From Continuous-Time Design to Sampled-Data Design of Nonlinear Observers

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1 Proceedgs o the 47th IEEE Coerece o Decso ad Cotrol Cacu Mexco Dec. 9-8 ThC.4 From Cotuous-Tme Desg to Sampled-Data Desg o Nolear Observers Iasso araylls ad Costas ravars Abstract I ths wor a sampled-data olear observer s desged usg a cotuous-tme desg coupled wth a tersample output predctor. The proposed sampled-data observer s a hybrd system. It s show that uder certa codtos the robustess propertes o the cotuous-tme desg are herted by the sampled-data desg as log as the samplg perod s ot too large. The approach s appled to tragular globally Lpschtz systems. T I. INTRODUCTION HE problem o desgg sampled-data olear observers s a very challegg problem that has attracted a lot o atteto the lterature. Cotuoustme olear observer desgs [8567] are meat to be used oly or very small samplg perods whereas ther potetal redesg or the purpose o dgtal mplemetato eve though straghtorward ad popular or lear systems [5] poses sgcat challeges the olear case. For ths reaso the ma le o attac has bee through the use o a exact or approxmate dscretetme descrpto o the dyamcs as the startg pot or observer desg [ ]. Ths s a reasoable pot o vew but aces two mportat dcultes: ( rom the momet that the cotuous-tme system descrpto s abadoed ad s substtuted by a dscretetme descrpto the ter-sample dyamc behavor s lost ( ay errors the samplg schedule get traserred to errors the dscrete-tme descrpto As a cosequece avalable desg methods ( do ot provde a explct estmate o the error betwee two cosecutve samplg tmes ad ( do ot accout or perturbatos o the samplg schedule. Moreover due to observablty ssues the magtude o the samplg perod caot be arbtrary (see [6]. Fally optmzato-based approaches or olear observer desg are provded [3349]. A hybrd observer desg approach was recetly proposed [6] whch bears smlartes to the above-metoed optmzato-based approach but the hybrd ature o ther observer oers certa advatages. I the preset wor our proposed sampled-data observer wll also be a hybrd I. araylls s wth the Evrometal Egeerg Departmet Techcal Uversty o Crete 73 Chaa Greece (e-mal: arayl@eveg.tuc.gr. C. ravars s wth the Chemcal Egeerg Departmet Uversty o Patras 65 Patras Greece (e-mal: ravars@chemeg.upatras.gr. system; however t wll drectly emerge rom a cotuoustme desg o a olear observer. Cosder a sgle-output cotuous-tme system: = ( x x y = h( x y where C ( R ; R h C ( R ; R wth ( = h ( =. For ths system suppose that a cotuous-tme observer desg s avalable z y z ( z where F C ( R R; R Ψ C ( R ; R wth F ( = Ψ ( =. The questo s whether ths desg would stll be useul the presece o sampled measuremets y(h = where h s the samplg perod or more geerally at some = ot ecessarly uormly spaced but satsyg < r or all =... or some r >. coutable set o tme stats π { } = The preset wor has bee motvated by the tutve expectato that a cotuous-tme olear observer desg would stll be useul the presece o medum-sze samplg perods as log as specal care s tae the tme-terval betwee measuremets. Holdg the most recet measuremet (zero-order hold s ot the most tutvely meagul strategy; stead the model ( could be used to predct the evoluto o the output up utl the ew measuremet s receved. I partcular the preset paper we propose a sampled-data observer cosstg o the cotuous-tme observer coupled wth a output predctor or the tme terval betwee two cosecutve measuremets: ( t [ = y( ( R w& ( = L h( Ψ( t [ ( ( /8/$5. 8 IEEE 548

