Approximate nonlinear filtering for a two-dimensional diffusion with one-dimensional observations in a low noise channel

APPROXIMATE NONLINEAR FILTERING FOR A TWO-DIMENSIONAL DIFFUSION WITH ONE-DIMENSIONAL OBSERVATIONS IN A LOW NOISE CHANNEL∗ PAULA MILHEIRO DE OLIVEIRA†AND JEAN PICARD‡ SIAM J. CONTROL OPTIM.c 2003 Society for Industrial and Applied Mathematics Vol. 41, No. 6, pp. 1801–1819 Abstract. The asymptotic behavior of a nonlinear continuous time filtering problem is studied when the variance of the observation noise tends to 0. We suppose that the signal is a two-dimensional process from which only one of the components is noisy and that a one-dimensional function of this signal, depending only on the unnoisy component, is observed in a low noise channel. An approximate filter is considered in order to solve this problem. Under some detectability assumptions, we prove that the filtering error converges to 0, and an upper bound for the convergence rate is given. The efficiency of the approximate filter is compared with the efficiency of the optimal filter, and the order of magnitude of the error between the two filters, as the observation noise vanishes, is obtained. Key words. stochastic differential models, nonlinear filtering, approximate filters AMS subject classifications. 93E11, 60G35, 60F99 PII. S0363012902363920 1. Introduction. Due to its vast application in engineering, the problem of filtering a random signal Xtfrom noisy observations of a function h(Xt) of this signal has been considered by several authors. In particular, the case of small observation noise has been widely studied, and several articles are devoted to the research of approximate filters which are asymptotically efficient when the observation noise vanishes. Among them, one notices a first group in which a one-dimensional system is observed through an injective observation function h(see [4, 5, 7, 1]); in this case, the filtering error is small when the observation noise is small, and one can find efficient suboptimal finite-dimensional filters. The multidimensional case appears later with [8, 9], but an assumption of injectivity of his again required; in particular, the extended Kalman filter is studied in [9]. See also previous work by Krener [6] for systems with linear observations. When his not injective, the process {Xt}cannot always be restored from the observation of {h(Xt)}, so the filtering error is not always small; such a case is studied in [3]. However, there are some classes of problems in which {Xt}can be restored from {h(Xt)}; in these cases, the filtering error is small, and one again looks for efficient suboptimal filters. For instance, {Xt}is sometimes obtained from {h(Xt)}and its quadratic variation; see [2, 10, 11, 13]. Here, we are interested in another case in which h(Xt) is differentiable with respect to the time t, and {Xt}is obtained from {h(Xt)}and its derivative. As opposed to [9], the existence of a Lipschitz inverse of his not assumed in this paper, as the dimension of the measurements that we consider is lower than that of the state. More precisely, we consider the framework