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Sanchez, Edinzo J. Iglesias.
Using fuzzy logic to enhance control performance of sliding mode control and dynamic matrix control
h [electronic resource] /
by Edinzo J. Iglesias Sanchez.
[Tampa, Fla] :
b University of South Florida,
ABSTRACT: Two application applications of Fuzzy Logic to improve the performance of two controllers are presented. The first application takes a Sliding Mode Controller designed for chemical process to reject disturbances. A fuzzy element is added to the sliding surface to improve the controller performance when set point change affects the control loop; especially for process showing highly nonlinear behavior. This fuzzy element, is calculated by means of a set of fuzzy rules designed based on expert knowledge and experience. The addition of improved the controller response because accelerate or smooth the controller as the control loop requires. The Fuzzy Sliding Mode Controller (FSMCr) is a completely general controller. The FSMCr was tested with two models of nonlinear process: mixing tank and neutralization reactor. In both cases the FSMCr improves the performance shown for other control strategies, as the industrial PID, the conventional Sliding Mode Control and the Stan dard Fuzzy Logic Controller. The second part of this research presents a new way to implement the Dynamic Matrix Control Algorithm (DMC). A Parametric structure of DMC (PDMC) control algorithm is proposed, allowing to the controller to adapt to process nonlinearities. For a standard DMC a process model is used to calculate de controller response. This model is a matrix calculated from the dynamic response of the process at open loop. In this case the process parameters are imbibed into the matrix. The parametric structure isolates the process parameters allowing adjust the model as the nonlinear process changes its behavior. A Fuzzy supervisor was developed to detect changes in the process and send taht [sic]information to the PDMCr. The modeling error and other parameters related were used to estimate those changes. Some equations were developed to calculate the PDMCr tuning parameter,lambda, as a function of the process parameters. The performance of PDMCr was tested using to model ^of nonlinear process and compare with the standard DMC; in most the cases PDMCr presents less oscillations and tracks with less error the set point. Both control strategies presented in this research can be implemented into industrial applications easily.
Dissertation (Ph.D.)--University of South Florida, 2006.
Includes bibliographical references.
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Variable Structure Controller.
Model Based Controller.
t USF Electronic Theses and Dissertations.
Using Fuzzy Logic to Enhance Control Performance of Sliding Mode Control and Dynamic Matrix Control by Edinzo J. Iglesias Sanchez A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Chemical Engineering College of Engineering University of South Florida Co-Major Professor: Carlos A. Smith, Ph.D. Co-Major Professor: Marco Sanjuan, Ph.D. Oscar E. Camacho, Ph.D. Aydin K. Sunol, Ph.D. John T. Wolan, Ph.D. Rangachar Kasturi, Ph.D. Date of Approval: February 7, 2006 Keywords: Artificial Intelligence, Variable Structure Controller, Model Based Controller, Nonlinear Process, Chemical Process. Copyright 2006, Edinzo J. Iglesias Sanchez
Dedication To my beloved wife, Silvia, without your love and encouragement this work could not have been possible. I hope to have enough lifetime and strength to compensate you for all the hard times that we shared. Finally we fulfill one of our dreams. To my family, Mom, Ivon and Brenda, your unconditionally love and trust always guide me to the right way. Your prayers always are listened. To Yaya, our guardian angel. To my others mothers, Haydee and Silvina, your affection always comfort me. To my friend and brother, Yohn, we shared good and hard times.
Acknowledgments I would like to thank to Dr. Carlos Smith, his dedication and patience were fundamental to the success of this to this work. I want to express my gratitude to Dr Marco Sanjuan, the friendly hand always willing to help and guide. Special thank to my friend and mentor, Jos Andrez. During all my professional life his opportune advice has meant the difference. My gratitude to the University of Los Andes, for help me to accomplish this goal. Finally, to my home country, Venezuela.
i Table of Contents List of Tables v List of Figures viii Abstract xii Chapter 1. Introduction 1 1.1 Contributions of the Research 1 1.2 Process Nonlinearities 3 1.3 Some Alternatives Control Stra tegies for Nonlinear Processes 10 1.3.1 Adaptive Control 10 1.3.2 Non-PID Based Control Approaches 11 1.4 Sliding Mode Control 14 1.5 Dynamic Matrix Control (DMC) 16 1.6 Summary 17 Chapter 2. Fuzzy Surface-Based Sliding Mode Control 18 2.1 Introduction 18 2.2 Sliding Mode Control 19 2.3 Fuzzy Logic in Process Control 21 2.4 Nonlinear Processes 22 2.5 The Surface-Based Fuzzy Sliding Mode Control 27 2.5.1 The Steady State Compensator 28
ii 2.5.2 Fuzzy Rules to Develop a Fuzzy Sliding Surface 31 2.6 Simulation Results 36 2.6.1 FS Behavior 37 2.6.2 Performance Comparison 38 188.8.131.52 Neutralization Process 38 184.108.40.206 Mixing Process 45 2.6.3 Sampling Time Effect 47 2.6.4 Effect of Noise 48 2.7 Conclusion 49 Chapter 3. Dynamic Matrix Control and Fuzzy Logic 50 3.1 Introduction 50 3.2 Conventional DMC Implementation 51 3.3 Nonlinear DMC in the Literature 55 3.4 Fuzzy Logic and MBC 58 3.5 A New Approach for DMC Structure 59 3.5.1 Parametric DMC 61 3.5.2 Tuning Equation for Suppression Factor 68 3.5.3 Fuzzy Supervisor 78 3.5.4 PDMCr Implementation 98 3.6 Simulation Results 102 3.7 Effect of Noise 109 3.8 Conclusions 111 Chapter 4. Summary 112
iii References 115 Appendices 123 Appendix A: Mathematical Model of Processes 124 A.1 Neutralization Reactor 124 A.2 Mixing Process 130 Appendix B: DMCr Control Law Deduction 134 Appendix C: Tuning Equations for DMCr 137 Appendix D: Modeling Error Indica tors (MEIs). Definition and Calculation 148 D.1 2nd, 6th and 10th Correlation Coefficients 149 D.2 Maximum em /cset 150 D.3 Minimum em /cset 150 D.4 Maximum em /Minimum em 151 D.5 Time for Maximum em / cset 151 D.6 Time for Minimum em /cset 151 D.7 Stabilization Time Between 0.1%/ TO 152 D.8 Difference in Time for Maximum Modeling Error and Time Minimum Modeling Error/ 152 D.9 Ratio Absolute Minimum Modeling Error/Absolute Maximum Modeling Error 152 D.10 Maximum Modeling Error Peak/cset 153 D.11 Minimum Modeling Error Peak/cset 153 D.12 Time for Maximum Peak/ 153 D.13 Time for Minimum Peak/ 154
iv D.14 Decay Ratio 154 D.15 Damping Ratio 154 D.16 Ratio Minimum em / Maximum em 155 D.17 Time Between Peaks/ 155 Appendix E: ANOVA Tables for Modeling Error Indicator ( MEI ) 156 Appendix F: Regression Equation for Modeling Error Indicators 179 About the Author End Page
v List of Tables Table 1.1. Steady State Values for Mixing Process 4 Table 2.1. Fuzzy Rules to Obtain FS Sets 35 Table 2.2. Results Comparison for Disturbance Test 39 Table 2.3. Results Comparison for Set Point Changes Test 42 Table 3.1. Factors Used to Perf orm the Designed Experiments 69 Table 3.2. ANOVA Table for Optimal Suppression Factor 72 Table 3.3. IAE Comparison for Te st Present in Figure 3.5 77 Table 3.4. Factors and Levels Used to Record Modeling Error and Develop the Regression Equations 81 Table 3.5. Modeling Error Indicators Sele cted to Predict Process Parameters Changes 83 Table 3.6. Summary of Modeling Error Indicator Selected and Correlation Coefficient of Regression Equation 85 Table 3.7. Parameters for Regression E quations Used in Supervisor Module 88 Table 3.8. Fuzzy Rules Designed to Calculate P K 93 Table 3.9. Fuzzy Rules Designed to Calculate 94 Table 3.10. Fuzzy Rules Designed to Calculate to 95 Table A.1. Steady State Parameters for Neutralization Reactor Model 129 Table A.2. Steady State Va lues for Mixing Process 133 Table C.1. IAE Values from Simu lation Presented in Figure C.1 140
vi Table C.2. Factors Used to Perf ormed the Designed Experiment 141 Table C.3. Constant Parameters and Correlation Value for Eq. C.1 143 Table C.4. Constant Parameters for Eq. C.2 144 Table E.1. ANOVA Table for S econd Correlation Coefficient 158 Table E.2. ANOVA Table for 6th Correlation Coefficient 159 Table E.3. ANOVA Table for 10th Correlation Coefficient 160 Table E.4. ANOVA Table for Ra tio Maximum Modeling Error/ Cset 161 Table E.5. ANOVA Table for Maximu m/Minimum Modeling Error Ratio 162 Table E.6. ANOVA Table for Minimum Modeling Error Over Cset 163 Table E.7. ANOVA Table for Modeling Error Angular Frequency Omega 164 Table E.8. ANOVA Table for Modeling Error Period T 165 Table E.9. ANOVA Table for Ratio Ti me for Maximum Modeling Error Occurrence Over Tau 166 Table E.10. ANOVA Table for Ratio Ti me for Minimum Modeling Error Occurrence Over Tau 167 Table E.11. ANOVA Table for Ratio Stabil ization Time for 10% Modeling Error Over Tau 168 Table E.12. ANOVA Table for Ratio Ti me for Maximum Minus Time for Minimum Modeling Error Over Tau 169 Table E.13. ANOVA Table for Ratio Ab solute Minimum Over Absolute Maximum Modeling Error 170 Table E.14. ANOVA Table for Ratio Maximu m Pick of Modeling Error Over Cset 171 Table E.15. Minimum Modeling Error Pick Over Cset 172 Table E.16. Ratio Time for Modeling E rror Maximum Pick Occurrence Over Tau 173
vii Table E.17. Ratio Time for Minimum Modeling Error Pick Over Tau 174 Table E.18. Decay Ratio for Modeling Error 175 Table E.19. Modeling Error Damping Ratio 176 Table E.20. Ratio Minimum Ov er Maximum Modeling Error 177 Table E.21. Ratio Distance Between First Two Picks Over Tau 178 Table F.1. 10th Correlation Coefficient 181 Table F.2. Ratio Maximum em Over Set Point Change 181 Table F.3. Ratio Minimum em Over Set Point Change 182 Table F.4. Modeling Error Period T 182 Table F.5. Modeling Erro r Angular Frequency 183 Table F.6. Ratio Time for Maximum em Over Process Time Constant 183 Table F.7. Ratio Time for Minimum em Over Process Time Constant 184 Table F.8. Ratio Time for Em Settle Be tween 0.1%TO Over Process Time Constant 184 Table F.9. Ratio Difference between Time for Maximum and Minimum em Over Process Time Constant 185 Table F.10. Ratio Absolute Minimum em Over Absolute Maximum em 186 Table F.11. Ratio Maximum em Peak Over Set Point Change 186 Table F.12. Ratio Minimum em Peak Over Set Point Change 187 Table F.13. Ratio Time for Maximum Peak Over Process Time Constant 187 Table F.14 Ratio Time for Minimum em Peak Over Process Time Constant 188 Table F.15. Modeling Error Damping Ratio 189
viii List of Figures Figure 1.1. Mixing Process Used to I llustrate Nonlinear Characteristics 3 Figure 1.2. PID Performance for Hot-Cold Model Facing +10% Up Change in Set Point and +10oC in T2( t ) 6 Figure 1.3. PID Performance for Hot-Cold Model Facing -10% Change in Set Point and +10oC in T2( t ) 7 Figure 1.4. Sequence of m ( t ) Changes to Perform Process Characterization 9 Figure 1.5. Open Loop Process Char acterization for Mixing Example 9 Figure 2.1. Schematic Representati on of Nonlinear Processes 23 Figure 2.2. Open Loop Response When Neutralization Process Faces Sequential Step Changes, With the Same Magnitude, in the Signal to the Control Valve 24 Figure 2.3 Kp and to Variations as m ( t ) Function for Neutralization Process 25 Figure 2.4. Kp and to Variations as m ( t ) Function for Mixing Process 26 Figure 2.5. Comparison Between SMCr a nd PID When Neutralization Process Faces a Change of +5% in q1( t ) as Disturbance 27 Figure 2.6. Comparison PID and SMCr When System Faces +10% Change in Set Point 28 Figure 2.7. S(t) Variation When SMCr and MSMCr Are Used Facing a 5% Set Point Change Down 29 Figure 2.8. Comparison Between System Response Using SMCr(a) and MSMCr(b), Facing a 5% Set Point Change Down 31 Figure 2.9. Example of Membership Functions Used to Perform Fuzzification and Defuzzyfication 33
ix Figure 2.10. Fuzzy Inference System Used to Determine FS 34 Figure 2.11. Schematic Representation of e ( n ) NB and NB in Physical Terms 36 Figure 2.12. Behavior of S ( t ), FS and SH(t) Behavior When the System Faces a Disturbance 38 Figure 2.13. Performance Comparison SMCr FLCr, FSMCr and PID, When a Sequence of q1( t ) Changes Affects the System 40 Figure 2.14. Performance Comparison among SMCr, FLCr, FSMCr and PID When System Faces a Set Point Change Sequence 41 Figure 2.15. Performance Comparison Wh en SMCr, FLCr, FSMCr and PID Face a Disturbance and Set Point Changes Simultaneously 43 Figure 2.16. PID Response When Tuning Para meters Are Adjusted to Enhance the Response to Disturbance Change 44 Figure 2.17. Controllers Performance Co mparison to Set Point Change and Disturbance in Acid Stream Concentration (pH Process) 45 Figure 2.18. FSMCr Performance When is Used Controlling the Mixing Process 46 Figure 2.19. Sampling Effect on FSMCr Performance 47 Figure 2.20. Noise Effect on FSMCr Performance 48 Figure 3.1. Discrete Data C ontended Inside Vector Av 63 Figure 3.2. Control Loop Block Diagram Used to Evaluate Experimental Conditions and Performance Parameter 70 Figure 3.3. Example of Set Point Change and Disturbance Used With FOPDT to Find Optimal Suppression Factor for DMCr 71 Figure 3.4. Performance Comparison Wh en DMCr Controlling a FOPDT System is Tuned Using Different Methods 75 Figure 3.5. Performing Comparison of DMCr Tuned Using Different Methods for Mixing Process 76 Figure 3.6. Example of Simulation Perf ormed to Record Modeling Error 82
x Figure 3.7. Supervisor Module Flow Diagram 89 Figure 3.8. Schematic Representation of Fuzzy Inference System Used to Calculate Weighting Factors 92 Figure 3.9. Membership Function Used to Fuzzified MEIs. 97 Figure 3.10. Membership Function Us ed to Defuzzified Output K 98 Figure 3.11. Example of Nonlinear Relation Among Two of the MEIs and K 99 Figure 3.12. Schematic Representation of Parametric DMC Working With Fuzzy Supervisor 100 Figure 3.13. Schematic Representation of PDMCr Algorithm Implementation 101 Figure 3.14. Comparison Standard DMCr and PDMCr Performance Handling Nonlinearities Emulated 103 Figure 3.15. Comparison Among Actual Pr ocess Parameters and Those Estimated by Fuzzy Supervisor Module for the Test Presented in Figure 3.14 104 Figure 3.16. Performance Comparison Wh en DMCr and PDMCr Are Used to Control Mixing Process 105 Figure 3.17. Process Parameters Estima tion Performed by Fuzzy Supervisor Module for the Test Presented in Figure 3.16 106 Figure 3.18. Performance Comparison Betw een Standard DMCr and PDMCr Handling Neutralization Process 107 Figure 3.19. Neutralization Reactor Te st for Disturbances Rejection 108 Figure 3.20. Cold Stream F2 Affected by Noise With Structure ARMA(1,1) 109 Figure 3.21. Noise Effect Over DMCr and PDMCr 110 Figure A.1. Schematic Representati on of Neutralization Reactor 124 Figure A.2. Schematic Representation Mixing Process 130 Figure C.1. Comparison of DMCr Performance When CH and Ts Change and Eq. 3.32 is Used to Calculate 139
xi Figure C.2. Graphical Representation of IAE as CH and Ts / Function 140 Figure C.3. Plot of Residual vs. Predicted Values Using Eq. C.2 145 Figure C.4. Comparison of Suppression F actor Prediction Using Eq. 3.32 and Eq. C.2 146 Figure D.1. Some Modeling Error Indicators (MEIs) 148
xii Using Fuzzy Logic to Enhance Control Performance of Sliding Mode Control and Dynamic Matrix Control Edinzo J. Iglesias Snchez ABSTRACT Two application applications of Fuzzy Logic to improve the performance of two controllers are presented. The first applica tion takes a Sliding Mode Controller designed for chemical process to reject disturbances A fuzzy element is added to the sliding surface to improve the controlle r performance when set point change affects the control loop; especially for process showing highl y nonlinear behavior. This fuzzy element,FS is calculated by means of a set of fuzzy rules designed based on expert knowledge and experience. The addition of FS improved the controller response because accelerate or smooth the controller as the control loop re quires. The Fuzzy Sliding Mode Controller (FSMCr) is a completely general controller. The FSMCr was tested with two models of nonlinear process: mixing tank and neutraliz ation reactor. In both cases the FSMCr improves the performance shown for other cont rol strategies, as the industrial PID, the conventional Sliding Mode Control and the Standard Fuzzy Logic Controller. The second part of this research pres ents a new way to implement the Dynamic Matrix Control Algorithm (DMC). A Parame tric structure of DMC (PDMC) control algorithm is proposed, allowing to the controlle r to adapt to proce ss nonlinearities. For a standard DMC a process model is used to calc ulate de controller response. This model is
xiii a matrix calculated from the dynamic response of the process at open loop. In this case the process parameters are imbibed into the ma trix. The parametric structure isolates the process parameters allowing adjust the m odel as the nonlinear process changes its behavior. A Fuzzy supervisor was developed to detect changes in the process and send taht information to the PDMCr. The modeli ng error and other parameters related were used to estimate those changes. Some equa tions were developed to calculate the PDMCr tuning parameter, as a function of the process parameters. The performance of PDMCr was tested using to model of nonlinea r process and compare with the standard DMC; in most the cases PDMCr presents less oscillations and tracks with less error the set point. Both control strategies presented in this research can be implemented into industrial applic ations easily.
1 Chapter 1 Introduction 1.1 Contributions of the Research This research focuses in combining two conventional control techniques, Sliding Mode Control (SMC) and Dynamic Matrix Control (DMC), with one of the most promising intelligent control technique, Fuzzy Logic. The reported problems of the Sliding Mode Controller (SMCr) when tracking set point changes with highly nonlinear processes motivated the idea to develop a SMCr with a robust response to changes in set point. To achieve this objective, a combination of SMC and Fuzzy Logic is proposed. The idea is to incorporate the human expert knowledge to the controller to react quickly or slowly depending of the process requirements. DMC is a linear controller. One way of improving its performance, when working with nonlinear processes, it is to adjust the model gain, time constant and dead time depending on control loop behavior. This adjustment can be performed in many ways; however, the use of human experience through Fuzzy Logic is an interesting alternative. As it is shown in Chapter 3, the DMC algorithm can be reformulated to isolate the process model parameters. This parametric algorithm for DMC can be adjusted according to the nonlinearities shown by the process.
