Intelligent Scheduling Control of Networked Control Systems with Networked-induced Delay and Packet Dropout

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1 Iteratoal Joural of Cotrol, Autoato, ad Systes, vol 6, o 6, pp , Deceber Itellget Schedulg Cotrol of Networked Cotrol Systes wth Networked-duced Delay ad Packet Dropout Hogbo L, Zegq Su*, Badog Che, Huapg Lu, ad Fuchu Su Abstract: Networked cotrol systes (NCSs) have gaed creasg atteto recet years due to ther advatages ad potetal applcatos he etwork Qualty-of-Servce (QoS) NCSs always fluctuates due to chages of the traffc load ad avalable etwork resources o hadle the etwork QoS varatos proble, ths paper presets a tellget schedulg cotrol ethod for NCSs, where the saplg perod ad the cotrol paraeters are sultaeously scheduled to copesate the effect of QoS varato o NCSs perforace For NCSs wth etwork-duced delays ad packet dropouts, a dscrete-te swtch odel s proposed By defg a saplg-perod-depedet Lyapuov fucto ad a coo quadratc Lyapuov fucto, the stablty codtos are derved for NCSs ters of lear atrx equaltes (LMIs) Based o the obtaed stablty codtos, the correspodg cotroller desg proble s solved ad the perforace optzato proble s also vestgated Sulato results are gve to deostrate the effectveess of the proposed approaches Keywords: Lear atrx equaltes (LMIs), etworked cotrol systes (NCSs), etworkduced delay, packet dropout, qualty-of-servce (QoS), stablty INRODUCION Networked cotrol systes (NCSs) are spatally dstrbuted cotrol systes wth cotrol loops closed va coucato etworks Usg NCSs has ay advatages over the tradtoal cotrol systes, such as reduced syste wrg, sple stallato, creased syste flexblty, ad the great beefts fro sharg of the resources herefore, NCSs have bee fdg ay applcato areas such as DC otors [,2], robots [3], vehcles [4], car suspeso syste [5], ball aglev syste [6,7], etc However, the use of a shared etwork, cotrast to usg dedcated wrg, rases ew challegg probles for the aalyss ad desg of NCSs such as etworkduced delay ad packet dropout As a result, covetoal cotrol theores wth assuptos such as o-delayed sgals ad o lost of forato, Mauscrpt receved Noveber 8, 2006; revsed Septeber 27, 2007; accepted Aprl 6, 2008 Recoeded by Edtoral Board eber B Jag uder the drecto of Edtor Youg- Hoo Joo hs work was jotly supported by the Natoal Scece Foudato of Cha (Grat No: , ) ad the Natoal Key Project for Basc Research of Cha (Grat No: 2002cb32205) Hogbo L, Zegq Su, Badog Che, Huapg Lu, ad Fuchu Su are wth Departet of Coputer Scece ad echology, State Key Laboratory of Itellget echology ad Systes, sghua Uversty, Bejg, P R Cha (eals: Hb-l04@alstsghuaeduc, szq-dcs@al tsghuaeduc, chebd04@alstsghuaeduc, hplu@ altsghuaeduc, fcsu@altsghuaeduc) * Correspodg author ust be re-evaluated before applyg to NCSs Recetly, NCSs have attracted uch atteto fro research coutes Issues such as etwork-duced delay [-3,6,7], packet dropout [2,3,6,7], etwork costrats [8,9], schedulg [0,], ad the sgal quatzato [2,3], have bee vestgated wth results reported the lterature I geeral, these results ca be classfed to two a approaches Oe approach s to desg the NCSs wthout cosderg the presece of the etwork, ad the study the etwork protocol or schedulg strategy to guaratee certa qualtatve propertes that the etwork should possesses [4] he other approach s to take the etwork effects to accout explctly ad study cotrol ethodologes to accoodate the upredctable ature of etwork forato trassso [-3] Network-duced delays ad packet dropouts are two ajor causes for the NCSs perforace deterorato ad potetal NCSs stablty I the lterature, soe portat cotrol ethodologes, such as stochastc optal cotrol [6], robust cotrol [7], predctve cotrol [], ad state feedback cotrol [5], have bee proposed to address the probles of etwork-duced delays or packet dropouts For ore detals o ths topc, please refer to [,6,7,5] ad the referece there It s otced that ost of the exstg cotrol ethodologes for NCSs adopt the costat cotrol paraeters ad saplg perod regardless of etwork Qualty-of-Servces (QoS) varatos I practcal crcustaces, the etwork QoS always fluctuates due to chages of the traffc load ad

2 96 Hogbo L, Zegq Su, Badog Che, Huapg Lu, ad Fuchu Su avalable etwork resources NCSs are fuctoally related syste ad ther perforace depeds ot oly o the cotrol algorth but also o the etwork QoS herefore, despte the progress ade NCSs cotroller desg, t has becoe evdet that advaced cotrol ethodologes wth QoS varato copesato are requred Ufortuately, there has oly bee a lted aout of research o cotrol ethodologes wth the QoS varato copesato takg to accout [] proposed ga schedulg cotroller for NCSs, where the cotrol paraeters are adjusted o-le based o etwork QoS varatos [0] optzed the cotrol paraeters for ga schedulg cotroller to prove the NCSs perforace It s otced that those works focused prarly o cotroller desg whle the NCSs stablty aalyss proble have ot bee addresses herefore, the stablty aalyss for NCSs wth scheduled cotrol paraeters stll reas as a ope research area Moreover, t s well kow that both cotrol paraeters ad saplg perod are closely related to the