Filtering of Differential Nonlinear Systems Via a Carleman

Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005

WeC11.1

Filtering of Differential Nonlinear Systems via a Carleman Approximation Approach 44th IEEE Conf. on Decision and Control & European Control Conference (CDC{ECC 2005) Alfredo Germani

Costanzo Manes

Abstract— This paper deals with the state estimation problem for a stochastic nonlinear differential system driven by a standard Wiener process. The solution here proposed is a linear filtering algorithm and is achieved by means of the Carleman approximation scheme applied to both the state and the measurement nonlinear equations. Such a procedure allows to define an approximate representation by means of a suitable bilinear system for which a filtering algorithm is available from literature. Numerical simulations support the theoretical results and show a rather interesting improvement in terms of sampled error covariance of the proposed approach with respect to the classical Kalman-Bucy filter applied to the linearized differential system. Index Terms—Nonlinear filtering, Extended Kalman Filter, Carleman approximation, Stochastic Systems.

I. INTRODUCTION Given the probability triple (l; F; P), in this paper it will be considered the ¯ltering problem for the following nonlinear stochastic di®erential system described by the Ito equations: ¡ ¢ dxt = Á xt dt + F dW s (t); ¡ ¢ dyt = h xt dt + GdW o (t);

xjt=0 = x0 ; yjt=0 = 0;

()

where xt is the state vector in IRn , yt is the measured output in IRq and W s (t) 2 IRp , W o (t) 2 IRq are independent standard Wiener processes with respect to some ª © increasing family of ¾-algebras, namely Ft ; t ¸ 0 . Á : IRn 7 ! IRn and h : IRn 7 ! IRq are smooth nonlinear maps. It is well known that the minimum variance state estimate requires the knowledge of the conditional probability density, whose computation, in the general case, is a di±cult in¯nite-dimensional problem [4, 20, 2, 22, 29]. Only in few cases the optimal ¯lter has a ¯nite dimension [27]. For this reason a great deal of work has This work is supported by MIUR (Italian Ministry for Education and Research) and by CNR (Italian National Research Council). A. Germani and C. Manes are with the Dipartimento di Ingegneria Elettrica, Universitµ a degli Studi dell'Aquila, Poggio di Roio, 67040 L'Aquila, Italy, [email protected], [email protected]. P. Palumbo is with Istituto di Analisi ed Informatica \A. Ruberti", IASI-CNR (National Research Council of Italy), Viale Manzoni 30, 0085 Roma, Italy, [email protected].

Pasquale Palumbo

been made to devise suboptimal implementable ¯ltering algorithms [9, 0, 2, 5]. Another approach consists in considering the time discretization of the original system and then to apply suboptimal ¯ltering procedures like the Extended Kalman Filter (EKF), the most widely used algorithm in nonlinear ¯ltering problems (see, e.g., [, 8, , 7]), particle ¯lters [23], Gaussian sum approximations [6] and more. Because of its local nature, the EKF performs well if the initial estimation error and the disturbing noises are small enough. An e®ective modi¯cation of the EKF is the Unscented Kalman Filter (UKF) [8], that uses the so-called unscented transform for the state and output prediction steps in the EKF scheme. More recently, in [3] has been proposed a polynomial extension of the EKF (denoted PEKF) which is based on the application of the optimal polynomial ¯lter of [5, 6] to the Carleman approximation of the nonlinear system (see [9, 24]). The aim of this paper is to overcome the drawbacks of the time discretization errors by means of the use of the Carleman approximation on the original stochastic di®erential system. The Carleman approximation of order º of a nonlinear system is achieved by suitably de¯ning an extended state made of the Kronecker powers of the original state up to a given order º. The result is a bilinear system (linear drift and multiplicative noise) with respect to the extended state. The tool of Carleman bilinearization has found some applications in problems of systems approximation [25, 26, 28], since there are many reasons for ¯nding a bilinear approximation of a nonlinear system (see [3]). In recent times such method has been successfully used in the problem of reduction of large scale systems [2]. Once the approximation is obtained, the recursive equations of the optimal linear ¯lter for bilinear stochastic di®erential systems are available and can be applied with no further approximations [7]. When º = the proposed ¯ltering algorithm reduces to the classical Kalman-Bucy ¯lter applied to the linearized di®erential system. With respect to this classical approach the method here proposed performs a better behavior by increasing the approximation degree of the nonlinear system. The paper is organized as follows: the next section

