FUNCTIONAL FRACTIONAL CALCULUS FOR SYSTEM IDENTIFICATION AND CONTROLS

By B. JATAK

CONTENTS INTRODUCTION TO FRACTIONAL CALCULUS INTRODUCTION BIRTH OF FRACTIONAL CALCULUS FRACTIONAL CALCULUS A GENERALIZATION OF INTEGER ORDER CALCULUS HISTORICAL DEVELOPMENT OF FRACTIONAL CALCULUS THE POPULAR DE?NITIONS OF FRACTIONAL DERIVATIVES/INTEGRALS IN FRACTIONAL CALCULUS ABOUT FRACTIONAL INTEGRATION DERIVATIVES AND DIFFERINTEGRATION FRACTIONAL INTEGRATION RIEMANN–LIOUVILLE (RL) FRACTIONAL DERIVATIVES RIEMANN–LIOUVILLE (RL) LEFT HAND DE?NITION (LHD) FRACTIONAL DERIVATIVES CAPUTO RIGHT HAND DE?NITION (RHD) FRACTIONAL DIFFERINTEGRALS GRUNWALD LETNIKOV (GL) COMPOSITION AND PROPERTY FRACTIONAL DERIVATIVE FOR SOME STANDARD FUNCTION SOLUTION OF FRACTIONAL DIFFERENTIAL EQUATIONS A THOUGHT EXPERIMENT QUOTABLE QUOTES ABOUT FRACTIONAL CALCULUS CONCLUDING COMMENTS FUNCTIONS USED IN FRACTIONAL CALCULUS FUNCTIONS FOR THE FRACTIONAL CALCULUS GAMMA FUNCTION MITTAG-LEF?ER FUNCTION AGARWAL FUNCTION ERDELYI’S FUNCTION ROBOTNOV–HARTLEY FUNCTION MILLER–ROSS FUNCTION GENERALIZED R FUNCTION AND G FUNCTION LIST OF LAPLACE AND INVERSE LAPLACE TRANSFORMS RELATED TO FRACTIONAL CALCULUS OBSERVATION OF FRACTIONAL CALCULUS IN PHYSICAL SYSTEM DESCRIPTION TEMPERATURE–HEAT FLUX RELATIONSHIP FOR HEAT FLOWING IN SEMI-IN?NITE CONDUCTOR SINGLE THERMOCOUPLE JUNCTION TEMPERATURE IN MEASUREMENT OF HEAT FLUX HEAT TRANSFER DRIVING POINT IMPEDANCE OF SEMI-IN?NITE LOSSY TRANSMISSION LINE PRACTICAL APPLICATION OF THE SEMI-IN?NITE LINE IN CIRCUITS APPLICATION OF FRACTIONAL INTEGRAL AND FRACTIONAL DIFFERENTIATOR CIRCUIT IN CONTROL SYSTEM SEMI-IN?NITE LOSSLESS TRANSMISSION LINE THE CONCEPT OF SYSTEM ORDER AND INITIALIZATION FUNCTION CONCEPT OF FRACTIONAL DIVERGENCE AND FRACTIONAL CURL CONCEPT OF FRACTIONAL DIVERGENCE FOR PARTICLE FLUX FRACTIONAL KINETIC EQUATION NUCLEAR REACTOR NEUTRON FLUX DESCRIPTION CLASSICAL CONSTITUTIVE NEUTRON DIFFUSION EQUATION DISCUSSION ON CLASSICAL CONSTITUTIVE EQUATIONS GRAPHICAL EXPLANATION ABOUT SURFACE FLUX CURVATURE STATISTICAL AND GEOMETRICAL EXPLANATION FOR NON-LOCAL DIVERGENCE FRACTIONAL DIVERGENCE IN NEUTRON DIFFUSION EQUATIONS SOLUTION OF CLASSICAL CONSTITUTIVE NEUTRON DIFFUSION EQUATION (INTEGER ORDER) SOLUTION OF FRACTIONAL DIVERGENCE BASED NEUTRON DIFFUSION EQUATION (FRACTIONAL ORDER) FRACTIONAL