NCSs perforace herefore, t sees ore sutable for us to costruct a cotroller wth the saplg perod ad cotrol paraeters sultaeously scheduled to further prove the NCSs perforace Motvated by the above observatos, we focus o vestgatg the tellget schedulg cotrol for NCSs ths paper, where the cotrol paraeters ad saplg perod are sultaeously adjusted o-le accordg to etwork QoS varatos to prove the cotrol perforace of NCSs For a class of NCSs wth etwork-duced delays ad packet losses, ths paper proposes a dscrete-te swtch odel to descrbe the closed-loop syste he proposed NCSs odel descrbes the NCS as a swtch syste ad eables us to apply the theory fro swtch systes to study NCSs the dscrete-te doa I the fraework of the gve odel, the stablty codtos are derved for NCSs ters of lear atrx equaltes (LMIs) Based o the obtaed stablty codtos, the correspodg cotroller desg proble s solved ad the perforace optzato proble s also vestgated hs paper s orgazed as follows he proble stateet ad soe deftos are forulated Secto 2 he structure of tellget schedulg cotroller s preseted Secto 3 he stablzato proble of NCSs s solved Secto 4 ad the perforace optzato proble s vestgated Secto 5 Nuercal exaples are provded Secto 6 ad coclusos are gve Secto 7 Notato: hroughout ths paper, R ad deote the desoal Eucldea space ad the set of all real atrces respectvely refers to the Eucldea or for vectors ad duced 2-or R for atrces he superscrpt deotes atrx trasposto; ad for syetrc atrces X ad Y, the otato X > Y eas that X Y s postve defte I s the detty atrces wth approprate + desos, ad the otato Z stads for the set of oegatve tegers Fally, syetrc block atrces, we use " as a ellpss for the ters troduced by syetry 2 PROBLEM SAEMEN he NCS cosdered ths paper s depcted Fg, where the sesor ad the cotroller are coected by coucato etworks he dyacs of the cotrolled plat s gve by xt () = Axt () + But () () yt () = Cxt (), p where xt () R deotes the plat state; y() t R represets the output of the plat; ut () R eas the cotrol put of the plat A, B, C are atrces wth approprate desos I the cosdered NCSs, the sesor s te-drve At each saplg perod, the output of the plat ad ts testap (e, the te the plat output s sapled) are ecapsulated to a packet ad set to the cotroller va the etwork he testap wll esure the cotroller copute the etwork-duced delay for the sesor packets, ad correspodgly select the rght oe to copute the cotrol sgal It s otced that, to obta better NCSs perforace, the saplg perod ay be adjusted o-le accordg to soe echas I practce, the data packets NCSs usually suffer etwork-duced delay ad packet dropout durg the etwork trasssos he etwork-duced delay here eas the sesor-to-cotroller delay ad t ay be loger tha the saplg perod h It wll be show that, for the k-th packet fro the sesor, f t ecouters a delay saller tha h, t wll be used to copute the cotrol sgal at te stat ( k + ) h Otherwse, at te stat ( k + ) h, the proposed cotroller wll use the predcted cotrol sgal to cotrol plat such a way to copesate log te delay ad packet dropout Note that for the packets fro the sesor, oly the sesor packets wth delay Cotroller Actuator Network Fg he structure of NCSs Plat Sesor y( k)

3 Itellget Schedulg Cotrol of Networked Cotrol Systes wth Networked-duced Delay ad Packet 97 saller tha h are used by the proposed cotroller herefore, the packets wth delay loger tha h ca be cosdered as dropped packet for the NCS uder vestgato Based the above observatos, we troduce the followg defto to capture the ature of packet dropouts NCSs Defto : he packet fro the sesor s called effectve packet f ts ecoutered delay s shorter tha S,, the saplg perod h Let { 2 } { 02,,, } deote the sequece of te dex of the effectve packets, the the packet dropout process s defed as { S} η( ) +,, (2) whch eas that, fro to +, the uber of dropped packets s η( ) Especally, whe two cosecutve sesor packets are effectve sesor packets, we have η( ) =, whch eas o packet s dropped because η( ) = 0 For the sake of aalyss, we defe Ndrop ax {η( )} S he we ca coclude that η( ) takes values a fte set Ω {, 2,, N drop } Let τ express the etwork-duced delay ecoutered by the th effectve packet Wthout loss of geeralty, ths paper, we assue τ, the lower boud of τ, s saller tha h By cosderg the defto of effectve packet, we have τ U [ τ, h) Obvously, wth the effectve packets NCSs, there exsts a sequece of pars {η( ), τ } herefore, the te delay ad packet dropout forato ca be eboded as { S U } {η( ), τ }:, τ, η( ) Ω (3) he objectve of ths paper s to costruct a tellget schedulg cotroller for the NCSs uder vestgato, where the cotrol paraeters ad saplg perod are sultaeously adjusted o-le accordg to etwork QoS varatos 3 INELLIGEN SCHEDULING CONROLLER SRUCURE hs secto wll preset the structure of the tellget schedulg cotroller he basc dea of costructg the tellget schedulg cotroller s as follows he etwork QoS of NCSs s dvded to dfferet levels frst he, for dfferet etwork QoS levels, we desg dfferet saplg perod ad cotrol paraeters Fally, the tellget schedulg Actuator u( k) u( k) Regulator Swtchg cotroller Plat cotroller wll schedule the saplg perod ad cotrol paraeters