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By Kunihiko Ichikawa

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Adaptive The, it will he omi%%ed V=F T > 8, fas% conuergenc=-. ively. %0 be found. 18) law limi%e4 i% does not prouide yields when ma]~e signals ere %he adap%iue bu% holds. exponential, adap%ive is %0 self, ~(%) 5u% 6(%)=[~(t)]" so as %o make e(4=) be zero, ohjec% auailable Since 8 _ ~(%), is ~:he expre-~sion of error Since in %0o, error uec%or. -'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.

D. , 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!

Howeoe~-, since T(s) is s%able, %his equation reduces ~o y(s) = gdr* (s) gdrd ( s ) p*(s) /;(s) = v(s). pd(S) Further, we ob%ain gdrd(s)p(s) Ub(S) = pd(s)r(s ) v(s). t2) 3O Sinc~ long r(s) is a-=~umed as v(%) %o be stable, is hounded. D. ed. The control law with raqard %0 u(s) can he written as u(s) = ±< k(s) (s)+T (s)r,(s)y(s)+gdO(S)}, h(s) g r(s)r~(s)U where k(s)=g~b(s) shown in %0 rather r*(s) Fiq. and h(s)=hb(s). 4. 13) considerable can is be reduced s~ahle polynomials freedom. [% is ~o he noticed %ha% the input dynamics introduced By ~olovich de- pend and hence for on the plan~ parameters, ex%enslon %0 adaptive control.

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