Page 88 - Biosystems Engineering

Biosystems Analysis and Optimization 69

Therefore, the feedback control system must have some immu-
nity with respect to these unmodeled dynamics. It is clear that the
smaller |(Ljω )| will be (<<1) at the phase crossover (i.e., where
)]
arg[ (Ljω= −180°), the more insensitive the feedback system will be
to unmodeled dynamic phenomena. This means that these dynamic
phenomena may add more gain to | (Ljω )| before instability occurs.
The gain margin (GM) is defined as the amount by which the system
gain can be increased before instability occurs and is normally quoted
in decibels. This can be calculated using the following equation:
⎛ 1 ⎞
GM = 20log ⎜ ⎟ dB (2.86)
⎝ )| c ⎠
10 |(Ljω
where| (Ljω )| is the magnitude of the loop gain L(s) corresponding to
c
a phase lag of 180° (i.e., at the phase crossover). The physical meaning
of gain margin has already been explained. Gain margin is also known
as amplitude margin.
The phase margin (PM) is defined as the amount by which the
open-loop lag falls short of 180° at the frequency where the open-loop
magnitude is unity (i.e., at the gain crossover). It is particularly sig-
nificant when investigating the effect on the stability of system
changes that primarily affect the phase shift arg[ (Ljω )]. The phase
margin also has a physical meaning. A sine wave with amplitude 1
and a frequency equal to the gain crossover frequency, which is
applied to the closed-loop system, will reach the error detector with
an unaltered amplitude of 1 but with a certain shift in phase. If this
phase shift (i.e., phase lag) is 180°, then the returning signal, when
inverted and added to the reference input sine wave, will just rein-
force the signal and the system will be marginally stable. The smaller
the phase margin (more negative), the larger the phase shift caused
by unmodeled dynamic phenomena can be before instability occurs.
This can be calculated using the following equation:

PM = arg[ (Ljω )] −180 ° (2.87)

It should be noted here that the gain of a transfer function is always
larger than or equal to zero.

2.5.3 Loop-Shaping Controller Design
In order to obtain the desired behavior of the feedback control sys-
tem, we can design the frequency domain response (Bode plot) of the
loop gain L(s) to satisfy the control system requirements in terms of
tracking performance, disturbance rejection, and stability. From this
designed loop gain L(s), the desired controller transfer function C(s)
can then be calculated. The reason for shaping the loop gain L(s) is
because the denominator of the closed-loop transfer function G (s),
c