Estimation A(p)y(t)=B(p)u(t) Example where p=d/ dt and Process model B(p)=boP" Reg ressor filter (t= Hyy(t) (t)=Hyu(t) t)=( )2 bn T D res po nse Model for filt ered p"y(t)=9(t)e s) cws+ Recursive least squares wit h ex po entia forgetting Observer poly nomial Ao(s)=s+ao de(t) t)-g(t)( Nominal values a=l, b=l,(=0.7 and dP(2-aP()-P()(t) p (t)P(t) Control di esg ugn fb≠0 Bn(s) cWs +w2 Dio phantine equation (10) Ident ificat io n of coefficients of powers of s C K.J. Ast ro m and B WittenmarkEstimation Process model (No disturbances!) A(p)y(t) = B(p)u(t) where p = d=dt and A(p) = p n + a1p n￾1 +


+ an B(p) = b0p n￾1 +


+ bn (7) Filtered signals yf (t) = Hf y(t) uf (t) = Hf u(t) '(t)=(￾p n￾1yf


￾ yf p n￾1 uf


uf )T  = (a1


an b1


bn)T (8) Model for ltered signals p n yf (t) = 'T (t) Recursive least squares with exponential forgetting d^ (t) dt = P (t)'(t)  p n yf (t) ￾ 'T (t)^ (t) dP (t) dt = P (t) ￾ P (t)'(t)'T (t)P (t) Example Process model G(s) = b s(s + a) Regressor lter Hf (s) = 1 Am(s) Desired response Gm(s) = !2 s2 + 2!s + !2 Observer polynomial Ao(s) = s + ao Nominal values a = 1, b = 1,  = 0:7, ! = 1 and ao = 2. Control Design G(s) = b s(s + a) Choose Ao (s) = s + ao and Bm(s) Am(s) = !2 s2 + 2!s + !2 Diophantine equation s(s + a)(s + r1) + b(s0s + s1 ) = (s 2 + 2!s + !2 )(s + ao ) Identication of coecients of powers of s a + r1 = 2! + ao ar1 + bs0 = !2 + 2!ao bs1 = !2ao (9) Control Design - Cont. If b 6= 0 r1 = 2! + ao ￾ a s0 = ao2! + !2 ￾ ar1 b s1 = !2ao b (10) c K. J. Åström and B. Wittenmark 7