Unformatted text preview: happen that there exists an input taking an initial state x into the origin in k time steps, but there does not exist one that does it in k-1. (b) Derive conditions for discrete-time controllability, modifying the def-inition accordingly so the transfer between states should be made in k n time steps. P6. Show that ( A, B ) is controllable if and only if W t is positive denite, i.e., W t &gt; 0, for all t &gt; 0. P7. Let A be stable. Show that ( A, B ) is controllable if and only if the solution P of the Lyapunov equation AP + P A * =-BB * is positive denite. Show that the matrix P is equal to the limit of the gramian W c ( t 1 ), when t 1 ....
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- Fall '08
- WT, time steps, positive definite, continuous time case, P1. Let