o-le for NCSs based o the etwork QoS varato As show Fg 2, the tellget schedulg cotroller ca be dvded to three parts: ) the QoS otor; 2) the swtchg cotroller; ad 3) the regulator Each copoet s descrbed the followg sectos 3 Network QoS partto ad QoS otor desg here are dfferet ways to defe the QoS for sesor-to-cotroller etwork I ths paper, oe of the ost popular QoS easures s used ad defed as follows delay deotes the pot-to-pot etworkduced delay sesor-to-cotroller etwork; t s used to dcate how log a packet s expected to be delvered fro the sesor to the cotroller Let the upper ad lower bouds of delay NCSs ax be ad, the the uverse of dscourse of etwork QoS ca be descrbed as ax Q {[, ]} Dvde Q to N dfferet subsets deoted as Q {[ Ť delay(), ˆ delay()]}, where s a pecewse costat fucto called a swtchg sgal takg values a fte set S {, 2,, N}, Ť delay () ad ˆ delay() are the upper ad lower bouds of Q he parttog of Q satsfes Q Q Q Q (4) 2 N = r QoS otor Fg 2 he cotroller structure hus far, we have dvded the uverse of dscourse of etwork QoS to N dfferet levels Correspodg to each QoS level Q, delay s varyg deterstc tervals defed by the swtch {}, e, delay [ Ť delay( ), ˆ delay( )] Moreover, slar to the dscusso Defto, we defe the axu cosecutve packet dropout Q as Ndrop ax {η( )} S he oe ca fer that, for each etwork QoS level Q, η( ) takes values a Sesor y( k) Network Itellget Schedulg Cotroller

4 98 Hogbo L, Zegq Su, Badog Che, Huapg Lu, ad Fuchu Su fte set Ω {, 2,, Ndrop} I the proposed tellget schedulg cotroller, the fucto of the QoS otor ut s to detere whch level the curret etwork QoS belogs to I the NCSs, each sesor packet s arked wth the te whe the plat output s sapled (e, the testap) herefore, uder the codto that the sesor ad the cotroller are sychrozed, the QoS otor ut ca easly calculate the etwork-duced delay for the arrvg sesor packet by sply coparg the testap of the sesor packet wth the clock of the cotroller ode Wth the detected delay forato, the QoS otor ca detere whch level the curret etwork QoS belogs to he forato o the curret etwork QoS level s the utlzed by the swtchg cotroller to schedule the saplg perod ad cotrol paraeters 32 he swtchg cotroller desg I the swtchg cotroller, there exsts a specal saplg perod h for each etwork QoS level Q, where S Deote the -th saplg perod relatve to tal te as h,, where S ad the value of s detered by the etwork QoS codto at the -th saplg perod o splfy wrtg, we deote the su of the frst k saplg perods as k h Let Q( h ) deote the etwork QoS h k, = k codto at te stat h k, tsc ( k ) deote the etwork-duced delay ecoutered by the k-th packet fro the sesor Notg that oly effectve packets are used by the cotroller, we ca assue that t sc = f a packet s dropped or ecouters delay larger tha the curret saplg perod he the swtchg cotroller s desged as Swtchg rule : If Q( h k) s Q, the Local cotroller rules : Rule : If tsc τ h ), the x( hk + h) = Fx ( hk) + Gu ( hk) K[ y( hk) Cx( hk)] u( hk + h) = Lx( hk + h) Rule 2 : If tsc =+, the x( hk + h) = Fx ( hk) + Gu ( hk) u( h k + h) = Lx ( h k + h ), where S, F ad h 0 G are gve as (5) Ah Aτ F = e, G = e dτ B (6) he swtchg cotroller s te-drve At each saplg perod, the swtchg cotroller copute the cotrol sgal for the plat ad the sed t to the actuator Iedately after recevg the cotrol sgal, the actuator wll use the arrvg cotrol sgal to cotrol the plat Dscretze the cotuous-te plat () at the saplg stats as Swtchg rule : If Q( h k) s Q, the x( hk + h) = Fx ( hk) + Gu ( hk) y( hk + h) = Cx( hk + h), (7) where S, C s of for (), F ad G are of for (6) Let e( h ) x( h ) x ( h ) ad troduce k k k k k k z( h ) = x( h ) e( h ) to (7) ad (5), the we have Swtchg rule : If Q( h k) s Q, the Rule If t : sc τ h ), the F GL GL z( hk + h ) = z( hk) 0 F KC + (8) Rule 2 : If tsc =+, the F GL GL z( hk + h ) = z( hk) 0 F It s otce that the sesor packets durg the two effectve packets are ot used to copute the cotrol sgal It eas that, for + k +, we have tsc ( k ) =+ Based o ths observato, t s obvous that (8) ca be represeted as Swtchg rule : If Q( h ) s Q the, z( h ) M ( ) z + + = h + 2 Π Π = z( 22 h ), + 0 Π where η( ) ( ) F GL 22 η( ) ( ) F KC F η( ) 2 η( ) F GL GL = 0 Π =, Π = +, Π = ( ) F (9) For splcty of otato, we rewrte (9) to the followg dscrete-te swtch syste: z( h ) = M z( h ), (0) + + r ( ) + where M r ( ) are atrces upo the swtch r ( ),

