A Construction of Hybrid Adaptive Control System Using a Fixed Compensator Shiro MASUDA*, Hiroshi OKAMOTO** and Akira INOUE* In this paper, we propose
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1 A Construction of Hybrid Adaptive Control System Using a Fixed Compensator Shiro MASUDA*, Hiroshi OKAMOTO** and Akira INOUE* In this paper, we propose a new design scheme of hybrid adaptive control system using a fixed compensator for single-input, single-output, continuous time invariant plant. In the proposed method, adaptive adjusting law can be simplified because parameters are updated by using estimated error at arbitrary sampling instants. Furthermore, boundedness of all the signal in the closed system is proved. In general, the stability of adaptive systems cannot be assured by only using sampled estimated error. However, in this paper, the difficulty is overcome by utilizing the fixed Compensator. Finally, a simulation result is illustrated in order to ensure the effectiveness of the proposed method. Key Words: hybrid adaptive control, fixed Compensator, stability to
2 I'H(s)~~-IIH(s-b/2)II~ pu(y(t)=sut ps~ 6(tk)=9((tk)+v(tk)--y(tk) (12) u(t=owt+srt (sy(t)ut+wt (6) [][l u(t)=-9(t)tw(t)+k(s)r(t) sy(t),...,asy(t)(7[l[l Q(s)y(t)-u(tet)w(t (8 ~(t)=1w(t),v(t)=1u(t) (11)
3 Q(s)=m{S]* (T>o) Yk1+I(t k)i2 V(k)~-17kbk[2I-k((tk)C(tk)T] X~(tk)((tk)Tqk ~2 ~k/'2i-7krkrkk (21 zw(k)=v(k+1)-v(k) k+~p~ (17)
4 cb(t)tw(t)=~b(t)ta1(s,a)[w(t)]sa1 +Ao(s,a)H(s)-'[~(t)TH(s)[w(t)]-Sx] (29) sal(s,a)=1-ao(s,a),ao(s,a) (30( s+a) ~b(t)tw(t) =~(t)ta1(s,a)[w(t)]-sa1 +Ao(s,a)H(s)1[~(t)TH(s)[w(t)]-Sx](36)
5 Iw(t)I<cmf(t) (38 ~Jw(t)JJ2't<cm1( (39 w(t)=hi(s)[u(t)y(t)]t+[00y(t)]t (40 JH(s)cbTW-~Hs)w11't~cImfzbJ' 47 (41) 1-Q(s)twty(t)()() PM(s)r(t) (43 y(t)c{~~q5wii+c ciiwi't+ (44 M(t+1)=[i3l(ti+i),...,rQq(ti+i)] 50 (tz+1)=ea(ti+1-p)bwj(p)dp (51 i3(ti+1)i2<c2p(ti+1-p)mf(p)2dp (54 IIWII:;C(Ilull+11Y11+CIIW2,t
6 ce-2p(ta+1-a)(2p+2c)ii3(u)i2 +e-2t2+1-a)i~?(~)iiw~(a.)i cce-2p(ti+1-a)mf(q)2 +Ce-2p(ta+1-a)mf(7){mf(a)+c ce-2p(ti+1-a)mf(q)2+ce-2p(ti+1-aimf(q) (63 Le-2p(ti+1-~)I~~I2'Ce-2p(t2+1-cY)?12 64 e-2p(ti+1-~~ja2 (57 e-2ptj+12peitj)el/3() <2pe-i()I+e-2p(ti+1-~)Iai(~ =I23(a)T3i(a) <2I/3j(o)l-/3j(a) ((58 ~j(~)=a~~(q)+bw~(~ (59 j(a)<ia~j(a)i+ibwj(a)i c(i,3j(a)i+iwj(ci (60 [e^2p(ti+1-a)iqj(~)i2] <2pe-2p(ti+1-a)Ii2~(~) e-2p(ti+1-a)-2i/3(~)ic(ii3?()i+wi(a IISHII2~t2_S(t-p))SH(P)I2 ll'dh(i2,t
7 Ts[(Ts+1)n*-1+(Ts+1)n* ] (Ts+1)n* =TsWiT( (76 TS+1)" (77)( fp[s]w(s)luii p[s~s,t+k(s)[r] 2 p[s]twi p fp[s]ss,t+(s)ilrll TssT ~)~~ 2 CTIIWIT(s)s[4'~W1II (78) u(t)=-3{s]1-q(s)tt~ m[s]ap[s]~ms 3p[s] mcsk(s){r(t)] (75 IIWi(s)s[PW <II~TWiT(s)s[ +IIW1T(s)s[4TW]-~W1TssW <ciiwii2't+ci)mfl0/ll (79 IIuII2't<CTIItI2't+CTIImfI~~III2' (80 IIyI2't<I1]WT(s)[cTW]+IIPM(s)[r]II2't~m[s 2 1aC TsIIW1T(s)II~II~TWII2't+C ii1m{8jo0 :5;CTIIY'TWII ($1) m1(t)5;;ctiitli2't+ctii~~~ii2't 8,t +ctllmflo~lll2' (82) H(s)=1 /3m{5] Ao(s'a) an* n*'a2(5+)m* WT(s)=1-Q(s)1~ _(Ts+1)n*1( Ts+1)n* m~sj q5(t)tw(t)=q5(t)ta1(s,a)[w(t)]--s~l +Ao(s,om[s][q5(t)Tfi(t)-SH] (83)
8 Il~TWll2't<c+Cm1(t)+canII~T~II (c+can*)ilmfiz II mf(t)<tc+-m1(t)+ccyn*ii~t~ii2't Cr +(C+ccan*)IImfIzIII2't +ctilmfio~iiiz't+ctmf(t)+c icta+1mt C+caIIqII+C+Ca)Tm~ m1(t)~c+nn*tii~tcll2't+(c+can*)tiimfizcbii2't cc+cftmf (c+can*)tiimfio~ii12't (87 1t<c+ce-~zeo(~dz (~(t)(t+089(t)2=t m(t)i(t)i2 (~b(t)t~(t) mf(t dp<c1d y(t)=s+2u(t)s2 -s-2 ym(t)=1r(t) s+1 ~p(q)2d~<c1(t-z mf(t)2<c+cea(t-z)qo(z)ec1(t-z) c+cea-d1)(t-z)o(z)
9 Fig.1 Simulation results of hybrid adaptive control system based on sampled estimated error with a fixed compensator Fig.2 Simulation results of hybrid adaptive control system based on sampled estimated error without a fixed compensator
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