Discrete Time Sliding Mode Flow Controllers for Connection-Oriented Networks with Lossy Links

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1 BULGARIAN ACADEMY OF SCIENCES CYBERNEICS AND INFORMAION ECHNOLOGIES Volume 3, No Sofi 3 Prit ISSN: 3-97; Olie ISSN: DOI:.478/it-3-9 Disrete ime Slidig Mode Flow Cotrollers for Coetio-Orieted Networks with Lossy Liks Adrze Brtoszewiz, Piotr Leśiewski Istitute of Automti Cotrol, ehil Uiversity of Łódź, 8/ Stefowskiego St., 9-94 Łódź, Pold Emils: drze.brtoszewiz@p.lodz.pl piotr.lesiewski@gmil.om Abstrt: I this pper we propose two disrete-time slidig mode flow otrollers for multisoure oetio-orieted etworks. Eh oetio i the osidered etworks is hrterized by time dely d pket loss rtio. he proposed otrollers tke dvtge of ppropritely desiged slidig hyperples, whih esure the losed-loop system stbility d fiite-time error overgee to zero. Moreover, the proposed otrollers esure full utiliztio of the bottleek lik vilble bdwidth, gurtee bouded trsmissio rtes, d elimite the risk of buffer overflow i the bottleek ode. Sie the primry otroller my led to exessive vlues of the trsmissio rte t the begiig of the otrol proess, the modified otroller is desiged usig the oept of the rehig lw, whih helps the removig of the udesirble effet. Keywords: Cogestio otrol, slidig mode otrol, disrete time systems.. Itrodutio I oetio-orieted ommuitio etworks the dt uits, set by soures pss through series of itermedite odes before rehig their destitios. If itermedite ode due to limited dt flow rte of its outgoig lik ot pss o ll the dt it reeives, the ogestio ours. Cosequetly, i order to mximize the throughput d miimize the queuig delys d itter i moder ommuitio etworks, ogestio otrol lgorithms re pplied. he mi diffiulty i pproprite ogestio otrol lgorithm desig is used by the lrge propgtio delys i the etworks. he delys re ievitble sie iformtio 7

2 bout the ogestio t speifi ode must be dispthed to ll soures trsmittig dt through this ode, i order to eble dustmet of the soure trsmissio rtes, d this tio does ot ffet the ogested ode immeditely, but oly with dely usully lled Roud rip ime (R). he problem of ogestio otrol i the oetio-orieted ommuitio etworks hs bee studied i my ppers (see for exmple [,, 5, 6]) d extesive review of the ppers be foud i the reet moogrph []. Furthermore, due to the robustess of the Slidig Mode (SM) otrol [7], vrious types of SM ogestio otrollers hve bee proposed. I [3] SM otroller with stte preditor ws used, d the mximum dely, eessry for the system stbility hs bee estblished. A fuzzy otroller ombiig the dvtges of lier d termil SM ws proposed i [4] for simplified dely-free etwork model. For DiffServ etwork dptive SM otroller (usig the bksteppig proedure) for model whih eglets the feedbk ltey ws preseted i []. O the other hd, for DiffServ etwork with dely, the seod order SM tehique hs bee pplied i [] i order to redue the htterig of the otrol sigl. I [] the problem of fir (i the mx-mi sese) dt rte distributio mog the soures is osidered. A biry ogestio sigl is used to otrol the dt output of the soures d the lysis of this lgorithm is performed for dely-free system. All otrollers preseted i the foremetioed publitios re desiged i the otiuous time domi. However, it is evidet tht y flow otrol lgorithm for dt trsmissio etwork must be implemeted s digitl otroller. herefore, i the followig works disrete-time pproh to the problem of the dt flow otrol ws used. A SM otroller ws preseted i [8], but the result of this pper ws derived without osiderig the system delys. I [9] it is show tht y mx-mi fir system with stble symmetri Jobi mtrix mitis symptoti stbility uder rbitrry diretiol delys. his mes tht if the otroller is desiged so tht the system hs symmetri Jobi mtrix, its stbility be exmied bsed o the orrespodig udelyed system. A dedbet SM otroller for multi-soure etworks with priori kow roud trip times is preseted i [3] d i [7] LQ optiml SM otroller for sigle-soure etworks is proposed. he sme pproh is the exteded for multi-soure etworks i [8], d i [9], similr optiml flow otroller is desiged for multisoure etworks with the roud trip times whih my hge durig the otrol proess. I most ppers published up to ow, oly pket losses due to bottleek lik buffer overflows re osidered, d the ourree of lossy liks i the etwork is egleted. Sie i rel etworks trsmissio losses re ievitble, i this pper we preset disrete-time slidig mode otrollers [, 4, 5, 6, ] for multisoure etworks, i whih pkets re lost durig the trsmissio proess.. Network model We osider oetio-orieted ommuitio etwork whih osists of persistet dt soures, itermedite odes d dt destitios. We ssume tht 8

