Lead and lag compensation
Shape the loop across frequency instead of increasing one gain everywhere.
Identify a lead network from its pole and zero and calculate one phase contribution.
Root locus changed poles with one gain. Lead and lag compensation add a zero and pole so the loop can change differently across frequencies.
Review the previous lesson: Root locus: follow the polesFirst, picture it
Changing one gain moves poles along an existing root locus. If that path cannot give the response we want, we can add a pole and a zero to the controller. Lead and lag compensators are two simple ways to change that path.
Build the idea step by step
In the form shown below, a lead compensator has its zero nearer zero than its pole. A lag compensator reverses that order. These added factors change the closed-loop characteristic equation. Another way to describe their effect is phase: how far a repeating output leads or lags its input in time. Lead adds positive phase over a range of frequencies; lag adds negative phase but can increase low-frequency gain relative to high-frequency gain. The next lessons explain this frequency view more fully.
Read the equations
The zero is closer to the origin than the pole. This arrangement adds positive phase over a frequency band.
What the symbols mean
- positive zero-location parameter, in inverse seconds
- positive pole-location parameter, in inverse seconds
- complex frequency, in inverse seconds
- controller transfer function
The worked angle calculation previews the next part. It does not prove that a whole loop is stable. The overall controller gain also matters, and extra high-frequency response can increase noise and actuator demand.
Lead adds positive phase over a band; lag adds negative phase. Exact curves and crossover depend on pole, zero and gain choices.
Phase added near two radians per second
Choose zero parameter 1 and pole parameter 4, both per second. This is a lead network.
At 2 radians per second, subtract the denominator angle from the numerator angle.
The result is about 36.9 degrees of positive phase. Its magnitude also changes, so a full design must check gain crossover and noise.
More detail and model assumptions
Move a zero and a pole for a reason
A lead network C=K(s+z)/(s+p), with 0<z<p, adds positive phase between its break frequencies. That can let the loop cross 0 dB with more phase margin or at a higher frequency. The same network also changes magnitude, so always find the new crossover rather than adding a phase number to an old one.
A lag network has 0<p<z. With a suitable gain choice it raises low-frequency loop gain relative to the crossover region and can reduce steady error. Its phase contribution is negative around the pole–zero band; slower settling or reduced margin may be the price. Neither network magically defeats a voltage limit.
Choose by the specification
Suppose your motor is accurate enough but too slow: lead is a candidate if the available voltage and noise bandwidth allow more aggressive response. Suppose speed is acceptable but loaded steady error is too large: lag or PI may help, with different effects on exact zero error, phase and windup. Write the required error, settling time and actuator limit before tuning.
In this Lab the lead/lag controller uses an explicit sampled implementation; the frequency panel analyzes the nominal sampled loop. Compare its time run too. A favorable small-signal margin does not describe intervals when the applied voltage clips at ±12 V.
Connect the example to the lab
If a speed loop needs more phase near its gain crossover, consider lead. If it needs lower steady error without lifting crossover as much, consider lag.
Compare lead, lag and PIMake a prediction before moving a control.
- PREDICTWhich controller would you try for error, which for speed?
- CHANGE ONE THINGPin the lead lesson, then run lag and PI with the same motor, target and load.
- OBSERVECompare final error, settling, applied voltage and the crossover region.
- EXPLAINRecommend a controller for one written requirement, including its cost.
Lead is not free speed: it can amplify high-frequency sensor noise. Lag is not a universal cure for drift or saturation.
Check your understanding+
In (s+1)/(s+4), which is closer to the origin: the zero or the pole? Is this lead or lag?
The zero at −1 is closer than the pole at −4. This is a lead compensator, which adds positive phase over a frequency band.Why the next lesson follows
The phase calculation used a sine-wave frequency. Next we define frequency response before reading its full plots.