2 In summary, this research presents ways of improving the performance of SMC and DMC using Fuzzy Logic. A Fuzzy Sliding Mode Controller (FSMCr) is presented which combines the best characteristics of SMC and Fuzzy Logic: robustness, stability and flexibility. This controller is suitable to be used in applications with highly nonlinear behavior. FSMCr is a completely general controller which incorporates human experience by means of a set of fuzzy rules. Chapter 2 presents a detailed discussion, and the results obtained. A parametric structure of DMC control algorithm is proposed. This new way to express DMC algorithm allows adapting the controller to process nonlinearities. A supervisor system able to determine, on line, if any of the characteristic process parameters gain, time constant and/or dead time has changed is developed. This supervisor system incorporates Fuzzy Logic to determine which parameter, and by how much, has changed. The integration of the parametric DMC algorithm with the supervisor system strategy constitutes a nonlinear controller with implicit dead time compensation able to handle highly nonlinear processes. A discussion is presented in Chapter 3. Tuning equations for the DMC algorithm are developed to determine the optimal suppression factor needed to minimize the error. These equations are incorporated into the supervisor system to enhance the parametric DMC algorithm.
31.2 Process Nonlinearities One of the most important problem every control engineer faces is the nonlinear characteristics of processes; almost all chemical processes are nonlinear. The following example demonstrates the meaning and importance of nonlinear characteristics. Consider the process shown in Figure 1.1, where streams F1 and F2 are fed to a tank and the contents of the tank are well mixed. F1 is a hot water stream (0.8 m3/s and 80oC) whereas F2 is a cold water stream (1.1 m3/s and 15oC). At steady state the temperature of the contents is 42.36oC. The cold stream temperature T2( t ) is considered the main disturbance, and the hot stream flow rate F1( t ) is the manipulated variable. Table 1.1 shows the steady state and other process information. Figure 1.1. Mixing Process Used to Illustrate Nonlinear Characteristics
4 Because the sensor/transmitter TT-E1 used to measure the output temperature is located a distance L from the tank, there is a time delay between the exit temperature from the tank, T ( t ), and the measured temperature, TÂ’ ( t) The sensor/transmitter range is 10-90oC. Table 1.1. Steady State Values for Mixing Process Variable Value Units VariableValue Units F1 0.8 m3/s 1000 kg/m3 F2 1.1 m3/s Cv 1 kcal/oC kg F 1.9 m3/s Cp 1 kcal/oC kg T1 80 oC h 3 m T2 15 oC Atank 1 m2 T 42.36 oC A PID controller is selected to control TÂ’ ( t ). There are several equations available to tune this controller , . These equations are based on a First Order Plus Dead Time (FOPDT) model of the process. () ()1tosCsKpe Mss (1.1) where C ( s ) is the Laplace Transform of the sensor/transmitter signal. %TO. M ( s ) is the Laplace Transform of the controller signal to the valve, %CO. Kp is the process gain, %TO/%CO. to is the process dead time, time units. is the process time constant, time units.
5 The method to obtain the FOPDT model of a process is simple. Under open-loop conditions a step change in the signal to the valve is introduced, and the signal from the sensor/transmitter is recorded. From this recording it is possible to determine the FOPDT terms; gain, time constant and dead time . Using this method, the following FOPDT model for the mixing tank is obtained around the steady state conditions: 25()0.365 ()5.051 s Cse Mss (1.2) where the time constant and dead time are in seconds. Using this information and tuning equations found in the literature , it is possible to determine PID tuning parameters: Kc = 0.889 %CO/%TO I = 19.26 s (1.3) D =15.23 s Figure 1.2 shows system response when a set point change and a disturbance affect the process.
6 0 100 200 300 400 500 600 700 800 10 15 20 25 30 time (sec)T2(t) (oC) 0 100 200 300 400 500 600 700 800 40 45 50 55 60 time(sec)c(t) (%TO) SP c(t) Figure 1.2. PID Performance for Hot-Cold Model Facing +10% Up Change in Set Point and +10oC in T2( t ) The set point change introduced was +10%TO (12oC), and T2( t ) was changed by 10oC. The PID controller is able to track the set point change and successfully reject the disturbance; maybe the overshoot when the disturbance is rejected could be criticized. Figure 1.3 shows the performance when a -10%TO set point change, and +10oC in T2( t ) are introduced into the system. The PID tuning parameters used for this test were the same as those used for the test described in Figure 1.1. It is obvious that this time the controller is not able to compensate for the changes; right after the set point change the system begins to oscillate until it finally becomes unstable.
7 0 100 200 300 400 500 600 700 800 10 15 20 25 30 time (sec)T2(t) oC 0 100 200 300 400 500 600 700 800 0 20 40 60 time (sec) c(t) %TO c(t) SP Figure 1.3. PID Performance for Hot-Cold Model Facing -10% Change in Set Point and +10oC in T2( t ) This simple test shows that although the PID controller is used in nearly 90% of industrial applications , some control applications need a different controller to handle their special characteristics. The obvious questions at this point are: What is special in these processes? How come the PID works for some conditions and it does not for others? To answer these questions we must first address the issue of process nonlinearities. A process is said to be nonlinear if its behavior changes with operating conditions. That is, in the controls nomenclature, if the process gain, and/or the time constant, and/or the dead time change with operating conditions. In these cases the process behaves differently at different operating conditions. As mentioned earlier, the
8 process parameters Kp and to are used to tune the PID controller. The values of these parameters in Eq. (1.2) were obtained at the steady state design operating condition. The controller tuning parameters in Eq. (1.3) were calculated using these values. When the process was subjected to the set point change and the disturbance depicted in Figure 1.2, the control response was very acceptable. At this new operating condition the process behavior is similar to the behavior at original conditions and thus, the numerical values of Kp and to have not changed much. However, when the process was subjected to the set point change and disturbance depicted in Figure 1.3, the process behavior changed enough, and the controller tuning calculated with the previous process parameters drove the controlled process unstable. To maintain stability and provide an acceptable control response, the tuning parameters need to be recalculated using the new process parameters. To further demonstrate the process nonlinearities consider the mixing process again. Sanjuan  suggested obtaining the pro cess parameters at different conditions, and graphing these parameters vs. the operating conditions. Specifically, the method consists in performing a sequence of process testing under open loop condition, and obtaining the process parameters for each test. Figure 1.4 shows a sequence of changes in signal to the valve, m ( t ). For each change the process response, as given by the signal from the sensor/transmitter, was recorded. From each recording the process parameters are obtained and graphed as a function of m ( t ); Figure 1.5 shows the plots.
9 0 0.5 1 1.5 2 x 104 0 10 20 30 40 50 60 70 80 90 100 time (sec)m(t) (%CO) m at steady state Figure 1.4. Sequence of m ( t ) Changes to Perform Process Characterization 0 50 100 20 25 30 35 40 45 50 55 to m(t) (%CO)Dead time (s) 0 50 100 0.2 0.4 0.6 0.8 1 1.2 Gain m(t) (%CO)Process Gain (%TO/%CO) 0 50 100 3 4 5 6 7 8 9 m(t) (%CO)Time constant (s) Figure 1.5. Open Loop Process Characterization for Mixing Example
10 Figure 1.5 shows that the process gain changes monotonically from 0.2 to 1.1 %TO/%CO (a factor of 5.5!) as the control valve position changes from almost opened to almost closed. The figures also shows that the process time constant and dead time exhibit similar behavior, large changes over the operating range; even more, these two parameters exhibit a hysteresis-type phenomena which is still another nonlinear behavior. The previous paragraph explained the phenomenon of process nonlinearities and its impact to the control of processes. The following sections present different approaches to handle nonlinear processes. 1.3 Some Alternatives Control St rategies for Nonlinear Processes The control literature is full of references of approaches to handle the control of nonlinear processes. To attempt a detailed classification of all of them is a large task. However, a simple way to attempt a general classification could be dividing the approaches between those that use PID controllers, and those which use other controllers. 1.3.1 Adaptive Control The PID controller has been used in industrial applications for more than 70 years, therefore it is usually the first choice. For this reason, many control strategies try to improve the PID performance by adjusting its tuning parameters. Different approaches have been used to accomplish this operation: Auto-tuning: This technique uses closed-loop information to determine new PID tuning parameters. The technique estimates the ultimate gain and ultimate period of the process manipulating the signal to the valve. The PID tuning parameters are
11 then calculated using equations similar to the Ziegler-Nichols method . Some authors  have proposed a modification where the PID parameters are determined based on tests performed manipulating the set point signal. Gain Scheduling: The idea behind gain scheduling is to change the PID parameters according to the operating conditions of the process. Gain scheduling philosophy is to generate tables of optimal tuning parameters for the different conditions where the process works. This approach generates particular solutions for each process where it is applied. Self-tuning: This control scheme is maybe the most known and used. There are several commercial products that work under this technology. The self-tuning methods are able to determine optimal PID parameters based on system response; the PID tuning parameters are changed continuously responding to changes in the process. In many cases the literature does not have a clear distinction among these controllers; this is especially true when these techniques are combined with Artificial Intelligence methods. 1.3.2 Non-PID Based Control Approaches The universe of controllers different fr om PID is vast, although paradoxically the number of application where they are used for industrial purposes is reduced. It is estimated that only 10% of industrial applications use controllers different to PID. Among this group of new technologies the followings are mentioned:
12 Robust Control: This control strategy main concern is the dependability or robustness of the control algorithm. The philosophy of this approach is to design controllers whose requirements of maintenance in practical environment are minima. Once the controller is designed, their parameters remain fixed and its stability and performance are assured. Robust Control methods require dynamic information about the process, detail or empirical models obtained by off-line identification. Usually the controller design is based on worst case scenario for process operation, therefore, under normal conditions the control system does not have the best performance that it could have. One of the best known variations of Robust Control is Sliding Mode Control. This nonlinear control technique based on Variable Structure Theory (VST) is robust and versatile. Section 1.4 and Chapter 2 provide more discussion about this technique. Predictive Control: This strategy, also called Model Predictive Control (MPC), is one of the few advanced control technologies with successful participation in industrial applications. MPC is based on three elements: a predictive model, an optimization law applied in a temporal window, and finally a feedback correction. The model is used to predict future behavior based on historical information and future inputs to the process. The Predictive Control algorithm calculates future control actions based on a penalty function or performance function; all the calculations are limited to a moving time interval. The feedback compensation of MPC allows compensating for disturbance and others uncertainties that can affect the process. The best known and successful version of MPC is Dynamic Matrix Control (DMC).
13 Section 1.5 and Chapter 3 provide more discussion about this control strategy. Optimal Control: This branch of control theory uses the systemÂ’s state equations and their initial conditions, as well as a defined objective function. The optimal control algorithm finds a path to lead the system from its initial conditions to a desired final condition by minimizing the error or other performance index. The main application of this control technique is in the aerospace field. Intelligent Control: This strategy includes all control techniques based on Artificial Intelligence. The following control strategies belong to this field: Learning Control: This technique uses pattern recognition techniques to determine the control loop status, as well as knowledge and/or previous experience to generate control decision . Expert Control: This technique uses a knowledge base to take control decisions. The knowledge base is created using human expertise and reproduced symbolically by means of an inference system . Fuzzy Control: This technique is based on Fuzzy Set theory. Its main characteristics and power is its ability to handle vagueness and incomplete information in mathematical terms. The technique has become very popular in industrial applications in recent years due to its flexibility and robustness. Further discussion about this technique is presented on Chapters 2 and 3. Neural Network Control: This control strategy uses artificial neural nets to generated control decisions. The artificial neural nets are trained to learn
14 the relationship between inputs and outputs. Artificial neural nets are comprised of neurons analogous to human nerve cells, which are able to generate complex nonlinear control signal in response to their inputs . Genetic Algorithms: These algorithms are designed to emulate the evolutionary behavior found in the nature; concepts such as chromosomes, crossover, mutations, etc. are used to emulate mathematically evolutionary processes. Genetic algorithm has been used in areas as multidimensional optimization and as an alternative method to help in decision making . The above discussion shows the wide and diverse field that process control is today. The following sections provide a brief introduction to Sliding Mode Control and to Dynamic Matrix Control; a more detailed review is presented in Chapters 2 and 3 respectively. 1.4 Sliding Mode Control Sliding Mode Control (SMC) was originally developed by Utkin in 1977 . It is a technique of derived from Variable Structur e Theory. This theory is used to design a controller whose structure and specifically their gains can change depending on the system condition. SMC was originally used almost exclusively to control simple electro-mechanic systems, such as electrical motors , , , . SMC has also been successfully used in robotic , and in flight control systems .
15 The application of SMC to chemical process control is not very extended; there are few references to applications in this field. The following are some of the more important works in this area. In 1987 Rao and Young  reported the successful application of SMC to retune a PID used to control the pH on a neutralization reactor. This work highlights the SMC robustness to handle highly nonlinear process. Fernandez and Hedrick  used SMC to control concentration and temperature in a CSTR with very good results. Kantor  designed a Sliding Mode C ontroller (SMCr) for a surge tank. In this case the proposed controller was used to control the output flow rate from the tank. Sira-Ramirez and Llanes-Santiago  showed a generalized method to design a SMCr for a CSTR and multiple effect evaporators. Hanczyc and Palazoglu  used SMC theory to control nonlinear chemical processes such as boilers and isothermal plug flow reactors. Colantino and coworkers  also used SMC to control the reaction temperature in a CSTR. In 1996 Camacho and Smith  developed a generalized form of SMCr for chemical processes. This work uses a First Order Plus Dead Time (FOPDT) model to develop the SMCr. This original point of view resulted in a very robust controller suitable for chemical processes. Later works [ 22] have shown that this controller is able to reject disturbances successfully in nonlinear processes with challenging characteristics such as inverse response. The robustness and stability shown by the SMCr designed by Camacho are desirable characteristics for any controller working in nonlinear processes. However, it
16 has been shown  that this controller has a slow response tracking set point changes in highly nonlinear processes. 1.5 Dynamic Matrix Control (DMC) Dynamic Matrix Control, as was previously expressed, is a one of the most successful and widely used Model Predictive Control strategy used in industrial applications. This technique is considered as a Model Based Control (MBC) because the prediction capability is based on the process model incorporated inside the algorithm. This control algorithm was originally developed by Cutler and Ramaker in 1979 . DMCÂ’s main characteristics are : Uses linear step response model to predict process behavior. A quadratic objective performance over a finite prediction horizon is employed. Future plant outputs are specified to follow the set point as closed as possible. Optimal outputs, to track set point, are calculated using least square method. DMC, and other MBC schemes, allows intrinsic dead time compensation because of the process model used to predict future behavior. The matrix operations, on which the DMC calculations are based, can easily be extended to any number manipulated and controlled variables. For any pair of manipulated-controlled variable, a unit step response vector is required. Each of these vectors is used to form the dynamic matrix. The individual dynamic matrices are sub matrices of the global matrix. A more extended discussion about the DMC algorithm and its implementation is presented in Chapter 3.
17 The process control literature is full of references of DMC applications in industrial applications . Nevertheless, DMC has a weak point: it is a linear controller. The entire DMC algorithm is based on the unit step vector response, which is information obtained at a given operating condition. Depending on the nonlinearities, the vector response may be different at another operating condition, resulting in performance deterioration of the controller. 1.6 Summary This chapter presents the research contribution focused in this work. Fuzzy Logic as a tool to improve the performance of two control strategies, Sliding Mode Control and Dynamic Matrix Control, is described. The effect of process nonlinearities over SISO control loops performance is also discussed. A general literature review on alternative control strategies to handle process nonlinearities is presented. Finally a general description of SMC and DMC is presented; their strength and weakness are discussed.
18 Chapter 2 Fuzzy Surface-Based Sliding Mode Control 2.1 Introduction Sliding Mode Control (SMC) is a procedure to design robust controllers for nonlinear processes. The usual approach to design SMC controllers (SMCrÂ’s) requires a model of the process, and usually the resulting controllers are complex, with many parameters to be tuned. Nevertheless, Camacho and Smith  have proved that it is possible to develop SMCrÂ’s using simple low-order models. In 1965 Professor Lotfi Zadeh introduced Fuzzy Logic, and since that time several successful applications, mostly in control, have appeared in the literature . The strength of Fuzzy Logic resides in its capacity to express in mathematical form the subjective knowledge based on experiences and analogies . Thus, Fuzzy Logic allows incorporating the Â“intelligenceÂ” and Â“experienceÂ” from experts into control strategies. The concept of linguistic variables, which is used to design the fuzzy elements in the controllers, confers the strategies a robustness and flexibility difficult to overcome . These characteristics make Fuzzy Logic a helpful tool to face difficult control problems, where conventional strategies do not work very well, or simply fail. The combination of SMC and Fuzzy Logic is not new; there are many references in the literature [43-46] . These earlier works have focused in two aspects. Some works have investigated using fuzzy rules for tuning SMCrÂ’s.
19 Other works have investigated using fuzzy rules to design the sliding surface . The SMCr has been shown robust and effici ent in rejecting disturbances   , however, this is not the case when facing set point changes, conditions under which the controller has been reported to show a very slow response . This research proposes modifying the sliding surface of a SMCr designed based on a first-order-plus-dead time (FOPDT) process model. The proposed sliding surface is composed of a modification of the conventional expression designed by Camacho and Smith  plus a term determined using fuzzy rules, based on the error and the change of the error of the controlled variable. 2.2 Sliding Mode Control SMC is a technique derived from Variable Structure Theory. The controllers designed using this technique have the capacity to handle nonlinear and time-varying systems. The SMC technique defines a surface, along which the process slides to its desired final value and a reaching function. The sliding surface, S ( t ), is a function of the order of the process model, as expressed in the equation proposed by Slotine : t 0 ) ( ) ( dt t e dt d t Sn (2.1) where e ( t ) is the error, the difference between the desired value, cset ( t ), and the actual value, c ( t ), for the variable of interest; is a tuning parameter of the surface; and n is the order of the process model.
20 Equation 2.1 shows that when the model is of high order the sliding surface S ( t ) also becomes of high order, with many parameters to tune. The works of Camacho and Smith  have demonstrated that by usi ng a First Order Plus Dead Time (FOPDT) empirical model it is possible to obtain a useful and versatile controller, with all the necessary characteristics for robustness when facing nonlinear systems. The equations for Sliding Mode Controller (SMCr) developed by Camacho and Smith are: ) ( ) ( ) (0 0 0t e t c t c K t m t mP ) ( ) ( t S t S KD (2.2) and (()) ()()Pdct StsignK dt 10 0()()tetetdt (2.3) where m ( t ) is the controller output to the final element of control; m is the controller output at the initial steady state; t0, and KP are the process dead time, process time constant and process gain respectively; KD, 0 1 and are tuning parameters of the SMCr; c is the sensor output at the initial steady state. The second term on the right side of Eq. 2.2 represents the sliding mode; this term is also called the continuous mode. This part of the controller is responsible for keeping the system at the desired value. The third term is called the reaching mode, or discontinue mode. This part of the controller is responsible for leading the system onto the sliding surface. This SMCr has shown good and robust performance when controlling several nonlinear chemical processes , including pr ocesses that have inverse response , or variable dead time .
21 However, when the system is extremely nonlinear, whereas the process parameters gain, time constant and/or dead time vary widely in the operation range, the SMCr shows a slow response in some cases or excessive overshoot in others when facing set point changes. In both cases the result is a long stabilization time for the system. It is precisely here where the symbiosis with Fuzzy Logic may be a solution. It is possible to increase the robustness and Â“intelligenceÂ” of the SMCr through fuzzy rules, making the response of the controller slower or faster when appropriate. This is the purpose of this research; to show the way to combine the best features of SMC and Fuzzy Logic to develop an efficient controller that can be used in processes difficult to control. 2.3 Fuzzy Logic in Process Control Fuzzy Logic is a relatively new technique that uses language and reasoning principles similar to the way humans solve problems. Fuzzy Logic began when Professor Lotfi Zadeh  proposed a mathematical way of looking at the intrinsic vagueness of human language. Observing that human reasoning often uses variables that are vague, Zadeh introduced the concept of linguistic variables. The values of these variables are words that describe a condition, such as High, Small, Big, Zero, Poor, Rich, Very Large, etc. These linguistic values are not single entities; they are a set of elements that have different degrees of membership in the set. This set of elements is called a fuzzy set. In conventional sets an element belongs to the set or not. In fuzzy sets, an element can belong completely to the set, belong partially to the set or not belong to the set.
