Predictive Filtering for Nonlinear Systems

Transcription

1 Pedictive Filteing fo Nonlinea Systems John L. Cassidis F. Landis Makley Abstact In this pape, a eal-time pedictive filte is deived fo nonlinea systems. he majo advantage of this new filte ove conventional filtes is that it povides a method of detemining optimal state estimates in the pesence of significant eo in the assumed (nominal) model. he new eal-time nonlinea filte detemines ( pedicts ) the optimal model eo tajectoy so that the measuement-minus-estimate covaiance statistically matches the known measuementminus-tuth covaiance. he optimal model eo is found by using a one-time step ahead contol appoach. Also, since the continuous model is used to detemine state estimates, the filte avoids discete state jumps. he pedictive filte is used to estimate the position and velocity of nonlinea mass-dampe-sping system. Results using this new algoithm indicate that the ealtime pedictive filte povides accuate estimates in the pesence of highly nonlinea dynamics and significant eos in the model paametes. Intoduction Conventional filte methods, such as the Kalman filte, have poven to be extemely useful in a wide ange of applications, including: noise eduction of signals, tajectoy tacking of moving objects, and in the contol of linea o nonlinea systems. he essential featue of the Kalman filte is the utilization of state-space fomulations fo the system model. Eo in the dynamics system can be sepaated into pocess noise eos o modeling eos. Pocess noise eos ae usually epesented by a zeo-mean Gaussian eo pocess with known covaiance (e.g., a gyo-eo model can be epesented by a andom walk pocess). Modeling eos ae usually not known explicitly, since system models ae not usually impoved o updated duing the estimation pocess. he theoetical deivation of the expession fo the estimate eo Assistant Pofesso, Catholic Univesity of Ameica, Dept. of Mech. Eng., Washington, D.C Membe AIAA. Staff Enginee, Goddad Space Flight Cente, Code 7, Geenbelt, MD 077. Associate Fellow AIAA.

2 covaiance in the Kalman filte is only available if one makes assumptions about the model eo. he most common assumptions about the model eo ae that it is also a zeo-mean Gaussian noise pocess. heefoe, in the filte-type liteatue, most often pocess noise and model eo ae teated equally. he Kalman filte satisfies an optimality citeion which minimizes the tace of the covaiance of the estimate eo between the system model esponses and actual measuements. Statistical popeties of the pocess noise and measuement eo ae used to detemine an optimal filte design. heefoe, model chaacteistics ae combined with sequential measuements in ode to obtain state estimates which ae moe accuate than both the measuements and model esponses. As stated peviously, eos in the system model of the Kalman filte ae usually assumed to be epesented by a zeo-mean Gaussian noise pocess with known covaiance. In actual pactice the noise covaiance is usually detemined by an ad hoc and/o heuistic estimation appoach which may esult in sub-optimal filte designs. Othe applications also detemine a steady-state gain diectly, which may even poduce unstable filte designs. Also, in many cases such as nonlineaities in the actual system esponses o non-stationay pocesses, the assumption of a Gaussian model eo pocess can lead to seveely degaded state estimates. In addition to nonlinea model eos, the actual assumed model may be nonlinea (e.g., theedimensional kinematic and dynamic equations 3 ). he filteing poblem fo nonlinea systems is consideably moe difficult and admits a wide vaiety of solutions than does the linea poblem. 4 he extended Kalman filte is a widely used algoithm fo nonlinea estimation and filteing. 5 he essential featue of this algoithm is the utilization of a fist-ode aylo seies expansion of the model and output system equations. he extended Kalman filte etains the linea calculation of the covaiance and gain matices, and it updates the state estimate using a linea function of the measuement esidual; howeve, it uses the oiginal nonlinea equations fo state popagation and in the output system equation. 5 But, the model eo statistics ae still assumed to be epesented by a zeo-mean Gaussian noise pocess. A new appoach fo pefoming optimal state estimation in the pesence of significant model eo has been developed by Mook and Junkins. 6 his algoithm, called the Minimum Model

