On the local controllability of a discrete-time inhomogeneous multi-input bilinear systems

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1 IJCSI Iteratoal Joral of Comter Scece Isses, Vol., Isse, o 3, Jaary 3 ISS (rt): ISS (Ole): O the local cotrollablty of a dscrete-tme homogeeos mlt-t blear systems Omar Balatf, Mohamed El ha, Jamal Boyaghrom, Mostafa Rach aboratory of Aalyss Modelg ad Smlato, Deartmet of Mathematcs ad Comter Scece, Faclty of Sceces Be M.S, Uversty Hassa II Mohammeda, B 7955, Sd Othma, Casablaca, Morocco Abstract hs aer stdes the local cotrollablty of a class of dscretetme homogeeos blear systems. A sffcet codto for the local cotrollablty s roosed ad the form of the otmal cotrol s also reseted. Frthermore, the establshed reslts are llstrated by a eamle ad mercal smlato. Keywords: Blear systems, dscrete tme, local cotrollablty, otmal cotrol.. Itrodcto Blear systems are a secal class of olear systems; they form a trastoal class betwee the lear ad the geeral olear systems. hrogh early half a cetry, they have receved great atteto by researchers. he mortace of sch systems les the fact that may mortat rocesses, ot oly egeerg [], bt also bology [], soco-ecoomcs [3], ad chemstry [4-5], ca be modeled by blear systems[6]. I the lteratre, several aers address the roblem of cotrollablty for blear systems. I [7], we rase two codtos for cotrollablty: oe for ecessty ad the other for sffcecy. Sch aroach s a local oe ad cossts decomosg the blear system model to a lear system ad a mltlcatve feedbac. However, t reqres that ra(q) = where Q mst be factorzed two vectors; other terms ths techc eeds orthogoalty roerty. he same roblem as cosdered [8] gves rse to a global ecessary ad sffcet codto. I addto to decomosg the system as [7], the aroach volves forward ad bacward comosto of the trasto fcto. It stll eses a codto of orthogoalty o the matr Q, ls a versblty codto o the matr A. etc he reset aer deals wth the qesto of local cotrollablty for dscrete tme homogeeos mltt blear systems. We adot a method based o the learzato of the system ad the defto of a arorate oerator that leads to the cotrol trasferrg the system to a desred gve state wth a mmm eergy. he aer s orgazed as follows. I secto, we reset a aromato of the fal state. Secto 3 s amed to the resetato of a sffcet codto for local cotrollablty of a homogeeos mlt-t blear dscrete-tme system. Secto 4 rovdes a eresso of a otmal cotrol that ca trasfer the system from the tal state to a fal desred state. Fally, a eamle of cotrollable blear systems s rovded secto 5.. A aromato of the fal state I ths artcle we cosder the followg homogeeos dscrete-tme blear system: ( + ) = A + B + B () = where ( ) s the -dmesoed state vector at tme, = ( ) s the -dmesoed cotrol vector at tme, B s a matr of dmeso, A ad B; ; B are sqare matrces of order. et deotes the fal state ad ( + ) = A + B + B = F(, ) = where F s a cotos vector fcto. et also B = ( b j ) wth bj for {,, } ad j,,, the the system () becomes { } ( + ) = A + B + V = F(, ) ( ) = where V ( ) b ( ) b ( = ) = = Cosder the followg fcto comosto ( ) = F F F ( ) () ( ) Coyrght (c) 3 Iteratoal Joral of Comter Scece Isses. All Rghts Reserved.

