A PARTICLE SWARM OPTIMIZATION APPROACH FOR TUNING OF SISO PID CONTROL LOOPS NELENDRAN PILLAY

A PARTICLE SWARM OPTIMIZATION APPROACH FOR TUNING OF SISO PID CONTROL LOOPS NELENDRAN PILLAY 2008

A PARTICLE SWARM OPTIMIZATION APPROACH FOR TUNING OF SISO PID CONTROL LOOPS By Nelendran Pillay Student Number: 19752683 Thesis submitted in comliance with the requirements for the Master s Degree in Technology: Electrical Engineering Light Current DURBAN UNIVERSITY OF TECHNOLOGY DEPARTMENT OF ELECTRONIC ENGINEERING This thesis reresents my own work N. Pillay APPROVED FOR FINAL SUBMISSION Suervisor: Dr. P. Govender Date Det. of Electronic Engineering Durban University of Technology ii

Table of Contents ABSTRACT... i ACKNOWLEDGEMENTS... ii LIST OF FIGURES... iii LIST OF TABLES... viii LIST OF ABBREVIATIONS... xi LIST OF SYMBOLS... xiii Chater 1 Introduction and Overview of the Study 1.1 Introduction... - 1-1.2 Motivation for the study... - 1-1.3 Focus of the study... - 2-1.4 Objectives of the study... - 3-1.5 Thesis overview... - 3 - Chater 2 Overview of PID Control 2.1 Introduction... - 5-2.2 Control Effects of Proortional, Integral and Derivative Action...- 5-2.2.1 Proortional control... - 5-2.2.2 Integral control (Reset control)... - 9-2.2.2.1 Integral action as automatic reset... - 10-2.2.2.2 Undesirable effects of Integral Control... - 13-2.2.3 Derivative control (Rate or Pre-Act control)... - 14-2.2.3.1 D-Action as Predictive Control... - 15-2.3 PID Algorithms... - 17-2.4 Performance evaluation criteria... - 20-2.5 Summary and conclusion... - 22 - i

Chater 3 Tyical Process Control Models 3.1 Introduction... - 23-3.2 Dynamics associated with the selected rocess models...- 23-3.3 A brief overview of integrating rocesses (Self-Regulating Processes)...- 25-3.4 Problems exerienced with tuning rocesses having unstable oles and dead-time... - 26-3.5 Summary and conclusion... - 28 - Chater 4 PID Tuning 4.1 Introduction... - 30-4.2 Ziegler-Nichols Tuning... - 31-4.2.1 ZN closed-loo tuning method (Ultimate gain and ultimate eriod method)... - 32-4.2.2 ZN oen-loo tuning method (Process reaction curve method)...- 34-4.2.3 Assessing the efficacy of Ziegler-Nichols tuning rules for dead-time dominant rocess... - 36-4.3 Cohen-Coon tuning (Oen-loo tuning)... - 38-4.3.1 Comarison between ZN and CC Tuning... - 39-4.4 Åström - Hägglund Gain and Phase Method (Closed-Loo Method)...- 40-4.5 Poulin-Pomerleau Tuning Method for Second-Order Integrating Process having Dead-Time (SOIPDT) - (Oen-Loo Tuning)... - 45-4.6 De Paor-O Malley Tuning for First-Order Oen-Loo Unstable Processes having Dead-Time (FODUP)... - 49-4.7 Venkatashankar-Chidambaram Tuning Method for First-Order Oen-Loo Unstable Processes Having Dead-Time (FODUP)... - 52-4.8 Summary and conclusion... - 55 - ii

Chater 5 Evolutionary Comutation and Swarm Intelligence Paradigms 5.1 Introduction... - 58-5.2 Evolutionary Comutation... - 58-5.3 An Overview of Genetic Algorithms... - 60-5.3.1 Premature convergence of Genetic Algorithms... - 62-5.4 Swarm Intelligence... - 62-5.4.1 Ant Colony Otimization... - 63-5.4.2 Background to Particle Swarm Otimization... - 63-5.4.2.1 The basic PSO algorithm... - 65-5.4.3 Variations to the PSO algorithm... - 68-5.4.4 Stes in imlementing the PSO method.... - 70-5.4.5 Selection of the search method.... - 73-5.4.6 Selection of termination method.... - 73-5.4.7 Factors affecting PSO erformance... - 73-5.5 Comarison between the GA and PSO... - 74-5.6 Summary and conclusion... - 75 - Chater 6 PSO Tuned PID Control 6.1 Introduction... - 76-6.2 Descrition of the PSO tuning methodology... - 76-6.2.1 Alication of PSO for PID Tuning... - 77-6.2.2 Position of the PSO algorithm within the selected control loo...- 78-6.3 Statistical Evaluation of the Dynamical Behaviour of Intelligent Agents...- 79-6.4 Summary and conclusion... - 80 - Chater 7 Simulation Study of PSO Performance for Process Control 7.1 Introduction... - 81-7.2 Preliminaries to the evaluation of PSO erformance...- 82-7.3 Process models used in the simulation tests... - 83 - iii

7.4 Results of PSO arameter variation... - 85-7.5. Observing the effects of varying PSO arameters...- 89-7.5.1 Variation in Swarm Size (See Figure 7.1)... - 89-7.5.2 Variation of Velocity Maximum (See Figure 7.2)...- 89-7.5.3 Variation of Social and Cognitive Acceleration Constants (See Figure 7.3 and Figure 7.4)... - 90-7.6 Summary and conclusion... - 90 - Chater 8 Simulation Studies to Comare the Performance of PSO vs. Other Tuning Techniques 8.1 Introduction... - 91-8.2 Preliminaries to the exeriments... - 92-8.3 Exeriments... - 95-8.3.1 Exeriment 8.1: Tuning of FOPDT rocess for otimal setoint tracking.- 95-8.3.2 Objective... - 95-8.3.3Methodology... - 95-8.3.4 Observations and analysis of results... - 95-8.4 Exeriment 8.2: Tuning of SOPDT rocess for otimal setoint tracking... - 100-8.4.1 Objective... - 100-8.4.2 Methodology for exeriment... - 100-8.4.3 Observations and analysis of results... - 101-8.5 Exeriment 8.3: Tuning of SOIPDT rocess for otimal setoint tracking.- 105-8.5.1 Objective... - 105-8.5.2 Methodology... - 105-8.5.3 Observations and analysis of results... - 105-8.6 Exeriment 8.4: Tuning of FODUP rocess for otimal setoint tracking..- 108-8.6.1 Objective... - 108-8.6.2 Methodology for exeriment... - 108-8.6.3 Observations and analysis of results... - 108-8.7 Exeriment 8.5: Tuning of FOPDT rocesses for setoint tracking and disturbance rejection.... - 111 - iv