of [12], which we now describe. We consider the two-dimensional process Xt=(x(1) t,x (2) t) given by the Itˆo equa- ∗Received by the editors February 4, 2002; accepted for publication (in revised form) June 26, 2002; published electronically February 6, 2003. This paper was partially supported by Funda¸c˜ao Calouste Gulbenkian and FCT–CEDEC. http://www.siam.org/journals/sicon/41-6/36392.html †Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, P–4200–465 Porto, Portugal (p[email protected]). ‡Laboratoire de Math´ematiques Appliqu´ees (CNRS–UMR 6620), Universit´e Blaise Pascal, F– 63177 Aubi`ere Cedex, France ([email protected]clermont.fr). 1801 1802 PAULA MILHEIRO DE OLIVEIRA AND JEAN PICARD tion      dx(1) t=f1(x(1) t,x (2) t)dt, dx(2) t=f2(x(1) t,x (2) t)dt +σ(x(1) t,x (2) t)dwt, (1.1) with initial condition X0=(x(1) 0,x (2) 0), and we are concerned by the problem of estimating the signal Xtwhen the observation process is modelled by the equation dyt=h(x(1) t)dt +εd¯wt,(1.2) where {wt}and {¯wt}are standard independent real-valued Wiener processes and εis a small nonnegative parameter. In particular, if f1(x1,x 2)=x2, then x(1) tis the position of some moving body on R,x(2) tis its speed, the body is submitted to a dynamical force described by f2and to a random force described by σ, and one has a noisy observation of the position. This class of problems arises in practice in tracking RADAR applications, for instance, as well as in control and communications engineering. The use of the method of proof introduced in [7] and later extended to [9] in the class of systems (1.1)–(1.2) is not covered by previous work. If ε= 0 and if the functions hand x2→ f1(x1,x 2) are injective, then the signal Xtcan (at least theoretically) be exactly restored from the observation; we are here interested by the asymptotic case ε→0, and we look for a good approximation of the optimal filter ˆ Xt=(ˆx(1) t,ˆx(2) t)=EXtys,0≤s≤t. This approximation should be finite-dimensional (a solution of a finite-dimensional equation driven by yt). The same problem has been dealt with in [12] (with σconstant) by means of a formal asymptotic expansion of the optimal filter in a stationary situation. Our aim is to work out a rigorous mathematical study of the filter proposed by [12], namely the solution Mt=(m(1) t,m (2) t)of dMt=f(Mt)dt +Rt[dyt−h(m(1) t)dt],(1.3) Rt def =   2σ(Mt)F12(Mt) h(m(1) t)ε σ(Mt) ε    ,(1.4) with F12 =∂f1/∂x2and with initial condition M0=E[X0]. This filter does in fact correspond to the extended Kalman filter with stationary gain if one neglects the contribution of the derivatives of fother than ∂f1/∂x2. The stability of this filter is not evident and requires some assumptions. When it is stable, we prove in this work that x(1) t−m(1) t=O(ε3/4),x (2) t−m(2) t=O(ε1/4),(1.5) and ˆx(1) t−m(1) t=O(ε),ˆx(2) t−m(2) t=O(√ε).