3 ll oetios pss through sigle bottleek ode. he totl mout of dt to be set (represeted by u) by ll of the soures is determied by otroller pled t the bottleek ode. We ssume tht this mout is distributed mog the oetios by y higher-level lgorithm. Eh oetio p reeives γ p of the totl rte, where γ p, d m p= γ = wher p e m is the umber of soures. I this wy the flow otrol d the totl rte distributio re deoupled. he blok digrm of the etwork is show i Fig.. Eh sigl γ p u rehes soure p fter bkwrd dely b p. he soure the seds the requested mout of dt, whih is pssed from ode ode util it rehes the bottleek queue fter the forwrd dely f p. Fig.. he etwork model he roud trip time of eh oetio the dely betwee geertig the sigl by the otroller, d the momet whe the requested dt from soure p rrives t the bottleek ode, be expressed s sum of the bkwrd d forwrd propgtio delys () R p = b p + f p. It is ssumed, tht durig the trsmissio some peretge of dt pkets re lost, so tht oly α p γ p u dt pkets from the soure p rrive t the buffer, where α p (, for p =,,..., m. We ssume tht eh R p is multiple of the disretiztio period, i.e., R p = m R p. Furthermore, y(k) deotes the bottleek queue legth t the time istt k. he mout of the dt whih my leve the bottleek ode t time k is modeled s priori ukow futio of time d(k). he mximum vlue of d(k) is represeted by d mx. he mout of dt tully levig the buffer t time k is deoted by h(k). For y k () h k d k d. mx ( ) ( ) It is ssumed thtt the buffer is empty prior to the dt trsmissio proess, i.e., y(k < ) =, dd tht the first otrol sigl is geerted t k =, i.e., u(k < ) =. he queue legth for y k be lulted from the followig expressio 9

4 (3) y( k) = αγ p pu( p) h( ) m k k R. p= = = o simplify the model we represet ll oetios with equl roud trip times s sigle oetio. he mout of dt tht will rrive t the buffer from this oetio is equl to i u, where = α γ, i =,,, d i p p pm : Rp = i = mx(m Rp ) +. Obviously, if o oetio hs the roud trip time i, the the orrespodig oeffiiet i equls zero. Now we rewrite (3) s follows: k k (4) y( k) = u ( i) h( ) i = = he etwork be desribed i the stte spe s x ( k + ) = Ax( k) + bu( k) + oh( k), (5) y( k) = q x( k), where x(k) = [x (k) x (k) x (k)] is the stte vetor d y(k) = x (k) is the queue legth. he stte vribles exept x re the delyed vlues of the otrol sigl, i.e. (6) xi ( k ) = u ( k + i ), i =,,. A is stte mtrix, (7) Α =, d b, o, d q re vetors (8) b= o= q =. Altertively, the stte spe equtio be writte s follows: x ( k + ) = x( k) + x( k) + x3( k) + + x( k) h( k) x ( k + ) = x3( k) (9), x ( k + ) = x( k) x ( k + ) = u( k).

