Fall 2001 16.3110-8 Is a zero at the frequency so if there exists a non-trivial solution of 1-A B 0 Compare with equation on page 9-1 Key point: Zeros have both a direction ao and a frequency so Just as we would associate a direction(eigenvector ) with each pole(fre- quency入) Example: G(s) 2+7s+ C=[12]D=0 +712-1 I-A-B det - det 1s00 120 =(s0+7)(0)+1(2)+1(s0)=s0+2=0 so there is clearly a zero at so =-2, as we expected. For the directions solve s0+712-1 512-1 01 1 1-20 gives o1 02 and wo=2 Co2 so that with ao2=1 2 anFall 2001 16.31 10–8 • Is a zero at the frequency s0 if there exists a non-trivial solution of det s0I − A − B C D = 0 – Compare with equation on page 9–1 • Key Point: Zeros have both a direction x0 u0 and a frequency s0 – Just as we would associate a direction (eigenvector) with each pole (fre￾quency λi) • Example: G(s) = s+2 s2+7s+12 A =   −7 −12 1 0   B = 1 0 C =  1 2  D = 0 det s0I − A −B C D = det   s0 + 7 12 −1 −1 s0 0 1 20   = (s0 + 7)(0) + 1(2) + 1(s0) = s0 +2=0 so there is clearly a zero at s0 = −2, as we expected. For the directions, solve:   s0 + 7 12 −1 −1 s0 0 1 20   s0=−2   x01 x02 u0   =   5 12 −1 −1 −2 0 120     x01 x02 u0   = 0? gives x01 = −2x02 and u0 = 2x02 so that with x02 = 1 x0 = −2 1 and u = 2e−2t