(1.6) APPROXIMATE NONLINEAR FILTERING 1803 We also verify that (1.6) can be improved when σis constant, his linear, and f1is linear with respect to x2. (This case will be referred to as the almost linear case.) The proofs follow the method of [9]. The contents are organized as follows. In section 2, we introduce the assumptions which will be needed in what follows, and we study the filtering error as εconverges to zero; more precisely, we obtain the rate (1.5). In section 3, the error between the approximate filter and the optimal filter is studied, and we prove (1.6). Section 4 is devoted to the almost linear case. Results of numerical simulations that illustrate the performance of this approach are included in section 5. Notation. The following notation is used: f=f1 f2,Σ=0 σ,H=h0; F= ˇF11 F12 F21 F22 ˘ and Σ= ˇ00 Σ 21 Σ 22 ˘ are the Jacobian matrices of fand Σ; ∇0Φ= ∂Φ ∂X0 is either a 2 ×2 matrix (if Φ is R2-valued) or a line-vector (if Φ is real-valued); see section 3. The symbol ∗is used for the transposition of matrices. When describing the behavior of approximate filters, we will write asymptotic expressions with the meaning given by the following definition. Definition 1.1. Consider a realor vector-valued stochastic process {ξt}.Ifβ is real and p≥1, we will write that ξt=O(εβ)in Lp when, for some q≥0,α>0, and some positive constants C1,C2,c3, Eξtp1/p ≤C1 εqe−c3t/εα+C2εβ for t≥0and εsmall. In this situation, the process {ξt}is usually said to converge to zero with rate of order εβ, in a time scale of order εα. 2. Estimation of Xt−Mt.The following assumptions will be used throughout this article. The last one depends on a parameter δ≥1. (H1) X0is a random variable, the moments of which are finite. (H2) {wt}and {¯wt}are standard independent Wiener processes independent of X0. (H3) The function his C3with bounded derivatives, and his positive. (H4) The function fis C3with bounded partial derivatives, and F12 =∂f1/∂x2is positive. (H5) The function σis C2with bounded partial derivatives. (H6.δ) One has 1 δ≤σ(x)≤δ, 1 δ≤h(x1)≤δ, 1 δ≤F12(x)≤δ for any x=(x1,x 2). Remark 2.1. In order to reduce the notation in (H6.δ), system (1.1)–(1.2) has been rescaled. Indeed, if we assume instead that one has 1 δ≤σ(x) ¯σ≤δ, 1 δ≤h(x1) ¯ H≤δ, 1 δ≤F12(x) ¯ F≤δ 1804 PAULA MILHEIRO DE OLIVEIRA AND JEAN PICARD for any x=(x1,x 2) and for some positive ¯σ,¯ H, and ¯ Fand if we replace the processes x(1) t,x(2) t, and ytby x(1) t/(¯σ¯ F), x(2) t/¯σ, and yt/(¯σ¯ F¯ H), then the functions f1,f2,σ, and hare replaced, respectively, by f1(¯σ¯ Fx1,¯σx2)(¯σ¯ F),f 2(¯σ¯ Fx1,¯σx2)¯σ, σ(¯σ¯ Fx1,¯σx2)¯σ, h(¯σ¯ Fx)(¯σ¯ F¯ H), and εis replaced by ε/(¯σ¯ F¯ H). We can apply the filter (1.3) to this new system, and we obtain m(1) t/(¯σ¯ F) and m(2) t/¯σ. This shows that the problem can be reduced to the case ¯σ=¯ F=¯ H=1. Assumption (H6.δ) says that the system does not contain too much nonlinearity; when it is not satisfied, there may be a small positive probability for the filter to lose the signal (see [10] for a similar problem). This is a rather restrictive condition, so we discuss at the end of the section the general case in which it does not hold. We consider the system (1.1)–(1.2) and the filter (1.3). We let Ftbe the filtration generated by (X0,w t,¯wt) and Ytthe filtration generated by (yt). Theorem 2.1. Assume (H1)–(H5).For1<δ<21/5,if(H6.