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 9.2
- 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
- Hint: Use $V(x) = \int_0^x (s + \eta(s)) ds$ as a Lyapunov function candidate for the reduced model.