3 Eo (MME) estimato, unlike most filte and smoothe algoithms, does not assume that the model eo is epesented by a Gaussian pocess. Instead, the model eo is detemined duing the MME estimation pocess. he algoithm detemines the coections added to the assumed model such that the model and coections yield an accuate epesentation of the system behavio. his is accomplished by solving system optimality conditions and an output eo covaiance constaint. heefoe, accuate state estimates can be detemined without the use of pecise system epesentations in the assumed model. Also, the MME estimato can be applied to systems with nonlinea models. he MME estimates ae detemined fom a solution of a twopoint-bounday-value-poblem (see Ref. [6-7]). heefoe, the MME estimato is a batch (offline) estimato which must utilize post-expeiment measuements. he filte algoithm developed in this pape can be implemented in eal-time (as can the Kalman filte). Howeve, the algoithm is not limited to Gaussian noise chaacteistics fo the model eo. Essentially, this new algoithm combines the good qualities of both the Kalman filte (i.e., a eal-time estimato) and the MME estimato (i.e., detemines actual model eo tajectoies). he new algoithm is based on a pedictive tacking scheme fist intoduced by Lu. 8 Although the poblem shown in Ref. [8] is solved fom a contol standpoint, the algoithm developed in this pape is efomulated as a filte and estimato with a stochastic measuement pocess. heefoe, the new algoithm is known as a pedictive filte. he advantages of the new algoithm include: (i) the model eo is assumed unknown and is estimated as pat of the solution, (ii) the model eo may take any fom (even nonlinea), and (iii) the algoithm can be implemented on-line to both filte noisy measuements and estimate state tajectoies. he oganization of this pape poceeds as follows. Fist, the basic equations and concepts used fo the filte development ae eviewed. hen, a pedictive filte is deived fo nonlinea systems. his appoach detemines optimal state estimates in eal-time by minimizing a quadatic cost function consisting of a measuement esidual tem and a model eo tem. hen, the concept of the covaiance constaint is intoduced fo detemining the optimal model eo weighting matix. Finally, an example involving the estimation of the position and velocity in a nonlinea mass-dampe-sping system is shown. 3

4 Peliminaies Nonlinea Pedictive Filte In this section, the nonlinea pedictive filte algoithm is deived. his development is based upon the duality which exists between the pedictive contolle fo nonlinea systems by Lu 8 and a geneal estimation poblem. In the nonlinea pedictive filte it is assumed that the state and output estimates ae given by a peliminay model and a to-be-detemined model eo vecto, given by whee f R n n cxt xt = f xt + Gxt dt (a) yt = (b) R is sufficiently diffeentiable, x R t is the state estimate vecto, dt R epesents the model eo vecto, Gxt : matix, cxt q n m R R is the measuement vecto, and y t n n n q R R is the model-eo distibution m R is the estimated output vecto. State-obsevable discete measuements ae assumed fo Equation (b) in the following fom whee ~ y k v k () ~ y = c x t + v k k k m R is the measuement vecto at time t k, xt k is the tue state vecto, and m R epesents the measuement noise vecto which is assumed to be a zeo-mean, Gaussian white-noise distibuted pocess with Ev k = 0 (3a) Evkvl = Rδ kl (3b) m m whee R R is a positive-definite measuement covaiance matix. A aylo seies expansion of the output estimate in Equation (b) is given by (4) yt + t yt + zxt, t + Λ t Sxt dt 4

