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Theoretical Models

This section focuses on the general equations and laws that PD Controller is based on.

Laplace Transform

Refname	TM:laplaceTransform
Label	Laplace Transform
Equation	${F_{\text{s}}}=\int_{\mathit{-∞}}^{∞}{{f_{\text{t}}}\,e^{-s\,t}}\,dt$
Description	${F_{\text{s}}}$ is the Laplace Transform of a function (Unitless) ${f_{\text{t}}}$ is the Function in the time domain (Unitless) $s$ is the Complex frequency-domain parameter (Unitless) $t$ is the time ( ${\text{s}}$ )
Notes	Bilateral Laplace Transform. The Laplace transforms are typically inferred from a pre-computed table of Laplace Transforms (laplaceWiki).
Source	laplaceWiki
RefBy	GD:gdPowerPlant, DD:ddPropCtrl, DD:ddProcessError, and DD:ddDerivCtrl

Inverse Laplace Transform

Refname	TM:invLaplaceTransform
Label	Inverse Laplace Transform
Equation	${f_{\text{t}}}=\mathit{L⁻¹[F(s)]}$
Description	${f_{\text{t}}}$ is the Function in the time domain (Unitless) $\mathit{L⁻¹[F(s)]}$ is the Inverse Laplace Transform of a function (Unitless)
Notes	Inverse Laplace Transform of F(S). The Inverse Laplace transforms are typically inferred from a pre-computed table of Laplace Transforms (laplaceWiki).
Source	laplaceWiki
RefBy	IM:pdEquationIM

Second Order Mass-Spring-Damper System

Refname	TM:tmSOSystem
Label	Second Order Mass-Spring-Damper System
Equation	$\frac{1}{m\,s^{2}+c\,s+k}$
Description	$m$ is the mass ( ${\text{kg}}$ ) $s$ is the Complex frequency-domain parameter (Unitless) $c$ is the Damping coefficient of the spring (Unitless) $k$ is the Stiffness coefficient of the spring ( ${\text{s}}$ )
Notes	The Transfer Function (from A:Transfer Function) of a Second Order System (mass-spring-damper) is characterized by this equation.
Source	abbasi2015
RefBy	GD:gdPowerPlant