UNSCENTED KALMAN FILTER REVISITED - HERMITE-GAUSS QUADRATURE APPROACH

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1 UNSCENED KALMAN FILER REVISIED - HERMIE-GAUSS QUADRAURE APPROACH Ja Štecha ad Vladimír Havlea Departmet of Cotrol Egieerig Faculty of Electrical Egieerig Czech echical Uiversity i Prague Karlovo áměstí 3 35 Prague, Czech Republic stecha@fel.cvut.cz havlea@fel.cvut.cz Abstract Kalma filter is a frequetly used tool for liear state estimatio due to its simplicity ad optimality. It ca further be used for fusio of iformatio obtaied from multiple sesors. Kalma filterig is also ofte applied to oliear systems. As the direct applicatio of bayesia fuctioal recursio is computatioally ot feasible, approaches commoly take use either a local approximatio - Exteded Kalma Filter based o liearizatio of the o-liear model - or the global oe, as i the case of Particle Filters. A approach to the local approximatio is the so called Usceted Kalma Filter. It is based o a set of symmetrically distributed sample poits used to parameterise the mea ad the covariace. Such filter is computatioally simple ad o liearizatio step is required. Aother approach to selectig the set of sample poits based o decorrelatio of multivariable radom variables ad Hermite- Gauss Quadrature is itroduced i this paper. his approach provides a additioal justificatio of the Usceted Kalma Filter developmet ad provides further optios to improve the accuracy of the approximatio, particularly for polyomial oliearities. A detailed compariso of the two approaches is preseted i the paper. the set of symmetrically distributed sample poits (so called sigma poits), which are used to parametrise the meas ad covariaces. Such filter is simple ad o liearizatio step is required. I this paper, aother approach to selectig the set of sample poits for obtaiig the mea ad covariace of the distributio is proposed, which is based o Hermite-Gauss Quadrature, 7, 8. his approach is simpler compared to that usig sigma poits ad exact for polyomial oliearities. hese two approaches to Usceted Filter are compared i the paper. he paper is orgaized as follows: i Sectio II, basic properties of Hermite-Gauss Quadrature are show; Sectio III shows how the mea ad variace of oliear fuctio ca be computed usig Hermite-Gauss Quadrature. I Sectio IV, sigma poit trasformatio for Usceted filter is compared with the results obtaied by Hermite-Gauss Quadrature. Fially, Sectio V describes how to use Hermite-Gauss Quadrature i Kalma filter. I. INRODUCION Wheever the state of the system eeds to be estimated from oisy measuremets, some kid of state estimator must be used. Miimum mea squared state error estimate for liear systems results i Kalma filter (KF). It is a excellet tool for iformatio fusio - processig data obtaied from differet sesors. Kalma filter is the best tool for trackig ad estimatio due to its simplicity ad optimality. However, its applicatio to oliear systems is difficult. he most commo approach to estimatig states of oliear systems is to utilize the Exteded Kalma filter, 3, which is based o the liearizatio of the o-liear model 4. It is kow that some difficulties with utilizatio of this approach may occur. First, the liearizatio requires computig Jacobia matrices, which is otrivial. Moreover, the resultig filters may be ustable. Kalma filter operates o meas ad covariaces of the probability distributio which may be o-gaussia. he so called Usceted Kalma filter 5 was developed, based o II. HERMIE-GAUSS QUADRAURE he objective of this sectio is to compute b a v(x)f(x)dx () where v(x) is a priori chose weightig fuctio ad f(x) is some oliear fuctio. For our purposes (Hermite Quadrature) the weightig fuctio is chose as v(x) = e x, ad the iterval of itegratio equals to (a, b) = (, ). We wat to aproximate such itegral by a quadrature formula of the form b a v(x)f(x)dx = A f(a )+A f(a )+...