FINITE-DIMENSIONAL ALGORITHMS OF NONLINEAR SYSTEM STATE ESTIMATION ALEXEI R. PANKOV Department of Applied Mathematics Moscow Aviation Institute Moscow,127080, Russia ALEXEI V. BOSOV Department of Applied Mathematics Moscow Aviation Institute Moscow,127080, Russia ANDREI V. BORISOV Department of Industrial & Operations Engineering University of Michigan Ann Arbor, MI 48109-2117 Technical Report 92-13 January, 1992. 1

12 FINITE-DIMENSIONAL ALGORITHMS OF NONLINEAR SYSTEM STATE ESTIMATION ALEXEI R. PANKOV,1 ALEXEI V. BOSOV, 2 ANDREI V. BORISOV3 ( Revised on January 28, 1992 ) Absract. In this paper we consider the method of conditionally-minimax nonlinear filtering (CMNF) of processes in nonlinear stochastic dynamic discrete-time systems. In order to compare CMNF with the optimal nonlinear filter (ONF), we also derive a special numerical algorithm which is based on the theory of spline approximation. This algorithm allows us to calculate the best mean-square estimate of the system state-vector. The results of numerical experiments with CMNF, ONF and the extended Kalman filter (EKF) show the favorable properties of the derived CMNF algorithm. Key Words. Optimal nonlinear filtering, conditionally-minimax filtering, extended Kalman filter, spline approximation. 1 Introduction. The problem of nonlinear state estimation is a very important part of the general control problem for systems with incomplete data. It is well-known that the problem of nonlinear filtering may be reformulated as the problem of solving the stochastic functional equations for the conditional distribution of the system state given all observations (Liptser, Shiryayev,1977), (Davis,Marcus, 1981). So the estimation problem in the general case is infinite-dimensional. Its complete solution is practically impossible. Recently various authors tried to determine nonlinear system properties that provide provide the finite-dimensionality of the optimal estimator (Sawitzki, 1981), (Benes, 1981), ( Wong, 1983), (Daum, 1986), (Tam, Wong and Yau, 1990). The investigations show that a finite-dimensional optimal nonlinear filter is not typical, and even in very simple cases may not take place ( Hazewinkel,Markus and Sussan, 1983). That is why we support the idea of obtaining the finite-dimensional estimators by means of appropriate approximations of the optimal nonlinear infinite-dimensional estimation algorithms. Some of these approximations are well-known and widely 1Department of Applied Mathematics, Moscow Aviation Institute, Volokolamskoye shosse, 4, Moscow, 127080, Russia 2Department of Applied Mathematics, Moscow Aviation Institute, Volokolamskoye shosse, 4, Moscow, 127080, Russia 3Department of Industrial & Operations Engineering, University of Michigan, Ann Arbor MI 48109-2117 1