5 Itellget Schedulg Cotrol of Networked Cotrol Systes wth Networked-duced Delay ad Packet 99 whch takes values S For each possble value of r ( ) =, we have Mr ( ) = M, where M s of for (9) Moreover, for + < l < + +, the dyacs of NCSs ca be expressed as z( hl) = M z( h + ), () r ( ) where M r ( ) are atrces upo the swtch r ( ) ad of followg for M wth ( l ) GL Π ( l ) 0 F ( F ) = (2) l 2 l 2 F GL GL = 0 Π = ( ) F 33 he regulator desg Based o the cotroller desg approach troduced the followg two sectos, h, L ad K ca be desged for each QoS level Q, where S Correspodgly, a table wth respect to Q, h, L ad K ca be fored he, lookup table techque ca be used the regulator to schedule the saplg perod ad cotrol paraeters o-le accordg to the etwork QoS codto forato provded by QoS otor 4 SABILIZAION OF NCS 4 Stablty aalyss of NCSs hs secto wll aalyze the stablty property of NCSs, where the stablty aalyss proble ca be stated as follows Proble (Stablty aalyss): Gve a plat () ad a etworked cotroller (5), check whether the closed-loop NCS (0) s asyptotcally stable over etworks wth gve paraeters D drop, where S I the followg theore ad corollary, suffcet stablty codtos are derved for NCS (0) va both saplg-perod-depedet Lyapuov fucto ad coo quadratc Lyapuov fucto heore : Let Q be a gve set of etwork QoS levels, ad let h, L, K be the gve paraeters for the tellget schedulg cotroller, where S he NCS (0) over etworks wth gve paraeters drop D s asyptotcally stable, f for, j S, there exst postve defte atrces 2 2, R satsfyg P 2 2, R Pj M PjM P < 0, (3) where M s of for (9) wth η( ) Ω Proof: For the closed-loop NCS (0), let a saplg-perod-depedet Lyapuov fucto caddate be: + + r ( ) + V( h ) = z ( h ) P z( h ), (4) where P r ( ) are atrces upo the swtchg r ( ) NCSs odel (0) ad r ( ) takes values S Let r( ) ad j r( + ), the we have V( h ) = z ( h ) Pz( h ) ad V ( h + + ) = M PjMz h z ( h ) ( ) + + Fro the codtos (3), oe ca easly show that V = V( h ) V( ) + + h + (5) = z ( h )[ M P M P] z( h ) < 0, + j + where z( h + ) 0 Frst, let us prove that the closed-loop NCS (0) s stable f the codtos (3) hold hat s, gve ay ε > 0, we ca fd a δ(ε) > 0 such that z (0) < δ(ε) ples z( h l) < ε for l Z For ths purpose, troduce α ax{ax S M, }, α ax, 2 S M 3 S / P fucto, we ca get 2 α ax S P ad α4 Fro the defto of Lyapuov 2 4 z h + V h + α ( ) ( ) α 3 z ( h ) + I NCS (0), three cases arse ad are dscussed as follows Case : 0 l < +, where s the te dex of the frst effectve packet Let the NCS tal states be x (0), x (0) ad z (0) = x(0) e(0) he the dyacs of the NCS ths case s gve by z( h ) = M z(0) Gve ay ε > 0, f let l r(0) z (0) < {/ α2, }ε, the we have z( h ) M z(0) < M {/ α,} ε < ε r(0) r(0) Case 2: l = + I ths case, we have z( h + ) = Mr(0) z(0) Fro prevous dscussos ad (5), we ca coclude that V( hl) < V( h + ) α z ( h ) α3 Mr(0) z (0) l + ad z( h ) /α V ( h ) herefore, for ay gve ε > 0, lettg l l

6 920 Hogbo L, Zegq Su, Badog Che, Huapg Lu, ad Fuchu Su z (0) < α /(α α )ε wll lead to z( h ) r(0) αα3 α4 α /α M z(0) ( ) / z(0) < ε Case 3: + < l < + + Gve ay ε > 0, f 2 2 let z (0) < α /(α α α )ε, the fro () we ca get z( h ) = M( r( )) z( h ) α2 z( h ) l α ( α α )/ α z(0) < ε Based o the above aalyss, f we let ,, 4 3, α {/ α } α /(α α ) α /(α α α ), the we ca coclude that z (0) < αε ples + z( h l) < ε for l Z We ow prove that l l z( h l) = 0 for NCS (0) Fro (5), we ca get l V ( h ) ( h + ) + = 0, whch ples l ( ) z( ) 0 h + h + = O the other had, otg that z( h ) = M ( ) z( h ) l l r α2 z( h ) for + < l < + +, we ca get l l z( h l) = 0 for l + Suarzg the above two cases leads to the cocluso that + l l z( h l) = 0 for NCS (0), where l Z Accordg to the defto of asyptotcally stable, we ca coplete the proof Va coo quadratc Lyapuov fucto, a spler codto that checks whether the NCS (0) s asyptotcally stable ca be obtaed as follows Corollary : Let Q be a gve set of etwork QoS levels, ad let h, L, K be the gve paraeters for the tellget schedulg cotroller, where S he NCS (0) over etworks wth gve paraeters D s asyptotcally stable, f for drop S, there exsts a coo postve defte atrx 2 2 P R satsfyg M PM P < 0, (6) where M s of for (9) wth η( ) Ω Proof: It drectly follows fro heore Apparetly, heore s based o the saplgperod-depedet Lyapuov fucto, whle Corollary s obtaed va the coo quadratc Lyapuov fucto hus, the stablty codtos of heore are less coservatve tha Corollary However, Corollary provdes a spler way to check whether the NCS (0) s asyptotcally stable Note that the codtos of heore ad Corollary are all LMIs, whch ca be readly checked by usg stadard uercal software such as the LMI toolbox MALAB 42 Stablzato of NCS he prevous secto presets the stablty codtos for NCSs A edate questo s whether the obtaed stablty codtos ca be further exteded to cope wth the cotroller desg proble, whch s stated as follows Proble 2 (he cotroller desg): Gve a gve set of etwork QoS levels Q ad a plat the for of (), where S, desg h, L, K for the tellget schedulg cotroller such that the closedloop NCS (0) over etworks wth gve paraeters drop D s asyptotcally stable Note that t s ot easly to calculate h, L ad K drectly fro the codtos of heore (or Corollary ) by covex optzato techques he a dffculty les the fact that, the selecto of the best saplg perod for NCSs s a coprose due to the teracto of etworks [6], ad the codtos of heore (or Corollary ) are olear atrx equaltes (NMI) due to ( F GL) j o crcuvet the sythess proble, we propose a three-step desg procedure for the cocered NCSs, whch eables us to desg h, L ad K oe by oe 42 he saplg perod selecto As dscussed [6], the NCSs perforace chart versus the saplg perod ca be derved as Fg 3 Pot A coes as o surprse sce NCSs are a type of dgtal cotrol syste As saplg perod gets saller, the traffc load the etwork becoes heaver Correspodgly, ore packets are dropped ad loger etwork-duced delays result hs stuato causes the exstece of Pot B NCSs As a result, there exsts a saplg perod rage ( h, h ) for the NCSs Whe the saplg perod s B Perforace Worse Better A Out of cotrol Uacceptable Perforace Acceptable Perforace Saller A Saplg perod B Larger Fg 3 he NCSs perforace vs the saplg perod