8.7.1 Objective... - 111-8.7.2 Methodology... - 111-8.7.3 Observations and analysis of results... - 111-8.8 Exeriment 8.6: Tuning of SOPDT rocesses for setoint tracking and disturbance rejection.... - 114-8.8.1 Objective... - 114-8.8.2 Methodology... - 114-8.8.3 Observations and analysis of results... - 114-8.9 Exeriment 8.7: Tuning of SOIPDT rocesses for setoint tracking and disturbance rejection.... - 116-8.9.1 Objective... - 116-8.9.2 Methodology... - 116-8.9.3 Observations and analysis of results... - 116-8.10 Exeriment 8.8: Tuning of FODUP rocesses for setoint tracking and disturbance rejection.... - 118-8.10.1 Objective... - 118-8.10.2 Methodology... - 118-8.10.3 Observations and analysis of results... - 118-8.11 Summary and conclusion... - 120 - Chater 9 Offline Tuning for Process Control 9.1 Introduction... - 122-9.2 Basic descrition of the Process Control Plant used in the study...- 122-9.3 Interfacing the lant to the PC based controllers... - 124-9.4 Preliminaries for the real-time exeriments... - 126-9.5 Pressure control loo... - 127-9.5.1 Results and observations... - 128-9.6 Flow Control... - 130-9.6.1 Results and observations... - 132-9.7 Level Control... - 134-9.7.1 Results and observations... - 135 - v

9.8 Summary and conclusion... - 137 - Chater 10 Online Tuning for Real-Time Positional Control 10.1 Introduction... - 138-10.2 Positioning servo-system... - 138-10.2.1 PID control structure used for the ositional servo-system...- 140-10.2.2 Positioning servo-system control loo... - 140-10.2.3 Model of the armature controlled DC motor and gear mechanism... - 141-10.3 Evaluating PSO Performance for Offline Tuning...- 143-10.3.1 Observations and analyses of results... - 143-10.3.1.1 Controller tuned for setoint tracking... - 143-10.3.1.2 System tuned for disturbance rejection... - 145-10.4 Evaluating PSO erformance using online PSO Tuning...- 147-10.4.1 Observations and analysis of results... - 149-10.4.1.1 System tuned for setoint tracking... - 149-10.4.1.2 System tuned for disturbance rejection... - 151-10.5 Summary and conclusion.......-152- Chater 11 Summary of Study, Recommendations and Conclusion 11.1 Introduction... - 153-11.2 PSO Tuning... - 154-11.3 Advantages of the PSO... - 155-11.3.1 Imroved Process Behaviour... - 155-11.3.2 Attractive features of PSO Based PID Tuning...- 155-11.3.3 Fixed PSO Oerating Parameters for Imroved Reeatability...- 156-11.4 Recommendations for further research... - 157-11.5 Summary and conclusion... - 157 - vi

References... - 158 - Aendix A... - 166 - Aendix B... - 167 - Aendix C... - 175 - Aendix D... - 182 - vii

ABSTRACT Linear control systems can be easily tuned using classical tuning techniques such as the Ziegler-Nichols and Cohen-Coon tuning formulae. Emirical studies have found that these conventional tuning methods result in an unsatisfactory control erformance when they are used for rocesses exeriencing the negative destabilizing effects of strong nonlinearities. It is for this reason that control ractitioners often refer to tune most nonlinear systems using trial and error tuning, or intuitive tuning. A need therefore exists for the develoment of a suitable tuning technique that is alicable for a wide range of control loos that do not resond satisfactorily to conventional tuning. Emerging technologies such as Swarm Intelligence (SI) have been utilized to solve many non-linear engineering roblems. Particle Swarm Otimization (PSO), develoed by Eberhart and Kennedy (1995), is a sub-field of SI and was insired by swarming atterns occurring in nature such as flocking birds. It was observed that each individual exchanges revious exerience, hence knowledge of the best osition attained by an individual becomes globally known. In the study, the roblem of identifying the PID controller arameters is considered as an otimization roblem. An attemt has been made to determine the PID arameters emloying the PSO technique. A wide range of tyical rocess models commonly encountered in industry is used to assess the efficacy of the PSO methodology. Comarisons are made between the PSO technique and other conventional methods using simulations and real-time control. i

ACKNOWLEDGEMENTS The work resented in this thesis was carried out under the suervision of Dr. P. Govender. My gratitude and sincere areciation goes out to him for his assistance, valuable contribution and guidance throughout the study. His challenging questions and imaginative inut greatly benefited the work. I also wish to exress my sincere gratitude and areciation to the following: The National Research Foundation (NRF) for their financial assistance. Mr. C. Reinecke from the Durban University of Technology (DUT): Deartment of Electronic Engineering, for his sound advice and valuable suggestions. Mr. K. Moorgas for his technical assistance regarding dedicated PC equiment. A secial note of thanks to my arents for their unwavering suort and guidance. Finally, thanks to Theresa Padayachee for your tolerance and atience while I sent many hours glued in front of the PC. You have shown great understanding and given selfless suort. ii

LIST OF FIGURES FIGURE 2.1: PROPORTIONAL CONTROLLER WITHIN A CLOSED-LOOP FEEDBACK CONTROL SYSTEM... - 7 - FIGURE 2.2: CONTROL EFFECT OF VARYING P-ACTION 1 G =... - 9-3 ( s + 1) FIGURE 2.3: PROPORTIONAL CONTROLLER WITH AN INTEGRATOR AS AUTOMATIC RESET... - 10 - FIGURE 2.4: CONTROL EFFECTS OF VARYING INTEGRAL ACTION 1 G =... - 12-3 ( s + 1) FIGURE 2.5: INTERPRETATION OF DERIVATIVE ACTION AS PREDICTIVE CONTROL... - 16 - FIGURE 2.6: SIMULATION OF A CLOSED-LOOP SYSTEM WITH PID CONTROL 1 G =... - 17-3 ( s + 1) FIGURE 2.7: NON-INTERACTING PID... - 18 - FIGURE 2.8: INTERACTING PID... - 18 - FIGURE 2.9: PARALLEL NON-INTERACTING PID... - 19 - FIGURE 3.1: SISO SYSTEM WITH UNITY FEEDBACK... - 23 - FIGURE 3.2: RESPONSE TRAJECTORIES FOR SELF-REGULATING (STABLE), MARGINALLY STABLE AND UNSTABLE PROCESSES... - 27 - FIGURE 4.1: RESPONSE CURVE FOR QUARTER WAVE DECAY RATIO... - 32-1 = ( s + 1) FIGURE 4.2: CLOSED-LOOP STEP RESPONSE OF G WITH K = [2,8], T = AND T = 0... - 33 - c i d FIGURE 4.3: OPEN-LOOP PROCESS REACTION CURVE FOR A STEP CHANGE... - 35 - FIGURE 4.4: STEP RESPONSE USING ZN OPEN-LOOP AND CLOSED-LOOP TUNING FOR A DEAD-TIME DOMINANT PROCESS 3 ex( 5s) G =... - 38-3 ( s + 1) iii

