Nonlinear Observers for Perspective Time-Varying Linear Systems

Transcription

1 Nonlinear Observers for Perspecive Time-Varying Linear Sysems Rixa Abursul an Hiroshi Inaba Deparmen of Informaion Sciences Tokuo Denki Universiy Hikigun, Saiama JAPAN Absrac Perspecive ynamical sysems arise in machine vision problems, in which only perspecive observaion is available, an he essenial problem is o esimae he sae an /or unknown parameers for a moving rigi boy base on he observe informaion This paper proposes an suies a Luenberger-ype observer for perspecive ime-varying linear sysems In paricular, assuming a given perspecive ime-varying linear sysem o be Lyapunov sable an o saisfy some sor of observabiliy coniion, i is shown ha he esimaion error converges exponenially o zero Finally, a simple numerical example is presene o illusrae he resul obaine Keywor: nonlinear observer, perspecive sysem, ime-varying sysems, observabiliy, exponenial sabiliy Inroucion Perspecive ynamical sysems arise in mahemaically escribing ynamic machine vision problems Such machine vision problems have been formulae an suie in a number of approaches in he framework of sysems heory See, eg, []-[9] One of ineresing approaches iscusse in he recen papers[7, 8], is o formulae such a problem by inroucing a noion of implici sysems in orer o ake ino accoun he consrains inrinsically involve in ynamic machine vision, an o evelop observers o esimae he unknown saes of such sysems The presen auhors wih B K Ghosh[] have propose a Luenberger-ype nonlinear observer for perspecive ime-invarian linear sysems an shown ha uner some suiable assumpions on such a sysem he esimaion error converges exponenially o zero In he presen paper he class of perspecive ime-varying linear sysems is consiere an an exension of he previous resul o his class is examine Such a perspecive ime-varying linear sysems consiere in his paper is escribe in he following form: { ẋ() A()x()+v(), x( )x R n y() h(cx()) () where x() R n is he sae, v() R n he exernal inpu, A( ) : R R n n a coninuous an boune marix-value funcion an y() R m he perspecive observaion of he sae x() Furher a nonlinear funcion h : R m+ R m ha prouces perspecive obser- This work was suppore in par by he Gran-Ai for General Scienific Research of he Japanese Minisry of Eucaion, Science an Culure uner Gran C vaion y() is given in he following special form [ ] T ξ ξ m h(ξ) :,ξ m+ ξ m+ ξ m+ () ξ [ ξ ξ m ξ m+ ] T R m+ an C R (m+) n is a marix wih m < n an rankc m + A simplifie 3-imensional vision problem consiere in he previous works[]-[] can be escribe as a special case of Sysem () In fac, since in such a machine vision problem he perspecive observaion is given essenially in he form [ ] T y [y y ] T x x, x 3 x 3 one can se n 3, m anc I 3 (he ieniy marix) in he Sysem () In he presen paper, he sae esimaion problem for Sysem () is invesigae In paricular, a nonlinear observer of he Luenberger-ype [] for Sysem () is propose, an he convergence problem of he nonlinear observer is examine More precisely, i is shown ha, uner suiable assumpions on Sysem (), incluing ha i is Lyapunov sable an saisfies some sor of observabiliy coniion, i is possible o consruc a nonlinear observer of he Luenberger-ype whose esimaion error converges exponenially o zero Finally, some numerical simulaion resuls using a simple example are presene o illusrae he resul obaine The numerical resuls show ha he observer works quie well Preliminaries In his secion, we inrouce basic noaions, an iscuss an imporan resul on he sabiliy of linear sysems