2 IJCSI Iteratoal Joral of Comter Scece Isses, Vol., Isse, o 3, Jaary 3 ISS (rt): ISS (Ole): wth F( )( ) = F(, ) Usg aylor's develomet to ead the rght-had sde of the revos eqato yelds: = F F F ( ) + ( ) ( ) = F ( ) ( ) F ( ) ( ) F ( ) ( ) ( ) ( ) + Ο wth ( ) as: = = = ( ) (), whch ca be rewrtte ( ) = = = F F F ( ) + + Ο where = - F ( ) ( ( )) F ( ) ( ( )) F ( ) ( ( )) = ( ) ( ) From the eqato () we have F ( ) = A + B ad ( ) = ( ) b F V ( ) = B + V wth V = = b for =,, So by comtg after fcto comosto, whe cotrols are assmed to be eqal to zero, we obta ( BA () + V BA () + V ) ( A( BA () + V) A( BA () + V )) = ( A ( BA () + V) A ( B() + V ) ) I other words, a aromato of the fal state whe eglectg hgher order cotrol terms ca be eressed as: = A + A ( BA + V ) (3) = = ote that, the rest of ths wor, we eglect hgher order cotrol terms. hs assmto gves a local crtero of cotrollablty. 3. A sffcet codto of local cotrollablty I ths secto we roose a sffcet codto of local cotrollablty for the system (). Frst recall the defto of the local cotrollablty for the systems (). Defto he system () s sad to be locally cotrollable o I = {,,, } for ay ad d from ; there ests a cotrol = (,,, ) as = d ; where ; gve by (3), s the aromate solto of () at stat corresodg to the tal state ad the cotrol. he, let cosder the oerator defed by H: ( ) ( ) = = = (),,( ) H= A ( BA () + V ) () wth = roosto he oerator H s lear, cotos ad ts adjot oerator H s gve by: H : ( ) H= he learty of H s obvos For the cotty of H we show the estece of a costat α > sch as H α,, H = A ( B A ( ) + V ) ( ) = = A ( BA + V ) = = = = = = α So H s lear For the adjot oerator we have H, =, =, =, Hece the eresso of H. (4) Coyrght (c) 3 Iteratoal Joral of Comter Scece Isses. All Rghts Reserved.

3 IJCSI Iteratoal Joral of Comter Scece Isses, Vol., Isse, o 3, Jaary 3 ISS (rt): ISS (Ole): roosto 3 If H s srjectve the () s locally cotrollable. et ; d. hs d A As H s srjectve, there ests a cotrol,, ( ) sch as H = d A, so d = A + H, the ( ) = d Hece the reslt accordg to the defto. roosto 4 If ra [] = ; the the system () s locally cotrollable. et H be the oerator defed by (4) ad H s the adjot oerator. We ow that Im H = er H = { } If ra [ ] = ra = he er = { } Hece er H = { } So f ra [] = the H s srjectve ad accordg to the roosto (3), the system () s locally cotrollable. 4. Otmal cotrol I ths secto we focs o the characterzato of otmal cotrol for the case of system (). et trodce the matr W defed by W We have the followg reslt = = (5) heorem 5 he system () s locally cotrollable f the matr has fll ra. Frthermore, the cotrol () whch ca trasfer the system from the tal state to the fal state d wth a mmm eergy, s gve by ( ) = W ( A d ) (6) {,,, } Before we rove ths theorem, we frst rove the followg lemma: emma 6 he matr has fll ra f ad oly f the matr W s ostve defte. We have = = =,,,,- W, =, =, = If W, = the { } he er = { } becase ra [] = Hece = ad therefore W s ostve defte. et er so = = = = the =, {,,,-} ad W = hece = (becase W s ostve defte) ths er = { } ad fally we get ra [] = roof (of theorem 5). Frst, sose the matr has fll ra the, accordg to the roosto 4, the system () s locally cotrollable. Frthermore, sg the revos lemma, the matr W s ostve defte whch mles t versblty. Coseqetly defed by (6) s well defed. he by relacg (3) by the eresso (6) oe ca easly chec that ( ) = d. Fally we show that = f U, wth U={ v / v s a cotrol that allows the trasfer of the system from to d }. Coyrght (c) 3 Iteratoal Joral of Comter Scece Isses. All Rghts Reserved.

