Formal Analysis of Fractional Order Systems in HOL Umair Siddique - - PowerPoint PPT Presentation

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Formal Analysis of Fractional Order Systems in HOL Umair Siddique - - PowerPoint PPT Presentation

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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

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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

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SLIDE 2

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

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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

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Fractional Order Systems

Physical systems are usually modeled with integral and differential equations Dnf(x) = dn dxn f(x) = d dx( d dx · · · d dx(f(x)) · · · )) · · ·

  • f(x1, x2, · · · xn)dx1, dx2 · · · dxn

Are these traditional concepts sufficient?

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 4 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Fractional Order Systems

Physical systems are usually modeled with integral and differential equations Dnf(x) = dn dxn f(x) = d dx( d dx · · · d dx(f(x)) · · · )) · · ·

  • f(x1, x2, · · · xn)dx1, dx2 · · · dxn

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

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Fractional Order Systems

Physical systems are usually modeled with integral and differential equations Dnf(x) = dn dxn f(x) = d dx( d dx · · · d dx(f(x)) · · · )) · · ·

  • f(x1, x2, · · · xn)dx1, dx2 · · · dxn

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

  • f non integer order (Fractional Calculus)
  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 4 / 37

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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

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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

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Fractional order Calculus - Why a Paradox?

Analogous to fractional exponents x3 = x • x • x x3.7 =? xπ =?

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 6 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Fractional order Calculus - Why a Paradox?

Analogous to fractional exponents x3 = x • x • x x3.7 =? xπ =? 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

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Fractional order Calculus - Why a Paradox?

Analogous to fractional exponents x3 = x • x • x x3.7 =? xπ =? 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

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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

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Mathematical Definitions of Fractional Calculus

Definition (Euler’s Fractional Derivative for Power Function xp) D0xp = xp, D1xp = pxp−1, D2xp = p(p − 1)xp−2 · · · can be generalized as follows: Dnxp = p! (p − n)!xp−n; n : integer Gamma function generalizes the factorial for all real numbers Γ(z) = ∞ tz−1e−tdt Thus Dnxp = Γ(p + 1) Γ(p − n + 1)xp−n; n : real

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 7 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Mathematical Definitions of Fractional Calculus

Definition (Euler’s Fractional Derivative for Power Function xp) D0xp = xp, D1xp = pxp−1, D2xp = p(p − 1)xp−2 · · · can be generalized as follows: Dnxp = p! (p − n)!xp−n; n : integer Gamma function generalizes the factorial for all real numbers Γ(z) = ∞ tz−1e−tdt Thus Dnxp = Γ(p + 1) Γ(p − n + 1)xp−n; n : real

Limited Scope (Only caters for power functions f(x) = xy )

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 7 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Mathematical Definitions of Fractional Calculus

Definition (Riemann-Liouville (RL) Fractional Integration) Jv

af(x) =

· · · t

a

f(x)dx = 1 Γ(v) x

a

(x − t)v−1f(t)dt Definition (Riemann-Liouville Fractional Differentiation) Dvf(x) = ( d dx)⌈v⌉J⌈v⌉−v

a

f(x) 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

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Mathematical Definitions of Fractional Calculus

Definition (Riemann-Liouville (RL) Fractional Integration) Jv

af(x) =

· · · t

a

f(x)dx = 1 Γ(v) x

a

(x − t)v−1f(t)dt Definition (Riemann-Liouville Fractional Differentiation) Dvf(x) = ( d dx)⌈v⌉J⌈v⌉−v

a

f(x) 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

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Mathematical Definitions of Fractional Calculus

Definition (Gr¨ unwald-Letnikov (GL) Fractional Diffintegral)

cDv xf(x) = lim h→0 h−v [ x−c

h ]

  • k=0

(−1)k v k

  • f(x − kh)

where v

k

  • represents the binomial coefficient expressed in terms of the

Gamma function

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 9 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Mathematical Definitions of Fractional Calculus

Definition (Gr¨ unwald-Letnikov (GL) Fractional Diffintegral)

cDv xf(x) = lim h→0 h−v [ x−c

h ]

  • k=0

(−1)k v k

  • f(x − kh)

where v

k

  • represents the binomial coefficient expressed in terms of the

Gamma function (0 < v): Fractional Differentiation (v < 0): Fractional Integration

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 9 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Mathematical Definitions of Fractional Calculus

Definition (Gr¨ unwald-Letnikov (GL) Fractional Diffintegral)

cDv xf(x) = lim h→0 h−v [ x−c

h ]

  • k=0

(−1)k v k

  • f(x − kh)

where v

k

  • represents the binomial coefficient expressed in terms of the

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

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Fractional order Calculus

Paradox Resolved!

