ANTI-WINDUP CONTROLLER PARAMETERIZATIONS

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1 Arif Syaih, Rohman, Anti-Windp Controllr Paramtrizations ANI-WINDUP CONROLLER PARAEERIZAIONS Arif Syaih-Rohman Shool of Eltrial Enginring & Informatis Institt knologi Bandng, Jl. Gansa, Bandng 43, Indonsia ABSRAC Following a linar ontrollr dsign, an anti-windp ompnsation is a poplar approah that may b takn to dal with inpt satration. hr hav bn many anti-windp thniqs proposd. Basd on a transfr fntion paramtrization of th rslting anti-windp ontrollr, ths antiwindp thniqs may b lassifid into two atgoris, whih may b alld -dgr of frdom (-DOF) and -dgr of frdom (-DOF) paramtrizations. Using nwly known qivaln btwn a mltivariabl nonlinar algbrai loop and a onstraind qadrati programming, two kind paramtrizations of som xisting anti-windp ompnsations ar xplaind. INRODUCION Atator satration is a biqitos onstraint that may ind advrs ffts in any ontrol systms. For linar systms, thr hav bn many approahs proposd to ovrom th ffts of ontrol inpt satration. Anti-windp approahs (inlding onditioning thniqs) ar widly poplar in whih an anti-windp ompnsator is only ativ in th nonlinar rgion or a linar ontrollr tak ontrol othrwis, s Fig.. r y ζ K P v Fig. An anti-windp shm h anti-windp ompnsator may b a stati gain matrix (.g. (ldr t.al., )) or a dynami transfr matrix (.g. (Grimm t.al., 3). Som of anti-windp shms may inld a nonlinar algbrai loop. It is illstratd in (ldr t.al., ) that a nonlinar algbrai loop in an anti-windp shm may improv th prforman of its losd loop systm ndr inpt satration. h rslting anti-windp ontrollr is a ombination of th nominal ontrollr and anti-windp ompnsator. Basd on a transfr fntion paramtrization, ths anti-windp approahs may b lassifid into two atgoris, whih may b alld -dgr of frdom (-DOF) and -dgr of frdom (-DOF) paramtrizations. In fat, a nifid viw of som anti-windp approahs that has bn prsntd in (Kothar t.al., 994) may b prsntd in a -DOF paramtrization. anwhil, th -DOF paramtrization may b fond in 8

2 JURNAL EKNIK GELAGAR Vol. 8, No., Oktobr 7 : 8 85 (Edwards and Postlthwait, 998). Howvr, a nonlinar algbrai strtr that may aris in th anti-windp shms has not takn into aont proprly in that paramtrization. Using th qivaln btwn a mltivariabl nonlinar algbrai loop and a onstraind qadrati programming that has bn rntly known (Syaih- Rohman t.al., 3), two kind paramtrizations, as in (Kothar t.al., 994) and (Edwards and Postlthwait, 998) of som xisting anti-windp ompnsations ar rvisitd and thn rformlatd hr. Svral stati anti-windp shms, i.. th anti-windp ompnsator that has a zro ordr or a stati matrix gain, ar spially onsidrd in this papr. h anti-windp (ompnsatd) ontrollr of th nominal on may b formlatd as follows: x& = A x B ζ () = C x D ζ = ζ () ζ Λ = Λv = ( ˆ) (3) ζ Λ Not that th anti-windp ompnsation is only ativ whn satration ors, i.. v. NONLINEAR ALGEBRAIC LOOP Considr a fdbak systm of Fig. in whih Ψ rprsnts a mltivariabl satration fntion with ± satration lvl. It is provd in Syaih-Rohman t.al, 3) that a nonlinar algbrai loop in Fig. is qivalnt to a onstraind qadrati programming (QP) problm (= >), that is ˆ = arg min ( ) ( ) (4) { } sbjt to th onstraint. (5) Ψ I Fig. A mltivariabl nonlinar algbrai loop Indd, th algbrai loop in Fig. is a wll-posd nonlinar algbrai loop and may b dnotd as Ψ with th following rlations: = ( I ) ˆ, (6) ˆ = Ψ. (7) In anti-windp shms, a nonlinar algbrai loop may aris and is gnrally a wll-posd non-symmtri loop, >. Using () and (3), th following rlation is obtaind. = ( I Λ ) ( I ( I Λ ) ) ˆ. (8) Hn, it is lar that (7) and (8) form a mltivariabl nonlinar algbrai loop with : = ( I Λ, (9) ) and th loop will hav a onstraind QP qivalnt problm if is symmtri. h paramtrization of som antiwindp shms (inlding som onditioning thniqs) to rgard th xistn of th qivalnt nonlinar algbrai loops within th shms may now b formlatd. h formlation will b pt into both -DOF and -DOF sing L and as paramtr matris. 8