δ)holds, then one has x(1) t−m(1) t=O(ε3/4),x (2) t−m(2) t=O(ε1/4) in Lpfor any p≥1. Consider a change of basis defined by a matrix Tand its inverse T−1, where Tdef = 2/ε −1 01  ,T −1= ε/2ε/2 01  . Then consider the process Zt def =T(Xt−Mt).(2.1) We are going to check that Ztis the solution of a linear stochastic differential equation; the study of the exponential stability of this equation will enable the estimation of both components of Zt, and the theorem will immediately follow. An equation for Zt. From (1.1)–(1.3), we have d(Xt−Mt)=f(Xt)−f(Mt)dt −Rth(x(1) t)−h(m(1) t)dt +     0−2εσ(Mt)F12(Mt) h(m(1) t) σ(Xt)−σ(Mt)     dwt d¯wt. In this equation, we introduce the Taylor expansions for the functions fand h, f(Xt)−f(Mt)=F(ξt,µ t)(Xt−Mt) and h(x(1) t)−h(m(1) t)=h(ηt)(x(1) t−m(1) t), APPROXIMATE NONLINEAR FILTERING 1805 where {ξt},{µt}, and {ηt}are R2and R-valued processes depending on {Xt}and {Mt}, and F(ξt,µ t)def = F11(ξt)F12(ξt) F21(µt)F22(µt) . We obtain a linear equation for Xt−Mt. By applying the transformation (2.1), we deduce for Ztan equation of the type dZt=AtZtdt +Utdwt d¯wt.(2.2) The precise computation shows that At=TF(ξt,µ t)−RtH(ηt)T−1=¯ At √2ε+ At, with ¯ A(11) t=−2h(ηt)F12(Mt)σ(Mt) h(m(1) t)+h(ηt)σ(Mt),¯ A(12) t=¯ A(11) t+2F12(ξt), ¯ A(21) t=¯ A(22) t=−h(ηt)σ(Mt), and where  Atisa2×2 matrix-valued process which is uniformly bounded as ε converges to 0; similarly, the matrix-valued process Utis also uniformly bounded. Stability of At.Ifδ= 1, then h=F12 =σ=1,so ¯ Atis the constant matrix ¯ At=−11 −1−1, and ¯ At+¯ A∗ t=−2I. In the general case δ>1, the coefficients of ¯ At+¯ A∗ tcan be controlled so that this matrix is uniformly close to −2Iif δis close to 1; in particular, for 1 <δ<21/5, there exists 0 <α<α <√2 such that ¯ At+¯ A∗ t≤−α√2I and, therefore, At+A∗ t≤−α √εI(2.3) if εis small. End of the proof of Theorem 2.1. Our goal is now to deduce an estimate of Ztin L2pfor the pinteger. From Itˆo’s formula and (2.2), the process Zt2=Z∗ tZtis the solution of dZt2=Z∗ t(At+A∗ t)Ztdt + trace(U∗ tUt)dt +2Z∗ tUtdwt d¯wt. 1806 PAULA MILHEIRO DE OLIVEIRA AND JEAN PICARD We deduce that the moment of order pof Zt2is finite and that d dtE[Zt2p]=pE[Zt2p−2Z∗ t(At+A∗ t)Zt]+pE[Zt2p−2trace(U∗ tUt)] +2 p(p−1) E[Zt2p−4U∗ tZt2]. From (2.3), one has Z∗ t(At+A∗ t)Zt≤−α √εZt2. As a consequence of the Cauchy–Schwarz inequality, one has U∗ tZt2≤trace(U∗ tUt)Zt2. Thus we obtain the inequality d dtE[Zt2p]≤−pα √εE[Zt2p]+p(2p−1)E[Zt2p−2trace(U∗ tUt)] ≤−pα √εE[Zt2p]+CpE[Zt2p−2]. Moreover, there exists C psuch that CpZt2p−2≤pα 2√εZt2p+C pε(p−1)/2, and so d dtE[Zt2p]≤− α 2√εpE[Zt2p]+C pε(p−1)/2. By solving this differential inequality, one obtains that, for some C p>0, E[Zt2p]≤C pεp/2+C pE[Z02p]e−αpt/(2√ε).(2.4) Thus Ztis O(ε1/4), and the order of magnitude of the components of Xt−Mtfollows from (2.1) and the form of T−1. We remark in (2.4) that the time scale of the estimation is of order √ε; one can compare it with the time scale εobtained when the observation function is injective (see, for instance, [7]). This means that here it takes more time to estimate the signal, and this is not surprising since the second component of the signal is not well observed. There are also other systems where the time scale is not the same for the different components of the signal (see [10]). In Theorem 2.1, we need the assumption (H6.