CDS 140b Spring 2014 Homework 4: Difference between revisions
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<li>'''Khalil, Problem 9.2'''</li> | <li>'''Khalil, Problem 9.2''' | ||
* Hint: Section 8.2 of Khalil gives information on how to find the upper bound for the region of attraction | |||
</li> | |||
<li>'''Khalil, Problem 9.3'''</li> | <li>'''Khalil, Problem 9.3'''</li> | ||
<li>'''Khalil, Problem 9.6'''</li> | <li>'''Khalil, Problem 9.6'''</li> | ||
<li>'''Khalil, Problem 9.17'''</li> | <li>'''Khalil, Problem 9.17'''</li> | ||
<li>'''Khalil, Problem 9.29'''</li> | <li>'''Khalil, Problem 9.29''' | ||
* For part b, let $\|\dot r(t)\| \leq \epsilon$, for all $t \geq 0$. Reason why there exists a Lyapanov function satisfying equations (9.41)-(9.44). Then explain why for some sufficiently small epsilon, solutions are uniformly ultimately bounded to a ball bound the equilibrium point $(\bar x, \bar z)$, with a radius of the ball in proportion to $\epsilon$, and that therefore the norm of the tracking error is smaller than $k \epsilon$ for some $k>0$. Also, what happens to the tracking error when $\dot r(t) \to 0$ as $t \to \infty$? | |||
</li> | |||
</ol> | </ol> | ||
Latest revision as of 15:33, 30 April 2014
| R. Murray, D. MacMartin | Issued: 30 Apr 2014 (Wed) |
| CDS 140b, Spring 2014 | Due: 8 May 2014 (Thu) |
__MATHJAX__
- Khalil, Problem 9.2
- Hint: Section 8.2 of Khalil gives information on how to find the upper bound for the region of attraction
- Khalil, Problem 9.3
- Khalil, Problem 9.6
- Khalil, Problem 9.17
- Khalil, Problem 9.29
- For part b, let $\|\dot r(t)\| \leq \epsilon$, for all $t \geq 0$. Reason why there exists a Lyapanov function satisfying equations (9.41)-(9.44). Then explain why for some sufficiently small epsilon, solutions are uniformly ultimately bounded to a ball bound the equilibrium point $(\bar x, \bar z)$, with a radius of the ball in proportion to $\epsilon$, and that therefore the norm of the tracking error is smaller than $k \epsilon$ for some $k>0$. Also, what happens to the tracking error when $\dot r(t) \to 0$ as $t \to \infty$?
