Page 429 - Mechanical Engineers' Handbook (Volume 2)

420 Basic Control Systems Design

Therefore, the system described by Fig. 39 is closed-loop stable.
The Nyquist theorem provides a convenient measure of the relative stability of a system.
A measure of the proximity of the plot to the 1 i0 point is given by the angle between
the negative real axis and a line from the origin to the point where the plot crosses the unit
circle (see Fig. 39). The frequency corresponding to this intersection is denoted . This
g
angle is the phase margin (PM) and is positive when measured down from the negative real
axis. The phase margin is the phase at the frequency where the magnitude ratio or ‘‘gain’’
g
of G(i )H(i ) is unity, or 0 decibels (dB). The frequency , the phase crossover frequency,
p
is the frequency at which the phase angle is 180 . The gain margin (GM) is the difference
in decibels between the unity gain condition (0 dB) and the value of G( )H( ) decibels
p
p
at the phase crossover frequency . Thus,
p
Gain margin G( )H( ) (dB) (34)
p p
A system is stable only if the phase and gain margins are both positive.
The phase and gain margins can be illustrated on the Bode plots shown in Fig. 41. The
phase and gain margins can be stated as safety margins in the design specifications. A typical
set of such specifications is as follows:
Gain margin 8 dB and Phase margin 30 (35)

In common design situations, only one of these equalities can be met, and the other margin
is allowed to be greater than its minimum value. It is not desirable to make the margins too
large, because this results in a low gain, which might produce sluggish response and a large
steady-state error. Another commonly used set of specifications is
Gain margin 6 dB and Phase margin 40 (36)
The 6-dB limit corresponds to the quarter amplitude decay response obtained with the gain
settings given by the Ziegler–Nichols ultimate-cycle method (Table 2).

9.2 Systems with Dead-Time Elements
The Nyquist theorem is particularly useful for systems with dead-time elements, especially
when the plant is of an order high enough to make the root-locus method cumbersome. A
delay D in either the manipulated variable or the measurement will result in an open-loop
transfer function of the form
G(s)H(s) e Ds P(s) (37)
Its magnitude and phase angle are
G(i )H(i ) P(i ) e i D P(i ) (38)
∠G(i )H(i )) ∠P(i ) ∠e i D ∠P(i ) D (39)
Thus, the dead time decreases the phase angle proportionally to the frequency , but it does
not change the gain curve. This makes the analysis of its effects easier to accomplish with
the open-loop frequency response plot.

9.3 Open-Loop Design for PID Control
Some general comments can be made about the effects of proportional, integral, and deriv-
ative control actions on the phase and gain margins. P action does not affect the phase curve
at all and thus can be used to raise or lower the open-loop gain curve until the specifications