLLER WITH INTERMEDIATE VARIABLE (FCIV) This chapter proposes a new Fuzzy Controller and it constitutes the first contribution of the dissertat ion. An intermediate variable and a new set of fuzzy logic rules are added to a conventional Fu zzy Logic Controller (FLC) to build the Fuzzy Controller with Intermediat e Variable (FCIV). This c ontroller is tested in the control of a nonlinear chem ical process, and its per formance is compared to several other controllers. 2.1 Introduction The wellknown PID controlle rs are still the most adop ted controllers in the process industries. These controllers hav e a simple structure and are easy to tune. However, real syst ems often have nonlinearitie s and contain highorder dynamics and dead time, all of which diminish the performance of these controllers. Fuzzy Logic is a technique that uses language and reasoning principles similar to the way in which humans solv e problems [Zadeh, 1965]. This technique provides means to deal with nonlinear func tions, and flexibility and simplicity that makes it suitable for many industrial app lications [Ming, 1994], [Martins, 1997] PAGE 28 9 [Sugeno, 1985]. In the process control field the boom started in 1974 when Mandani controlled a steam en gine using fuzzy logic [R esnick, 1997]. In recent years, the technique has been applied successfully in the area of nonlinear process control [De Silva, 1995], [Chen and Kuo, 1995]. Feedback Control (FC) is the simplest form of automatic process control. However, its disadvantage is t hat it reacts only after the process has been upset. Even with this disadvantage, over 80 % of all strategies used in industrial practice are FC. In many processes with slow dynamics and with too many upsets, the control performance provided by feedback control often becomes unacceptable. It is necessary in these cases to use other strategies to provide the required performance. Cascade control is a strategy that improves, in some applications significantly, the performance provided by conventional feedback control. Recently, some works have applied fuzzy logi c into cascade control strategies to control mechanical suspension systems. In these cases, fuzzy identifications are developed to establish the TakagiSuge no fuzzy model for predictive purposes [Leptic, 2002], [Leptic, 2003]. Fuzzy Logic ha s also been applied as combination in cascade with a proportional integral (PI) controller to control the temperature of glass melting furnace [Moon and Lee, 2000]. This paper proposes a new fuzzy cont roller in which an intermediate process variable and a new set of fu zzy rules are added to the conventional Fuzzy Logic Controller (FLC); we refer to this controller as a Fuzzy Controller with Intermediate Variable (FCIV). The c ontroller resembles a cascade strategy PAGE 29 10 in that it uses an intermediate variable; however, it is a single controller with a single set point. This controller is tested in the control of a nonlinear chemical process, and its performance is compared to that of a PI D controller, FLC controller, PIDs in cascade, and FLCs in cascade. Pleas e note that actually the controllers used in this chapt er are PI controllers; we are using the term PID in a generic sense. 2.2 Fuzzy Controller With Inte rmediate Variable (FCIV) Figure 1 shows a control system with the FCIV as the controller. The controller consists of two fuzzy logic units as shown in Fig 2. The first unit (FLC) is a regular fuzzy logic controller with th e inputs being the error of the primary controlled variable, e(n) and its change, e(n) The second unit (FI) handles the intermediate variable. The input to this unit is the change in the intermediate variable, c2(n) The output to the valve, m(n) depends on the contributions from the FLC, mFB(n) and from the FI, mINT(n), units Controller Disturbances PROCESS FCIVSignal to valve, m Set Point, C1 setIntermediate variable, C2Main variable, C1 Controller Disturbances PROCESS FCIVSignal to valve, m Set Point, C1 setIntermediate variable, C2Main variable, C1 Controller Disturbances PROCESS FCIVSignal to valve, m Set Point, C1 setIntermediate variable, C2Main variable, C1 Controller Disturbances PROCESS FCIVSignal to valve, m Set Point, C1 setIntermediate variable, C2Main variable, C1 Figure 1. Cascade Control Loop Using the FCIV PAGE 30 11 K C 2 K e KeFLC FI KFBKINT FCIV e(n)e(n)c2(n)m(n)mFB(n)mINT(n) K C 2 K e KeFLC FI KFBKINT FCIV e(n)e(n)c2(n)m(n)mFB(n)mINT(n) K C 2 K e KeFLC FI KFBKINT FCIV e(n)e(n)c2(n)m(n)mFB(n)mINT(n) K C 2 K e KeFLC FI KFBKINT FCIV e(n)e(n)c2(n)m(n)mFB(n)mINT(n) Figure 2. Scheme of the FCIV The input and output terms for this controller are defined as follows: ) n ( c ) n ( r ) n ( e1 ) 1 n ( e ) n ( e ) n ( e ) 1 n ( c ) n ( c ) n ( c2 2 2 where: ) n ( r is the desired response, or set point ) n ( 1 c is the main, or primary, variable response ) n ( 2 c is the intermediate variable response n is the present value. 1 n is the previous value. PAGE 31 12 Tuning the FCIV requires five scaling factors: three for the inputs (Ke, K e and K C 2) and two for the outputs (KFB and KINT). We also refer to these scaling factors as tuning parameter s. These parameters are used to scale the inputs and outputs to match the range [1, 1] which is needed in the fuzzy units. 2.2.1 Fuzzy Rules Set for FCIV The rule matrix used by the FLC unit is based on the MacvicarWhelan matrix [MacvicarWhelan, 1976]. The meanings of t he linguistic variables involved are: negative big (NB), negative medium (NM) negative small (NS), zero (Z), positive small (PS), positive medi um (PM) and positive big (PB). Table 1 shows the distribution rules to obtain mFB(n). Table 1 Fuzzy Rules for the First Unit, the FLC NBNMNSZPSPMPB NBNBNBNBNBNMNSZ NMNBNBNBNMNSZPS NSNBNBNMNSZPSPM ZNBNMNSZPSPMPB PSNMNSZPSPMPBPB PMNSZPSPMPBPBPB PBZPSPMPBPBPBPB e(n) \ e(n) mINT(n) is obtained using another set of rule s shown in Table 2. These rules were chosen to correct the changes of the intermediate variable, c2(n) independent of the error, e(n) and its change, e(n ) in the primary variable. PAGE 32 13 Table 2. Basic Rules Used for the FCI Unit NBPB NMPM NSPS ZZ PSNS PMNM PBNB c(n) m(n) Five triangular membership functi ons and two trapezoi dal membership functions are used for both inputs and output s. Fig. 3 represents the membership functions for the inputs, while Fig.4 repr esents the membership functions for the outputs. 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 e(n), e(n), c2(n)Degree of membershipNBNSZPSPB NMPM Figure 3. Membership Functions for the Inputs of the FCIV PAGE 33 14 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 mFB(n), mINT(n)Degree of membershipNBNSZPSPB NMPM Figure 4. Membership Functions for the Outputs of the FCIV PAGE 34 15 2.3 Testing the FCIV The chemical reactor with a preheating tank described in Appendix 1 is used to illustrate the FCIV performance. The control loop consists of the reactor concentration as the controlled variable, the flow of steam as the manipulated variable, and the temperatur e in the preheating tank as the intermediate variable. A First Order Plus Dead Time (FOPDT) equation is used to simulate the process behavior. The FOPD T equation is the empiri cal model most commonly used for chemical process. The equation 2. 3.1 is a representation of a FOPDT in Laplace terms: 1 K ) s ( M ) s ( C ) s ( p Gs s t0 pe (2.3.1) where: M(s) is the Laplace transform of the controller output C(s) is the Laplace transform of the transmitter output Kp is the process gain and it indica tes how much the output changes per unit change in the input. is the process time constant; it i ndicates how fast the output changes once it started to chance. t0 is the process dead time. This param eter indicates how much time the takes the output to start changi ng once the input has changed. PAGE 35 16 Using the method recommended by Corri pio & Smith (fit 3), the process characteristics obtaine d for the model are: Kp = 0.67 %TO/%CO, = 29 min and t0 = 7.6 min. Figure 5 shows t he process and model responses. 0 50 100 150 50 51 52 53 54 55 56 57 58 time, minc, %TO Process Model(FOPDT) 1 29 67 06 7 s e FOPDTs .. 0 50 100 150 50 51 52 53 54 55 56 57 58 time, minc, %TO Process Model(FOPDT) 0 50 100 150 50 51 52 53 54 55 56 57 58 time, minc, %TO Process Model(FOPDT) 1 29 67 06 7 s e FOPDTs .. 1 29 67 06 7 s e FOPDTs .. Figure 5. Responses from the Process and from the Empirical Model When the Controller Output Signal is Increased by 10 %CO Figure 5 shows a good fitting bet ween the empirical model and the process at the steady state. Figures 6 and 7 illustrate how the nonlinear process response differs from the empirical model due to changes in operating conditions under openloop. PAGE 36 17 Figure 6 shows how the pr ocess model response dev iates from the model response when the controller output signal increases. This deviation gives an idea of the nonlinear nature of the process. Figure 7 illustrates a much bigger deviation from the process model when t he controller output decreases, showing a highly nonlinear behavior at low values of the controller output. 0 300 600 900 1200 50 60 70 80 90 100 time, minController output, %CO 0 300 600 900 1200 1500 40 50 60 70 80 90 time, minControlled variable, c, %TO Nonlinear Process Empirical Model (FOPDT) Controller Output (a) (b) 0 300 600 900 1200 50 60 70 80 90 100 time, minController output, %CO 0 300 600 900 1200 1500 40 50 60 70 80 90 time, minControlled variable, c, %TO Nonlinear Process Empirical Model (FOPDT) Controller Output (a) (b) Figure 6 Responses from the Process and fr om the Empirical Model (b) When the Controller Output Signal Increases (a) PAGE 37 18 0 200 400 600 800 1000 1200 1400 1600 1800 0 10 20 30 40 50 60 70 time, minController output, %CO 0 200 400 600 800 1000 1200 1400 1600 1800 10 20 30 40 50 60 time, minControlled variable, c, %TO Controller Output Nonlinear Process Empirical Model (FOPDT) (a) (b) 0 200 400 600 800 1000 1200 1400 1600 1800 0 10 20 30 40 50 60 70 time, minController output, %CO 0 200 400 600 800 1000 1200 1400 1600 1800 10 20 30 40 50 60 time, minControlled variable, c, %TO Controller Output Nonlinear Process Empirical Model (FOPDT) (a) (b) Figure 7 Responses from the Process and fr om the Empirical Model (b) When the Controller Output Signal Decreases (a) PAGE 38 19 2.4 Partial Results (FCIV Results) The results reported in this chapte r are based on simulations done using Simulink 5.0. A sampling time of 0.25 mi n was used for all the controllers. The input temperature to the prehe ating tank is assumed to be the main disturbance. Five control strategies were implement ed: PID feedback, PIDs in cascade, FLC feedback, FLCs in cascade, and the proposed FCIV. All tuning parameters were optimized to obtain the best control performance of each controller. The Integral of the Abso lute Value of the Error (IAE) was used as the optimiz ation criterion. A constr aint on the signal to the valve to avoid excessive oscillations wa s also used. The application of this constraint is explained on Section 2.8. The optimization method used for this purpose was Fminimax from Matlab 6.5. Figure 8 shows the responses when the input temperature to the preheating tank increases by 10 oF (5.56 K). The controlled variable, Cc(t), is recorded from the transmitter in %TO1. The IAE is reported for each control strategy PAGE 39 20 0 20 40 60 80 100 120 140 160 180 200 49.5 50 50.5 51 51.5 52 52.5 53 time,min%TO1 FLC PID PIDs Cascade FLCs Cascade FCIV Controller IAE PID PIDs Cascade FLCs Cascade FLC FCIV 36.46 9.62 95.47 8.89 2.54 Figure 8. Process Responses Under Different Controllers Figure 9 shows the responses when the input temperature to the preheating tank is changed at different times and for different values. The response for the control under FLC feedback is not shown because in all cases was much worse. The figure also shows the IAE values. PAGE 40 21 0 100 200 300 400 500 600 700 800 48 50 52 %TO1 0 100 200 300 400 500 600 700 800 48 50 52 %TO1 0 100 200 300 400 500 600 700 800 48 50 52 %TO1 0 100 200 300 400 500 600 700 800 48 50 52 time, min%TO1IAE=144 IAE=42.76 IAE=26.19 IAE=41.42 FCIV PIDsCascade PID +10F 20F 25F +15F FLCsCascade 0 100 200 300 400 500 600 700 800 48 50 52 %TO1 0 100 200 300 400 500 600 700 800 48 50 52 %TO1 0 100 200 300 400 500 600 700 800 48 50 52 %TO1 0 100 200 300 400 500 600 700 800 48 0 100 200 300 400 500 600 700 800 48 50 52 %TO1 0 100 200 300 400 500 600 700 800 48 50 52 %TO1 0 100 200 300 400 500 600 700 800 48 50 52 %TO1 0 100 200 300 400 500 600 700 800 48 50 52 time, min%TO1IAE=144 IAE=42.76 IAE=26.19 IAE=41.42 FCIV PIDsCascade PID +10F 20F 25F +15F FLCsCascade Figure 9. Process Responses Un der Different Changes of +10 oF (+5.56 K), 20 oF (11.11 K), +15 oF (+8.33 K), and 25 oF (13.89 K) in Ti(t) For another comparison, Fig 10 superim poses the response of the three schemes. The figure shows that the c ontrol provided by FCIV reaches the desired steady state value fa ster than the other strategies, and it also maintains its set point once it is reached, without undesired oscillations. The IAE obtained by FCIV (26.19) is less than the other two controllers, PID Cascade (42.76) and FLC Cascade (41.42). PAGE 41 22 0 100 200 300 400 500 600 700 800 49.4 49.6 49.8 50 50.2 50.4 50.6 50.8 time, min%TO1 +10F 20F +15F 25F PIDsCascade FLCsCascade FCIV 0 100 200 300 400 500 600 700 800 49.4 49.6 49.8 50 50.2 50.4 50.6 50.8 time, min%TO1 +10F 20F +15F 25F PIDsCascade FLCsCascade FCIV Figure 10. Responses of Cascade Control Strategies to Control the Output Concentration for the Mentioned Disturbances 2.5 Performance of the FCIV Figure 11 is a detailed scheme that s hows a) the responses of the main and the secondary variables, b) the fuzzy inputs variables a nd c) the fuzzy outputs variables. The data for Fig 11 was taken for a disturbance on the temperature to the preheating tank (+10 oF or +5.56 K). Three zones are considered to analyze the controller performance. PAGE 42 23 Figure 11. Scheme for the FCIV Performance 2.5.1 Zone 1 There is an immediate contribution fr om the FI unit because of the sudden change in the interm ediate variable, c2(t). The contribution from the FLC unit is minimum because the main variable has not yet been affected much (changes on e(n) and e(n) are very small). On this zone th e signal to the valve is mostly provided by the action of the FI unit. Both signals (from FI and FLC units) contribute to decrease the signal to the va lve in the same direction. For a better 0 10 20 30 40 50 C1 and C2, %TO 0 10 20 30 40 50 0.5 0 0.5 Fuzzy inputs 0 10 20 30 40 50 2 1 0 1 time, minFuzzy outputs e FI FLCa) b) c) Zone 1 Zone 2 Zone 3c1c2 e c2 mFB mINT m m PAGE 43 24 idea on how the FCIV controller works, the following data is taken from the response at 12 min: For FI unit: c2 = 0.1988 mFI = 0.9681 For FLC unit: e = 0.0139 & e = 0.1342 mFB = 0.3730 Thus the total signal to the valve is: m = (0.9681) + (0. 