Difference between revisions of "CDS 240, Spring 2016: HW 1"
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{{CDS homework | {{CDS homework | ||
| instructor = R. Murray, | | instructor = R. Murray, J. Doyle | ||
| course = CDS | | course = CDS 240 | ||
| semester = Spring | | semester = Spring 2016 | ||
| title = Problem Set # | | title = Problem Set #1 | ||
| issued = | | issued = 31 Mar 2016 (Thu) | ||
| due = | | due = 11 Apr 2016 (Mon) | ||
}} __MATHJAX__ | }} __MATHJAX__ | ||
<ol> | <ol> | ||
<li>'''Khalil, Problem 9.6'''</li> | <li>'''Khalil, Problem 9.6'''</li> | ||
<li>'''Khalil, Problem 9.17'''</li> | <li>'''Khalil, Problem 9.17'''</li> | ||
Latest revision as of 00:38, 18 April 2016
| R. Murray, J. Doyle | Issued: 31 Mar 2016 (Thu) |
| CDS 240, Spring 2016 | Due: 11 Apr 2016 (Mon) |
__MATHJAX__
- 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 $t \to \infty$?
- Khalil, Problem 10.11
- Hint: first analyze the portion of the system without forcing
- Khalil, Problem 10.12
- Khalil, Problem 11.22
- Hint: Use $V(x) = \int_0^x (s + \eta(s)) ds$ as a Lyapunov function candidate for the reduced model.
- Khalil, Problem 11.25, part (a)