22 The practical applications for this theory are multiple. In the process control field the boom started when in 1974 Mamdani controlled a steam engine using fuzzy logic . Many references can be found in the lite rature explaining the application of fuzzy logic to process control  . 2.4 Nonlinear Processes The control of highly nonlinear processes is one of the most difficult task in the process control field. To show the performan ce of the proposed controller this work uses two nonlinear process: a neutralization react or presented by Henson, Seaborg and Hall , and a modification of the mi xing process presented by Camacho and Smith . Appendixes A.1 and A.2 show mathemati cal models of both processes. Figure 2.1 shows a schematic representation of them. Figure 2.1(a) shows the neutralization pro cess. Stream q1(t), an aqueous solution of HNO3, is introduced in tank 2; the exit flow from this tank, q1e(t), is manipulated using a manual valve. Stream q2(t) is a buffer stream, an aqueous solution of NaHCO3. Stream q3(t) is a basic solution, an aqueous solution of NaOH and NaHCO3. It is assume that all the species are completely dissociated.
23 Figure 2.1. Schematic Representation of Nonlinear Processes The process neutralizes the acid stream q1(t) manipulating the flow of basic stream q3(t). The three streams are introduced to the neutralization reactor where they are assumed to be perfectly mixed. Constant density is assumed, as well as complete solubility of the ions involved. Henson and Seaborg  have compared the mathematical model and the actual process, finding excellent agreement between them. Note that in practice ratio control between the flow from tank 2 (disturbance) and the flow of stream 3 (manipulated variable) w ould probably be implemented. The AC would still be present to compensate for other disturbances. This controller would probably set the required ratio between the two flows. This ratio control with feedback compensation from the AC would provide better compensati on for flow disturbances than the scheme shown in Figure 2.1(a). However, in this research we are interested in comparing the performance provided by a PID controller, by a SMCr designed using low-order models (Camacho's), and by the controller proposed in this work.
24 Figure 2.1(b) shows the mixing process. A hot water stream F1(t) is manipulated to mix with a cold water stream F2(t) to obtain an output flow FÂ’(t) at a desired temperature TÂ’(t). The temperature transmitter is located at a distance L from the mixing tank bottom. The volume of the tank varies freely without overflowing. Figure 2.2 shows the open-loop behavior for the neutralization process. With the controller in manual mode, three step changes of the same magnitude (+5%CO) are introduced in the signal to the valve. The figure shows that although each step change has the same magnitude, the process response is different in every case. As the signal to the valve increases, as more base is added to the system, the change in the output stream pH is smaller. These results indicate the nonlinear nature of the process gain. 0 1000 2000 3000 50 55 60 65 70 75 time (sec)m(t) (%CO) 0 1000 2000 3000 50 55 60 65 70 75 time (sec)c(t) (%TO) Figure 2.2. Open Loop Response When Neut ralization Process Faces Sequencial Step Changes, With the Same Magnitude, in the Signal to the Control Valve To quantify the system nonlinear behavior, Sanjuan  proposes to calculate how KP, and t0 change when the signal m ( t ) to the valve varies as a series of successive step changes. KP is defined as the ratio of the change in the process output c ( t ), in percent of transmitter output (%TO), divided by the change in the process input m ( t ), in percent of
25 controller output (%CO). Figure 2.3 shows this relation when the signal to the control valve increases from 10% to 90%CO for neutralization process. Changes of 5%CO in m ( t ) were made to obtain the information. 0 50 100 20 40 60 80 100 to Signal to Valve (%CO)Dead Time (s) 0 50 100 0 0.5 1 1.5 2 2.5 3 3.5 Gain Signal to Valve (%CO)Process Gain (%TO/%CO) 0 50 100 100 150 200 Tau Signal to Valve (%CO)Time Constant (s) Figure 2.3. Kp and t o Variations as m ( t ) Function for Neutralization Process It is evident the highly nonlinear behavior of the process gain, its value changes from 0.2 to 3.3%TO/%CO. The presence of two maximum values, and a minimum value, in a narrow range of m ( t ) worsens the situation. A change in the valve signal around the operating point, for instance from 52%CO (s teady state value) to 60%CO, produces a large change in the process gain, from 0.7 to 3.3%TO/%CO approximately. However, if the signal to valve changes from 52%CO to 65%CO, the process gain changes slightly. This non-monotonic behavior shown by the process gain reduces the performance of linear controllers such as the conventional PID; in some cases the control loop can even become unstable. The process time constant and dead time graphics also show a highly nonlinear behavior; they also show a hysteresis-type phenomenon. The values of these process
26 parameters are different when m ( t ) increases to those obtained when m ( t ) decreases. The process time constant varies from 72 to 210 s, whereas the process dead time varies from 22 to 95 s. Figure 2.4 shows the same information for the mixing process when the signal to the valve changes from 5% to 95%. These graphs were generated using the same procedure described for the neutralization process. In this case the process shows a monotonic increment in the process gain, from 0.2 to 1.1, when the signal to the valve decrease. The time constant and the dead time also vary monotonically when m ( t ) changes; the hysteresis-type phenomenon is also present for this process. These results show the nonlinear character of the mixing process. 0 50 100 20 30 40 50 to m(t) (%CO)Dead time (s) 0 50 100 0 0.5 1 1.5 Gain m(t) (%CO)Process Gain (%TO/%CO) 0 50 100 4 6 8 Tau m(t) (%CO)time constant (s) Figure 2.4. KP and to Variations as m ( t ) Function for Mixing Process
272.5 The Surface-Based Fuzzy Sliding Mode Control The SMCr proposed by Camacho and Smith  shows good performance working as a regulatory controller for nonlinear processes. Figure 2.5 shows a comparison between SMCr and PID responses when the neutralization process faces a change of +5% in q1(t). SMCr compensates faster and with less overshoot than the PID controller. However, when the SMCr is used in servo control, it shows a slow response tracking the set point. Figure 2.6 compares the system response when the neutralization process faces a change of +10% in set point. This test was performed using the same tuning parameters used for the test shown in Figure 2.5. In this case the SMCr slowly tracks the new set point, and with a substantially larger overshoot than the one reached by the PID controller. 0 500 1000 1500 2000 6.75 6.8 6.85 6.9 6.95 7 7.05 time (sec)pH(t) pH reference SMCr PID Figure 2.5. Comparison Between SMCR and PI D When Neutralization Process Faces a Change of +5% in q1(t) as Disturbance
28 0 500 1000 1500 2000 7 7.5 8 8.5 9 time (sec)p H(t) PID SMCr pH Reference Figure 2.6. Comparison PID and SMCr When System Faces +10% Change in Set Point These simple examples show the strengths and weaknesses of the SMCr. The main goal of this research is to precisely enhance its performance when facing set point changes. To accomplish this objective we analyze the reasons why the SMCr exhibits a slow response when tracking changes in set point. Once these reasons are uncovered, we use fuzzy logic to incorporate intelligence, and knowledge based on experience. 2.5.1 The Steady State Compensator Analyzing the behavior of the variables involved in the SMCr operation it is possible to observe that the main reason for the controllerÂ’s slow response when facing set point changes is that the sliding surface S ( t ) has an inverse response behavior. Figure 2.7 shows how S ( t ) varies (solid line) when the neutralization process experiences a
29 decrease of 5% in set point. Initially S ( t ) decreases, reaching a minimum value, then starts to increase reaching its final value. The difference between the minimum value reached and the final value is 0.216 units. This behavior results in a long stabilization time for the system. 0 500 1000 1500 2000 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 time (sec)S(t) MSMC SMC Figure 2.7. S ( t ) Variation When SMCr and MSMCr Are Used Facing a 5% Set Point Change Down One way to enhance the SMCr performance is to consider changes in Eq. (2.2), to try to remove the Â“inverse responseÂ” shown by S ( t ). After extensive search, it was found that by altering the continuous mode, second term on the right side in Eq. 2.2, the inverse response disappears. This continuous term controls the way the system approaches its new final value. Eliminating the error e ( t ) in the second term of Eq. (2.2) the equation for the Modified Sliding Mode Controller (MSMCr) is:
30 ) ( ) ( ) ( ) ( t S t S K c t c K m t mD S (2.4) The continuous mode is transformed into a steady state compensator. KS is a tuning parameter equivalent to 1/ KP. The dotted line in Figure 2.7 shows how S ( t ) change when the MSMCr faces a decrease in 5% set point. The behavior is less erratic than the one shown by the SMCr; it starts to decrease smoothly and in the same direction where the final value is placed. The difference between the minimum value and the final value is now 0.04 units. Additionally the time to reach the new steady state value is 1100 s, and the one for SMCr is around 1590 s (a 30% faster time). The proposed steady state compensator introduces a stable and non oscillatory term into the controller equation. The error variation now does not affect the way in that the system Â“slipsÂ” over the surface once it has been reached. These advantages are reflected in the time needed to reach a new steady state when the system is affected by a set point change. More importantly, the system response is enhanced as shown in Figure 2.8. Figure 2.8(a) exhibits the system response when the SMCr is used as controller and Figure 2.8(b) shows the system response when MSMCr is employed. The MSMCr leads the system to the new steady state faster than the SMCr. SMCr takes 1730 s to do so, and MSMCr takes 1390 s (almost 20% faster time). It is also good to highlight the reduction in the overshoot accomplished using the MSMCr; approximately 0.05 pH units.
31 0 500 1000 1500 2000 6.55 6.6 6.65 6.7 6.75 6.8 6.85 6.9 6.95 7 7.05 time (sec)pH 0 500 1000 1500 2000 6.55 6.6 6.65 6.7 6.75 6.8 6.85 6.9 6.95 7 time (sec)pH pHset pH(t) pHset pH(t) (a) SMCr (b)MSCr Figure 2.8. Comparison Between System Response Using SMCr(a) and MSMCr(b), Facing a 5% Set Point Change Down However, the time to reach set point for the first time for the MSMCr is greater than for the SMCr, meaning that the MSMCr controller is less aggressive at the beginning. If this feature could be enhanced the performance of the MSMCr should even be better. This problem can be overcome using the other element proposed in this research, Fuzzy Logic. 2.5.2 Fuzzy Rules to Develop a Fuzzy Sliding Surface Fuzzy Logic confers the controller enough intelligence to react quickly and aggressively, or slowly and smoothly, when necessary. The previous section showed how any change that contributes to S ( t ) reaching quickly its final value is reflected in a faster
32 and less erratic process response. Therefore, the ideal place to introduce the fuzzy element is in S ( t ). This research proposes to add a fuzzy element, FS to the original surface equation to enhance the controller performance. The proposed fuzzy element has a discrete character, in other words, its value is a function of the sample time used to calculate it. For this reason the expression for the sliding-fuzzy surface is written in terms of discrete n as: ()()((),())HFSnSnSenen or 10 0(()) ()()()(,)((),())t HPFdcn SnsignKenentdtSenen dt (2.5) where SH( n ) is the sliding-fuzzy surface. Equation (2.5) expresses that the surface is a combination of two terms, the classical expression for SMCr, plus a term whose value is determined by means of fuzzy rules. This new term is a function of the error, e ( n ), and the variation of the error, () en of the controlled variable. The error is defined as ()() encsetcn in %TO units; and the change in error as ()(1)() enenen where n is the present value, and n -1 is the previous value. Figures 2.9 and 2.10 show the calculation of FS The equation for the Fuzzy-Sliding Mode Controller (FSMCr) is then written as: () ()() ()H SD HSn mnmKcncK Sn (2.6) The addition of FS to the controller equation adds the intelligence and robustness desired. The e ( n ) and () en inputs to the FS are scaled to a value between
33 Â–1 and 1, using the scaling factor FKe and FKDe (see Figure 2.10), which are tuning parameters for the FSMCr. These values allow assigning more or less relative weight to the FSMCr inputs, given more or less importance to the error or to the change in the error depending on the particular process. There are not tuning equation availables to calculated FKe and FKDe values, this matter could be object of further research. Each of these scaled crisp values is then translated into fuzzy variables, fuzzification, (see Figure 2.10) using the fo llowing five membership functions: Negative Big (NB), Negative Small (NS), Zero (Z), Positive Small (PS) and Positive Big (PB). The shapes of membership functions used in this work are the most often used in the literature: triangular and trapezoidal . NB and PB are trapezoidal membership functions, while the rest are triangular (see Figure 2.9). The fuzzy variables are then evaluated in the fuzzy rules (MamdaniÂ’s inference system) to determineFS ; Table 2.1 shows the fuzzy rules. The output fuzzy variable is then translated back into a crisp value, defuzzification, using the same membership functions. The centroid calculation is used in all three membership functions (2 inputs and 1 output). -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 errorDegree of membershipNBNSZPSPB Figure 2.9. Example of Membership Functions Used to Perform Fuzzification and Defuzzyfication
34 NB NSZPSPB NB NSZPSPB NB NSZPSPB Fuzzy Rules-11 1 -1 -1 1 FKe e(n) FKDe e(n) KFS SF(n) Scaling Scaling Scaling FuzzificationFuzzy inferenceDefuzzification NB NSZPSPB NB NSZPSPB NB NSZPSPB NB NSZPSPB NB NSZPSPB NB NSZPSPB Fuzzy Rules-11 1 -1 -1 1 FKe FKe e(n) FKDe FKDe e(n) KFS KFS SF(n) Scaling Scaling Scaling FuzzificationFuzzy inferenceDefuzzification Figure 2.10. Fuzzy Inference System Used to DetermineFS It is possible to analyze the physical meaning of the fuzzy rules in Table 2.1. Using the neutralization reaction as example, assume that e ( n ) is NB, and () en is also NB. If e ( n ) is negative it means that the pH is greater than the desired value; if it is NB it means that the pH is much greater than the desire value. If () en is also negative it means that the present error is greater then th e past error; if it is NB it means that the present error is much greater than the past error. Thus, the pH is above the desired value and the trend indicates that it continues rising very fast; Figure 2.11 shows this condition. To solve this situation, the controller should close the valve to reduce the base flow and consequently, the pH. Because the trend is fast, the controller should act aggressively, meaning that the valve should close by a sizeable amount. Thus, the decision from the
35 rules should be NB as shown in Table 2.1. All other rules were obtained using similar reasoning. Because the control valve is fail closed, the signal to the valve must reduce to close the valve. Equation (2.6) shows that in order to reduce the signal to the valve, ()HSn should decrease. Eq. (2.5) shows that to reduce()HSn FS should reduce. The FS value found is defuzzyfied and multiplied by a scaling factor, KFS before used in Eq. (2.5). This scaling factor is another FSMCr tuning parameter; its function is to transfer the defuzzyfied value, FS into a scale comparable to the value of S ( n ). If KFS is not properly chosen the effect of FS over the overall S ( n ) could be too strong, affecting the controller performance. Table 2.1. Fuzzy Rules to Obtain FS Sets e(n) e(n) NB NS Z PS PB NB NB NB NB NB NS NS NB NS NS Z PS Z NS NS Z PS PS PS NS Z PS PS PB PB PS PB PB PB PB
36 e(2) e (-) pH(t) time pH Reference e(n) = pH Reference Â–pH(t) e(2) = e(1) Â–e(2) e(n) = e(n-1) Â–e(n) e(1) e(2) e (-) pH(t) time pH Reference e(n) = pH Reference Â–pH(t) e(2) = e(1) Â–e(2) e(n) = e(n-1) Â–e(n) e(1) Figure 2.11. Schematic Representation of e ( n ) NB and () en NB in Physical Terms 2.6 Simulation Results This section presents the control performance when the Fuzzy Sliding Mode Controller (FSMCr) is used to control the neutralization reactor and the mixing process. The results of CamachoÂ’s Sliding Mode C ontroller (SMCr), a standard Fuzzy Logic Controller (FLCr) from the literature , and a PID controller are also presented for the neutralization process. Although tuning equa tions are available for the PID controller and for the SMCr, the tuning parameters for these two controllers were optimized to minimize the Integral of the Absolute value of the Error (IAE); the tuning parameters for the FSMCr and for the FLCr were also optimized using the same criterion.
37 2.6.1 FS Behavior Figure 2.12 shows the S ( t ), FS and SH( t ) behavior when an increase of 10% in acid flow enters the process, and the FSMCr controls the process. The figure shows how quickly FS increases during the first instants, it remains constant for a while, and finally it decreases first quickly and smoothly afterwards until it reaches zero again. The explanation for this behavior is clear, when the disturbance affects the system, e ( n ) and () en start to increase rapidly the fuzzy rules infer that FS should act fast and aggressively. When the error and its difference begin to decrease, the fuzzy contribution is more passive. At the moment that the fuzzy rules infer that the error stars to decrease, FS decreases gradually. Thus, the addition of th e fuzzy rules results in a surface that is aggressive and fast when it is necessary, or smooth and slow when the system requires it.
38 0 200 400 600 800 1000 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 time (sec)S(t) SH(t) S(t) SF(t) Figure 2.12. Behavior of S ( t ), FS and SH( t ) Behavior When the System Faces a Disturbance 2.6.2 Performance Comparison 220.127.116.11 Neutralization Process To evaluate the performance of the four controllers, the process was exposed to a series of 5% step changes in acid flow. Th e tuning parameters for the PID were found to be: KC = 0.1 %CO/%TO, I = 40s and D = 10s. The tuning parameters for the SMCr were found to be: 0 = 0.00062, 1 =0 .04996, KD = 400 and = 68.75. FLCr tuning parameters were found to be: Ke = 0.08, KDe =1.7 and Km = 0.3; these tuning parameters were calculated using equation available in literature . The tuning parameters used
39 for the FSMCr were 0 = 0.00065, 1 =0 .047, KD=380, 65.20, FKe =0.5, KFDe = 0.5 and KFS =0.05. The process response, the total error as given by the Integral of the Absolute value of the Error (IAE), the overshoot and the time to reach steady state are used as criteria to compare the performance. Figure 2.13 shows the acid flow change and the process response. The figure shows that the FSMCr provides a faster response with less overshoot. Table 2.2 shows the IAE values and maximum overshoot percentage for these tests. Table 2.2. Results Comparison for Disturbance Test Controller IAE Maximum overshoot (%TO) SMCr 6197 3.0% FLCr 4814 4.2% FSMCr 2136 2.2% PID 9278 5.1%
40 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 104 14 16 18 20 time (sec)q1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 104 6.8 7 7.2 7.4 time (sec)pH 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 104 6.8 7 7.2 7.4 time (sec)pH 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 104 6.8 7 7.2 7.4 time (sec)pH 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 104 6.5 7 7.5 time (sec)pH PID SP FSMCr SP FLCr SP SMCr SP Figure 2.13. Performance Comparison SMCr, FLCr, FSMCr and PID, When a Sequence of q1(t) Changes Affects the System The process was also exposed to a series of set point changes. Figure 2.14 shows the changes and the process response. The FSMCr again shows the best performance. Table 2.3 compares the numerical values obtained from these tests. Once more, the FSMCr shows shorter time to reach steady state.
