PROBABILITY OF STABILITY AND RELIABILITY OF DISCRETE DYNAMICAL SYSTEMS UDC

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1 FACTA UNIVERSITATIS Srs: Automatc Cotro ad Robotcs Vo. 8, N o, 009, pp ROBABILITY OF STABILITY AND RELIABILITY OF DISCRETE DYNAMICAL SYSTEMS UDC Bojaa M. Zatovć, Bjaa Samardžć Th Facuty of Occupatoa Safty, Ns, Srba, E-ma: bojaa.zatovc@adau.com Th Facuty of Scc ad Mathmatcs, E-ma: vdojovc@yahoo.com Abstract. Mthod for th probabty of stabty stmato of dscrt systms wth radomy chos paramtrs s prstd ths papr. Varous formuas for dffrt typs of paramtr dstrbuto ar obtad ad usd to cacuat probabty of stabty for a arbtrary ordr systm. Sc th ma goa s to obta th systm maxmum probabty of stabty, usg ths mthod ad choosg th adquat vaus of paramtrs t s possb to do so. For ts smpcty ad ffccy ths mthod ca b usd practca appcatos ad t s approprat for rabty aayss of dscrt systms wth varab paramtrs. Ky words: robabty of stabty, rabty, mprfct systm. INTRODUCTION Systms wth radom paramtrs xst may dustrs, for xamp: procss, chmca, rubbr, pastc matras dustry, tc. Oft ths systms hav som dffcuts worg propry sc thr paramtrs vaus, usuay, dffr from th watd os. I ths cas stmatg th fuc of ths stochastc paramtrs o th systm prformacs advac s of grat mportac. Ths stmato s cssary for th aayss of systm stabty, rabty, quaty of th systm, tc. Ths systms ar cad mprfct systms. Stabty probm of systms wth mprfctos s w ow. I ths papr stabty probm of ar dscrt systms wth paramtrc mprfcto s cosdrd. Stabty of th systm s dtrmd by vau of systm paramtrs. If paramtrs hav costat vaus, systm ca b stab or ustab. If paramtrs ar stochastc, systm s stab wth som probabty cad probabty of stabty. Stochastc stabty pays a mportat ro rsarchg th bhavour of dyamc ad cotro systms udr radom paramtrc varatos. Svra mods of stochastc stabty ar ow: stabty of probabty, stabty of th K th momt, amost sur sta- Rcvd Fbruary 8, 00

2 8 B. M. ZLATKOVIĆ, B. SAMARDŽIĆ bty, Lyapuov avrag stabty, mootoc tropy stabty, tc. For a dftos of stochastc stabty t s cssary for probabty of stabty to b qua to. Caughy ad Gray, [], dtrmatd amost sur stabty of ar dyamc systms wth stochastc coffcts. Khasms, [], Koz, [3, 4], ad sy, [5], gav dffrt dftos ad charactrstcs of stochastc stabty of ordary dffrta quatos. ortr ad Crossy, [6], aayzd th probabty of stabty of a cass of ar dyamca systms ad gav som thortca rsuts. Stochastc stabty of systms wth radom mprfctos s aayzd [7]. Th basc mthods for th probabty of stabty stmato of cotuous systms ar gv [8-]. I [] th mthod for th probabty of stabty stmato of dscrt systms wth radom paramtrs s prstd. Som thorms from th thory of th radom procsss ad th basc codtos for th dscrt systms stabty ar usd [3] [4], rspctvy. I ths papr w cosdrd radomy chos ad tm varab paramtrs. Th probabty of stabty stmato ca b usd for rabty ad robustss stmato, as w. Th probabty of stabty s corrato wth rabty. If paramtrs hav statoary vaus tm trva t, th that tm trva th systm s rab. Th vau of paramtr ca b chagd udr th fuc of dffrt factors, xtror ad tror,.., war, corroso, agg, tc. Th ffcts of ths chags dpd o compots quaty, [5], ad ca dcras th systm rabty causg th faur of th compot ad systm as w. Robustss stmato ca b accompshd usg th probabty of stabty stmato, [6]. I th cas of costat paramtrs, th rsuts obtad usg th mthod for probabty of stabty stmato, ar quvat to th rsuts obtad by Khartoov's mthod, [7]. Mthod for probabty of stabty stmato rprsts th grazato of Khartoov's mthod wh th systm paramtrs ar uformy dstrbutd radom varabs. Th probabty of stabty stmato of ar dscrt systms wth radom paramtrs s gv [3], [8], [9]. I [0] th faur probabty of th systm has b cosdrd. Th prstd mthod, [], ca b appd for th aayss of th cotuous systms too []. Th grat mportac of ths mthod s ts appcato practc. Th scto of adquat vaus of paramtrs provds th maxmum probabty of stabty. Ths way may probms ca b avodd, for xamp faurs of compots ad systm. Ths mthod ca b appd for dffrt paramtrs dstrbutos such as uform, orma, xpota, osso dstrbuto.. ROBABILITY OF STABILITY ESTIMATION Th mathmatca mod of th th ordr dscrt systm s gv by: x( T ) 0, T, 0 () 0 whr paramtrs ar radom varabs wth probabty dstrbuto dsts p ( ) ad T s th sampg tm.

