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Unformatted text preview: J.M. Gon¸ calves Fall 2003 CDS 212 - Introduction to Modern Control Homework # 7 Date Given: November 20th, 2003 Date Due: December 4th, 2003 P1. For the continuous-time system ˙ x ( t ) = Ax ( t ) + Bu ( t ) , x ∈ IR n , u ∈ IR m y ( t ) = Cx ( t ) (a) Show that the subspace of states reachable from the origin is the smallest A-invariant subspace that contains the image of B . (b) Give a dual statement for the case of observability, and prove it. P2. Show that ( A, B ) is stabilizable if there exists a positive definite matrix X such that: AX + XA *- BB * < . Given such an X , find a stabilizing state feedback. You are not required to prove it, but the converse of this is also true. P3. In this problem we investigate some invariance properties of controllability. (a) Show that ( A, B ) is controllable iff ( T AT- 1 , T B ) is. (b) Show that ( A, B ) is controllable iff ( A + BK, B ) is. (c) Show that ( A, B ) is controllable iff ( A, BB * ) is. (d) State the equivalent conditions for observability....
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