373) = 1.3441%CO 2.5.2 Zone 2 On the second zone, both the FI and FLC units are significantly contributing to the control effort. The second variable is return ing to its set point; c2 is negative in this zone, thus the FI unit sends positi ve values to valve signal. On the other hand, the FLC unit is send ing negative values to the valve signal due to the error, e(n), and its change, e(n) Both contributions make the manipulated variable, m(n), approach its new steady state value ( m(n) close to zero). 2.5.3 Zone 3 There is still some small compensati on from the FI a nd the FLC units, avoiding undesired oscillations and pr oviding stability to the response. Figure 12 shows the si gnal to the valve, m(n), from the FCIV controller, the PID controller, and 2PID controllers in cascade. PAGE 44 25 Figure 12. Signal to the Va lve from a PID, 2PIDs and the FCIV Controllers for Controlling the Output C oncentration for a Disturbance of Temperature by +10 oF (+5.56 K) 2.6 FCIV Surfaces As previously mentioned, the FCIV c ontroller is composed of the FLC and the FI fuzzy units. Figures 13 and 14 show their corresponding surfaces; U1 and U2 are the normalized outputs (before to the output scalin g factors) in the range of [1, 1]. 0 5 10 15 20 25 30 35 40 45 50 30 35 40 45 50 55 60 65 time, minController Output, m, %CO FCIV 2PIDs PID Zone 1 Zone 2 Zone 3 PAGE 45 26 1 0.5 0 0.5 1 1 0.5 0 0.5 1 1 0.5 0 0.5 1 error, e FCIV FIS surface error change, e U1 Figure 13. FLC Surface PAGE 46 27 1 0.5 0 0.5 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 c2U2FI Unit 1 0.5 0 0.5 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 c2U2FI Unit Figure 14. FI Nonlinear Function Figure 13 shows how U1 is related to the error and its change, and Fig. 14 shows that U2 is a function (nonlinear) of the intermediate variable change, c2. 2.7 Other Disturbances Figure 15 shows the control perform ance using a PID controller, PID controllers in a cascade environment, and the FCIV controller for different disturbances; Figure 16 shows the mani pulated variable signal. The FCIV controller reaches the steady state faster and offers better stability in all cases. PAGE 47 28 0 200 400 600 800 1000 1200 1400 1600 1800 48.5 49 49.5 50 50.5 51 51.5 52 time, min%TO1 PID PIDs FCIV Disturbances: 100min) 10 oF 300min) 20% inlet conc. up 600min) +5 oF 800min) 20% inlet flow up 1200min) 5 oF 1500min) +5 oF Figure 15. Responses of PID, PIDs in Cascade and the FCIV to Control the Output Concentration for the Mentioned Disturbances 0 200 400 600 800 1000 1200 1400 1600 1800 0 10 20 30 40 50 60 70 80 time, minController Output, m, %CO PID PIDs FCIV Disturbances: 100min) 10 oF 300min) 20% inlet conc. up 600min) +5 oF 800min) 20% inlet flow up 1200min) 5 oF 1500min) +5 oF Figure 16. Signal to the Va lve from PID, 2PIDs and the FCIV to Control the Output Concentration for t he Disturbances of Fig. 15 PAGE 48 29 To briefly study the effect of noise, an Auto Regressive Moving Average noise (ARMA(1,1) noise) with standard deviation of 0.4%TO was added to the signal from the analyzer transmitter. Fi g. 17 shows both curves with and without noise when the FCIV controller is facing the disturbances me ntioned in Fig 15. The presence of this particular noise does not make a significant difference on the performance of the FCIV. 0 200 400 600 800 1000 1200 1400 1600 1800 49 49.5 50 50.5 51 51.5 52 52.5 time, min%TO1FCIV controller noise no noise Figure 17. Signals of the Main Variable (the Output Concentration) With and Without Noise for the FCIV Controller PAGE 49 30 2.8 Optimization Method To obtain the optimized parameters for all controllers a program was developed using optimization tools from MATLAB. Fig. 18 shows the scheme of this program, which is named OptController. The OptController program calls the Fm inimax optimization routine and it also loads the required parameters for Fminimax execution: the closed loop model, the set of fuzzy rules and the guessing values. The optimal conditions obtained from this optimization routine are saved on this main program. For our purpose, Fminimax was fixed in order to find the set of controller parameters which lead to minimum I AE under one constraint applied over the controller output signal, m. The constraint was designed to avoid the presence of undesired oscillations on the controller output sign al at the new stead y state. In other words, when the steady state is reac hed again, the controlle r output signal (m) must be constant; therefor e, the following condition should be accomplished: m(n) = m(n) m(n1) = 0 (2.8.1) where m(n) Is the present value of t he controller output signal. m(n1) Is the previous value of the controller output signal. PAGE 50 31 Since the Fminimax pr ogram requires a constraint in the form C 0 for optimization purposes, the fo llowing equation is used as a constraint on the controller output signal: Cm(n) (2.8.2) where C is the constraint value and represents the valve noise tolerance from the controller output. For the simulations in this chapter has been established as 0.01. Thus, when the new steady state is reached, m(n) should be 0 (if there is no valve noise) or m(n) (because of the minimum valv e noise), and from Equation 2.8.2 C always will be 0, which is allowed by Fminimax program as a constraint m(n) > implies either high level of noise in the controller output signal or oscillating contro ller output signal. This constr aint is applied for each controller response. For all iterations, Fminimax calls two additional programs: trackmmobj and trackmmcon. The trackmmobj program executes the simulation and maintains the values of the IAE signal, yout(n), and the controller output signal, m(n). The trackmmcon program verifies that m(n) has been obtained under the fixed constraint for this signal, C 0. If this is true, Fminimax obtains the set of parameters for a minimum yout(n) value. PAGE 51 32 Fminimax(Routine program)trackmmobj(program)trackmmcon(program)C 0? Model Simulation yout(n), m(n)Optimum Parameters OptControllerMain Program Optimization Routine for minimum IAE ( minimum yout ) min yout ? No Yes Fminimax(Routine program)trackmmobj(program)trackmmcon(program)Model Simulation yout(n), m(n) OptControllerMain Program Optimization Routine for minimum IAE ( minimum yout ) ? No Yes Fminimax(Routine program)trackmmobj(program)trackmmcon(program)C 0? Model Simulation yout(n), m(n)Optimum Parameters OptControllerMain Program Optimization Routine for minimum IAE ( minimum yout ) min yout ? No Yes Fminimax(Routine program)trackmmobj(program)trackmmcon(program)Model Simulation yout(n), m(n) OptControllerMain Program Optimization Routine for minimum IAE ( minimum yout ) ? No Yes Figure 18. OptController Program Scheme 2.9 Summary This chapter has presented a new fu zzy controller called, FCIV, Fuzzy Controller with Intermediate Variable, us eful for cascade control purposes. An intermediate variable and a new se t of fuzzy logic rules were added to a conventional Fuzzy Logic Controller (FLC). The new controller was tested in the control of a non linear chemical process, and its performance was compared PAGE 52 33 to several other controllers. The FCIV shows the best control performance regarding stability and robus tness. The new controller also has an acceptable performance when noise is added to the sensor signal. An optimization program has been used to determine the optimum tuning parameters for all controllers to control t he chemical process of this chapter. This program allows obtaining the tuning par ameters for a minimum IAE (Integral absolute of the error). PAGE 53 34 CHAPTER 3 A FUZZY ADAPTIVE INTERNAL MO DEL CONTROLLER (FAIMCr) This chapter proposes an improved In ternal Model Controller using Fuzzy Logic; we refer to this controller as the FAIMCr. The body of the FAIMCr consists of adding three response analysis stru ctures to the c onventional IMC: A fuzzy module for updating the proce ss parameters in the IMC. This fuzzy module is the second contri bution of this dissertation. A tuning equation for the filter time constant based on the updated process parameters. This equation consti tutes the third contribution of this dissertation. A fuzzy module to modify the filter time constant when it is required. This module constitutes the fourth cont ribution of this dissertation. The FAIMCr is tested on a couple of nonlinear systems; the controller improves the IMC performance and successf ully avoids stability problems. 3.1 Introduction Fuzzy logic is an artificial intelligent technique which provides a way to face the problem of proce ss nonlinearities. Internal Model Control (IMC) is a PAGE 54 35 Model Based Control (MBC) techniqu e that has shown good performance regarding disturbance rejection and r obustness [Morari and Zafiriou, 1989]. Recent developments have shown t hat combining Fuzzy Logic with IMC improves the control strategy performanc e on a variety of linear systems [Gormandy and Postlethwaite, 2002]. Looking for improving the IMC performance in nonlinear systems, fuzzy nonlinear models have been c onstructed by applying leastsquares identification techniqu e to past process pH data, and the performance of this fuzzy controller is considered good in pH control [Edgar and Postlethwaite, 2002]. This Postlethwa ites IMC fuzzy controller was designed using fuzzy relational models equivalent to the 0th order Takagi Sugeno fuzzy inference system, and also uses two fi lter constants to improve the IMC performance in pH control. Due to the nonlinear nature of most processes, and searching for a general industrial application, this re search pursues improving the performance of the Internal Model Cont rol strategy on nonlinear sy stems by including the use of Fuzzy Logic. The proposed IMC controller uses a 1st order TakagiSugeno fuzzy inference system (more accurate than a 0th order) to update the process model parameters, and also uses a fuzzy filter to overcome the stability problems when required. The new controller is referred to as Fuzzy Adaptive Internal Model Controller, (FAIMCr). The FAIMCr consists of two fuzzy inference systems (FIS) added to the conventional IMC. The first FIS, the IMC Fuzzy Adaptive Model, (IMCFAM), PAGE 55 36 determines based on the modeling error response the changes on the process characteristics, and updates the process model parameters values in the IMC. If no changes are detected on t he process characteristics, then the process model parameters conserve the current values. The second fuzzy inference system, t he "IMC Fuzzy Filter," (IMCFF), only acts after a second peak appears on the controlled variable response. At this time, the IMCFF tends to increase the f ilter value. This action reduces the controller aggressiveness; ther efore, it is useful for avoiding stability problems due to excessive oscillations. If no oscillat ions are present then the filter time constant remains unchanged until the new steady state is reached, at that time the value is recalculated based on the process model parameters and updated into the IMC. PAGE 56 37 3.2 The Conventional Intern al Model Control (IMC) Figure 19 represents the structure of the conventional IM C. Three transfer functions constitute the scheme for this control strategy. The process model transfer function is a First Order Plus Dead Time (FOPDT) equation. The FOPDT model is commonly used for chemical processes and widely used for tuning purposes [Smith and Corripio, 2005]. modeling error,em Plant Model Inverse Filter + _Set Point, Gf+ + _CControlled variable,FinvF F* *CsetCset adjM Cm Disturbances modeling error,em Plant Model Inverse Filter + _Set Point, Gf+ + _CControlled variable,FinvF F* *CsetCset adjM Cm Disturbances modeling error,em Plant Model Inverse Filter + _Set Point, Gf+ + _CControlled variable,FinvF F* *CsetCset adjM Cm Disturbances modeling error,em Plant Model Inverse Filter + _Set Point, Gf+ + _CControlled variable,FinvF F* *CsetCset adjM Cm Disturbances Figure 19. Scheme of the Conventional IMC Control Strategy The transfer functions involved are: Process model: 10 s eK Fm st m *m (3.2.1) Inverse: m m invK s F 1 (3.2.2) Filter: 1 1 s Gf f (3.2.3) where Km, m and t0 m are the process model paramet ers (process characteristics) and f is the filter time const ant of the IMC controller. PAGE 57 38 3.3 The FAIMCr Structure Figure 20 shows the bl ock diagram of the pr oposed controller. The scheme includes the three st ructures: the tuning equat ion module, plus the two fuzzy inference systems (the IMC FF unit and the IMCFAM unit). em Plant Model Inverse Filter + _Set Point + + IMCFAMFuzzy Adaptive ModuleemMControlled Variable, CGfFinv* F* F M CmModel Parametersm new IMCFFFuzzy Filter Module ee IMCf fTuning Equation C setCsetm newt0m newDisturbances C set adj em Plant Model Inverse Filter + _Set Point + + IMCFAMFuzzy Adaptive ModuleemMControlled Variable, CGfFinv* F* F M CmModel Parametersm new IMCFFFuzzy Filter Module ee IMCf fTuning Equation C setCsetm newt0m newDisturbances C set adj em Plant Model Inverse Filter + _Set Point + + IMCFAMFuzzy Adaptive ModuleemMControlled Variable, CGfFinv* F* F M CmModel Parametersm new IMCFFFuzzy Filter Module ee IMCf fTuning Equation C setCsetm newt0m newDisturbances C set adj em Plant Model Inverse Filter + _Set Point + + IMCFAMFuzzy Adaptive ModuleemMControlled Variable, CGfFinv* F* F M CmModel Parametersm new IMCFFFuzzy Filter Module ee IMCf fTuning Equation C setCsetm newt0m newDisturbances C set adj Figure 20. Scheme of the FAIMC PAGE 58 39 3.3.1 The IMCFAM Unit (I MC Fuzzy Adaptive Module) Once a new steady state is reached, the IMCFAM unit determines the process parameters changes and updates the process model parameters in the IMC structure. The IMCFAM unit consis ts of adding one analyticalfuzzy module to the conventional IMC (see Figure 21). em Plant Model Inverse Filter + _Set Point + + IMCFAMFuzzy Adaptive ModuleControlled Variable, CGfFinv* F* F M CmModel Parameters em Plant Model Inverse Filter + _Set Point + + emM Controlled Variable, CGfFinv* F* F M Cm C setC set m newm newt0m newm newm newt0m newC set adj Disturbances em Plant Model Inverse Filter + _Set Point + + IMCFAMFuzzy Adaptive ModuleControlled Variable, CGfFinv* F* F M CmModel Parameters em Plant Model Inverse Filter + _Set Point + + emM Controlled Variable, CGfFinv* F* F M Cm C setC set m newm newt0m newm newm newt0m newC set adj Disturbances em Plant Model Inverse Filter + _Set Point + + IMCFAMFuzzy Adaptive ModuleControlled Variable, CGfFinv* F* F M CmModel Parameters em Plant Model Inverse Filter + _Set Point + + emM Controlled Variable, CGfFinv* F* F M Cm C setC set m newm newt0m newm newm newt0m newC set adj Disturbances em Plant Model Inverse Filter + _Set