Page 669 - Mechanical Engineers' Handbook (Volume 2)

660 Controller Design

compensation can be usefully applied. A PID controller offers a convenient method for
combining the electronic integration with lead compensation. As discussed in Section 6.1,
the PI scheme provides first-order lead, while the complete PID scheme offers second-order
lead.

6.3 Systems Having an Inherent Integration
Section 2.1 explains that an electronic integrator in the forward loop minimizes static servo
errors. However, the addition of an electronic integrator to a system that already has an
inherent integration will usually cause dynamic instability. To prevent this type of instability,
lead compensation must be combined with the electronic integrator. This can be accom-
plished with a PI controller.
As shown in Fig. 36, a PI controller contributes nearly 90 of phase lag at low fre-
quencies. If this is added to the 90 of lag already contributed by the integrator inherent in
the system, the total low-frequency lag approaches 180 . Since the other system dynamics
will add even more phase lag at high frequencies, the PI break frequency must be set low
enough to ensure an intermediate frequency range over which the phase lag is reduced. The
open-loop frequency responses of Fig. 37a illustrates this effect for a system whose inherent
characteristics consist of an integrator and a second-order lag. Generally, a PI break fre-
quency greater than 10% of the system’s lowest lag frequency will substantially reduce
closed-loop stability. This is shown by the phase plots of Fig. 37a, by the root loci of Fig.
37b, and by the closed-loop frequency responses of 37c.
Using the 10% rule of thumb for the PI controller, it is interesting to examine some
time histories of the closed-loop system. Figure 38 shows time responses to step and ramp
commands for the system of Fig. 37. As illustrated in Fig. 38a, the PI controller degrades
the step response. Figure 38b shows that the PI’s double integration at low frequencies
eliminates the following error but causes larger overshoot of the steady state.
The 10% rule also applies to systems whose inherent characteristics consist of an in-
tegrator and a first-order lag. Frequency responses, root loci, and time histories for such a
system are given in Fig. 39. It should also be noted that the effective lag break frequency
of the system can be increased by using lead compensation techniques, as described in
Section 3. For the system of Fig. 39, this can be accomplished by using a PID, rather than
a PI, controller. Reexamining the example of Fig. 39 using the PID characteristics of Fig.
36, a 10-fold increase in effective system bandwidth can be achieved by setting 1/
pid 1
and 1.0. In this case, one of the two PID lead terms cancels the system lag at 1/ .
pid 1
The resulting open-loop frequency response is shown in Fig. 40.

7 EFFECTS OF NONLINEARITIES
The previous sections have concentrated on the design of controllers for linear systems. In
practice, physical systems are never truly linear. If the nonlinearities are not large, design of
the controller using linear techniques is very useful. However, it is important to understand
the limitations of this approach. The following sections discuss these limitations and offer
several approaches to dealing with nonlinearities.
Figure 41 illustrates idealized forms of several nonlinearities that are commonly en-
countered in systems controlling mechanical loads. Saturating nonlinearities can occur in
transducers, electronics, and the servoactuator itself. Deadzone is the lack of output for small
changes in input and is generally most significant in servoactuators and transducers. Reso-