FIGURE 4.5A: NYQUIST PLOT OF STABLE SYSTEM SHOWING GAIN AND PHASE MARGINS... - 41 - FIGURE 4.5B: NYQUIST PLOT OF UNSTABLE SYSTEM SHOWING GAIN AND PHASE MARGINS... - 41 - FIGURE 4.6: RELAY FEEDBACK SYSTEM... - 43 - FIGURE 4.7: TYPICAL OPEN-LOOP FREQUENCY RESPONSE FOR SECOND-ORDER INTEGRATING PROCESS WITH TIME DELAY IN CASCADE WITH A PI CONTROLLER... - 46 - FIGURE 4.8: OPTIMAL M r, ACCORDING TO THE ITAE CRITERION FOR SOIPDT PROCESS AS A FUNCTION OF THE FIGURE 4.9: NYQUIST DIAGRAM FOR OPEN-LOOP UNSTABLE PROCESS L T RATIO... - 47 - (EQUATION 4.22)... - 49 - FIGURE 4.10: ITERATIVE ALGORITHM FOR DETERMINATION OF δ c... - 50 - FIGURE 5.1: CONCEPT OF MODIFICATION OF A SEARCHING POINT BY PSO... - 67 - (KENNEDY AND EBERHART, 1995)... - 67 - FIGURE 5.2: STEPS IN PSO (EBERHART AND KENNEDY, 1995)... - 72 - FIGURE 6.1: POSITION OF SWARM AGENT WITHIN A 3-D SEARCH SPACE... - 76 - FIGURE 6.2: POSITIONING OF THE PSO OPTIMIZATION ALGORITHM WITHIN A SISO SYSTEM... - 79 - FIGURE 7.1: ADJUSTMENT OF SWARM SIZE (2; 5; 10; 20; 40; 50)... - 87 - FIGURE 7.2: ADJUSTMENT OF VELOCITY MAXIMUM (0.1; 1; 5; 10)... - 87 - FIGURE 7.3: ADJUSTMENT OF COGNITIVE ACCELERATION (1; 2.05; 3; 4; 5)... - 88 - FIGURE 7.4: ADJUSTMENT OF SOCIAL ACCELERATION (1; 2.05; 3; 4; 5)... - 88 - FIGURE 8.1: PROCESS CONTROL LOOP USED IN THE EXPERIMENTS... - 93 - FIGURE 8.2: FOPDT SYSTEM RESPONSE FOR EXPERIMENT 8.1 ex ( 0.2s) G =... - 97 - ( s + 1) FIGURE 8.3: PSO VS. GA - EXP. 8.1 RESULTS FOLLOWING 10 TRIALS ex ( 0.2s) G =... - 98 - ( s + 1) iv

FIGURE 8.4: SOPDT SYSTEM RESPONSES FOR EXPERIMENT 8.2 ex( 0.5s) G =... - 103-2 s + 2s + 1 FIGURE 8.5: ITAE CONVERGENCE FOR PSO VS. GA ex( 0.5s) G =... - 103-2 s + 2s + 1 FIGURE 8.6: STATISTICAL ANALYSIS FOR PSO VS. GA ex( 0.5s) G =... - 104-2 s + 2s + 1 FIGURE 8.7: SOIPDT SYSTEM RESPONSES FOR EXPERIMENT 8.3 ex( 0.2s) G =... - 107 - s( s + 1) FIGURE 8.8: FODUP SYSTEM RESPONSES FOR EXPERIMENT 8.4 ex( 0.2s) G =... - 109 - ( s 1) FIGURE 8.9: FOPDT SYSTEM RESPONSES FOR SETPOINT TRACKING AND DISTURBANCE REJECTION (EXPERIMENT 8.5) ex( 0.2s) G =... - 112 - ( s + 1) FIGURE 8.10: SOPDT SYSTEM RESPONSES FOR EXPERIMENT 8.6 ex( 0.5s) G =... - 115-2 s + 2s + 1 FIGURE 8.11: SOIPDT SYSTEM RESPONSES FOR EXPERIMENT 8.7 ex( 0.2s) G =... - 117-2 ( s + s) FIGURE 8.12: FODUP SYSTEM RESPONSES FOR EXPERIMENT 8.8 ex( 0.2s) G =... - 119 - ( s 1) FIGURE 9.1: PROCESS PLANT USED FOR THE TESTS... - 123 - FIGURE 9.2: P&ID OF THE PLANT UNDER STUDY... - 124 - FIGURE 9.3: INTERFACE BETWEEN PLANT AND PC... - 125 - v

FIGURE 9.4: MATLAB SIMULINK BASED PID CONTROLLER FOR REAL-TIME CONTROL... - 126 - FIGURE 9.5: CLOSED-LOOP STEP RESPONSES OF THE PRESSURE CONTROL LOOP USING ZN, CC, GA AND PSO TUNING PARAMETERS 0.62ex( 0.1s) =... - 129 - (0.5s + 1) G ressure FIGURE 9.6: CLOSED-LOOP STEP RESPONSE OF THE FLOW CONTROL LOOP WITH K = 7, T = AND T = 0 c i d 0.5ex( 6.5s) =... - 130-2 1.24s + 3.5s + 1 G flow FIGURE 9.7: CLOSED-LOOP STEP RESPONSES OF THE FLOW CONTROL LOOP USING ZN, AH, GA AND PSO TUNING PARAMETERS 0.5ex( 6.5s) =... - 133-2 1.24s + 3.5s + 1 G flow FIGURE 9.8: CLOSED-LOOP STEP RESPONSES OF THE LEVEL CONTROL LOOP USING PP, GA AND PSO TUNING PARAMETERS 0.02ex( 3s) =... - 136 - s(0.76s + 1) G level FIGURE 10.1: SCHEMATIC OF SERVO CONTROL SYSTEM... - 139 - FIGURE 10.2: SCHEMATIC OF THE POSITIONAL SERVO-MECHANISM... - 141 - FIGURE 10.3: FEEDBACK CONTROL LOOP FOR THE POSITIONAL SERVO-MECHANISM.. - 142 - FIGURE 10.4: CLOSED-LOOP SETPOINT RESPONSE OF THE POSITIONAL SERVO-MECHANISM USING OFF-LINE TUNING 9.65ex( 0.1s) G =... - 145 - s(0.01s + 1) FIGURE 10.5: CLOSED-LOOP SETPOINT AND DISTURBANCE RESPONSE OF THE POSITIONAL SERVO-MECHANISM USING OFF-LINE TUNING 9.65ex( 0.1s) G =... - 147 - s(0.01s + 1) vi

FIGURE 10.6: CLOSED-LOOP SETPOINT RESPONSE OF THE POSITIONAL SERVO-MECHANISM. (ON-LINE TUNING)... - 150 - FIGURE 10.7: CLOSED-LOOP DISTURBANCE REJECTION RESPONSE OF THE POSITIONAL SERVO-MECHANISM. (ON-LINE TUNING)... - 151 - vii

