Page 266 - Modern Control Systems

240 Chapter 4 Feedback Control System Characteristics

smaller sensitivity, S(s). The question arises, how do we define sensitivity? Since our
goal is to reduce system sensitivity, it makes sense to formally define the term.
The system sensitivity is defined as the ratio of the percentage change in the sys-
tem transfer function to the percentage change of the process transfer function. The
system transfer function is
Y(s)
T(s) = (4.10)
R(sY

and therefore the sensitivity is defined as
*ns)/T( S)
=
(4.11)
kG(s)/G(sY
In the limit, for small incremental changes, Equation (4.11) becomes

(4.12)

System sensitivity is the ratio of the change in the system transfer function
to the change of a process transfer function (or parameter) for a small
incremental change.

The sensitivity of the open-loop system to changes in the plant G(s) is equal to 1.
The sensitivity of the closed-loop is readily obtained by using Equation (4.12). The
system transfer function of the closed-loop system is
G c(s)G(s)
T(s) =
1 + G c(s)G(s)'

Therefore, the sensitivity of the feedback system is

T arc
S r=^ 2
dG' T (1 + G CG) GG C/(1 + G CG)
or
1
Sf; = (4.13)
1 + G c(s)G(s)'
We find that the sensitivity of the system may be reduced below that of the open-
loop system by increasing L(s) = G c(s)G(s) over the frequency range of interest.
Note that S£ in Equation (4.12) is exactly the same as the sensitive function S(s)
given in Equation (4.5). In fact, these functions are one and the same.
Often, we seek to determine S£, where a is a parameter within the transfer
function of a block G. Using the chain rule, we find that
vT _ cTcG (4.14)