Nonlinear observation over erasure channel

Nonlinear observaion over erasure channel Ami Diwadkar Umesh Vaidya Absrac In his paper, we sudy he problem of sae observaion of nonlinear sysems over an erasure channel. The noion of mean square exponenial sabiliy is used o analyze he sabiliy propery of observer error dynamics. The main resuls of his paper prove, fundamenal iaion arises for mean square exponenial sabilizaion of he observer error dynamics, expressed in erms of probabiliy of erasure, and posiive Lyapunov exponens of he sysem. Posiive Lyapunov exponens are a measure of average expansion of nearby rajecories on an aracor se for nonlinear sysems. Hence, he dependence of iaion resuls on he Lyapunov exponens highlighs he imporan role played by non-equilibrium dynamics in observaion over an erasure channel. The iaion on observaion is also relaed o measure-heoreic enropy of he sysem, which is anoher measure of dynamical complexiy. The iaion resul for he observaion of linear sysems is obained as a special case, where Lyapunov exponens are shown o emerge as he naural generalizaion of eigenvalues from linear sysems o nonlinear sysems. I. INTRODUCTION The problem of sae esimaion of sysems over erasure channels has araced a lo of aenion laely, given he imporance of his problem in he conrol of sysems over a nework ]. The problem of sae esimaion wih inermien observaion was firs sudied in 2], 3]. In 4], 5], sae esimaion over an erasure channel wih differen performance merics on he error covariance is sudied. In 4], under some assumpions on sysem dynamics, i is proved ha here exiss a criical non-erasure probabiliy below which he error covariance is unbounded. A Markov jump linear sysem framework is used o model he sae esimaion problem wih inermien measuremen and o provide condiions for he convergence of error covariance in 6]. In 7], sae esimaion over erasure channel wih Markovian packe loss is sudied. However, all he above resuls are developed for linear ime invarian LTI) sysems. There is no sysemaic resul ha addresses he sae esimaion problem for nonlinear sysems over erasure channels. Thus here is a need for exension and developmen of such resuls for nonlinear sysems, wih regard o heir applicaions in nework sysems consising of nonlinear componens, such as power sysem neworks, biological neworks, and Inerne communicaion neworks. In his paper, we sudy he problem of sae observaion of nonlinear sysems over an erasure channel, wih he objecive o develop iaion resuls for sae observaion. We expec he iaion resuls for he sae observaion problem, o provide useful insigh ino he more challenging problem of sae esimaion over an erasure channel. The erasure channel is modeled as an on/off ernoulli swich. We use mean square exponenial MSE) sabiliy o sudy he sae observaion problem over an erasure channel. The main resul of his paper shows, ha a fundamenal iaion arises in MSE sabilizaion of he observer error dynamics. This iaion is expressed in erms of erasure probabiliy and global insabiliy of he nonlinear sysem. In paricular, under a cerain ergodiciy assumpion, we show he insabiliy of a nonlinear sysem can be expressed in erms of he sum of posiive Lyapunov exponens of he sysem. Using Ruelle s inequaliy from ergodic heory of a dynamical sysem 8], he sum of he posiive Lyapunov exponens can be relaed o he enropy of a nonlinear sysem. Hence, he iaion resul can be A. Diwadkar is a graduae suden wih he Deparmen of Elecrical and Compuer Engineering, Iowa Sae Universiy, Ames, IA, 5 diwadkar@iasae.edu U. Vaidya is wih he Deparmen of Elecrical and Compuer