3 robabty of Stabty ad Rabty of Dscrt Dyamca Systms 9 Frst w hav to dtrm stabty rgo of th systm () th paramtrc spac. Th charactrstc poyoma of th dffrc quato () s: z z L 0 () Th cssary ad suffct codto for th stabty of systm () s that a zros of ts charactrstc poyoma ar ocatd sd th ut crc th z pa. To tst ths codto th bar trasform mthod s usd mappg th sd of th ut crc to th ft haf of th compx pa. Th w quato s obtad wth coffcts ϕ (,, K, ) ϕ : s ϕ L ϕ (3) s 0 0 Hurwtz crtro s appd to obta stabty rgo S. Th cssary ad suffct codto that a zros of quato (3) ar ocatd th ft haf of th compx pa s that a dagoa mors D, of Hurwtz matrx D ar gratr tha zro so th stabty rgo ϕ ϕ3 S s dtrmatd from th xt ratos D ϕ > 0; D > 0, tc. ϕ ϕ For th scod ordr systm th stabty rgo, S, s gv by th xt st of quats: Th stabty rgo S s prstd Fg., whr N prsts th ustab rgo. I th cas of th scod ordr systm, th stabty rgo s a trag. 0 0 (4) N - - S - Fg. Stabty rgo, S, of th scod ordr dscrt systm For th thrd ordr systm, th stabty rgo, S 3, s gv by th xt st of quats: 3 Th stabty rgo s show Fg > > < (5) 3

4 30 B. M. ZLATKOVIĆ, B. SAMARDŽIĆ 3 (-,-,) (-3,3,-) (,-,-) (3,3,) Fg. Stabty rgo, S 3, of th thrd ordr dscrt systm Th tota dstrbuto dsty s gv by: ad probabty of stabty of systm () s: p(, K, ) p ( ) (6) L p(, K, ) d L d (7) S I th cas wh stabty rgo s dtrmd for mor paramtrs probm bcoms compx. Hc, surfacs mtg th stabty rgo ar usuay dfd by vry compx mathmatca ratos. It s, aso, cssary to tgrat by th ara S, (7), whch s vry compcatd cosdrg th compx dstrbuto dsts of radom paramtrs. I th cas of th th ordr systm, th stabty rgo s dtrmd by th ow codtos for th stabty of th th ordr dscrt systms. Ths way cosd body s obtad, prstg th stabty rgo th th ordr paramtrc pa. Ths body s vry compx ad th cacuato of th probabty of stabty, (7), s vry dffcut. So t s mportat to stmat th probabty of stabty for th practca appcatos. For hghr ordr systm ths stmato s vry rough, but sgfcat practc ad t s th bst possb stmato so far. For th probabty of stabty stmato xt thorms ca b appd ffctvy. Thorm. Th stabty rgo, S, of dffrc quato () bogs to th rgo (hypr parappd) gv by:,,, K, (8) Th stabty rgo s mtd abov by th rgo th paramtrc pa, S.