δ), which is a restriction to the nonlinearity of the system; otherwise, it is difficult to ensure that the filter does not lose the signal. (This problem also occurs in [10].) Actually, we have chosen the filter (1.3) because it gives a good approximation of ˆ Xt(see the next section), but it is not the most stable one. If in (1.4) we replace the processes σ(Mt), F12(Mt), and h(m(1) t) by constant numbers ¯σ,¯ F, and ¯ H, then we obtain a filter with constant gain; we can again work out the previous estimations and prove that the result of Theorem 2.1 holds for this filter without (H6.δ)assoonas max F12 ¯ F<2min h ¯ H. Thus we have two filters—a filter which is stable and tracks the signal under rather weak assumptions and the filter (1.3) which seems more fragile but gives (under good stability assumptions) a better approximation of the optimal filter. APPROXIMATE NONLINEAR FILTERING 1807 3. Estimation of ˆ Xt−Mt.The main result contained in this section is Theorem 3.1, which states the rate of convergence of the approximate filter considered in this paper toward the optimal filter. In order to give a proof of this theorem, a sequence of steps is needed: a change of probability measure, the differentiation with respect to the initial condition, and an integration by parts formula. A similar method of proof is adopted in [9]. As in Theorem 2.1, we may have a problem of stability in the general nonlinear case. Theorem 3.1. Consider a finite time interval [0,τ]. Assume (H1)–(H6.δ)and the following: (H7) The law of X0has a C1positive density p0with respect to the Lebesgue measure and ∇p0(X0)/p0(X0)is in L2. If δin (H6.δ)is close enough to 1, in the sense that 1<δ<22/9, then the filter Mt given by (1.3) satisfies ˆx(1) t−m(1) t=O(ε),ˆx(2) t−m(2) t=O(√ε) in L2. The rest of this section is devoted to the proof of this theorem. Consider the matrix Pt def =         1 h(m(1) t)2σ(Mt)F12(Mt) h(m(1) t)ε3/2σ(Mt) h(m(1) t)ε σ(Mt) h(m(1) t)εσ(Mt)2σ(Mt) h(m(1) t)F12(Mt)ε1/2         , which depends only on Mt. Notice that Ptis the solution of the stationary Riccati equation −1 ε2PtH∗(Mt)H(Mt)Pt+ F(Mt)Pt+Pt F∗(Mt)+Σ(Mt)Σ∗(Mt)=0(3.1) with  F(Mt)=0F12(Mt) 00  and that the process Rtof (1.4) is Rt=Pt ε2H∗(Mt).(3.2) We will also need the inverse of Pt, namely, P−1 t=         h(m(1) t)2h(m(1) t) σ(Mt)F12(Mt)ε−3/2−h(m(1) t) σ(Mt)ε−1 −h(m(1) t) σ(Mt)ε−11 σ(Mt)2h(m(1) t)F12(Mt) σ(Mt)ε−1/2         . Change of probability measure. Our random variables can be viewed as functions of the initial condition X0and of the Wiener processes wand ¯w. We are going to 1808 PAULA MILHEIRO DE OLIVEIRA AND JEAN PICARD make a change of variables; in view of the Girsanov theorem, this can be viewed as a change of probability measure; however, all the estimations will be made under the original probability P. Thus consider the new probability measure which is given on Ftby d˙ P dP Ft =L−1 t, where L−1 t= exp −1 εt 0 h(x(1) s)d¯ws−1 2ε2t 0 h2(x(1) s)ds. The