GEOMETRICAL BUCKLING AND NON-POINT REACTOR KINETICS CONCEPT OF FRACTIONAL CURL IN ELECTROMAGNETICS DUALITY OF SOLUTIONS FRACTIONAL CURL OPERATOR WAVE PROPAGATION IN UNBOUNDED CHIRAL MEDIUM FRACTIONAL DIFFERINTEGRATIONS: INSIGHT CONCEPTS SYMBOL STANDARDIZATION AND DESCRIPTION FOR DIFFERINTEGRATION REIMANN–LIOUVILLE FRACTIONAL DIFFERINTEGRAL SCALE TRANSFORMATION CONVOLUTION PRACTICAL EXAMPLE OF RL DIFFERINTEGRATION IN ELECTRICAL CIRCUIT ELEMENT DESCRIPTION GRUNWALD–LETNIKOV FRACTIONAL DIFFERINTERATION UNI?CATION OF DIFFERINTEGRATION THROUGH BINOMIAL COEF?CIENTS SHORT MEMORY PRINCIPLE: A MOVING START POINT APPROXIMATION AND ITS ERROR MATRIX APPROACH TO DISCRETIZE FRACTIONAL DIFFERINTEGRATION AND WEIGHTS IN?NITESIMAL ELEMENT GEOMETRICAL INTERPRETATION OF FRACTIONAL DIFFERINTEGRATIONS INTEGRATION DIFFERENTIATION ADVANCE DIGITAL ALGORITHMS REALIZATION FOR FRACTIONAL CONTROLS CONCEPT OF GENERATING FUNCTION DIGITAL FILTER REALIZATION BY RATIONAL FUNCTION APPROXIMATION FOR FRACTIONAL OPERATOR FILTER STABILITY CONSIDERATION LOCAL FRACTIONAL DERIVATIVES INITIALIZED DIFFERINTEGRALS AND GENERALIZED CALCULUS NOTATIONS OF DIFFERINTEGRALS REQUIREMENT OF INITIALIZATION INITIALIZATION FRACTIONAL INTEGRATION (RIEMANN–LIOUVILLE APPROACH) TERMINAL INITIALIZATION SIDE INITIALIZATION INITIALIZING FRACTIONAL DERIVATIVE (RIEMANN–LIOUVELLE APPROACH) INITIALIZING FRACTIONAL DIFFERINTEGRALS (GRUNWALD–LETNIKOV APPROACH) PROPERTIES AND CRITERIA FOR GENERALIZED DIFFERINTEGRALS TERMINAL CHARGING SIDE CHARGING THE FUNDAMENTAL FRACTIONAL ORDER DIFFERENTIAL EQUATION THE GENERALIZED IMPULSE RESPONSE FUNCTION GENERALIZED LAPLACE TRANSFORM FOR FRACTIONAL DIFFERINTEGRALS RECALLING LAPLACE TRANSFORM FUNDAMENTALS LAPLACE TRANSFORM OF FRACTIONAL INTEGRALS DECOMPOSITION OF FRACTIONAL INTEGRAL IN INTEGER ORDER DECOMPOSITION OF FRACTIONAL ORDER INTEGRAL IN FRACTIONAL ORDER LAPLACE TRANSFORMATION OF FRACTIONAL DERIVATIVES DECOMPOSITION OF FRACTIONAL ORDER DERIVATIVE IN INTEGER ORDER DECOMPOSITION OF FRACTIONAL DERIVATIVE IN FRACTIONAL ORDER EFFECT OF TERMINAL CHARGING ON LAPLACE TRANSFORMS START POINT SHIFT EFFECT FRACTIONAL INTEGRAL FRACTIONAL DERIVATIVE LAPLACE TRANSFORM OF INITIALIZATION FUNCTION EXAMPLES OF INITIALIZATION IN FRACTIONAL DIFFERENTIAL EQUATIONS PROBLEM OF SCALAR INITIALIZATION PROBLEM OF VECTOR INITIALIZATION