Engineering, Iowa Sae Universiy, Ames, IA, 5 ugvaidya@iasae.edu inerpreed in erms of he enropy of a nonlinear sysem. Our resul involving Lyapunov exponens of a non-rivial oher han equilibrium poin) invarian measure is also he firs o highligh he imporan role played by he non-equilibrium dynamics in he iaions on nonlinear observaion. There are wo main conribuions of his paper. Firs, i adops and exends he formalism from erogodic heory of random dynamical sysems o sudy he problem of nonlinear observaion over an erasure channel. Second, he resul provides an analyical relaionship beween he maximum olerable channel uncerainy i.e., he maximum erasure probabiliy) and he inabiliy of he sysem o mainain mean square exponenial sabiliy of he observer error dynamics. The organizaion of his paper is as follows. In secion II, we discuss he problem and provide necessary assumpions and sabiliy definiion. In secion III, we prove he main resuls of his paper. A simulaion example is presened in secion IV, followed by conclusions in secion V. II. PRELIMINARIES The se-up for nonlinear observaions wih a unique erasure channel a he oupu is described by he following equaions: x + = fx ), y = ξ hx ), ) where x X R N is he sae, y Y R M is he oupu, and ξ {, } is a ernoulli random variable wih probabiliy disribuion Probξ = ) = p for all, wih < p <, and independen of ξ τ for τ. The IID independen idenically disribued) random variable, ξ, models he erasure channel beween he plan and he observer hrough which all he oupus are sen o he observer simulaneously. Remark : To make he problem ineresing, we assume ha M < N and < p <. The < p assumpion implies ha he sysem dynamics, x + = fx ), is unsable and hence requires some nonzero probabiliy of erasure for he observer o work. We now provide he following definiion of an observabiliy rank condiion for nonlinear sysems 9]. Definiion 2 Observabiliy Rank Condiion): Consider he map θ N x) : X Y... Y }{{} N θ N x) := hx), hfx)),..., hf x)) N. 2) The sysem ) is said o saisfy he observabiliy rank condiion a x, if ) θ N x) rank = N. We make following assumpion on he sysem dynamics. Assumpion 3: The sysem mapping, f, and oupu funcion, h, are C r funcions of x, for r, wih f) =, h) =, and he Jacobian f x) is uniformly bounded above and below for all x X. Furhermore, he sysem saisfies he observabiliy rank condiion Definiion 2) and here exis α θ > and β θ >, such ha α θ I N < x) θn θn x) < β θi N 3) for all x X and, I N is he N N Ideniy marix. Remark 4: Assumpion 3 and in paricular he observabiliy rank condiion are essenial for he observer design for he sysem wih no erasure a he oupu. The sochasic noion of sabiliy we use o analyze he observer error dynamics is defined in he conex of a general random dynamical sysem RDS) of he form x + = Sx, ζ ), where x X R N, ζ W = {, } for, are IID random variables wih

probabiliy disribuion Probζ = ) = p. The sysem mapping S : X W X is assumed o be a leas C wih respec o x X and measurable w.r. ζ. We assume x = is an equilibrium poin, i.e., S, ζ ) =. The following noion of sabiliy can be defined for RDS ], ]. Definiion 5 Mean Square Exponenial MSE) Sable): The soluion, x =, is said o be MSE sable for x + = Sx, ζ ), if here exis posiive consans L < and β <, such ha E ζ x+ 2] Lβ x 2, for Lebesgue almos all iniial condiion, x X, where E ζ ] is he expecaion aken over he sequence {ζ,..., ζ }. III. MAIN RESULTS The main resuls of his paper are derived under he following assumpion on he observer dynamics. Assumpion 6: The observer gain, K, is assumed deerminisic and no an explici funcion of he channel erasure sae ξ nor is hisory i.e., ξ ). The observer dynamics is assumed o be of he form: ˆx + = fˆx ) + Ky ) Kŷ ), ŷ = ξ hˆx ), 4) where ˆx X is he observer sae, ŷ Y is he observer oupu, and K : Y X is he observer gain and assumed o be a C r funcion of y, for r, and saisfies K) =. Thus he propery K) = and ξ {, }, allows us o rewrie he observer dynamics 4) as follows: ˆx + = fˆx ) + ξ Khx )) ξ Khˆx )). 