Page 92 - Biosystems Engineering

Biosystems Analysis and Optimization 73

Once the return ratio L(s) has been designed, the transfer function of
the controller C(s) can be derived from this.

2.5.4 Controller Design Example: Depth Control
for Slurry Injection
The stability analysis and design principles that have been described in
the previous sections will now be applied to the design of the depth
control system for slurry injection for which the model has been identi-
fied in Sec. 2.3.4. The estimated transfer function was expressed as V/V,
but we would like to know the measured depth in centimeters, there-
fore Eq. (2.79) is multiplied by a gain factor of 5 cm/V to get
2
Gs () = Ys () = 11 .7 s + 30 .9 s + 3315 (2.94)
2
3
4
Xs () s + 15 .1 s + 139s + 1560s
9
The closed-loop gain L(s) for this system can now be designed through
the stepwise procedure described in the previous section. This can be
done quite easily using the SISO design toolbox in MATLAB (The
Mathworks, Natick, Massachusetts). The design requirements are the
following:
1. The maximal actuator speed is determined by the maximal oil
flow that can be generated through the proportional hydraulic
valve and sent to the hydraulic actuators. The actuator rate
limit a for this system was found to be 5 cm/s. If we consider
rl
an input step size of 1 cm, the magnitude crossover frequency
ω can be calculated using Eq. (2.93) to be 5 rad/s.
co
2. The desired disturbance attenuation rate at 0.5 rad/s is a fac-
tor of 10, which means that the desired magnitude of the
closed-loop gain is 20 log (10) = 20 dB.
10
3. An integrator action is desired to avoid steady-state error.
4. High-frequency noise and unmodeled dynamics should be
sufficiently filtered.
5. The phase margin should be at least 45° at crossover and the
gain margin at least 10 dB.
Because of the integrator action, which is desired to avoid steady-state
error, the bode plot decreases with 20 dB per decade. This makes that
the disturbance attenuation requirement of 20 dB at 0.5 rad/s cannot
be met without introducing a low-frequency zero when the crossover
frequency is restricted to the actuator rate limit of 5 rad/s. Therefore,
a single zero with a natural frequency of 0.1 rad/s is introduced to
keep the midfrequency gain of the closed-loop gain more than 20 dB
in the region around 0.5 rad/s. Introducing this zero causes the cross-
over frequency to move to a higher frequency, which violates design
requirement (1). Because lowering the gain is not an option, where