Filtering for a Duffing-van der Pol stochastic differential equation

Applied Mathematics and Computation 226 (2014) 386–397

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Filtering for a Duffing-van der Pol stochastic differential equation Hiren G. Patel ⇑, Shambhu N. Sharma Department of Electrical Engineering, National Institute of Technology, Surat, India

a r t i c l e

i n f o

Keywords: Continuous state-discrete measurement system The Fokker–Planck equation Modified second-order filter Stochastic differential equations

a b s t r a c t The Stochastic Duffing-van der Pol (SDvdP) system is a non-trivial case in stochastic system theory, since it involves linear, non-linear vector fields and state-dependent stochastic accelerations. Notably, analyzing specific and non-trivial cases brings greater theoretical insights as well as reveals joining points between different topics. In the theory of stochastic differential systems, the filtered estimates are more accurate in contrast to the predicted, since the observation correction terms associated with the filtering equations contribute to the effectiveness of the estimation procedure. Thus, it becomes reasonable to analyze the stochastic Duffing-van der Pol system from the filtering viewpoint. In this paper, we wish to revisit the stochastic Duffing-van der Pol ‘filtering’ in the Fokker–Planck setting in lieu of the filtering in the Kushner setting. In the Fokker–Planck setting, observations are accounted at discrete-time instants. As a result of this, we arrive at continuousdiscrete filtering equations. The Duffing-van der Pol filtering equations of this paper can be regarded as a consequence of the two-stage estimation procedure, since this paper utilizes stochastic differential equation formalism for the stochastic system of this paper. On the other hand, a discrete-time stochastic evolution of the observation is accounted for in this study. Notably, filtering equations of the paper explain explicit contributions of the predicted estimate and the observation noise to the filtered estimate. As a result of this, the filtering approach of this paper offers greater convenience for examining the filtering efficacy via simulations for specific cases. The filtering theory of this paper will be of interest to applied mathematicians, control theorists aspiring for understanding the continuous-discrete filtering better, especially for stochastic problems arising from technology, where observation rates are less. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction The filtering model in the theory of dynamical systems encompasses the stochastic differential system in combination with the observation equation. Here, we describe some of the previous investigations discussing noisy dynamical systems and their filtering. In classical mechanics, the two-body problem, i.e. the satellite-earth system, is formulated deterministically. However, when uncertainties in initial conditions and acting forces are present, one has to formulate the dynamics in terms of stochastic differential equations (SDEs), which lead to a Fokker–Planck equation (FPE) (Bierbaum et al. [1]). The Fokker–Planck equation, i.e. the evolution of the conditional probability density for Markov processes for given initial states, is useful for the problem of realizing the prediction algorithm (Challa and Bar-Salom [2]). Colored noise-driven stochastic ⇑ Corresponding author. E-mail addresses: [email protected] (H.G. Patel), [email protected] (S.N. Sharma). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.10.038

H.G. Patel, S.N. Sharma / Applied Mathematics and Computation 226 (2014) 386–397

387

differential equations lead to the notion of non-Markov processes. In this regard, Charles [3] will be useful. The problem of estimating unknown parameters, diffusion parameters can be accomplished by taking ‘observations at discrete-time instants and maximizing the conditional probability density given diffusion parameters’, see Dacunha-Castelle et al. [4] and Timmer [5]. A second-order Gaussian filter for the van der Pol system involving the Ordinary Differential Equation (ODE) with random initial conditions can be found in Kushner ([6], p. 551), see Chang [7] as well. A stochastically dust-perturbed two-body dynamics to model and study the effect of stochastic dust particles on the orbiting particle is available in Sharma and Parthasarathy [8]. Birkoff [9] can be consulted for understanding the theory of dynamical systems accounting random perturbations as well. The popular examples of randomly perturbed dynamical systems, which assume the structure of SDEs, are autonomous systems, electrical systems, biological and physical systems, see Brancik and Kolarova [10] as well as Kolarova [11]. Baxendale and Goukasian [12] have derived Lyapunov exponents for small perturbations of two-dimensional Hamiltonian systems. In their paper, the stochastic Duffing-van der Pol system was considered as a ‘standard test case’. The stochastic version of the Duffing-van der Pol system deserves considerable attention on the part of investigators from the filtering point of view as well. The stochastic Duffing-van der Pol system is a non-trivial and standard case in dynamical systems because of a generalized structure involving system linearity, system non-linearity, state-dependent as well as state-independent stochastic perturbations. Accounting observation terms in the stochastic analysis of dynamical systems gives rise to the concept of filtering theory. The problem of designing a non-linear filter for stochastic systems becomes quite difficult, especially involving investigations on vector random differential equations. Approximate filtering equations for the ‘continuous state-discrete measurement system’ involve a greater complexity in contrast to filtering theory for the ‘continuous statecontinuous measurement system’. Note that the continuous state-discrete measurement system involves the concept of prediction as well as filtering densities. For this reason, the noise analysis of appealing, non-trivial cases by utilizing the continuous-discrete filtering model is scarce in literature. This paper is inspired from the A N Kolmogorov recommendation that investigating specific and non-trivial cases reveals many-sided relationships in multi-varied branches of mathematics. In this regard, a greater detail can be found in a celebrated paper published in the 1954 International Congress of Mathematicians (ICM) [13], see references therein as well. With these motivations, we wish to develop and analyze continuous-discrete non-linear filtering equations for the Duffing-van der Pol system, an ‘Itô non-linear stochastic system’. The term continuous-discrete filtering’ of the paper accounts for the process noise in the dynamical equation and the observation noise in the discrete measurement equation. This paper can be regarded as a discrete counterpart of a paper [14]. In [14], the stochastic differential system in combination with a continuous measurement system was the subject of investigation. In contrast to the stochastic Duffing-van der Pol ‘filtering’ in the Kushner setting published in [14], the filtering theory of this paper adopts the discrete measurement system. The discrete measurement system gives rise to the notion of a two-stage estimation procedure. The two-stage estimation procedure accounts prediction equations between the observations and filtering equations at the observation instant. It is interesting to note that the notion of the discrete measurement system is useful for two cases: (i) observation rates are less (ii) lack of communications between the dynamical system and the tracking station measurement, e.g. satellite trajectory estimation problems. In lieu of the continuous filtering, realizing the continuous-discrete filtering becomes reasonably harder, since it involves the concept of the prediction as well as filtering theory. Because of two different estimation stages, the notations must be chosen carefully as well. For simplified discussions, we attempt for the notational brevity and define them explicitly in succeeding sections. It is hoped that Researchers with understanding of minimum essential formalisms associated with ‘matrix theory’ will find the paper revealing. Notations: For convenience, we introduce an additional notation hi, especially for the action of the conditional expectation operator on ‘quite lengthier terms’. On the other hand, we adopt the notation h; i for the inner product for finite-dimensional vectors. Throughout the paper, the notation for the state stochastic process xt is adopted from a celebrated book authored by Jazwinski ([16], p. 119). Alternatively, the notation Xt has found its application in stochastic processes literature as well. 2. Duffing-van der pol continuous-discrete filtering In this section, we explain a continuous-discrete filtering theory beginning from the exact filtering equations. Subsequently, the approximate continuous-discrete filtering equations are derived. Moreover, the filtering equations are exploited to accomplish the continuous-discrete filtering for the stochastic Duffing-van der Pol system. Consider the Itô stochastic system described by

