Difference between revisions of "CDS 212, Homework 1, Fall 2010"
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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  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  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 {timedelay invariant} if for  
every bounded transfer function <amsmath>\widehat G</amsmath> and every <amsmath>\tau > 0</amsmath> we have  
<ol type="">  
A norm  <amsmath>\textstyle \ \widehat D \widehat G \ = \ \widehat G \ </amsmath>  
every bounded transfer function  </ol>  
\  Determine if the 2norm and <amsmath>\infty</amsmath>norm are timedelay invariant.  
Determine if the 2norm and  
</li>  </li>  
<li> [DFT 2.5, page 30] <br>  <li> [DFT 2.5, page 30] <br>  
Compute the 1norm of the impluse response corresponding to the  Compute the 1norm of the impluse response corresponding to the  
transfer function  transfer function <amsmath> \frac{1}{\tau s + 1} \qquad \tau > 0 </amsmath>.  
\  
</li>  </li>  
<li> DFT 2.7, page 30] <br> Derive the  <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  proper plant <amsmath>\widehat G</amsmath>. (Hint: write <amsmath>\widehat G = c + \widehat G_1</amsmath> where <amsmath>c</amsmath> is a constant  
and  and <amsmath>\widehat G_1</amsmath> is strictly proper.)  
</li>  </li>  
<li> [DFT 2.8, page 30] <br> Let  <li> [DFT 2.8, page 30] <br> Let <amsmath>\widehat G</amsmath> be a stable, proper plant (but not necessarily strictly proper).  
<ol type="a">  
<li> 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>.</li>  
\widehat G \_\infty  <li> 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).</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  
<amsmath>\widehat G(s) = \frac{s+2}{4s + 1}</amsmath>  
and input <amsmath>u</amsmath> and output <amsmath>y</amsmath>. Compute  
<ol>  
and input  <amsmath>\ G \_1 = \sup_{\u\_\infty = 1} \ y \_\infty</amsmath>  
</ol>  
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  For a linear system with input <amsmath>u</amsmath> and output <amsmath>y</amsmath>, prove that  
<ol>  
<amsmath>\sup_{\u\ \leq 1} \ y \ =  
\sup_{\u\ = 1} \ y \  \sup_{\u\ = 1} \ y \</amsmath>  
</ol>  
where  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  
<ol>  
<amsmath> \widehat G(s) = \frac{1}{s^2 + 2 \omega_n \zeta s + \omega_n^2}</amsmath>  
</ol>  
(  (<amsmath>\omega_n</amsmath> is the natural frequence of the system and <amsmath>\zeta</amsmath> is the  
damping ratio). Setting  damping ratio). Setting <amsmath>\omega_n = 1</amsmath>, write a short MATLAB  
program to generate a plot of the  program to generate a plot of the <amsmath>\infty</amsmath>norm as a function of the  
damping ratio  damping ratio <amsmath>\zeta > 0</amsmath>.  
</li>  </li> 
Revision as of 17:54, 18 September 2010
 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 \em {timedelay 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 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} \qquad \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>.