0195136047.pdf

794 BASIC CONTROL SYSTEMS

amplifier is so designed that it provides an output signal containing a term proportional to the

derivative of the input, as well as one proportional to the input itself, error-rate damping will be

introduced. For the system that includes error rate, the only modification needed is in the transfer

function of the servoamplifier. Instead of the gainKa, the new transfer function will beKa+sKe,

whereKedenotes the error-rate gain factor of the amplifier. The complete block diagram is shown

in Figure 16.2.11. In this case the direct transmission function becomes

G(s)=

KpKm(Ka+sKe)

Js^2 +Fs

(16.2.28)

and the closed-loop transfer function is given by

M(s)=

C(s)

R(s)

=

G(s)

1 +H G(s)

=

K+sQe

Js^2 +(F+Qe)s+K

(16.2.29)

whereK=KpKaKmandQe=KpKeKmare known as theloop proportional gain factorand the

loop error-rate gain factor, respectively.

Note that the steady-state solution for a step inputr 0 is the same whether or not error-rate

damping is present. The advantage of error-rate damping lies in the fact that it allows higher

gains to be used without adversely affecting the damping ratio, and thereby makes it possible to

satisfy the specifications for the damping ratio as well as for the steady-state performance. Also,

the system’s natural frequency is increased, which in turn implies smaller settling times.

OUTPUT-RATECONTROL

A system is said to have output-rate damping when the generation of the output quantity in some

way is made to depend upon the rate at which the controlled variable is varying. Output-rate control

often involves the creation of an auxiliary loop, making the system multiloop. For the system of

Figure 16.2.7, output-rate damping can be obtained by means of a tachometer generator, driven

from the servomotor shaft. The complete block diagram of the servomechanism with output-rate

damping is depicted in Figure 16.2.12, whereKois the output-rate gain factor (V·s/rad) and the

output-rate signal is given byKo(dc/dt). By applying the feedback relationship of Equation

(16.2.3) to the minor (inner) loop, one gets

C(s)

E(s)

=

KaKm

Js^2 +sF

1 +

sKoKaKm

Js^2 +Fs

=

KaKm

Js^2 +(F+Qo)s

(16.2.30)

whereQo=KoKaKmis known as the loop output-rate gain factor. Figure 16.2.12 may then be

simplified, as shown in Figure 16.2.13. The closed-loop transfer function for the complete system

is then given by

− Potentiometer Amplifier Servomotor

Js^2 + Fs

REK^1 C

p Km

Ea Td

Ka + sKe

Figure 16.2.11Block diagram of servomechanism with error-rate damping.