Observer Gain Adaptation of Output Feedback Sliding Mode Controller with Support Vector Machine Regression

Transcription

1 WSEAS RASACIOS o SYSEMS ad COROL Seza oat Serdar Iplc Ltf Ulso Observer Ga Adaptato of Otpt Feedbac Sldg Mode Cotroller wth Spport Vector Mache Regresso SEZAI OKA SERAR IPLIKCI LUFI ULUSOY Compter Egeerg epartmet Pamale Uverst 7 Kl ezl URKEY Electrcal ad Electrocs Egeerg epartmet Pamale Uverst 7 Kl ezl URKEY {stoat plc}@pa.ed.tr ltflso@hotmal.com Abstract: - he covetoal sldg mode cotroller eeds the eact owledge of sstem state measremets. I ths std olear secod order sstems wth measred sstem states ad boded eteral dstrbaces are cosdered. he sldg mode observer based o olear observato error damcs s cosdered ad the observer ga s adsted b sg a spport vector mache based plat model. From the otpt of the spport vector mache model -step ahead predctos are obtaed. herefore the vale of s frst aalzed to search for a proper vale. It s also show wth the smlatos that the stablt codtos are satsfed for the chose observer gas. Compter smlatos are preseted to show the effect of the proposed ga adstmet mechasm o the performace of otpt feedbac sldg mode cotroller. It s see that the traector tracg performace s mproved wth respect to a covetoal otpt feedbac sldg mode cotrol scheme havg costat sldg mode observer gas. Ke-Words: - Sldg mode cotrol Sldg mode observer Otpt feedbac sldg mode cotrol Spport vector mache regresso Observer ga Boded eteral dstrbaces. Itrodcto Egeers alwas search for better cotrol methods to atta hgher prodctvt tha classcal methods ad to prodce qalt prodcts at compettve prces. Whe ow bt boded eteral dstrbaces are cosdered sldg mode cotrol s a promsg area of std for both theoretcal ad applcato oreted robst cotrol problems []. Sldg mode cotrol s based o varable strctre sstems theor []. It s a olear cotrol method wth a hgh-freqec chatterg pheomeo that alters damcs of a olear sstem. he state-feedbac cotrol law swtches from oe cotos strctre to aother based o the crret posto of sstem traector the state space. he cotrol law mst provde the sstem to move alwas towards a swtchg codto. he moto of the sstem as t sldes alog these bodares s called a sldg mode ad geometrcal locs cosstg of the bodares s called sldg srface []. Sldg mode cotrol strctres have bee appled to varos egeerg problems a wde varet of applcato areas sch as electrcal motors [] moble robots [3] mcro-electromechacal sstems [] chemcal processes ad space sstems. For the mplemetato of the well-ow covetoal sldg mode cotroller SMC strctre eact owledge of sstem state measremets s eeded. However for most applcatos t s ether mpractcal or approprate to se sesors for o-le measremet of all state varables cosderg dfferet reasos as cost relablt harsh evromet or eve dced errors from the sesors [5]. hs ecesst of completel measrg the states of a sstem ca be regarded as a mportat drawbac of covetoal SMCs. Observers ca be sed to replace sesors a cotrol sstem. herefore a cosderable amot of wor has bee doe the feld of state estmato of damc sstems b observers as t s a mportat reqremet for safe ad cost-effectve operato of dstral ts [6]. he observers are frst proposed ad developed for lear sstems [7]. However all practcal sstems heret some degree of oleart ad some cases the lear appromatos based o eact learzato or psedo-learzato ma ot be accrate eogh. ISS: Isse Volme 5 Febrar

