Observer design and identification methods for hybrid systems

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(1)Observer design and identification methods for hybrid systems Citation for published version (APA): Juloski, A. L. (2004). Observer design and identification methods for hybrid systems. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR580956. DOI: 10.6100/IR580956 Document status and date: Published: 01/01/2004 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne. Take down policy If you believe that this document breaches copyright please contact us at: openaccess@tue.nl providing details and we will investigate your claim.. Download date: 14. Sep. 2021.

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(121) xwp"}kl~vregfdk|"e:yxwp)dklqp5"xw"} ½ dklqse1}klqdbyb|"ev8qlp,etuf õ¸V iö ²xwf |"l~}1egfbdke |"essetw}krfbegsl~}Megdkp5|"d e h)}kedRygtudbdbxwe1f¾hietuygp"dbv xwf l~}dk|"e1lqprrrdMh)eygl~db}`xwft }klqdbyb|rlqprl~n9} ³dkr|"p"eygp)dkl~rxw`p&dkeg|,ftudxuMstux)"vr}sedk}lq s{e|"lqp5e8dbxË}ktulqdbp&yb|rlqp"lqprvrne«È}be}bÐ)eg"d egp"yel~} tufkrlqdkfRtufkÇ)|"regrfbed ½LDIBnÍ±D }krlÆlqeVdbyb|rdklq|"preZn.hVxwtu",}bege fkhiegfz²xwl~}MfztdkhV|"tue`lqå}ktuÇr}k~dbe eg tudMõ¸eVtw ybi|.öMdkylqtuspKe e` dRtujiegpKlqp'dk|"e ²xwq~xlqprn ²xwfk.´ õ¸V Ì ö { |"e-xw"}begfkh)egfvre}klqnVp yxwp"}kl~}kdb}slqp vregdbegfklqprlqprn!xw"}begfkh)egfn)tulqp"} ²xwflqp"vl~ye} { |"ezv"eg}klqnVpxuì dk|"ezxw"}begfkh)egfZõ¸V Ì öCl~}}kdkfRtulqnV|)d¸²xwfktufbvN { |"e:vÇ5p,tul~y}xuìdk|"e e}kdklqtudkl~xwp-egfkfbxwf l~}nVlqhiegp.)ÇP´ õ¸V 5ö { |"eè}kdklqtudkl~xwpKegfkfbxwfzl~}Mnix$hiegfkp"ev')Ç-t}klqdbyb|"ev'}kÇ}kdbeg. { xegp"}krfbe`}kdRturlqqlqd¿Ç ½ fRxutuìegdb|,dkeg|"tqÇih5eZ)l~egxwtwfkf`}fbxwõ³dk¯

(122) f|"efRvtuÇ5}kprp,l~ybtulqdbj5ybÇ9|rl~ylqõ¸pr}n`Vlqd tuÍl~s}ökö«xwp" prxwn1d}b}kegY h)egfRygtul~egrp5}kddRtudbrx ~e}kdR}ktuÇr}klqdbqeglqseZ}¾etwybt|.Ç`xu}kÃdklqdkq|"rezÇ5l~seg~v x5vrrep"}}k}bdRegtu,rt ~e { Ð)}bxw,p-twvtufRxp"tuv.dkegl~p"yZ z}kÀNturp)fbÇ5e8dktureg}krdRf`tup"õ¸rxlqhVqlq do³ÇËrÍ tip"rökygöp"dkl~vrxuxw¾egpËfdk|"õåtu²ezxwfkrf²xwlqvrdkfkfReg tudRfktuÇ lq~}}k}belqeZdbyRÀN|rlqlqpregnfkxwe8p-}begedMtutufbyb²;| õ¸²xwVfsVit'ö~Áryµixwxw|,stup,xw} p ½ x(t) ˙ = Aσ(t) x(t) + Bσ(t) u(t) y(t) = Cσ(t) x(t). t≥0 y(t) ∈ Rm. x(t) ∈ Rn. u(t) ∈ Rp σ : [0, ∞) 7→ {1, 2, . . . , N } {1, 2, . . . , N }. N > 1. σ(t). t. x ˆ˙ (t) = Aσ(t) x ˆ(t) + Bσ(t) u(t) + Lσ(t) (ˆ y (t) − y(t)), yˆ(t) = Cσ(t) x ˆ(t). t≥0. Li. i ∈ {1, 2, . . . , N }. e := x − x ˆ. e(t) ˙ = (Aσ(t) − Lσ(t) Cσ(t) )e(t).. |"egfbe. P = P> > 0. r}k"yb|.dk|,tud. V (e) = e> P e,. }kut lq~dkxwl~prxwnsp;´ dk|"e:dkfRtum¸eygdbxwfkl~e}Mxu`õ¸V 5ög|"egp. { |rl~}Ç5l~eg~vr}dk|,ez²xwq~x$lqprnsrfbxwx ½. V˙ (e(t)) < 0, e 6= 0.

