Fall 2001 16.318-4 ● Consider case3with y bos+0182+628+b3 a152+a28+a3 1s2+2s+/3 +D C15 Gi()+D where (s3+a1s2+a2s+a3) +(+/1s2+B2s+3) bos+0182+b2S +63 so that, given the bi, we can easily find the Bi D 61=01-Dar Given the Bi, can find G1(s) Can make a state-space model for G1(s)as described in case 2 Then we just add the"feed-through"term Du to the output equa tion from the model for Gi(s) e Will see that there is a lot of freedom in making a state-space model because we are free to pick the x as we wantFall 2001 16.31 8–4 • Consider case 3 with y u = G(s) = b0s3 + b1s2 + b2s + b3 s3 + a1s2 + a2s + a3 = β1s2 + β2s + β3 s3 + a1s2 + a2s + a3 + D = G1(s) + D where D( s3 +a1s2 +a2s +a3 ) +( +β1s2 +β2s +β3 ) = b0s3 +b1s2 +b2s +b3 so that, given the bi, we can easily find the βi D = b0 β1 = b1 − Da1 . . . • Given the βi, can find G1(s) – Can make a state-space model for G1(s) as described in case 2 • Then we just add the “feed-through” term Du to the output equa￾tion from the model for G1(s) • Will see that there is a lot of freedom in making a state-space model because we are free to pick the x as we want