2 WSEAS RASACIOS o SYSEMS ad COROL Seza oat Serdar Iplc Ltf Ulso herefore observer theor has bee eteded to clde olear process models [8]. he metoed reqremet for state estmato based o olear observato error damcs ad smple strctre ad robst stablt of SMC prompted the std of sldg mode observers SMOs [9]. I SMOs stead of sg a otpt error feedbac betwee the observer ad sstem learl a olear dscotos term s ected to the observer depedg o the otpt estmato error []. SMOs have a heret robstess the face of eteral dstrbaces ad model certates []. he eqvalet cotrol cocept s proposed [] for lear sstems where the observer states coverge to the sldg srface step b step fte tme. he the eqvalet otpt ecto term that s defed as a coterpart of eqvalet cotrol term of SMC s appled to lear sstems wth ow pts [3]. A SMC that ses the state estmates obtaed from a observer strctre [] or a SMC that ol ses sstem otpts [5] costtte the cocept of otpt feedbac sldg mode cotroller OFSMC. State estmato of olear sstems the presece of eteral dstrbaces or model certates s a actve feld of std [6-8]. he dea [ 3] s eteded [9] to OFSMC desg whch a olear sstem wth ow dstrbaces s cosdered. he parallel processg capabltes of artfcal eral etwor A archtectres provdes a vable meas for costrctg the states of comple damc sstems from pt otpt measremets []. herefore sg soft comptg methodologes order to mprove the performace of SMOs or SMCs s a actve area of research. For stace the speed cotrol of a dcto motor sg a SMC s cosdered [] ad a feedforward A archtectre s sed to estmate the rotor speed. For the SMO case the modelg error of the A observer s compesated b the SMO []. Also a radal bass fcto A ad SMO are sed parallel order to cosder dfferet sstem states or evrometal varables []. I [3] o the other had the A observer ad SMO are coected seral ad the A s sed to obta a olear model of the sstem. For the SMC case a A based observer s sed order to mprove the SMC performace []. For the worst case errors As provde better performace tha lear regresso techqes [5]. However As have a local mma problem whch s a mportat drawbac for most cotrol problems. herefore ths std a spport vector maches SVMs based strctre s chose. SVMs were orgall created to solve classfcato problems. I SMC problems SVMs are sed for dfferet prposes. For stace the desg parameters of a tme-varg sldg srface for a gve tal codto are obtaed b sg SVMs [6]. he the fcto appromato propert of SVMs s sed to desg a ew sldg srface wth addtoal damc states [7]. Also the dscotos cotrol law of SMC s costrcted sg the otpt of SVMs based model order to elmate chatterg [8]. OFSMCs the presece of ow dstrbaces s eamed [9] b provg the stablt der a set of orestrctve assmptos ad t s show that the desged cotroller esres asmptotc traector tracg behavor. o acheve ths am the gas of the state observer mst be properl selected for a acceptable traector tracg performace for the observato error to coverge towards zero. herefore selecto of the observer gas s mportat for the stablt ad performace of the cotroller. A observer that estmates all of the state varables s called a fll-order observer. Whereas a observer that estmates a part of the state varables s referred to be a redced-order observer [9]. I ths std the OFSMC wth a fll-order observer preseted [9] s cosdered ad SVM based plat model ad cotroller tg scheme gve [3] that s developed for tg PI parameters s eteded order to mprove the performace of the SMO ad also to compesate the SMC otpt. he covetoal SMC strateg s orgall desged for cotos-tme operato ad t s more dffclt to choose a sthess for dscrete-tme case [3]. he dscrete-tme SMC s qte dfferet from the covetoal coterpart ad s also called as qas sldg mode. screte-tme SMC desg s sall based o a appromate sldg mode sstem evolto de to the o-qe attractvt codto ad appromate evolto o sldg srface [3]. I or std the parameter adaptato wth SVM s dscrete-tme. However the statefeedbac cotrol scheme based o the SMO SMC ad the plat are all cotos-tme. he strctre of the paper s as follows: I the et secto two ma compoets of the OFSMC scheme the SMC ad SMO are brefl descrbed ad SMC law wth sstem state estmates s preseted. he SVM based modelg predcto ad Jacoba calclatos preseted [3] are gve Secto 3 ad the SMO based ga adaptato scheme s preseted Secto. he smlatos to demostrate the valdt ad advatage of the ga adaptato scheme are gve Secto 5. ISS: Isse Volme 5 Febrar

