Chapter 9.1.2 - Transient Response with Integral Control

9.1.2 Transient Response with Integral Control

The elimination of steady-state error comes at a price in that integral control
slows system response. The reason for this is that the integrator adds an open-loop
pole at z = 1. Thus, the closed-loop system has one more pole than the openloop
transfer function G(z). This additional pole is typically closer to the unit
circle than any of the open-loop poles, and hence results in a slower closed-loop
response.

Example 9.2: Closed-loop poles of the IBM Lotus Domino Server with
integral control Consider the closed-loop system in Figure 9.1 in which
G(z) = 0.47/(z − 0.43). The closed-loop system has two poles, one from the
IBM Lotus Domino Server and one from the integrator. Instead of computing
their locations directly, we can plot the root locus—all possible locations of the
closed loop poles as K_I varies from 0 to ∞. In Figure 9.2, the root locus of the
IBM Lotus Domino Server alone is shown on the left, and the root locus of the
IBM Lotus Domino Server plus an integrator is shown on the right. Since the
integrator adds an open-loop pole at z = 1, there are two branches to the root
locus. One branch ends at the finite zero (z = 0), and the other at the zero at
infinity. The largest closed-loop pole is always closer to the unit circle than the
open-loop pole 0.43.

Another way to view the effect of integral control is to plot the magnitude of
the largest closed-loop pole versus the control gain. Figure 9.3 shows this plot
for both the IBM Lotus Domino Server with P control (top) and the IBM Lotus
Domino Server with I control (bottom). Recall that the settling time of the system
depends on the magnitude of the dominant closed-loop pole, as k_s = −4/ log p,
where p is the dominant pole. Since the magnitude of the largest closed-loop pole
with I control is always greater than the magnitude of the open-loop pole (0.43),
and the settling time increases with the pole magnitude, the closed-loop response
will always be slower than the open-loop response, no matter what value of K_I
is chosen. This trade-off between accuracy and speed is one of the many choices
that must be made in control design.

Consider three different choices for K_I : K_I = 1, 3, 5. According to Figure 9.3,
each of these choices results in a stable closed-loop system. The closed-loop poles
are shown in Table 9.1, along with the predicted settling time and maximum
overshoot. The predicted settling time is the same for all cases but is twice

Fig. 9.2 Root locus of the IBM Lotus Domino Server without and with an integrator in the loop.

Figure 9.3 and Table 9.1

as long as the settling time of the open-loop system [−4/ log(0.43) = 4.7].
The actual maximum overshoot is computed from the simulations as shown in
Figure 9.4, for a reference change of 20 (increasing RIS from 325 to 345). The
estimated and actual do not always agree, due to the presence of a zero in the
transfer function. As noted in Chapter 6, a zero in the transfer function can
increase the overshoot substantially.

Control design for an integral controller can be accomplished by plotting the
expected settling time and overshoot for a range of possible integral gains K_I that
result in a stable closed-loop system (we omit plotting the steady-state error since
it is always zero if the closed-loop system is stable). From this plot, a reasonable
value of K_I can be chosen. If M_P is a critical factor in the design, the estimate
from the closed-loop poles should be validated by simulation, since the zero in
the integrator can cause the overshoot to increase. As an example, the expected
settling time ks and expected overshoot M_P are plotted in Figure 9.5 for the IBM
Lotus Domino Server with integral control.

Figures 9.4 & 9.5

Example 9.3: Disturbance rejection in the IBM Lotus Domino Server with
integral control Consider the IBM Lotus Domino Server with integral control,
and choose K_I = 1. Suppose that as in Section 8.1, the desired reference is r_ss =
10, but the initial condition is y = 0. That is, the system starts at its operating
point, and we want RIS to increase by 10. Further, there is a step increase in the
disturbance input, with d_ss = 20.

The time response of all signals in this system can be seen in Figure 9.6.
Note that the steady-state error to the step reference is zero and that the output
y reaches its steady-state value after k_s = 10 sample times, as predicted. At
k = 10, the disturbance occurs. The control u reacts to this changed error, and
the output y again returns to its desired value after the settling time k_s= 10.
Integral control achieves a zero steady-state error because the sum is
nonzero even though e(k) = 0.

Example 9.4: Moving-average filter plus integral control In Section 8.4.3
we show how a moving-average filter can be used to smooth the output of the

Fig. 9.6 Reference tracking (rss = 10) and disturbance rejection (dss = 20) in the IBM Lotus

system before it is fed back to the controller. Such an approach is depicted in
Figure 9.7(a) for the IBM Lotus Domino Server. One consequence of a moving-
average filter is to slow the system response so that it reacts only to sustained
changes in the reference and disturbance inputs.

Since an integral controller also slows down the system response, the combination
of a moving-average filter with integral control can lead to undesirable
or overly slow behavior. To see this, consider the root locus of the IBM Lotus
Domino Server plus a moving-average filter and an integral controller, as shown
in Figure 9.7(b). There are in total three closed-loop poles: the integral controller
contributes a pole at z = 1, the moving-average filter contributes a pole at z = c
(here c = 0.9), and the IBM Lotus Domino Server contributes a pole at z = 0.43.
As can be seen in the root locus plot, the region of controller gains K_I that result
in a stable closed-loop system is very small. In addition, since in the stable region
the dominant closed-loop pole is very close to the unit circle (magnitude greater
than 0.95, or halfway in between c and 1), the settling time will necessarily be
very long (here, k_s> 78).

In essence, an integral controller acts like a moving-average filter. A short
transient disturbance contributes only slightly to the integral of the error, while a
sustained change in the output has a more substantial effect on the integral of the
error. The difference is that an integral controller can drive the steady-state error
to zero, whereas a moving-average filter, with a steady-state gain of 1, cannot
change the steady-state behavior of the system.

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Chapter 9.1.2 - Transient Response with Integral Control

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