3 Arif Syaih, Rohman, Anti-Windp Controllr Paramtrizations Rfrring to (Kothar t.al., 994), th anti-windp approahs that ar onsidrd hr ar th on in (ldr t.al., ) and onditioning thniqs in (Hans t.al., 987), (Hans & Kinnart, 989) and th mltivariabl vrsion of (Walgama t.al., 99). his vrsion is a shm with non-filtrd ralizabl rfrn, s also (Png t.al., 998) that ss optimal ralizabl rfrn omptation or optimal dirtion hangr as part of th anti-windp shms. Analyzing thos anti-windp shms or approahs, th assoiatd L and paramtr matris may b obtaind as th following. In (ldr t.al., ), th paramtr matris ar L := Λ ( I Λ ) and := ( I Λ ). [] anwhil, th onditioning thniq of (Hans & Kinnart, 989) has L := B D and := D D. [] Similarly, th shm in (Png t.al., 998) ss th sam L as in (Hans & Kinnart, 989) bt with := D Π D, [] for any sr-hosn Π = Π >. As for th onditioning thniq of (Walgama t.al., 99), it has L B D ρi := ( ) and := ( ρ ) ( ρ ), [3] D I D I with ρ <. It is lar that dfining = I mans disrgarding th algbrai loop or making Ψ = Ψ. -DOF PARAEERIZAION Considr a -DOF paramtrization as illstratd by Fig.3 in whih K(s) is a nominal (linar) ontrollr and X(s) is an anti-windp ompnsator K( s) X ( s) Ψ Fig.3 -DOF paramtrization of antiwindp ontrollr In th -DOF paramtrization, th antiwindp ontrollr is paramtrizd by a linar transfr fntion X ( s ) (s [Campo, Posthlthwait]), i.., = ( I X ( s)) K( s) ( I X ( s)) X ( s). ˆ [4] h stat spa ralization of th nominal ontrollr K( s ) and th transfr fntion paramtr X ( s ) ar as follows: A B A L K( s) := ; X ( s) := C D. C I Using th rlatd L and matris to th antiwindp approahs, abl prsnts th dtails of stat spa ralization of X(s). Not that if =I (withot algbrai loop), thn th paramtrization rslts in (Edwards & Postlthwait, 998) will b obtaind. Othrwis, th sond olmn of th tabl is obtaind. -DOF PARAEERIZAION anwhil, in th -DOF paramtrization (s Fig.4), th ontrollr is paramtrizd by two transfr fntions K ( s ) and 8

4 JURNAL EKNIK GELAGAR Vol. 8, No., Oktobr 7 : 8 85 K ( s ) (s (Png t.al., 998)), or K ( s) and K ( s), i.., = K ( s) K ( s) ˆ := Κ ( s), ˆ [5] with th following stat spa ralizations: A LC B LD L Κ ( s) := C D I, [6] whr A LC B LD K( s) := C D,and A LC L K s := C I. [7] K( s) Ψ K ( s) Fig.4 -DOF Anti-windp Paramtrization abl that is similar to abl may thn b dvisd for th as of -DOF paramtrization. Again, if nonlinar algbrai loops ar not to onsidr (i.. =I), som shms basd on onditioning thniq will fit to th framworks as rportd in (Kothar t.al, 994). In addition, as prsntd in (Kothar t.al., 994), th nominal ontrollr may also b writtn as th following a lft oprim paramtrization : K( s) := V ( s) U ( s) = V ( s) U ( s) [8] h assoiatd -DOF paramtrization transfr fntions ar K ( s) := U ( s), and K ( s) := I V ( s). [9] Now, if th nonlinar algbrai loop is to onsidr, th following transfr fntions of th -DOF paramtrization will b obtaind. ( s) K ( s) U ( s) K = :=, ( s) I ( I K ( s)) I V ( s) K = := [] whr ( s) [ ( s) ( s) ] Κ = K K. It is thn lar that matrix, whih is involvd in th nonlinar algbrai loop, is th ommon fator of th lft oprim paramtrization of th nominal ontrollr. CONCLUDING REARKS Using nwly known qivaln btwn a mltivariabl nonlinar algbrai loop and a onstraind qadrati programming, two kinds of paramtrizations of som xisting antiwindp ompnsations ar xplaind. In partilar, th s of qadrati programming (QP) in som anti-windp shms may now b modld as a nonlinar algbrai loop that nabls a mor appropriat paramtrization of th shms. 83

