# 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. - 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.) - [DFT 2.8, page 30]

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

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

- [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. - [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 $\zeta > 0$.