Studies in Computational Intelligence 438

Editor-in-Chief

Prof. Janusz KacprzykSystems Research InstitutePolish Academy of Sciencesul. Newelska 601-447 WarsawPolandE-mail: kacprzyk@ibspan.waw.pl

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Indranil Pan and Saptarshi Das

Intelligent Fractional OrderSystems and Control

An Introduction

ABC

AuthorsIndranil PanDepartment of Power EngineeringJadavpur UniversityWest BengalIndiaE-mail: indranil@pe.jusl.ac.in

Saptarshi DasDepartment of Power EngineeringJadavpur UniversityWest BengalIndiaE-mail: saptarshi@pe.jusl.ac.in

ISSN 1860-949X e-ISSN 1860-9503ISBN 978-3-642-31548-0 e-ISBN 978-3-642-31549-7DOI 10.1007/978-3-642-31549-7Springer Heidelberg New York Dordrecht London

Library of Congress Control Number: 2012942174

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Dedication

This book is dedicated to our spiritual father Sri Gurudev and the supreme mother Devi Bhavatarini

pa hanti veda str i bibadanti parasparam|

na j nanti para tattva darvi p karasa yath || iro bahati pu p i gandha j n ti n sik |

pa hanti veda str i durlabho bh bavedaka || ...

agrata pri hata kecita par vayorapi kecana| tattvam d k tad giti bibadanti parasparam|

sadvidy d na uradyaigu airbikhy tam nava || ...

ida jñ namida jñeya sarvata rotumicchati| devi var asahasrayuh str nta naiva gacchati||

vedadyaneka str i svalpayurvighnakotaya | tasm t s ra bij n y ta k h r ha sa ib sambhasa ||

- Jiva-Sthiti-Kathanam (Ch. 1), Verse 94-102, Kularnava Tantram

Translation: People read Vedas (literatures) and involve in petty bickering among themselves. But like the ladle that does not know the taste of the treacle it holds, they do not know the ultimate truth. The flowers may lie on the head but it is the nose and not the head that gets their fragrance. Many are they who chant the Veda-Scriptures but rare is he who is one with their spirit... Among the people who are renowned for pedagogic skills, some see the truth from the front, some from back and others from two other sides and quarrel among themselves about their observations in-terpreting the esoteric theory from their view point... You may spend a thousand years hearing the knowledge in the Sastras (literatures), but you will never reach their end. Endless is the expanse of the Sastras but the life-duration is limited and there are a multitude of obstacles. It is wise to go straight to the essence of the scriptures, like the swan sipping the pure milk from that diluted with water.

Preface

Fusion of computational intelligence and control theory has already emerged as a new discipline known as intelligent control which has strengthened the field of automatic control with biologically inspired systems. These intelligent control systems can learn, modify and adapt the control action using feedback from the environment and can also use some sort of linguistic description of an uncertain system like a human expert. These intelligent systems can be programmed to work in an optimum fashion with respect to the designer’s choice of a specific perform-ance measure. In a nutshell, the Neuro-Fuzzy systems and the Swarm and Evolu-tionary optimization techniques together have unleashed a new era in the field of industrial automation and control. These intelligent control systems have inherent capability to model and control nonlinear dynamical systems and time varying systems. They can be used for fault detection and can be designed to have a level of fault tolerance. Besides the well-studied linear systems, the nonlinear phenom-ena present in a control system may be due the actuator/sensor or the inherent non-linear governing laws. These have been less understood and the mathematical tools to deal with them are few in number. Though stability analysis and stabiliz-ing control scheme design for nonlinear systems have been attempted in past, the-se techniques are often difficult, if not impossible, by someone with lesser depth in pure mathematics. Moreover, inspite of these analyses, optimization for meet-ing performance specifications is still an open problem for nonlinear and time var-ying systems. Artificial Neural Network based intelligent systems are good in mimicking nonlinearity and adaptation but they are often complained for low ro-bustness which means the design cannot be guaranteed to work in a perturbed condition. Its hybridization with approximate reasoning or fuzzy logic produces the popular Neuro-Fuzzy systems which have the capability to overcome these is-sues of both preciseness and robustness in control. The Neuro-Fuzzy inference systems which mimic the working principle of the human brain and the linguistic variables used by humans to describe and reason in an approximate sense, have proved expedient in modeling such non-linear and time varying systems. These neuro-fuzzy systems in conjunction with bio-inspired optimization techniques like Darwinian evolution or foraging behavior of micro-organisms have together pro-vided an efficient tool to model and control highly complex systems which are almost impossible with linear estimation and control techniques. One of the major impediments of using such systems is that there might be some hidden heuristics to design intelligent control systems and almost no guideline can be provided for their design, since it will differ from case to case. However there have been many industrial applications of these intelligent computational schemes and the initial