2 47th IEEE CDC Cacu Mexco Dec. 9-8 ThC.4 Fgure depcts the structure o the sampled-data observer (3 compared to the cotuous-tme observer (. The sampled-data observer uses the cotuous-tme observer as a ey gredet coupled wth a ter-sample output predctor. The latter s talzed by the most recet measuremet ad tegrates the rate o chage o the output calculated by the model ( L h( x : = h( x ( x s the Le dervatve o the output map. y( y( zw ˆx =Ψ(z Cotuous-tme Observer ˆx( ˆ & = L h(x( t [ zw = y( Iter-sample Output Predctor ˆx =Ψ(z Cotuous-tme Observer Fgure : Cotuous-tme observer ( (top versus sampled-data observer (3 (bottom. It s mportat to pot out that the etre system ( wth (3 s a hybrd system whch does ot satsy the classcal semgroup property. However the wea semgroup property holds (see [56] ad cosequetly t ca be aalyzed usg the recet results [567]. The ma result o the preset paper s that the propertes o the observer ( uder cotuous measuremet are herted by the observer (3 uder sampled measuremets as log as the samplg perod s ot too large. ˆx( By R we deote the set o o-egatve real umbers. We deote by the class o postve C uctos deed o R. We say that a o-decreasg cotuous ucto γ : R R s o class N γ ( =. We say that a ucto ρ : R R s postve dete ρ ( = ad ρ ( s > or all s >. By we deote the set o postve dete creasg ad cotuous uctos. We say that a postve dete creasg ad cotuous ucto ρ : R R s o class lm ρ ( s =. By L we deote the set o all s cotuous uctos σ = σ ( s : R R R wth the propertes: ( or each t the mappg σ ( s o class ; ( or each s the mappg σ ( s s ocreasg wth lm σ ( s =. t l Let D R be a o-empty set ad I R a terval. By L ( I; D we deote the class o all Lebesgue measurable ad ally bouded mappgs d : R D. Notce that by sup d( we do ot mea the essetal [ t] supremum o d : D o [ t ] but the actual supremum o d : R D o [ t ]. R Let C ( R ; R h C ( R ; R. By L h( x : = h( x ( x we deote the Le dervatve o the ucto h C ( R ; R alog the vector eld C ( R ; R. II. MAIN ASSUMPTIONS AND NOTIONS I the preset wor we study systems o the orm ( uder the ollowg hypotheses: Notatos Throughout ths paper we adopt the ollowg otatos: Let A R be a ope set. By C ( A ; Ω we deote the class o cotuous uctos o A whch tae values Ω R. By C l ( A ; Ω where l {...} we deote the class o cotuous uctos o A wth cotuous dervatves o order l whch tae values Ω. For a vector x R we deote by x ts usual Eucldea orm ad by x ts traspose. By { Ax ; x } A : = sup x = we deote the duced m orm o a matrx A R ad I deotes the detty matrx. By B = dag( b... b we deote the dagoal matrx B R wth b... b ts dagoal. (H System ( s Robustly Forward Complete (see [5].e. there exst uctos μ ad a such that or every x the soluto x ( o ( wth tal codto x ( = x satses ( x( μ( a x t (4 The ollowg deto o the oto o robust observer or system ( wth respect to measuremet errors s crucal to the developmet o the ma results o the preset wor. Deto.: Cosder the ollowg system z y z z (5 549

3 47th IEEE CDC Cacu Mexco Dec. 9-8 where F C ( R R; R Ψ C ( R ; R wth F ( = Ψ ( =. System (5 s called a robust observer or system ( wth respect to measuremet errors the ollowg codtos are met: there exst uctos σ L γ p N μ ad a such that or every v L ( R ; R the soluto ( x ( o = ( x z h( x z ( x z R ad wth tal codto ( x ( = ( x z correspodg to v L ( R ; R satses the ollowg estmates: ( x z t sup γ ( v( (6 x( σ t (7 μ( a ( x z sup p( v( t (8 or every x there exsts z such that the soluto ( x ( o (6 wth tal codto x ( = ( x z correspodg to v satses ( x( or all t. Remar.: I system (5 s a robust observer or system ( wth respect to measuremet errors the system (6 wth output Y z x satses the Uorm Iput-to-Output Stablty property rom the put v L ( R ; R wth ga γ N (see [7]. We ext dee the correspodg oto o robust sampleddata observer. Notce that cotrary to usual observers or whch the output sgal y ( o system ( s avalable ole a sampled-data observer uses oly the output values y( at certa tme staces π = { } = wth < r or