M3-T4 — The Nyquist Criterion and Stability Margins

Learning objectives

After this lesson, you should be able to:

• State the Nyquist criterion and apply it to a loop with an unstable open-loop pole.
• Read the gain margin, the phase margin and the delay margin from a loop's frequency response.
• Compute the delay margin from the phase margin and the crossover frequency.
• Relate the margins to the peak of the sensitivity function.

The intuition

The closed loop of M3-T2: What Feedback Buys and What It Costs has the characteristic equation $1 + L(s) = 0$. It is unstable if that equation has a root in the right half-plane. The Nyquist criterion counts such roots without finding them, using the frequency response of $L$ from the last lesson.

Why the point −1

The loop is closed with negative feedback. A signal that goes once round the loop comes back multiplied by $-L$. If, at some frequency, $L(j\omega) = -1$, the signal comes back unchanged: it feeds itself, and a sinusoid at that frequency can sustain itself with no input. That is the edge of instability. So the point $-1$ in the complex plane is critical, and the question is how the curve $L(j\omega)$ is placed relative to it.

The criterion

Draw $L(j\omega)$ for $\omega$ from $-\infty$ to $+\infty$. The negative frequencies give the mirror image of the positive ones, so the curve is symmetric about the real axis and closed. Count how many times it goes round $-1$.

• If the open loop is stable, the loop is stable if the curve does not go round $-1$.
• If the open loop has $P$ unstable poles, the closed loop is stable only if the curve goes round $-1$ counterclockwise exactly $P$ times. The feedback has to supply the encirclements that the unstable poles would otherwise deny.

Hopper's pitch axis has one unstable pole, so the Nyquist plot of the pitch loop must encircle $-1$ once counterclockwise. The figure shows that it does.

Margins: how far from the edge

A stable loop is not necessarily safe. A real plant differs from its model, and the question is how large a discrepancy the loop can take. Three numbers answer it, each reading a different part of the curve.

• The gain margin is the factor by which the loop gain could change before the curve passes through −1 on the real axis. It guards against a wrong gain, such as a loss of authority.
• The phase margin is the extra phase lag that would make the curve pass through −1 at the crossover. It guards against lags that were not in the model.
• The delay margin is the pure time delay that would do the same. A delay of $\tau$ adds a lag of $\omega\tau$, so the delay margin is the phase margin divided by the crossover frequency. It guards against computation, sensor and communication delays, and is the margin that real systems violate most often.

Hopper's loop has a phase margin of 55.5° at 6.07 rad/s, a delay margin of 0.160 s, and a lower gain margin of −30 dB. What each of these numbers means, and how to compute them, is the subject of the rest of the lesson.

The Nyquist plot answers one yes-or-no question (does the loop go round −1 the right number of times?) and the margins answer the question that matters in practice: how far from −1 does it pass, in gain, in phase and in time?

Definitions

Nyquist plot

The curve traced by $L(j\omega)$ in the complex plane as $\omega$ goes from $-\infty$ to $+\infty$.

Encirclements $N$

The net number of times the curve goes round $-1$, counted clockwise as positive.

Open-loop unstable poles $P$

The number of poles of $L(s)$ in the right half-plane. For Hopper's pitch loop, $P = 1$.

Phase margin $\mathrm{PM}$

The angle between the curve at the crossover (where $|L| = 1$) and the negative real axis: $\mathrm{PM} = 180^\circ + \angle L(j\omega_{gc})$.

Gain margin $\mathrm{GM}$

The factor by which the gain can be changed before the loop reaches the critical point: $1/|L(j\omega_{pc})|$ at the frequency where the phase is −180°. A loop with an unstable pole also has a lower margin: how far the gain can fall.

Delay margin $\tau_m$

The largest added time delay that keeps the loop stable.

The Nyquist criterion relates three counts. The number of right-half-plane poles of the closed loop $Z$ is:

The delay margin follows from the phase margin and the crossover frequency, since a delay changes only the phase:

The peak of the sensitivity bounds how small the phase margin can be:

Derivation: the criterion, the delay margin and the sensitivity peak

The Nyquist criterion is an application of the argument principle of complex analysis. The idea fits in four lines.