3 WSEAS RASACIOS o SYSEMS ad COROL Seza oat Serdar Iplc Ltf Ulso Otpt Feedbac Sldg Mode Cotroller he OFSMC cossts of a SMC to geerate the cotrol law ad a SMO to obta the sstem state estmates from measred sstem otpt ad cotrol pt. hese corerstoes of the preseted strctre are emphaszed ths secto.. Cotos-tme Sldg Mode Cotroller he state space represetato of a secod order sgle-pt olear sstem caocal form wth state vector [ ] ca be gve as & & f b t d t where t s the cotrol pt dt s the eteral dstrbaces ad f b are olear fctos that determe the sstem characterstcs [ 9]. he SMC scheme volves selecto of a sldg srface sch that the sstem traector ehbts desrable behavor whe cofed to ths mafold ad fdg feedbac gas so that the sstem traector tersects ad stas o the gve mafold. herefore for sstem assmg the traector tracg problem the error damcs for the secod order sstem gve ca be wrtte as e& & d e& f b t d t & d ad from ths damcs the covetoal lear sldg srface wth costat desg parameters ca be wrtte as s e t c e c e e 3 where e[e e ] s the error state vector ad e s the th error state varable d d s the desred traector of the th state ad c[c ] s the costat sldg srface parameter that determe the sstem behavor the error phase plae. It s ecessar ad sffcet to dfferetate 3 oce for t to appear. hs ths s a frst order stablsato problem based o set. Lapov's drect method cold be sed to obta t that wold eep set at zero. Cosder a Lapov fcto caddate as V s s t wth V Vs> for st> []. A effcet codto for sstem stablt ca be gve as d V& s s t η s t 5 dt where η s a strctl postve real costat that determes the covergece veloct of the traector to the sldg srface. Obtag the eqalt 5 meas that the dstace to the srface decreases alog all traectores ad ths meas that the sstem s stable. herefore 5 s called as the reachablt codto for the sldg srface. B sbstttg 3 to 5 ad omttg the argmets of the depedet varables oe obtas s. f b. b. d& c e& η s 6 d herefore a cotrol pt satsfg the reachg codto ca be chose as f & c e& d b sg s 7 where g s a strctl postve real costat wth a lower bod depedg o the boded eteral dstrbaces. he fcto sg. deotes the sgm fcto defed as follows g f s < sg s f s 8 f s > he cotrol pt 7 cossts of two parts. he frst part eq s the cotos term that s ow as eqvalet cotrol based o estmated sstem parameters ad t compesates the estmated desrable damcs of the sstem. he secod part wth the sgm fcto s the dscotos cotrol law ds that reqres fte swtchg o the part of the cotrol sgal ad actator at the tersecto of error state traector ad sldg srface. I ths wa the traector s forced to move alwas towards the sldg srface [].. Sldg Mode Observer he state estmato problem for a sstem sbect to ow eteral dstrbaces der otpt feedbac sldg mode cotrol wth a eqvalet otpt ecto sldg mode observer s cosdered [9]. I ths std the sldg mode observer strctre preseted [9] s sed. For the sstem gve ol the sstem otpt s measred. herefore the error damcs cold ot be obtaed. he sstem states ad also the error damcs ca be obtaed from b sg a observer of the form gve as & λ sg & f b E λ sg ~ eq ds 9 ISS: Isse Volme 5 Febrar