dxt ¼ f ðxt ; tÞdt þ Gðxt ; tÞdBt ;

ð1aÞ

where xt is an n-dimensional state vector, f is an n-dimensional non-linear function of the state vector xt, G(xt, t) is the dispersion matrix, var(dBt) = Idt. Suppose the non-linear discrete noisy measurement system is of the form

ytk ¼ hðxtk ; t k Þ þ v tk ;

ð1bÞ

where ytk is m  1 observation vector, hðxtk ; tk Þ is an m-dimensional non-linear function of the state vector xtk and the observation noise v tk is Nð0; Rk Þ: Eq. (1b) contributes to the observation correction term for the filtered estimate, the updated moment equation. More precisely,

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hðxtk ; tk Þ ¼ ðha ðxtk ; tk ÞÞ16a6m ;

v ðxt ; tk Þ ¼ ðv a ðtk ÞÞ16a6m : k

_

The Kolmogorov–Fokker–Planck equation (Feller [15], p. 338) in conjunction with the definition of the evolution d /ðxs Þ of the conditional expectation of /(xs) for Eq. (1a) leads to _

d /ðxs Þ ¼

Z

/ðxÞdpðx; sjY tk1 Þdx ¼

* + * + * +! X X @/ðxs Þ 1 X @ 2 /ðxs Þ @ 2 /ðxs Þ þ þ ds; fp ðxs ; sÞ ðGGT Þpp ðxs ; sÞ ðGGT Þpq ðxs ; sÞ 2 2 @xp 2 p @xp @xp p p
where the scalar function /(xs) is twice continuously differentiable and _

/ðxs Þ ¼ Eð/ðxs ÞjY tk1 Þ ¼ h/ðxs Þi;

Y tk ¼ fytn ; 0 6 n 6 kg; t k1 < s < t k :

More clearly, the terms tk1 ; tk are the observation instants and the term s denotes the instant at which the observation is not available. The proof of the above equation is quite straightforward. One can arrive at the above equation by exploiting the definition of the conditional expectation, integration by part formula, the adjoint property of the Fokker–Planck operator, see Eq. (4.122) of Jazwinski ([16], p. 130). For /(xs) = xi(s) and xi(s)xj(s), we derive the evolutions of conditional mean and variance respectively, i.e. _

dx i ðsÞ ¼ hfi ðxs ; sÞids;

ð2aÞ



 dPij ðsÞ ¼ hxi fj i  hxi ihfj i þ hfi xj i  hfi ihxj i þ hðGGT Þij ðxs ; sÞi ds;

ð2bÞ

where,

Pij ¼ Eððxi  Eðxi ðsÞjY tk1 ÞÞðxj  Eðxj ðsÞjY tk1 ÞÞjY tk1 Þ: The analytical and numerical solutions to the above exact evolutions are not possible, since they become infinite dimensional and involve higher-order moments. Subsequently, after introducing ‘second-order partials of the system, measurement non-linearities and the diffusion coefficient evaluated at xs = hxsi’ into Eqs. (2a), (2b) and some simplifications, we obtain the following approximate evolutions: _

_

! _ 1 X @ 2 fi ðx s ; sÞ ds; P pq _ _ 2 p;q @ xp@ x q

dx i ¼

fi ðx s ; sÞ þ

dPij ¼

! _ _ _ T 2 X @fj ðx X @fi ðx _ 1 X @ ðGG Þij ðx s ; sÞ s ; sÞ s ; sÞ T ds: P ip þ Pjp þ ðGG Þij ðx s ; sÞ þ Ppq _ _ _ _ 2 p;q @ xp @ xp @ x p@ x q p p

ð3Þ

ð4Þ

The above evolution equations are regarded as the prediction equations for the stochastic differential system, since observation terms are not accounted for. Here, this paper explains the filtering equations at the observation instant tK. The ith component of conditional mean vector evolution and the (i,j)th element of conditional variance matrix evolution are the following: _t xik

t

_t

_t

t

_t

_ T

_

1

_ T

¼ x i k1 þ ðhxi hi  x i k1 h Þ ðhðh  h Þðh  h Þ i þ RÞ _ T

_

_ T

1

_t

 ðytk  y tk1 Þ; k __

t

Pijk ¼ P ijk1  ðhxi hi  x i k1 h Þ ðhðh  h Þðh  h Þ i þ RÞ ðhhxj i  h x j k1 Þ þ D; where _t xik

