TU/e University of Technology Eindhoven Mechanical Engineering

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1 TU/ Unvrsty of Tcnology Endovn Mcancal Engnrng Stablty of Ntword Control Systms: Do smart olds mprov robustnss for ntwor dlays? Rolf Gaasb D&C.5 Fnal Baclor Projct (4W) Projct suprvsor: Projct coac: Dr. r. N. van d Wouw M.Sc. N.W. Baur R.I. Gaasb Fbruary

2 Do smart olds mprov robustnss for ntwor dlays? Pag I bstract In ts rport w study stablty of a control systm wc communcats masurmnt data ovr a ntwor. T prsnc of t ntwor causs nformaton on sampld masurmnts to b snt n dgtal pacts; owvr all practcal actuators rqur a contnuous-tm nput sgnal. It s wdly accptd to apply a zro-ordr-old to convrt t dscrt-tm control sgnal to a contnuous on. Our contrbuton s a comparson btwn t zro-ordr-old (ZOH) a convntonal old wc olds ac sampld valu for on samplng ntrval to t systm-matcd-old (SMH) a spcfc typ of old wc mploys prdcton of t contnuous-tm closd-loop systm. T goal s to dtrmn wtr t systm-matcd-old mprovs robustnss for ntwor dlays and tm-varyng samplng ntrvals and trby gvs t NCS a largr stablty rgon. In ts rport w consdr tat t ntwor ntroducs two ffcts: varyng samplng ntvals and varyng dlays. W dstngus two cass: constant samplng ntrvals and transmsson dlays and t cas of on or bot of tm varyng. In t constant cas stablty analyss as bn don usng t gnvalu-basd stablty cc and by usng Lyapunov stablty tory. In t varyng cas a polytopc ovrapproxmaton of t uncrtan dscrt-tm NCS modl s mad wc nabls to formulat Lnar Matrx Inqualty (LMI) condtons guarantng (robust) asymptotc stablty. W compar t two olds by a numrcal xampl. W can conclud tat for constant ntwor dlays and samplng ntrvals a systm quppd wt t SMH (a smart old) as a sgnfcantly largr stabl ara. For varyng ntwor dlays and samplng ntrvals w also xpct a largr stabl ara wc w can prov for most cass. Howvr for t cas of small varyng ntwor dlays and samplng ntrvals mor systm stablty for t SMH s xpctd but cannot b provd.

3 Do smart olds mprov robustnss for ntwor dlays? Pag II Tabl of Contnts bstract... I Tabl of Contnts... II. Introducton... Motvaton... B m for ts projct... C Structur of t rport.... Ntwor Control Systm modl... 3 Dscrpton of t Ntwor Control Systm... 3 B T zro-ordr old (ZOH)... 4 C Systm matcd old (SMH)... 4 D Closd-Loop Modl Stablty analyss for constant samplng ntrvals and dlays... 6 Prov stablty usng an gnvalu-basd stablty tst... 6 B Prov stablty usng Lyapunov tory... 7 C Illustratv Exampl Stablty analyss for tm varyng samplng ntrvals and dlays... 9 Polytopc ovrapproxmaton... 9 B Prov stablty for t varyng cas usng Lyapunov tory... C Illustratv xampl... D T study of stablty by smulaton Rsults... 6 Rsults for constant samplng ntrvals and transmsson dlays... 6 C Varyng cas smulaton-stablty-mtod Dscusson Concluson and Rcommndatons Rfrncs Dscrtsaton of t NCS... 4 Dtrmn... 4 B Dtrmn Rwrtng... 7 Rwrtng for t SMH... 7 B Rwrtng for t ZOH Matlab fls... 3

4 Do smart olds mprov robustnss for ntwor dlays? Pag. Introducton Motvaton Ntword control systms (NCS) ar convnnt to us du to many advantags: as of nstallaton low mantnanc and modularty to nam a fw. Howvr t stablty of t systm can b lost by t prsnc of a wrd or wrlss ntwor n t control loop. Tr ar many communcaton ffcts wn consdrng a ntwor; n ts wor w consdr tat ts communcaton nducs varyng samplng ntrvals and transmsson dlays []. NCS ar sampld-data control systms. Wn dalng wt a sampld-data systm t dscrt-tm sgnal basd on t sampld masurmnt data s not an accptabl sgnal for t actuator and trfor nds to b convrtd to a contnuous-tm control sgnal. T convrson stp s don n t old; typcally a zro-ordr-old s usd wc olds t masurd sgnal at a constant valu for on samplng ntrval. Wat t controllr rcvs s vry dffrnt from t contnuous-tm output of t plant; t output s constant for a samplng ntrval and ts sampld valus ar rcvd by t controllr only aftr t transmsson dlay. To possbly mprov t contnuous-tm sgnal fd to t controllr a dffrnt typ of old could b usd wc s tund to t dynamcs of t plant. T da s to us a modl of t closd loop systm to mprov t nput sgnal to t controllr. Snc n practc a NCS always as varyng samplng ntrvals and transmsson dlays t s vry rasonabl to mnmz mpact of ts two uncrtan paramtrs by modfyng t nput sgnal of t controllr and trby ncras stablty of t NCS. In sort a smartr old could b abl to mnmz t nflunc of samplng ntrvals and transmsson dlays on t dffrnc btwn output sgnal of t plant and nput sgnal of t controllr. smartr old s trfor xpctd to gv t NCS mor robustnss aganst tm-varyng and uncrtan samplng ntrvals and transmsson dlay wc wll rsult n mor stablty. B m for ts projct In ts projct w wll focus on t mpact of t ntwor n t control loop usng a zroordr-old and a systm-matcd-old []. W wll loo at dffrnc btwn ts two typs of olds. T man part of our comparson wll b a stablty analyss comparson btwn t cas of constant samplng ntrvals and transmsson dlays and t cas wn samplng ntrvals and/or transmsson dlays vary. C Structur of t rport T outln of t rport s as follows. In Captr a dscrpton of t NCS a dscrpton of t olds and t closd-loop modl s prsntd. In Captr 3 t cas of constant samplng ntrvals and dlays s consdrd; ccng stablty for constant samplng ntrvals

