CDS 212, Homework 1, Fall 2010

J. Doyle	Issued: 28 Sep 2010
CDS 212, Fall 2010	Due: 7 Oct 2010

Reading

• DFT, Chapters 1 and 2
• Dullerud and Paganini, Ch 3

Problems

1. [DFT 2.1, page 28
   Suppose that <amsmath>u(t)</amsmath> is a continuous signal whose derivative <amsmath>\dot u(t)</amsmath> is also continuous. Which of the following quantities qualifies as a norm for <amsmath>u</amsmath>:
   1. <amsmath>\textstyle \sup_t |\dot u(t)|</amsmath>
   2. <amsmath>\textstyle |u(0)| + \sup_t |\dot u(t)|</amsmath>
   3. <amsmath>\textstyle \max \{ \sup_t |u(t)|,\, \sup_t |\dot u(t)| \}</amsmath>
   4. <amsmath>\textstyle \sup_t |u(t)| + \sup_t |\dot u(t)|</amsmath>

   Make sure to give a thorough answer (not just yes or no).

2. [DFT 2.4, page 29]
   Let <amsmath>D</amsmath> be a pure time delay of <amsmath>\tau</amsmath> seconds with transfer function <amsmath> \widehat D(s) = e^{-s \tau} </amsmath>. A norm <amsmath>\|\cdot\|</amsmath> on transfer functions is time-delay invariant if for every bounded transfer function <amsmath>\widehat G</amsmath> and every <amsmath>\tau > 0</amsmath> we have <amsmath>

   \| \widehat D \widehat G \| = \| \widehat G \|
   

   </amsmath>

   Determine if the 2-norm and <amsmath>\infty</amsmath>-norm are time-delay invariant.

3. [DFT 2.5, page 30]
   Compute the 1-norm of the impluse response corresponding to the transfer function <amsmath> \frac{1}{\tau s + 1}, \quad \tau > 0 </amsmath>.
4. [DFT 2.7, page 30]
   Derive the <amsmath>\infty</amsmath>-norm to <amsmath>\infty</amsmath>-norm system gain for a stable, proper plant <amsmath>\widehat G</amsmath>. (Hint: write <amsmath>\widehat G = c + \widehat G_1</amsmath> where <amsmath>c</amsmath> is a constant and <amsmath>\widehat G_1</amsmath> is strictly proper.)
5. [DFT 2.8, page 30]
   Let <amsmath>\widehat G</amsmath> be a stable, proper plant (but not necessarily strictly proper).
   1. Show that the <amsmath>\infty</amsmath>-norm of the output <amsmath>y</amsmath> given an input <amsmath>u(t) = \sin(\omega t)</amsmath> is <amsmath>|\widehat G(jw)|</amsmath>.
   2. Show that the 2-norm to 2-norm system gain for <amsmath>\widehat G</amsmath> is <amsmath>\| \widehat G \|_\infty</amsmath> (just as in the strictly proper case).
6. [DFT 2.11, page 30]
   Consider a system with transfer function <amsmath>\widehat G(s) = \frac{s+2}{4s + 1}</amsmath> and input <amsmath>u</amsmath> and output <amsmath>y</amsmath>. Compute <amsmath>

    \| G \|_1 = \sup_{\|u\|_\infty = 1} \| y \|_\infty
   

   </amsmath>

   and find an input which achieves the supremum.

7. [DFT 2.12, page 30]
   For a linear system with input <amsmath>u</amsmath> and output <amsmath>y</amsmath>, prove that <amsmath>

    \sup_{\|u\| \leq 1} \| y \| =
      \sup_{\|u\| = 1} \| y \|
   

   </amsmath>

   where <amsmath>\|\cdot\|</amsmath> is any norm on signals.