4 WSEAS RASACIOS o SYSEMS ad COROL Seza oat Serdar Iplc Ltf Ulso where ~ λ sg ad the eqvalet otpt ecto term eq λ sg eq s obtaed b sg a low pass flter [5 9]. he term E f ad E otherwse [9]. Wth proper λ λ observer gas the observer state frstl coverges to ad the coverges to. For the gve sstem fte tme covergece of sstem state estmates to actal plat states s proved the lteratre [9]. herefore stead of 3 obtaed from oe ca se s e t c e c e e where e [ e e ] s estmated error state vector ad e d s th estmated error state varable. If the sstem states are ot measrable the covetoal form of eq sg state estmates ca be rewrtte as t b f & c e & eq d he the overall cotrol law based o estmated sstem states ca be desged as t t sg s eq Choosg the Lapov fcto caddate as V/ ŝ sg estmated state varables ad tag the dervatve of the Lapov fcto alog the traectores of the estmated sstem states the dscotos cotrol ga g mst be chose as [9] g g b c λ λ η 3 order to satsf the reachg codto. 3 Spport Vector Mache based Modelg Predcto ad Jacoba Calclatos Cosder a olear sstem damcs of whch ca be represeted b ARX model f L L where s the cotrol sgal appled to the plat at tme de s the correspodg otpt of the plat ad ad deote the mber of past cotrol sgals ad mber of the past otpts volved the model respectvel. It s assmed that o-lear fcto f s ow ad that a trag data set s obtaed the form gve as { L set { } L } 5 where X s the th pt data pot pt space ad Y s the correspodg otpt vale. It s desred to obta a model that represets the relatoshp betwee pt ad otpt data pots. he trag data set set s to be sed to obta a appromate model of the plat damcs. he prmal form of a SVM regresso model s gve b 6 whch s lear a hgherdmesoal featre space F w Φ bas 6 where w s a vector the featre space F Φ s a mappg from the pt space to the featre space bas s the bas term ad <> stads for er prodct operato F [3]. he SVR algorthms regard the regresso problem as a optmzato problem dal space whch the model s gve b bas α K 7 where α s are the coeffcets of each trag data ad K s the erel fcto gve b K Φ Φ K [3] he erel fcto K hadles er prodct featre space ad hece the eplct form of Φ does ot eed to be ow. I the model 7 a trag pot correspodg to a ozero α vale s referred to as the spport vector. I [3] ε-svr algorthm emplog Vap s ε- sestve loss fcto L ε gve as ε L ε 8 > ε s sed whch formlates the prmal form of the regresso problem as follows: m P w C 9 ε wbasξ ξ sbect to the costrats ξ ξ ι wφ τ ε ξ w Φ τ ε ξ ξ ξ... where ε s the pper vale of tolerable error ξ are slac varables. s the Ecldea orm ad C s a reglarzato parameter that provdes a compromse betwee model complet ad degree ξ ISS: Isse Volme 5 Febrar

5 of tolerace to the errors larger tha ε [3]. al form of the optmzato problem becomes a QP problem as K m ε ε 3 sbect to the costrats C... solto of the QP problem 3 ad gves the optmm vales of ad s. he vale of bas the model s determed as follows: the codto ε s satsfed for each spport vector for whch the codto C holds. If α s defed to be the ew coeffcet of for as α the a SVM model as gve b 7 s obtaed. Moreover whe ol spport vectors are cosdered the model becomes K bas #SV SV α 5 where #SV s the mber of spport vectors. If we follow the procedre gve [3] the we costrct the crret state vector as ] [ L L v 6 the the correspodg otpt of the SVM model becomes K bas #SV SV v α 7 I SVM-based observer ga adaptato a radal bass adopted erel fcto s sed that s gve as - - ep K K 8 where s the wdth parameter [3]. If s defed as Ecldea dstace betwee th spport vector ad crret state vector v as v v 9 the the erel fcto ca be rewrtte as ep K v 3 ad the SVM regresso model becomes bas #SV ep α 3 ow 3 ca be sed to predct -step ahead ftre traector of the plat as b K bas... ep #SV α 3 where < m 33 the frst order partal dervatves ca be wrtte as SV # ep α 3 where ep ep ep d 35 ad ] [ ] [ m δ δ 36 where. δ stads for t step fcto [3]. ow the frst-order terms ca be sed to calclate the Jacoba matr. WSEAS RASACIOS o SYSEMS ad COROL Seza oat Serdar Iplc Ltf Ulso ISS: Isse Volme 5 Febrar