LAPLACE TRANSFORM S ?W PLANE FOR FRACTIONAL CONTROLS STABILITY RATIONAL APPROXIMATIONS OF FRACTIONAL LAPLACE OPERATOR APPLICATION OF GENERALIZED FRACTIONAL CALCULUS IN ELECTRICAL CIRCUIT ANALYSIS ELECTRONICS OPERATIONAL AMPLI?ER CIRCUITS OPERATIONAL AMPLI?ER CIRCUIT WITH LUMPED COMPONENTS OPERATIONAL AMPLI?ER INTEGRATOR WITH LUMPED ELEMENT OPERATIONAL AMPLI?ER INTEGRATOR WITH DISTRIBUTED ELEMENT OPERATIONAL AMPLI?ER DIFFERENTIAL CIRCUIT WITH LUMPED ELEMENTS OPERATIONAL AMPLI?ER DIFFERENTIATOR WITH DISTRIBUTED ELEMENT OPERATIONAL AMPLI?ER AS ZERO-ORDER GAIN WITH LUMPED COMPONENTS OPERATIONAL AMPLI?ER AS ZERO-ORDER GAIN WITH DISTRIBUTED ELEMENTS OPERATIONAL AMPLI?ER CIRCUIT FOR SEMI-DIFFERINTEGRATION BY SEMI-IN?NITE LOSSY LINE OPERATIONAL AMPLI?ER CIRCUIT FOR SEMI-INTEGRATOR 8 2 10 OPERATIONAL AMPLI?ER CIRCUIT FOR SEMI-DIFFERENTIATOR 8 2 11 CASCADED SEMI-INTEGRATORS 8 2 12 SEMI-INTEGRATOR SERIES WITH SEMI-DIFFERENTIATOR CIRCUIT BATTERY DYNAMICS BATTERY AS FRACTIONAL ORDER SYSTEM BATTERY CHARGING PHASE BATTERY DISCHARGE PHASE TRACKING FILTER OBSERVATIONS FRACTIONAL ORDER STATE VECTOR REPRESENTATION IN CIRCUIT THEORY APPLICATION OF GENERALIZED FRACTIONAL CALCULUS IN OTHER SCIENCE AND ENGINEERING FIELDS DIFFUSION MODEL IN ELECTROCHEMISTRY ELECTRODE–ELECTROLYTE INTERFACE IMPEDANCE CAPACITOR THEORY FRACTANCE CIRCUIT FEEDBACK CONTROL SYSTEM CONCEPT OF ISO-DAMPING FRACTIONAL VECTOR FEEDBACK CONTROLLER OBSERVER IN FRACTIONAL VECTOR SYSTEM MODERN ASPECTS OF FRACTIONAL CONTROL VISCOELASTICITY (STRESS–STRAIN) VIBRATION DAMPING SYSTEM 10 SYSTEM ORDER IDENTI?CATION AND CONTROL FRACTIONAL ORDER SYSTEMS CONTINUOUS ORDER DISTRIBUTION DETERMINATION OF ORDER DISTRIBUTION FROM FREQUENCY DOMAIN EXPERIMENTAL DATA ANALYSIS OF CONTINUOUS ORDER DISTRIBUTION VARIABLE ORDER SYSTEM 10 6 1 RL DE?NITION FOR VARIABLE ORDER 10 6 2 LAPLACE TRANSFORMS AND TRANSFER FUNCTION OF VARIABLE ORDER SYSTEM 10 6 3 GL DE?NITION FOR VARIABLE ORDER GENERALIZED PID CONTROLS CONTINUUM ORDER FEEDBACK CONTROL SYSTEM TIME DOMAIN RESPONSE OF SINUSOIDAL INPUTS FOR FRACTIONAL ORDER OPERATOR 10 10 FREQUENCY DOMAIN RESPONSE OF SINUSOIDAL INPUTS FOR FRACTIONAL ORDER OPERATOR 10 11 ULTRA-DAMPED SYSTEM RESPONSE 10 12 HYPER-DAMPED SYSTEM RESPONSE 10 13 DISADVANTAGE OF FRACTIONAL ORDER SYSTEM 10 14 CONCLUDING COMMENTS BIBLIOGRAPHY