Formalization of Transform Methods using HOL Light


Algebraic techniques based on transform methods are widely used for solving differential equations and evaluating transfer function, and frequency response of signals while analyzing physical aspects of many safety-critical systems. To facilitate formal analysis of these systems, we present the formalization of transform methods (Laplace and Fourier transforms) using the multivariable calculus theories of HOL Light. In particular, we use integral, differential, transcendental and topological theories of multivariable calculus to formally define transform methods in higher-order logic and reason about the correctness of their properties, such as existence, linearity, frequency shifting, modulation, time shifting, time scaling, differentiation and integration in time domain, relationship of Laplace transform with Fourier transform and relationship of Fourier transform with Fourier Cosine and Fourier Sine transforms. In order to demonstrate the practical effectiveness of this formalization, we use it to formally verify some commonly used electrical circuits, an automobile suspension system, an audio equalizer, a MEMs accelerometer, controllers and compensators, and pitch control of an unmanned free swimming submersible vehicle.









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