7 Itellget Schedulg Cotrol of Networked Cotrol Systes wth Networked-duced Delay ad Packet 92 wth ( hb, ha), the cotrol perforace of NCSs s acceptable Based o the above dscussos, oe ca fer that, to guaratee the cotrol perforace of NCSs, h should be selected wth the saplg perod rage ( ha, hb ) herefore, the h desg proble s casted to the h A ad h B coputg proble, whch s addressed as follows he coputg of h A Let w bw deote the cotrol syste badwdth, whch s defed as the axu frequecy at whch the output of a syste wll track a put susod a satsfactory aer Alteratvely, the cotrol syste badwdth s the frequecy of put at wtch the output s atteuated to a factor of 0797 tes the put (or dow 3 db) [7] I order to guaratee the cotrol perforace, the rule of thub for selectg saplg perod dgtal cotrol s that the desred saplg ultple s [7] ws 20 40, (7) w bw where w s s the saplg frequecy (or saplg rate) dgtal cotrol he, by cosderg (7) ad by cosderg the effect of etwork-duced delay, the saplg perod of Pot A for h A Q, e, h A, ca be coputed by = ˆ delay(), (8) 20w bw where ˆ delay() s the upper boud of etworkduced delay Q, as dscussed Secto 3 he coputg of h B As dscussed [6], h B ca be coputed by ttt h B =, (9) 0696 where ttt deotes the total trassso te of all cyclc essages the etwork I a ore cocrete sese, let be the uber of devces the etwork, ad defe the trassso te of each essage as j 2 j tx, the we have ttt =Σ = j tx for strobe coecto etwork, ad have ttt =Σ ( + ) j j= tx for poll coecto etwork For ore detals o ths topc, please refer to [6] For etwork wth other coecto type or wth ttt beg uavalable, a estato of by ttt s gve ttt = 05( Ť delay() + ˆ ()) τ, (20) delay where τ s the lower boud of etwork-duced delay NCS, as dscussed Secto 2; Ť delay () ad ˆ delay() are the lower ad upper bouds of etwork-duced delay Q, as dscussed Secto 3 hus far, we have provded the ethod to copute the h A ad h B for NCSs Based o the preseted ethod, we ca obta the saplg perod rage ( hb, ha) for NCSs, where S he we ca select the saplg perod h wth the saplg perod rage ( h, h ) B A 422 he feedback ga atrx desg Wth h desged fro the frst stage, ths secto wll address the feedback ga atrx desg proble Note that a atrx s called schur stable f all ts egevalues are placed the ut crcle he the ecessary codtos uder whch the tellget schedulg cotroller wll stablze the NCSs (0) are establshed as follows heore 2: For a set of gve paraeter D drop, f for S, the codtos of heore or Corollary are satsfed (e, there exst postve defte atrces 2 2 P R > 0 such that (3) hold, or there exsts a 2 2 coo postve defte atrx P R > 0 such that (6) holds), the the atrces ( F GL) j are schur stable, where j Ω Proof: hs ca be proved by cotradcto Assue that oe egevalue of ( F GL) j s outsde the ut crcle, where j Ω By cosderg (9), the characterstc polyoal for M j ca be gve by M j 2 2 Πj Π j zi Πj Π j = zi = j 0 zi Π Πj (2) Sce M j s a block tragular atrx, ts deterat ca be wrtte as 22 j = Π j Πj η( ) η( ) j j j j j M zi zi = zi ( F G L ) zi ( F + K C) F (22) By cosderg the assupto that oe egevalue

8 922 Hogbo L, Zegq Su, Badog Che, Huapg Lu, ad Fuchu Su of ( F GL s outsde the ut crcle, oe ca coclude that there do ot exst postve defte 2 2 atrces P R > 0 such that (3) hold, or there does ot exst a coo postve defte atrx 2 2 P R > 0 such that (6) holds hat cotradcts wth the codtos of ths theore herefore, the atrces ( F GL) j ust be schur stable heore 2 elghtes us to desg L by guarateeg that ( F GL) j are schur stable for all j Ω Note that A beg schur stable ples j ) j A beg schur stable herefore we ca desg the L just by guarateeg that F G L s schur stable hs eables us to eploy pole placeet or other cotrol desg ethods to desg L hose ethods are avalable the lterature, ad so they are otted 423 he observer ga atrx desg Wth h ad L desged va the above stages, the observer ga atrx K desg techque s provded as follows heore 3: For a gve NCS (0) ad a set of gve paraeters L ad h, f for S, there exst syetrc atrces P R, P22 R ad atrces P2 R, X R, X2 R, p Y R, satsfyg P P 2 P 22 P P 2 P22 2 ϒ XΠ ϒ2 0 ϒ3 P2 j ϒ4 where > 0, (23) < 0, η( ) X F GL 2 Pj X ϒ = ( ), ϒ = X, (24) η( ) η( ) 3 X2F YCF 4 P22j X2 2 ϒ = +, ϒ = X, the the NCS (0) uder the tellget schedulg cotroller s asyptotcally stable, where K s gve by K = X2 Y Proof: Let X dag { X, X 2} ad P P (25) P 2 P 22 It s obvous that (24) ca be wrtte as P Γ where Pj X X 2 XΠ ϒ3 ϒ Γ = 0 < 0, (26) (27) Lettg ϒ, ϒ 3, 2 Π, ad Y = X 2K to (27) yelds Γ = X M, (28) where M s of for (9) Fro the vewpot of Γ = X M, (26) ca be rewrtte as P X M Pj X X < 0 (29) Pre- ad Post-ultplyg (29) by [ I M ] ad ts traspose respectvely leads to j M P M P < 0 (30) Accordg to heore, (30) ples that the closed-loop NCS (0) s asyptotcally stable hs copletes the proof Based o the codtos of Corollary, whch s obtaed va the coo quadratc Lyapuov fucto, a spler observer ga atrx desg ethod ca be obtaed as follows Corollary 2: For a gve NCS (0) ad a set of gve paraeters L ad h, f for S, there exst syetrc atrces P R, P22 R ad atrces P2 R, X R, X2 R, p Y R, satsfyg P > 0, P 2 P 22 (3) P P2 P 22 < 0, ϒ XΠ ϒ2 0 ϒ3 P2 ϒ 4 (32) where ϒ = ( ) ( ) X F GL η, ϒ 2 = P X X, ( ) ( ) X2 F η + Y CF η, ϒ 3 = ϒ 4 = P22 X 2 X2,