+A a f(a a ) () where A i are the weightig coefficiets, a i < a, b > are the odes ad a is the order of the quadrature formula. Quadrature formula has the algebraic accuracy m, if polyomials up to order m are itegrated exactly. Quadrature formula with a odes has a parameters A i, a i which leads to the 495

2 algebraic accuracy m = a, because a polyomial of order m = a has a coefficiets. he solutio of Gauss Quadrature is give i the followig theorem 9. heorem: Quadratic formula is Gaussia if the product of roots factors a ω(x) = (x a i ) (3) is a orthogoal polyomial with weight v(x) ad whe coefficiets of quadrature formula equal A i = b a v(x)l i (x)dx, i =,,... a (4) where l i are elemetary Lagrage iterpolatig polyomials. Elemetary Lagrage iterpolatig polyomials l i (x) are equal to Hece, l i (x) = a j =, j i l i (a j ) = δ i,j = x a j a i a j, i =,,... a (5) {, i = j 0, i j Hermite orthogoal polyomials o the iterval (, ) with respect to weightig fuctio v(x) = e x are defied by the recurret relatio H + (x) = xh (x) H (x), () where H 0 (x) =, H (x) = x, H (x) = 4x. hese polyomials are called physical Hermite orthogoal polyomials. Nodes a i are roots of Hermite polyomials. hus, we are able to approximate the itegral e x f(x)dx = A f(a ) A a f(a a ). (7) If the fuctio f(x) is polyomial of order m a, the the result is exact. he weights A i ca be computed by (4) or by the formula A i =! π H (a i ) (8) I the followig table there are odes a i ad coefficiets A i for differet orders of approximatio a.. a a i A i 0 π a = A = π a = A = π a = a = a 3 = A = π 3 A = π 3 A 3 = π a =.507 A = a =.507 A = a 3 = 0.54 A 3 = a 4 = 0.54 A 4 = a = 0 A = a =.00 A = a 3 =.00 A 3 = a 4 = A 4 = a 5 = A 5 = a =.350 A = a =.350 A = a 3 = A 3 = 0.57 a 4 = A 4 = 0.57 a 5 = A 5 = 0.74 a = A = 0.74 Note: Some refereces (e.g. 4) use the weightig fuctio v(x) = e x ; related Hermite orthogoal polyomials H S (x) are called statistical. Such polyomials are defied by the recurret relatio H S +(x) = xh S (x) H S (x), (9) where H S 0 (x) =, H S (x) = x, H S (x) = x. he odes a S i are the roots of Hermite polyomials H S (x) ad the weights A S i are equal to A S i =! π H S (as i ) (0) Usig statistical Hermite orthogoal polyomials the quadrature formula has the form f(x)e x dx = A S i f(a S i ) () Simple trasformatio s = x ca be used to relate quadrature of physical ad statistical Hermite polyomials f(x)e x dx = f( s)e s ds () Similar results are obtaied usig Hermite-Gauss physical Quadrature: where Ãi = A i ad ã i = a i. f(x)e x dx = Ã i f(ã i ) (3) 49

