Filtering of Nonlinear Systems with Sampled Noisy Data by Output Injection

Transcription

1 Filtering f Nnlinear Systems with Sampled Nisy Data by Output Injectin Filipp Cacace, Francesc Cnte, Alfred Germani, Givanni Palmb Abstract This paper cncerns with the state estimatin prblem f nnlinear systems with sampled nisy measurements. The main idea is t explit a result f the analysis f nnlinear systems, which cnsiders a class f state-space nnlinear mdels admitting the design f a deterministic bserver with linear errr dynamics. Such systems are suppsed t prvide sampled nisy measurements. A cnsistent filtering technique is then suitably develped, using the same apprach f the mentined classical result. I. INTRODUCTION During the last three decades, a part f the research n autmatic cntrl has been largely dedicated t the study f the state estimatin prblem fr nnlinear systems. In particular, significant results have been btained fr deterministic systems with the design f several nnlinear bservers. One f the pssible strategies uses a suitable change f crdinates by which the system is made linear except fr a nnlinear utput injectin. In this case, the errr dynamics f the bserver is exactly linearizable and the bserver design thery fr linear system can be applied. The use f utput injectin in rder t btain bservers with linear errr dynamics has been independently suggested by [1 and [2 fr single-utput systems. A cmplete analysis f the crrespnding prblem fr multi-utput systems can be fund in [. Further develpments have been presented in [4, [5, [6, [7, [8, [9, [1, and [11. All mentined literature cncerns with the cntinuus-time case. In [12 the effect f time-sampling n the design f nnlinear cntinuus-time based bservers with linear errr dynamics is investigated. A dual analysis n the feedbac linearizatin prblem can be fund in [1. A discrete-time cunterpart f the listed results has been parallely btained by [14 and then extended by [15 and [16. A great deal f attentin has been als devted t the state estimatin prblem fr nnlinear stchastic systems, als nwn as the nnlinear filtering prblem, bth in the cntinuus- and discrete-time cases (e.g. [17, [18, [19, [2, [21). In this framewr, the system is suppsed t be pssibly driven by a stchastic prcess and the measurements t be crrupted by a disturbing nise. In a real scenari, this F. Cacace is with Università Campus Bi-Medic di Rma, Via Álvar del Prtill 21, 128 Rma, Italy ( f.cacace@unicampus.it). F. Cnte is with Intelligent Electric Systems Labratry, Università degli studi di Genva, Via all Opera Pia 11A, Genva, Italy ( fr.cnte@unige.it). A. Germani and G. Palmb are with Dipartiment di Ingegneria e Scienze dell Infrmazine e Matematica, Università degli studi dell Aquila, Via Veti (Cppit 1), 671 Cppit (AQ), Italy ( {alfred.germani-givanni.palmb}@univaq.it. always is the case. Observers perfrmances are indeed ften cnsidered als in the presence f nise, in rder t analyze the rbustness f the methd and its significance fr real applicatins. In this sense, the principal difference amng deterministic and stchastic appraches cnsists in the explicit cnsideratin f nise used in the latter methdlgy. Hwever, strategies f nnlinear filtering maybe strngly different frm thse used by deterministic appraches. The idea f this paper, n the ther hand, is t explit the mentined results f the analysis f deterministic nnlinear systems in rder t derive a nnlinear filtering technique. Mre precisely, cntinuus-time systems which result t be lienarizable with (nnlinear) utput injectin (e.g. by ne f the cnditins prvided by [1, [2, r [) are cnsidered. Such systems are then suppsed t be measured with sampled nisy signals. The presence f nise is explicitly taen int accunt fr realizing a derivative-free filtering technique which fllws a strategy similar t the ne used in the design f nnlinear bservers with linear errr dynamics. The advantage f the prpsed technique is the cnsideratin f nise with respect t deterministic appraches and the avidance f linearizatin with respect t classical nnlinear filters. Even if designed fr cntinuus-time systems with sampled measurements, the prpsed methd can be als applied t discrete-time systems which admit linearizatin by utput injectin (e.g. satisfying ne f the cnditins in [14 r [15). The paper is rganized as fllws. Sectin II gives the prblem frmulatin. The prpsed apprach is described in Sectin III. A numerical example is illustrated in Sectin IV. Cnclusins are summarized in Sectin V. II. PROBLEM FORMULATION Cnsider a cntinuus-time stchastic autnmus system having the frm ẋ(t) = f(x(t)), x() = x (1) dr t = h(x(t))dt + σdw t (2) where x(t) R n is the system state prcess, x R n is the initial state, r t R is the stchastic integral frm f the measurement prcess y t R, w t is a Wiener prcess, σ R, and f : R n R n and h : R n R are analytic functins. The aim f this paper is the estimatin f the system state x(t) under the fllwing hypthesis: Hp. 1: The deterministic part f system (1)-(2) ẋ(t) = f(x(t)) () λ(t) = h(x(t)) (4)

