Page 90 - Biosystems Engineering

Biosystems Analysis and Optimization 71

This can be reformulated to obtain design criteria for the con-
troller C(s):
| (Cjω )|<< | + G (jω ) (Cjω ) (jω )|≤ + | (jω ) (Cjω )H(jω )|
H
1
(
1
G
≤+1 |(Gjω )||(jω )||(jω )| (2.92)
C
H
Equation (2.92) can be interpreted as follows. As long as | (Gjω )|
is not much larger than 1, | ( )| must be as small as possible to
Cs
keep U(s) small in that frequency band. For most physical sys-
tems, this is at high frequencies, as they become insensitive to
high-frequency disturbances (think about a tractor crossing a
field, small ripples do not induce tractor vibrations any more,
whereas large clods and field undulations do cause tractor
vibrations and movements). At low frequencies, where |(Gjω )|
is normally large, a high gain for |( )|Cs is allowed.
Thus, we may state that to avoid excitation of unmodeled high-
frequency dynamics of the system, U(s) must be kept as small as pos-
sible at high frequencies. The phenomenon where high-frequency
modes are excited by the feedback control system so that they desta-
bilize the feedback system is called control spillover.
Design rules (1) and (2) are conflicting so a trade-off has to be made
between disturbance rejection [suppression of the negative influence of
W(s) on the performance] and noise filtering [suppression of sensor noise
V(s) and unmodeled high-frequency modes in order to keep the feedback
system stable]. However, in most practical systems, W(s) is dominated by
low-frequency signals (low-frequency spectrum), whereas V(s) is domi-
nated by high-frequency signals (noise). Therefore, design rule (1) must be
satisfied at low frequencies and design rule (2) at high frequencies.
Design rules (2) and (3) are conflicting too, so that there also exists
a trade-off between suppression of sensor noise and unmodeled high-
frequency modes and the improvement of reference input tracking.
However, R(s) typically only has a low-frequency spectrum so that (3)
must only be satisfied at low frequencies.
Kopasakis (2007) defines a desired shape for the loop gain by
combining these design rules, and making a trade-off between them.
This desired return ratio starts with a pole at the origin for high-loop
gain and zero steady-state error, followed by a zero to maintain a
relatively high gain at the midfrequency range for disturbance atten-
uation and reference tracking, followed by a pole to attenuate the
gain at higher frequencies to avoid exceeding the available actuator
rate (input limitation) and to retain sufficient noise filtering. Such a
design would also result in adequate stability margins with a phase
margin of more than 90° and an infinite gain margin because the
phase does not cross 180°. The trade-off between design criteria will
then be made by the placement of the magnitude crossover frequency,
which defines the bandwidth of the controlled system. For frequencies