Decentralized Sliding Mode Control and Estimation for Large-Scale Systems

Transcription

1 HE UNIVERSIY OF HULL Decenralzed Sldng Mode Conrol and Esmaon for Large-Scale Sysems eng a hess submed for he Degree of hd n he Unversy of Hull by ZHENG HUANG Sc n Engneerng Chna February 3

2 Acknowledgemens I s a pleasure o hank hose people who have helped me o fnsh hs hess. Frs of all I would lke o show my deepes graude o my academc supervsor rofessor Ron J aon a respecable responsble and resourceful scholar who has guded and helped me n every maer durng my sudy. He has always suppored me o undersand how o fnd he drecon of my research wh hs broad knowledge creave hnkng and deep nsgh n he subjec of conrol. I s really a wonderful feelng o do research under hs supervson and I would reasure hs perod of me as my mos valuable memory. I would lke o hank all colleagues a he School of Engneerng especally people n he Conrol and Inellgen Sysems Engneerng Conrol Group he Unversy of Hull. I wan o epress my graude o my colleagues Fengmng Sh and Monadher Sam Shaker for her useful advce and Xaoyu Sun for her frendshp durng all hree years. I graefully acknowledge he fnancal suppor of my hd sudy from he Chna Scholarshp Councl CSC and he Hull-Chna scholarshp. Fnally a specal acknowledgemen o my parens and my frend Jng Qu for her undersood and encouragemen durng all hese years.

3 Absrac hs hess concerns he developmen of an approach of decenralsed robus conrol and esmaon for large scale sysems LSSs usng robus sldng mode conrol SMC and sldng mode observers SMO heory based on a lnear mar nequaly LMI approach. A complee heory of decenralzed frs order sldng mode heory s developed. he man developmens proposed n hs hess are: he novel developmen of an LMI approach o decenralzed sae feedback SMC. he proposed sraegy has good ably n combnaon wh oher robus mehods o fulfll specfc performance and robusness requremens. he developmen of oupu based SMC for large scale sysems LSSs. hree ypes of novel decenralzed oupu feedback SMC mehods have been developed usng LMI desgn ools. In conras o more convenonal approaches o SMC desgn he use of some complcaed ransformaons have been obvaed. A decenralzed approach o SMO heory has been developed focused on he Walco-Żak SMO combned wh LMI ools. A dervaon for bounds applcable o he esmaon error for decenralzed sysems has been gven ha nvolves unknown subsysem neracons and modelng uncerany. Sraeges for boh acuaor and sensor faul esmaon usng decenralzed SMO are dscussed. he hess also provdes a case sudy of he SMC and SMO conceps appled o a nonlnear annealng furnace sysem modelderved from a dsrbued parameer paral dfferenal equaon hermal sysem. he sudy commences wh a lumped sysem decenralsed represenaon of he furnace derved from he paral dfferenal equaons. he SMO and SMC mehods derved n he hess are appled o hs lumped parameer furnace model. Resuls are gven demonsrang he valdy of he mehods proposed and showng a good poenal for a valuable praccal mplemenaon of faul oleran conrol based on furnace emperaure sensor fauls.

4 able of Conens Ls of Fgures... v Ls of ables... v Ls of Symbols and Abbrevaon... Ls of ublcaons... Chaper Inroducon.... Inroducon.... Man Dffcules and fauls n LSSs Faul oleran conrol and sldng mode heory n LSS hess Srucure and Conrbuons... 8 Chaper Revew of Large Scale Sysem Conrol.... Inroducon.... Mullevel conrol srucure....3 Sngle Level decenralzed conrol Dsjon decomposon Overlappng decomposon....4 Decenralzed esmaon for LSSs Decenralzed observer based conrol Faul esmaon n LSS Concluson... 8 Chaper 3 Annealng Furnace Modellng: A Dfferenal Quadraure Approach Inroducon Furnace Sysem Mahemacal Modellng Srp hermal Model Furnace wall hermal model wo mehods of dfferenal quadraure Lagrangan Inerpolaon olynomal Cubc splne dfferenal quadraure Sysem model denfcaon Model Valdaon... 5

5 3.5. Sysem smulaon Zero npu response Seady sae soluon ID furnace conrol Concluson... 6 Chaper 4 Sldng Mode Conrol for Large Scale Sysems Inroducon Revew of ypcal sldng mode conrol heory Regular form and mached perurbaons rejecon Reachably roblem and Reachng hase Conrol law desgn Decenralzed SMC desgn usng LMI approach Conrol law desgn wh LMI approach Feasbly dscusson ole assgnmen and quadrac mnmzaon Improvemen Smulaon resuls Concluson Chaper 5 Decenralzed Oupu based Sldng Mode Conrol Inroducon Sac oupu feedback Dynamc compensaor desgn approach Oupu Inegral sldng mode conrol Mul-machne power sysem case sudy Sysem descrpon Sac oupu feedback sldng mode conrol Observer based negral sldng mode conrol Concluson Chaper 6 Decenralzed Sldng Mode Observer and Faul Esmaon Inroducon Sldng mode observer Walco-Żak observer v

6 6.. Edwards & Spurgeon observer LMI approach for decenralzed Walco-Żak observer Decenralzed SMO faul esmaon Acuaor faul reconsrucon Sensor faul reconsrucon Smulaon resul Concluson... 7 Chaper 7 Sldng Mode Conrol and Esmaon for Furnace Sysem Model Inroducon Lnearzaon and conroller desgn ssues Lnearzaon Furnace conroller desgn ssues Nonlnear closed-loop smulaon ID conroller performance and fauls descrpon Sae feedback sldng mode performance Oupu feedback sldng mode conrol Faul esmaon and sensor faul hdden for furnace model Concluson... 9 Chaper 8 Concluson and Fuure Work Concluson and Summary Fuure work Reference v

7 Ls of Fgures Fgure -. Faul classfcaon wh respec o her locaon n LSS... 6 Fgure -. he hree dscplnes of FC aon Fgure -3. Faul oleran conrol mehods adaped from aon Fgure -. Mullevel conrol srucure for LSSs adaped from Mahmoud Fgure -. wo-level neracon predcon conrol srucure... 6 Fgure -3. Dsjon lnear sysem conrol desgn: a overall sysem; b decompose no nerconneced subsysem; c decenralzed conrol desgn.... Fgure -4. Overlappng conrol desgn: a overall sysem; b epanded sysem; c conrol desgn; d conraced closed loop sysem. akule 8... Fgure -5. Observer based decenralzed conrol whou neracons beween observers... 5 Fgure -6. Observer based decenralzed conrol wh neracons beween observers 6 Fgure 3-. Cross-secon of he furnace McGunness and aylor Fgure 3-. Furnace dmensons Fgure 3-3. Energy balance n a sngle un of furnace Fgure 3-4. Cross secon of furnace and wall emperaure model Fgure 3-5. Chebyshev-Gauss-Lobao dsrbuon... 4 Fgure 3-6. Srp emperaure response whou power npu Fgure 3-7. Furnace wall emperaure response whou power npu Fgure 3-8. Srp emperaure wh ID conrol Fgure 3-9. Absolue value of he errors beween he DE soluon and he sysem smulaon resuls under dfferen grd spacng... 6 Fgure 4-. A nonlnear dsconnuous boundary layer funcon Fgure 4-. A smooh boundary layer funcon Fgure 4-3. Egenvalue cluserng on he lef hand sde of Fgure 4-4. Egenvalue cluserng nsde he dsk Fgure 4-5. Sae responses wh only lnear conrol Fgure 4-6. Saes responses wh decenralzed SMC Fgure 4-7. Influence from unwaned sgnals beween =3s and =5s Fgure 4-8. Convergence speed dfference represened by sae from =3s Fgure 5-. hree-machne power sysem... 6 Fgure 5-. Sablzed power sysem saes for 3 nerconneced sysems... 9 v

8 Fgure 5-3. Sablzed power sysem saes for 3 nerconneced sysems... 3 Fgure 5-4. Sablzed power sysem saes for 3 nerconneced sysems... 3 Fgure 5-5. Sablzed power sysem saes for 3 nerconneced sysems... 3 Fgure 5-6. Saes responses wh lower and whou upper he sldng mode nonlnear gan for a sep faul of a... 3 Fgure 5-7. Sae responses of all hree subsysems usng he observer based ISMC.. 33 Fgure 5-8. Errors beween orgnal subsysems and local observers Fgure 5-9. Sae responses wh lower and whou upper he OISMC non-lnear gan erm for a sep faul of a Fgure 6-. Lnear observer case of sae responses for boh he sysem dash and observer sold when fauls occur Fgure 6-. SMO case of sae responses for boh he sysem dash and observer sold when fauls occur Fgure 6-3. Fauls dash and her esmaons sold usng equvalen oupu njecon Fgure 6-4. Fauls sold and her esmaons dashed wh resrced neracons.. 68 Fgure 6-5. Sensor faul esmaon upper and he faul esmaon error lower... 7 Fgure 7-5. Furnace e emperaure rackng performance wh ID conroller Fgure 7-6. ower npu for each heang zone wh ID conroller Fgure 7-7. Fauls unceranes n furnace model sysem Fgure 7-8. Furnace e emperaure wh dfferen ypes of fauls usng ID conroller Fgure 7-9. Absolue values of he srp e emperaure rackng error wh hckness varaon fauls Fgure 7-. ower npu for each heang zone when hckness uncerany occurs Fgure 7-. Absolue values of rackng error of srp e emperaures wh velocy uncerany for ID and SMC respecvely... 8 Fgure 7-. Absolue value of srp e emperaure rackng error wh fauls... 8 Fgure 7-3. Absolue values of srp e emperaure rackng error wh 3 mehods 8 Fgure 7-4. Absolue values of epor emperaure rackng error wh I conroller and I-OSMC conroller v

9 Fgure 7-5. Absolue values of srp e emperaure rackng error wh oupu feedback SMC and I-OSMC conroller Fgure 7-6. Sensor faul esmaon for all he heang zones Fgure 7-7. Sae responses wh I-OSMC when here s a sensor faul n he s heang zone faul occurs a Fgure 7-8. Sensor faul compensaon srucure for a sngle heang zone Fgure 7-9. Sae response wh sensor faul compensaon faul occurs a... 9 Fgure 7-. Srp e emperaure of he 3 rd heang zone wh sensor faul compensaon faul occurs a... 9 Ls of ables able 3-. roperes of seel n dfferen emperaure able 3-. roperes of brck of he furnace wall able 3-3. Furnace Sysem parameers able 3-4. ID conroller gans able 3-5. Modelng error a seady saes wh nerval able 3-6. Error beween DE and Smulaon under dfferen nerval dsance able 5-. he parameers of he power sysem wh hree nerconneced machnes. 8 v

10 Ls of Symbols and Abbrevaon Symbols Eucldean norm vecors or nduced specral norm marces he absolue value of he real number Smalles and larges egenvalues Feld of real numbers and he se of srcly posve real numbers he se of comple number wh negave real par he mar orhogonal o he mar A Mar Kronecker produc Abbrevaons AFC CS DM FDI FC ISM LI LMI LQR LSS ODE OISMC DE FC I-OSMC SMC SMO SOF VSC VSS s.p.d. Acve faul oleran conrol Cubc Splne Decson maker Faul deecon and solaon Faul oleran conrol Inegral sldng mode Lagrangan Inerpolaon olynomal Lnear mar nequaly Lnear-quadrac regulaor Large scale sysem Ordnary dfferenal equaon Oupu feedback ISM conrol aral dfferenal equaon assve faul oleran conrol roporonal negral conrol and oupu feedback sldng mode conrol Sldng mode conrol Sldng mode observer Sac oupu feedback Varable srucure conrol Varable srucure sysem Symmerc posve defne

11 Ls of ublcaons Whn he perod of hs research he followng works were publshed: Huang Z. & aon R. J. a. An adapve sldng mode approach o decenralzed conrol of unceran sysems. UKACC Inernaonal Conference Conrol on Conrol Cardff UK Sep. Huang Z. & aon R. J. b. Decenralzed Conrol of Unceran Sysems Va Adapve Sldng and Overlappng Decomposon. 7h IFAC Symposum on Robus Conrol Desgn Aalborg Denmark June.

12 Chaper Inroducon. Inroducon Wh he fas developmens of modern echnologes he compley of ndusral sysems keeps ncreasng and as a consequence sysem applcaons become more nerconneced and dsrbued. hese sysems are ofen referred o as Large Scale Sysems snce poenally a sgnfcan number of varables can be nvolved wh nonlnear ner-relaonshps. As a consequence of compley mahemacal models of her dynamcs may be hard o defne precsely and hence modellng uncerany s a sgnfcan challenge f model-based mehods of conrol or esmaon are o be used. In fac he erm Large scale sysemslsss does no represen a sngle ype of sysem havng specal srucure e.g. dsrbued bu sysems whch canno be solved by onesho approaches akule 8. As a consequence of ncreased compley resulng n an ever ncreasng range of applcaons he research neres n LSS does no decrease even afer hree decades. However can be noed ha he ermnology has changed over he years. In he early years he erm LSS was frequenly used Sngh and l 978; Sandell e al 978; Ikeda 989; Šljak 996. In subsequen years he ermnology and emphass have changed wh he recen leraure focussng more on decenralzed conrol and conrol of ner-conneced sysems drven very much by he needs of ever changng applcaons. Sankovć Sanojevc and Šljak ; Akar and Özgüner ; Yan Spurgeon and Edwards 3; aglla and Zhu 4; Shyu Lu and Hsu 5; Kals Lan and Żak 9; Lu Ln and eeman 9; Sankovć and Šljak 9; ll and raek 9; Yau and Yan 9; Kals Lan and Żak ; aruka ; Zhu and L ; Lu ; Mahmoud ; Mukadan We and Jn ; Wu ec.. he IFAC Large Scale Sysems symposum n France saes he applcaons usng LSS heory ncludng: aerospace engneerng envronmen sysems power sysems ransporaon sysems medcal sysems busness sysems engneerng ec