6 WSEAS RASACIOS o SYSEMS ad COROL Seza oat Serdar Iplc Ltf Ulso. Spport Vector Mache Based Observer Ga Adaptato he proposed SVM based sldg mode observer ga adaptato scheme s gve Fg.. It s adopted from the std proposed [3] whch s frst sed for tg PI cotroller parameters. he dea s mal based o obtag the -step ahead predctos of the plat otpt b sg a SVM model ad a Jacoba bloc for tg SMO gas. he SVM model s obtaed b applg radoml chose boded cotrol sgals to the plat. After the trag process -step ahead predctos [... ] are obtaed from the otpt of the SVM model wth t d tme dratos as show Fg.. he order to mmze the SVM predcto error ad to pealze the wated rapd chages the cotrol pt a obectve fcto s chose as [3] φ ε ρ 37 d where ε s the predcto error of SVM at th d step s the ow desred otpt at th step ad ρ determes the amot of pealt o the cotrol devatos. I ths std the proposed dea [3] s appled to the OFSMC case b choosg the observer ga λ as the pdated parameter. I order to have a mercal solto to the problem of mmzg 37 Leveberg- Marqardt learg rle whch terpolate betwee Gass-ewto ad steepest descet algorthms ca be wrtte as J J I J ε λ λ κ 38 ew old where κ s a bledg factor whch determes a mg rato betwee gradet-descet ad Gass- ewto algorthms ad ε s the predcto error vector whch s defed as ε [ ε ε L ε ρ ] 39 he plat s desred otpt traector does ot deped o observer gas. herefore Jacoba matr J 38 ca be obtaed from 37 as ρ J L λ λ λ λ Usg the preseted scheme the observer ga λ s pdated dscretel at ever samplg perod t d ad sed to pdate the SMO that s cotos tme. AC SMO Eq.9 Fgre. Schematc dagram of the SVM based observer ga adaptato ad cotrol law compesato scheme. he cotrol law compesato mechasm ca also be obtaed b splttg the Jacoba matr to two dfferet parts b sg secod order alor appromato of 37 [3]. hs applg the gve dea to the observer ga adaptato scheme the Jacoba matr ca be wrtte as follows J ρ λ he partal dervatve of wth respect λ cold be drectl obtaed b solvg the eqatos from the SMO ad SMC blocs whch have both olear strctres. hs olear strctre rases dffcltes obtag the mathematcal solto. herefore ths std the mercal soltos are obtaed b sg the appromato gve as λ λ From the stablt aalss gve [9] order to provde the fte tme covergece of the estmated states to the actal states the observer gas mst satsf the codtos gve as λ > J Eq. SMC Eq. µ d Otpt Feedbac SMC Scheme t d ŷ SVM Model Eq.3 Plat Eq. λ > f b d f b µ û AC 3 where µ µ are small postve real costats [9]. herefore these bods mst be provded whe tg the λ observer ga. Itall λ s set to acceptable vales that provde 3. Proper choce of the gas λ ad λ wll garatee that the redced order damcs are stable o the sldg srface ad t d d ISS: Isse Volme 5 Febrar

7 WSEAS RASACIOS o SYSEMS ad COROL Seza oat Serdar Iplc Ltf Ulso ths wll esre asmptotc stablt of the referece traector. o have a better observer performace ad ths to provde a better otpt tracg performace ths preset vale shold be ted properl. 5 Smlato Stdes o show the performace of the ew tg scheme compter smlatos are performed o a olear mass-sprg-damper sstem o a horzotal srface der the effect of a horzotal force. he damc eqatos of the sstem s descrbed as m && v & t t t d t v & t v & v & & 3 t where m s the mass t s the dsplacemet & t s the veloct v & t ad t are olear terms wth respect to the damper ad sprg respectvel. B tag & ad b rewrtg the sstem eqatos the form of oe ca obta f v & t t t d t m b / m 5 he sstem parameters are chose as m v o v.35 ad o.55. he tal state vales are chose as.5. he traector tracg problem s cosdered ad the desred state traectores are chose as t.5cos πt/5 d t.