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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
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
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
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
Formalization of Laplace Transform using HOL-Light
Laplace Transform - Example
O. Hasan
5
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
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
O. Hasan 6
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
O. Hasan 7
?
Formalization of Laplace Transform using HOL-Light
Proposed Approach for Verifying Transfer Functions
O. Hasan 8
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
Formalization of Laplace Transform using HOL-Light
Formal Definition of Laplace Transform q Mathematical definition
O. Hasan 9
Definition : Laplace Transform
Definition : Conditions for Laplace Existence
Formalization of Laplace Transform using HOL-Light
Formalized Laplace Transform Properties
O. Hasan 10
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
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
O. Hasan 12
Differential Equation:
Transfer Function:
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:
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
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
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
O. Hasan 16
Formalization of Laplace Transform using HOL-Light
Thanks!
q For More Information q Visit our website
§ http://save.seecs.nust.edu.pk
q Contact § osman.hasan@seecs.nust.edu.pk
O. Hasan 17
Formalization of Laplace Transform using HOL-Light
Additional slides
O. Hasan 18
Formalization of Laplace Transform using HOL-Light
Formalized Laplace Transform
O. Hasan 19
Definition 3: Exponential Order Function
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
O. Hasan 20
Transformation
Laplace
t
A
Sinusoidal Function
σ
j𝝎
x
x
Unit Circle
s-plane (s=angular frequency)
Poles
Formalization of Laplace Transform using HOL-Light
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
O. Hasan 21
Formalization of Laplace Transform using HOL-Light
Limit Existence of Laplace Transform q Proof Steps
O. Hasan 22
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
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