2 is lcally state-equivalent t a linear system with nnlinear utput injectin, ([5, [12) i.e. there exist a crdinates transfrmatin z = Φ(x) defined in a neighbrhd U f the initial state x, a matrix Ā Rn n, a mapping β : h(u ) R n, and a rw vectr C R 1 n s.t. 1) ż(t) = ϕ(x) x f(x) x=φ 1 (z(t)) = Āz(t) + β (Cz(t)), 2) λ(t) = h ϕ 1 (z(t)) = Cz(t), C CĀ ) ran. = n, CĀn 1 fr all z(t) Φ(U ). Hp. 2: Fr a given sampling time R +, the sampled measurements y() := y, Z + are available. Hp. : The initial state x is a randm vectr independent f the measurement nise prcess w t. Ntice that [1 and [2 gives cnditins fr stating when a deterministic nnlinear system satisfies Hp. 1. In this case, the state bservatin can be carried ut by simply substituting the argument Cz f β( ) with the utput λ and design a Luemberger bserver with an exactly linear errr dynamics. It is wrth remaring that, when hypthesis Hp. 1 is satisfied, system (1)-(2) has the fllwing state-equivalent linear frm with nnlinear utput injectin: ż(t) = Āz(t) + β (Cz(t)), z() = Φ(x ), (5) dr t = Cz(t) + σdw t. (6) III. PROPOSED APPROACH The apprach here prpsed cnsists in designing a state estimatr fr the transfrmed system (5)-(6). Because f the availability f sampled measurements, such a system is firstly discretized. Then, a recursive filtering technique is intrduced fr carrying ut the estimatin f the sampled state. Finally, the estimate f the riginal state can be btained by using the inverse mapping Φ 1. A. System Discretizatin Despite the transfrmatin, system (5)-(6) is nnlinear. It is well nwn that the discretizatin f nnlinear systems cannt be exactly btained. Hwever, in the case f (5)-(6) the explicit presence f a linear part in the state dynamic can be used t btain a sufficient apprximatin f the crrespnding discrete-time system. Fr cnventin, the sampled values f x(t), z(t) and y(t) will be hereafter indicated with x() := x( ), z() := z( ), and y() := y, Z +, respectively. In rder t btain the discrete-time dynamic f the state vectr z, it is sufficient t get the slutin f the rdinary differential equatin (5) using and ( + 1) as initial and current times, respectively, and suppse the state z(t) t be cnstant within the sampling interval, i.e. z(t) = z( ) fr all t [, ( + 1) ): z( + 1) eā z() + where = eā z() + (+1) = Az() + B β(cz()). A := eā R n n, B := eā((+1) τ) β (Cz(τ)) dτ eāθ dθ β(cz()) eāθ dθ R n n. With the same assumptins, frm (6) fllws r +1 r (+1) Cz(t)dt + (+1) = Cz() + σ ( w (+1) w ) σdw t frm which, by suppsing that y t = y() within the sampling interval [, ( + 1) ), ne has y() r +1 r Cz() + σ w +1 w = Cz() + σ n = Cz() + n, where {n } := {σ/ n } results t be a zer-mean white Gaussian sequence with variance σ 2 /, as clearly fllws by the definitin prperties f the Wiener prcess. Mrever, frm Hp. 1 fllws that {n } is independent f the initial cnditin z(). The discretized system hence has the frm: z( + 1) = Az() + β(cz()) (7) y() = Cz() + n, (8) with β( ) := B β( ). Finally, the initial cnditin z() is suppsed here t be a randm vectr with nwn mean value z and cvariance matrix Ψ z. B. The Filtering Algrithm Despite the prblem frmulatin cnsiders cntinuustime systems with sampled scalar measurements, the filtering algrithm develped in this sectin can be generally applied t discrete-time systems having the frm (7)-(8), als with q- dimensinal utput. Fr example, such systems may belng t the class f multi-utput cntinuus-time systems cnsidered in [, r t the class f multi-utput discrete-time systems analyzed in [15. Then, the utput will be suppsed t be y R q, as well as n R q, which is suppsed t be a statinary zer-mean white Gaussian sequence with cvariance matrix R R q q. In the deterministic case, the utput injectin is classically explited in rder t btain an bserver with linear errr dynamics [1, [2, [5, just substituting the term Cz with the utput y. The aim f this paper is t prpse a slutin fr the stchastic discrete-time case fllwing the same fashin.

