Bode and Nyquist stability criteria, gain and phase margins

Why it matters

Routh tells you whether a loop is stable; gain and phase margins tell you how far it is from instability, which is what matters when process gain drifts, valves stick and models are wrong. Frequency-response stability tests also handle dead time exactly, so for most real process loops they are the practical way to find the ultimate gain and set safe controller gains.

Key ideas

Open-loop frequency response. All tests use the open-loop transfer function G_OL(s) = G_c·G_v·G_p·G_m with s = jω, i.e. the loop broken just before the comparator. Its amplitude ratio AR_OL and phase φ_OL come from the Bode rules of the previous topic.

Bode stability criterion. For open-loop stable systems whose phase crosses −180° only once:

• Find the phase-crossover (critical) frequency ω_co, where φ_OL = −180°.
• If AR_OL(ω_co) < 1 the closed loop is stable; if AR_OL(ω_co) > 1 it is unstable; AR_OL(ω_co) = 1 means sustained oscillation at ω_co.
• Physical idea: a signal going around the loop at ω_co is inverted by the −180° lag and inverted again by the comparator, so it returns in phase; if it also returns larger, it grows.

Ultimate gain and period. For proportional control, AR_OL = K_c·AR_p, so the ultimate gain is K_cu = 1/AR_p(ω_co) (with AR_p including valve and sensor gains) and the ultimate period is P_u = 2π/ω_co. These are the same values Routh gives for rational transfer functions, but the Bode method also works with dead time.

Gain margin (GM). GM = 1/AR_OL(ω_co). It is the factor by which the loop gain could increase before instability; in dB, GM_dB = −20·log₁₀AR_OL(ω_co). Typical design: GM = 1.7–2.0 (about 4.6–6 dB).

Phase margin (PM). Find the gain-crossover frequency ω_g, where AR_OL = 1. PM = 180° + φ_OL(ω_g). It is the extra phase lag (for example, extra dead time) the loop could tolerate. Typical design: PM = 30–45°. Extra dead time Δθ that would destabilise the loop is about PM (in rad)/ω_g.

Nyquist criterion. The Nyquist (polar) plot traces G_OL(jω) in the complex plane as ω goes from −∞ to +∞. With P open-loop poles in the right half-plane and N clockwise encirclements of the point (−1, 0), the number of unstable closed-loop poles is Z = N + P. For the usual open-loop-stable process (P = 0), the closed loop is stable if the plot does not encircle −1. Nyquist is more general than Bode: it handles open-loop unstable processes and phase curves crossing −180° several times. On the polar plot, GM is 1/(distance from origin to where the curve crosses the negative real axis), and PM is the angle between the negative real axis and the point where the curve crosses the unit circle.

Practical reading. Increasing K_c lifts the AR curve without changing phase, so it lowers GM and PM. Integral action adds phase lag at low frequency (lowers margins); derivative action adds lead (raises PM). Dead time lowers ω_co sharply and therefore K_cu.

Formulas

φ_OL(ω_co) = −180°, GM = 1/AR_OL(ω_co), GM_dB = −20·log₁₀ AR_OL(ω_co)

AR_OL(ω_g) = 1, PM = 180° + φ_OL(ω_g)

K_cu = 1/AR_p(ω_co), P_u = 2π/ω_co

• P control; AR_p is the AR of G_v·G_p·G_m; ω_co in rad per unit time.

Z = N + P

• Nyquist: Z unstable closed-loop poles, N clockwise encirclements of (−1, 0), P open-loop right-half-plane poles.

φ = −ω·θ − tan⁻¹(ω·τ), AR = K/√(1 + ω²·τ²)

• FOPDT process K·e^(−θs)/(τs + 1).

Worked examples

Example 1 (standard): margins of a three-lag loop. G_OL = K_c/((s + 1)(2s + 1)(4s + 1)), time in min. Find K_cu, P_u, and the GM and PM for K_c = 4.

1. Phase crossover: tan⁻¹ω + tan⁻¹2ω + tan⁻¹4ω = 180°. By trial ω_co = 0.935 rad/min (43.1° + 61.9° + 75.0° = 180.0°).
2. AR_p(ω_co) = 1/(1.369·2.121·3.873) = 1/11.25, so K_cu = 11.25 and P_u = 2π/0.935 = 6.72 min (the same as the Routh result).
3. GM at K_c = 4: GM = K_cu/K_c = 11.25/4 = 2.81, i.e. 20·log₁₀2.81 = 8.98 dB.
4. Gain crossover: 4/(√(1 + ω²)·√(1 + 4ω²)·√(1 + 16ω²)) = 1. By trial ω_g = 0.542 rad/min (1.137·1.474·2.386 = 4.000).
5. φ_OL(ω_g) = −(28.4° + 47.3° + 65.2°) = −140.9°; PM = 180 − 140.9 = 39.1°.

K_cu = 11.25, P_u ≈ 6.72 min; at K_c = 4: GM ≈ 2.81 (8.98 dB), PM ≈ 39°

Example 2 (GATE level): first-order-plus-dead-time process. G_p = 2·e^(−s)/(5s + 1), time in min, with P control and unity valve and sensor gains. Find ω_co, K_cu, P_u and the K_c giving GM = 2.

1. Phase condition: ω·1 + tan⁻¹(5ω) = π rad.
2. Trial: ω = 1.6 gives 1.600 + 1.446 = 3.046; ω = 1.7 gives 1.700 + 1.454 = 3.154. Interpolate: ω_co ≈ 1.689 rad/min (96.8° + 83.2° = 180.0°).
3. AR_p(ω_co) = 2/√(1 + (5·1.689)²) = 2/8.503 = 0.2352.
4. K_cu = 1/0.2352 = 4.25; P_u = 2π/1.689 = 3.72 min.
5. GM = 2: K_c = K_cu/2 = 2.13.

ω_co ≈ 1.69 rad/min; K_cu ≈ 4.25; P_u ≈ 3.72 min; K_c ≈ 2.13 for GM = 2

Common mistakes

• Using the closed-loop transfer function instead of the open-loop one in Bode or Nyquist tests.
• Swapping the two crossover frequencies: GM is read where the phase is −180°, PM where AR = 1.
• Writing PM = −180° − φ instead of 180° + φ (a phase of −150° gives PM = +30°).
• Forgetting the valve and transmitter gains in AR_p when computing K_cu.
• Applying the simple Bode criterion to open-loop unstable systems; use Nyquist (Z = N + P).
• Mixing radians and degrees in the dead-time phase term.

For GATE CH

This topic regularly produces numericals: the crossover frequency of a loop with dead time (by trial), the ultimate gain and period, gain margin (as a ratio or in dB) for a given K_c, the K_c for a specified GM, and phase margin. Conceptual questions test the Nyquist encirclement rule. Practise solving tan⁻¹ sums equal to 180° quickly and carry four significant figures.

Quick check

1. The open-loop AR at the phase-crossover frequency is 0.25. What is the GM in dB?
2. At the gain crossover the open-loop phase is −150°. What is the PM?
3. A loop has K_cu = 6 and runs at K_c = 2. What is the GM?
4. Can a first-order process with no dead time under P control have a finite gain margin?

Answers: 1. 12.0 dB (GM = 4). 2. 30°. 3. 3. 4. No: its phase never reaches −180°, so GM is infinite.