used: the linearized and the extended Kalman filters, the second order filter and others ( Sage,Melsa, 1972). Their practical utilization shows that they often obtain very unpleasant properties, i.e. give a seriously biased estimate which usually diverges. So it is important to continue efforts to obtain the finite-dimensional nonlinear filters which provide an estimate with improved properties. In this paper we consider two different approaches for finite-dimensional nonlinear filtering. The first approach is based on the idea of a local conditional optimization of the filter structure given the class of admissible filters. This idea was first realized by V.S. Pugachev (Pugachev, 1979), (Pugachev and Sinitsyn, 1990), who derived the conditionally-optimal nonlinear filter. To utilize this method one needs to know the joint characteristic function of the system state and the estimate vectors. This function may be obtained by solving the special functional equation, which is nonrandom but rather complicated. It takes substantional effort to obtain the desired solution (Raol,Sinha 1987). Using the idea of the conditionally-optimal filter we derive the conditionally-minimax filter (CMNF) which may be determined without information about the above-mentioned characteristic function by some a priori information (Pankov, 1990). 'The second method is based on spline approximation of the conditional system state density given an observation process. We call this method the optimal nonlinear filter (ONF) because it may be shown that the ONF provides the optimal ( in a mean square sense ) estimate approximation with the desired accuracy. In order to test properties of the obtained algorithms we compare the CMNF and the ONF with the extended Kalman filter (EKF). The corresponding computer simulation results are also considered in this paper. 2 CMNF algorithm. We shall use the following notations: E{x} - the mathematical expectation of x; cov{x,y} - the covariance of x and y; P(m,S) - the set of random vectors with E{x} = m, cov{x,x} < S; A+ - the pseudoinverse matrix with respect to A; y = col(xl,...,,,) if y is a vector-column of the form y = col(xt,...,xT)T; I||I| (xTWx)l/2 for some weight matrix W: W = WT, W > 0. Let us consider the following discrete-time stochastic dynamic observation model: Y yn= q(yn-l,Wn), n= 1,2,...; o/Y = (1) Zn =t (Ynon) where Yn E RI is a state vector; Wn E Rg is a vector of random disturbances; r{ E RP is a vector of initial conditions; Zn ~. R' is a vector of observations; Vn E R' is a vector of random observation errors, and On(Y, w) and On(y, v) are known nonlinear functions. Let Z" = col(z,..., zl), i.e. Zn is a vector of all observations up to the moment n. 2

Consider the structure of the CMNF for the processes {Yn} given Zn. Let gn(y) and 7rn(y, z) be some fixed nonlinear vector functions ( the choice of these functions will be considered in a sequel ) and Yn-1 be the CMNF-estimate of yn-1 given Zn-'. Let yn = (n(yn-i). Then the conditionally-minimax prediction Yn for yn takes the form: Yn = *(Yn), (2) where (yn) = argmin max E{IIYn - (Yn)|12} (3) eEiD P(mi,Si) and x1 = col(Yn,Yn) E P(ml, Si), D is a set of measurable functions, E I|](yn)l l2 < oo. The estimate Yn is a conditionally-minimax correction of yn given the new observation Zn: Yn = Yn + b*(irn(Yn, zn)), (4) where )-(vn) = argmin max E{llYn - -n - k(vn) 12} (5) nkEI P(m2,S2) and x2 = col(Yn - Yn), vn) E P(m2, S2), vn = 'rn(Yn, Zn). If we solve (3),(5) and find the functions q*(.) and 0b (.) then equations (2),(4) will define the recursive CMNF. It may be shown that under some rather general conditions there exists a solution of (3),(5) where the corresponding optimal functions q5 (.) and b n(.) are linear. Consider the following conditions for the model (1), ~n(y) and lrn(y, z). 