5 Do smart olds mprov robustnss for ntwor dlays? Pag and transmsson dlays usng t gnvalu stablty tory and t Lyapunov tory [3]. lso an llustratv xampl s consdrd. Captr 4 dals wt varyng samplng ntrvals and/or transmsson dlays. To prov stablty frst a polytopc ovrapproxmaton of an uncrtan dscrt-tm NCS modl s ntroducd and tn stablty analyss for t varyng cas s consdrd usng Lyapunov tory. Ts agan s word out n an llustratv xampl followd by smulatons. In Captr 5 t rsults of all stablty analyss ar gvn and compard. Captr 6 wll dscuss ts rsults and Captr 7 wll gv a concluson and rcommndatons. ll scrpts usd to ma stablty plots can b found as dgtal attacmnts.

6 Do smart olds mprov robustnss for ntwor dlays? Pag 3. Ntwor Control Systm modl In a classc control stup t controllr s consdrd to b drctly wrd to t plant. T output data of t plant and controllr s snt xactly at a spcfc constant samplngntrval wt no dlay. Howvr wn a controllr s mplmntd va a sard communcaton ntwor ts condtons ar not ralstc. In t control stup n ts projct varyng samplng-ntrvals and varyng dlays ar consdrd. Dscrpton of t Ntwor Control Systm For ts projct w consdr a closd-loop systm as dpctd n Fgur wr t plant dynamcs ar gvn by x x Bu() t () and t contnuous-tm stat-fdbac of t controllr s gvn by u( t) Kx ( t ). () T stat of t plant s assumd to b masurd at tm nstant s wc ylds t sampld masurmnt data x : x( s ). Ts sampld masurmnt data ar snt ovr t ntwor and consquntly dlayd by a dlay. T old mcansm convrts t sampld data x to a contnuous-tm functon x () t wc s t nput of t contnuous-tm controllr; f a nw sampl s rcvd x s rst to tat nw valu: x x t for t [ s s ) () x ( s ) x (3) wr x ( s ) dnots a stat rst at tm s. W consdr a cas n wc t samplng ntrval dfnd by s s satsfs and t sampld output arrvs at t old mcansm aftr a dlay τ wc s allowd to ta a valu n t rang. W spcfcally consdr t cas wn t dlay s assumd to always b smallr tan t samplng ntrval for all (.. ).

7 Do smart olds mprov robustnss for ntwor dlays? Pag 4 Fgur : Scmatc ovrvw of t NCS. Ts spcfc stup wc s scmatcally dpctd n Fgur s most valuabl wn snsors and actuators ar far apart. Ts s.g. t cas n watr dstrbuton ntwors. B T zro-ordr old (ZOH) Nowadays t most commonly usd typ of old s t zro-ordr old (ZOH). T zroordr old convrts a dscrt tm sgnal to a contnuous tm sgnal by oldng ac sampl valu for on samplng ntrval. Kpng (3) n mnd ts bavor can b modld wn x wc mans :. (4) C Systm matcd old (SMH) T systm matcd old (SMH) s a spcfc old tund to t dynamcs of t closd-loop systm mor spcfcally w wll dfn : BK. (5) W wll nvstgat n ts rport wtr ts spcfc old producs a systm wt mor robustnss n t prsnc of varyng samplng ntrvals and dlays. D Closd-Loop Modl Lt us dfn x x x as t stat of t closd-loop systm. Tang () () and (3) nto account t closd-loop dynamcs can b dscrbd as x BK x x for t [ s s ) x x (6) and xprncs a rst at tm s accordng to

8 Do smart olds mprov robustnss for ntwor dlays? Pag 5 x ( s ) xs ( ) x. (7) In ordr to analyz robust stablty of t systm gvn n (6) (7) w wll us tcnqus dvlopd for dscrt-tm paramtr varyng systms. Usng (6) (7) a dscrtsaton can b don. T dscrtsaton rsults n a dscrt-tm systm of t form x ( ) x (8) wr ( ) s t dscrt tm matrx. By tang two dffrnt approacs of dscrtsaton two ( ) matrcs ar drvd and usd for stablty analyss. To dstngus btwn tm ty ar dnotd ( ) and ( ). ( ) cl ( ) I I cl (9) ( ) ( ) r ( r) r ( r) BK dr BK dr ( ) () wr cl s gvn by cl BK. () T drvaton of ts two modls s ncludd n ppndx.. Of cours snc ( ) and ( ) dscrb t sam dynamcs w av tat ( ) ( ) ( ). ()

9 Do smart olds mprov robustnss for ntwor dlays? Pag 6 3. Stablty analyss for constant samplng ntrvals and dlays In ts captr w consdr t cas tat and () s a constant matrx wc w dnot by ( ) ar constant. Trfor ( ) n and t NCS s dscrbd by t dscrt-tm LTI systm x ( ) x. W can now xplot two tools for t purpos of stablty analyss; gnvalu-basd stablty tory and Lyapunov stablty tory. Prov stablty usng an gnvalu-basd stablty tst s dscrbd n Captr x ( ) x dscrbs t total modl. Snc ts s a dscrt-tm LTI systm t gnvalus of t matrx can dtrmn wtr t stats of t systm x wll convrg to t qulbrum x or not as gos to nfnty. T gnvalus of can b calculatd by solvng dt( I ). (3) If all gnvalus of av a modulus smallr tan.. t systm s asymptotcally stabl. In otr words f t gnvalus ar wtn t unt crcl as dsplayd n Fgur t NCS s asymptotcally stabl. s can b sn n () as svral ntgral componnts. To calculat t gnvalus of ts ntgrals nd to b solvd. Trfor w cc t gnvalus of snc ts form of t matrx dos not contan ntgral componnts. Fgur : Stabl gnvalus ar wtn t dpctd unt crcl.