¼ Eðxi ðt k ÞjY tk Þ;

_t

x i k1 ¼ Eðxi ðt k ÞjY tk1 Þ;

_

h ¼ Eðhðxtk ; tk ÞjY tk1 Þ;

t

Pijk ¼ Eððxi  Eðxi ðt k ÞjY tk ÞÞðxj  Eðxj ðt k ÞjY tk ÞÞjY tk Þ; t

Pijk1 ¼ Eððxi  Eðxi ðtk ÞjY tk1 ÞÞðxj  Eðxj ðtk ÞjY tk ÞÞjY tk1 Þ:

ð5aÞ

ð5bÞ

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Note that Eq. (5) is a system of two equations, (5a) and (5b). The term R of Eq. (5) is the observation noise variance. The above set of evolutions for the scalar case is given in the famous book authored by Jazwinski ([16], p. 344). The random forcing term D of the conditional variance evolution poses the possibility of negative variances, which are not allowed in the theory of stochastic processes. For this reason, the notion of ‘modified variance evolution’ is introduced in which the random forcing term vanishes. This assumption gives rise to the notion of ‘modified’ filtering equations. The modified variance evolution is t

_t

t

_ T

_

_ T

1

__

t

Pijk ¼ Pijk1  ðhxi hi  x i k1 h Þ ðhðh  h Þðh  h Þ i þ RÞ ðhhxj i  h x j k1 Þ:

ð6Þ

The analytical solution to the conditional mean and variance evolutions, Eqs. (5)–(6), is not possible because of their infinite dimensionalities. For this reason, we explore the usefulness of the modified second-order approximate filter. The Carleman linearization has proven useful to transform the non-linear ODE as well as SDE into their approximate counterparts (Kowalski and Steeb [17]). However, the Carleman linearization for the exact mean and variance evolutions involving the ordinary differential equation as well as stochastic differential equation formalisms becomes difficult. The difficulty is attributed to the expectation operator associated with non-linearities. For this reason, in this paper, approximate filtering equations are derived by introducing the second-order partials of the non-linearities evaluated at the conditional mean into exact evolution equations. This method of approximation has found its applications for analyzing specific cases in dynamical systems literature as well, e.g. [8]. As a result of this approximation, the variance term is accounted in the mean trajectory, see Eqs. (3) and (4). Since the procedure is lengthy, here we state the coupled filtering equations. The detail discussing the intermediate steps to arrive at the filtering equations can be found in the appendix of the paper. The modified coupled filtering equations at the observation instant tk are _t xik

_t

¼ xi

k1

þ

XX a;b

t

t

Pijk ¼ Pijk1

_t

Pip

@ha ðx tk1 ; tk Þ k

p

_t

@ x pk1

_

XX @ha ðx ttk1 ; tk Þ k Hab dhb ðxtk ; t k Þ þ Pip Hab v b ðt k Þ; _t @ x pk1 a;b p

ð7Þ

_ 1 0 ! t _ ; t X X @ha x tk1 X @hb ðx ttk1 ; t k Þ k k k @ A ; H ab  P ip Pjp _t _t @ x pk1 @ x pk1 p p a;b

ð8Þ

where ha(.) is the ath component of the measurement non-linearity, vb(.) is the bth component of the observation noise and _t @ 2 hð x tk1 ;t k Þ P _t 1 k dhðxtk ; t k Þ ¼ ðdhb ðxtk ; tk ÞÞ16b6m ¼ hðxtk ; tk Þ  hðx tk1 ; t Þ  P þ v ðtk Þ, H _t _t pq k pq 2 k1 k1 k @ xp

00 ¼ ðHab Þ ¼ @@

X p1 ;p2

_t

Pp1 p2

@hðx tk1 ; tk Þ k _t

@ x pk1 1

T _t @h ðx tk1 ; tk Þ k _t @ x pk1 2

þ

@ xq

!1 1 1 _t T _t @ 2 hðx tk1 ; t k Þ @ 2 h ðx tk1 ; tk Þ 1X k k A A: P P þ R pr qs _t _t _t _t p;q;r;s 2 @ x pk1 @ x qk1 @ x rk1 @ x sk1 ab

Interestingly, Eqs. (7), (8) constitute the modified second-order filtering equations at the observation instant tk. The terms ‘modified’ and ‘second-order’ are attributed to ignoring the random forcing term D and introducing second-order partials of the measurement non-linearity hðxtk ; tk Þ into the variance evolution respectively. Here, we explain how the structure of filtering equations of the paper allows a greater convenience, which is different from the filtering results available in literature. First, we arrive at filtering Eqs. (7) and (8) of the paper using different notations and interpretation in contrast to the filtering results available, see the conditionally exact discrete filtering result in Pugachev and Sinitsyn [18]. Eqs. (7) and (8) explain the filtering equations for the vector observation equation rather than the scalar. Secondly, Eq. (7) of this paper discusses explicitly contributions from the measurement non-linearity and the observation noise to the filtered estimate. For this reason, filtering Eqs. (7), (8) offer greater mathematical convenience in contrast to evolution equations stated in Jazwinski ([16], p. 345), Pugachev and Sinitsyn ([18], pp. 517–518). The filtering equations, (7) and (8), in combination with prediction equations, (3) and (4), would be further applied to realize a continuous-discrete filter for the Duffing-van der Pol system. Most notably, the extended Kalman filter, a non-linear filter, is useful especially for the estimation procedure of non-linear stochastic differential systems involving the ‘state-independent’ perturbation (Jazwinski [16], pp. 338–339). On the other hand, the stochastic problem of this paper involves the nth order state-dependent perturbation, see Eq. (10). For this reason, the extended Kalman filtering equations for the stochastic problem of concern here were not the subject of investigation. Note that the problem of deriving the expression, which describes the contributions of neglected non-linearities to high_

