Observer Design for Nonlinear Systems using Linear Approximations

Transcription

1 Observer Desgn for Nonlnear Ssems sng Lnear Appromaons C. Navarro Hernandez, S.P. Banks and M. Aldeen Deparmen of Aomac Conrol and Ssems Engneerng, Unvers of Sheffeld, Mappn Sree, Sheffeld S 3JD. e-mal: s.banks@sheffeld.ac.k Dep of Elecrcal and Elecronc Engneerng The Unvers of Melborne Parkvlle VIC 3 Asrala Kewords: Observer, Nonlnear Ssems, Lnear Appromaons. Absrac: In hs paper we presen he desgn of an observer for nonlnear ssems. The nonlnear ssem s represened as a seqence of lnear me varng appromaons and he convergence of he seqence s shown. In hs wa, we can se he classcal echnqes for he desgn of lnear observers.. Inrodcon There es several approaches o he desgn of observers for nonlnear ssems, ncldng he separaon of he nonlnear ssem no a lnear par and a nonlnear perrbaon of he ssem wh a bonded condon 2, he se of Le dervaves and he nverson of he Jacoban of a coordnae ransformaon o oban he gan of he nonlnear observer and he se of a Lapnov eqaon o desgn he observer for a nonlnear ssem represened n a specal canoncal form 8. The desgn of observers for lnear ssems s beer ndersood see, 4 snce n nonlnear heor here s a necess o se more comple mahemacs. Therefore, here es neres n he developmen of smpler and general mehods o solve he problem of nonlnear sae reconsrcon. Ths paper deals wh he desgn of observers for nonlnear ssems b sng a recen echnqe n whch he nonlnear dnamcal ssem s represened as he lm of a seqence of lnear me-varng appromaons ha converge o he solon of he nonlnear ssem nder a local Lpschz condon. The framework of hs desgn s based on a prevos applcaon of he lnear appromaons scheme n opmal conrol heor 2, n he heor of chaos 4 and n he heor of nonlnear dela ssems 5. The paper s organzed as follows. Secon 2 presens he prncple of he lnear appromaon scheme see 2, 6, n secon 3 a lnear mehod for he desgn of mevarng observers s referenced, secon 4 nrodces he mehod for he desgn of he observer for nonlnear ssems and n secon 5 he convergence of he seqence s proven. Some eamples are gven n secon 6. The mehod can be easl generalzed o dfferen knds of observers, lke he redcer order and he algorhm descrbed for he desgn s smple as s based onl on lnear me-varng ssems, allowng he se of dfferen lnear echnqes.

2 2. Lnear Appromaon Scheme The appromaon scheme was developed b Banks 2 and s a mehod for redcng non-lnear problems o seqences of lnear non-aonomos eqaons. The mehod s descrbed n he followng Consderng he non-lnear dfferenal eqaon n A, R Now, we nrodce he seqence of lnear non-aonomos eqaons: A A,., for. I can be shown ha he solon of seqence 2.2 converges o he solon of he orgnal nonlnear ssem 2.. The proof of local convergence of hs mehod can be fond n 2 and he global resl s presened n 6. The proof of convergence s based on he assmpon of local Lpschz conn on he marces of he nonlnear ssem, and shows ha he seqence of appromaons { } s a Cach seqence of appromaons 2.2 ha converges o he real solon of he ssem. 3. Lnear Mehods A known mehod o desgn observers for lnear me-varng ssems s gven n 5. In he followng, we gve a bref olne of hs mehod. Gven an nforml observable and lecograph fed me-varng ssem A B C, 3. where s n -dmensonal navalable sae vecor, s p -dmensonal np vecor and s q -dmensonal op vecor of he ssem. I s possble o fnd a sae esmaor of he form: F G H 3.2 The relaonshp beween he re sae and s esmae s T e, and he ssem 2.2 wll be an asmpoc den sae esmaor f H B and f F A G C s a consan mar wh sable egenvales.

