CDS 212, Homework 1, Fall 2010
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 REDIRECT HW draft
J. Doyle  Issued: 28 Sep 2010 
CDS 112, Fall 2010  Due: 7 Oct 2010 
Reading
 DFT, Chapterss 1 and 2
 Dullerud and Paganini, Ch 3
Problems
 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>: <amsmath>\textstyle \sup_t \dot u(t)</amsmath>
 <amsmath>\textstyle u(0) + \sup_t \dot u(t)</amsmath>
 <amsmath>\textstyle \max \{ \sup_t u(t),\, \sup_t \dot u(t) \}</amsmath>
 <amsmath>\textstyle \sup_t u(t) + \sup_t \dot u(t)</amsmath>
Make sure to give a thorough answer (not just yes or no).
 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 <amsmath> \emph {timedelay invariant}</amsmath> if for every bounded transfer function <amsmath>\widehat G</amsmath> and every <amsmath>\tau > 0</amsmath> we have
<amsmath>\textstyle \ \widehat D \widehat G \ = \ \widehat G \ </amsmath>
Determine if the 2norm and <amsmath>\infty</amsmath>norm are timedelay invariant.
 [DFT 2.5, page 30]
Compute the 1norm of the impluse response corresponding to the transfer function <amsmath> \frac{1}{\tau s + 1}, \quad \tau > 0 </amsmath>.  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.)  [DFT 2.8, page 30]
Let <amsmath>\widehat G</amsmath> be a stable, proper plant (but not necessarily strictly proper). 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>.
 Show that the 2norm to 2norm system gain for <amsmath>\widehat G</amsmath> is <amsmath>\ \widehat G \_\infty</amsmath> (just as in the strictly proper case).
 [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.
 [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.
 Consider a second order mechanical system with transfer function

<amsmath> \widehat G(s) = \frac{1}{s^2 + 2 \omega_n \zeta s + \omega_n^2}</amsmath>
(<amsmath>\omega_n</amsmath> is the natural frequence of the system and <amsmath>\zeta</amsmath> is the damping ratio). Setting <amsmath>\omega_n = 1</amsmath>, write a short MATLAB program to generate a plot of the <amsmath>\infty</amsmath>norm as a function of the damping ratio <amsmath>\zeta > 0</amsmath>.