Systematic Configuration Procedure of LMI-Based Linear Anti-windup Synthesis

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1 Sysemac Cofgrao Procere of LMI-Base Lear A-p Syhess a a a Jgcheg Wag Absrac I hs paper, a ovel sysemac cofgrao procere choosg parameers s presee for he syhess of lear a-p scheme by revsg he orgal goal of he fe a-p frameor. he a-p coroller s erve from he parameers, hch are characere by he sffce coos for he sably a performace objecve as o proec he sysem from averse effecs prese of sarao a recover he eergy efc afer sarao. Moreover, he ere syhess s cas as sb-opmao problems over Lear Marx Ieqales (LMI a he effecveess of he resl s sho va smlao examples h comparso o oher exsg a-p schemes. I. IROUCIO HE oo of lear a-p esg has playe a very mpora role he sy of sysems h acaor sarao. he frs paragm hs fel as sae [5], here all prevos lear a-p esgs ere fe a geeral frameor, sho specal cases erms of o marx parameers as choces for lef coprme facoraos of he lear coroller. he, a frher resl poe o [6], hch evelope sffce coos o garaee sably for he geeral frameor. Recely, more lear a-p schemes have bee propose, provg esrable sably properes as ell as performace achevemes (see e.g., [3,7,8,9,]. Who gog o he eal abo hese approaches, or or has focse o he orgal goal of he fe geeral frameor, a obae a seres of ecoragg resls (see, []. I [], by assmg oe of he o parameers o be ero, sably coo a performace objecves are erve he form of LMIs. hese resls gve lgh o sysemac proceres choosg parameers for he syhess of a-p corollers. I hs paper, base o he resls [], he ovel sysemac cofgrao proceres choosg parameers Mascrp receve Sepember, 3. hs or as sppore by aoal aral Scece Foao of Cha (64, aoal aral Scece Foao of Cha (6374, aoal 863 Projec (AA43, SRF for ROCS, SEM. a a s h he Aomao eparme, Shagha Jao og Uversy, Shagha 3, P. R. Cha, (e-mal: aea@sj.e.c. Jgcheg Wag s h he Aomao eparme, Shagha Jao og Uversy, Shagha 3, P. R. Cha, (phoe: ; e-mal: jcag@sj.e.c. are egrae, hch s characere by he sffce coos for sably a performace objecves as o proec he sysem from averse effecs prese of sarao a recover he eergy efc afer sarao. Moreover, he a-p coroller s erve by he parameers, a he ere syhess s cas as a sb-opmao problem over LMIs. oao. he sarao oleary s escrbe as ag( σ ( m σ max ( σ m ( σ m ( max σ max,,m; m <, a max > II. PROBLEM EFIIIO Recall from [4] ha all o lear a-p schemes ca be fe as he mofe coroller: K ˆ ( ( here [ ] ( H C B H H H ( H C I H ( a I- correspo o lef coprme facors of he orgal lear coroller K( as: B K( ( I ( C. Iser he sarao fco he coroller as sho Fg.. Whe here s o sarao, (I, coroller ll be he r + - ( + P( lear coroller. Whe sarao occrs, ( I, he + sarao Fg.. he a-p coroller srcre y

