Minimum phase systems The ldea Consider the equation Process mode 八(t)=B-(Ru(t)+Sy(t) Ay(t= Bu(t Mirimumphase system(B-=bo)!! respoNse AmOy(t=bo(Ru(t+ Sy(t))= Ru(t+ Sy(t Amym(t=Bmc(t Natual to choase Diophantine eq uation A,Am= AR+B-s Introduce parameter vector let this operate ony(t)! A. Amy(t)= R'Ay(t+B- Sy(t and regression vector RBu(t)+B- Sy(t) But Hence p(t)=(u(t) u(t-e y(t y(t-) AoAmy(t=B(Ru(t+ Sy(t)) Estimate parameters in this ea uation instead n(t)=A%(9)Am(9 )y(t)=p(t-d )e An Alternat As bef an direct str Am Aoy(t=bo(Ru (t+ Sy(t)= Ru(t)+ Sy(t dly nomials Am, Bm, and A, and Filt (deg A. 2. Estimate the coeffiaents of the poly nomi (t) A2(q-1)Amn(q-1) (t as r and s in y(t) A3()4(-y( y(t)=Ru,(t-d)+S"y(t-d) Introduce parameter vector by recursive least squares. 8=(ro Compute the contro signa from and regression vector Ru(t=Tuc(t-Sy(t p(t)=(u(t)…ur(t-)gy(t)…y(t-) were The mode then becomes T=A, Am(1) y(t ADAm(Ru(t+ Sy(t) R",(t-do)+s"y,(t-d)=p (t-d, )e Repeat Steps 2 and 3 at each sampling A standard regression modd C K.J. AstO m and B wittenmarkThe Idea Process model Ay(t) = Bu(t) Desired response Amym(t) = Bmuc(t) Diophantine equation AoAm = AR0 + B￾ S let this operate on y(t)! AoAmy(t) = R0Ay(t) + B￾ Sy(t) = R0Bu(t) + B￾ Sy(t) But Hence AoAmy(t) = B￾ (Ru(t) + Sy(t)) (4) Estimate parameters in this equation instead. Minimum Phase Systems Consider the equation AoAmy(t) = B￾ (Ru(t) + Sy(t)) Minimum phase system (B￾ = b0)!! AmAoy(t) = b0 (Ru(t) + Sy(t)) = Ru~ (t) + Sy~ (t) Natural to choose Bm = q d0Am(1) Introduce parameter vector  = ( r0 ::: r` s0 ::: s` ) and regression vector '(t)=( u(t) ::: u(t ￾ `) y(t) ::: y(t ￾ `) ) Hence (t) = A o ￾ q￾1
Am ￾ q￾1
y(t) = 'T (t ￾ d0) An Alternative As before AmAoy(t) = b0 (Ru(t) + Sy(t)) = Ru~ (t) + Sy~ (t) Filter signals so that we can take derivatives uf (t) = 1 A o (q￾1 )Am(q￾1 ) u(t) yf (t) = 1 A o (q￾1 )Am(q￾1 ) y(t) Introduce parameter vector  = ( r0 ::: r` s0 ::: s` ) and regression vector '(t)=( uf (t) ::: uf (t ￾ `) yf (t) ::: yf (t ￾ `) ) The model then becomes y(t) = 1 AoAm (Ru(t) + Sy(t)) = Ruf (t ￾ d0) + Syf (t ￾ d0)= 'T (t ￾ d0) A standard regression model!! An direct STR 1. Data: Polynomials Am, Bm, and Ao and relative degree d0 (deg Ao = d0 ￾ 1) 2. Estimate the coecients of the polynomi￾als R and S in y(t) = Ruf (t ￾ d0) + Syf (t ￾ d0) by recursive least squares. 3. Compute the control signal from Ru(t) = T uc(t) ￾ Sy(t) where T = A oAm(1) 4. Repeat Steps 2 and 3 at each sampling period. c K. J. Åström and B. Wittenmark 5