Page 351 - Modern Control Systems

Section 5.6 The Steady-State Error of Feedback Control Systems 325

Table 5.5 Summary of Steady-State Errors
Number of
Integrations Input
in GJs)G(s), Type Step, r(t) = A, Ramp, At, Parabola,
Number R(s) = A/s A/s 2 A?/2, A/s 3
A
0 tss Infinite Infinite
~ 1 + K p
A
1 Infinite
K,
A
2 0

The steady-state error is infinite for one integration. For two integrations, N = 2,
and we obtain

A
Ccc K (5.31)
KIU/IIP* «

is designated the acceleration error constant. The acceleration error con-
where K a
stant is
2
K a = lim s G c(s)G(s).
s—*0
When the number of integrations equals or exceeds three, then the steady-state
error of the system is zero.
Control systems are often described in terms of their type number and the error
constants, K p, K v, and K a. Definitions for the error constants and the steady-state
error for the three inputs are summarized in Table 5.5. The usefulness of the error
constants can be illustrated by considering a simple example.

EXAMPLE 5.3 Mobile robot steering control
A mobile robot may be designed as an assisting device or servant for a severely dis-
abled person [7]. The steering control system for such a robot can be represented by
the block diagram shown in Figure 5.19. The steering controller is

G c(s) = K, + K 2/s. (5.32)

Controller Vehicle dynamics
Y{s)
FIGURE 5.19 Desired o GAs) K - • Actual
Block diagram of G(s) TS + \ heading angle
steering control heading angle
system for a mobile
robot.