Page 408 - Mechanical Engineers' Handbook (Volume 2)

5 Control Laws 399

continues to move upward until the tension in the feedback spring counteracts the force
produced by the electromagnetic actuator, thus returning the capsule to its equilibrium po-
sition.
A decrease in the dc input signal causes the opposite actions to occur, and the valve
moves downward.

5 CONTROL LAWS
The control logic elements are designed to act on the error signal to produce the control
signal. The algorithm that is used for this purpose is called the control law, the control
action, or the control algorithm. A nonzero error signal results from either a change in
command or a disturbance. The general function of the controller is to keep the controlled
variable near its desired value when these occur. More specifically, the control objectives
might be stated as follows:

1. Minimize the steady-state error.
2. Minimize the settling time.
3. Achieve other transient specifications, such as minimizing the overshoot.
In practice, the design specifications for a controller are more detailed. For example, the
bandwidth might also be specified along with a safety margin for stability. We never know
the numerical values of the system’s parameters with true certainty, and some controller
designs can be more sensitive to such parameter uncertainties than other designs. So a pa-
rameter sensitivity specification might also be included.
The following control laws form the basis of most control systems.

5.1 Proportional Control
Two-position control is the most familiar type, perhaps because of its use in home thermo-
stats. The control output takes on one of two values. With the on–off controller, the controller
output is either on or off (e.g., fully open or fully closed). Two-position control is acceptable
for many applications in which the requirements are not too severe. However, many situations
require finer control.
Consider a liquid-level system in which the input flow rate is controlled by a valve. We
might try setting the control valve manually to achieve a flow rate that balances the system
at the desired level. We might then add a controller that adjusts this setting in proportion to
the deviation of the level from the desired value. This is proportional control, the algorithm
in which the change in the control signal is proportional to the error. Block diagrams for
controllers are often drawn in terms of the deviations from a zero-error equilibrium condition.
Applying this convention to the general terminology of Fig. 6, we see that proportional
control is described by
F(s) KE(s)
P
where F(s) is the deviation in the control signal and K is the proportional gain. If the total
P
valve displacement is y(t) and the manually created displacement is x, then
y(t) Ke(t) x
P
The percent change in error needed to move the valve full scale is the proportional band.
It is related to the gain as