Formalization of Laplace Transform using the Multivariable...

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Formalization of Laplace Transform using the Multivariable Calculus Theory of HOL-Light Hira Taqdees and Osman Hasan System Analysis & Verification (SAVe) Lab, National University of Sciences and Technology (NUST), Islamabad, Pakistan LPAR-19 Stellenbosch, South Africa December 15, 2013

Transcript of Formalization of Laplace Transform using the Multivariable...

Page 1: Formalization of Laplace Transform using the Multivariable ...ohasan.seecs.nust.edu.pk/talks/lpar_2013.pdf · Formalization of Laplace Transform using the Multivariable Calculus Theory

Formalization of Laplace Transform using the Multivariable Calculus Theory of HOL-Light

Hira Taqdees and Osman Hasan

System Analysis & Verification (SAVe) Lab,

National University of Sciences and Technology (NUST), Islamabad, Pakistan

LPAR-19 Stellenbosch, South Africa

December 15, 2013

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Formalization of Laplace Transform using HOL-Light

Outline

q Introduction

q Motivation

q Formalization Details

q Case Study q Linear Transfer Converter (LTC) circuit

q Conclusions O. Hasan 2

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Formalization of Laplace Transform using HOL-Light

Laplace Transform

q Integral transform method q Pierre Simon Laplace 1749–1827

q Mathematically represented by the following improper integral

q A linear operator q Input: Time varying function, i.e., a function f(t) with a

real argument t (t ≥ 0) q Output: F(s) with complex argument s

O. Hasan 3

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Formalization of Laplace Transform using HOL-Light

Laplace Transform – Key Benefits and Utilizations

q Solve linear Ordinary Differential Equations (ODEs) using simple algebraic techniques

q Obtain concise and useful input/output relationships (Transfer Functions) for systems q Widely used in Control System and Analog Circuit

Design

O. Hasan 4

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Formalization of Laplace Transform using HOL-Light

Laplace Transform - Example

O. Hasan

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Inverse Laplace

Taking Laplace Transform on Both sides

Using the Laplace of a differential and the Linearity of Laplace Properties

Transfer Function

Laplace of sine

Solution in time domain 5

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Formalization of Laplace Transform using HOL-Light

Real-World Applications for Laplace Transforms q Integral part of analyzing many physical systems

q Aerodynamic systems

q Circuit Analysis q Control systems

q Mechanical networks q Analogue filters

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Formalization of Laplace Transform using HOL-Light

Criteria Paper-and-

Pencil Proof

Simulation/Symbolic Methods

Automated Formal

Methods (MC, ATP)

Computer Algebra Systems

Higher-order-logic Proof Assistants

Expressiveness

Accuracy

Automation

Laplace Transform based Analysis

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?

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Formalization of Laplace Transform using HOL-Light

Proposed Approach for Verifying Transfer Functions

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Higher-order Logic

HOL-Light Multivariable

Calculus Theories

Formalized Definition of

Laplace Transform

Supporting Theorems (Integral Comparison

Test etc)

Formally Verified Properties of Laplace Transform

Differential Equation

Transfer Function

Formal Model

Theorems

HOL-Light Theorem Prover

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Formalization of Laplace Transform using HOL-Light

Formal Definition of Laplace Transform q Mathematical definition

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Definition : Laplace Transform

Definition : Conditions for Laplace Existence

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Formalization of Laplace Transform using HOL-Light

Formalized Laplace Transform Properties

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Formalization of Laplace Transform using HOL-Light

Formalized Laplace Transform Properties

§  5000 lines of HOL-Light code and approximately 800 man-hours

O. Hasan 11

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Formalization of Laplace Transform using HOL-Light

Case Study: Linear Transfer Converter (LTC) circuit q Converts the voltage and current levels in power

electronics systems

q Functional correctness of power systems depends on design and stability of LTC

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Differential Equation:

Transfer Function:

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Formalization of Laplace Transform using HOL-Light

Linear Transfer Converter (LTC) circuit

O. Hasan 13

Definition : Differential Equation of LTC

Definition : Differential Equation

Differential Equation:

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Formalization of Laplace Transform using HOL-Light

Linear Transfer Converter (LTC)

q 650 lines of HOL-Light code and the proof process took just a couple of hours

O. Hasan 14

Theorem : Transfer Function of LTC

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Formalization of Laplace Transform using HOL-Light

Conclusions

q Formalization of Laplace transform theory using higher-order logic q Multivariable Calculus Theory of HOL-Light

q Advantages q Accurate Results

q Reduction in user-effort while formally analyzing Physical Systems that involve Differential Equations

q Case Study: Transfer function verification of LTC circuit

O. Hasan 15

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Formalization of Laplace Transform using HOL-Light

Future Directions

q Application of Laplace transform theory in Analog and Mixed Signal circuits and controls engineering

q Formalization of Inverse Laplace transform

q Formalization of Fourier transform

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Formalization of Laplace Transform using HOL-Light

Thanks!

q For More Information q Visit our website

§  http://save.seecs.nust.edu.pk

q Contact §  [email protected]

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Additional slides

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Formalized Laplace Transform

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Definition 3: Exponential Order Function

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Formalization of Laplace Transform using HOL-Light

Laplace Transform

q Provides compact representation of the overall behavior of the given time varying function

q s-plane representation depicts frequency and phase of sinusoidal signal

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Transformation

Laplace

t

A

Sinusoidal Function

σ

j𝝎

x

x

Unit Circle

s-plane (s=angular frequency)

Poles

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HOL-Light

q Multivariable calculus theories q Integral theory

q Differential theory q Transcendental theory

q Topological theory q Complex analysis theory

q Real number theory

q Natural number theory

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Limit Existence of Laplace Transform q Proof Steps

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Split the complex integrand into real and imaginary parts

Convert both complex integrals to their corresponding real integral and split the complex limit to both integrals

Using formalized integral comparison test, prove the convergence of each integral

In our case, g is Mexp(αt)

Lemma 3: Comparison Test for Improper Integrals

Lemma 1: Relationship between the Real and Complex Integral Lemma 2: Limit of a Complex-Valued Function