Ex ample 3.10 Modification of estimator Process model Process model G(s) s(s+1) Ay= B(a+u) Disturbance st Disturbance model Nominal cont roller Notice e small. hence Modified cont roller Ad=g-1, X AAy= Baru+ BAv= BAu+ Be R=q(g+r1)+(b+b)=(q-1)(q-b23) nt roduce the filtered signals Estimator A t a-d Then +a19+a,y Estimat io Ayy= Bu+ Be y(t)+a13(t-1)+a2y(t-2) (2) bu(t-1)+bu,(t-2) Simulation y Direct str Com pariso ns wit h ot her algo rit hms The Id phase Syster Feedforward cont rol Phase b 1 C K. J. Ast ro m and B. WittenmarkModication of Estimator Process model Ay = B(u + v) Disturbance model Adv = e Notice e small. Hence AdAy = BAdu + BAdv = BAdu + Be Introduce the ltered signals yf = Ady uf = A ￾ du Then Ayf = Buf + Be Example 3.10 Process model G(s) = 1 s(s + 1) Disturbance step i.e. Ad = q ￾ 1 Nominal controller R0 = q + r1 S0 = s0q + s1 Modied controller Ad = q ￾ 1, X = q R = q(q + r1) + y0(b0 + b1)=(q ￾ 1)(q ￾ b1y0) S = q(s0q + s1) ￾ y0(q 2 + a1q + a2) Estimator uf = q ￾ 1 q2 + a1q + a2 u yf = q ￾ 1 q2 + a1q + a2 y Estimation model yf (t) + a1yf (t ￾ 1) + a2yf (t ￾ 2) (2) = b0uf (t ￾ 1) + b1uf (t ￾ 2) (3) Simulation 0 20 40 60 80 100 −1 0 1 0 20 40 60 80 100 −4 −2 0 2 Time Time uc y u 0 20 40 60 80 100 −2 −1 0 0 20 40 60 80 100 0.0 0.1 0.2 Time Time ^a2 ^a1 ^ b1 ^ b0 Direct STR  Why? { Avoid design calculation { Comparisons with other algorithms  The Idea  Minimum-phase Systems  Examples  Feedforward Control  Non-minimum Phase Systems c K. J. Åström and B. Wittenmark 4