LIST OF TABLES TABLE 2.1: SUMMARY OF PERFORMANCE INDICES... - 22 - TABLE 4.1: ZIEGLER-NICHOLS CLOSED-LOOP TUNING PARAMETER (ZIEGLER AND NICHOLS, 1942)... - 34 - TABLE 4.2: ZIEGLER-NICHOLS OPEN-LOOP TUNING PARAMETER (ZIEGLER AND NICHOLS, 1942)... - 35 - TABLE 4.3: ZIEGLER-NICHOLS OPEN-LOOP AND CLOSED-LOOP TUNING PARAMETERS FOR ex( 5s) G =... - 37-3 ( s + 1) TABLE 4.4: COHEN COON TUNING FORMULA (OPEN-LOOP)... - 40 - TABLE 4.5A: SUMMARY OF TUNING RULES... - 56 - TABLE 4.5B: SUMMARY OF TUNING RULES... - 57 - TABLE 7.1: EMPIRICALLY DETERMINED PSO PARAMETERS... - 83 - TABLE 7.2A: FOPDT MODELS... - 84 - TABLE 7.2B: SOPDT MODELS... - 84 - TABLE 7.2C: SOIPDT MODELS... - 85 - TABLE 7.2D: FODUP MODELS... - 85 - TABLE 7.3: FIGURE REFERENCES TO SHOW THE EFFECTS OF VARYING S S, V MAX, C 1 AND C 2 PARAMETERS FOR THE SELECTED PROCESSES... - 86 - TABLE 8.1: PSO PARAMETERS... - 93 - TABLE 8.2: GA PARAMETER SETTINGS... - 94 - TABLE 8.3: PID PARAMETERS AND CLOSED-LOOP RESPONSE SPECIFICATIONS FOR EXPERIMENT 8.1 ex( 0.2s) G =... - 96 - ( s + 1) TABLE 8.4: PSO VS. GA EXP. 8.1 STATISTICAL ANALYSIS FOLLOWING 10 TRIALS ex ( 0.2s) G =... - 98 - ( s + 1) viii

TABLE 8.5: PID PARAMETERS AND CLOSED-LOOP RESPONSE SPECIFICATIONS FOR EXPERIMENT 8.2 ex( 0.5s) G =... - 102-2 s + 2s + 1 TABLE 8.6: STATISTICAL ANALYSIS OF THE 10 TRIAL RUNS FOR PSO VS. GA FOR EXPERIMENT 8.2 ex( 0.5s) G =... - 104-2 s + 2s + 1 TABLE 8.7: PID PARAMETERS AND CLOSED-LOOP RESPONSE SPECIFICATIONS FOR EXPERIMENT 8.3 ex( 0.2s) G =... - 106 - s( s + 1) TABLE 8.8: STATISTICAL ANALYSIS OF THE 10 TRIAL RUNS FOR PSO VS. GA FOR EXPERIMENT 8.3 ex( 0.2s) G =... - 107 - s( s + 1) TABLE 9.1: TUNING PARAMETERS FOR THE PRESSURE CONTROL LOOP 0.62ex( 0.1s) =... - 127 - (0.5s + 1) G ressure TABLE 9.2: STATISTICAL ANALYSIS OVER THE 10 TRIALS FOR PSO AND GA FOR PRESSURE CONTROL LOOP 0.62ex( 0.1s) =... - 128 - (0.5s + 1) G ressure TABLE 9.3: CLOSED-LOOP PERFORMANCE OF THE PRESSURE CONTROL LOOP USING ZN, CC, GA AND PSO TUNING METHODS 0.62ex( 0.1s) =... - 129 - (0.5s + 1) G ressure TABLE 9.4: TUNING PARAMETERS FOR THE FLOW CONTROL LOOP 0.5ex( 6.5s) =... - 131-2 1.24s + 3.5s + 1 G flow TABLE 9.5: STATISTICAL ANALYSIS OF THE 10 TRIAL RUNS FOR PSO AND GA FOR THE FLOW CONTROL LOOP 0.5ex( 6.5s) =... - 131-2 1.24s + 3.5s + 1 G flow ix

TABLE 9.6: CLOSED-LOOP PERFORMANCE OF THE FLOW CONTROL LOOP USING ZN, AH, GA AND PSO TUNING METHODS 0.5ex( 6.5s) =... - 133-2 1.24s + 3.5s + 1 G flow TABLE 9.7: TUNING PARAMETERS FOR THE LEVEL CONTROL 0.02ex( 3s) =... - 134 - s(0.76s + 1) G level TABLE 9.8: STATISTICAL ANALYSIS OF THE 10 TRIAL RUNS FOR PSO AND GA FOR THE LEVEL CONTROL LOOP 0.02ex( 3s) =... - 134 - s(0.76s + 1) G level TABLE 9.9: CLOSED-LOOP PERFORMANCE CHARACTERISTICS FOR LEVEL CONTROL LOOP USING PP, GA AND PSO TUNING 0.02ex( 3s) =... - 136 - s(0.76s + 1) G level TABLE 10.1: PID PARAMETERS OF THE POSITIONAL SERVO-MECHANISM FOR SETPOINT TRACKING 9.65ex( 0.1s) G =... - 144 - s(0.01s + 1) TABLE 10.2: CLOSED-LOOP RESPONSE SPECIFICATIONS FOR SETPOINT TRACKING 9.65ex( 0.1s) G =... - 144 - s(0.01s + 1) TABLE 10.3: PID PARAMETERS OF THE POSITIONAL SERVO-MECHANISM FOR DISTURBANCE REJECTION 9.65ex( 0.1s) G =... - 146 - s(0.01s + 1) x

LIST OF ABBREVIATIONS ACO AH AI ANN CC DCS DO EC EP ES FODUP FOPDT GA GP IAE ISE ITSE ITAE MIMO MISO P Ant Colony Otimization Åström and Hägglund Artificial Intelligence Artificial Neural Network Cohen and Coon Distributed Control System De Paor and O Malley Evolutionary Comutation Evolutionary Programming Evolutionary Strategies First order delayed unstable rocess First order lus dead time Genetic Algorithms Genetic Programming Integral absolute-error criterion Integral square-error criterion Integral-of-time multilied square-error criterion Integral-of-time-multilied absolute-error criterion Multile-Inut-Multile-Outut Multile-Inut-Single-Outut Proortional controller xi

PI PID PP PLC PSO SOIPDT SOPDT SI SIMO SISO VC ZN Proortional-integral controller Proortional-integral-derivative controller Poulin and Pomerleau Programmable Logic Controller Particle Swarm Otimization Second order integrating lus dead time Second order lus dead time Swarm Intelligence Single-Inut-Multile-Outut Single-Inut-Single-Outut Venkatashankar and Chidambaram Ziegler and Nichols xii

