Automatic Control 1 - Reachability Analysis
Automatic Control 1 - Reachability Analysis
Automatic Control 1 - Reachability Analysis
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Lecture: <strong>Reachability</strong> <strong>Analysis</strong>Canonical reachability decompositionCanonical reachability decompositionPropositionA c ∈ n c×n c and Bc ∈ nc×m are a completely reachable pairProof:Consider the reachability matrix˜R = ˜B ØB . . . à n−1˜B 0 0 . . . 0=B c A c B c . . . A n−1cB cand˜R = T −1 B T −1 ATT −1 B . . . T −1 A n−1 TT −1 B = T −1 RSince T is nonsingular, rank(˜R) = rank(R) = n c , sorank B c A c B c . . . A n c−1c B c = ncthat is (A c , B c ) is completely reachable□Prof. Alberto Bemporad (University of Trento) <strong>Automatic</strong> <strong>Control</strong> 1 Academic year 2010-2011 16 / 23
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