View source for CDS 240, Spring 2016: HW 1

{{CDS homework
 | instructor = R. Murray, J. Doyle
 | course = CDS 240
 | semester = Spring 2016
 | title = Problem Set #1
 | issued = 31 Mar 2016 (Thu)
 | due = 11 Apr 2016 (Mon)
}} __MATHJAX__

<ol>
<li>'''Khalil, Problem 9.6'''</li>
<li>'''Khalil, Problem 9.17'''</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>
<li>'''Khalil, Problem 10.11'''</li>
* Hint: first analyze the portion of the system without forcing
<li>'''Khalil, Problem 10.12'''</li>
<li>'''Khalil, Problem 11.22'''</li>
* Hint: Use $V(x) = \int_0^x (s + \eta(s)) ds$ as a Lyapunov function candidate for the reduced model.
<li>'''Khalil, Problem 11.25, part (a)'''</li>
</ol>