Chapter 9.2.3 - PI Control Design Using Root Locus

9.2.3 PI Control Design Using Root Locus

In pole placement design, the desired closed-loop poles are determined based
on an a priori specification of desired properties of the closed-loop system (e.g.,
settling time and maximum overshoot). However, it could be that even shorter
settling times and a smaller maximum overshoot are possible based on the relationship
between control gains and closed-loop poles. Root locus design provides
a way to explore the possible pole placements and hence gain insight into the
trade-offs in controller design.

As described in Chapter 8, the root locus of the system allows all possible
closed-loop pole locations to be considered at one time. However, the root locus
allows only one parameter to be varied, which creates a problem since PI controllers
have two parameters: K_P and K_I. One possible solution is to first fix K_P
and then draw the root locus with respect to K_I, choose a value of K_I, then
draw the root locus with respect to K_P, and so on. This is basically a trial-and-error
approach. Care must be taken to write the denominator of the closed-loop
transfer function in the correct form so that it matches the root locus definition
of Equation (8.4); some algebra is usually required to get it in this form.

A more effective approach is to transform K_P and K_I into a more convenient
form. For the first parameter, we use the location of the zero of the PI controller,
which is K_P/(K_P + K_I). For the second parameter we use K_P + K_I, a quantity that
we refer to as the overall gain of the PI controller. Placing the zero effectively
fixes the ratio between the proportional and integral parts of the controller and
leaves only one parameter to vary. Then we vary the overall gain to assess control
performance. This method is outlined below.

The zero of the PI controller must be between 0 and 1 on the positive real
axis. Determine the possible locations relative to other poles and zeros on
the positive real axis.
For each relative location of the zero, draw the root locus. Study the possible
closed-loop poles. Determine which relative location is best to meet the
design constraints.
For the most promising relative location, choose a few possible exact locations
(e.g., near the ends of the segment, in the middle of the segment),
and either draw an exact root locus or plot the expected settling time and
overshoot. Choose the zero location K_P/(K_P + K_I) and the overall gain
(K_P + K_I).
Simulate to verify the design, since the zero introduced by the PI controller
and any zeros present in G(z) can affect the system overshoot.

This design method is illustrated in the following example.

Example 9.6: PI control using root locus Consider again the IBM Lotus
Domino Server. The open-loop pole is at z = 0.43. The zero can be placed
either to the right or to the left of the open-loop pole; the two different root locus
plots are shown in Figure 9.11. If the zero is to the right of the pole, there is
always a closed-loop pole between the zero location and z = 1. This closed-loop
pole near the unit circle results in a slow closed-loop response. If the zero is
placed to the left of the open-loop pole, the closed-loop response can be faster

Fig. 9.11 Root locus of the IBM Lotus Domino Server and PI control. The zero can be placed

than the open-loop response (the closed-loop poles can be closer to the origin).
However, since the closed-loop poles can be complex, there could be a significant
amount of overshoot in the closed-loop response, which may be undesirable.

We consider three different locations of the zero on the segment between the
origin and the open-loop pole: 0.1, 0.2, 0.4. The first is very near the origin,
the second is in the middle of the segment, and the third is near the open-loop
pole. For each of these zero locations, and a range of overall gains, we compute
the expected settling time and overshoot as predicted by the closed-loop poles.
We also compute the actual overshoot by simulation. The results are shown in
Figure 9.12.

The minimum settling time occurs with the zero at 0.2 and the overall gain of
4. However, the overshoot at this point is large (almost 90%). We choose to place
the zero at 0.4, since it gives a larger region of gains with no overshoot. A choice
of overall gain at 1.5 results in overshoot M_P = 0 and settling time k_s = 4.2. From
the zero location and overall gain, we can find the gains K_P and K_I as

The simulation results of Figure 9.13 show the response to a step reference of
10. As predicted, the settling time is less than 5 and the maximum overshoot is
zero. Comparing these results with those in Figure 9.10, we see that the larger
proportional gain results in a faster response. The contribution of the proportional
controller u_P (k) is also much larger, as can be seen in the figures.

< Previous Excerpt Next Excerpt >

Purchase This Book

TABLE OF CONTENTS

Chapter 9.2.3 - PI Control Design Using Root Locus

Contact Preferences

This is embarrasing...

Customize Your GlobalSpec Experience

Select Your Free Newsletters

Industry Newsletters

Select Your Free Product Alerts

This is embarrasing...