Point + + IMCFAMFuzzy Adaptive ModuleControlled Variable, CGfFinv* F* F M CmModel Parameters em Plant Model Inverse Filter + _Set Point + + emM Controlled Variable, CGfFinv* F* F M Cm C setC set m newm newt0m newm newm newt0m newC set adj Disturbances Figure 21. Scheme of the IMC Working With the IMCFAM Unit The inputs to the IMCFAM are: the modeling error signal ( em), the set point change (C set), the previous proc ess model parameters ( Km, m and t0 m) and the signal to the plant ( M ). The outputs are the new process model parameters: Km new, m new and t0 m new. Figure 22 shows the internal structure of t he IMCFAM unit. The figure illustrates the IMCFAM consisting of two internal ca lculation modules: PAGE 59 40 the Module K which determines using ma thematical analysis the new process model gain value ( Km new), and the Module TS which pr edicts, using fuzzy logic, the fraction of change of the process time constant () and the fraction of change of the process dead time (). These changes are added to the previous model values (m and t0 m) resulting in m new and t0 m new Module K acts after a set point change or a disturbance occurs, while M odule TS only acts after a set point change. Model P arametersC setemMnemratio++ + + +Km newm newt0m newKmmt0m K TSp Model P arametersC setemMnemratio++ + + +Km newm newt0m newKmmt0m K TSp Model P arametersC setemMnemratio++ + + +Km newm newt0m newKmmt0m K TSp Model P arametersC setemMnemratio++ + + +Km newm newt0m newKmmt0m K TSp Figure 22. IMCFAM Internal Structure PAGE 60 41 3.3.1.1 Module K Module K uses the modeling error signal ( em), the signal to the plant ( M ), and the previous model gain ( Km) to calculate the new model gain ( Km new). Figure 23 shows the bold delineated blocks diagram useful fo r this purpose, and Figure 24 illustrates the plant in a bit more detail. Plant Model + _CControlled variable,F F*Cm + _CControlled variable,F F*M Cmmodeling error,em Inverse Filter Finv* Gf Set Point,CsetCset adj Disturbances Plant Model + _CControlled variable,F F*Cm + _CControlled variable,F F*M Cmmodeling error,em Inverse Filter Finv* Gf Set Point,CsetCset adj Disturbances Plant Model + _CControlled variable,F F*Cm + _CControlled variable,F F*M Cmmodeling error,em Inverse Filter Finv* Gf Set Point,CsetCset adj Disturbances Plant Model + _CControlled variable,F F*Cm + _CControlled variable,F F*M Cmmodeling error,em Inverse Filter Finv* Gf Set Point,CsetCset adj Disturbances Figure 23. IMC Structure Used by Module K Plant Model + _CControlled variable, F* + GpMmodeling error,em Disturbance, D Gd Cm+ Plant Model + _CControlled variable, F* + GpMmodeling error,em Disturbance, D Gd Cm+ Plant Model + _CControlled variable, F* + GpMmodeling error,em Disturbance, D Gd Cm+ Plant Model + _CControlled variable, F* + GpMmodeling error,em Disturbance, D Gd Cm+ Figure 24. IMC Plant Internal Structure PAGE 61 42 Once the new steady state is reached, the change in the modeling error is calculated as: ) s ( M ) s ( F s lim ) s ( D ) s ( G ) s ( M ) s ( G s lim C C es d p s m m 0 0 (3.3.1.1) where: em is the change in the modeling error. C is the change in the controlled variable. Cm is the change in the model variable. Gp(s) is a process transfer function (unknown) that describes how the signal to the valve affects the controlled variable. Gd(s) is a transfer function (unknow n) that describes how the disturbance affects the controlled variable. F*(s) is the process model transfer function, 10 s e K Fm s t m *m (3.3.1.2) M(s) is the change on the signal from the controller to the plant. D(s) is the change on a disturbance. Using step changes for M(s) and D(s): s M ) s ( F s lim s D ) s ( G s lim s M ) s ( G s lim e* s d s p s m 0 0 0 (3.3.1.3) PAGE 62 43 Taking the limit, the change on the modeling error is: M K D K M K em d p m (3.3.1.4) where Kp is the process gain, Km is the process model gain and Kd is the disturbance transfer function gain. Dividing by M and arranging terms: M e K M D K Km m d p (3.3.1.5) Defining an apparent gain, (Kap) as: M D K Kd p M e K M D K K Km m d p ap (3.3.1.6) For a set point change (0 D ) the apparent gain results in: M e K K Km m p ap (3.3.1.7) For a disturbance change (0 D ) the apparent gain results in: M e K M D K K Km m d p ap (3.3.1.8) In both cases, the apparent gain, Kap, is estimated as M e Km m and its value represents the new model gain (new mK), which includes the nonlinear effects on PAGE 63 44 the process gain (Kp) and the contribution (unknown) from disturbances on the process ( M D Kd ). Equation (3.3.1.8) is used in Modul e K to determine the new process model gain value after each disturbance or a set point change. 3.3.1.1.1 Testi ng the Module K Assume a process with the next transfer function, 1 5 2 130 s e s e K Gs p s t p pp (3.3.1.9) The disturbance transfer function as, 1 5 1 10 s s e K ) s ( Gd s t d dd (3.3.1.10) and the process model transfer function, F*(s), is: 1 5 2 130 s e s e K ) s ( Fs m s t m *m (3.3.1.11) To test the Module Ks performance, several set point changes and disturbance values were introduced, and changes in the process gain (Kp) were made at the same time a set point chang e or a disturbance enter the process. The set point changes and the disturbances values (D values) are shown in Figure 25, and the process gain changes are shown in Figure 26. PAGE 64 45 Figure 25 shows the improvement of the IMC performance by updating the process model gain in the IMC structure. The figure illustrates how the Module K helps the IMC to overcome the stability problems. The f ilter time constant has been initially calculated using the tuning equation from Sectio n 3.4, and its value is 3.445. 0 100 200 300 400 500 600 700 800 40 50 60 70 80 90 100 time, minControlled variable, c, %TO IMC constant Km (IAE=2082) IMC updating Km (IAE=382.4) Disturbances (D): 200min) D=2 400min) D=2 600min) D=10 Figure 25. IMC Performances With and Without Updating the Process Model Gain in the IMC Structure Figure 26 shows the different val ues of the process model gain (Km new) calculated by the Module K, once the new steady state is reached for each set point change and also a fter the disturbances. PAGE 65 46 0 100 200 300 400 500 600 700 800 0 1 2 3 4 5 6 7 8 9 10 time, minProcess and Model gains Process gain ( Kp ) Model gain ( Kp new) Figure 26. Process Model Gain Values () Calculated by the Module K, Tracking the Changes on the Process Gain Values (___) 3.3.1.2 Module TS Module TS predicts whether the dy namic process parameters time constant ( ) or dead time (0t) have changed. Specifically the module predicts, using fuzzy logic, the fraction change of the time constant () and the fraction change of the dead time ( ). Once these fractions ar e known, and knowing the current values of the param eters, their new values can be easily calculated. Module TS acts when the new steady state is reached after a set point change. This module uses the new process model gain value calculated by Module K. PAGE 66 47 The calculations of the and fractions are not as straightforward as the calculation for the new gain in Module K. The first thing that needed to be decided on was what terms would indica te a change in the time constant and dead time parameters. No previous information was available to help on this decision. After an extensive search, and mu ch iteration, it was decided that the response curve of the normalized modeling error (mne) provides an indication that the process characteristics ma y have changed. The normalized modeling error (mne) is defined as the ratio between the modeling error and the set point change, set m mC e ne /, and it is a dimensionless quantity Specifically, the research showed that from the response curve the terms P1 (maximum peak of the mneresponse curve), P2 (time to reach P1 ), and P3 (the inv erse peak (if it exists) to P1), are very good indicators of the changes in the dynamic process characteristics; Fig. 27 shows the mneresponse curve and the P1, P2, and P3 terms. PAGE 67 48 0 10 20 30 40 50 60 70 80 90 100 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 timenem P2 P1 P3 0 10 20 30 40 50 60 70 80 90 100 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 Normalized modeling error, nem P2 P1 P3 0 10 20 30 40 50 60 70 80 90 100 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 timenem P2 P1 P3 0 10 20 30 40 50 60 70 80 90 100 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 Normalized modeling error, nem P2 P1 P3 Figure 27. Normalized Modeling Error Response Showing P1, P2 and P3 Obviously, the issue of how to relate and 0t, or their fraction change, to P1, P2, and P3 still remained. Many simulation experiments were conducted to establish the relations between the proce ss characteristics and the indicators P1, P2, and P3. These experiments yielded that the following five terms could be used: The fraction of change of the process model gain, m m new mK K K / ) ( The previous process model time constant,m The ratio between the previous proc ess model dead time and the previous process model time constant, m m / t ratio0 The and fractions PAGE 68 49 A significant number of equations were developed (shown in this chapter) and integrated into a TakagiSugeno fu zzy inference system, relating P1, P2, and P3 to the five terms mentioned. Let us define some new nom enclature that will help in the explanation that follows. The values of the three indicators obtained from the mneresponse curve are called P1, P2, and P3. The values obtained from the equations are referred to P1*, P2*, and P3*, or ) , , ( 11 ratio f Pm ) , , ( 22 ratio f Pm ) , , ( 33 ratio f Pm An analysis of the terms involv ed in the equations show that ,m and ratio are based on information known. However, and are not known, and actually, these are the terms of interest. If these terms the fraction changed by the time constant ( ), and the fraction changed by the dead time ( ) could be obtained, then the new values of the ti me constant and dead time could be calculated by, ) 1 ( m new m and ) 1 (0 0 m new mt t. It is proposed to obtain the values of and by minimizing the follo wing objective function, 2 2 2 *3 3 2 2 1 1 P P P P P P f (3.3.1.2.1) PAGE 69 50 The minimization of this function yields and That is, the objective is to search for the values of and that minimize the functionf. Figure 28 illustrates the internal st ructure, the inputs and the outputs of Module TS. FindingP1 P2 P3 and Fuzzy_TSIMC TSIMC_ConFmincon_TSIMC min f ? Optimization Routine for minimum f Optimum values! Call TSIMCfis Calculate P1 *P2 *P3 2 iiPP f If > 0 and P3 < 0 then > 0 If < 0 and P3 > 0 then < 0nemmratioNo Yes FindingP1 P2 P3 and Fuzzy_TSIMC TSIMC_ConFmincon_TSIMC min f ? Optimization Routine for minimum f Optimum values! Call TSIMCfis Calculate P1 *P2 *P3 2 iiPP f If > 0 and P3 < 0 then > 0 If < 0 and P3 > 0 then < 0nemmratioNo Yes FindingP1 P2 P3 and Fuzzy_TSIMC TSIMC_ConFmincon_TSIMC min f ? Optimization Routine for minimum f Optimum values! Call TSIMCfis Calculate P1 *P2 *P3 2 iiPP f 2 iiPP f If > 0 and P3 < 0 then > 0 If < 0 and P3 > 0 then < 0nemmratioNo Yes FindingP1 P2 P3 and Fuzzy_TSIMC TSIMC_ConFmincon_TSIMC min f ? Optimization Routine for minimum f Optimum values! Call TSIMCfis Calculate P1 *P2 *P3 2 iiPP f 2 iiPP f If > 0 and P3 < 0 then > 0 If < 0 and P3 > 0 then < 0nemmratioNo Yes Figure 28. Scheme of the Module TS As mentioned earlier, the inputs to Module TS are: The normalized modeling error (nem). Note that nem is a vector containing the normalized modeling erro r from the moment the variable deviates from the set point until it returns back to steady state. The fraction of change on the process model gain, The previous process model time constant (m) PAGE 70 51 A ratio (ratio), defined as the ratio between the previous process model dead time and the previous process model time constant (ratio = t0 m /m). The outputs from this fuzzy module are th e fraction change of the process time constant () and the fraction change of the process dead time (). Four programs constitute the structure of this module: The FindingP1P2P3 program, which determines three parameters from the normalized modeling error response, P1, P2 and P3. The Fmincon_TSIMC program. This program optimizes the process time constant and the process dead time fraction changes, and. The Fmincon_TSIMC program uses two additional programs: the fuzzyTSIMC and the TSIMC_Con. The fuzzyTSIMC program. This program ca lls the Takagi Sugeno fuzzy inference system (TSIMC.fis), obtains the P1* P2* P3* values, calculates the objective f unction (Equation. 3.3.1.2.1) and sends its value to the main program. The TSIMC_Con program, contains the constraints for the main program execution. Th is program is used by the optimization routine for changes on the process dead time. The following sections present in deta il the different programs in Module TS. PAGE 71 52 3.3.1.2.1 The FindingP1P2P3 Program When the new steady sate is established, the FindingP1P2P3 program calculates three parameter s from the normalized modeling error response, P1 (maximum peak of the nem response), P2 (time to r each P1) and P3 ("the inverse peak" (if this exists) to P1); If any of these three parameters are not present in the nem response, the FindingP1P2P3 program assigns a zero value to the missing variables. Appendix 2 contains the Matlab structure of this program. Figure 28 shows the parameters P1, P2 and P3 in a nem response. 3.3.1.2.2 The Fmincon_TSIMC Program Fmincon from Matlab 6.5, is an op timization program that finds a constrained minimum of a func tion of several variables. Fmincon_TSIMC is the program that obtains the values of and that minimize the objective function given by Equation 3.3.1.2.1. The Fmincon_TSIMC program uses two additional programs, the fuzzyTSIMC and the TSIMC_Con. Once the parameters P1, P2 and P3 are determined by the FindingP1P2P3 program, the Fmincon_TSIMC program sends to the fuzzyTSIMC program, the guessing values for and the fraction of change on the process gain (), the previous process model time constant (m), and the ratio (ratio); using this information fuzzyTSIMC calculates the predictive parameters P1*, P2* and P3* using a fuzzy inference syst em (the TSIMC.fis). The initial guess values of and are always zero (0), whic h assumes that no changes in and 0thave occurred. PAGE 72 53 For each iteration routine (a ma ximum of 500 iterations), the Fmincon_TSIMC program changes the values of and to minimize the objective function, f. The optimization is carried out according to constraints in the TSIMC_Con program (to be explained later). 