Fall 2001 6.3111-3 There is an analogous problem on the input side as well. Consider Model #1 =2a + 2u with a=a a2 2 #3 0-1 32 which is also clearly different than model #1, and has a different form from the second model 20 [32]sr 0-1 2 6 0 s+2 Once again the dynamics associated with the pole at s=-1 are cancelled out of the transfer function But in this case it occurred because there is a o in the b matrix So in this case we can"see"the state a2 in the output C=3 2 but we cannot "influence that state with the input since b So we say that the dynamics associated with the second state are uncontrollable using this actuator(defines the b matrix)Fall 2001 16.31 11—3 • There is an analogous problem on the input side as well. Consider: Model # 1 x˙ = −2x + 2u y = 3x with ¯x = [ x x2] T Model # 3 x¯˙ =   −2 0 0 −1   x¯ + ∙ 2 0 ¸ u y = £ 3 2 ¤ x¯ which is also clearly different than model #1, and has a different form from the second model. Gˆ(s) = £ 3 2 ¤  sI −   −2 0 0 −1     −1 ∙ 2 0 ¸ = £ 3 s+2 2 s+1 ¤ ∙ 2 0 ¸ = 6 s + 2 !! • Once again the dynamics associated with the pole at s = −1 are cancelled out of the transfer function. — But in this case it occurred because there is a 0 in the B matrix • So in this case we can “see” the state x2 in the output C = £ 3 2 ¤ , but we cannot “influence” that state with the input since B = ∙ 2 0 ¸ • So we say that the dynamics associated with the second state are uncontrollable using this actuator (defines the B matrix)