Most of the Mathematical Fractional Calculus theory was developed prior to the turn of the 20th century

Useful Consequences?

First book on modeling Engineering systems using Fractional Calculus was published in 1974 by Oldham and Spanier Recent monographs and symposia proceedings have highlighted the application of Fractional Calculus in

Continuum Mechanics Signal Processing Electro-magnetics Control Engineering Electronic Circuits Biological Systems

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 10 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Analysis of Fractional Order Systems

Fractional order Systems are widely used in safety-critical domains like medicine and transportation

Example: Cardiac tissue electrode interface

Analysis inaccuracies may even result in the loss of human lives

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 11 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Analysis of Fractional Order Systems

Fractional order Systems are widely used in safety-critical domains like medicine and transportation

Example: Cardiac tissue electrode interface

Analysis inaccuracies may even result in the loss of human lives Usage of Fractional Calculus guarantees correct models What about the accuracy of Analysis techniques?

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 11 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Analysis of Fractional Order Systems: Comparison

Criteria Paper-and- Pencil Proof Simulation Automated Formal Methods (MC, ATPs) Higher-

  • rder-logic

Theorem Proving Expressiveness Scalability Accuracy FOS Fundamentals Automation

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 12 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Analysis of Fractional Order Systems: Comparison

Criteria Paper-and- Pencil Proof Simulation Automated Formal Methods (MC, ATPs) Higher-

  • rder-logic

Theorem Proving Expressiveness Scalability Accuracy

?

FOS Fundamentals Automation

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 12 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Analysis of Fractional Order Systems: Comparison

Criteria Paper-and- Pencil Proof Simulation Automated Formal Methods (MC, ATPs) Higher-

  • rder-logic

Theorem Proving Expressiveness Scalability Accuracy

?

FOS Fundamentals Automation

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 13 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Analysis of Fractional Order Systems: Comparison

Criteria Paper-and- Pencil Proof Simulation Automated Formal Methods (MC, ATPs) Higher-

  • rder-logic

Theorem Proving Expressiveness Scalability Accuracy

?

FOS Fundamentals Automation

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 14 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Analysis of Fractional Order Systems: Comparison

Criteria Paper-and- Pencil Proof Simulation Automated Formal Methods (MC, ATPs) Higher-

  • rder-logic

Theorem Proving Expressiveness Scalability Accuracy

?

FOS Fundamentals Automation

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 14 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Analysis of Fractional Order Systems: Comparison

Criteria Paper-and- Pencil Proof Simulation Automated Formal Methods (MC, ATPs) Higher-

  • rder-logic

Theorem Proving Expressiveness Scalability Accuracy

?

FOS Fundamentals Automation

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 14 / 37

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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 15 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Proposed Framework

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 16 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Proposed Framework

Fractional

  • rder System

Properties of System Higher-order Logic

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 16 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Proposed Framework

Fractional

  • rder System

Properties of System Higher -order Logic Description (Theorem) Higher-order Logic Real Analysis & Integer

  • rder

calculus

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 16 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Proposed Framework

Fractional

  • rder System

Properties of System Higher -order Logic Description (Theorem) Higher-order Logic Real Analysis & Integer

  • rder

calculus Gamma function

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 16 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Proposed Framework

Fractional

  • rder System

Properties of System Higher -order Logic Description (Theorem) Higher-order Logic Real Analysis & Integer

  • rder

calculus Gamma function Differintegrals

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 16 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Proposed Framework

Fractional

  • rder System

Properties of System Higher -order Logic Description (Theorem) Higher-order Logic Real Analysis & Integer

  • rder

calculus Gamma function Differintegrals Formally Verified Properties

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 16 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Proposed Framework

Fractional

  • rder System

Properties of System Higher -order Logic Description (Theorem) Higher-order Logic Real Analysis & Integer

  • rder

calculus Gamma function Differintegrals Formally Verified Properties Formal Model

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 16 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Proposed Framework

Fractional

  • rder System

Properties of System Higher -order Logic Description (Theorem) Higher-order Logic Real Analysis & Integer

  • rder

calculus Gamma function Differintegrals Formally Verified Properties Formal Model Theorems Theorems Theorems

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 16 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Proposed Framework