3.3.1.2.3 The fuzzyTSIMC Program The fuzzyTSIMC program obtains the predictiv e parameters P1*, P2* and P3* using a Takagi Sugeno fuzzy inferenc e system, the TSIMC.fis. To obtain these values, the TSIMC.fis uses 5 inputs (, m, ratio, and ) and 3 sets of linear equations, one set for each parameter. For each iteration, the fuzzyTSIMC program obtains the predictive parameters and evaluates t he objective function, f sending its value to the main program. For all iterations, the param eters P1, P2 and P3 obtained from the FindingP1P2P3 program remain constant. 3.3.1.2.3.1 The TSIMC Fuz zy Inference System A first order Sugeno (or TakagiSugeno) fuzzy inference system (TSIMC.fis) is designed to obtain the val ues of P1*, P2*, and P3*. Higher order Sugeno fuzzy models are possible; howe ver, they introduce significant complexity with little obvious merit. The inputs to the TSIMC.fis are the values of and the fraction of change on the process gain (), the previous model time constant (m), and the ratio (ratio). The outputs are the P1*, P2* and P3* values. That is, PAGE 73 54 P1*= f (, m, ratio, ) P2*= f (, m, ratio, ) P3*= f (, m, ratio, ) Figure 29 shows the scheme of the TSIMC.fis. TSIMC.fismratio P1* P2* P3* (Sugeno) 27 rules and TSIMC.fismratio P1* P2* P3* (Sugeno) 27 rules and Figure 29. TSIMC.fis Scheme The following example illus trates the meaning of a fuzzy logic function in a Sugeno fuzzy inference system. For a Sugeno fuzzy first order model wi th 2 inputs and 1 output, a typical rule has the form: If Input 1 = x and Input 2 = y, then Output is z = ax + by + c The constants a, b and c are obtained from the fitted data between the output and the inputs as a linear regression. For a zero order model, the output z is a PAGE 74 55 constant (a = b = 0). For several output rules, the output level of each rule, zi, is weighted by the rule weigher, wi. For example, for AND rule with input 1 = x and input 2 = y, the rule weigher is: w1 = andMethod [F1(x), F2(y)] = min [F1(x), F2(y)] where F1 and F2 are the membership functions for the inputs 1 and 2. andMethod is a fuzzy implic ation method to obtain the minimum value from two membership functions evaluated at thei r respectively inpu ts. orMethod (not used in this example) is a fuzzy aggr egation method to obtain the maximum value between them. The final output (the fuzzy logic function) is the weighted average of all outputs rules: N i i N i i iw z w Output1 1 (3.3.1.2.3.2) Next, Figure 30 illustrates how a Sugeno rule operates. PAGE 75 56 AND z = ax + by + cx yF2(y) F1(x) Input 1 Input 2 Output level, z rule weigher, w Input MF Input MF Output MF AND z = ax + by + cx yF2(y) F1(x) Input 1 Input 2 Output level, z rule weigher, w Input MF Input MF Output MF AND z = ax + by + cx yF2(y) F1(x) Input 1 Input 2 Output level, z rule weigher, w Input MF Input MF Output MF AND z = ax + by + cx yF2(y) F1(x) Input 1 Input 2 Output level, z rule weigher, w Input MF Input MF Output MF Figure 30. A Sugeno Rule Operation Scheme Each input is used for two purposes: 1) to provide a weight for the linear equation to be used, and 2) to be a vari able in a linear equation. Section 3.3.1.2.3.2 presents an example t hat illustrates in detail how a Sugeno rule operates in the TSIMC.fis. 3.3.1.2.3.2 TSIMC.fis Rules A set of 27 rules constitutes the fu zzy inference system of the TSIMC.fis. These rules relate 10 input variables and 3 output variables. The 5 initial input variables (, m, ratio, and ) have been increased to 10 variables to include the interactions t hat show more influe nce over the output variables. The useful interactions are determined using Analysis of Variance (ANOVA) and linear regressions analysis to obtain the best fitting (maximum R2). Appendix 4 contains 81 ANOVAs used for this purpose. Table 3 shows all the PAGE 76 57 inputs variables determined for the TSIMC.fis. Each input variable is renamed to help understanding the structure of the fuzzyTSIMC program. The inputs variables are classified in two sets of variables; the fuzzy variables (x1, x2 and x3) and the nonfuzzy variables (x4, x5, x6, x7, x8, x9, and x10). Table 3. The 10 Inputs Variables for the TSIMC Variable TSIMC input x1 m x2 ratio x3 x4 2 x5 3 x6 x7 2 x8 x9 ( *)2 x10 The fuzzy variables provide the rule we ighers to decide which set of linear equations should be used. The nonfuzzy vari ables constitute the variables for the linear equations. To generate the set of linear equat ions, normalized modeling error data were collected from 2025 simulations using Simulink 5.0 from Matlab 6.5. Table 4 shows the process model parameters and t he fractions of change on the process parameters used for the simulations. PAGE 77 58 Table 4. Process Model Parameters Values for the 2025 Simulations Variable Values Km[ 0.5 1.5 2.5] m [ 1 3 5] ratio [ 0.5 1 1.5] (fraction of change on Kp) [0.5 0 0.5] (fraction of change on p) [ 0.5 0.25 0 0.25 0.5] (fraction of change on t0p) [ 0.5 0.25 0 0.25 0.5] For simulating the process responses, a FOPDT is used as the plant, with the following process parameters: Process gain, Kp = Km *(1+ ) Process time constant, p = m *(1+) Process dead time, t0p = ratio*m *(1+) All the simulations were made for a set point change of +10%TO and no disturbances. Every 75 simulations a data matrix is saved and labeled. The matrix labels are: NSS, NSM, NSB, NM S, NMM, NMB, NBS, NBM, NBB, ZSS, ZSM, ZSB, ZMS, ZMM, ZMB, ZBS, ZB M, ZBB, PSS, PSM, PSB, PMS, PMM, PMB, PBS, PBM, and PBB. The matrices nam es develop from the values of the first three columns of each matr ix, the fuzzy variables values: [x1, x2m, and x3ratio]. When the fuzzy variables are intr oduced into the TSIMC, every fuzzy input takes a membership function. T he selected membership function depends on the fuzzy inputs values: PAGE 78 59 1. from the change on process gain value, x1 : the member ship function is N if x1 is negative the membership function is Z if x1 is zero the membersh ip function is P if x1 is positive 2. from the process model time constant value, x2 : the membership function is S if x2 is small (1) the membership function is M if x2 is Medium (3) the membership function is B if x2 is Big (5) 3. from the ratio value, x3 : the membership function is S if x3 is small (0.5) the membership function is M if x3 is Medium (1) the membership function is B if x3 is Big (1.5) The nonfuzzy variables (from x4 to x10) use ONE as membership function for any value. Table 5 shows the results of the first 75 simulations. This matrix is named as NSS matrix. The first three co lumns are the fuzzy variables (x1,x2 and x3), columns 4 and 5 are the fraction of c hange on the process time constant (x4) and the fraction of change on the process dead time (x7), columns 7, 8 and 9 are process variables; and the last five columns are the parameters previously selected from the normalized modeling error response (nem): P0 (the change on nem), P1 (maximum peak of the nem response), P2 (time to reach P1), P3 ("the inverse peak" (if this exists) to P1) and P4 (time to reach P3). PAGE 79 60 Table 5. Results of the First 75 Simulations (NSS Matrix) x1 x2 m)x3(ratio)KmKp pt0pP0P1P2P3P4 051050.50.5050.250.50250.99980.999913030470.55 051 050.50.25050.250.50.3750.99980.999912.50.17830.55 051 050.50050.250.50.50.99980.999911.9500 051 050.50.25050.250.50.6250.99980.999911.3500 051 050.50.5050.250.50.750.99980.999910.800 051 050.250.5050.250.750250.99980.999911.65021740.55 051 050.250.25050.250.750.3750.99980.999911050.1208055 051 050.250050.250.750.50.99980.999910.4500 051 050.250.25050.250.750.6250.99980.99999.800 051 050.250.5050.250.750.750.99980.99999.1500 051 0500.5050.2510250.99980.99999.80.16760.55 051 0500.25050.2510.3750.99980.99999.10.090.55 051 0500050.2510.50.99990.99998.300 051 0500.25050.2510.6250.99990.99997.4500 051 0500.5050.2510.750.99990.99996.500 051 050.250.5050.251250251.00011.00086.90.13550.55 051 050.250.25050.251250.37511.00336.05007080.55 051 050.250050.251250.511.00915.2500 051 050.250.25050.251250.6250.99991.02074.5500 051 050.250.5050.251250.750.99991.04163.9500 051 050.50.5050.251.50250.99981.03185.20.11310.55 051 050.50.25050.251.50.3750.99991.04834.75005770.55 051 050.50050.251.50.50.99991.07084.3500 051 050.50.25050.251.50.6250.99991.1012400 051 050.50.5050.251.50.7511.14123.6500 051 050.50.5150.750.50250.99980.999913030470.55 051 050.50.25150.750.50.3750.99980.999912.50.17830.55 051 050.50150.750.50.50.99980.999911.9500 051 050.50.25150.750.50.6250.99980.999911.3500 051 050.50.5150.750.50.750.99980.999910.800 051 050.250.5150.750.750250.99980.999911.65021740.55 051 050.250.25150.750.750.3750.99980.999911050.1208055 051 050.250150.750.750.50.99980.999910.4500 051 050.250.25150.750.750.6250.99980.99999.800 051 050.250.5150.750.750.750.99980.99999.1500 051 0500.5150.7510250.99980.99999.80.16760.55 051 0500.25150.7510.3750.99980.99999.10.090.55 051 0500150.7510.50.99990.99998.300 051 0500.25150.7510.6250.99990.99997.4500 051 0500.5150.7510.750.99990.99996.500 051 050.250.5150.751250251.00011.00086.90.13550.55 051 050.250.25150.751250.37511.00336.05007080.55 051 050.250150.751250.511.00915.2500 051 050.250.25150.751250.6250.99991.02074.5500 051 050.250.5150.751250.750.99971.04163.9500 051 050.50.5150.751.50250.99981.03185.20.11310.55 051 050.50.25150.751.50.3750.99961.04834.75005770.55 051 050.50150.751.50.50.99991.07084.3500 051 050.50.25150.751.50.6250.99991.1012400 051 050.50.5150.751.50.7511.14123.6500 051 050.50.5251.250.50250.99980.999913030470.55 051 050.50.25251.250.50.3750.99980.999912.50.17830.55 051 050.50251.250.50.50.99980.999911.9500 051 050.50.25251.250.50.6250.99980.999911.3500 051 050.50.5251.250.50.750.99980.999910.800 051 050.250.5251.250.750250.99980.999911.65021740.55 051 050.250.25251.250.750.3750.99980.999911050.1208055 051 050.250251.250.750.50.99980.999910.4500 051 050.250.25251.250.750.6250.99980.99999.800 051 050.250.5251.250.750.750.99980.99999.1500 051 0500.5251.2510250.99980.99999.80.16760.55 051 0500.25251.2510.3750.99980.99999.10.090.55 051 0500251.2510.50.99990.99998.300 051 0500.25251.2510.6250.99990.99997.4500 051 0500.5251.2510.750.99990.99996.500 051 050.250.5251.251250251.00011.00086.90.13550.55 051 050.250.25251.251250.37511.00336.05007080.55 051 050.250251.251250.511.00915.2500 051 050.250.25251.251250.6250.99991.02074.5500 051 050.250.5251.251250.750.99971.04163.9500 051 050.50.5251.251.50250.99981.03185.20.11310.55 051 050.50.25251.251.50.3750.99961.04834.75005770.55 051 050.50251.251.50.50.99991.07084.3500 051 050.50.25251.251.50.6250.99991.1012400 051 050.50.5251.251.50.7511.14123.6500 PAGE 80 61 From the NSS matrix, three linear equations are obtained to predict the P1, P2 and P3 values: NSSP1, NSSP2 and NSSP3. Analyses of variance determine the main variables that show th e most influence over each parameter. Table 6 reports an analysis of variance pr ovided by Matlab 6.5 for the parameter P1 in the NSS matrix. The table shows t he pvalues in the last column, where and the interaction between them, *, have pvalues lower th an 0.05. Therefore, these variables are considered as the ma in variables to predict P1. The next paragraphs show the procedure to obtain the NSSP1 equation to predict P1. Table 6. Analysis of Variance for P1 in the NSS Matrix The main variables for P1: and *, and powers of these variables, are used in order to obtain t he best fitting (the maximum R2). After several testing using a linear regression (r egress) from Matlab 6.5, the variables that allow obtaining the best fitting are: , and *[ x4, x5, x6, x7 and x9 from Table 3 ]. The following Matlab commands help achieve the fitting: PAGE 81 62NSSP1: load NSS x4=NSS(:,4); %[ x5=x4.^2;%[ x6=x4.^3;%[ x7=NSS(:,5);%[ x9=x4.*x7;%[ Y=NSS(:,11); X=[ones(size(x4)) x4 x5 x6 x7 x9]; [b,bint,r,rint,stats]=regress(Y,X); NSSP1: load NSS x4=NSS(:,4); %[ x5=x4.^2;%[ x6=x4.^3;%[ x7=NSS(:,5);%[ x9=x4.*x7;%[ Y=NSS(:,11); X=[ones(size(x4)) x4 x5 x6 x7 x9]; [b,bint,r,rint,stats]=regress(Y,X); where the vector b contains the coefficient terms: bo = 0.9984 b4 = 0.0143 b5 = 0.1627 b6 = 0.2579 b7 = 0.0297 b9 = 0.1028 and the fitting reported is R2 = 0.9420 Finally the best linear equation for NSSP1 is: NSSP1 = 0*x1 + 0*x2 + 0*x3 0.0143*x4 0.1627*x5 0.2579*x6 0.0297*x7 + 0*x8 0.1028*x9 + 0*x10 0.9984 Next, Tables 7, 8 and 9 show the se t of linear equations for P1*, P2* and P3*. PAGE 82 63 Table 7. Set of Linear Equations to Determine P1* y x1x2x3x4x5x6x7x8x9x10bo R2NSSP10000.01430.16270.25790.029700.102800.99840.942 NSMP10000.002300.04830.0070.00620.02760.16651.00010.6923 NSBP10000.000400.00630.00090.00050.00370.02510.99980.5625 NMSP10000.01430.16270.2580.029700.102800.99840.9419 NMMP10000.002300.04830.0070.00620.02760.16651.00010.6923 NMBP10000.000400.00630.00090.00050.00370.02510.99980.5625 NBSP10000.01430.16270.25790.029700.102800.99840.9419 NBMP10000.002300.04830.0070.00620.02760.16651.00010.6903 NBBP10000.000200.00060.00040.00050.00050.00010.99980.9527 ZSSP10000.45450.477500.13580.4670.51480.96470.03320.9208 ZSMP10000.31620.293100.19980.49250.6040.83940.03390.9294 ZSBP10000.22830.239800.27360.74550.53091.66770.02440.9449 ZMSP10000.45650.481500.1370.47140.51660.9890.03330.9205 ZMMP10000.31670.293300.20040.49460.60510.8440.0340.9291 ZMBP10000.22850.239500.27440.74770.53171.67140.02450.9446 ZBSP10000.45670.481700.13710.47220.51670.99270.03340.9203 ZBMP10000.31680.293200.20050.49510.60520.8440.03390.929 ZBBP10000.23470.235600.28920.69310.50711.5690.02650.939 PSSP10000.51130.837200.0580.66870.01290.85430.36540.9875 PSMP10000.30730.521400.03090.71740.27280.58980.3670.986 PSBP10000.20030.346200.02150.74860.36250.46090.36610.9779 PMSP10000.51690.845700.05450.6690.00680.85090.36660.9881 PMMP10000.30780.522300.03020.71880.27310.58610.36710.986 PMBP10000.20060.346800.02120.7490.36230.46180.36620.9779 PBSP10000.51740.846700.05420.66920.00680.8490.36660.9881 PBMP10000.30780.522300.03010.7190.27310.58570.36710.986 PBBP10000.20630.345300.04330.55950.33960.41770.37330.9433 Table 8. Set of Linear Equations to Determine P2* y x1x2x3x4x5x6x7x8x9x10bo R2NSSP20008.056002.50400.3608.061 NSMP20009.39673.53181.54676.87603.248016.00341 NSBP20008.19333.18861.706710.5605.032023.84461 NMSP200024.128007.50801.056024.0381 NMMP200028.223310.57144.533320.58409.704047.87141 NMBP200024.829.46294.1631.684015.04071.39091 NBSP200055.012.205769.612.49601.736039.74431 NBMP200047.033317.67.573334.316016.208079.7321 NBBP200038.313316.674311.306743.068022.1520120.81231 ZSSP20003.1048.811402.485.49712.72833.37140.83490.7681 ZSMP20004.23612.533903.89211.50532.30458.12241.45730.7511 ZSBP20005.02412.46206.15610.91923.01642.64492.43970.7418 ZMSP20009.21625.805707.42416.0686896.91432.490.7661 ZMMP200012.93635.6571011.65233.97716.88167.31434.3860.7505 ZMBP200015.20836.818018.47232.6588.952126.4987.2580.7433 ZBSP200015.48843.3176012.36827.08913.488164.0494.02840.7701 ZBMP200021.60859.3796019.4456.568211.408278.46537.26960.7515 ZBBP20002595261.1053029.22448.968217.64203.036712.25730.734 PSSP20002.043.534701.341.7110.16812.73471.8130.6882 PSMP20002.4843.639202.8321.42370.94412.93063.00110.8089 PSBP20002.9123.426904.3160.85881.17614.43274.20850.8603 