2 Le R an C enoe he fiels of real an complex numbers, respecively, an R n an C n he Eucliean vecor spaces wih norm x For a marix M R m n or M C m n, he marix norm, again enoe M, is efine by M : max{ Mx x } (3) For a vecor or marix ζ, enoe he ranspose by ζ T The se of funcions f : R R wih he norm given by { /p f p : f() } p, p [, ) (4) sup{ f() R}, p is enoe by L p (R) Finally, an ieniy marix is enoe by I where is orer is usually unersoo from he conex Now he following lemma is prove, which plays an essenial role o prove our main heorem Is proof is sraigh forwar, bu for compleeness a brief proof is given here Lemma Le A( ) L n n (R) an assume ha A() is coninuous Consier he linear sysem of ifferenial equaions ẋ() A()x(), x( )x R n, (5) an assume ha (5) has a unique smooh soluion for any R an any x( ) R n Then, (5) is exponenially sable, ha is, for each R here exis some α>anβ>such ha for any x( ) R n he soluion x() saisfies x() β x( ) e α( ) (6) if an only if for each R here exiss some γ> such ha for any x( ) R n he soluion x() of(5) saisfies x() γ x( ) (7) Proof: The necessiy is obvious, an hence only he sufficiency is prove Firs, assume ha (7) is saisfie Le he sae ransiion marix for (5) be enoe by Φ(, τ), an express i in he column vecor from Φ(, τ) [ ϕ (, τ) ϕ n (, τ) ] Then for each uni vecor [ e k k h ] T, k,,n, an for any R one obains from (7) ha Φ(, )e k Furher for Φ k (, )e k γ, k,,n Φ(, ) max Φ(, )x x n max x kφ k (, ) x + +x n k n Φ k(, ), k (8) which ogeher wih (8) gives Φ(, ) n k nγ < Φ k (, ) Finally, i follows from Theorem 3 on page 9 in[] ha Sysem (5) is exponenially sable, ha is, (6) is saisfie 3 Nonlinear Observers In his secion, we propose a Luenberger-ype observer for a perspecive linear sysem of he form (), an presen our main heorem, which saes ha uner some reasonable assumpions on he sysem i is possible o consruc a nonlinear observer of he Luenberger-ype whose error converges exponenially o zero Firs, noice ha a mos general full-orer sae observer for Sysem () generally has he form ˆx() ϕ(ˆx(),v(),y(),), ˆx()ˆx R n (9) wih he propery ha if ˆx( )x( ) hen he soluion ˆx() of (9) coincies compleely wih he soluion x() of Sysem () for any v( ), ie, ˆx() x() for all an any v( ) Therefore, uner he coniion ha boh sysem ()an (9) have heir unique soluions, i is possible o assume ha he funcion ϕ(ˆx, v, y, ) in(9)is of he form ϕ(ˆx, v, y, ) A()ˆx + v + r(ˆx, y, ) where r(ˆx, y, ) is any funcion saisfying he coniion ha r(x, h(cx),) for all an all x R n Furher, i is possible o assume ha he funcion r(ˆx, y, ) has he form r(ˆx, y, ) K(y, ˆx, )[y h(c ˆx)], where K(y, ˆx, ) is any sufficienly smooh funcion Hence, as an observer for Sysem (), i is possible o consier a nonlinear observer of he Luenberger-ype[], ha is, ˆx() A()ˆx()+v() +K(y(), ˆx(),)[y() h(c ˆx())], () x( )ˆx R n where ˆx() R n is an esimae of he sae x() an K(y, ˆx, ) is a suiable marix-value funcion K : R m R n R R n m, calle an observer gain marix Nex, we nee o choose a suiable form of he gain marix K(y, ˆx, ) To o so, firs se ξ [ ] T ξ ξ m ξ m+ : Cx, ˆξ [ ] T ˆξ ˆξm ˆξm+ : C ˆx, C [ ] C T C T m C T T m+ R (m+) n,

