Fall 2001 16.3110-9 Further observations: apply the specified control input in the frequency domain so that 1(s)=G(s)U(s) where u=2e-2t, so that U(s)=2 +2 2 s2+7s+12s+2s2+7s+12 Say that s=-2 is a blocking zero or a transmission zero The response Y(s) is clearly non-zero, but it does not contain a component at the input frequency s=-2. That input has been "blocked Note that the output response left in Yi(s) is of a very special form-it corresponds to the(negative of the) response you would see from the system with u(t)=0 and To=[-2 Y2 s)=C(sI-A s+712 2 1 s 12 -2 1s+7」1」s2+7s+12 s2+7s+12 So then the total output is Y(s)=Y(s)+Y2(s)showing that Y(s)=0- y(t)=0Fall 2001 16.31 10–9 • Further observations: apply the specified control input in the frequency domain, so that Y1(s) = G(s)U(s) where u = 2e−2t , so that U(s) = 2 s+2 Y1(s) = s + 2 s2 + 7s + 12 · 2 s + 2 = 2 s2 + 7s + 12 Say that s = −2 is a blocking zero or a transmission zero. • The response Y (s) is clearly non-zero, but it does not contain a component at the input frequency s = −2. That input has been “blocked”. • Note that the output response left in Y1(s) is of a very special form – it corresponds to the (negative of the) response you would see from the system with u(t) = 0 and x0 =  −2 1 T Y2(s) = C(sI − A) −1 x0 =  1 −2  s + 7 12 −1 s −1 −2 1 =  1 −2  s −12 1 s + 7 −2 1 1 s2 + 7s + 12 = −2 s2 + 7s + 12 • So then the total output is Y (s) = Y1(s) + Y2(s) showing that Y (s)=0 → y(t) = 0, as expected