3 Therefre, (7) is rewritten by substituting the argument f β( ) frm (8): z( + 1) = Az() + β (y() n ). (9) This simple substitutin maes the system linear with respect t the state vectr z. The nnlinear map β( ) is indeed applied t the nwn nisy quantity y() and t the randm term n. The s btained vectr β (y() n ) can be tugh treated as a nngaussian state nise. The idea is similar t that used in [22 fr slving the planar tracing prblem: t transfrm nnlinearity in nngaussianity fr aviding linearizatin. Because f the particular frm f the randm term β (y() n ), it is nt bvius that applying a standard Kalman-lie filter t the transfrmed system will return any characterizable estimate (e.g. sub-ptimal, linear ptimal etc.). As a cnsequence, a prper filtering algrithm is here directly derived. This will be dne using a gemetrical apprach, as in [2. Fr cmpleteness and clarity, sme cncepts abut Hilber spaces and estimate thery are recalled in the next subsectin. 1) The Gemetrical Apprach fr Estimatin Prblems: Let (Ω, F, P ) be a prbability space. Fr any given sub σ- algebra G f F, dente by L 2 G the Hilbert space f the G- measurable, randm vectrs with finite secnd mment as { } L 2 G := x : Ω R n, G-meas., x(ω) 2 dp (ω) <, This space is endwed with the nrm x L 2:= x T (ω)x(ω)dp (ω) = E [ x 2. Ω It is imprtant t stress here that, when x is zer-mean, the squared L 2 nrm crrespnds t its variance. It is als pssible t define the rtgnality amng tw elements f this space as: x y E [ xy T = (1) which, in the case f zer-mean randm vectrs, crrespnds t uncrrelatin. Then, if M is a subspace f L 2 G, the symbl Π(x M) indicates the rthgnal prjectin f x L 2 G nt M, which is defined as the (unique) vectr in M such that the difference x Π(x M) is rthgnal t M, i.e. (x Π(x M)) y, y M. If N is anther sub-space, N M indicates that x y fr all x N and y M, whereas x M indicates that x y fr all y M. Mrever, when G is the σ-algebra generated by a randm vectr y : Ω R m, that is G = σ(y), the ntatin L 2 y indicates L 2 σ(y). Frm the Hilbert prjectin therem [24 fllws that the minimum variance estimate f a randm vectr x L 2 G with respect t a randm vectr y crrespnds t Π(x L 2 y). When x and y are nt Gaussian, such a prjectin cannt be cmputed with finite time. Therefre, subptimal Ω estimates are btained by prjecting x nt smaller subspaces. The mstly used (and usable) is L y := {x : Ω R n M R n (m+1) s.t. x = M[1 y } which is cmpsed by all affine transfrmatins f the randm vectr y : Ω R m. The prjectin Π(x L y ) is the linear minimum variance estimate f x w.r.t. y. It is pssible t shw that this prjectin can be cmputed as Π(x L y ) = E[x + E [ xȳ T E [ ȳȳ T 1 ȳ (11) where ȳ := y E[y [25. A secnd useful sub-space is L y := { x : Ω R n M R n m s.t. x = My } which is cmpsed by all linear transfrmatins f the randm vectr y : Ω R m. In this case, the prjectin is given by Π(x L y ) = E [ xȳ T E [ ȳȳ T 1 ȳ. (12) 2) Filtering Derivatin: The filtering prblem (frmulated fr system (7)-(8)), requires the recursive estimatin f the state z() frm the measurements sequence {y(τ), τ =, 1,..., }, which can be cllected in the ( + 1)q vectr Y := [ y T () y T (1) y T () T. Fllwing the intrduced gemetrical apprach, the ptimal minimum variance estimate