probability ˙ Pis the so-called reference probability, and one checks easily from the Girsanov theorem that yt/ε and wtare standard independent Wiener processes under ˙ P. Let us define now the probability measure  Pon Ftby d P d˙ PFt =Λ −1 t, where Λ−1 t= exp t 0 Σ∗(Ms)P−1 s(Xs−Ms)dws−1 2t 0 (Σ∗(Ms)P−1 s(Xs−Ms))2ds. Then the processes wt=wt−t 0 Σ∗(Ms)P−1 s(Xs−Ms)ds and yt/ε are standard independent Wiener processes under  P. On the other hand, one has dXt=f(Xt)dt +Σ(Xt)Σ∗(Mt)P−1 t(Xt−Mt)dt +Σ(Xt)dwt (3.3) and log(LtΛt)= 1 ε2t 0 h(x(1) s)dys−1 2ε2t 0 h2(x(1) s)ds −t 0 Σ∗(Ms)P−1 s(Xs−Ms)dws (3.4) −1 2t 0 (Σ∗(Ms)P−1 s(Xs−Ms))2ds . Differentiation with respect to the initial condition and an estimation. The random variables involved in our computation can now be viewed as functions of X0, {wt}, and {yt}; let us denote by ∇0the differentiation with respect to the initial condition X0(computed in Lp). In particular, we can see on (3.3) and (3.4) that the processes Xtand log(LtΛt) are differentiable, and we obtain matrixand vector-valued processes, respectively. Our aim is to estimate the process Vt def =(∇0log(LtΛt)(∇0Xt)−1+(Xt−Mt)∗P−1 t)U(3.5) APPROXIMATE NONLINEAR FILTERING 1809 with Udef =11 02/ε ,U −1=1−ε/2 0ε/2. Then an integration by parts will enable us to conclude. By applying the operator ∇0to (3.4), one gets ∇0log(LtΛt)= 1 ε2t 0 h(x(1) s)∇0x(1) s(dys−h(x(1) s)ds) −t 0 Σ∗(Ms)P−1 s∇0Xs(dws+Σ ∗(Ms)P−1 s(Xs−Ms)ds)(3.6) =1 εt 0 h(x(1) s)∇0x(1) sd¯ws−t 0 Σ∗(Ms)P−1 s∇0Xsdws. We can also differentiate (3.3), and, if Σis the Jacobian matrix of Σ, we obtain d(∇0Xt)=[F(Xt)+Σ(Xt)Σ∗(Mt)P−1 t]∇0Xtdt +Σ (Xt)∇0Xtdwt. The matrix ∇0Xtis invertible, and Itˆo’s calculus shows that d(∇0Xt)−1=−(∇0Xt)−1[F(Xt)+Σ(Xt)Σ∗(Mt)P−1 t−Σ2(Xt)] dt (3.7) −(∇0Xt)−1Σ(Xt)dwt. From this equation and (3.6), one can write that d∇0log(LtΛt)(∇0Xt)−1=1 εH(Xt)d¯wt−Σ∗(Mt)P−1 tdwt −∇0log(LtΛt)(∇0Xt)−1Σ(Xt)dwt −∇0log(LtΛt)(∇0Xt)−1 (3.8) .[F(Xt)+Σ(Xt)Σ∗(Mt)P−1 t−Σ2(Xt)]dt +Σ∗(Mt)P−1 tΣ(Xt)dt since one has h(x(1) t)∇0x(1) t(∇0Xt)−1=H(Xt). On the other hand, from the equations of Xtand Mt((1.1) and (1.3), respectively), one has d(Xt−Mt)=[f(Xt)−f(Mt)] dt −Rt[h(x(1) t)−h(m(1) t)] dt −Rtεd¯wt+Σ(Xt)dwt. By writing the differential of P−1 tin the form dP−1 t=J(1) tdt +J(2) td¯wt, we obtain d(Xt−Mt)∗P−1 t=[f∗(Xt)−f∗(Mt)−R∗ t(h(x(1) t)−h(m(1) t))]P−1 tdt +Σ∗(Xt)P−1 tdwt−εR ∗ tP−1 td¯wt (3.9) +(Xt−Mt)∗[J(1) tdt +J(2) td¯wt]−εR ∗ tJ(2) tdt . 1816 PAULA MILHEIRO DE OLIVEIRA AND JEAN PICARD -10 -5 0 5 10 15 20 25 30 0 10 20 30 -15 -10 -5 0 5 10 15 20 25 X−M X−¯ M time (s) errorerror (a) 0 20 40 60 80 100 120 140 0 10 20 30 -20 0 20 40 60 80 100 120 X−M X−¯ M time (s) errorerror (b) Fig. 5.1.Estimation errors for the (a) first and (b) second components of Xcomputed on a single trajectory. 5. Numerical simulation results. Let us consider the following example illustrating the case of free fall of a body through the atmosphere: dx(1) t=x(2) tdt, dx(2) t=(ρ0e−x(1) t/k(x(2) t)2/(2β)−g)dt +σdwt and dyt=!