3 an hen, using Sysem () an (), evaluae ifference y h(c ˆx) as follows: ξ ˆξ ξ m+ ˆξ m+ y h(c ˆx) h(cx) h(c ˆx) ξ m ˆξ m ξ m+ ˆξ m+ ξ ξ m+ ξ ξ ˆξ ξ ˆξ m+ ξ ˆξ m+ ξ ξ m m+ ˆξ m+ ξ m+ y C (x ˆx) y C (x ˆx) C m+ ˆx y m C m+ (x ˆx) C B(y)C(x ˆx) B(y)Cρ () m+ˆx C m+ˆx where B(y) is he marix-value funcion efine as B(y) : [ I m y ] R m (m+) () an ρ R n is he esimaion error vecor efine as ρ : x ˆx (3) Now, consiering he form of (), we propose a gain marix K(y, ˆx, ) of he form: K(y, ˆx, ) C m+ˆxp ()C T B T (y) (4) where P () R n n is an appropriaely chosen marixvalue funcion, which is consiere o be a free parameer for he gain marix In fac, (4) urns ou o be a goo choice as seen in he sequel Nex, we make various assumpions on Sysem (), which seem o be necessary an/or reasonable from he viewpoin of machine vision Assumpion Sysem () is assume o saisfy he following coniions (i) (ii) Sysem () is Lyapunov sable, ha is, here exiss b> such ha Φ(, s) b<,, s R where Φ(, s) is he ransiion marix for A() The perspecive observaion vecor y() isaconinuous an boune funcion of, hais, y( ) C m [, ) L m [, ) (iii) There exis T>anε> such ha Φ T ( + τ,)c T B T (y( + τ)) B(y( + τ))cφ( + τ,)τ εi, (5) Remark All he coniions given in Assumpion are reasonable assumpions from he viewpoin of machine vision (i) (ii) (iii) The coniion (i) is impose o ensure ha if v() hen he moion of a moving boy akes place wihin a boune region The coniion (ii) is impose o ensure ha he moion is smooh enough an akes place insie a conical region cenere a he camera o prouce a coninuous an boune measuremen y() on he image plane The coniion (iii) ensures some sor of observabiliy for he perspecive ime-varying sysem () In fac, he inequaliy (5) (C, A()) is an uniformly observerble pair This is verifie in Proposiion below Proposiion Consier Sysem (), an assume ha Assumpion (iii) is saisfie Then he pair (C, A()) is uniformly observable, ha is, here exis T > an ˆε > such ha Φ T ( + τ,)c T CΦ( + τ,)τ ˆεI, Proof: Using (), efine a symmeric nonnegaive marix by S(y) : B T (y)b(y) [ I y ] T [ ] [ I ] y I y y T y R (m+) (m+) Then i is no ifficul o see ha e(λi m+ S(y)) λ(λ ) m (λ y ) an hence eigenvalues λ(s(y)), an + y for any y R m Thus ( + y( ) )I S(y()) ( ), an hence i follows from (5) ha Φ T ( + τ,)c T CΦ( + τ,)τ T Φ T ( + τ,)c T S(y( + τ)) + y( ) CΦ( + τ,)τ εi ˆεI,, + y( ) where ˆε : ε, + y( ) which verifies he saemen For a general ime-varying linear sysem wih sysem marix A() an oupu marix C(), he observabiliy is 3

4 compleely eermine by he pair (C(),A()), ancompleely inepenen of he inpu an iniial sae However for a nonlinear sysem he observabiliy is generally epenen on he inpu an iniial sae The inequaliy (5) requires ha no only he pair (C, A()) is uniformly observable, bu also ha he sysem () is uniformly observable (more precisely uniformly eecable) along he rajecory x() ha is generae by a given inpu v() an iniial sae x( ) This fac is no easy o irecly verify, bu i is seen from he nex heorem ha saes exisence of a nonlinear observer uner Assumpion (iii) because observabiliy (more precisely, eecabiliy) is a necessary coniion for such an exisence Now, i is reay o sae our main heorem ρ T ( )P ( )ρ( ) B(y(s))Cρ(s) s, > Theorem (Luenberger-ype Nonlinear Observers) which immeiaely leas o Assume ha Sysem () saisfies Assumpion an he ρ T ()P ()ρ() ρ T ( )P ( )ρ( ) marix ifferenial inequaliy A T ()P ()+P()A()+ P P ( ) ρ( ), (), (6) B(y(s))Cρ(s) s ρt ( )P ( )ρ( ) (3) has a symmeric posiive-efinie marix-value soluion P ( ) : [, ) R n n such ha P () ρ( ) P () δi >, (7) for some δ> Consier a nonlinear observer of he Luenberger-ype (), ha is, ˆx() A()ˆx()+v() +K(y(), ˆx(),)[y() h(c ˆx())], (8) ˆx( )ˆx R n wih a gain marix of he form (4) K(y, ˆx, ) :C m+ˆxp ()C T B T (y) (9) where B(y) is given by () Then, he following saemens hol (i) (ii) The esimaion error ρ() : x() ˆx() saisfies he ifferenial equaion ρ() [A() P ()C T B T (y()) B(y())C]ρ(), ρ( ) R n () ρ() converges exponenially o zero, ha is, here exis α>,β > such ha ρ() : x() ˆx() βe α ρ( ), () Proof The saemen (i) can be easily verifie by iffereniaing ρ() : x() ˆx() an subsiuing (), (8) an (9) o he resulan Therefore, only he saemen (ii) will be prove in eail To prove his, i suffices o show by virue of Lemma ha he esimaion error ρ() saisfies he inequaliy ρ() γ ρ( ) () 4 where γ is a consan, possibly epenen on R, bu inepenen of ρ( ) Firs, using () an (6), one can easily obain (ρ()t P ()ρ()) ρ() T (A T ()P ()+P()A()+ P ())ρ() ρ T ()C T B T (y())b(y())cρ() B(y())Cρ() Hence, inegraing he above from o gives, ρ T ()P ()ρ() Nex, using () an Assumpion (ii), one can obain B(y( )) : sup B(y()) [I m sup sup y()] {+ y() } + y( ) (4) Furher, expressing () as ρ() A()ρ() θ() (5) where θ() :P ()C T B T (y())b(y())cρ(), (6) one can wrie is soluion of (5)in he form ρ( + s) Φ( + s, )ρ() +s (7) Φ( + s, τ)θ(τ)τ, s Now, using (5) in Assumpion an (7) one evaluae ρ() ρ T ()(εi)ρ() ε [ { ρ T () Φ T ( + s, )C T B T (y( + s)) ε B(y( + s))cφ( + s, )s} ρ()] [ ] T B(y( + s))cφ( + s, )ρ() s ε [ B(y( + s))c {ρ( + s) ε +s } ] + Φ( + s, τ)θ(τ)τ s ε [Q + Q ], (8)

5 where Q : Q : B(y( + s))cρ( + s) s B(y( + s))c +s Φ( + s, τ)θ(τ)τ s Before evaluing Q an Q, an inequaliy is prove Consier a funcion f L (R), an le R an T> be given Then +T f (τ)τ T +T +T τ τ f (τ) + f (τ)(τ )τ + +T f (τ)τ + T +T +T +T τ τ τ T f (τ) f (τ)(τ τ + T )τ f (τ)τ T f (τ)τ (9) Now, i is reay o evaluae Q Using (9) an (3), Q +T B(y( + s))cρ( + s) s B(y(τ))Cρ(τ) τ T B(y())Cρ() (3) T P () ρ( ) : γ ρ( ) where γ > is a consan, possibly epenen on R, bu inepenen of ρ( ) Nex, using (4), (i) in Assumpion an (9), Q is evaluae as follows: Q B(y( + s))c +s Φ( + s, τ)θ(τ)τ s ( + y( ) ) C +s Φ( + s, τ)θ(τ)τ s ( + y( ) ) C b T +T θ(τ) τs (+ y( ) ) C b T { +T θ(τ) τ (+ y( ) ) C b T +T θ(τ) τ ( + y( ) ) C b T 3 } s θ(), (3) an furher, using (6), (3) an (7), one obains θ() P ()C T B T (y())b(y())cρ() δ ( + y( ) ) C B(y())Cρ() δ ( + y( ) ) C P ( ) ρ( ) (3) Thus, (3) an (3) lea o Q δ ( + y( ) )4 C 4 b T 3 P ( ) ρ( ) : γ ρ( ) (33) where γ > is a consan, possibly epenen on R, bu inepenen of ρ( ) Finally, i follows from (8), (3) an (33) ha ρ() ε (γ + γ ) ρ( ) : γ ρ( ) which is he esire inequaliy () 4 Compuer Simulaion A very simple perspecive linear sysem of he form () is use for numerical simulaions o illusrae he resul obaine in he previous secions The sysem we use is given by he following aa: A() C sin(π) sin(π) +sin(π) sin(π), v() π [ sin(π) cos(π) ] T, x [ ] T, Now, we roughly check he coniions (i)-(iii) in Assumpion Firs, noice ha A() is perioic an hence by virue of Floque heorem Φ(, ) is also perioic wih 5