5) We assume ha he observer oupu ŷ is an explici funcion of channel sae, ξ. This assumpion is jusified by assuming a TCP-like proocol, where he observer receives an immediae acknowledgemen of he channel erasure sae 4]. Remark 7: In 4], he problem of sae esimaion for an LTI sysem over an erasure channel is sudied. The opimal esimaor gain ha minimizes he error covariance is shown o be a funcion of he channel erasure sae hisory. Wih he esimaor gain, a funcion of he channel erasure sae hisory, he resuls in 4] only prove he error covariance will remain bounded and no converge o a seady sae value, unlike he regular Kalman filering problem for an LTI sysem wih no loss of measuremen. Hence, we conjecure Assumpion 6) on he observer gain, no being a funcion of he channel erasure sae or is hisory, is necessary for he error dynamics o be MSE sable. We firs prove Lemma 8 ha provides a necessary condiion for MSE sabiliy of he error dynamics x ˆx in erms of MSE sabiliy of he linearized error dynamics. Lemma 8: Consider he observer dynamics in Eq. 5) and le he error dynamics i.e., e = x ˆx ) be MSE sable Definiion 5). Then, he following linearized error dynamics, η R N, ) f η + = x) K h ξ x) η, x + = fx ) 6) is also MSE sable, i.e., here exis posiive consans L < and β <, such ha E ξ η+ 2] Lβ η 2. The funcions K and h in 6) are he observer gain and oupu funcion, respecively, from Eq. 4). Proof: Define gx, ξ ) := fx ) ξ Khx )) and Ax, ξ ) := g x, ξ). Then using Mean Value Theorem for he vecor valued funcion, he error dynamics, can be wrien as ) g e + = gx, ξ ) gx e, ξ ) = x se, ξ)ds e ) = Ax k se k, ξ k )ds e, k= Here e is an implici funcion of he iniial error e, iniial sae x, and he sequence of uncerainies ξ. We define k x, ξ k, e ) := Ax k se k, ξ k )ds and x, ξ, e ) := k= kx, ξ k, e ). This gives E ξ e+ 2] = E e ] +e + = e E ξ x, ξ, e ) x, ξ, e ) ] e. Using Assumpion 3, we know here exiss a posiive consan L <, such ha k x, ξ k, αe ) < L for Lebesgue almos all x X and for some scalar, α >. Le k x, ξ k, αe ) ij denoe he i h row j h column enry in k x, ξ k, αe ). Now consider a sequence, {α l } l=, such ha l α l =. Then, we have by Dominaed Convergence Theorem 2] and coninuiy of Ax k se k, ξ k ), l k x, ξ k, α l e ) ij = k x, ξ k, ) ij which implies l k x, ξ k, α l e ) = k x, ξ k, ). Hence, we have l k x, ξ k, α l e ) = k x, ξ k, ). 7) From MSE sabiliy of he error, we obain e E ξ x, ξ, e ) x, ξ, e ) ] e Lβ e e, for some posiive consans L < and β <. Since he above inequaliy is rue for any iniial error, his will be rue if he iniial error vecor used o compue he produc of marices is scaled by α l, where l α l =. Subsiuing α l e for e, we can wrie e E ξ x, ξ, α l e ) x, ξ, α l e ) ] e Lβ e e. Now, leing l and by Faou s Lemma, we have ] e E ξ l x, ξ, α l e ) x, ξ, α l e ) e e l E ξ x, ξ, α l e ) x, ξ, α l e ] ) e Lβ e e. 8) Thus, using 7) and 8), we obain e E ξ x, ξ, ) x, ξ, ) ] e Lβ e e, where x, ξ, ) is he produc of he Jacobian marices Ax, ξ ), wih zero iniial error and compued along he nominal rajecory, x + = fx ). Hence, ) ) ] E ξ e Ax k, ξ k ) Ax k, ξ k ) e Lβ e e. k= k= Since he marices in he above equaion are independen of e, we can