The nominal description of a SISO plant in a unity feedback system with a constant gain K is
given by:
G=a/(s-a)
The actual plant Gs
contains time delay uncertainty which can be described as the following
multiplicative error:
Em=(pi/30)s/((pi/60)s+1)
.
a. Calculate the nominal loop (LN(s)), sensitivity (SN(s)), and closed-loop (CN(s)) transfer
functions of the feedback system. What is the range of K to stabilize the nominal system?
Perhaps you have already done this, but the first place to start is to examine the characteristic equation of the closed-loop system: determine the values of k that will keep the roots o this equation in the left half of the complex plane. That is the range of values you are looking for.
For a unity feedback system, the LN(s), sensitivity, SN(s) and closed-loop CN(s) transfer functions are normally derived and defined in most classical control and feedback books, so this question is simply straightforward, just plug the given to the formulas.
I assume that Gs = G and that Em is in series with this plant, so your open loop transfer function should be GOL = G*Em. From then, you can work it out.
You can use root locus or Routh-Hurwitz criterion to solve for the values of K that will make your unity feedback system stable.
You can refer to this site for review.
http://www.cds.caltech.edu/~murray/courses/cds101/fa02/caltech/astrom-ch5.pdf
Note: I think this is part of your problem set so I rather not go into the details of solution. :)
The book A "Unified Algebraic Approach to Linear control Design" published by Springer 1997 contains the set of all stabilizing output feed back control gains.
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