A Modelg Mehod of SISO Dscee-Eve Syses Max Algeba Jea-Lous Bood, Laue Hadou, P. Cho To ce hs veso: Jea-Lous Bood, Laue Hadou, P. Cho. A Modelg Mehod of SISO Dscee-Eve Syses Max Algeba. Euopea Cool Cofeece ECC 95, Sep 1995, Roe, Ialy. pp.223-226, 1995. <hal-843551> HAL Id: hal-843551 hps://hal.achves-ouvees.f/hal-843551 Subed o 11 Jul 213 HAL s a ul-dscplay ope access achve fo he depos ad dsseao of scefc eseach docues, whehe hey ae publshed o o. The docues ay coe fo eachg ad eseach suos Face o aboad, o fo publc o pvae eseach cees. L achve ouvee pludscplae HAL, es desée au dépô e à la dffuso de docues scefues de veau echeche, publés ou o, éaa des éablssees d esegee e de echeche faças ou éages, des laboaoes publcs ou pvés.
A MODELIN METHOD OF SISO DISCRETE-EVENT SYSTEMS IN MAX-ALEBRA J.-L. Bood, L. Hadou, P. Cho L.I.S.A. 62, aveue Noe Dae du Lac 49 ANERS Face ea-lous.bood@sa.uv-ages.f laue.hadou@sa.uv-ages.f pascale.cho@sa.uv-ages.f Keywods : Dscee-eve syses, (ax, +) algeba, defcao, asfe fuco, ARMA odel. Absac Ths pape deals wh he odelg of he e behavo of dscee-eve syses (ax, +) algeba. The syses we cosde ae lealy odeled hs algeba. The poposed ehod s sped by coveoal lea syse heoy : fo a ARMA fo of he odel ad he syse pulse espose, he odel paaees ae copued by usg a basc esul of Resduao Theoy ode o ze a eo ceo. 1 Ioduco I auoac cool we eed a aheacal odel o chaaceze syse popees (coadably, sably,...) o o desg he cool law of a syse. Theefoe, s poa o oba a odel of he syse whch should, as fa as possble, be boh ealsc ad sple. I coveoal lea syse heoy ay wos deal wh hs poble. These sudes have allowed o develop ue sple ad elavely effecve ehods ode o defy coveoal lea syses. The pupose of hs pape s o popose a odelg ehod of dscee-eve syse (DES). Ths ehod uses he ax-plus algebac oao ode o do aaloges wh he coveoal lea syse heoy. To have a lea epeseao of a syse (ax, +) algeba (fo shoess we wll oe ax-algeba), we cosde DES whch oly sychozao pheoea appea. Moeove, hey ae deesc ad we esc ouselves o sgle-pu sgle-oupu (SISO) syses. I he secod seco we befly ecall he axalgebac opeaos ad he a epeseaos of DES. A coplee oduco o ax-algeba ca be foud [2], [3]. The hd seco s coceed wh he defcao ehod ax-algeba whch s aly based o he Resduao Theoy [1], [2]. I he fouh seco, we popose a paccal odelg pocedue. 2 Noaos ad Descpos of DES Max- Algeba We cosde he seg ( { + } laws, ae defed by R,,, ) whee he a b= ax ( a, b), a b= a+ b The elee s eual fo he law ad absobg fo, he elee oed e s eual fo. We ca oe ha s depoe,.e., a a = a. Fsly we ecall he usual ecue lea euaos axalgeba x( + 1) = A x( ) B u( + 1) y( ) = C x( ) x = ( x1 x ) s he desoal odel sae. u, x ad y ae called cool, h odel sae ad odel oupu especvely. The odel descbed by E. (1) ca also be epeseed by a ed eve gaph whee u, ad y ae x (1)
asos ; u( ), x ( ) ad y( ) epese he daes whe u, x ad y ae especvely fed fo he h e. I he followg, he sg wll be oed. The use of he γ-asfo [2], [4], whee γ opeaes as he z 1 opeao of coveoal syse heoy, leads o he followg epeseao of E. (1) X ( γ ) = Aγ X ( γ ) BU ( γ ) Y ( γ ) = C X ( γ ) whee U( γ ), X ( γ ) ad Y( γ ) ae he γ-asfo of u, x ad y especvely. A basc heoe [2], [4] shows ha he leas soluo of E. (2) s gve by Y( γ) C( Aγ) BU( γ) = wh ( A γ ) = ( Aγ ) + = ad hus he pu-oupu behavo of E. (2) ca also be descbed by he asfe fuco h( γ) = C( Aγ) B