1) There exists C~(w) < oo, w E Rq such, that Il(y, w)112 < Ct(w)(1 + IIylI2) and E{C~(wn)} < o, y E RP; 2) There exists Cn < oo, lln(Y)I)2 < Cn(1 + |lly|2), y E RP; 3.a) There exists Cn < oo: for all y E RP, z E Rm Ilarn(y,z) 12 < Cn(l + yll 2); 3.b) The processes {vn} is independent of {wn} and t7: There exists Cn < oo: llrn(y, z)112 < C(1 + Ilyll2 + IzII2); There exists C (v) <: llbn(y,v)112 < C(1 + IlylI2) and E{C'(vn)} < oo Proposition 1.Let 1),2) and 3.a) or 3.b) hold. Let also E{l I7112} < oo and yo = E{r7}, then *(y) = Fny + fn; O/b(v) = Hnv + hn, (6) where f Fn =COv(yn, Yn)cov+(yn, yn); fn = E{n - Fn}; (7 Hn = E{yn - ynv }cov+(v, v); hn =-HnE{vn}; ( 3

The estimate ', is unbiased and has the error covariance matrix n= K- Fncov(y,, y.) - H~cov(vn, yn - Y-n)7 (8) where Kn = cov(yn - _,, - Yn), and Kn = cov(ynyn) is the covariance matrix of the process {yn}. Proof of Proposition 1: see Appendix 1. From Proposition 1 it follows that CMNF is defined by (2),(3),(6)-(8). If we consider a dynamic system which is not general as (1), then restrictions 1)-3) may be weakened. Let the dynamic observation system be described by the discretetime diffusion equation Yn = an(yn-1) + b(yn-l)Wn7 n = 1) 2,...; yo ";(9) Zn = kn(Yn) + V2, where {wn}, {vn} are independent gaussian white noises; 77 is a gaussian vector independent of {wn}, {wj}. Model (9) is widely used in applications ( Sage,Melsa 1972). Proposition 2.Let {y,, zn} be described by (9) and: 1)there exists C1 Qa < 00: Ilan(Y)II + Ijbn(y)II ~ Cn(1 + IlYJIan); 2)there exists CQ < oo: IIn(Y)II ~ Cn(1 + IIYIIOf); II~n(Y)II ~ Cn(1 + IyI a n) Ilin(y, z)II ~ C(1 + Ilylclan + IIZIIan); 3)AO = Ejqj. Then the result of Proposition 1 holds. Proof of Proposition 2: see Appendix B. It should be mentioned that the conditions of Proposition 2 allow the polynomial growth of all functions. However in Proposition 1 the growth is only linear. Consider some obvious types of the structural functions Sn(y) and ln (y, z). a) ' (Yn-1) = yn-1 -linear prediction; b) ~n(An-1) = On(An-1, Efwn})-prediction via dynamic system equations (1); c) 7n On zn) = Zn - On 1y, V\ - residual, where vo = E I n} d) irn (Ynz) = Kn n ((I n Ifn + Rn) )+(zn - n 7 ~n where Kn = cov(yn - 'n7, Yn - Yn), (9 = V)( V )/O, = [ObPni v0)/8VA]Co0V(Vn.Vn)[[O/(Pn, V')/9V,01]- transformed residual. 7rn(-n,7zZ) in r~ L ~~\y, n/ VJ n n Y n n YLlOVII~U IDIUCI the given above form is used in the EKF as a correction term. The only difference is that in the EKF instead of kn is used some random approximation ( given by the Riccati-type equation ) of the real prediction error covariance matrix Kn. There are some other types of 4n(y) and 7,n(y, Z). In the general case we choose 4

