0195136047.pdf

16.2 FEEDBACK CONTROL SYSTEMS 787

for which the corresponding time solution is given by

c(t)=K( 1 −et/τ) (16.2.10)

which is clearly unstable, since the response increases without limit as time passes. By placing a
negative feedback pathHaround the direct transfer function, as shown in Figure 16.2.4(b), the
closed-loop transfer function is then

M=

C

R

=

K

1 −pτ

1 +

KH

1 −pτ

=

K

1 −pτ+HK

(16.2.11)

By choosingH=apandaK > τ, it follows that

M=

K

1 −pτ+paK

(16.2.12)

and the corresponding time response of the closed-loop system, when subjected to a unit step,
becomes

cf(t)=K( 1 −e−t/(aK−τ)) (16.2.13)

wherecf(t) denotes response with feedback. The system is now clearly stable withaK > τ. Thus,
the insertion of feedback causes the unstable direct transmission system of Figure 16.2.4(a) to
become stable. Such a technique is often used to stabilize space rockets and vehicles, which are
inherently unstable because of their large length-to-diameter ratios.
Note that the direct transfer function of Equation (16.2.8) has a pole located in the right half
p-plane atp= 1 /τ, whereas the pole of the closed-loop transfer function of Equation (16.2.12)
withaK > τis located in the left halfp-plane.

EFFECT OFFEEDBACK ONDYNAMICRESPONSE ANDBANDWIDTH

Let us consider the block-diagram representation of the open-loop system shown in Figure
16.2.5(a), whose direct transfer function is given by

G=

C

E

=

K

1 +pτ

=

K/τ

p+ 1 /τ

(16.2.14)

corresponding to which, the transient solution of the system is of the form given by

c(t)=Ae−t/τ (16.2.15)

The transient in this system is seen to decay in accordance with a time constant ofτseconds. By
placing a feedback pathHaround the direct transfer function, as shown in Figure 16.2.5(b), the
closed-loop transfer function is then

H

(a) (b)

R E C

−

K

G = 1 + pτ

Note: p ≡

EC K

1 + pτ

d

dt

Figure 16.2.5Block-diagram representation of system.(a)Without feedback.(b)With feedback.