03 Control Systems · #11 of 46

Nyquist Stability

One squiggle in the complex plane decides whether your loop holds or screams

Feed a system's own output back into itself and you have made a bet. Either the loop settles down and holds a value, or it grabs its own tail and spirals into a howling, self-amplifying mess.

Here is the unsettling part: you can call that bet before closing the loop, from a single curve. And the whole verdict comes down to one lonely point.

Every feedback controller has an open-loop response, which is what the plant-plus-controller does to a signal before you wire the output back to the input. Sweep a sine wave through it at every frequency, plot each result as a point in the complex plane (real part across, imaginary part up), and you get a curve. Nyquist's insight was that the shape of that curve, relative to a single magic point at minus one, tells you whether the closed loop will be stable. No solving for roots, no factoring polynomials. Just: does the curve loop around minus one, or not?

Wrap the point, lose the loop

Drag the plant's poles in the left plane and slide the loop gain. The right panel draws the open-loop curve. Watch the red point at −1.

Encircle the −1 point, lose the loop

K (loop gain) 1.00

gain margin —

phase margin —

encirclements (N) —

closed-loop —

what to try

Turn K up slowly. The Nyquist curve grows radially. At some critical K it kisses the (−1, 0) point. That's the stability boundary.
Drag a pole rightward (toward unstable). Watch the curve start encircling (−1) even at small K; some plants need negative compensation.
At the gain-crossover frequency (where |L| = 1, i.e. the curve crosses the unit dashed circle), the angle to (−1, 0) is the phase margin. 30°–60° is the comfort zone.

The rule is almost rude in its simplicity. As long as the curve does not encircle the −1 point, the closed loop is stable. Push the gain up and the curve balloons outward; at some critical gain it brushes right against −1: that is the knife-edge, the stability boundary. Past it, the curve swallows the point, and your loop is now an oscillator you did not order.

Why −1 specifically? Because that is the value where feedback flips from corrective to catastrophic. If the loop transfer hits a gain of 1 and a phase lag of 180° at the same frequency, the fed-back signal comes around perfectly inverted and full-strength, reinforcing the very wiggle it was supposed to cancel. That magic combination of "gain 1, phase −180°" lives at exactly the point −1 on the plot. Encircle it and you have built a machine that amplifies its own echo.

A 1919 AT&T installation of telephone repeater amplifiers in Providence, Rhode Island. — Why Nyquist cared: AT&T's long-distance network chained repeater amplifiers to keep a voice alive across the country. Stack enough feedback amplifiers and the line can burst into a howl. Nyquist's 1932 curve says exactly when, before you ever build it. · AT&T (Gherardi & Jewett), public domain

The two panels are doing different jobs. The left one is the pole-zero map: drag a pole rightward, toward the unstable right half-plane, and you are simulating a plant that is already misbehaving on its own. The right one is the Nyquist curve of that plant scaled by your loop gain. Together they let you feel a deep truth of control: an open-loop-unstable plant can be stabilized by feedback, and a perfectly tame plant can be destabilized by too much of it. Stability is a property of the loop, not the parts.

The man who fed the output back

The whole problem Nyquist was solving belonged to a colleague down the hall. In 1927, a Bell engineer named Harold Black had a sudden idea for taming the distortion in the amplifiers that carried phone calls across the continent: feed a piece of the output back to the input, but inverted, so the amplifier fights its own errors. He sketched the negative-feedback amplifier on a page of his morning newspaper while riding the ferry across the Hudson to work, then signed and dated the page. It was a wonderful idea with one terrifying catch. Feed the signal back wrong, or with too much gain, and the amplifier stops correcting itself and starts howling. Black had the cure for distortion; he did not have a way to know in advance whether a given design would sing. That question is exactly what Nyquist answered, and it is why the criterion was born in a telephone lab and not a math department.

The math: the encirclement criterion and Z = N + P

Nyquist's criterion is really a piece of complex analysis in disguise, the argument principle applied to the loop. Count the clockwise encirclements $N$ that the open-loop response $L(j\omega)$ makes around the point $(-1, 0)$ as $\omega$ sweeps the whole frequency axis. Let $P$ be the number of open-loop poles already in the unstable right half-plane. Then the number of closed-loop poles in the right half-plane is

Z = N + P

A stable closed loop is exactly the case $Z = 0$ . So for a plant that is already open-loop stable, $P = 0$ , and the whole condition collapses to $N = 0$ : simply do not encircle the point. But for an unstable plant (say one with a single right half-plane pole, $P = 1$ ), stability demands one counterclockwise encirclement ( $N = -1$ ) so that $Z = N + P = 0$ . That is the precise, countable statement behind "feedback can rescue an unstable plant," and it is why dragging a pole into the right half-plane changes which curves are safe.

One edge case is worth filing away. If the curve does not merely loop near −1 but passes exactly through it, the count breaks down and the criterion goes quiet. All you can conclude is that the system has roots sitting right on the imaginary axis, the boundary between settling and oscillating, with no firm verdict either way. In practice that is the universe telling you to redesign rather than ship.

The clean version is not Nyquist's. It came later.

The tidy argument-principle proof you just read is not the one in Nyquist's 1932 paper. His original derivation was, by the polite admission of later textbooks, "less elegant." The clean version, the one that ties the encirclement count directly to Cauchy's argument principle, was worked out around 1945 by two other Bell Labs people: Leroy MacColl, in his book Fundamental Theory of Servomechanisms, and Hendrik Bode, in Network Analysis and Feedback Amplifier Design. (Yes, the Bode of the Bode plot.) Their reframing is what almost every modern control course teaches, with Nyquist's name still on it.

There is a practical payoff to that complex-analysis framing, and it explains why engineers still reach for this test over root-finding. The criterion never asks you to factor a polynomial. It only needs the open-loop frequency response, a curve you can measure on a real machine. So it works even when there is no polynomial to factor: a system with a pure time delay like $e^{-sT}$ , where the input is fed back after some travel time, is not a rational function and has no finite set of poles to chase. A root-finder chokes. The Nyquist curve does not care. You sweep the frequency, plot the response, and count loops around −1 exactly as before. That tolerance for delays and other awkward dynamics is the reason it outlived its cleaner-looking rivals.

Key takeaways

The Nyquist criterion judges closed-loop stability from a single plot of the open-loop response in the complex plane, no root-finding required.
The entire verdict is topological: how many times the curve encircles the point −1.
Gain margin and phase margin measure how far the curve stays from −1; aim for 30°–60° of phase margin so a real machine has slack.
Feedback is a loop property: it can stabilize an unstable plant or destabilize a tame one, depending on how the curve sits relative to −1.

A century of control theory hangs on one point and one question: does the curve wrap around −1?

It is a strangely visual way to settle a question that sounds purely algebraic: will this loop hold? You do not crank through characteristic polynomials; you look at a shape and check whether it lassoed a single dot. The same Harry Nyquist who told us how fast to sample a signal also handed control engineers the cleanest go/no-go test in the field, and it is still the first thing you reach for when a loop starts to sing.