5 whee the i th element of zxt, t is given by k t zix t, t= L k f ci k! p i k = whee p i, i =,,, m, is the lowest ode of the deivative of ci x t in which any component of dt fist appeas due to successive diffeentiation and substitution fo x it on the ight side. k Lf ci is a k th ode Lie deivative, defined by (see Ref. [9]) L c = c fo k = 0 L k f k f i i k Lf ci ci = f fo k x m m Λ t R is a diagonal matix with elements given by Sxt m q λ ii R is a matix with each i th ow given by i p t i = i = m p!,,,, (7) p p i g f i g f i (5) (6) i i s = L L c,, L L c, i =,,, m (8) q whee the Lie deivative with espect to L g j in Equation (8) is defined by i L L c g j p f i p i Lf ci gj, j =,,, q (9) x Equation (8) is in essence a genealized sensitivity matix fo nonlinea systems. Nonlinea Filteing A cost functional consisting of the weighted sum squae of the measuement-minus-estimate esiduals plus the weighted sum squae of the model coection tem is minimized, given by J d t = ~ y t+ t y t + t R ~ y t + t y t+ t + d t W d t (0) 5

6 whee W ~ yt+ t ~ y k q q R is positive semidefinite. Also, a constant sampling ate is assumed so that +. Substituting Equation (4), and minimizing Equation (0) with espect to dt leads to the following model eo solution d t = Λ t S x R Λ t S x + W Λ t S x R z x, t ~ y t+ t + y t By using the matix invesion lemma, 0 the model eo in Equation () can be e-witten as () dt = Mt zx, t ~ yt+ t + yt () whee + Mt = W I Λ tsx Λ tsxw Λ tsx R Λ tsx W Λ tsx R (3) his fom will late be used to show the elationship of the pedictive filte to a linea estimato fo linea systems. Equation () is used in Equation (a) to pefom a nonlinea popagation of the state estimates to time t k, then the measuement is pocessed at time t k + to find the new dt in tk, tk+, and then the state estimates ae popagated to time t k +. he matix W seves to weight the amount of model eo added to coect the assumed model in Equation (). As W deceases, moe model eo is added to coect the model, so that the estimates moe closely follow the measuements. As W inceases, less model eo is added, so that the estimates moe closely follow the popagated model. Covaiance Constaint he weighting matix W in Equation () can be detemined on the basis that the measuement-minus-estimate eo covaiance matix must match the measuement-minus-tuth eo covaiance matix (see Ref. [6]). his condition is efeed to as the covaiance constaint, shown as N N ek eek e R (4) k = 0 6

7 whee ek ~ y y k k, e is the sample mean of ~ y y, and N is a lage numbe. A test fo whiteness can be based upon the autocoelation function matix of the measuement esidual. 5 he maximum likelihood estimate of the m m autocoelation function matix fo N samples is given by C N k = ee i i k N i= k (5) A 95% confidence inteval fo whiteness using a finite sample length is given by 5 ρ ii k 96. / N / (6) whee ρ ii coesponds to the diagonal elements esulting by nomalizing the autocoelation matix by the zeo-lag elements, given by ρ ii k c = c ii ii k 0 (7) If the confidence inteval in Equation (6) and the covaiance constaint in Equation (4) ae met, then the weighting matix is optimal. heefoe, the pope balance between model eo and measuement esidual has been achieved. If the measuement esidual covaiance is highe than the known measuement eo covaiance, R then W should be deceased to less penalize the model eo. Convesely, if the esidual covaiance is lowe than the known covaiance, then W should be inceased so that less unmodeled dynamics ae added to the assumed system model. he sample measuement covaiance can be detemined fom a ecusive elationship given by (see Ref. []) k (8a) k R R e e e e R k+ = k + k+ k k+ k k + k + he covaiance constaint is met when R e = e + e e k + k+ k k+ k k (8b) R, afte the filte has conveged (i.e., the estimate eaches a stochastic steady-state so that the effects of tansients become negligible). 7