6 x iniial poin x x -5 - Figure : The rajecory of x() [x () x () x 3 ()] T y 4 3 iniial poin Figure : The perspecive obsevaion y() [y () y ()] T y respec o for each,sohaa() is Lyapunov sable, verifying ha (i) is saisfie Nex, noice ha v() is again perioic an hence x() is boune an saisfies inf C 3x() inf x 3() > (In fac, his is seen from Fig an Fig above) Thus i is clear ha he perspecive observaion y() is coninuous an boune, showing ha (ii) is saisfie Finally, noice ha rank C 3he pair (C, A()) is observable bu i seems o be no easy o irecly check he inequaliy (5) However i woul be quie reasonable o expec form he observabiliy of pair (C, A()) ha (iii) is saisfie In fac, his seems o be verifie from he numerical simulaion resul given in Fig 3 ha he esimaion error of a esigne nonlinear observer converges o zero The aa use for our nonlinear observer ˆx() ofhe Luenberger-ype (8) are given by ˆx [ 3 ] T, P() iag{,, } where he free parameer P () is compue so as o saisfy he inequaliy ifferenial equaion (6) In fac, in his example, since A() is ani-symmeric ie, A T () A(), A T ()P () +P ()A() an hence P () can be chosen o be any consan symmeric posiive-efinie marix P> The ime evoluions of each componen of he rue sae x() an he esimae sae ˆx() areshowninfig 3 The resuls seem o show ha he propose nonlinear 4 x xê - True sae Esimae sae x xê x 3 xê ime[s] Figure 3: The ime evaluions of each componen of x() an ˆx() observer works quie well 5 Concluing Remarks This paper suie a nonlinear observer for perspecive ime-varying linear sysems arising in machine vision Firs a Luenberger-ype nonlinear observer was propose, an hen i was shown uner some reasonable assumpions on a given perspecive sysem ha i is possible o consruc such a nonlinear observer whose esimaion error converges exponenially o zero Furher, o illusrae effeciveness of he propose observer, a simple numerical example was presene, an his resul seems o inicae ha he propose nonlinear observer works quie well There are several fuure problems o be examine Firs, noice ha (5) in Assumpion (iii) is a sufficien coniion for he rajecory x() generae by a given inpu v() an iniial sae x( ) o be uniformly observable, bu i has no been examine ha he coniion (5) is how close o a necessary coniion for he observabiliy Thus seeking some necessary an sufficien coniions woul be an improan problem o be examine in he fuure Furher, we have no examine how o effecively check he inequaliy (5) This is a very imporan prolbem, bu as in he case of usual nonlinear observers i is very ifficul o expec ha here is a reasonably effecive meho for checking he inequaliy (5) because i conains he oupu y() which is a funcional of he inpu v() an iniial sae x( ) References [] R Abursul, H Inaba an B K Ghosh, Luenberger-ype observers for perspecive linear sysems, Proceeings of he European Conrol Conference, CD-ROM, Poro, Porugal, Sepember 6

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