9 Itellget Schedulg Cotrol of Networked Cotrol Systes wth Networked-duced Delay ad Packet 923 the the NCS (0) uder the tellget schedulg cotroller s asyptotcally stable, where K s gve by K = X2 Y Proof: he proof s aalogous to that of heore 3, ad so t s otted I suary, the proposed cotroller desg approach ca be forulated as the followg threestep procedure Step : Accordg to (8) ad (9), copute h A ad h B for each etwork QoS level Q, where for S he select h wth ( ha h B ) Step 2: Desg L by guarateeg that F GL s schur stable Step 3: Desg K by solvg the LMIs heore 3 or Corollary 2 5 HE NCS PERFORMANCE OPIMIZAION Sce the cotrol paraeters ad the saplg perod are closely related to the cotrol perforace of NCSs, the NCSs perforace ca be proved by usg optu saplg perod ad cotrol paraeters Ufortuately, a closed-for relatoshp aog the etwork QoS, the saplg perod, the cotrol paraeters ad the cotrol perforace s ot avalable o obta the optal h, L ad K for NCSs, we wll trasfor the optal paraeter desg proble to a optzato proble ad the solve the optzato proble he optzato varables I the optzato proble, the optzato varables are the eleets of h, L ad K : V = [ h K L h K L ], (33) N N N where S, N s the uber of dfferet QoS levels Note that L ad K ay be atrces For future use, we trasfor the optzato varables to a vector: V = v, v2,, v S ( ), (34) where S s the deso of V he ftess fucto I ths paper, we defe the ftess fucto as N J J = r( k) y( k), (35) k= 0 where N J s the approprate te dex such that the trackg has arrved at the steady state, rk ( ) ad y( k ) are the easureets o referece put ad plat output at te dex k Saplg Perod Costrat (SPC) o esure the physcal feasblty of the saplg perod ad the syste perforace, we troduce the saplg perod costrat subjected by the optzato proble as follows: B A SPC : h < h < h, (36) where S, h B ad h A are the lower ad upper bouds of h, ad they are desged accordg to (8) ad (9) Stable Doa Costrat (SDC) S A set Ψ R s called stable doa (SD) of NCSs (0), f ay V Ψ satsfes the stablty codtos of heore or Corollary Obvously, ay V that ca provde satsfactory cotrol perforace for NCSs ust be wth the defed Stable doa, sce satsfactory cotrol perforace ot oly requres a NCS to be asyptotcally stable, but also requre the NCS to eet soe perforace specfcatos herefore, t s suffcet to search the optal paraeters wth the Stable doa, whch ca effectvely reduce the search space ad guaratee the stablty of NCSs durg the optzato procedure Based o ths dea, Stable Doa Costrat (SDC) s troduced as follows: SDC : V Ψ, V, (37) whch eas that the optzato varables are costraed to the defed Stable Doa throughout the optzato procedure I suary, the optzato proble ca be expressed as OP : J st SPC: hb < h < ha SDC : V Ψ, V (38) (38) asks for V that zes the value of the cost fucto wth the defed Stable Doa Obvously, the SDC ca guaratee the stablty of NCSs durg the optzato procedure O the other had, by zg the cost fucto, the cotrol perforace of NCSs s optzed herefore, both stablty ad cotrol perforace ca be guarateed by solvg the optzato proble (38) I ths paper, a populato-based, heurstc search algorth aely Estato of Dstrbuto Algorth (EDA) s used to solve the optzato proble (38) EDA has recetly bee recogzed as a ew coputg paradg evolutoary coputato Ulke other Evolutoary Algorths (EAs), EDA does ot use crossover or utato Istead, t

10 924 Hogbo L, Zegq Su, Badog Che, Huapg Lu, ad Fuchu Su explctly extracts global statstcal forato fro the prosg solutos, ad subsequetly bulds a probablty dstrbuto odel of prosg solutos based o the extracted forato New populato are geerated by saplg fro the probablty dstrbuto odel rather tha rely o rado search operators (such as crossover ad utato) hus, oe of the ost appealg advatages of EDAs over classcal EAs s the reducto the uber of paraeters to be tued or assessed by the user For ore detals o EDA, please refer to [8-20] ad the refereces there I ths paper, a typcal EDA algorth s used to solve the optzato proble (38), where the optzato varables the search space are odeled as a ultvarate oral S dstrbuto pv ( ) =Π = pv ( ), whch s a product of S depedet uvarate oral dstrbutos p( v ) = (µ, σ ) he EDA algorth used ths paper ca be suarzed as follows: Geerate dvduals eetg Stable Doa costrat the tal search space radoly to for a tal populato, where the postve teger deotes the populato sze of EDA, the geerated j j j j dvduals are deoted as V = ( v, v2,, v ), S j =,, 2 Repeat the followg steps utl the terato crtero s et a Select the best 2 ( 2 < ) dvduals fro the paret geerato, where the postve teger 2 deotes the populato sze of prosg dvduals, the selected 2 ga caddates are deoted as j j j j 2 V ˆ = ( vˆ, vˆ,, vˆ S ), j =,, 2 b Update the probablty dstrbuto odel pv ( ) usg the selected 2 prosg dvduals accordg