Page 264 - Modern Control Systems

238 Chapter 4 Feedback Control System Characteristics

Then, in terms of F(s), we define the sensitivity function as

Similarly, in terms of the loop gain, we define the complementary sensitivity function as
L{s)
cw (46)
= TTjfe-

In terms of the functions S(s) and C(s), we can write the tracking error as
E(s) = S(s)R{s) - S{s)G{s)T d(s) + C(s)N(s). (4.7)

Examining Equation (4.7), we see that (for a given G(s)), if we want to minimize the
tracking error, we want both S(s) and C(s) to be small. Remember that S(s) and C(s)
are both functions of the controller, G c(s), which the control design engineer must
select. However, the following special relationship between S(s) and C(s) holds

S(s) + C(s) = 1. (4.8)
We cannot simultaneously make S(s) and C(s) small. Obviously, design compromises
must be made.
To analyze the tracking error equation, we need to understand what it means for
a transfer function to be "large" or to be "small." The discussion of magnitude of a
transfer function is the subject of Chapters 8 and 9 on frequency response methods.
However, for our purposes here, we describe the magnitude of the loop gain L(s) by
considering the magnitude |L(/o>)| over the range of frequencies, a>, of interest.
Considering the tracking error in Equation (4.4), it is evident that, for a given
G(s), to reduce the influence of the disturbance, T d(s), on the tracking error, E(s),
we desire L(s) to be large over the range of frequencies that characterize the distur-
bances. That way, the transfer function G(s)/(1 + L(s)) will be small, thereby re-
ducing the influence of T d(s). Since L(s) = G c(s)G(s), this implies that we need to
design the controller G c(s) to have a large magnitude. Conversely, to attenuate the
measurement noise, N(s), and reduce the influence on the tracking error, we desire
L(s) to be small over the range of frequencies that characterize the measurement
noise. The transfer function L(s)/(1 + L(s)) will be small, thereby reducing the in-
fluence of A^(^). Again, since L(s) = G c(s) G(s), that implies that we need to design
the controller G c(s) to have a small magnitude. Fortunately, the apparent conflict
between wanting to make G c(s) large to reject disturbances and the wanting to
make G c(s) small to attenuate measurement noise can be addressed in the design
phase by making the loop gain, L(s), large at low frequencies (generally associated
with the frequency range of disturbances), and making L(s) small at high frequen-
cies (generally associated with measurement noise).
More discussion on disturbance rejection and measurement noise attenuation
follows in the subsequent sections. Next, we discuss how we can use feedback to re-
duce the sensitivity of the system to variations and uncertainty in parameters in the
process, G(s).This is accomplished by analyzing the tracking error in Equation (4.2)
when T d(s) = N(s) = 0.