aa
Section 5–7 / Effects of Integral and Derivative Control Actions on System Performance 223
+
R(s) C(s)
(a)
(b)
Kp^1
Js^2
c(t)
1
0 t
Figure 5–43
(a) Proportional
control of a system
with inertia load;
(b) response to a
unit-step input.
Although derivative control does not affect the steady-state error directly, it adds
damping to the system and thus permits the use of a larger value of the gain K, which
will result in an improvement in the steady-state accuracy.
Because derivative control operates on the rate of change of the actuating error and
not the actuating error itself, this mode is never used alone. It is always used in combi-
nation with proportional or proportional-plus-integral control action.
Proportional Control of Systems with Inertia Load. Before we discuss further
the effect of derivative control action on system performance, we shall consider the
proportional control of an inertia load.
Consider the system shown in Figure 5–43(a). The closed-loop transfer function is
obtained as
Since the roots of the characteristic equation
are imaginary, the response to a unit-step input continues to oscillate indefinitely, as
shown in Figure 5–43(b).
Control systems exhibiting such response characteristics are not desirable. We shall
see that the addition of derivative control will stabilize the system.
Proportional-Plus-Derivative Control of a System with Inertia Load. Let us
modify the proportional controller to a proportional-plus-derivative controller whose
transfer function is The torque developed by the controller is proportional
to Derivative control is essentially anticipatory, measures the instantaneous
error velocity, and predicts the large overshoot ahead of time and produces an
appropriate counteraction before too large an overshoot occurs.
KpAe+Td e
B.
KpA 1 +Td sB.
Js^2 +Kp= 0
C(s)
R(s)
=
Kp
Js^2 +Kp