to 2 j µ = vˆ, (39) 2 j= 2 j 2 σ = ( vˆ µ ) (40) 2 j= c Geerate the ext populato: he best dvdual the curret populato s coped to the ext populato; geerate dvduals (we use rather tha because the best dvdual the curret populato has already bee selected as a dvdual) eetg Stable Doa costrat based o the updated probablty dstrbuto odel S pv ( ) =Π = (µ, σ ) Although we do ot kow the exact Stable Doa of a NCS, we ca realze the Stable Doa Costrat based o the stablty codtos descrbed (3) or (6) akg Step for exaple, we ca geerate a dvdual caddate the search space radoly, ad the check whether the dvdual caddate satsfes the stablty codtos descrbed (3) or (6) If t does, the dvdual caddate wll be selected as a dvdual of the populato Otherwse, we dscard the dvdual caddate pot ad repeat the above steps utl we obta oe satsfyg the stablty codtos descrbed (3) or (6) Slar arrageets are for Step 2-c, except that the ga caddate s geerated accordg to the probablty dstrbuto odel p(v) at Step 2-c 6 ILLUSRAIVE EXAMPLES I order to llustrate the effectveess of the proposed approaches, let us cosder a etworked servo otor cotrol syste wth setup show Fg he paraeters of the otor used ths paper are lsted able Let x [ θω] p =,, where θ ad ω are the output agle ad the agular speed respectvely, the the DC otor dyacs ca be expressed as: 0 0 x p() t = xp () t + u() t (4) I the presece of etwork, the etwork-duced delay correspodg to edu ad hgh etwork loads rages [2s 22s] ad [25s 43s] radoly, wth packet dropout rates of 2% ad 4%, respectvely I the followg text, we provde a coparatve study of the proposed ethod wth two publshed ethods Case : he state feedback cotroller Let us cosder the etworked cotroller [5], whch s a eoryless state feedback cotroller of for u = Kx Notg that ths etworked cotroller ca oly be appled to NCSs wth etwork-duced delay less tha a saplg perod, the saplg perod of etworked DC otor cotrol syste s set to 005s he we solve the LMIs of heore 2 [5] ad able he paraeters of the etworked DC otor J Ierta kg L Iductace H R Resstace 232Ω K orque Costat N / A Gear reducto rato /38 K e Back-EMF Costat Vs/rad

11 Itellget Schedulg Cotrol of Networked Cotrol Systes wth Networked-duced Delay ad Packet 925 obta K = [ ] Wth the tal state x 0 = [ 2 2], the sulato result uder the etworked cotroller u = [ ] x s depcted by dotted le Fg 4 Case 2: he observer-based cotroller wth fxed saplg perod ad fxed cotrol paraeters Let us cosder the etworked cotroller [2], whch s a observer-based cotroller wth fxed saplg perod ad fxed cotrol paraeters We apply the cotroller desg ethod [2] to the etworked DC otor cotrol syste, ad therefore L = ad K = [ obta [ ] 00000] Wth the sae tal state, the sulato result of etworked otor cotrol syste uder ths case s depcted by sold le Fg 4 Case 3: he tellget schedulg cotroller Let Q { delay [0 02, 0 22]} ad Q2 { delay [0 25, 0 43]} deote the etwork QoS codtos correspodg to the edu ad hgh etwork loads he we apply the proposed three-step cotroller desg procedure to the etworked DC otor cotrol syste, ad therefore obta h = 002, L = [ 0486 ] [ ] L [ ] 0023, K = for Q ad h 2 = 0 039, 2 = , K 2 = [ ] for Q 2 Wth the sae tal state, the sulato result of syste uder the proposed tellget schedulg cotroller s depcted by dottedsold le Fg 4 Case 4: he tellget schedulg cotroller wth optzed saplg perod ad cotrol paraeters he sulato used a populato sze of 60 for the EDA, wth the best 20 dvduals selected fro the paret geerato to update the ultvarate Gaussa dstrbuto for the ext geerato By usg EDA to solve the optzato proble (38), we obta h O NKR N F É ê Ö MKR É = E Ç äé Ö M å ^ JMKR JN JNKR M MKR N NKR O OKR P PKR Q qáãé=eëf = 002, L = [ ], [ `~ëé=n `~ëé=q `~ëé=p `~ëé=o Fg 4 ypcal syste perforaces usg dfferet etworked cotrollers K = ] for Q, ad h 2 = 004, L 2 = [ ], [ ] K 2 = for Q 2 Wth the sae tal state, the sulato result of syste uder the tellget schedulg cotroller wth optzed saplg perod ad cotrol paraeters s depcted by dotted-dashed le Fg 4 he sulato results deostrate that for the cocered etworked DC otor cotrol syste, the syste usg the eoryless state feedback cotroller has ore overshoot ad s ore oscllatory hs dcates that the eoryless state feedback cotroller s ot capable of cotrollg the etworked syste very well hs pheoea s reasoable sce t uses less forato about the syste tha other cosdered etworked cotrollers Whe a observerbased etworked cotroller wth fxed sa-plg perod ad fxed cotrol paraeters s used, the syste has o overshoot, but stll has log settlg te However, usg the tellget schedulg cotroller, especally usg the tellget schedulg cotroller wth optzed saplg perod ad cotrol paraeters, the syste ot oly has o over-shoot, but also has sgfcatly shorter settlg te Moreover, aalyss of the saplg perod of the tellget schedulg cotroller dcates a terestg ature For hgh etwork loads, the cotroller saplg perod s relatve log However, for low etwork loads, the tellget schedulg cotroller creases the saplg rate to obta better perforace hs echas eables the NCSs to acheve hgher resource utlzato rate ad avod overloads 7 CONCLUSIONS hs paper presets a tellget schedulg cotrol