Why Use Lead Compensators? - Goals For This Lesson
Root Locus Effects
Example
Bode' Plot Effects
Problems
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Lead compensators are sometimes the best controller to use to get a system to do what you want it to do.  It's as simple as that.  They are an option that you may need if you cannot use anything in the PID family to bring a system's performance within specifications.  There's no guarantee that a lead compensator will do the trick, but it is another weapon in the arsenal.

Goals Of This Lesson

Before we get very far, we need to establish what our goals are for lead compensators.

Given a Compensator Transfer Function:
Know which compensators are leads and which are lags.
Determine the effects of the lead compensator on closed loop system behavior using Bode' plot methods,
Determine the effects of the lead compensator on closed loop system behavior using Root Locus methods,
Given a system to be controlled,
Determine if a lead compensator can be used to satisfy closed loop system specifications,
Given a system in which a lead compensator can be used,
Determine the parameters of a lead compensator to produce a closed loop system that meets specifications.

A lead compensator can be thought of in several different ways.

• First, a lead compensator is a device that provides phase lead in its' frequency response.
• If the compensator has phase lead - and never a phase lag - then there are implications about where the corner frequencies are in the Bode' plot.
• Other implications are that the phase lead compensator will have only certain types of pole-zero patterns in the s plane.
Next, we will examine those implications.  A lead compensator will have a transfer function of the form:
• Since a lead compensator has only positive phase angle, we must have:
• wz < wp
• A Bode' plot will make this clearer.
Here's a Bode' plot for a transfer function, G(s), with:

G(s) = (10s + 1)/(s + 1)

• Notice:
• wz < wp
• since:
This Bode' plot shows the essential characteristic of a phase lead compensator.
• There is one pole and one zero.  Both are real.
• Phase is always positive!
• wz < wp
If, for example,
• wz > wp ,
• we would have a lag network, not a lead network since the phase would always be negative.
Here note the following in the plot of the example we encountered earlier.
• Phase angle is always positive.  The zero at wz = .1, or f = .0159 Hz, causes the magnitude plot to bend up, and the phase to become positive near f = .159 Hz.
• Later, the pole at 1.59 Hz. brings the Bode' plot's slope back to zero for high frequencies, and the phase back to zero.
On this page we have a video that shows how lead networks behave as the ratio of the pole to the zero changes.

• In the video, the ratio of pole to zero is Alpha.  The pole is at -1, so the corner frequency - in hertz - is .159.
• The zero starts out equal to the pole, so the zero corner is at .159 and then, as Alpha increases, the zero moves lower in frequency.
• Notice the phase behavior especially on the plot.  Phase is always positive, but you can get larger phase angles when the pole-to-zero ratio is larger.
You can see the features of a lead network.
• A lead network has one pole and one zero - although you can add multiple lead networks in a system.  The pole corner frequency (Bode' plot) is higher than the zero corner frequency.
In the next section, we'll look at how you can use a lead network in a system.

Usually, a lead network is used as part of the controller in a feedback system.

• Here's where the lead network would be used.
• In this system, we assume that you have a lead network, and an adjustable gain in the lead network.
We'll use this system as an example system to investigate the effects of using a lead network.

Now, let's look at how the lead network affects how the system behaves.  We'll look at root locus effects first and Bode' plot effects later.

Root Locus Effects

Now consider the system just above.  It has open loop poles at -1 and -4.

• How does the lead network affect the system?
• The green button controls whether the uncompensated system's root locus is visible.
• Click to show the root locus.
• Release to hide the root locus.
• Release outside the button to keep the root locus showing.
• The amber button controls the compensated system's root locus (with a lead compensator).
• The red button controls both together.

Note the following for these root loci.

• The closed loop poles that emanate from the open loop poles at -1 and -4 are further into the left half of the s-plane using the lead compensator.
• That implies the system may be faster!
• The centroid (center of gravity) shifts to the left when the lead compensator is added because the pole is larger than the zero.
• The asymptotes for the branches going to infinity will be shifted to the left because of the centroid's shift to the left.
• If there is root locus activity near the zero, then the locus will have a tendency to bend toward the zero.
Check this system again.