1. $$\text{poles of } \frac{L}{1 + L} \;=\; \text{zeros of } 1 + L$$

   The closed-loop poles are the roots of the characteristic equation $1 + L = 0$. The closed loop is stable when none of them is in the right half-plane.

2. $$\text{(clockwise turns of } 1 + L \text{ round } 0\text{)} = Z - P$$

   The argument principle says that as $s$ goes round the boundary of the right half-plane (the imaginary axis closed by a large semicircle), the function $1 + L(s)$ goes round the origin once clockwise for every zero inside and once counterclockwise for every pole inside. The zeros of $1 + L$ are the closed-loop poles $Z$, its poles are the open-loop poles $P$.

3. $$1 + L \text{ round } 0 \;\Longleftrightarrow\; L \text{ round } -1$$

   Shifting a curve by 1 does not change how many times it goes round a point: going round the origin by $1 + L$ is going round $-1$ by $L$.

4. $$N = Z - P \;\Rightarrow\; Z = N + P$$

   Write the count as $N$, the clockwise encirclements of −1 by $L$. On the large semicircle $L$ is almost zero for any proper loop, so only the imaginary axis contributes, and that is the Nyquist plot.

The delay margin

1. $$L_\tau(j\omega) = L(j\omega)\,e^{-j\omega\tau}, \qquad |L_\tau| = |L|, \quad \angle L_\tau = \angle L - \omega\tau$$

   A delay $\tau$ multiplies the loop by $e^{-j\omega\tau}$, which has magnitude 1: it does not move the gain crossover, only the phase.

2. $$\angle L_\tau(j\omega_{gc}) = -180^\circ + \mathrm{PM} - \omega_{gc}\tau = -180^\circ \;\Longleftrightarrow\; \tau = \frac{\mathrm{PM}}{\omega_{gc}}$$

   The delay makes the curve pass through −1 when the phase at the crossover has fallen to −180°. This is the delay margin (with PM in radians).

The sensitivity peak and the phase margin

1. $$|S(j\omega)| = \frac{1}{|1 + L(j\omega)|} = \frac{1}{\text{distance from } L(j\omega) \text{ to } -1}$$

   The sensitivity is the reciprocal of the distance between the curve and the critical point.

2. $$\text{at the crossover: } |1 + L| = \left|1 + e^{j(\mathrm{PM} - 180^\circ)}\right| \ \text{ is the chord } 2\sin\tfrac{\mathrm{PM}}{2}$$

   At the crossover $L$ lies on the unit circle at an angle PM from the negative real axis. Its distance to −1 is the chord of that angle.

3. $$\min_\omega |1 + L| \le 2\sin\tfrac{\mathrm{PM}}{2} \;\Rightarrow\; M_s \ge \frac{1}{2\sin(\mathrm{PM}/2)}$$

   The closest approach cannot be farther than the crossover point, so the peak of the sensitivity is at least the reciprocal of that chord. A phase margin of 30° forces $M_s \ge 1.93$; one of 60° only $M_s \ge 1$.

The delay margin is the phase margin in time: PM divided by the crossover frequency. It is usually the smallest of the three margins that matters, because the phase margin shrinks and the crossover rises as a design gets faster.

A worked example

Find the margins of Hopper's pitch loop with its servo, $L(s) = (5.6\,s + 16.49)/\bigl((s^2 - 0.49)(s/40 + 1)\bigr)$, and check each by an independent calculation.

