Change and calculus

1812

Laplace transform and poles

Pierre-Simon Laplace; made practical by Oliver Heaviside in the 1890s

How can you tell whether a machine will settle down or shake itself apart?

Laplace introduced a transform in his work on probability. Decades later, Oliver Heaviside used similar tricks to solve telegraph-line equations with algebra instead of calculus. Today every control engineer reads a system’s behaviour from a few points, its poles, on the complex plane.

First, make a guess

Drag a pole across the vertical line to the right. What happens to the response?

How they figured it out

Laplace used his integral in probability theory. The engineering story belongs to Oliver Heaviside, a self-taught English telegraph engineer, who in the 1880s and 1890s treated “d/dt” as if it were an ordinary number and solved circuit equations with algebra.

It worked brilliantly, but he offered no proofs, and mathematicians sneered. Heaviside replied: “Shall I refuse my dinner because I do not fully understand the process of digestion?” Decades later, mathematicians showed his methods were justified by Laplace’s transform.

The equation

F(s)=∫0∞f(t) e−st dtF(s) = \int_0^\infty f(t)\,e^{-st}\,dt

Turn a signal into a function of s; the points where it blows up (the poles) tell you how the system rings, fades or grows.

Everyday echoWhether a car feels floaty or firm over a bump is decided by where its poles sit.

See it in real life

Floaty or firm

A soft car suspension floats and bounces after a bump; a sporty one thumps once and settles. A thermostat that overshoots, a drone holding steady in wind, your phone camera’s stabiliser: each is tuned by deciding where its poles should sit.

Where engineers use it

  • Control systems

    Cruise control, drones and autopilots are tuned by placing poles.

  • Circuits

    Filters are designed by choosing pole positions, then building them from resistors and capacitors.

  • Vehicle suspension

    Damping is chosen to put the poles where the ride feels right.

Worked example: Mass on a spring with damping

  1. mx¨+cx˙+kx=0m\ddot x + c\dot x + kx = 0 becomes ms2+cs+k=0ms^2 + cs + k = 0.
  2. Poles: s=−ζωn±i ωn1−ζ2s = -\zeta\omega_n \pm i\,\omega_n\sqrt{1-\zeta^2}
  3. Real part sets the fade, imaginary part the pitch.

Where it came from

What it led to

  • The swaying footbridge 10 June 2000

    Led to

    In pole language: walkers in step dragged the bridge’s pole across the line into growth.

  • Hearing black holes merge 14 September 2015

    Same technique

    A ringing black hole is described by complex frequencies, like poles: pitch and fade in one number.

    Where the comparison stops: The black hole’s frequencies come from Einstein’s equations, not from a circuit or spring, and only the late ringdown looks like this.