41 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 104 6 6.5 7 7.5 8 8.5 time (sec)pH 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 104 6 6.5 7 7.5 8 8.5 time (sec)pH 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 104 6 6.5 7 7.5 8 8.5 time (sec)pH 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 104 6 6.5 7 7.5 8 8.5 time (sec)pH SP PID FSMCr SP FLCr SP SMCr SP Figure 2.14. Performance Comparison among SMCr, FLCr, FSMCr and PID When System Faces a Set Point Change Sequence
42 Table 2.3. Results Comparison for Set Point Changes Test Controller IAE Maximum overshoot (%TO) SMCr 5411 6.7% FLCr 6966 0.5% FSMCr 2541 2.0% PID 8207 1.2% Finally, the process was exposed to a test that consists first in a 10% reduction in set point, a severe reduction (38%) in flow q1(t), and finally an additional 15% reduction in set point. The same tuning parameters used in the previous tests were used. Figure 2.15 shows the results obtained. The FSMCr shows again a better performance facing both set point changes and the strong disturbance. The FLCr shows an oscillatory behavior when the last set point change affects the system. The PID controller shows a high overshoot when the disturbance affects the process. This can be attributed to the conservative set of tuning parameter used to keep stable the PID in the tests described in Figures 2.13 and 2.14. However, if the PID tuning parameters are readjusted to perform a fast response to the disturbance, when the second set point change affects the process, the control loop becomes unstable as it is shown in Figure 2.16.
43 0 1000 2000 3000 4000 5000 6000 10 15 20 time (sec)q1 0 1000 2000 3000 4000 5000 6000 4 6 8 10 time (sec)pH 0 1000 2000 3000 4000 5000 6000 4 6 8 10 time (sec)pH 0 1000 2000 3000 4000 5000 6000 4 6 8 10 time (sec)pH 0 1000 2000 3000 4000 5000 6000 4 6 8 10 time (sec)pH FLCr SP PID SP FSMCr SP SMCr SP Figure 2.15. Performance Comparison When SMCr, FLCr, FSMCr and PID Face a Disturbance and Set Point Changes Simultaneously
44 0 1000 2000 3000 4000 5000 6000 4 5 6 7 8 9 10 time ( sec ) pH(t) SP PID Figure 2.16. PID Response When Tuning Parameters Are Adjusted to Enhance the Response to Disturbance Change Figure 2.17 shows the controller performance when changes in the acid stream concentration affect the process as disturbance. At time 500 and 5000 s the acid stream concentration increases 20% consecutively; at time 6500 s the acid concentration decreases 20%. Additionally, the set point is increased 20% at time 2500 s. Figure 2.17 shows the FSMCr tracking set point faster and rejecting concentration disturbance effectively, keeping the neutralization proce ss at set point. FLCr and SMCr show slow responses compare with FSMCr. PID shows oscillatory behavior when reject the disturbances.
45 0 1000 2000 3000 4000 5000 6000 7000 8000 6 8 10 pH(t) 0 1000 2000 3000 4000 5000 6000 7000 8000 6 8 10 time (s)pH(t) 0 1000 2000 3000 4000 5000 6000 7000 8000 6 8 10 time s)pH(t) 0 1000 2000 3000 4000 5000 6000 7000 8000 6 8 10 time (sec)pH(t) SP SMCr SP FLCr SP FSMCr SP PID Figure 2.17. Controllers Performance Comparison to Set Point Change and Disturbance in Acid Stream Concentration (pH Process) 18.104.22.168 Mixing Process Figure 2.18 shows the performance comparison when SMCr and FSMCr are used to control the mixing process described in Section 2.4 and Apendix A.2. The tuning parameters used for SMCr in these test were found to be: 0 = 0 .01407, 1 = 0.237, KD = 410 and = 78.42. The tuning parameters used for the FSMCr were: 0 = 0.03, 1 = 0.9, KD = 42, = 67.5, FKe =0.7, KFDe =0.7 and KFS =0.06.
46 Figure 2.18(a) shows the system behavior when the set point is increased 10oC (12.5%TO) and later (at 400 s) the cold temperature T2(t) is increased 10oC. The SMCr track slowly the set point change until finally reach the new steady state, but when the disturbance affects the process, the SMCr becomes oscillatory. Under the same test the FSMCr reaches faster the new set point and faces successfully the disturbance. 0 200 400 600 800 40 45 50 55 60 c(t) %TO(a) 0 200 400 600 800 0 50 100 c(t) %TO(b) 0 200 400 600 800 40 45 50 55 60 time (sec)c(t) %TO 0 200 400 600 800 0 50 100 time (sec)c(t) %TO SP FSMCr SP FSMCr SP SMCr SP SMCr Figure 2.18. FSMCr Performance When is Used Controlling the Mixing Process Figure 2.18(b) shows SMCr and FSMCr be havior for a set point change of -10oC, and a disturbance in T2(t) of +10oC. In this case, SMCr is unable to track the set point change and the system becomes oscillatory. The FSMCr can track the set point and compensate for the disturbance without major problems. These results show that FSMCr is a general controller able of handle highly nonlinear processes, even when the system has a marked dead time like the mixing process.
47 2.6.3 Sampling Time Effect Figure 2.19 shows how the FSMCr performance is affected when the sampling time is increased; the same test used in Figure 2.15 for the neutralization process is used for this purpose. The sampling time used in the previous results was equal to the integration time of the simulation. Therefore, the controllers behave as continuous controllers. The sampling time used for each case in Figure 2.19, is a fraction of the smaller process time constant shown by the process, around 70 s, see Figure 2.3. The curve labeled as not sampled is obtained sampling the signal from the sensor once for every integration step. Figure 2.19 shows that the process response becomes oscillatory when the sampling time is 0.5 The responses obtained with faster sampling time are not very different. 0 1000 2000 3000 4000 5000 6000 2 3 4 5 6 7 8 9 time (sec)pH(t) 3000 3200 3400 3600 3800 6.5 7 7.5 8 8.5 SP Not sampled Sampled 0.25 Sampled 0.5 Figure 2.19. Sampling Effect on FSMCr Performance
48 2.6.4 Effect of Noise Figure 2.20 shows how the presence of noise in the feed flow affects the performance of FSMCr. The test used to show the noise effect is the same described in Figure 2.15. Noise with an ARMA (1,1) structure was used in this test: 1111 nnnn Z Zaa (10.6 10.3 20.08a ). The FSMCr tracks the sequence of set point changes and also rejects the disturbance. This result is not unexpected because SMCr is a controller with chattering avoidance; therefore, FSMCr inherits the same characteristic. 0 1000 2000 3000 4000 5000 6000 7000 8000 8 10 12 14 16 18 time (sec)q1(t) ml/s 0 1000 2000 3000 4000 5000 6000 7000 8000 4 5 6 7 8 time (sec)pH(t) and pHset SP FSMCr Figure 2.20. Noise Effect on FSMCr Performance
492.7 Conclusion The proposed controller combining conventional Sliding Mode Control with Fuzzy Logic is efficient and robust in handling highly nonlinear processes. It significantly improves the performance of SMCrÂ’s, designed using low-order models, in servo control applications. The symbiosis between both techniques confers the FSMCr the necessary intelligence to adapt to process conditions. The presence of noise does not have a significant effect over FSMCr performa nce. Because there are not tuning equation availables to calculated FKe, FKDe and KFS values, this matter could be object of further research.
50 Chapter 3 Dynamic Matrix Control and Fuzzy Logic 3.1 Introduction Model Based Control (MBC) techniques make use of process models to obtain a control signal by minimizing an objective function. The basic idea is: Explicit use of a model to predict process outputs at future times (horizon). Calculation of a control sequence minimizing an objective function. Receding strategy, each instant the horizon is displaced towards the future. MBC has a series of advantages: It can be used to control a great variety of processes. It is not difficult to implement the multivariable case. It intrinsically has dead time compensation . The first MBC works appeared at the end of the 1970Â’s. At that time there was a timid interest in industry, however; MBC strategies are now popular in industrial applications. Dynamic Matrix Control (DMC) is an MBC technique developed by Cutler and Ramaker in the seventies . There are several other commercial strategies similar to the DMC design such as: Identification and Command (IDCOM) . Model Algorithmic Control (MAC) .
51 Internal Model Control (IMC) . Predictor Based Self Tuning . Extended Horizon Adaptive Control (EHAC)  Extended Predictive Self-Adaptiv e Control (EPSAC) , . Generalized Predictive Control (GPC) [68-70]. Multistep Multivariable Adaptive Control (MUSMAR) [71-73]. The difference among these algorithms is the model used to predict the system behavior and the function to be minimized. 3.2 Conventional DMC Implementation There are many ways to describe the DMC algorithm; for purpose of this research, the implementation suggested by Sanjuan  will be used. The following discussion describes the DMC implementati on for Single-Input-Single-Output (SISO) systems. Initially, the process is identified using a step change in the signal to the valve input, m ; sampling the sensor signal provides the process response. This data can be expressed as the vector S: 1 1 2 3 1S .f f SSc c c c c c (3.1)
52 where 1c is the steady state signal from the sensor before the step affects the process. fc is the sensor signal when the process reaches the new steady state. How often samples are taken is a designer decision. However, a sampling time between one tenth and one fifth the process time constant is usually recommended, resulting in a Sampling Size (SS) between 25 and 50 samples. The length of the sample vector is directly related to the Prediction Horizon (PH), which is the number of future steps where the process output variable will be predicted. Subtracting 1c from each element of the vector S provides the deviation matrix Sv: 10 0 S .v SSC C (3.2) and dividing Sv by m provides Av: 10 A .v SSKp (3.3) The process gain Kp is the last element of vector Av. For the DMC algorithm it is necessary to define the matrix A, which is referred to as the dynamic process matrix:
53 .0 A0. A A. . .A .v v v P HCHKp Kp Kp KpKp (3.4) The number of columns of A corresponds to the number of control moves that will be calculated every iteration; this number is called Control Horizon ( CH ). The selection of CH must be done considering that the larger the control horizon is, the slower the controller (less aggressive) will be and vice versa. If CH is too large, the controller will delegate most compensation action for steps into the future. If CH is too small, the controller becomes aggressive because it will try to compensate for the entire error in the first few iterations. For industrial applications usually a Control Horizon between 5 and 10 is used . The relation between Prediction Horizon ( PH ) and Control Horizon ( CH ) is expressed as: 1PHSSCH (3.5) Therefore, once the designer chooses the CH the PH can be determined. The equation to calculate the control moves is based on the fact that the controller output must compensate for present and future errors: AME P (3.6) 1 1 2 2 1 1.0 A0. A .. . .. . .A .v v vCH PH PH PHCHCHm e m e Kp Kp Kp m e KpKp
54 where M is the control output vector and EP is the predicted error vector. The goal is to determine M, to eliminate the error EP. Because PH is greater than CH there are more equations than unknowns, and an exact solution can not be obtained. For that reason an optimization model must be used: AMERP (3.7) where R is the residual for non-compensated erro rs. The goal of the optimization is to find M to minimize the sum of square of the residual, RTR. To find the appropriate controller move, it is necessary to take the derivative with respect to M and set the resulting matrix expression to zero: RR 0 MT (3.8) The solution for the minimization problem is : AAMAETT (3.9) The controller output can be calculated from Eq. (3.9) as: 1M(AA)AETT (3.10) Appendix B shows a mathematical development to obtain Eq. (3.10). Usually a suppression factor is used as a tuning parameter to change the aggressiveness of the controller; in that case the control move is expressed as: 21M(AAI)AETT (3.11) where is the suppression factor and I is the iden tity matrix. Eq. (3.11) is the control law proposed for Cutler and Ramaker for DMC.
55 3.3 Nonlinear DMC in the Literature DMC was conceived as control strategy to work with linear systems, or with processes with slight deviation from linear behavior. Unfortunately, industrial processes are complex with nonlinear characteristics. This fact limits the use of DMC to those where linear behavior is held, or do not move far away from their original operating condition. Several articles report that when DMC is used in nonlinear processes the performance goes from very slow responses to oscillatory responses  [75-77]. For this reason, several approaches have been proposed to modify the DMC algorithm to improve its performance with nonlinear processes. McDonald and McAvoy  proposed a gain and time scheduling technique to update the DMC algorithm and enhance its control performance. These authors used a moderate and high purity distillation column to show their strategy. The applicability of this approach has some limitations; the computational resources required to perform online process parameter evaluation and updating, as well as the non-general character of the solution are some of them. Georgiou, et. al.  proposed to use a variable transformation to derive a Nonlinear DMC (NDMC). The NDMC is also a controller less complex than gain and time scheduling DMC, or adaptive DMC. NDMC is focused in model modifications instead of a DMC algorithm reformulation. Brengel and Seider  presented a MIMO control scheme based in a multi-step predictor that can be extended to IMC and DMC techniques. The algorithm is based on a linearization of ordinary differential equations several times during a sampling interval;
56 recursive algebraic equation are derived in this way to relate predicted outputs to future and past of manipulated inputs. The algorithm is able to handle constraints in state variables and manipulated variables. Chang, et. al.  proposed an average convolution model to improve DMCÂ’s performance. These authors showed a comparison among conventional DMC, GeorgiouÂ’s NDMC and their design; their approach shows less oscillatory behavior than the others, when used in a high purity distillation. Peterson, et. al.  proposed another nonlinear DMC by modifying the conventional algorithm to include a disturbance vector to take into account the effect of nonlinearities in the prediction horizon. This work uses an iterative method to determine a time varying disturbance vector that captures the future disturbance thus, updating the nonlinear model. While the algorithm is iterating, the conventional DMC with linear model is working as controller strategy. When the disturbance vector calculation converges, the nonlinear model is used in the DMC scheme. In 1991 Bequette  published a major review of nonlinear control techniques for chemical process, including predictive control. Bequette indicated that the major disadvantages of nonlinear predictive techniques are: convergence and robustness in real applications. Computational time is also mentioned as an issue when the process has fast dynamics. This author highlights that the successful implementation of nonlinear predictive strategies is strongly influenced by the correct determination of initial condition for the state variables. In 1998 Henson published another review about nonlinear model predictive control techniques . He analyzed the advantages and disadvantages of all nonlinear
57 MBC techniques available at that time. He found that the major practical challenge in the implementation of nonlinear MBC is the online solution of nonlinear optimization. Some of the approaches that have been used for this optimization are Successive linearization of model equations using the Jacobian linearization about the operating point resulting in a linear model. Quadratic optimization technique is used to find the controller outputs. This approach only provides indirect compensation for process nonlinearities. Sequential model solution and optimization. A nonlinear algorithm computes the manipulated variable value, while an ordinary differential equation solver is used to integrate the nonlinear model equations. Finite elements and orthogonal collocation methods have been used to solve in discrete terms, the differential equations. Simultaneous model solution and optimization. This method requires a discretization of the model equations. The decision variables are the inputs on each finite element and the state variable at each collocation point. Alternative Nonlinear Model Based Control (NMBC) formulations. Some other approaches have been used in NMBC. Polynomial ARMAX models have been used, which allow solution of global optimum. Transformations of nonlinear problems into linear ones have also been used; the transformation is performed using a feedback linearization control law. Recently Aufderheide and Bequette  proposed a modification of the DMC using a multiple model structure. One model is based on step response as is used in the standard DMC; while other the model is First Order Plus Dead Time (FOPDT). The basic
58 idea behind this approach is to have a bank of linear models to describe the dynamic behavior on a wide operating range. A Bayesi an scheme assigns weights to each model to find a combined model which is used to predict the optimal control movement. These authors found good results when applying this approach to the control of the Van de Vusse reaction. This section has shown some of the most important references available in the literature to adapt DMC and MBC strategies, to work with nonlinear processes. The above review only covers regular approaches, not those based on Artificial Intelligence, and specifically Fuzzy Logic strategy. The next section describes the main works combining Fuzzy Logic and MBC. 3.4 Fuzzy Logic and MBC The idea of incorporating Fuzzy Logic into MBC techniques is not new, Oliveira and Lemos  presented a comparison between some fuzzy MBC control strategies. They used a fuzzy process model to predict the process behavior. They also used a new parameter estimation algorithm in their work. The use of fuzzy model in combination with MBC has been an area of research for some years now. It has been used in many different applications: chemical industries ,      , robotic s  , and nuclear plants . Some authors have used the Takagi-Sugeno method to obtain the fuzzy model, and to combine it with the MBC techniques . Ben Ghalia  has presented some different alternatives to enhance the stability and performance of fuzzy-MBC strategies based on a new defuzzification approach.
59 De Neyer, et. al. , have proposed a special combination of fuzzy and MBC strategies where they implement a fuzzy integral action into the controller. As well as others authors  , they have used some fuzzy strategies to reject measurable disturbances. Melin and Castillo  have presented their results using a combination of neural networks, fuzzy logic and fractal theory as a new approach to control nonlinear plants; they applied their approach to a biochemical reactor. Abonyi, et. al.  have studied the use of a new linearization technique for product-sum crisp-type fuzzy model. They have found that this approach works very well when applied to a highly non-linear process such as a neutralization reactor. The works mentioned represent the main and more recent trends in Fuzzy Model Based Control. All of them take advantage of the best of both techniques to handle nonlinear process with dead time. The next section presents a proposal for a new controller based on a combination of Fuzzy Logic and DMC. 3.5 A New Approach for DMC Structure Section 3.2 presented the conventional implementation of the DMC control algorithm, showing that all the characteristic information of the process, i.e. process gain, time constant and dead time, are embedded inside matrix A (see Equations 3.1 to 3.4). This characteristic is the main weakness of the DMC strategy, its response is based on a fixed linear model stored in matrix A.
60 A more convenient way to express the DMC control algorithm would be in a parametric form, where Eq. 3.11 be expressed as a function of the process parameters, gain, time constant, dead time, and the suppression factor: M(,,,)E fKpto (3.12) This parametric form of DMC algorithm has the advantage of isolating the effect of every process parameter in the control law. The DMC algorithm can be then adjusted to take into account changes in the process gain, and/or process time constant, and/or dead time. The complete expression for Eq. 3.12 is developed analytically for SISO systems in Section 3.5.1. Equation 3.12 expresses that, additionally to the process parameters; the control law is also a function of the suppression factor The purpose of the suppression factor is to regulate the aggressiveness of the controller; it is a tuning parameter of the DMC algorithm. Section 3.5.2 describes a procedure to find the tuning equation for the suppression factor as a function of the process parameters, Kp, and to. An important part of the proposed controller is a supervisor system to determine changes in process parameters. The supervisor system determines if any of the process parameters have significantly change from the original values; if so, it decides if it is necessary to adjust the DMC parameters of Eq. 3.12. The supervisor system works online with the close loop system. The modeling error is used as the factor to determine changes in the process parameters. Analysis of variance (ANOVA) and regression techniques are used to find nonlinear equations which relate modeling error and process parameters. The nonlinear equations are solved simultaneously using optimization methods to determine process parameters changes. The supervisor system incorporates
61 Fuzzy Logic to adjust the process parameters changes previously calculated by optimization. Finally the parametric DMC is adjusted to incorporate the process parameters changes detected by the Fuzzy supervisor. Section 3.5.3 describes in detail this procedure. Section 3.5.4 shows the simulation results when the proposed controller is used controlling some nonlinear processes. 3.5.1 Parametric DMC As previously mentioned, the main weakness of the DMC controller is that its response is based on a fixed linear process model, which is stored in matrix A (see Eq. 3.4). The parametric structure of DMC proposed is designed to include variable terms whose values are changed as necessary adapting to variation in the process behavior. This can be performed without changing the essential characteristics of the DMC. The proposed modification is as follows. In Eq. 3.3 the process gain Kp can be factored out as a common term, then Av is expressed as: 1100 .. AV .. .. 1v SSSSKpKp Kp (3.13) where V is a vector whose elements are between 0 and 1. These elements contain dynamic information (time constant and dead time) of the process.
