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 k1. (b) Derive conditions for discretetime controllability, modifying the definition 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
 Tarraf,D
 WT, time steps, positive definite, continuous time case, P1. Let

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