CDS 212, Homework 1, Fall 2010

1. 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

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 $u$:
   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 $D$ be a pure time delay of $\tau$ seconds with transfer function \begin{displaymath} \widehat D(s) = e^{-s \tau}. \end{displaymath} A norm $\|\cdot\|$ on transfer functions is {\em time-delay invariant} if for every bounded transfer function $\widehat G$ and every $\tau > 0$ we have \begin{displaymath} \| \widehat D \widehat G \| = \| \widehat G \| \end{displaymath} Determine if the 2-norm and $\infty$-norm are time-delay invariant.
3. [DFT 2.5, page 30]
   Compute the 1-norm of the impluse response corresponding to the transfer function \begin{displaymath} \fract{1}{\tau s + 1} \qquad \tau > 0.
4. DFT 2.7, page 30]
   Derive the $\infty$-norm to $\infty$-norm system gain for a stable, proper plant $\widehat G$. (Hint: write $\widehat G = c + \widehat G_1$ where $c$ is a constant and $\widehat G_1$ is strictly proper.)
5. [DFT 2.8, page 30]
   Let $\widehat G$ be a stable, proper plant (but not necessarily strictly proper).
   1. Show that the $\infty$-norm of the output $y$ given an input
   $u(t) = \sin(\omega t)$ is $|\widehat G(jw)|$.
   1. Show that the 2-norm to 2-norm system gain for $\widehat G$ is $\|
   \widehat G \|_\infty$ (just as in the strictly proper case).
6. [DFT 2.11, page 30]
   Consider a system with transfer function \begin{displaymath} \widehat G(s) = \fract{s+2}{4s + 1} \end{displaymath} and input $u$ and output $y$. Compute \begin{displaymath} \| G \|_1 = \sup_{\|u\|_\infty = 1} \| y \|_\infty \end{displaymath} and find an input which achieves the supremum.
7. [DFT 2.12, page 30]
   For a linear system with input $u$ and output $y$, prove that \begin{displaymath} \sup_{\|u\| \leq 1} \| y \| = \sup_{\|u\| = 1} \| y \| \end{displaymath} where $\|\cdot\|$ is any norm on signals.
8. Consider a second order mechanical system with transfer function \begin{displaymath} \widehat G(s) = \fract{1}{s^2 + 2 \omega_n \zeta s + \omega_n^2} \end{displaymath} ($\omega_n$ is the natural frequence of the system and $\zeta$ is the damping ratio). Setting $\omega_n = 1$, write a short MATLAB program to generate a plot of the $\infty$-norm as a function of the damping ratio $\zeta > 0$.