Control Oriented LFT Modeling of a Non Linear MIMO system
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1 I E E E C Intnton on of Ectc, Ectoncs ISSN No. (Onn : n Cot Ennn (: 5-( Sc Eton fo Bst Ps of c F IE In St-, FIIS- Conto Ont LF on of Non Ln IO sst Ro* n Rnt K B** *CKV Insttt of Ennn, L, How, (WB ** Unst, Kot, (WB (Rc 5 Octob, Acct Dcb, ABSRAC: s snts oc fo tc nctnt on of nonn t- nt t- ott (IO coss co sst fo t os of Ln Fcton nsfoton (LF o zton. A no tcnq s bn cons to snt t conto ont LF on of n tc os. A ssttc oc s to fn t qnt snttons of nton of ton nnt tc tcs. o nctnt ss wn t sst n n t ts not cs nown, o o n n. On so nstct nctnts, b wc on ns cox t tons stsfn n nt bons. Ln Fcton nsfoton s obcts of st fo obst n Ln Pt Vn (LPV conto. Fo obst conto LF o s sb. A nonn ston co ncs boto twn oto IO sst, wc consttts cns fo n cssc n conto tcnqs, s bn cons s cnt sst. A two--of-fo (DOF sn fwo s bn ot fo t foton of t LF on. Inx s Dnc tbton, LF on, IO sst, nstct nctnt. I. INRODUCION In t f of conto to n tb ssts s t sbct wt t of sccssf cton. Ato t nonn conto to s qt ttct n so n scs cnt sow n ct ntst n t ont n cton of nonn conto tooos. It s ffct to sn sc conto fo nonn coss co tb nc sst. Robst conto s nnt bot o nctnt, tc focsn on t ctons of o nctnt fo csons n t o of t sst s bn snt n t fo of LF. s concns wt t sstc conto ont on of nctn non-n sst wos o to cns n t sst confton n otn contons. ffncs btwn t nz ttc o n t ct sst, snc of stbnc sn, n t o o cton bn cons fo t o nctnt. A nonn ston co boto sc twn oto t-nt t-ott sst s bn cons s cnt sst. os of t s to o sst o to cctz sst tons s nctnt. o nctnt n s non sst o n t nnown tnsf fncton tx. s on oc s to t sst n t fo of n fcton tnsfoton (LF. A n scto t LF sntton of ton tc tcs s nz snt of bt ton nnt tt fnctons n LF-fo []. A tcnq s bn oos to o nctn nonn ssts wos os to cns n t sst confton n otn contons []. A nf f wo fo t stton obs wc s n sst ntfcton contxt. ts to b stt n LF wt nown constnt tx [3]. A n oc s to oxt tc n fcton sntton (LFR wt c o LFR [4]. E sts snt ssttc oc fo t nton of nctnt os scb b LF n ot on cnt o sboc n nc softw to ssst t nton of ow o LF-bs nctnt os. It contbts ssttc oc to snt nonn coss co nc ssts to tx ono of n o n n nb of ts s n LF [5]. Psc ssts n t nt t ott sst n x ssts non-n. In non n IO sst coxt s ncs b t ncs of nt ott. Hcot s n cft wc s ft n o b on o o ozont otos consstn of two o o oto bs. It cn t off n n tc n to ntn st o n t o sn ont on t on. wo of fo cot qnt (wn Roto IO Sst o b Fbc b n t botos of t Anc Conto Ssts Rsc. s o s oo tb conto bnc w s n t tt. It ows sttn t conto of cot cncs wt two s of fo tt otts on two ctons.
