First- and Second-Order Responses

Tune the damping for the fastest settle

The upper plot is the s-plane: the gold × marks are the two poles of a second-order system, and the dashed semicircle marks the points at distance ωn from the origin. When underdamped, the two poles leave the real axis as a complex pair on this semicircle and the response swings as it follows; when overdamped, the two poles split far apart on the real axis and the response is sluggish. The lower plot is the unit step response, with the dashed line at the target 1. You start at ζ = 0.3 and ωn = 1.4, so the response overshoots the target by about 37%. Move ζ toward 1 to find critical damping, where the two poles meet on the real axis, and you get the response that reaches the target fastest without overshooting; raise ωn and the same shape simply speeds up as a whole.

First order: a single time constant

A first-order system H(s) = 1/(τs + 1), with only one pole on the real axis, has that pole at s = −1/τ. Its step response 1 − e^(−t/τ) is an exponential curve that rises smoothly toward the target without oscillating. The time constant τ is the time the response takes to reach about 63% of its final value, and after 4τ it is a bit over 98%, effectively settled. The closer the pole is to the origin (the larger τ), the slower the response, and the farther away, the faster, so a single pole location decides the speed.

Second order: a transient response set by two poles

The poles of the standard second-order system H(s) = ωn²/(s² + 2ζωn s + ωn²) are the two roots of the characteristic equation s² + 2ζωn s + ωn² = 0, which work out to s = −ζωn ± ωn√(ζ² − 1). When the damping ratio ζ is less than 1, the quantity under the square root is negative, so the poles become a complex pair −ζωn ± jωd, where ωd = ωn√(1 − ζ²) is the angular frequency at which the response actually swings. The pole's real part −ζωn sets how fast the amplitude dies down, and its imaginary part ωd sets how fast it oscillates. The natural frequency ωn is the distance from the origin to the pole and measures overall speed, while ζ is the cosine of the angle θ the pole makes with the negative real axis (ζ = cos θ) and decides the shape of the response. That is why underdamped poles always sit on the dashed semicircle of radius ωn in the plot above.

Over, critical, and underdamped behavior

The damping ratio ζ alone sorts the behavior into types. Overdamped, ζ > 1, has two real roots and moves slowly without overshoot; critically damped, ζ = 1, has a double root and is the fastest with no overshoot; underdamped, ζ < 1, has complex poles and overshoots before dying down. At ζ = 0 the poles sit on the imaginary axis and the response oscillates forever. The size of the underdamped overshoot depends on ζ alone: about 5% at ζ = 0.7 and about 37% at ζ = 0.3. The 2% settling time is roughly 4/(ζωn), so the larger the pole's real part, the faster it settles. Designers therefore tune overshoot with ζ and speed with ωn separately, and they often choose ζ around 0.7, accepting a little overshoot in exchange for a faster rise.

Back to the first screen

On the first screen, with a small damping ratio ζ the poles were a complex pair tilted far from the real axis, so the response overshot a lot and kept swinging for a long time; raising ζ to 1 brought the two poles together on the real axis and the response settled fastest with no overshoot. Raising ζ further into overdamping split the poles again and made the response sluggish. Raising ωn moved the poles away from the origin and sped up the same shape as a whole. The angle the poles tilt away from the real axis decided the overshoot, and their distance from the origin decided the speed. The shape and speed of the response were written plainly in where the poles sat on the s-plane.

This completes the story of continuous-time signals and systems. The Laplace transform turned differential equations into algebra in s, and the pole locations let you read the response. Computers and digital devices, however, cannot handle a signal continuously; they take its value at fixed intervals. The first unit of the next chapter, The Sampling Theorem, covers the condition under which a continuous signal can be recovered even when you take samples only at intervals. There you meet the Nyquist criterion, which says you must sample faster than twice the highest frequency in the signal, along with the aliasing that appears when you sample too slowly.