LIST OF SYMBOLS λ ε φ m Unstable ole Self regulating index / Controllability ratio Phase margin θ (s) Angular dislacement ω c Standard deviation Standard deviation of the ITAE index for the trial Gain crossover frequency ω Phase crossover frequency χ A m Mean Mean value of the ITAE index for the trial Mean number of iterations used to erform a search Mean time taken by PSO to comlete a search Constriction factor Gain margin b c 1 Controller bias Cognitive acceleration constants (self confidence) c 2 Social acceleration constant (swarm confidence) C (s) Angular dislacement of motor shaft d Relay amlitude D(s), d (t) Disturbance E (s), e (t) Error xiii

e ss Steady-state error f Viscous friction coefficient of the motor and load G c (s) Controller transfer function G (s) Process model transfer function gbest gbest n h iter Global best of the oulation Global best of the oulation for n dimension Unit ste function Current iteration iter max Maximum number of iterations J K K b Moment of inertia of motor and load Motor torque constant Back emf constant K c Proortional gain K d Derivative gain K i Integral gain K Process gain K u Ultimate gain L a Armature inductance L Process dead time L (s) Loo transfer function. M r Maximum eak resonance xiv

M (%) Maximum ercentage overshoot n N PB Number of dimensions to roblem Gear ratio Number of articles in oulation Proortional band best Personal best of agent best, Personal best of agent ifor n dimension i n P u Ultimate eriod q Number of arameters being otimized by PSO R a Armature resistance R (s), r (t) Setoint rand Random number between 0 and 1 1, 2 S (s) Sensitivity function S s Swarm size k +1 s Modified searching oint k s i, n Current osition of agent iat iteration k for n dimension s Position of agent iat iteration ( k + 1) for n dimension ( k+ 1) i, n T t c Samling interval Period of relay T d Derivative time constant t dist Time of unit ste disturbance xv

T i Integral time constant T Process time constant t r Rise-time (10% to 90%) t s Settling time (2%) t Time of inut unit ste uste U (s), u (t) Controller outut u Process inut roc V max Velocity maximum v best Velocity based on best v gbest Velocity based on gbest k +1 v Modified velocity k v i, n Velocity of agent iat current iteration k for n dimension v Velocity of agent i at iteration ( k + 1) for n dimension ( k+ 1) i, n w Inertia weight w max Initial weight w min Final weight Y (s), y (t) Process outut xvi

Chater 1 Introduction and Overview of the Study 1.1 Introduction The PID controller is regarded as the workhorse of the rocess control industry (Pillay and Govender, 2007). Its widesread use and universal accetability is attributed to its simle oerating algorithm, the relative ease with which the controller effects can be adjusted, the broad range of alications where it has reliably roduced excellent control erformances, and the familiarity with which it is erceived amongst researchers and ractitioners within the rocess control community (Pillay and Govender, 2007). In site of its widesread use, one of its main short-comings is that there is no efficient tuning method for this tye of controller (Åström and Hägglund, 1995). Given this brief background, the main objective of this study is to develo a tuning methodology that would be universally alicable to a range of oular rocesses that occur in the rocess control industry. 1.2 Motivation for the study Several tuning methods have been roosed for the tuning of rocess control loos, with the most oular method being that of Ziegler and Nichols (1942). Other methods include the methods of Cohen and Coon (1953), Åström and Hägglund (1984), De Paor and O Malley (1989), Zhuang and Atherton (1993), Venkatashankar and Chidambaram (1994), Poulin and Pomerleau (1996) and Haung and Chen (1996). In site of this large range of tuning techniques, to date there still seems to be no general consensus as to - 1 -

which tuning method works best for most alications (Liták, 1995). Some methods rely heavily on exerience, while others rely more on mathematical considerations (Liták, 1995). The Ziegler-Nichols method (1942) is the method most referred by rocess control ractitioners and alternate methods are often not alied in ractice because of the reluctance of control ersonnel to learn new techniques which they erceive as being comlicated, time consuming and laborious to imlement (Pillay and Govender, 2007). Also, some commonly used techniques do not erform sufficiently well in the resence of strong nonlinear characteristics within the control channel (Åström and Hägglund, 2004, Shinskey, 1994). 1.3 Focus of the study This study rooses the develoment of a tuning technique that would be suitable for otimizing the control of rocesses oerating in a single-inut-single-outut (SISO) rocess control loo. The SISO toology has been selected for this study because it is the most fundamental of control loos and the theory develoed for this tye of loo can be easily extended to more comlex loos. The research focuses on utilizing a softcomuting strategy, namely the article swarm otimization (PSO) technique that was first roosed by Kennedy and Eberhart (1995), as an otimization strategy to determine otimal controller arameters for PID control and its variants. The control erformance of loos tuned with the roosed PSO technique will also be comared to that of loos tuned using another soft-comuting technique, namely the genetic algorithm (GA) lus - 2 -

the methods mentioned reviously in the discussions. The GA was selected for comarison with the PSO because both are oulation based soft-comuting techniques. 1.4 Objectives of the study The objectives of the study are to: i) Develo a PSO based PID tuning methodology for otimizing the control of SISO rocess control loos. ii) Determine the efficacy of the roosed method by comaring the control erformance of loos tuned with the PSO method to that of loos tuned using the GA and the other so-called conventional methods of Ziegler-Nichols (1942), Cohen and Coon (1953), Åström and Hägglund (1984), De Paor and O Malley (1989), Venkatashankar and Chidambaram (1994) and Poulin and Pomerleau (1996). 1.5 Thesis overview This document is arranged as follows: Chater one gives an introduction and general overview of the study. It focuses on the research roblem and motivation for the study. Chater two rovides a brief outline on PID control and classical control theory. Chater three highlights tyical rocess models that are commonly encountered in rocesses control loos. Tyical nonlinear characteristics commonly found in most rocess control loos are reviewed and their effects on controller tuning and closed-loo erformance are also exlored in this chater. - 3 -

Chater four reviews selected PID controller tuning algorithms roosed in the literature. Chater five discusses soft comuting techniques such as evolutionary comutation (EC) and comares the intrinsic characteristics of GA s to that of the PSO. Chater six discusses the PSO tuning aroach. Chater seven describes a simulation that study focuses on the effects of PSO arameter variation. Chater eight describes a simulation study that comares the control erformance of PSO tuned systems to that of systems tuned using methodologies roosed in the literature. This chater also comares the control erformance of PSO tuned systems to GA tuned systems. In Chater nine the PSO method is alied offline to tune rocess control loos. Chater ten describes the real-time control of a ositional servo-mechanism. Chater eleven summarizes the findings of the study and rovides direction for further research that could be ursued in the field. Aendix A rovides the PSO source code used in all the exeriments. Aendix B gives details of the exeriments conducted in Chater 9. Aendix C rovides the loo diagram associated with the rocess control lant and details all the exeriments conducted for the PSO and GA tuning methods. Aendix D resents two conference aers and a draft journal aer arising from the work conducted in this study. - 4 -