10 Do smart olds mprov robustnss for ntwor dlays? Pag 7 B Prov stablty usng Lyapunov tory Provng stablty usng Lyapunov tory s t asst to undrstand by tnng of a pyscal systm and consdrng t Lyapunov functon to dscrb t stord nrgy of suc a systm. If t nrgy (Lyapunov functon) dcrass ovr tm t systm wll nd up n t qulbrum pont. T stat x of a systm s usd n a functon V x wc s postv dfnt and V. T functons V w consdr ar all of t followng quadratc form T T V x x Px wt P P. (4) If V x always dcrass tn w now x convrgs towards as gos to nfnty and t systm (8) wt s asymptotcally stabl. To cc f V s always dcrasng w can comput t ncrmnt V of t Lyapunov functon along solutons of t dscrt-tm systm and s wat t condtons ar to nsur tat t s ngatv. T condton w want to satsfy for stablty s trfor T T V V V x Px x Px. Substtutng (8) (wt ) for x V s qual to V V V x ( ) P ( ) x x Px T T T V V V x ( ) P ( ) P x. T T T abov nqualty s satsfd wn t followng lnar matrx nqualty (LMI) olds: T T ( ) P ( ) P wt P P. (5) W can now mploy an LMI solvr to sarc for a matrx P satsfyng (5). If t LMI-solvr fnds suc a P for wc (5) s vald stablty of t NCS s provd. For constant samplng ntrvals and dlays a good LMI solvr s always abl to fnd a vald P f t systm s stabl. Trfor wn usng a quadratc Lyapunov functon systm nstablty can also b provd. C Illustratv Exampl T tors xpland n scton 3. and 3.B bot us matrcs ar formd usng B to dtrmn stablty. Frst t and K 8 6 wc s a NCS wc uss t ZOH. By coosng dffrnt combnatons of and can b dtrmnd

11 Do smart olds mprov robustnss for ntwor dlays? Pag 8 for all combnatons. T sam can b don for a NCS wc uss t SMH tn B and K 8 6. s can b sn n bot cass t sam valus of 8 6 B and K ar usd. By mang plots of provd stablty for t dffrnt combnatons of constant and w can compar t ZOH and t SMH and a gnral da of t stablty nflunc of t old can b gand. T plots ar dsplayd n Fgurs and 6. It can b sn tat t gnvalu cc xactly matcs t rsults of t Lyapunov functon cc. Blu = stablty Rd = nstablty Blu = stablty Rd = nstablty Fgur 3: Stablty cc usng gnvalu stablty tory for ZOH. Fgur 4: Stablty cc usng Lyapunov tory for ZOH. Blu = stablty Rd = nstablty Blu = stablty Rd = nstablty Fgur 5: Stablty cc usng gnvalu stablty tory for SMH. Fgur 6: Stablty cc usng Lyapunov tory for SMH. Fgurs and 6 gv a good comparson of t systm usng t SMH and t systm usng t ZOH. T rsults of t SMH ar mprssv; t actuator s fd wt a bttr nput sgnal wc translats nto a sgnfcantly largr stablty ara. Not only t systm wt t SMH s bttr capabl of dalng wt larg samplng ntrvals t also gvs mor robustnss aganst ntwor dlays. Howvr w nd to ta n mnd tat ts stablty analyss s only vald for constant and. In Fgur 5 and 6 w can also spot tat t systm usng t SMH sms to b stabl for arbtrarly larg samplng ntrvals n cas of no ntwor dlays. Snc t SMH s tund to t dynamcs of t closd-loop systm wc w av a prfct dscrpton of ts could b suspctd. It can also b provn usng t gnvalu stablty tory for t SMH. Wn w ta t xampl systm usng t SMH w fnd t followng gnvalus for ( ) : Snc t gnvalus ar always btwn and so n cas of no ntwor dlays t systm wt t SMH s always stabl. 4

12 Do smart olds mprov robustnss for ntwor dlays? Pag 9 4. Stablty analyss for tm varyng samplng ntrvals and dlays For t cas tat and ar tm-varyng and uncrtan s not a constant matrx. Snc and l n a compact st ( ) also ls n a compact matrx st. Trfor t gnvalu stablty tory of scton 3. cannot b usd. lso Lyapunov tory as sown n scton 3.B cannot b usd drctly bcaus of t uncrtants causd by t varyng and. Howvr t s possbl to ma a polytopc ovrapproxmaton of t uncrtan dscrt-tm NCS modl wc tn allows for LMI basd ccs to prov stablty. Polytopc ovrapproxmaton Lt us construct a polytopc ovrapproxmaton of systm (8). T polytopc ovrapproxmaton can b don n a fw stps. Frst w rwrt ( ) n t followng form: n ( ) ( ) S (6) wr ( ) ar scalar functons and S ar constant matrcs for... n. T systm matrx as dtrmnd n Scton.D: ( ) ( ) r ( r) r ( r) BK dr BK dr ( ). Tr ar two componnts n ts matrx wc contan or trms. Ts componnts ar s s and... ds. W can us t Jordan form of t contnuous-tm systm matrcs and (.g. QJQ wt J J J n ) to rwrt t s trms as follows Js s Q Q () s R J ns. (7)

