2. A third-order system is described by -2 2 dx 0 1 + u, x (to) = xo dt 0 0 the (a) Show, by hand calculation, that the system is uncontrollable but is stabilizable. (b) Use Matlab to verify the conclusion in (a). Hint: from linear algebra, for a matrix P = Rnxn, n-dimensional vector space can be decomposed as R = R(P) ON (PT). Use the command orth to obtain an orthonormal basis for the range space of the controllability matrix P; use command null(P') to obtain the other vectors that complete the basis. The following is a starting set of codes: 1% ME 547, UW 2% Prof. Xu Chen 3% Kalman decomposition example 4 5 A = [-2 1 2; 0 0 1; 0 1 0]; 6 B 7 [101]'; 8 Pctrb (A, B); 9% check rank of the controllability matrix 10 k = rank (P) 11 12% obtain the linearly independent columns 13 help orth 14 Mc= 15 16% complete the basis by obtaining the orthonormal completement of Mc 17 help null 18 Muc = 19 20% check that the columns of Muc are linearly independent with the columns 21% of Mc 22 23 Mc (:,1) *Muc 24 25 Mc (:,2) '*Muc 26 27% you should get zero or a very very small number in the above computation 28 29% construct M 30 M= 31 32 % the similarity transform 33 A bar [Mc Muc]; inv (M) *A*M 34 B bar = inv (M) *B 35 36 A barc 37 B barc 38 A bar (1:k, 1:k) B_bar (1:k) 39 Pc = ctrb (A_barc, B_barc) 40 rank (Pc)