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

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=== Problems === | === Problems === | ||

+ | |||

+ | <ol> | ||

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

+ | Suppose that $u(t)$ is a continuous signal whose derivative $\dot | ||

+ | u(t)$ is also continuous. Which of the following quantities qualifies | ||

+ | as a norm for $u$: | ||

+ | * $\sup_t |\dot u(t)|$ | ||

+ | * $|u(0)| + \sup_t |\dot u(t)|$ | ||

+ | * $\max \{ \sup_t |u(t)|,\, \sup_t |\dot u(t)| \}$ | ||

+ | * $\sup_t |u(t)| + \sup_t |\dot u(t)|$ | ||

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

+ | </li> | ||

+ | |||

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

+ | 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. | ||

+ | </li> | ||

+ | |||

+ | <li> [DFT 2.5, page 30] <br> | ||

+ | Compute the 1-norm of the impluse response corresponding to the | ||

+ | transfer function | ||

+ | \begin{displaymath} | ||

+ | \fract{1}{\tau s + 1} \qquad \tau > 0. | ||

+ | </li> | ||

+ | |||

+ | <li> DFT 2.7, page 30] <br> 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.) | ||

+ | </li> | ||

+ | |||

+ | <li> [DFT 2.8, page 30] <br> Let $\widehat G$ be a stable, proper plant (but not necessarily strictly proper). | ||

+ | # Show that the $\infty$-norm of the output $y$ given an input | ||

+ | $u(t) = \sin(\omega t)$ is $|\widehat G(jw)|$. | ||

+ | # Show that the 2-norm to 2-norm system gain for $\widehat G$ is $\| | ||

+ | \widehat G \|_\infty$ (just as in the strictly proper case). | ||

+ | </li> | ||

+ | |||

+ | <li>[DFT 2.11, page 30] <br> | ||

+ | 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. | ||

+ | </li> | ||

+ | |||

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

+ | 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. | ||

+ | </li> | ||

+ | |||

+ | <li> | ||

+ | 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 $\zeta > 0$. | ||

+ | </li> |

## Revision as of 16:28, 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 $u(t)$ is a continuous signal whose derivative $\dot u(t)$ is also continuous. Which of the following quantities qualifies as a norm for $u$:- $\sup_t |\dot u(t)|$
- $|u(0)| + \sup_t |\dot u(t)|$
- $\max \{ \sup_t |u(t)|,\, \sup_t |\dot u(t)| \}$
- $\sup_t |u(t)| + \sup_t |\dot u(t)|$

- 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. - [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. </li> <li> DFT 2.7, page 30] <br> 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.) </li> <li> [DFT 2.8, page 30] <br> Let $\widehat G$ be a stable, proper plant (but not necessarily strictly proper). # Show that the $\infty$-norm of the output $y$ given an input $u(t) = \sin(\omega t)$ is $|\widehat G(jw)|$. # Show that the 2-norm to 2-norm system gain for $\widehat G$ is $\| \widehat G \|_\infty$ (just as in the strictly proper case). </li> <li>[DFT 2.11, page 30] <br> 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. - [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. - 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 $\zeta > 0$.