13 lss.ulbsbu.ro. hese are ypcal applcaon areas of LSS ha presen sgnfcan challenges o conrol sysems heory and o conrol sysem desgners. he praccal physcal LSSs are ofen characerzed by geographcal separaon or large dmensonaly so ha ssues such as he ner-connecon epense and relably e.g. possbly and frequency of nerconnecon falure delays nformaon consrans for each subsysem ec. have o be aken no accoun. Šljak 99 concluded ha he compley of real sysems may no be well organzed whls for conrol o be effecve good srucural organzaon of a sysem s requred. Hence well-organzed compley s he man challenge of large scale nerconneced sysem desgn ncludng he noons of subsysems neracons neural neworks parallel processng ec. I should be noed ha hese complees brng n some confusng and somemes over-lappng ermnology e.g. he words Large scale sysems Dsrbued sysems Decenralzed sysems and Inerconneced sysems whch can have smlar meanng. o develop hs subjec properly hese erms need o be carefully defned. A Dsrbued sysem s a sysem conanng a collecon of auonomous subsysems whose componens and resources may no be shared by all local decson makers. I s ofen used n compuer scence. he word dsrbued orgnally referred o compuer neworks where ndvdual compuers were physcally dsrbued whn some geographcal area. However he erm s nowadays used n a much wder sense. he common defnng properes of dsrbued sysem are:. here are several ndvdual subsysems each of whch has s own local decson maker and. hey communcae wh each oher. On he oher hand a Decenralzed sysem s more concerned wh he subjec of akng he conrol acons from a cenral funcon o conrol acons a decenralzed locaons of he sysem. A sysem ha s decenralzed lacks a conroller nucleus as s usually composed of many subsysems whch are workng n unson o form a sable srucure. ha means ha he emphass of hs noon s ha he sysem lacks cenralzed decson makers or coordnaors. he noon Inerconneced sysem s concerned more wh neracons. In hs case f here are nerconnecons beween subsysems he overall sysem can be referred o as an Inerconneced Sysem. Some leraure use he noon Large scale nerconneced sysem e.g. Kals Lan and Żak only o sae ha he subsysems of he LSS are nerconneced. hese hree noons have her own emphass and should no be med up. hey all belong o he

14 concep of a LSS bu each n urn has a dfferen classfcaon. And no maer wha knd of classfcaon he sysem s n he challenges descrbed n he ne Secon are wha LSS conrol sysem desgners need o deal wh.. Man Dffcules and fauls n LSSs Some researchers sae ha he compley and dffcules of LSSs arse manly from he dmensonaly uncerany delay and nformaon consrans Šljak 99; akule 8. hese are defned as follows: Dmensonaly. he dmenson dynamcal order of a sysem can be very large. For a sngle LSS sysem here are a large number of saes and npus ha canno be handled easly by usng a one-sho conrol mehod. Some LSSs ha are already decomposed conss of many subsysems ha requre srucure and robusness analyss before effecve conrol sysems a herarchcal and/or local levels can be desgned. Uncerany. he overall sysem canno be precsely descrbed by a lnear mahemacal model. Unceranes come from ncomplee denfcaon of he sysem and some unknown dsurbances/conrol sgnals. Moreover model aggregaon or smplfcaon whch s delberaely desgned o make he sysem manageable may also lead o unceranes. Informaon Consrans. ecause of he dmensonaly problem s necessary o desgn many decson makers DMs o manage he subsysems. None of hese DMs knows he sysem compleely. A conroller s an eample of a DM for a subsysem ha can only use he local nformaon.e. saes/oupus of hs subsysem o sablze he subsysem. As a consequence of hese dffcules he analyss and synhess asks canno be solved effcenly n a sngle sep conroller. Many conrol epers ake he pragmac vew of LSS as a sysem ha canno be managed by convenonal mehods akule 8. he developmen of LSS decomposon heory s devoed o he problems arsng from he dmensonaly problem. he heory answers he queson of how o decompose he gven conrol problem no manageable sub-problems. In hs case he sysem s no longer conrolled by a sngle conroller bu several ndependen local conrollers whch ogeher perform he conrol funcon of he overall sysem. 3

15 A sgnfcan number of publcaons focus on approaches o he remanng challenges of handlng modellng uncerany and nformaon consrans. Varous conrol mehods have been used o address hese challenges for eample varable srucure conrol Yan sa and Kung 997; Hu and Zhang ; Yan Edwards and Spurgeon 3 4a 4b 9; Shyu Lu and Hsu 3 egensrucure assgnmen Labb e al 3 vecor Lyapunov funcon Lunze 989; Marynyuk 998; Nersesov and Haddad 6 adapve conrol Jan Khorram and Fardanesh 994; Hansheng ; Rcca-ype conrol akule and Rodellar 996 model predcve conrol Lavae Momen and Aghdam 8; Ocampo-Marnez e al ; ec. In LSSs he performance of subsysems afer decomposon may be affeced by: neracons from oher subsysems eernal dsurbances and modellng uncerany arsng from srucure uncerany or parameer varaons. he dfference beween eernal dsurbance or eogenous dsurbance and modellng uncerany s ha he former perurbaon does no vary wh he sysem parameer saes npu or oupu ec.. However should be noed ha n regulaon problems he goal of he sysem s o drve he sysem error o zero wh proper conrol desgn he effec of he modellng uncerany maybe sgnfcanly reduced when he conrol objecve has been reached. Unlke eernal dsurbance and uncerany radonally neracons have been reaed as a par of he sysem and are aken care of n sysem desgn a a cenralzed level Aok 97; l Lefevre and Rchen 973; Smh and Sage 973; Sngh and amura 974; Sngh Hassan and l 976 Ikeda 98; Ikeda 983. Wh he developmen of he compley of a dynamc sysem aemps were made o deal wh LSS desgns usng decenralzed conrol akule and Lunze 988; Gavel and Sljak 989 Feng and Jang 995; Hsu 997; Chou and Cheng ; Hu and Zhang ; Šljak and Zečevć 5; ll and raek 9; ec.. However should be clear ha he book of Sngh and l 978 makes a clear defnon of he dfferences beween herarchcal and de-cenralzed conrol he conceps were around a long me before beng fully aken up n he leraure. I became apparen ha he conrol of an LSS wh decenralzed conrol requres ha he neracons be reaed as specal sgnals. As he neracons nvolve couplng 4

16 beween oher subsysems s seldom possble o have complee nformaon abou hem when desgnng local conrollers. In oher words he neracons may have unceran srucure/parameers. I s clearly desrable o aemp o reduce he effec of he neracons beween he subsysems. From hs follows ha he desgns of ndvdual conrol sysems can be made whou assumng knowledge of he neracons bu akng accoun of her unceran effecs as a robusness problem. If all he subsysems are combned no an aggregae sysem he neracons are affeced by varaons of he combned se of all sysem parameers. However usng he robusness saemen above can be assumed ha for a sngle subsysem pon of vew he neracons from oher subsysems can be reaed as eernal dsurbances. hs s sll an open problem Rosnováand Veselý he noon of uncerany descrbed above does no fully represen he unwaned changes n he LSS. Fauls also affec he dynamc behavour of he sysem n unceran ways. Varous publcaons provde alernave defnons for he erm faul. For eample Isermann 984 defnes a faul as a non-permed devaon of a characersc propery whch leads o he nably o fulfl he nended purpose lanke e al 6 defned a faul as a devaon n he sysem srucure or he sysem parameers from he nomnal suaon. Moreover f fauls are no aken care of carefully hey mgh become falures aon Frank and Clark 989. A falure descrbes he condon when he sysem s no longer performng he requred funcon and canno be correced by a conroller. Iserman 6 defnes falure as a permanen nerrupon of a sysem s ably o perform a requred funcon under specfed operang condons. ha s eacly wha s o be avoded. Chen and aon 999 classfes clearly dfferen fauls by he locaon of a faul where acs n he sysem. Accordng o hs classfcaon he faul can be recognzed as. Acuaor fauls. Sensor fauls and. Componen fauls. An acuaor faul and a sensor faul appear n an acuaor and a sensor of he sysem respecvely and are normally consdered as addve effecs whls he componen faul shows up hrough srucural and/or parameer varaons of he sysem.e. as mulplcave or parameer-varyng effecs. However n LSS decenralzed conrol from a subsysem pon of vew one addonal faul should be consdered as an abnormal behavour n he neracon beween wo subsysems. One can hus hnk of 5

17 an neracon faul Fgure -. he local conroller desgn ha seeks o ake accoun of neracon fauls mus do so by aempng o mnmze he sensvy of he local conrol o hese fauls. hs s an eenson of he dea of he concep of robusness o parameer varaons and modellng uncerany. In hs respec a faul can be consdered as a form of uncerany Chen and aon 999. However he reverse s no rue he uncerany s no a form of faul! Fgure -. Faul classfcaon wh respec o her locaon n LSS.3 Faul oleran conrol and sldng mode heory n LSS Several nvesgaors have consdered mnmzaon of he local conrol funcon o sysem fauls as a so-called faul oleran conrol FC problem aon e al 7. Compared wh he number of publcaons on robus conrol of LSS he number of publcaons ha nclude FC aspecs of LSS s much lower especally before cenury. Ffeen years ago robus conrol was no wdely consdered as a par of he FC problem. aon 997; Chen and aon 999; lanke e al 6 have poned ou ha FC ncludes hree major research felds.e. Faul Deecon and Isolaon FDI/Faul Deecon and Idenfcaon Esmaon Robus conrol and Reconfgurable conrol. Fgure - Fgure -. he hree dscplnes of FC aon 997 6

18 Generally wo man approaches o FC are known:. assve faul oleran conrol FC and. Acve faul oleran conrol AFC. eard 97; aon 997; Chen and aon 999 Fgure -3. Faul oleran conrol mehods adaped from aon 997 Fgure -3 shows he generally acceped aonomy of acve and passve FC mehods. he man dfference beween hese wo mehods s wheher or no a reconfguraon/adapon procedure s requred. In he passve approach robus conrol mehods are used o ensure ha small and bounded fauls are oleraed n a fed gan conroller desgn however hs approach s lmed o mnor faul effecs. On he oher hand acve FC requres onlne faul nformaon o reconfgure/re consruc he conroller. Acve FC mehods are furher classfed accordng o wheher or no hey make drec use of FDI resdual sgnals o provde faul nformaon. A specal ype of acve FC sysem use faul esmaon and compensaon o hde he effec of a faul n a conroller however hs approach s manly applcable o FC for sensor faul effecs. A specal case of FC classfcaon s he use of sldng mode heory. Sldng mode conrol SMC s a ype of conrol ha provdes nheren robusness properes of sldng modes o a ceran class of fauls. I has he ably o drecly handle acuaor fauls whou requrng he faul o be deeced and whou requrng conroller 7

19 reconfguraon. Classcal SMC s acually a form of FC wh fed srucure ha can de-couple he effecs of he fauls n he feedback conrol durng sldng moon as long as he fauls are bounded and sasfy a so-called machng condon. Several sudes have consdered applyng SMC o LSS problems. However mos of hese sudes assume ha he neracons sasfy he machng condon consrans. Unforunaely hs assumpon does no hold for mos praccal sysem applcaons.. However SMC has good compably wh oher mehods. For eample adapve SMC can be caegorzed as an AFC mehod aon ura and Klnkheo whch combnes a sldng mode observer o esmae he faul wh an SMC; he faul esmae causes he SMC non-lnear gan o swch accordng o he magnude of he faul esmae. hs hess s concerned wh he challenges of applyng SMC and sldng mode observer SMO heory o handle he jon problem of local conroller desgn and uncerany compensaon n LSS. Whls he hess does no focus on approaches o FC as such he work conans an applcaon eample of a hermal annealng furnace sysem whch s shown o be faul oleran o emperaure sensor fauls usng SMC and a smple concep of faul esmaon..4 hess Srucure and Conrbuons he remander of he hess s arranged n he followng manner: Chaper provdes an nroducory movaon for he hess by revewng he varous known approaches o conrol and esmaon mehods for LSSs. Afer descrbng he generalzed mullevel srucure of he LSS problem a furher specalzaon o he sngle level case s gven n whch varous approaches o sysem decomposon for LSS conrol are descrbed focussng on aemps o mnmze he effecs of he subsysem neracons. he LSS properes presened lead o a dscusson of mehods for robus sablzaon of LSS. Furhermore decenralzed esmaon approaches are also revewed and dfferen ypes of observer based esmaon mehods are brefly nroduced based on he LSS concep. Chaper 3 nroduces a seel annealng furnace model aken from a New Zealand Seel projec. he furnace model s se up as an eample of a LSS o whch he mehods developed n Chapers 4 5 and 6 can be appled and descrbed more fully as a robus 8

20 conrol eample n Chaper 7. hs s an orgnal sudy usng a lle known bu very powerful way of ransformng he hermal model no a suable framework for decenralzed conrol and esmaon. A mahemacal model of he furnace sysem s derved sarng from he non-lnear paral dfferenal equaon hermal sysem aken from he projec repor by McGunness and aylor 4. he dea s o ransform he nfne dmensonal hermal sysem no a lumped represenaon of a sysem wh nerconneced hermal subsysems. A sraegy called dfferenal quadraure s used o derve he correspondng non-lnear ordnary dfferenal equaon sysem usng wo dfferen nerpolaon mehods whch akes he requred boundary condons no accoun as well. Several smplfcaons o he non-lnear sysem are proposed o prepare a model sysem ha s suable for he decenralzed conrol desgn. hen he hermal properes of he model are dscussed based on a zero-npu response of he nonlnear model o qualavely valdae he smplfed model. o dscuss he modellng accuracy he seady sae soluons o he paral dfferenal equaons DEs.e. consderng hea balance are compared wh he resuls of he nonlnear ordnary dfferenal equaon ODE sysem usng ID conrol. Chaper 4 sars wh an nroducon o sldng mode heory. For he sldng surface desgn wo approaches are descrbed for paronng he sysem no dfferen regular form srucures Znober 99; Cho 997. hese regular form decomposons lead o wo clearly dfferen approaches o he SMC problem. Followng hs he SMC reachably problem s hen presened. Several approaches used o reduce or remove he reachng phase are also dscussed. A he end of he nroducon ypcal mehods for bound consran relaaon and chaerng reducon are also dscussed and proved. A novel decenralzed SMC approach based on lnear mar nequaly LMI heory s hen proposed n Chaper 4. Wh hs approach he unmached neracons unceranes are consdered n he sldng surface desgn procedure and he sably of he overall sysem s guaraneed. he pole assgnmen heory and quadrac mnmzaon are hen combned wh hs mehod o provde mproved robusness o uncerany and neracons. A uoral eample of a non-lnear nerconneced sysem s used o llusrae he mehod a he end of hs Chaper. 9

21 Chaper 5 focuses on he oupu feedback approach o he decenralzed SMC as an eenson o he sae feedback approaches descrbed n Chaper 4. As conrbuons o formulae a sysemac LMI-based decenralzed SMC heory novel decenralzed sac oupu feedback as well as dynamc oupu feedback are presened and compared based on a common SMC represenaon. An oupu feedback negral SMC desgn mehod s also nroduced n hs Chaper as a new conrbuon o hs research. Wh hs mehod he SMC reachng phase can be elmnaed. A he end of hs Chaper a mul-machne problem s nroduced o llusrae he proposed mehods. he sysem neracons are adaped o sasfy he so called quadrac consran. I s shown ha boh sac oupu feedback and observer-based negral sldng mode gve good robus regulaon performance. Chaper 6 focuses on he desgn decenralzed observer sysems usng sldng mode observer SMO heory. oh he Walco-Żak observer and he Edwards & Spurgeon observer are revewed. o acheve decenralzed sysem sae esmaon an LMI-based decenralzed Walco-Żak observer s developed from he heory used for sngle or cenralzed sysems. he chosen SMO approach s an eenson of he Walco-Żak observer usng a novel mprovemen o he conrol law n whch he oupu errors are guaraneed o be zero when he sldng surfaces are reached n he presence of bounded unmached unceranes and bounded unceran neracons arsng from non-lneary. he SMO mehods developed n hs Chaper are appled o boh acuaor and sensor faul esmaon. he nfluence from neracons/unceranes o he acuaor faul esmaon s dscussed wh hs decenralzed SMO desgn. A uoral eample of an nerconneced non-lnear sysem s used a he end of hs Chaper o llusrae he proposed SMO approach provdng robus sae and faul esmaon. Chaper 7 furher dscusses he furnace problem proposed n Chaper 3 o llusrae he SMC and SMO mehods descrbed n Chapers 4 5 and 6. he Chaper sars wh a smulaon of he furnace model developed n Chaper 3. hen a lnearzaon sraegy s appled o lnearzed he nonlnear furnace model. hree ypes of fauls are chosen o es he robusness of he SMC sraegy proposed n Chapers 4 and 5. In comparson wh he ID conroller sae feedback SMC s frs proposed. However snce no all

22 he saes are measurable sac oupu SMC s hen used o conrol he sysem. Moreover o smplfy he mplemenaon of he SMC a ID-OSMC algorhm s proposed whch gves more desgn freedom and makes he sysem nsensve o he mached fauls. In hs Chaper sensor fauls due o he hermocouple deeroraon are also consdered. If he hermocouples gve lower measuremens ID-OSMC canno conrol he furnace emperaure appropraely. Afer usng a sae augmenaon fler n he SMO s shown ha he deeroraon of he hermocouple faul can be esmaed precsely even f he faul s of he mulplcave ype. he faul esmaon sgnal s hen used o compensae he effec of he hermocouple n he ID-OSMC conrollers for each heang zone subsysem. he resul demonsraes he robusness of he faul esmaon and compensaon and enhances he value of he proposed ID-OSMC mehod. Chaper 8 summarzes and concludes he overall work descrbed by he hess and makes suggesons and recommendaons as o how he research can be furher developed n he fuure.