62 Eq. 3.4 can now be expressed using the expression for Av as: .0 A0.V0..0 A1V. A.U ..1. .1.. .A .11..Vv v v PHCH PHCHKp KpKp Kp Kp KpKp (3.14) Using this definition of A and the properties of matrices, the controller output can be expressed as: 1 11 M(AA)AEUUUETTTTKp (3.15) When the DMC algorithm includes the suppression factor, the expression becomes: 1 21 MUUIUETTKp (3.16) This way to express the DMC algorithm allows manipulating the parameter 1/ Kp to adjust the controller according to the nonlinear behavior of the process gain. The foregoing discussion is focused in some modifications to the DMC algorithm to compensate for changes in the process gain, the question now is: what about changes in process dead time and process time constant? One way to adjust DMC for changes in process dead time ( to ) and process time constant () arises when Eq. 3.3 is studied. As it was mentioned before, vector Av is
63 composed of a set of values obtained from the process response to a step change in the valve signal at open loop. It contains all the dynamic behavior of the process ( to and ) as well as the static process gain. Figure 3.1 shows that the data recorded to build vector Av is taken from the process using a sampling period Ts, which in this case is 0.1 Because of the discrete nature of the data inside Av, it is possible to express the dead time in terms of the sampling time: s tonT (3.17) where n represents how many sampling periods correspond to the dead time. Therefore, vector V in Eq. 3.12 can be written as: 0 10 20 30 40 -4 -2 0 2 4 Sample kAvProcess dead time Reaction curve Figure 3.1. Discrete Data Contended Inside Vector Av
64 1 ()1 1 10 Z 0 AV D : 1n v SSn SS SSu KpKpKp u (3.18) where vector Z u represents a vector which elements are zero. The Z u dimensions, 1 n is directly related to the process dead time as Eq. 3.17 expresses. Vector D u is composed by the remaining no null elements in vector V. Using Eq. 3.18, it is possible to rewrite Eq. 3.14 as: V00..0Z00000 1V0...DZ0... A .1V...1D.... .......1...Z 111..V11111D P HCHPHCHu uu KpKp u u u (3.19) ()Z AU DnCH PHnCH P HCHKpKp (3.20) where Z is a matrix composed by Z u vectors, one in each Z column; as many as CH has been defined for the designer. D is a matrix composed for the remaining elements in matrix U. Expressing matrix U as indicated in Eq. 3.20, it is possible to isolate the effect of process dead time on DMC algorithm. If Eq. 3.15 is considered, the terms which matrix U is involved can be factorized as:
65 111 MUUUEKETT upsKpKp (3.21) where: 1K(UU)UTT ups (3.22) If matrix U is expressed in terms of Z and D, as is indicated in Eq. 3.20, and substituting in Eq 3.22: ()1 1 () ()Z K(UU)UZDZD DCHPHnnCH T TTTTT upsCHnCHnCHPHn PHnCH (3.23) Performing the multiplication inside the inverse operation: 1 ()KZZDDZDTTTT upsCHCHCHCHCHnCHPHn (3.24) Because all terms of square matrix ZTZ are zero, adding the two matrixes inside the inverse operation results in just one matrix: 1 ()KDDZDTTT upsCHnCHPHn CHCH (3.25) Multiplying both matrixes, Eq. 3.25 becomes: ()1KZDDDZKDPS CHPHnTT upsCHnCHn CHPH (3.26) note that ()1KDDDCHPHnTT DPS
66 Eq. 3.26 allows introducing compensation for changes in dead time into the DMC algorithm; if the dead time changes it is only necessary to modify the number of columns (n) of matrix Z to adjust the process dead time effect. Eq. 3.26 was developed using a null suppression factor However, for practical applications it is necessary to include the suppression factor as a tuning parameter. Following a similar development to that used to obtain Eq. 3.26, it is possible to find the following expression: 1 2KZDDIDTT upsCHnP CHPH (3.27) where the suppression factor can be calculated as: PKp (3.28) Eq. 3.27 allows adjusting DMC algorithm for changes in dead time and also includes the suppression factor to manipulate the aggressiveness of the controller. The next step is to find how changes in process time constant can be compensated in the DMC algorithm. Eq. 3.20 shows that matrix D contain the process dynamic information after the dead time. Therefore, the process time constant is inside matrix D. Because the dynamic behavior of real processes can be very complex, there is not an easy way of isolating the information concerning the process time constant, as it was previously done for the process gain and dead time. For this reason a nonlinear correction to adjust changes in the process time constant is used: 1 DD 1i ikTs new kTs preve adj e (3.29)
67 where: ik is the i-th term in D row. Ts is the sampling time used to record process data. new is the new process time constant p rev is the previous process time constant. The correction factor is a ratio of tw o exponential terms with the typical FOPDT form; the numerator is function of the new process time constant, whereas the denominator is function of the previous process time constant value. The exponential for the time constant correction was chosen based on the good agreement obtained when empirical models with this form are used to fit real process dynamic behavior. It is common practice to use a FOPDT model to determine the processes characteristic parameters. The idea is to adjust each term of the vector D proportionally to its position, and thus, adjusting the dynamic information of the process contained in vector D. Every i-th term in a D column is multiplied by the corresponding i-th correction factor in order to adjust that particular term by the adequate amount. Although the proposed time constant correction may not be a perfect way to adjust process time constant changes, the results shown in the following sections confirm that works very well for practical purposes. The process time constant correction completes the development of the Parametric DMC controller (PDMCr) proposed in this research. The control law of this new controller is then expressed as:
68 1 2 ()1 MZDDIDETT CHnP CHPHnadjadjadj Kp (3.30) Eq. 3.30 expresses that the PDMCr, as well as the standard DMC, needs a tuning parameter; the suppression factor P .This tuning parameter is a function of the characteristic process parameters Section 3.5.2 shows the development of tuning equations to calculate P 3.5.2 Tuning Equation for Suppression Factor Although there are some references    that propose equations to determine the adequate values of the suppression factor, it is common industrial practice to use a trial and error procedure to choose the value . The tuning equations proposed in the literature were tested using nonlinear processes finding tuning values that generate very aggressive behavior on the standard DMCr. Therefore, it is necessary to develop a set of more reliable equations to determine values, depending of the process characteristics. To reach this goal a factorial experiment was designed and an analysis of variance was performed to determine the variables that have a significant influence on the optimal suppression factor. The experiment consisted in the use of FOPDT systems, as process in a SISO control loop, and determined using constrained optimization the best value to minimize a cost function.
69 A total of 35 simulations were performed for the experiment, corresponding to the 243 possible combinations of values chosen for the study. No replicates were necessary because this experiment is a deterministic computational test where repetitions of factor levels provide the same result every time. Table 3.1 shows the three levels used in the factorial experiment for each factor. Table 3.1. Factors Used to Perform the Designed Experiments Level KP /to /Ts Low 0.5 1 0.2 0.05 2 Medium 1.5 3 0.6 0.1 5 High 2.5 5 1 0.15 8 The factors considered for the experiment where KP, /to ,/Ts (ratio of sampling time divide by the time constant) and is a weighted parameter used in the cost function. The cost function used was defined using a combination of the Integral of the Absolute Value of the Error (IAE) and the Integral of the Absolute Value of the Change in Manipulated Valve signal (IMV). This cost function or Performance Parameter (PP) is expressed as: 0 0()()ssPPetdtmmtdt (3.31) where mss is the signal to the valve at final steady state condition. The optimal suppression factor for each experiment condition is defined as the value which minimizes Eq. 3.31. This cost function was selected after some attempts of
70 using only IAE as performance parameter; for many experiment conditions the minimum IAE resulted in a non desirable oscillatory behavior. Adding the IMV, the suppression factors selected as optimal minimize oscillations in the response. To run the experiment and determine the optimal suppression factor for each condition, a Matlab program was developed using the Optimization and Statistical toolboxes, available in Matlab Release 6.5. The experimental conditions described in Table 3.1 were evaluated in a control loop with a FOPDT model as process, implemented in Simulink. The Matlab code called the Simulink model to evaluate the Performance Parameter. Figure 3.2 shows the control loop block diagram previously mentioned. 1 1 s 1tos PKe s + + M(s) C(s) D(s) Process Disturbance DMCr Cset(s) + Figure 3.2. Control Loop Block Diagram Used to Evaluate Experimental Conditions and Performance Parameter
71 For every experiment condition, a set point change equal to +10%TO at time 10 s was introduced. Later, at time 40 s a disturbance of +10%TO was introduced into the process. Figure 3.3 shows an example of control loop response. These two changes were made for the purpose of finding optimal values of useful for set point changes and disturbances affecting the process. 0 10 20 30 40 50 60 70 80 55 60 65 70 75 80 time (sec)cset(t) and c(t) 0 10 20 30 40 50 60 70 80 -5 0 5 10 15 time(sec)Disturb %TO c(t) cset Figure 3.3. Example of Set Point Change and Disturbance Used With FOPDT to Find Optimal Suppression Factor for DMCr Once the complete set of optimal values of the suppression factor was found, an analysis of variance (ANOVA) was performed. The ANOVA allows determining the most significant factors for the optimal t uning. Only main effects and second order interactions were considered. Table 3.2 shows the ANOVA table for the experiment.
72 Table 3.2. ANOVA Table for Optimal Suppression Factor Source Sum Sq.DoF Mean Sq. F P value KP 151.54782 75.7739 447.9556 0 4.4170002 2.2084 13.06000 0 /to 14.789302 7.3946 43.71510 2.2204e-016 / Ts 7.640302 3.8202 22.58380 1.5546e-009 1.685502 0.84273 4.982000 0.0077736 *PK 2.282004 0.5705 3.372700 0.0107730 */PKto 2.524104 0.63102 3.730400 0.0060036 */PKTs 1.184804 0.29619 1.751000 0.1404500 *PK 0.127354 0.031837 0.188210 0.9443400 */to 2.921504 0.73037 4.317700 0.0022875 */ Ts 1.179404 0.29485 1.743100 0.1421300 0.363834 0.090959 0.537720 0.7081900 /*/toTs 3.984404 0.9961 5.888700 0.00017208 /*to 0.476144 0.11904 0.703710 0.59030000 /*Ts 0.260254 0.065063 0.384630 0.81947000 Error 32.4778192 0.16915 Total 227.8611242 where DoF: Degree of Freedom F: Test Statistic F The significant factors are those with a P value less than 0.05. Therefore, the significant factors are: KP, /to /Ts , *PK */PKto */to and /*/ toTs Using this information and the set of optimal suppression factor available, nonlinear regressions were performed using many possible combinations of the significant factors, until a good correlation coe fficient was obtained. The resulting tuning equation that best fits the optimal values of the suppression factor (R2 = 0.9595) is: 0.40941.631Pto K (3.32)
73 For all experiment conditions, the DMCr was implemented using a Control Horizon (CH) equal to 5, a sampling time equivalent to 0.1 and a Sampling Size (SS) equal to4to Appendix C shows an evaluation of tuning equation performance when other conditions are used. To validate the new tuning equation, s uppression factors were calculated using the Shridhar and Cooper tuning equations  and Eq. 3.32 ; the DMCr performance was compared using those values. Shridhar and Cooper have proposed the following tuning equations for SISO DMCr: if 1 0 3.51 if >1 2 5002sM f MM M T (3.33) 2 P SC f K (3.34) where M is the control horizon, an integer number usually from 1 to 6. Ts is the sampling time, the largest value that satisfies 0.1sT and 0.5sTto For the following FOPDT process: 0.2()0.5 ()1 s Cse Mss (3.35) The suppression factor using Shridhar and Cooper SC is: 0.0875SC (3.36)
74 Eq. 3.32 gives: 0.4220 (3.37) Using optimization methods, the value for is: 0.325Opt (3.38) Figure 3.4 shows the performance comparison using the three suppression factors. The figure shows that the controller response using the Shridhar/ Cooper equations generates oscillatory behavior. The suppression factor predicted from these equations is the smaller one, which implies the most aggressive controller behavior. This result seems to be a general tendency of these equations, because for all the combinations of FOPDT tested the suppression factor calculated always generated the smaller Figure 3.4 shows that the response obtained using the suppression factor calculated with Eq. 3.32 is almost the same when the optimal value of is used.
75 0 10 20 30 40 50 60 70 80 60 62 64 66 68 70 72 74 76 78 80 time (sec)cset and c(t) 10 15 20 74 74.5 75 75.5 76 76.5 77 time (sec)cset and c(t) Shridhar/Cooper SP Eq. 3.32 Optimization Figure 3.4. Performance Comparison When DMCr Controlling a FOPDT System is Tuned Using Different Methods As a second test to evaluate Eq. 3.32, the mixing process described in Appendix A.2 was chosen as nonlinear process. As a first step of DMCr implementation, the process was identified as FOPDT in order to determine its characteristic parameters. Fit 3 method  was used to perform the identification; introducing changes in the signal to the valve of +10%CO and -10%CO. The results where: 25.35 10%()0.365 ()5.061 s MTOCse Mss (3.39)
76 25.59 10%()0.432 ()5.601 s MCse Mss (3.40) Using Eq. 3.40 to calculate the suppression factor the results are: 1.3125 and0.0653SC If the suppression factor is determined by optimization methods the result is 1.092Opt Figure 3.5 shows the control performance provided by each suppression factor. The Shridhar/ Cooper tuning value generates a very aggressive controller behavior, a non desirable operation condition. The tuning obtai ned using Eq. 3.32 produced a stable and smooth behavior. Table 3.3 shows the IAE values for the test presented in Figure 3.5. 0 50 100 150 200 250 300 350 400 40 45 50 55 time (sec)cset and c(t) SP Shridhar/Cooper Optimamization Eq. 3.32 Figure 3.5. Performing Comparison of DMCr Tuned Using Different Methods for Mixing Process
77 Table 3.3. IAE Comparison for Test Presented in Figure 3.5 Tuning Method IAE Shridhar/ Cooper 168.8 Eq. 3.32 150.6 Optimization 142.1 The results just presented corroborate the convenience of Eq. 3.32 to tune DMCr based on the process parameters. Because of these results, it was decided to find a more general tuning equation to include the effect of the Control Horizon ( CH ) and the Sampling Time ( Ts ) variations over the suppression factor. Appendix C describes the procedure followed to design that equation and shows as final result the Eq. C.2: 3 4 2 51 PtoCH K Ts (C.2) where 1 to 5 are constantans values reported in Table C.4. Appendix C also compares the suppression factor calculation using Eq. 3.32 and Eq. C.2, showing that under same Control Horizon and sampling time conditions, CH = 5 and Ts = 0.1, both equations predict very similar values. Equation C.2 is a completely general equation to calculate suppression factors for standard DMCr in SISO loops.
78 3.5.3 Fuzzy Supervisor Section 3.5.1 developed a parametric DMC controller (PDMCr). The parameters of this controller can be changed to compensate for changes in process parameters. In Section 3.5.2 a reliable tuning equation for was developed based on the process parameters. The effectiveness of these two developments is directly related to the capacity of detecting variations in the process parameters. For this reason it is necessary to design a tool to detect and quantify changes in process parameters on-line, able of providing this information to the PDMCr and tuning equation to adjust the control system to compensate for nonlinearities. There are reported evidence  about the possibility of using modeling error or other related parameters (i.e. maximum modeling error, minimum modeling error, etc.) as a reliable factor to detect changes in process parameters. Based on these evidences a supervisor module was developed to accomplish this task. Modeling error,em, is defined as the difference between the actual value of c ( t ), and the predicted value cp ( t ) from the model resident inside the controller. This concept applies for all MBC controllers. The main idea behind the supervisor module designed in this research is to determine how the modeling error, or associated information, is affected by changes in the process parameters. Once these relations were established, the goal was to develop a set of regression equations that relates modeling error or associated information to the changes in process parameters; 21 regression equations were developed as result of this work. Sets of three regression equations were then tested until finding those which gave the best result in predicting changes in process parameters. Those equations were used to
79 implement the supervisor to predict, on-line, changes in process parameters and readjust the process model, as well as the suppression factor in the PDMCr. The supervisor adjusts the prediction provided by the regression equations using a Fuzzy Inference System (FIS). The following paragraphs describe in detail how the fuzzy supervisor was developed and how it works. To determine how the modeling error is affected by changes in the process parameters an experiment was designed. A Matlab routine was written to run a large number of simulations where a FOPDT process is controlled by a standard DMCr. Parameters of the FOPDT and set point were modified for every experimental condition, recording into a vector the modeling error for every condition. Table 3.4 shows the factors and their levels used in the simulations. A total of 61,236 simulations, corresponding to all possible combination of factor levels, were performed. Figure 3.6 shows a typical simulation result found using the Matlab routine used to generate the modeling error at the experiment condition. Additionally Figure 3.6 compares the actual process output c ( t ) with the controller predictive value Cp ( t ); the divergence between both variables is small because the process used in the test is a FOPDT with constant parameters. For that reason the modeling error, also shown in Figure 3.6, is almost negligible compared with the values of c ( t ) and Cp ( t ). Once the modeling error from all the simulations were collected, it became necessary to develop a procedure to detect if the process parameters have changed. It was decided to develop some indicators to signal these changes. There is nothing in the literature to help in choosing these indicators. It was decided to graph many of the modeling errors to see if anything could be obtained from them such as time for peaks,
80 time to reach steady-state, ratio of maximum peak to minimum peak, etc. The result of this extensive search is shown in Table 3.5. The table shows 21 proposed indicators. These indicators were called Modeling Error Indicators (MEI). Appendix D presents a detailed discussion of their significance and calculation. The next step consisted in learning which of these indicators are signi ficantly affected by changes in process parameters, To do so, the information was analyzed using Analysis of Variance (ANOVA); only main and second order interac tions were considered. Appendix E shows the 21 ANOVA tables generated based on the information. On the basis of this analysis, it was decided that only 15 of these indicators yield the correct result. Table 3.6 shows the selected.
81 Table 3.4. Factors and Levels Used to Record Modeling Error and Develop the Regression Equations Cset (%TO) KP (%TO/%CO) (s)/ to P K (%) (% ) to (%) -15 0.5 1 0.2 -40 -40 -15 -7.5 1.5 3 0.6 -30 -30 -10 7.5 2.5 5 1 -20 -20 -5 15 -10 -10 0 0 0 5 10 10 10 20 20 15 30 30 40 40
82 9 9.5 10 10.5 11 11.5 12 12.5 13 60 65 70 75 80 time (sec)c(t), Cp(t) and cset(t) 9 9.5 10 10.5 11 11.5 12 12.5 13 -1 0 1 2 time (sec)em(t) %TO SP c(t) Cp(t) Figure 3.6. Example of Simulation Performed to Record Modeling Error
83 Table 3.5. Modeling Error Indicators Selected to Predict Process Parameters Changes Modeling Error Indicator Calculation Form 2nd 6th and 10th Correlation Coefficient 2 1 Nk iik ii k N i iYYYY r YY Maximum em /Cset () M axem Cset Maximum em /Minimum em () () M axem M inem Minimum em /Cset () M inem Cset T ()()2()MaximumemMinimumemtimetime 2 T Time for Maximum em / () M aximumemtime Time for Minimum em / () M inimumemtime Stabilization time between 0.1%/TO 0.1% emtime Difference time for Maximum and time for Minimum/ ()() M aximumemMinimumemtimetime Ratio Absolute Minimum/Abs Maximum () () M inem M axem Maximum peak/Cset M axem Cset Minimum peak/Cset () M inem Cset
84 Table 3.5. Continued Time for Maximum Peak/ M axemtime Time for Minimum Peak/ M inemtime Decay ratio second peak() first peak() em em Damping ratio 2 2second peak() log first peak() second peak() 4log first peak() em em em em Ratio Minimum em / Maximum em () () M inem M axem Time between peaks/ second peak()first peak()ememtimetime Appendix F shows the complete the set of regression equations found. All the regression equations developed have as indepe ndent variables the factor shown in Table 3.4: P K ,/ to P K and to The general form of all the available regression equation is: ,,,,,PPto M EIFKKto (3.41) where F is a nonlinear expression of the independent variables. Table 3.6 summarizes the modeling error indicators selected and correlation factor found for each regression equation.
