Fall 2001 163112-9 The new model in the state z is diagonal. There is no coupling in the dynamics matrix A But by definition TB and CT . Thus if it turned out that B≡0 then that element of the state vector zi would be uncontrollable by the input u ●Also.if then that element of the state vector z; would be unobservable with this sensor Thus, all modes of the system are controllable and ob servable if it can be shown that B≠0 an Cv;≠0VFall 2001 16.31 12—9 • The new model in the state z is diagonal. There is no coupling in the dynamics matrix Λ. • But by definition,   wT 1 . T −1 B =  . .  B wT n and £ ¤ CT = C v1 · · · vn • Thus if it turned out that T wi B ≡ 0 then that element of the state vector zi would be uncontrollable by the input u. • Also, if Cvj ≡ 0 then that element of the state vector zj would be unobservable with this sensor. • Thus, all modes of the system are controllable and ob￾servable if it can be shown that wi TB 6= 0 ∀ i and Cvj 6= 0 ∀ j