2 Ro n B 6 In ttc on of RS, t sst s o n ts of tc on--of-fo (DOF, ozont (DOF, n two--of-fo (DOF ncs sn Nwtonn s w s Lnn tos [6]. Stb t o fo ct conto oc s bn oos fo constn nonn ssts. s to s nown s tst Nwton-t conto stts ow, t foton ffs fo t on on [7]. A ttc o of wn Roto Ao nc Sst contnn xnt cctstcs s bn s to sn t contos fo tcn [8]. Gntc on n bton conto of twn oto IO sst s ntoc ob sc tcnq of GA s s to ntf t ts of t RS bs on on-st- cton [9]. Rcnt st sows nnown nonnts of RS s bn stt b n ntwo wos wts t st []. A t n n non n on of t RS s so oxton s s n o to t so ncss now. D to ts oxton so nctnt s ntoc nto t sst n wc s nob n conto sst. A ts nctnt s nto sn boc. Unctnt boc ffcts t nt ott tons of t LF. s LF on s ssnt fo t obst Conto. II. ODEL UNCERAINY ost conto sns bs on t s of t ttc on of t ssts. A ttc o ocs fo nts to sonss. qt of t o ns on ow cos ts sons tcs to t sst. Snc no sn fx o cn son xct t nt. s scnc ntoc to t no ncs, nctn t nonnts n t on, ffcts of c o o n sst t ton to nonnt cns. s scnc btwn t o n t ct nt s bn cons s o nctnt n obst conto to []. Stbt n fonc of t conto sst s c nfnc b t o nctnt. o nctnt s cssf n two ctos, stbnc sns n nc tbtons. Dstbnc sns occ to t nt ott stbnc, snso nos ctto nos, tc. Dnc tbtons snt t scnc btwn ttc o n t ct ncs of t sst n nc tbtons tt occ n ffnt ts of sst cn, ow, b nto on sn tbton boc, fo nstnc, so no, fqnc ncs. s nctnt sntton s f to s nstct nctnt.in t cs of n, t-nnt ssts, t boc Δ b snt b n nnown tnsf fncton tx. n ffnt ts of nctn sst o sntton [] n t fo of o to b s ns on t of nctnt xct n t tctbt of obst conto ob cosonn to ts nctn sst o. Gnton of LF os Ln fcton nsfoton (LF s n otnt o n on tc nctnt n n ssts. LF bs o sntton [] sf to o tc nctnt ntn ton n t sst tx. s os s n t obst conto cton H conto o conto. Cons cox tx s n t = C q C (( + q + q q n C tcs. ow LF wt sct to F (, : C C ( b two ot cox s q q Wt ttc xsson F (, :( = + I ( o tt t ns ( I s xsts. LF wt sct to s F (, : C C q q Wt F (, :( = + I (3 so o tt t ns ( I s xsts. tx s nown s t co- ffcnt tx. ow n LF s sown n t f. F.. Low LF Rsnttons. F F.. U LF Rsnttons.
3 ow LF sntton xss b t foown st of qtons: z w w = =, (4 = (5 An t LF sntton xss b = =, (6 z w w = (7 III. WIN ROOR IO SYSE Aon tonoos fn ssts, cots tc nc fts. n ffcts n snn contos fo t foow fo nonnts n coss cons. A -DOF o s cons, n n n ost of t cnt wos, o nc focs n toqs ntoc nto t on of t sst. sst s ntstn bcs t s t ossb to fo os xnts n t f of on, ntfcton n conto to. F 3 sows boto o of t wn oto IO ssts (RS, bsc t nts t otts sst (IO. In ctn scts ts bo sbs tt of cot. It conssts of b ot on ts bs n sc w tt t cn o f bot n t ozont n tc n. At t bot n of t b, t two os n b t two c otos. RS sst s n n t oto fo ntn tc n ozont tst. n oto ocs tst to ft t b n tc n n t oto oc foc to t b tn ft o t n ozont n. RS sst s two of fo (DOF ont, on of fo ( DOF n t tc n n on of fo ( DOF n t ozont n ont. sst s q wt contwt nn fo t b, wc s s fo bncn t n ont n st stt o wt o. Ro n B 7 oncs focs of RS conto b n t s of t otos. fo, s ots of t DC otos t conto nt of t sst. It s non n sst n ocss ston coss con btwn t tc n w xs. In ts st Lnn ttc o s tn n to t conston. IV. LF ODELING OF RS ttc on of RS, t sst s o n ts of tc on--of-fo (DOF, ozont (DOF, n two--of-fo (DOF ncs sn Nwtonn s w s Lnn tos [6]. Lnn bs ttc o s s fo t n fcton tnsfoton (LF on. Dffnt ts of snsos s n t ct sst so stbnc sns ntoc nto t sst. ffnc btwn t ttc o n t ct o s to t snc of t nc tbton n nzton of t non-nt. s nctnt cn b nto boc. ((( cos + sn + + sn( + 3 cos( sn ( cos( + [ cos( sn(] [(sn( + + cos(] + + [ ]sn(cos( [ + cos( sn(] + cos( o, + = t o, (8 o, D(,(,,( C,, + + = o, (9 Aft nzton of t (8 n t sstons of s ton of t ozont n tc oston, on obtn t foown qton ( ( o, + + = t o, D + C + E + K = ( = D C D E D K + D ( F. 3. wn Roto IO sst. F. 4. Boc of wn oto IO sst.