PMSP20006.06410.723303.9285.29960.4836.96335.29960.6847 PMMP20007.52811.312708.3563.32732.88839.11848.69710.7999 PMBP20008.90410.8849012.6642.48654.0447.216312.4060.8593 PBSP200010.11618.426106.5727.6310.7665.76338.62620.6805 PBMP200012.7618.4653013.7965.5695.23266.35114.4670.8018 PBBP20001489216.8441018.89212.18457.63265.175520.98190.8491 PAGE 83 64 Table 9. Set of Linear Equations to Determine P3* y x1x2x3x4x5x6x7x8x9x10bo R2NSSP30000.0605000.19150.31060.195400.01940.9436 NSMP30000.0528000.19520.31120.168600.02070.9504 NSBP30000.0466000.19530.30950.147100.0210.9547 NMSP30000.0597000.19750.31710.193200.02050.9438 NMMP30000.0528000.19520.31120.168600.02070.9504 NMBP30000.0465000.19740.31190.146700.02140.9545 NBSP30000.0595000.19860.31840.192700.02080.9439 NBMP30000.0524000.19880.31550.167500.02130.9504 NBBP30000.0464000.19790.31230.146600.02150.9544 ZSSP30000.0175000.46980.26930.399400.05780.8951 ZSMP30000.0563000.48270.29610.363300.05120.9096 ZSBP30000.0472000.43810.46150.261400.03930.9067 ZMSP30000.0265000.47860.26510.386600.06040.894 ZMMP30000.059000.48580.29810.360100.05140.9091 ZMBP30000.0502000.4410.46050.256300.04030.9058 ZBSP30000.0283000.48040.26410.384100.0610.8938 ZBMP30000.0596000.48640.29830.359500.05150.909 ZBBP30000.0496000.360.22730.260400.04920.8323 PSSP30000.011000.33680.58470.030700.02761 PSMP30000.0045000.35980.58450.013400.03630.9737 PSBP30000.0037000.36470.60240.010400.03510.9766 PMSP30000.0036000.35120.58150.010100.03360.977 PMMP30000.0015000.36460.59010.004400.03720.973 PMBP30000.0012000.36970.60130.003400.03720.9739 PBSP30000.0021000.3540.58080.00600.03470.9755 PBMP30000.0009000.36560.59120.002700.03730.9729 PBBP30000.0007000.2880.36440.002100.04650.905 Figure 31 shows the internal structure of the TSIMC .fis using the 10 inputs; Figures 32, 33, 34 illustrate the member ship functions for the fuzzy variables inputs; and Figures 35, 36, 37, 38 and 39 illustrates the membership function of the nonfuzzy variables inputs. Three tri angular membership functions are used for the fuzzy variables and one trapezoi dal membership function for the nonfuzzy variables. Table 10 shows the se t of 27 fuzzy rules applied on the TSIMC.fis. PAGE 84 65 TSIMC.fismratio P1* P2* P3*27 rules(Sugeno)and x1x3x2x4x5x6x7x8x9x10 TSIMC.fismratio P1* P2* P3*27 rules(Sugeno)and x1x3x2x4x5x6x7x8x9x10 TSIMC.fismratio P1* P2* P3*27 rules(Sugeno)and x1x3x2x4x5x6x7x8x9x10 TSIMC.fismratio P1* P2* P3*27 rules(Sugeno)and x1x3x2x4x5x6x7x8x9x10 Figure 31. TSIMC.fis Internal Structure PAGE 85 66 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 x1Degree of membershipNZP Figure 32. Membership F unctions for the Input x1 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.2 0.4 0.6 0.8 1 x2Degree of membershipSMB Figure 33. Membership F unctions for the Input x2 PAGE 86 67 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0 0.2 0.4 0.6 0.8 1 x3Degree of membershipSMB Figure 34. Membership F unctions for the Input x3 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 x4 x7Degree of membershipONE Figure 35. Membership Function for the Inputs x4 and x7 PAGE 87 68 0 0.05 0.1 0.15 0.2 0.25 0 0.2 0.4 0.6 0.8 1 x5 x8Degree of membershipONE Figure 36. Membership Function for the Inputs x5 and x8 0.125 0.05 0 0.05 0.125 0 0.2 0.4 0.6 0.8 1 x6Degree of membershipONE Figure 37. Membership Function for the Input x6 PAGE 88 69 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 0 0.2 0.4 0.6 0.8 1 x9Degree of membershipONE Figure 38. Membership Function for the Input x9 0 0.01 0.02 0.03 0.04 0.05 0.0625 0 0.2 0.4 0.6 0.8 1 x10Degree of membershipONE Figure 39. Membership Function for the Input x10 PAGE 89 70 Table 10. 10 TSIMC.fis Fuzzy Rules rulex1x2x3x4x5x6x7x8x9x10P1*P2*P3*1 if NSSONEONEONEONEONEONEONE then NSSP1NSSP2NSSP3 2 if NSMONEONEONEONEONEONEONE then NSMP1NSMP2NSMP3 3 if NSBONEONEONEONEONEONEONE then NSBP1NSBP2NSBP3 4 if NMSONEONEONEONEONEONEONE then NMSP1NMSP2NMSP3 5 if NMMONEONEONEONEONEONEONE then NMMP1NMMP2NMMP3 6 if NMBONEONEONEONEONEONEONE then NMBP1NMBP2NMBP3 7 if NBSONEONEONEONEONEONEONE then NBSP1NBSP2NBSP3 8 if NBMONEONEONEONEONEONEONE then NBMP1NBMP2NBMP3 9 if NBBONEONEONEONEONEONEONE then NBBP1NBBP2NBBP3 10 if ZSSONEONEONEONEONEONEONE then ZSSP1ZSSP2ZSSP3 11 if ZSMONEONEONEONEONEONEONE then ZSMP1ZSMP2ZSMP3 12 if ZSBONEONEONEONEONEONEONE then ZSBP1ZSBP2ZSBP3 13 if ZMSONEONEONEONEONEONEONE then ZMSP1ZMSP2ZMSP3 14 if ZMMONEONEONEONEONEONEONE then ZMMP1ZMMP2ZMMP3 15 if ZMBONEONEONEONEONEONEONE then ZMBP1ZMBP2ZMBP3 16 if ZBSONEONEONEONEONEONEONE then ZBSP1ZBSP2ZBSP3 17 if ZBMONEONEONEONEONEONEONE then ZBMP1ZBMP2ZBMP3 18 if ZBBONEONEONEONEONEONEONE then ZBBP1ZBBP2ZBBP3 19 if PSSONEONEONEONEONEONEONE then PSSP1PSSP2PSSP3 20 if PSMONEONEONEONEONEONEONE then PSMP1PSMP2PSMP3 21if PSBONEONEONEONEONEONEONE then PSBP1PSBP2PSBP3 22 if PMSONEONEONEONEONEONEONE then PMSP1PMSP2PMSP3 23 if PMMONEONEONEONEONEONEONE then PMMP1PMMP2PMMP3 24 if PMBONEONEONEONEONEONEONE then PMBP1PMBP2PMBP3 25 if PBSONEONEONEONEONEONEONE then PBSP1PBSP2PBSP3 26 if PBMONEONEONEONEONEONEONE then PBMP1PBMP2PBMP3 27 if PBBONEONEONEONEONEONEONE then PBBP1PBBP2PBBP3 The following example illustrates how TSIMC.fis operates. Lets suppose: A fraction of change on the model gain, x1 = 0.2 A process model time constant, x2 = 3 A ratio, x3 = 1.5 A fraction of change on process time constant, x4 = 0.25, and A fraction of change on process dead time, x7 = 0.25 Figures 40, 41 and 42 show the eval uation of the membership functions for the TSIMC.fis fuzzy inputs, and Fi gure 43 shows the evaluation of the membership function for a TSIMC.fis nonfuzzy input. This procedure is known as Fuzzification step on fuzzy systems. PAGE 90 71 0.5 0.4 0.3 0.2 0.1 0 0.1 02 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 x1Degree of membershipNZP F1(0.2)ZF1(0.2)P F1(0.2)ZF1(0.2)P 0.5 0.4 0.3 0.2 0.1 0 0.1 02 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 x1Degree of membershipNZP 0.5 0.4 0.3 0.2 0.1 0 0.1 02 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 x1Degree of membershipNZP F1(0.2)ZF1(0.2)P F1(0.2)ZF1(0.2)P Figure 40. Evaluation of the Membership Functions for x1 = 0.2 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.2 0.4 0.6 0.8 1 x2Degree of membershipSMB F2(3) F2(3) 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.2 0.4 0.6 0.8 1 x2Degree of membershipSMB 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.2 0.4 0.6 0.8 1 x2Degree of membershipSMB F2(3) F2(3) Figure 41. Evaluation of t he Membership Function for x2 = 3 05 0.6 0.7 0.8 09 1 1.1 1.2 1.3 1.4 1.5 0 0.2 0.4 0.6 0.8 1 x3Degree of membershipSMB F3(1.5) F3(1.5) 05 0.6 0.7 0.8 09 1 1.1 1.2 1.3 1.4 1.5 0 0.2 0.4 0.6 0.8 1 x3Degree of membershipSMB 05 0.6 0.7 0.8 09 1 1.1 1.2 1.3 1.4 1.5 0 0.2 0.4 0.6 0.8 1 x3Degree of membershipSMB F3(1.5) F3(1.5) Figure 42. Evaluation of t he Membership Function for x3 = 1.5 PAGE 91 72 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 x4, x7Degree of membershipONE F7(0.25) F4(0.25) F7(0.25) F4(0.25) x7 = 0.25x4 = 0.25 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 x4, x7Degree of membershipONE 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 x4, x7Degree of membershipONE F7(0.25) F4(0.25) F7(0.25) F4(0.25) x7 = 0.25x4 = 0.25 Figure 43. Evaluation of t he Membership Function for x4 = 0.25 and x7 = 0.25 All the nonfuzzy variables have the same function value [ Fi(xi) = 1.0 ]. The fuzzy variables decide what set of li near equations should be used. Figure 40 shows two membership functions for x1 = 0.2 (Z and P); Figure 41 shows one membership function for x2 = 3 (M); and Figure 42 shows one membership function for x3 = 1.5 (B). Therefore, the set of linear equations are ZMB and PMB. From Table 10 rules 15 and 24 ar e applied to these fuzzy variables. The parameters P1*, P2*, and P3* are obtained using equation 3.3.1.2.3.2: ) w w ( 1 PMBP w 1 ZMBP w 1 P2 1 2 1 3.3.1.2.3.2.a where ZMBP1 and PMBP1, are linear equations from Table 7; and w1 and w2 are the rule weighers. PAGE 92 73 ) w w ( 2 PMBP w 2 ZMBP w 2 P2 1 2 1 3.3.1.2.3.2.b where ZMBP2 and PMBP2 are linear equations from Table 8. ) w w ( 3 PMBP w 3 ZMBP w 3 P2 1 2 1 3.3.1.2.3.2.c where ZMBP3 and PMBP3, are linear equations from Table 9. The rule weighers, are estimated as: w1 = min([ F1(0.2)Z, F2(3), F3(1.5), F4(0.25), F5(0.0625 ), F6(0.0156), F7(0.25), F8(0.0625), F9(0.0625), F10(0.0039) ]) substituting for the mem bership functions values: w1 = min([0.6, 1, 1, 1, 1, 1, 1, 1, 1, 1]) = 0.6 w2 = min([ F1(0.2)P, F2(3), F3(1.5), F4(0.25), F5(0.0625 ), F6(0.0156), F7(0.25), F8(0.0625), F9(0.0625), F10(0.0039) ]) substituting for the mem bership functions values: w2 = min([0.4, 1, 1, 1, 1, 1, 1, 1, 1, 1]) = 0.4 substituting x1 = 0.2 x2 =3, x3 = 1.5, x4 = 0.25, x5 = 0.0625 x6 = 0.0156 x7 = 0.25, x8 = 0.0625 x9 = 0.0625 and x10 = 0.0039 into the linear equations: PAGE 93 74 from Table 7: from Table 8: from Table 9: ZMBP1=0.0350 ZMBP2=11.3621 ZMBP3= 0.1759 PMBP1=0.4657 PMBP2= 13.4340 PMBP3= 0.1667 Finally from equations 3.3.1.2.3.2.a, 3.3.1.2.3.2.b and 3.3.1.2.3.2.c: ) . ( 0.4657 0.0350 P*4 0 6 0 4 0 6 0 1 = 0.2073 ) . ( 1 11.3621 P*4 0 6 0 4340 3 4 0 6 0 2 = 12.1909 ) . ( . P*4 0 6 0 1667 0 4 0 1759 0 6 0 3 = 0.1722 The following Matlab co mmands are used by the fuzzyTSIMC program to obtain P1*, P2* and P3*using the TSIMC.fis: TSIMC=readfis('TSIMC'); Par=evalfis([x1 x2 x3 x4 x5 x6 x7 x8 x9 x10],TSIMC) where Par is a vector which contains the P1*, P2* and P3* values. The results using the TSIMC.fis are: Par = [0.2045 12.1666 0.1723] where the difference with the previous P1*, P2*, and P3* is due to the round off in reading Fig. 40. PAGE 94 75 3.3.1.2.4 The TSIMC_Con Program The TSIMC_Con program contains the constraints for the main program execution. This program is used by the opt imization routine to detect changes on the process dead time. The TSIMC_Con program is only useful for two particular cases (to be explained later). The sign of the fraction of change on the process gain ( ) and the sign of the parameter P3 dete rmines the sign on the fraction of change of the process dead time change faction ( ). Starting with Equation 3.3.1.4 with a set point change and no disturbance (D = 0 ) M K M K em p m (3.3.1.2.3.3) Assuming Kp = Km(1+) : M K M ) ( K em m m 1 (3.3.1.2.3.4) From the IMC stru cture (Figure 24): m m setK / ) e C ( M (3.3.1.2.3.5) Substituting Equation 3.3.1.2.3. 5 into Equation 3.3.1.2.3.4: ) e C ( ) e C )( ( em set m set m 1 (3.3.1.2.3.6) Dividing by setCand arranging terms: emn 1 (3.3.1.2.3.7) PAGE 95 76 From Equation 3.3.1.2.3.7: if = 0 then nem = 0 if > 0 then nem > 0 if < 0 then nem < 0 Thus, the normalized modeling error change (nem) and the fraction of change on the process gain ( ) have the same sign. The following constitutes the two constraints: case 1) If > 0 and the process dead time increases ( ); the normalized modeling error response, nem, presents a first peak with negative sign (P3<0); thus, constraint 1 is if > 0 and P3 < 0 then . case 2) If < 0 and the process dead time decreases ( ); the normalized modeling error response, nem, presents a first peak with positive sign (P3>0); thus, constraint 2 is if < 0 and P3 > 0 then . Figures 44 and 45 show the two typical normalized modeling error responses (nem) where the constraints are applied. PAGE 96 77 0 10 20 30 40 50 60 70 80 90 100 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 timenem P3 < 0> 0 0 10 20 30 40 50 60 70 80 90 100 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 Normalized modeling error, nem P3 < 0> 0 0 10 20 30 40 50 60 70 80 90 100 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 timenem P3 < 0> 0 0 10 20 30 40 50 60 70 80 90 100 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 Normalized modeling error, nem P3 < 0> 0 Figure 44. Normalized Modeli ng Error Response Showing and P3 Signs, When the Process Dead Time Increases ( >0) 0 10 20 30 40 50 60 70 80 90 100 4 3 2 1 0 1 2 time P3 > 0< 0Normalized modeling error, nem 0 10 20 30 40 50 60 70 80 90 100 4 3 2 1 0 1 2 P3 > 0< 0nem Figure 45. Normalized Modeli ng Error Response Showing and P3 Signs, When the Process Dead Time Decreases ( <0) PAGE 97 78 3.3.1.3 Testing Module TS Module TS is tested with Equations 3. 3.1.9 and 3.3.1.11. No disturbances are applied. Several set point changes were introduced. At the same time, changes on process parameters were intr oduced to simulate nonlinearities. The filter time constant is calculated using the tuning equa tion from Section 3.4, and its initial value is 3.445. The new process model param eters and the filter time constant are calculated and saved when the new steady state is reached after each set point change. Figure 46 shows the performance of the conventional IMC and the IMC working with Modules K and TS facing different set point changes, and Figure 47 shows the process and the process model parameters changes. PAGE 98 79 0 100 200 300 400 500 600 700 800 900 1000 30 40 50 60 70 80 90 100 timeControlled variable, c, %TO IMC IMC (K,TS) Figure 46. Performances of the IMC and the IMC Working With Module K and Module TS for Several Set Point Changes Figure 46 illustrates that the new controller successfully overcomes the stability problems. Figure 47 shows t he changes on: a) the process and the model gains, b) the process and model time constants, c) t he process and model dead times, and d) the f ilter time constant. PAGE 99 80 0 100 200 300 400 500 600 700 800 900 1000 2 4 6 8 timeFilter time constant,f [time units] 0 100 200 300 400 500 600 700 800 900 1000 1 2 3 4 5 Process and Model Gain [%TO/%CO] 0 100 200 300 400 500 600 700 800 900 1000 2 4 6 8 Process and Model Time Constant [time units] 0 100 200 300 400 500 600 700 800 900 1000 2 4 6 Process and Model Dead Time [time units] Kp Km new p m new t0 p t0 m new f a) b) c) d) Figure 47. Values of the Process Parameters (___), Values Obtained for the Process Model Parameters () and Val ues Calculated for the Filter Time Constant ( _ ) Testing Module TS PAGE 100 81 3.3.1.3.1 Testing the IMCFAM Unit As previously mentioned, the IMCF AM unit consists of Module K and Module TS. This unit is tested with t he same transfer functions used by Module K, Equations 3.3.1.9, 3.3. 