13 Do smart olds mprov robustnss for ntwor dlays? Pag T ntgrals can b rwrttn usng (7): b a b a s ( s) BK ds ( s) W BK ( s) T ds j j j b j a ( s) ( s) ds W BKT. (8) j j T rprsntaton of ( ) as n (6) can now b constructd by xplotng (7) (8). Ts as bn don compltly for an xampl cas s Scton.. W now av a systm dscrbd by (6) wc s not yt an ovrapproxmaton. T ovrapproxmaton of t uncrtan matrx ( ) for all and to b of t followng form: as ( ) co (9) N n wc co stands for convx ull. convx ull s dfnd by ts vrtcs N ; ts vrtcs ar t cornr ponts of a polytopc ovrapproxmaton as vsualzd n Fgur 7. ( ) as wrttn n (9) and vry otr matrx n t ara contand n t convx ull can b wrttn as convx combnatons of t vrtcs as follows: N ( ) for any and. N and () Fgur 7: Vsualzaton polytopc st wt 5 vrtcs. Consdrng (6) t mnmal and maxmal valus of ( ) can b calculatd by

14 Do smart olds mprov robustnss for ntwor dlays? Pag mn : ( ) max ( ). () Usng ts mnmal and maxmal valus t vrtcs can b cratd. T vrtcs concrn all combnatons of mnmal and maxmal valus of t functons so n total n N vrtcs rsult wc can b dscrbd as follows: V V V V V V n n n n () V V V N n n Ts matrcs N ar calld t ovrapproxmaton vrtcs wc dfn t polytopc st as formulatd n (9) and (). Ts polytopc st s also calld t ovrapproxmaton systm snc all t dynamc bavors of t orgnal systm (8) ar dscrbd by t ovrapproxmatd systm. B Prov stablty for t varyng cas usng Lyapunov tory s can b rcalld from (5) of scton 3.B stablty can b provd wt t Lyapunov tory T by provng tat tr xsts a P P suc tat t followng LMI s satsfd T ( ) P ( ) P. s mntond bfor w cannot drctly us ts formula n t varyng cas snc s not constant. Howvr rcallng() all ar unnown constants but all ar constant matrcs wc ar nown. Wn w substtut () nto (5) w obtan N N N T P P wr and. (3) T nqualts n (3) can also b wrttn as N N T P P wr and. (4) Snc (4) s satsfd f t followng LMIs ar satsfd T P P N. (5)

15 Do smart olds mprov robustnss for ntwor dlays? Pag T Ts mans tat w can us a LMI-solvr to fnd a matrx P P for wc (5) s vald and trfor prov stablty of our NCS wt varyng and varyng. If suc a matrx can b found for all t ovrapproxmaton vrtcs stablty s provd for t wol polytopc ovrapproxmaton (wc of cours contans ). Provng stablty of by mang a polytopc ovrapproxmaton and us t Lyapunov tory to prov stablty of t ovrapproxmaton vrtcs s furtr rfrrd to as LMI- OvrprxVrt-mtod. downsd of ts mtod s tat f t ovrapproxmaton s too larg muc consrvatsm could b addd and t may appn tat t uncrtan dscrt-tm NCS modl s stabl wl t cannot b provd du to t consrvatsm n t ovrapproxmaton. C Illustratv xampl Provng stablty for t cas of tm-varyng and can b don as dscrbd n scton 4.B. In ts scton t llustratv xampl of scton 3.C s laboratd for t varyng cas. Rcall for t NCS wt t ZOH t followng systm matrcs ar vald B B and K 8 6. For t NCS wt t SMH w av tat and K T frst stp of mang t polytopc ovrapproxmaton s rwrtng from () n a form of a product of constant matrcs and tm-varyng functons wc dpnd on and. Rcall tat: ( ) r ( r) r ( r) n BK dr BK dr ( ) ( ) S. ( ) W start by rwrtng t ( ) trm usng t Jordan form dcomposton of t matrx : s s S () s S s. (6) s Ts mtod can b usd for t otr trms nvolvng. W can also us t mtod as gvn n (8) to rwrt t ntgral trms ts s compltly word out n Scton.. Ts gvs us for t ZOH:

16 Do smart olds mprov robustnss for ntwor dlays? Pag ZOH ( ) V ( ) ( ) (7) and for t SMH can b rwrttn as ( ) 4( ) SMH ( ) V ( ) ( ) 4 ( ) 4 4 (8) s word out n (9) () t polytopc st wc mbds t uncrtan matrx st ( ) s caractrzd by vrtcs wr ac vrtx s a

17 Do smart olds mprov robustnss for ntwor dlays? Pag 4 combnaton of mnmal and maxmal valus of s (so n total n N vrtcs s (9)). If tn and all vrtcs ar qual (.. tr s only on vrtx). If and ZOH (ts bcaus n bot cass ts rsults n 8 = 56 vrtcs for t SMH and 4 = 6 vrtcs for t s constant). Basd on t LMI cc (5) stablty plots can b mad; ts can b found n t rsults n scton 5.B. D T study of stablty by smulaton s alrady pontd out n Scton 4.B a downsd of t LMI-OvrprxVrt-mtod s tat tr ar som scnaros n wc t NCS s stabl but stablty cannot b provd usng t LMI-basd mtod dscussd abov. For nstanc f w us t NCS wt t paramtrs of t llustratv xampl for t SMH: B and K 8 6 ; and us and. tn t LMI-OvrprxVrt-mtod cannot prov stablty. Howvr ts pont can also b smulatd usng t dscrt-tm NCS modl x ( ) x. (9) By startng wt a crtan ntal condton x w can smulat t valu of x on stp furtr n tm. By calculat ac tmstp by tang a random.. and.... T componnts of x can b plottd aganst t tm. Fgur 8 s suc a plot for t dscussd cas of.. and.... Dspt t lac of proof of stablty va t LMI-OvrprxVrt-mtod Fgur 8 s a plot of a stabl systm. Fgur 8: Smulatd componnts of systm output n tm.