5 Arif Syaih, Rohman, Anti-Windp Controllr Paramtrizations abl Stat spa ralization of X(s) Anti-windp Shm X ( s ) I = I (ldr t.al., ) a A Λ A Λ C Λ C Conditioning thniqs b A B D A B D C D D I C (Walgama t.al., 99) A B D ρi A B ( D ρi) C D ρi D ρi I C a Λ = (withot algbrai loop) b (Hans, 989) or (Png t.al., 998) with (Hans, 987) Anti-windp Shm (ldr t.al., ) (Hans & Kinnart, 989) a (Walgama t.al., 99) Π = I abl Stat spa ralization of Κ ( s) Κ ( s) : I A Λ ( I Λ ) C B Λ ( IΛ ) D Λ ( IΛ ) ( IΛ ) C ( IΛ) D Λ ( IΛ) A B D C B D D D C D A B D ρi C ρb D ρi B D ρi ( D ) ρi D ρi C D ρi D ρi D Anti-windp Κ ( s) : = I Shm (ldr t.al., A ) b ΛC B ΛD Λ C D (Hans t.al., A B D C B D 987) C D (Walgama t.al., A B ( D ρi) C ρb ( D ρi) B ( D ρi) 99) C D a or (Png t.al., 998) with Π = I b Λ = (withot algbrai loop) 84

6 JURNAL EKNIK GELAGAR Vol. 8, No., Oktobr 7 : 8-85 REFERENCES C. Edwards and I. Postlthwait, Anti-windp and Bmplss-transfr Shms, Atomatia, Vol.34, No., pp.99-, 998. G. Grimm, J. Hatfild, I. Postlthwait, A. R. l,. C. rnr and L. Zaarian, Antiwindp for Stabl Linar Systms With Inpt Satration: An LI-Basd Synthsis, IEEE ransations on Atomati Control, Vol.48, No.9, pp.59-55, Sptmbr 3. R. Hans and. Kinnart, Control of onstraind mltivariabl systms sing th onditioning thniq, Prodings of th Amrian Control Confrn, Pittsbrg, PA, USA, pp.7-78, 989. R. Hans,. Kinnart, and J.L. Hnrott, Conditioning thniq, a gnral antiwindp and bmplss transfr mthod, Atomatia, Vol.3, No.6, pp , V. Kothar, P. J. Campo,. orari and C. N. Ntt, A nifid framwork for th stdy of anti-windp dsigns, Atomatia, Vol.3, No., pp , 994. E. F. ldr and. V. Kothar and. orari, ltivariabl anti-windp ontrollr synthsis sing linar matrix inqalitis, Atomatia, Vol.37, No.9, pp.47-46,. Y. Png, D. Vrani, R. Hans and S. Wllr, Conditioning thniq: a gnral antiwindp and bmplss transfr mthod, Atomatia, Vol.34, No., pp , 998. Syaih-Rohman, R. H. iddlton and.. Sron, A ltivariabl Nonlinar Algbrai Loop as a QP with Appliation to PC, Prodings of th 7 th Eropan Control Confrn, Cambridg, UK, Sptmbr 3. K. S. Walgama, S. Ronnbak and J. Strnby, Gnralization of onditioning thniq for anti-windp ompnsator, IEE Prodings on Control hory and Appliations, Vol.39, pp.9-8,

Lecture 14 (Oct. 30, 2017)

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