Phase Lead and Lag: What Pole-Zero Order Changes

A first-order compensator can contain one real zero and one real pole. Placing the zero at a lower corner frequency than the pole gives a positive phase contribution. Reversing their order gives a negative contribution. Phase lead and lag name these two responses. The names describe what the compensator contributes across frequency. To understand what happens to a robot control loop, the compensator response must then be combined with the rest of the loop and evaluated at the resulting gain crossover.

Read both corner frequencies on a positive frequency axis

Use a continuous-time compensator with transfer function C(s) = K(1 + s/wz)/(1 + s/wp). The constants wz and wp are positive corner frequencies in radians per second and K is a positive gain. The actual zero of the transfer function is at s = −wz, and its pole is at s = −wp. On a Bode plot the horizontal axis shows the positive angular frequency w. A zero listed as −10 on the real s axis has its corner at 10 radians per second on that plot. The axis conversion matters when someone says the zero comes first.

At a chosen frequency, substitute s = jw. The numerator contributes an angle of arctan(w/wz). The denominator subtracts arctan(w/wp). MIT's lead-compensation material illustrates how these two contributions combine. At very low frequencies both angles are close to zero. At very high frequencies both approach ninety degrees and their difference returns toward zero. The phase between those limits depends on which factor turns first.

Putting the zero first makes a phase hump

If wz is lower than wp, the zero term acquires positive angle earlier. Over intermediate frequencies its angle exceeds that of the pole term, so the net phase is positive. The result is a hump whose peak lies at the geometric mean of the two corner frequencies. The two corners mark changes in the individual factors' responses, while the resulting phase contribution is smooth across the range. That is why the highest phase contribution can sit between the named pole and zero frequencies.

The same ordering affects gain. At low frequency, each normalized factor is close to one, so the gain approaches K. At high frequency, the factors approach jw/wz and jw/wp, so their magnitude ratio approaches wp/wz. Lead therefore has higher high-frequency gain than low-frequency gain within this form. Rescaling K moves the gain curve without changing the phase curve. A plot of the lead response needs both curves to show what multiplication by the compensator will do.

As a calculated example, consider corners at one and ten radians per second. Their geometric mean is about 3.16 radians per second, where the difference of the two arctangents is about 55 degrees. Swapping the pole and zero leaves the center frequency unchanged and reverses the sign. If the two corners coincide, their factors cancel and the compensator reduces to its constant gain.

Putting the pole first makes a phase dip

With wp below wz, the denominator builds angle before the numerator catches up, so the compensator contributes negative phase at intermediate frequencies. MIT's Feedback Systems recitation shows this phase dip for a lag compensator and notes the effect of its placement relative to crossover. The same high-frequency limit now gives wp/wz less than one. In this normalization the gain at high frequency is lower than the gain at low frequency. One can instead rescale K to keep a desired high-frequency gain, which makes the relative gain increase at low frequency more evident. The relative difference remains the same.

Low-frequency loop gain matters when assessing tracking of slowly varying references and some steady-state errors. A lag element can help change that part of the loop response while keeping its negative phase contribution away from the intended crossover. Whether this happens depends on the full loop, including any shift of crossover. The compensator's phase dip and its relative gain change describe the available behavior before it is connected to the plant.

Check phase margin at the loop's actual crossover

In a frequency-response description, loop gain magnitudes multiply and loop phases add. Gain crossover occurs where the magnitude of the resulting loop equals one. A compensator can move that frequency because it also changes magnitude. For the usual negative-feedback comparison near minus 180 degrees, read the resulting loop phase at its actual crossover when examining phase margin. A positive phase contribution at the old crossover by itself gives only part of that result.

A documentation example can label the transfer function, each positive corner frequency, the normalization and the crossover being discussed. This makes it clear whether a plotted curve belongs to the compensator or to the compensated loop. The robot learning archive and technical article directory provide related control-system examples.

University references

See the MIT 6.310 lead-compensation material and the MIT 6.302 recitation on compensation.