Transfer Functions, Poles and Stability

Transfer functions, poles and zeros, how pole positions set the response, stability, second-order specifications and dominant poles

7 November 1940 — Galloping Gertie

The Tacoma Narrows Bridge had moved in the wind from the day its deck was built; the construction workers called it "Galloping Gertie". On the morning of 7 November 1940, in winds of about 40 mph (64 km/h), a new motion appeared: the deck began to twist, one side rising as the other fell. Frederick Farquharson, an engineering professor who had been testing models of the bridge, reported the destructive mode at 12 cycles a minute, 0.2 Hz, in a steady wind. Cables failed one after another, and at about 11 a.m. the central span fell into the strait (Tacoma Narrows Bridge (1940)). Textbooks often call it resonance. It was not. The wind produced aeroelastic flutter, self-exciting and unbounded: at any sustained wind speed above about 35 mph the twisting grew continuously, with a negative damping factor (Tacoma Narrows Bridge (1940)). In this lesson's language, the bridge's poles had crossed into the right half of the s-plane.

The transfer function of a linear time-invariant system is the transform of its impulse response,

H(s)

, or equivalently

Y(s)/X(s)

with zero initial conditions. For systems described by differential equations it is rational,

H(s) = \frac{b_m s^m + \dots + b_0}{a_n s^n + \dots + a_0} = K\,\frac{(s - z_1)\cdots(s - z_m)}{(s - p_1)\cdots(s - p_n)},

and it is described completely by its zeros

z_i

, its poles

p_i

and a gain

K

— a few points in the s-plane.

💡Where the poles are, what the system does

Each pole

p

contributes

e^{pt}

to the impulse response:

a real pole at $-a$ : a decay with time constant $1/a$ ; the farther left, the faster;
a complex pair $\sigma \pm j\omega_d$ : an oscillation at $\omega_d$ inside an envelope $e^{\sigma t}$ — decaying if $\sigma < 0$ , steady if $\sigma = 0$ , growing if $\sigma > 0$ ;
a pole at the origin: an integrator, whose step response is a ramp.

Distance from the imaginary axis sets how fast a term decays; height above the real axis sets how fast it oscillates.

💡Stability

A causal system is stable — every bounded input gives a bounded output — if and only if

\int_0^\infty |h(t)|\,dt

is finite, which for a rational

H(s)

means every pole lies in the open left half-plane. Simple poles on the imaginary axis make it marginally stable: its free oscillations neither grow nor decay, but an input at the same frequency drives it without bound. Any pole in the right half-plane makes it unstable:

1/(s^2 - 2s + 5)

, with poles at

1 \pm 2j

, has a step response that has already reached 17 after five seconds and keeps growing like

e^t

Derivation

The second-order system

The standard second-order system

H(s) = \omega_n^2/(s^2 + 2\zeta\omega_n s + \omega_n^2)

has poles

p = -\zeta\omega_n \pm j\omega_n\sqrt{1 - \zeta^2} = -\zeta\omega_n \pm j\omega_d ,

at distance

\omega_n

from the origin and at an angle

\arccos\zeta

from the negative real axis. For

0 < \zeta < 1

its step response is

y(t) = 1 - e^{-\zeta\omega_n t}\left(\cos\omega_d t + \frac{\zeta}{\sqrt{1 - \zeta^2}}\sin\omega_d t\right).

Setting

\dot y = 0

puts the first peak at

t_p = \pi/\omega_d

, where

y(t_p) = 1 + e^{-\zeta\omega_n\pi/\omega_d}

. Hence the overshoot and, from the envelope, the 2 % settling time:

M_p = e^{-\pi\zeta/\sqrt{1 - \zeta^2}}, \qquad t_s \approx \frac{4}{\zeta\omega_n} .

📐Specifications from a pole map

With

\omega_n = 2

rad/s, python-control's step_info measures: |

\zeta

| Overshoot | Peak time | 2 % settling | |---|---|---|---| | 0.2 | 52.6 % (formula 52.7 %) | 1.63 s (1.60 s) | 9.89 s (4/ζωₙ: 10.0 s) | | 0.5 | 16.3 % (16.3 %) | 1.81 s (1.81 s) | 4.05 s (4.0 s) | | 0.8 | 1.5 % (1.5 %) | 2.62 s (2.62 s) | 1.92 s (2.5 s) | The formulas are exact for overshoot and peak time; the settling rule is an envelope estimate that overstates the time for heavy damping.

💡Dominant poles

Poles far to the left decay quickly; the poles nearest the imaginary axis linger and dominate the response. A real pole five to ten times farther left than a dominant pair can usually be dropped, keeping the DC gain. In this lesson's lab, adding a pole at

s = -10

to a pair at

-1 \pm 2j

barely changes the overshoot (20.2 % against 20.7 % for the pair alone); moved to

s = -2

, it halves it (9.3 %), and the approximation fails.

Zeros shape the terms. Zeros do not affect stability, but they set the residues: a zero close to a pole nearly cancels that pole's term, and a zero in the right half-plane makes the step response start off in the wrong direction — the non-minimum-phase behaviour that showed up as extra phase lag in Lesson 4-3.

For a successful technology, reality must take precedence over public relations, for nature cannot be fooled.

– Richard Feynman, appendix to the Report of the Presidential Commission on the Space Shuttle Challenger Accident (1986), quoted on Wikiquote

🕰Wind tunnels and a lasting lesson

Farquharson and his students had built a 1:200 model of the whole bridge and a 1:20 model of a section of its deck; their first studies concluded on 2 November 1940, five days before the collapse. The disaster boosted research into bridge aerodynamics and aeroelasticity, which has shaped the design of every long-span bridge since (Tacoma Narrows Bridge (1940)).

⚠Never cancel an unstable pole

A zero placed exactly on a right-half-plane pole removes it from

H(s)

on paper, but the unstable mode is still inside the system: any initial condition or the slightest mismatch excites it, and it grows. Stability has to be achieved by moving poles — with feedback, the subject of Lesson 7-4 — not by hiding them.

💡Where you will use this

Feedback design (Lesson 7-4) is the art of moving poles. In discrete time the stable region becomes the inside of the unit circle (Lesson 8-2), and IIR filter design places poles deliberately (Lesson 9-2). Aircraft flutter clearance, launch-vehicle control and power-grid stability are all pole-location questions.

📚Further reading

Oppenheim and Willsky, Signals and Systems, sections 6.5 and 9.7 — first- and second-order systems, and LTI systems analysed with the Laplace transform.
Norman S. Nise, Control Systems Engineering (8th ed., Wiley, 2019), chapter 4 — time response from poles and zeros.
Tacoma Narrows Bridge (1940) on Wikipedia.

↓But there is a problem

A system with poles in the right half-plane is not condemned. Measure its output, compare it with what you want, and feed the difference back: the poles of the combined system can be put wherever you choose — or, done carelessly, made worse. One morning in August 1927, an engineer on his way to work at Bell Laboratories sketched that idea on a page of The New York Times.