Page 142 - Mechatronics with Experiments

128 MECHATRONICS
T
u(z) 2 .a(1 + z)
=
e(z) (z − 1) + T.a (1 + z)
2
a
= (2.228)
2 z−1
+ a
T z+1
The equivalent substitution relationship between s and z is
a a
⟷ (2.229)
s + a 2 z−1 + a
T z+1
2 z − 1
s ⟷ (2.230)
T z + 1
A summary of finite difference based digital appoximation of analog filters is given
below.
method approximation
FWD − rule s ⟵ z − 1
T
z − 1 (2.231)
BWD − rule s ⟵
Tz
2 z − 1
Trapezoidal − rule s ⟵
T z + 1

2.14 PROBLEMS

1. Consider a time domain signal y(t) = 1.0 ⋅ sin(2 ⋅ 10t) which is periodic. Select proper sampling
frequencies to illustrate the following sampling effects.
1. A sampling period that is fast enough for accurate sampling and does not violate the sampling
theorem.
2. A sampling period that illustrates aliasing problem as a result of sampling.
3. A sampling period that illustrates the beat phenomenon as a result of sampling.
4. A sampling period that illustrates the hidden oscillations problem as a result of sampling.
Explain your time domain results with the frequency content of the sampled signals. Plot the original
signal and sample signals in the time domain. Plot the magnitude component of the Fourier series of
the original signal and the Fourier transform of the sampled signals as a function of frequency.
2. Consider a mass-force system. Let m = 5 kg. A controller decides on the force using a PD
controller on position error. Select the PD controller gains such that the step response of the closed
loop system has no more than 5% overshoot, and the settling time is less than 2.0 s. Confirm your
®
®
results with a Simulink or MATLAB simulation.
3. What is the effect of switching the sign on the controller, that is u(s) = (K + K s)(x(s) − x cmd (s))
d
p
(this can easily happen in practice by swapping the signal input lines to the analog PD controller
between the command signal and the sensor signal) or by switching the polarity of the controller
output (sign change once in the controller)? What happens if we switch the polarity of both the
command signal and sensor signal as well as the output polarity of the controller (sign change twice
in the controller)? Use a simple mass-force system model under a PD control algorithm to simulate
your analysis and present results.
4. Consider a mass-force system. Let m = 5 kg. A controller decides on the force using a PID type
controller on velocity error. Let velocity commands be r(t) = 10 ⋅ t. It is assumed that the feedback
sensor provides the velocity measurement of the mass. The controller acts on the velocity error.
1. What is the steady-state error when a P-type controller is used?