5 robabty of Stabty ad Rabty of Dscrt Dyamca Systms 3 Thorm. Th stabty rgo, S, of dffrc quato () comprss th rgo (smpx poyhdro) gv by: Th stabty rgo S s mtd owr by th rgo K (9), S. Ths mas that th stabty rgo S s sd th rgo, ad th rgo s sd th stabty rgo S. Th proofs of thorms ad ar gv []. Usg ths thorms, th probabty of stabty of th ordr systm ca b stmatd th foowg way: L p(, L, ) d Ld < < L p(, L, ) d L d (0).. < < () s th probabty that th stabty rgo s sd th rgo ad s th probabty that th stabty rgo s sd th rgo. Sc t s dffcut to stmat probabty of stabty bcaus of th compxty of rgo t s bttr to choos costat mts of rgo, whch s supportd by th xt thorm. Thorm 3. Th th ordr poyoma f ( z) z z L z s gv. A zros 0 / of th poyoma ar th crc of radus R max, whr λ ar ra umbrs λ satsfyg th codto λ, []. Sc th stabty rgo of dscrt systms s th ut crc, usg thorm 3, for R th xt ratos ar obtad: max λ / () λ /, L, (3) λ (4) λ (5)

6 3 B. M. ZLATKOVIĆ, B. SAMARDŽIĆ Summg th foowg rato s obtad: λ (6) Ths rsut s qua to th rato (9). Usg thorm 3, costat mts for rgo s chos,.. λ /. I ths cas th rgo s hypr parappd dfd as /, whr s ordr of th systm. I prvous wor, [], usg rato (9), formua for cacuatg probabty for th th ordr dscrt systm, coud hav b obtad oy for th uform dstrbuto of paramtrs. For othr probabty dstrbutos, probabty coud hav b cacuatd oy for th scod ad thrd ordr dscrt systms. For hghr ordr systms cacuato of was vry compcatd bcaus of th compxty of th rgo. Now, usg thorm 3 ad choosg th costat mts of rgo t s possb to obtad formuas for cacuatg for othr paramtrs dstrbutos bsds uform. Ths s th ovty rato to th rsuts obtad [] ad th grat cotrbuto th papr. I th cas of th uform dstrbuto of th paramtrs: ; a < a < a ( a a ) p 0; a > a ; a < a Th xt probabts ar obtad for th th ordr dscrt systm: (7) a a a ( a a ) a (8) 0 f j h( f ) j ( a a ) (9) whr a ad a prst th uppr ad owr mt of th trva, ad f j s obtad usg th formua: j j ( ) ( ) fj aj a j (0) j Ths formua s propr for f j. For othr cass, probabty s obtad from th trscto of th rgo ad th uform dstrbuto (graphca prstato of th

7 robabty of Stabty ad Rabty of Dscrt Dyamca Systms 33 uform dstrbuto for th th ordr systm s a hypr parappd) whch s vry compcatd bcaus of th shap of th trsctos. For th th ordr systm whos paramtrs ar radom, ormay dstrbutd varabs, wth ( ) m p σ πσ, probabts ad ca b cacuatd from th foowg quatos: m m Φ Φ σ σ () m m Φ Φ σ σ () whr Φ s th Lapac fucto. For xpota dstrbuto of paramtrs probabty of stabty ca b stmatd usg ratos: (3) ` (4) For osso dstrbuto of paramtrs, th xt ratos ar obtad: ( ) ;! Γ Γ (5) ( ) ;! Γ Γ (6) whr ( ) Γ s gamma fucto, ad Γ ; (or Γ ; ) compt gamma fucto. For th frst, scod ad thrd ordr dscrt systm probabty of stabty ca b cacuatd xacty, but for th hghr ordr dscrt systms usg ths formuas th probabty of stabty ca b stmatd.