4 Ro n B 8 Bs on t ctc constons, t tons of t ont of nt of t f-f b, ont of nt of t contbnc b n scos fcton coffcnt n cons. It s ss tt t onts of nt constnts bt wt ossb t o of 3% on t non s; s, t fcton coffcnts wt 5% t os [3]. fo, t ct onts of nt snt s = (, +, = (3 W s t non of t cosonn ont of nt, =.3 s t x t nctnt n c of ts onts n, =,. D = D + D (4 w t nts D tn b t non s of t ont of nt D=, D = n = + tx D cn b snt s n n fcton tnsfoton (LF. D = F (,( = + I (5 U = D D, = D, = D D n = D D D D = = D D D (6 fo, t ct scos fcton coffcnt snt s = (, + s, = (7 w s t non of t cosonn scos fcton, s =.5 s t x t nctnt n c of ts coffcnt n, =,. C = C + C s (8 w t nts C tn b t non s of t scos fcton coffcnt. s C =, Cs = n = s tx C cn b snt s n n fcton tnsfoton (LF. C = F (,( = + I (7 U = C C, = C, = C C n = C s s C C s C = = (8 C Cs C o snt t RS o s LF of t nctn ts, n,.w fst xtct ot t nctn ts n tn not t nts n otts of n s, n, sct. K K K b - Non s of t ts Sbo(Unt Sbo(Unt ( t ( t ( ts ( s ( ( ( b ( cb ( ( ( ( ( ( ( Non s of t RS tn fo t Fbc nstnt ns [4]. Lt s now cons t nctnts of t fcton coffcnt. H n scos fcton coffcnts n ozont n tc oston of t RS. t b cb ( s ts F.5. Sst boc wt nctn t D D D = D D ( K D + C Cs C = C Cs C (9 (9b = (
5 Ro n B 9 = (b RS stt cto x = [ x x x3 x4 ] b x =, x =, x 3 =, x 4 =. = x x, = x x Hnc [ ] [ ] 3 4 Ott s fn = ( = (b Now ntocn t ott cto = = E ( So W E = Fo t stt qtons of t RS t nt ott tons s sz s x x x x x 3 x 3 (3 x 4 x4 = Π I Π = C C C E I D E D C D D D C Cs D D K D E D C D D D C Cs D D K s = (4 nts( on oo o of t RS s fo,,, n fo otts(,,,. = G (5 ts stt sc sntton of t RS s A B B Gts = C D D C D D nt ott ton of t tb RS s scb b t LF = FU ( Gts, ts (6 wt t on, nctn tx ts = (7 t n F.6. LF sntton of tb wo nt two ott ssts Sn V Pot of RS Now cons t o of otos of t twn oto IO sst. Fo t Fbc nstnt ns cons t n tnsf fncton of t t n n otos Kt Gt = = t s +.384s + K G = = s +.43s + It s ss tt t ct n coffcnt Fqnc (/sc F: 7 Sn s of t wn oto IO sst Kt n K (8 (8b constnt wt t o % on t non s n t t constnts t n wt t o % [3 ]. nctn tnsf fncton t t n n oto G = ( + w Gt (9 t t t G = ( + w G (9b
6 Ro n B W. W, t Wt n nctnt wts. nsf fncton of t wts Wt n W s + 4. Wt = (3 s W s + 4 = (3b s Intocn t nt cto [ ] = tn = G ( I + w. W Gt wt t G =, w = G w n = Lt t nt n ott of t nctnt boc not b t, n t, sct. [ ], [ ] = = t t Fqnc sons of t n oto n t t oto of t RS s n n f 8 n f 9 sct. F.. LF sntton of t tb. otos. Ott of t tb oto s sn t os foc to t tb RS. Not tt t s cox nctnt, w n K nctnts. Bo D F.. Unctnt on of wn Roto IO sst wt otos (n n t oto. B ( t n ( s P Fqnc (/sc F. 8. Fqnc sons of t n oto. o of t tot wn Roto IO sst s n b t qtons n bow W = G ss Gss s tn b t tcs G n G ts. (33 5 B ( -5 n t - ( s P Bo D = FU ( Gss, Wt t on tx = (34 (35-8 Now t oto o bcos - 4 Fqnc (/sc F. 9. Fqnc sons of t t oto. = G = (3 (3 \ F.. LF sntton of RS sst wt nctnt.