85 Table 3.6. Summary of Modeling Error Indicat or Selected and Correlation Coefficient of Regression Equation Modeling Error Indicator R2 10th Correlation Coefficient 0.843 Maximum em /Cset 0.988 Minimum em /Cset 0.939 T 0.562 0.879 Time for Maximum em / 0.995 Time for Minimum em / 0.878 Stabilization time between 10%/TO 0.766 Difference between time for maximum and time for minimum/ 0.620 Ratio Abs minimum/Abs maximum 0.610 Maximum peak/Cset 0.988 Minimum peak/Cset 0.684 Time for Maximum Peak/ 0.995 Time for Minimum Peak/ 0.872 Damping ratio 0.898
86 Because the purpose of the regression equation is to predict changes in the three process parameters, the supervisor module must solve a set of three equations and three unknowns. To select the three MEI and their respective regression equations, among the 15 available, that would be used in the supervisor module, the equations were selected in groups of three to test their capacity to predict changes in process parameters. These tests were performed using the following procedure: Using a FOPDT as process a control loop was implemented. A standard DMCr was used to close the loop. A set of process parameters were selected for the FOPDT. A set point change was induced to the control loop. At the same time the set point change is introduced, known changes were applied to the process parameters. When the set point change is applied, a Matlab routine starts to record the modeling error until the system reaches the new steady state. Using the collected information from modeling error, another Matlab routine calculates the corresponding three MEICalc. Using the calculated MEI and appropriated regression equations F1, F2 and F3, a Matlab optimization routine was used to obtain the P K and to that minimize a cost function ( CF ). The cost function (CF) used was: 112233(,,)(,,)(,,)CalcPCalcPCalcPCFMEIFKtoMEIFKtoMEIFKto (3.42)
87 where F1, F2 and F3 are the appropriate regression equations corresponding to each MEICalc calculated from the actual modeling error. The reason to calculate the process parameters changes, through optimization methods instead of solving directly the set of three equations and three unknowns, as it was initially anticipated, is because from the computational point of view the optimization routines available in Matlab are faster that those designed to solve nonlinear equation systems. This is a particul ar way of solving the three nonlinear equations with three unknowns general problem. Once changes in process parameters were predicted using Eq. 3.42 the results were compared with the actual values to determine the goodness of prediction. Many combinations of available regression equations were evaluated using the test just described, until a set of three equations that satisfactorily predicted changes in process parameters was found. The three regression equation selected and their corresponding MEI are: Time for Maximum Peak/ Time for Minimum Peak/ 10th Correlation Coefficient of em. Coincidentally, the three regression equations corresponding to these MEI have the same form: 12345678910 PPtoto M EIKtoKto (3.43) Table 3.7 shows the coefficient for each MEI
88 Table 3.7. Parameters for Regression Equations Used in Supervisor Module Time for Maximum Peak/ Time for Minimum Peak/ 10th Correlation Coefficient of em 1 12.54941 7.963501 -0.64086 2 -2.329702 -1.202002 -0.00542 3 0.7890603 5.310503 0.26446 4 0.0031254 0.197604 0.01052 5 -0.0037505 -0.030705 0.00154 6 -0.0337506 0.140626 0.05056 7 0.2109407 0.181647 0.08960 8 0.2109408 0.005468 0.000924 9 -0.00031259 -0.044849 0.000354 10 0.03750010 -0.0031210 -0.00409 Once the regression equation were chosen, it was possible to design the structure of a functional Supervisor Module to detect and estimate changes in process parameters. Figure 3.7 shows the Supervisor Module flow diagram.
89 Figure 3.7. Supervisor Module Flow Diagram
90 The Supervisor Module was designed to work only when a set point change is detected. This design could seem peculiar because set points are very commonly constants; however the idea behind this design is to use the set point change as a tool to test the system for changes in process parameters that could cause controller aging. From a practical point of view this mean that when a plant operator notes that the control loop is not responding adequately when rejecting disturbances, the operator would perform as small set point in one direction and later a set point in the opposite direction to reach the initial conditions. In this way the supervisor has two opportunities to evaluate the process parameter and adjust the PDMCr to the new conditions. The supervisor records em until the system reaches the new set point. Once the supervisor stops recording em it calculates the three MEI : time for modeling error maximum peak/ time for modeling error minimum peak/ and 10th modeling error correlation coefficient. This information and the regression equation is used to estimate changes in process parameters: P K and to by minimization of Eq. 3.42. These three amounts are expressed as percentage of change in the respective parameter. Then, the new process parameters are estimated using the following expressions: 1 100PKP PAdjPK KK (3.44) 1 100Adj (3.45) 1 100to Adjto toto (3.46)
91 where Adj subscript denotes that variables are the initial process parameters adjusted by a correction factor, with values between 0 and 1. There is an uncertainty in the values predicted using the optimization of the regression equations, given by the error involved in using regression equations with no perfect original data fit. The idea behind the factor is to complement the prediction obtained using the optimization of regression equations, employing the experience gained through the results of more than 60,000 simulations used to develop the Supervisor Module. Fuzzy Logic and the experience gained provide a way to calculate each individual giving more or less importance to the correction predicted. The fuzzy inference system is shown in Figure 3.8, and Tables 3.8, 3.9 and 3.10 show the fuzzy rules.
92 Fuzzy Rules Table 3.8 Scaling Scaling Scaling ZSL 01 ZSL 01 ZSL 01 ZSL 01 ZSL 01 ZSL 01 max timeforpeak min timeforpeak10 Corr CoefthPK to Fuzzy Rules Table 3.9 Fuzzy Rules Table 3.10 See Fig.3.9 See Fig.3.9 See Fig.3.9 See Fig.3.10 See Fig.3.10 See Fig.3.10 Fuzzy Rules Table 3.8 Scaling Scaling Scaling Scaling Scaling Scaling ZSL 01 ZSL 01 ZSL 01 ZSL 01 ZSL 01 ZSL 01 ZSL 01 ZSL 01 ZSL 01 ZSL 01 ZSL 01 ZSL 01 max timeforpeak min timeforpeak10 Corr CoefthPK to Fuzzy Rules Table 3.9 Fuzzy Rules Table 3.10 See Fig.3.9 See Fig.3.9 See Fig.3.9 See Fig.3.10 See Fig.3.10 See Fig.3.10 Figure 3.8. Schematic Representation of Fuzzy Inference Used to Calculate Weighting Factors
93 Table 3.8. Fuzzy Rules Designed to Calculate P K Time Maximum Peak Time Minimum Peak 10th Correlation coefficient P K L L S Z L L L Z Z Z Z Z Z Z L Z L L L S L L Z S Z Z S S Z Z L S L L L L Z S Z L Z S S L Z S L L
94 Table 3.9. Fuzzy Rules Designed to Calculate Time Maximum Peak Time Minimum Peak 10th Correlation coefficient L L S Z L L L Z L L L S L L S S Z Z Z L Z Z L L L L Z S Z Z S L Z Z L L Z S Z L Z S S L Z S L L
95 Table 3.10. Fuzzy Rules Designed to Calculateto Time Maximum Peak Time Minimum Peak 10th Correlation coefficient to L L S Z L L L S L L L L L L L Z L L S L Z Z Z Z Z Z L S Z Z L L L L Z Z Z Z S Z Z S Z Z Z S S S Z S L L
96 Every input MEI is fuzzified using three membership functions: Zero (Z), Small (S) and Large (L); Figure 3.9 shows a representation of the triangular membership functions used for every MEI in the calculation of P K The membership functions used to fuzzify and to are identical to that shown in Figure 3.9. Once the fuzzy rules are evaluated for each input, the Fuzzy inference system gives a set of fuzzy values as result. Those fuzzy values are expressed using a set of membership functions design for each output. Figure 3.10 shows the membership functions used to defuzzified P K ; the membership functions used to perform the same operation with and to are identical to that shown in Figure 3.10. The final defu zzification operation to convert fuzzy values of into crisp values is performed using the centroid method.
97 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 time for maximum em over Degree of membershipZ S L 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 time for minimum em over Degree of membershipZ S L 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 10th correlation coefficient Degree of membershipZ S L Figure 3.9. Membership Function Used to Fuzzified MEIs
98 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 KpDegree of membershipZ S L Figure 3.10. Membership Function Used to Defuzzified Output P K Figure 3.11 shows the kind of nonlinear relationship among MEIs and P K that can be obtained evaluating the fuzzy rules. It would be very complex to express mathematically the relationship among the variables, but using 12 Fuzzy Logic rules is easy to do so.
99 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 time max em/ time min em / Kp Figure 3.11. Example of Nonlinear Relation Among Two of the MEIs and P K Once the initial process parameters are corrected based on the information provided by the modeling error, the adjusted parameters are sent to the PDMCr to recalculate the matrices involved in the control law and determine the best tuning parameter. Figure 3.12 shows a schematic representation of the controller.
100 Fuzzy Supervisor Tuning Equations c(t) Parametric DMC em Kpto cset(t)m(t) Figure 3.12. Schematic Representation of Parametric DMC Working With Fuzzy Supervisor 3.5.4 PDMCr Implementation One of the main concerns when developing the PDMCr was the necessary computer time for the matrix operations. For this reason the algorithm for the PDMCr was structured to save computer time; the matrix calculations are performed only when the Fuzzy Supervisor updates the process parameters sent to the PDMCr. Figure 3.13 shows a flow diagram of how PDMCr was implemented. Figure 3.13 shows that the first step is to initialize all the internal variables and load the store information, the initial value of matrix Kups, as well as the initial set of process parameters. The algorithm start to read the inputs: c cset the process parameters and the sampling time Ts
101 Figure 3.13. Schematic Representation of PDMCr Algorithm Implementation
102 At every sampling time the algorithm recalculates m ; between sampling times the PDMCr holds the last m calculated. For every sampling time, if any of parameters are changed in the process, the algorithm calculates a new suppression factor using Eq. 3.32. If the time constant changes Eq 3.29 is used to adjust vector D. If the process dead time changes, the zeros matrix Z is adjusted (see Eq 3.30). If the process gain changes, the matrix Kups is recalculated without modifying matrixes Z or D, just the new KP is changed to perform the calculation (see Eq.3.30). This algorithm was designed to avoid long and complex calculation for every sampling time, only when needed the matrix calculations are performed. 3.6 Simulation Results To evaluate the PDMCr a set of tests were performed. The first of them was designed to use a known FOPDT system as the process in the control loop. A sequential set of set point changes were induced to the control loop at different times. Simultaneously, the process parameters were modified by +25%, in order to emulate the nonlinear behavior of the process. At time 400 s a disturbance affects the process. Figure 3.14 shows the results when standard DMCr and PDMCr are used. The standard DMCr can not compensate for the changes in FOPDT parameters and becomes oscillatory. The PDMCr using the Fuzzy Supervisor Module can detect and estimate the process parameters changes and compensate for them, allowing stable control.
103 0 100 200 300 400 500 40 60 80 100 time (sec)c(t) and cset(t) %TO 0 100 200 300 400 500 40 60 80 100 time (sec)c(t) and Cset(t) %TO DMCr SP PDMCr SP Figure 3.14. Comparison Standard DMCr and PDMCr Performance Handling Nonlinearities Emulated Figure 3.15 shows the comparison of process parameters estimation performed by the Fuzzy Supervisor Module during the test presented in Figure 3.14. Figure 3.15 shows that the prediction are calculated after every set point change is detected and the modeling error is recorded by the fuzzy supervisor to perform the necessary calculation; for this reason the actualization of process parameter in PDMCr is not immediate. The time to update the process parameter varies and it depends how long takes to the modeling error to settle down. The process parameters predicted by the Fuzzy Supervisor
104 are not 100% accurate, there are discrepancies with the actual values used in the simulation, but the predicted values are close enough to allow to the PDMCr to adapt to the changing process conditions and keep the process under control. At time 400 s a disturbance affects the process; Fig 3.14 shows the process response under DMCr and PDMCr, but Fig 3.15 shows that the supervisor does not change the process estimation parameters as expected. 0 50 100 150 200 250 300 350 400 450 500 2 2.5 3 3.5 4 time (sec)Process gain 0 50 100 150 200 250 300 350 400 450 500 4 6 8 10 time (sec)Process time constant 0 50 100 150 200 250 300 350 400 450 500 3 4 5 6 7 time (sec)Process dead time Actual to Updated to Actual Updated Actual KPUpdated KP Figure 3.15. Comparison Among Actual Process Parameters and Those Estimated by Fuzzy Supervisor Module for the Test Presented in Figure 3.14
105 As a second test the PDMCr was test controlling the mixing process described in Appendix A.2 Figure 3.16 shows the comparison when standard DMCr and PDMCr are used controlling the process when facing consecutive set point changes. Figure 3.16 shows that every time the set point is decreased by 5%, the standard DMCr becomes more and more oscillatory until finally a completely oscillatory behavior is observed. This is the result of the nonlinear characteristic observed in the mixing process (see Figure 2.3). The PDMCr shows a smooth response and is able of tracking the set point during the test; it also rejects a cold temperature increment used as disturbance at time 700s. Figure 3.17 shows how the model parameters are changed by the Fuzzy Supervisor. 0 100 200 300 400 500 600 700 800 10 20 30 40 50 time (sec)c(t) and cset %TO 0 100 200 300 400 500 600 700 800 10 20 30 40 50 time (sec)c(t) and cset %TO SP PDMCr SP DMCr Figure 3.16. Performance Comparison When DMCr and PDMCr Are Used to Control Mixing Process
106 0 100 200 300 400 500 600 700 800 0 2 4 6 8 10 12 14 time (sec)Kp, and to updated Kp updated updated to updated Figure 3.17. Process Parameters Estimation Performed by Fuzzy Supervisor Module for the Test Presented in Figure 3.16 Every time a set point change is detected by the Fuzzy Supervisor, the model parameters are estimated and adjusted to adapt the controller to the new process conditions. When the disturbances affect the process, the PDMCr tracks the set point avoiding large deviation; but the Fuzzy Supervisor Module does not update the model process parameters. Figure 3.17 shows no change in updated parameters at time 700 s when the disturbance affects the process. The PDMCr also was tested using the neutralization reactor describe in Appendix A.1. A series of consecutive set point changes were induced, and a reduction of 15% in
107 acid stream concentration was used as disturbance. Figure 3.18 shows the results for described test; PDMCr tracks the set point with less overshoot that the standard DMCr, no matter if faces set point changes or rejects disturbances. The total Integral of the Absolute value of the Error (IAE) for the discussed test were: 4284 for standard DMCr and 3214 for PDMCr; a reduction around 23%. 0 1000 2000 3000 4000 5000 6000 7000 8000 50 55 60 65 70 75 time (sec)c(t) and cset (%TO) 0 1000 2000 3000 4000 5000 6000 7000 8000 50 55 60 65 70 75 time (sec)c(t) and cset (%TO) PDMCr SP DMCr SP Figure 3.18. Performance Comparison Between Standard DMCr and PDMCr Handling Neutralization Process Figure 3.19 shows a test performed using the neutralization reactor as process. Initially a reduction on acid stream affects the process as disturbance; later, two consecutive set point changes in opposite directions (reaching the initial value again), are
108 induced into the control loop, to allow the PDMCr estimate and update the process parameters. Later two consecutives changes in acid stream affect again to the process. Figure 3.19 shows that PDMCr tracks set point with less deviation than standard DMCr. The total IAE for discussed test were: 8080 for standard DMCr and 6717 for PDMCr; a reduction about 17% on IAE when PDMCr is used. The last two disturbances were compensated in less time and with less overshoot when PDMCr is used. This test could represent the way the control engineer should use the consecutive set point changes to allow to the PDMCr adapts to varying operating conditions. 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 2.5 3 3.5 4 4.5 x 10-3 time (sec)q1(t) mol/lt 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 40 50 60 70 time (sec)c(t) and cset (%TO) 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 40 50 60 70 time (sec)c(t) and cset PDMCr SP DMCr SP Figure 3.19. Neutralization Reactor Test for Disturbances Rejection
109 3.7 Effect of Noise To test the effect of presence of noi se over DMCr and PDMCr performance, the mixing process describe in Appendix A.1 was modified to include a noisy signal in cold stream F2. The noise used had an ARMA (1,1) structure: 1111nnnn Z Zaa (10.6 10.3 ). Two different values of variance were used to generate the noise: 20.02a and 20.03a Figure 3.20 shows cold stream affected by the noise. 0 500 1000 0 0.5 1 1.5 2 2.5 3 (a) 2 =0.02 time (sec)F2 (m3/s) 0 500 1000 0 0.5 1 1.5 2 2.5 3 time (sec)F2 (m3/s)(b) 2=0.03 Figure 3.20. Cold Stream F2 Affected by Noise With Structure ARMA(1,1) Figure 3.21 shows the effect of noise on the DMCr and PDMCr performance; the test described in Figure 3.16 is now shown w ith the noisy cold water flow shown in Fig 3.20. Figure 3.21 (a) and (b) compares standard DMCr and PDMCr when noise has a
110 variance equal to 0.02, whereas Figure 3.21 (c) and (d) shows a noise with variance 0.03. The test shows that the PDMCr is sensible to the presence of noise as all controllers with discrete nature are. A slight variation on noise variance can cause oscillatory behavior on PDMCr, see Figure3.21 (c). However, compar ed with the standard DMCr, the PDMCr have more tolerance to noise presence. 0 200 400 600 800 1000 15 20 25 30 35 40 45 (b) time (sec)c(t) and cset (%TO) 0 200 400 600 800 1000 15 20 25 30 35 40 45 time (sec)c(t) and cset (%TO)(a) 0 200 400 600 800 1000 15 20 25 30 35 40 45 (d) time (sec)c(t) and cset (%TO) 0 200 400 600 800 1000 15 20 25 30 35 40 45 time (sec)c(t) and cset(c) PDMCr SP PDMCr SP DMCr SP DMCr SP Figure 3.21. Noise Effect Over DMCr and PDMCr
111 3.8 Conclusions The PDMCr is able to overcome the main problem of the standard DMCr, that of handling highly nonlinear processes. The Fuzzy Supervisor Module, determines changes in the process parameters, estimates the magnitude of the detected changes, and sends this information to the PDMCr adjusting the embedded model to the new operation conditions. The tuning equation developed to calculate the optimal suppression factor, allows adjusting the controller aggressiveness according to the operation conditions. Although the calculations required by the PDMCr algorithm are complex and involve a significant amount of data, the calculation time does not affect the PDMCr performance. Finally, it is necessary to men tion that presence of noise could affect the PDMCr performance.
112 Chapter 4 Summary This research focuses in combining two modern control techniques, Sliding Mode Control (SMC) and Dynamic Matrix Control (DMC), with one of the most promising intelligent control technique: Fuzzy Logic. The goal is to overcome the problems, reported in the literature, of these controllers when handling highly nonlinear chemical processes. The reported problems of the Sliding Mode Controller (SMCr), when tracking set point changes with highly nonlinear processes, motivated the idea to obtain a SMCr with quick response to changes in set point, but conserving its characteristic stability and robustness. To achieve this objective, a combination of SMC and Fuzzy Logic is proposed. The main idea is to incorporate the human expert knowledge to the controller to react quickly or slowly depending of the process requirements. DMC is a linear controller. One way of improving its performance, when working with nonlinear processes, it is to adjust the model gain, time constant and dead time depending on control loop behavior. This adjustment can be performed in many ways; however, the use of human experience through Fuzzy Logic is an interesting alternative. As it is shown in Chapter 3, the DMC algorithm can be reformulated to isolate the process model parameters. This parametric algorithm for DMC can be adjusted according to the nonlinearities shown by the process.
