# 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__

**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 $t \to \infty$?