Stochastic State Estimation and Control for Stochastic Descriptor Systems

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1 Sochasc Sae smaon and Conro for Sochasc Descrpor Sysems hwe Gao and aoyan Sh Schoo of ecrc and ecronc ngneerng ann Unversy ann 372, Chna e-ma: bsrac In hs paper, a sochasc observer s proposed, whch can make he esmaon error dynamcs o be sochascay asympoca sabe and mpuse-free. Moreover, an negraed esmaon and feedback conro mechansm s presened, whch can make boh he sae and error dynamcs sochascay asympoca sabe and mpuse-free. Smuaed resus demonsrae he effcency of he proposed approach. Fnay, hs echnque s exended o fuzzy sochasc sysems. eywords sochasc sysems, decrpor sysems, sgna esmaon, fuzzy sysems. I. INRODUCION Sae and sgna esmaon and reconsrucon are aways ho n sgna processng and conro communy. here have been huge resus repored [-3]. Sochasc descrpor sysem s a more compex sysem compared wh he convenona deermnsc or sochasc modes. hs s because hose sochasc descrpor sysems possess snguar naure and sochasc behavors. herefore, nvesgaon on sgna esmaon and conro for sochasc sysems s of sgnfcance, bu chaengng. However, here are few resus for sochasc descrpor sysems, excep for some med work n aman ferng [4-5]. In hs sudy, we consder a descrpor sysem mode wh Iô formua. descrpor sochasc esmaor s proposed. Moreover, an negraed esmaon and conro mechansm s deveoped. II. Consder he foowng sochasc sysem PROLM SMN dx x x u y Cx n x R s he sae vecor, u R s he conro p npu, y R s he oupu vecor, and w s he onedmenson rownan moon. Now we gve he foowng defnons: Defnon. Sochasc sysem s caed sochasc sabe, f for any ε, and r >, here s aways a posve number δ δ ε, r, > such ha P { x ;, x r for a } ε x δ. Defnon 2. Sochasc sysem s sochasc asympocay sabe, f sysem s sochasc sabe, and for any ε,, here s δ δ ε, such ha > P { m x ;, x x δ. Defnon 3. Sochasc sysem s admssbe f he sysem s sochasc asympocay sabe and mpuse-free. Lemma [6]. Sochasc sysem s admssbe f here s marx such ha he foowng nequaes hod: } ε s he Moore-Penrose nverse of. he goa of he sudy s o desgn an esmaor such ha he esmaon error s admssbe. m /8 /$ I RM 28

2 III. SIMOR DSIGN FOR SOCHSIC SSMS. smaor Desgn Desgn he foowng esmaor: dxˆ xˆ u G yˆ y 2 yˆ Cxˆ Subracng 2 from, and eng e x xˆ, he error dynamcs are governed by he foowng equaon: de GC e x 3 x Le ξ, and hen consruc he foowng e augmened sysem ~ ~ ~ dξ e ξ 4 ~ ~ ~,,. 5 GC ~ ~ ~ I s obvous ha rank rank, whch means ha he nose erm does no change he sysem srucure. heorem. here s a sochasc observer n he form 2 for sysem, f here s marx such ha ~ ~ 6a ~ ~ ~ ~ ~ ~ ~ 6b ~, ~ and ~ are defned as 5. Proof: hs resu can be obaned drecy by usng Lemma. Remark. heorem ndcaes ha ξ, and hen means e and x equvaeny. hs means ha he proposed observer above requres sysem s admssbe. However, hs conon does no hod aways n pracca cases. herefore, hs movaes us o make some mprovng for he esmaor desgn.. smaor-ased Conroer Nex, we w desgn a mechansm o make boh he observed sysem and esmaon error o be admssbe. ppyng he foowng feedback u xˆ 7 o sysem, one has dx x x xˆ x x e 8 x Denoe byξ. Usng 3 and 8, one has e dξ e ξ 9. GC he foowng heorem s gven. heorem 2. Sysem 9 s admssbe f here s marx such ha a b, and are defned as. Proof: he resu can be obaned drecy n erms of Lemma. heorem 3. Consder sysem, sae esmaor 2, and sae feedback 7. If here are marces and such ha 2a 2b 2c GC GC 2d hen he cosed-oop sysem s admssbe. Proof: For and sasfyng 2a-2d, e, s a posve number. hen, one has In adon, accordng o GC GC, one has