whch ca also be expessed as a polyoal expesso of he fo whee p( γ) ν h( γ) = p( γ) γ ( γ)( sγ ) (3) ν 1 = = p γ 1 = =, ( γ) γ (2) ad s s a ooal [2]. Le us ed ha h(γ) ca be cosdeed as he γ-asfo of he pulse espose of he odel. Theefoe, he h(γ) expesso ca expess ha he pae epeseed by (γ) s defely epoduced because he ulplcao by s γ sybolzes a u abscssa shf ad a s u odae shf. Ths peodc behavo begs afe a ase behavo whch ay be epeseed by p(γ). We us oe ha such a ealzao s o ecessaly al [2, 6.5.4], howeve allows a sple epeao of h(γ). Fo exaple, he asfe fuco 2 3 2 3 h( γ) = e 1γ 3γ γ ( 6 8γ 9γ )( 7γ ) leads o he pulse espose llusaed Fg. 1. The pupose s o defy paaees of he asfe fuco h( γ ) descbed by E. (3),.e., paaees of polyoals p( γ ) ad ( γ ) ad ooal s, by usg he syse pulse espose. The ( s γ ) e of he asfe fuco h( γ ) ples a fe epeo of he ( γ ) pae whch eas a fe ube of paaees o copue. Hece as coveoal syse heoy, we asfo he asfe fuco h( γ ) o a ARMA fo o oba a fe ube of paaees. A soluo s gve [2, 9.2.2] o oba hs ARMA fo, howeve we pefe a sple ehod whch s : Assug ha h( γ ) s defed by E. (3), he Y( γ ) ad U( γ ) sasfy he ARMA euao Y( γ) sγ p( γ) U( γ) = p( γ) γ ( γ) U( γ) sγ Y( γ) Poof s easly obaed by expessg E. (3) as Y( γ) = p( γ) γ ( γ)( e sγ ( sγ ) ) U( γ) whch ca be we as Y ( γ ) = 2 [ ] [ ] a1 p( γ ) γ ( γ ) U ( γ ) γ ( γ ) sγ ( sγ ) U ( γ ) a2 a3 O he ohe had he ulplcao of E. (3) by s γ leads o sγ Y( γ) = sγ p( γ) U( γ) sγ γ ( γ)( sγ ) U( γ) a4 Because a a5 a6 = a, hece we have 3 6 a1 a5 = a2 a3 a5 = a2 a5 a6 = a2 a4 whch leads o he esul. 2 16 15 13 9 8 6 3 1 h (γ) ase behavo ase ν pae s peodc behavo γ 1 2 3 4 5 6 7 8 9 1 11 Fgue 1 : Ipulse espose coespodg o h( γ ). ( ν = 3, = 3, s = 7) (4)
3 Idefcao Mehod I hs seco we develop a defcao ehod of a ARMA odel ax-algeba whch wll be used o copue he paaees of he asfe fuco h( γ ). The poposed appoach offes a aalogy wh coveoal dscee e syse heoy. Le us ed ha a lea dscee e syse ca be expessed as he ARMA euao y ( ) + a y ( ) = b u( ) = 1 = To esae he paaees of hs ARMA euao,.e., a1,, a, b,, b, soe defcao ehods cosde he followg pedco eo ε( ) = y ( ) + a y ( ) b u( ) = 1 = whee y s he easued syse oupu ad a,, a, b,, b 1 ae he esaed paaees. Such a pedco eo allows o defe a ceo whch s zed fo he se of paaees seached. Fo exaple a uadac ceo s used he well ow leas suae ehod [5, chap. 7]. ε = Y ( M θ ) whee ε ε ε = ( ) ( N) s he pedco eo veco, Y = y ( ) y ( N) s he easued syse oupu veco ad M s he ax ϕ possble o defe a ceo J as N J( θ ) = ( ) = ε ϕ N. Veco ε aes To deee he esaed paaees veco θ whch zes hs ceo, we cosde a basc esul of Resduao Theoy [1], [2] whch saes ha ( ) Y θ = M (6) s he geaes subsoluo of Y = M θ whee efes o he ulplcao of wo aces whch he opeao s used ahe ha he ax-opeao [2]. Thus, by usg hs geaes subsoluo, each elee of he pedco eo veco ε s always posve ad J( θ ) s zed. 3.1 Idefcao Mehod of Max-Algeba ARMA Model Le us suppose ha odel ca be descbed axalgeba by he ARMA euao 1 Y( γ) = b b γ U( γ) a γ a γ Y( γ) The defcao ehod we popose s based o he syse pulse espose,.e., u ( ) = Slaly o ohewse he coveoal syse heoy, we defe he pedco eo ε( ) y ( = ) ( ϕ θ ) whee y s he easued syse oupu, ϕ = u ( ) u ( ) y ( 1 ) y ( ) s he egesso veco a eve ad θ= bb a a s he esaed paaees veco. 1 By usg he daa of u( ) ad y ( ) wh = o N ad N > +, oe ca oba he ax expesso (5) 3.2 Idefcao Mehod of Tasfe Fuco h(γ) The defcao ehod poposed he pevous seco eeds o have a pacula lea ARMA fo (see E. (5)). Because hs codo s o vefed by he ARMA euao (4) (owg o he sγ p( γ) U( γ) e), we popose o defy ase pa ad peodc pa sepaaely. Ths pocedue leads us o cosde he wo followg euaos ahe ha E. (4). Whe we cosde a pulse espose whou peodc behavo,.e., ( γ ) =, E. (3) s educed o he sple MA euao Y( γ ) = p( γ ) U( γ ) (7) Slaly, a pulse espose whou ase behavo,.e., p( γ ) =, leads by usg E. (4) o Y( γ) = γ ( γ) U( γ) sγ Y( γ) (8)
Le us ed ha ν 1 ad 1 ae he polyoal odes of p( γ ) ad ( γ ) especvely. Theefoe we oba a esao of he p( γ ) paaees by applyg he esul of seco 3.1 o E. (7) whee θ p p θ= ϕ ϕ Y ν 1 (9) = ν 1 s he esaed veco of he p( γ ) paaees ad ϕ = u ( ) u ( ν+ 1 ) Y = y ( ) y ( ν 1) Le us oe ha we aually eed he fs daa of he easued syse oupu o coecly defy he ase pa. Slaly, we oba a esao of he paaees of ( γ ) ad s by applyg he esul of seco 3.1 o E. (8) θ= ϕ ϕ wh N Y ν N ν + whee θ= 1 s (1) s he esaed veco of he paaees of ( γ ) ad s ad Y = y ( ν) y ( N) ϕ = u ( ν) u ( ν + 1 ) y ( ) Accodg o he assupo whch leads o E. (8) p( γ ) =, we cosde ha y ( ) = = y ( ν 1) = o copue ϕ. 4 Paccal Modelg of h ( γ ) As coveoal lea syse heoy [5, chap. 16], he defcao of he asfe fuco h( γ ) by usg Es. (9) ad (1) poses o ow he ube of coeffces of polyoals p( γ ) ad ( γ ),.e., ν ad especvely. I pacce he owledge of hese values s a poa poble of he odelg. Ideed he odel sucue should as fa as possble be boh coplex eough o coecly ualfy he syse ad sple eough o educe he ube of paaees o be defed. To oba hs copose we eed a pelay aalyss o have a esao of ν oed l p, ca be coase bu us be geae ha he exac ase legh. The we popose a odelg ehod based o he aalyss of he ceo J (see seco 3.1) as a fuco of l p ad l (defed as he esao of he pae legh ). I ode o esae : - we ae l o 1 ; - we copue ( =,, 1) ad s by usg E. (1) ; we epea he lae calculao wh l J s al. l = l +1 ul ceo To esae ν we fx o s pevous esao. The we decease l p ad we calculae ( =,, 1) ad s by usg E. (1) ul ceo J ceases. Ths ccal value of l p coespods o he esao of ase pa legh (ν). 5 Cocluso We have poposed a defcao ehod of DES ax-algeba. Ths ehod aes possble he defcao of SISO syses. I offes a aalogy wh he coveoal lea syse heoy : s based boh o he aalyss of he syse pulse espose ad o a lea ARMA fo of he odel. To oba hs ARMA fo we sepaae he ase behavo ad he peodc oe. Such popey aes possble he use of a basc esul of Resduao Theoy whch leads o a sple ehod of paaees odel esao. Howeve we cao guaaee he aly of he odel due o he cosdeed expesso of he asfe fuco, we eep wog o hs poble. O he ohe had wll be eesg o chaaceze he aco of dsubaces o he ehod. Refeeces [1] R. Cughae-ee, Max Algeba, Lecue Noes Ecoocs ad Maheacal Syses, Spge- Velag, 166, 1979. [2] F. Baccell,. Cohe,.J. Olsde, J.P. Quada, Sychozao ad Leay. A algeba fo Dscee Eve Syses, New Yo : Wley, 1992. [3]. Cohe, D. Dubos, J.P. Quada ad M. Vo, A Lea-Syse-Theoec Vew of Dscee-Eve Pocesses ad s Use fo Pefoace Evaluao Maufacug, IEEE Tasacos o Auoac Cool, 3, 3, ach 1985, pp. 21-22. [4]. Cohe, P. Molle, J.P. Quada ad M. Vo, Algebac Tools fo he Pefoace Evaluao of Dscee Eve Syses, IEEE Tasacos o Auoac Cool, 77, 1, auay 1989, pp. 39-58. [5] L. Lug, Syse Idefcao : Theoy fo he Use, Pece-Hall, Ic., Eglewood Clffs, New Jesey, 1987.