A Robust H Filter Design for Uncertain Nonlinear Singular Systems

Transcription

1 A Robus H Filer Desig for Ucerai Noliear Sigular Sysems Qi Si, Hai Qua Deparme of Maageme Ier Mogolia He ao College Lihe, Chia College of Mahemaics Sciece Ier Mogolia Normal Uiversiy Huhho, Chia Absrac For a ype of ucerai geeralized ime-delay sysems wih Noliear perurbaio, he robus H filer desig of such forms he focus of his paper I is proved ha he filerig error dyamic sysem o be robusly asympoically sable ad has a sufficie codiio of H orm bouded wih Lyapuov fucioal mehod ad liear marix iequaliies wih ukow parameers Expressios of he robus H filer of such geeralized sysem are give o he basis of he correspodig feasible soluio of liear marix iequaliy A umerical example shows he feasibiliy ad validiy of his mehod Key words - geeralized sysem oliear liear marix iequaliy filer I INODUCION I he pas few decades, H filerig problem has always bee a ho research opic i corol heory ad applicaio 3 Whe he oise is ay eergy limied sigal, H filer ca esure ha he level of oise suppressio is give Lieraure [4] uses H filer o elimiae he wave perurbaio so ha he sable filerig effec is obaied he geeralized sysem widely exiss i egieerig sysems, such as elecric power sysems, uclear reacors, limied robos ad so o While he uceraiy, ime delay ad olieariy i he acual sysem is ieviable Uceraiy ad ime delay are always he mai facors which lead o he sysem isabiliy or he performace degradaio While he oliear brigs greaer difficuly for sysem sabilizaio, filerig, faul deecio ad so o herefore, o sudy ucerai robus filerig problem of oliear geeralized sysems wih ime delay has impora heoreical sigificace ad applicaio value Lieraure [5-6] sudies H filer desig problem for geeralized sysems, bu i does ivolve he ime-delay ad ucerai siuaio Lieraure [7] sudie he robus H reduced-order filer problem of he ucerai muliple ime-delay sigular sysems bu i does cosider H filer problem of geeralized sysems which coais he oliear perurbaio A prese, we do o see he relaed repors abou his kid of problems We adop he ool of Lyapuov fucioal mehod ad he liear marix iequaliy o sudy he robus H filer desig problem of ucerai geeralized sysems wih oliear perurbaio i he aricle Ad we give he sufficie codiio whe robus H filer problem has he soluio We also give he srucural mehod of robus H filer based o liear iequaliies II PROBLEM DESCRIPION AND PREPARAION We ca cosider he followig ucerai oliear geeralized ime-delay sysems Ex A AxAAx d f, x B y CCx CCx d D () z HHx HHx d D x, d,0, Cd,0, R x Amog hem, R is he sae vecor of sysem q z R is corol oupu vecor L 0, is p -dimesioal perurbaio ipu vecor f f, xr is oliear vecor fucio, ad i f saisfies,0 0 ad he followig Lipschiz codiio f x, f x, F x x, x, x R is a posiive umber, () Amog hem, 0 F R is cosa marix he orm represes he disace fucio i R -dimesioal space Accordig o he las equaio, we ge f x, Fx (3) DOI 003/IJSSSa7 ISSN: x olie, pri

2 he marix E, A, A, B, C, C, D, H, H ad D are kow cosa marixes wih proper dimesios d is delay ime ad d 0 rake r A, A, C, C, H, H are ucerai imevaryig marixs wih proper dimesios Suppose ha he followig forms exsi A AL C C L F M M (4) H H L 3 Amog hem, L, L, L, M, 3 M are kow cosa F marixes wih proper dimesios is bouded F F I ucerai fucio marix ad saisfies he purpose of his aricle is o desig a full order liear ime ivaria filer E c Ac Bcy, 00 (5) zˆ Cc z So we ca obai he esimaio of sigal Combie he saes of sysem () ad (5) Ad he dyamic sysem of augmeed filer error is Ex Ax Ax d f, x B (6) z Cx CxdD x,0 d,0 Amog hem, z z zˆ x x, f f x, f x,,0 0 E f, x Fx E 0 E c A 0 A 0 B A A BC c AcBC B c 0 Bc D C H C c C H 0 D D F 0 F 0 0 A AA A A A C CC CC C H H H H H H (7) he purpose of his aricle is o desig he robus filer as he equaio (5) herefore, for he uceraiy which saisfies he allowable codiios, he dyamic sysem (6) of augmeed filer error is asympoically sable Uder zero iiial codiios, for all o-zero L 0,, here is z, L 0, Lemma 8 sigular sysem Ex Ax I is regular ad impulse -free I will exsi whe ad oly whe he marix P s E P P E 0 APPA 0 As far as Lemma 9 F for ay marix is cocered, F F I if, for ay 0, here is x PBF Ex x PBB Px x E Ex For he real marixes, H, wih appropriae dimesio give by lemma 3, amog hem is FH H F 0 symmerical, he If F F F I all he marixes which saisfy is esablished, whe ad oly whe a cosa 0 exsis, i will make H H 0 III HE PERFORMANCE ANALYSIS OF ROBUS H We sudy H filerig problem i his secio, H performace adjusme of he dyamic sysem ( 6 ) of filerig error ca be give i he followig heorem heorem For ucerai oliear geeralized imedelay sysem () ad he give posiive cosa 0, if he symmeric posiive defiie marix P exsis, S R ad he scalar 0, i should saisfy he iequaio E P P E 0 (8a) APPAS PA PB C F F P P 0 * S 0 C * * I D * * * I (8b) he he dyamic sysem (6) of filerig error is asympoically sable, ad i saisfies H oise suppressio level DOI 003/IJSSSa7 ISSN: x olie, pri