23 Chaper Revew of Large Scale Sysem Conrol. Inroducon he developmen of conrol for LSSs can be recognzed from publcaons Sandell 978; Ikeda 989; Šljak 99; Šljak 996; Šljak and Zečevć 5; akule 8. Nong ha he concep of drvng an LSS by a cenralzed conroller s no longer aracve s beer o desgn a decenralzed sysem Šljak 996. he advanages of usng decenralzed conrol can be found from eher economy or relably sandpons. When he sysem s oo large o be deal wh by cenralzed conrol s compuaonally effcen o use only local nformaon.e. local saes or oupus o make he conrol decson. hs mehod s also economcal snce s easer o mplemen and can effecvely reduce he communcaon cos Šljak 996. Decenralzed conrol also faclaes he developmen of good robusness. I makes he sably of he closed-loop sysem oleran o a broad range of unceranes regardless of he unceranes n he subsysems or n he nerconnecons Šljak and Zečevć 5. Followng Šljak 996. decenralzed conrol sraeges are nherenly robus wh respec o a wde varey of srucured and unsrucured unceranes n he nerconnecons. he sraeges can be made relable o boh neracons and conrol falure nvolvng ndvdual subsysems. As descrbed n Chaper here are several dffcules n desgnng conrol sraeges for LSSs. Dfferen conrol srucures and dfferen decomposon mehods are revewed n hs Chaper o overcome hese dffcules.. Mullevel conrol srucure he research abou mullevel conrol sared n he 96s and araced sgnfcanly more aenon from he 97s Mahmoud 977 Sngh and ll 978. Afer a furher 4 decades of research nvesgaors sll work wh hs conrol srucure. Now has become que a maure conrol sraegy wh applcaon sudes on several praccal sysems Mesel 98; Van Cusem Howard and Rbbens-avella 98; Rubaa 99; Okou 5; Gómez-Epóso and Vlla Jaén 9; Chen. Some neresng

24 research abou FC usng SMC conceps and wo-level srucures have been publshed Ln aon and Zong 9; Larbah and aon. he cenral dea of he mullevel sraegy s o form a conrol srucure n a pyramdlke form Fgure - Mahmoud 977. he problems a he base of he pyramd are smpler hough numerous. Each of he problems can be solved accordng o some decson rules local decson maker whch should be manpulaed by problems locaed hgher n he pyramd. In a hree or more level srucure hs model of parameerzed sub-problems repeas self over many levels whn he organzaon. I can be referred o as a level comprsng a group of decson problems performng smlar knds of problems n he srucure. here s one decson problem upon whch he overall objecve of he sysem depends sandng a he op of he pyramd. In hs sense a mullevel sysem s a herarchy of goal-seekng subsysems or decson problems. As we can see he more levels he conrol srucure has he more complcaed he sraeges become. he compley of he srucure whch lms he mplemenaon of herarchcal conrol s one of he man dsadvanages. On he oher hand he pyramd srucure mples cenralzaon. I has he dsadvanages of cenralzed conrol.e. communcaon delay ec. Coordnaor Herachcal level Coordnaon Decson Maker Decson Maker erformance Resuls ase conrol level Conroller Conrol acon Conroller Conroller Conroller Feedback Informaon Large Scale Sysem Fgure -. Mullevel conrol srucure for LSSs adaped from Mahmoud 977 3

25 In he srucure of he wo-level LSS conrol sraegy he base level s a group of local conrollers dealng wh local nformaon whls he second level s a coordnaor dealng wh neracons. he algorhm of quadrac opmzaon wo-level conrol s proposed n Sngh Hassan and l 976 hs s he so called neracon predcon mehod. Assumng ha afer proper decomposon he -h subsysems of he LSS are n he form of: where are he saes and npus of he -h subsysem respecvely represens he neracons from oher subsysems. I can be noed ha s a lnear combnaon of he saes of he overall sysem. Local conrol desgn he local conroller s desgned followng he LQR heory. he subsysem conrol performances are measured va he subjec o he solaed subsysem.e. neglecng he neracons: - where are posve sem-defne and posve defne weghng marces respecvely. he local conrollers can be obaned by solvng he Rcca equaon: Usng he MALA Robus conrol oolbo he conrol low for he base level s gven by seng : - 4

26 Global conrol he objecve of usng global conrol s o compensae he nfluence from he neracon erms. he global conrol performance s measured by he quadrac performance nde: he analyss gves he followng Lagrangan mnmzed n order o deermne he sysem conrol npus: whch mus be where are he Lagrange mulplers are he co-saes. he Hamlonan for each subsysem can be wren as: he he correspondng necessary condons for opmaly are: Defne and he global conrol. Followng he algorhm n Sngh Hassan and l 976 he conrol gan can be obaned by solvng: -3 wh. he conrol law for each subsysem s hus gven n erms of he local conrol and as: 5

27 -4 he srucure of hs wo level conrol sraegy s shown n Fgure -. Snce he cenralzed coordnaor requres nformaon from every conroller mgh have he same problem as we have n cenralzed conrol. However mgh provde beer resuls han he oal decenralzed sngle level conrol sraegy snce has a coordnaor o deal wh he neracons. In he sngle level sraeges s more lke desgnng local conrollers whch are robus o he neracons as perurbaons comng from oher subsysems. Coordnaor u g u g u N g N N + Conroller- + Conroller- + Conroller-N u u un Subsysem- Subsysem- Subsysem-N Ineracons Large Scale Sysem Fgure -. wo-level neracon predcon conrol srucure.3 Sngle Level decenralzed conrol he sngle-level decenralzed conrol sysem has a much smpler srucure han he mul-level sraegy. he conrol srucure only conans he base conrol level see Fgure -. he neracons are aken care of by he local conrollers. he more robus he local conrollers he beer he performance he LSSs wll have. As descrbed n Chaper dmensonaly uncerany nformaon consrans are he hree man dffcules n desgnng LSSs conrol. Several general mehodologes have been and are beng researched durng 3 decades of developmen of for he sngle level srucure. Mos of hem belong o one or more of he followng hree groups Šljak 978; Šljak 99; akule 8: 6

28 Decenralzaon; Decomposon; Robusness and model smplfcaon. Decenralzaon concerns he nformaon srucure nheren o he gven problem. he desred goal s o acheve as closely as possble compleely ndependen mplemenaon of he LSS conrol n each subsysem. here are varous movaons of decenralzaon of he desgn process such as weak couplng beween subsysems conradcory goals of dfferen subsysems or hgh dmensonaly of he overall sysem. akule 8 Decomposon s anoher par of he conrol desgn. I concerns he smplfcaons of he analyss and synhess asks for LSS by decomposng he problem no several subproblems. he goals of decomposon are he reducon of compuaonal compley and weakenng he neracon nfluence. Dfferen decomposon mehods gve dfferen base conrol level srucures. Robusness concerns he robus propery of a conrol desgn when dealng wh unceranes on he bases of he sably analyss of coupled sysems. Robusness analyss becomes more serous n LSS snce he neracons mgh ac as unceranes and hey are unavodable. Model smplfcaon manly ncludes model reducon mehods and appromaons Šljak and Zečevć 5. For robus feedback conrol sraeges Šljak 996 nroduce a bordered blockdagonal form for he gan mar and eended hs dea n Šljak and Zečevć 5. In her work such a srucure can sgnfcanly mprove he decenralzed sablzaon of LSSs a he epense of only mnmal communcaon overhead. Šljak and Zečevć 5. Chen Ikeda and Gu 5; Chen Gu and Zha. 6 used a homoopy mehod o desgn an decenralzed dynamc conrol and nerconneced descrpor sysem. Rosnováand Veselý 7 proposed an LMI based decenralzed ID conrol. he adapve sablzed decenralzed conrol sraegy was frs proposed by Gavel and Šljak 989. Sh and Sngh 99 hen provded an adapve algorhm for srong nonlnear neracons wh sngle npu. Wu 3 furher proposed adapve sraeges for unceran nerconnecons. hen eruka eruka gave a good survey of decenralzed adapve conrol sang ha ha can be useful o combne he robus and adapve conrol ogeher. 7

29 Anoher bg mpac o sngle-level LSS s he quck developmen of varable srucure heory sldng mode heory. Rcher and Lefebvre 98 frs combned he decenralzed conrol wh varable srucure heory and apply n a wo-pendulum sysem eample. Followng hs many researchers pad aenon o a parcular ype of varable srucure he sldng mode Furua 99; Drakunov 99; Young 996; Ukn 993; Levan 998 ec.. In he sldng mode heory here are wo pars n he conrol law. he frs par s used o sablze he sysem and he second dsconnuous par s used o drve he sysem o he so called sldng surface. Once he sysem operaes on he sldng manfold he sysem s nsensve o he mached perurbaons he perurbaons comng from npu channel wh he concep of machng frs defned by Draženovć 969 I becomes obvous ha f he neracons sasfyng he machng condon hey can be compensaed compleely by he SMC. Laer Edwards and Spurgeon 998 sysemacally eended he sldng mode conceps o nclude conrol and esmaon. Meanwhle he decenralzed sldng mode conrol sared beng popular Xu 99; Wang 993; Feng 995; Hsu 997; Yan 997; Koan-Yuh 997. However some ceran resrcve condons such as he machng condon for neracons and known upper bounds were always assumed n a smplsc way n hese research sudes. From abou he sldng mode as a powerful dsurbance rejecon mehod has been consdered more and more ofen n sudes on decenralzed sysems Hu ; Yan Spurgeon and Edwards 3; Yan Edwards and Spurgeon 4; Shyu 5; Cheng and Chang 8; Kals 9; Yan Spurgeon and Edwards 9; Kals ; Zhu and L. However no sysemac way of usng sldng mode heory n LSS has as ye been proposed. Some sudes n he leraure even furher complcae he problem! Hence a need o buld up a sysemac concep of neracon mnmzaon SMC heory and o furher eend hs mehod n LSSs have been he man movaons for hs curren hd research. hs hess also provdes some new less resrcve and easer o mplemen conceps n robus conrol and esmaon for LSSs. In he sngle-level conrol srucure decomposon plays a sgnfcan par of he conrol desgn process. roper decomposon no only smplfes he compuaon process bu also weakens he nfluence from nerconnecons and a he same me provdes an opporuny for mprovng he sysem performance. I s well known ha some LSSs 8

30 have naural spaal decomposons. Spaal decomposon here means ha he subsysems are defned accordng o her dfferen locaons/dsrbuons. For eample power dsrbuon neworks and raffc regulaon. However a leas on a heorecal bass or as a sysem plan some of he LSSs can be decomposed by conrol sysem desgners. he decomposon mehods can be classfed by wheher or no he subsysems shared sae varable nformaon accordng o:. Dsjon decomposon or. Overlappng decomposon..3. Dsjon decomposon he dsjon earng of he sysem may be performed for eher physcal or numercal reasons. he physcal reason s manly because of he spaal separaon of he subsysems. Numercal condonng reasons requre he developmen of a unversal conrol echnque for applcaon o LSSs akule 8. hs secon s only concerned abou he numercal reasons snce he decomposon mehod canno be made f he sysem s already spaally separaed. A ypcal dsjon decomposon s descrbed by Šljak 99; Šljak 996 he so called Nesed epslon decomposon. he dea of hs ype of decomposon can be llusraed by a smple lnear eample: where he mar s n he form of a block dagonal and he mar has all elemens < and s a prescrbed small number represenng he srengh of he neracons. hen: he algorhm gves freedom o convenenly choose he srengh of couplng beween he subsysems and conrol he sze and domnance of he subsysems. Šljak 996. As he mos used decomposon mehod he algorhm of dsjon conrol desgn s llusraed by a lnear LSS n Fgure -3. he conrol sraeges for dsjon decenralzed sysems manly focus on he reducon of he nfluence from neracons and unceranes. Many of he researchers keep seekng he possbly of reang he neracons among he subsysems as perurbaons Jang ; Šljak and Spanovć and Zečevć ; Casaños and Frdman 5 Hung 7; Zhu and aglla 7; Shyu Lu and Hsu 5; Yan Spurgeon and Edwards 9. Some nvesgaors do no clarfy hs dea n her work bu hey rea he neracons and perurbaons n he same way. hs dea means ha he sysem desgners can rejec or a leas mnmze he dsurbance comng from 9

31 neracons as well as he fauls acng n each subsysem by dsconnecng or reconnecng subsysems. hs s he so-called plug and play problem of LSS aon e al 7. a 3 D A D D D u D b 3 A u 3 j A j 3 j c C C C3 u K 3 3 Fgure -3. Dsjon lnear sysem conrol desgn: a overall sysem; b decompose no nerconneced subsysem; c decenralzed conrol desgn. In mos publcaons abou decenralzed conrol he subsysem sae equaon s descrbed n he form: -5 where are he saes and npus of he -h subsysem respecvely. s he number of he subsysems. he erm represens he neracons from oher subsysems. In lnear LSS. he advanage of usng s ha can represen no only he lnear neracons bu also nonlnear neracons and he uncerany of he subsysem self. Šljak and Spanovć gves a quadrac consran for hese neracons as follows:

32 where s he boundng parameer and s a consan mar. hs assumpon also offers a possbly o apply a varey of sraeges avalable n he LMI framework. Some of he research based on hs assumpon has been done Šljak and Spanovć and Zečevć ; Zhu and aglla 7; ll and raek 9; Kals as well as some work of hs hess. More deals are gven n Chaper Overlappng decomposon Dfferen from Dsjon decomposon overlappng decomposon of an LSS allows he decomposed subsysems o share some common pars and gves more flebly n he choces of he subsysems. he movaon of hs ype of decomposon s he necessy of buldng decenralzed conrol and esmaon schemes usng overlappng nformaon ses on realsc applcaons such as power sysems large space srucures ec. Moreover many large scale sysems e.g. see Özgüner Khorram and İfar 988 may conss of subsysems whch are srongly conneced hrough ceran dynamcs he overlappng par bu weakly conneced oherwse İfar 993. For hose sysems dsjon decenralzed conrol may easly fal whls overlappng decomposon may produce feasble soluons. he conrol desgn sraegy s llusraed n Fgure -4. here are four seps n desgnng overlappng conroller: a Decde whch pars of he sysem are he overlappng pars; b Epand he sysem by usng he Incluson rncple Ikeda 98; c Desgn he decenralzed conroller based on he epanded sysem; d Conrac he conroller o form he decenralzed conroller for he orgnal sysem. he research abou overlappng srucures sared nhe 98s. Ikeda 98 proposed he Incluson rncple as he basc heory underlyng he sraegy of overlappng decomposon whch jusfes he ransformaon of a lower dmensonal orgnal sysem o a hgher dmensonal epanded sysem. Šljak 99 gves a very clear saemen of overlappng decomposon. wo dfferen desgn mehods have been brough ou n he 98s and 99s. İfar 99; İfar 993 desgns he conroller based on he epanded sysem and conraced o he smaller spaces for mplemenaon on he

33 orgnal sysem. Ikeda and Šljak 986 on he oher hand hey defne he conrol law n he orgnal space and oban he npu srucure n he epanded space. Laer he srucure of he epanson-conracon relaons ncludng he conracbly of conrollers s analyzed n Šljak ana and Spanovć ; Sankov Sankovć and Šljak ; Chu and Šljak 5 for LI sysems and akule Rodellar and Rossell ; Sankovć and Šljak 3 for LV sysem. a b c 3 S S S C C 3 d S S C 3 3 S Fgure -4. Overlappng conrol desgn: a overall sysem; b epanded sysem; c conrol desgn; d conraced closed loop sysem. akule 8 However he compley of epanson and conracon s que hgh. he epanson and conracon operaon mus be performed wh non-square ransformaon marces and he conroller desgn mus be performed carefully. Laer n 5 Zečevć and Šljak 5 proposed a convenen LMI approach o desgn he overlappng conrol on he orgnal sysem drecly. In hs approach a symmerc posve defne s.p.d. Lyapunov mar should be pre-srucured n block-dagonal form accordng o he srucure of he sysem. he conrol gan mar s also pre-deermned accordng o a procedure defned n Zečevć and Šljak 5; Šljak and Zečevć 5:

34 K K K K K 3 K 33-6 he dea proposed by Zečevć and Šljak 5 s dfferen from he dea n İfar 993. he sysem model n Zečevć and Šljak 5 consders ha he overlappng pars do no have her own conrollers. In hs way he approach can consran he overlappng pars o be sable. In İfar 993 he overlappng pars have her own conrollers. hus he gan mar for he İfar s mehod s n he form: K K K 3 K 4 K K K 3 K K 4-7 Que a number of sudes have been done o apply he overlappng decomposon mehods o realsc sysems for eample power sysems by Šljak 99; Chen and Sankovć 5 7 a plaoon of vehcles by Sankovć Sanojevc and Šljak formaon of aeral vehcles by Spanovć e al 4 ec. Also he more comple mul-overlappng decomposon srucure s dscussed by Chen and Sankovć 5. o combne wh oher robus mehods akule e al 5 consder he approach o mnmze he neracons n he overlappng srucure. A quadrac opmzaon approach has been represened by akule and Rossell 8. Akar and Özgüner proposed a sldng mode mehod based on he overlappng srucure. Huang and aon b used negral sldng mode combned wh İfar s srucure o reduce he nfluence from neracons. here are sll a lo of problems lef n hs overlappng srucure area such as robus faul oleran conrol uncerany n he neracons or oupu based overlappng decomposon usng LMI approach ec. 3

35 .4 Decenralzed esmaon for LSSs Very few publcaons focused only on he sae esmaon problem usng decenralzed esmaon mehods unl he presence of modern robus esmaon mehods Šljak and Vukcevc 976; Edwards and Menon 8. he dffculy of sae reconsrucon for LSSs s obvous when can be seen ha he unavodable neracons preven he esmaon error from reachng zero value. he decenralzed esmaon sraeges for LSSs manly focus on: wo aspecs: Decenralzed observer based conrol Faul deecon/esmaon In he frs aspec he dsurbance from neracons can be handled by boh conrol and observer. he objecve s o acheve some conrol goal. For he faul deecon/esmaon neracons ac as unceranes whch can be oleraed or compensaed n he faul esmaon..4. Decenralzed observer based conrol A srong research effor has been made n he leraure owards he developmen of decenralzed conrol schemes based on oupu feedback va consrucon of decenralzed observers. Assume ha he -h subsysem s n he form: -8 where are he saes and npus of he -h subsysem respecvely. s he number of he subsysems. he erm represens he neracons from oher subsysems. hree broad mehods are hen used o desgn observer-based decenralzed oupu feedback conrollers for LSSs as follows: Desgn a local observer and conroller for each subsysem ndependenly and check he sably of he overall closed-loop sysem. In hs mehod he nerconnecon erms acng n each subsysem are regarded as an unknown npu fauls. A ypcal eample of hs s gven by Kals 9 an SMO o make he observer nsensve o he neracons. Wh hs mehod he neracons have o sasfy ceran condons. For subsysem -8 assumng ha he neracons he condons are 4

36 and any nvaran zeros of he rple are n he open lefhand comple plane. In hs case he observer can esmae he saes precsely whou any oher unceranes. he conrol srucure of he sysem s llusraed by n Fgure -5 wh wo nerconneced subsysems. Subsysem Ineracons Subsysem y y Observer Observer ˆ ˆ u Conroller Conroller u Fgure -5. Observer based decenralzed conrol whou neracons beween observers Desgn local observer and conroller for each subsysem ndependenly and check he sably of he overall closed-loop sysem. Compared wh n hs mehod he neracon nformaon s assumed known. For eample f he neracon erm n subsysem -8 s lnear and n he form of hen he gan marces are known. hus he observer for he -h subsysem s n he form of: Or f he neracon s nonlnear should sasfy he Lpschz condon and he observer s n he form of: A number of publcaons use hs mehod snce consders he neracons n he desgn of he observer Šljak and Vukcevc 976; Looze e al 978; Sandareshan and Huang 984; Dae and Chow 989; Hu 994; Uang and Chen 5

37 ; Zhang and olycarpou and arasn. he srucure of hs mehod s shown n Fgure -6. Subsysem Ineracons Subsysem y ˆ Observer Observer y ˆ ˆ ˆ u Conroller Conroller u Fgure -6. Observer based decenralzed conrol wh neracons beween observers 3 Desgn he observer and conroller by posng he oupu feedback sablzaon problem as an opmzaon problem based on he overall sysem. he framework of he opmzaon approach usng LMIs can be found n Šljak and Spanovć. he dea of he decenralzed conroller and observer desgn problems were formulaed n he LMI framework for LSSs wh non-lnear nerconnecons sasfyng quadrac consrans descrbed n Secon.3.. he esence of a sablzng conroller and observer depends on he feasbly of solvng a seres of LMIs. he opmzaon problem wll resul n he selecon of conroller and observer gans ha wll no only sablze he overall LSSs bu also mamze he nerconnecon bounds Zhu and aglla 7. he conrol srucure s n he same form of Fgure -5. he neracons are reaed as unceranes and can be oleraed by proper choce of conrol and observer gan marces. he recen research can be found n Šljak and and Spanovć ; aglla and Zhu 5; Zhu and aglla 7; Swarnakar and Marquez 8; Kals Lan and Żak 9 ; Shafa Ghadam and Saf. hs mehod s also he basc dea behnd he oupu feedback mehods proposed n Chaper Faul esmaon n LSS he hsory of faul deecon and solaon FDI can be raced back o he 97s. From boh heorecal and applcaon-based perspecves FDI has araced consderable aenon Clark 978; Hmmelblau 978; Chow and Wllsky 984; Isermann 984; 6

38 Gerler 988; aon Frank and Clark 989; Chen and aon 999; aon Frank and Clark ; Isermann 6; Dng 8. he man dea of he model-based approach o FDI s o generae sgnals ha reflec nconssences beween nomnal and fauly sysem operaon. Such sgnals ermed resduals are usually generaed usng analycal approaches such as observers Chen and aon 999; aon Frank and Clark parameer esmaon Isermann 994 or pary equaons Gerler 998 based on analycal or funconal redundancy. Among all of he FDI approaches observer-based mehods are he mos popular mehod o be researched and appled Edwards Spurgeon and aon b for eample see Chen and aon 999; Shelds and Du 3; Xu and Zhang 4. All he leraure oulned above focus on cenralzed sysems. When consderng he FDI of LSSs raher less research has been done Yan and Edwards 8. he reason s he neracons ac as an era dsurbance/uncerany for he faul esmaon. he nfluence from neracons leads o naccurae esmaon of fauls whle mos research on FDI mehod can only deal wh sngle ypes of fauls whou unceranes or wh very small unceranes. Early work on FDI for LSSs can be raced back o Hassan Sulan and Aa 99 who used a Kalman fler based on overlappng decomposon o deec and solae he faul n dscree me. In Chung and Speyer 998 a game heorec faul deecon fler combnng wh decenralzed fler approach s proposed. Shankar Darbha and Daa dscusses he decenralzed observer based faul deecon for nerconneced LI subsysems. More recenly Farrar 9 deecs fauls wh a decenralzed adapve esmaor based on overlappng srucure. Zhang and olycarpou and arasn represen he decenralzed faul deecon under canoncal form and usng an adapve hreshold for robus faul deecon. Wh he developmen of sldng mode heory he dea of ulzng sldng mode heory n an observer has proved o be very effecve n he feld of FDI. However compared wh he resdual generaon approach he sldng mode observer SMO forces he esmaon of he oupus o be dencal o he oupus of he plan wh he so called swchng funcon. hus he convenonal resdual generaed by he oupu esmaon error would be zero. he acuaor and sensor faul esmaon problems have been proposed n Edwards and Spurgeon 998 and eended n an and Edwards 3. Laer n Yan and Edwards 8 he faul esmaon sldng mode 7

39 approach for LSS s proposed. In hs mporan research he Edwards & Spurgeon SMO s used o esmae he acuaor faul. Informaon abou he suaons under whch he nfluence from neracons and unceranes are mnmzed n he faul esmaon s also ncluded. However Yan and Edwards assumed ha he srucure of he neracons s known he second suaon n secon.4. and he unceranes sasfy ceran condons. here seems o be no oher leraure on he applcaon of decenralzed SMO heory for nonlnear-nerconneced sysems o oban a decenralzed and precse faul reconsrucon. hs movaed he mehod proposed n Chaper 6..5 Concluson hs Chaper gves a revew of decenralzed conrol and esmaon mehods for LSSs. Accordng o he conrol srucure he conrol sraeges can be classfed as mul-level and sngle-level srucures. Mul-level conrol consrucs a pyramd srucure usng local conrol level o deal wh he ndependen nformaon of subsysems whls usng hgher level o gve he decson rules o he lower lever. he wo-level conrol sraegy as a ypcal mul-level s revewed n hs Chaper. hs srucure gves a clear vson of he mul-level srucure. However he hgher level of mul-level conrol sll requres cenralzaon.e. a requremen for a coordnaor. hs condon ncreases he compley of hs mehod. he rapd ncrease n compuer echnology n he las decade means ha he sngle-level conrol srucure s much smpler and more wdely used han mul-level LSS srucures. As hs Chaper descrbes he neracons are aken care of by robus local conrollers. here are numerous decenralzed mehods for LSSs. Mos of hem can be classfed by her decomposon mehods. hs Chaper revews wo man decomposon mehods when usng sngle-level conrol srucure: dsjon decomposon and overlappng decomposon. o concenrae on he man opc of hs hess he quadrac consran assumpon for neracons s nroduced for dsjon sae space srucures. Also a bref dea abou decenralzed sldng mode heory s gven. he reamen of hese opcs s useful n Chapers 4 and 5. Alhough hs hess manly focuses on he dsjon decomposon some work on overlappng decomposon Huang and aon b has also been done. For he 8

40 fuure work here are sll a lo of open problems o be addressed based on he overlappng srucure. For he esmaon appled o LSSs he decenralzed observers are ofen n he role of observer based conrol or faul esmaon. In hs Chaper hree observer based conrol mehods are revewed whch gve a good bass for Chapers 5 and 6. Decenralzed faul esmaon mehods based on he use of decenralzed observers are also dscussed. 9

41 Chaper 3 Annealng Furnace Modellng: A Dfferenal Quadraure Approach 3. Inroducon Furnaces have been wdely used n ndusral applcaons as heang devces o rase he emperaure of parcular sysem processes such as for power sysems or n he annealng process used for preparaon of a meal for rollng n he meal ndusry. hs Chaper proposes he developmen of a model of a seel annealng furnace sysem as an eample of a LSS applcaon. he work s based on a well-known model formulaon ncluded n he McGunness and aylor s repor of he MISG projec. New Zealand Seel use a unque process o conver New Zealand ron-sand no seel shee producs a s Glenbrook mll near Auckland. radonal galvansed seel and he new produc Zncalume are produced n a range of dmensons grades and coang weghs. McGunness and aylor 4. he seel srp s annealed before beng coaed. he furnace heas he seel srp o a predeermned emperaure n well defned me producng desrable changes n he crysallne srucure of he seel srp o alor s srengh and ducly. he Cross-secon of he furnace s shown n Fgure 3-. he seel srps pass hrough he furnace o ge heaed up. he furnace lengh s denoed by m. he velocy of he srp m/s s consan. hus f he seel acheves he desred emperaure a he end of he heang zone he me of heang up can be calculaed by.he emperaure of he furnace s conrolled by several heang elemens fed n he wall. In hs Chaper he coolng par of he furnace s no consdered alhough ess n he real sysem McGunness and aylor 4. I s mporan ha he seel es he furnace wh he correc emperaure because he coang process s appled a he e pon. 3

