PD Control Sampled Second Order Type 1 Root Locus
Cuthbert Nyack
This Applet compares 2 methods of obtaining a sampled type 1 second
order system with PD control.
The first method combines the transfer functions of the controller
Gc(s) and system Gs(s) with a sampler and ZOH to obtain a forward
transfer function:-
G(z) = (1 - z-1)Z{Gc(s)Gs(s)/S}
The closed loop transfer function is then:-
Gcs(z) = G(z)/(1 + G(z).
The second method digitises the controller to get D(z) using the back
difference approximation to the derivative and combines the result
with Gs(z) = (1 - z-1)Z{Gs(s)/S} to get
a forward transfer function G(z) = D(z)Gs(z) and a closed loop
transfer function Gd(z) = G(z)/(1 + G(z))
The Applet shows the closed loop pole locations and the step
response of the 2 methods.
eg parameters (5.0, 1.0, 0.1, 0.1, 1.0), (5.0, 1.0, 0.1, 0.04, 0.5). These shows good correspondence
between the 2 approaches.
eg parameters (5.0, 1.0, 1.0, 0.1, 0.5). Here there are significant
differences in the pole locations and the second method reaches
its final value sooner.
eg parameters (15.0, 1.0, 1.0, 0.13, 1.0). Here the first method is
on the verge of becoming unstable at 1/2 the sampling frequency,
while the second has become unstable at close to 1/4 of the
sampling frequency.
eg parameters (K, 1.0, 0.5, 0.3, 1.0). The second method is stable
for K < 9.9, while the first is stable for K < 13.3.
eg parameters (K, 2.0, 0.1, 0.4, 1.0). The second method is stable
for K < 8.1(osc at 0.13ws), while the first is stable for K < 10.8(osc at 0.145ws).
Gif image below show how the applet should appear.
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COPYRIGHT © 2006 Cuthbert Nyack.