<n(yn-1) and 7rn(Yn, zn) to approximate ( may be in the minimax sense ) the conditional expectations E{yn Y.n-1} and E{yn - Ynyn, Zn} respectively. All unknown parameters (7) of the CMNF algorithm may be determined a priori by a computer simulation. The corresponding algorithm for this is considered in detail in (Pankov, 1990). 3 ONF spline algorithm. It is well-known that if {y,} is the second order process, then the conditional expectation yn = E{(ylZn} minimizes the mean-square criterion Jn = E(ly n - y ll2} (Davis,Marcus 1981). The estimate yn may be calculated recursively if {yn} is the Markov process. It may be shown that if q7, w1, w2,... are independent random vectors, E{ll77112} < o and E{llqn(y,w,)ll2} < cn(l + ully|2) < oo, n > 1, then the process {yn} has finite second-order moments and obtain the Markov property. We assume existence of the following probability characteristics: the conditional density gn-i(y) of Yn-l given Zn-'; the transition density a(n - l,.; n, y) of the process {yn}; the conditional density pn(y,z) of Zn given yn = y. Denote wn(y) the unnormalized conditional density of yn given Zn (Davis,Marcus 1981). then as in (Takeuchi,Akashi 1981) wf n(y) = pn(Zn, Y) fRp I(n- 1,x;n, y)Wn(x)dx, (10) Wo ( y) Pn (Y), where p,(y) is the af probability density. Conditional densities gn(y) and wn(y) are connected by the formula gn(y) = Wn(Y)[JWn(x)dx]- (11) From (11) it follows that y% is obtained by y;= /R xn(x)dx[J wn(x)dz]l (12) We are going to use wn(y) spline approximation for the numerical treatment of (10),(12). The spline approximation methods are widely used for solving the statistical estimation problems (Andrade Netto,Gimeno and Mendes 1978), (Wegman,Wright 1983), (Villalobos,Wahba 1987). We shall consider the case yn E R1 for simplicity. Let us consider the sequence of partitions A(n) = {O.., X)} of the intervals [a(n), b(n)]: n () +h( ^ = a(n) Xk -- NXk_1 Tgk k 3;0 X xn) = b(n) A() may be extended in the following way: x( n) x ()- h)i (n) (n) +, i ( i 2,.... TN+1 p= N Tof dege 2m 1. To approximate wn-i (y), wn(y) we shall use polynomial splines of degree 2m - 1. The 5

general representation of the spline is defined in (Grebennikov, 1983) by the formula N+p p N+p S2m'p(X) = E [ E ajjfI+yj131m,2m(X) = > CIs1im,2m(X), x E [a, b], (13) l=-p 1=-p l=-p where {aj3} are unknown paremeters; fk = f (Xk): f (x) is functional to be approximated on [a, b]; P is an arbitrary integer number from the set {o, 1,...,I [(N - 1)/2]} { Si,2m(x)} is a system of B-splines. The recursive procedure of the numerically stable determination of B-splines at the point x E [a, b] is defined by the formula (Cox 1978): sig(X) = [(x - XX)(xi+g.... - X,)]si'u...(x) + [(Xi+1 - x)/xi+I - i1i1'1X) {Sij (x) = 1, if X E [xi, Xi+j], i =2,. m Si'(x) = 0, otherwise Let us describe the procedure of y* determination given the spline-approximation of Wn1()onN N+p A (y -S(n 1() = Cn-1snj-,m(),(14) l=-p where {s'~'} is a system of B-splines on n1) Let us introduce the following notations (n) =f~oo,qs(n - 1x; n ~) n-)(~x { (n) - fjoo xsnj-I(x) dx; (5 = fie. Slrm,2m(x)dx. Then from (11), (12), (14), (15) it follows that, at the knots {x.'k } and N the -following conditions are fulfiled: l=-p Using the values of A (X~n))given by (16), we can define unknown parameters {C/rn)} of the spline S~n)(y), which is an approximation of the Wn(Y).The corresponding estimate A* of the conditional expectation y* is given by To obtain { Cfn)} we shall use a special method, which was called the method of direct approximation (Grebennikov, 1983). At the points { X~n} we use the approximation conditions of the following form 6

