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

From Murray Wiki

Jump to navigationJump to search(8 intermediate revisions by 2 users not shown) | |||

Line 1: | Line 1: | ||

− | |||

{{CDS homework | {{CDS homework | ||

| instructor = J. Doyle | | instructor = J. Doyle | ||

− | | course = CDS | + | | course = CDS 212 |

| semester = Fall 2010 | | semester = Fall 2010 | ||

| title = Problem Set #1 | | title = Problem Set #1 | ||

Line 10: | Line 9: | ||

=== Reading === | === Reading === | ||

− | * {{DFT}}, | + | * {{DFT}}, Chapters 1 and 2 |

* Dullerud and Paganini, Ch 3 | * Dullerud and Paganini, Ch 3 | ||

Line 16: | Line 15: | ||

<ol> | <ol> | ||

− | <li>DFT 2.1, page 28<br> | + | <li>[DFT 2.1, page 28<br> |

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 <amsmath>u</amsmath>: | 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> |

− | <li> <amsmath>\textstyle |u(0)| + \sup_t |\dot u(t)|</amsmath> </li> | + | <li> <amsmath>\textstyle |u(0)| + \sup_t |\dot u(t)|</amsmath> </li> |

− | <li> <amsmath>\textstyle \max \{ \sup_t |u(t)|,\, \sup_t |\dot u(t)| \}</amsmath> </li> | + | <li> <amsmath>\textstyle \max \{ \sup_t |u(t)|,\, \sup_t |\dot u(t)| \}</amsmath> </li> |

− | <li> <amsmath>\textstyle \sup_t |u(t)| + \sup_t |\dot u(t)|</amsmath> </li> | + | <li> <amsmath>\textstyle \sup_t |u(t)| + \sup_t |\dot u(t)|</amsmath> </li> |

</ol> | </ol> | ||

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

</li> | </li> | ||

− | <li> DFT 2.4, page 29] <br> | + | <li> [DFT 2.4, page 29] <br> |

Let <amsmath>D</amsmath> be a pure time delay of <amsmath>\tau</amsmath> seconds with transfer function | 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> \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 | every bounded transfer function <amsmath>\widehat G</amsmath> and every <amsmath>\tau > 0</amsmath> we have | ||

− | < | + | <center><amsmath> |

− | + | \| \widehat D \widehat G \| = \| \widehat G \| | |

− | </ | + | </amsmath></center> |

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

</li> | </li> | ||

Line 41: | Line 40: | ||

<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 <amsmath> \frac{1}{\tau s + 1} \ | + | transfer function <amsmath> \frac{1}{\tau s + 1}, \quad \tau > 0 </amsmath>. |

</li> | </li> | ||

− | <li> DFT 2.7, page 30] <br> Derive the <amsmath>\infty</amsmath>-norm to <amsmath>\infty</amsmath>-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 <amsmath>\widehat G</amsmath>. (Hint: write <amsmath>\widehat G = c + \widehat G_1</amsmath> where <amsmath>c</amsmath> 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 <amsmath>\widehat G_1</amsmath> is strictly proper.) | and <amsmath>\widehat G_1</amsmath> is strictly proper.) | ||

</li> | </li> | ||

− | <li> [DFT 2.8, page 30] <br> Let <amsmath>\widehat G</amsmath> 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). | ||

<ol type="a"> | <ol type="a"> | ||

<li> Show that the <amsmath>\infty</amsmath>-norm of the output <amsmath>y</amsmath> given an input | <li> Show that the <amsmath>\infty</amsmath>-norm of the output <amsmath>y</amsmath> given an input | ||

Line 62: | Line 63: | ||

<amsmath>\widehat G(s) = \frac{s+2}{4s + 1}</amsmath> | <amsmath>\widehat G(s) = \frac{s+2}{4s + 1}</amsmath> | ||

and input <amsmath>u</amsmath> and output <amsmath>y</amsmath>. Compute | and input <amsmath>u</amsmath> and output <amsmath>y</amsmath>. Compute | ||

− | < | + | <center><amsmath> |

− | <amsmath>\| G \|_1 = \sup_{\|u\|_\infty = 1} \| y \|_\infty</amsmath> | + | \| G \|_1 = \sup_{\|u\|_\infty = 1} \| y \|_\infty |

− | </ | + | </amsmath></center> |

and find an input which achieves the supremum. | and find an input which achieves the supremum. | ||

</li> | </li> | ||

Line 70: | Line 71: | ||

<li> [DFT 2.12, page 30] <br> | <li> [DFT 2.12, page 30] <br> | ||

For a linear system with input <amsmath>u</amsmath> and output <amsmath>y</amsmath>, prove that | For a linear system with input <amsmath>u</amsmath> and output <amsmath>y</amsmath>, prove that | ||

− | < | + | <center><amsmath> |

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

− | \sup_{\|u\| = 1} \| y \|</amsmath> | + | \sup_{\|u\| = 1} \| y \| |

− | </ | + | </amsmath></center> |

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

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

</li> | </li> |

## Latest revision as of 17:42, 28 September 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

- [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*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.

- [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>. - [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 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).

- [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.