all =.... The umber r > s called the upper dameter o the samplg partto. Deto.3: The system ( = g( y( t [ = G lm t y( where g C ( R R R; R G C ( R R; R Ψ C ( R ; R wth g ( = G ( = Ψ ( = s called a robust sampled-data observer or ( wth respect to measuremet errors the ollowg codtos are met: (9 there exst uctos σ L γ p N μ ad a such that or every ( x z d R R L ( R ; R L ( R ; R the soluto ( x ( o ( = ( x( ( = g( y( v( t [ = G lm y( v( t = r exp( d( ( wth tal codto ( x ( = ( x z correspodg to d L ( R ; R v L ( R ; R satses the ollowg estmates: x( σ x z t sup γ ( v( t ( μ( a ( ( x z sup p( v( t ( or every x there exsts z such that or all d L ( R ; R the soluto ( x ( o ( wth tal codto x ( = ( x z correspodg to ( d L ( R ; R ad v satses x( or all t. Remar.4: For each ( t x z R R R ad or each d L ( R ; R v L ( R ; R the soluto ( x ( o ( wth tal codto ( x ( = ( x z correspodg to d L ( R ; R v L ( R ; R s produced by the ollowg algorthm: Step : Gve ad d L ( R ; R calculate usg the equato = r exp( d( Compute the state trajectory ( x ( t [ as the soluto o the deretal equato x &( = ( x( ad ( = g( h( x( v( 3 Calculate z usg the equato ( ThC.4 = G lm h( x( ( v. t For = we tae = t ad x ( = x (tal codto. Hybrd systems o the orm ( were studed [567] where the wea semgroup property or such systems was exploted. Tag to accout hypothess (H 54

4 47th IEEE CDC Cacu Mexco Dec. 9-8 or system ( regularty propertes o the rght had-sdes o (9 ad usg the results o [5] we may coclude that ( system ( has the Boudedess-Imples- Cotuato (BIC property.e. or each ( t x z d R R L ( R ; R L( R ; R there exsts t ( t ] such that the soluto max ( x ( o ( wth tal codto ( z ( R ; R x ( t t = ( x correspodg to d L v L ( R ; R exsts or all t t t. I addto t < the or every [ max C > there exsts t [ t t max wth z ( > C. ( R s a robust equlbrum pot rom the put ( d L ( R ; R L ( R ; R.e. or every max ε > T there exsts δ : = δ ( ε T > such that or all ( t x z d R R L ( R ; R L( R ; R wth x z sup d( sup v( < δ t holds that the t t t soluto ( x ( o ( wth tal codto ( z d ( R ; R L x ( t t = ( x correspodg to ( L ( R ; R exsts or all t [ t t T] ad sup{ ( x ( ; t [ t t T] t [ T] } < ε ( system ( s autoomous.e. or each t x z d R R L ( R ; R L ( R ; ( R t t ad or each ( t ] θ t holds that the soluto ( x ( o ( wth tal codto ( z ( R ; R x ( t t = ( x correspodg to d L v L ( R ; R cocdes wth ( x ( t θ z ( t θ where ( x ( z ( s the soluto o ( wth tal codto x ( t θ z ( t θ = ( x correspodg to ( z P L ( R ; R Pθ θ d ad v L ( R ; R where ( P θ d ( = d( t θ ad ( P θ ( = v( t θ or all t θ. Remar.5: The reader should otce that the samplg perod s allowed to be tme-varyg. The actor exp( d( wth d ( some o-egatve ucto o tme s a ucertaty actor the ed-pot o the samplg terval. Provg stablty or ay o-egatve put d L ( R ; R wll guaratee stablty or all samplg schedules wth r (robustess to perturbatos o the samplg schedule. To uderstad the mportace o robustess to perturbatos o the samplg schedule cosder the ollowg stuato. Suppose that hardware lmtatos restrct the samplg perod to be s. I we maage to desg a sampled-data observer wth r s the the applcato o the sampled-data observer wll guaratee covergece o the state estmates eve we mss measuremets or we have delayed measuremets (or example due to mproper operato o the sesor. I such a case robustess to perturbatos o the samplg schedule becomes crtcal. The troducto o the actor exp( d( s a mathematcal way o troducg perturbatos to the samplg schedule; however t s ot uque. Other ways o troducg perturbatos o the samplg schedule ca be cosdered. III. MAIN RESULTS We are ow a posto to state our ma result. Theorem 3.: Cosder system ( uder hypothess (H ad suppose that system (5 s a robust observer or system ( wth respect to