Chater 2 Overview of PID Control 2.1 Introduction The PID controller is by far the most commonly used controller strategy in the rocess control industry (Åström and Hägglund, 1995; Åström et al., 2004). Its widesread use is attributed to its simle structure and robust erformance over a wide range of oerating conditions (Gaing, 2004). PID control is imlemented as either stand-alone control, or on DCS, SCADA and PLC control systems. The oularity and widesread use of PID control in the rocess control industry necessitates a detailed discussion on the fundamental theory that underins this tye of three-term rocess control. The dynamics associated with each control mode will also be discussed and the advantages and shortcomings associated with each tye of control will also be given. 2.2 Control Effects of Proortional, Integral and Derivative Action 2.2.1 Proortional control Proortional control is defined as the control action that occurs in direct roortion with the system error. The outut of a roortional controller varies roortionally to the system error according to (2.1): u ( t) = K e( t) b Equation (2.1) c + - 5 -

With regards to (2.1), u (t ) is the controller outut, e (t) is the error, b is the controller bias and K c is the controller gain (referred to as the roortional gain). Proortional control action resonds to only the resent error. For a small value of roortional gain, a large error yields a small corrective control action. Conversely, a large roortional gain will result in a small error and hence a large control signal. The controller bias is necessary in order to ensure that a minimum control action is always resent in the control loo. The gain of a roortional controller is usually described in terms of its roortional band (PB). The concet of the roortional band is inherited from neumatic controller and is defined as: 1 PB = 100% Equation (2.2) K c From (2.2), a large roortional gain Kc corresonds to a small roortional band PB, while a large PB imlies a small gain K. A ure P controller reduces error but does not c eliminate it (unless the rocess has naturally integrating roerties). With ure P control an offset between the actual and desired value will normally exist. This is illustrated as follows: Consider Figure 2.1: - 6 -

Ste R(s) E(s) Kc U(s) G(s) 1 Y (s) Proortional Gain Plant (Process) 1 Outut Figure 2.1: Proortional controller within a closed-loo feedback control system With regards to Fig. 2.1: The closed-loo transfer function of this control system is reresented by (2.3): Y KcG = Equation (2.3) R( s) 1 + K G c where G (s) is the transfer function of the rocess, R(s) and Y(s) reresents the inut and outut of the rocess, resectively and the error signal E(s) is: R( s) E( s) = Equation (2.4) 1 + K G c The action of the roortional controller usually results in an offset i.e. the difference between the desired outut and the actual outut of the system for rocesses that do not have any inherent integrating roerties. Under these conditions the steady-state error for the control system can be calculated using the final value theorem (2.5): - 7 -

e ss ( lim s + ) = [ se ( )] Equation (2.5) s o For a unit ste inut: e ss 1 1 s 1 1 ( + ) = lim = = s o s KcG s s KcGs 1 + ( ) lim 0 1 + 1 + KcGs Equation (2.6) This indicates the resence of a steady state error for (s) ± G, which is the case for systems with no inherent integrating roerties. From (2.6), the absolute value of the steady-state error can be reduced by sufficiently increasing K. However since K affects system stability and its dynamics, it will be limited by the stability constraints of the overall control system. A high value of which could result in instability (See Figure 2.2). c K c may lead to oscillations and large overshoots c It is for this reason that roortional control is often combined with integral control in order to eliminate offset, while alying the smaller values of the gain K. A tyical examle of system resonse using only roortional control is illustrated in Figure 2.2. c - 8 -

1.4 Closed loo ste resonse of G(s)=1/((s+1) 3 ) 1.2 Setoint Kc=2 Kc=5 1 Process Outut 0.8 0.6 0.4 0.2 Kc=1 0 0 2 4 6 8 10 12 14 16 18 20 Time (s) Figure 2.2: Control effect of varying P-action G 1 ( s + 1) = 3 2.2.2 Integral control (Reset control) Integral control is used in systems where roortional control alone is not caable of reducing the steady-state error within accetable bounds. Its rimary effect on a rocess control system is to ermanently attemt to gradually eliminate the error. The action of the integral controller is based on the rincile that the control action should exist as long as the error is different from zero, and it has the tendency to gradually reduce the error to zero. The integrator control signal (u i (t)) is roortional to the duration of the error and is given by: K t f t c f u i ( t) = e( t) dt = K i t e( t) dt Equation (2.7) T i t i i With regards to (2.7): T i is the integral time constant, K c is the roortional gain, K c /T i = K i is the gain of the integral controller, e (t) is the instantaneous error signal and the limits t i and t reresent the start and end of the error, resectively. The smaller the f integral time constant, the more often the roortional control action is reeated, - 9 -

therefore resulting in greater integral contribution toward the control signal. For a large integral time constant, the integral action is reduced. Integral control can be seen as continuously looking at the total ast history of the error by continuously integrating the area under the error curve and reducing any offset. The greater the error signal the larger the correcting action from the integral controller will be. 2.2.2.1 Integral action as automatic reset Integral action may be erformed as a kind of automatic reset (see Figure 2.3) and is equivalent to ermanently adjusting the bias of the roortional controller. Ste R(s) E(s) Kc Ui(s) G(s) Y (s) U(s) Proortional Gain 1 Plant (Process) Y (s) 1 Outut Ui(s) 1 T i.s+1 Integral Gain Y (s) Figure 2.3: Proortional controller with an integrator as automatic reset With regards to Figure 2.3, the control signal alied to the rocess is: U and = K E U Equation (2.8) i c + i U i( s) Ui = Equation (2.9) 1 + T s i - 10 -

Substituting (2.9) into (2.8) yields: U U U U i( s) U i( s) = KcE( s) + = U i( s) KcE( s) 1 + T s 1 + T s i = 1 1 KcE( s) 1 Ti s + i = 1 + T s 1 i KcE( s) 1 Ti s 1 Ti s + + i = i i U i 1 + Ti s Kc = K c = E( s) Ti s Ti s + K T s c i i T s Kc = T s i + K c = K s i + K c and U i Ki = + Kc E( s) Equation (2.10) s K where i. E( s) and KcE(s) reresents the control action of the integral and roortional s controller on the error signal, resectively. Proortional action comes into effect immediately as an error different from zero occurs. If the roortional gain is sufficiently high it will drive the error closer to zero. Integral control accomlishes the same control effect as the roortional control but with an infinitely high gain. This results in the offset eliminating roerty of integral action which can be illustrated by alying the final value theorem to the control structure of Figure 2.3. With regards to Figure 2.3: R( s) E( s) = Equation (2.11) 1 + G G c - 11 -

where K i Gc = K c + and s 1 R =. From (2.12) the integral controller drives the s error to zero: e ss ( + ) = lim s 0 1 1 Gc s + ( K [ se( s) ] = lim s = lim = 0 s 0 s 1 + G s 0 c s s + K ) G i Equation (2.12) e ( + ) = 0 indicates that the offset is zero and roves that integral action eliminates any ss offset. The control effects of integral action are illustrated in Figure 2.4. With regards to Figure 2.4, the roortional gain is ket constant ( K = 1 ) and the integral time is adjusted to illustrate the effects of the integral time constant. c 1.6 1.4 1.2 Closed loo ste resonse of G(s)=1/((s+1) 3 ) Ti=1 Ti=2 Ti=5 Process Outut 1 0.8 0.6 0.4 0.2 Ti=infinity 0 0 2 4 6 8 10 12 14 16 18 20 Time (s) Figure 2.4: Control effects of varying integral action G 1 ( s + 1) = 3-12 -