8 Even though the model eo is detemined by Equation () o (), it still involves stochastic pocesses. heefoe, a covaiance of the model eo can be deived. Fist, the covaiance constaint is e-witten as = k k k k (9) E ~ y y ~ y y R Substituting Equation () into Equation (9), and using leads to k k k k k k k k E y v = E v y = E y v = E v y = 0 (0) E ~ y ~ y y y = R () If Equation (4) is satisfied at steady-state, then the following equation is also tue k k E ~ y ~ y y y = R k+ k+ k+ k+ Fo a constant sampling inteval, Equation () is equivalent to k k () (3) E ~ y t+ t ~ y t + t = y t + t y t+ t + R As long as the pocess emains stationay, Equation (3) is valid even if the covaiance constaint is not satisfied. Also, since the optimal model eo solution in Equation () is a function of the stochastic measuement noise pocess, a test fo the whiteness of the detemined model eo can be found by using the coelation function in Equations (5)-(7), eplacing e with d. If the model eo is sufficiently white, then the covaiance of the model eo can also be detemined using a ecusive fomula shown in Equation (8), again eplacing e with d. Anothe fom fo the model eo covaiance can be detemined by using Equation (), and assuming that which leads to E y t v t + t = E v t + t y t = 0 (4a) E z x, t v t + t = E v t + t z x, t =0 (4b) 8

9 E d t d t = M t y t y t+ t + z x, t + R M t (5) whee a aa fo any a (6) heefoe, the elative magnitude of the model eo can now be detemined. In fact, if the detemined model eo pocess is tuly white, then the invese of the weighting matix W can be shown to be the maximum likelihood estimate of the model eo. his can be used to detemine an adaptive scheme fo detemining W to satisfy the covaiance constaint (which will be epoted at a late time). Stability Filte Stability he effect of W on filte stability and bandwidth can be detemined by applying a discete eo analysis. he filte esidual is given by Substituting Equation (4) into Equation (7) leads to whee et+ t = ~ yt+ t yt + t (7) et+ t = I Λ t S x Mt et + t (8) (9) et + t ~ yt+ t yt zx, t which is the pedicted measuement esidual at t + t assuming d = 0. If S is squae and full ank, then Λ SM is also full ank. As W 0, then Λ SM I, and I Λ S M 0. his appoaches a deadbeat esponse fo the filte dynamics. As W, then M 0, and I Λ S M I. his yields a filte esponse with eigenvalues appoaching the unit cicle. As long as the covaiance matix is positive, the eigenvalues of the filte will lie within the unit cicle. heefoe, the filte emains contactive. 9

10 Robustness In the pevious section, the filte stability was shown fo the lineaized system. In this section, filte obustness and stability is shown fo the nonlinea system with unmodeled dynamics. his situation may aise when W is not chosen popely. Lu 3 has shown that the dual contol poblem achieves input/output lineaization, and asymptotic tacking of any given tajectoy if p i 4. An analysis of the obustness popeties in the face of unmodeled dynamics fo p i =, W = 0, and squae and nonsingula Sx has also been shown in Ref. [3]. In this pape, the case of p i =, W 0, and Sx output estimate fo p i = is given by R m y = L c + S x d f 3, whee m 3 is consideed. he continuous (30) whee L f c L L f f c (3) cm Suppose that the unmodeled eos ae intoduced into the output estimate by y = L c + L c + S x + S x d (3) f and suppose that Lf c and Lf c ae bounded by f L c n L c n fo all x X (33) f, f, Futhemoe, assume that S x is epesented by S x = δ xsx (34) whee δ x is a scala, continuous function with bound given by < δ x < n3. Assuming that the model eos and measuement eos ae isotopic leads to W = wi, and R= I. hen, the matix invese in Equation () can be witten as (suppessing aguments) 0

11 ΛS R ΛS+ W = v σ I + C (35) whee t C S S (36a) σ = t ts S (36b) v = w+σ (36c) By the Cayley-Hamilton theoem, any meomophic function of C can be expessed as a quadatic in C (see Ref. [4]), yielding whee v σ I + C = I + C+ C γ α β (37) α = v σ + k (38a) β σ (38b) = v + (38c) γ = v σ α + = wα + t k = tadjc 4 = t adjs S (38d) 6 = t S S 3 det (38e) heefoe, the eo dynamics become whee (39) t ~ t e = + δ Qe+ y L L QL ~ f f + + δ f y γ γ