(x(1) t)2+a2dt +εd¯wt, where x(1) tis the position of the moving body and x(2) tis its speed, ρ0being the reference air density, kthe atmosphere thickness, βthe ballistic coefficient of the body, gthe acceleration due to gravity, and athe horizontal distance between the body and the measuring device (ρ0=3.4×10−3lbs2/ft4,k=22×103ft,β=1.6×103lb2/ft4, g=32.2ft/s2,σ=5ft/s, and a=10 4ft). Figure 5.1 shows the estimation errors obtained from applying the two approximate filters (filter (1.5), noted Mt, and the constant gain filter mentioned at the end of section 2 with ¯ H=0.02, noted ¯ Mt)toa single trajectory of the state with measurements taken each 0.001 s. The parameter εis equal to 1 and X0∼N3×105 −103,900 0 02×104. It illustrates the fact that the errors get small very quickly, and one notices that the constant gain filter needs more time than filter (1.5) to attain small errors in the second component. APPROXIMATE NONLINEAR FILTERING 1817 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 00.5 1 1.5 2 2.5 3 3.5 4 RMS error −log ε X−M X−˜ M M−˜ M (a) 1e-05 0.0001 0.001 0.01 0.1 0 0.5 1 1.5 2 2.5 3 3.5 4 RMS error −log ε X X−M X−˜ M M−˜ M (b) Fig. 5.2.Estimation errors for the (a) first and (b) second components of X. 1e-05 0.0001 0.001 0.01 -0.5 0 0.5 1 1.5 RMS error −log ε X−M X−˜ M M−˜ M (a) 0.0001 0.001 0.01 -0.5 0 0.5 1 1.5 RMS error −log ε X X−M X−˜ M M−˜ M (b) Fig. 5.3.Estimation errors for the (a) first and (b) second components of X(G=0.5). Figure 5.2 illustrates the asymptotic behavior of the estimation errors when system (1.1)–(1.2) with f(x1,x 2)=[x2−1.5×10−3x2 1]∗,σ= 2, and h(x1)= x2 1+10 8is considered. Although fand hfail to verify assumption (H3) and, in fact, inf h= 0, we will assume that the state remains in a bounded domain with high probability, thus assuming that inf h>1/√120. The root mean square error between the two approximate filters (with ¯ H=0.18) was computed for ε=1,10−1,... ,10−4 over 200 simulations for both components in the time interval [0,5]. The solid lines exhibit approximate slopes of −0.76 (first component) and −0.28 (second component) which agree with the results in section 2. The error associated with the constant gain filter and that associated to filter (1.5) are very similar. Figures 5.3–5.5 illustrate the van der Pol oscillator example presented in [12, section 6]: f(x1,x 2)=[x2−x1−x2]∗,σ= 1, and h(x1)=0.606(1 −G)x1+ 1818 PAULA MILHEIRO DE OLIVEIRA AND JEAN PICARD 0.0001 0.001 0.01 -0.5 0 0.5 1 1.5 RMS error −log ε X−M X−˜ M M−˜ M (a) 0.0001 0.001 0.01 -0.5 0 0.5 1 1.5 RMS error −log ε X X−M X−˜ M M−˜ M (b) Fig. 5.4.Estimation errors for the (a) first and (b) second components of X(G=0.8). 0.0001 0.001 0.01 -0.5 0 0.5 1 1.5 RMS error −log ε X−M X−˜ M M−˜ M (a) 0.0001 0.001 0.01 -0.5 0 0.5 1 1.5 RMS error −log ε X X−M X−˜ M M−˜ M (b) Fig. 5.5.Estimation errors for the (a) first and (b) second components of X(G=0.9). Gx3 1with G=0.5,0.8,0.9, respectively. The time interval [0,100] was considered. One can observe the increasing benefit of using filter (1.5) as the nonlinearity in the observations gets stronger. The results obtained by using the extended Kalman filter (EKF) are also shown in [12] for comparison. REFERENCES [1] A. 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