3 ** I hnk mgh beer o defne e as e T n sead of wha s shown above** The condon of nform observabl s accomplshed b checkng ha he rank of he observabl mar s eqal o n for all, 2 ; and s lecograph fed on, 2, f s possble o oban a nn non-snglar mar for all, 2 b elmnang he rows of he observabl mar ha are lnearl dependen. The seps n he desgn of he observer are as follows, see 5.. Canoncal ransformaon A B C a Consrc he observabl mar of he ssem b sng a mar operaor defned as: W M M A d M. d b Check nform observabl of he ssem b calclang he rank of he observabl mar. c Consrc an n n mar wh rank n b elmnang he rows of he observabl mar ha are lnearl dependen. d Consrc a ransformaon mar b sng a second mar operaor defned as: L M A M d M. d e Transform he orgnal ssem 2. no he eqvalen ssem. 2. Consrcon of he asmpoc esmaor F G B 3.3 a Choose n sable egenvales for he sae esmaor. b Desgn a mar G sch ha F A G C s a consan mar b sng he chosen egenvales and some relaonshps beween he ssem marces. c Consrc he sae esmaor d Calclae he esmae sng he ransformaon mar 3.4 P Observer for nonlnear ssems In hs secon, he algorhm o desgn he nonlnear observer s presened. Ths s done b combnng he desgn of an observer for lnear me-varng ssems and he appromaon scheme presened n 2 and 6, n whch he nonlnear ssem s represened as a seqence of lnear me-varng appromaons. We consder he followng nonlnear ssem

4 ; C B A 4. We nrodce he appromaon scheme see 2, 6, C B A 4.2 and for,, C B A 4.3 Therefore, he orgnal nonlnear ssem 2. s appromaed b he seqence of eqaons 2.2, 2.3. As each appromaon s onl a lnear me-varng eqaon, he echnqes avalable o analse lnear ssems can be appled. The desgn mehod sed o oban he lnear me-varng observer s he one descrbed n secon 3, whch s based on he mehod repor b Ngen and Lee 5. Then, for he frs appromaon we oban he followng sae esmaor, B G F 4.4 and for, B G F 4.5 Fnall, b nrodcng he ransformaon mar P, we can oban he esmaes of he ssem for he frs appromaon as, P 4.6 and for as, P 4.7 The saes of he nonlnear ssem 4. are hen esmaed b he solon of he fnal appromaons gven n eqaons 4.6 and Convergence Proof

5 A local convergence proof for he represenaon of a nonlnear ssem as a seqence of lnear me-varng ssem s presened n 2. The global convergence proof of he seqence appears n 6. Here he convergence of he desgned observer s presened, hs s done sng he resls of 6 and he heor of lnear observers see for eample. Consder he followng nonlnear ssem A B ; C 5. From 6, he basc assmpon s he one of local Lpschz conn of mar A. Gven ha we have a nonlnear ssem of form A n, R 5.2 he seqence of appromaons { } s a Cach seqence ha converges o he real solon of he nonlnear ssem. These resls are generalsed for a nonlnear ssem of form 5.. The seqence of appromaons { } of he nonlnear ssem 5.2 s a Cach n seqence ha wll converge nforml on,τ for τ> n he space C, τ, R o he real solon of he nonlnear ssem 5. f he followng condons are sasfed A A B B C C α, β, δ, n, R, 5.3 The seqence of appromaons { } s also a Cach seqence provded ha marces F, G, B and P of he sae esmaor depend on marces A, B and C whch are locall Lpschz. From observer s heor, he followng lm ms be sasfed lm 5.4 Therefore, follows ha lm 5.5 And a each appromaon,

6 lm 5.6 Ths s re, snce } { and } { are Cach seqences, and and are bonded. For he fnal appromaon lm 5.7 and lm 5.8 Therefore, lm ; as Eamples In hs secon we presen he desgn of an observer for dfferen cases of nonlnear ssems. Frsl, we shall desgn an observer for he followng nonlnear ssem The ssem s p n form 5. o drecl appl he heor. The mehod sed o desgn he lnear me-varng observer reqred a each lnear appromaon s he one descrbed a secon 3. Fgres, 2 show he esmaes a he fnal appromaon and he error of he esmaes for he case of nal condons.5; ; and -; 2;.5 respecvel. The esmaon error s drven o zero afer some shor seps n me provdng adeqae sae esmaon.