42 un: mm Fgure 3-. Cross-secon of he furnace McGunness and aylor 4 Assume ha he lne speed and he hckness of he seel are consans mgh be vared as a form of paramerc uncerany and he wdh of he seel srp s a consan so ha he heang process s connuous. I s also assumed ha he whole srp acheves he requred emperaure as eher a sngle emperaure or as a defned varaon of emperaures. he emperaure measuremens of he furnace are made usng hermocouples and nonconacng pyromeers. he hermocouples used o measure he furnace emperaure are fed o he furnace wall and he non-conac pyromeers are used o measure he emperaure a hree dsnc srp locaons If here s no varaon n srp dmensons and annealng sengs hen he sysem wll run n a seady sae McGunness and aylor 4. he furnace emperaure wll reman seady a he desred hermocouple sengs. However parameer varaons can be consdered as unceranes. he approprae conrol mehod s hen chosen o aenuae or compensae he nfluence of he uncerany. he Chaper has he followng srucure: Secon 3. descrbes he man equaons of he furnace model n wo pars he srp emperaure and furnace wall emperaure models 3

43 respecvely. In order o fully develop he model several furher assumpons mus be made. Secon 3.3 nroduces wo alernave mehods for dscresng he DEs no ODE forma. hese mehods are known as dfferenal quadraure mehods followng he work nroduced by ellman ellman Kashef and Cas 97: One quadraure approach uses cubc splne nerpolaon n whch he paral dervaves are represened over one daa nerval usng a separae cubc splne. In conras he dfferenal quadraure usng orhogonal nerpolaon polynomals are chosen as Lagrangan Inerpolaon polynomals spannng he whole range of he quadraure. hese wo approaches are appled separaely o he srp and furnace wall models respecvely. Secon 3.4 hen descrbes an organzaon and smplfcaon of he srucure of he equaons leadng o a non-lnear sae space sysem where he non-lneary comes manly from consderng Sefan s radaon law appled o he furnace sysem. he wo boundary condons of he orgnal paral dfferenal sysem are handled va a smple rck n he use of he Lagrange nerpolaon. he modellng procedure produces a nonlnear furnace model for smulaon whch s shown n Chaper Furnace Sysem Mahemacal Modellng As oulne n Secon 3. he furnace model conans wo hea ransfer componens: he srp model and he furnace wall model respecvely ogeher makng up he srpfurnace model. hs Chaper descrbes he developmen of an approprae non-lnear sae space model o llusrae he man dynamc properes of he srp-furnace sysem. Several assumpons are made for he denfcaon of hs sysem McGunness and aylor 4: Assumpon : he emperaure of he srp over each cross-secon s consdered consan. Wh hs assumpon he emperaure varaon n he cross-secon area can be gnored. Assumpon : I s assumed ha he meal s conveyed only n he longudnal drecon. Consder ha he furnace s perfecly sragh wh recangular cross secon as shown n he 3-dmensonal model of Fgure 3- Consder he longudnal drecon va he as he laudnal drecon wh as and he vercal drecon va as. hen assumng 3

44 all srp moon s only n he longudnal drecon he emperaure varaons orhogonal o he conveyor drecon wll be neglgble.e. and. Fgure 3-. Furnace dmensons Assumpon 3: he nner surface emperaure of he furnace walls depends on me and dsance along he furnace measured from he enry pon of he srp. Assumpon 4: he emperaure of a heang elemen s he same as he emperaure of he nner surface of he wall adjacen o he elemen. Assumpons 3 and 4 conrbung o he smplfcaon of he hermal equaons of he furnace model are descrbed n he Secon 3.4. Assumpon 5: emperaure changes whn he furnace are so gradual ha he radave or convecve hea ransfer componens along he lengh of he furnace can be gnored. hs assumpon leads o a reducon n compley snce here s no no ne radaon hea ransfer as he srp ravels from one zone no anoher.e. here s no neracons beween he furnace wall models. Assumpon 6: hs model only concerns he heang zone of he furnace. Normally he furnace conans coolng ubes afer he heang zone. However hs work s only concerned wh he heang funcon of he furnace. eween he heang and coolng zones Assumpon 5 s no vald. 33

45 3.. Srp hermal Model Frsly se up a regon of space: S y z : l y w z h} where s he lengh of he furnace of he srp all dmensons n m. are ypcal values of he hckness and wdh For he srp he hea conducon equaon of he conveyed maeral s gven by he classcal hea conducon equaon modfed by a convecve erm and a erm whch accouns for he hea source q scs v s y z wh 3- s he srp emperaure are he srp densy hermal conducvy and specfc hea capacy respecvely. s he radave hea ransfer beween he furnace wall and he srp. s he conveyor speed n. Assumpon mples ha he emperaure varaon n he cross secon of he srp can be gnored. hs means ha he hea conducon n he laeral and vercal drecons can be removed from he srp model. And followng Assumpon here s no conveyor movemen n he laeral and vercal drecons. Accordng o hs Assumpon he erm descrbes he emperaure varaon n he longudnal drecon nfluenced by he conveyor speed. In hs case he hea equaon can be furher smplfed o reduce he compuaon burden as follows: s C s s v q wh C s s 3- he hermal properes of seel change wh emperaure are shown n able 3-. aken from McGunness and aylor 4: 34

46 able 3-. roperes of seel n dfferen emperaure Accordng o hs able wo suable polynomal nerpolaons can be acheved for as: 5e s c s e 6 5.5e ecause he annealng process has a large range of emperaure varaon he properes of seel canno be gnored n he sysem. Hence Eqs. 3-3 and 3-4 should be consdered durng he lnearzaon n Chaper 7. y consderng he hermal combnaon of he srp and furnace wall McGunness and aylor gves he equaon for he radaon per un lengh of he hea source of he furnace walls o he srp as: w s q s w w p w 4 w s he srp wdh s he appromaely he sum of he vercal lengh and wdh of he nsde of he furnace. are he hermal emssvy of he srp and furnace wall respecvely. s he Sefan-olzmann consan whch s. y assumng he Eq. 3-5 s fnally smplfed o: 4 4 q w 3-6 s 3.. Furnace wall hermal model Now consder he energy balance for a sngle un of he heang elemens and nner wall surface o consruc he hermal model for he furnace walls. Assume ha he un lengh s he specfc hea capacy of he un s and he mass of he un s. y w 35

47 consderng he combned nner wall surface and heang elemens as a sngle lumped sohermal objec he approprae energy balance can be epressed as McGunness and aylor 4: mc u d d w p q 3-7 Where s he hea flu no he walls s he energy suppled o he un s he wdh and hegh of he nner surface of he furnace so ha denoes he oal area of he nner surface. he erms n Eq. 3-7 are llusraed n Fgure 3-3: ower suppled Hea flu o he wall Energy o he srp by radaon Fgure 3-3. Energy balance n a sngle un of furnace Assumng ha he heang elemens have lle hermal nera no energy sorage Eq. 3-7 can be furher smplfed o: mc u d d w p q q 3-8 p A furher smplfcaon can be made by reang each heang elemen and furnace wall as a separae one dmensonal brck hermal model consderng only hea conducon.e. no convecon or radaon. McGunness and aylor 4: wcw w r d 3-9 r hs s combned wh he followng boundary condons: w 3-36

48 w 3- r r w H d 3- r rd Where are he densy specfc hea capacy and hermal conducvy of he walls respecvely. s he emperaure of he nernal wall brck of he furnace a a dsance and a deph no he wall. s he emperaure of he nner surface of he furnace. m s he hckness of he wall. s he eernal amben emperaure. s a convecon coeffcen. he srucure of furnace wall s shown n Fgure 3-4. w Fgure 3-4. Cross secon of furnace and wall emperaure model Moreover he hermal properes of he wall as gven n McGunness and aylor 4 are shown n able 3- able 3-. roperes of brck of he furnace wall I can be seen ha he varaon of he hermal properes wh emperaure are small whch means ha can be assumed ha hese brck properes are consan. 37

49 o conclude he nonlnear model of he whole sysem can be wren as: s C s s v q wh C s s q w 3-6 s w q 3-8 p wcw w r d 3-9 r w 3- w 3- r r w H d 3- r rd 3.3 wo mehods of dfferenal quadraure he mehod of dfferenal quadraure developed by Rchard ellman n he 97s ellman Kashef and Cas 97 s a numercal soluon echnque for dfferenal sysems by means of a polynomal-collocaon approach a a fne number of pons. wo dfferenal quadraure approaches have been adoped n hs work. Frsly he connuous dervave funcon paral dervave s appromaed usng an orhogonal polynomal over a fne number of collocaon pons he roos of he polynomal. hs mehod has a dsadvanage ha he node spacng n he lumped parameer sysem spacng n he drecon s non-unform. he approach s compared wh he use of ellman s second quadraure whch uses a se of cubc splne polynomals wh each splne funcon effecve over a collocaon nerval. I can be seen ha boh srp and furnace wall dynamcal models conan DEs whch make hese models hard o be conrolled. hs Secon nroduces wo dfferenal quadraure mehods: Lagrangan Inerpolaon olynomal LI mehod Hsu 9 38

50 and Cubc Splne CS Inerpolaon ellman 97 Shampne and Allen 973. Wh hese wo mehods he appromae models whch are n he form of ODE can be obaned. However once he model s cas n he sae space form becomes possble o develop suable robus conrol and esmaon desgns o acheve he requred furnace performance objecves. he concep of dfferenal quadraure sars wh he noon of applyng m-dmensonal dfferenal operaor o a connuous and dfferenable funcon where. he m- dmensonal vecor dfferenaon can be wren n he followng form: m m f f D f n m f f f n 3-3 where s he funconal value a grd pon and s a mar defnng he operaons requred o acheve dfferenal quadraure. In he followng subsecons LI and CS are nroduced respecvely. he furher applcaon of boh hese mehods n furnace sysem s proposed n Secon Lagrangan Inerpolaon olynomal hs subsecon descrbes a Lagrangan Inerpolaon olynomal approach for he dscrezaon of DE sysems no ODE sysem form. hs nerpolaon s vald over a range of he space varable for whch he range s dvded no collocaon pons ha mach he roos of he orhogonal polynomal used. For eample he Chebyshev- Gauss-Lobao polynomal roos are dsrbued accordng o Hsu 9. cos for N Nw w 3-4 where s he order of he polynomal for eample for he roos are dsrbued as shown n Fgure 3-5: 39

51 Fgure 3-5. Chebyshev-Gauss-Lobao dsrbuon he concep of he Lagrangan Inerpolaon s gven as follows: where N w M f f 3-5 M l M N w j j Nw M l j j j for... N w In hs case he mar D n 3-3 can be epressed as: D j M l for... N w M j l j j 3-6 also: N D D for... N j j j w 3-7 Once he grd pons are seleced he coeffcens of he weghed mar can be acqured usng equaons 3-6 and 3-7. Accordng o 3-3 quadraure marces represenng second-order dervave operaons on a funcon can also be acqured usng mar mulplcaon: 4

52 D j N k D k D kj Cubc splne dfferenal quadraure I s somemes he case ha equally spaced collocaon pons are desrable or acually requred or freedom o choose he spacng s necessary. CS provdes a convenen nerpolaon approach for whch he spacng of he collocaon pns can be chosen arbrarly whn he doman of he space varable. he Lagrangan approach o nerpolae he connuous and dfferenable funcon requres a sngle polynomal for whch he polynomal order s ncreased accordng o he requred number of collocaon pons. In he CS nerpolaon ndvdual CS funcons are appled o each nerval of he nerpolaon wh specal properes as follows. Suppose he funconal values are known a each grd pon of neres over he nerval wh. he CS funcon mus sasfy he followng condons:. s connuous along wh s frs and second dervaves on.. s a cubc polynomal on each nerval. v.. For convenence use he same nerval n he srp model hence le: h h consan S S [ ] s S Noe he dfference beween and : s he overall nerpolaory splne funcon a he collocaon pon whls s he sub-funcon n he nerval. 4

53 4 From condon snce s a cubc polynomal s a lnear polynomal ha can be epressed n he form:... n h s h s S 3-9 o oban he epresson of negrang 3-9 wce: c c h s h s S 3- he negraon consans and are deermned accordng o he propery :. I follows ha: h c h s f S 6 h c h s f S 6 he consans and can hus be represened by wo funcons: 6 6 h s h f c h s h f c hen Eq. 3- can be rewren as: h s h f h s h f h s h s S... n for 3- Dfferenang 3- wh respec o me follows ha: 6 s s h h f f h s h s S 3- Usng condon and s frs dervave are connuous funcons.e.: S S 3-3 Eqs. 3- and 3-3 can be combned and hus a new condon s gven by:

54 6 f f f s 4s s h h... n 3-4 he se of Eq. 3-4 s a sysem of lnear equaons wh unknown varables. wo addonal condons should be added o solve hs equaon se. Normally hese wo condons are he boundary condons of he DEs. For some DE problems e.g. furnace emperaure problem he second dervaves of he frs and las pons mgh be consdered consan..e. 3-5 he mehod used o solve 3-4 and 3-5 s aken from Shampne and Allen 973. hs mahemacal procedure s appled here o he dscrezaon of he DE sysem no a se of non-lnear ODEs. I s mporan o noe ha he lnearzaon operaon s appled o he non-lnear ODE sysem and hs s descrbed n Chaper 7. y furher defnng: 6 d f f f h 3-6 From 3-4 follows ha: s 4s s d... n 3-7 Recallng ha he splne s always of order 3 hen follows whou loss of generaly ha he second dervave has he form: s s 3-8 On subsung 3-8 no 3-7 and afer some manpulaon follows ha: s s 4 d d 4 4 s 3-9 Eq. 3-9 has he same form as Eq. 3-8 hus he followng defnons can be derved: 43

55 d 3-3 Snce and. Noe ha he scalar mulpler only depends on and hence as can be calculaed by recursve eraon. Furhermore o compue he he followng procedure s consdered: Rewrng 3-6 as: F f f h h h f f f h d n n where 6 6 n h h h f n f F. y furher eraon s calculaed as: ]F F [ F F d 4 4 F F d Hence s gven by: F where F F d j j j k k j

56 j j j k k j Snce by combnng all he and usng a smlar eraon o he one gven n 3-9 can be rewren as: F D s s where has a smlar form o he and can be calculaed smply by eraon. In hs case he second dervave can be wren as: D F F D D s s s s n s s n 3-3 Subsung 3-3 no 3- he frs dervaves of he splne funcons have he form: F D S S f n Sysem model denfcaon Recallng he furnace wall model and he srp model 3- wh he dfferenal quadraure mehods descrbed n Secon 3.3 hese DEs could be ransferred no ODEs. However he mehods should be chosen carefully for dfferen models. Furnace wall model For he furnace wall model only wo spacal grds are of neres a he nner wall surface emperaure and b he ouer surface emperaure boh of whch are assumed measured. he grd pons nsde he furnace wall do no need o be equally spaced n he grd and hence LI s suable for generang he requred dfferenal quadraure. Anoher advanage of applyng hs mehod n he furnace wall model s ha he boundary condons could be easly handled. Usng dfferenal quadraure he heang equaons are ransferred o:

