0195136047.pdf

786 BASIC CONTROL SYSTEMS

EFFECT OFFEEDBACK ONSENSITIVITY TOPARAMETERCHANGES

A good control system, in general, should be rather insensitive to parameter variations, while it

is still able to follow the command quite responsively. It is apparent from Figure 16.2.1 that in an

open-loop system the gain of the system will respond in a one-to-one fashion to the variation in

G. Let us now investigate what effect feedback has on the sensitivity to parameter variations.

The sensitivitySof a closed-loop system with Equation (16.2.3) to a parameter changepin

the direct transmission functionGis defined as

SpM=

per-unit change in closed-loop transmission

per-unit change in open-loop transmission

=

∂M/M

∂Gp/Gp

(16.2.4)

whereGpdenotes that the derivative is to be taken with respect to the parameterpof the function

G. From Equation (16.2.3) it follows that

∂M=

−HGp∂Gp

( 1 +HGp)^2

+

∂Gp

1 +HGp

=

∂Gp

( 1 +HGp)^2

(16.2.5)

Substituting Equations (16.2.5) and (16.2.3) into Equation (16.2.4), one gets

SpM=

∂M/M

∂Gp/Gp

=

[

∂Gp/( 1 +HGp)^2

][

( 1 +HGp)/Gp

]

∂Gp/Gp

=

1

1 +HGp

(16.2.6)

or

∂M

M

=

(

1

1 +HGp

)

∂Gp

Gp

(16.2.7)

Equation (16.2.6) shows that the sensitivity function can be made arbitrarily small by increasing

HGp, provided that the system remains stable.

EFFECT OFFEEDBACK ONSTABILITY

Let us consider the direct transfer function of the unstable first-order system of Figure 16.2.4(a).

The transfer function is given by

G=

C

E

=

K

1 −pτ

=

−K/τ

p− 1 /τ

(16.2.8)

wherep=d/dtis the differential operator. If a unit-step function is applied as the input quantity

E, the output becomes

C(p)=

−K/τ

p− 1 /τ

E(p)=

−K/τ

p(p− 1 /τ )

(16.2.9)

H

(a) (b)

R E C

−

K

G = 1 − pτ

Note: p ≡

EC K

1 − pτ

d

dt

Figure 16.2.4(a)Block-diagram representation of an unstable first-order system.(b)System of part (a)

modified with a feedback path.