Modern Control Engineering

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224 Chapter 5 / Transient and Steady-State Response Analyses

Consider the system shown in Figure 5–44(a). The closed-loop transfer function is

given by

The characteristic equation

now has two roots with negative real parts for positive values of J,Kp, and Thus

derivative control introduces a damping effect. A typical response curve c(t)to a unit-

step input is shown in Figure 5–44(b). Clearly, the response curve shows a marked

improvement over the original response curve shown in Figure 5–46(b).

Proportional-Plus-Derivative Control of Second-Order Systems. A compromise

between acceptable transient-response behavior and acceptable steady-state behavior may

be achieved by use of proportional-plus-derivative control action.

Consider the system shown in Figure 5–45. The closed-loop transfer function is

The steady-state error for a unit-ramp input is

The characteristic equation is

Js^2 +AB+KdBs+Kp= 0

ess=

B

Kp

C(s)

R(s)

=

Kp+Kd s

Js^2 +AB+KdBs+Kp

Td.

Js^2 +Kp Td s+Kp= 0

C(s)

R(s)

=

KpA 1 +Td sB

Js^2 +Kp Td s+Kp

+–

R(s) C(s)

Kp+Kds s(Js^1 +B)

Figure 5–44

(a) Proportional-plus-derivative control of a system with inertia load; (b) response to a unit-step input.

R(s) C(s)

(a) (b)

Kp(1+Tds)

c(t)

1

(^0) t
1
Js^2
+–
Figure 5–45
Control system.
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