9.3 PROPORTIONAL–DERIVATIVE CONTROL
The control actions of the proportional or integral controllers are based on the
current error or past errors. In derivative control the controller output is proportional
to the rate of change of the error. The idea behind derivative control is that
the controller should react immediately to a large change in the control error;
in essence, predicting that the error will continue to increase (or decrease) and
act accordingly. Although this quick reaction can result in fast response times,
it can also result in undesirable overreaction, especially if the system output has
significant stochastics.
The derivative control law has the form
| |  |
where the derivative control gain KD defines the ratio of the input magnitude
to the change in the error (Figure 9.18). Since the derivative controller adjusts
the control input according to the speed of error variation, it is able to make an
adjustment prior to the appearance of even larger errors. Practically, the derivative
controller is never used by itself since if the error remains constant, the output
of the derivative controller would be zero.
The transfer function of a derivative controller can be found by taking the
Z-transform of Equation (9.14) with zero initial conditions to get
| |  |
Note that the steady-state gain of a derivative controller is equal to zero. As noted
above, a derivative controller cannot react to a constant error. Thus, derivative
control is always used in conjunction with proportional control and sometimes
also with integral control.
The proportional–derivative (PD) control law has two terms: one proportional
to the current error, and the other proportional to the change in error:
Its transfer function can be found by taking the Z-transform with zero initial
conditions and rearranging terms to get
| |  |
The controller has a pole at z = 0; this pole is fast and hence does not slow
down the transient response like the pole at z = 1 does in integral control. The
control transfer function also has a finite zero at z = KD/(KP + KD). If KP and
KD > 0 have the same sign, the zero is always on the real line between 0 and 1.
When the zero is exactly at 0, it cancels the pole at 0 and PD control reduces to
the pure proportional control case. When the zero is exactly at 1, it reduces to
the pure derivative control case.
PD controllers can be designed using the root locus method as in Section 9.2.3.
However, PD controllers are not appropriate for first-order systems because pole
placement is quite limited. For example, Figure 9.19 shows the root locus plots
of a first-order system with PD control with two different zero locations. Observe
that the poles are restricted to a limited section of the real axis. Compare these
plots with the root locus of a first-order system and P control, as shown in
Figure 8.12. Note that P control allows the closed-loop pole to be placed anywhere
on the real axis to the left of the open-loop pole.
With their predictive ability, PD controllers can be used effectively to reduce
the overshoot for a system that exhibits a significant amount of oscillation with
P control. For example, consider a second-order system with transfer function
| |  |
The root locus of this system is shown in Figure 9.20. As the proportional gain
increases, the closed-loop poles move farther away from the origin. Both the
settling time and overshoot increase as KP increases.
| |  |
| |  |
If we add a PD controller to this system, we add a zero and a pole to the
open-loop system. The pole is at z = 0, and the zero can be placed anywhere
along the positive real axis. Figure 9.21 shows the root locus for two different
zero locations. When the zero is farther to the right, the root locus moves farther
to the left.
For each of these two choices of zero location, settling time and overshoot are
a function of overall gain (KP + KD), as shown in Figure 9.22. When the zero is
at 0.8, the root locus is pulled farther toward the origin, and thus the expected
overshoot and settling time are lower. However, the actual overshoot is greater.
This difference between the estimated and actual overshoot is due to the effect
of the zero. The zero has the largest effect on the actual overshoot when it is to
the right of the closed-loop poles.
Choosing the zero at 0.5 and the overall gain at 0.18, we solve for the gains
KP and KD as follows:
| |  |
The closed-loop transfer function from the reference input to the output can be
found as
the closed-loop poles are located at 0.55 ± 0.39j and 0.20. Using the dominant
poles of 0.55 ± 0.39j = 0.67e±jπ/5, the expected settling time is ks = 10 and
the estimated overshoot is MP = 0.13. The steady-state gain of the closed-loop
system is
which gives a steady-state error of almost 70% (1 − 0.31/100). The response
of the system to a step reference of magnitude 10 is shown in Figure 9.23. PD
control cannot eliminate the steady-state error.
9.3 PROPORTIONAL–DERIVATIVE CONTROL
The control actions of the proportional or integral controllers are based on the
current error or past errors. In derivative control the controller output is proportional
to the rate of change of the error. The idea behind derivative control is that
the controller should react immediately to a large change in the control error;
in essence, predicting that the error will continue to increase (or decrease) and
act accordingly. Although this quick reaction can result in fast response times,
it can also result in undesirable overreaction, especially if the system output has
significant stochastics.
The derivative control law has the form
| | |
where the derivative control gain KD defines the ratio of the input magnitude
to the change in the error (Figure 9.18). Since the derivative controller adjusts
the control input according to the speed of error variation, it is able to make an
adjustment prior to the appearance of even larger errors. Practically, the derivative
controller is never used by itself since if the error remains constant, the output
of the derivative controller would be zero.
The transfer function of a derivative controller can be found by taking the
Z-transform of Equation (9.14) with...
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