0-7803-9568-9/05/$20.00 ©2005 IEEE

5917

presents the Carleman approximation of stochastic nonlinear di®erential systems of the type (); in section three the linear minimum variance ¯lter for the Carleman approximation is derived; section four displays some numerical results where the performances of the proposed algorithm are compared with those of the classical approach (the case of º = ). II. CARLEMAN APPROXIMATION © ª Let (l; F ; P ) be a probability space, Ft with t 2 [0; ] be a¢family of nondecreasing ¾-algebras of F and ¡ T ¡ ¢ W s (t); Ft , W o (t); Ft be standard Wiener processes taking values in IRp and IRq respectively. Consider the stochastic nonlinear di®erential system described by (), where xt is the state vector in IRn , yt is the measured output in IRq and Á : IRn 7 ! IRn , h : IRn 7 ! IRq are smooth nonlinear maps. By de¯ning the standard ¤T £ taking values in Wiener process W = W sT W oT IRb , b = p + q, system () may be rewritten as: b X ¡ ¢ Fi dWi (t); dxt = Á xt dt +

xjt=0 = x0 ;

i=1

(2)

b X ¡ ¢ Gi dWi (t); dyt = h xt dt +

yjt=0 = 0;

i=1

£ ¤ with Fi , Gi , i = ; : : : ; b columns of matrices F = F 0 , £ ¤ G= 0 G .

The initial state x0 is an F0 -measurable random variable, independent of W , with ¯nite and available moments up to degree 2º, namely: £ [i] ¤ ³i = IE x0 ;

k³i k < +1;

i = ; : : : ; 2º;

(3)

with the square brackets denoting the Kronecker powers (see [6] for a quick survey on the Kronecker product and its main properties). Under standard analyticity hypotheses, both the state and the output equations can be written by using the Taylor polynomial expansion around a given state x~. According to the Kronecker formalism, the di®erential system in (2) becomes:

dxt =

1 X

¡ ¢[i] ©i (~ x) xt ¡ x~ dt +

1 X

b X ¡ ¢[i] Hi (~ x) xt ¡ x~ dt + Gj dWj (t);

i=0

dyt =

i=0

b X

Fj dWj (t);

j=1

(4)

j=1

with: ©i (x) =

´ ³ [i] rx l Á ; i!

Hi (x) =

´ ³ [i] rx l h : (5) i!

5918

[i]

The operator rx l applied to a function Ã = Ã(x) : IRn 7 ! IRp is de¯ned as follows ¡ ¢ r[i+1] l Ã = rx l r[i] i ¸ ; r[0] x l Ã = Ã; x x lÃ ; (6) with rx = [@=@x1 ¢ ¢ ¢ @=@xn ]. Note that rx l Ã is the standard Jacobian of the vector function Ã. The idea of the paper is to use the º-degree Carleman approximation applied to system (2). Such an approach allows to obtain a bilinear di®erential system, which is then ¯ltered according to corresponding optimal linear ¯lter [7]. For the reader's convenience, below are reported some useful propositions, concerning the Kronecker product properties and the di®erential operator (6). Recall that the Kronecker product is not commutative: given a pair of integers (a; b), the symbol Ca;b denotes a commutation matrix, that is a matrix in f0; ga¢b£a¢b such that, given any two matrices A 2 IRra £ca and B 2 IRrb £cb B l A = CrTa ;rb (A l B)Cca ;cb ;

(7)

where Cra ;rb , Cca ;cb are de¯ned so that, denoted [Cu;v ]h;l their (h; l) entries: 8 ¸ ¶ µ· h¡ > < ; if l = (jh ¡ j )u + + ; v v [Cu;v ]h;l = > : 0; otherwise: (8) In the following the symbol In will denote the identity matrix of order n. Proposition 1: [7, Lem. 5.1] For any integer h ¸ and x 2 IRn , it results that: ¡ ¢ (9) rx l x[h] = Unh In l x[h¡1] ; and, for any h > :

¡ ¢ [h] r[2] = Onh In2 l x[h¡2] ; x lx

(0)

where: Unh

=

h¡1 X

T Cn;n h¡1¡¿ l In¿ ;

¿ =0

Onh =

h¡1 X h¡2 X ¿ =0 s=0

¡

T Cn;n h¡1¡¿ l In¿

¢¡ ¢ T In l Cn;n h¡2¡s l In : ()

Lemma 2: Consider system (2) with the nonlinear maps Á; h analytical, so that the power expansion around a generic point x ~ in (4) may be used to describe the system. Then, for k ¸ 2, the di®erential of the k-th order Kronecker power of the state xt is given by: 1 b X ¡ ¢[r] ¡ [k] ¢ X [k¡1] Akn x) xt ¡ x ~ dt + Bjk xt dWj (t) d xt = r (~ r=0

j=1

(2)

with r X

Akn x) = r (~

i=(r¡k+1)_0

¡ ¢ Unk ©i (~ x) l Ink¡1

(3)

¢¡ ¢ ¡ k¡1 Inr l x~[k¡r+i¡1] ; ¢ Ini l Mr¡i

and

¡ ¢ Bjk = Unk Fj l Ink¡1 :

with Bjk as in (4). Then, taking into account the power expansion of the nonlinear map Á and applying (8), the ¯rst term in (7) becomes: 1 X

(4)

Pb [2] Proof. De¯ne F0 = j=1 Fj . According to the vector Ito formula [7, Thm. 5.2]: ´¯ ¡ [k] ¢ ³ ¯ dxt d xt = rx l x[k] ¯ x=xt ´¯ ¡ ¢[2] (5) ³ [2] ¯ rx l x[k] ¯ d¤(t) ; + 2 x=xt with ¤(t) the following martingale Z tX b Fj dWj (¿ ): (6) ¤(t) = 0 j=1

Then, thanks to Proposition : ¡ ¡ [k] ¢ ¡ [k¡1] ¢ [k¡2] ¢ dxt + Onk In2 l xt F0 dt d xt = Unk In l xt 2 0 1 b X ¡ [k¡1] ¢ @ = Unk In l xt Á(xt )dt + Fj dWj (t)A j=1

¡ [k¡2] ¢ F0 dt + Onk In2 l xt 2 ¡ ¡ [k¡1] ¢ [k¡2] ¢ = Unk In l xt Á(xt )dt + Onk In2 l xt F0 dt 2 b X ¡ [k¡1] ¢ Fj dWj (t); Unk In l xt +

i=0

(A ¢ B) l (C ¢ D) = (A l C) ¢ (B l D);

b X j=1

¡ [k¡1] ¢ (Fj l )dWj (t) Unk In l xt =

=

¡ [k¡1] ¢ dWj (t) Unk Fj l xt

b X

¡ [k¡1] ¢ Unk (Fj ¢ ) l (Ink¡1 ¢ xt ) dWj (t)

b X

Bjk xt

j=1

=

[k¡1]

Unk

=

1 X

´ ¡ ¢³¡ ¢[i] [k¡1] dt xt ¡ x Unk ©i (~ x) l Ink¡1 ~ l xt

=

1 X

¡ ¢ Unk ©i (~ x) l Ink¡1

i=0

¢

³¡ ¢[i] ¡ ¢[k¡1] ´ ~ l (xt ¡ x ~) + x ~ xt ¡ x dt:

(20) Recall that the Kronecker power of a binomial allows the following expansion: (a + b)[k] =

k X l=0

¡ ¢ Mlk a[l] l b[k¡l] ;

8a; b 2 IRn ; (2)

with Mlk suitably de¯ned matricial coe±cients in IRn£n , [6]. By applying (2) to the (k ¡ )-th Kronecker power in (20): 1 X

¡ ¢ Unk ©i (~ x) l Ink¡1

i=0

¢ =

³¡ X ¢[i] k¡1 ¡ ¢´ xt ¡ x ~ l Mlk¡1 (xt ¡ x ~)[l] l x ~[k¡1¡l] dt l=0

1 k¡1 X X i=0 l=0

¢ =

¡ ¢¡ ¢ Unk ©i (~ x) l Ink¡1 Ini l Mlk¡1

´ ³¡ ¢[i+l] ~ l x~[k¡1¡l] dt xt ¡ x

1 k¡1 X X i=0 l=0

¡ ¢¡ ¢ Unk ©i (~ x) l Ink¡1 Ini l Mlk¡1

¢¡ ¢[i+l] ¡ ~ dt: ¢ Ini+l l x~[k¡1¡l] xt ¡ x

(22) According to the change of index r = i + l, the sums in (22) become:

b X j=1

¶ ¡ ¢[i] ´ [k¡1] ©i (~ dt l xt x) xt ¡ x~

1 X

i=0

(8)

which holds for any four matrices A, B, C, D suitably dimensioned according to the standard matricial product, the bilinear noise corresponding to the last term in (7) can be written as:

µ³

=

i=0

j=1

(7) see also the proof of Theorem 6., reference [7] for more details. By using the following property of the Kronecker product:

¡ ¡ ¢[i] [k¡1] ¢ ©i (~ Unk In l xt x) xt ¡ x~ dt

1 i+k¡1 X X ¡ ¢ Unk ©i (~ x) l Ink¡1 i=0

dWj (t)

j=1

(9)

5919

r=i

¢¡ ¢¡ ¢[r] ¡ k¡1 I nr l x ~[k¡r+i¡1] xt ¡ x ~ dt; ¢ Ini l Mr¡i (23) so that (2) with (3) is achieved by changing the order of the sums in (23). t u

By neglecting in (4) and (2) the higher order terms, greater than a chosen degree º, the above mentioned dif¡ [k] ¢ ferentials d xt , k = ; 2; : : :, and dyt are approximated as follows: º ¡ [k] ¢ X ¡ ¢[r] d xt ' Akn x) xt ¡ x~ dt r (~

with Akj (~ x) =

r=j

b X ¡ ¢ [k¡1] + Bjk xt + Fjk dWj (t); j=1

dyt '

i=0

b X ¡ ¢[i] Hi (~ x) xt ¡ x ~ dt + Gj dWj (t);

º X i=0

º X ¡ ¢[i] [j] Hi (~ x) xt ¡ x ~ dt = Cj (~ x)xt dt;

j=1

º X i=j

Fjk = 0;

8j = ; : : : ; b; 8j = ; : : : ; b;

8k > ; (25)

The º-degree Carleman bilinearization of the stochastic di®erential system (2) consists in: i) exploiting the powers of the binomial (xt ¡ x ~) in the sums of (24); [k] xt

with ii) substituting in (24) the powers of the state a vector Xkº (t) of the same dimension (recall that k [k] xt 2 IRn ), and considering them up to k = º; iii) substituting in (24) the output yt with a vector Y º (t) 2 IRq ; iv) consider the stochastic Ito equations in (24) after items i-iii) as the bilinear generation model for the extended vector: 2 X º (t) 3 1

6 X2º (t) 7 nº 7 X º (t) = 6 4 .. 5 2 IR ; . Xºº (t)

nº =

º X

ni : (26)

i=1

r=0

=

=

r=0 j=0 º X º X

¡ [j] ¢ Akn x)Mjr xt l (¡~ x)[r¡j] dt r (~ Akn x)Mjr r (~

j=0 r=j

=

º X

dX º (t) = Aº (~ x)X º (t)dt + Nº (~ x)dt +

b X ¡ º º ¢ Bj X (t) + Fºj dWj (t);

X º (0) = X0º;

j=1

dY º (t) = Cº (~ x)X º (t)dt + Dº (~ x)dt +

b X

¡ ¢ [j] Inj l (¡~ x)[r¡j] xt dt

Gºj dWj (t);

j=1

(3) ¢T k ¡ [k] , x = x , k = ; : : : ; º, and: with X0º = xT0 ¢ ¢ ¢ xºT 0 0 0 2 1 2 13 3 A1 ¢ ¢ ¢ A1º A0 6 .. 7 ; Nº = 6 .. 7 ; .. (32) Aº = 4 ... 4 . 5 . . 5 Aº1

¢ ¢ ¢ Aºº

Aº0

Cº = [ C1 ¢ ¢ ¢ Cº ] ; Dº = C0 ; (33) 3 3 2 2 Bj1 Fj1 6 . 7 6 . 7 º º . Bj = 4 . 5 ; Fj = 4 .. 5 ; Gºj = Gj : (34) Bjº

Fjº

In the following it will be assumed that Dº = 0. It is not a loss of generality, in that it can always be de¯ned an auxiliary output Ye º such that: dYe º (t) = dY º (t) ¡ Dº (~ x)dt:

¡ ¢[r] Akn x) xt ¡ x~ dt r (~ r º X X

(30)

III. THE FILTERING ALGORITHM

The ¯rst step is achieved according to the Kronecker binomial formula (2): º X

¡ ¢ Hi (~ x)Mji Inj l (¡~ x)[i¡j] :

Then, according to the items previously mentioned, the generation model for the pair (X º ; Y º ) is given by:

8r = 0; : : : ; º;

= ©r (~ x);

(29)

with

(24)

Bj1 = 0; Fj1 = Fj ;

(28)

j=0

Cj (~ x) =

with x) A1n r (~

¡ ¢ Akn x)Mjr Inj l (¡~ x)[r¡j] : r (~

Analogously, it comes that:

r=0

º X

º X

(27)

[j]

Akj (~ x)xt dt

j=0

5920

(35)

The state xt is estimated as a linear transformation of b º (t) described by (3), the extended state estimate X that is: ¤ º £ b (t): x^ºt = In On£(nº ¡n) X (36)

b º (t) would be As is well known, the optimal choice for X the conditional expectation w.r.t. all the Borel transformations of the measurements: £ ¤ b º (t) = IE X º (t)jFtY º ; X (37)

whose computation in general can not be obtained through algorithms of ¯nite dimension. Nevertheless, from an applicative point of view, it is useful to look for ¯nite-dimensional approximations of the optimal ¯lter, that is estimates described by stochastic di®erential equations of the form: d»t = ®(»t )dt + ¯(»t )dY º (t);

(38) b º (t) = °(»t ); X ª © where »t ; t ¸ t0 is a process taking its values on a ¯nite-dimensional space. The ¯nite-dimensional system (38) is an optimal linear ¯lter for the random process X º (t) if: £ ¤ b º (t) = ¦ X º (t)jL(Ytº ) ; X (39) £ ¤ º where ¦ ¢jL(Yt ) denotes the projection onto the space L(Ytº )© linearly spanned by ª the family of random variables Y º (¿ ); t0 · ¿ · t . £ ¤ Theorem 3: Let mºX (t) = IE X º (t) and ªºX (t) = Cov(X º (t)) be the mean value and the covariance matrix of X º (t) respectively, whose evolutions are given by the following equations: ¢T ¡ m _ ºX = Aº mºX (t) + Nº (~ x); mºX (0) = ³1T ¢ ¢ ¢ ³ºT ; (40) b X _ º (t) = Aº (~ ª x)ªºX (t) + ªºX AºT (~ x)+ Bºi ªºX (t)BºT X i i=1

+

b X

with P (0) = ªºX (0). Proof. The proof is a straightforward consequence of [7, Thm. 4.4]. t u IV. SIMULATION RESULTS Some simulation results are here reported in order to show the e®ectiveness of the proposed algorithm. Consider the following nonlinear system: ¡ ¢ dx1 (t) = ¡ x1 (t) + x1 (t)x2 (t) dt + adWt ; ¡ ¢ dx2 (t) = ¡ 2x2 (t) ¡ 2x1 (t)x2 (t) dt + bdWt ; (45) ¡ ¢ dy(t) = x1 (t) ¡ x1 (t)x2 (k) dt + °dWt ; with a = 4, b = , ° = 2. The initial state x0 is a Gaussian random variable, with zero mean and unitary variance. x0 has been chosen equal to [0: 0:2]T . Numerical simulations are obtained through the construction of an exact stochastic realization of system (45) at discrete times tk = k¢, with ¢ = 0: on the interval [0; T ] with T = 00, according to the Euler-Maruyama method [4]. In the following plots, the estimates obtained with the proposed ¯ltering algorithm with º = 2 are compared to those obtained by applying the classical Kalman-Bucy ¯lter to the linearized di®erential system (º = ).