π s πt/5 d 6 rg the smlatos order to show the robstess agast boded eteral dstrbaces dt s modeled wth a ssodal sgal tae as d t.5.5cos3πt 7 he SMO for all OFSMCs s tae as 9 ad to obta ~ frst order low pass flter wth badwdth w rad/s s sed. For all of the cotrollers the sldg srface parameter s tae as c 7. Smlatos have bee carred ot Matlab evromet ad ordar dfferetal eqato solver mplemetg Rge-Ktta mercal tegrato method has bee selected for smlatg the dscotos atre of sldg mode cotroller ad observer. For the smlato evromet a fed samplg tme of e-s has bee appled for smlatg the cotos tme observer cotroller ad plat. O the other had the SVM bloc wors a dscrete atre b tag observatos ad calclatg the pdate vale at ever t d e- s tme dratos. All smlatos are performed the tme terval betwee [ 5] s. he sstem performace s fleced b the selecto of the observer tal codtos. herefore assmg that the tal vales of sstem states ad are at the org average ad. he SVM predcts -step ahead sstem behavor ad s a desg parameter. o aalze the effect of o the performace dces ad cotrol pt magtde the sstem s smlated for dfferet vales of betwee [:] ad the reslts are gve Fg.-3. As ca be see from Fg. the performace has ts best vales for ad the performace s the smlar for. However from Fg.3 t s see that for ths performace mprovemet has a trade-off as a creased cotrol pt magtde. herefore ad are chose for comparso. he traector tracg ad state estmato performaces are gve Fg. ad the cotrol pts are gve Fg.5 for ad respectvel. ta ts treachs treachŝ b a c d Fgre. Performace dces for dfferet vales. ISS: Isse Volme 5 Febrar

8 WSEAS RASACIOS o SYSEMS ad COROL Seza oat Serdar Iplc Ltf Ulso ma Fgre 3. Cotrol pt magtde ma. Fgre. Actal estmated ad referece sstem otpt for a b d a.5 d b tme [s] - - Fgre 5. Cotrol pts for a a b tme [s] b a.8.6 λ m λ b tme [s] Fgre 6. Observer gas ad ther stablt bods 3 for : a λ m λ b λ m λ. λ m λ.5.5 Fgre 7. Observer gas ad ther stablt bods 3 for : a λ m λ b λ m λ. he performace for Fg. s better tha the case for. However the cotrol pt magtde Fg.5 reaches ear 6 for whch s far more tha the covetoal case. For the stablt of the sstem the codtos gve 3 mst be provded. he λ λ vales ad ther mmm vales that are obtaed from 3 are plotted Fg.6 ad 7 for ad respectvel. It s see that the observer gas do ot cocde wth the gve stablt codtos. Cosderg above aalss o sstem stablt ad performace for detaled comparsos cotrol pt magtde s also cosdered ad for the -step ahead predcto otpt of the SVM model s chose. he smlatos are mplemeted for the proposed OFSMC wth SVM based ga adaptato scheme OFSMC-SVM ad for the covetoal OFSMC preseted [9] OFSMC-C. Bearg md the stablt codtos [9] three dfferet cases for OFSMC-C s cosdered: OFSMC-C : λ. λ. 8 ad g 3.6 OFSMC-C : λ. λ. 8 ad g OFSMC-C 3 : λ. 93 λ. 8 ad g he three cases for OFSMC-C are desged order to show the effect of adsted vales obtaed b the SVM scheme. For OFSMC-C OFSMC-C 3 ad OFSMC-SVM g s calclated from 3 wth µ.. For OFSMC-C o the other had g s chose as the mamm vale obtaed wth OFSMC-SVM. OFSMC-C 3 has costat λ vale that s obtaed at last wth OFSMC-SVM ad g s calclated from 3 for costat λ.93. he tal vale of λ. s chose for OFSMC-SVM. he bodares for λ s tae as λ m. λ m λ a.8.6 λ. m λ b tme [s] ISS: Isse Volme 5 Febrar