18 Do smart olds mprov robustnss for ntwor dlays? Pag 5 Of cours for t sam ntal condton x ts plot s dffrnt vry run snc t valus of and ar cosn randomly vry run. To gv suc smulatons somwat mor rlablty ty can b rpatd (ac tm wt nwly gnratd random and ). Howvr t s mpossbl to dtrmn t 'worst cas' squnc of and trfor dtrmnng stablty by smulaton s xtrmly unrlabl. ltoug suc a smulaton s not a good way to vrfy stablty t s an accptabl way to dsprov stablty. In t sns tat f t LMI ccs say t systm s stabl t only tas on unstabl smulaton to say tat t systm s n fact not guarantd to b stabl. To cc for stablty by smulaton n smulaton runs ar don. Evry run s as long as tm stps. If for all n smulaton runs at tm stp t absolut valu of vry componnt of x s smallr tan t maxmum absolut valu m t combnaton of and s suggstd to b stabl. Stablty of cours can nvr b rlabl provd n ts way snc t n smulaton could av dvrgd from zro. lso an nstablty suggston sn t rlabl; bcaus f m s tan too strct (or to small) nstablty could b suggstd wn t systm s stabl. Ts smulaton basd mtod of vrfyng stablty wc of cours s not a rlabl way to prov stablty can also b usd as a 'santy cc' to ma stablty plots. Ts can b found n t Rsults (Captr 5). T mtod wll b rfrrd to as smulaton-stabltymtod and can gv an ndcaton of stablty.

19 Do smart olds mprov robustnss for ntwor dlays? Pag 6 5. Rsults W want to dtrmn wtr for a crtan NCS t SMH as a bttr robustnss for varyng ntwor dlays and varyng samplng ntrvals tan t ZOH. To b abl to compar t two olds stablty plots nd to b mad on ntwor dlays bot for varyng and nonvaryng samplng ntrvals and transmsson dlays. Trfor mtods of Captr 3 and 4 can b usd. ltoug all mtods ar not lmtd to on systm t rsults ar gnratd usng a spcfc NCS wt t followng paramtrs for t ZOH: B and K 8 6. nd for t SMH t paramtrs: B and K By usng t sam systm mtods can b compard. Rsults for constant samplng ntrvals and transmsson dlays For constant and LTI systm x ( ) x dscrbs t dynamcs of t total modl. Snc ts s a dscrt-tm LTI systm t gnvalu-basd stablty cc as dscrbd n scton 3. gvs us t stablty plots as gvn n Fgur 9 and. Blu = stablty provd Rd = nstablty provd Blu = stablty provd Rd = nstablty can t provd b provd Fgur 9: Stablty cc ZOH. Fgur : Stablty cc SMH. Snc t SMH s tund to t dynamcs of t closd-loop systm t as a sgnfcant largr stabl ara. Not only t systm wt t SMH s bttr capabl of dalng wt larg samplng ntrvals t also gvs mor robustnss aganst ntwor dlays. lso t systm usng t SMH sms to b stabl for arbtrarly larg samplng ntrvals n cas of no ntwor dlays ; proof can b found n Scton 3.C. Howvr w nd to ta n mnd tat ts stablty analyss s only vald for constant and.

20 Do smart olds mprov robustnss for ntwor dlays? Pag 7 B Cas tm-varyng dlays and samplng ntrvals LMI-OvrprxVrt-mtod Wn and/or ar not constant s not a constant matrx. By mang a polytopc ovrapproxmaton as dscussd n Scton 4. systm stablty can b provn for crtan bounds and on t tm-varyng dlays and samplng ntrvals. If w substtut t ovrapproxmaton vrtcs nto t ncrmnt of a Lyapunov functon (as xpland n Scton 4.B) w can prov stablty by solvng an LMI. Trfor stablty of t ovrapproxmaton systm can b provd wt usng a LMI-solvr. Stablty plots mad wt t LMI-OvrprxVrt-mtod ar gvn n Fgur 3 and 4. Ts ar gnratd wt a Matlab scrpt gvn n.3. Dpctd n Fgur and ar t stablty plots wt.3. Wn w loo at Fgur 9 and t can b spottd tat for.3 and constant dlays w av bot a stabl and an unstabl ara for bot of t olds. Trfor ts cas s ntrstng snc w can spot robustnss and stablty around t boundary of t stabl ara. Fgur and 3 ar t stablty plots wt.. Wn w loo at Fgur 9 and for. t SMH s always stabl. Howvr w don t now anytng about t robustnss aganst varyng samplng ntrvals wc w can plot usng t LMI-OvrprxVrtmtod. Blu = stablty provd Rd = stablty can t b provd Blu = stablty provd Rd = stablty can t b provd Fgur : Stablty cc ZOH =.3. Fgur : Stablty cc SMH =.3. Blu = stablty provd Rd = stablty can t b provd Blu = stablty provd Rd = stablty can t b provd Fgur 3: Stablty cc ZOH τ =.. Fgur 4: Stablty cc SMH τ =..