3 III. MEAN AND VARIANCE BY HERMIE-GAUSS QUADRAURE I this sectio we use Hermite-Gauss quadrature to compute the mea ad the covariace of fuctio f(x) of a radom variable x. he mea µ f = E {f(x)}, where x is a radom variable with probability desity fuctio p(x), equals µ f = f(x)p(x)dx. (4) First, we will treat the oe dimesioal case; subsequetly, the extesio to multi-dimesioal cases will be made. A. Oe dimesioal case If a radom variable x is of Gaussia distributio with mea µ ad variace σ, the µ f equals µ f = f(x) e (x µx) σ dx (5) π σ If the substitutio x µ σ = v is used, the formula for µ f equals µ f = π f( σv + µ)e v dv () he Hermite-Gauss Quadrature ca be used i the form µ f = π A f(ā ) A a f(ā a ) (7) where ā i = σa i + µ, i =,..., a ad A i are coefficiets of Hermite-Gauss Quadrature ad a i are odes of Hermite orthogoal polyomials of order a. For the order of Hermite polyomial a =, the formula for µ f = E {f(x)} is simple: µ f = f(µ + σ) + f(µ σ) (8) For the order of Hermite polyomial a = 3, the formula for µ f = E {f(x)} becomes µ f = 4f(µ) + f(µ + 3σ) + f(µ 3σ) (9) For variace σf of fuctio f(x), which equals σ f = E { (f(x) µ f ) }, a similar formula for Hermite-Gauss Quadrature ca be obtaied σf = { A f(ā ) µ f +... A a f(ā a ) µ f } π (0) where the odes ā i ad coefficiets A i are the same as i the previous case. Example : For f(x) = x, formula (8) yields the mea of f(x) as µ f = (µ + σ) + (µ σ) = µ + σ, () which is a well-kow formula. he same result is obtaied if the order of Hermite polyomial equals a = 3. For the variace of radom fuctio f(x) = x it is ecessary to use the order of Hermite polyomial a = 3. It is because the variace σ f = E { (f(x) µ f ) } ; the order of the fuctio for which the mea is computed equals m = 4 ad a = = 5 > 4. he variace σ f equals σ f = π 3 A i f( a i + µ) µ f. () After the substitutio of coefficiets A i ad odes a i, the formula for σf = E { (f(x) µ f ) } is σf = 4(µ µ f ) + ((µ + 3σ) µ f ) + ((µ 3σ) µ f ) = σ 4. Example : If f(x) = x for x 0 ad f(x) = 0 for x < 0 the, for the order of Hermite polyomial a =, formula (8) for the mea yields µ f = µ + σ + µ σ If the mea of radom variable x equals µ = 0 ad variace σ = 3, the mea of the fuctio f(x) equals µ f = = 3.57 If a = 3, the mea of f(x) = x uder the same coditios equals µ f = 4 µ + µ + 3σ + µ 3σ = 3.3. Sometimes approximate formula µ f = f(µ) = µ is used, which i our case gives the result µ f = 3.3. he same result is obtaied by Hermite-Gauss Quadrature if the order of quadrature equals a =. he mea value obtaied by Mote Carlo simulatio (with 0 7 samples) gives the result µ f = he polyomial approximatio is ot exact i this case, but the results are satisfactory. B. Extesio to multidimesioal cases Let us assume dimesioal radom vector x with mea µ ad covariace matrix P. We are lookig to compute first two momets of multidimesioal fuctio f(x) of radom vector x. Vector mea µ f = E {f(x)} of fuctio f(x) equals µ f = f(x) (π) P e (x µ) P (x µ) dx (3) where P is the determiat of covariace matrix P. o simplify the expoet i the previous formula, let us make the substitutio x µ = P v, where the square root of covariace matrix P = P ( P ). he square root of a covariace matrix ca be obtaied by Cholesky factorizatio. Realize that P = ( P ) ad accordig the substitutio theorem dx = P dv. 497