The integral time (T i ) constant is varied within the range T i = [1,2,5, ]. The case when T =, corresonds to ure roortional control and is identical to K=1 in Figure 2.2, i where the steady-state error is 50%. The steady-state error is removed when T i has finite value. For large values of the integration time constant, the resonse gradually moves towards the setoint. For small values of T i, the resonse is faster but oscillatory. 2.2.2.2 Undesirable effects of Integral Control Although integral control is very useful for removing steady-state errors it is also resonsible for sometimes introducing undesirable effects into the control loo in the form of increase settling time, reduced stability and integral windu (Govender, 1997). A short exlanation of each of these undesirable effects is discussed. Increased settling time: An increase of the closed-loo system settling time is usually caused by the increased oscillations as a consequence of the resent integral action. Reduced stability: The resence of the integral action may lead to increased oscillations within the control loo. These oscillations generally have a tendency to move the system towards the boundary of instability. In some cases these oscillations will result in the loo becoming unstable. Integral windu: Integrator windu occurs when the integral controller calls for a control action that the rocess actuator cannot roduce because of its saturated state. This socalled integrator windu state results in severe overshoots in the controlled variable. - 13 -

2.2.3 Derivative control (Rate or Pre-Act control) In linear roortional control the strength of the control action is directly roortional to the magnitude of the error signal and P-action becomes assertive only when a significant error has occurred. The integral controller erforms corrective action for as long as an error is resent but its gradual ram shaed resonse means that significant time exires before it roduces a sizeable control resonse. Both these control modes are incaable of resonding to the rate of change of the error signal. D-control action ositively enhances system closed-loo stability (Åström and Hägglund, 1995). When oerating in the forward ath, the derivative controller resonds to the rate at which system error changes according to (2.13a): de( t) de( t) ud ( t) = K ctd = K Equation (2.13a) d dt dt With regards to (2.13a): K c = K d T is the derivative gain, d d T denotes the derivative time de( t) constant and = De( t) is the rate of change of the error feedback signal. From dt (2.13a) and (2.13b) it is obvious that D-action is only resent when the error is changing; for any static error the contribution of the D-controller will be zero. Derivative action on its own will therefore allow uncontrolled steady-state errors. It is for this reason that derivative control is usually combined with either P-control or PI control. - 14 -

Another shortcoming of the D-controller is its sensitivity. The D-controller can be regarded as a high-ass filter that is sensitive to set-oint changes and rocess noise when oerating in the forward ath (Liták, 1995). To reduce this sensitivity, it is quite common to find the D-controller oerating in the feedback loo enabling it to act on the feedback signal according to (2.13b): K c dy( t) dy( t) ud ( t) = = K d = K d Dy( t) Equation (2.13b) T dt dt d With regards to (2.13b) reresents the rate of change of the feedback signal; all the other terms have the same meaning as was defined for (2.13a). 2.2.3.1 D-Action as Predictive Control The control action of a PD-controller can be interreted as a tye of redictive control that is roortional to the redicted rocess error. The rediction is erformed by extraolating the error from the tangent to the error curve in Figure 2.5. PD controllers oerate according to control law (2.14): u d de( t) ( t) = Kc e( t) + Td Equation (2.14) dt A Taylor series exansion of e ( t + T ) gives: d - 15 -

de( t) e( t + Td ) e( t) + T Equation (2.15) d dt The PD control signal is thus roortional to an estimate of the control error at time T d seconds ahead, where the estimate is obtained through linear extraolation. From Figure 2.5, the longer the derivative time constant T d is set, the further into the future the D-term will redict. Derivative action deends on the sloe of the error, hence if the error is constant the derivative action has no effect. The effects of derivative action on control erformance are illustrated in Figure 2.6. The controller roortional gain and integrating time constant are ket constant, K = 3 and T = 2, and the derivative time is varied according to T d = [0.1;0.7;4.5]. For T d = 0 we have a ure PI control. c i Error (e) Present error Actual error Predicted error e(t) e + ( t Td ) e( t) + T d de( t) dt t t + Td Time (t) Figure 2.5: Interretation of derivative action as redictive control - 16 -

1.6 1.4 1.2 Td=0.1 Td=0.7 Closed loo ste resonse of G(s)=1/((s+1) 3 ) Process Outut 1 0.8 0.6 0.4 Td=4.5 0.2 0 0 2 4 6 8 10 12 14 16 18 20 Time (s) Figure 2.6: Simulation of a closed-loo system with PID control G 1 ( s + 1) = 3 From Figure 2.6, we observe that system resonse is oscillatory for low values for Td and highly damed for higher derivative time settings. 2.3 PID Algorithms The transfer functions for PID algorithms are classified as follows: standard noninteracting (2.16), series interacting (2.17) and arallel non-interacting PID (2.18). U 1 = Kc [1 + + Td s] + b E( s) T s i Equation (2.16) Most tuning methods are based on (2.16) (Liták, 1995). U( s) 1 = K E s 1 + Ti s 1 ( ) ( + T s) b c d + Equation (2.17) - 17 -

U( s) ki = kc + + kds + b E( s) s Equation (2.18) With regards to (2.16) (2.18): U (s) reresents the control signal; E (s) is the error signal; K denotes the roortional gain; c T i and T d refers to the integral and derivative time constants; b denotes the controller bias. The imlementation strategy for (2.16), (2.17) and (2.18) is shown in Figure 2.7, Figure 2.8 and Figure 2.9. E(s) T i 1s + + + K c + + U (s) s T d b Figure 2.7: Non-interacting PID E(s) T i 1s + + + + K c + + U (s) s T d b Figure 2.8: Interacting PID - 18 -

k c E(s) k i + + + + + U (s) k d b Figure 2.9: Parallel non-interacting PID Historically, neumatic controllers based on (2.17) were easier to build and tune (Åström and Hägglund, 1995). Note that the interacting and non-interacting forms are different only when both integral and derivative control actions are used. (2.16) and (2.17) are equivalent when the controller is utilized for P, PI or PD control. It is evident that in the interacting controller the derivative time does influence the integral art, hence the reasoning that it is interacting. The reresentation for the arallel non-interacting PID controller is equivalent to the standard non-interacting controller with the excetion that the arameters are exressed in a different form. The relationshi between the standard and arallel tye is given by k c = K c, ki = Kc/Ti and k d = K c T d. The arallel structure has the advantage of often being useful in analytical calculations since the arameters aear linearly. The reresentation also has the added advantage of being referred for ure P, I or D control by the selection of finite tuning arameters (Åström, 1995). - 19 -

