Difference between revisions of "CDS 212, Homework 1, Fall 2010"

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Line 19: Line 19:
 
Suppose that <amsmath>u(t)</amsmath> is a continuous signal whose derivative <amsmath>\dot
 
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
 
u(t)</amsmath> is also continuous.  Which of the following quantities qualifies
as a norm for $u$:
+
as a norm for <amsmath>u</amsmath>:
 
<ol type="a">
 
<ol type="a">
 
<li> <amsmath>\textstyle \sup_t |\dot u(t)|</amsmath></li>
 
<li> <amsmath>\textstyle \sup_t |\dot u(t)|</amsmath></li>
Line 30: Line 30:
  
 
<li> DFT 2.4, page 29] <br>
 
<li> DFT 2.4, page 29] <br>
Let $D$ be a pure time delay of $\tau$ seconds with transfer function
+
Let <amsmath>D</amsmath> be a pure time delay of <amsmath>\tau</amsmath> seconds with transfer function
\begin{displaymath}
+
<amsmath> \widehat D(s) = e^{-s \tau} </amsmath>. A norm <amsmath>\|\cdot\|</amsmath> on transfer functions is \em {time-delay invariant} if for
  \widehat D(s) = e^{-s \tau}.
+
every bounded transfer function <amsmath>\widehat G</amsmath> and every <amsmath>\tau > 0</amsmath> we have
\end{displaymath}
+
<ol type="">
A norm $\|\cdot\|$ on transfer functions is {\em time-delay invariant} if for
+
<amsmath>\textstyle \| \widehat D \widehat G \| = \| \widehat G \| </amsmath>
every bounded transfer function $\widehat G$ and every $\tau > 0$ we have
+
</ol>
\begin{displaymath}
+
Determine if the 2-norm and <amsmath>\infty</amsmath>-norm are time-delay invariant.
  \| \widehat D \widehat G \| = \| \widehat G \|
 
\end{displaymath}
 
Determine if the 2-norm and $\infty$-norm are time-delay invariant.
 
 
</li>
 
</li>
  
 
<li> [DFT 2.5, page 30] <br>
 
<li> [DFT 2.5, page 30] <br>
 
Compute the 1-norm of the impluse response corresponding to the
 
Compute the 1-norm of the impluse response corresponding to the
transfer function
+
transfer function <amsmath> \frac{1}{\tau s + 1} \qquad \tau > 0 </amsmath>.  
\begin{displaymath}
 
  \fract{1}{\tau s + 1} \qquad \tau > 0.
 
 
</li>
 
</li>
  
<li> DFT 2.7, page 30] <br> Derive the $\infty$-norm to $\infty$-norm system gain for a stable,
+
<li> DFT 2.7, page 30] <br> Derive the <amsmath>\infty</amsmath>-norm to <amsmath>\infty</amsmath>-norm system gain for a stable,
proper plant $\widehat G$.  (Hint: write $\widehat G = c + \widehat G_1$ where $c$ is a constant
+
proper plant <amsmath>\widehat G</amsmath>.  (Hint: write <amsmath>\widehat G = c + \widehat G_1</amsmath> where <amsmath>c</amsmath> is a constant
and $\widehat G_1$ is strictly proper.)
+
and <amsmath>\widehat G_1</amsmath> is strictly proper.)
 
</li>
 
</li>
  
<li> [DFT 2.8, page 30] <br> Let $\widehat G$ be a stable, proper plant (but not necessarily strictly proper).
+
<li> [DFT 2.8, page 30] <br> Let <amsmath>\widehat G</amsmath> be a stable, proper plant (but not necessarily strictly proper).
# Show that the $\infty$-norm of the output $y$ given an input
+
<ol type="a">
   $u(t) = \sin(\omega t)$ is $|\widehat G(jw)|$.
+
<li> Show that the <amsmath>\infty</amsmath>-norm of the output <amsmath>y</amsmath> given an input
# Show that the 2-norm to 2-norm system gain for $\widehat G$ is $\|
+
   <amsmath>u(t) = \sin(\omega t)</amsmath> is <amsmath>|\widehat G(jw)|</amsmath>.</li>
   \widehat G \|_\infty$ (just as in the strictly proper case).
+
<li>  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).</li>
 +
</ol>
 
</li>
 
</li>
  
 
<li>[DFT 2.11, page 30] <br>
 
<li>[DFT 2.11, page 30] <br>
 
Consider a system with transfer function
 
Consider a system with transfer function
\begin{displaymath}
+
<amsmath>\widehat G(s) = \frac{s+2}{4s + 1}</amsmath>
  \widehat G(s) = \fract{s+2}{4s + 1}
+
and input <amsmath>u</amsmath> and output <amsmath>y</amsmath>.  Compute
\end{displaymath}
+
<ol>
and input $u$ and output $y$.  Compute
+
<amsmath>\| G \|_1 = \sup_{\|u\|_\infty = 1} \| y \|_\infty</amsmath>
\begin{displaymath}
+
</ol>
  \| G \|_1 = \sup_{\|u\|_\infty = 1} \| y \|_\infty
 
\end{displaymath}
 
 
and find an input which achieves the supremum.
 
and find an input which achieves the supremum.
 
</li>
 
</li>
  
 
<li> [DFT 2.12, page 30] <br>
 
<li> [DFT 2.12, page 30] <br>
For a linear system with input $u$ and output $y$, prove that
+
For a linear system with input <amsmath>u</amsmath> and output <amsmath>y</amsmath>, prove that
\begin{displaymath}
+
<ol>
    \sup_{\|u\| \leq 1} \| y \| =
+
  <amsmath>\sup_{\|u\| \leq 1} \| y \| =
     \sup_{\|u\| = 1} \| y \|
+
     \sup_{\|u\| = 1} \| y \|</amsmath>
\end{displaymath}
+
</ol>
where $\|\cdot\|$ is any norm on signals.
+
where <amsmath>\|\cdot\|</amsmath> is any norm on signals.
 
</li>
 
</li>
  
 
<li>
 
<li>
 
Consider a second order mechanical system with transfer function
 
Consider a second order mechanical system with transfer function
\begin{displaymath}
+
<ol>
  \widehat G(s) = \fract{1}{s^2 + 2 \omega_n \zeta s + \omega_n^2}
+
<amsmath>  \widehat G(s) = \frac{1}{s^2 + 2 \omega_n \zeta s + \omega_n^2}</amsmath>
\end{displaymath}
+
</ol>
($\omega_n$ is the natural frequence of the system and $\zeta$ is the
+
(<amsmath>\omega_n</amsmath> is the natural frequence of the system and <amsmath>\zeta</amsmath> is the
damping ratio).  Setting $\omega_n = 1$, write a short MATLAB
+
damping ratio).  Setting <amsmath>\omega_n = 1</amsmath>, write a short MATLAB
program to generate a plot of the $\infty$-norm as a function of the
+
program to generate a plot of the <amsmath>\infty</amsmath>-norm as a function of the
damping ratio $\zeta > 0$.
+
damping ratio <amsmath>\zeta > 0</amsmath>.
 
</li>
 
</li>

Revision as of 17:54, 18 September 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 <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 \em {time-delay invariant} 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 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} \qquad \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.

  8. 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>.