5 with the output sigl y(k) = x (k). he desired stte of the system is deoted by x d = [x d x d x d ]. he first stte vrible x d is the demd queue legth, d further i the pper it is represeted by x d. It be otied from (9) tht for h(k) = ll other ompoets of the demd stte vetor re equl to zero. 3. Proposed otrol strtegy Further i the pper we pply the etwork model preseted i the previous setio to desig two slidig mode ogestio otrollers: lssil ded-bet otroller d its rehig lw bsed extesio. 3.. Primry slidig mode otroller I this setio the flow otrol lgorithms for the desribed etwork re desiged d essetil properties of the lgorithms re proved. For the purpose of the otroller desig we ssume h(k) = d we itrodue slidig hyperple desribed by equtio () s( k) = e ( k) =, where vetor = [ ] stisfies b. he losed-loop system error is deoted by e(k) = x d x(k). Substitutig (5) ito e[(k + )] = we obti the followig otrol lw () u( k) ( b) x Ax ( k) = d, with the pplitio of this otrol sigl, the losed-loop system stte mtrix hs the form A = [ b( b) ]A. he hrteristi polyomil of this mtrix be foud s () det + = z + z, ( zi A ) z z whih leds to the oditio. A disrete-time system is symptotilly stble if d oly if ll its eigevlues re loted iside uit irle. Furthermore, i order to esure error overgee to zero i fiite time, the hrteristi polyomil () hs to stisfy (3) det ( zi A ) = z. Oe esily fid tht (3) is stisfied with the followig vetor =, i (4) i = for i =,,. = Substitutig (7) d (8) ito () we obti

6 (5) ( ) u k xd x k = ( ) ( ) x x k x k d i i = = = ( ) u ( k ) i + i. = his ompletes the desig of the primry flow otrol lgorithm for the osidered etwork. Further i this setio, importt properties of the proposed lgorithm will be disussed. Let us first otie tht the mout of dt to be set t the iitil otrol istt is xd (6) u( ) =. Next we determie the reltio betwee the utilized bdwidth d the trsmissio rte geerted by the otroller. his reltio is stted i the followig lemm. Lemm. If the proposed otroller is pplied, the its output sigl for y k > is give by (7) u( k) = h ( k ). = P r o o f: From (4), for y i = 3,,, we obti (8). i = i + + i he usig (8) d (5) we get u ( k + ) = xd x ( k ) ixi ( k ) u( k) + + = (9) = xd x ( k ) ixi ( k ) + + xd x ( k ) ixi ( k ) h( k ) h ( k ). = = his equtio hs bee derived ssumig k. If we ssume k > we rewrite it to be idetil to (7). his eds the proof. = =

7 Remrk. Beuse h(k), d mx for y k, d (, this lemm with (6) imply tht the vlue of the otrol sigl is lwys oegtive d upper-bouded. Further importt properties of the proposed otroller re give i the followig two theorems. heorem. With the pplitio of the proposed otrol lgorithm the queue legth ever exeeds its demd vlue. P r o o f: rsformig (5) we obti () x x ( k) u( k) u ( k i ) = + +. d i Sie i for i (, ), usig Remrk we olude tht the right hd side of () is oegtive, whih implies tht x (k) x d for y k. his eds the proof. heorem. If the proposed otroller is pplied d the demd queue legth stisfies iequlity () x d > ( ) i i+ dmx, = the the queue legth is stritly positive for y k > mx(m Rp ) +. Proof: We observe, tht u(k < ) =. his llows us to rewrite (4) for k > mx(m Rp ) + s k k y( k) = iu ( i) h( ) = = = i k k = iu ( i) + u i ( ) + i u ( i) h( ) = = = i+ = k k x d = i + i u ( i) h( ) = = i+ = () = k i k = xd + i h( ) h( ) = = = = k i h( ) i ( i+ ) dmx = k i x d = = = = xd + >. 3