21 Do smart olds mprov robustnss for ntwor dlays? Pag 8 In gnral t SMH as a bggr stablty rgon for constant and and gvs bttr robustnss aganst tr of tm varyng. Ts s as xpctd and t LMI-OvrprxVrtmtod gvs proof for tat owvr wn and ar small and trfor t varyng n t dlays rlatvly small t ZOH sms to b mor robust aganst dlays. Ts mans tat t LMI-OvrprxVrt-mtod s unabl to fnd a proof for stablty wt t SMH wn and ar small. T systm could tn tr b unstabl or t s stabl but t LMI-OvrprxVrt-mtod cannot prov stablty. Ts could b causd by varous rasons furtr xpland n Captr 6. C Varyng cas smulaton-stablty-mtod Snc t LMI-OvrprxVrt-mtod gvs som unxpctd stablty plots for small and t smulaton-stablty-mtod as dscrbd n scton 6.D can tr partly confrm or rjct ts rsults. Of cours snc t smulaton-stablty-mtod as no provng caractr ts plots wll not provd proof for anytng. Fgur 5- gv stablty plots wc ar mad usng t smulaton-stablty-mtod. Wn t ttl ndcats unform a unform dstrbuton s usd for cratng t random and for ac stp. Evn so normal ndcats a normal dstrbuton. Wn comparng Fgur 5- wt Fgur -4 som ponts ar mard unstabl wn usng t smulaton-stablty-mtod but ar provd to b stabl by t LMI-OvrprxVrtmtod. Ts s du to t paramtrs of t smulaton-basd stablty cc. s xpland n scton 6.D a smulaton s tm stps long and wll b rpatd n tms. If for all n smulaton runs at tm stp t absolut valu of vry componnt of x s smallr tan maxmum absolut valu m t combnaton of and s suggstd to b stabl. In Fgur 5- t xampl systm s bng analyzd usng t smulaton-stablty-mtod usng t paramtrs m.and n 5. Blu = stablty suggstd Rd = nstablty suggstd Blu = stablty suggstd Rd = nstablty suggstd Fgur 5: Stablty cc ZOH =.3. Fgur 6: Stablty cc SMH =.3.

22 Do smart olds mprov robustnss for ntwor dlays? Pag 9 Blu = stablty suggstd Rd = nstablty suggstd Blu = stablty suggstd Rd = nstablty suggstd Fgur 7: Stablty cc ZOH =.3. Fgur 8: Stablty cc SMH =.3. Blu = stablty suggstd Rd = nstablty suggstd Blu = stablty suggstd Rd = nstablty suggstd Fgur 9: Stablty cc ZOH τ =.. Fgur : Stablty cc SMH τ =.. Blu = stablty suggstd Rd = nstablty suggstd Blu = stablty suggstd Rd = nstablty suggstd Fgur : Stablty cc ZOH τ =.. Fgur : Stablty cc SMH τ =.. In som cass at tm stp t absolut valu s bggr tan maxmum absolut valu m altoug t valus do slowly convrg to. Du to ts fact som ponts ar mard unstabl wl n ralty ty ar not. Ts can b sn n Fgur 9 and for larg tr stablty as alrady bn provd usng t LMI-OvrprxVrt-mtod but t spcfc run of smulaton-stablty-mtod gvs an nstablty suggston. Of cours t s also possbl tat at tm stp t absolut valu s smallr tan m but t systm s unstabl. Trfor t smulaton-stablty-mtod as no provng caractr.

23 Do smart olds mprov robustnss for ntwor dlays? Pag Howvr t smulaton-stablty-mtod dos ndcat a sgnfcantly largr stablty ara for t SMH. Trfor t s xpctd tat t SMH s also mor robust aganst small ntwor dlays and samplng ntrvals but t LMI-OvrprxVrt-mtod s unabl to prov tat.

24 Do smart olds mprov robustnss for ntwor dlays? Pag 6. Dscusson s rportd n Captr 5 som of t rsults wr unxpctd or can t b provn. Ts captr wll gv som ntrprtaton to ts cass by suggstng crtan tors. lso t probablty of ts tors s xamnd. Wn loong at t constant cas (Captr 3) wr ( ) s constant t gnvalu-basd stablty tory gvs t sam stablty ara as t Lyapunov tory. In t cas tat and as dscussd n captr 4 t LMI- OvrprxVrt-mtod provs tat t SMH also as a bttr robustnss. Howvr on low valus of dlays t ZOH sms to av bttr robustnss for varyng dlays. T smulaton-stablty-mtod on t otr and dos suggst a bttr robustnss for t SMH also n cas of small varyng dlays. Howvr snc t only smulats t dscrt-tm NCS modl usng matrcs ( ) wr and ta random valus for ac ts can nvr guarant stablty for vry squnc of and. Tr ar tr tngs wc could b causng ts rsults. T frst s tat t LMI-solvr dos not succd n fndng t propr Lyapunov functon and trfor can t prov stablty wn tr rally s stablty. Possbly a paramtrdpndant Lyapunov functon [4] would b mor ffctv n provng stablty. scond suspcon s tat t rsults of t smulaton-stablty-mtod ar unrlabl. Dspt t fact tat t can nvr b provd tat t stablty suggstons of t smulatonstablty-mtod ar corrct t paramtrs for t m and n usd n mang t plots ar vry strct (a larg numbr of runs for low tm stps and a strct margn) wc ncrass t valu of ts mtod. Lastly t s possbl tat t polytopc ovrapproxmaton s too larg. Wn tr s unstabl ara n t polytopc ovrapproxmaton but t systm tslf s stabl stablty can t b provd by t LMI-OvrprxVrt-mtod. s xpland n Scton 4. all matrcs wtn t ovrapproxmaton systm can b wrttn as lnar combnatons of t vrtcs n t convx ull. In cas of t SMH t convx ull xsts of 56 unqu vrtcs. In cas of t ZOH t convx ull xsts of only 6 vrtcs. T larg amount of vrtcs for t SMH may ndcat a poor ovrapproxmaton. If ndd t ovrapproxmaton s poor ts could b t rason for an nfasbl soluton. ltoug tr s no guarant tat a tgt ovrapproxmaton rsults n fasblty of t LMI condtons a tgtr ovrapproxmaton wll provd a bttr rprsntaton of t NCS and tus mor accurat LMI-basd stablty rsults. Futur wor could focus on loong nto ow to tgtn t ovrapproxmaton.