4 After the substitutio, the formula for the mea of f(x) becomes µ f = f( P v + µ)e v v dv (π) Usig the so called stochastic decorelatio techique, vector P v ca be expressed as P v = ( P ) v + ( P ) v ( P ) v (4) where ( P ) i is the ith colum of matrix P. Accordig to the Fubiia theorem, a multidimesioal itegral ca be chaged to the product of oe dimesioal itegrals of the form π f( P i v i + µ)e v i vi dv i hese itegrals ca be solved by Gauss-Hermite Quadrature ad so the fuctio mea µ f equals µ f = π A f( ( P ) i a + µ) A a f( ( P ) i a a + µ). (5) For the order of quadrature a =, the formula for fuctio mea takes the simple form µ f = f(( P ) i + µ) + f( ( P ) i + µ) ; () the correspodig formula for a = 3 is give by µ f = 4f(µ)+ f(µ + 3( P ) i ) + f(µ 3( P ) i + µ). Covariace matrix P f of vector fuctio f(x) equals P f = E { (f(x) µ f )(f(x) µ f ) } ad ca be expressed by itegral formula P f = (f(x) µ f )(f(x) µ f ) (7) (π) P e (x µ) P (x µ) dx his relatio is simplified usig the substitutio x µ = P v, where P is agai obtaied via Cholesky factorizatio. he resultig formula for covariace matrix P f is obtaied as P f = (π) (f( P v + µ) µ f ) (f( P v µ) µ f ) e v v dv his itegral ca be solved by Gauss-Hermite Quadrature P f = A (f( ( ) P ) i a + µ) µ f π ( f( ( ) P ) i a + µ) µ f A a (f( ( P ) i a a + µ) µ f ) (f( ( P ) i a a + µ) µ f ) (8) For the order of quadrature a = the formula for covariace matrix is obtaied as P f = s i s i + w i wi (9) where s i = (f(( P ) i + µ) µ f ) ad w i = (f( ( P ) i + µ) µ f ). For order of Hermite polyomial a = 3, the formula for covariace matrix takes the form P f = 4s i s i + w i wi + z i zi (30) where s i = (f(µ) µ f ), w i = (f( 3( P ) i + µ) µ f ) ad z i = (f( 3( P ) i + µ) µ f ). IV. SIGMA POIN RANSFORMAION FOR UNSCENED FILER Usceted Kalma filter based o the so called reduced sigma poits is described i 5. hese poits are mapped by a oliear trasformatio to obtai the mea ad the covariace of oliear fuctio. he algorithm will be described, usig the otatio itroduced i 5. A symmetrically distributed set of poits which match the mea ad covariace is obtaied as X 0 = ˆx X i = ˆx + ( + k)p i X i+ = ˆx ( + k)p i ad a set of weights is chose as W 0 = k + k, W i = W +i = ( + k). here, k R is a tuig parameter ad i =,...,. Further, ˆx ad P deote the mea ad the covariace of x, respectively; P i is ith colum of covariace matrix P. Let the radom vectors x ad y be related by a kow oliear fuctio y = f(x). he problem is to calculate the mea ad the covariace matrix of vector y ad the crosscovariace matrix P xy, i.e., ŷ = E {y} = E {f(x)} P yy = E { (y ŷ)(y ŷ) } P xy = E { (x ˆx)(y ŷ) } 498

5 he solutio of this problem by Usceted rasformatio is based o the selectio of symmetrically distributed set of poits X 0, X i ad X i+ ad trasformed poits Y i related to X i as Y i = f(x i ), i = 0,,...,. he solutio is give by p ŷ U = W i Y i P U yy = P U xy = i=0 p W i (Y i ŷ)(y i ŷ) i=0 p W i (X i ˆx)(Y i ŷ) i=0 where p =. he costat k is chose so that ( + k) = 3. If the otatio itroduced i the previous sectios is used, the set of sigma poit vectors is give by a = µ a,i = µ + ( + k)( P ) i a 3,i = µ ( + k)( P ) i where ( P ) i is the ith colum of square root of covariace matrix ad the weights A = k + k, A = A 3 = ( + k) he mea of fuctio f(x) usig Usceted trasformatio equals µ U f = A f(µ) + A f(µ + ( + k)( P ) i ) + A 3 f(µ ( + k)( P ) i ) ad after the substitutio the formula simplifies to µ U f = k 3 f(µ) + f(µ + 3( P ) i ) + f(µ 3( P ) i ) where + k = 3 is used. he covariace matrix ad crosscovariace matrix computed by sigma poits result i Pf U = A ss + A w i wi + A 3 z i zi P U xy = A ( ( + k)( P ) i )wi + A 3 ( ( + k)( P ) i )z i where s = (f(µ) µ f ), w i = (f(µ + ( + k)( P ) i ) µ f ) ad z i = (f(µ ( + k)( P ) i ) µ f ). Good performace of this algorithm is show i 5. Example 3: Let us compare the sigma poit filter with Hermite-Gauss Quadrature (HGQ) algorithm for oe dimesioal case with fuctio f(x) = x. he fuctio mea usig HGQ is solved i Example obtaiig µ f = µ + σ. Usig sigma poit trasformatio the fuctio mea is obtaied as µ f = k 3 µ + (µ + 3σ) + (µ 3σ) ( k = 3 + ) µ + σ. 3 For k = the same result is obtaied as for HGQ, equal to the exact value. For variace σf of fuctio f(x) = x, the result usig HGQ was obtaied i Example as σf = σ4. Usig sigma poit trasformatio, the variace approximated as σ f = A µ µ f + A (µ + 3 σ) µ f + A3 (µ 3 σ) ad, after the substitutio of the weight values, the same result σ f = σ4 are recovered (agai for k = ). For multidimesioal cases, due to the stochastic decorelatio techique, the situatio is similar. Usceted trasformatio based o sigma poits gives similar results as Hermite-Gauss Quadrature for order of quadrature a = 3. For more complicated fuctios f(x), it is possible to use a higher order of quadrature resultig i more accurate results. So it ca be stated that computatio the mea ad variace of fuctio f(x) usig Hermite-Gauss-Quadrature is more geeral. V. UILIZAION OF HERMIE-GAUSS QUADRAURE IN KALMAN FILER State estimatio problem for discrete time oliear stochastic system is solved. State equatios of the system have the form x(t + ) = f(x(t), u(t)) + v(t), y(t) = g(x(t), u(t)) + e(t), (3) where v(t) is zero mea white system oise sequece idepedet of the past ad curret states. Usually, ormality of the oise is cosidered v(t) N (0, R v (t)). Similarly e(t) is also zero mea white measuremet oise sequece of kow probability desity fuctio (p.d.f.), idepedet of past ad preset state ad system oise. Normality of the measuremet oise is also usually assumed, e(t) N (0, R e (t)). A. Bayesia approach to state estimatio Assume we observe the iputs u(τ) ad outputs y(τ) for τ =,..., t ad our kowledge of the parameters ad state of the process based o the data set D t = {u(), y(),..., u(t ), y(t )} is described by a coditioal probability desity fuctio (c.p.d.f.) p ( x(t) D t ). he problem is how to update the kowledge described by c.p.d.f. p ( x(t) D t ) to p (x(t+) D t ) after ew iput-output data {u(t), y(t)} have bee measured. he output equatio (3b), defiig the c.p.d.f. 499

6 p (y(t) x(t), u(t)) ad state trasitio equatios (3a) defiig the c.p.d.f. p (x(t+) x(t), u(t), y(t)) are give. he solutio ca be give i the followig steps: ) C.p.d.f. p ( x(t) D t ) is give. ) Usig the output model p (y(t) x(t), u(t)) determie the joit c.p.d.f. p ( y(t), x(t) D t, u(t) ) = (3) p (y(t) x(t), u(t)) p ( x(t) D t, u(t) ) Natural coditio of cotrol p ( x(t) D t, u(t) ) = p ( x(t) D t ) is used to complete this step. Natural coditio of cotrol expresses the fact that all iformatio about state is i iput-output data oly ad cotroller which geerates the cotrol u(t) has o extra iformatio about the state. 