# CDS 240, Spring 2016: HW 1

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

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

**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)**