0195136047.pdf

16.2 FEEDBACK CONTROL SYSTEMS 791

TABLE 16.2.2Steady-State Error Response for Unit Inputs

System Type Unit Step

(

1

s

)

Unit Ramp

(

1

s^2

)

Unit Acceleration

(

1

s^3

)

0 Finite Infinite Infinite

1 0 Finite Infinite

2 0 0 Finite

• Anintegrating device,such as an operational amplifier with feedback.

Many industrial controllers utilize two or more of the basic control elements. The classification
by control action is summarized in Table 16.2.3.

Error-Rate Control, Output-Rate Control, and Integral-Error (Reset)

Control

Let us consider a typical second-order servomechanism (containing two energy-storing elements)
whose defining differential equation for obtaining the dynamic behavior of the system is given
by

J

d^2 c

dt^2

+F

dc

dt

+Kc+TL=Kr (16.2.22)

whereJ, F,andTLrepresent, respectively, effective inertia, equivalent viscous friction, and
resultant load torque appearing at the motor shaft;K=KpKaKm, whereKpis the potentiometer
transducer constant (V/rad),Kais the amplifier gain factor (V/V), andKmis the motor-developed
torque constant (N·m/V) of the physical servomechanism shown in Figure 16.2.7;ris the input
command andcis the output displacement, both in radians.
The block diagram of the servomechanism of Figure 16.2.7 is given in Figure 16.2.8, in
which the transfer function of each component is shown. Since the actuating signal is given by
E=Kp(R−C), the block diagram may be simplified, as shown in Figure 16.2.9, in which
TLis assumed to be zero for simplicity. The direct transmission function (forward path transfer
function) for the system can be seen to be

G(s)=

KpKaKm

Js^2 +Fs

=

K

Js^2 +Fs

(16.2.23)

The corresponding closed-loop transfer function is given by

M(s)=

C(s)

R(s)

=

G(s)

1 +H G(s)

=

K/(J s^2 +Fs)

1 +K/(J s^2 +Fs)

=

K

Js^2 +Fs+K

(16.2.24)

TABLE 16.2.3Classification by Control Action

Control Action G(s)

Proportional Kp
Differential sKd
Integral Ki/s
Proportional and differential (PD) Kp+sKd
Proportional and integral (PI) Kp+Ki/s
Proportional and integral and differential (PID) Kp+Ki/s+sKd