VIII Preface

skepticism has given way to the foundation of a new off shoot in the systems and control community. The synergism of the three computational intelligence para-digms i.e. Neural Network, Fuzzy Logic and Swarm/Evolutionary optimization has opened new avenues in the field of systems and control engineering.

Recently fractional order control systems are becoming increasingly popular in the control community. Fractional systems are governed by fractional order de-rivatives and integrals which are basically infinite order linear operators. These fractional order systems extend the notion of our integer order concepts in control and improvise on the existing results. So in essence, both fractional calculus and control systems have been applied to improve the performance of existing control theories. This book tries to see if a symbiotic approach between fractional calculus and computational intelligence can further increase the performance or flexibility of a control system. To the best of author’s knowledge, this is the first book of its kind, which particularly focusses on hybridization of intelligence based paradigms and fractional systems to get performance enhancement of control systems. The authors are well aware that often these hybrid intelligent control techniques are frowned upon for not providing a mathematical basis for guaranteed stability and insight like linear control systems. Nonetheless history has proved time and again that it is the necessity which attracted the human race for the progress of a particu-lar scientific field. Necessity is the mother of invention as they call it. If hybridiza-tion of two paradigms improves the performance in a significant manner, we believe these should be encouraged and someday we may have full mathematical understanding of highly complex nonlinear infinite-dimensional (fractional) sys-tems. This is the way how technology has evolved in the past. We consider the fusion of two different disciplines of computational intelligence and fractional cal-culus as the first stepping stone to more sophisticated control systems of the fu-ture. We also believe that it has huge potential in various fields like Chemical, Computer and Electrical, Mechanical, Biological, Financial systems etc.

This book firstly introduces the basic concepts of fractional calculus and frac-tional order systems along with basic tools in computational intelligence for the uninitiated researchers. Rest of the chapters presents diverse application examples where the synergistic coupling of fractional order systems and computational in-telligence has proved to be beneficial. The authors are indebted to Elsevier for permitting to reuse the following materials from their published works in modified form.

Indranil Pan, Saptarshi Das, and Amitava Gupta, Tuning of an optimal fuzzy PID controller with stochastic algorithms for networked control systems with random time delay, ISA Transactions, vol. 50, no. 1, pp. 28-36, Jan. 2011, doi: 10.1016/j.isatra.2010.10.005. Saptarshi Das, Suman Saha, Shantanu Das, and Amitava Gupta, On the selec-tion of tuning methodology for FOPID controllers for the control of higher or-der processes, ISA Transactions, vol. 5, no. 3, pp. 376-388, July 2011, doi: 10.1016/j.isatra.2011.02.003.

