Difference between revisions of "CDS 140b Spring 2014 Homework 4"

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(Created page with "{{CDS homework | instructor = R. Murray, D. MacMartin | course = CDS 140b | semester = Spring 2014 | title = Problem Set #4 | issued = 30 Apr 2014 (Wed) | due = 8 May 20...")
 
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<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 $t \to \infty$?
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Revision as of 03:46, 30 April 2014

R. Murray, D. MacMartin Issued: 30 Apr 2014 (Wed)
CDS 140b, Spring 2014 Due: 8 May 2014 (Thu)

__MATHJAX__

  1. Khalil, Problem 9.2
  2. Khalil, Problem 9.3
  3. Khalil, Problem 9.6
  4. Khalil, Problem 9.17
  5. 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 $t \to \infty$?