(123) .. _DIE1 fB¡2¢L£E1¤1BD Þ Ý. à ê ç æ3à é ä3å è â3ã á ç ß>à. Ú ÛOÜ. ¥ ¤c¦2¢n fF7§]¨ ØL©CëC¢n Lì FÏJ.JEGFH° ;1õ¸Vý i2ö -ÇDI (¡í E1«IíF2²BL®nÿ Jf²1FHïî ÄnFHªQ ¿1Fyõ¸JHVJ~ ¸Ì DIö ¨í¡] ¯¤»_º¸DIE1«I¡´FH fACF#BF¯¿1FòYE¯¤E_J¸ÆE7¸óî(¶YðtJYðt°§¸°&ñ¯ñ¼FE1AC ¯¤1B£'D,J¯¹¤1¡.BJ¯FH¤v 'E1¤vE1Än«IFF7¡.JFH.Å.JD.EGFH¤v°EGF J¢L£«:E1«¸LEH© °3¸LE1 ¯¤cò ¸DI¡¼BL®nJfFH ÄFH ¦Y¸¤D,J õ¸V Ïiö E1«IFHDôE1«IFõFyJE1¤°3¸LE1¤1BDóFH ¯ fB õ¸V 5ö ¤J»¦]¿1BL®]¸¿c#¸JY°7¹EGBE1¤1£f¸¿c¼JE_¸Y®¿1Fö¨ P. L 1 , . . . , LN. (Ai − Li Ci )> P + P (Ai − Li Ci ) < 0,. i = 1, 2, . . . , N,. fbxwPxV}klqdkl~xwp V Ì q1nVlqh)e}ztupKtup"}kegfMdbxsdk|"e Ð)"e}kdkl~xwp-|"xÎdbxvre}klqnVp-dk|"e xw ½ }bvregefk}kh)lqnVegpfzdk|"|"e8egpKxw"dk|"}bege fkhiygegrf fkfbegp)|"dZegp sx)dk|"vre8eygxurfkdkfb|"ege p5d |5Ç5srx5fkvrl~vKe8}kxuÇM}kdbdkeg|" e8xwl~}z"}bj5egp"fkxh)ev»p;1}kÇ¯

(124) }kdbregdS|"l~x$}`ºp"xwdbx d j5p"x${ xp÷ n)tulqp}bxwse lqp"}klqnV|5d:lqp)dbxdk|rl~}zrfbxwr~eg e lqqyxwp"}kl~vregf1tsqlqp"etufZ}kÇ}kdbeg l"qp5nVdbregfbfbe9yxwVprp"Ì eö«yg dbe{ v |rlql~p}sygtKåtw²}be}sevxu,Ztw}kyRÇj}kdbyegxwsp} "nVl~}rfRj5tup"dkl~x$xwp p&twlq}sdk|4ÀNrt9f}kd e-dRtu}kdkÇl~y.}kdbp"egsxwpr}.qlqp"õ¡Äzetul~vfklqÇ)d¿twÇº}Rtuõ³n)}betuef õ¸V ¹ËÌ ökeMö«}kdRtufkd5Çsyxwp"}kl~vregfklqprn`dk|"ezytw}beM}kdk"vl~evlqpMfbytujtup"3v øMxwj)xwdbxh5l ù õ²VV$ö«

(125) xwp"}kl~vregfdk|"e ¿1B£f¸¿c¼ë ¤ ¹Jf£«.¤vE1ú p"xwprqlqp"etufz}kÇ}kdbeg.´ ¸õ V iö u t " p K v k d " |. e k } R d u t k d ~ l ` y " p w x r p q q l " p e u t f }Rtudkl~}¸,e}Mt ¯J ¿1Bf¹±F Pxwrp"v.yxwp"vlqdkl~xwp;´ õ¸V óVö. x(t) ˙ = Ax(t) + Bu(t) + Gγ(Hx(t)). ³ r|"p"egygfbdke l~xwp. y(t) = Cx(t). x(t) ∈ Rn , y(t) ∈ Rm , u(t) ∈ Rp H ∈ R1×n γ : R 7→ R γ(v) − γ(w) ≤ b, v−w. û1ücýyþ±ÿ ý

(126) Ïýyü

(127) ý

(128) nÿYÿ vý _ÿ

(129) !" #ÿvÿ fÿHþ%$&(')'

(130) *)+ 0≤. ∀v, w ∈ R, v 6= w..

