Page 432 - Mechanical Engineers' Handbook (Volume 2)

9 Graphical Design Methods 423

Figure 44 Root-locus plot for PI control of a second-
order plant.

K (s 1/T )
P
I
G(s)H(s) (43)
s(s r )(s r )
1 2
One design approach is to select T and plot the locus with K as the parameter. If the zero
I P
at s 1/T is located to the right of s r , the dominant time constant cannot be made
I 1
as small as is possible with the zero located between the poles at s r and s r (Fig.
1 2
44). A large integral gain (small T and/or large K ) is desirable for reducing the overshoot
I P
due to a disturbance, but the zero should not be placed to the left of s r because the
2
dominant time constant will be larger than that obtainable with the placement shown in Fig.
44 for large values of K . Sketch the root-locus plots to see this. A similar situation exists
P
if the poles of the plant are complex.
The effects of the lead compensator in terms of time-domain specifications (character-
istic roots) can be shown with the root-locus plot. Consider the second-order plant with the
real distinct roots s , s
. The root locus for this system with proportional control
is shown in Fig. 45a. The smallest dominant time constant obtainable is , marked in the
1
figure. A lead compensator introduces a pole at s 1/T and a zero at s 1/aT, and
the root locus becomes that shown in Fig. 45b. The pole and zero introduced by the com-
pensator reshape the locus so that a smaller dominant time constant can be obtained. This
is done by choosing the proportional gain high enough to place the roots close to the as-
ymptotes.
With reference to the proportional control system whose root locus is shown in Fig.
45a, suppose that the desired damping ratio and desired time constant are obtainable
1 1
with a proportional gain of K , but the resulting steady-state error
/(
K ) due to
P1 P1
a step input is too large. We need to increase the gain while preserving the desired damping
ratio and time constant. With the lag compensator, the root locus is as shown in Fig. 45c.
By considering specific numerical values, one can show that for the compensated system,
roots with a damping ratio correspond to a high value of the proportional gain. Call this
1
value K . Thus K K , and the steady-state error will be reduced. If the value of T is
P2 P2 P1
chosen large enough, the pole at s 1/T is approximately canceled by the zero at s
1/aT, and the open-loop transfer function is given approximately by
aK
G(s)H(s) P (44)
(s )(s
)