System Control - Proportional Gain and a Delay in Series

[Logio]

Hi,

Just wondering if anyone could give me a few pointers in the right direction for this... thanks!

Homework Statement

Incredibly long problem statement so I'll summarise and ask more about the concept:
I have a Plant and Controller set up in the form of a usual control system. The controller is a proportional gain in series with a time delay.
a) What is the Laplace transform of this controller.
b) Draw a bode plot (using a computer package)
c) Estimate the phase margin
d) How much extra time delay can be added to the control loop before instability.
e) Using the bode plot, draw a Nyqiust diagram.

Homework Equations

N/A

The Attempt at a Solution

a) Transfer function = k exp(-iD) where D is the time delay in seconds
b) Done on a computer
c) Phase margin read off the graph as the phase between -180 and the phase for the frequency with 0db gain (fairly confident this is accurate)... ANSWER = 75 degrees (for a given k and D)
d) Not sure where to start... I suspect I could consider another series delay of duration say, P, and then use some form of analysis to find P.
e) I know the phase margin and gain margin from the bode plot... is this enough to plot the Nyq. diagram or can I get more information simply?

Many thanks for any pointers on this!

 

[swraman]

the Nyquist diagram really only needs the BODE plot.

what you are trying to do is plugging in values of s into the open loop transfer function corresponding to a large semicircle surounding the positive right half plane.

What this essentially ends up being is starting at low frequency s=0 and increasing s intil you get to very high frequencies, plot the vector represented by G(s).

that is, start at w=0. look at the magnitude of the open loop transfer function at (or close to) w=0 (call it A). now look at the phase of the open loop TF at (or close to) 0 (call it theta). now on your nyquist diagram, mark the point represented by A*exp(i*theta).

do this again for higher and higher frequency values. they should converge to a single point eventually.

that represents half the graph, the other half (representing negative values of s) is the mirror image of that across the real axis.

also, the MATLAB command nyquist(tf) is useful.

 

[Logio]

Thanks for that - it makes sense.

If I have a phase margin of say X degrees, then X degrees of phase lag can be added before instability. How can I process this to calculate the amount of extra time delay the system could have before instability?

Thanks