The C-poly nomial The model xam C(a=z+ (+)=A1(q (t) Spect ral density Distur bances e=dc(e c(e-in 1(q e() But A2(q) C(z)C(z-)=(z+2)(z-+2) 4(z+0.5)(z-1+0.5) y(+)=m(t)+v(t) B1 q C u(t)+ The disturbance We can write t his as y(t)=e(1)+2e(t-1) A(g)y(t)=b(gu(t)+ C(ae(t) with Ee2= l can thus be represent ed as The st and ard model y(t)=∈(t)+0.5e(t-1) ith ee Prediction The general case . Model c st a ble Process model c(a) y(k+m)=A(g) e(k+m) (q-1) (k+m) A(gy(h)= b(gu(k)+c(g)e(k) deg a-deg b=d, deg C=n F(a e(h+ m)+9 4(q-1) (k+m) C stable SIso. Innovat io n model Predict Desig n criteria: Mini mize G under the condition that the closed loo p Predict ion error stable May ass ume any causal no nlinear con- (k+m|k)=F(q-1)e(k+m) Optimal predictor dynamics C(a) C K. J. Ast ro m and B. WittenmarkThe Model Process dynamics x(t) = B1(q) A1(q) u(t) Disturbances v(t) = C1(q) A2(q) e(t) Output y(t) = x(t) + v(t) = B1(q) A1(q) u(t) + C1(q) A2(q) e(t) We can write this as A(q)y(t) = B(q)u(t) + C(q)e(t) The standard model!!! The C-polynomial Example C(z) = z + 2 Spectral density (e i!h) = 1 2C(e i!h)C(e￾i!h) But C(z)C(z￾1 )=(z + 2)(z￾1 + 2) = 4(z + 0:5)(z￾1 + 0:5) The disturbance y(t) = e(1) + 2e(t ￾ 1) with Ee2 = 1 can thus be represented as y(t) = (t)+0:5(t ￾ 1) with E2 = 4 The General Case  Process model A(q)y(k) = B(q)u(k) + C(q)e(k) deg A ￾ deg B = d, deg C = n C stable SISO, Innovation model  Design criteria: Minimize E(y 2 + u2 ) under the condition that the closed loop system is stable  May assume any causal nonlinear con￾troller Prediction  Model C stable y(k + m) = C(q) A(q) e(k + m) = C (q￾1 ) A (q￾1 ) e(k + m) = F (q￾1 )e(k + m) + q￾m G (q￾1 ) A (q￾1 ) e(k + m)  Predictor y^(k + mjk) = G (q￾1 ) C (q￾1 ) y(k) = qG(q) C(q) y(k)  Prediction error y~(k + mjk) = F (q￾1 )e(k + m)  Optimal predictor dynamics C(q) c K. J. Åström and B. Wittenmark 2