Computational Delays Choice of sampling interval ub≈0.2-0.6 Wo- the nat ural fre q uency of the dominating R"(q-1)u(t)=T(q-1)2(t)-S"(q1)y() he rule im plies a bo ut 5-20 sam ples in a ste p Im plement as follows res ponse of the c losed-loo p system 1. Make A-D con Version of y(t)and u(t) Different rates in controller and estim ator may be useful 2. Com pute u(t=touc(t)-soy(t)+ui(t) Special hold circ uits 3. Make D-A con Version of (t) Post filters te u1(t+1=(1-R")u(t+1+(r”-t)y(t+1 Effect of Anti-Aliasing Filter G(s s Antialiasing filter Fixed parame er pole placement controlle Fourth order bessel filter wit h the bandwidt h Eliminate all freq uencies a b fre quency(wn=T/h) before sam pling (t=y(t+ ad sin(wat Use 2-6 order B utter Wort h or Bessel filters (b Sam pling interVal de pendence Gaa(s 82+2cs+ Bad neWs: The antia liasing filter will influence t he process and t he design i y Good Can often be time de lay. good in t he ada ptive case Parameters wd= 11.3 rad/s and ad=0.1 6.28rad/ cOm pensa ted for a delay of 1. 7h O K.J. Astomand B w itterakComputational Delays y y(tk−1) y(tk ) y(tk+1) Time u t k−1 t k tk+1 u(tk− 1) u(t k) Time Control Variable Measured Variable Case B y(tk−1) y(tk ) y(t k+1) y t k−1 t k t k+1 Time u(tk ) u(tk+ 1) u Time Control Variable Measured Variable Case A Computa￾tional lag τ = h Computa￾tional lag τ R(q￾1)u(t) = T (q￾1)uc(t) ￾ S(q￾1)y(t) Implement as follows: 1. Make A-D conversion of y(t) and uc(t) 2. Compute u(t) = t0uc(t) ￾ s0y(t) + u1(t) 3. Make D-A conversion of u(t) 4. Compute u1(t + 1) = (1 ￾ R )u(t + 1) + (T ￾ t0)y(t + 1) ￾ (S ￾ s0)y(t + 1) Choice of sampling interval Rule of thumb (deterministic design): !oh  0:2 ￾ 0:6 !o { The natural frequency of the dominating poles of the closed-loop system The rule implies about 5{20 samples in a step response of the closed-loop system Dierent rates in controller and estimator may be useful Special hold circuits Post lters Antialiasing lter Eliminate all frequencies above the Nyquist frequency (!N = =h) before sampling. Use 2{6 order Butterworth or Bessel lters Sampling interval dependence Gaa(s) = !2 s2 + 2!s + !2 Bad news: The antialiasing lter will in uence the process and the design Good news: Can often be approximated by a time delay. Good in the adaptive case Eect of Anti-Aliasing Filter Process G(s) = 1 s(s + 1) Fixed parameter pole placement controller. Fourth order Bessel lter with the bandwidth !B. ym(t) = y(t) + ad sin(!d t) 0 5 10 15 20 0 1 0 5 10 15 20 0 1 0 5 10 15 20 0 1 0 5 10 15 20 0 1 Time Time Time Time (a) uc y u (b) uc y u (c) uc y u (d) uc y u Parameters !d = 11:3 rad/s and ad = 0:1. (a) !B = 25 rad/s; (b) !B = 6:28 rad/s; (c) !B = 2:51 rad/s and the regulator compensated for a delay of 1:7h. c K. J. Åström and B. Wittenmark 3