2 feebac by lear coroller oes o or, a he a-p problem has ece o selec sable parameer, H a H, a-p forar coroller ( a yamc compesaor o sable he close-loop sysem a prove gracefl performace egraao. For frher aalyss, sbse Fg. h Fg. (a. he Fg. (a s eqvale o he Fg. (b. he he ercoeco of Fg. (b ca be cocsely re as x& Ax + B + B ˆ H C x + H ˆ + ( I H C x + + ˆ ˆ Where A, B, B^, C,, ^, C,, a ^ are marces of sable mesos. I parclar, H acs epeely o mofyg he sae eqao va A, B, a B^ a H oes ha o he op eqao. ˆ ˆ III. LMI-BASE AI-WIUP SYHESIS A. A-p performace Frsly, coser a ypcal case ha he sysem of Fg. s sarae by sep srbace a/or sep referece p. Sppose ha sarao sars a, a he effecs of he eral saes o he respose are eoe as P ( x p ( a K ( (. Afer he sarao occrs, he respose x sarao of he fferece beee ^ a s represee by r ( ˆ( ( P( ˆ( + P( x p ( s + ( I ˆ( + K( x ( P( (a P ( s ( sarao (b Fg. Saar ercoeco for he A-p srcre For he prpose of proecg he sysem from severe overshoo pla op, he cosrae sgal ms be proece from creasg aflly. Sce ^ s boe a oher varables are fe, a eal solo s (ε a I-ε (f ε, hch sasfe ((I-*K(. So he performace objecve s efe as he lmao of y m (3 (4 he L orm of he a-p forar coroller ( from s p,, o s op, y sp y heorem. he a-p forar coroller ( Fg. s asympoc sablable a has a eghe ce y l ga < ε, hch s a pre-efe pper bo ε, f here exs marx QQ >, sch ha he follog LMI h respec o Q, Y, H s sasfe: A Q + QA C Y YC QB Y C H ( QB Y ε I H < H C H ε I (6 If he LMI s feasble, he H Q - Y a H are he expece parameers o cosrc he a-p coroller. Proof. See he appex. I ao o proec he sysem from severe overshoo pla op, he a-p coroller s also reqre o garaee he sably of he close-loop sysem, a recover he eergy efc prese of sarao, so as o prove sasfe performace egraao. herefore, by choosg he eergy relae form l orm, he performace objecve s efe as he eghe ce l orm from he exogeos p,, o he evao ˆ beee ^ a : sp. efo. (Memoryless me-varyg oleare efe he se V of all alloable srcre memoryless me-varyg oleares as follos: { : R R R (,, V, {,,, }, sec [,] } ag K or. Obvosly, V ypcally cles he sarao oleary presee by oao. heorem. (l ga crero. he a-p sysem (3 Fg.(b s l sablable for all V a has a ˆ eghe ce l ga λ, hch s a pre-efe pper bo, f here exs δ >, PP >, ( W > h,,, R W ag W W K, a WH sch ha he follog LMI h respec o P,, W, δ s sasfe: (5

3 A P + PA PB PBˆ + C B B ˆ P + C I ˆ ˆ δ C W ˆ (7 C < W ˆ he pper bo o he eghe l ga ca be obae from he efo of λ. H W - s he propose solo o he a-p problem. Proof. See he appex. B. Sysemac Procere A-p Syhess Alhogh coos (6 a (7 gve ecoragg resls o avo averse effecs he sarao occrs, a pmp ecessary eergy o recover he efc afer sarao, here are some lmaos. Coserg H acg o he sae eqao he close-loop sysem (3, s ffcl o arse LMI formlao for performace reqremes (7; o he oher ha, he solos (6 oly focs o mprovg he behavor of he a-p coroller a lac of he coserao of he behavor of he ere close-loop sysem. herefore, a cofgrao procere s presee o solve he problems a prove sasfe combao (6 a (7. Sep. Solve he a-p coroller coo Gve he lear coroller K( a pla h a proper smaller real scalar ε, eerme a solo H ha sasfes coo (6. Sep. Mofy he sae eqao of he close-loop sysem Gve oe parameer H from sep, mofy A, B, a B^ he sae eqao of he close-loop sysem (3. Sep 3. Solve he sably a performace coos LMIs efe he pper bo l ga λ, a eerme a solo H ha sasfes coo (7, hch garaees esrable