(131) §]¨ ØL¨¬Ù®nJfFH ÄFH >¡.FyJ¯¤c¦]D¼º_B «]0® ¯¤1¡3J.JEGFH°µJ ÿ. 3 > ; :73 = 879 < 576 4 ; 23 1. , -/.. 0. ¥ ¤c¦2¢n fF7§]¨ ?Y©A@>FH¿¸~ J.JEGFH° ñ"xwfdk|rl~}}kÇ}kdbegôt ÀN"egp5egfkniegf ½ d¿Ç5Pe xw"}begfkhiegf`õ¸V iöxu¾dk|"e:²xwfk.´ ¸õ V Í ö tw}M{ yb|"xV}begp; Í öyxwp"}kl~}kdb}:xuC"p"vlqprn8n){ tulp,} tup"v P}k"yb| " | ` e r v e k } q l V n p u x

(132) u t K p w x " b } g e k f ) h g e f ¸ õ V dk)|,Çtudkd|"dke:|"ez²xweq~}kxdklqlqprtundkl~xwrp8fbxwegfkxVfb}kxwlqfdkl~vxwÇ5p;p, tul~y} l~}

(133) }kdRtur~eV |"ez}bxwqrdkl~xwpl~}CnVlqh)egp ;1ý2-C ( í2² ÿ ²( îªQ f£f¸½3¸DI¡CBõB½.BEGBnÄ¤Dsî(¶Yðtðt§~ñ¯ñ_º

(134) ¸°3¸LE1 ¯¤cò E x ˆ˙ (t) = Aˆ x(t) + Bu(t) + L(ˆ y (t) − y(t)) + Gγ(H x ˆ(t) + K(ˆ y (t) − y(t))) yˆ(t) = C x ˆ(t). L. K. e = x−x ˆ. ¸µ£BD,JE_¸DIE . ¸DI¡¼BL®nJfFH ÄFH ¦Y¸¤D,J ¸DI¡ £f¸D6®YFQº_B¢nDI¡'J¢L£«:E1«¸LEH©. P = P> > 0. ν>0. L. K. (A + LC)> P + P (A + LC) + νI P G + (H + KC)> G> P + (H + KC) −2/b. E1«IFHDôE1«IF>FyJE1¤°3¸LE1¤1BD:FH ¯ fB 9¤J»¦]¿1BL®]¸¿cÃF¯ò¹BDIFHDIE1¤¸¿c¼JE_¸Y®¿1F¨. . ≤0. õ¸V iö. y5xwegehivregrfRe~tu}begygsfkyglqegåtwp)e}bdRv!}btwefk}1twlqdo}`xuÇ

(135) ÀN}kr|5ÇfÇ5}kddRre e«½fksdol~vÇ5}}ktue8Çp"}k}kvdbÇeg}kegsdb~ege}syg;dkfk}}kl~"Ãyybrtu|»Nrd`ygtwlq}:fbdkyg|"fRree«lqådbtp"}

(136) ÇKxwpr}klqqÇdklqp"}k|8dbeeg}ktusfklqlq}dkdbl~yReõå|r"} lqnVprlqrp»n fbe.dkeg|"~egVess 5²egö«ep5;egdbvrq}l,p,ytwetuybtupj f ,tw}Ãtudkeg|.h)egtup`fbezdk|"p"e

(137) eglqe«dkÈ|"l~eg}kfdbeg}kp"syex5xwtudkp"|;v 5rp"prxwl~fÐ)~"x)egyp"tueqq}bÇ}¾xuÀNlq"}bxw}byRqr|rdklqdkl~xwip" }Ã{ l~|r}Ãl~p"}

(138) xwPdÃxVnV}b,etu}fR}btuegp5fkdbl~xwee"v}

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