25 Do smart olds mprov robustnss for ntwor dlays? Pag 7. Concluson and Rcommndatons In stablty analyss of a ntword control systm (NCS) wt tm-varyng samplng ntrvals and transmsson dlays t old s oftn assumd as gvn; typcally bng a zro-ordr old. Howvr t can b dsgnd n dffrnt ways and t plays an mportant rol n t stablty of t NCS. In ts rport w compar t zro-ordr-old (ZOH) a convntonal old wc olds ac sampld valu for on samplng ntrval and t systm-matcd-old (SMH) a spcfc old tund to t dynamcs of t closd-loop systm. Wn and ar constant. T stablty ara of t SMH s sgnfcantly largr tan t stablty ara of t ZOH. Ts as bn provd usng t gnvalu-stablty-cc. Wn and/or ar not constant t systm matrx of t uncrtan dscrt-tm NCS modl contans trms wc dpnd nonlnarly on t uncrtanty paramtrs and. Howvr a polytopc ovrapproxmaton systm can b constructd and t stablty of t systm can b analyzd basd on t ovrapproxmaton vrtcs. T stablty of ts vrtcs can b vrfd by a LMI cc snc t ovrapproxmatd systm contans t dynamcs of t orgnal NCS. Trfor stablty of t NCS can b provd wt Lyapunov tory usng an LMI-basd tcnqu. s xpctd t SMH as a bggr stablty ara tan t ZOH. Howvr wn t varyng ntrval of and ar rlatvly small t ZOH sms to b mor robust aganst varyng and. W can also do ts cc smulatng t systm x ( ) x wr w calculat ac tmstp by tang a random and. T stablty plots rsultng from ts smulaton basd approac ndcat tat t SMH not only as a bggr ara but s also mor robust aganst varyng and wn t varyng ntrval s rlatvly small. T rason w suspct tat t smulaton basd stablty tst sows a largr stablty rgon for t SMH tan t ovrapproxmaton stablty tst s tat t polytopc ovrapproxmaton for t SMH s too larg. Wn t polytopc ovrapproxmaton s too larg n spt of t fact tat t NCS s stabl t polytopc ovrapproxmaton also contans an unstabl ara; tn stablty cannot b provd by t LMI-OvrprxVrt-mtod. In furtr rsarc on ts mattr t cas of varyng and sould b furtr studd. Ts can b don by usng anotr stablty cc or by optmzng t LMI-OvrprxVrtmtod. Possbly a paramtr-dpndant Lyapunov functon [5] would b mor ffctv n provng stablty. lso a tgtr polytopc ovrapproxmaton can mprov t mtod. Hopfully wt ts masurs bttr robustnss aganst varyng dlays can also b provd for rlatvly small varyng ntrvals.

26 Do smart olds mprov robustnss for ntwor dlays? Pag 3 8. Rfrncs [] W Zang Mcal S. Brancy and Stpn M. Pllps () Stablty of Ntword Control Systms IEEE Control Systms Magazn. [] Gawtrop Ptr J. and Wang Lupng (9) Constrand ntrmttnt modl prdctv control Intrnatonal Journal of Control 8: [3] João P. Hspana (7) Cours n Lnar Systms Tory [4] M.B.G. Cloostrman L. Htl N. van d Wouwa W.P.M.H. Hmlsa J. Daafouz H. Njmjr () Controllr syntss for ntword control systms utomatca

27 Do smart olds mprov robustnss for ntwor dlays? Pag 4. Dscrtsaton of t NCS To b abl to dtrmn stablty of t NCS a dscrtsaton of t sampld-data NCS dynamcs nds to b don. Tang nto account () () and (3) t closd-loop dynamcs of t NCS can b dscrbd wt t stat x x x. W rcall tat t sampld-data NCS dynamcs s dscrbd as follows: x BK x x for t s s x x x ( s ) xs ( ) x. (3) Usng (3) a dscrtsaton can b don n two ways ladng n a dscrt-tm NCS modl of t form: x ( ) x T two ways dffr n t matrx ( ). Trfor t matrx can b wrttn n two ways wc wll b rfrrd to as and. Dtrmn Frst w loo at. From (3) w can conclud tat: x s x s x cl ( ) ( ) ( ) and ( s ) I x s cl ( ) ( ) cl ( ) ( ) x I x( s ) I x s (3) x s x s (3) ( ) ( ) cl ( ) wr cl s gvn by

28 Do smart olds mprov robustnss for ntwor dlays? Pag 5 cl BK. Combnng (3) and (3) gvs us t followng xprsson: x x s x ( ) cl ( ) cl ( ) x x I I n wc w rcognz t dscrt-tm NCS modl x ( ) x wt I cl ( ) cl ( ) I. B Dtrmn Frst lt us valuat t soluton of t old n on samplng ntrval. x t x s for t s s ( t s) ( ) ( ) x ( s ) x x t x for t s s ( t s ) ( ) x ( s ) x ( ) Now w can dtrmn voluton of t plant stat xt () at t s by s ( s s ) ( s ) s x( s ) x BKx ( ) d s ( ) ( s ) ( s ) s x BK x ( s ) d s s ( s ) ( s ) BK xd. To solv t ntgrals w cang t boundars by coosng r s :

29 Do smart olds mprov robustnss for ntwor dlays? Pag 6 ( ) ( ) ( ) r r ( ) x s x BK x s dr r ( r) BK xdr wc can b wrttn n matrx form as: x ( ) r ( r) r ( r) BK dr BK dr ( ) x Wn w rwrt ts n t convoluton x x( s ) x w can also fnd ( ) r ( r) r ( r) BK dr BK dr ( ).

