analysis techniques for fractional order systems: a...

$: Analysis Techniques for Fractional Order Systems: A Surveyohasan.seecs.nust.edu.pk/talks/icnaam_2012.pdf · Analogous to fractional exponents x3 = x x x x3:7 =? xˇ=? Integrals and$
Introduction and MotivationAnalysis Techniques

Conclusions and Discussions

Analysis Techniques for Fractional OrderSystems: A Survey

Umair Siddique Osman Hasan

System Analysis and Verification (SAVE) LabNational University of Sciences and Technology (NUST)

Islamabad, Pakistan

ICNAAM 2012

U. Siddique and Osman Hasan Analysis of Fractional Order Systems 1 / 35



Outline

1 Introduction and Motivation

2 Analysis Techniques

3 Conclusions and Discussions




Outline







Fractional Order Systems

Physical systems are usually modeled with integral anddifferential equations

Dnf(x) =dn

dxnf(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 aResistor (v = iR) and an Inductor (v = L di

dt )

Fractional Order Systems involve derivatives and integralsof non integer order (Fractional Calculus)




Fractional order Calculus

Fractional Calculus was born in 1695

Why a paradox?

Useful Consequences?




Fractional order Calculus - Why a Paradox?

Analogous to fractional exponents

x3 = x • x • x

x3.7 =?

xπ =?

Integrals and Derivatives are certainly more complex thanmultiplication

Fractional Integrals and Derivatives can be defined innumerous ways

Fractional Calculus started off as a study for the bestminds in mathematics

Leibniz, Euler, Lagrange, Laplace, Fourier, Abel, Liouville,Riemann




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) =

∫ ∞0

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 )





Definition (Riemann-Liouville (RL) Fractional Integration)

Jvaf(x) =

1

Γ(v)

∫ x

a

(x− t)v−1f(t)dt

Definition (Riemann-Liouville Fractional Differentiation)

Dvf(x) = (d

dx)dveJdve−va f(x)

where v is the order and dve is its ceiling (largest and closest integer).

General definition that caters for all functions that can beexpressed in a closed mathematical form

Usage requires expertise and rigorous mathematical analysis





Definition (Grunwald-Letnikov (GL) Fractional Diffintegral)

cDvxf(x) = lim

h→0h−v

[ x−ch ]∑

k=0

(−1)k(v

k

)f(x− kh)

where(vk

)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 summationinvolved




Fractional order Calculus

Paradox Resolved!

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

Useful Consequences?

First book on modeling Engineering systems usingFractional Calculus was published in 1974 by Oldham andSpanierRecent monographs and symposia proceedings havehighlighted the application of Fractional Calculus in

Continuum MechanicsSignal ProcessingElectro-magneticsControl EngineeringElectronic CircuitsBiological Systems




Analysis of Fractional Order Systems

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

Example: Cardiac tissue electrode interface

Analysis inaccuracies may even result in the loss of humanlives

Usage of Fractional Calculus guarantees correct models

What about the accuracy of Analysis techniques?




Outline







Analysis Techniques

Paper-and-Pencil Proof Methods

Computer Simulation

Formal Methods




Paper-and-Pencil Proof Methods

Construct a mathematical model of the given fractionalorder system on paper

Using the Riemann-Liouville (RL) definition

Use this model to verify that the system exhibits thedesired properties using mathematical reasoning on a paper

Accurate

Scalability Very Cumbersome for Large systems

Error Prone




Erroneous Analysis Using Paper-and-Pencil Proof

Analysis of Waves in planar waveguide by Cheng (2007)Journal of Optics and Communication (IF 1.316)

It was found to be erroneous by Naqvi (2010)

Cheng 2007

Naqvi2010




Simulation

Construct a discretized system model using computerarithmetic

Analyze the system using Numerical Methods

Using the Grunwald-Letnikov definition

Easy to use

Approximate

Can never be termed as 100% accurate due to the infinitesummations involved in the Grunwald-Letnikov definitionand the usage of computer arithmetic




Simulation Problem

Computation of Gamma function Γ(x) in State-of-the-artsimulation tool, MATLAB

x = 171 it returns 7.26e306

x = 172 it returns Inf




Consequences of using Traditional Techniques inSafety-Critical System Analysis

2009 Air France Flight447 Crash

Pilot probe sensorsdid not accuratelymeasure air speed

Loss of 228 lives

Arianne 5 Crash (1996)

Erroneous dataconversion from a64-bit floating pointto 16-bit signedinteger

Loss of 500 millionUS dollars




Formal Methods

Bridges the gap between Paper-and-pencil proof methodsand simulation

As precise as a mathematical proof can be

Scalable since computers are used for bookkeeping

Immature




Formal Methods

Construct a computer-based mathematical model of thesystem

Use mathematical reasoning to check if the model satisfiesthe properties of interest in a computerized environment

Based on Mathematical techniques




Formal Verification Techniques

Model Checking

Theorem Proving




Model Checking




Model Checking

Advantages

Automatic (Push button type analysis tools)No proofs involvedDiagnostic counter examples

Disadvantages

State-space explosion problemLimitted Expressiveness Only discrete models of Fractionalorder systems can be handled




Theorem Proving




Logic

Study of drawing conclusions (reasoning)

Types of LogicPropositional logic

Supports statements that can be true or false

First-order logic (Predicate logic)

Quantification over variables (∀: For all, ∃: there exists)

Higher-order logic

Quantification over sets and functions




Theorem Prover

Its core consists of a handful of

Fundamental axioms (facts)Inference rules

Soundness is assured as every new theorem must becreated from

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

The availability of Harisson’s seminal work on Realanalysis and Integer order Calculus greatly facilitateFractional order system analysis




Theorem Proving

Advantages

High expressiveness: Fractional order Systems can beanalyzedNo risk of mistakes (human errors)Some parts of the proofs can be automated

Disadvantages

Detailed and explicit human guidance requiredThe state-of-the-art is limited




Theorem Proving based Analysis of Fractional orderSystems

FMCAD 2011: Riemann-Liouville (RL) Definition






Fractional order System

Properties of System

Higher-order Logic








Higher -order Logic Description

(Theorem)

Higher-order Logic

Real Analysis

&Integer order

calculus









(Theorem)

Higher-order Logic

Real Analysis

&Integer order

calculus

Gamma function









(Theorem)

Higher-order Logic

Real Analysis

&Integer order

calculus

Gamma function

Differintegrals









(Theorem)

Higher-order Logic

Real Analysis

&Integer order

calculus

Gamma function

Differintegrals

Formally Verified Properties









(Theorem)

Higher-order Logic

Real Analysis

&Integer order

calculus

Gamma function

Differintegrals


Formal Model









(Theorem)

Higher-order Logic

Real Analysis

&Integer order

calculus

Gamma function

Differintegrals


Formal Model

TheoremsTheorems

Theorems









(Theorem)

HOL Theorem Prover

Higher-order Logic

Real Analysis

&Integer order

calculus

Gamma function

Differintegrals


Formal Model

TheoremsTheorems

Theorems









(Theorem)

HOL Theorem Prover

Formal Proof of System Properties

Higher-order Logic

Real Analysis

&Integer order

calculus

Gamma function

Differintegrals


Formal Model

TheoremsTheorems

Theorems





Applications

ResistoductanceFractional Differentiator circuitFractional Integrator circuit

Verification ProcessGamma function and Differintegrals

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

Applications

300 lines of HOL codeApproximately 6 man-hours




Outline







Analysis of Fractional Order Systems: Comparison

Criteria Paper-and-Pencil Proof

Simulation

Automated Formal

Methods (MC, ATPs)

Higher-order-logic Theorem Proving

Expressiveness

Scalability

Accuracy

FOS Fundamentals

Automation






Simulation

Automated Formal

Methods (MC, ATPs)


Expressiveness

Scalability

Accuracy

?

FOS Fundamentals

Automation




Conclusions

Fractional Calculus provides accurate and robust modelingcapabilities

Can the current analysis techniques guarantee accuracy?

Traditional techniques cannot cope with this challengeTheorem proving provides accuracy but is not mature

Law of ExponentsRelationship with the Beta functionComplex numbers based Fractional Calculus Libraries

We recommend a hybrid framework for the analysis offractional order systems.

Higher-order-logic theorem proving wherever possibleAutomated theorem proving and model checking ondiscretized modelsSemi-Formal Techniques like Computer Algebra Systems

Enrich Higher-order-logic theorem proving capabilities




Conclusions

Fractional Calculus provides accurate and robust modelingcapabilities

Can the current analysis techniques guarantee accuracy?Traditional techniques cannot cope with this challengeTheorem proving provides accuracy but is not mature

Law of ExponentsRelationship with the Beta functionComplex numbers based Fractional Calculus Libraries

We recommend a hybrid framework for the analysis offractional order systems.

Higher-order-logic theorem proving wherever possibleAutomated theorem proving and model checking ondiscretized modelsSemi-Formal Techniques like Computer Algebra Systems

Enrich Higher-order-logic theorem proving capabilities




Thank You!

www.save.seecs.nust.edu.pk


analysis techniques for fractional order systems: a...

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