subsiue η for e. Now, using he evoluion of η from Eq. 6), we obain he desired resul. Our nex heorem provides he necessary condiion for MSE sabiliy of he linearized error dynamics. Theorem 9: Le he η dynamics for he sysem 6) be MSE sable Definiion 5). Then, here exiss a marix funcion of x, P x ), such ha γ I P x ) γ 2I and E ξ A x, ξ )P x +)Ax, ξ ) ] < P x ), 9) for some posiive consans γ, γ 2, where x + = fx ) and Ax, ξ ) = f K h x) ξ hx)) x) from 6). y Proof: To prove he necessary par, assume he sysem is MSE sable and consider he following consrucion of P x ). k ) j)) k P x ) = Ax j, ξ j) Ax j, ξ, k= E ξ k j= where E j ξ ] is he expecaion over he random sequence i {ξ i,..., ξ j}. The exisence of posiive consans γ, γ 2 follows from he fac ha η dynamics is MSE sable and he Jacobian f is j=

bounded from above and below. The inequaliy 9) follows from he consrucion of P x ). We have Corollary o he Theorem 9. Corollary : Le he RDS 6) be MSE sable. Then, here exiss a marix funcion of x, Qx ) and posiive consans γ and γ 2, such ha γ I Qx ) γ 2I and E ξ Ax, ξ )Qx )A x, ξ ) ] < Qx +). ) Proof: The proof follows from Theorem 9 and by consrucing Qx ) = P x ). Remark : We will refer o marix Qx ), saisfying he condiions ) of Corollary as he marix Lyapunov funcion. Our goal is o derive a necessary condiion for he MSE sabiliy of he linearized error dynamics; hereby, providing a necessary condiion for MSE sabiliy of he rue error dynamics. Lemma 2: The necessary condiion for exponenial mean square sabiliy of he linearized error dynamics 6) is given by p) M deax ))) 2 deq x )) <, ) deq x +)) for Lebesgue almos all x X. In ) Q x ) is a soluion of he following Riccai equaion, Q x +) = Rx )+Ax )Q x )A x ) Ax )Q x )C x ) I M + Cx )Q x )C x ) ) ) Cx )Q x )A x ), 2) where Rx ) is some symmeric posiive semi-definie marix. Furhermore, Q x ) is uniformly bounded above and below wih Ax ) := f h x), Cx) := x), x+ = fx), IM is M M ideniy marix, and p) is he probabiliy of erasure. Proof: Using he resul of Corollary, he necessary condiion for MSE sabiliy of 6) can be expressed in erms of he exisence of γ I Qx ) γ 2I, such ha γ, γ 2 > and, E ξ Ax, ξ )Qx )A x, ξ ) ] < Qx +), 3) where Ax, ξ ) = Ax ) ξ Kx)Cx ) and K Kx ) := hx)). Minimizing race of he lefhand side of 3) wih respec o Kx), we obain y K x ) = Ax )Qx )C x ) Cx )Qx )C x )) and Qx ) o saisfy Qx +) > Ax )Qx )A x ) p Ax )Qx )C x ) Cx )Qx )C x ) ) ) Cx )Qx )A x ). 4) I is imporan o noice ha he inequaliy 4) is independen of any posiive scaling i.e., if Qx ) saisfies he above inequaliy hen cqx ) also saisfies he above inequaliy for any posiive consan c. Since Qx ) is a marix Lyapunov funcion and hence lower bounded, i follows from Remark, ha here exiss a posiive consan > such ha Cx )Qx )C x ) p) I p M. Hence 4) implies following inequaliy o be rue Qx +) > Ax )Qx )A x ) Ax )Qx )C x ) I M + Cx )Qx )C x ) ) ) Cx )Qx )A x ). 5) Now define Q x ) := Qx), hen using he fac ha 5) is independen of posiive scaling, we obain following inequaliy for Q x ) Q x +) > Ax )Q x )A x ) Ax )Q x )C x ) IM + Cx )Q x )C x ) ) ) Cx )Q x )A x ). 6) Inequaliy 6) implies here exiss Rx ), such ha he following equaliy is rue. Q x +) = Rx )+Ax )Q x )A x ) Ax )Q x )C x ) I M + Cx )Q x )C x ) ) ) Cx )Q x )A x ). 