sably properes a a-p performace he close-loop sysem. Sep 4. Cosrc he a-p compesaor Gve he parameers eerme sep a 3, cosrc he a-p forar coroller ( a feebac compesaor he a-p coroller as K ˆ [ ( ]. Sep 5. Valae hs a-p coroller Gve ( a eerme sep 4, compe he scalar ε aga, hch s ormally o ecal h ha pre-efe ε sep, a prove ha s properly small as expece coo (4. IV. COMPARISO O OHER AI-WIUP MEHOS A. Example Coser he follog pla a lear coroller ae 4 5 from [7,] as: P (, s K ( ( +. s.5 A se-po chage of [.63.79] s apple o he sysem a, h a sarao lms of ± o he coroller op. o compare or meho o oher a-p mehos sch as he sac compesaor [7] a he a-p IMC [4,], he performace ex s lse able.. ABLE I PERFORMACE IE IUCE BY IFFERE AI-WIUP SRUCURES. UCOSRAIE;. COVEIOAL IMC; 3. SPECIAL IMC; 4. SAIC COMPESAOR; 5. PROPOSE MEHO I HIS PAPER Meho p ˆ p / / / / / / / / / / By seg ε ag(.,. a λ. 3 p e p prevosly, parameers H a H are eerme from (6..64 a (7, respecvely. Fally, compe he scalar ε aga,.5 hch s eerme as ε. I s small.7 eogh a sasfe he coo (4 as expece. From he aalyss of able., he meho hs paper s speror o IMC mehos [4,] obvosly a s performace resls are parallel o he sac a-p compesaor. Hoever, he lmao of he sac compesaor syhess echqe s ha he LMI cosras are o alays feasble. Or meho s more flexble o ge feasble sb-opmal solos from he coos a he expese of he opmal solo. hs ll be prove he follog example here he lear sac a-p

4 compesaor s o feasble for. B. Example he ypcal example as a ampe mass-sprg sysem s ae from [3]. By selecg ε., H [.5. ] s eerme coo (6. he by seg r, e, a a smaller pper bo as λ.4, parameer H.99 * s eerme from (7. he respose of cosrae, cosrae a a-p se meho are sho Fg.. Or meho performs sasfacorly, parallel o he meho [3, ], a cofrms he effecveess of he sysemac proceres o choose parameers. (a Op of he Pla (b Ip of he Pla Fg.3 Example pla respose: ashe, cosrae; oe, cosrae ho a-p; sol, a-p meho hs paper V. COCLUSIO Movae parly from he eas fe a-p frameor a exee he or [], he lear a-p paragm propose hs paper focses o he evelopme of sysemac proceres o choose he o marx parameers for he syhess of a-p coroller. A he performace of a-p syhess s characere by he l orm of he evao beee he coroller op a pla p, hch s relae o he eergy of he cosrae sgal a he cosrae sgal. herefore, he propose a-p coroller ca 5 proec he sysem from creasg aflly he cosrae sgal (coroller op, a recover he eergy efc he cosrae sgal (pla p relao o he cosrae corol. Moreover, he performace objecves are sho o ce sb-opmal problem over LMIs. From he comparso beee he propose meho a oher exsg a-p mehos va smlao examples, or meho shos gracefl performace egraao, parallel o f o more effecve ha oher schemes, he presece of sarao, a cofrms he sccess of he sysemac proceres o choose parameers. APPEI Proof of heorem. By applyg a smple cogrece / / / rasformao ag{ ε I, ε I, ε I } o (6, a efg M ε Q, a a sable small real scale < γ <, coo (6 s garaee f here exss a marx M>, sch ha ( A M + M ( A M ( B H ( B H M ε ( γ I H C H C H H < I (A I mples ( A M + M ( A <. Sce M>, he forar coroller ( s asympocally sable a V ( x x ( Mx( s a Lyapov fco of he sysem. If > a he al coo s assme as x, ge J ( ( γ ε ( [ ( ( ( γ ε ( ( + V ( x] V ( ( x( ( H C ( [ H C ( ( H H ] ( A M + M ( A + ( B H M V ( x( M ( B ε If coserg (A, a sg he al coo, ( γ ε ( < ε ( ( H x( ( I ( γ, ge Proof of heorem. Lemma (Mlloop crcle crero []. Le P ( eoe he rasfer fco relag o Fg.(b. he, he close-loop Fg. s l sablable for V f. P ( s Fg.(b s asympocally sable; a