3) Usig the output measuremet y(t), determie the c.p.d.f. p ( x(t) D t) = p ( y(t), x(t) D t, u(t) ) p (y(t) D t (33), u(t)) where p ( y(t) D t, u(t) ) = (34) p ( y(t), x(t) D t, u(t) ) dx(t) 4) Usig the state trasitio model (3a), for which p ( x(t+) x(t), D t) = (35) determie the predictive c.p.d.f. B. Kalma filter p (x(t+) x(t), u(t), y(t))) p ( x(t+) D t) = (3) p ( x(t+) x(t), D t) p ( x(t) D t) dx(t) Kalma filter operates o the first two momets of radom variable x(t). Our aim is to estimate the state mea of the system which ca be deoted as ˆx(t, i) ad state covariace matrix P (t, i). It is the state mea ad covariace estimate i time t based o the data u(τ), y(τ) till time i. Kalma filter cosists o two steps: Data update step: Let us have state mea estimate ˆx(t, t ) ad state covariace matrix P (t, t ) ad ew data y(t), u(t) are obtaied. Data update step or predictio step of Kalma filter is the followig: Predictive state mea equals ˆx(t, t) = ˆx(t, t ) + P xy (t, t )P yy (t, t ) (y(t) ŷ(t, t )) (37) where ŷ(t, t ) = E {g(x(t, t ), u(t )} ad state covariace matrix P (t, t)=p (t, t ) P xy (t, t )P yy (t, t ) P yx (t, t ) (38) Covariaces ad cross-covariaces are give by P yy (t, t ) = E {(y(t) ŷ(t, t )) (y(t) ŷ(t, t )) } + R e (39) P xy (t, t ) = E {(x(t) ˆx(t, t )) (y(t) ŷ(t, t )) } (40) ime update step: Let us have the state mea estimate ˆx(t, t) ad we ca proceed further to obtai ˆx(t +, t), which is the state mea estimate i time (t+), based o the same set of data. Such estimate is called time update or model update step, because such update is based oly o the model of the system ad o ew data are give. So state time update mea equals ˆx(t +, t) = E(f(x(t), u(t)) + v(t)) (4) ad state covariace matrix P (t +, t) = E {(f(.) ˆx(t, t))((f(.) ˆx(t, t))} + R v. All meas ad covariaces ca be obtaied by Hermite-Gauss Quadrature or Usceted trasformatio as it is show i the ext sectio. C. Kalma filter by Hermite-Gauss Quadrature Data update step: State ˆx(t, t) is computed accordig to (37) where ŷ(t, t ) = E {g(x(t, t ), u(t )} = π A g( ( P ) i a + ˆx(t, t ), u(t)) A a g( ( P ) i a a + ˆx(t, t ), u(t)), ad the covariaces P yy (t, t )= A s i, s π i, +... A a s i,a s i, a +R e, (4) where s i,j = g( ( P ) i a j + ˆx(t, t ), u(t)) ŷ(t, t ), P xy (t, t ) = π A w i, s i, +... A a w i,a s i, a, where w i,j = f( ( P ) i a j + ˆx(t, t ), u(t)) ˆx(t, t ). he solutio for time update step by Hermite-Gauss Quadrature equals: ˆx(t +, t) = π ad the state covariace P (t+, t) = π A f( ( P ) i a + ˆx(t, t), u(t)) A a f( ( P ) i a a + ˆx(t, t), u(t)), A z i, z i, A a z i,a z i, a +R v, where z i,j = f( ( P ) i a j + ˆx(t, t), u(t)) ˆx(t +, t). I data update step we set P i = P (t, t ) i, while i time update step it is P i = P (t, t) i. 500

7 D. Kalma filter by Cholesky factors of covariace matrices From the previous formulas it is obvious that oly Cholesky factors P of covariace matrices are eeded. For the data update step accordig the formula (39) ad (4) the covariace matrix P yy (t, t ) ca be expressed as P yy (t, t ) = NN where matrix N is of the form N = s,,..., s,,..., s,a, I m m A / π... A / π Re It is clear from (37) that for Kalma gai K = P xy Pyy, (the argumets beig omitted) there is P xy = KP yy. he mutual y covariace matrix P c = cov equals x Pyy P P c = yx NN NN = K KNN MM P xy P xx where matrix M is the Cholesky factor of the covariace matrix P xx. he mutual covariace matrix P c ca be expressed i the form P c = QQ where Q is give by s,... s Q =,... s,a, I m m 0 v,... v,... v,a, 0 I A / π... A a / π Re 0 0 Rv Applyig some orthogoal trasformatio, matrix Q ca be trasformed to a lower triagular matrix such that P c = QQ = H 0 H G G F 0 F HH HG = GH GG + F F where matrices H ad F are lower triagular matrices. It ca be proved that accordig to formula (38) the state covariace matrix P (t, t) equals P (t, t) = F F (43) Comparig two previous matrices reveals that GG +F F = MM, ad hece F F = MM GG. he state covariace matrix P (t, t) which is the result of data update step accordig to (38) equals P (t, t) = MM KNN K = MM KHH K ad therefore, G = KH. Updatig state mea i the data update step is simple, usig the formula ˆx(t, t) = ˆx(t, t ) + Gs (44) where the vector s is obtaied as the solutio of liear algebraic equatio Hs = y(t) ŷ(t, t ). where H is a lower triagular matrix. Previous formulas follow from equatio for Kalma gai K = GH. I the time update step, state ˆx(t +, t) is obtaied i the stadard way usig the Hermite-Gauss Quadrature. he state covariace matrix is give by P (t +, t) = SS, where the auxiliary matrix S equals S = z,,..., z,,..., z,a, I A / π Cholesky factorizatio P (t +, t) = SS = Q... A / π 0 Q 0 Rv results i lower triagular matrix Q the desired Cholesky factor of state covariace matrix QQ = P (t +, t). his cocludes the algorithm of Kalma filter operatig etirely o Cholesky factors of covariace matrices. VI. CONCLUSION We described i this paper how to use Hermite-Gauss Quadrature for computig the mea ad variace of fuctio f(x), where x is a radom variable of ormal distributio whose mea ad variace are kow. Kalma filterig ivolvig oliear systems results i a o-ormal distributio of the radom state x. Applyig the proposed procedure thus yields a approximatio of this radom variable distributio by the first two momets. Usceted filter based o sigma poits selectio was itroduced i 5. I this paper it was show that this filter gives the same results as that usig Hermite-Gauss Quadrature itroduced here, if the order of quadrature equals a = 3. Hermite-Gauss-Quadrature ca be used also for higher or lower order of quadrature ad if the fuctio f(x) is a polyomial fuctio of ormal radom variable x, the results obtaied by Hermite-Gauss Quadrature of sufficiet order are exact. While attempts have bee made to improve the accuracy of the Usceted filter by sigma poit radomizatio 0, Hermite-Gauss Quadrature provides a alterative approach with a soud mathematical backgroud. 50

8 ACKNOWLEDGMEN his work was supported by the grat P03//353 of the Czech Sciece Foudatio. REFERENCES R. E. Kalma, A ew approach to liear filterig ad predictio problems, i ras. ASME J. Basic. Eg., vol. 8, 90, pp G. C. Goodwi, Adaptive Filterig, Predictio ad Cotrol. Pretice- Hall, Eglewood Cliffs, F. L. Lewis, Optimal Estimatio. Joh Wiley & Sos Ic., New York, V. Havlea, Estimatio ad Filterig (i Czech). Publishig Co. CU Prague, S. Julier ad J. K. Uhlma, Reduced Sigma Poit Filters for the Propagatio of Meas ad Covariaces hrough Noliear rasformatio, i Proceedigs of the America Cotrol Coferece, 00, pp E. W. Weisstei, Hermite polyomial, 0. Olie. Available: 7 S. R. MCReyolds, Multidimesioal Hermite Gauss Quadrature Formulae ad their Applicatio to Noliear Estimatio, i Proceedigs of the Symposium o Noliear Estimatio heory a its Applicatio, 975, pp I. Arasaratha, S. Hayki, ad R. Elliott, Dicrete-time oliear filterig algorithms usig gauss-hermite quadrature, i Proceedigs if IEEE ras. ASME J. Basic. Eg., vol. 95, o. 5, 007, pp W. Gautchi, Orthogoal Polyomials: computatio ad approximatio. Oxford Uiversity Press, New York, J. Duik, O. Straka, ad M. Simadl, he Developmet of a Radomised Usceted Kalma Filter, i Preprits of the 8th IFAC World Cogress Milao (Italy), 0, pp