8 his eds the proof. his theorem shows tht if oditio () is stisfied, the the queue legth is stritly greter th zero for y k > mx(m Rp ) + whih implies tht the vilble bdwidth is fully used. 3.. Rehig lw bsed slidig mode otroller A disdvtge of the slidig mode otroller preseted i the previous hpter is the lrge vlue of the otrol sigl t the iitil time istt. herefore, i this subsetio we itrodue rehig lw bsed otrol strtegy i order to miimize this effet. he properties of the modified strtegy re the formulted d proved. I order to desig the modified otroller, we dpt the rehig lw proposed i [6]. It be formulted i the followig wy (3) s ( k + ) s( k) = φ s( k), where φ[s(k)] = mi[ s(k), δ]sg[s(k)] d δ >.With this lw pplied to the ded-bet type otroller like the oe proposed i the previous subsetio the system represettive poit is gurteed to reh the hyperple s(k) = mootoilly i fiite umber of steps. Equtio (3) be ltertively expressed s follows (4) s ( ) k = e( k ) + f ( k ), where s (k) is ew slidig vrible, vetor is give by (4) d futio f(k) is defied s follows f ( k ) = f ( ) sg{ ( ) } for, (5) k + δ s k k < k f ( k ) = for k k. We ssume, tht s () =, whih results i the followig oditio (6) f ( ) = e ( ) = x d = xd. I this wy the represettive poit will move towrds the origil slidig hyperple desribed by (), tti it fter k, d remi o it fterwrds. Now we will determie the reltio betwee the desig prmeter δ d the time istt k. From (5) we obti (7) f ( k ) = f ( ) + δ( k ) sg s( ) = xd + δ ( k ). If f(k ) =, the f[(k )] δ. From this reltio it follows tht x (8) d k =, δ where futio ξ deotes the smllest iteger greter th its rgumet ξ. his shows how the hoie of prmeter δ ffets the time istt whe the represettive poit rehes the slidig hyperple. With the pplitio of the proposed rehig lw, the otrol sigl be derived by substitutig (5) ito equtio e[(k + )] + f[(k + )] =. his leds to (9) u( k) = ( b) { x ( k) + f ( k + ) d Ax } 4 Substitutig (7), (8) d (4) ito (9) we obti.

9 (3) u( k) = xd x ( k) i xi( k) + f ( k ). + his ompletes the desig of the rehig lw bsed otrol strtegy. Further i this setio, we stte d prove importt properties of the proposed strtegy. Lemm shows tht the otrol sigl is lwys oegtive d upper bouded. heorems 3 d 4 (logous to heorems d ) show tht the queue legth ever exeeds its demd vlue d tht fter some iitil time the queue legth is lwys stritly positive, whih mes tht the vilble bdwidth is fully used. Lemm. If the desiged slidig mode otroller is pplied, the its output for y k stisfies h ( k ) + f ( k + ) f ( k) (3) u( k) =. Proof: he proof is similr to the proof of Lemm. We rewrite (3) s u( k) = xd x ( k) ixi ( k) + f ( k ) u ( k ) + = = xd x ( k ) ixi ( k ) + f ( k ) + + (3) xd x ( k ) i xi ( k ) f ( k) + = h ( k ) + f ( k + ) f ( k) =. = = = his eds the proof. Remrk. We otie tht f(k) is o-deresig, h(k), d (,. From this it follows, tht the otrol sigl is lwys oegtive. Furthermore h(k) d mx, d mx{f[(k + )] f(k)} = δ. his mes tht the otrol sigl is lwys upper-bouded, i.e., dm x + δ (33) u( k) for y k. By pproprite hoie of futio f(k) we obtied ostt upper boud o the otrol sigl, whih is prtil for pplitio i rel etwork. heorem 3. If the proposed otrol strtegy is pplied, the the queue legth ever exeeds its demd vlue. Proof: Usig (6) we trsform (3) d obti = 5

