DESIGN OF OBSERVER-BASED CONTROLLER FOR LINEAR NEUTRAL SYSTEMS. M. N. Alpaslan Parlakçı

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1 EIGN OF OBEE-BE ONOLLE FO LINE NEUL YEM M. N. lpala arlakçı parm of ompur cc abul Blg Uvry olapr 3444 abul urky -mal: brac: I papr problm of obrvr-ba a-fback corollr g for lar ural ym vga. Employg Lyapuov mo a quarac ably ory a w lay-p ably crro oba form of a lar mar qualy wc ca b aly olv by wll-kow ror-po algorm. umrcal ampl rouc o mora ffcv of propo mo roug mulao u. opyrg 5 IF Kywor: m-lay full a-obrvr clo-loop ably lar mar qual.. INOUION m lay vably yamcal a pycal ym bcau of lack of uffc formao procg ra a aa ramo capably rougou ffr par of ym. ma ffc of m lay ar a ably of ym bavour a of cau poor a rora prformac. uao mova rarcr o uy ablzao of m lay ym. parcular cla of m lay ym wc m lay alo rvav of a call m lay ym of ural yp. robu ablzao problm of ural m lay ym a b vga lraur Xu al. ; Xu al. 3; ark 3a; ark 3b; Xu al. 4. ablzao mo rpor by abov auor ar all ba o mmoryl afback corollr. uao praum a all a ar avalabl or accbl roug maurm. I orr o avo rqurm of all a o b avalabl wc o o ralc pracc for may pycal ym oly rmy o g a obrvr-ba corollr. g problm of a-obrvr for m lay ym a alo b u by a umbr of rarcr r 999; Bouayb ; Hou al. ; Wag ; Mamou a rb 3; r al. 4. Howvr auor av grally cor lay ffral ym of oly rar yp. r qu a fw umbr of work Wag al. ; al. 4; ark 4 wc a obrvr g for ural yp of m lay ym a b vga. problm of obrvr g for a cla of m lay olar ym w paramr ucra cor by Wag a Ubau. algbrac paramrz approac ba o a cca mar quao plo o caracrz c coo a of pc robu olar obrvr. Howvr paramrz formulao of obrvr ga mar volv compuaoal comply u o rqurm of prrm g paramr. full-orr obrvr a guara poal ably of rror yamc ym a b cor by Wag al. wr y vlop a mar quao approac o olv problm. al. 4 ar problm of guara co corol for a cla of ural lay ym by ug a obrvr-ba mmory a-fback corollr. ark 4 vlop a obrvr-ba corollr for a cla of lar ffral ym of ural yp. Howvr o of ablzao crro gv ark 4 for c of

2 obrvr-ba corollr ca o b pr form of lar mar qual. mpl a mar qual for ablzao crra cao b olv mulaouly by ug cov opmzao mo. oly way o pu om aoal coo orr o covr mar qual o lar mar qual LMI. Howvr uao may uc aoal corvam o ym. I papr w cor obrvr-ba mmoryl a-fback corollr g problm for a cla of m lay ym of ural yp. lay a a a rvav aum o b coa. full-orr a-obrvr coruc a bo rror yamc a clo-loop ym yamc ar ak o corao. layp ablzao crro oba rm of a lar mar qualy. Ulk mo of ark 4 our ablzao crro ca b aly olv by LMI opmzao cqu LMI corol oolbo w o rqurm o mak furr aumpo. umrcal ampl gv orr o mora ulzao of obrvr-ba ablzao mo propo o. om umrcal mulao ar alo pr.. OBLEM EMEN L u cor a cla of lar ural m lay ym of followg form: Bu y F w al coo fuco wr [ ] θ Φ θ θ a vcor; corol pu vcor; y u m p oupu vcor; m p B F ar coa kow ym marc; ar kow ogav coa calar og ural a cr lay rpcvly; ma a Φ gv cououly ffrabl fuco o [ ]. umpo. W aum a par B a ar corollabl a obrvabl rpcvly. L u f followg yp of a opraor a 3 follow from Lmma gv by Ivacu al. 3 a opraor guara o b abl f mar abl of cur- o. Morovr uao alo abl o aur ably of 3 for ay ural m-lay afyg <. W ca cor followg yp of a full-orr mmory a-obrvr for ym crb Bu L[ y F 4 ] a w coo a-fback corol law a follow u K 5 wr ma a vcor; m K coa fback ga mar a p L coa obrvr-ga mar o b lc. I orr o oba rror yamc w ca f mao rror a 6 wr mao rror vcor. w ca formula rror yamc a follow Bu [ 7 Bu ] or w ca rwr 7 form of ural yp lay-ffral quao a 8 Morovr ubug corol law 5 o ym gv clo-loop ym yamc a [ ] objcv of papr o g a obrvrba a-fback corollr u for ural ym uc a rror yamc 7 a rulg clo-loop ural ym 9 ar aympocally abl. 3. MIN EUL 9 I co w vlop a lay-p crro pr form of a lar mar qualy aurg aympoc ablzably of ural lay-ffral ym w mmory obrvr-ba a-fback corollr 5. followg orm ummarz ma rul of o. orm. Gv a ogav coa calar α > f r ymmrc a pov f marc X a a arbrary mar W a arbrary vcor K all w appropra mo afyg Σ < wr Σ X X α αk B α Σ Σ α X Σ 3 3 Σ X α α Σ Σ α Σ Σ α Σ Σ XB Σ Σ K Σ Σ Σ K 33 Σ Σ 44 F W WF Σ Σ WF Σ Σ 45 Σ Σ K Σ 55 Σ 66 K Σ Σ 6 6