12 4 3 t t Q α SS + β SS + β SS (40) Now, define a Lyapunov function V Appendix) = e. Using the nom inequality, 5 and the fact that (see e Qe Q e e (4) leads to (4) t + δ V V e ~ t + δ y + L L e Q L ~ f f + f y γ Q γ 4 Next, using the well known inequality ab za + b z yields ξ t + δ γ V ξ + 4 zv+ Q Substituting 4 z = ξ Q leads to fo any a, b, and z > 0, and defining ~ ~ y L L + t ξ Q L y f f f 4 z (43) V ξ V + b Q (44) whee ~ ~ b Q y Lf Lf + t ξ Q Lf y (45) ξ heefoe, Equation (44) can be solved to yield V V 0 bq ξ ξ t Q e bq + ξ (46)

13 whee V 0 b Q ξ is equied to maintain the inequality. Defining the bounds (47a) θ = inf + δ > 0 ~ y m = max ~ y t 0, t i f i= (47b) and using the matix nom inequality again leads to e γ Q µ + tq Qµ (48) t θ whee ~ y n n (49a) µ = µ = θ n ~ y (49b) Assuming that SS s fo all SS leads to the following bound on Q t t Q αs + β s 4 + s 3 3 Also, ts S 3s and dets S s, which leads to (50) σ 3 t s (5a) t k 9 4 s (5b) 6 t s 3 3 (5c) Substituting Equation (5) into Equations (38) and (50) leads to the following bounds on Q and γ 3

14 Q 4w t s 3 t s w s (5a) w t s 9 t s t s γ w (5b) A bound on Q is found by witing it as Q!! t = γ S S S+ wi S!! t γ SS S S+ wi t γ SS S S + w!!!! (53) Assuming that SS s fo all SS leads to Q t γ s s + w (54) heefoe Equations (48), (5), and (54) define the bound fo the eo dynamics unde unmodeled uncetainty. Simila esults can be obtained fo < p i 4. he case whee q > 3 can also be detemined using a Cayley-Hamilton expansion, but becomes inceasingly moe complicated. Numeical Stability If the system is unobsevable then S S is not full ank. Howeve, the filte can compensate fo this by adding moe model coection. It can be shown that the filte emains fo bounded model uncetainties as long as If S S v σα + > 0 (55) is not full ank, then =0, which leads to the following condition 4

15 t v > ts S (56) heefoe, the filte emains contactive as long as w > 0. his condition is always met, but Equation (56) can be used to help detemine any numeical difficulties (i.e., lage values of σ may poduce numeical difficulties). One possible solution is to make as lage as possible. Howeve, then w will be adjusted to meet the covaiance constaint, so that the numeical difficulties emain. Anothe solution to this poblem is to use smalle sampling inteval, but this may not be possible. A moe pactical solution is to utilize a U D factoization of Equation (3) (see Ref. [6]). Cases Case. Let p i = fo both the state and output systems. Equations (5), (7) and (8) educe to z = t H x f x (57a) Hx y x (57b) S Λ = ti (57c) = H x G x (57d) heefoe, the model eo tajectoy in Equation () is given by d = t I W G H HGW G H + t R HG W G H R y t ~ y t+ t + t H f (58) Equations (57-58) can be used to develop a pedictive filte fo a linea system, given by x = F x + Gd (59a) y = Hx (59b) Fo the linea case Equation (58) is simila to a linea estimato. his can be shown be conveting Equation (59) into discete fom 5

16 xk+ = Φxk + Γ dk (60) If the sampling inteval in the discete convesion in Equation (60) is equal to the measuement sampling inteval ( t ), and if the fist-ode appoximations of Φ made, then the following equation fo d k is given I + t A, and Γ tg ae d = I W Γ H HΓW Γ H + R HΓ W Γ H R ~ y + HΦx k Case. Conside the following system x f x, x k + k (6) = (6a) (6b) x = f x + G x d y = c x (6c) with p i =. Equation (6a) usually defines the kinematics, and Equation (6b) usually defines the dynamics of a system. Equations (5), (7) and (8) now become t L f L f z = t Lf + f x x + f x x, x (63a) L f c x f x x c f x, + x (63b) Λ = t I (63c) L S = x f G x (63d) Example In this section, a simple example which illustates the application of the pedictive filte to a nonlinea mass-dampe-sping system is shown. Conside the following system 7 6