9 WSEAS RASACIOS o SYSEMS ad COROL Seza oat Serdar Iplc Ltf Ulso ad λ ma 5. he bledg factor 38 s tae as κ. ad the pealt o the cotrol devatos s ρ.. he tme resposes of estmated sstem otpt ad actal sstem otpt are gve Fg.8. For all the cotrollers there s some traset observato error at the begg of observato as observer tal codtos are cosstet wth those of the plat. Bt observer states ad ths sstem otpt estmate approach to ts actal vale after a fte tme. he sldg srface s ad estmated sldg srface ŝ are plotted Fg.9. he sldg moto Fg.9 provdes a estmate of the sstem states. For OFSMC-C creasg g mproves t reach ŝ. However ths does ot mprove the observer behavor as ca be easl see from the vale of t reach s. he cotrol pts are also plotted Fg. ad chatterg s a reslt of sgm fcto ad ca be avoded b sg a satrato fcto. he tme-varg behavor of the pdated λ t ad calclated g t for the proposed OFSMC-SVM cotroller s plotted Fg.. he g s calclated from 3 b sg the tme-varg λ vale whch s calclated wth the proposed method at each t d tme tervals. At tme t.57 s the parameters reach ther optmm vales ad sta costat as λ.93 ad g after that tme stat..5 d -.5 a.5 d -.5 b.5 d -.5 c.5 d -.5 d tme [s] Fgre 8. Actal estmated ad referece sstem otpt : aofsmc-c bofsmc-c cofsmc-c 3 dofsmc-svm. a Fgre 9. Actal ad estmated sldg srface varables: aofsmc-c bofsmc-c cofsmc- C 3 dofsmc-svm. 6 6 b 6 c 6 d - a - b - c - d Fgre. Cotrol pts: a OFSMC-C b OFSMC-C c OFSMC-C 3 d OFSMC-SVM. λ a 8 g 6 tme [s] tme [s] b tme [s] Fgre. a λ ad b g for OFSMC-SVM s ŝ s ŝ s ŝ s ŝ ISS: Isse Volme 5 Febrar

10 WSEAS RASACIOS o SYSEMS ad COROL Seza oat Serdar Iplc Ltf Ulso able. Performace dces of the cotrollers. OFSMC- OFSMC- OFSMC- OFSMC- C C C 3 SVM t a t a t s t s t reach s t reach ŝ ma{ } Ol oe of the observer gas s cosdered ths std. However the observer ga adstmet mechasm preseted ths std ca be appled to both observer gas. Also the case of hgher order sstems the form of the gve strctre proposed method ca be eteded b cosderg the stablt codtos. Acowledgemet hs pblcato s a reslt of a proect carred ot over the perod 8-. he athors wsh to tha he Scetfc ad echologcal Research Cocl of re UBIAK for facal spport Proect o: 7E86. Fall the performace dces of the related cotrollers are gve able. he error bod for the settlg tme s tae as 5% of the stead state vale. I able t a s the tme that estmated state approach ts actal vale t s s the settlg tme for state ad t reach s the reachg tme of estmated ad actal sldg srface varables. It s see that ol creasg g does ot have a postve effect o t a. he OFSMC-C 3 represets the obtaed vales geerated b sg OFSMC-SVM adaptato scheme. hs t has the best observato behavor ad settlg tme performace whch shows the costrctve tg strateg of the preseted SVM model. 6 Coclso I ths std otpt feedbac sldg mode cotrol of a olear secod order sstem sbect to boded eteral dstrbaces s cosdered. he ovelt of ths std s that the spport vector mache regresso algorthm s frstl sed wth the otpt feedbac sldg mode cotroller strctre. I ths std the spport vector mache regresso algorthm s sed to adst the sldg mode observer gas. B sg compter smlatos t s see that the mber of ftre data pots predcted b the spport vector mache based model flece both the performace of the sstem ad the magtde of the cotrol pt. herefore a proper vale for the mber of ftre data pots s selected. From the smlato reslts t was coclded that the proposed method mproves the sstem traector tracg performace ad the observer gas. Also t s show wth the smlatos that the stablt codtos for the observer gas are satsfed. Refereces: [] A.S.I. Zober A trodcto to sldg mode varable strctre cotrol I Varable Strctre ad Lapov Cotrol Zober A.S.I. ed. Lectre otes Cotrol ad Iformato Sceces Vol.93/99 Sprger Verlag Lodo 99 pp [] O. Baramboes F.J. Maseda A.J Garrdo ad P. Gomez A Sldg Mode Cotrol Scheme for Idcto Motors Usg eral etwors for Rotor Speed Estmato Proceedgs of the 5 th WSEAS Iteratoal Coferece o Istrmetato Measremet Cotrol Crcts ad Sstems Cac Meco Ma - 5 pp [3] A. Flpesc A.L. Stac S. Flpesc ad G. Stamatesc O-le Parameter Estmato Sldg-mode Cotrol of Poeer 3-X Wheeled Moble Robot Proceedgs of the 7 th WSEAS Iteratoal Coferece o Sstems heor ad Scetfc Comptato Athes Greece Agst -6 7 pp [] A. Kz S. Bogosa ad M. Goasa Cotrol Strateges for Icreased Relablt MEM Comb rves Proceedgs of the d WSEAS Iteratoal Coferece o Appled ad heoretcal Mechacs Vece Ital ovember - 6 pp.3-6. [5] I. Hasara U. Ozger ad V. Ut O sldg mode observers va eqvalet cotrol approach Iteratoal Joral of Cotrol Vol.7 o pp [6] G. Ells Observers cotrol sstems: a practcal gde Academc Press Sa ego CA. [7].G. Leberger A trodcto to observers IEEE rasactos o Atomatc Cotrol Vol.AC-6 97 pp ISS: Isse Volme 5 Febrar