113 In summary, this research presents ways of improving the performance of SMC and DMC using Fuzzy Logic. Contribution 1: A Fuzzy Sliding Mode Controller (FSMCr) is presented which combines the best characteristics of SMC and Fuzzy Logic: robustness, stability and flexibility. This controller is suitable to be used in applications with highly nonlinear behavior. FSMCr is a completely general controller which incorporates human experience about process control by means of a set of fuzzy rules. Chapter 2 presented a detailed discussion about this approach. The FSMCr was tested with two models of nonlinear process: mixing tank and ne utralization reactor (Appendix A). In both cases the FSMCr improves the performance shown for control strategies as the industrial PID, the conventional Sliding Mode Control and the Standard Fuzzy Logic Controller. The performance of FSMCr can be affected when sampling time is too large. Tests in Section 2.7.3 showed that for sampling time grater than 0.5 the process time constant, the FSMCr behaves oscillatory. The presence of noise in process signal does not appreciably affect the FSMCr performance. Section 2.7.4 presents in detail this result.
114 Contribution 2: A parametric structure of DMC control algorithm is proposed. This new way to express the DMC algorithm allows adapting the controller to process nonlinearities. It was statistically established that the modeling error ( em ) and related information such as maximum em time for maximum em etc. can be effectively used as an indicative index of process parameters changes. On the basis of this information a Fuzzy Supervisor module was developed. The integration of the parametric DMC algorithm with the Fuzzy Supervisor Module constitutes a nonlinear controller with implicit dead time compensation able to handle highly nonlinear processes. Chapter 3 described in detail the Parametric Dynamic Matrix Controller (PDMCr) implementation in Matlab environment. Tuning equations for DMC algorithm were developed to determine the optimal suppression factor needed to manipulat e DMCr aggressiveness (see Appendix C). These equations were incorporated into the Fuzzy Supervisor Module to enhance PDMCr algorithm. Every time that process parameter changes are detected by the Fuzzy Supervisor Module an optimal suppression factor is calculated and used to recalculate the appropriate matrices.
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124 Appendix A: Mathematical Model of Processes A.1 Neutralization Reactor Tank 2 Neutralization reactor q1e(t) q2(t) q4(t) q3(t) q1(t) AT AC c(t) %TO m(t) %CO SP Figure A.1. Schematic Representation of Neutralization Reactor The purpose of the process shown in Figure A.1 is to neutralize the acid stream q1(t), manipulating the flow of basic stream q3(t); while q2(t), a buffer solution, remains constant. The main disturbances for this process are the acid flow and acid stream concentration. The three streams are introduced to the neutralization reactor, where they are assumed to be perfectly mixed. It is also assumed constant density and complete solubility of the ions involved. A manual valve is used to manipulate the output flow from the reactor. The following chemical reactions take place inside the reactor: H HCO CO H3 3 2 (A.1) H CO HCO2 3 3 (A.2)
125 Appendix A (Continued) H OH O H2 (A.3) The equilibrium expressions can be written as: ] [ ] ][ [3 2 3 1CO H H HCO Ka (A.4) ] [ ] ][ [3 2 3 1 HCO H CO Ka (A.5) ] ][ [ OH H KW (A.6) The chemical equilibrium is modeled using the definition of two reaction invariants, Wa and Wb. The first invariant, Wa is a charge related quantity, and Wb is related to the concentration of the ion 2 3CO Unlike pH, these invariants are conserved quantities. The invariants are expressed as: i i i i aiCO HCO OH H W ] [ 2 ] [ ] [ ] [2 3 3 (A.7) i i i biCO HCO CO H W ] [ ] [ ] [2 3 3 3 2 (A.8) where i represents every stream involved in the process, from 1 to 4. Substituting Eq. (A.6) into (A.7) and using Eq. (A.4) and (A.5), it is possible to derive an expression to find the pH: 112 2 112 22 ()() ()0 () 1 ()()aaa W ba aaaKKK K HtHt WWHt KKK Ht HtHt (A.9) and ) ]( log[ ) ( t H t pH (A.10)
126 Appendix A (Continued) The dynamic model for the process is developed through mass balance in every individual element of the process. For tank 1, the unsteady state mass balance is: dt t dh A t q t qe) ( ) ( ) (2 2 1 1 (A.11) where A2 is the cross section of the tank 1, h2 ( t ) is the liquid high and is the density of the streams. Valve 2, which manipulates the output fl ow from tank 2, gives another equation: ) ( ) (2 1 1t h Cv t qe (A.12) Cv1 is the valve constant for valve 1. The total unsteady state mass balance on the reactor is: ) ( ) ( ) ( ) (4 3 2 1t q t q t q t qe dt t dh A ) (1 1 (A.13) A1 is the cross section of the reactor, and h1( t ) is the liquid high inside the reactor. The valve placed in the reactor output, which manipulates the flow q4( t ), contributes another equation: nt h Cv t q )) ( ( ) (1 4 4 (A.14) Cv4 is the valve coefficient; n is a constant valve exponent. This equation is modified from the original expression presented by Henson and Seborg. For this research, the vertical distance between the bottom reactor and the outlet for q4 was took as equal to zero; this to avoid drainage of th e reactor when the input flows are to small.
127 Appendix A (Continued) This consideration modified the steady state liquid level in the reactor presented by Henson and Seborg. Writing unsteady state mass balance on each ionic species, it is possible derivate expressions for the reaction invariants Wa and Wb: 3 3 2 2 1 1) ( ) ( ) (a a a eW t q W t q W t q dt t W t h d A t W t qa a) ( ) ( ) ( ) (4 1 1 4 4 (A.15) 3 3 2 2 1 1) ( ) ( ) (b b b eW t q W t q W t q dt t W t h d A t W t qb b) ( ) ( ) ( ) (4 1 1 4 4 (A.16) These two last equations are also different to those presented by Henson and Seborg. For this work, the variation of the volume inside the reactor, and its influence over the reaction invariants, Wa and Wb, it is considered. The pH transmitter is modeled as first order transfer function: ) ( ) ( )) ( (1 1t pH K t c dt t c dT T (A.17) where c ( t ) is the sensor output, 1 T and KT 1 are the time constant and sensor gain respectively. Additionally, because the pH transmitter is located downstream from the reactor, it is necessary to consider a time delay t0( t ) in the measurement: ) ( ) (4 0t q LAp t t (A.18) where L and Ap are the distance from the bottom of the reactor to the measurement point, and the pipe cross-section, respectively.
128 Appendix A (Continued) Therefore, the expression for pH(t) taking into account the time delay is: ) ( ) (0t t t pH t pH (A.19) Respect to the control valve, it is assumed that also can be modeled by a first order transfer function: ) ( ) ( )) ( (3 3t m K t q dt t q dV V (A.20) where V and KV are the time constant and valve gain respectively; and m ( t ) is the controller output. This control valve is a fail close valve. The steady state values and parameters for the process model are shown in the Table A.1.
129 Appendix A (Continued) Table A.1. Steady State Parameters for Neutralization Reactor Model Parameters Steady State Values [ q1] = 0.003M HNO3 q1 = 16.6 ml/s [ q2] = 0.03M NaHCO3 q2 = 0.55 ml/s [ q3] = 0.003M NaOH + q3 = 15.6 ml/s + 0.0005M NaHCO3 q4 = 32.75 ml/s A1 = 207 cm2 h1 = 25.5 cm A2 = 42 cm2 h2 = 3 cm n = 0.607 Wa1= 1.003 M Ka1 = 4.47x 10-7 Wb1 = 0 M Ka2 = 5.62 x 10-11 Wa2 = 0.03 M KW = 1x 10-14 Wb2 = 0.03 M T = 15 s Wa3 = -3.05x10-3 M V = 6 s Wb3 = 5x10-5 M KT = 7.1429 %TO/pH Wa4 = -4.36x10-4 M K V = 0.3 (ml/s)/(%CO) Wb4 = 5.276x10-4 M Cv2 = 9.584 (ml/s)/(cm0.5) pH = 7.025 Cv1 = 4.5861 (ml/s)/(cm0.607) 52% CO m Vp=Lp*Ap= 327.5 ml 50.18% TO c
130 Appendix A (Continued) A.2 Mixing Process Figure A.2. Schematic Representation Mixing Process Figure A.2.1 shows the mixing process. A hot water stream F1( t ) is manipulated to mix with a cold water stream F2( t ) to obtain an output flow F ( t ) at a desired temperature T Â’( t ). The temperature transmitter is located at a distance L from the mixing tank bottom. The volume of the tank varies freely without overflowing. The unsteady state mass balance can be expressed as: 12()[()] ()()() dVtdht FtFtFtA dtdt (A.21)
131 Appendix A (Continued) where is the flow density, A is the tank cross section and h(t) is the liquid level inside the tank. The output flow F ( t ) can be modeled as a function of the liquid level and the manual valve used in bottom of tank: ()()V F tCht (A.22) In order to relate controlled variable, T ( t ), and manipulated variable, F1( t ); it is necessary to write an energy balance: 1122(()()) ()()()()()() dhtTt FtCpTtFtCpTtFtCpTtACv dt (A.23) where Cp and Cv are the heat capacity of the liquid at pressure constant and volume constant respectively. T1( t ) is the hot water stream temperature, T2(t) is the cold water stream temperature. T ( t ) is the temperature just in the bottom of the tank. Because the sensor/transmitter TT is located at a distance L from the tank bottom, there is a delay time between T ( t ) and the temperature registered by the sensor/transmitter TÂ’ ( t ). That delay time to ( t ) can be calculated as: () () LAt tot Ft (A.24) where At is the pipe cross section an L is the distance between the tank bottom and the sensor/transmitter position.
132 Appendix A (Continued) The temperature registered by the sensor/transmitter can be related to the output temperature as: '()(()) TtTttot (A.25) Regarding the sensor/transmitter it could be modeled as a first order differential equation: min() ()('())TTdct ctKTtT dt (A.26) where T and KT are the sensor/transmitter time constant and gain respectively. Tmin is the minimum reading of the sensor/transmitter. c(t) is the signal output from the sensor/transmitter sent to the controller. The control valve used to manipulate stream F1( t ) also can be modeled as a first order differential equation: 1 1() ()()vvdFt FtKmt dt (A.27) where v and Kv are the time constant and gain of the valve respectively. Table A2.1 shows all steady state values used for the mixing process.
133 Appendix A (Continued) Table A.2. Steady State Values for Mixing Process Parameter Steady State Values Units F1 0.8 m3/s F2 1.1 m3/s F 1.9 m3/s T1 80 oC T2 15 oC T 42.36 oC 1000 kg/m3 V 10 m3 Cv 1 kcal/oC kg Cp 1 kcal/oC kg CV 0.6 m3/m0.5 L 3 m At 0.005 m2 V 0.5 s Kv 0.016 (m3/s)/(%CO) KT 1.25 %TO%C T 0.5 s
134 Appendix B: DMCr Control Law Deduction The equation to calculate the control moves is based on the fact that the controller output must compensate for present and future errors: AME P (B.1) 1 1 2 2 1 1.0 A0. A .. . .. . .A .v v vCH PH PH PHCHCHm e m e Kp Kp Kp m e KpKp where M is the control output vector and EP is the predicted error vector. The goal is to determineM to eliminate the error EP. Because the Prediction Ho rizon (PH) is greater than the Controller Horizon (CH), there ar e more equations than unknowns, and an exact solution can not be obtained. For that r eason an optimization model must be used: AMERP (B.2) where R is the residual for no compensated erro r. The goal of the optimization is to find M that minimize the sum of square of re sidual. In terms of the vector R, the summation of the square can be expressed as RTR ,where RT denotes the R transpose. Equation (B.2) can be written as: AMERP (B.3) Therefore, the sum of square of residual in terms of A, M and EP is: RRAMEAMET T P P (B.4)
135 Appendix B (Continued) Applying the properties of matrix algebra: RRMAEAMETTTT P P (B.5) RRMAAMEAMMAEEETTTTTTT P PPP (B.6) To find M that minimizes RTR, it is necessary take the derivate with respect to M and set the equation equal to zero: RR 0 MT (B.7) Differentiating Eq. (B.6) and setting the result matrix equation equal to zero: AAMMAAEAEA00T TTTTT PP (B.8) Applying matrix transpose prope rties and grouping terms: MAAMAA2EA0TTTTT P (B.9) MAAEATTT P (B.10) Transposing both sides Eq. (B.10) MAAEATT TTT P (B.11) AAMAETT P (B.12)
136 Appendix B (Continued) Solving for M : 11AAAAMAAAETTTT P (B.13) 1MAAAETT P (B.14) The equation (B.14) is the expression proposed by Cutler and Ramaker as the control law for the DMC algorithm. When the suppression factor is used to change the DMCr response, the control move can be expressed as: 21M(AAI)AETT P (B.15) where is the suppression factor and I is th e identity matrix. The mathematical development that leads to Eq. (B.15) is analogous to that shown above.
137 Appendix C: Tuning Equations for DMCr Section 3.5.2 describes the procedure to find a tuning equation for the DMCr suppression factor. The equation presented, Eq 3.32, was tested with good results. The equation was developed using constant values of two design variables: Control Horizon ( CH ) and Sampling Time ( Ts ); the values used were CH = 5 and Ts = 0.1 The tests shown in Section 3.5.2 were also performed at these conditions. Therefore, it is necessary to test whether Eq. 3.32 can be used at different CH and Ts conditions. Figure C.1 presents the DMCr performance when is calculated using Eq. 3.32, in a control loop identical to that shown in Figure 3.2. The process parameters used were: KP = 0.5, = 1 and to = 0.2. Every Figure C.1 graph presents the DMCr performance when a particular set of CH and Ts is used in the control loop. CH was changed from 2 to 7, and Ts values were 0.05 0.10 and 0.15 respectively. Figure C.1 shows that values calculated using Eq. 3.32 do not generate unstable behavior in the control loop. However, as the CH increases and Ts is 0.15 the systems shows high frequency oscillations that finally settle down; this behavior indicates that the suppression factor is too small for those conditions. This observation is confirmed if the system response for a particular Ts is observed: for larger CH the response becomes more oscillatory, but less deviation is observed Table C.1 shows the total IAE for every test condition used in Figure C.1. IAE values ratify the observation discussed before. As CH increases, IAE decreases indicating less deviation from set point.
138 Appendix C (Continued) Figure C.1 also shows that as CH increases the controller response becomes more aggressive, showing high frequency oscillations; this is particularly evident for CH = 7 and Ts = 0.15 This form of response in DMCr is associated to small values of suppression factor adjusting the controller aggressiveness. Therefore, these results imply a direct relation between CH and suppression factor; the larger CH is the larger should be to smooth the controller response. This observation gives a key about the suppression factor dependency with CH and can be use to formulate the regression equation that is required. Another interesting fact observed from Table C.1 is the presence of the minimum IAE at CH = 5 and Ts = 0.05 contrary to that expected at CH = 5 and Ts = 0.1 (design conditions for Eq. 3.32). Figure C.2 represents graphically the IAE behavior as a function of CH and sampling ratio Ts / This result indicates that the sampling time affects the performance of suppression factors calculated using Eq. 3.32.
139 Appendix C (Continued) 0 50 60 75 0 50 60 75 0 50 60 75 0 50 60 75 0 50 60 75 0 50 60 75 0 50 60 75 0 50 60 75 0 50 60 75 0 50 60 75 0 50 60 75 0 50 60 75 0 50 60 75 0 50 60 75 0 50 60 75 0 50 60 75 0 50 60 75 0 50 60 75 CH=2 CH=3 CH=5 CH=4 CH=6 CH=7 Ts=0.05 Ts=0.1 Ts=0.15 Figure C.1. Comparison of DMCr Performance When CH and Ts Change and Eq. 3.32 is Used to Calculate
140 Appendix C (Continued) Table C.1. IAE Values from Simulation Presented in Figure C.1 CH Ts 2 3 4 5 6 7 0.05 15.5126 11.3177 9.5991 9.0807 9.1442 9.1377 0.10 18.2059 13.6164 11.4914 10.8556 10.6474 10.5765 0.15 24.5199 18.3731 15.2515 13.7699 13.2963 13.6881 2 3 4 5 6 7 0.05 0.1 0.15 5 10 15 20 25 Sampling Time Ts/ IAEMinimum IAE Control Horizon Figure C.2. Graphical Representation of IAE as CH and Ts/ Function
141 Appendix C (Continued) Because of the results showing the effects of CH and Ts on the suppression factor, it was decided to find a new tuning equation that incorporates these design parameters. To accomplish this goal, a similar procedure to that shown in Section 3.5.2 was developed, but this time incorporating CH and Ts in the experimental conditions. Table C.2 shows the factors used in the experiment. A total of 486 optimizations were performed using Eq. 3.31 as objective function: 0()()ssPPetmmtdt (3.31) A constant value of 5 was used for experimental conditions, because in the tests describe in Section 3.5.2 gave the best results. The FOPDT process implemented to run the Simulink and Matlab simulations is identical to that shown in Figure 3.2. Table C.2. Factors Used to Performed the Designed Experiment CH Ts/ P K /to 2 0.05 0.5 1 0.2 3 0.10 1.5 3 0.6 4 0.15 2.5 5 1 5 6 7
142 Appendix C (Continued) Once the optimal values of were found for all experimental conditions, the result were sorted by CH and Ts / values, in order to find regression equations for every combination of this design parameters. The general equation form that best fits the data is: 23 1 Pto K (C.1) Table C.3 presents the constant parameters for the 18 regression equations found, as well as the correlation factors, R2 and R2 adjusted, for each equation. With some exceptions, R2 and R2 adjusted values are over 80%, implying a good agreement between real and predicted values. Tuning equations with Eq. C.1 form were tested using FOPDT processes as well as nonlinear processes under appropriated CH and Ts conditions with very good results.
143 Appendix C (Continued) Table C.3. Constant Parameters and Correlation Values for Eq. C.1 CH Ts/ 1 2 3 R2 R2 adj 2 0.05 0.622210.648790.429930.9165 0.9013 2 0.10 0.819300.725490.572370.9284 0.9164 2 0.15 0.861270.843330.617850.9845 0.9821 3 0.05 1.163180.546660.526710.9325 0.9175 3 0.10 1.238330.664450.598810.9336 0.9225 3 0.15 1.263470.823770.622840.9664 0.9597 4 0.05 1.149570.469110.5242 0.481 4 0.10 1.880940.837310.656570.9353 0.9246 4 0.15 1.486630.84320.514730.9687 0.9631 5 0.05 1.396560.54690.6014 0.5616 5 0.10 2.095910.844770.592250.9397 0.9311 5 0.15 1.591570.862610.439360.9925 0.991 6 0.05 2.287460.676780.517980.8708 0.8473 6 0.10 2.089970.851750.484160.9118 0.9008 6 0.15 1.247290.749130.8058 0.7937 7 0.05 1.902770.784460.6928 0.6709 7 0.10 2.228370.908310.395860.8856 0.8713 7 0.15 1.260640.816120.8015 0.7898
144 Appendix C (Continued) Because the good results found, it was decided try to find a unique general equation including CH and Ts as design parameters. Using the data, a new regression analysis was performed with good results. The regression equation that best fits the data is: 3 4 2 51PtoCH K Ts (C.2) where the i are the constant values shown in Table C.4. The correlation factor R2 for the regression is 0.9308 and R2 adj is 0.9298. Table C.4. Constant Parameters for Eq. C.2 1 2 3 4 5 0.3234 0.78174 0.51627 0.93052 0.09253 Figure C.3 shows the plot of Residuals vs. Predicted values. The absence of recognizable patterns in that figure, added to the high value of R2 is statistical indicators that the regression equation can predict optimal suppression factors adequately. Eq. C.2 was tested tuning for control loops with FODPT, and nonlinear processes with good results; it can be used to tune either standard DMCr or implemented in the PDMCr to find the best values.