Fractional

  • rder System

Properties of System Higher -order Logic Description (Theorem) HOL Theorem Prover Higher-order Logic Real Analysis & Integer

  • rder

calculus Gamma function Differintegrals Formally Verified Properties Formal Model Theorems Theorems Theorems

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 16 / 37

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SLIDE 39

Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Proposed Framework

Fractional

  • rder System

Properties of System Higher -order Logic Description (Theorem) HOL Theorem Prover Formal Proof of System Properties Higher-order Logic Real Analysis & Integer

  • rder

calculus Gamma function Differintegrals Formally Verified Properties Formal Model Theorems Theorems Theorems

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 16 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

HOL4 Theorem Prover

Higher-order-logic Theorem Prover developed at the University of Cambridge Its core consists of

5 fundamental axioms (facts) 8 Inference rules

Soundness is assured as every new theorem must be created from

The basic axioms and primitive inference rules Any other already proved theorems (Theory Files)

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 17 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

HOL4 Theorem Prover

Higher-order-logic Theorem Prover developed at the University of Cambridge Its core consists of

5 fundamental axioms (facts) 8 Inference rules

Soundness is assured as every new theorem must be created from

The basic axioms and primitive inference rules Any other already proved theorems (Theory Files)

The availability of Harisson’s seminal work on Real analysis and Integer order Calculus has been the primary motivation for this choice

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 17 / 37

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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 18 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formalization of Gamma function

Γ(z) = ∞ tz−1e−tdt The integrand tz−1e−t becomes unbounded on the lower limit (t = 0) for z < 1 The upper limit ∞ is undefined

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 19 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formalization of Gamma function

Γ(z) = ∞ tz−1e−tdt The integrand tz−1e−t becomes unbounded on the lower limit (t = 0) for z < 1 The upper limit ∞ is undefined Γ(z) = limn→∞

  • limb→∞

b

1 2n

tz−1e−tdt

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 19 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formalization of Gamma function

Γ(z) = ∞ tz−1e−tdt The integrand tz−1e−t becomes unbounded on the lower limit (t = 0) for z < 1 The upper limit ∞ is undefined Γ(z) = limn→∞

  • limb→∞

b

1 2n

tz−1e−tdt

  • Definition

⊢ ∀ z. gamma z = lim(λn.(lim(λb. b

1 2n t rpow (z-1)exp(-t) dt))

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 19 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formal Verification of Gamma function properties

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 20 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formal Verification of Gamma function properties

Γ(z + 1) = zΓ(z) Theorem: Pseudo-recurrence Relation ⊢ ∀ z . (0 < z) = ⇒ (gamma (z + 1)= z gamma (z))

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 20 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formal Verification of Gamma function properties

Γ(z + 1) = zΓ(z) Theorem: Pseudo-recurrence Relation ⊢ ∀ z . (0 < z) = ⇒ (gamma (z + 1)= z gamma (z)) The paper-and-pencil based proof is based on the integration-by-parts property We also had to utilize the concepts of limits of a real sequence, differentiability and integrability The formal proof required 10 main lemmas. e.g.,

(∀n.∃k.(λb. b

1 2n tz−1e−tdt) −

→ k) (∀b.∃p.(λn. b

1 2n tz−1e−tdt) −

→ p)

It took approximately 2000 lines of ML code

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 20 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formally Verified Properties of Gamma Function

Property HOL Formalization Pseudo-Recurrence Relation ⊢ ∀ z.(0 < z) = ⇒ (gamma (z + 1)= z gamma (z)) Functional Equation ⊢ gamma 1 = 1 Factorial Generalization ⊢ ∀ n ∈ N. gamma(n + 1) = n!

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 21 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Proposed Framework

Fractional

  • rder System

Properties of System Higher -order Logic Description (Theorem) HOL Theorem Prover Formal Proof of System Properties Higher-order Logic Real Analysis & Integer

  • rder

calculus Gamma function Differintegrals Formally Verified Properties Formal Model Theorems Theorems Theorems

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 22 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formalization of Fractional Integration

We follow Riemann-Liouville Definition Jv

af(x) =

1 Γ(v) x

a

(x − t)v−1f(t)dt The integrand (x − t)v−1f(t) becomes undefined on upper limit (x) of integration

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 23 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formalization of Fractional Integration

We follow Riemann-Liouville Definition Jv

af(x) =

1 Γ(v) x

a

(x − t)v−1f(t)dt The integrand (x − t)v−1f(t) becomes undefined on upper limit (x) of integration Jv

af(x) = limn→∞

  • 1

Γ(v) x− 1

2n

a

(x − t)v−1f(t)dt

  • Definition

⊢ ∀ f v a x.frac int f v a x = if (v = 0) then f else lim(λn.