1.10 and 3.3.1.11. To simula te nonlinearities, changes on process parameters were introduced at the same time disturbances or set point changes occur. Module K works w hen either disturbances or set point changes affect the process, while Module TS works only when set point changes are introduced. Based on the new process par ameters, the filter time constant is recalculated using the tuning equation from Section 3.4. The new process parameters and the filter ti me constant are updated at the same time the next disturbance or the next set point occurs. Figure 49 shows the performance of the conventional IMC and the IMCFAM facing disturbances and set point changes, and Figure 50 shows the process and the process model parame ters changes. The set point changes and the disturbances values (D values) are shown in Figure 48. PAGE 101 82 0 100 200 300 400 500 600 700 800 20 30 40 50 60 70 80 90 100 timeControlled variable, c, %TO IMC IMC with IMCFAM Disturbances values (D): 200min) D = 2 400min) D = 2 600min) D = 3 Figure 48. Performances of the IMC and the IMCFAM Working for Several Disturbances and Set Point Changes Figure 48 illustrates the improv ed performance of the IMC with the IMCFAM unit. The figure shows that the IMC using the IMCFAM unit is capable of overcoming stability problems. Fi gure 49 shows the c hanges on: a) the process and the model gains, b) the process and model time constants, c) the process and model dead times, and d) the filter time constant. PAGE 102 83 0 100 200 300 400 500 600 700 800 2 4 6 8 timeFilter time constant,f [time units] 0 100 200 300 400 500 600 700 800 2 4 6 8 Process and Model Gain [%TO/%C] 0 100 200 300 400 500 600 700 800 0 2 4 6 Process and Model Time Constant [time units] 0 100 200 300 400 500 600 700 800 2 4 6 8 Process and Model Dead Time [time units] Kp Km new p m new t0 p t0 m new f Figure 49. Values of the Process Parameters (___), Values Obtained for the Process Model Parameters () and Val ues Calculated for the Filter Time Constant ( _ ) Testing the IMC Working With the IMCFAM Unit PAGE 103 84 All along this chapter we have indi cated that the K and TS modules calculate the process characteristi cs once a steady state is reached. The following variables are used to determine t he new steady state: the error of the controlled variable, e(n), the change on the error of the controlled variable, e(n), and the change on the cont roller output signal, M(n). These changes are defined as the difference between the pr esent value and the previous value: e(n) = e(n) e(n1) M(n) = M(n) M(n1) For small deviations due to noise, a noise parameter ( ) is used to help defining the steady state. This par ameter depends on the level of process noise. The new steady state is reached when all of the following are accomplished: abs(e(n)) < abs(e(n)) < and abs (M(n))< To briefly study the effect of noise on the IMC working with IMCFAM unit, an Auto Regressive Moving Average noi se (ARMA(1,1) noise) with standard deviation of 0.4%TO was added to the signal from the tr ansmitter. For testing the FAIMCr, has been set as 0.1%TO for e(n) and e(n); and 0.1%CO for M(n). Due to the presence of noise, set point changes or disturbances are detected when the error is higher than 1.5 %TO or e(n) > 15 Fig. 50 shows both curves with and without noise when the controller is facing disturbances and set point changes. Figure 51 shows the signal from the PAGE 104 85 controller output. The presence of this noise does not make a significant difference on the performance of the controller. 0 100 200 300 400 500 600 700 800 55 60 65 70 75 80 85 Controlled variable, c, %TO without noise with noise time 90 100 110 120 130 140 68.5 69 69.5 70 70.5 71 71.5 90 100 110 120 130 140 68.5 69 69.5 70 70.5 71 71.5 0 100 200 300 400 500 600 700 800 55 60 65 70 75 80 85 Controlled variable, c, %TO without noise with noise time 90 100 110 120 130 140 68.5 69 69.5 70 70.5 71 71.5 90 100 110 120 130 140 68.5 69 69.5 70 70.5 71 71.5 90 100 110 120 130 140 68.5 69 69.5 70 70.5 71 71.5 90 100 110 120 130 140 68.5 69 69.5 70 70.5 71 71.5 Figure 50. Performance of the IMC Working With the IMCFAM Unit When Noise is Added to the Transmitter fr om the Contro lled Variable PAGE 105 86 0 100 200 300 400 500 600 700 800 56 58 60 62 64 66 68 timeController output signal, m, %CO Figure 51. Signal from the Controller Out put from the IMC With the IMCFAM Unit in the Presence of Noise PAGE 106 87 3.4 The IMC Filter Tuning Equation This research develops an equation to tune the IMC filter time constant based on the process parameters. Assume a plant with the transfer func tion shown in Figure 52. This figure also contains a set of curves showing the performance of the IMC for different values of the filter time constant (f ) when the set point is changed by +10%TO. To simulate a nonlinear effe ct, the process dead time ( t0 p) has been increased by 50% at 10min. The parameters of the model transfer function ( Km, m and t0 m) remain constant in the IMC structure. 0 5 10 15 20 25 30 35 40 45 50 60 62 64 66 68 70 72 74 timeControlled Variable, c, %TO f= 1.5f= 2.5f= 3.5f= 4.5f= 5.5 s e Fs s es Plant ( time >10 ) Plant ( time <10 ) 0 5 10 15 20 25 30 35 40 45 50 60 62 64 66 68 70 72 74 f= 1.5f= 2.5f= 3.5f= 4.5f= 5.5 15 254 s e Fs 15 23 s e Fs Plant ( time >10 ) Plant ( time <10 ) 0 5 10 15 20 25 30 35 40 45 50 60 62 64 66 68 70 72 74 timeControlled Variable, c, %TO f= 1.5f= 2.5f= 3.5f= 4.5f= 5.5 s e Fs s e Fs s es es Plant ( time >10 ) Plant ( time <10 ) 0 5 10 15 20 25 30 35 40 45 50 60 62 64 66 68 70 72 74 f= 1.5f= 2.5f= 3.5f= 4.5f= 5.5 15 254 s e Fs 15 254 s e Fs 15 23 s e Fs 15 23 s e Fs Plant ( time >10 ) Plant ( time <10 ) Figure 52. IMC Responses for Different Values of f PAGE 107 88 The curves on Figure 52 show that a small f yields an aggressive controller (larger overshoot and more o scillatory response). As a consequence, small values of f could mean stability problems w hen facing nonlinearities. This research recommends (tuning criterion) that the filter value sh ould not allow the controlled variable to overshoot more t han 10% for a set point change. According to this criterion, the best value of the filt er time constant for the initial transfer function on Figure 52 is 3.5. To generate the tuning equation, process data was collected from 27 optimizations. The optimization program for this purpose, the Opt_IMC_Tf program, uses Fminimax as optimization routine from Matlab 6.5 to obtain the optimum value.The Opt_IMC_ Tf program uses two additional programs: the trackmmobj_IMC_Tf that contains the objective f unction to minimize (the IAE) and the trackmmcon_IMC_Tf containing the optimizat ion constraint (maximum overshoot of 10% of the controlled vari able change). Table 11 shows the process model parameters used for the process simulations. Table 11. Process Model Parameters Values Variable Values Km[ 0.5 1.5 2.5] m [ 1 3 5] ratio [ 0.2 0.6 1] For simulating the process responses, a FOPDT is used as the plant, with the following process parameters: PAGE 108 89 Process gain, Kp = Km Process time constant, p = m Process dead time, t0p = ratio*m *(1.5) A set point change of 10%TO is introduced at 10 min. A disturbance (D=5) is introduced at 100 min. The di sturbance transfer function have a gain equal to 1, time constant equal to the process time constant, and no dead time. Table 12 shows the results from the 27 optimizations. Table 12. Process Parameters and Optimum f Values Kmmratio t0mKppt0pf0.510.20.20.510.30.22502 1.510.20.21.510.30.22502 2.510.20.22.510.30.22502 0.530.20.60.530.91.1501 1.530.20.61.530.90.67669 2.530.20.62.530.90.67669 0.550.210.551.51.387 1.550.211.551.51.1284 2.550.212.551.51.1284 0.510.60.60.510.90.67669 1.510.60.61.510.90.67669 2.510.60.62.510.90.67669 0.530.61.80.532.72.0314 1.530.61.81.532.72.0314 2.530.61.82.532.72.0314 0.550.630.554.53.3857 1.550.631.554.53.3857 2.550.632.554.53.3857 0.51110.511.51.1284 1.51111.511.51.1284 2.51112.511.51.1284 0.53130.534.53.3857 1.53131.534.53.3857 2.53132.534.53.3857 0.55150.557.55.6429 1.55151.557.55.6429 2.55152.557.55.6429 Kmmratio t0mKppt0pf0.510.20.20.510.30.22502 1.510.20.21.510.30.22502 2.510.20.22.510.30.22502 0.530.20.60.530.91.1501 1.530.20.61.530.90.67669 2.530.20.62.530.90.67669 0.550.210.551.51.387 1.550.211.551.51.1284 2.550.212.551.51.1284 0.510.60.60.510.90.67669 1.510.60.61.510.90.67669 2.510.60.62.510.90.67669 0.530.61.80.532.72.0314 1.530.61.81.532.72.0314 2.530.61.82.532.72.0314 0.550.630.554.53.3857 1.550.631.554.53.3857 2.550.632.554.53.3857 0.51110.511.51.1284 1.51111.511.51.1284 2.51112.511.51.1284 0.53130.534.53.3857 1.53131.534.53.3857 2.5Kmmratio t0mKppt0pf0.510.20.20.510.30.22502 1.510.20.21.510.30.22502 2.510.20.22.510.30.22502 0.530.20.60.530.91.1501 1.530.20.61.530.90.67669 2.530.20.62.530.90.67669 0.550.210.551.51.387 1.550.211.551.51.1284 2.550.212.551.51.1284 0.510.60.60.510.90.67669 1.510.60.61.510.90.67669 2.510.60.62.510.90.67669 0.530.61.80.532.72.0314 1.530.61.81.532.72.0314 2.530.61.82.532.72.0314 0.550.630.554.53.3857 1.550.631.554.53.3857 2.550.632.554.53.3857 0.51110.511.51.1284 1.51111.511.51.1284 2.51112.511.51.1284 0.53130.534.53.3857 1.53131.534.53.3857 2.53132.534.53.3857 0.55150.557.55.6429 1.55151.557.55.6429 2.55152.557.55.6429 To determine the variables that show the most influence on f, an Analysis of Variance (ANOVA) is applied to the re sults of Table 12. Table 13 shows the result of this ANOVA. PAGE 109 90 Table 13. Analysis of Variance for f The last column of Table 13 shows that m, ratio and the interaction between them,m *ratio have pvalues lower than 0.05. Therefore, these variables are used to predict f. Using a nonlinear regression command (nlinfit) from Matlab 6.5, the follo wing expression is achieved after several trials (maximum R2=0.9966): ) (ratio .m f9574 06 1.123 (3.4.1) 3.5 The IMCFF Unit (IMC With Variable Fuzzy Filter) The IMCFF unit is designed to avoi d excessive oscillations in the controlled variable response. This unit consists of on e fuzzy filter module added to the conventional IMC in or der to modify the constant filter value of the IMC (see Figure 53). The inputs to the IMCFF are the error of the controlled variable ( e ) and its change (e ). The output is the required in crement of the filter value (f ). For PAGE 110 91 tuning the IMCFF three scaling factors are required: two for the inputs (Ke and K e) and one for the output (K F). modeling error, Plant Model Inverse Filter + + _Set Point Gf+ + _Set PointcControlled variableFinvF F* FK eK eKeFuzzification DefuzzificationInference Engine Fuzzy Rule BaseFuzzy Filter Module+ IMCFF Z1 Z1 Z1 f eK eK eK eKFuzzification DefuzzificationInference Engine Fuzzy Rule Base+ _+ Z1 Z1+ +f e e em Plant Model Inverse Filter + + _Set Point Gf+ + _Set PointcControlled variableFinvF F* FK eK eKeFuzzification DefuzzificationInference Engine Fuzzy Rule BaseFuzzy Filter Module+ IMCFF Z1 Z1 Z1 Z1 Z1 Z1 f eK eK eK eKFuzzification DefuzzificationInference Engine Fuzzy Rule Base+ _+ Z1 Z1+ +f e e IMC modeling error, Plant Model Inverse Filter + + _Set Point Gf+ + _Set PointcControlled variableFinvF F* FK eK eKeFuzzification DefuzzificationInference Engine Fuzzy Rule BaseFuzzy Filter Module+ IMCFF Z1 Z1 Z1 f eK eK eK eKFuzzification DefuzzificationInference Engine Fuzzy Rule Base+ _+ Z1 Z1+ +f e e em Plant Model Inverse Filter + + _Set Point Gf+ + _Set PointcControlled variableFinvF F* FK eK eKeFuzzification DefuzzificationInference Engine Fuzzy Rule BaseFuzzy Filter Module+ IMCFF Z1 Z1 Z1 Z1 Z1 Z1 f eK eK eK eKFuzzification DefuzzificationInference Engine Fuzzy Rule Base+ _+ Z1 Z1+ +f e e IMC Figure 53. Scheme of the IMC Worki ng With the IMCFF Unit To reduce the aggressiveness of the IMC controller, the IMCFF unit increases the filter time constant val ue when the controlled variable moves away from its set point and after the first peak on the controlled variable. (See Figure 54). The IMCFF actions are useful for avoi ding stability problems when excessive oscillations occur. For small deviations on the controlled variable, the contribution of the IMCFF is minimum; t herefore the control is carr ied out by the IMC itself. PAGE 111 92 Figure 54 shows the regions where the fuzzy filter module acts. Above the set point value, the error and its change have negative signs. Below the set point, both of them have positive signs. This is the condition applied on the fuzzy filter module. 0 50 100 150 200 250 145 150 155 160 165 170 175 180 185 190 Regions Regions e<0 e<0 e>0 e>0 0 50 100 150 200 250 145 150 155 160 165 170 175 180 185 190 Regions Regions e<0 e<0 e>0 e>0 Controlled variable, c(t )time 0 50 100 150 200 250 145 150 155 160 165 170 175 180 185 190 0 50 100 150 200 250 145 150 155 160 165 170 175 180 185 190 Regions Regions e<0 e<0 e>0 e>0 0 50 100 150 200 250 145 150 155 160 165 170 175 180 185 190 0 50 100 150 200 250 145 150 155 160 165 170 175 180 185 190 Regions Regions e<0 e<0 e>0 e>0 Controlled variable, c(t )time First peak Compensating actions 0 50 100 150 200 250 145 150 155 160 165 170 175 180 185 190 0 50 100 150 200 250 145 150 155 160 165 170 175 180 185 190 Regions Regions e<0 e<0 e>0 e>0 0 50 100 150 200 250 145 150 155 160 165 170 175 180 185 190 0 50 100 150 200 250 145 150 155 160 165 170 175 180 185 190 Regions Regions e<0 e<0 e>0 e>0 Controlled variable, c(t )time 0 50 100 150 200 250 145 150 155 160 165 170 175 180 185 190 0 50 100 150 200 250 145 150 155 160 165 170 175 180 185 190 Regions Regions e<0 e<0 e>0 e>0 0 50 100 150 200 250 145 150 155 160 165 170 175 180 185 190 0 50 100 150 200 250 145 150 155 160 165 170 175 180 185 190 Regions Regions e<0 e<0 e>0 e>0 Controlled variable, c(t )time First peak Compensating actions Figure 54. Regions in the Response Wher e the Fuzzy Filter Module Acts Table 14 shows the rule matrix used by the IMCFF to obt ain the increment on the filter value (f). The meanings of the linguistic variab les involved are: negative big (NB), negative medium (NM), negative small (NS), zero (Z), positive small (PS) positive medium (PM) and positive big (PB). These rules were chosen to increase the filter value when required. PAGE 112 93 Table 14. Fuzzy Rules for the Fuzzy F ilter Inference System NBNMNSZPSPMPB NB PBPBPMZZZZ NM PBPMPSZZZZ NS PMPSZZZZZ Z ZZZZZZZ PS ZZZZZPSPM PMZZZZPSPMPB PB ZZZZPMPBPB e \ eNBNMNSZPSPMPB NB PBPBPMZZZZ NM PBPMPSZZZZ NS PMPSZZZZZ Z ZZZZZZZ PS ZZZZZPSPM PMZZZZPSPMPB PB ZZZZPMPBPB e \ e Figures 55 and 56 show the membership functions required for inputs and the output of the IMCFF respectively. Fi ve triangular membership functions and two trapezoidal membership functions are used for the inputs ( e, e ). Three triangular membership functions and one tr apezoidal are required for the output (f ). Figure 57 is a surface representation of the IMCFF output versus both of its inputs. PAGE 113 94 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 e, eDegree of membershipNBNSZPSPB NMPM Figure 55. Membership Function s for the Inputs of the IMCFF 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 fDegree of membershipZ PSPB PM Figure 56. Membership Functions for the Output of the IMCFF PAGE 114 95 1 0 1 1 0.5 0 0.5 1 0 0.2 0.4 0.6 e e f Figure 57. IMCFF Module Surface 3.5.1 Introducing the Fuzzy Filter into the IMC Referring to Figure 19, the implementation of* invFby itself is difficult because the order of the numerator is hi gher than that of the denominator. This issue is easily solved by multiplying invF times the filter fGbefore the implementation. Figure 58 shows this modification. PAGE 115 96 modeling error,em Plant Model + + _Set Point, + + _CControlled variable,F M F* (Gf )(Finv )* Fuzzy filter, fset adjCsetC modeling error,em Plant Model + + _Set Point, + + _CControlled variable,F F* (Gf )(Finv )* (Gf )(Finv )* Fuzzy filter, fset adjCsetC modeling error,em Plant Model + + _Set Point, + + _CControlled variable,F M F* (Gf )(Finv )* (Gf )(Finv )* Fuzzy filter, fset adjCset adjCsetCsetC modeling error,em Plant Model + + _Set Point, + + _CControlled variable,F F* (Gf )(Finv )* (Gf )(Finv )* Fuzzy filter, fset adjCset adjCsetCsetC Figure 58. Conventional IMC Briefly Modified The transfer function 1 1 11 1 1 s s K K s s FGf m m m m f invf ))((* Converting into Partial Fractions: f f mf f m m invfs K FG )( ))((*1 1 Arranging: 1 12 1s KK sGsGFGffm mf fm m invf )()())((* where fm mK sG )(1 ; and 1 12s K sGf fm mf )( G1(s) and G2(s) are most convenient for updating the process model parameters and the fuzzy filter time constant. A ppendix 3 shows the implementation using Simulink. PAGE 116 97 3.5.2 Testing the IMCFF To test the performance of the IM C working with the IMCFF unit, Equations 3.3.1.9, 3.3.1. 