C.K. Chui, G. Chen,"Kalman Filtering with Real-Time Applications," Springer-Verlag, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... (3.3) Now, the following inequality is claimed: 1 1 1 1 1 3 2 k mh k k mh k k mh k mh F H R H H H F q tr G G G G . 0 ) 1 1 1 1 1 5 2 G G k k k mh k k k mh F H R H H H F q tr (3.4) 3.4) can be easily shown by using Lemma 1.7 and Lemma 1. 10 of Chui and Chen [5] . The first term of (3.4) can be rewritten as follows: 1 1 1 1 1 2 3 2 mh k mh k k mh k k mh k mh F H R H H H F tr q G G G G G 1 1 1 1 2 3 2 3 2 G G G G G k k mh k k k mh mh k mh mh H R H H H tr F F tr tr q . 1 1 1 1 1 2 3 2 G G G k k mh k ....

C. K. Chui and G. Chen, Kalman Filtering with Real-time Applications, Springer-Verlag, New York, 1991.

....hidden Markov model [26] for a speech recognition task. Our previous work on joint state and parameter estimation of the target directed nonlinear model with switching states, which we will refer to as the hidden dynamic model (HDM) investigated the use of the extended Kalman filter (EKF)[1, 13, 21] and EM algorithms [2, 29, 28] for estimation of the parameters in the state equation only [30] In this paper we extend the results to include joint state and parameter estimation of both the state and observation equation parameters, and perform more rigorous simulation experiments with ....

C. K. Chui and G. Chen, Kalman Filtering with Real-time Applications, Springer-Verlag, 1991, Chapter 8, pp. 108-130.

....of speech except a i are replaced by c i and the model order changes form p to r. In general r is smaller than p. 15 CHAPTER III KALMAN FILTER III.1 General Kalman lter is an optimal lter in the minimum mean squared error (MMSE) sense for a linear system corrupted by random perturbations [1, 2, 10, 11, 24]. Given the noisy observations and the system model, the unknown state vector is estimated recursively. The optimality of the estimate is restricted to the case where the random processes involved are white. It has found many applications because of sequential operation and applicability to ....

....w(n 1) is represented as a white Gaussian noise with variance = d w . The system order is not increased as in the state augmentation method, however the system driving noise and the measurement noise are correlated. This requires somewhat more complex formulas in the ltering [1, 2]. Initialization: x(0) E fx(0)g (III.20) E f (n)d(k)g = d nk (III.22) K = D d 21 P(njn 1) F KC)P(n 1jn 1) F KC) KK x(njn 1) F x(n 1) K [y(n 1) C x(n 1) III.27) III.3 Smoothing When the estimation of the state is based on future ....

C. K. Chui and G. Chen. Kalman Filtering with Real-Time Applications. Springer Series in Information Sciences. Springer-Verlag, Berlin, Gemany, third edition, 1999.

....the future position of the rectangles a few steps in advance. Every side of a rectangle is modelled by a dynamic model. This gives us model independence from the hand shapes. The position, velocity and acceleration of each side are considered in this model. These parameters are related together [18] based on the Equation 1 for i=1, 2 for the two hands and j=1, 2 for the left and right or top and bottom sides. Fig. 2. A rectangle is formed around the hands or the big blob in occlusion a d H1 H2 H2 c H1 H1 H2 g H1 H2 H2 b H1 H2 e H1 H2 f H2 H1 h t t 1 ....

....hands or the big blob in occlusion a d H1 H2 H2 c H1 H1 H2 g H1 H2 H2 b H1 H2 e H1 H2 f H2 H1 h t t 1 = k j i k j k j i k j i k j k j i x h x x x h x h x x , 1 , 2 , 1 , 1 # # # # # # # . 1) where h 0 is the sampling time [18], k is the time index, x is the position, x # the velocity and x # # the acceleration [18] of every side of a rectangle. In the normal Kalman filtering loop [19] the estimation of a state vector representing a rectangle side is updated with the current measurement of the position of the ....

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Chui, C.K., Chen, G.: Kalman Filtering With Real Time Applications. Springer-Verlag (1999)

....objects on the field will be at some given time in the future. Although there are many statistical tracking algorithms that have been developed, the Kalman Bucy filter has proved to be one of the most capable filters. The KBF and its extended variant for EKBF non linear systems, are widely used [3], have a solid and well understood mathematical basis, and are computationally tractable when used in such a system as the one discussed here. As such, the EKBF represents a good choice for a tracking mechanism and was the one used for the robots described in this paper and is discussed next. ....