2.4 Performance evaluation criteria Quantification of system erformance is achieved through a erformance index. The erformance selected deends on the rocess under consideration and is chosen such that emhasis is laced on secific asects of system erformance. Performances indices referred by the control engineering disciline include the Integral Square-Error (ISE) index (2.19), Integral-of-Time multilied by Square-Error (ITSE) index (2.20), Integral Absolute-Error (IAE) index (2.21) and the Integral-of-Time multilied by Absolute-Error (ITAE) index (2.22). ISE Index: = 2 e 0 ISE ( t) dt Equation (2.19) An otimal system is one which minimizes this integral. The uer limit may be relaced by T which is chosen sufficiently large such that e (t) for T < t is negligible and the integral reaches a steady-state. A characteristic of this erformance index is that it enalizes large errors heavily and small errors lightly. A system designed by this criterion tends to show a raid decrease in a large initial error. Hence the resonse is fast and oscillatory leading to a system that has oor relative stability (Ogata, 1970). ITSE Index: = 2 te 0 ITSE ( t) dt Equation (2.20) - 20 -

This criterion laces little emhasis on initial errors and heavily enalizes errors occurring late in the transient resonse to a ste inut. IAE Index: IAE = e( t) dt Equation (2.21) 0 Systems based on this index enalize the control error. ITAE Index: ITAE = t e( t) dt Equation (2.22) 0 System s designed using this criterion has small overshoots and well damed oscillations. Any large initial error to a ste-resonse is enalized lightly whilst errors occurring later in the resonse are enalized heavily. The ITAE erformance index is used in this study. A summary of the erformance indices and their resective roerties is shown in Table 2.1. - 21 -

Performance Index ISE ITSE ISE ITSE Equation = 2 e 0 = 2 te 0 ( t) dt ( t) dt Proerties Penalizes large control errors. Settling time longer than ITSE. Suitable for highly damed systems. Penalizes long settling time and large control errors. Suitable for highly damed systems. IAE IAE = e( t) dt Penalizes control errors. 0 ITAE ITAE = t e( t) dt Penalizes long settling time and control errors. 0 Table 2.1: Summary of erformance indices 2.5 Summary and conclusion Tyical PID algorithms that form the building blocks of controllers have been discussed. The control actions of roortional, integral and derivative terms and some of their adverse effects have also been reviewed. The roortional controller rovides a corrective action that is roortional to the size of the error and also has an effect on the seed of a system s resonse; integral control rovides corrective action roortional to the time integral of the error and is resent for the entire duration of the error; the derivative controller rovides a corrective action roortional to the time derivative of the error signal and resonds to the rate at which the error is changing. The effects of rocess dynamics on controller tuning are discussed in the next chater. - 22 -

Chater 3 Tyical Process Control Models 3.1 Introduction This chater resents a discussion on the transfer function models of systems commonly encountered in rocess control. These lant models will be used to comare the control erformance of loos tuned with the PSO versus that of loos tuned using methodologies roosed in the literature. The dynamics associated with each rocess model is also discussed. 3.2 Dynamics associated with the selected rocess models The SISO control loo used in this study is given in Figure 3.1. The SISO configuration has been chosen because it forms the fundamental building block of all rocess control loos and the dynamics associated with it are universally alicable to configurations such as SIMO, MISO and MIMO control loos. D(s) R(s) + E(s) PID controller U(s) Process Y(s) - + + Figure 3.1: SISO system with unity feedback - 23 -

Tyical real-world rocess models that have been selected for this study are listed in (3.1) to (3.4): A Stable First Order Plus Dead-Time Process (FOPDT): ( L s) K ex G = Equation (3.1) ( T s + 1) A Stable Second Order Plus Dead-Time Process (SOPDT): ( L s) K ex G = Equation (3.2) 2 ( T s + 1) A Stable Second Order Integrating Process with Dead-Time (SOIPDT): ( L s) K ex G = Equation (3.3) s( T s + 1) A First Order Delayed Unstable Process (FODUP): ( L s) K ex G = Equation (3.4) ( T s 1) - 24 -

Equations (3.1)-(3.4) cature the tyical dynamics that are resent in most real-world rocess control systems, with the excetion that the L T ratios may vary (Åström et al., 2004). Equation (3.2) characterizes systems that are rich in dynamics and include systems such as underdamed, critically damed and overdamed systems. These systems usually follow an S-shae closed-loo resonse. The L T ratio, or controllability ratio, is used to characterize the difficulty or ease of L controlling a rocess. Processes having small controllability ratios (i.e. 0 < 1) are T easier to control and the difficulty of controlling the system increases as the L controllability ratio increases (Åström and Hägglund, 1995). Processes with 1 T corresond to dead-time dominant rocesses that are difficult to control with conventional PID control (Åström, 1995). 3.3 A brief overview of integrating rocesses (Self-Regulating Processes) Most real-world rocess control systems are characterized by offset or steady-state error which can arise from load friction, intrinsic steady state nonlinearities or uncertainties in modeling (Haung et al., 1996). If the forward branch of a feedback control system contains an integrator, the resence of an error will cause a rate of change of outut until the error has been eliminated (Chen et al., 1996; Poulin and Pomerleau, 1996). - 25 -

The dynamics of certain real-world rocess control systems are such that an inherent integrating control effect could naturally arise during normal oeration of the lant. This natural integrator is urely error driven and will ensure that any steady-state error is driven to zero following either a setoint change or disturbance. There is no static error to a setoint change for ure roortional control. However this is not the case when nonzero mean disturbances act at the rocess inut. Therefore in order to ensure that there will be no static error, a control with an integrator must be used (Poulin and Pomerleau, 1996). 3.4 Problems exerienced with tuning rocesses having unstable oles and deadtime Processes having only right-hand oles are inherently unstable under oen-loo conditions (Poulin and Pomerleau, 1996; Majhi and Atherton, 1999). The undesirable effects of dead-time will contribute towards the instability inherently resent in systems of this nature. The tuning of these oen-loo unstable rocesses having dead-time delay becomes more challenging than for stable rocesses (Poulin and Pomerleau, 1996). The Ziegler-Nichols (1942) and Cohen-Coon (1953) tuning techniques are unsuitable for tuning loos that have only unstable ole/s lus dead-times because: The oen-loo ste resonse of systems having unstable oles will be unbounded (Poulin and Pomerleau, 1996; Haung et al., 1996). The Ziegler-Nichols and Cohen-Coon oenloo methods rely on a stable oen-loo resonse for determining the controller s tuning arameters. - 26 -