10 and 3.3.1.11 are used as pr ocess, disturbance and process model transfer functions. The filter time constant is initially calculated by Equation 3.4.1 and its value is 3.445. The IMCFF scalin g factors used for the inputs are Ke = 0.4509 and K e = 15.001; and for the output is K F = 0.330. These scaling factors were obtained using the optimization rout ine described on Section 2.8, and they are used for all simulations presented in this section. For all cases a set point change of +10%TO is introduced at 50min, and two disturbances are introduc ed to the process, the firs t disturbance at 250min (D=+10) and the second disturbance at 500min (D=+10). To observe the performance of the IMC working with the IMCFF unit, the IMCFAM unit has been disconnected. Figures 59, 60, and 61 show the per formance of the IMC with the IMCFF unit, facing process gain changes, proc ess time constant changes and process dead times changes respectively. Figure 62 shows the performance of the controller facing all parameters changes shown in Figures 59, 60, and 61. PAGE 117 98 Figure 59 shows a) changes on the proces s gain: +70% of the initial value at 50min, +80% of its previ ous value at 250min and +20% of its previous value at 500min, b) Responses of the Conventio nal IMC and the IMC working with the IMCFF unit, and c) the filters va lues for both controllers. 0 100 200 300 400 500 600 700 800 900 1000 0 2 4 6 8 Process gain, Kp [%CO/%TO] 0 100 200 300 400 500 600 700 800 900 1000 55 60 65 70 75 80 Controlled variable, c, %TO 0 100 200 300 400 500 600 700 800 900 1000 3 4 5 6 7 timeFilter time constant, f IMC with IMCFF Conventional IMC Kp Fuzzy filter Constant filter a) b) c) Figure 59. Performances of the Controllers IMC and the IMC Working With the IMCFF Unit Facing Process Gain Changes PAGE 118 99 Figure 60 shows a) changes on the proc ess time constant: 50% of the initial value at 50min, 60% of its previous value at 250min and 45% of its previous value at 500min, b) Respons es of the Conventional IMC and the IMC working with the IMCFF unit, and c) Cons tant and fuzzy filter values for these controllers. 0 100 200 300 400 500 600 700 800 900 1000 0 2 4 6 8 Process time constant,p 0 100 200 300 400 500 600 700 800 900 1000 55 60 65 70 75 80 Controlled variable, c, %TO IMC with IMCFF Conventional IMC p 0 100 200 300 400 500 600 700 800 900 1000 2 4 6 8 timeFilter time constant, f Fuzzy filter Constant filter a) b) c) Figure 60. Performances of the Controllers IMC and the IMC Working With the IMCFF Unit Facing Process Time Constant Changes PAGE 119 100 Figure 61 shows a) changes on the process dead time: +50% of the initial value at 50min, +55% of its previous value at 250min and +60% of its previous value at 500min, b) Responses of t he Conventional IMC and the IMC working with the IMCFF unit, and c) the fuzzy filter and the constant filter values. 0 100 200 300 400 500 600 700 800 900 1000 0 5 10 15 Process dead time, t0p 0 100 200 300 400 500 600 700 800 900 1000 60 65 70 75 80 Controlled variable, c, %TO 0 100 200 300 400 500 600 700 800 900 1000 0 5 10 timeFilter ime constant, f IMC with IMCFF Conventional IMC t0 p Fuzzy filter Constant filter a) b) c) Figure 61. Performances of the Controllers IMC and the IMC Working With the IMCFF Unit Facing Process Dead Time Changes PAGE 120 101 Figure 62 shows a) the performance of the conventional IMC b) the performance of the IMC work ing with the IMCFF unit, and c) the filter value to compensate the nonlinearities effects (fuzzy filter). 0 100 200 300 400 500 600 700 800 900 1000 0 50 100Controlled variable, c, %TO 0 100 200 300 400 500 600 700 800 900 1000 0 50 100Controlled variable, c, %TO 0 100 200 300 400 500 600 700 800 900 1000 0 10 20 30timeFilter time constant, f IMC IMC with IMCFF Fuzzy filter a) b) c) Figure 62. Performances of the Controllers IMC and the IMC Working With the IMCFF Unit Facing the Parameters Chang es Shown in the Figures 59, 60, and 61 PAGE 121 102 3.6 Testing the FAIMCr As we described at the beginning of this chapter, the FAIMCr controller consists of adding the IMCFAM and the IMCFF units to the conventional IMC controller. This section compares the performance of the FAIMCr and of an IMC controller with the IMCFAM unit only. Equations 3.3. 1.9, 3.3.1.10 and 3.3.1.11 are the process, process model and the disturbance transfer functions. Several set point changes and distur bances are introduced for this test, and at the same time process parameters changes are introduc ed to simulate the nonlinearities. Figure 63 shows the performance of t he controllers facing disturbances and set point changes, and Figure 64 shows the process and the process model parameters changes. The set point changes and the disturbances values (D values) are shown in Figure 63. Fi gure 63 shows that the controllers performances are quite similar. PAGE 122 103 0 100 200 300 400 500 600 700 800 55 60 65 70 75 80 85 90 timeControlled variable, c, %TO FAIMCr IMC with IMCFAM Disturbances values (D): 200min) D = 2 400min) D = 2 600min) D = 3 Figure 63. Performances of the IMC Working With the IMCFAM Unit and of the FAIMCr PAGE 123 104 0 100 200 300 400 500 600 700 800 2 4 6 FAIMCr Process and Model Gain [%TO/%C] 0 100 200 300 400 500 600 700 800 2 4 6 8 Process and Model Time Constant [time units] 0 100 200 300 400 500 600 700 800 0 5 10 Process and Model Dead Time [time units] 0 100 200 300 400 500 600 700 800 2 4 6 8 timeFilter time constant,f p m new t0 p t0 m new f Kp Km new a) b) c) d) Figure 64. Process Parameter Changes (___) and the Process Parameters Changes Calculated by the FAIMCr () PAGE 124 105 Figure 65 shows the performances of the controllers for more severe process parameters changes. The tracking of the process parameters changes calculated by the IMC with IM CFAM unit is shown in Fi gure 66, while Figure 67 shows the tracking of the proce ss parameters change by the FAIMCr. 0 100 200 300 400 500 600 700 800 50 55 60 65 70 75 80 85 90 95 100 timeControlled variable, c, %TO FAIMCr IMC with IMCFAM Figure 65. Performances of the Conventional FAIMCr (___) and the IMC Working With the IMCFAM Unit () PAGE 125 106 0 100 200 300 400 500 600 700 800 2 4 6 IMC with IMCFAMProcess and Model Gain [%TO/%C] 0 100 200 300 400 500 600 700 800 2 4 6 8 Process and Model Time Constant [time units] 0 100 200 300 400 500 600 700 800 2 4 6 8 Process and Model Dead Time [time units] 0 100 200 300 400 500 600 700 800 2 4 6 8 timeFilter time constant,f Kp Km new p m new t0 p t0 m new a) b) c) d) Figure 66. Process Parameters Changes Calculated by the IMC With IMCFAM Unit PAGE 126 107 0 100 200 300 400 500 600 700 800 2 4 6 FAIMCrProcess and Model Gain [%TO/%C] 0 100 200 300 400 500 600 700 800 2 4 6 8 Process and Model Time Constant [time units] 0 100 200 300 400 500 600 700 800 2 4 6 8 Process and Model Dead Time [time units] 0 100 200 300 400 500 600 700 800 2 4 6 8 timeFilter time constant,f Kp Km new p m new t0 p t0 m new a) b) c) d) Figure 67. Process Parameters Changes Calculated by the FAIMCr PAGE 127 108 Fig. 68 shows the performance of t he FAIMCr in a process with and without noise. The presence of this noise does not make a significant difference on the performance of the controller. 0 100 200 300 400 500 600 700 800 55 60 65 70 75 80 85 90 95 timeControlled variable, c, %TOFAIMCr without noise with noise Figure 68. FAIMCr Performanc es With and Without Noise PAGE 128 109 3.6.1 Testing the FAIMCr on a Nonlinear Process The mixing tank, a nonlin ear system shown in Append ix 2, is used once more to test the FAIMCr. Based on the process testing around the steady state operating point, the proc ess parameters are: Kp = 0.785 %CO/%TO p = 2.0838min and t0p = 3.746min. The tuning param eters for the PID are: Kc = 0.5905 %CO/%TO I = 2.0838min y D = 1.8730min; the filter time constant is recalculated using Equation 3.4.1. The scaling factors used by the IMC module in this nonlinear process are: Ke = 0.25, K e = 15 and K F = 0.25. These scaling factors were obtained using the optimization program described on Section 2.8. Figure 69 shows the performance of t he FAIMCr for the following set point and disturbances changes: +10 oF in set point at 100min, 5 oF in the hot flow temperature ( T1(t)) at 200 min, +20 oF in set point at 350min, 40% of reduction on the hot stream mass flow rate ( W1(t)) at 450min, 40% of increment of the initial hot str eam mass flow rate ( W1(t)) at 650min 20 oF in set point at 800min +5 oF in the hot flow temperature ( T1(t)) at 950 min, 10 oF in set point at 800min. Figure 70 shows the process model par ameters calculated by the FAIMCr and Figure 71 compares the performances of the PID, IMC and FAIMCr controllers. PAGE 129 110 0 200 400 600 800 1000 1200 145 150 155 160 165 170 175 180 185 190 195 timeTemperature, T4, oF FAIMCr Set point Figure 69. Performance of the FAIMCr for Set Point and Disturbances Changes PAGE 130 111 0 200 400 600 800 1000 1200 1.2 1 0.8 0.6 0 200 400 600 800 1000 1200 0 2 4 6 0 200 400 600 800 1000 1200 0 2 4 6 8 10 0 200 400 600 800 1000 1200 0 5 10 15 time, minFilter time constant, min t0 m new Km new m new f Process Model Gain [%TO/%C] Process Model Time Constant, min Process Model Dead Time, min Figure 70. Process Model Parameters C hanges and the Filter Time Constant Values Calculated by the FAIMCr for the Mixing Tank System PAGE 131 112 0 200 400 600 800 1000 1200 140 160 180 200 220 time, minTemperature, T4 oF 0 200 400 600 800 1000 1200 140 160 180 200 220 Temperature, T4 oF 0 200 400 600 800 1000 1200 140 160 180 200 220 Temperature, T4 oF PID (IAE=8777) IMC (IAE=6119) FAIMCr (IAE=1119) Figure 71. Responses from the PID, IMC and FAIMCr Controllers, Facing the Set Points and Disturbances Introduced to the Mixing Tank Process Figure 71 shows how the conventional PID and IMC are unable to provide a good control performance, wh ile the FAIMCr is the on ly maintaining stability PAGE 132 113 and reporting the lowest value of IAE. Figure 72 superimposes the IMC and the FAIMCr responses. 0 200 400 600 800 1000 1200 130 140 150 160 170 180 190 200 210 220 time, minTemperature, T4 oF IMC(IAE=6119) FAIMCr(IAE=1119) Figure 72. Performances of t he IMC and FAIMCr Controllers PAGE 133 114 3.7 Summary In this chapter a new controller is developed using fuzzy logic to improve the performance of the conv entional internal model c ontroller (IMC). This controller is called FAIMCr (Fuzzy A daptive Internal Model Controller). Two fuzzy modules plus a filter tuning equation are added to the conventional IMC to achieve the objecti ve. The first fuzzy module, the IMCFAM, determines the process parameters changes. The second fuzzy module, the IMCFF, provides stability to the c ontrol system, and a tuning equation is developed for the filter time const ant based on the process parameters. The results show the FAIMCr provid ing a robust response and overcoming stability problems. Adding noise to the sensor signal does not affect the performance of the FAIMCr. PAGE 134 115 CHAPTER 4 CONCLUSIONS AND FURTHER RESEARCH 4.1 Conclusions This principal objective of this disse rtation was to develop and to improve process controllers using fuzzy logic as an artificial intelligent tool. 4.1.1 The FCIV Controller A new fuzzy logic controller, calle d Fuzzy Controller with Intermediate Variable (FCIV) was designed for cascade control purposes. This controller has an improved performance, r egarding stability and robustness, than conventional cascade control strategies. 4.1.2 The IMCFAM Fuzzy Module A fuzzy adaptive module, the IMCFAM was designed to update the IMC process parameters based on its modeling e rror. The process gain is calculated by an analytical module; a first order Takagi Sugeno fuzzy in ference system is specially designed to predict the changes of the process time constant and process dead time. The predicted values from the IMCFAM unit successfully PAGE 135 116 follow the process parameters changes, and consequently avoid the instability sometimes provided by the conventional IMC. 4.1.3 The IMC Filt er Tuning Equation An IMC filter tuning equat ion was developed as a function of the process model parameters. This equation allows calc ulating the filter time constant value and updates its value in the IMC struct ure. A FOPDT is used as a process model. The main advantage of this equati on is recalculating the filter time constant once the IMCFAM provides t he new process model parameters. This action improves the performance of the IM C controller working in highly nonlinear systems. 4.1.4 The IMCFF Fuzzy Module A fuzzy filter module, the IMCFF, was creat ed and used in the IMC controller. This unit uses fuzzy logic trans forming the IMC filter into a variable filter. The IMCFF is a useful unit helpi ng the IMC controller avoiding stability problems when excessive oscillations occur. 4.2 Further Research The further research should be addressed to: Developing a tuning equation for the FC IV controller based on the process parameters. The FCIV requires five tuning parameters: Ke, K e, K C 2, KFB PAGE 136 117 and KINT. The optimum values of the FCIV parameters in this research were obtained by an opt imization routine. Extending the knowledge developed in th is research to advanced control strategies such as Multivariable Process Control to predict process parameters changes by using fuzzy logi c, especially Takagi Sugeno fuzzy inference systems. PAGE 137 118 REFERENCES 1. Burden, A., Tantalean R., and Deshpande, P., [2003]. Control and Optimize Nonlinear Systems, Measurement and Contro l (CEP), pp 6373. 2003. 2. Chen, CL., and Kuo, FC., [1995]. Design and Analysis of a Fuzzy Logic Controller, Int. J. Systems Sci., Vol. 26, pp12231248,1995. 