Modern Control Engineering

24 Chapter 2 / Mathematical Modeling of Control Systems

From Figure 2–9, we notice that the amplitude of the output oscillation can

be reduced by decreasing the differential gap. The decrease in the differential

gap, however, increases the number of on–off switchings per minute and reduces

the useful life of the component. The magnitude of the differential gap must be

determined from such considerations as the accuracy required and the life of

the component.

Proportional Control Action. For a controller with proportional control action,

the relationship between the output of the controller u(t)and the actuating error signal

e(t)is

or, in Laplace-transformed quantities,

whereKpis termed the proportional gain.

Whatever the actual mechanism may be and whatever the form of the operating

power, the proportional controller is essentially an amplifier with an adjustable gain.

Integral Control Action. In a controller with integral control action, the value of

the controller output u(t)is changed at a rate proportional to the actuating error signal

e(t). That is,

or

whereKiis an adjustable constant. The transfer function of the integral controller is

Proportional-Plus-Integral Control Action. The control action of a proportional-

plus-integral controller is defined by

u(t)=Kp e(t)+

Kp

Ti 3

t

0

e(t)dt

U(s)

E(s)

=

Ki

s

u(t)=Ki

3

t

0

e(t)dt

du(t)

dt

=Ki e(t)

U(s)

E(s)

=Kp

u(t)=Kp e(t)

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