Preface IX

Indranil Pan, Saptarshi Das, and Amitava Gupta, Handling packet dropouts and random delays for unstable delayed processes in NCS by optimal tuning of PI D controllers with evolutionary algorithms, ISA Transactions, vol. 50, no. 4, pp. 557-572, Oct. 2011, doi: 10.1016/j.isatra.2011.04.002. Saptarshi Das, Indranil Pan, Shantanu Das, and Amitava Gupta, A novel frac-tional order fuzzy PID controller and its optimal time domain tuning based on integral performance indices, Engineering Applications of Artificial Intelligence, vol. 25, no. 2, pp. 430-442, March 2012, doi: 10.1016/j.engappai.2011.10.004. Saptarshi Das, Indranil Pan, Shantanu Das, and Amitava Gupta, Improved model reduction and tuning of fractional order PI D controllers for analytical rule extraction with genetic programming, ISA Transactions, vol. 51, no. 2, pp. 237-261, March 2012, doi: 10.1016/j.isatra.2011.10.004.

The authors are indebted to Trans Tech Publications for permitting to reuse the following material from their published works in modified form.

Indranil Pan, Saptarshi Das, Ayan Mukherjee, and Amitava Gupta, Gain and order scheduling of optimal fractional order PID controllers for random delay and packet dropout in networked control systems, Advanced Materials Research, vol. 403-408, pp 4814-4820, 2012, doi: 10.4028/www.scientific.net/AMR.403-408.4814

The authors are also indebted to Springer for permitting to reuse the following ma-terial from their published works in modified form.

Saptarshi Das, Indranil Pan, Shantanu Das, and Amitava Gupta, Master-slave chaos synchronization via optimal fractional order PI D controller with bacte-rial foraging algorithm, Nonlinear Dynamics, vol. 69, no. 4, pp. 2193-2206, 2012, doi: 10.1007/s11071-012-0419-x.

An outline of the contents of the individual chapters is described next.

Chapter 1 introduces the motivation behind coupling two different disciplines of fractional calculus based control and computational intelligence. An overview of the advantages of each of these two paradigms have been given along with exist-ing and other possible hybridizations for further performance improvement in the control of complex systems.

Chapter 2 gives a brief overview of the definitions of fractional calculus and mathematical representations of fractional dynamical systems etc. Popular rational approximation techniques for truncating the infinite dimensional representation of fractional systems are reviewed then. Analysis tools for fractional systems and popular fractional order control techniques are also introduced in this chapter for the beginners in this field.

X Preface

Chapter 3 reviews the well practiced computational intelligence tools in control engineering. The three major pillars of intelligent systems i.e. the Artificial Neural Network, Fuzzy Logic and Swarm/Evolutionary Optimization are presented in the context of control of complex systems where analytical solutions are difficult to obtain. State of art intelligent control techniques are reported in this chapter which may be easily extended in fractional control systems as well.

Chapter 4 introduces optimum parameter selection of fractional order control-lers using Swarm and Evolutionary Optimization techniques. Swarm intelligence based optimization has been shown for a sample test-bench function and then ap-plied to a couple of process control applications viz. multivariable fractional order control and networked fractional order control.

Chapter 5 shows a multi-objective fractional order controller design example in the context of power systems. The multi-objective evolutionary algorithm is aug-mented with a chaotic map for greater effectiveness. Such kind of application is particularly important when design trade-offs among different performance objec-tives are to be studied rather than getting a particular solution with fixed weights associated with the conflicting objectives.

Chapters 6 illustrates the concept of gain and order scheduling of fractional or-der controllers so as to adapt with changing operating conditions of the systems that are being controlled. Scheduling of the integro-differential orders along with the controllers gains is capable of providing better performance and has been elu-cidated with design examples.

Chapter 7 shows enhancement of fuzzy PID controllers with fractional calculus. Few structural variants of fractional order fuzzy PID controllers are also shown in this chapter. Simulation studies show the potential of the new fractional order fuzzy PID controller in comparison with other structures like fuzzy PID, fractional PID and PID controllers.