3. Chiu, S., [1998]. Using Fuzzy Logic in Control Aplications: Beyond Fuzzy PID Control, IEEE. Control System Magazi ne, Vol. 18 No. 5, October 1998, pp 100104. 4. De Silva, C.W., [1995]. Intelligent ControlFu zzy Logic Applications, CRC Press, New York,1995. 5. Edgar C.R., and Post lethwaite B.E., [2002].] Model based control using fuzzy relational models, IEEE International Conference on Fuzzy Systems v 3 2002 (IEEE cat n 01ch37297) pp 15811584. 6. 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Passino, K. [2001]. Intelligent Control: An Overview of Techniques, in T. Samad (Ed.), Perspectives in Cont rol Engineering: Technologies, Applications, and New Directions. Pisca taway, NJ: IEEE Press, 2001. 19. Resnick, L., [1997]. Fuzzy Controllers, Oxford, Newnes, 1997. 20. Smith, C., and Corripio A. [2005]. Principles and Practice of Automatic Process Control, John Wiley and Sons.2005. 21. Sugeno, M., [1985]. Industrial Applications of Fuzzy Control, NorthHolland, Amsterdam, 1985. 22. Zadeh. L., [1965]. Fuzzy Sets, Information and Control, vol 8, pp 338353, 1965. PAGE 139 120 APPENDICES PAGE 140 121 Appendix 1 Process Model to Test the FCIV The process selected to test the FC IV is shown Figure 73. The process consists of a preheating tank followe d by a chemical reactor where the endothermic reaction A 2B+C takes place. This process is quite nonlinear and therefore useful for our purposes. Figure 73. Process Diagram The controlled variable is the output concentration of component C, CC(t), the manipulated variable is the input flow of steam, w(t), and the intermediate variable is the temperatur e in the preheating tank, T1(t). The recycle is carried out by means of a pump which supp lies a constant recycle fluid, fr. Because of length of pipe, the variables involved in this flui d have a delay time when they arrive at ReactorA 2B + C Px Pa CA1(t) CB1(t) CC1(t)1(t) T1(t) f1(t) CA(t) CB(t) CC(t)(t) T(t) f(t) TT AT c2(t),%TO2c1(t),%TO1 m(t),%CO w(t) h4 h3 h2(t)Preheating TankCAi(t) fi(t) Ti(t)A A B BfofrRecycleSat. vap. Sat. liq. h1(t) ReactorA 2B + C Px Pa CA1(t) CB1(t) CC1(t)1(t) T1(t) f1(t) CA(t) CB(t) CC(t)(t) T(t) f(t) TT AT c2(t),%TO2c1(t),%TO1 m(t),%CO w(t) h4 h3 h2(t)Preheating TankCAi(t) fi(t) Ti(t)A A B BfofrRecycleSat. vap. Sat. liq. h1(t) PAGE 141 122 Appendix 1 (Continued) the preheating tank. A second pump removes a constant flow from the bottom of the preheating tank. The flows through the valves ar e given by the following equations: Equation for valve A: O H x At h t h g t h t h g t P Pa Cv t f2 1 3 2 4 1 1 1 ) ( ] ) ( [ ) ( ] ) ( [ ) ( ) ( (A1.1) Equation for valve B: O H x Bt P gh t Cv t f2 2 ) ( 7 14 ) ( ) ( (A1.2) The reaction rate is given by : R t T E B A Be t C t C k t r) ( 0) ( ) ( ) ( (A1.3) The equation for density of the flui d in the reactor is obtained as: ) ( ) ( ) ( ) ( 3 2 1 0t C t C t C tC B A (A1.4) The final control element is an equal percentage valve with a maximum flow of 3.6 times the steady state flow and time const ant of 0.2 minutes. The equation for this valve is: PAGE 142 123 Appendix 1 (Continued) 1 100 ) (25 329 3 ) ( ) ( 2 0t mt w t w dt d (A1.5) The analyzer transmitter has a first or der dynamics with a time constant of 0.35 minutes and a range on CC(t) from 6.413 to 32.066 kgmoleC/m3. Thus, the equation for the analyzer transmitter is: 413 6 ) ( 653 25 100 ) ( ) ( 35 01 1 t Cc t c t c dt d (A1.6) Finally, the temperature transmitter al so has a first order dynamics with a time constant of 0.25 minutes and a range on T1(t) from 50F (283.33K) to 250 F (394.44K). The respective equation is: 33 283 ) ( 11 111 100 ) ( ) ( 25 01 2 2 t T t c t c dt d (A1.7) The steady state values and constant s for the process variables are shown in Tables A1.1, A1.2, and A1.3. PAGE 143 124 Appendix 1 (Continued) Table 15. Constants and Steady State Va lues for Preheating Tank Variables VARIABLE VALUE UNITS fi 0.03776 m3/s fo 0.014158 m3/s fr 0.014158 m3/s w 0.92313 kg/s Pa 1.01325*105 Pa Px 1.2410*105 Pa 0 1058.17 kg/ m3 CAi 27.23139 kgmoleA / m3 Cp 3.9747 kJ/(kg.K) Cv 3.6818 kJ/(kg.K) Ti 325 K 2243.43 kJ/kg H2O999.55 kg/ m3 CVA 2.024*104 ( m3/s)Pa0.5 AHT 7.432 m2 U 1.7361 kJ/( m2.K.s) Ac 37.161 m2 Cm 531.31 kJ/(K) 1 2.4 kg/kgmoleA 2 1.2 kg/kgmoleB 3 1.8 kg/kgmoleC h4 0.914 m PAGE 144 125 Appendix 1 (Continued) Table 16. Steady State Values for the Reactor VARIABLE VALUE UNITS h3 0.61 m AR 5.5741 m2 CVB 1.73378*104 ( m3/s)Pa0.5 k0 1.1861 x107 m3/(kgmole.s) E 6.461*104 kJ/kgmole R 8.31451 kJ/(kgmole.K) HR 2786.87 kJ/kgmoleB Cp 3.9747 kJ/(kg.K) Cv 3.6818 kJ/(kg.K) Px 1.241*105 Pa PAGE 145 126 Appendix 1 (Continued) Table 17. Steady State Values for Some Variables in the Process VARIABLE VALUE UNITS CA 8.8414 kgmoleA / m3 CB 38.5679 kgmoleB / m3 CC 19.2846 kgmoleC / m3 CA1 22.2160 kgmoleA / m3 CB1 10.5185 kgmoleB / m3 CC1 5.2597 kgmoleC / m3 1159.42 kg/ m3 11132.62 kg/ m3 f0.03688 m3/s h16.74089 m h22.6169 m T313.94 K T1330.77 K Tw362.87 K r2.28382*104 m3/(kgmoleB s) c1 50.241 %TO1 c2 42.698 %TO2 m 60.149 %CO PAGE 146 127 Appendix 2 Process Model to Test the FAIMCr The following system is selected to test the FAIMCr. The tank shown in Figure 74 receives two streams, a hot stream, W1(t), and a cold stream, W2(t). The outlet temperature is measured 125 ft downstream from the tank bottom. The following assumptions are consi dered in developing the mathematical model: 1) Constant volume of liquid in the tank, 2) perfect mixing and 3) tank and pipes are well insulated. The temperature transmitter is calibrated for a range of 100 oF to 200 oF and Table A2.1 contains the steady state conditions and other process conditions. TT W1(t) W2 (t) W3 (t) T1(t) T2 (t) T3 (t) T4(t)4 Hot Water TT IMCW1(t) W2 (t) W3 (t) T1(t) T2 (t) T3 (t) T4(t) Set Point, T4 Cold WaterL = 125 ft TT W1(t) W2 (t) W3 (t) T1(t) T2 (t) T3 (t) T4(t)4 Hot Water TT IMCW1(t) W2 (t) W3 (t) T1(t) T2 (t) T3 (t) T4(t) Set Point, T4 Cold WaterL = 125 ft Figure 74. The Process: A Mixing Tank PAGE 147 128 Appendix 2 (Continued) Table 18. Mixing Tank Operating Conditions VARIABLE VALUE VARIABLE VALUE W1 250.00 lb/min V 15 ft3 W2 191.17 lb/min TO 50 %TO Cp1 0.8 Btu/lboF Vp 0.4779 Cp2 1.0 Btu/lboF CVL 12 gpm/psi05 Cp3, Cv 0.9 Btu/lboF Pv 16 psi SP(T4) 150 oF T 0.5 min T1 250 oF Vp 0.1 min T2 50 oF A 0.2006 ft2 T3 150 oF L 125 ft 62.4 lb/ft3 m 47.79%CO The following equations constitute the mathematical model of the process: Total Mass Balance (mixing tank): 0)()()(3 2 1 tWtWtW (A2.1) Energy Balance around the tank: )]([ )()()()()()(3 33 3 22 2 11 1tT dt d CvVtTCptWtTCptWtTCptW (A2.2) Relationship between tank temperature and sensor location: )()(0 3 4ttTtT (A2.3) Transportation lag (delay time): )( 3 0tW LA t (A2.4) PAGE 148 129 Appendix 2 (Continued) Temperature transmitter equation: ) 100 ) ( ( 1 ) ( ) ( 4 t T t TO t TO dt dT (A2.5) Valve position equation: ) ( 01 0 ) ( ) ( t m t V t V dt dp p p V (A2.6) Valve equation: V f p VLP G t V C t W) ( 60 500 ) (2 (A2.7) where: W1(t) = hot stream mass flow rate, lb/min W2(t) = cold stream mass flow rate, lb/min W3(t) = outlet mass flow rate, lb/min Cp = liquid heat capacity at c onstant pressure, Btu/lboF Cv = liquid heat capacity at c onstant volume, Btu/lboF T1(t) = hot flow temperature, oF T2(t) = cold flow temperature, oF T3(t) = temperature of liqui d in the mixing tank, oF T4(t) = temperature T3(t) delayed by t0, oF t0 = dead time, min = density of the mixing tank liquid, lbm/ft3 V = liquid volume into the tank, ft3 PAGE 149 130 Appendix 2 (Continued) TO(t) = temperature transmitter output signal, %TO, scale from 1 to 100%. Vp(t) = Valve position, from 0 (clos ed valve) to 1 (open valve) m(t) = controller output signal, %CO, scale from 1 to 100% VLC = valve flow coefficient, gpm/psi0.5 Gf = specific gravity, dimensionless Pv = pressure drop across valve, psi T = time constant of the temperature sensor, min pV = time constant of the actuator, min A = pipe cross section, ft2 L = pipe length, ft. PAGE 150 131 Appendix 3 Simulink Implementation for ) )( (* inv fF G Inverse Model Parameters Fuzzy Filter 1 M(t) Km Tm 1 s [Tm] [Tf] [Km] 1 1 Km [Tf] [Tf] Km Tm [Tf] Tm 2 Tf 1 C_set adj 1 12s K s Gf f m p f ) ( f m pK s G ) (1 Inverse Model Parameters Fuzzy Filter 1 M(t) Km Tm 1 s [Tm] [Tf] [Km] 1 1 Km [Tf] [Tf] Km Tm [Tf] Tm 2 Tf 1 C_set adj 1 12s K s Gf f m p f ) ( f m pK s G ) (1Figure 75. Implementation of the Transfer Function) )( (*inv fF GUsing Simulink PAGE 151 132 Appendix 4 Analyses of Variances for P1, P2, and P3 Table 19. Analysis of Variance for P1 in the NSS Matrix Table 20. Analysis of Variance for P2 in the NSS Matrix Table 21. Analysis of Variance for P3 in the NSS Matrix PAGE 152 133 Appendix 4 (Continued) Table 22. Analysis of Variance for P1 in the NSM Matrix Table 23. Analysis of Variance for P2 in the NSM Matrix Table 24. Analysis of Variance for P3 in the NSM Matrix PAGE 153 134 Appendix 4 (Continued) Table 25. Analysis of Variance for P1 in the NSB Matrix Table 26. Analysis of Variance for P2 in the NSB Matrix Table 27. Analysis of Variance for P3 in the NSB Matrix PAGE 154 135 Appendix 4 (Continued) Table 28. Analysis of Variance for P1 in the NMS Matrix Table 29. Analysis of Variance for P2 in the NMS Matrix Table 30. Analysis of Variance for P3 in the NMS Matrix PAGE 155 136 Appendix 4 (Continued) Table 31. Analysis of Variance for P1 in the NMM Matrix Table 32. Analysis of Variance for P2 in the NMM Matrix Table 33. Analysis of Variance for P3 in the NMM Matrix PAGE 156 137 Appendix 4 (Continued) Table 34. Analysis of Variance for P1 in the NMB Matrix Table 35. Analysis of Variance for P2 in the NMB Matrix Table 36. Analysis of Variance for P3 in the NMB Matrix PAGE 157 138 Appendix 4 (Continued) Table 37. Analysis of Variance for P1 in the NBS Matrix Table 38. Analysis of Variance for P2 in the NBS Matrix Table 39. Analysis of Variance for P3 in the NBS Matrix PAGE 158 139 Appendix 4 (Continued) Table 40. Analysis of Variance for P1 in the NBM Matrix Table 41. Analysis of Variance for P2 in the NBM Matrix Table 42. Analysis of Variance for P3 in the NBM Matrix PAGE 159 140 Appendix 4 (Continued) Table 43. Analysis of Variance for P1 in the NBB Matrix Table 44. Analysis of Variance for P2 in the NBB Matrix Table 45. Analysis of Variance for P3 in the NBB Matrix PAGE 160 141 Appendix 4 (Continued) Table 46. Analysis of Variance for P1 in the ZSS Matrix Table 47. Analysis of Variance for P2 in the ZSS Matrix Table 48. Analysis of Variance for P3 in the ZSS Matrix PAGE 161 142 Appendix 4 (Continued) Table 49. Analysis of Variance for P1 in the ZSM Matrix Table 50. Analysis of Variance for P2 in the ZSM Matrix Table 51. Analysis of Variance for P3 in the ZSM Matrix PAGE 162 143 Appendix 4 (Continued) Table 52. Analysis of Variance for P1 in the ZSB Matrix Table 53. Analysis of Variance for P2 in the ZSB Matrix Table 54. Analysis of Variance for P3 in the ZSB Matrix PAGE 163 144 Appendix 4 (Continued) Table 55. Analysis of Variance for P1 in the ZMS Matrix Table 56. Analysis of Variance for P2 in the ZMS Matrix Table 57. Analysis of Variance for P3 in the ZMS Matrix PAGE 164 145 Appendix 4 (Continued) Table 58. Analysis of Variance for P1 in the ZMM Matrix Table 59. Analysis of Variance for P2 in the ZMM Matrix Table 60. Analysis of Variance for P3 in the ZMM Matrix PAGE 165 146 Appendix 4 (Continued) Table 61. Analysis of Variance for P1 in the ZMB Matrix Table 62. Analysis of Variance for P2 in the ZMB Matrix Table 63. Analysis of Variance for P3 in the ZMB Matrix PAGE 166 147 Appendix 4 (Continued) Table 64. Analysis of Variance for P1 in the ZBS Matrix Table 65. Analysis of Variance for P2 in the ZBS Matrix Table 66. Analysis of Variance for P3 in the ZBS Matrix PAGE 167 148 Appendix 4 (Continued) Table 67. Analysis of Variance for P1 in the ZBM Matrix Table 68. Analysis of Variance for P2 in the ZBM Matrix Table 69. Analysis of Variance for P3 in the ZBM Matrix PAGE 168 149 Appendix 4 (Continued) Table 70. Analysis of Variance for P1 in the ZBB Matrix Table 71. Analysis of Variance for P2 in the ZBB Matrix Table 72. Analysis of Variance for P3 in the ZBB Matrix PAGE 169 150 Appendix 4 (Continued) Table 73. Analysis of Varianc e for P1 in the PSS Matrix Table 74. Analysis of Varianc e for P2 in the PSS Matrix Table 75. Analysis of Varianc e for P3 in the PSS Matrix PAGE 170 151 Appendix 4 (Continued) Table 76. Analysis of Variance for P1 in the PSM Matrix Table 77. Analysis of Variance for P2 in the PSM Matrix Table 78. Analysis of Variance for P3 in the PSM Matrix PAGE 171 152 Appendix 4 (Continued) Table 79. Analysis of Varianc e for P1 in the PSB Matrix Table 80. Analysis of Varianc e for P2 in the PSB Matrix Table 81. Analysis of Varianc e for P3 in the PSB Matrix PAGE 172 153 Appendix 4 (Continued) Table 82. Analysis of Variance for P1 in the PMS Matrix Table 83. Analysis of Variance for P2 in the PMS Matrix Table 84. Analysis of Variance for P3 in the PMS Matrix PAGE 173 154 Appendix 4 (Continued) Table 85. Analysis of Variance for P1 in the PMM Matrix Table 86. Analysis of Variance for P2 in the PMM Matrix Table 87. Analysis of Variance for P3 in the PMM Matrix PAGE 174 155 Appendix 4 (Continued) Table 88. Analysis of Variance for P1 in the PMB Matrix Table 89. Analysis of Variance for P2 in the PMB Matrix Table 90. Analysis of Variance for P3 in the PMB Matrix PAGE 175 156 Appendix 4 (Continued) Table 91. Analysis of Varianc e for P1 in the PBS Matrix Table 92. Analysis of Varianc e for P2 in the PBS Matrix Table 93. Analysis of Varianc e for P3 in the PBS Matrix PAGE 176 157 Appendix 4 (Continued) Table 94. Analysis of Variance for P1 in the PBM Matrix Table 95. Analysis of Variance for P2 in the PBM Matrix Table 96. Analysis of Variance for P3 in the PBM Matrix PAGE 177 158 Appendix 4 (Continued) Table 97. Analysis of Varianc e for P1 in the PBB Matrix Table 98. Analysis of Varianc e for P2 in the PBB Matrix Table 99. Analysis of Varianc e for P3 in the PBB Matrix PAGE 178 ABOUTH THE AUTHOR Yohn Euclides Garca Zambrano wa s born on December 08, 1965 in Triba, Tchira, Venezuela. The author received his bachelor in Chemical Engineering from Universidad de Los Andes, Mrida, Vene zuela in 1993. In 1994 started the Master of Science in C hemical Engineering at Un iversidad de Los Andes. He graduated in 1997 with a Thesis in Distribution of Nonionic Surfactants in Emulsions. In 1998 He cooperated with Pequiven (Petroquimi ca de Venezuela) in automatic process control area. In 2000 Yohn entered the program in the Chemical Engineering Department at USF, where he served as a Teaching Assistance of the courses Automatic process Control I, Thermody namics and Fluids, Process Engineering I and Process Engineering II, during his studies. His future interests include doing research in Intelligent Control and Optimization applied on Chemical Process. 