17 mx + bx x + kx+ k x 3 = 0 (64) whee bx x epesents the nonlinea damping, and kx+ k x 3 epesents a linea sping with a nonlinea hadening effect. his system can be shown to be asymptotically stable by choosing the following Lyapunov function 7 V= x mx + k x + k x 4 (65) 4 which leads to V = x b x 3 (66) heefoe, the mechanical enegy of the system conveges to zeo fo any initial condition. Using the pedictive filte appoach, the system model is modified by the addition of a to-be-detemined unmodeled effect. he state-space epesentation is given by x x = x c x x c x c x dt + (67) whee c b m, c k m, c3 k m, and x and x epesent position ( x ) and velocity ( x ), espectively. Fo this system, the model eo is epesented as an input to the mass-dampesping system. he measuements ae given by (68) ~ y = x t + v k k k whee the vaiance of v k is defined as. he lowest ode time-deivative of Equation (68) in which the model eo fist appeas is two. heefoe, the pedictive filte equations ae given by Equations (6-63), which is Case. Fo this example, the detemined model eo is given by 3 (69) t d t w x t = c x x + c x + c x ~ y x + 4 t whee ~ y ~ y k +. he case whee w = 0 coesponds to the feedback lineaization case, yielding 7

18 x 0 x 0 x = t t x t + ~ y (70) he eigenvalues of the state matix ae given by s, = ± j t (7) heefoe, the filte s dynamics ae dependent only on the sampling inteval. Also, this case epesents a linea filte on the measuements only, so that no model eo coection is added. = 0 fo all t, he tue state histoy fo this example is given using Equation (67) with dt c = 0, c = 0., and c 3 = 3, and initial conditions of xt0= 0., and xt0= 0.. A plot of the tue states with these paametes is shown in Figue. Measuements ae obtained by using a sampling inteval of 0. seconds, and the standad deviation of v k in Equation (68) is Model eo is intoduced into the system by petubing c and c 3, which ae chosen to be c = 00, and c 3 = 40. Also, a weighting facto of w = 009. was detemined by satisfying the covaiance constaint once the filte eached steady-state. Even though a significant amount of eo is pesent in the assumed model, the pedictive filte is able to accuately estimate fo the states, as shown by Figue. A plot of the actual and detemined model eo histoies is shown in Figue 3. his example shows that the model eo fo this example cannot be epesented by a zeo-mean Gaussian pocess, as is assumed in the Kalman filte. Howeve, the pedictive filte is clealy able to coectly detemine the actual model eo in the system. Finally, the pedictive filte is tested fo initial conditions eos. Fo this test, the assumed initial conditions in the filte ae set to zeo. Figue 4 depicts the filte convegence fo this case. he pedictive filte is able to convege vey quickly (within 0.06 seconds). his example clealy shows that the pedictive filte scheme povides obust pefomance in a nonlinea system fo both significant eos in the assumed model and in the initial conditions. Conclusions In this pape, a pedictive filte was pesented fo nonlinea systems. Advantages of the new algoithm ove the extended Kalman filte include: (i) the model eo is assumed unknown and 8

19 is estimated as pat of the solution, (ii) the model eo may take any fom (even nonlinea), and (iii) the model eo is used to popagate a continuous model which avoids discete jumps in the state estimate. An example of this algoithm was shown which estimated the position and velocity of nonlinea mass-dampe-sping system. Results using this new algoithm indicated that the eal-time pedictive filte povides accuate estimates in the pesence of highly nonlinea dynamics and significant eos in the model paametes. Appendix In this section, the inequality given by Equation (4) is poved. he vecto e is fist epesented by m e= α i u i= i (A) whee u i ae the eigenvectos of Q, and α i ae some scala coefficients. heefoe, the poduct Qe is given by m Qe= α i λ iu i= i (A) whee λ i ae the eigenvalues of Q. Using the fact that e m = α i i= (A3) leads to the following inequality m m min i max i= i= λ α e Qe λ α (A4) i heefoe, using the following identity λ min Q = = λ Q Q max (A5) 9