4 IJCSI Iteratoal Joral of Comter Scece Isses, Vol., Isse, o 3, Jaary 3 ISS (rt): ISS (Ole): et U, the d = = A + A BA + V = the = = = = ( )( ) A B A + V = ( () )( () (); ) ( () d) A BA + V W A = = = ( ); ( d ) W A = ( ); ( d ) ; = = ; = W A = ; = = ; Hece the reslt. 5. Eamle Cosder the dyamcal system (Mohler, 973) = A + B + B (7) Where Ra K a a a A = ; B = D K y J J a a B = ; θ e = = ad = = υ a 3 ω J s the momet of erta, D s the vscos damg rato, R a s the armatre resstace, a s the aled armatre dctace, K y, K a are motor characterstcs, K a s the motor cost, a s the armatre crret, e s the feld crret, υ a s the armatre voltage, ω s the aglar velocty, ad θ s the aglar osto. Eqato (7) ca be dscretzed by se of a frst-order Eler easo to gve ( + ) = + A + B + B (8) where s the samlg terval. Eqato (8) ca be rewrtte as ( + ) = A + B + B (9) Wth A = I + A, B B = ad B = B he arameter vales chose for the model are tae from [9] ad are =., K a =.56, K y =37.7, a =.5, J = 4. 4, D =.3 ad. he the system (9) becomes. 88 ( + ) = () Cosder = the tal state ad d = the desred state. We reset mercal reslts obtaed sg Matlab. For = ; we have ra[] = 3, so the system () s locally cotrollable ad the otmal cotrol s gve by the followg table. able : otmal cotrol () () Coyrght (c) 3 Iteratoal Joral of Comter Scece Isses. All Rghts Reserved.

5 IJCSI Iteratoal Joral of Comter Scece Isses, Vol., Isse, o 3, Jaary 3 ISS (rt): ISS (Ole): [8] M.E. Evas ad D... Mrthy. Cotrollablty of a class of dscrete tme blear systems. IEEE ras o Atomatc Cotrol, AC-, 78-83, Febrary 977. [9] B. Gerard. Observers ad cotrol based o a observer for blear systems. Doctoral thess of Her ocare versty -acy, ovember Coclso I ths aer, we have stded the local cotrollablty of a blear dscrete-tme system. he method that we reset ths aer s based o a learzato of the system ad the the defto of a stable oerator whch ca lead to cotrol trasferrg the system to a desred gve state wth a mmm eergy. Acowledgmets hs wor was sorted by: "e Résea de la héore des Systèmes". Refereces [] R.R. Mohler. Blear Cotrol rocesses, volme 6 of Mathematcs Scece ad Egeerg. Academc ress, ew Yor, 973. [] D. Wllamso. Observato of blear systems wth alcato to bologcal cotrol. Atomatca, 3 :43 54, 977. [3] R.R. Mohler. olear systems : Alcatos to Blear Cotrol, volme. retce Hall, Eglewood Cls, ew Jersey, 99. [4] M. Esaña ad I.D. ada. Redced order blear models for dstllatos colms. Atomatca, 4 : , 977. [5] M. V. Bas ad A. Alcorta-Garca, Otmal flterg for blear system states ad ts alcato to olymerzato rocess detfcato, roceedgs of the Amerca Cotrol Coferece, , Dever, Colo, USA, Je 3. [6] M. Ema : Modelg ad Cotrol of Blear Systems: Alcatos to the Actvated Sldge rocess. Wrtte Eglsh. ACA UIVERSIAIS USAIESIS. Usala Dssertatos from the Faclty of Scece ad echology Usala, Swede, 5. ISB [7]. Goa,.J. ar ad J. Zaborszy. O the cotrollablty of a class of dscrete blear systems. Atomatca, vol 9, 973. Coyrght (c) 3 Iteratoal Joral of Comter Scece Isses. All Rghts Reserved.