Formalization of Laplace Transform using the Multivariable Calculus Theory of HOL-Light
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1 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
2 Outline q Introduction q Motivation q Formalization Details q Case Study q Linear Transfer Converter (LTC) circuit q Conclusions O. Hasan Formalization of Laplace Transform using HOL-Light 2
3 O. Hasan Formalization of Laplace Transform using HOL-Light 3 Laplace Transform q Integral transform method q Pierre Simon Laplace 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
4 O. Hasan Formalization of Laplace Transform using HOL-Light 4 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
5 O. Hasan Formalization of Laplace Transform using HOL-Light 5 Laplace Transform - Example Taking Laplace Transform on Both sides Using the Laplace of a differential and the Linearity of Laplace Properties Transfer Function Laplace of sine Inverse Laplace Solution in time domain
6 O. Hasan Formalization of Laplace Transform using HOL-Light 6 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
7 Laplace Transform based Analysis Criteria PaperandPencil Proof Simulation/ Symbolic Methods Automated Formal Methods (MC, ATP) Computer Algebra Systems Higher-orderlogic Proof Assistants Expressiveness Accuracy? Automation O. Hasan Formalization of Laplace Transform using HOL-Light 7
8 O. Hasan Formalization of Laplace Transform using HOL-Light 8 Proposed Approach for Verifying Transfer Functions Differential Equation Transfer Function 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 Formal Model Theorems HOL-Light Theorem Prover
9 O. Hasan Formalization of Laplace Transform using HOL-Light 9 Formal Definition of Laplace Transform q Mathematical definition Definition : Laplace Transform Definition : Conditions for Laplace Existence
10 Formalized Laplace Transform O. Hasan Formalization of Laplace Transform using HOL-Light 10 Properties
11 Formalized Laplace Transform O. Hasan Formalization of Laplace Transform using HOL-Light 11 Properties 5000 lines of HOL-Light code and approximately 800 man-hours
12 O. Hasan Formalization of Laplace Transform using HOL-Light 12 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 Differential Equation: Transfer Function:
13 Linear Transfer Converter (LTC) O. Hasan Formalization of Laplace Transform using HOL-Light 13 circuit Differential Equation: Definition : Differential Equation of LTC Definition : Differential Equation
14 O. Hasan Formalization of Laplace Transform using HOL-Light 14 Linear Transfer Converter (LTC) Theorem : Transfer Function of LTC q 650 lines of HOL-Light code and the proof process took just a couple of hours
15 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 Formalization of Laplace Transform using HOL-Light 15
16 O. Hasan Formalization of Laplace Transform using HOL-Light 16 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
17 O. Hasan Formalization of Laplace Transform using HOL-Light 17 Thanks! q For More Information q Visit our website q Contact osman.hasan@seecs.nust.edu.pk
18 O. Hasan Formalization of Laplace Transform using HOL-Light 18 Additional slides
19 O. Hasan Formalization of Laplace Transform using HOL-Light 19 Formalized Laplace Transform Definition 3: Exponential Order Function
20 O. Hasan Formalization of Laplace Transform using HOL-Light 20 Laplace Transform q Provides compact representation of the overall behavior of the given time varying function A Laplace x jω Unit Circle t Transformation x σ Poles Sinusoidal Function s-plane (s=angular frequency) q s-plane representation depicts frequency and phase of sinusoidal signal
21 O. Hasan Formalization of Laplace Transform using HOL-Light 21 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
22 Limit Existence of Laplace O. Hasan Formalization of Laplace Transform using HOL-Light 22 Transform q Proof Steps 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 Lemma 3: Comparison Test for Improper Integrals Using formalized integral comparison Lemma test, 1: prove 2: Relationship Limit the of a Complex-Valued between the Real Function and Complex Integral convergence of each integral In our case, g is Mexp(αt)
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BINOMIAL TRANSFORMS OF QUADRAPELL SEQUENCES AND QUADRAPELL MATRIX SEQUENCES
Journal of Science and Arts Year 17, No. 1(38), pp. 69-80, 2017 ORIGINAL PAPER BINOMIAL TRANSFORMS OF QUADRAPELL SEQUENCES AND QUADRAPELL MATRIX SEQUENCES CAN KIZILATEŞ 1, NAIM TUGLU 2, BAYRAM ÇEKİM 2