_ T

er-order filtering equations, becomes harder for two reasons: (i) approximations to the term hðh  h Þðh  h Þ i of Eqs. (5)–(6), involving higher-order partials, become difficult, since the term hðxtk ; tk Þ of the measurement system is a vector non-linear function of the n-dimensional state vector (ii) for further simplifications, the expectation operator associated with the term _

_ T

ðh  h Þðh  h Þ requires the relationship between ‘higher-order conditional moments and conditional variance terms’ under the ‘nearly’ Gaussian assumption as well. These explain why the second-order filtering equations are popular in filtering

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theory in contrast to the higher-order and exact filtering. The approximate filtering equations seem to be intractable theoretically, especially their convergence analysis, since their global properties are replaced with the local. Alternatively, the efficacy of approximate filtering equations is adjudged on the basis of numerical experiments. Approximate filtering equations have proven useful in estimation theory, since they offer simplified analysis as well as preserve some of the qualitative characteristics of the exact filtering (Kushner [6]). Here, we briefly describe the stochastic Duffing-van der Pol system. The stochastic version of the Duffing-van der Pol system can be found in a probability Journal (Baxendale and Goukasian [12]), i.e.

€xt ¼ axt þ bx_ t  ax3t  bx_ t x2t þ rB xt

dBt dut þ ru ; dt dt

ð9Þ

2

t xt ¼ ddtx2t ; a P 0; b P 0; a < 0; b < 0; the Brownian motion process B ¼ fBt ; t0 < t < 1g: The term ut has the where x_ t ¼ dx ;€ dt interpretation as the Brownian motion process as well. Note that the Brownian motion process is differentiable nowhere. The nth order Taylor series expansion of the stochastic term Gðxt ; tÞdBt around its mean value involves a term rB xnt dBt ; a generalized structure of the stochastic perturbation. Eq. (9) is modified to

€xt ¼ axt þ bx_ t  ax3t  bx_ t x2t þ rB xnt

dBt dut þ ru : dt dt

ð10Þ

Thanks to a Theorem of Arnold ([19], p. 60), the above fluctuation equation can be replaced with the following system of dynamical equations:

x_ 1 ¼ x2 ; x_ 2 ¼ ax1 þ bx2  ax31  bx2 x21 þ rB xn1

dBt dut þ ru ; dt dt

where xt = x1 is the solution of Eq. (10). The vector nt = (x1, x2)T is a solution vector of the above system of dynamical equations. In the matrix–vector format, the above can be formalized as

dnt ¼ f ðnt ; tÞdt þ Gðnt ; tÞdBt þ Lt dut ;

ð11Þ

where T

T

nt ¼ ðx1 ; x2 ÞT ; f ðnt ; tÞ ¼ ðx2 ; f2 ðt; x1 ; x2 ÞÞ ¼ ðx2 ; ax1 þ bx2  ax31  bx2 x21 Þ ; T

Gðnt ; tÞ ¼ ð0; G2 ðt; x1 ; x2 ÞÞT ¼ ð0; rB xn1 Þ ;

Lt ¼ ru :

Eq. (11) describes a Duffing-van der Pol SDE in the Itô setting. The problem of realizing non-linear filters involves the observation equation. Here, we explain a mathematical method to design the structure of a non-linear discrete observation equation. The measurement non-linearity hðxtk ; tk is a physical quantity and a non-linear function of the phase variable xtk The choice of the measurement non-linearity depends on the problem under the subject of investigation. The stochastic system of this paper is a non-conservative system that can be demonstrated by using the multi-dimensional Itô differential rule. The multi-dimensional Itô differential rule is summarized in Theorem (3.6) of Karatzas and Shreve ([20], p. 153). A greater detail about examining the non-conservative property of stochastic differential systems can be found in the paper [14], see references therein. For a non-conservative system, Lvf, the Lie derivative of a scalar non-linear function of the phase variables of the system does not vanish, see the definition of the first integral in Arnold ([19], p. 75). Because of the non-conservative nature of the stochastic system of this paper, the term Lvf, which is a non-linear function of the phase variable nt = (x1, x2)T as well as a physical quantity, can be taken as the measurement non-linearity. Thus 2

Lv f ¼ hðx1 ; x2 ; tÞ ¼ bx22  bx2 x21 ; and in the ‘discrete-time’ setting, the observation equation becomes 2

ytk ¼ bx22 ðt k Þ  bx2 ðt k Þx21 ðt k Þ þ v tk :

ð12Þ

This paper considers the non-linear discrete observation equation, which contributes to higher-order partials of the nonlinearity and allows ‘examining the efficacy of non-linear filters’. This paper accomplishes filtering analysis of an SDE. Importantly, the theoretical interpretations of the white noise SDE are the Stratonovich and Itô SDEs (Mannella and McClintock [21]). In this paper, we restrict our discussions to the filtering analysis of an Itô SDE in lieu of the Stratonovich. Making the use of Eqs. (3), (4) and Eq. (11), we get the mean and variance evolutions for the Duffing-van der Pol SDE between the observations, i.e.