8 34 B. M. ZLATKOVIĆ, B. SAMARDŽIĆ 3. THE ROBABILITY OF STABILITY ALICATION FOR RELIABILITY ANALYSIS Th mthod for probabty of stabty stmato ca b usd for rabty aayss. Thr ar may rasos for rabty dcras ad ths papr w cosdr th cas wh systm stabty s th caus of t. It s assumd that th systm s rab as og as t s stab. For th hghr ordr systms stabty rgo s vry compx ad t s mtd from abov by hypr parappd, ad from bow by smpx poyhdro. A systm s rab f t s stuatd th stabty rgo, so t s cssary to dtrmat th codtos that w ab th systm to rma stab durg crta tm trva. I ordr to fd adquat probabty dsty fucto of radom varab, whch s our cas th vau of systm paramtr, w hav usd th xamp of odmsoa Browa moto of partcs. It ca b provd that radom varab has approxmaty N(0;σ t) probabty dstrbuto, [3]. For th scod ad th thrd ordr dscrt systms t s possb to cacuat rabty xacty, wth rsuts gv th xt fgurs. Fg. 3 Rsuts for th scod ordr dscrt systm wth N(0;σ t) dstrbuto of paramtrs Vaus, ad, whr s th probabty of systm wor wthout faurs,.. systm rabty, corrspod to rato (). For th hghr ordr dscrt systms t s dffcut to cacuat rabty bcaus of compxty of stabty rgo. I that cas, rabty must b stmatd usg th mthod for th probabty of stabty stmato. Ths stmato has sgfcat vau practca appcatos.

9 robabty of Stabty ad Rabty of Dscrt Dyamca Systms Fg. 4 Rsuts for th thrd ordr dscrt systm wth N(0;σ t) dstrbuto of paramtrs 4. CONCLUSION Th mthod to prform th probabty of stabty stmato of dscrt systms wth radomy chos paramtrs s dscrbd. For dffrt typs of paramtr dstrbutos, dffrt formuas for th probabty of stabty stmato ar obtad. Usg ths formuas th probabty of stabty of th arbtrary ordr systms ca b cacuatd. Ths papr cosdrs ra tm tchqu, basd o th probabty of stabty, for rabty aayss of dscrt systms wth paramtr mprfctos. Rsuts ar ustratd wth th xamp of th scod ad thrd ordr dscrt systms. REFERENCES. T. K. Caughy, A. H. Jr. Gray, O th amost sur stabty of ar dyamc systms wth stochastc coffcts, ASME J. App. Mch., 3, pp , R. Khasms, Ncssary ad suffct codtos for asymptotc stabty of ar stochastc systms, Thor. rob. Ad Apps., pp.44-47, F. Koz, S. rodromou, Ncssary ad suffct codtos for amost sur samp stabty of a Itô quatos, 97, SIAM J App. Math., (3), F. Koz, Stabty of stochastc dyamca systms, Lctur ots mathmatcs 94. I roc IUTAM Symp., pp.73-85, M. A. sy, Stochastc stabty ad th Drcht probm, Commu. ur App. Math., 7, pp.3-350, B. ortr, T. R. Crossy, robabty of stabty of ar dyamca systms, Kybrts, Vo., No. 4, pp.07-09, 97.

