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Control System Design based on Exact Model Matching by Kunihiko Ichikawa

By Kunihiko Ichikawa

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Differential equations containing small parameter and relaxation oscillations,: Lectures delivered at the University of Michigan, September-December 1964

A large number of paintings has been performed on usual differential equations with small parameters multiplying derivatives. This publication investigates questions on the topic of the asymptotic calculation of rest oscillations, that are periodic strategies shaped of sections of either slow-and fast-motion elements of part trajectories.

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Pd(s) is Let r*(s) and p*(s) be any stable m and n deqree polynomials respectively, and rewrite td(s) es follows. qdr*(s) rd(s)p*(s) %d (s) : p"(s) r~(S)Pd(S) . 4) 25 v~s, ~~~precom,~en-lU ~s'i ~l'i+~~sat°r kts) Fig. 2 ! riP'ant h(s) Exact model matching system in frequency domain 26 Lee us exam{ne %iN(=)=~d(S)p*(s)[r*(s)Pd(s)]-1. Since a It* (s)pcl(S) ] -a [rd(s)p~ (s) ]--(nd-md)- (n-m)>_@ i ~:1~(s) must be proper and stable, which we call input dynamics. Now, par~:ion %d(s) as shoOn in Fig.

T. 31) as follows. 47) is nonnega%ive end tends %0 infinity I1~<%)[I ~. The ~ime ra~e only when of change of V is evaluated as V(~(%)) = -2 ~2(%)/[c+QT(%)Q(%)] < 8 Since V decreases monotonically, ~(%) Also, since V decreases monotonically must conuerge %o some zero. 48) is always bounded. and bounded below, V constant and hence V conuerges to is, lim s2(%) = 8. oci%y of is 5ounded. 45). ize with velocity i% izes If ~(t) Q(%), as is nonzero %hen [[Q(t)[[, I~(%)I which wi~h ~(~) with being and hence lim Let us now examine and does will diuerge no% or%hogonalwi~h %he same is the contradiction.

5) w(s). T(s)p~(s) stable, e(t) Also, when w<%) w(%). In %~. the contonues %o be zero, following, e(t)~@ even so that bounded for bounded is if w(t) the is controller is applled to the plan% persistently. 2 Suppression matching. 1 u(s)_I< -~ is effect either but we assume of disturbance. will be dealt with Theorem disturbance in exact model unknown nor here the knowledge The unknown disturbance in context to adaptive control. The control law h(s) k(s) T(s)r~- (s) U(S) + T(S)V~( y ()s ) ) s qw ~(S)rw(S) ....

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