Compensation Techniques

Why it matters

Compensation modifies loop gain and phase to meet tracking, disturbance rejection and stability-margin requirements. A compensator must be evaluated together with its plant and feedback path.

Lead and lag

For G_c(s) = K_c(s+z)/(s+p), with positive z and p, the zero lies at −z and pole at −p. Lead requires p > z; lag requires z > p. Phase lead can increase phase margin near crossover. Lag can increase low-frequency loop gain relative to crossover to improve tracking, but its added phase lag must be included in stability checks.

A normalized lead network is C(s) = K(1+Ts)/(1+αTs), with 0 < α < 1. Its maximum phase is φ_max = asin((1−α)/(1+α)), at ω_m = 1/(T√α). Its magnitude at that frequency is K/√α. Changing the magnitude shifts crossover, so adding φ_max does not automatically increase the final phase margin by exactly that amount.

Worked example

Design a lead network providing maximum phase lead 30° at 10 rad/s and unity magnitude at that frequency.

α = (1−sin30°)/(1+sin30°) = 1/3. T = 1/(10√(1/3)) = 0.173205 s. Choose K = √α = 0.577350 so |C(j10)| = 1.

C(s) = 0.577350(1+0.173205s)/(1+0.057735s). Its zero is at −5.7735 rad/s and pole at −17.3205 rad/s. At 10 rad/s the phase is atan(1.73205)−atan(0.57735) = 60°−30° = 30°, and magnitude is one.

This completes the network design. It is not a claim about an unspecified plant’s phase margin. For an existing loop with unity magnitude at 10 rad/s, this network retains that crossing; check for any additional crossings and verify overall closed-loop stability.

Design procedure

Determine the uncompensated loop’s actual crossover and margins. Choose a target crossover consistent with bandwidth and noise limits. Place lead or lag poles and zeros, adjust gain using the magnitude condition, then recompute all crossings and closed-loop poles. Check time response, actuator limits and model uncertainty.

Common mistakes

Calling (s+1)/(s+0.1) a lead network; interpreting z and p as the signed pole locations; assuming a phase margin without finding a unity-gain crossing; and neglecting crossover movement after compensation.

Quick check

1. Is (s+1)/(s+10) lead or lag?
2. Is (s+10)/(s+1) lead or lag?
3. Does 30° maximum network lead guarantee 30° improvement in every loop’s phase margin?

Answers: 1. Lead. 2. Lag. 3. No; the crossover and complete loop must be checked.