5 .here exs (,,, R W ag W W K W >, h PP >, δ >, a WH, sch ha he follog LMI h respec o P,, δ s sasfe A P + PA PB ˆ C < (B Bˆ P C δi ˆ ˆ Remar. Obvosly, (B s garaee by he ma resl (7. Lemma. (Lyapov sably crero [6]. efe Lyapov fco, hch s eqvale o (B as: V ( x x Px + ˆ ( τ δˆ( τ τ (B + W ( ˆ ( τ ( τ ˆ ( τ ˆ ( τ τ he, he close-loop sysem Fg. (b s L sablable for all V, f here exs PP >, δ >, a W ag( W, W, K, R > W, sch ha: V(x> for x a V(; V(x sasfes V ( x <, for x. Remar. V(x> for x s garaee by he P>, δ >, a W>; V( s obvosly; V ( x <, for x s eqvale o he coo (B, a hs garaee by (7. herefore, (7 garaees he sably of he close-loop sysem. o, he ma resl abo performace coo (7 s sae. Coo (7 s eqvale o he exsece of a marx ( W > W ag W, W, K, R, h δ >, PP >, a λ sch ha x Px + ˆ δ ˆ τ + W ( ˆ {( ˆ ˆ λ } < ˆ ˆ τ + W Iegrae (B3 from o h he al coo x, a ge V ( x + W Sce V(x>, mples [( ˆ ˆ λ ] τ ˆ λ. ˆ ˆ ˆ λ hch sasfes he performace objecve. Coo (B3 s garaee for all (x,, ^ f J x Px + ˆ δ ˆ τ + W ( ˆ + W {( ˆ ˆ λ } ˆ ˆ τ (B3 [ x ˆ ] A P + PA PB B WH C B P WH + ˆ C H H ( ( I H ˆ PB C H W ˆ + H W β x < C H ˆ H ( I H ˆ β δi H ˆ ˆ H W Becase J <, he above marx s egave efe, he eqvale coo s obae as follos by applyg Schr compleme: A P + PA PB PBˆ + C H W B H W B ˆ P + WH C WH β H C H I H ˆ (B4 C H I H ˆ H < By applyg a smple cogrece rasformao ag(i, I, I, W o (B4 a efg WH, (7 ll be obae. REFERECES [] Aerso B O a Moore J B. Lear opmal corol. Egla Clffs, J: Prece Hall.97. [] a,., a Wag, J.C. A-p Compesaor esg Usg LMI-base yamc Feebac. IEEE Ieraoal Coferece o Sysems, Ma & Cyberecs. Washgo, USA, Oc. 5-8, 3 (o be appeare. [3] Grmm, G., eel, A.R. a Zaccara, L. Robs LMI-base lear a-p esg:opmag he cosrae respose recovery. Proc. 4s IEEE Cof. o ecso a Corol. evaa, USA,, [4] Has, R. a Kaer, M. Corol of cosrae mlvarable sysems sg he coog echqe. Proceegs of he 989 Amerca Corol Coferece, Psbrgh, PA, 989,7-78. [5] Kohare, M. V., Campo, P. J., Morar, M. a e, C.. A fe frameor for he sy of a-p esgs. Aomaca, 994, 3(: [6] Kohare, M. V. a Morar, M. Mlpler heory for sably aalyss of a-p corol sysems. Aomaca, 999, 35(5: [7] Mler, E. F., Kohare, M. V., a Morar, M. Mlvarable a-p coroller syhess sg lear marx eqales. Aomaca,, 37(: [8] Waa,. a Sae, M. esg of a sac a-p coroller base o marx eqales. Proc. 35h IEEE Cof. o ecso a Corol. Kobe, Japa, 996, 6-6. [9] Weso, P. F. a Poslehae, I. Lear coog for sysems coag sarag acaors. Aomaca,, 36(: [] Zheg, A., Kohare, M. V. a Morar, M. A-p esg for eral moel corol. Ieraoal Joral of Corol, 994, 6(5: 5-4. [] Zaccara, L. a eel, A.R. A commo frameor for a-p bmpless rasfer a relable esgs. Aomac,, 38(: