Lead, lag and lead-lag compensators

Why it matters

Often a loop cannot meet both its accuracy and its damping specifications by adjusting gain alone: the gain needed for small steady-state error leaves too little phase margin. A compensator — a simple pole-zero network, built from an RC circuit with an op-amp or as a few lines of controller code — reshapes the loop's frequency response so both can be met. Lead, lag and lead-lag networks are the classical tools for this, and their design is a standard GATE topic.

Key ideas

Lead compensator — zero closer to the origin than the pole.

• Transfer function Gc(s) = (1 + aTs)/(1 + Ts), with a > 1 (zero at −1/(aT), pole at −1/T).
• Gives positive phase (lead) between the corner frequencies, with a maximum φm at their geometric mean ωm.
• Raises high-frequency gain by the factor a. Used to increase phase margin and bandwidth: less overshoot, faster response. Cost: more noise amplification; and a single stage gives at most about 60° (larger leads need two stages).
• Root-locus view: the zero pulls the locus left, toward better damping. It behaves like a filtered PD controller.

Lag compensator — pole closer to the origin than the zero.

• Gc(s) = (1 + Ts)/(1 + bTs), with b > 1 (zero at −1/T, pole at −1/(bT)).
• Attenuates high frequencies by 1/b while leaving low frequencies untouched. With the loop gain raised by b, the low-frequency gain (and so Kp, Kv or Ka) rises by b while the crossover region is unchanged — steady-state error falls by about b times with little change in transient response.
• Its phase lag must be kept well below the gain crossover: place the zero about one decade below the new ωgc, so it costs only about 5° of phase margin. Cost: lower bandwidth and a slow "tail" in the response from the near-cancelling pole-zero pair. It behaves like a filtered PI controller.

Lead-lag compensator: a lag section for accuracy in series with a lead section for margin and speed, Gc(s) = [(1 + T1s)/(1 + bT1s)]·[(1 + aT2s)/(1 + T2s)], with the lag corners well below the lead corners.

Another common notation: many textbooks write the lead network as (1 + Ts)/(1 + αTs) with α < 1 (so α = 1/a) and the lag network as (1 + Ts)/(1 + βTs) with β > 1. The physics is identical; always identify which corner is the zero and which is the pole before using a formula.

Passive RC networks: a passive lead network has DC gain 1/a (an attenuator), so an amplifier of gain a is needed to restore the gain; a passive lag network has unity DC gain.

Frequency-domain lead design (outline)

1. Choose loop gain K from the steady-state error specification.
2. Find the uncompensated phase margin.
3. Required lead: φm = PM(required) − PM(uncompensated) + 5° to 12° (the extra allows for the crossover moving right).
4. Get a from φm.
5. Place ωm at the new gain crossover, where the uncompensated magnitude equals −10·log10(a) dB.
6. T = 1/(ωm·√a); check the final margin.

Formulas

• Lead: Gc(s) = (1 + aTs)/(1 + Ts), a > 1.
• φm = sin⁻¹[(a − 1)/(a + 1)]; inverse a = (1 + sin φm)/(1 − sin φm).
• ωm = 1/(T·√a) — geometric mean of the corners 1/(aT) and 1/T.
• |Gc(jωm)| = √a, i.e. 10·log10(a) dB.
• Lag: Gc(s) = (1 + Ts)/(1 + bTs), b > 1; maximum lag sin⁻¹[(b − 1)/(b + 1)] at ω = 1/(T·√b); high-frequency gain 1/b (−20·log10 b dB).
• Gc(jω) phase = tan⁻¹(ω·(zero time constant)) − tan⁻¹(ω·(pole time constant)).

Symbols: T time constant (s); a, b dimensionless ratios of corner frequencies; φm degrees; ω, ωm rad/s.

Worked examples

Example 1 — characterising a lead network Given: Gc(s) = (1 + 0.1s)/(1 + 0.02s).

1. Zero corner 1/0.1 = 10 rad/s, pole corner 1/0.02 = 50 rad/s; zero first, so this is a lead network with T = 0.02 s and a = 0.1/0.02 = 5.
2. φm = sin⁻¹[(5 − 1)/(5 + 1)] = sin⁻¹(0.667) = 41.8°.
3. ωm = 1/(0.02 × √5) = 22.4 rad/s (also √(10 × 50)).
4. Gain at ωm: √5 = 2.24, i.e. 6.99 dB; high-frequency gain a = 5 (14 dB).

Result: maximum lead 41.8° at 22.4 rad/s, with 7.0 dB of gain added there.

Example 2 — lead design for a velocity servo (GATE level) Given: plant G(s) = 4/[s(s + 2)], unity feedback. Specifications: Kv = 20 s⁻¹, PM ≥ 45°.

1. Gain: Kv = lim s·K·G = K × 4/2 = 2K = 20 → K = 10, so the loop is 40/[s(s + 2)].
2. Uncompensated crossover: ω·√(ω² + 4) = 40 → ω² = 38.05, ω = 6.17 rad/s; PM = 90° − tan⁻¹(6.17/2) = 18.0°.
3. Lead needed: 45° − 18° + about 5° → choose a = 4, giving φm = sin⁻¹(3/5) = 36.9°.
4. New crossover where |G| = 1/√4 = 0.5: ω·√(ω² + 4) = 80 → ω² = 78.0, ωm = 8.83 rad/s.
5. T = 1/(8.83 × 2) = 0.0566 s; aT = 0.226 s. Compensator: Gc(s) = (1 + 0.226s)/(1 + 0.0566s), zero at −4.42, pole at −17.7.
6. Check: uncompensated phase at 8.83 rad/s is −90° − tan⁻¹(4.42) = −167.2°; adding 36.9° gives −130.4°, so PM = 49.6°.

Result: Gc(s) = (1 + 0.226s)/(1 + 0.0566s) with loop gain 10 gives Kv = 20 s⁻¹ and PM ≈ 49.6° at 8.83 rad/s.

Common mistakes

• Calling a network "lag" or "lead" without checking which corner is the zero; the zero comes first (lower frequency) in a lead network.
• Mixing the a > 1 and α < 1 conventions in the same calculation.
• Placing ωm at the old crossover: the lead's own gain moves the crossover to the right.
• Using a lag network to add phase margin — lag adds phase lag; it helps margin only indirectly by lowering the crossover.
• Putting a lag zero too close to the crossover, which eats the phase margin.
• Forgetting the DC attenuation of a passive lead network.

For GATE IN

Expect questions on identifying lead or lag networks from a transfer function or RC circuit, computing maximum phase lead/lag and the frequency at which it occurs, the gain added at ωm, and single-step lead or lag design to meet Kv and phase-margin specifications. Practise the sin⁻¹[(a − 1)/(a + 1)] and geometric-mean results until they are automatic.

Quick check

1. Is (1 + 0.5s)/(1 + 0.1s) a lead or lag network?
2. What is the maximum phase lead of a network with a = 9?
3. At what frequency does (1 + 10s)/(1 + 50s) give its maximum lag?
4. Which compensator increases bandwidth?

Answers: 1. Lead (zero at 2 rad/s before pole at 10 rad/s); 2. sin⁻¹(0.8) = 53.1°; 3. 1/√(10 × 50) = 0.0447 rad/s; 4. Lead.