57 d wcw wdw d 3-34 w 3-35 w 3-36 r r w H N r w 3-37 rd where s he emperaure vecor for grds n he furnace wall of he -h furnace model. Snce here s no need o specfy a dfferen number of grds for he furnace model n dfferen zones s a consan and represens he number of grds for he furnace model of all subsysems. Moreover he nerval beween he nodes n he furnace wall sasfes Chebyshev-Gauss-Lobao dsrbuon: d cos for N Nw w 3-38 where d s he hckness of he furnace wall. I should be noed ha and 3-37 are he boundary condons for he furnace model. hey are handled laer n hs Secon. Srp model he reason for choosng CS as he nerpolaon mehod for hs srp model s ha n he srp he same nerval dsance beween wo nodes s adoped. In he CS mehod splnes beween wo nearby pons are desgned separaely. he dsrbuon of he collocaon pons does no much affec he accuracy of he mehods. hs s dfferen from LI mehod nroduced n Secon Moreover wh he CS mehod he larger he nerval beween wo grds he smaller he neracons hey have. hs s much closer o he real suaon han LI mehod whch mgh have very large neracons beween wo long-dsance pons. Accordng o Secon 3.3. he DE 3- of he srp model can be ransferred o: 46

58 d d s s Ds vdf s qs sc 3-39 s whscs where s he emperaure of he srp s he vecor of heang source. I should be noed ha 3-39 s he model for he overall sysem. he nerval beween he nodes s consan. Ineracons beween furnace wall model and srp Model he nerconnecon beween he furnace wall model and srp model s radave hea ransfer from he furnace wall o he srp. I s represened by of Eq hey can be epressed by: of Eq and q 3-8 p 4 4 q w 3-4 s w s where s he npu heang power for he sysem. s he nner furnace wall emperaure whch. I can be seen ha he srp model and he furnace wall model have dfferen coordnaes. he dsrbuon of nodes n he srp s n he drecon and he dsrbuon of nodes n he wall model s n he drecon. hus n he drecon he srp model s seperaed no zones wh nodes and each zone has s own heaer furnace heang model. he neracons beween he subsysems heang zones are hea conducve and he movemen of he srp s gven by he decomposon of he mar. o brefly llusrae he procedure of modellng usng a sngle zone he sae vecor for each subsysem s chosen as: Snce LI s used n he furnace wall model Eqs and 3-37 could be rewren as: 47 s

59 48 w N k k k w r w w D r 3-4 H D r w w w N w N k k k w N d r w 3-4 And he sae equaon 3-34 for he furnace model can be wren as: D D c w w w w w 3-43 he dea of dealng wh he boundary condons s o subsue he rgh sde of 3-4 and 3-4 no hus frs defnng wo marces: Q Q and rewrng 3-43 as: w w w w w w w HQ H Q D D D c y furher defnng: HQ D Q D D D D c A w w w w w w w w w w w w w w D c H D c E w w w w gves he furnace wall model n he sae space: w w w E A 3-44

60 On he oher hand he srp model can be smplfed as: A q 3-45 s s s s s where and. Snce he npu s he power n 3-8 Combnng Eqs and 3-45 gves he sysem equaon s n he form of: s As s s u q 3-46 A w w w p Ew p Now ha he paral dfferenal equaons are ransferred o ordnary dfferenal equaons and combned wh he boundary condons n he model equaons he problem remanng s he nonlnear par. he nonlnear par n Eq s no easy o be handled. As can be seen n he srp model every srp grd pon wh s emperaure represened by receves he heang energy from he furnace wall. In hs case he erm of Eq should be a vecor. However n he furnace wall model s a scalar whch can only be used n he boundary condon 3-. Ideally s a vecor where are he hea sources receved n he srp model and s he scalar hdden n he boundary condon 3- of he furnace wall model. As a maer of fac emperaures a dfferen pons of he srp are dfferen. hus s necessary o make a vecor wh dfferen elemens. here are 3 dfferen ways o deal wh he scale n he wall model. Usng q w w s 4 w 4 s m Where Usng s he emperaure of he mddle pon of he srp grds. q w w s 4 w 4 s aver 49

61 5 Where s he average emperaure of he srp grds.. 3 Usng n s w s w n w q 4 4 o ge he average energy o he wall. Inuvely he s mehod seems he eases approach o use. However can be seen ha wh he 3 rd mehod he only nformaon needed s he saes emperaures of grd pons of he srp model nsead of grd selecon or more compuaon. hus he heang model can be wren as: n s s n s n n I w q 3-47 Subsue 3-47 no he sysem equaons 3-46 he new sysem equaon s gven by: E u A n s s where w s A A A w p n n I p I w n w n s s E w E Assume ha he emperaure of he npu pon of he furnace s fed. hus f he nal condons for he srp are known. he srp model can be furher smplfed and he order of can be reduced o. Snce he frs sae of he srp model s fed.e. he dervave of hs emperaure s zero he sysem mar of he srp model can be modfed by:

62 A s A: n A s * hus he new sysem mar becomes: * As A Aw Meanwhle he nfluence from he npu pon should no be gnored. he consan E of he sysem should be modfed as: where Es A : n s. E E E s w Noe ha f he nonlneares of he properes of seel are negleced he sysem conans hree lnear pars and one nonlnear par 4 4 s. 4 4 s n he above descrpon llusraes he modellng procedure of he furnace sysem wh only one heaer. o model a large scale furnace sysem whch has several heaers assumng ha he large scale furnace sysem has N zones he saes of he overall sysem should be n he form of: s 3-49 N s N where s he number of pons n he -h srp model. s he srp emperaures of he -h srp model. s he furnace wall emperaures of he -h furnace heang model. I should be noed ha he frs zone conans he npu pon hus he dmenson of s 5

63 . Normally choose. he sysem mar of he overall sysem s n he form of 3-48 bu wh dfferen srucure: As As A As N A w A s A s N N A sn s N N A A s NN Aw 3-5 where. s he submarces of he overall srp model mar n hus he neracons beween dfferen zones are he hea conducon and he emperaure movemen n he srp model Oher marces should be modfed n order o sasfy he sae srucure Model Valdaon Accordng o he descrpon n Secon 3.4 he sae space model of he large scale furnace model has been esablshed. o verfy he modellng procedure hs secon frs descrbes he smulaon of he furnace model wh MALA usng he parameers gven n McGunness and aylor 4. Followng hs he model properes can be valdaed by usng he zero npu response.e. by nspecng he furnace sysem s naural open-loop hermal behavour whou emperaure regulaon. hen he seady sae soluon of he DE sysem can be calculaed by gvng some of he reference emperaures of he srp model. A ID conrol for he nonlnear sysem model s hen desgned o provde good rackng performance of he same references. y comparng he resuls of he seady sae soluon of he DEs and he nonlnear model wh conrol he modellng accuracy can be dscussed Sysem smulaon o smulae he furnace sysem wh he number of zones chosen o be he number of grd pons for each srp model s accordng o Secon 3.4 and he number of grd pons for he furnace wall model s. hese parameers are approprae for he furnace model sysem derved n Chaper 3. 5

64 53 Accordng o Chaper 3 he aggregae model sysem srucure has he followng form: E f u A 3-5 where: w s s N s s w s s s s w s A A A A A A A A A A A A A w w w p p p 3 f f f f n s s n w n s s n n I p I w q f w s N w s E E E E E In he frs srp model by consderng he npu pon emperaure as consan he dmenson of he frs srp model s reduced o 5. hus he dmenson of he overall sysem s 9. he sysem parameers are akng from McGunness and aylor 4 and hey are shown n able 3-3. able 3-3. Furnace Sysem parameers Srp Densy 7854 [Kg m -3 ] hckness.5 [mm] Wdh.94 [m] Velocy [m s - ] Emssvy. Inerval of he srp model.5 [m] Furnace Wdh+hegh 3.4 [m] Densy [Kg m -3 ] Hea capacy 9 [J Kg K - ] hermal conducvy.8 [W m K - ] hckness of he wall.4 [m] Hea convecon coeffcen 4.7 [W m K - ]

65 3.5. Zero npu response he furnace sysem s assumed o sable. Whou any npu when he emperaure of he nner wall s hgher han he srp he emperaures of he srp grds are. ncreased because he nner wall emperaure s hgher hen. decreased slowly because here s no power supply and he emperaure of he wall s decreasng. Meanwhle he emperaure of he furnace wall should keep decreasng. earng hese n mnd by seng he nal condons of he sysem: he srp emperaure response wh zero npu can be gven: Fgure 3-6. Srp emperaure response whou power npu Fgure 3-6 shows he sysem response of he srp emperaure wh zero npu. As dscussed above he srp emperaure frs ncreases as he nner wall heas up. As he srp moves furher he furher he grd s away from he npu he more hea ges from he wall. From Fgure 3-7 he emperaure of he furnace wall s decreasng durng he operang snce here s no power supply. hus he nner wall s cooled by he radaon from he srp and he ouer wall s heaed by he hea conducon from he nner wall. u one can epec ha afer a long me he nner and ouer wall emperaures as well as he srp emperaure wll converge o he amben emperaure whch s n hs model. 54

66 Fgure 3-7. Furnace wall emperaure response whou power npu Snce only he nner and ouer wall emperaures are concerned he order of he furnace wall emperaure model can be reduced by a model reducon funcon usng MALA. he fnal furnace model for each zone can be represened by a smple second order sysem Seady sae soluon Consder all he emperaures are nvaran wh me.e. follows ha he seady sae soluon from he sysem Eqs are McGunness and aylor 4: s C s s v q wh C s s ds d q 3-5 vwh C s s wcw w r d r dr 3-53 w w

67 56 d H r d r w d H r d r w 3-55 he reason ha he erm s negleced n 3-5 s ha he nfluence from hs second order paral dfferenal erm s so small n seady saes. Hence he seady sae furnace wall emperaure can be calculaed as follows: d H r d d dr d w w w hen he wall emperaure s gven by: r H d w w w 3-56 where H d w w s he graden of he wall emperaure. Now consder 3- whch s: r w r 3- gves: H d w w w Consderng 3-6 and 3-8: 4 4 w q w s 3-6 p q 3-8 he seady-sae power npu of he -h zone s gven as: 4 4 s w s w w w w H d p 3-57 Compared wh wha s assumed n Secon 3.4 by subsung he emperaure of he npu pon of he -h zone no 3-6 an appromaon soluon of 3-5 s obaned:

68 s 4 4 s w s vh C s s s 3-58 he wall emperaure can be wren as: w 4 s s vh scs n h s 4 s ID furnace conrol ID conrol s he mos common conrol srucure used for process sysems and s easy o mplemen n he furnace sysem. I s known ha he negral conroller can remove he sac oupu error. hus can be appled as a good nner-loop conroller or a conroller for lnearzaon. he model oupus are he end pons of he zones. hus consderng he furnace model 3-5 he conroller power npu can be desgned n he form of: u K ref s end K I ref s end d K D ref s end ' Seng he desred emperaures of each zones as. able 3-4 shows he choces of he gans of he conroller. able 3-4. ID conroller gans

69 Fgure 3-8. Srp emperaure wh ID conrol Fgure 3-8 shows he srp emperaures of he furnace model. Assume ha he npu emperaure s 3 and he zone e srp e emperaures for heang zones are. he grd nerval of he srp model s chosen as. Usng 3-58 and 3-59 he emperaures of he nner wall surface and he srps n he seady sae can be calculaed. hen by drvng he nonlnear model o he requred reference emperaures he emperaure of each sae can be obaned. Comparng he emperaures calculaed from he DEs 3-58 and 3-59 and from smulaon he errors are shown n able 3-5. able 3-5. Modelng error a seady saes wh nerval Heang Zone Heang Zone emperaure [ ] emperaure [ ] Heang Zone 3 emperaure [ ] oson DE Model Error DE Model Error DE Model Error Srp Srp Srp Srp Srp

70 Srp Inner wall I can be seen ha he modellng error s accepable. ecause of he srong numercal accuracy of he ellman dfferenal quadraure nerpolaon mehods used o derve he appromae ODE sae space model sysem can be deduced ha f one reduces he grd nerval he error s also reduced. he resuls are gven by seng and. able 3-6. Error beween DE and Smulaon under dfferen nerval dsance oson Heang Zone Heang Zone Heang Zone 3 emperaure [ ] emperaure [ ] emperaure [ ] Error Error Error Error Error Error Error Error Error Srp Srp Srp Srp Srp Srp Inner wall

71 Absolue Error beween calculaon and smulaon h=.8m h=m h=m Grds n he srp Fgure 3-9. Absolue value of he errors beween he DE soluon and he sysem smulaon resuls under dfferen grd spacng From able 3-6 and Fgure 3-9 can be seen ha reducng he srp model grd spacng mproves he appromaon accuracy of he DE soluon as epeced. However here are sll resdual errors even wh a small enough choce of he nerval. hese errors are nroduced as a consequence of some resrcons used n he compuaon of he cubc splne coeffcens and hence n he coeffcens of he dfferenal quadraure marces. Furher errors may be due o he appromaon of hea source used n Secon 3.4. hs may be he subjec of furher work on hs eample. 3.6 Concluson In hs Chaper he concep of a seel annealng furnace he model equaons and he denfcaon of hs furnace model are presened. Generally speakng he am of hs furnace s o hea he seel up o a ceran emperaure a he end hs heang zone. Several heang elemens are se n he wall o conrol he emperaure of he furnace wall. And hea up he seel by hea radaon. Several DEs are nroduced o represen he heang procedure. However n order o apply modern conrol n hs furnace DEs model should be appromaed by ODEs frs. In hs Chaper wo nerpolaon mehods for dfferenal 6

72 quadraure are proposed o conver he DE problem o ODE problem. he nfne nodes of DEs dsrbued parameer sysem whch can be appromaed by fne nodes ODEs make hs furnace sysem large scaled. Some smplfcaon and assumpons are furher made o denfy he nonlnear model. In hs case he sae space model for hs large scale furnace model s proposed. In he fnal par of hs Chaper he nonlnear model s smulaed and valdaed qualavely compared usng zero-npu response. hen he seady sae soluons o he paral dfferenal equaons.e. consderng hea balance are compared wh he resuls of he nonlnear ODE sysem usng ID conrol o dscuss he modellng accuracy. hs model s conrolled and furher dscussed n Chaper 7. 6