(Bºi mºX (t) + Fi )(Bºi mºX (t) + Fi )T ;

i=1

(4) with ªºX (0) = Cov(X0º ). Assume that rank(G) = q. Then, the optimal linear estimate of the state process b º (t), is given by: X º (t), namely X Ã b X b º (t) = Aº (~ b º (t)dt + dX x)X (Bº mº (t) + Fi )GT i

+ P (t)C Pb

ºT

!

i

¡ ¢ b º (t) dt (~ x) R¡1 dY º (t) ¡ Cº (~ x)X

T i=1 Gi Gi

with R = matrix:

X

i=1

(42) and P (t) the error covariance

h¡ ¢¡ ¢ i b º (t) X º (t) ¡ X b º (t) T P (t) = IE X º (t) ¡ X

Fig. 4. { True and estimated state: x1 .

(43)

evolving according to the following equation

P_ (t) = Aº (~ x)P (t) + P (t)AºT (~ x) + R Ã p ! X º º T ºT ¡ (Bi mX (t) + Fi )Gi + P (t)C (~ x) i=1

¢R

¡1

Ã p X

(Bºi mºX (t)

+

Fi )GTi

+ P (t)C

ºT

(~ x)

!T

i=1

(44)

5921

Fig. 4.2 { True and estimated state: x2 .

The improvements obtained by increasing the index º can be recognized by comparing the sampling variances of the estimation errors: ¾12 (º = ) = 4:5592;

¾12 (º = 2) = 2:5684;

¾22 (º = ) = 3:208;

¾22 (º = 2) = 2:4942:

(46)

[] A. Gelb, Applied Optimal Estimation, Cambridge, MA: MIT Press, 984. [2] A. Germani and M. Piccioni, \Finite dimensional approximation for the equations of nonlinear ¯ltering derived in mild form," Applied Math. and Optim., Vol. 6, pp. 5{72, 987. [3] A. Germani, C. Manes and P. Palumbo, \Polynomial extended Kalman ¯ltering for discrete-time nonlinear stochastic systems," 42nd IEEE Conf. on Decision and Control, pp. 886{89, Maui, Hawaii, 2003.

V. CONCLUSIONS The problem of state estimation for a nonlinear di®erential system driven by a standard Wiener process has been investigated in this paper. The ¯ltering algorithm here proposed is based on two steps: ¯rst the nonlinear system is approximated by using the Carleman bilinearization approach, taking into account all the powers of the series expansion up to a ¯xed degree º; next, the optimal linear ¯lter of the approximating system is achieved. This step is based on a well known literature concerning suboptimal estimates for bilinear state space representations [5, 6]. When the index º = , the proposed algorithm gives back the classical Kalman-Bucy ¯lter applied to the linearized di®erential system.

[4] D. J. Higham, \An algorithmic introduction to numerical simulation of stochastic di®erential equations," SIAM Review, Vol. 43, No. 3, pp. 525{546, 200. [5] K. Ito, \Approximation of the Zakai equation for nonlinear ¯ltering theory," SIAM J. Control Optim., Vol. 34, No. 2, pp. 620{634, 996. [6] K. Ito and K. Xiong, \Gaussian ¯lters for nonlinear ¯ltering problems," IEEE Trans. Autom. Contr., Vol. 45, pp. 90{927, 2000. [7] A. H. Jazwinski, Stochastic Processes and Filtering Theory, New York: Academic, 970. [8] S. J. Julier and J. K. Uhlmann, \Unscented ¯ltering and nonlinear estimation," Proc. of IEEE, Vol. 92, pp. 40{ 422, 2004. [9] K. Kowalski and W. H. Steeb, Nonlinear Dynamical Systems and Carleman Linearization, World Scienti¯c, Singapore, 99.

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