Automatic Control 1 - Reachability Analysis

Lecture: Reachability AnalysisReachability AnalysisMinimum-energy controlLet (A, B) reachable and consider steering the state from x(0) = x 1 intox(k) = x 2 , k > n⎡ ⎤u(k − 1)x 2 − A k x} {{ } 1 = B AB . . . A k−1 B u(k − 2)⎢} {{ }⎥⎣ . ⎦XR ku(0)} {{ }U(R k ∈ n×km is the reachability matrix for k steps)Since rank(R k ) = rank(R) = n, ∀k > n (Cayley-Hamilton), we getrank R k = rank[R k X] = nHence the system X = R k U admits solutions UProblemDetermine the input sequence {u(j)} k−1j=0that brings the state fromx(0) = x 1 to x(k) = x 2 with minimum energy 1 ∑k−1‖u(j)‖ 2 = 1 22 U′ Uj=0Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 11 / 23