Chapter 8 has two distinct parts viz. evolutionary algorithm based model reduc-tion and its use in tuning rule generation using symbolic regression. A wide test-bench of higher order processes are reduced in second order plus time delay template. The reduced order process parameters are then used to construct analyti-cal expression for controller gains using an evolutionary technique known as ge-netic programming. The evolved expressions, being analytical in nature, can be easily computed by the process system engineer and are useful in the process con-trol industry. Trade-offs between accuracy and complexity of such tuning rules and the achievable performances are also shown by credible simulation studies.

Chapter 9 extends the concept of sub-optimum model reduction from the inte-ger order templates to the fractional order templates using evolutionary algo-rithms, which are reliably used to find the global minimum. Complex Nyquist domain optimization is framed for representing very high order complex dynami-cal systems as fractional order systems for compact representation and control.

Chapter 10 shows iso-damped fractional order controller design for integer and fractional order plants using swarm/evolutionary algorithms. User specified frequency domain specifications on the accuracy and speed of control is used to formulate the controller design task. This is then cast into a global optimization problem which can be efficiently solved using Swarm/Evolutionary algorithms.

Preface XI

Chapter 11 presents a global optimization framework for master-slave synchro-nization in two chaotic systems. Bacterial foraging optimization algorithm has been used for the simulation study to minimize integral performance index of synchronization errors. Extensive simulation examples have been given to show the effectiveness of fractional order controllers in such an application over the integer order PID controllers.

Writing a monograph like this is a tedious and time consuming process and in-volves a lot of sacrifice from the people associated with the authors in their per-sonal lives. The authors have deprived their respective families, the association and company for countless hours, which have been spent working on this manu-script. The authors are really indebted to their respective family members for relieving them of their daily family chores and letting them concentrate on their academic pursuits. Indranil would like to acknowledge his parents Prof. Dr. Tapan Kumar Pan and Sefali Pan for their encouragement and support. He would also like to thank his cousin brother, Soumyadeep Ghosh, for helping him out with a few translations. Saptarshi would like to acknowledge the constant inspiration, re-ceived from his parents Supriya Das and Arunima Das and his brother Subhajit Das without whom the book would not have been a reality.

Indranil Pan Saptarshi Das

Contents

1 Motivation for Application of Computational Intelligence Techniques to Fractional Calculus Based Control Systems ..............................................1 1.1 Introduction ................................................................................................1 References ..........................................................................................................8 2 Applied Fractional Calculus for Computational Intelligence Researchers .......................................................................................................9 2.1 Requirement of Fractional Order Calculus .................................................9 2.2 Some Important Functions in Context of Fractional Calculus ..................10 2.2.1 Gamma Function............................................................................10 2.2.2 Beta Function .................................................................................11 2.2.3 Mittag-Leffler Function .................................................................11 2.2.4 Miller-Ross Function .....................................................................12 2.3 Definitions of Fractional Differ-Integrals .................................................13 2.3.1 Grünwald-Letnikov (G-L) Definition ............................................13 2.3.2 Riemann-Liouville (R-L) Definition..............................................15 2.3.3 Caputo Definition ..........................................................................15 2.3.4 Equivalence between the Definitions.............................................16 2.3.5 Some Properties of Fractional Differ-Integrals..............................16 2.4 Laplace Transform of Fractional Differ-Integrals.....................................17 2.4.1 Basics of Laplace Transform .........................................................17 2.4.2 Laplace Transform for Fractional Integrals ...................................18 2.4.3 Laplace Transform for Fractional Derivatives ...............................19 2.5 Fourier Transform of Fractional Differ-Integrals .....................................20 2.5.1 Basics of Fourier Transform ..........................................................20 2.5.2 Fourier Transform for Fractional Integrals ....................................20 2.5.3 Fourier Transform for Fractional Derivatives................................20 2.6 Realization of Fractional Order Differ-Integrators ...................................21 2.6.1 Continuous Time Realization ........................................................21 2.6.1.1 Carlson’s Method.............................................................21 2.6.1.2 Charef’s Method ..............................................................25 2.6.1.3 Oustaloup’s Method .........................................................27 2.6.2 Discrete Time Realization..............................................................30 2.6.2.1 Based on Discretization Method ......................................30 2.6.2.2 Series Expansion for FO Element Realization .................35 2.6.3 Time Domain Simulation Methods for Fractional Order Systems ..........................................................................................37