73 Chaper 4 Sldng Mode Conrol for Large Scale Sysems 4. Inroducon he so called sldng mode conrol SMC emphaszes he mporan role of sldng when desgnng varable srucure conrol VSC. VSC can be consdered as a se of conrol laws drven by a decson rule. he conrol law for he sysem swches from one conrol law o anoher f he sysem sasfes a ceran decson rule. Normally usng he swchng funcon he decson rule consders some properes or behavour of he curren sysem and helped change he conrol law nsanly. he sysem wh hs specfc srucure s so called varable srucure sysem VSS Hung 993. I s easy o fnd ha he benef of usng VSC s o combne useful properes of each of he compose srucures of he sysem and he sysem may conan new properes Edwards and Spurgeon 998. Varable srucure conrol wh a sldng mode was frs descrbed n Russan n he early 93s. I dd no appear ousde of Russa unl he md-97s. In he lae 7s Iks 976 and Ukn 977 nroduced hs mehodology n Englsh. Durng VSC dd no arac much aenon snce people prefer oher smpler lnear conrol desgn echnques and he robusness properes of VSC were no ye fully recognzed. In 98s engneers sared o pay aenon o hs very robus mehod wh he developmen of general VSC desgn mehods Hung 993. y 993 general applcaon areas ncluded: roboc conrol moor conrol fleble srucure conrol and power sysems Hung 993. oday research and developmen connue o apply VSC o a wde varey of engneerng sysems. Durng he 99s and he begnnng of cenury VSC heory for lnear sysems became a raher complee subjec. Some oher ypes of VSC such as negral sldng mode hgh-order sldng mode ec. also receved consderable aenon. Curren applcaons also nclude: hree-as Opcal ckup Chao and Shen 9 ermanen-magne Synchronous Moor Conrol Sysem Feng and Jang 9 Saelle sysem Lee and Km Spacecraf sysem ukdeboon Znober and hen ec. Secon 4.3 conans maeral ha has been presened as An adapve sldng mode approach o decenralzed conrol of unceran sysems. UKACC Inernaonal Conference on Conrol Cardff UK Sep. 6

74 he concep of sldng mode can be raced back o 934 Nkolsk frs used he erm sldng mode moon n Russan Hung 993. Durng he developmen of VSC heory n 9s and s he erm sldng mode conrol was quckly acceped by mos of researchers because of s vsual descrpon o VSC. One of he mos aracve aspecs of sldng mode heory s he dsconnuous propery of he conrol acon whose prmary funcon of each of he feedback channels s o swch beween wo dsncvely dfferen sysem srucures such ha a new ype of sysem moon called sldng moon ess n a manfold. hs specal sysem characersc s clamed o resul n superb sysem performance whch ncludes nsensvy o parameer varaons and complee rejecon of he mached dsurbances Young Ukn and Özgüner 999. hs robus propery makes he sldng mode aracve from a desgn perspecve. Whn wo decades SMC has become more lke a branch of VSC. I has a well-defned desgn procedure conanng wo componens. he frs s he sldng surface desgn n whch some desred desgn specfcaons should be consdered. Some oher robus conrol dea can be negraed no hs desgn framework. he second componen s concerned wh he selecon of a conrol law whch drves he sysem o he sldng surface. Survey and uoral papers have been wren on sldng mode n Ukn 977; Hung 993; DeCarlo Żak and Mahews 988; Edwards and Spurgeon 998; Young Ukn and Özgüner 999; Casaños and Frdman 6; ec. Alhough here are many publcaons dscussng sldng mode prncples he leraures conan much less nformaon o descrbe sldng mode mehods applcable o LSSs. As a consequence of he nsensvy propery afer reachng he sldng surface sldng mode heory s mosly combned wh dsjon and decenralzed conrol. One can noe hs hrough he publcaons n years Feng and Jang 995; Hsu 997; Chou and Cheng 3; Shyu Lu and Hsu 5; Yau and Yan 9; Zhu and L. However mos of hese leraures are based on he assumpon ha he neracons sasfy an approprae so-called machng condon. Yan Spurgeon and Edwards propose several papers dscussng decenralzed sldng mode usng he Edwards & Spurgeon canoncal form. However n hs approach he raher complcaed use of sae ransformaons and calculaon procedure are he man problem. In hs Chaper he developmens of sldng mode heory are proposed ncludng regular form reachably and conrol law desgn. Sysemac sldng mode conrol 63

75 mehods for LSS are nroduced as well. Combned wh he properes of LSS ypcal decenralzed sae feedback and oupu based SMC mehods are represened. A numercal eample s also used o llusrae he mehodology. 4. Revew of ypcal sldng mode conrol heory he sldng mode has proved o be a very powerful ool for dsurbance rejecon. In hs Secon he sldng mode conrol s nroduced and he properes of are oulned based on he wo seps of sldng surface desgn and conrol law desgn. he regular form of decomposon s frs dscussed. 4.. Regular form and mached perurbaons rejecon Consder he followng lnear me nvaran LI sysem: A u 4- s he sysem sae vecor are he sysem npus and he sysem marces are. he npu mar s assumed o have full rank and he par s sablzable. Normally he classcal sldng surface funcon s desgn as: S 4- Each sldng surface funcon descrbes a lnear surface he so called sldng surface. Oher erms lke sldng swchng manfold swchng hyperplane are also used. he classcal sldng mode surface desgnng algorhms are mosly based on he so called regular form. he man dea of hs form s o decompose he sysem saes equaons no wo par: he saes conrolled by npus drecly and he saes conrolled by npus ndrecly..e. where z A z A z 4-3 z A z A z u

76 A A z A M 4-5 A A he man advanage of hs regular form s ha dsurbances or fauls are decomposed no mached appear n Eq. 4-4 and unmached pars appear n Eq he mached par can be compensaed by he conrol npu. Hence he problem remanng s how o aenuae he unmached par of dsurbances appearng n Eq o make he ransformaon mar nverble he mar should be desgned properly such ha s full rank. he smples way o desgn s by seng. he followng descrbes wo oher approaches ha appear n he leraure. he mehod used n Znober 99; Edwards and Spurgeon 998 s QR decomposon he advanage of usng hs decomposon s ha here s no longer a need o consder he desgn of a mar. Afer usng QR Orhogonal-rangular decomposon he mar s decomposed no an upper-rangular mar and he ransformaon mar s an orhogonal mar.e. full rank and nverble. y modfyng hs ransformaon mar he srucure 4-5 can be shown o be sasfed. In he new coordnaes he sldng surface funcon becomes: where. S z S z 4-6 Once he sldng surface s reached.e. follows ha: z S Sz Nz 4-7 Moreover z A A N z 4-8 So he problem becomes desgnng he mar so ha s sable. Noe ha hs problem s smlar o he sae feedback conrol problem: ha of desgnng he mar o make sable. For hs problem several mehods are avalable based on hs regular form e.g. pole-placemen H Lnear-quadrac regulaor LQR 65

77 ec. hree sldng surface funcon desgn approaches are descrbed n Edwards and Spurgeon 998:. Robus pole-placemen Ryan and Corless 984. Quadrac mnmzaon Ukn and Young 978 and. Egensrucure assgnmen Znober 99 Anoher approach o desgn he regular form s o pay more aenon o specfc desgn of he mar M. Cho 997 descrbes an approach ha defnes ogeher wh he sldng surface funcon. he new coordnae vecor afer ransformaon hen follows as: z A z A 4-9 A z A u 4- In hs case he assocaed vecor desgnng approach can be appled. and a LMI based sldng surface funcon As menoned above f here are some dsurbances or unceranes n he sysem he sldng mode can compensae all he perurbaons appear n 4-4. he reason s ha f he sysem has he followng form: A u Df 4- where funcon and hen he so called machng condon s sasfed. y usng he sldng f he sldng surface s reached and can be mananed hen. he conrol law s hen equvalen o: u eq S SA S SDf 4- One should noe ha mus be non-sngular s requred. hen he deal sldng mode s gven by subsung 4- no he sysem 4-: I S S A I S S Df 4-3 As for hs case follows ha: 66

78 I S S D I S S R 4-4 hus by subsung 4-4 no 4-3 here s no perurbaon.e. he conroller rejeced all he mached perurbaons afer he sldng surface s reached and mananed. hs s he man propery of sldng mode ha he deal sldng moon s oally nsensve o he unceran funcon f Ryan and Corless 984; Dorlng and Znober 986; DeCarlo 998; Znober 99; Edwards and Spurgeon 998. Now he essenal desgn challenges as dscussed n he followng secons are o deermne:. he sysem response before reachng he sldng surface reachably;. he conrol law desgn; and 3 he naure of he unmached perurbaons. 4.. Reachably roblem and Reachng hase Once he sldng surface funcon s desgned a conrol law should be carefully desgned n order o drve he saes rajecory o he sldng surface. hs problem s he so called Reachably problem. From he descrpon n Secon 4.. he sysem s sable and nsensve o he mached perurbaons only f he sldng surface s reached.e.. In boh Hung 993 and Edwards and Spurgeon 998 he soluon o he sngle npu sngle oupu reachably problem s eplaned clearly. If a sysem can reach he sldng surface he sldng surface should be a leas locally aracve. hs can be epressed mahemacally as: lm and lm 4-5 for some doman. For he MIMO sysem Lyapunov heory s used. Consder he Lyapunov funcon for he sldng surface: V 4-6 he dervave of he Lyapunov funcon should sasfy he condon: V

79 Also smlar o he SISO case descrbed by Edwards and Spurgeon 998 defne - reachably condon n he MIMO case as: V 4-8 where s a posve desgn scalar. o prove hs rewre 4-8 as: dv d V Usng chan rule he nequaly can be smplfed o V V d V d.e. d V d And by negrang boh sdes from o follows ha: V V s s hus he me aken o reach represened by sasfes Snce s proporonal o he sldng funcon we can conclude ha he sldng surface can be reached n fne me. Oher reachng condons can be found n Hung 993 o specfy he characerscs of he sysem durng he reachng phase and guaranee he fne me reachably. In mul-sae sysems one can desgn he sldng surface funcon usng he so called Reachng law approach. hs s done by desgnng he dervave of he sldng surface funcon as: 68

80 a b c he sysem can hen reach he sldng surface wh specfed characersc me behavour. he reachably problem s one of he man dsadvanages of he sldng mode. Durng he reachng phase he sysem does no have he benef and properes of a sldng mode conroller e.g. durng he reachng phase he sysem s sensve o he mached perurbaons. Hence s of neres o drve he sysem o he sldng surface as soon as possble. Ukn and Sh 996 proposed a new sldng surface funcon desgn approach named negral sldng mode conrol n whch he sldng surface s reached from nal me and mananed here durng he enre sysem operaon by addng an negral par n he sldng surface funcon. hs mehod brngs up a new concep ha he sldng mode can be a conrol componen o rejec he mached perurbaons and some oher conrollers can be desgned o handle he unmached perurbaons. Wh hs concep numerous mehods can be combned wh sldng mode o acheve beer performance. hs Inegral sldng mode conrol s dscussed and eended o he oupu based approach n Chaper Conrol law desgn As anoher mporan par of sldng mode heory conrol law s manly desgned wh wo pars: u 4-9 u l u n where s he lnear componen and s he dsconnuous swchng conrol componen and usually has he form Ryan and Corless 984: u u n 4-69

81 hese wo componens and form he sldng mode conrol law. In dfferen sldng mode conrol sraeges hese wo pars have dfferen funcons. For eample f he sldng surface funcon s desgned as as s descrbed n Secon 4.. boh of hese wo componens are used o drve he sysem saes o he sldng surface and he choce of sldng surface funcon helps sablze he sysem afer he sldng surface s reached. However n he negral sldng mode he swchng componen s used o rejec he mached perurbaons whls he lnear par s used o sablze he sysem whle he sysem s runnng n he sldng surface. hs Secon consders he former case n whch boh lnear and nonlnear conrol laws are used o drve he sysem o he sldng surface wh he sldng surface gan mar used o sablze he sysem afer reachng he sldng surface. Consder he sysem n he form of: A u f 4- where represens a generalzed perurbaon funcon. he sldng surface funcon s desgned as: S 4- If boh lnear and dsconnuous pars of he SMC are desgned o ensure ha he sldng surface s reached wh he Lyapunov funcon he me dervave of hs Lyapunov funcon s gven by: V SA S u f 4-3 If he lnear conroller s chosen as s desgned accordng o can be furher rewren as: and he dsconnuous conroller V SA S u f u f 4-4 I s clear o see from 4-4 ha f he sldng surface can be reached he gan has a lower bound gven by: 7

82 u f 4-5 s he upper bound of he dsurbance and s a user seleced posve scalar. Followng hs he dervave of he Lyapunov funcon 4-5 sasfes whch means he sldng surface s reached n fne me. Furhermore he reachng me s gven by negrang as proposed n Secon 4... Moreover by dvdng he sysem no he mached par and unmached par Eq. 4-3 and 4-4 follow. I can be assumed ha n hs coordnae sysem he sldng gan mar. hus wh proper choce of S he unmached par of he sysem s sable.e. s sable and furhermore: ma A A S S 4-6 Adapve mechansm Usng he nequaly 4-5 he dervave of he Lyapunov funcon 4-4 s negave. However he bound of he unknown dsurbance should be known. hs s a very resrcve condon because he bounds of he dsurbances or fauls are no usually known n pracce. Some research has been done o dscuss hs resrcon. For eample Yu and Kaynak 9 summarzes several sof-compung mehods ncludng neural neworks and fuzzy sysems whch when combned wh he sldng mode oban paral nformaon abou he dsurbance bounds. A ypcal use of hs adapve mechansm can be o esmae he bound of he dsurbance auomacally. Wh he adapve mechansm he conroller gan s gven by: ˆ 4-7 where s chosen by he desgner o specfy he reachng speed. Assume ha here es an unknown bound for he mached dsurbance or unceranes such ha where s a vrual consan whch ess bu s unknown hen n Eq. 4-7 s assumed o be an esmae of. he adapve law can be proposed as: ˆ 4-8 7

83 where. Wh hs adapve gan he sldng moon s nsensve o he mached dsurbances/unceranes. he sably of hs knd of adapve mechansm can be proved usng anoher Lyapunov funcon wh me-dervave: V f ˆ 4-9 ˆ here s some desgn freedom o choose and. he eases way s o defne for whch sasfyng 4-9. hus he sldng surface s reached n fne me as descrbed above and he known upper bound consran s relaed. Alernavely and can be chosen as and o decrease he reachng me. he newes research on adapve mechansm for frs order sldng mode s gven by lesan e al n whch he adapve law s desgned as: ˆ sgn f f 4-3 where are small posve consans faclang he decrease of when he sldng surface funcon remans whn a small regon. In fac f 4-3 s equvalen o 4-8. he advanage of hs algorhm compared wh he frs algorhm 4-7 and 4-8 s ha prevens from ncreasng. An ncrease n he values of may lead o serous chaerng or rapd dsconnuous moon around he swchng boundary self. In lesan e al can also be desgned usng adapve mechansm. he parameer s used o keep he gan posve. oundary Layer When he sysem s runnng eacly n he sldng surface s called he deal sldng mode.e. for whch s sasfed eacly. However n mos suaons s dffcul o keep he sysem runnng n he sldng surface. Also he dsconnuous conrol law mgh cause chaerng moon due o hgh gan operaon snce he gan has a lower bound bu no upper bound. o ncrease he reachng speed and o 7