391

H.G. Patel, S.N. Sharma / Applied Mathematics and Computation 226 (2014) 386–397 _

_

dx 1 ðsÞ ¼ x 2 ds; _

_

ð13aÞ _

_

_ _

_

_

_

dx 2 ðsÞ ¼ ða x 1 þ bx 2  ax 31  bx 2 x 21 þ ð3ax 1  bx 2 ÞP 11 þ ð2bx 1 ÞP12 Þds;

ð13bÞ

dP11 ðsÞ ¼ 2P12 ds;

ð13cÞ _

_ _

_

dP12 ðsÞ ¼ dP 21 ðsÞ ¼ ðP11 ða  3ax 21  2bx 1 x 2 Þ þ P12 ðb  bx 21 Þ þ P22 Þds; _

_ _

_

ð13dÞ

_

_

2n2 2 dP22 ðsÞ ¼ ð2P12 ða  3ax 21  2bx 1 x 2 Þ þ 2ðb  bx 21 ÞP 22 þ r2B x 2n þ r2u Þds: 1 þ nð2n  1ÞrB P 11 x 1

ð13eÞ

_

Note that Eq. (13) is a system of equations, i.e. (13a)–(13e). The right-hand side term x i of Eq. (13) denotes _ x i ¼ Eðxi ðsÞjY tk1 Þ0 : The right-hand side term Pij of Eq. (13) denotes 0 Pij ¼ Eððxi  Eðxi ðsÞjY tk1 ÞÞðxj  Eðxj ðsÞjY tk1 ÞÞjY tk1 Þ0 ; where 1 6 i 6 2; 1 6 j 6 2: After combining Eq. (12) in combination with Eqs. (7), (8), we are led to

0

_t x 1k

_t

_ _

_

_ _

_

2

_ _

_

¼ x 1k1 þ ðP11 ð2bx 1 x 22 Þ þ P12 ð2bx 2  2bx 21 x 2 ÞÞ  ðHxx þ RÞ1  ðbx22  bx2 x21  bx 22 þ bx 22 x 21 þ P11 ðbx 22 Þ _

_ _

þ P22 ðbx 21  bÞ þ P12 ð4bx 1 x 2 Þ þ v tk Þ; _t x 2k

_t

_ _

_

ð14aÞ _ _

_

2

_ _

_

¼ x 2k1 þ ðP12 ð2bx 1 x 22 Þ þ P22 ð2bx 2  2bx 21 x 2 ÞÞ  ðHxx þ RÞ1  ðbx22  bx2 x21  bx 22 þ bx 22 x 21 þ P11 ðbx 22 Þ _

_ _

þ P22 ðbx 21  bÞ þ P12 ð4bx 1 x 2 Þ þ v tk Þ;

ð14bÞ

t

t

_ _

_

_ _

_ _

_

_ _

t

t

_ _

_

_ _

_ _

_

_ _

t

t

_ _

_

_ _

_ _

_

_ _

k k1 P11 ¼ P11  ðP11 ð2bx 1 x 22 Þ þ P12 ð2bx 2  2bx 21 x 2 ÞÞ  ðHxx þ RÞ1  ðP11 ð2bx 1 x 22 Þ þ P12 ð2bx 2  2bx 21 x 2 ÞÞ;

k k1 P12 ¼ P12  ðP11 ð2bx 1 x 22 Þ þ P12 ð2bx 2  2bx 21 x 2 ÞÞ  ðHxx þ RÞ1  ðP12 ð2bx 1 x 22 Þ þ P22 ð2bx 2  2bx 21 x 2 ÞÞ;

k k1 P22 ¼ P22  ðP12 ð2bx 1 x 22 Þ þ P22 ð2bx 2  2bx 21 x 2 ÞÞ  ðHxx þ RÞ1  ðP12 ð2bx 1 x 22 Þ þ P22 ð2bx 2  2bx 21 x 2 ÞÞ:

ð14cÞ ð14dÞ ð14eÞ

More precisely, Eq. (14) is a system of five equations, i.e. (14a)–(14e), as well. The term Hxx of Eq. (14) can be stated as 2_ _

_

_

2

_ _

_

2_

2_ _

2_ _

Hxx ¼ 4b x 21 x 42 P11 þ 4x 22 ðb  bx 21 Þ P22  8bx 1 x 32 ðb  bx 21 ÞP12 þ 2b x 42 P211 þ 16b x 1 x 32 P11 P12 þ 16b x 21 x 22 P212 _

_

2_ _

_ _

_

_

2

þ 4bx 22 ðbx 21  bÞP212 þ 16b x 21 x 22 P11 P22 þ 16bx 1 x 2 ðbx 21  bÞP12 P 22 þ 2ðb  bx 21 Þ P 222 : The above system of stochastic difference equations describes the conditional mean and variance evolutions ‘at the observation’. Importantly, the ‘notation’ for the conditional expectation of the random variable must be chosen carefully, espe_ cially for the continuous state-discrete measurement system. The right-hand side term x i of Eq. (14) denotes _ ’x i ¼ Eðxi ðtk ÞjY tk1 Þ’. The right-hand side term Pij of Eq. (14) denotes 0

P ij ¼ Eððxi ðt k Þ  Eðxi ðt k ÞjY tk1 ÞÞðxj ðtk Þ  Eðxj ðtk ÞjY tk1 ÞÞjY tk1 Þ0 :

Note that Eqs. (13)–(14) describe the continuous-discrete filtering equations for the Duffing-van der Pol SDE that are derived the first time. 3. Numerical simulations The numerical simulation of filtering equations can be accomplished by discretizing the evolution equations. The initial conditions, system parameters and the observation noise variance are chosen as _

_

a ¼ 0:1; b ¼ 0:4; a ¼ 2; b ¼ 0:5; ru ¼ 0:07; rB ¼ 0:008; R ¼ 0:0025; n ¼ 3; x 1 ð0Þ ¼ 0:1; x 2 ð0Þ ¼ 0:5; P11 ð0Þ ¼ 1; P22 ð0Þ ¼ 2; Pij ð0Þ ¼ 0; i–j; T

where R denotes the observation noise variance. The system parameter vector ðn; a; b; b; a; rB ; ru Þ is associated with the dynamical equation (11). Notably, the parameters a; b have interpretations as the Duffing and van der Pol parameters respectively. For the bound state trajectory, we consider a > 0; b > 0: Furthermore, the parameters a; b have interpretations as the oscillatory and damping terms respectively that take negative values for the bounded state trajectory. Here, the initial variances are chosen ‘non-zero’ and covariances take zero values, which illustrate uncertainties in initial conditions and uncertainties are initially uncorrelated respectively. The order of the state-dependent perturbation, n; is chosen as three, since this choice of the order contributes to the higher-order partials of the diffusion coefficient ðGGT Þðxt ; tÞ. The ‘state-dependent perturbation order’ can be chosen larger. As far as the Authors know, there is no alternative published ‘continuous-discrete filtering equations’ for the stochastic problem of concern here available in literature. For this reason, comparison with

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Position trajectory

0.4 0.2 0 -0.2 -0.4

0

4

8

12

16

Time (t) Fig. 1. True and filtered position.

Velocity trajectory

0.5

0

-0.5

0

5

10

15

Time (t)

Variance trajectory of position

Fig. 2. True and filtered velocity.

1 0.8 0.6 0.4 0.2 0 0

5

10

15

Time (t) Fig. 3. A conditional variance trajectory.

the published filtering results is not given. As a result of this, the numerical testing associated with the filtering equations of this paper is accomplished using two different sets of system parameters, random initial conditions, and observation noise variances. Secondly, the comparison between the true and filtered states is demonstrated as well. It is believed that the numerical experiments of this paper will suffice to examine the efficacy of the filtering results derived in the previous section. A good source for understanding ‘filtering via simulations’ can be found in a probability Journal [22], see a seminal paper authored by Kushner ([6], p. 552) as well. The discrete evolutions of the conditional mean and variance involve the parameter R; see Eq. (14) of the paper. For R ¼ 0; the terms involving the parameter R of Eq. (14) become somewhat undefined. For R ¼ 1; the terms of Eq. (14) vanish. As a result of this, non-linear filtering equations become prediction equations. It becomes reasonable to examine the effectiveness of non-linear filters for 0 < R < 1: In this paper, the filter efficacy is adjudged on the basis of a relatively larger intensity of the observation noise as well as the less intensity.

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Variance trajectory of velocity

2

1.5

1

0.5

0

0

5

10 Time(t)

15

Fig. 4. A conditional variance trajectory.

Position trajectory

0.4

0.2

0

-0.2

-0.4 0

5

10

15

Time(t) Fig. 5. True and filtered position.

Velocity trajectory

0.6 0.4 0.2 0 -0.2 -0.4

0

5

10

15

Time(t) Fig. 6. True and filtered velocity.

Note that the dotted line (–), solid line (-) associated with the graphs of this paper, Figs. 1 and 2, correspond to true simulated states, second-order filtered states respectively. Figs. 1–4 illustrate the numerical testing for the modified continuousdiscrete filtering equations, i.e. the modified second-order, under less intensity of the noise introduced into observations. The numerical testing demonstrated in Figs. 1–4 illustrates that the second-order Duffing-van der Pol filtering algorithm of the paper displays bounded mean and variance trajectories respectively. The modified second-order filter accounts for additional correction terms, which come from second-order partials of the system non-linearity f(xs, s) and the diffusion coefficient GGT ðxs ; sÞ‘ between the observations’ and second-order partials of the measurement non-linearity f ðxtk ; tk Þ ‘at the observation’. These correction terms suggest a greater usefulness of the filtering equations of the paper. For accomplishing numerical simulations, the MATLAB programs are developed. We simulate the filtering equations of the paper in the MATLAB environment by choosing two sets of initial conditions, system parameters. Since this paper demonstrates numerical simulations for the vector van der Pol stochastic differential equation in lieu of scalar deterministic cases, the computational time is relatively larger.

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Variance trajectory of position

2

1.5

1

0.5

0

0

5

10

15

Time(t)

Variance trajectory of velocity

Fig. 7. A conditional variance trajectory.

4 3 2 1 0

0

5

10

15

Time (t) Fig. 8. A conditional variance trajectory time (t).

Here, we accomplish the numerical testing for the filtering equations, Eqs. (13)–(14), utilizing the second set of system parameters, random initial conditions, observation noise variance, i.e. a ¼ 0:1; b ¼ 0:4; a ¼ 2; b ¼ 0:5; ru ¼ 0:07; rB _ _ ¼ 0:028; R ¼ 0:045; n ¼ 3; x 1 ð0Þ ¼ 0:1; x 2 ð0Þ ¼ 0:5; P11 ð0Þ ¼ 2; P 22 ð0Þ ¼ 4; Pij ð0Þ ¼ 0; i–j: For the graphical notations, we utilize the dotted line (–), solid line (-) for Figs. 5 and 6 that correspond to true simulated states, second-order filtered states respectively. Figs. 5–8 illustrate the numerical testing for the modified continuous-discrete filtering equations, i.e. the modified second-order, under larger intensity of the noise introduced into observations. Figs. 5 and 6 confirm the usefulness of the second-order Duffing-van der Pol filtering algorithm of this paper in the sense that it produces bounded estimates. It is believed that the term Hxx of Eq. (14), involving multiplicative variance terms, introduces a corrective effect into discrete mean and variance evolutions. The approximate filtering Eqs. (13)–(14) would be useful for the estimation procedure of the Duffing-van der Pol system, since the filtering equations preserve non-linearities via _ P @ 2 ðGGT Þij ð x s ;sÞ accounting the contributions coming from the terms 12 p;q P pq . Figs. 1 and 2 resulting from the first set of system _ _ @ x p@ x q

parameters and Figs. 5 and 6 from the second set of parameters show the bounded state vector trajectory with decrease in oscillations. The estimated state trajectory follows the pattern of the true state trajectory as well. Interestingly, the difference between the true simulated state and filtered estimate becomes larger at some instants, which is attributed to the measurement noise variance [14]. Notably, the filtering equations of the stochastic problem of concern here are derived by choosing the measurement system described by Eq. (12). In this paper, we tried to list and cite the related seminal papers on the topic. Some of the important papers are not explicitly, but indirectly credited as well. 4. Conclusion In this paper, we have developed a continuous-discrete filter for a stochastic Duffing-van der Pol SDE. That can be regarded as a discrete counterpart of the stochastic Duffing-van der Pol continuous filtering. The continuous-discrete filtering equations, Eqs. (13)–(14) of this paper, will be greatly useful to develop the ‘stochastic adaptive control law’ of the Duffingvan der Pol SDE as well, where observations are available at the discrete-time instants. Secondly, this paper discusses a mathematical method to design the structure of non-linear discrete observation equation. The mathematical method hinges on the qualitative analysis of the stochastic differential system in the context of continuous-discrete filtering. The method will be useful for the choice of the observation equation for continuous-discrete filtering of other stochastic problems as well.