is the prjectin Π(z() L 2 Y ). As mentined, because f nngaussianity, this cannt be cmputed. The aim here is t develp a filter able t return the linear 1 minimum variance estimate f z(), i.e. Π ( z() L y), where L y dentes the sub-space L Y. In rder t clearly present the derivatin f the filter, sme definitins fllw: state estimate: ẑ( ) := Π ( z() Ly) ; state predictin: ẑ( 1) := Π ( ) z() L 1 y ; estimate errr: ê( ) := z() ẑ( ); predictin errr: ê( 1) := z() ẑ( 1); utput innvatin: ν () := y() Π ( ) z() L 1 y. Mrever P ( ) and P ( 1) dente the estimate and predictin errr cvariance matrices, respectively. Let us fcus n the utput innvatin sequence {ν ()}. By definitin, it is easy t shw that E[ν () = and E[ν ()ν T (τ) = fr τ, i.e. it is a zer-mean white randm sequence. Mrever, ν () L 1 y, and, by denting L ν () with L ν, it results that L y = L 1 y L ν and L 1 y L ν. (1) Cnsider nw that frm (8), the whiteness f {n }, and the given definitins, fllws that ν () = y() Cẑ( 1) = Cz() + n Cẑ( 1). (14) 1 The term linear is usually used t indicate general affine estimates.

4 Nte that, frm (14) fllws that ν () = Cê( 1) + n, (15) The argument Cz f the nnlinear functin β( ) in (7) can be then substituted using (14) instead f directly use the utput equatin (8), btaining z( + 1) = Az() + β (Cẑ( 1) + ν () n ). (16) This substitutin eeps the linearity w.r.t. z but maes easier t statistically characterize the randm nngaussian term β (Cẑ( 1) + ν () n ), where Cẑ( 1) is a nwn quantity at time and {n } and {ν ()} and are white randm sequences. The cvariance matrix f {n } is R, whereas the cvariance matrix E [ ν ()ν T () is time varying and can be recursively cmputed as Ψ ν () := E [ ν ()ν T () = CP ( 1)C T + R. (17) The prf f (17) is left t end f the sectin. Nw, let us indicate the randm vectr β (Cẑ( 1) + ν () n ) with β() and define n s := β() Π ( β() L y), which is a is zer-mean nngaussian randm vectr. It is here assumed that the sequence {n s } is white with the cvariance matrix Q() := E [ n s n st. (18) Ntice that Q() is time-varying since it depends n the current value f the predicted state ẑ( 1). Finally, it is clear that the state and utput sequences are crrelated each ther, and therefre the crrelatin matrix J() must be defined as J() := E [ n s n T. (19) Fr clarity, the cmputatin f Π ( β() Ly), Q(), and J(), as well as a discussin abut the whiteness f n s are left t end f the sectin. Using the given definitins, the system equatins are rewritten as z( + 1)=Az() + Π ( β() L y) + n s, (2) y()=cz() + n, (21) where Π ( β() L y) plays the rle f the deterministic input. It is nw pssible t prvide the filter equatins: with ẑ( ) = ẑ( 1) + K() (y() Cẑ( 1)) (22) ẑ( + 1 ) = Az( ) + Π ( β() L y), (2) K() = P ( 1)C T ( CP ( 1)C T + R ) 1, (24) P ( ) = (I n K()C) P ( 1) (I n K()C) T +K()RK T (), (25) P ( + 1 ) = AP ( )A T + Q() AK()J T () J()K T ()A T, (26) and initial cnditins ẑ( 1) = z, P ( 1) = Ψ z. (27) The fllwing therem states the main result f this paper. Therem 1: Under the hypthesis that {n s } is a white sequence, the recursive cmputatin f equatins (22)-(27) returns the linear minimum variance estimate ẑ( ) = Π(z() L y) f the state z() f system (2)-(21). Prf. Frm prperty (1) it results that ẑ( ) = Π ( z() L 1 ) ( y + Π z() L ν ) = ẑ( 1) + Π ( z() L ) ν. (28) The term Π ( z() L ) ν can be cmputed using (12) and nting that Π ( ẑ( 1) L ) ν = : Π ( z() L ) ( ) ν = Π z() ẑ( 1) L ν = Π ( ê( 1) L ) ν where, by (15) and (17) = K()ν () K() = E [ ê( 1)ν T = K() (y() Cẑ( 1)) (29) E [ ν ()ν T () 1 = P ( 1)C T ( CP ( 1)C T + R ) 1, which is the well nwn expressin f the Kalman gain matrix K() prvided in (24). Equatin (22) is then btained by substituting (29) in (28). As far as the predictin term ẑ( + 1 ) is cncerned, expressin (2) trivially fllws by applying the prjectin Π( L y) t (2). It remains t prve (25) and (26). Fr this purpse, let us cnsider the recursive relatin amng the state and predictin errrs: ê( ) = z() ẑ( ) (by (22)) = z() ẑ( 1) K()ν () (by (14)) = (I n K()C) ê( 1) K()n, () ê( + 1 ) = z( + 1) ẑ( + 1 ) (by (2)) = Az() + Π ( β() L y) + n s ẑ( + 1 ) (by (2)) = Aê( ) + n s, (1) frm which it can be nted that, because f the whiteness f {n s } and {n }, E [ ê( 1)n st =, (2) E [ ê( 1)n T =. () Mrever, by (), and taing int accunt (19), E [ ê( )n st = (In K()C) E [ ê( 1)n st K()E [ n n st (by (2)) = K()J T (). (4) By taing int accunt (), (25) readily fllws frm (). Whereas, (26) fllws frm (1) by using (4) and recalling definitin (18).

5 It remains nw t prve (17) and discuss the cmputatin f Π ( β() Ly), Q(), and J(), and the whiteness f {n s }. Equatin (17) is readily prved by taing int accunt prperty (15): Ψ ν () = CE [ ê( 1)ê T ( 1) It is als interesting that + CE [ê( 1)n T +E [ n n T +E [ n ê T ( 1) C T = CP ( 1)C T + R. E [ ν ()n T = R. (5) As far as Π(β() L y) is cncerned, frm (1) fllws that Π ( β() L y ) ( ) ( ) = Π β() L 1 y + Π β() L ν. (6) In rder t cmpute these tw quantities, tw assumptins are required t be admitted: {ν ()} is apprximately a Gaussian distributed sequence with the cvariance matrix Ψ ν (); the term ẑ( 1) is available at time, i.e. it is a deterministic quantity. It can be remared that the first assumptin is true less then the nngaussianity f the system dynamic, nly due t the stchastic initial cnditin z(). As a cnsequence, when the evlutin f the system has the prperty f stability f frgetting the initial cnditin, it fllws that the innvatin prcess becmes asympttically Gaussian. Because f Gaussianity, ν () becmes independent f y(τ), τ =, 1,..., 1, and als f n τ and the same innvatin ν (τ) at a different time, τ. These tw last prperties imply that ν () is uncrrelated with β(τ), fr τ. Therefre, under the abve intrduced hypthesis, and taing int accunt the prjectin rules (11) and (12), the expressins in the next hld true: Π ( β() L 1 ) y = E [β() (7) Π ( β() L ) ν = M()ν () (8) with M() = [β()ν () Ψ 1 ν () (9) and where the mean values are cmputed ver the distributins f ν () and n s. As a cnsequence it results that: Π ( β() L y) = E [β() + M()ν (), (4) and E [n s n s τ = E [(β() E [β()) (β(τ) E [β(τ)) T E [ (β() E [β()) ν T (τ) M T (τ) M()E [ν () (β(τ) E [β(τ)) T [ +M()E ν ()ν T (τ) M T (). (41) When τ last expressin is equal t zer. Mrever the cvariance matrix Q() can be cmputed by (41) with = τ. Finally, matrix J() has the frm: J() = E [n s n = E [ β()n T E [ β()n T where prperty (5) has been used. IV. EXAMPLE [ M()E ν ()n T M()R, (42) In rder t validate the prpsed