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>Linear algebra recallsAlgebraically equivalent systemsConsider the linear system x(k + 1) = Ax(k) + Bu(k)y(k) = Cx(k) + Du(k)x(0) = x 0Let T be invertible and define the change of coordinates x = Tz, z = T −1 x z(k + 1) = T −1 x(k + 1) = T −1 (Ax(k) + Bu(k)) = T −1 ATz(k) + T −1 Bu(k)y(k) = CTz(k) + Du(k)z 0 = T −1 x 0and hence z(k + 1) = Ãz(k) + ˜Bu(k)y(k) = ˜Cz(k) + ˜Du(k)z(0) = T −1 x 0The dynamical systems (A, B, C, D) and (Ā, ¯B, ¯C, ¯D) are called algebraicallyequivalentProf. Alberto Bemporad (University of Trento) <strong>Automatic</strong> <strong>Control</strong> 1 Academic year 2010-2011 4 / 23
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