1.1 Laplace Transforms

  • Laplace Transform for Control System Studies: Dynamics of chemical process systems are often modelled by differential equations. By using Laplace transforms, these differential equations can be converted to algebraic equations. Solving the algebraic equations is relatively easier than the direct solution of differential equations. Hence, Laplace transform finds wide applications in control system studies.

  • Definition: Laplace transform of a function \(f(t)\) is given by \[\mathcal{L}[f(t)] = F(s) = \int_0^{\infty}e^{-st} f(t)\, dt\] Laplace transforms of simple functions are given in the following table.

    Function \(\boldsymbol{f(t)}\) \(\boldsymbol{F(s)}\)
    Constant \(a\) \(\dfrac{a}{s}\)
    Step \(\displaystyle f(t) = \left\{\begin{array}{ll}a & t>0 \\ 0 & t<0\end{array}\right.\) \(\dfrac{a}{s}\)
    Ramp \(at\) \(\dfrac{a}{s^2}\)
    \(t^n\) \(\dfrac{n!}{s^{n+1}}\)
    Exponential \(\displaystyle e^{-at}\) \(\dfrac{1}{s+a}\)
    \(e^{at}\) \(\dfrac{1}{s-a}\)
    \(t^n e^{-at}\) \(\dfrac{n!}{(s+a)^{n+1}}\)
    Sinusoidal \(\sin(\omega t)\) \(\dfrac{\omega}{s^2+\omega^2}\)
    \(\cos(\omega t)\) \(\dfrac{s}{s^2+\omega^2}\)
    Sin with exponential \(\displaystyle e^{-at}\sin(\omega t)\) \(\dfrac{\omega}{(s+a)^2+\omega^2}\)
    \(\displaystyle e^{-at}\cos(\omega t)\) \(\dfrac{s+a}{(s+a)^2+\omega^2}\)
    Hyperbolic \(\sinh(\omega t)\) \(\dfrac{\omega}{s^2-\omega^2}\)
    \(\cosh(\omega t)\) \(\dfrac{s}{s^2-\omega^2}\)
    Dead time \(f(t - t_o)\) \(e^{-st_o}F(s)\)
    Impulse \(\delta(t)\) 1
    Rectangular pulse \(\displaystyle f(t) = \left\{\begin{array}{ll} 0 & t<0 \\ A & 0 < t < T \\ 0 & t>T \end{array} \right.\)      \(\displaystyle \frac{A}   {s}\left(1-e^{-sT}\right)\)