apprach, the filtering algrithm has been tested n the Van der Pl Oscillatr, as an example f system admitting linearizatin by utput injectin (see [9). The equatins are: ẋ 1 (t)= x 2 (t) ẋ 2 (t)= x 1 (t) + x 2 (t) x 2 1(t)x 2 (t) y(t)= x 1 (t) As shwn in [9, the mapping Φ(z) = applied t the system leads t [ x1 x 2 + x 1 ż 1 (t) = z 2 (t) z 1 (t) ż 2 (t) = z 2 (t) z 1 (t) z 1 (t) (4) which has the frm in (5), with the dynamic matrix and the nnlinear utput injectin: [ 1 Ā = 1 1 β(cz(t)) = 1 z 1(t) [ 1 1 Ging ahead by discretizatin, fllwing the prcedure described in Sectin III-A, the system assumes the frm (7)-(8) with the discrete-time utput injectin β(cz()) = 1 [ 1 (Cz()) B (44) 1 The nisy measurement prcess has been simulated based n mdel (2) using Euler-Maruyama integratin with σ =.28. The sampled utput sequence {y()} has been acquired with sampling time =. s. The filtering algrithm (22)-(27) has been implemented using (4), (41), and (42) fr cmputing Π(β() L y), Q() and J(), respectively. The initial estimate has been generated with a Gaussian distributin in the z crdinates with the real initial state as mean value and unitary cvariance matrix (Ψ z = I 2 ). Figure 1 shws the time evlutin f the measured state x 1 tgether with the crrespnding measurements and the estimatin result btained with the prpsed filtering technique. It can be nted that the estimatr succeeds in filtering the signals with a fast transient. It is wrth remaring that, in this example, the estimatr is required t face with the discretizatin errr als..

6 4 measurements estimated state bserved estimated state Time [s Time [s Fig. 1. Real, estimated and measured state cmpnent x 1. Fig. 2. Real, bserved and estimated state cmpnent x 2. Figure 2 cmpares the estimates f the state cmpnent x 2 btained with the prpsed technique and a discrete-time bserver based n the utput injectin [15. Because f the presence f nise, the bserver gain was difficult t be tuned in rder t btain a gd trade-ff between the transient interval and the estimatin precisin. In particular, nte that the bserver seems t start frm a different initial estimate. Actually, the filter and the bserver start frm the same initial cnditins, but, in the illustrated case, the bserver gain was set t a very lw value in rder t btain a sufficiently smth estimated signal. Therefre, while the first crrectin f the filter mves the estimate clse t the real state, this des nt happen in the bserver case. It is clear that the filter definitively have better perfrmances with respect t the bserver because f the explicit cnsideratin f nise. V. CONCLUSIONS The use f utput injectin in rder t btain bservers with linear errr dynamics in the deterministic framewr has been explited fr deriving a cnsistent filtering technique fr nnlinear systems with sampled nisy measurements. The utput injectin has been used t transfrm nnlinearity in nngaussianity. This allwed aviding linearizatin and getting a linear minimum variance filter. A numerical example cnfirmed the effectiveness f the prpsed apprach. REFERENCES [1 A. J. Krener and A. Isidri, Linearizatin by utput injectin and nnlinear bservers, Syst. & Cntr. Letters, vl., pp , June 198. [2 D. Bestle and M. Zeitz, Cannical frm bserver design