Page 352 - Modern Control Systems

326 Chapter 5 The Performance of Feedback Control Systems
Therefore, the steady-state error of the system for a step input when K 2 = 0 and
G e(s) = K x is

(5.33)
Cc c
i + K;
= When K 2 is greater than zero, we have a type-1 system,
where K p KK V
K xs + K 2
G e(s) =

and the steady-state error is zero for a step input.
If the steering command is a ramp input, the steady-state error is

e« = (5.34)
K n
where

= lim sG c(s)G(s) K 7K.
K v
s—»0
The transient response of the vehicle to a triangular wave input when
G c(s) = (Kis + K 2)/s is shown in Figure 5.20. The transient response clearly shows
the effect of the steady-state error, which may not be objectionable if K v is suffi-
ciently large. Note that the output attains the desired velocity as required by the
input, but it exhibits a steady-state error. •

The control system's error constants, K p, K m and K a, describe the ability of a
system to reduce or eliminate the steady-state error. Therefore, they are utilized as
numerical measures of the steady-state performance. The designer determines the
error constants for a given system and attempts to determine methods of increasing
the error constants while maintaining an acceptable transient response. In the case
of the steering control system, we want to increase the gain factor KK 2 in order
to increase K v and reduce the steady-state error. However, an increase in KK 2
results in an attendant decrease in the system's damping ratio I and therefore a

FIGURE 5.20
Triangular wave
response.