Automatic Control 1 - Reachability Analysis

Lecture: Reachability AnalysisReachability problemReachabilityConsider the linear discrete-time systemx(k + 1) = Ax(k) + Bu(k)with x ∈ n , u ∈ m and initial condition x(0) = x 0 ∈ n∑k−1The solution is x(k) = A k x 0 + A j Bu(k − 1 − j)Definitionj=0The system x(k + 1) = Ax(k) + Bu(k) is (completely) reachable if ∀x 1 , x 2 ∈ n thereexist k ∈ and u(0), u(1), . . ., u(k − 1) ∈ m such that∑k−1x 2 = A k x 1 + A j Bu(k − 1 − j)j=0In simple words: a system is completely reachable if from any state x 1 we canreach any state x 2 at some time k, by applying a suitable input sequenceProf. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 2 / 23