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 ...
Formalization of Laplace Transform using theMultivariable 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-19Stellenbosch, South Africa
December 15, 2013
Formalization of Laplace Transform using HOL-Light
Outline
qIntroduction
qMotivation
qFormalization Details
qCase StudyqLinear Transfer Converter (LTC) circuit
qConclusionsO. Hasan 2
Formalization of Laplace Transform using HOL-Light
Laplace Transform
qIntegral transform methodqPierre Simon Laplace 1749–1827
qMathematically represented by the following improper integral
qA linear operatorqInput: Time varying function, i.e., a function f(t) with a
real argument t (t ≥ 0)qOutput: F(s) with complex argument s
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Laplace Transform –Key Benefits and Utilizations
q Solve linear Ordinary Differential Equations(ODEs) using simple algebraic techniques
qObtain concise and useful input/output relationships (Transfer Functions) for systemsqWidely used in Control System and Analog Circuit
Design
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Formalization of Laplace Transform using HOL-Light
Laplace Transform - Example
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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 domain5
Formalization of Laplace Transform using HOL-Light
Real-World Applications for Laplace TransformsqIntegral part of analyzing many physical systems
qAerodynamic systemsqCircuit Analysis
qControl systemsqMechanical networks
qAnalogue filters
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Formalization of Laplace Transform using HOL-Light
Criteria Paper-and-
Pencil Proof
Simulation/SymbolicMethods
AutomatedFormal
Methods(MC, ATP)
Computer Algebra Systems
Higher-order-logic Proof Assistants
Expressiveness
Accuracy
Automation
Laplace Transform based Analysis
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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
Formalization of Laplace Transform using HOL-Light
Formal Definition of Laplace TransformqMathematical definition
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Definition : Laplace Transform
Definition : Conditions for Laplace Existence
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
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Formalization of Laplace Transform using HOL-Light
Case Study: Linear Transfer Converter (LTC) circuitqConverts the voltage and current levels in power
electronics systems
qFunctional correctness of power systems depends on design and stability of LTC
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Differential Equation:
Transfer Function:
Formalization of Laplace Transform using HOL-Light
Linear Transfer Converter (LTC) circuit
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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
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Theorem : Transfer Function of LTC
Formalization of Laplace Transform using HOL-Light
Conclusions
qFormalization of Laplace transform theory using higher-order logicqMultivariable Calculus Theory of HOL-Light
qAdvantagesqAccurate ResultsqReduction in user-effort while formally analyzing
Physical Systems that involve Differential Equations
qCase Study: Transfer function verification of LTC circuit
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Future Directions
qApplication of Laplace transform theory in Analog and Mixed Signal circuits and controls engineering
qFormalization of Inverse Laplace transform
qFormalization of Fourier transform
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Thanks!
qFor More InformationqVisit our website
§ http://save.seecs.nust.edu.pk
qContact§ [email protected]
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Formalization of Laplace Transform using HOL-Light
Additional slides
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Formalization of Laplace Transform using HOL-Light
Formalized Laplace Transform
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Definition 3: Exponential Order Function
Formalization of Laplace Transform using HOL-Light
Laplace Transform
qProvides compact representation of the overall behavior of the given time varying function
qs-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
Formalization of Laplace Transform using HOL-Light
HOL-Light
qMultivariable calculus theoriesqIntegral theoryqDifferential theory
qTranscendental theoryqTopological theory
qComplex analysis theory
qReal number theory
qNatural number theory
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Limit Existence of Laplace TransformqProof 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 IntegralLemma 2: Limit of a Complex-Valued Function