1 gamma v(

x− 1

2n

a

((x - t) rpow (v-1)) f(t) dt)

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 23 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formalization of Fractional Differentiation

Dvf(x) = ( d dx)⌈v⌉J⌈v⌉−v

a

f(x) Definition ⊢ ∀ f v a x. frac diff f v a x = n order deriv (clg v) (frac int f (clg v - v) a x)

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 24 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formally Verified Properties of Differintegrals

Property HOL Formalization Identity ⊢ ∀ f a x. (a < x) = ⇒ (frac int f 0 a x = f)∧ (frac diff f 0 a x = f) Generalized Integral ⊢ ∀ f a x v ∈ N. (a < x)∧ (1 < v) = ⇒ frac int f v a x = lim(λn.

1 (v-1)!

x− 1

2n

a

(x - t) rpow (v-1)f(t) dt) frac int Linearity ⊢ ∀ f v x a b. (frac exists f x v)∧ (frac exists g x v) = ⇒ frac int (a f + b g) v 0 x = a(frac int f v 0 x)+ b(frac int g v 0 x) frac diff Linearity ⊢ ∀ f v x a b. (frac exists f x v)∧ (frac exists g x v)∧ (∀ m. (m <= clg v) ⇒ (n order deriv m (frac int f v 0 x)) differentiable x)∧ (∀ m. (m <= clg v) ⇒ (n order deriv m (frac int g v 0 x)) differentiable x)= ⇒ ( frac diff (a f + b g) v 0 x = a(frac diff f v 0 x)+ b(frac diff g v 0 x))

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 25 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

HOL Formalization of Fractional Caclulus

Fractional

  • rder System

Properties of System Higher -order Logic Description (Theorem) HOL Theorem Prover Formal Proof of System Properties Higher-order Logic Real Analysis & Integer

  • rder

calculus Gamma function Differintegrals Formally Verified Properties Formal Model Theorems Theorems Theorems

The formalization took around 7500 lines of ML code and approximately 600 man hours

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 26 / 37

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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 27 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Case Studies

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 28 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Case Studies

We apply our framework to analyze three real world fractional order systems

Resistoductance Fractional Differentiator circuit Fractional Integrator circuit

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 28 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Resistoductance

An electrical component with characteristics between

  • hmic resistor and an Inductor

v(t)= K D i(t)

α

i(t)= J v(t)

α

1 Κ

+ −

α = 0 : Purely resistive behavior with K = R ohms α = 1 Purely inductive behavior with K = L henrys

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 29 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formal Model of Resistoductance

The governing current-voltage relationship is given as follows: i(t) = 1 K Jαv(t)

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 30 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formal Model of Resistoductance

The governing current-voltage relationship is given as follows: i(t) = 1 K Jαv(t) Definition (Resistoductance Current) ⊢ ∀ K v i alpha x. i t K v i alpha x = (1/K)frac int v i(t) alpha 0 x v i = Input voltage i t = Resistoductance current alpha = Order of integration

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 30 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Verification of Resistoductance properties

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 31 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Verification of Resistoductance properties

i t for constant voltage V 0 ⊢ ∀ K V 0 alpha x. (0 < x) ∧ (0 < alpha) = ⇒ (i t K V 0 alpha x = (1/(K Gamma (alpha + 1))) (V 0(x rpow alpha)))

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 31 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Verification of Resistoductance properties

i t for constant voltage V 0 ⊢ ∀ K V 0 alpha x. (0 < x) ∧ (0 < alpha) = ⇒ (i t K V 0 alpha x = (1/(K Gamma (alpha + 1))) (V 0(x rpow alpha))) Theorem: Special Cases for i t ⊢ ∀ x. (0 < x) = ⇒ (alpha = 0) ⇒ i t K V 0 alpha x = V 0 / K ∧ (alpha = 1) ⇒ i t K V 0 alpha x = (V 0 / K) x

