P Control Second Order Type 0 Step response and Bode Plot

Cuthbert Nyack
Applet shows the step response and Bode plot for a type 0 second order system with poles at -a and -b. The continuous system is always stable.
The closed loop transfer function is shown below:-
with the following coefficients:-
The applet shows that the phase for the continuous system remains greater than -180° while that of the sampled system goes below -180° and the sampled system can therefore become unstable.
eg parameters (5.0, 0.5, 0.5, 0.03, 0.5). The sampled and continuous systems response are very similar. The gain margin is ~ 20dB.
eg parameters (K, 0.5, 0.2, 0.15, 1.0). As K is increased, the sampled system becomes oscillatory at ~ 3rad/s( half the sampling frequency is ~21rad/s) for K = 9.4 and unstable for K > 9.4. At ~3rad/s, the magnitude of the gain is 0dB, the phase is -180° and the closed loop magnitude reaches a maximum.
eg parameters (K, 0.1, 0.1, 0.15, 1.0). As K is increased, the sampled system becomes oscillatory at ~ 1.6rad/s for K = 2.7.




Gif image below show how the applet should appear.

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COPYRIGHT 2006 Cuthbert Nyack.