ethod for NCSs uder effects of etworkduced delay ad packet dropout, where the cotrol paraeters ad saplg perod are sultaeously adjusted to prove the NCSs perforace For the cocered NCSs, a dscrete-te swtch odel s proposed, whch eables us to apply the theory fro swtch systes to study NCSs dscrete-te doa I the proposed fraework, we obtaed two types of stablty codtos ters of LMIs, whch ca be easly checked by usg stadard uercal software Based o the obtaed stablty codtos, the correspodg cotroller desg proble s solved ad the perforace optzato proble s also vestgated Sulato results are gve to deostrate the effectveess of the proposed approaches REFERENCES [] S Cha, G P Lu, D Rees, ad Y Xa, Desg ad practcal pleetato of teret-based predctve cotrol of a servo syste, IEEE ras o Cotrol Systes echology, vol 6,

12 926 Hogbo L, Zegq Su, Badog Che, Huapg Lu, ad Fuchu Su o, pp 58-68, 2008 [2] H L, Z Su, H Lu, ad M-Y Chow, Predctve observer-based cotrol for etworked cotrol systes wth etwork-duced delay ad packet dropout, Asa Joural of Cotrol, vol 0, o 6, pp -3, 2008 [3] Y psuwa ad M-Y Chow, Ga scheduler ddleware: A ethodology to eable exstg cotrollers for etworked cotrol ad teleoperato-part II: teleoperato, IEEE ras o Idustral Electrocs, vol 5, o 6, pp , 2004 [4] P Seler ad R Segupta, A H approach to etworked cotrol, IEEE ras o Autoatc Cotrol, vol 50, o 3, pp , 2005 [5] M Gad, A Çela, ad Y Haa, Optal tegrated cotrol ad schedulg of etworked cotrol systes wth coucato costrats: Applcato to a car suspeso syste, IEEE ras o Cotrol Systes echology, vol 4, o 4, pp , 2006 [6] K J ad W-J K, Stochastc optal cotrol ad etwork co-desg for etworked cotrol systes, Iteratoal Joural of Cotrol, Autoato ad Systes, vol 5, o 5, pp , 2007 [7] K J ad W-J K, Robust cotrol for etworked cotrol systes wth adssble paraeter ucertates, Iteratoal Joural of Cotrol, Autoato ad Systes, vol 5, o 4, pp , 2007 [8] L A Motestruque ad P J Atsakls, O the odel-based cotrol of etworked systes, Autoatca, vol 39, o 0, pp , 2003 [9] P V Zhvoglyadov ad R H Mddleto, Networked cotrol desg for lear systes, Autoatca, vol 39, o 4, pp , 2003 [0] M-Y Chow ad Y psuwa, Ga adaptato of etworked DC otor cotrollers based o QoS varatos, IEEE ras o Idustral Electrocs, vol 50, o 5, pp , 2003 [] Y psuwa ad M-Y Chow, O the ga schedulg for etworked PI cotroller over IP etwork, IEEE/ASME ras o Mechatrocs, vol 9, o 3, pp , 2004 [2] K L ad J Balleul, Robust quatzato for dgtal fte coucato badwdth (DFCB) cotrol, IEEE ras o Autoatc Cotrol, vol 49, o 9, pp , 2004 [3] L A Motestruque ad P J Atsakls, Statc ad dyac quatzato odel-based etworked cotrol systes, Iteratoal Joural of Cotrol, vol 80, o, pp 87-0, 2007 [4] W Zhag, M S Bracky, ad S M Phllps, Stablty of etworked cotrol systes, Cotrol Systes Magaze, vol 2, o, pp 84-99, 200 [5] M Yu, L Wag, G Chu, ad G M Xe, Stablzato of etworked cotrol systes wth data packet dropout ad etwork delays va swtchg syste approach, Proc of the 43th IEEE Coferece o Decso ad Cotrol, pp , 2004 [6] F L La, Aalyss, Desg, Modelg, ad Cotrol of Networked Cotrol Systes, PhD Dssertato, Uversty of Mchga, 200 [7] G F Frakl, M L Worka, ad J D Powell, Dgtal Cotrol of Dyac Systes, 3rd edto, Addso-Wseley, 998 [8] C Gozález, Cotrbutos o heoretcal Aspects of Estato of Dstrbuto Algorths, PhD Dssertato, Uversty of the Basque Coutry, Spa, 2005 [9] Y Ca, X Su, ad P Ja, Probablstc odelg for cotuous EDA wth Boltza selecto ad Kullback-Lebeler dvergece, Proc of the 8th Aual Cof o Geetc ad Evolutoary Coputato, pp , 2006 [20] P A Bosa ad J Grahl, Matchg ductve search bas ad proble structure cotuous Estato-of-Dstrbuto algorths, Europea Joural of Operatoal Research, vol 85, o 3, pp , 2008 HogBo L receved the BS degree 2004 fro Northeaster Uversty, Sheyag, Cha He s ow pursug a PhD degree the Departet of Coputer Scece ad echology, sghua Uversty, Bejg, Cha Hs curret research terests clude etworked cotrol systes ad tellget cotrol ZegQ Su receved the PhD degree Cotrol Egeerg 98 fro the Chalas Uversty of echology Swede He s curretly a Professor the Departet of Coputer Scece ad echology, sghua Uversty, Cha Hs curret research terests clude tellget cotrol, robotcs, fuzzy systes, eural etworks ad evoluto coputg etc adaptve cotrol Badog Che receved the BS ad MS degrees Cotrol heory ad Egeerg fro Chogqg Uversty, Chogqg, Cha, 997 ad 2003 He s curretly pursug a PhD degree Coputer Scece ad echology fro sghua Uversty, Bejg, Cha Hs curret research terests are sgal processg ad

13 Itellget Schedulg Cotrol of Networked Cotrol Systes wth Networked-duced Delay ad Packet 927 Huapg Lu receved the PhD degree 2004 fro the Departet of Coputer Scece ad echology, sghua Uversty, Bejg, Cha Curretly he s a Assocate Professor Departet of Coputer Scece ad echology, sghua Uversty Hs research terests clude tellget cotrol ad robotcs Fuchu Su receved the PhD degree 998 fro the Departet of Coputer Scece ad echology, sghua Uversty, Bejg, Cha Curretly, he s a Professor the Departet of Coputer Scece ad echology, sghua Uversty, Bejg, Cha Hs research terests clude tellget cotrol, eural etworks, fuzzy systes, varable structure cotrol, olear systes ad robotcs