3 he Proof: Firs we prove ha he dyamic sysem(6 )of filerig error is regular ad pulse-free he equaio (8a) ad equaio(8b) are esablished ad S 0 herefore, E P P E 0 APPA 0 From he Lemma we kow ha he dyamic sysem(6 )of filerig error is regular ad pulse-free Nex, we will prove ha he dyamic sysem(6)of filerig error is asympoically sable So we ake he Lyapuov fucio, V x x PEx x ssxs ds d V x We ake he derivaive of alog he dyamic sysem(6)of filerig error, so we ge V x x PEx x PEx x A PPAS x x Sx x d Sx d x PAx d x Pf x PB x d Sx d From he Lemma we ge V x x A PPAS F F P P x x PAx d x PB x d Sx d 0 Whe,here is x PA x V x x d * S xd Amog hem A PPAS F F P P From he Schur compleme lemma of marix we kow ha if marix iequaliy ( 8b ) is esablished, here is APPAS FF PP PA 0 * S ha V x 0 is o say, egaive defiiio Accordig o he Lyapuov heorem, he dyamic sysem(6)of filerig error is asympoically sable A las we prove ha uder zero iiial codiio H oise suppressio level of dyamic sysem (6)of filerig error makes he give 0 be brough i H performace idex: J z z d 0 We make use of he srucural Lyapuov fucio V x ad zero iiial codiio For ay ozero L 0,, here is J z z V x 0 d V 0 V 0 z z V x d Ad x C Cx x C Cx d x dc DD D * z z Cx C x d D Cx C x d D x C D x d C C x d So x PA C C J x d * S C 0 C * * PB C D x C D x d d I D D Amog hem A PPAS F F P PC C Accordig o he Schur compleme lemma of marix we kow ha if marix iequaliy(8b)is esablished, here is PA C C PB C D * SC C C D 0 * * I D D So ha J 0 z,ha is o say, herefore, he dyamic sysem ( 6 ) of filerig error saisfies H oise suppressio level he proof is over DOI 003/IJSSSa7 3 ISSN: x olie, pri

4 IV HE DESIGN OF ROBUS FILER Filer desig mehod is give i he followig heorem heorem If, for he give cosas, symmeric posiive defiie marix ad appropriae dimesio marix exsi so as o make he followig marix iequaliy be esablished EX X E (9a) E Y Y E (9b) X Y (9c) * 88 (9d) Amog hem ˆ X AXA AH A 0 B 6 * H * * S S 0 H * * * S3 0 0 * * * * I D * * * * * I XF 0 I 0 X M XM L F 0 Y N I 0 M M L3 I * I * * I * * * I * * * * S S 0 0 * * * * * S * * * * * * I 0 * * * * * * * I 6 X H G ˆ Y A A Y ˆ ˆ FC C F 3 Y A FC ˆ 5 Y BFD ˆ 5 Y BFD ˆ 4 Y L FL he he filer problem of sysem () has he soluios ad filer parameers safisfy Bc N Fˆ Cc GM X A X X AX X A XB * S 0 * * I * * * * * * * * * * * * XC X F X X C D I * I 0 0 * * I 0 * * * S (4) I he equaio(4), we ake S S S S S 0 * S S 0 3 * S 3 A he same ime, we make ˆ F NB c G CcM ˆ H Y AX NBCX c NAM c (5) We ake he forms of marix S 0, S 0, X ad X be subsiued io he equaio (7) ad lemma 3 Ad he we ca ge he equaio(9d) from he equaio(7) ad lemma 3 A las we give he desig seps for H filer of sysem () Ad we show hem as follows We solve for he marix X, Y, G, F ad Ĥ by marix iequaliy(9d) We ake he righ side of equaio () o make marix decomposiio o solve for he osigular marix M ad N We make X, Y, G, F, H ˆ, M ad N be subsiued A he equaio (5) so as o ge he soluios,, c Bc, Cc ad E c he proof is over DOI 003/IJSSSa7 4 ISSN: x olie, pri