fr nnlinear time variable systems, Int. J. f Cntr., vl. 1, pp. 7 81, 198. [ A. J. Krener and W. Respnde, Nnlinear bserver with linearizable errr dynamics, SIAM J. f Cntr. and Opt., vl. 2, n. 2, pp , [4 X. Xia and W. Ga, Nnlinear bserver design by bsever errr linearizatin, SIAM J. f Cntr. and Opt., vl. 27, pp , [5 A. Isidri, Nnlinear Cntrl Systems. Springer-Verlag, 2nd ed., [6 A. Banaszu and W. M. Sluis, On nnlinear bservers with apprximately linear errr dynamics, in IEEE ACC, vl. 2, pp , [7 A. J. Krener and M. Xia, A necessary and sufficient cnditin fr the existence f a nnlinear bserver with linearizable errr dynamics, in IEEE CDC, n. December, pp , 21. [8 A. F. Lynch and S. A. Brtff, Nnlinear bservers with apprximately linear errr dynamics: The multivariable case, IEEE Trans. n Autm. Cntr., vl. 46, n. 6, pp , 21. [9 A. J. Krener and M. Xia, Observers fr linearly unbservable nnlinear systems, Syst. & Cntr. Letters, vl. 46, pp , July 22. [1 J. Deutscher, Numerical design f nnlinear bservers with apprximately linear errr dynamics, 26 IEEE Cnference n Cmputer- Aided Cntrl Systems Design, pp , Oct. 26. [11 A. J. Krener and M. Xia, Nnlinear bserver design fr smth systems, in Chas in Autmatic Cntrl (W. Perruquetti and J. P. Barbt, eds.), ch. 1, pp , Taylr & Francis Grup, 26. [12 S. T. Chung and J. W. Grizzle, Sampled-data bserver errr linearizatin, Autmatica, vl. 26, n. 6, pp , 199. [1 A. Arapstathis, B. Jaubczy, H. G. Lee, S. I. Marcus, and E. D. Sntag, The effect f sampling n linear equivalence and feedbac linearizatin, Syst. & Cntr. Letters, vl. 1, n. 5, pp. 7 81, [14 H.-G. Lee and S. I. Marcus, Apprximate and lcal linearizability f nn-linear discrete-time systems, Int. J. f Cntr., vl. 44, pp , Oct [15 W. Lin and C. Byrnes, Remars n linearizatin f discrete-time autnmus systems and nnlinear bserver design, Syst. & Cntr. Letters, vl. 25, pp. 1 4, [16 N. Kazantzis, C. Kravaris, A. J. Krener, and M. Xia, Nnlinear discrete-time bserver design with linearizable errr dynamics, IEEE Trans. n Autm. Cntr., vl. 48, n. 4, pp , 2. [17 A. V. Balarishnan, Kalman filtering thery. New Yr: Optimizatin Sftware, Inc., [18 M. S. Arulampalam, S. Masell, N. Grdn, and T. Clapp, A tutrial n particle filters fr nline nnlinear/nn-gaussian bayesian tracing, IEEE Trans. n Sign. Prc., vl. 5, n. 2, 22. [19 S. J. Julier and J. K. Uhlmann, Unscented filtering and nnlinear estimatin, in Prceedings f the IEEE, vl. 92, 24. [2 A. Germani, C. Manes, and P. Palumb, Plynmial extended alman filter, IEEE Trans. n Autm. Cntr., vl. 5, n. 12, pp , 25. [21 F. E. Daum, Nnlinear filters: Beynd the alman filter, IEEE Aerspace and Electrnics Systems Magazine, vl. 2, n. 8, pp , 25. [22 F. Cnte, V. Cusiman, and A. Germani, Rbust planar tracing via a virtual measurement apprach, Eurp. J. f Cntr., vl. 19, n. 2, pp , 21. [2 F. Carravetta, A. Germani, and M. Raimndi, Plynmial filtering fr linear discrete time nn-gaussian systems, SIAM J. f Cntr. and Opt., vl. 4, n. 5, pp , [24 A. V. Balarishnan, Applied Functinal Analysis. Springer-Verlag: New Yr, [25 F. Carravetta, A. Germani, and M. Raimndi, Plynmial filtering f discrete-time stchastic linear systems with multiplicative state nise, IEEE Trans. n Autm. Cntr., vl. 42, n. 8, pp , 1997.