N+P-1 S() (n)) n((n")) + d( n)6l[^ ((n )], (17) U2m,p k \^ k + n L (17) 1=1 where S2[.] is a central divided difference of degree 21 of the function W(x) at the knot (n). The coefficients {d(n)} together with {C()} (see formula (13)) can be found from (17) (Grebennikov, 1983). It should be mentioned that this method of spline-approximation differs from the standard spline-interpolation method (Ahlberg,Nilson and Walsh, 1967). The coefficients {C(n)} satsfy some system of linear equations, which may be solved analytically in some particular but important cases. For example, for the cubic spline defined on the uniform partition A\( (i.e. h(n = h(n), k = 1,..., N), it may be shown that for = -P,...,N+P C} ) = -1/6&_1 + 4/3t1 - 1/6/+1j if P = 1; C?( = 1/36(,1_2 + 1+2) - 5/18(1-_l + cI+l ) + 3/2at if P = 2, where wk =,n(Xk)) The similar formulas are valid for the five-degree spline-approximation. Using the B-spline representation of the approximating spline allows us to organize the effective numerical integration in (15) because {sl,2m(x)} is a system of functions with compact support. It should be mentioned that the obtained method may be extended to the case of p > 1 by using the B-splines of the vector argument and the corresponding direct spline-approximation (Grebennikov, 1983). 4 Numerical example. Consider the following nonlinear dynamic observation system Yn Yn-1(1 + (Yn-1 )2)-1 +0.7, YO = / (18) n = 0.6y, + 0.4yn + 0.2Vn, where {wn} and {vn} are independent standard gaussian white noises and r7 is a standard gaussian random variable. The model (18) satisfies the conditions of Proposition 2 with an = 2. For n = 1, 2,..., 10 this example was simulated on a computer. M = 2000 realizations {I,}Y were obtained of the CMNF-estimate Y, {y.}iM of the ONF-estimate y lYn ~ i~l rrv, y,~ li=f the ONF-estimateY y* and {yI}iM of the EKF-estimate yE (the EKF is described in (Sage,Melsa, 1972)). The ONF-estimate yn was calculated by the spline-filtering algorithm with N = 20 a = -0.3, b = 3.0, p = 1. The structural functions for the CMNF were taken as follows: 4n(Yn-1) = n-l(l + (Yn-)2)-1 7rn(Yn, zn) = Knn(Kn($,n)2 + 0.04)-1(n - 0.6n - 0.4(n-)2), n = 0.6 + 0.8.yn 7

We have compared the following values of the mean-square errors of the above described estimates: A -1 f l (y _ yt )2; Jn = 1 M Zl=(Yiyn Yn J* M-1 EyM l(y _ y*)2; -n - 2 n n Z~ M -1 — M 1/ i 2. (Yin- yn); where {yh }1i is a set of realizations of the process {yn} simulated in accordance with (18). The following table shows the numerical results. n J J J J/J JE/j* n n Jn n rJn/ Jn n n/ 1 0.261 0.278 0.563 1.07 2.16 2 0.238 0.257 0.450 1.08 1.89 3 0.236 0.253 0.419 1.07 1.77 4 0.224 0.237 0.392 1.06 1.75 5 0.269 0.279 0.487 1.04 1.81 6 0.224 0.238 0.387 1.06 1.73 7 0.215 0.233 0.373 1.08 1.73 8 0.227 0.244 0.387 1.07 1.70 9 0.236 0.256 0.446 1.08 1.89 10 0.239 0.258 0.421 1.08 1.76 Table 1. Values of the mean-square criterion for the estimates of ONF, CMNF and EKF. The obtained results show that the CMNF-estimate and the ONF-estimate have practically the same accuracy, and the EKF gives a much less accurate estimate. It should be mentioned that the algorithms CMNF and EKF require nearly the same computation time which is