CDS 212, Homework 4, Fall 2010

J. Doyle	Issued: 19 Oct 2010
CDS 212, Fall 2010	Due: 28 Oct 2010

Reading

• DFT, Chapter 6

Problems

1. [DFT 6.1]
   Show that any stable transfer function can be uniquely factored as the product of an all pass function and a minimum phase function, up to a choice of sign.
2. [DFT 6.4]
   Let <amsmath>

   P(s) = 4\frac{s-2}{(s+1)^2}

   </amsmath>

   and suppose that <amsmath>C</amsmath> is an internally stabilizing controller such that <amsmath>\| S \|_\infty = 1.5.</amsmath> Give a positive lower bound for

   <amsmath>

    \max_{0 \leq \omega \leq 0.1} |S(j\omega)|.
   

   </amsmath>
3. [DFT 6.5]
   Define <amsmath>\epsilon = \|W_1 S\|_\infty</amsmath> and <amsmath>\delta = \| C S \|_\infty</amsmath>, so that <amsmath>\epsilon</amsmath> is a measure of tracking performance and <amsmath>\delta</amsmath> measures control effort. Show that for every point <amsmath>s_0</amsmath> with Re <amsmath>s_0 \geq 0</amsmath>, <amsmath>

    |W_1(s_0)| \leq \epsilon + |W_1 (s_0) P(s_0)|\, \delta.
   

   </amsmath>

   Hence <amsmath>\epsilon</amsmath> and <amsmath>\delta</amsmath> cannot both be very small and so we cannot get good tracking without exerting some control effort.

4. [DFT 6.6]
   Let <amsmath>\omega</amsmath> be a frequency such that <amsmath>j \omega</amsmath> is not a pole of <amsmath>P</amsmath>. Suppose that <amsmath>

    \epsilon := |S(j\omega)| < 1.
   

   </amsmath>

   Derive a lower bound for <amsmath>C(j\omega)</amsmath> that blows up as <amsmath>\epsilon \to 0</amsmath>. Hence good tracking at a particular frequency requires large controller gain at this frequency.

5. [DFT 6.7]
   Consider a plant with transfer function <amsmath>

    P(s) = \frac{1}{s^2 - s + 4}
   

   </amsmath>

   and suppose we want to design an internally stabilizing controller such that

   1. <amsmath>|S(j\omega| \leq \epsilon</amsmath> for <amsmath>0 \leq \omega \leq 0.1</amsmath>
   2. <amsmath>|S(j\omega| \leq 2</amsmath> for <amsmath>0.1 \leq \omega \leq 5</amsmath>
   3. <amsmath>|S(j\omega| \leq 1</amsmath> for <amsmath>5 \leq \omega \leq \infty</amsmath>

   Find a (positive) lower bound on the achievable <amsmath>\epsilon</amsmath>.