11 WSEAS RASACIOS o SYSEMS ad COROL Seza oat Serdar Iplc Ltf Ulso [8] F.E. ha Observg the state of olear damcal sstems Iteratoal Joral of Cotrol Vol pp [9] A. McCa M.S. Islam ad I. Hssa Applcato of a sldg-mode observer for posto ad speed estmato swtched relctace motor drves IEEE rasactos o Idstr Applcatos Vol.37 o. pp [] C. Edwards S.K. Sprgeo C.P. a ad. Patel Sldg mode observers Lectre otes Cotrol & Iformato Sstems Vol pp.-. [] S.K. Sprgeo Sldg mode observers: a srve Iteratoal Joral of Sstems Scece Vol.39 o.8 8 pp [] S. raov ad V. Ut Sldg mode observers: a ttoral Proceedgs of the 3th IEEE Iteratoal Coferece o ecso ad Cotrol 995 pp [3]. Floqet ad J.-P. Barbot A caocal form for the desg of ow pt sldg mode observers I Advaces Varable Strctre ad Sldg Mode Cotrol C. Edwards C.E. Fossas L. Frdma eds. Lectre otes Cotrol ad Iformato Sceces Vol.33 Sprger Berl 6 pp [] C. Usal ad P. Kachroo Sldg mode measremet feedbac cotrol for atloc brag sstem IEEE rasactos o Cotrol Sstems echolog Vol.7 o. 999 pp.7-8. [5] M.C. Pa screte-tme otpt feedbac sldg mode cotrol for certa sstems Joral of Mare Scece ad echolog Vol.6 o. 8 pp [6] E.H.E. Baom Speed sesor-less sldg mode cotrol of dcto motor drve WSEAS rasactos o Crcts ad Sstems Vol.3 o. 8 pp [7] W. Sagtgtog ad S. Stor Adaptve Sldg-Mode Speed-orqe Observer. WSEAS rasactos o Sstems Vol.5 o.3 6 pp [8] W. Sagtgtog ad S. Stor Stablt Aalss of a Sldg-Mode Speed Observer drg raset State Proceedgs of the 5 th WSEAS Iteratoal Coferece o Istrmetato Measremet Crcts ad Sstems Hagzho Cha Aprl pp.35-. [9] J.M. al ad.w.l. Wag Otpt feedbac sldg mode cotrol the presece of ow dstrbaces Sstems ad Cotrol Letters Vol.58 9 pp [] A. Baz A eral observer for damc sstems Joral of Sod ad Vbrato Vol.5 o. 99 pp.7-3. [] I. Charez A. Poza ad. Poza ew sldg-mode learg law for damc eral etwor observer IEEE rasactos o Crcts ad Sstems II Epress Brefs Vol.53 o. 6 pp [] W. Y Stablt aalss of vsal servog wth sldg mode estmato ad eral comptato Iteratoal Joral of Cotrol Atomato ad Sstems Vol. o.5 6 pp [3] J. Resedz W. Y ad L. Frdma wo-stage eral observer for mechacal sstems IEEE rasactos o Crcts ad Sstems II Epress Brefs Vol.55 o. 8 pp [] M. Lee A sldg mode cotroller wth eral etwor ad fzz logc Proceedgs of the IEEE Iteratoal Coferece o eral etwors Vol pp.-7. [5] V. Krova ad M. Saget Comparso of worst case errors lear ad eral etwor appromato IEEE rasactos o Iformato heor Vol.8 o. pp [6] S. oat me-varg sldg srface desg wth spport vector mache based tal codto adaptato Joral of Vbrato ad Cotrol Vol. o.8 6 pp [7] F.G. Wag S.K. Par M.C. Km S.J. Cho ad.s. Yoo A ovel sldg srface desg b sg spport vector maches Proceedgs of the Iteratoal Coferece o Comple Sstems ad Applcatos Ja Cha 7 pp.5-9. [8] J.. L Y.B. Zhag ad H.P. Pa Chatterg free LS-SVM Sldg Mode Cotrol Proceedgs of the 5 th Iteratoal Smposm o eral etwors Beg Cha 8 pp [9] E.M. Jafarov A ew Redced-order sldg mode observer desg method: A trple trasformatos approach Proceedgs of the 9 th WSEAS Iteratoal Coferece o Sstems Athes Greece Jl -3 5 pp.-8. [3] S. Iplc A comparatve std o a ovel model-based PI tg ad cotrol mechasm for olear sstems Iteratoal Joral of Robst ad olear Cotrol do../rc.5 9. ISS: Isse Volme 5 Febrar