# 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__ | ||

## Revision as of 14:25, 31 March 2016

R. Murray, J. Doyle | Issued: 31 Mar 2016 (Thu) |

CDS 240, Spring 2016 | Due: 11 Apr 2016 (Mon) |

__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 9.6****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)**