5 V HE SIMULAION EXAMPLE We cosider ucerai oliear geeralized imedelay sysems () Amog hem all he coefficies ca be see as follows E A 4 A 0 B C C 3 D 0 D H H L 0 00 L 0 00 L3 0 0 M M I si x f, x si x x x x x he here is f x, si x si x x x x x x Fx Amog hem F We ake 0 Accordig o heorem, we use he oolbox of MALAB o ge a feasible soluio I is X Y G ˆ F H ˆ We ake he righ side of equaio () o make marix decomposiio o solve for he osigular marix M ad N M N We ake X, Y, G, F, H ˆ, M ad N be subsiued A he equaio (5) so as o ge he soluios,, c Bc, Cc ad E c B c C c A c E c VI CONCLUSION Aimig a a kid of ucerai geeralized ime-delay sysems wih oliear perurbaio, we sudy he desig mehod of H filer i his aricle We use he iequaliy ool of Lyapuov fucio ad liear marix o ge ha he robus of filerig error dyamic sysem is asympoically sable ad have he sufficie codiio ha H orm is bouded Ad he we make i be rasformed io he liear marix iequaliy wihou parameer ucerai marix Based o he correspodig feasible soluio of liear marix iequaliy, we give his kid of expressios abou geeralized sysem robus H filer he give resuls all be give i he form of sric liear marix iequaliy so ha he soluio process will be coveie ad simple if LMI oolbox of Malab is adoped REFERENCES [] YU L, CHU J, A LMI approach o guaraeed cos corol of liear ucerai ime-delay sysems,auomaica, vol 35, pp 55-59, 999 [] MU KAIDANI H, A LMI approach o guaraeed cos corol for ucerai delay sysems, IEEE ras,circuis ad Sysems I : Fudameal heory ad Applicaios, vol 50, No 06, pp , 003 [3] L IEN C H, HOU YY, Guaraeed cos observer-based corol for a class of ucerai ime- delay sysems,joural of Dyamic Sysems, Measureme, ad Corol, vol 7, No, pp 73-78, 005 [4] Wu L G, Wag C H, Zeg Q S, e al, Robus slidig mode filerig for a class of ucerai oliear discree ime sae delayed sysems,aca Auomaica Siica, vol 3, No 0, pp 96-00, 006 DOI 003/IJSSSa7 5 ISSN: x olie, pri

6 [5] Xu Sheg yua, Lam Jams, Reduce -order H filerig for sigular sysem,sysems ad corol Leers, vol 48, No 0, pp 48-57, 007 [6] CHEN Lig, ZEN Jia-pig, Reduced-order H filerig desig for sigular sysem,joural of Xiame Uiversiy (Naural Sciece), vol 47, No 04, pp , 008 [7] WU Bao-wei, ONG Yu-xu, CHEN Mi, A desig of robus H reduced-order filer for ucerai sigular sysems wih muliple ime-varyig sae delays,joural of Shaaxi Normal Uiversiy (Naural Sciece), vol 36, No 0, pp 7-4, 008 [8] Lu Re-qua, Su Hog-ye, Chu Jia, e al, Robus corol heory of sigular sysem,beijig: Sciece Publishig House,008 [9] BAO Ju-dog, asympoic sabiliy o a class of oliear eural sysems,joural of Jishou Uiversiy (Naural Sciece Ediio), vol 6, No 30, pp -7, 009 [0] Wag ia-cheg, GAO Zai-rui, e al, corol for a class of ucerai oliear sigular sysems wih ime-delay,corol Eegieerig of Chia, vol 5, No 6, pp 64-67, 009 [] WO Sog-li, SHI Guo-dog, ZOU Yu, e al, Robus H corol for sigular sysems wih oliear perurbaio, Corol ad Decisio, vol 3, No 4, pp , 009 Abou he auhors: Si Qi, Xig'a, League People i Ier Mogolia, lecurer, maser, maily egaged i he sudy of geeralized sysem corol ad exac soluio of oliear differeial equaio siqi@63com Hai Qua, Kulu, Baer People i Ier Mogolia, lecurer, maser, maily egaged i he sudy of geeralized sysem corol baohq@imueduc DOI 003/IJSSSa7 6 ISSN: x olie, pri