7 Fgre. Observer desgn for he nonlnear ssem wh nal condon.5; ; sng he proposed mehod: a Esmaes ; b Esmaon error a fnal appromaon.

8 Fgre 2. Observer desgn for he nonlnear ssem wh nal condon -; 2;.5 sng he proposed mehod: a Esmaes ; b Esmaon error a fnal appromaon. In he ne eample, he observer shall be desgned for he followng nonlnear ssem

9 Two cases are shown n fgres 3 and 4, wh nal condons.5; 2; -.5 and.5; -; 2 respecvel. a b

10 Fgre 3. Observer desgn for he nonlnear ssem wh nal condon.5; 2; -.5 sng he proposed mehod: a Esmaes ; b Esmaon error a fnal appromaon. a b

11 Fgre 4. Observer desgn for he nonlnear ssem wh nal condon.5; -; 2 sng he proposed mehod: a Esmaes ; b Esmaon error a fnal appromaon. 7. Conclsons In hs paper we presened a new mehod for he desgn of observers for nonlnear ssems. The mehod emplos a recen echnqe, whch models a nonlnear ssem as a seqence of lnear me-varng appromaons. The appromaons are shown o converge o he solon of he nonlnear ssem. I s shown, hrogh smlaon sdes, ha he observer adeqael performs ha ask of sae esmaon of nonlnear ssems. The seqences of saes and esmaes converge nder a mld local Lpschz condon sng classcal echnqes for lnear observers. Ths resl also confrms he possbl of sng he appromaon mehod on a new knd of nonlnear problem, provded ha he correspondng lnear me-varng problem can be solved. 8. References S.P. Banks, 98, A noe on non-lnear observers, In. J. Conrol, 34, S.P. Banks and K. Dnesh, 2, Appromae Opmal Conrol and Sabl of Nonlnear Fne-and Infne-Dmensonal Ssems, Annals of Operaons Research 98, S.P. Banks and K.J. Mhana, 993, A Smple Observer Desgn for Nonlnear Ssems, Research Repor No. 498, Unvers of Sheffeld. Deparmen of Aomac Conrol and Ssems Engneerng. 4 S.P. Banks and McCaffre.D, 998, Le Algebras, Srcre of Nonlnear Ssems and Chaoc Moon, In. J. Bfrcaon & Chaos, 8, No. 7, S.P. Banks, 22, Nonlnear dela Ssems, Le Algebras and Lapnov Transformaons, IMA J. Mah Con & Inf, 9,

12 6 S.P. Banks and M. Tomas-Rodrgez, Lnear appromaons o Nonlnear Dnamcal Ssems wh Applcaons o Sabl and Specral Theor 7 G. Basn and M. Gevers, 988, Sable Adapve Observers for Nonlnear Tme Varng Ssems, IEEE. Trans. on Aomac Conrol, 33, No. 7, K. Bsawon, M. Farza and H. Hammor, 998, Observer desgn for a specal class of nonlnear ssems, P. Cook, 985, Nonlnear Dnamcal Ssems, London, Prence Hall. J.P. Gaher, H. Hammor and S. Ohman, 992, A smple Observer for Nonlnear Ssems Applcaons o Boreacors, IEEE. Trans. Aomac Conrol, 37, No. 6, J. O Rell, 983, Observers for Lnear Ssems, Mahemacs n scence and engneerng, London, Academc Press Inc. 2 Khall, H. 996, Nonlnear Ssems, New Jerse, Prence Hall. 3 S.R. Ko, D.L. Ello and T.J. Tarn, 975, Eponenal Observers for Non-lnear Dnamcal Ssems, Informaon and Conrol, 29, D.G. Lenberger, 964, Observng he Saes of a Lnear Ssem, IEEE Trans. Ml. Elecroncs, C. Ngen and T. Lee, 985, Desgn of a Sae Esmaor for a Class of Tme- Varng Mlvarable Ssems, IEEE Trans. Aomac Conrol, 3, No. 2, L. Perko, 99. Dfferenal Eqaons and Dnamcal Ssems, New York, Sprnger-Verlag Inc.