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2.7 Fractional Order Systems..........................................................................39 2.7.1 Fractional Order LTI Systems .......................................................39 2.7.1.1 Transfer Function Representation ....................................39 2.7.1.2 State Space Representation ..............................................42 2.7.1.3 Controllability and Observability.....................................44 2.7.2 Fractional Order Nonlinear Systems ..............................................44 2.8 Stability of Fractional Order Systems .......................................................45 2.8.1 Stability of FO LTI Systems ..........................................................45 2.8.2 Stability of FO Nonlinear Systems ................................................47 2.9 Fractional Order Controllers .....................................................................48 2.9.1 Family of Fractional Order PID Controllers ..................................48 2.9.2 CRONE Controllers .......................................................................50 2.9.3 Fractional Lead-Lag Compensator.................................................50 2.9.4 Concept of Iso-damping Using FO Controllers .............................51 2.9.5 Effect of Controller Knob Variation of a Well-Tuned Control Loop...............................................................................................53 2.9.6 Controller Tuning Methods for FOPID Controllers.......................56 References ........................................................................................................59 3 Brief Introduction to Computational Intelligence Paradigms for Fractional Calculus Researchers ...................................................................63 3.1 Introduction ..............................................................................................63 3.2 Artificial Neural Networks .......................................................................64 3.2.1 Network Architecture and Training Methods ................................65 3.2.2 Neural Networks in Control...........................................................65 3.2.2.1 ANN in Model Predictive Control Techniques ................66 3.2.2.2 ANN in Inverse Model Based Techniques.......................67 3.2.2.3 ANN in Adaptive Control Techniques.............................68 3.2.2.4 System Identification, Filtering and Prediction................69 3.3 Fuzzy Logic ..............................................................................................69 3.3.1 Use of Fuzzy Logic in Control Applications .................................73 3.4 Global Optimization Using Evolutionary and Swarm Algorithms ...........73 3.4.1 Why Bio-inspired Algorithms over Other Conventional Techniques? ...................................................................................74 3.4.2 Genetic Algorithms: Key Concepts and Attributes........................75 3.4.2.1 Chromosome Encoding and Population Initialization......76 3.4.2.2 Fitness Evaluation ............................................................77 3.4.2.3 Selection...........................................................................77 3.4.2.4 Crossover .........................................................................77 3.4.2.5 Mutation...........................................................................79 3.4.2.6 Elitism ..............................................................................80 3.4.3 Brief Theory of Genetic Algorithms ..............................................80 3.4.4 Genetic Programming ....................................................................81 3.4.5 Use of Bio-inspired Algorithms in Control Applications ..............82 3.5 Synergism between Various Paradigms....................................................83 References ........................................................................................................84

Contents XV

4 Fractional Order Controller Tuning Using Swarm and Evolutionary Algorithms .......................................................................................................87 4.1 Introduction ..............................................................................................87 4.2 Optimizing a Multimodal Function with Particle Swarm Optimization ...88 4.2.1 Outline of Particle Swarm Optimization........................................89 4.2.2 Optimizing a Function with PSO ...................................................90 4.3 A MIMO Process Control Application Example ......................................97 4.3.1 Background of the Problem ...........................................................97 4.3.2 Objective Function and Optimization Algorithm ..........................99 4.3.3 Results and Discussions ...............................................................100 4.4 Application to Networked Control Systems ...........................................101 4.4.1 Background of the Problem .........................................................101 4.4.2 Control over Communication Network and Scope for Stochastic Optimization Based Controller Tuning........................................104 4.4.2.1 Networked FOPID Control Scheme...............................104 4.4.2.2 Time Domain Integral Performance Index Based Tuning of Networked Process Controllers via Stochastic Optimization for Randomly-Varying Objective Function.........................................................106 4.4.2.3 Fractional Order PID Controller in NCS Applications ...................................................................107 4.4.2.4 Handling Unstable Processes over NCS ........................109 4.4.3 Evolutionary Algorithms Based PID/FOPID Controller Tuning..........................................................................................110 4.4.3.1 Brief Description of Genetic Algorithm (GA) for Controller Tuning...........................................................110 4.4.3.2 Differential Evolution (DE) and Its Variants for Controller Tuning...........................................................111 4.5 Simulations and Results..........................................................................113 4.5.1 MATLAB Based Simulation Study of a NCS with Packet Dropout and Variable Delay ........................................................113 4.5.1.1 Performance Degradation of Well Tuned Control Loops with Stochastic Consideration of the Network Delay ....115 4.5.1.2 Performance Degradation of Well Tuned Control Loops due to Out of Order Packets and Handling Packet Drop Out with Buffers ............................................................116 4.5.2 Optimal PID and FOPID Controller Tuning for Unstable Processes with the Consideration of Randomness in Network Induced Delays and Packet Dropouts ..........................................116 4.5.3 Effect of Delay Distribution on the Tuned Networked FOPID Control Loops ..............................................................................124 4.5.4 Validation of the Tuning Methodology for Lesser Complicated FOPTD Processes .......................................................................126 4.6 Summary.................................................................................................127 References ......................................................................................................128

XVI Contents

5 Multi-objective Fractional Order Controller Design with Evolutionary Algorithms .....................................................................................................133 5.1 Introduction to the Optimization Problem ..............................................133 5.2 The AVR System with FO Controller ....................................................134 5.3 Contradictory Objective Functions for Optimization .............................135 5.4 Multi-objective Chaotic Non-dominated Sorting Genetic Algorithm II (NSGA-II)...............................................................................................136 5.5 Results and Discussions..........................................................................139 5.6 Robustness Analysis of Obtained Solutions ...........................................144 5.7 Conclusions ............................................................................................145 References ......................................................................................................146 6 Gain and Order Scheduling for Fractional Order Controllers ................147 6.1 Introduction ............................................................................................147 6.2 A Networked Control System Application .............................................149 6.2.1 Problem Formulation ...................................................................150 6.2.1.1 Test Plant Considered ....................................................150 6.2.1.2 Time Domain Performance Index ..................................150 6.2.1.3 Application of Fractional Order PID Controller in NCS................................................................................150 6.2.1.4 Network Model Used for Simulation Study...................151 6.2.1.5 Genetic Algorithm for Optimal FOPID Tuning.............152 6.3 Results and Discussions..........................................................................153 6.4 Summary.................................................................................................155 References ......................................................................................................156 7 Enhancement of Fuzzy PID Controller with Fractional Calculus............159 7.1 Introduction ............................................................................................159 7.2 Review of the Existing Intelligent Tuning Techniques of FO Controllers ..............................................................................................161 7.3 New Fractional Order Fuzzy PID Controller and Its Time Domain Optimal Tuning.......................................................................................163 7.3.1 Structure of Fractional Order Fuzzy PID Controller....................163 7.3.1.1 Fractional Order Fuzzy PI+PD Controller .....................164 7.3.1.2 Fractional Order Fuzzy P+ID Controller .......................165 7.3.1.3 Fractional Order Fuzzy PI+D Controller .......................166 7.3.1.4 Fractional Order Fuzzy PD+I Controller .......................167 7.3.2 Membership Functions and Rule Base.........................................168 7.3.3 Formulation of the Objective Functions for Time Domain Optimal Controller Tuning ..........................................................171 7.3.4 Optimization Algorithm Used for the Tuning of Optimal Controllers ...................................................................................173 7.4 Simulations and Results..........................................................................175 7.4.1 Nonlinear Process with Time Delay ............................................175 7.4.2 Unstable Process with Time Delay ..............................................181

Contents XVII

7.4.3 Comparative Performance Analysis of the Different Controllers and Few Discussions....................................................................185 7.5 Conclusion ..............................................................................................189 References ......................................................................................................190 8 Model Reduction and Analytical Rule Extraction with Evolutionary Algorithms .....................................................................................................195 8.1 Background.............................................................................................195 8.2 An Improved Sub-optimal Model Reduction Technique ........................198 8.2.1 New Optimization Framework for Model Reduction in Nyquist Plane ...............................................................................198 8.2.2 Model Reduction of a Test-Bench of Higher Order Processes ....200 8.3 Generation of Time Domain Optimal Controller Tuning Rule...............205 8.3.1 Controller Structure and Objective Function for Tuning .............205 8.3.2 Application of Genetic Algorithm for Optimal Controller Tuning..........................................................................................207 8.3.3 Genetic Programming Based Analytical Tuning Rule Extraction for PID/PI D Controllers...........................................208 8.4 Visualization of the Optimal PID/FOPID Tuning Rules ........................227 8.4.1 Optimal FOPID Tuning Rules .....................................................227 8.4.2 Optimal PID Tuning Rules ..........................................................230 8.5 Performance of the Analytical Tuning Rules..........................................232 8.5.1 Effect of Plant Perturbation on the Tuning Rules ........................233 8.6 Summary.................................................................................................236 References ......................................................................................................237 9 Model Reduction of Higher Order Systems in Fractional Order Template ........................................................................................................241 9.1 Introduction ............................................................................................241 9.2 Reduced Order Modeling: Review of the Existing Methodologies ........242 9.3 New Approach towards Reduced Parameter FO Modeling of Higher Order Processes Using H2 Norm Based Method.....................................244 9.4 FO Model Reduction of Higher Order Processes Using Nyquist Based Technique.....................................................................................249 9.5 Summary.................................................................................................254 References ......................................................................................................254 10 Global Optimization Based Frequency Domain Design of Fractional Order Controllers with Iso-damping Characteristics .............................257 10.1 Introduction.........................................................................................257 10.2 Frequency Domain Design of PID/FOPID Controllers Using Global Optimization ...........................................................................258 10.3 Frequency Response of the Reduced Order Process Models and Controllers ..........................................................................................261

XVIII Contents

10.3.1 First Order Plus Time Delay (FOPTD) Model.......................261 10.3.2 Second Order Plus Time Delay (SOPTD) Model ..................262 10.3.3 One Non-Integer Order Plus Time Delay (NIOPTD-I) Model .....................................................................................263 10.3.4 Two Non-Integer Orders Plus Time Delay (NIOPTD-II) Model .....................................................................................264 10.3.5 Integer Order Proportional Integral Derivative (IOPID) Controller ...............................................................................265 10.3.6 Fractional Order Proportional Integral Derivative (FOPID) Controller ...............................................................................266 10.4 Illustrative Examples ..........................................................................267 10.4.1 Control of FOPTD Plant ........................................................268 10.4.2 Control of NIOPTD-II Plant ..................................................270 10.5 Summary.............................................................................................272 References ....................................................................................................273 11 Chaos Synchronization with a Fractional Order Controller and Swarm Intelligence ....................................................................................275 11.1 Introduction.........................................................................................275 11.2 Master-Slave Synchronization between Two Chaotic Lu Systems.....277 11.3 Optimal PID and PI D Controller Design for Chaos Synchronization ..................................................................................280 11.4 Basics and Customization of Bacterial Foraging Optimization Algorithm............................................................................................283 11.4.1 Chemotaxis ............................................................................284 11.4.2 Swarming...............................................................................284 11.4.3 Reproduction..........................................................................285 11.4.4 Elimination-Dispersal ............................................................285 11.5 Results and Discussions......................................................................287 11.6 Conclusion ..........................................................................................293 References ....................................................................................................293