Page 414 - Mechanical Engineers' Handbook (Volume 2)

6 Controller Hardware 405

implement if signal noise is present. The gains K and K can be chosen to yield the desired
2
1
damping ratio and steady-state deviation, as was done with K and K .
P
I
5.5 PID Control
The position servomechanism design with PI control is not completely satisfactory because
of the difficulties encountered when the damping c is small. This problem can be solved by
the use of the full PID control law, as shown in Fig. 23 with G(s) K K s K /s.
P
D
I
A stable system results if all gains are positive and if (c K )K IK 0. The
D
I
P
presence of K relaxes somewhat the requirement that K be large to achieve stability. The
D
P
steady-state errors are zero, and the transient response can be improved because three of the
coefficients of the characteristic equation can be selected. To make further statements requires
the root-locus technique presented in Section 9.
Proportional, integral, and derivative actions and their various combinations are not the
only control laws possible, but they are the most common. PID controllers will remain for
some time the standard against which any new designs must compete.
The conclusions reached concerning the performance of the various control laws are
strictly true only for the plant model forms considered. These are the first-order model
without numerator dynamics and the second-order model with a root at s 0 and no nu-
merator zeros. The analysis of a control law for any other linear system follows the preceding
pattern. The overall system transfer functions are obtained, and all of the linear system
analysis techniques can be applied to predict the system’s performance. If the performance
is unsatisfactory, a new control law is tried and the process repeated. When this process fails
to achieve an acceptable design, more systematic methods of altering the system’s structure
are needed; they are discussed in later sections. We have used step functions as the test
signals because they are the most common and perhaps represent the severest test of system
performance. Impulse, ramp, and sinusoidal test signals are also employed. The type to use
should be made clear in the design specifications.

6 CONTROLLER HARDWARE
The control law must be implemented by a physical device before the control engineer’s
task is complete. The earliest devices were purely kinematic and were mechanical elements
such as gears, levers, and diaphragms that usually obtained their power from the controlled
variable. Most controllers now are analog electronic, hydraulic, pneumatic, or digital elec-
tronic devices. We now consider the analog type. Digital controllers are covered starting in
Section 10.

6.1 Feedback Compensation and Controller Design
Most controllers that implement versions of the PID algorithm are based on the following
feedback principle. Consider the single-loop system shown in Fig. 1. If the open-loop transfer
function is large enough that G(s)H(s) 1, the closed-loop transfer function is approxi-
mately given by

G(s) G(s) 1
T(s) (24)
1 G(s)H(s) G(s)H(s) H(s)
The principle states that a power unit G(s) can be used with a feedback element H(s)to
create a desired transfer function T(s). The power unit must have a gain high enough that