3 2 2 2 GC GC Usng he Schur compemen, one has f and ony f For posve number and 2 suffcen sma such ha,one can fnd a suffcen arge 2 ha s, for and sasfyng 2b and 2d, one can choose a suffcen arge such ha mees In he meanwhe, when and sasfy 2a and 2c, he marx make he foowng hod hs compees he proof. Remark 2. heorem 3 has shown ha he separaon prncpe of esmaor and sae feedback conroer. ha s, one can desgn he esmaor gan G and sae feedback gan, respecvey. In hs case, we do no need he sysem dynamcs o be admssbe. IV. SIMOR-SD CONROLLR FOR FU SOCHSIC SSMS Fuzzy mode s an effecve oo for handng nonnear sysems [7-9]. In hs secon, we w dscuss he esmaorbased desgn probem for sochasc fuzzy sysems. Consder he foowng sysem descrbed by IF-HN rues: Rues : IF z s M and and z p s M p, HN dx x x u y C x n 3 wher x R s he sae, u R s he npu, z [ z z p ] are premse varabes, M M p are fuzzy ses., 2,,, s he number of he fuzzy rues. he whoe sysems s obaned by akng he weghed average of a subsysems: dx h z x x u d y h z C x sae esmaor s gven as: Rues : IF z s M and and z p s M p, HN dxˆ xˆ u G yˆ y y C xˆ he whoe esmaor s express as: dxˆ h z xˆ u G yˆ y yˆ h z C xˆ he esmaor-based conroer can be gven as: Rues : IF z s M and and z p s M p, HN m u xˆ 7 he conro for he whoe sysem s he foowng: u h z xˆ 8 Subracng 5 from 4, and usng 8, and eng e x xˆ, one can ge he foowng error dynamc equaon: e x de h z h z 9

4 Le e x ξ, and consruc he foowng augmened pan: z h z h d ξ ξ ξ 2 2 Lemma 2. fuzzy sochasc sysem 4 can be admssbe by usng he esmaor-based conroer 6-7, f here exss a marx such ha 22 23, and are defned by 2. Proof. he resu can be obaned drecy by usng Lemma. Now we dscuss he separaon propery of he desgn for he sae-feedback gan and he observer gan. heorem 4. fuzzy sochasc sysem 4 can be admssbe by usng he esmaor-based conroer 6-7, f here exs marces and such ha 24a 24b 24c 24d Proof. Suppose here are marces and o sasfy 24a-24d. Le, and be any posve number. herefore, one has Denoe by. One can derve ha C G ccordng o he Schur compemen, s obvous ha f and ony f From he nequaes, one can concude ha for and, one can fnd a suffceny arge posve number such ha In oher words, for and sasfyng 24b and 24d, one can aways fnd a suffceny arge posve number such ha.

5 Moreover, for and sasfyng 24a and 24c, one can oban:. s a resu, he fuzzy sochasc sysem 4 can be sabzed by he esmaor-based conroer 6-7 by Lemma 2. Remark 3. heorem 4 has usraed he separaon prncpe of fuzzy esmaor and fuzzy sae feedback conroer. herefore, we can desgn he fuzzy esmaor-based conroer conveneny. V. MPL ND SIMULION Consder a sochasc sysem n he form of, Fgure. Sae x : esmaor-based conro C 2 Desgn he foowng sae esmaor-based conroer dxˆ xˆ u G yˆ y yˆ Cxˆ u xˆ From heorem 3, one can oban he gans Fgure 2. Sae x : esmaor-based conro G Le x.5. From he smuaed curves, he esmaon and conro performance are desred. Fgure 3. Sae x : esmaor-based conro. 3

6 VI. CONCLUSION In hs sudy, a descrpor sochasc esmaor has been proposed. Moreover, an negraed esmaon and conro mechansm has aso been deveoped. he fuzzy sochasc case has been aso nvesgaed. he proposed esmaon and conro echnques w fnd many appcaons n sgna processng and conro ssues, such as robus sgna esmaon, sgna change deecon and sgna compensaons. CNOWLDGMN he auhors woud ke o hank he fnanca suppor from he Naura Scenfc Foundaon of Chna NSFC under he gran RFRNCS [] R. Nkoukhah, S. L. Campbe, and F. Deebecque, aman ferng for genera dscree-me near sysems, I rans. uom. Conro, vo.44, no , 999 [2]. Mao, xponena Saby of Sochasc Dfferena quaons, Marce Dekker, New ork, 994. [3]. Gao, and D. W. C. Ho, Sae/nose esmaor for descrpor sysems wh appcaon o sensor fau dagnoss, I ransacons on Sgna Processng, vo. 54, no.4, , 26. [4] L. Da, Ferng and LQG probems for dscree-me sochasc snguar sysems,.i rans. uom. Conro, 989, 34: 5-8 [5] R. Nkoukhah,. S. Wsky, and. C. Levy, aman ferng and Rcca equaons for descrpor sysems, I rans. uom. Conro, 992, 379: [6] D. W. C. Ho,. Sh,. Wang and. Gao, Ferng for a cass of sochasc descrpor sysems, Proc. Inernaona Conference on DCDIS, vo.2, , Canada, 25. [7]. anaka, and H. O. Wang, Fuzzy Conro Sysems Desgn and nayss: Lnear Marx Inequay pproach, ohn Wey & Sons, Inc., New ork, 2. [8]. Gao,. Sh,. and S.. Dng, ``Observer desgn for -S fuzzy sysems wh measuremen oupu noses,'' IFC Word Conro Congress, Prague, ugus, 25. [9]. Sh, and. Gao, ``S fuzzy conroer and observer desgn: augmened sysem approach,'' Proc. of I Inernaona Conference on Conro and uomaon, Guangzhou, une, 27.