Modern Control Engineering

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222 Chapter 5 / Transient and Steady-State Response Analyses

If this control system is stable—that is, if the roots of the characteristic equation

have negative real parts—then the steady-state error in the response to a unit-step

disturbance torque can be obtained by applying the final-value theorem as follows:

Thus steady-state error to the step disturbance torque can be eliminated if the controller

is of the proportional-plus-integral type.

Note that the integral control action added to the proportional controller has

converted the originally second-order system to a third-order one. Hence the control

system may become unstable for a large value of Kp, since the roots of the characteristic

equation may have positive real parts. (The second-order system is always stable if the

coefficients in the system differential equation are all positive.)

It is important to point out that if the controller were an integral controller, as in

Figure 5–42, then the system always becomes unstable, because the characteristic

equation

will have roots with positive real parts. Such an unstable system cannot be used in

practice.

Note that in the system of Figure 5–41 the proportional control action tends to

stabilize the system, while the integral control action tends to eliminate or reduce steady-

state error in response to various inputs.

Derivative Control Action. Derivative control action, when added to a

proportional controller, provides a means of obtaining a controller with high

sensitivity. An advantage of using derivative control action is that it responds to the

rate of change of the actuating error and can produce a significant correction before

the magnitude of the actuating error becomes too large. Derivative control thus

anticipates the actuating error, initiates an early corrective action, and tends to

increase the stability of the system.

Js^3 +bs^2 +K= 0

= 0

=lim

sS 0

- s^2

Js^3 +bs^2 +Kp s+

Kp

Ti

1

s

ess=lim

sS 0

sE(s)

Js^3 +bs^2 +Kp s+

Kp

Ti

= 0

+

• 
  

++ C

E

D

R= (^0) K T
s
1
s(Js+b)
Figure 5–42
Integral control of a
load element
consisting of moment
of inertia and viscous
friction.
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