7) For any fixed rajecory {x } generaed by he sysem, x + = fx ), he above equaliy resembles he Riccai equaion obained for he minimum covariance esimaor design problem for he linear ime varying sysem, where he marices Q x ) and Rx ) can be idenified wih he error and inpu noise covariance marices, respecively 3] wih oupu noise variance marix equal o ideniy marix. The difference beween he regular Riccai equaion obained from he minimum variance esimaor problem for he linear ime varying sysem and Eq. 7) is ha, he various marices appearing in 7) are parameerized by x insead of ime. Furhermore Q x ) as he soluion of Riccai-like equaion 7) is boh bounded above and below and is proved as follows. The sysem marices Ax ) and Cx ) saisfy Assumpion 3 along any given rajecory. Hence, he linearized sysem, η + = Ax )η, ζ = Cx )η, along any fixed rajecory is uniformly compleely reconsrucible as defined in 3] Definiion 6.6). I hen follows from 4] Lemmas 7. and 7.2) ha he covariance marix Q x ) is uniformly bounded above and below for all x X. The marix Q x ) saisfies 4) follows from he definiion of Q x ) i.e., Q x ) := Qx)) and he fac ha 4) is independen of posiive scaling. We obain, Q x +) > Ax )Q x )A x ) p Ax )Q x )C x ) Cx )Q x )C x ) ) ) Cx )Q x )A x ). 8) This proves ha Q x ) obained as a soluion of Riccai-like equaion is a valid marix Lyapunov funcion. To derive he required necessary condiion ), we ake deerminans on boh sides of 8) o obain de I N pc x ) Cx )Q x )C x ) ) ) Cx )Q x ) deax ))) 2 deq x )) <. 9) deq x +)) y Sylveser s deerminan Theorem i.e., dei N +GJ) = dei M + JG), G R N M, J R M N ), we obain de I N pc x ) Cx )Q x )C x ) ) ) Cx )Qx ) = p) M. 2) We obain he required inequaliy ) by combining Eqs. 9) and 2). The resuls of Lemma 2 will now be used o prove he main resuls of he paper under various assumpions on he sysem dynamics. Theorem 3 Linear Sysems): Le fx) = Ax wih x R N and hx) = Cx R M. Assume ha all eigenvalues λ k for k =,..., N of A have absolue value greaer han one. The necessary condiion

for he observer error dynamics o be MSE sable is given by N 2 p) M λ k ) <. 2) k= Proof: For he linear sysem, he soluion of Riccai-like equaion 2) from Lemma 2 leads o a consan marix Q independen of x. Hence he necessary condiion ) for he sabiliy will reduce o p) M dea 2 ) <. The required necessary condiion 2) hen follows by subsiuing dea 2 N 2. ) = k= k ) λ Remark 4: A careful examinaion of he proofs for Lemma 8 and 2, and Theorem 9 for he special case of linear sysems wih single oupu, reveals he necessary condiion 2) is also sufficien for MSE sabiliy of he linear sysem. Theorem 5 Nonlinear sysems on unbounded space): Consider sysem ) wih sysem mapping f and oupu h saisfying Assumpion 3 and sae space X possibly unbounded. The necessary condiion for MSE sabiliy of he observer error dynamics 4) is given by p) M deax ))) 2 deq x )) <, 22) deq x +)) for Lebesgue almos all x X, where Ax) = f x) and Qx) saisfy he Riccai-like Eq. 2). Proof: The proof follows by combining resuls from Lemmas 8 and 2, and Theorem 9. In Theorem 2, we show, for a nonlinear sysem evolving on a deqx )) deqx + )) compac sae space, he erm deax ))) 2 from 22) relaes o he sum of posive Lyapunov exponens of he sysem. For Theorem 2 we provide he following definiions 5]. Definiion 6 Physical measure): Le MX) be he space of probabiliy measures on X. A measure µ MX) is said o be invarian for x + = fx ) if µf )) = µ) for all ses X) orel σ-algebra generaed by X). An invarian probabiliy measure, µ, is said o be ergodic if any coninuous bounded funcion ϕ ha is invarian under f, i.e., ϕfx)) = ϕx), is µ almos everywhere consan. Ergodic invarian measure, µ, is said o be physical if n n n k= ϕf k x)) = X ϕx)dµx) for posiive Lebesgue measure of he iniial condiion x X and all coninuous funcion ϕ : X R. Definiion 7 Lyapunov exponens): For a deerminisic sysem x + = fx ), le Λx ) = D x fx ) D xfx ) ) 2, 23) where D xfx) = f x) and D xfx ) := D xfx ) D xfx ). Le λ i exp for i =,..., N be he eigenvalues of Λx ), such ha λ exp λ 2 exp λ N exp. Then, he Lyapunov exponens Λ i exp are defined as Λ i exp = log λ i exp for i =,..., N. Furhermore, if de Λx ) ), hen log de D x fx ) N ) = log k= λ k expx). 24) Remark 8: The echnical condiions for he exisence of is in 23) and 24) are provided by he Muliplicaive Ergodic Theorem 6] Theorem.6), 8] Theorem.4), 7] Secion D). The is in 23) and 24) are known o be independen of he iniial condiion and are unique under he assumpion of unique ergodic invarian measure for sysem dynamics. For a compac sae space, he exisence of an invarian measure is always guaraneed 8] Corollary 6.9.). Furhermore, every invarian measure admis ergodic decomposiion 8] Remarks pp. 53), 5] Theorem 6.4). We now make Assumpion 9 on he sysem dynamics. Assumpion 9: We assume he nonlinear sysem, x + = fx ), has a unique physical measure wih all Lyapunov exponens posiive. The assumpion of a unique physical measure is no resricive and i allows us o prove he main resul in Theorem 2, ha is independen of iniial condiions. Wih ergodic invarian measures ha are guaraneed o exis Remark 8), he main resul in Theorem 2 will be a funcion of a paricular ergodic measure under consideraion. The assumpion of all Lyapunov exponen being posiive is analogous o he assumpion made in he LTI case ha all eigenvalues are posiive. We verify hrough simulaion resuls in secion IV ha he resul of Theorem 2 also applies o he case where one of he Lyapunov exponen is negaive. Theorem 2 Nonlinear sysems on compac space): Consider he sysem ) wih sysem mapping f and oupu h saisfying Assumpions 3 and 9 and sae space X compac. The necessary condiion for MSE sabiliy of he observer error dynamics 4) is given by N 2 p) M λexp) k <, 25) k= where λ k exp = e Λk exp, and Λ k exp is he k h posiive Lyapunov exponen of x + = fx ). Proof: We follow he noaions from Lemma 2. The necessary condiion for MSE sabiliy Eq. ) is rue for almos all poins x X, and, hence in paricular for x evaluaed along he sysem rajecory x + = fx ). Evaluaing ) along he sysem rajecory and aking he produc, we wrie he necessary condiion as p) M ) n deqx )Q x n+)) n deax )) 2 <. Taking ime average for he log of he expression and in he i as n, we obain he following necessary condiion for MSE sabiliy, n n log = p) M ) n deqx )Q x n+)) ) n deax )) 2 <. 26) Using he fac ha boh Q x ) and Q x) are almos always uniformly bounded and using 24) from Definiion 7, 26) gives he required necessary condiion 25) for MSE sabiliy. Remark 2: The necessary condiion for MSE sabiliy in Theorems 3, 5, and 2 for single inpu case is igher however for < M < N, we expec he condiion o be improved furher. The necessary condiion for MSE sabiliy from our main resuls provides a criical dropou rae, i.e., he erasure probabiliy, q = p, above which he sysem is guaraneed MSE unsable. In paricular, he criical dropou rae for a nonlinear sysem wih single oupu, evolving on compac space from Theorem 2 is given by q = A. Enropy and iaion for observaion = N k= λk exp) 2. Measure-heoreic enropy, H µf), for he dynamical sysem, x n+ = fx n), is associaed wih a paricular ergodic invarian measure, µ, and is anoher measure of dynamical complexiy. While he measure-heoreic enropy couns he number of ypical rajecories for heir growh rae, he posiive Lyapunov exponens measure he rae of exponenial divergence of nearby sysem rajecories. For more deails on enropy refer o 8]. These wo measures of dynamical complexiy are relaed by Ruelle s inequaliy.