10 36 B. M. ZLATKOVIĆ, B. SAMARDŽIĆ 7. Y. C. Schorg, C. Buchr, Stochastc stabty of structurs wth radom mprfctos, I Stochastc Structura Dyamcs, Bama, Rottrdam/Broofd, pp , S. A. Ajsagav, & G. S. Crsj, Oca vrojatost ustojcvost jh sstm so sucajm paramtram, Thcsaja brta, No. 5, pp.9 0, A. M. Mhajco, K vyboru optmajogo mtoda oc acstva sstmy pr djstv paramtrcsh vozmoscj, (Isttut matmat a USSR, Kv, 989). 0. B. Daovc, Th probabty stabty stmato of th systms wth mor radom paramtrs, Hpf, pp , B. Daovc, & M. Jvtc, O th stmato of worg capabty of th automatc cotro systm, Hpf, Bgrad, Yugosava, pp.33 38, B. Daovc, B. M. Vdojovc & B. Vdojovc, Th probabty stabty stmato of dscrt tm systms wth radom paramtrs, Cotro ad Itgt Systms, Vo. 35, Numbr, pp.34-39, B. Daovc, B. M. Vdojovc, Z. Jovaovc, & B. Vdojovc, Th probabty stabty stmato of dscrt systms wth radom paramtrs, XXXVII Itratoa Sctfc Cofrc o Iformato, Commucato ad Ergy Systms ad Tchoogs, Ns, Yugosava, pp.57 60, J. A. Borr, Stochastc Systms for Egrs, Nw Yor, rtc Ha, B. M. Vdojovc, Z. Jovaovc & M. Mojovc, Th probabty stabty stmato of th systm basd o th quaty of th compots, FACTA UNIVERSITATIS, (NIS), Sr.: Ec. Erg. vo.9, o.3, pp , Dcmbr B. Daovc, B. M. Vdojovc & B. Vdojovc, A way for th dscrt ar systms robustss stmato, 6 th Itratoa Cofrc o Tcommucatos Modr Satt, Cab ad Broadcastg Srvcs, Ns, Srba ad Motgro, pp , Octobr 3, V. L. Hartoov, Ustojcvost odogo smjstva razosth sstm, pp.38 45, Y.C.Schorg, T. Most & C. Buchr, Stabty aayss for mprfct systms wth radom oadg, rocdgs of th 8 th Itratoa Cofrc o Structura Safty ad Rabty, Nwport Bach, USA, pp. 9, Z. Jovaovc, B. Daovc, O th probabty stabty of dscrt tm cotro systms, FACTA UNIVERSITATIS (NIS), vo.7, pp. 0, Apr T. Most, C. Buchr & Y.C.Schorg, Dyamc stabty aayss of oar structurs wth gomtrca mprfctos udr radom oadg, Joura of Soud ad Vbrato, pp , D. Atc, B. M. Vdojovc & B. Vdojovc, Th probabty stabty of cotuous systms wth radomy sctd paramtrs, XL Itratoa Sctfc Cofrc o Iformato, Commucato ad Ergy Systms ad Tchoogs, Ns, Srba ad Motgro, pp , Hrc, Appd ad computatoa compx aayss, Joh Wy & Sos, vo., Nw Yor, J. Masc, Stochastc procsss, thory ad appcatos, Gradjvsa jga, Bgrad, 989. VEROVATNOĆA STABILNOSTI I OUZDANOST DISKRETNIH DINAMIČKIH SISTEMA Bojaa M. Zatovć, Bjaa Samardžć U ovom radu j prdstavj mtod za procu vrovatoć stabost dsrth sstma sa sučajo zabram paramtrma. Izvd su formu za zračuavaj vrovatoć stabost sstma prozvojog rda za razčt raspod paramtara. ošto j gav cj dobt masmau vrovatoću stabost, oršćjm ovog mtoda zborom advath vrdost paramtara to j moguć ostvart. Zbog svoj jdostavost fasost ovaj mtod s mož orstt u pratčm prmama pogoda j za aazu pouzdaost dsrth sstma sa promjvm paramtrma. Kjuč rč: Vrovatoća stabost, pouzdaost, savrš sstm.