# CDS 140b Spring 2014 Homework 4

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

__MATHJAX__

1. Khalil, Problem 9.2
• Hint: Section 8.2 of Khalil gives information on how to find the upper bound for the region of attraction
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 $t \to \infty$?