Page 91 - Biosystems Engineering

72 Chapter Two

less than the crossover frequency, the reference signal is well tracked,
and disturbance rejection is good. For frequencies higher than this
crossover frequency, the controlled system can no longer follow the
reference signal, and measurement noise is increasingly filtered out.
The maximal bandwidth that can be reached with a given system is
determined by the available hardware actuation speed or the actua-
tor rate limit a (in units of magnitude per second). Therefore, the
rl
maximal crossover frequency ω in radians per second can be derived
co
from the actuator rate limit and the input step size r as
m
a
ω = K rl (2.93)
co r
m
where K is the gain factor between the actuator control rate and the
system response rate.
The natural frequency of the closed-loop control system also
depends on the lowest frequency zero of the return ratio, L(s). Thus,
both the magnitude crossover frequency and the time constant of the
lowest frequency zero will influence the overall response time of the
system. The magnitude crossover frequency of the system, which
would be normally higher than the frequency of the lowest frequency
zero, will therefore influence the initial response of the system. The
lowest frequency zero will dictate the final response of the system
before the system response settles. If it turns out that the return ratio
has no zero less than the crossover frequency, the system time response
will only depend on the magnitude crossover frequency. However, a
zero placed before (actually, well before) the magnitude crossover fre-
quency is a good design practice because it helps to boost the midfre-
quency gain for disturbance rejection and, increase the phase margin
for stability, as well as increasing the control bandwidth.
A design procedure for the shaping of the return ratio would then
consist of the following steps (Kopasakis 2007):
1. Choose the bandwidth of the controlled system based on the
available hardware actuation speed (input limitation).
2. Choose the midfrequency gain based on the midfrequency
disturbance rejection requirements.
3. Evaluate the phase margin at magnitude crossover. If the phase
margin is not sufficiently large, the design should be adjusted
(e.g., by lowering the midfrequency gain).
4. Compute the desired lower zero frequency by the settling time
requirements of the response.
5. Calculate the gain of the closed-loop gain transfer function
based on the midfrequency gain at the lower frequency zero.
6. Simulate the Bode plots of the return ratio and the closed-loop
system response and make adjustments if necessary.