Tham khảo tài liệu 'challenges and paradigms in applied robust control part 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Loop Transfer Recovery for the Grape Juice Concentration Process 289 From 36 Brn-1 - 2Ke zfc ffc. But for SISO systems qk 1 and from 37 we have that BỊ I 2Ke zk zkI A 1 B 1 then if 37 is satisfied for k 1 the result follows. Corollary Ẹk zkI A 1Mk_1B 39 Mo I 40 M I 2 e z1 z1Z A -1 41 Theorem Consider a non-minimum-phase system A B c and its minimum-phase counterpart A Bm C with Bm computed according to lemma . Let L and Lm be the optimal observer gains for these two systems then Lm L. Proof See Zhang Freudenberg Zhang 1990 . Loop transfer recovery and non-minimum phase plants Assume now that G s is a non-minimum phase plant and that it is factorized as in equation 33 . If the standard LTR procedure is applied to recover the input sensitivity then when q OT the sensitivity function satisfies Sobs s So s l E s 42 where E s F sl A 1 B qBmW I qCỘBmW 1CỘBmBz s 43 then E s F sl A 1 B BmBz s 44 It has been also shown that E s El s F B s 45 s zk From equation 44 it is evident that for this type of plants the amount of recovery at a frequency w depends on the value of w where . II is a suitable norm. As in equation 41 E s corresponds to the error of the sensitivity in loop with the LTR observer. The results of the previous two sections can be appreciated if we consider a SISO system with one zero in s z e fi . If the standard LTR procedure is applied we have that . Sobs s So s 1 H z 46 where H z F zl A 1B 47 One can then notice that if B z is small . when the LQR design bandwidth is small in comparison with the magnitude of the c zero then the recovery is almost complete. This 290 Challenges and Paradigms in Applied Robust Control case will be also the situation for high frequencies since the factor is low-pass filter. The sensitivity resulting from a LQR LTR applied to a non-minimum phase plant is very significant at low frequencies and decreases as the frequency increases. The inability of the LQR LTR scheme to recover sensitivity is consistent with .
