By Kunihiko Ichikawa
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Extra resources for Control System Design Based on Exact Model Matching Techniques
Decreases 5u% implies unlimitedly. 4T(t)F-l@(t)~ const, the ellipsoid helnq octhogonal to spans interval. is 2n Then, identi-fication only, bu~ identifics~ion space of ~(%) wh ich ~(t). the is not some such tI+T] for all the condition components. is that the The adaptive by adaptive is needed for 2n dimensional Clearly, the pmoper%y the p r ' o p e r ~ y a Of U(t). , 4(t)-~ %hat interval. that these [t I, seen spans condition contains objective, that rich". oci~y o~ property of ~(~;) that vec~oc ~(t) and in 2n dimensional s p a c e , original have 4(%) @T(%)~(t)~8 @(t) must be "suff icient ly is that Since ~e does not imply motion of u e c % o c fact = const, depends on bestows The small, in a fini~:e [8(t)]'~@ that the di-mensiona!
V canno~ be negative. = -2~T(%)~(t)e(%) From %his is positive def"i n i "~= ~ for all F-I(%) 0 = 24T(~)F-I(%)~(t) On the o~he~ hand, + 4T(%)[F-I]'4(%) + 4T(%)~(%)~T(t)4(%) = -e2(%) < 0. Therefore, ~o Thecefore, hounded which %o if and 4(%) below, is if ]Idl] %ends o~ %he V %ends ~o infinity. 1~). cegardless Fur%hermoce, uniformly bounded. 23) V deczeases mono%onically. infini%y lim zeco. or_, e(~) implies %hat ~T(t)~(%) conuecges The argument up boundedness %0 %his poin% o~ ~(%). 21). D. useful adap%ive law is now pcesen~ed, does no~ manife_~ a so ~as% c o n w e c g e n c e .
E /~- finite. fo~" ~(r)~T(r) This ~ormula, an adap%ive form dr law, of adap%ive f ~ ( r ) : c ( r ) dr. ~o be is a-~sumed however, does and hence we law. 13) positive de- no% a-~sume the fo~'m proceed %o o5-hain the By differen%ia%ing bo-~h si4es o~ ~he re la%ion f~ aJ('c)a~T(r) d r . - ou~:pu% ~(t). ~'s~ion of inverse and in-~eg:'a~- F(:c) = [ / ~ ( r ) ~ T ( r ) Then, dr] -I ~(%) e(%). Then, ma%i'ix ['(%) by (Z 1Y) dr] -I c=_n be wr-i%~en as 84%) - - F(~:) ~)(%) e(~:). F(%) - I. %he above equation [~(t) = -[/~ ~(~)~T(r) Since by %~ we obtain dr] -I ~(t)~T(%)~(%) = - F(%)~(%)~T(~)7(%).
Control System Design Based on Exact Model Matching Techniques by Kunihiko Ichikawa