3 Ι Σ 77 Ι Σ 88 Ι Σ 99 Ι Σ Ι Σ. a rmag r ar zro ym aympocally ablzabl w obrvrba corollr K u. roof. L u coo a caa Lyapuov- Kraovk fucoal a 6 wr w bg ymmrc a pov f marc all o b lc appropraly. ompug m rvav of 6 alog a a rror rajcor of 8 a 9 rpcvly yl 3 ubug 9 a rlao of o 3 gv { } { } 4 a w alo compu followg a a mlar mar w oba 4 Ug 7 a rplacg w abov quao gv { } { } 4 7 Fally w g followg rvav ummg up o g a W ca pr quarac rm volvg a rm of a rpcvly. 3 u ubug -3 o yl 4 W ca rwr 4 quarac form a follow χ Ωχ 5 wr [ ] χ Ω Ω Ω 3 3 Ω Ω Ω Ω 4 4 Ω Ω 6 6 Ω 33 Ω Ω 44 Ω Ω Ω Ω Ω Ω. a rmag r ar all zro. Hc orr o guara aympoc ably of ym w mmory obrvr-ba corollr 5 o o afy followg qualy < Ω χ χ

4 wc mpl a Ω < 6 No a mar qualy gv 6 o form of lar mar qual. W ca mploy mar compoo cqu Barolomu al. 997 o b abl o rpr 6 form of a lar mar qualy. Gv a coa ogav calar α > w aum a co uc ab compo a α Ι X 7 wr X a ymmrc a pov f mar. w ca rcompu r of Ω a volv by ubug 7 appropraly X X α αk B α K B X X a X α α X a α X a α X a α X. Now w ca rwr 6 by ubug abov pro o oba Ω Φ Γ Γ < 8 wr Φ X X α αk B α Φ Φ α X Φ Φ 3 3 Φ Φ α Φ Φ α Φ 33 X α α Φ 6 Φ 44 Φ Φ Φ Φ Φ 55 Φ Γ X 66 Γ 3 X Γ 4 X Γ 6 X a rmag r of Φ a Γ ar all zro. No a w ca rpr Γ followg um of prouc rm 4 Γ Γ Γ 9 wr Γ B X K [ ] [ ] [ ] [ K] Γ Γ K Γ K 3 4 Γ. W ca ulz Lmma gv Wag a follow: Γ Γ 4Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ ubug 3 appropraly o 8 allow o g Φ Γ Γ < Φ 4Γ Γ wr Π Γ Γ Γ Γ Γ Γ Γ Γ Π < X X α αk B α 4 XBB X K K Π Φ Π Φ Π Φ Π Φ 6 6 Π Φ Π Φ K K Π Φ K K Π Φ Π Φ Π Φ Π Φ K K. oog a arbrary mar W uc a L W 3 a ug cur complm Boy al. 994 w ca rpr 3 a a mar qualy form gv by. If coo af ym 7 9 ar guara o b aympocally abl. Hc proof compl. mark. No a qualy form of a lar mar qualy wc ca b aly olv ug ror po algorm Boy al mark. problm of ow o coo calar paramr α ca b al by coog rpaly arbrary pov valu for α ul LMI corol oolbo a gv fabl oluo for LMI. 4. NUMEIL EXMLE I co w cor a umrcal ampl orr o mora ffcv of propo g approac. Eampl. or followg lar ural ym: Bu 33 y F w B F [ ]..3 W fr coo α 5 a olv w rorpo algorm LMI corol oolbo. fabl oluo ar oba a follow X W K [ ] L..397 For mulao ca uy w co ural a cr lay a c a 3c

5 rpcvly a ug g paramr K a L prov by LMI corol oolbo w oba plc yamc quao for clo-loop ym a mao rror ra pro qu low a raply covrg o zro aya. acual a ma gal for ar ow Fgur 3 wr a mlar prformac w a of oc mao of. rfor mulao rul ca a propo lay-p obrvr-ba corollr b a afacory prformac. a mlarly w g a-obrvr yamc quao mulao ca uy a b carr ou w ma a-fback corol pu gal wc ca b Fgur. Fgur pc rpo of acual a ma gal for. Fgur 3. po of acual a ma gal rpcvly. 5. ONLUION Fgur. Ipu gal u. I papr mmory obrvr-ba ablzably problm of a cla of ural mlay ym ar. Ba o Lbz- Nwo mol raformao a Lyapuov quarac ably ory a w lay-p lar mar qualy form of ablzao crro a ur aympoc ably of bo rror yamc a clo-loop ym yamc oba. Ulk a rcly gv mo by ark 4 propo ablzably coo for c of a mmory obrvr-ba corollr ca b rcly olv ug ffcv ror-po algorm. EFEENE Fgur. po of acual a ma gal rpcvly. Bouayb M.. Obrvr g for lar m lay ym. ym a corol Lr Boy. L. El-Gaou E. Fro a. Balakra 994. Lar Mar Iqual ym a corol ory IM u ppl Mamac Nw York. B. J. Lam a. Xu 4. Nw obrvrba guara co corol for ural lay ym. roc. I. Mc. Egr: Joural of ym a orol Egrg Hou M.. k a.j. ao. obrvr g for lar m-lay ym. IEEE raaco o uomac orol Goub-Barolomu. M. ambr a J.. car 997. ably of prurb ym

6 w m-varyg lay. ym a orol Lr Ivacu..-I. Nculcu L. ugar J-M. o a E.I. rr 3. O lay-p ably for lar ural ym. uomaca Mamou M.. a M. rb 3. Guara co obrvr-ba corol of ucram-lag ym. ompur a Elcrcal Egrg ark J.H. 4. O g of obrvr-ba corollr of lar ural lay-ffral ym. ppl Mamac a ompuao ark J.H. 3a. Guara co ablzao of ural ffral ym w paramrc ucray. Joural of ompuaoal a ppl Mamac ark J.H. 3b. obu guara co corol for ucra lar ffral ym of ural yp. ppl Mamac a ompuao r H. M. l a. Naava 4. obrvr g procur for a cla of olar m-lay ym. ompur a Elcrcal Egrg r H Lar fucoal a obrvr for m-lay ym. Iraoal Joural of orol Xu. J. Lam J. Wag a G.-H. Yag 4. No-fragl pov ral corol for ucra lar ural lay ym. ym a corol Lr Xu. J. Lam. Yag a E.I. rr 3. LMI approac o guara co corol for ucra lar ural lay ym. Iraoal Joural of obu a Nolar orol Xu. J. Lam a. Yag. obu ably aaly a ablzao for ucra ural lay ym. Iraoal Joural of ym cc Wag. a H. Ubau. Nolar obrvr for a cla of couou-m ucra a-lay ym. Iraoal Joural of ym cc Wag... Gooall a K.J. Buram. O gg obrvr for m-lay ym w olar urbac. Iraoal Joural of orol Wag. J. Lam a K.J. Buram. ably aaly a obrvr g for ural lay ym. IEEE raaco o uomac orol