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395

Thirdly, the structure of filtering equations of the paper explains the contributions of the predicted estimate and observation noise ‘explicitly’ to the filtered estimate. For this reason, the general structure of filtering equations, Eqs. (7), (8) of the paper, are more convenient for filtering analysis of specific cases in contrast to results available in literature. More precisely, Eqs., (3), (4), (7), (8) and (13)–(14), are the main analytic results of the paper. Despite the continuous-discrete filtering equations of the paper were derived under restrictions for a specific case, research communities in ‘control and stochastic processes’ will find the paper suggestive in the sense that describing the continuous-discrete filtering procedure in detail. That will be of use to other potential engineering problems as well. A potential extension of the filtering results of the paper is to develop higher-order continuous-discrete filtering theory. Another important extension of the paper is to accomplish control of van der Pol SDE by utilizing filtering equations of van der Pol SDE as well. Acknowledgments The authors are grateful to ananymous reviewers for their useful suggestions that helped refine the paper. Appendix A Here we explain a simple but a formal approach to derive the modified coupled filtering equations at the observation instant tk. The modified filtering equations are derived by introducing the second-order partials of the non-linearities evaluated at the conditional mean into exact evolution equations. The matrix–vector format and the inner product will be the standard formalisms to derive the filtering equations at the observation instant. After introducing second-order partials of _t the measurement non-linearity hðxtk ; tk Þ evaluated at xtk ¼ x tk1 ; we have k

ðhxi hi 

_ T _t x i k1 h Þ

_

X @hT ðx ttk1 ; tk Þ k ¼ Pip : _t @ x pk1 p

ðA:1Þ

_

_ T

After taking the second-order approximate to the term hðh  h Þðh  h Þ i of Eqs. (5)–(6) as well as using the definition of the conditional characteristic function for the Gaussian random vector (Papoulis [23], p. 115), we get _

_ T

hðh  h Þðh  h Þ i ¼

X

_t

Pp1 p2

T _t

@hðx tk1 ; t k Þ @h ðx tk1 ; tk Þ k k _t

_t

@ x pk1 1

p1 ;p2

@ x pk1 2

_

þ

_

T t t @ 2 hðx tk1 ; t k Þ @ 2 h ðx tk1 ; tk Þ 1X k Ppr Pqs _t k _t : _t _t k1 k1 k1 2 p;q;r;s @ xp @ xq @ x r @ x sk1

ðA:2Þ

Most notably, the approximate filtering equations of this paper hinge on the ‘nearly’ Gaussian assumption. After a simple calculation and utilizing the definition of conditional characteristic function for the Gaussian random vector with zero conditional mean vector, we can show that the odd powers of the conditional moment vanish. Moreover, even powers of the conditional moment can be replaced with conditional variance terms. After introducing the second-order approximation to the measurement non-linearity, we have _t

_t

_t

ytk  y tk1 ¼ hðxtk ; tk Þ  hðx tk1 ; tk Þ  k k

2 ; tk Þ 1 X @ hðx tk1 Ppq _t k _t þ v ðt k Þ ¼ dhðxtk ; tk Þ þ v ðt k Þ: k1 2 p;q @ x p @ x qk1

ðA:3Þ

Eq. (A.3) is a consequence of the observation ytk ¼ hðxtk ; t k Þ þ v tk and the definition of conditional expectation. Eq. (5a) in conjunction with Eqs. (A.1), (A.2) is rewritten as _t xik

_t

¼ x i k1 þ

! !1 _ _t _t T _t T _t X @hT ðx ttk1 ; tk Þ X @hðx tk1 ;tk Þ @h ðx tk1 ; tk Þ 1 X @ 2 hðx tk1 ;tk Þ @ 2 h ðx tk1 ; tk Þ k k k k k P ip P þ P P þ R  ðdhðxtk ;tk Þ þ v ðtk ÞÞ;  p p pr qs _t _t _t _t _t _t _t 1 2 2 p;q;r;s @ x pk1 @ x pk1 @ x pk1 @ x pk1 @ x qk1 @ x rk1 @ x sk1 p p1 ;p2 1 2

ðA:4Þ and

! _ _ X @ha ðx ttk1 ; t k Þ X @hðx ttk1 ; tk Þ k k ¼ Pip Pip ; _t _t @ x pk1 @ x pk1 p p X p1 ;p2

_t

P p1 p2

_t

@ha ðx tk1 ; t k Þ @hb ðx tk1 ; tk Þ k k _t

@ x pk1 1

_t

@ x pk1 2

! ¼

X p1 ;p2

ðA:5aÞ _t

Pp1 p2

T _t

@hðx tk1 ; t k Þ @h ðx tk1 ; tk Þ k k _t

@ x pk1 1

_t

@ x pk1 2

;

ðA:5bÞ

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00 H ¼ ðHab Þ ¼ @@

X

_t

P p1 p2

T _t

@hðx tk1 ; t k Þ @h ðx tk1 ; tk Þ k k _t

@ x pk1 1

p1 ;p2

_t

@ x pk1 2

_

_

T t t @ 2 hðx tk1 ; t k Þ @ 2 h ðx tk1 ; tk Þ 1X k þ Ppr Pqs _t k _t þR _t _t 2 p;q;r;s @ x pk1 @ x qk1 @ x rk1 @ x sk1

!1 1 1 A A:

ðA:5cÞ

ab

Note that Eq. (A.5) is a set of three equations, (A.5a), (A.5b), (A.5c). Eq. (A.4) can be further recast in a convenient form using the notion of the inner product. Thus, we have _t xik

¼

_t x i k1

_t

¼ x i k1

* + _ X @hðx ttk1 ; tk Þ k þ Pip ; Hðdhðxtk ; t k Þ þ v ðt k ÞÞ _t @ x pk1 p * + * + _ _ X @hðx ttk1 ; tk Þ X @hðx ttk1 ; tk Þ k k þ þ Pip ; Hdhðx ; t Þ P ; H v tk k tk : ip _t _t @ x pk1 @ x pk1 p p

After combining Eq. (A.5) with the above equation, we get Eq. (7) of the paper. _

_t xik

_t

¼ x i k1 þ

_

XX @ha ðx ttk1 ; t k Þ XX @ha ðx ttk1 ; tk Þ k k P ip Hab dhb ðxtk ; t k Þ þ Pip Hab v b ðt k Þ: _t _t k1 k1 @ x @ x p p a;b a;b p p

From Eq. (6) in conjunction with Eqs. (A.1)–(A.3), we get t

t

Pijk ¼ P ijk1 



X p1 ;p2



! _ X @hT ðx ttk1 ; t k Þ k P ip _t @ x pk1 p _t

P p1 p2

T _t

@hðx tk1 ; tk Þ @h ðx tk1 ; tk Þ k k _t

@ x pk1 1

! _ X @hðx ttk1 ; t k Þ k P jp : _t @ x pk1 p

_t

@ x pk1 2

_

þ

_

T t t @ 2 hðx tk1 ; t k Þ @ 2 h ðx tk1 ; tk Þ 1X k Ppr Pqs _t k _t þR _t _t 2 p;q;r;s @ x pk1 @ x qk1 @ x rk1 @ x sk1

!1

ðA:6Þ

Eq. (A.6) can be further recast by utilizing ‘the notion of the inner product and Eq. (A.5)’, i.e. t

t

Pijk ¼ P ijk1

_ 1 0 * + ! t _ _ @ha x tk1 ; tk X @hðx ttk1 ; t k Þ X @hðxt ; tk Þ X X X @hb ðx ttk1 ; t k Þ k t k1 k k k @ A H ab ¼ Pij  :  P ip ; H P jp P ip Pjp _t _t _t _t @ x pk1 @ x pk1 @ x pk1 @ x pk1 p p p p a;b

The above equation is Eq. (8) of the paper. This completes the proof of the coupled filtering Eqs. (7), (8) of the paper. References [1] M.M. Bierbaum, R.I. Joseph, R.L. Fry, J.B. Nelson, A Fokker–Planck model for a two-body problem, in: AIP Conference Proceedings, vol. 617, 2002, pp. 340–371. [2] S. Challa, Y. Bar-Salom, Non-linear filter design using Kolmogorov–Fokker–Planck probability density evolutions, IEEE Trans. Aerosp. Electron. Syst. 36 (1) (2000) 309–314. [3] W.M. Charles, Application of coloured noise as a driving force in the stochastic differential equations, in: Chris Myers (Ed.), Stochastic Control, Sycio Science Publisher, Vienna, Rizeka, 2010, pp. 43–58. [4] D. Dacunha-Castelle, D. Florens-Zmirou, Estimations of the coefficients of diffusion from discrete observations, Stochastics 19 (1986) 263–284. [5] J. Timmer, Parameter estimations in non-linear stochastic differential equations, Chaos Solitons Fractals 11 (2000) 2571–2578. [6] H.J. Kushner, Approximations to optimal non-linear filter, IEEE Trans. Automat. Control 12 (5) (1967) 546–556. [7] R.L. Chang, Pre-computed gain non-linear filters for non-linear systems with state-dependent noise, J. Dyn. Syst. Meas. Contr. 112 (1990) 270–275 (an ASME Journal). [8] Shambhu N. Sharma, H. Parthasarathy, Dynamics of a stochastically perturbed two-body problem, Proc. R. Soc. A Math. Phys. Eng. Sci. 463 (2007) 979– 1003. [9] G.D. Birkoff, Dynamial systems, Colloq. Publ. IX, Amer. Math. Soc., Providence R.I., 1966. [10] L. Brancik, E. Kolarova, Analysis of higher-order electrical circuits with stochastic parameters via SDEs, Advances in Electrical and Computer Engineering 13 (1) (2013) 17–22. [11] E. Kolarova, An application of stochastic integral equations to electrical networks, Acta Electrotech. Inform. 8 (3) (2008) 14–17. [12] P.H. Baxendale, L. Goukasian, Lyapunov exponents for small random perturbations of Hamiltonian systems, Ann. Probab. 30 (1) (2002) 101–134. [13] A.N. Kolmogorov, General theory of dynamical systems and classical mechanics, Proc. Intern. Congress of Mathematicians, Amsterdam, 1, Noordhoff, Groningen, 1954, pp. 315–333. [14] Shambhu N Sharma, A Kushner approach for small random perturbations of a stochastic Duffing-van der Pol system, Automatica (a Journal of IFAC, International Federation of Automatic Control), vol. 45, pp. 1097–1099, 2009. [15] W. Feller, An Introduction to Probability Theory and Its Applications, second edition., John Wiley and Sons, New York, 1988. [16] A.H. Jazwinski, Stochastic Processes and Filtering Theory, Academic Press, New York and London, 1970. [17] K. Kowalski, W.-H. Steeb, Non-linear Dynamical Systems and Carleman Linearization, World Scientific, Singapore, New Jersy, 1988.

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