hw6 - happen that there exists an input taking an initial...

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J.M. Gon¸ calves Fall 2003 CDS 212 - Introduction to Modern Control Homework # 6 Date Given: November 14th, 2003 Date Due: November 21st, 2003, 5pm P1. Let A and B be matrices, not necessarily square, of compatible dimension. Show that: B ( I + AB ) - 1 = ( I + BA ) - 1 B ( I + A ) - 1 = I - A ( I + A ) - 1 P2. Let k A k < 1. Show ( I - A ) - 1 = I + A + A 2 + ··· • k ( I - A ) - 1 k≤ 1+ k A k + k A k 2 + ··· = 1 1 -k A k • k ( I - A ) - 1 k≥ 1 1+ k A k P3. Problem 2.7 from the text book (Dullerud and Paganini). P4. In the proof of lemma 2.8 we used the fact that given two subspaces V 1 and V 2 , then V 1 V 2 if and only if V 1 V 2 . Prove this fact. [Hint: ±rst show that for any subspace V = ( V ) ]. P5. Consider the discrete-time linear system: x ( k + 1) = Ax ( k ) + Bu ( k ) , x IR n , u IR m In principle, it is possible to give a de±nition of controllability very similar to the one given in class for continuous systems. However, some di²erences appear. (a) Show by example that as opposed to the continuous time case, it may
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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 &amp;gt; 0, for all t &amp;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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This note was uploaded on 01/04/2012 for the course CDS 212 taught by Professor Tarraf,d during the Fall '08 term at Caltech.

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