measuremet errors. Moreover suppose that there exsts a costat ad a ucto σ L such that or every ( x z R ad v L ( R ; R the soluto ( x ( o (6 satses the ollowg estmate or all t : L h( Ψ( L σ h( x( ( x z t sup v( (3 Fally suppose that r < where r > s the upper dameter o the samplg partto ad s the costat volved estmate (3. The (3 s a robust sampled-data observer or system ( wth respect to measuremet errors. Remar 3.: The reader should otce the structural dereces betwee the cotuous tme observer (5 ad the sampled-data observer (3 whch are show Fgure. The sampled-data observer uses the estmate o the state x ˆ( ad the measuremet y( order to geerate a addtoal sgal w ( : the sgal w ( wll approxmate the output sgal y ( ad actually replaces the output sgal y( the observer. The proo o Theorem 3. utlzes the recet Small-Ga Theorem or hybrd systems [7] ad s omtted due to space lmtatos. IV. APPLICATIONS I ths secto we preset the applcato o Theorem 3. to tragular globally Lpschtz systems leadg to cocrete sampled-data observer desgs. Cosder the system ThC.4 54

5 47th IEEE CDC Cacu Mexco Dec. 9-8 ThC.4 M = ( x x y = x = ( x... x = ( x... x x (4 where : R ( =... wth ( = ( =... are globally Lpschtz uctos.e. there exsts a costat L such that the ollowg equaltes hold or =... : R ( x... x ( z... z L ( x z... x z x ( x... ( z... (5 The reader should otce that all lear observable systems ca be wrtte the orm (4 wth : R R ( =... beg lear uctos. Notce that systems o the orm (4 are Robustly Forward Complete ad satsy hypothess (H sce or every x the soluto o (4 wth tal codto x ( = x satses the estmate: x( exp( c x t (6 where c : = L. Iequalty (6 s obtaed by evaluatg the dervatve o the ucto W ( x = x alog the solutos o (4 ad usg equaltes (5. A hgh-ga observer desg s descrbed []: rst a vector = (... s oud so that the matrx ( A c s Hurwtz where c : = (... ad A R s the matrx A = { a : =.. j... } wth z j = a or =... ad a otherwse. The = j = exstece o the requred vector = (... s guarateed sce the par o matrces ( A c s observable. The proposed observer s o the orm: = ( z... z z = z = ( z... z θ ( c z y z = ( z... z θ ( c z y =... (7 where θ s a costat sucetly large. The proo s based o the quadratc error Lyapuov ucto V ( e : = e Δθ PΔθ e where e : = z x Δ θ : = dag( θ θ... θ ad P R s a symmetrc postve dete matrx that satses P( A c ( A c P μ I or certa costat μ > (see [] or detals. Ater some computatos we guaratee that or all x z R L ( R ; ( R ad P L θ max the soluto o (4 wth μ = ( z... z z = z = ( z... z θ ( c z c x z = ( z... z θ ( c z c x =... (8 ad tal codto ( x( = ( x z R correspodg to v L ( R ; R satses the ollowg estmates or all t ad =... : θ μ x( θ exp z x 4 P (9 P θ sup v( μ θ μ z ( x ( θ exp z x 4 P ( P θ sup v( μ where are costats such that > x x Px x or all x R. It ollows rom (9 ad (6 that system (7 s a robust observer or system (4 wth respect to measuremet errors. Moreover usg equalty (5 or = ad ( we obta that or all ( x z R L ( R ; R the soluto o (4 wth (8 ad tal codto ( x( = ( x z R correspodg to v L ( R ; R satses the estmate or all t : ( z ( z ( ( x ( x ( ( L θ P ( L θ μ θ μ exp z 4 P sup v( x ( It ollows rom Theorem 3. ad equalty ( that the ollowg system 54

6 47th IEEE CDC Cacu Mexco Dec. 9-8 ThC.4 ( = ( z (... z ( z = z z = ( z... z ( = ( z (... z ( θ ( c w& ( = ( z ( z ( = y( ( θ ( c =... t [ ( s a robust sampled-data observer or system (4 wth respect to measuremet errors provded that the upper dameter o the samplg partto r > satses the equalty: P r ( L θ < (3 μ Notce that sce P L θ max t ollows rom μ (3 that the upper dameter o the samplg partto r > must ecessarly be less tha μ P ( Lμ max( μ P L V. CONCLUDING REMARS The preset wor developed a desg method or olear sampled-data observers based o a avalable cotuoustme desg coupled wth a ter-sample output predctor. I addto to beg tutvely meagul ey attractve eatures o the proposed sampled-data observer clude that t provdes checable sucet codtos or robustess wth respect to measuremet errors. t provdes a explct ormula or estmatg the maxmum allowable samplg perod. 