20 then the following inequality must hold tue e Qe Q e e (A6) Acknowledgments he fist autho s wok was suppoted by a National Reseach Council Postdoctoal Fellowship tenued at NASA-Goddad Space Flight Cente. he autho geatly appeciates this suppot. Also, the autho wishes to thank D. D. Joseph Mook of the State Univesity of New Yok at Buffalo, and D. Ping Lu of Iowa State Univesity fo many inteesting and helpful discussions. Refeences Kalman, R.E., A New Appoach to Linea Filteing and Pediction Poblems, ansactions of the ASME, Jounal of Basic Engineeing, Vol. 8, Mach 960, pp Mason, P.A.C., and Mook, D.J., Scala Gain Intepetation of Lage Ode Filtes, Poceedings of the Flight Mechanics/Estimation heoy Symposium, NASA-Goddad Space Flight Cente, Geenbelt, MD, 99, pp Kane,.R., Likins, P.W., and Levinson, D.A., Spacecaft Dynamics, McGaw-Hill, NY, Gelb, A., Applied Optimal Estimation, MI Pess, MA, Stengel, R.F., Optimal Contol and Estimation, Dove Publications, NY, Mook, D.J., and Junkins, J.L., Minimum Model Eo Estimation fo Pooly Modeled Dynamic Systems, Jounal of Guidance, Contol and Dynamics, Vol., No. 3, May-June 988, pp Cassidis, J.L., Mason, P.A.C., and Mook, D.J., Riccati Solution fo the Minimum Model Eo Algoithm, Jounal of Guidance, Contol and Dynamics, Vol. 6, No. 6, Nov.-Dec. 993, pp

21 8 Lu, P., Nonlinea Pedictive Contolles fo Continuous Systems, Jounal of Guidance, Contol and Dynamics, Vol. 7, No. 3, May-June 994, pp Hunt, L.R., Luksic, M., Su, R., Exact Lineaizations of Input-Output Systems, Intenational Jounal of Contol, Vol. 43, No., 986, pp Bieman, G.J., Factoization Methods fo Discete Sequential Estimation, Academic Pess, FL, 977. Lewis, F.L., Optimal Estimation, John Wiley & Sons, NY, 986. Vidyasaga, M., Nonlinea Systems Analysis, Pentice Hall, NJ, Lu, P., Optimal Pedictive Contol of Continuous Nonlinea Systems, Intenational Jounal of Contol, Vol. 6, No. 3, Sept. 995, pp Shuste, M.D., and Oh, S.D., Attitude Detemination fom Vecto Obsevations, Jounal of Guidance and Contol, Vol. 4, No., Jan.-Feb. 98, pp Hon, R.A., and Johnson, C.R., Matix Analysis, Cambidge Univesity Pess, Cambidge, honton, C.L., and Jacobson, R.A., Linea Stochastic Contol Using the UDU Matix Factoization, Jounal of Guidance and Contol, Vol., No. 4, July-Aug. 978, pp Slotine, J.J.E, and Li, W., Applied Nonlinea Contol, Pentice Hall, NJ, 99.

22 Figue ue States Figue Estimated States Figue 3 Actual and Detemined Model Eo Histoies Figue 4 Sensitivity to Initial Condition Eos

23 0.5 Plot of ue States 0. X (Position) X (Velocity) ime (Sec)

24 0.5 Plot of Estimated States 0. X (Position) X (Velocity) ime (Sec)

25 0 ue and Detemined Model Eo ue Model Eo Detemined Model Eo ime (Sec)

26 0 Initial Condition Sensitivity: Estimated (Dashed), uth (Solid) X (Position) X (Velocity) ime (Sec)