30 Do smart olds mprov robustnss for ntwor dlays? Pag 7 In ts attacmnt. Rwrtng s rwrttn nto a form of ( ) V. Ts only for t spcfc llustratv xampl as dscussd n t rport. T llustratv xampl for t ZOH s: B and K 8 6 nd for t NSC wt t SMH: B and K s provd bfor ( ) r ( r) r ( r) BK dr BK dr ( ). (33) Rwrtng for t SMH ll functons wtn can b rwrttn n t form ( ) V. Startng wt t trm ( ) w av tat: s s S () s S wr S I S and () s s. ( ) Ts can also b don for : s ( s) T ( s) T wr s T 4 () s s and T. 4 () 4s T trm n t ntgral can b rwrttn as follows:

31 Do smart olds mprov robustnss for ntwor dlays? Pag 8 r BK ( r) S ( r) S BK ( r) T ( r) T S BKT ( r) S BKT ( r) S BKT ( r) ( r) S BKT ( r) ( r). Ts can b substtutd n t frst ntgral n (33): r ( r) BK dr S BKT ( r) dr S BKT ( r) dr S BKT ( r) ( r) dr S BKT ( r) ( r) dr ( r ( r ) ) 8 r ( r ) 4( r ) ( ) r ( ) ( ) 4( ) 4( ). T scond ntgral can b rwrttn nto: r ( r) ( r) BK dr S BKT dr 4( r) ( r) S BKT dr S BKT r dr S BKT r dr 8 4( r) ( r )

32 Do smart olds mprov robustnss for ntwor dlays? Pag 9 4( r ) r ( r ) ( r ) 6 8 4r ( ) ( ) Snc now all trms ar word out ty can b flld nto t total dscrt-tm matrx as follows: ( ) r ( r) r ( r) BK dr BK dr ( ) () () () (). (34) T sparat trms n (34) can b wrttn as: 4 4 ( ) 4( ) () ( ) () ( ) 4 4 () 4 4 () 4 ( ) 4 wc gvs for t SMH: 4 V ( ) ( ) SMH 4( ) 4 ( ) 4 4 4

33 Do smart olds mprov robustnss for ntwor dlays? Pag ( ) ( ). B Rwrtng for t ZOH Wn w ta t sam approac as wt t SMH as dscussd n t prvous scton; w can also us t systm wt t ZOH: B and K 8 6. Tn w obtan t followng matrx for : ZOH ( ) V ( ) ( ).

34 Do smart olds mprov robustnss for ntwor dlays? Pag 3.3 Matlab fls Ts CD contans all Matlab fls usd to ma t plots wc ar dsplayd n ts rport. Tabl gvs an ovrvw of t fls and tr dscrpton. Tabl Matlab fls gn.m systm_on_zoh.m systm_on_smh.m smulat_plot.m smu.m stab_cc.m runnr_const.m runnr_tauconst.m runnr_mnmn.m % Ts functon gnrats t matrx dt wc dscrbs % t systm. T gnvalus of ts matrx ar ccd % f t modulus of t gnvalus ar all smallr tan %. Stablty valus ar. Ts s don for bot t % ZOH (mard n t scrpt) and t SMH (mard ). % Ts functon crats an OvrprxVrt (a st % of vrtcs wc s gratr tan t stuat- % onmatrx w want to cc for stablty). T % functon s only usabl n t cas of systm % wt t zro-ordr old: % = [ ; ]; B = [ ; ]; K = [8 6]; = [ ; ]; % Ts functon crats an OvrprxVrt (a st % of vrtcs wc s gratr tan t stuat- % onmatrx w want to cc for stablty). T % functon s only usabl n t cas of systm % wt t systm matcd old: % = [ ; ]; B = [ ; ]; K = [8 6]; = [ - B* ] % Ts functon gvs out n plots smulatng X for ZOH % and SMH. It smulats X X... Xstps_. Plots ar % savd n worspac. Bcaus of random varyng dlays % vry plot sould b dffrnt. If output == _ % and tau_ ar dsplayd. If lgnds == t lgnds % ar dsplayd. % Ts functon smulats valus of x n tm usng t % convoluton X+=dtX. Usng a dt formd of random % componnts and tau. Smulaton s don for as long as % tmstps and rpat t n tms. If t absolut valu % of vry componnt of x s smallr tan x for tmstp % for ac of t n tms. Stablty s suggstd and % stablty output of ts functon s otrws t % output s. Stablty s onc ccd usng unform % dstrbuton for tang random valus and onc for usng % normal dstrbuton. % Stab_cc uss and tau (wc conssts of mnmal and % maxmal and tau). nd for tat spcfc combnaton of % _mn _max tau_mn tau_max t calculats stablty % usng svral dffrnt mtods. ll mtods ar addd to a % -cll. % Ts functon gnrats stablty plots for % fxd and varyng tau_mn and tau_max. % It uss t functon stabl_cc to gt all % stablty data for svral stablty-dtr- % mnaton mtods. Ts data s usd to ma % t plots plots ar also savd as *.png. % Ts functon gnrats stablty plots for % fxd tau and varyng _mn and _max. % It uss t functon stabl_cc to gt all % stablty data for svral stablty-dtr- % mnaton mtods. Ts data s usd to ma % t plots plots ar also savd as *.png. % Ts functon gnrats stablty plots for % fxd _mn and tau_mn and varyng maxmals. % It uss t functon stabl_cc to gt all % stablty data for svral stablty-dtr- % mnaton mtods. Ts data s usd to ma % t plots plots ar also savd as *.png.