145 Appendix C (Continued) Figure C.3. Plot of Residual vs. Predicted Values Using Eq. C.2
146 Appendix C (Continued) Now, it can be compared the suppression factor predicted using Eq. C.2, with those obtained with the equation presented in Section 3.5.2, Eq. 3.32; of obviously, valid condition of CH and Ts for Eq. 3.32 should be used, which are 5 and 0.1 respectively. Figure C.4 shows the comparison between values obtained using both equations for a FOPDT model. 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Process Gain (%CO/%TO)Supression Factor Eq. 3.32 Eq. C.2 Figure C.4. Comparison of Suppression Fact or Prediction Using Eq. 3.32 and Eq. C.2
147 Appendix C (Continued) It is evident that both equations predict very similar values of under the same conditions. Therefore, Eq. 3.32 could be considered equivalent to Eq. C.2 when CH = 5 and Ts = 0.1
148 Appendix D: Modeling Error Indicators ( MEIs ). Definition and Calculation Section 3.5.3 describes the process to develop the fuzzy supervisor. The main goal of the supervisor is to estima te changes in process parameters: KP, and to based on the modeling error behavior. The way change s in the process parameters affect the modeling error were studied through 21 Mode ling Error Indicators originally proposed. One of the concepts involved in the development of these MEIs was that they should be dimensionless to avoid dependency on pr ocess condition. For th at reason all the MEIs that involved time units were divided by the process time constant, and those with %TO units were divided by the set point change us ed in the test. Figure D.1 shows graphically some of the MEIs proposed. The following paragraphs present the significance and calculation of the MEIs used. Maximum em Minimum peak em Maximum peak em Minimum em Time for Maximum em Time for Minimum em Stabilization time Difference: time for Maximum and time for Minimum X2 X1 0.1%TO -0.1%TOem (%TO)time (time units) Maximum em Minimum peak em Maximum peak em Minimum em Time for Maximum em Time for Minimum em Stabilization time Difference: time for Maximum and time for Minimum X2 X1 0.1%TO -0.1%TO Maximum em Minimum peak em Maximum peak em Minimum em Time for Maximum em Time for Minimum em Stabilization time Difference: time for Maximum and time for Minimum X2 X1 0.1%TO -0.1%TOem (%TO)time (time units) Figure D.1. Some Modeling Error Indicators ( MEIs )
149 Appendix D (Continued) D.1 2nd 6th and 10th Correlation Coefficients These three MEIs refer to the 2nd, 6th and 10th terms in the succession calculated using Eq. D.1: 2 1 Nk iik ii k N i iYYYY r YY (D.1) where Y1, Y2, ..., YN are observation recorded at time t1, t2, ..., tN, respectively. Observations are equispaced in time. Th is succession is called the Autocorrelation function, and it is a measure of linear asso ciation among observations. The greater the auto correlation value for a particular obser vation, the more related the value is to adjacent and near-adjacent observations. Gene rally, the autocorrelation function can be used to detect non-randomness in data or to identify an appropriate time series model if the data is not random. These applications of the correlation coefficien ts are not used in this research; they were tested only as possi ble index to detect ch anges in modeling error due to changes in process parameters. For the purpose of the Supervisor Modul e, the modeling error values were taken as the observations to ca lculate, using Eq. D.1, r for k equal to 2, 6 and 10; corresponding to the 2nd, 6th and 10th correlations coefficients of the modeling error. These values are a measure of how far are rela ted the modeling er ror values, and how far in time the relationship is present. The order of the autocorrelation coefficients (2nd, 6th and 10th) were chosen having in mind to calculate coefficients for equally spaced positions.
150 Appendix D (Continued) It was expected that the three correlation coeffi cients were different of zero, because they correspond to 2, 6 and 10 sampling periods after a set point change affects the process; which is not enough time to reach steady state, and the controller is taking decisions based precisely in how the modeling error is changing. D.2 Maximum em /cset This MEI is defined as the ratio between the maximum modeling error observed and the set point change used. This MEI is independent of the magnitude of the change in set point. The mathematical expression is: max() em cset (D.2) D.3 Minimum em /cset This MEI is defined as the ratio between the minimum modeling error observed and the set point change used. The MEI is independent of the magnitude of the set point change, the mathematical expression is: min() em cset (D.3)
151 Appendix D (Continued) D.4 Maximum em /Minimum em This MEI is defined as the ratio between the maximum modeling error observed and the minimum modeling error observed and the mathematical expression is: max() min() em em (D.4) D.5 Time for Maximum em / This MEI is defined as the ratio between th e time to reach the maximum modeling error and the process time constant The mathematical expression is: max() emtime (D.5) D.6 Time for Minimum em / This MEI is defined as the ratio between th e time to reach the minimum modeling error and the process dead time. The mathematical expression is: min() emtime (D.6)
152 Appendix D (Continued) D.7 Stabilization Time Between 0.1%/ TO This MEI is defined as the ratio between the stabilization time, which is defined the time that takes the modeling error to settle between 0.1%TO and the process time constant. The mathematical expression is: 0.1% emTOtime (D.7) D.8 Difference in Time for Maximum Mo deling Error and Time for Minimum Modeling Error/ This MEI is the ratio between the time di fference between the maximum and minimum modeling error occurren ces, and the process time c onstant. It is expressed as: ()() M aximumemMinimumemtimetime (D.8) D.9 Ratio Absolute Minimum Modeling Error/Absolute Maximum Modeling Error This MEI is the ratio between the absolute minimum modeling error over the absolute maximum modeling error; it is expressed as: () () M inem M axem (D.9)
153 Appendix D (Continued) D.10 Maximum Modeling Error Peak/cset This MEI is defined as the ratio between the maximum modeling error reached (positive or negative), and the set poin t change used. It is expressed as: M axem cset (D.10) D.11 Minimum Modeling Error Peak/cset It is the ratio between the absolute va lue of the smaller modeling error recorded and the set point change; it is expressed as: () M inem cset (D.11) D.12 Time for Maximum Peak/ This MEI is defined as the ratio between the time at maximum modeling error peak and the process time constant; it is expressed as: M axemtime (D.12)
154 Appendix D (Continued) D.13 Time for Minimum Peak/ This MEI is defined as the ratio betw een the time to reach the minimum modeling error peak and the process time consta nt; the mathematical expression is: M inemtime (D.13) D.14 Decay Ratio The decay ratio is the ratio by which the amplitude of the modeling error is reduced during a complete cycle. It is calculated as the ratio between two first consecutives peaks in the same direction: second peak() first peak() em em (D.14) D.15 Damping Ratio The Damping ratio is defined as the ratio between the actual damping to critical damping. It is indicative of how oscillatory is the signal behavior. It is calculated by the following expression: 2 2second peak() log first peak() second peak() 4log first peak() em em em em (D.15)
155 Appendix D (Continued) D.16 Ratio Minimum em / Maximum em This MEI is defined as the ratio between the minimum modeling error over the maximum modeling error; taking into account th e sign of those values. It expressed as: () () M inem M axem (D.16) D.17 Time Between Peaks/ This MEI is defined as the ratio between to consecutives two first peaks in the same direction and the process time constant. The mathematical expression is: second peak()first peak()ememtimetime (D.17)
156 Appendix E: ANOVA Tables for Modeling Error Indicator ( MEI ) From ANOVA tables it can be inferred wh at are the input variables with more influences over the response variable. For the experiment designed to determine the MEIs suitable to be used in the fuzzy supe rvisor, more than 60,000 simulations were performed. For every experimental conditi on the input variable s were the process parameters KP, and to as well as percentage vari ations on those variables, P K and to The change in set point,cset made for every experimental condition was also considered an input variable. The re sponse variables we re each of the 21 MEIs discussed in Appendix C. Therefore, it was possible pe rformed Analysis of Variance to each of them, to determine what of the input variables had more effect over the MEI considered. Tables E.1 to E.21 show the ANOVA tables for all MEI tested. Every ANOVA table presents a set of sta tistic values, as sum of squares, F probability, P value, etc. associated to each input variable, and their interactions; only main and second order interactions were consid ered. If a particular input variable have a statistical influence on the MEI its F probability value appears in the ANOVA table showing a large value, and its P value is smaller than 0.05, which means that with a 95% of certainty the input variab le influence the considered MEI For instances, if Table E.1 is considered, it can be noted that for the 2nd correlation coefficient, among the input variables which have more statisti cal influence can be mentioned: cset KP, / to etc. But the interaction between *PKto do not have influence over this MEI Thus, only when the MEI is influence by P K , to and their interaction with the other input
157 Appendix E (Continued) variables, it can be consider ed useful for the Fuzzy Supervisor Module. Based on this analysis some of the MEIs were discarded. For instance, Table E.12 shows that th e ratio, time for maximum menus time for minimum modeling error over pr ocess time constant, does not can be used for the Fuzzy Supervisor Module because statistically th e changes in process parameters do not influence it. Table 3.6 shows the MEIs selected to be used in the Fuzzy Supervisor Module; their correspondi ng ANOVA tables are presen ted in this appendix: 10th Correlation Coefficient, Table E.3. Maximum em /Cset Table E.4. Minimum em /Cset Table E.6. Modeling error period T, Table E.8 Modeling error angular frequency Table E.7. Time for Maximum em / Table E.9. Time for Minimum em / Table E.10. Stabilization time between 10%/ TO Table E.11. Difference between time for maximum and time for minimum/ Table E.12. Ratio Abs minimum/Abs maximum, Table E.13. Maximum peak/Cset Table E.14. Minimum peak/Cset Table E.15. Time for Maximum Peak/ Table E.16. Time for Minimum Peak/ Table E.17. Damping ratio, Table E.19
158 Appendix E (Continued) Table E.1. ANOVA Table for S econd Correlation Coefficient
159 Appendix E (Continued) Table E.2. ANOVA Table for 6th Correlation Coefficient
160 Appendix E (Continued) Table E.3. ANOVA Table for 10th Correlation Coefficient
161 Appendix E (Continued) Table E.4. ANOVA Table for Ra tio Maximum Modeling Error/ Cset
162 Appendix E (Continued) Table E.5. ANOVA Table for Maximu m/Minimum Modeling Error Ratio
163 Appendix E (Continued) Table E.6. ANOVA Table for Minimum Modeling Error Over Cset
164 Appendix E (Continued) Table E.7. ANOVA Table for Modeling Error Angular Frequency Omega
165 Appendix E (Continued) Table E.8. ANOVA Table for Modeling Error Period T
166 Appendix E (Continued) Table E.9. ANOVA Table for Ratio Time for Maximum Modeling Error Occurrence Over Tau
167 Appendix E (Continued) Table E.10. ANOVA Table for Ratio Time for Minimum Modeling Error Occurrence Over Tau
168 Appendix E (Continued) Table E.11. ANOVA Table for Ratio Stabiliza tion Time for 10% Modeling Error Over Tau
169 Appendix E (Continued) Table E.12. ANOVA Table for Ratio Time for Maximum Minus Time for Minimum Modeling Error Over Tau
170 Appendix E (Continued) Table E.13. ANOVA Table for Tatio Absolu te Minimum Over Absolute Maximum Modeling Error
171 Appendix E (Continued) Table E.14. ANOVA Table for Ratio Maxi mum Pick of Modeling Error Over Cset
172 Appendix E (Continued) Table E.15. Minimum Modeling Error Pick Over Cset
173 Appendix E (Continued) Table E.16. Ratio Time for Modeling E rror Maximum Pick Occurrence Over Tau
174 Appendix E (Continued) Table E.17. Ratio Time for Minimum Modeling Error Pick Over Tau
175 Appendix E (Continued) Table E.18. Decay Ratio for Modeling Error
176 Appendix E (Continued) Table E.19. Modeling Error Damping Ratio
177 Appendix E (Continued) Table E.20. Ratio Minimum over Maximum Modeling Error
178 Appendix E (Continued) Table E.21. Ratio Distance Between First Two Picks Over Tau
179 Appendix F: Regression Equation for Modeling Error Indicators To determine how the modeling error is affected by changes in process parameters, a total of 61,236 simulations were performed; a set of 21 Modeling Error Indicators ( MEIs ) were proposed to establish the relationship (see section 3.5.3). From the original 25 only 15 MEIs were selected using Analysis of Variance (ANOVA) as possible variables suitable to predict changes in process parameters (see Appendix E). The next step was to perform a nonlinear regression analysis using the data available to find nonlinear equation that relate the MEIs with the process parameters changes. Equations with the following form were found: (,,,,,)iPP M EIfKtoKto (F.1) These equations can be used to calculate the value of each MEI when the process parameters and their changes are known. However, these equations can be also used to perform the calculation backwards: if the MEI values and process parameters are known, the changes in process parameters (,,PKto ) can be calculated simultaneously, using three of these equations. This is the way that a set of three of these equations are used in Fuzzy Supervisor Module, to predict changes in process parameters. To chose the three equations to be used into the Fuzzy Supervisor Module, many combinations of them were tested until obtain a set with an appropriated prediction capacity. The three equations selected were those corresponding to the 10th Correlation Factor, Time for Maximum Peak/ and Time for Minimum Peak/ as Table 3.7 in Chapter 3 shows.
180 Appendix F (Continued) The summary of the 15 regression equations corresponding to each MEI their correlation coefficient (R2) and constant values (i) are shown in Tables F.1 to F.15 as follows: 10th Correlation Coefficient, Table F.1. Maximum em /Cset Table F.2. Minimum em /Cset Table F.3. Modeling error period T Table F.4 Modeling error angular frequency Table F.5. Time for Maximum em / Table F.6. Time for Minimum em / Table F.7. Stabilization time between 10%/ TO Table F.8. Difference between time for maximum and time for minimum/ Table F.9. Ratio Abs minimum/Abs maximum, Table F.10. Maximum peak/Cset Table F.11. Minimum peak/Cset Table F.12. Time for Maximum Peak/ Table F.13. Time for Minimum Peak/ Table F.14. Damping ratio, Table F.15
181 Appendix F (Continued) Table F.1. 10th Correlation Coefficient 1234567891010 th PPtoto CoefKtoKto R2= 0.8433 1 -0.64086 6 0.05056 2 -0.00542 7 0.08960 3 0.26446 8 0.000924 4 0.01052 9 0.000354 5 0.00154 10 0.00409 Table F.2. Ratio Maximum em Over Set Point Change 12345678910()PPMaxemtoto KtoKto cset R2= 0.9886 1 0.07135 6 -0.0030774 2 -0.017417 7 0.011212 3 -0.093384 8 -5.99x10-5 4 0.0006615 9 7.77x10-5 5 -0.0006468 10 0.0003675
182 Appendix F (Continued) Table F.3. Ratio Minimum em Over Set Point Change 12345678910()PPMinemtoto KtoKto cset R2= 0.9397 1 -0.022713 6 0.00037711 2 0.007638 7 -0.0065452 3 0.036232 8 -4.932x10-5 4 0.00059836 9 3.651x10-5 5 -0.00037471 10 -6.491x10-5 Table F.4. Modeling Error Period T 1234567****Pto TcsetcsetcsetcstKcsetcsetto R2= 0.562 1 2.8937 6 -0.007 2 -0.53758 7 -17.862 3 -2.1104 4 -0.11123 5 0.11403
183 Appendix F (Continued) Table F.5. Modeling Error Angular Frequency 12345678910 PPtoto KtoKto R2= 0.879 1 -6.7693 6 0.014782 2 2.4533 7 -1.575 3 8.7758 8 -0.050432 4 0.27285 9 0.027248 5 -0.1648 10 0.0059402 Table F.6. Ratio Time for Maximum em Over Process Time Constant max() 12345678910 em PPtime toto KtoKto R2= 0.9954 1 6.2648 6 -0.03375 2 -2.3297 7 0.21094 3 0.78906 8 -0.003125 4 0.003125 9 0.00375 5 -0.00375 10 0.03375
184 Appendix F (Continued) Table F.7. Ratio Time for Minimum em Over Process Time Constant max() 12345678910 em PPtime toto KtoKto R2= 0.8781 1 3.9813 6 0.14062 2 -1.202 7 0.18164 3 5.3105 8 0.0054685 4 0.19766 9 -0.044844 5 -0.030781 10 -0.003125 Table F.8. Ratio Time for em Settle Between 0.1%TO Over Process Time Constant 0.1% 12345678910 stabTO PPtime toto KtoK R2= 0.7661 1 8.5574 6 -0.47313 2 -2.7961 7 0.33984 3 -2.3711 8 -0.0015625 4 0.092813 9 0.0125 5 -0.08375 10 0.0660625
185 Appendix F (Continued) Table F.9. Ratio Difference Between Time for Maximum and Minimum em Over Process Time Constant max()min() 1234567***emem Ptimetime to csetcsetcsetcsetKcsetto R2= 0.6231 1 0.36683 6 -0.0051 2 -0.018792 7 -2.6812 3 -0.35521 4 -0.017617 5 0.013817
186 Appendix F (Continued) Table F.10. Ratio Absolute Minimum em Over Absolute Maximum em 12345678910() ()PPMinem toto KtoKto Maxem R2= 0.6101 1 0.19503 6 0.0074262 2 -0.071998 7 0.063166 3 -0.26392 8 -0.0013592 4 0.012384 9 0.001169 5 0.0097118 10 0.0015085 Table F.11. Ratio Maximum em Peak Over Set Point Change 12345678910 PPMaxPeacktoto KtoKto cset R2= 0.989 1 0.1425 6 -0.0031 2 -0.0173 7 0.012 3 -0.0934 8 -6.01x10-5 4 0.0006615 9 7.79x10-5 5 -0.0006468 10 0.00042
187 Appendix F (Continued) Table F.12. Ratio Minimum em Peak Over Set Point Change 1234567****PMinPeackto csetcsetcsetcsetKcsetcsetto cset R2= 0.684 1 0.002053 6 -1.35x10-5 2 -0.00025055 7 0.0066185 3 -0.0013673 4 -3.66x10-5 5 2.19x10-5 Table F.13. Ratio Time for Maximum Peak Over Process Time Constant max Peack 12345678910 PPtime toto KtoKto R2= 0.9954 1 6.2648 6 -0.03375 2 -2.3297 7 0.21094 3 0.78906 8 -0.003125 4 0.003125 9 0.0375 5 -0.0375 10 0.03375
188 Appendix F (Continued) Table F.14 Ratio Time for Minimum em Peak Over Process Time Constant minPeack 12345678910 PPtime toto KtoKto R2= 0.9886 1 0.07135 6 -0.0030774 2 -0.017417 7 0.011212 3 -0.093384 8 -5.99x10-5 4 0.0006615 9 7.77x10-5 5 -0.0006468 10 0.0003675
189 Appendix F (Continued) Table F.15. Modeling Error Damping Ratio 123456 7891011 121314151617 1819202122* ***P P PP PPto DampRatiocsetKto to csetcsetcsetKcsetcsetto tototo KtoK to toKKtoto R2= 0.898 1 -0.0063477 12 -0.040707 2 0.029939 13 -0.00043751 3 0.4256 14 0.00062774 4 -0.0087799 15 0.0020847 5 -0.023314 16 -0.0030808 6 0.002547 17 -0.0048546 7 0.00094375 18 -0.015759 8 0.0047188 19 0.0013646 9 9.1494x10-5 20 0.00044147 10 -0.00010288 21 -0.00061241 11 -0.00013752 22 -0.14192
About the Author Edinzo Jess Iglesias Snchez received his Bachelor in Chemical Engineering from Universidad de Los Andes (ULA), Me rida, Venezuela in 1993. In the same year start his activities as Instruct or in the department of Chem ical Engineering at ULA. In 1998 Edinzo obtains his Master Degree in Chem ical Engineering at ULA. In 2000 Edinzo is selected by ULA to complete doctoral studi es. Edinzo is sent to University of South Florida to the Ph. D. program in the Chemi cal Engineering Department, to work in the Automatic Process Control area. His future interests include doing resear ch in automatic process control of chemical process using Fuzzy Logic applications As well as start a research laboratory in his home country.