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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: first 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 definition of controllability very similar to the one given in class for continuous systems. However, some differences 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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