7 Ro n B V. DISCUSSION s contbts n xcnt tooo to snt t o nctnts to t ffnc btwn t nz ttc o n t ct sst. In on tc nctnts n n ssts t n fcton tnsfoton (LF s n otnt o. LF-bs snttons sf to o tc nctnts ntn ton n t sst tcs. s os to b s n obst conto toos. boto o of t wn oto IO ssts (RS, bsc t nts t otts sst (IO n n ctn scts ts bo sbs tt of cot. ctc cot s t of fo (DOF ont bt t RS sst s two of fo (DOF ont, on of fo ( DOF n t tc n n on of fo ( DOF n t ozont n ont. nt ott ton of t tb RS s scb b t LF. In LF on of sst, t nstct nctnt conss n sn boc wc b snt b n nnown tnsf fncton tx n s nown s n nctnt tx. LF o sn ts oc s t nt of bn sc nnf n s b to cct snt t nctn nonn o. s LF on s ot n t obst conto st fo nctnt on. LF on of RS s ssnt fo t H conto sn. sts obtn b nz o on fo sffcnt s-n ont n t ozont n t tc n. REFERENCES [] S. Hc n A. V, Gnz LF- Bs Rsntton of Ptc Unctn os Eon on of Conto,Vo./4, , 4. [] G. E. Bo n. H. Cow, on of Nonn Sst Unctnts sn Ln Fcton nsfoton Aoc, Pocns of t Acn Conto Confnc, Vo. 5, , 998. [3] G. Woon, S. Rnn n K. Poo, An LF Aoc to Pt Estton, Pocns of t Acn Conto Confnc, Vo. 3,. 88-9, 997. [4] H. Pff n S. Hc, x Ptc/ nctnt LF on fo Robst Conto Dsn, Pocns of t Acn Conto Confnc, ,. [5] A. V n G. Loo, Sboc n Nc Softw oo fo LF Bs Low O Unctnt on, Poc. CACSD 99 Sos, Co,. -6, 999. [6] A. R n. H. S, ttc nc on of twn-oto t nt t ott sst, Pocns of t IcE, Pt-I: on of Ssts n Conto Ennn,,. 89-, 6. [7] A. R,. H. S n H..C. Hbts, Stb At o Pct Conto fo Nonn Ssts, Acn Conto Confnc, Stt, Wsnton, , 8. [8] P. Goczc n K. H, cn Conto Aots fo Lboto Ao nc Sst, Int.. A. t. Cot. Sc.,Vo. 4, No. 4, , 4. [9] F.. Abz,.S. A,.O. on n. H. S, Gntc on n bton conto of twn oto sst, Pocns of UKACC Intnton Confnc on Conto, Bt, UK, 4. [] B. Pt n S. Pw, N Ntwo Obs fo wn Roto IO Sst: An LI Bs Aoc, Pocns of t Intnton confnc on on, Intfcton n Conto, ,. []K. Zo n.c. Do, Essnts of Robst Conto, st., Pntc H, Nw s,999. [] I. R. Ptsn,V. A. Unos, n A. V. Sn, Robst Conto Dsn sn H tos, st., Sn- V Lonon Lt, Lonon,. [3] Y. C. Pw n G.. Bs, Ptc nctnt on fo LF o zton, Pocn of Cot-A Conto Ssts CACSD, , 8. [4] Fbc Instnts Lt, Cowboo, UK, wn Roto IO Sst 33- Us n, 998.
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