3 t provdes explct bouds or the estmato error betwee samplg stats. 4 t provdes robustess wth respect to perturbatos o the samplg schedule. REFERENCES [] Aeyels D. O the Number o Samples Necessary to Acheve Observablty Systems ad Cotrol Letters ( [] Alamr M. Optmzato Based Nolear Observers Revsted Iteratoal Joural o Cotrol 7( [3] Alessadr A. M. Bagletto T. Pars ad R. Zoppol A Neural State Estmator wth Bouded Errors or Nolear Systems IEEE Trasactos o Automatc Cotrol 44( [4] Arca M. ad D. Nesc A Framewor or Nolear Sampled-Data Observer Desg va Approxmate Dscrete-Tme Models ad Emulato Automatca [5] Astrom. J. ad B. Wttemar Computer-Cotrolled Systems Theory ad Desg Pretce Hall New Jersey [6] By E. ad M. Arca A Hybrd Redesg o Newto Observers the Absece o a Exact Dscrete-Tme Model Systems ad Cotrol Letters [7] Besaco G. H. Hammour ad S. Beamor State Equvalece o Dscrete-Tme Nolear Cotrol Systems to State Ae Form up to Iput/Output Ijecto Systems ad Cotrol Letters [8] Boutayeb M. ad M. Darouach A Reduced-Order Observer or Nolear Dscrete-Tme Systems Systems ad Cotrol Letters [9] Calao C. S. Moaco ad D. Normad-Cyrot O the Observer Desg Dscrete-Tme Systems ad Cotrol Letters [] Cccarela G. M. Dalla Mora ad A. Germa A Robust Observer or Dscrete-Tme Nolear Systems Systems ad Cotrol Letters [] El Assoud A. E. H. El Yaagoub ad H. Hammour Nolear Observer Based o the Euler Dscretzato Iteratoal Joural o Cotrol 75( [] Gauther J. P. ad I. upa Determstc Observato Theory ad Applcatos Cambrdge Uversty Press. [3] ag W. Movg Horzo Numercal Observers o Nolear Cotrol Systems IEEE Trasactos o Automatc Cotrol 5( [4] araylls I. ad C. ravars O the Observer Problem or Dscrete- Tme Cotrol Systems IEEE Trasactos o Automatc Cotrol 5( 7-5. [5] araylls I. A System-Theoretc Framewor or a Wde Class o Systems I: Applcatos to Numercal Aalyss Joural o Mathematcal Aalyss ad Applcatos 38( [6] araylls I. A System-Theoretc Framewor or a Wde Class o Systems II: Iput-to-Output Stablty Joural o Mathematcal Aalyss ad Applcatos 38( [7] araylls I. ad Z.-P. Jag A Small-Ga Theorem or a Wde Class o Feedbac Systems wth Cotrol Applcatos SIAM Joural Cotrol ad Optmzato 46( [8] azatzs N. ad C. ravars Nolear Observer Desg Usg Lyapuov s Auxlary Theorem Systems ad Cotrol Letters [9] azatzs N. ad C. ravars Dscrete-Tme Nolear Observer Desg Usg Fuctoal Equatos Systems ad Cotrol Letters [] reer A.J. ad A. Isdor Learzato by Output Ijecto ad Nolear Observers Systems ad Cotrol Letters [] Lee W. ad. Nam Observer Desg or Autoomous Dscrete- Tme Nolear Systems Systems ad Cotrol Letters [] L W. ad C. I. Byres Remars O Learzato o Dscrete-Tme Autoomous Systems ad Nolear Observer Desg Systems ad Cotrol Letters [3] Moraal P.E. ad J.W. Grzzle Observer Desg or Nolear Systems wth Dscrete-Tme Measuremets IEEE Trasactos o Automatc Cotrol [4] Rao C. V. J. B. Rawlgs ad D. Q. Maye Costraed State Estmato or Nolear Dscrete-Tme Systems: Stablty ad Movg-Horzo Approxmatos IEEE Trasactos o Automatc Cotrol 48( [5] Respode W. A. Pogromsy ad H. Njmejer Tme Scalg or Observer Desg wth Learzable Error Dyamcs Automatca [6] Sotag E.D. "Mathematcal Cotrol Theory" d Edto Sprger- Verlag New Yor 998. [7] Tsas J. "Further Results o the Observer Desg Problem Systems ad Cotrol Letters [8] Xao M. N. azatzs C. ravars ad A.J. reer Nolear Dscrete-Tme Observer Desg wth Learzable Error Dyamcs IEEE Trasactos o Automatc Cotrol 48( [9] Zmmer G. State Observato by o-le Mmzato Iteratoal Joural o Cotrol 6(