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 31 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Verification of Resistoductance properties

i t for constant voltage V 0 ⊢ ∀ K V 0 alpha x. (0 < x) ∧ (0 < alpha) = ⇒ (i t K V 0 alpha x = (1/(K Gamma (alpha + 1))) (V 0(x rpow alpha))) Theorem: Special Cases for i t ⊢ ∀ x. (0 < x) = ⇒ (alpha = 0) ⇒ i t K V 0 alpha x = V 0 / K ∧ (alpha = 1) ⇒ i t K V 0 alpha x = (V 0 / K) x Proof heavily relies upon the formally verified properties of Gamma function and Differintegrals 350 lines of HOL code Approximately 2 man-hours by an expert user

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 31 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Fractional integrator and differentiator circuits

(a) (b)

Used in fractional-order PID and PI controllers Offer more flexibility for gain adjustment

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 32 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formal Models

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 33 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formal Models

The output voltage equations for a fractional integrator vo(t) = − 1 RC Jµvi(t)

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 33 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formal Models

The output voltage equations for a fractional integrator vo(t) = − 1 RC Jµvi(t) Definition (Fractional Order Integrator) ⊢ ∀ R C v i mu x. v I 0 R C v i mu x =

  • (1/RC)frac int v i(t) mu 0 x
  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 33 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formal Models

The output voltage equations for a fractional integrator vo(t) = − 1 RC Jµvi(t) Definition (Fractional Order Integrator) ⊢ ∀ R C v i mu x. v I 0 R C v i mu x =

  • (1/RC)frac int v i(t) mu 0 x

The output voltage equations for a fractional differentiator v0(t) = −RCDµvi(t)

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 33 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formal Models

The output voltage equations for a fractional integrator vo(t) = − 1 RC Jµvi(t) Definition (Fractional Order Integrator) ⊢ ∀ R C v i mu x. v I 0 R C v i mu x =

  • (1/RC)frac int v i(t) mu 0 x

The output voltage equations for a fractional differentiator v0(t) = −RCDµvi(t) Definition (Fractional Order Differentiator) ⊢ ∀ R C v i mu x. v D 0 R C v i mu x=

  • (RC)frac diff v i(t) mu 0 x
  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 33 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formal Analysis: For Unit Step signal

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 34 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formal Analysis: For Unit Step signal

Theorem: Output of Fractional Integrator Circuit ⊢ ∀ R C mu x. (0 < x) ∧ (0 < mu) ∧ (mu < 1) = ⇒ (v I 0 R C (unit t) mu x =

  • 1/(RC Gamma (mu + 1)) (x rpow mu)
  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 34 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formal Analysis: For Unit Step signal

Theorem: Output of Fractional Integrator Circuit ⊢ ∀ R C mu x. (0 < x) ∧ (0 < mu) ∧ (mu < 1) = ⇒ (v I 0 R C (unit t) mu x =

  • 1/(RC Gamma (mu + 1)) (x rpow mu)

Theorem: Output of Fractional Differentiator Circuit ⊢ ∀ R C mu x. (0 < x) ∧ (0 < mu) ∧ (mu < 1) = ⇒ (v D 0 R C (unit t) mu x = (-(RC (Gamma (1 - mu)))(x rpow -mu))

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 34 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Formal Analysis: For Unit Step signal

Theorem: Output of Fractional Integrator Circuit ⊢ ∀ R C mu x. (0 < x) ∧ (0 < mu) ∧ (mu < 1) = ⇒ (v I 0 R C (unit t) mu x =

  • 1/(RC Gamma (mu + 1)) (x rpow mu)

Theorem: Output of Fractional Differentiator Circuit ⊢ ∀ R C mu x. (0 < x) ∧ (0 < mu) ∧ (mu < 1) = ⇒ (v D 0 R C (unit t) mu x = (-(RC (Gamma (1 - mu)))(x rpow -mu)) The proof relies heavily upon the proposed formalization 400 lines of HOL code Approximately 2.5 man-hours

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 34 / 37

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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 35 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Conclusions

Summary

Formalization of Gamma function and Differintegrals Formal Analysis of Fractional order Systems

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 36 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Conclusions

Summary

Formalization of Gamma function and Differintegrals Formal Analysis of Fractional order Systems

Future Work

Enriching the library of the formally verified Fractional Calculus properties

Law of Exponents Relationship with the Beta function

Development of the current framework using Complex Numbers More Case Studies

Fractional Electromagnetic Systems (Fractional Rectangular Waveguides)

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 36 / 37

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Introduction and Motivation Proposed Methodology Formalization Details Case Studies Conclusions

Thank You!

www.save.seecs.nust.edu.pk

  • U. Siddique and Osman Hasan

Fractional Order Systems in HOL 37 / 37