84 compensae he unknown upper bound perurbaon he gan should be desgned o be as large as possble. However he larger he gan s he larger he chaerng comes. For many ndusral applcaons hgh frequency swchng conrol s unaccepable due o hardware physcal consrans bandwdh of acuaors and converson ranges ec and ndeed hgh frequency moon canno acually resul from band-lmed sysems. Hence for real applcaons he concep of he boundary layer s used o overcome hs problem. hs dea has been dscussed for many years and sysemacally descrbed n Edwards and Spurgeon 998 alhough hese auhors do no provde a proof for he sably of a SMC sysem ha ncorporaes a boundary layer. In he followng a proof s proposed o gve condons under whch he SMC sysem wh boundary layer remans sable.. Non smooh boundary layer. Insead of usng he conrol law 4- he sldng mode conrol wh boundary layer can be wren as: f u n u sa sa 4-3 sgn f he boundary layer s. - Fgure 4-. A nonlnear dsconnuous boundary layer funcon Fgure 4- shows he boundary layer funcon 4-3 wh a sngle-npu case. o prove he reachably of he sldng regon and he sably of he sysem we have o consder wo suaons: when he sysem runnng ousde of he boundary layer we have o 73

85 prove he sysem s runnng oward he sldng regon and when he sysem runnng nsde of he boundary layer he sysem s sable. When he sysem operaes ousde he boundary layer Lyapunov funcon: hen by usng he V wh he gan 4-5. I follows ha s srcly decreasng unl he sysem reaches he boundary layer and sably s sasfed subjec o hs bound. When he sysem rajecores are nsde he boundary layer. Moreover he conrol law becomes. Wh he consran 4-5 follows whou loss of generaly ha can be chosen as. hen consderng he separaon of he sysem no mached and unmached pars follows ha: z Az Az z Az Az u f he sldng surface mar S n hs new coordnaon s wh Edwards and Spurgeon 998. When he sysem s n he boundary layer he moon s gven by and hence: z z A 4-3 A AS S Consder he Lyapunov funcon for sysem 4-3 V z.5z z wh me dervave: V z z z z A ma ma A A A S A A S z S S A S S z z A A 4-33 I can now be proved by conradcon ha he sysem sae s bounded. If s unbounded.e. as from 4-6 we have ha V whch mples 74 z

86 ha s bounded and leads o a conradcon. hus s bounded and he orgnal sysem s bounded. Moreover by defnng A A S ma A he nequaly 4-33 can be rewren as: ma S V z z A z z z A S z S z A whch mples ha he sldng moon s ulmaely bounded wh respec o: z : z 4-34 where s an arbrarly small posve scalar. From 4-34 we know ha he sze of he se s dependen on he choces of sldng surface gan mar S and he boundary layer..e. smaller more negave larges egenvalue of and lead o beer regulaon performance. Hence f he boundary layer mus be used he egenvalues of he mar should be chosen o be as small more negave as possble.. Smooh boundary layer. Anoher choce of boundary layer s make he conrol law a connuous funcon uron and Znober 986: u n u 4-35 where s a small group and he sze of boundary layer s appromaely. he advanage of hs mehod s ha he funcon 4-35 s smooh.e. here s no dsconnuy n he dervave funcon of Chaper 6 descrbes he use of he boundary layer formulaon of 4-35 as he mos suable choce for he faul esmaon problem. Fgure 4- shows he smooh boundary layer funcon wh a sngle npu case. I s easy o verfy hs mehod usng a proof smlar o Non-smooh boundary layer. A formal 75

87 eamnaon of he properes of hs boundary layer funcon s gven n uron and Znober Fgure 4-. A smooh boundary layer funcon When consderng combnng adapve mechansm and boundary layer heory ogeher he conrol law can be desgned as: u n u ˆ ˆ sgn f f 4-36 where and are chosen by he desgner. s he parameer chosen o deermne he accuracy of he SMC; deermne he duraon of he reachng phase and resrc he magnude of he gan. In fac all of hese parameers affec he chaerng of he SMC. I should be noed ha he orgnal SMC conrol law can be replaced by 4-36 wh he sably of he sysem remanng unaffeced. 4.3 Decenralzed SMC desgn usng LMI approach hs Secon descrbes a novel desgn mehod for decenralzed ypcal sldng mode conrol for LSSs. In hs hess he mehod used for sldng surface funcon desgn s LMI approach based on he Lyapunov funcon. he regular form of hs mehod for a sngle sysem was frs proposed by Cho 997 usng he orgnal sysem mar. 76

88 Compared wh he algorhm proposed by Edwards & Spurgeon 998 here s no need o ransfer he sysem no regular form n he sldng funcon desgn process. Moreover even n he proof of hs heorem he regular form only requres one ransformaon. As dscussed n Secon 4..3 he conrol law n hs mehod uses boh lnear and swchng conrol pars o drve he sysem o he sldng surface. he sably of he unmached par of he sysem s lef o he choce of sldng surface funcon Conrol law desgn wh LMI approach Consder a large scale sysem conans N small subsysems afer decomposon he -h subsysem has he form: A u f u h 4-37 where are he saes and npus of hs subsysem respecvely. are he sysem marces. I mus be assumed ha he followng are vald: A: he pars are conrollable. A: he local saes are avalable. A3: here are some known bounded posve consans for he mached dsurbance: f u f f u f3 All he mached unceranes mulplcave fauls and eernal dsurbance addve fauls should sasfy hs consran. hey should be bounded bu he consran of known consans can be relaed by he adapve mechansm. A4: he neracons of he subsysem sasfy he quadrac consran Šljak and Spanovć : h h H H Where s a boundng consan. he overall sysem can be wren n a compac form as: 77

89 A u f u h 4-38 where and. Wh he assumpon A4 he nerconnecons are bounded as follows: N h h H H H H 4-39 hs quadrac consran s frs proposed by Šljak and Spanovć n he framework of sngle sysem and eended o decenralzed sysem usng he S-procedure Lemma. hs consran could no only represen he nonlnear neracons bu also he uncerany of he -h subsysem. he sablzaon problem for LSS s also solved n Šljak and Spanovć s work. hey proposed ha he decenralzed conrol could be obaned by solvng he followng LMI: Mnmze subjec o A A H L L H N I H I H I N N I I 4-4 he decenralzed conrol gan mar K s obaned by. he objecve of he mehod proposed n hs Secon s o desgn a oally decenralzed SMC ha robusly regulaes he sae of he overall sysem whou any nformaon echange beween he conrollers. Dfferen o Šljak and Spanovć whn conrol desgn procedure n hs secon he overall decenralsed sysem mus be robus o all he unceranes and nsensve o mached perurbaons. heorem 4.. For he overall sysem 4-38 he sysem s asympocally sable afer he sldng surface s reached f here ess an s.p.d. mar sasfyng he followng LMIs: 78

90 79 Mnmze subjec o I X H I X H XH XH I XA AX N N N 4-4 where s he orhogonal complemen of he npu mar. wh sldng surface funcon n he form: S S dag X N N N roof: Defne a ransformaon mar for he overall sysem X S 4-4 And he assocaed vecor z: z z z 4-43 where. I can be seen ha he sldng surface funcon s. As s he orhogonal mar of.e. he ransformaon mar s non-sngular and has full rank.e. s nverble. he nverse mar can easly be deermned from: S X X 4-44 hen he assocaed vecor s gven as: h u f u z A z 4-45

91 8 y subsung he ransformaon 4-4 and 4-44 no 4-45 he assocaed vecor equaon can be rewren as: h S u u f S z z S SA X SAX S A X AX z z 4-46 When he sysem s runnng n he sldng surface and becomes h z X AX z 4-47 hs s he so called sldng moon. I conans he unmached par of he sysems. Snce he sldng surface s reached.e. mached par of he sysem has been compensaed he remanng problem s o fnd he sldng surface mar whch can sablze he sldng moon ecause s an s.p.d mar s also an s.p.d. mar. In hs case a suable Lyapunov funcon can be defned as: Yz z z V 4-48 he me dervave of s: Yz h h Y z z X AX Y Y X AX z z V 4-49 Wh a useful lemma ha s nroduced n oyd e al 993: Y Y X X X Y Y X 4-5 From 4-5 follows ha: h h Yz Y z Yz h h Y z 4-5 And from assumpon A4:

92 h h z H H z X XH H H z HX X z 4-5 Subsue 4-5 and 4-5 no 4-49 hen gves: AX X Y Y AX X Y Y V z z z X XH HX X he sablzaon of he sysem requres for all whch leads o AX X Y Y AX X X XH HX X Y Y 4-53 Snce can be easly found ha and s nverse mar are boh s.p.d. marces as long as X s a s.p.d. mar defne and pre- and pos-mulply 4-53 by gves: AX XA I XH HX 4-54 Recallng Assumpon A4 rewre he nequaly 4-54 as: AX XA I N XH H X y defnng and usng he well known Schur complemen lemma he above nequaly can be rewren n he form of nequaly 4-4. hus f here ess a soluon mar X o 4-4 he dervave of he above Lyapunov funcon s negave:.e. he assocaed sysem s asympocally sable afer he sysem reachng sldng surface and hence he proof s complee. he proof for heorem 4. shows ha afer reachng he sldng surface he remanng sysem sldng moon s sable. As and can easly be seen ha heorem 4. concerns he unmached par of he sysem. Hence he ne sep s o ensure he reachably of he sldng funcon unmached par of he sysem whch s gven by heorem 4.. 8

93 8 heorem 4.. For he -h LSS subsysem 4-37 f he sldng surface funcon could be obaned from heorem 4. he conrol law 4-55 can drve each subsysem o he sldng surface and compensae he mached perurbaons. A S S sgn A S S u 4-55 where S S ˆ ˆ 4-56 wh a posve consan chosen by he desgner. roof: From 4-47 he sldng moon s gven by: h X AX 4-57 he dervave of he sldng funcon for he -h subsysem s: S h u f u S A S S 4-58 Assume ha here ess some consans large enough.e. o sasfy. he value of hese unknown consans are esmaed by he adapve erms. Defne he esmaon errors s easy o verfy ha he dervave of hese errors are. Now defne he Lyapunov funcon for he sldng funcon as N S S S V 4-59 I can be seen ha s an s.p.d. mar as. he me dervave of he Lyapunov funcon 4-59 s gven by:

94 83 N N S S h S u f u A S S S S S d d V ˆ 4-6 I can be verfed ha hence by subsung he conrol law 4-55 and 4-56 no 4-6: N N N S S S S S S h S S u f V ˆ ˆ ˆ 4-6 whch mples ha he sldng surface can be reached n fne me as proved n Secon hus he sysem rajecores can reach he sldng surface n fne me and reman on wh sable sldng moon. hs complees he roof. From he frs par of he proof can be found ha LMI 4-4 focuses on he unmached par of he sysem. hus he ransformaon nroduced n Edwards and Spurgeon 998 s no necessary. he adapve par nroduced n hs mehod can rela he common bound consran. Moreover here s no need o assume he known bound of he mached perurbaon.e. Assumpon A3. he adapve mechansm can esmae as long as s bounded. hus one of he man consrans of sldng mode s relaed Feasbly dscusson heorem 4. n Secon 4.3. shows he suffcen condon for he SMC desgn. Alhough he assumpons A-A4 are nroduced s no enough o prove he feasbly of he LMIs 4-4. I s valuable o deermne under wha condons soluons o he LMIs es as dscussed n hs Secon.

95 he sldng surface nvarance propery guaranees nsensvy o mached fauls. Hence s now feasble o consder an overall sysem whou mached fauls: A u h 4-6 All he parameers of 4-6 are n he same form as he well-known consran for he sysem sably s: here ess a sae feedback conrol law for he sysem f and only f here s an s.p.d. mar such ha: XA AX XK KX 4-63 Furhermore 4-63 can be resrced o: XA AX XK KX I N XH H X 4-64 he suffcen condon for he solvably of hs nequaly 4-64 n he decenralzed sysem s ha he overall sysem 4-6 s conrollable. hs can be ensured by he conrollably of each subsysem. As a consequence of he sysem conrollably one can always fnd a gan mar and a p.s.d mar X sasfyng 4-64 wh large enough. Usng he projecon lemma by Gahne and Apkaran 994 he followng nequaly s feasble wh large enough : XA AX I N XH H X 4-65 whch mples ha he LMIs 4-4 s feasble.e. f all he subsysems are conrollable one can always fnd a feasble soluon o 4-4 and consruc an SMC as dscussed n Secon If he neracon erm of he overall sysem can be wren n he form of where some esng resuls could be used o dscuss he feasbly. If he sysem can be wren n he form: A DFE u he resuls from Khargonekar eersen and Zhou 99 show ha he above sysem s quadracally sable va a consan lnear sae feedback conrol f and 84

96 only f here ess an s.p.d. mar such ha for some gan marces Cho 998: A A DD E E K K Usng he projecon lemma and gves: AX XA DD E E Choosng he above nequaly s n he same form of ole assgnmen and quadrac mnmzaon Improvemen he LMI approach presened n secon 4.3. can be easly eended. In hs case he SMCs are reaed as mached perurbaon componens and can be combned wh oher robus mehods. I s shows ha he proposed LMI based decenralzed SMC sraegy has such good compably ha can be combned wh oher robus mehod and acheve specfc robus performance. Alhough he mehods proposed n hs secon are no llusraed n Secon 4.4 s sll valuable o ls some of he sraeges snce hs dea of combnaon wh oher robus conrol mehod s mporan n he res of hs hess. hs Secon nroduces several mehods o eend he desgn feaures of he basc lnear conrol oulned above. hese eensons mehods nclude egenvalue assgnmen and quadrac mnmzaon. Egenvalues assgnmen he LMIs 4-4 provdes some degrees of freedom. he egenvalues assgnmen can be used wh D-sably heory. Here wo D sably regons are proposed:. I s well known ha o assgn all he egenvalues of he unmached overall sysem n he lef hand sde of he lne Fgure 4-3 s necessary o fnd a soluon o he followng LMIs Mnmze subjec o X dag X... X N 85

97 86 I X H I X H XH XH X I XA AX N N N 4-66 Fgure 4-3. Egenvalue cluserng on he lef hand sde of. o assgn all he egenvalues o le n a dsk of radus and cener. Consder he sldng moon 4-47: h z X AX z Defne. Assume ha he neracons erm sasfes and he assumpon A4 s sasfed by. Accordng o he LMI regon funcon he followng nequaly should be feasble: H Y YA Y H AY Y r q q r.e. Y H H Y ry YA qy AY qy ry Y H Y H ry YA qy AY qy ry

98 87 Y H H Y ry YA qy AY qy ry Defne he above nequaly can be rewren as: HX XH rx XA qx AX qx I rx Furhermore usng Shur Complemen he LMIs are gven by: Mnmze subjec o... N X X dag X I X H I X H XH XH X r XA qx AX qx I rx N N 4-67 If he above LMIs 4-67 have feasble soluon hen he poles of he sldng moon are assgned n he dsk D as shown n Fgure 4-4. Fgure 4-4. Egenvalue cluserng nsde he dsk opmzaon Consder he sldng moon: