Transfer the given model into Jordan form x1 x2 0 1 2 3 x1

Transfer the given model into Jordan form [x_1 x_2] = [0 1 -2 -3] [x_1 x_2] + [0 1]u, y = [0 1] [x_1 x_2] + 3u Transfer the given model into Controllable canonical form [x_1 x_2] = [-2 0 0 -1] [x_1 x_2] + [12 6]u, y = [1 1] [x_1 x_2] + u Transfer the given model into Observable canonical form [x_1 x_2] = [0 1 -2 -3] [x_1 x_2] + [0 1]u, y = [0 1] [x_1 x_2] Transfer the given model into Jordan form [x_1 x_2] = [-6 1 0 -2] [x_1 x_2] + [1 1]u, y = [0 1] [x_1 x_2]

Solution

what if A cannot be diagonalized? any matrix A Rn×n can be put in Jordan canonical form by a similarity transformation, i.e. T 1AT = J = J1 ... Jq where Ji = i 1 i ... ... 1 i Cni×ni is called a Jordan block of size ni with eigenvalue i (so n = Pqi=1 ni) Jordan canonical form 12–2 • J is upper bidiagonal • J diagonal is the special case of n Jordan blocks of size ni = 1 • Jordan form is unique (up to permutations of the blocks) • can have multiple blocks with same eigenvalue Jordan canonical form 12–3 note: JCF is a conceptual tool, never used in numerical computations! X (s) = det(sI A) = (s 1)n1 · · ·(s q)nq hence distinct eigenvalues ni = 1 A diagonalizable dim N (I A) is the number of Jordan blocks with eigenvalue more generally, dim N (I A)k = X i= min{k, ni} so from dim N (I A)k for k = 1, 2, . . . we can determine the