Automatic Control 1 - Reachability Analysis

Lecture: Reachability AnalysisControllabilityControllabilityThe linear system of equations⎡−A k x 0 = B AB . . . A k−1 B } {{ }⎢⎣R ku(k − 1)u(k − 2).u(0)admits a solution if and only if A k x 0 ∈ Im(R k ), ∀x 0 ∈ nTheoremThe system is controllable to the origin in k steps if and only ifIm(A k ) ⊆ Im(R k )⎤⎥⎦If a system is controllable in n steps, it is also controllable in k steps for eachk > n (just set u(n) = u(n + 1) = . . . = u(k − 1) = 0)For the same reason, if a system is controllable in k steps with k < n, it is alsocontrollable in n steps (just set u(k) = u(k + 1) = . . . = u(n − 1) = 0)Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 18 / 23