1. Crossover and phase margin. Scan the frequency for $|L| = 1$: it happens at 6.07 rad/s. The phase there is −124.5°, so the phase margin is 55.5°, and the delay margin is $0.9686\ \text{rad}/6.07 = 0.160$ s.
2. Count encirclements. The open loop has one pole at +0.7, so $P = 1$. For gain factors of 0.05 and above, the Nyquist plot goes round −1 once counterclockwise ($N = -1$), so $Z = 0$: stable. For 0.01 and 0.02 there is no encirclement, $Z = 1$: unstable. The count matches the actual roots of $1 + kL$ in every case.
3. Gain margin. Scanning $k$ with the exact roots, the loop is stable from $k = 0.030$ up. The lower margin is therefore $-30.4$ dB: the gain can fall to 3% before the divergence of the pitch axis wins. The proportional gain is $K_p = (\omega_n^2 + \lambda^2)/b$ and must exceed $\lambda^2/b$, so the exact lower limit is $\lambda^2/(\omega_n^2 + \lambda^2) = 0.0297$.
4. The upper margin. There is none to find. With an ideal derivative the phase approaches −180° at high frequency and never crosses it, so the gain can be raised indefinitely in the model. The model does not include the delay that a real loop has, which is why the delay margin is the stricter measure.
5. Check the delay margin by brute force. Add a delay to the loop and search for the largest one for which the Nyquist count still gives $Z = 0$: 0.160 s, the same as $\mathrm{PM}/\omega_{gc}$ to the digits shown.
6. Compare with the sensitivity peak. The bound is $1/(2\sin 27.7^\circ) = 1.07$, and the loop's peak $|S|$ is 1.16, above it as it must be.

import cmath, math
import gnckit as g

b, lam, wn, zeta = 19.62, 0.7, 4.0, 0.7
Kd, Kp = 2 * zeta * wn / b, (wn ** 2 + lam ** 2) / b
den = g.poly_mul([1, 0, -lam ** 2], [1 / 40, 1])
def loop(k=1.0, delay=0.0):
    return g.TF([k * b * Kd, k * b * Kp], den, delay)

L = loop()
ws = g.logspace(-2, 3, 20000)
wgc = [ws[i] for i in range(len(ws) - 1) if (abs(L.freq(ws[i])) - 1) * (abs(L.freq(ws[i + 1])) - 1) < 0][0]
pm = 180 + math.degrees(cmath.phase(L.freq(wgc)))
print(f'gain crossover {wgc:.2f} rad/s, phase margin {pm:.1f} deg, delay margin {math.radians(pm) / wgc:.3f} s')

print('open-loop unstable poles P =', sum(1 for p in L.poles() if complex(p).real > 0))
print('gain k    N (clockwise)   Z = N + P   closed-loop poles in the right half plane')
for k in (0.01, 0.02, 0.05, 0.5, 1.0, 50.0):
    Lk = loop(k)
    truth = sum(1 for r in g.poly_roots(g.poly_add(Lk.den, Lk.num)) if complex(r).real > 0)
    print(f'{k:6.2f}   {g.nyquist_encirclements(Lk):8d}   {g.closed_loop_unstable_poles(Lk):10d}   {truth:6d}')

ks = [10 ** (i / 100) for i in range(-400, 401)]
stable = [k for k in ks if all(complex(r).real < 0 for r in g.poly_roots(g.poly_add(L.den, [k * c for c in L.num])))]
print(f'stable for gains from {min(stable):.4f} ({20 * math.log10(min(stable)):.1f} dB) up to at least {max(stable):.0f} (the largest tried)')

lo, hi = 0.0, 1.0
for _ in range(30):
    mid = (lo + hi) / 2
    lo, hi = (mid, hi) if g.closed_loop_unstable_poles(loop(1.0, mid)) == 0 else (lo, mid)
print(f'largest added delay that keeps it stable (Nyquist search): {lo:.3f} s')

gain crossover 6.07 rad/s, phase margin 55.5 deg, delay margin 0.160 s
open-loop unstable poles P = 1
gain k    N (clockwise)   Z = N + P   closed-loop poles in the right half plane
  0.01          0            1        1
  0.02          0            1        1
  0.05         -1            0        0
  0.50         -1            0        0
  1.00         -1            0        0
 50.00         -1            0        0
stable for gains from 0.0302 (-30.4 dB) up to at least 10000 (the largest tried)
largest added delay that keeps it stable (Nyquist search): 0.160 s

In robotics

Margins are the common currency of control reviews, and the criteria are written in them.

