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Unformatted text preview: EEL5669 Autonomous Robotic Systems Homework #1 January 19, 2010 Problem 1. Consider the following linear system: ˙ x = Ax + Bu = "- 1 1 1- 1 # x + " 1 2 # u. • Find eigenvalues and eigenvectors of A . • Find e At . • Is the system Lyapunov stable or asymptotically stable? Problem 2. Consider the following discrete time-varying system: x k +1 = D k x k , where E = " c 1 c 2 0 0 . 1 # , F = " . 9 0 c 2 c 3 # , and D k = ( E, if k is even F, if k is odd . • Find a set values of c i so that matrices E and F are stable but matrix ( EF ) is not. • Find a set values of c i so that matrices E and F are unstable but matrix ( EF ) is stable. • What are the implications of these choices of D k on stability of time-varying discrete systems (or piecewise-constant continuous systems)? Problem 3. Reconsider the system in Problem 1. • Find the control and state transformations so that the system is mapped into the controllable canonical form....
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- Spring '10
- Autonomous Robotic Systems, Robotic Systems Homework