Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions Formal Analysis of Fractional Order Systems in HOL Umair Siddique Osman Hasan System Analysis and Verification (SAVE) Lab National University of Sciences and Technology (NUST) Islamabad, Pakistan FMCAD, 2011 U. Siddique and Osman Hasan Fractional Order Systems in HOL 1 / 37
Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions Outline 1 Introduction and Motivation 2 Proposed Methodology 3 Formalization Details 4 Case Studies 5 Conclusions U. Siddique and Osman Hasan Fractional Order Systems in HOL 2 / 37
Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions Outline 1 Introduction and Motivation 2 Proposed Methodology 3 Formalization Details 4 Case Studies 5 Conclusions U. Siddique and Osman Hasan Fractional Order Systems in HOL 3 / 37
Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions Fractional Order Systems Physical systems are usually modeled with integral and differential equations D n f ( x ) = d n dx n f ( x ) = d dx ( d dx · · · d dx ( f ( x )) · · · )) � � � · · · f ( x 1 , x 2 , · · · x n ) dx 1 , dx 2 · · · dx n Are these traditional concepts sufficient? U. Siddique and Osman Hasan Fractional Order Systems in HOL 4 / 37
Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions Fractional Order Systems Physical systems are usually modeled with integral and differential equations D n f ( x ) = d n dx n f ( x ) = d dx ( d dx · · · d dx ( f ( x )) · · · )) � � � · · · f ( x 1 , x 2 , · · · x n ) dx 1 , dx 2 · · · dx n Are these traditional concepts sufficient? Example Resistoductance: Exhibits intermediate behavior between a Resistor ( v = iR ) and an Inductor ( v = L di dt ) Cannot be modeled using an integer order Differential Equation U. Siddique and Osman Hasan Fractional Order Systems in HOL 4 / 37
Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions Fractional Order Systems Physical systems are usually modeled with integral and differential equations D n f ( x ) = d n dx n f ( x ) = d dx ( d dx · · · d dx ( f ( x )) · · · )) � � � · · · f ( x 1 , x 2 , · · · x n ) dx 1 , dx 2 · · · dx n Are these traditional concepts sufficient? Example Resistoductance: Exhibits intermediate behavior between a Resistor ( v = iR ) and an Inductor ( v = L di dt ) Cannot be modeled using an integer order Differential Equation Fractional Order Systems involve derivatives and integrals of non integer order (Fractional Calculus) U. Siddique and Osman Hasan Fractional Order Systems in HOL 4 / 37
Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions Fractional order Calculus Fractional Calculus was born in 1695 U. Siddique and Osman Hasan Fractional Order Systems in HOL 5 / 37
Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions Fractional order Calculus Fractional Calculus was born in 1695 Why a paradox? Useful Consequences? U. Siddique and Osman Hasan Fractional Order Systems in HOL 5 / 37
Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions Fractional order Calculus - Why a Paradox? Analogous to fractional exponents x 3 = x • x • x x 3 . 7 =? x π =? U. Siddique and Osman Hasan Fractional Order Systems in HOL 6 / 37
Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions Fractional order Calculus - Why a Paradox? Analogous to fractional exponents x 3 = x • x • x x 3 . 7 =? x π =? Integrals and Derivatives are certainly more complex than multiplication Fractional Integrals and Derivatives can be defined in numerous ways U. Siddique and Osman Hasan Fractional Order Systems in HOL 6 / 37
Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions Fractional order Calculus - Why a Paradox? Analogous to fractional exponents x 3 = x • x • x x 3 . 7 =? x π =? Integrals and Derivatives are certainly more complex than multiplication Fractional Integrals and Derivatives can be defined in numerous ways Fractional Calculus started off as a study for the best minds in mathematics Leibniz, Euler, Lagrange, Laplace, Fourier, Abel, Liouville, Riemann U. Siddique and Osman Hasan Fractional Order Systems in HOL 6 / 37
Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions Mathematical Definitions of Fractional Calculus U. Siddique and Osman Hasan Fractional Order Systems in HOL 7 / 37
Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions Mathematical Definitions of Fractional Calculus Definition (Euler’s Fractional Derivative for Power Function x p ) D 0 x p = x p , D 1 x p = px p − 1 , D 2 x p = p ( p − 1) x p − 2 · · · can be generalized as follows: p ! D n x p = ( p − n )! x p − n ; n : integer Gamma function generalizes the factorial for all real numbers � ∞ t z − 1 e − t dt Γ( z ) = 0 Thus Γ( p + 1) D n x p = Γ( p − n + 1) x p − n ; n : real U. Siddique and Osman Hasan Fractional Order Systems in HOL 7 / 37
Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions Mathematical Definitions of Fractional Calculus Definition (Euler’s Fractional Derivative for Power Function x p ) D 0 x p = x p , D 1 x p = px p − 1 , D 2 x p = p ( p − 1) x p − 2 · · · can be generalized as follows: p ! D n x p = ( p − n )! x p − n ; n : integer Gamma function generalizes the factorial for all real numbers � ∞ t z − 1 e − t dt Γ( z ) = 0 Thus Γ( p + 1) D n x p = Γ( p − n + 1) x p − n ; n : real Limited Scope (Only caters for power functions f ( x ) = x y ) U. Siddique and Osman Hasan Fractional Order Systems in HOL 7 / 37
Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions Mathematical Definitions of Fractional Calculus Definition (Riemann-Liouville (RL) Fractional Integration) � t � x � � 1 J v ( x − t ) v − 1 f ( t ) dt a f ( x ) = · · · f ( x ) dx = Γ( v ) a a Definition (Riemann-Liouville Fractional Differentiation) D v f ( x ) = ( d dx ) ⌈ v ⌉ J ⌈ v ⌉− v f ( x ) a where v is the order and ⌈ v ⌉ is its ceiling (largest and closest integer). U. Siddique and Osman Hasan Fractional Order Systems in HOL 8 / 37
Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions Mathematical Definitions of Fractional Calculus Definition (Riemann-Liouville (RL) Fractional Integration) � t � x � � 1 J v ( x − t ) v − 1 f ( t ) dt a f ( x ) = · · · f ( x ) dx = Γ( v ) a a Definition (Riemann-Liouville Fractional Differentiation) D v f ( x ) = ( d dx ) ⌈ v ⌉ J ⌈ v ⌉− v f ( x ) a where v is the order and ⌈ v ⌉ is its ceiling (largest and closest integer). General definition that caters for all functions that can be expressed in a closed mathematical form Usage requires expertise and rigorous mathematical analysis U. Siddique and Osman Hasan Fractional Order Systems in HOL 8 / 37
Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions Mathematical Definitions of Fractional Calculus Definition (Gr¨ unwald-Letnikov (GL) Fractional Diffintegral) [ x − c h ] � v � � c D v h → 0 h − v ( − 1) k x f ( x ) = lim f ( x − kh ) k k =0 � v � where represents the binomial coefficient expressed in terms of the k Gamma function U. Siddique and Osman Hasan Fractional Order Systems in HOL 9 / 37
Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions Mathematical Definitions of Fractional Calculus Definition (Gr¨ unwald-Letnikov (GL) Fractional Diffintegral) [ x − c h ] � v � � c D v h → 0 h − v ( − 1) k x f ( x ) = lim f ( x − kh ) k k =0 � v � where represents the binomial coefficient expressed in terms of the k Gamma function (0 < v ): Fractional Differentiation ( v < 0): Fractional Integration U. Siddique and Osman Hasan Fractional Order Systems in HOL 9 / 37
Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions Mathematical Definitions of Fractional Calculus Definition (Gr¨ unwald-Letnikov (GL) Fractional Diffintegral) [ x − c h ] � v � � c D v h → 0 h − v ( − 1) k x f ( x ) = lim f ( x − kh ) k k =0 � v � where represents the binomial coefficient expressed in terms of the k Gamma function (0 < v ): Fractional Differentiation ( v < 0): Fractional Integration Facilitates Numerical Methods based computerized analysis Approximate solutions due to the infinite summation involved U. Siddique and Osman Hasan Fractional Order Systems in HOL 9 / 37
Recommend
More recommend