10 (34) 6 d ( ) x x k ( ) = u k + = = ( ) ( ) iu k + i f k +. Futio f(k) is lwys smller th or equl to zero, >, d from = Remrk the otrol sigl is lwys oegtive. his mes, tht the right hd x k x d eds the proof. side of equtio (34) is oegtive, whih gives ( ) d heorem 4. If the proposed otrol lgorithm is pplied, d the demd queue legth stisfies the followig oditio (35) x d > i ( + ) i = the for y k > k + mx(m Rp ) + the queue legth is stritly greter th zero. P r o o f: Usig (3) d reltio u(k < ) =, we rewrite (4) for y k > k + mx(m Rp ) + s follows: (36) his eds the proof. ( ) ( ) d mx k k { i } ( ) y k = u i h = = = k k = i u ( i) h( ) = = i = k i h ( ) f ( ) f ( ) k = i + + h ( ) = = = r r r= r= k i h ( ) ( ) k =+ i f h ( ) = = = r r= k i k = xd + i h( ) h( ) = = = r r= k ( ) ( ) = =, i h i i+ dmx = k i = xd x. d

11 his theorem shows tht if the demd queue legth stisfies the sme oditio s for the bsi versio of the ded-bet slidig mode otroller, the the queue legth is stritly greter th zero for y k > k + mx(m Rp ) +, whih implies tht the vilble bdwidth is fully used. 4. Simultio exmples I order to verify the properties of the proposed flow otrol strtegy, omputer simultios of the etwork with 4 dt soures were performed. As stted i Setio, we ssume tht the distributio of dt rte mog the oetios is performed by some higher-level lgorithm. he disretistio period is seleted s ms. he prmeters of the oetios re show i ble. ble. Lik prmeters i the simulted etwork p 3 4 m Rp b p, ms 3 5 γ p α p Beuse mx(m Rp ) = 9, we get =. Aordig to Setio, the stte spe etwork model prmeters re: =, =.98, 3 =, 4 =.97, 5 =, 6 =.38, 7 =, 8 =, 9 =.86. he mximum vilble bdwidth of the bottleek ode is d mx = 3 kb. Aordig to heorems d 4 i order to esure full bdwidth utiliztio, the demd queue legth must be greter th 97 kb for both proposed otrollers. herefore, the demd queue legth hs bee hose s kb. he vilble bdwidth hges rpidly from low to high vlues, whih reflets the most dverse possible oditios i the etwork. he vilble bdwidth is show i Fig.. For the rehig lw bsed slidig mode otroller, prmeter δ hs bee hose s 35 kb, whih ordig to (8) gives k = 6. Fig.. Avilble bdwidth Fig. 3 shows the bottleek lik queue legth whe the primry ded-bet otroller is pplied. We otie from the figure tht the queue legth ever exeeds its demd vlue, d fter the iitil time derived i heorem it is lwys stritly positive. herefore, there is o risk of buffer overflow, d full bdwidth 7

12 utiliztio is esured. he vlue of the otrol sigl d the mout of dt levig the buffer re show i Fig. 4. he vlue of the otrol sigl t the first time istt is equl to 8. kb. As oe see from the figure, the otrol sigl is lwys stritly positive d upper bouded. he trsmissio rtes of idividul soures with the pplitio of the bsi otrol strtegy re show i Fig. 5. Fig. 3. Queue legth with pplitio of the primry slidig mode otroller Fig. 4. Amout of dt levig the buffer d the otrol sigl with pplitio of the primry slidig mode otroller 8 Fig. 5. rsmissio rtes of idividul soures i the etwork with the primry slidig mode otroller

13 Figs 6, 7 d 8 preset the simultio results obtied for the rehig lw bsed slidig mode otroller. Fig. 6 shows the queue legth i the bottleek lik buffer. It be see from the figure, tht the queue ever exeeds its demd vlue d fter the iitil time predited by heorem 4, it is lwys stritly positive. Fig. 7 shows the mout of dt levig the buffer d the vlue of the otrol sigl. Comprig Figs 4 d 7 oe otie, tht the itrodutio of the rehig lw sigifitly redues the mximum vlue of the otrol sigl t the begiig of the dt trsmissio proess. he trsmissio rtes of idividul soures re show i Fig. 8. Fig. 6. Queue legth with pplitio of the rehig lw bsed slidig mode otroller Fig. 7. he mout of dt levig the buffer d the otrol sigl with pplitio of the rehig lw bsed otroller Fig. 8. rsmissio rtes of idividul soures i the etwork with the rehig lw bsed slidig mode otroller 9

14 5. Colusios I this pper two disrete time slidig mode flow otrollers for oetioorieted etworks with multiple oetios hve bee preseted. he possible losses durig the trsmissio hve bee expliitly tke ito out i desigig the slidig hyperple. he first lgorithm hs bee desiged to esure fiite-time error overgee. he it hs bee modified by itroduig rehig lw to redue the mximum vlue of the otrol sigl t the begiig of the dt trsmissio. It hs bee proved tht the flow rtes geerted by both otrol lgorithms re lwys oegtive d upper-bouded. Furthermore, both proposed otrollers esure full bottleek bdwidth osumptio d elimite the risk of buffer overflow. Akowledgemet: his work hs bee performed i the frmework of Proet Optiml Slidig Mode Cotrol of ime Dely Systems, fied by the Ntiol Siee Cetre of Pold Deisio No DEC //B/S7/ Referees. Bdyopdhyy, B., S. Jrdh. Disrete-ime Slidig Mode Cotrol: A Multirte Output Feedbk Approh. Berli, Heidelberg, Spriger-Verlg, 6.. B r t o s z e w i z, A. Nolier Flow Cotrol Strtegies for Coetio-Orieted Commuitio Networks. I: IEE Proeedigs o Cotrol heory d Applitios, Vol. 53, 6, Brtoszewiz, A., J. Ż u k. Disrete ime Slidig Mode Flow Cotroller for Multi-Soure Sigle-Bottleek Coetio-Orieted Commuitio Networks. Jourl of Vibrtio d Cotrol, Vol. 5, 9, No, F u r u t, K. Slidig Mode Cotrol of Disrete System. Systems & Cotrol Letters, Vol. 4, 99, G o, W., Y. W g, A. H o m i f. Disrete-ime Vrible Struture Cotrol Systems. IEEE rstios o Idustril Eletrois, Vol. 4, 995, G o l o, G., Ć. Milosvlević. Robust Disrete-ime Chtterig Free Slidig Mode Cotrol. Systems & Cotrol Letters, Vol. 4,, I g i u k, P., A. B r t o s z e w i z. Lier Qudrti Optiml Disrete-ime Slidig-Mode Cotroller for Coetio-Orieted Commuitio Networks. IEEE rstios o Idustril Eletrois, Vol. 55, 8, Igiuk, P., A. Brtoszewiz. Lier Qudrti Optiml Slidig Mode Flow Cotrol for Coetio-Orieted Commuitio Networks. Itertiol Jourl of Robust d Nolier Cotrol, Vol. 9, 9, I g i u k, P., A. B r t o s z e w i z. Disrete-ime Slidig-Mode Cogestio Cotrol i Multisoure Commuitio Networks with ime-vryig Dely. IEEE rstios o Cotrol Systems ehology, Vol. 9,, I g i u k, P., A. B r t o s z e w i z. Cogestio Cotrol i Dt rsmissio Networks Slidig Mode d Other Desigs. Lodo, Spriger-Verlg,.. J g t h, S., J. l l u r i. Preditive Cogestio Cotrol of AM Networks: Multiple Soures/Sigle Buffer Serio. Automti, Vol. 38,, J i, J., W. H. W g, M. P l i s w m i. A Simple Frmework of Utility Mx-Mi Flow Cotrol usig Slidig Mode Approh. IEEE Commuitio Letters, Vol. 3, 9,

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