**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

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where the derivative control gain *K*_{D} 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

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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

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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* = *K*_{D}/(*K*_{P} + *K*_{D}). If *K*_{P} and

*K*_{D} > 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

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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 *K*_{P} increases.

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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 (*K*_{P} + *K*_{D}), 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

*K*_{P} and *K*_{D} as follows:

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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 ± 0.39*j* and 0.20. Using the dominant

poles of 0.55 ± 0.39*j* = 0.67e^{±jπ/5}, the expected settling time is *k*_{s} = 10 and

the estimated overshoot is *M*_{P} = 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 *K*_{D} 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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