Theorem 22 Ruelle s Inequaliy): 7] Eq. 4.4); 8] Theorem 2) Le x n+ = fx n) be he dynamical sysem, f : X X be a C r map, wih r, of a compac meric space X and µ an ergodic invarian measure. Then, H µf) k Λ k exp) +, 27) where a + = max{, a}, H µf) is he measure-heoreic enropy corresponding o he ergodic invarian measure µ, and Λ k exp are he Lyapunov exponens of he sysem. The Ruelle inequaliy 27) can be used o relae he iaion for observaion wih sysem enropy. Theorem 23: Consider he sysem ) wih sysem mapping f and oupu h saisfying Assumpions 3 and 9 and sae space X compac. The necessary condiion for MSE sabiliy of he observer error dynamics 4) is given by M log p) + 2H µf) < 28) where µ is he physical invarian measure of f Definiion 6 and Assumpion 9) and H µf) is he measure-heoreic enropy corresponding o measure µ. Proof: The proof follows by applying he resuls of Theorems 2 and 22. IV. SIMULATION RESULTS Henon map is one of he widely sudied examples of wodimensional chaoic maps. The small random perurbaion of a wodimensional Henon map is described by following equaions: x + = ax 2 + x 2 + r, x 2+ = bx + r 2, y = ξ x, 29) where a =.4, b =.3 are consan parameers, and r i, E-6], i {, 2}, are uniform random variables. The small amoun of exernal noise, r i, is essenial o see he effec of mean square insabiliy. The sysem has Lyapunov exponens given by λ =.426 and λ 2 =.63. Alhough he main resuls of his paper are proved under he assumpion ha all Lyapunov exponens are posiive, he simulaion resuls verify ha he resuls hold rue even for his example wih one Lyapunov exponen negaive. The criical probabiliy p is compued, based on he posiive exponen and is equal o p = =.5734. The observer is designed exp 2λ such ha error dynamics wih no erasure is asympoically sable. In Figs. a) and b), we plo he error norm for he observer dynamics, averaged over 5 realizaions of he erasure sequence, a probabiliies below and above he criical probabiliy p, respecively. We clearly see he average error norm for non-erasure probabiliy, p =.7 > p, is negligible compared o flucuaions in he average error norm for p =.55 < p, which are four orders of magniude higher han he uniform noise in he sysem. In Fig. c), we plo he peak error variance for linearized error dynamics vs. non-erasure probabiliy. The dashed line indicaes he criical probabiliy, p =.5734. We observe he peak linearized error variance is unbounded below criical probabiliy. V. CONCLUSIONS In his work, he problem of sae observaion for a nonlinear sysem over erasure channel is sudied. The main resuls of his paper prove ha iaion arises for MSE sabilizaion of observer error dynamics. We show ha insabiliy of he non-equilibrium dynamics of he nonlinear sysem, as capured by posiive Lyapunov exponens, plays an imporan role in obaining he iaion resul for nonlinear observaion. The iaion resul for LTI sysems is 4 3 2 5 5 2 25 Time Maximum Linearized Error Variance 2 5 5 a) Error Norm 2.5 x!5 2.5.5 2 4 6 8 Time b).2.4.6.8 Non erasure Probabiliy