significantly less than the computation time of the ONF. This feature of the CMNF is that the optimal values Fn, Hn, fn and hn can be calculated before the process of filtering itself as in the case of the linear Kalman filter or the EKF. However, in the ONF it is necessary to integrate a complicate enough system each time with a new coming measurement. It takes substantional computational efforts in real time in comparison with the CMNF. 5 Appendix A. Lemma. Let z = col(x,y), m = E{z} = col(m, my), s = cov(z,z) is an unknown matrix which is upper bounded by the known matrix S 8

s < T hS Then 1) *(yy) = Sy S+ +(m-S yS+ my) = m,+SyS+ (y-my) = argmin max E{IIx-(y)II)2}; =EO P(m,S) 2) E{x - } =- 0, cov(x -,x - ) < SX- SySSyXs where x = 0*(y) is the estimate of x given y and the upper inequality holds as an equality in the case of s = S. Proof. Let us denote f()(.),pz) = Epz{llx - q(y)ll2, pz is a gaussian distribution with expectation m and covariance matrix S, q*(y) = mX + SxyS+(y - my). Then it is sufficiently to prove the following inequality f(0*(.),pz) < f(q*(.),p*) < f(q(.),p) (19) for all q(.), p, E P(m, S). q*(y) = Ep*{xiy} is known to be the best mean-square estimate of x given y and from the normal correlation theorem (Liptser,Shiryayev, 1977)it follows that the right inequality (19) is true. Let pz E P(m, S) is an arbitrary distribution with the expectation m and the covariance matrix s < S. It should be mentioned that O < C= S - s = c CY [ Cyx Cy J Ep{(x - q*(y))(x - *(y))T} =, - SySy+ySy+yS -ySy+ysy, - sxys+y syx Ep{(x - /*(y))(x - *(y))T} = -+ SySyySyx s + c - S~ S+Y(syx + c) - (sy + cXy)SyY + SxySy( + cy)SyyS.y Then E{(x- *(y))(x- *(y))'}-E{(x-*(y(y(x- (y))} = qcqT > O, where q = [SxyS+y I]. It follows that f(0*(), p*) - f(q*(.), pz) = trWqcqT > 0, i.e. the left inequality (19) is true for all pz E P(m, S). Lemma is proved. Proof of Proposition 1. Suppose that Ellyn_|ll2 < oo. Let us check that EIIYnll2 < oo0 E{ll0n(y, wn)ll2} E{Ct(wn)(1 + IIy112)} < E{Ct(wn)}(l + IIyII2) for any y E RP 9

Hence E{lly|=2} E={E{||n(yn-l, wn)|2/yn-l}} < E{Ct(wn)(1 + E{llyn-i||2})} < oo. we have yo = 77 and E{||17||2} < oo for all n > 0. Let yn-I be the CMNF estimate given Z"-1. Suppose that E{IIly_-112} < oo. Let us show that {IYnII2} < oo. Consider the basic predictor n =- n(n-1) E{llYn|I12} = E{lln(.Yn-l)112} < Cn(1 + E{llyn-1112}) < o00 From the obtained results it follows that x1 = col(y, yn) is a random vector with a finite second order moment. Applying Lemma to z = x1 with x = yn, y = yn we obtain the result Yn = FnYn + fn, where Fn = cov(yn,,,y)cov+(n, n), f E{yn} - FnE{yn}. Then:E{|I|112} = tr{W(FnE{ynyT}Fn) + fE{nF2)} + I 2 < o0. Now let us show the existence of Hn and hn. Let AYn = y -,n. From Lemma it follows E{Ayn} = 0 and Kn = cov(AYn,, An) = Kn-cov(yn, yn)cov+(Yn, yn)cov(yn, y,) = Kn - Fncov (,, Yn) where IKn = cov(yn, Yn). Let condition 3.a) be fulfiled, then E{|IInII2} = E{lwn(n,zn)[I2} < C(1 + E{ll{y.n-il2}) < 00. Let now 3.b) be fulfiled, then taking into account the independence of {vn} and {yn} we have E{llzn112} = E{I|Ibn(Yn, n) 112} < E{CW(vn)}(1 + E{ll yn12}) < oo. Then we obtain E{II7rn(yn,zn)12} < Cn(1 + E{llynI12 + IIZn2}) < oo. So if 3.a) or 3.b) take place then E{|IIvlI2} < o0. Applying Lemma to x2 = col(A On,vn) we obtain Y = Yn + Hnv, + hn, where Hn = cov(Ayn,,y)cov+(vn,,vn) and hn = E{An - Hnvn} = -HnE{vn}sinceyn is unbiased. E{llynll2} < 3(E{lIynI12} + IIHnI2E{lIIv112} + I|hn||2) < oo. Besides this E{Ayn} = E{yn-Yn} = 0 and KI = cov(An, AYn) = Kn-cov(An,, vn)cov+(vn, v,)cov(vn, AY) = Kn - Hncov(v,, An,). Now let n = 1, then yn-1 = rl and o = E{l}. E{IIYoII2} = IIE{}|112 < E{l1171i|2 < 00, hence Y1 exists and E{ 11 yi2} < oo. Now the proof follows from the mathematical induction principle. 6 Appendix B. Proof of Proposition 2. Suppose that E{IIy-n-lIk} < oo and E{IIin-lIk} < oo for all k > 1 and some n > 1. Let us show that yn and Yn have the moments of arbitrary order. We shall use the following inequality: (11 x1 + Ilyl) < Cr(IIXIIr + IIYlIr), r > 1, Cr = 2-. It follows from the Cr-inequality (Loeve,1960): (Ixl+Iyl)' < Cr(lIX+lyl) and the fact that 11IlX = (xTW)1/2 = Il, where x = W/2x. Ilynllk = Ilan(n-1) + bn(yn-xl)Wnl I < Ck(llan(Yn-1)1)k + Ilbn(yn-l)Wnllk) < (k(CN)k[(l + Ilyn-11 li)k + (1 + Ilyn,-lllYn)kllwnllk] < Do[1 + llyn-il1n + ( + + Ilyn- Pn)llIwnlk], where, = ank. Hence E{||yn||I} < Dl(1 + E{llyn-iIP})(1 + E{lIWnIlk}) < 00, since E{IlwnlIk} < oo and yn-l and Wn are independent. 10

Yn = gn(Yn-1), then E{ll ynIk} = E{(n(_n-i)l k} < (C2)kE{(1 + IIyn,_lll1)k} < D2(1 + E{||yll.1n-1 }) < 00. From the obtained inequalities it follows that E{llxi;llk} < oo for k > 1, where xl = col(yn, yn). Then by applying Lemma we obtain Y = Fnn + fn, IIFn|l + IIfn|l <00 and E{||I[nIk} < Ck(llFllk E{llyn1k} + IIfnl k) < 0 II7rn(y z)llk < (C2)k(1 + IIyllan + IIZlan)k < D3(1 + \\y\\n + IIzIIn). Then E{IIn(yn, zn)|lk} < D3(1 + E{llynIln} + E{IIznlPn}) < oo since E{llznln} = E{II|n(yn) + Vn|lln} < Cn[E{I In(yn) In} + E{llvnlIPn}] < Cpn(Cn2)nE{1 + Ilynll "n)} + CnnE{lvnI lIn } < Dn(1 }) + E{i }) + CnE{Ivnll < oo where 7y, = an/n; E{IlvnIIln} < o0 since v, is a gaussian vector. Hence the vector x2 = col(\xn, vn) has finite second-order moments and consequently Hn, hn exist. From this, the desired result follows for n > 1. Now, if n = 1, then Yn-i = = and yo = E{7i}. If T7 is a gaussian vector then E{llyollk} < oo and E{ y01ok} = IlE{1}lIk < oo. Now the desired result follows from the mathematical induction principle. References 1. Ahlberg,J.H., Nilson,E.N., and Walsh,J.L., 1967 The theory of splines and their applications. N.-Y., Academic Press. 2. Andrade Netto,M.L., Gimeno,L., and Mendes,M.J., 1978 A new spline algorithms for nonlinear filtering of discrete-time systems. 7th IFAC World Congress. Preprint 3, London, Pergamon Press, 2123-2130. 3. Benes,V.E., 1981 Exact finite-dimensional nonlinear filters for certain diffusions with nonlinear drift. Stochastics, 5, 65-92. 3. Cox,M.G., 1978 The numerical evaluation of a spline from its B-spline representation, J. Inst. Maths. Applies., 21,135-143. 4. Davis,M.H.A., and Marcus,S.I., 1981 An introduction to nonlinear filtering. Stochastic systems: the mathematics of filtering and identification and applications. Dotrecht, the Netherlands, 565-572. 5. Daum,F.E., 1986 Exact finite-dimensional nonlinear filters. IEEE Trans. on Automat. Control, AC-31, 616-622. 11

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