Compensation Techniques

Compensation purpose

Compensators shape loop gain and phase to meet stability, accuracy and speed requirements. Design must consider the complete loop including sensor, actuator and delay dynamics.

Lead and lag forms

A lead form is C(s) = K(1 + Ts)/(1 + αTs), with 0 < α < 1. The pole is farther left than the zero, and the phase contribution is positive over a frequency band. Maximum phase lead is sin⁻¹[(1 − α)/(1 + α)] at ωm = 1/(T√α). Its high-frequency gain exceeds its DC gain, increasing noise sensitivity. A lag form is C(s) = K(1 + Ts)/(1 + βTs), β > 1. Its pole is closer to the origin than its zero, and it adds phase lag. Together with gain adjustment it can raise low-frequency loop gain relative to crossover, improving steady-state accuracy while limiting crossover change. Phase margin must be rechecked. Lead-lag compensation combines both effects; cancellation of uncertain or unstable plant dynamics is unreliable.

Worked example

Choose α = 0.25 and target maximum phase lead at 4 rad/s. Then T = 1/(4√0.25) = 0.5 s. With K = 1, C(s) = (1 + 0.5s)/(1 + 0.125s). Zero is −2 rad/s and pole is −8 rad/s. Maximum phase lead is sin⁻¹(0.6) = 36.87°. At 4 rad/s the magnitude is 1/√α = 2, or 6.02 dB. Therefore the new gain crossover can move; adding 36.87° to the old phase margin without recomputing crossover is incorrect.

Design checks

Confirm closed-loop poles, margins at all relevant crossovers, reference and disturbance responses, control effort and sensitivity to model error. A lead compensator does not create unlimited bandwidth.

Quick check

1. Where is the pole of a lead relative to its zero? Farther left for the stated form.
2. Does lag add positive phase? No.
3. Why recalculate crossover? Compensation changes magnitude as well as phase.