• Requirements. Flight control requirements are commonly written as a minimum gain margin and phase margin, for example 6 dB and 45°, evaluated at every flight condition and often at each actuator, and a loop that meets them has a modest sensitivity peak (about 1.5 to 2 or below). They are measured on the loop broken at a chosen point, usually the actuator input.
• Delay margin in practice. A multirotor's attitude loop has a delay margin of a few tens of milliseconds; a software change that adds a 20 ms filter or a slower bus can turn a stable vehicle into an oscillating one. Timing budgets, the subject of UAV M7-T6: End-to-end latency, are budgets against the delay margin.
• Unstable plants. For Hopper's pitch loop the critical point is approached from two sides: too little gain and the divergence wins (the lower margin of −30 dB), too much lag and the loop oscillates. The window shrinks as the delay grows, which the next lesson turns into a limit of performance.
• Measuring margins. Margins can be measured on the real vehicle: inject a small sinusoid into the loop, record the response, and read the loop gain and phase at each frequency. Flight tests do this with sweeps before every envelope expansion.
• Tools. margins, nyquist_encirclements and closed_loop_unstable_poles in gnckit compute them, and python-control's stability_margins and nyquist_plot do the same. For loops with unstable poles use the encirclement count, since the simple rule of reading the margin from the crossing can be misleading.

Hopper's lab on this lesson, M3-L1: Rescue the Phase Margin, lets you break the loop of a heavier vehicle and measure its margins before fixing them.

Common misconceptions

Key takeaways

1. The loop is unstable if the closed-loop characteristic equation $1 + L = 0$ has a root in the right half-plane; the Nyquist criterion counts such roots from the frequency response: $Z = N + P$.
2. With $P$ unstable open-loop poles, a stable loop needs $P$ counterclockwise encirclements of −1. Hopper's pitch loop has $P = 1$ and one encirclement.
3. The phase margin is the angle from −180° at the crossover, the gain margin the factor to −1 on the real axis, and the delay margin $\mathrm{PM}/\omega_{gc}$ in seconds.
4. Hopper's loop: crossover 6.07 rad/s, phase margin 55.5°, delay margin 0.160 s, lower gain margin −30 dB, with no upper limit in the model.
5. The sensitivity peak satisfies $M_s \ge 1/(2\sin(\mathrm{PM}/2))$, so a small margin means a high peak.
6. Check the margins at every operating point, and prefer the delay margin as the measure of how much unmodelled lag the loop can take.

References and further reading

Each source is marked free or paid. Free to read does not mean free to reuse: the course resources page lists every source with its licence.

• K. J. Åström and R. M. Murray, Feedback Systems, 2nd edition, Princeton University Press, 2021 , the chapter on frequency domain analysis, with the Nyquist criterion and margins. Free to read as the authors' PDF; the book is copyrighted.
• B. Douglas, “Nyquist Stability Criterion” (video series) . A visual introduction to the criterion and the margins. Free to watch.
• J. How and E. Frazzoli, MIT 16.30 Feedback Control Systems (OpenCourseWare) , the lectures on the Nyquist criterion and margins. Free; licence CC BY-NC-SA 4.0.
• Python Control Systems Library documentation , the functions nyquist_plot and stability_margins. Free; BSD licence.

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What’s next

Related content

Nothing below is required to finish this lesson.

• Before
  M3-T3: Frequency Response and Bode Plots

  Sketch the Bode plot of a transfer function from its poles and zeros, and read bandwidth and steady-state error from it.

• Lab
  M3-L1: Rescue the Phase Margin

  Hopper's pitch loop rings after a heavier engine was fitted. Measure the margins from the open-loop response and add a lead compensator that restores 45 degrees and 6 dB.

Other courses on this site treat neighbouring topics from their own angle:

• Lab: Tune a PID Controller
• Lab: Fix the Oscillating Robot
• UAV M5-T6: PID controllers
• UAV M7-T6: End-to-end latency

This course leaves the following to the pages that already teach them:

• What P, I and D do, and tuning them by hand on a step response
• PID cascades for a multirotor