c) Fig. : a) Error norm as a funcion of ime for p =.55; b) Error norm as a funcion of ime for p =.7; c) Maximum linearized covariance vs non-erasure probabiliy for Henon map obained as a special case, where Lyapunov exponens emerge as he naural generalizaion of eigenvalues from linear sysems o nonlinear sysems. The proof echnique presened in his paper can be easily exended o prove resuls for he esimaion of linear ime varying sysems over erasure channels. VI. ACKNOWLEDGMENT The research work was suppored by Naional Science Foundaion CMMI 87666) and ECCS 253) gran. The auhors would like o hank Prof. Nicola Elia for useful discussion. REFERENCES ] P. Ansaklis and J. aillieul, Special issue on echnology of neworked conrol sysems, Proceedings of IEEE, vol. 95, no., pp. 5 8, 27. 2] N. Nahi, Opimal recursive esimaion wih uncerain measuremens, IEEE Transacions on Informaion Theory, vol. 5, no. 4, pp. 457 462, 969. 3] M. Hadidi and S. Schwarz, Linear recursive esimaiors under uncerain measuremens, IEEE Transacions on Informaion Theory, vol. 24, no. 6, pp. 944 948, 979. 4]. Sinopoli, L. Schenao, M. Franceschei, K. Poolla, M. I. Jordan, and S. S. Sasry, Kalman filering wih inermien observaions, IEEE Transacions on Auomaic Conrol, vol. 49, pp. 453 464, 23. 5] M. Epsein, L. Shi, A. Tiwari, and R. M. Murray, Probabilisic performance of sae esimaion across a lossy nework, Auomaica, vol. 44, no. 2, pp. 346 353, 28. 6] O. Cosa, Saionary filer for linear minimum leas square error esimaior of discree-ime Markovian jump sysems, IEEE Transacions on Auomaic Conrol, vol. 47, no. 8, pp. 35 356, 22. 7] S. Smih and P.Seiler, Esimaion wih Lossy measuremens: jump esimaor for jump sysems, IEEE Transacions on Auomaic Conrol, vol. 48, no. 2, pp. 263 27, 23. 8] P. Walers, An Inroducion o Ergodic Theory. New York: Springer- Verlag, 982.

9] H. Nijmeijer, Observabiliy of auonomous discree ime nonlinear sysems: a geomeric approach, Inernaional Journal of Conrol, vol. 36, no. 5, pp. 867 874, 982. ] R. Z. Has minskiĭ, Sabiliy of differenial equaions. Germanown,MD: Sijhoff & Noordhoff, 98. ] D. Applebaum and M. Siakalli, Asympoic sabiliy of sochasic differenial equaions driven by levy noise, Journal of Applied Probabiliy, vol. 46, no. 4, pp. 6 29, 29. 2] G.. Folland, Real Analysis: Modern Techniques and Their Applicaions. New York: John Wiley and Sons, Inc., 999. 3] H. Kwakernaak and R. Sivan, Linear Opimal Conrol Sysems. New York: Wiley Inerscience, 972. 4] A. H. Jazwinski, Sochasic Processes and Filering Theory. Mineola, New York: Dover, 27. 5] R. Mane, Ergodic Theory and Differeniable Dynamics. New York: Springer-Verlag, 987. 6] D. Ruelle, Ergodic heory of differeniable dynamical sysems, Publicaions mahémaiques de L I.H.É.S., vol. 5, pp. 27 58, 979. 7] J. P. Eckman and D. Ruelle, Ergodic heory of chaos and srange aracors, Reviews of Modern Physics, vol. 57, pp. 67 656, 985. 8] D. Ruelle, An inequaliy for he enropy of differeniable maps, ullein of he razalian Mahemaical Sociey, vol. 9, pp. 83 87, 978.