Nyquist criterion, gain and phase margins

Why it matters

Routh needs a polynomial, but real machines include transport delays, measured frequency-response data and open-loop unstable plants (a magnetic levitator, an inverted pendulum). The Nyquist criterion decides closed-loop stability from the open-loop frequency response in all these cases, and the gain and phase margins it produces are the standard robustness numbers written into controller specifications.

Key ideas

Nyquist plot. The polar plot of the open-loop frequency response G(jω)H(jω) as ω runs from −∞ to +∞ (the negative-frequency half is the mirror image of the positive half about the real axis). It is the image of the Nyquist contour — the whole jω-axis closed by a large semicircle around the right half-plane, with small detours around any open-loop poles on the jω-axis.

Why −1 matters. Closed-loop poles are the zeros of 1 + G(s)H(s). Encirclements of the origin by 1 + GH are the same as encirclements of the point (−1, j0) by GH. By the argument principle:

Z = P + N

• P: open-loop poles of G(s)H(s) in the right half-plane (known from the model).
• N: net number of clockwise encirclements of (−1, j0) by the full Nyquist plot (anticlockwise counts negative).
• Z: closed-loop poles in the right half-plane. Stable if and only if Z = 0.

So for an open-loop stable system (P = 0) the closed loop is stable if the plot does not encircle −1. For an open-loop unstable system the plot must encircle −1 anticlockwise P times (N = −P).

Poles at the origin. Type-N systems start at infinite magnitude. The small detour around s = 0 maps to N clockwise half-circles of infinite radius that close the plot — needed to count encirclements correctly.

Relative stability — margins. For minimum-phase, open-loop stable systems, how close the plot comes to −1 measures robustness.

• Phase crossover frequency ω_pc: where ∠GH = −180° (the plot cuts the negative real axis).
• Gain margin GM = 1/|GH(jω_pc)|, or in dB GM = −20·log₁₀|GH(jω_pc)|. It is the factor by which the gain can be raised before instability. If the plot cuts the negative real axis at −a (a < 1), GM = 1/a.
• Gain crossover frequency ω_gc: where |GH| = 1 (0 dB; the plot cuts the unit circle).
• Phase margin PM = 180° + ∠GH(jω_gc). It is the extra phase lag (for example from a delay) that would bring the system to the verge of instability.
• Positive GM (dB) and positive PM ⇒ stable (for minimum-phase systems). Typical design targets: GM ≥ 6 dB and PM between 30° and 60°.
• First- and second-order systems without delay never reach −180°, so their GM is infinite.

Link to time response. For a dominant second-order loop, PM ≈ 100ζ (degrees) for PM up to about 60°, so a 45° PM corresponds to ζ ≈ 0.45 and roughly 20 % overshoot.

Delay. A delay e^(−sT_d) rotates every point of the plot clockwise by ωT_d radians without changing its magnitude. It reduces PM by ω_gc·T_d (converted to degrees), so the largest tolerable delay is PM/ω_gc (PM in rad).

Formulas

Z = P + N — Nyquist criterion (N clockwise positive).

∠G(jω_pc)H(jω_pc) = −180° — phase crossover. GM = 1/|G(jω_pc)H(jω_pc)|; GM_dB = −20·log₁₀|G(jω_pc)H(jω_pc)|

|G(jω_gc)H(jω_gc)| = 1 — gain crossover. PM = 180° + ∠G(jω_gc)H(jω_gc) (degrees)

T_d,max = PM(rad)/ω_gc — largest added delay before instability (s); ω_gc in rad/s.

PM ≈ 100·ζ — rough rule for a dominant second-order loop, PM ≤ about 60°.

Worked examples

Example 1 (standard — margins). A unity-feedback loop has G(s) = 12/((s + 1)(s + 2)(s + 3)). Find ω_pc, GM, ω_gc and PM, and state whether the closed loop is stable.

1. P = 0 (all open-loop poles in the LHP).
2. Phase: ∠G = −[tan⁻¹ω + tan⁻¹(ω/2) + tan⁻¹(ω/3)]. Setting this to −180° gives ω² = 11, so ω_pc = 3.32 rad/s (check: the angles 73.2° + 58.9° + 47.9° = 180°).
3. |G(jω_pc)| = 12/√((1 + 11)(4 + 11)(9 + 11)) = 12/√3600 = 12/60 = 0.2. The plot cuts the negative real axis at −0.2.
4. GM = 1/0.2 = 5 = 13.98 dB.
5. Gain crossover: (1 + ω²)(4 + ω²)(9 + ω²) = 144, solved numerically: ω_gc = 1.22 rad/s.
6. ∠G(jω_gc) = −(50.7° + 31.4° + 22.2°) = −104.4°, so PM = 75.6°.
7. The plot does not encircle −1 (N = 0), so Z = 0: stable. The gain could be raised fivefold, to 60, before instability.

Example 2 (GATE level — open-loop unstable plant). A unity-feedback loop has G(s) = K/(s − 1), K > 0. Use the Nyquist criterion to find the range of K for stability.

1. Open-loop pole at s = +1, so P = 1. Stability needs Z = 0, i.e. N = −1: one anticlockwise encirclement of −1.
2. G(jω) = K/(jω − 1) = K(−1 − jω)/(1 + ω²).
3. ω = 0: G = −K. ω → ±∞: G → 0. For ω > 0 the imaginary part is negative, for ω < 0 positive.
4. The plot is a circle with diameter from −K to 0 (centre −K/2, radius K/2). Going from ω = −∞ to +∞, it passes from the origin through the upper half to −K and back through the lower half: an anticlockwise circle.
5. If K > 1 the circle contains −1 and is traversed once anticlockwise: N = −1, Z = 1 − 1 = 0, stable. If K < 1 it does not contain −1: N = 0, Z = 1, unstable.
6. Stable for K > 1. Check: closed-loop characteristic equation s − 1 + K = 0 gives the pole s = 1 − K, which is negative only for K > 1.

Common mistakes

• Using "no encirclement means stable" for an open-loop unstable plant; always find P first.
• Counting encirclements of the origin instead of −1.
• Mixing the sign conventions: in Z = P + N, N counts clockwise encirclements positive.
• Reading GM from |GH| at the gain crossover, or PM at the phase crossover — they are the other way round.
• Quoting 10 dB GM as a tenfold gain increase: 10 dB is a factor of 10^(10/20) = 3.16.
• Applying GM/PM rules to non-minimum-phase or open-loop unstable systems without checking with the full Nyquist plot.

For GATE ME

Expect: GM from the point where a polar plot crosses the negative real axis; PM from the phase at gain crossover; phase crossover of a third-order type-0 or type-1 loop; Z = P + N counting for a given sketch; the effect of delay on phase margin; and the range of K for stability using the Nyquist plot. Practise solving tan⁻¹ sums to −180° quickly — for (s + a)(s + b)(s + c) type-0 loops the crossover is at ω² = ab + bc + ca.

Quick check

1. The Nyquist plot crosses the negative real axis at −0.5. What is GM in dB?
2. The open-loop phase at gain crossover is −120°. What is PM?
3. P = 0 and the plot encircles −1 twice clockwise. How many closed-loop RHP poles are there?
4. What gain factor does a GM of 20 dB allow?
5. A loop has PM = 45° at ω_gc = 2 rad/s. What added delay makes it marginally stable?

Answers: 1. 6.02 dB. 2. 60°. 3. Two. 4. A factor of 10. 5. 0.785 rad / 2 rad/s = 0.393 s.