....the robot EKBF and the ball EKBF, where the difference is due to the different dynamics of the object being tracked. We briefly describe the EKBF equations that are the basis for the filters, which are an extension on our previous EKBF s [8] For a more detailed discussion of EKBF s refer to [3, 9, 6, 11]. The symbols used here are identical to [11] In the following discussions, the state estimate will be represented by k x and the state covariances by P k . Two stochastic difference equations model the system dynamics and observations, respectively, as: 1 1 , k k k k w u x f x ....

Chui, C. K. Kalman Filtering: with real-time applications, 2 nd Ed., New York, Springer-Verlag, 1991.

....and disambiguating occlusions in pedestrian scenes. Currently there are several popular methods for tracking moving targets, which include: Active Shape Models [4] which are flexible shape models, allowing iterative refinements of estimates of the objects pose, scale and shape; the Kalman filter [15, 3], which has been used in many tracking applications due to its computational efficiency and its ability to estimate future states; and Isard and Blake s newer method of CONDENSATION [11] which is a powerful technique allowing the propagation of conditional densities over time, and has been used ....

C. K. Chui and G. Chen. Kalman Filtering with Real-Time Applications. Springer, 1999.

....(2) we get (4) Now that the noise component has become a low pass signal , we may take the AR(1) process for its modeling, i.e. 5) where denotes the AWGN whose average and variance are 0 and 1, respectively, and and are weighting parameters. If we apply the KF equations for the colored noise [9], then we obtain the following results. Initialization (6) For (7) where , is the estimated value of , the variance of the difference , and the Kalman gain. D. Determination of Initial Conditions and AR(1) Parameters The initial conditions of KF, and , that we need in implementing the KALP ....

C. K. Chui and G. Chen, Kalman Filtering With Real-Time Applications, 2nd ed. New York: Springer-Verlag, 1990.

....as an optimization problem, which can be solved in many ways because of different optimization criteria and several possible parameterizations. Generally, the literature on fitting can be divided into three general techniques: least squares fitting (e.g. 1, 3, 9, 11, 13] Kalman filtering (e.g. [4, 5, 6]) and robust clustering techniques (e.g. 2, 7] While the clustering methods are based on mapping data points to the parameter space, such as the Hough transform and the accumulation methods, the least squares methods are centered on finding the sets of parameters that minimize some distance ....

Chui, C. K. and G. Chen. Kalman filtering with real time applications. Springer, BerlinHeidelberg -New York, 1987.

....1997, Ben Akiva, Bierlaire, Koutsopoulos and Mishalani, forthcoming, BenAkiva, Bierlaire, Burton, Koutsopoulos and Mishalani, forthcoming) The main drawback of the Kalman filter algorithm appears to be its inability to handle large scale problems. Indeed, even if e#cient implementations are used (Chui and Chen, 1991), the analytical computation of the normal equations and the variance propagation require intensive linear algebra computation. Moreover, the sparsity of the least square problem is not exploited by the algorithm, and a lot of fill in is taking place. Another limitation of the Kalman filter is its ....

Chui, C. and Chen, G. (1991). Kalman Filtering with Real-Time Applications, Spinger-Verlag.

....the signal has been screened out, a hole is created in the continuous 3D coordinate stream. Such holes are filled by using the predictions of Kalman filters (see below) and a constant signal stream is still maintained. 6. 2 Kalman Filtering and Adaptive Filtering A set of linear Kalman filters [16] are used both to suppress the Gaussian noise in the 3D co ordinates signal and make predictions for the holes in the signal. This is the second step of our post processing task. Preliminary experiments show that the Kalman filtering results are not stable. This is partly due to the fact that ....

Chui, C. K. and Chen, G., Kalman Filtering with Real-Time Applications, SpringerVerlag, 1987. 14

....a simplified version of the filter equations and, where relevant, describes how they are implemented in the tracker design. B.2. 1 Background An introduction to the Kalman filter algorithm can be found in [11] the historical background and mathematical foundations of the filter are described in [12]. The State Description A process state is represented as an n vector x. With no noise or control input, the state at time k 1 is related to the state at time k by the n Theta n matrix A: x k 1 = A k x k Control input to the state is represented by an l vector u, which is related to the ....

....G k is the Kalman gain, chosen to minimise the error covariance P k .The difference (z k Gamma H k x Gamma ) is called the measurement residual. The Kalman gain is an n Theta m matrix G of Equation B.7. Extensive justification for this matrix as minimising P k is given by Chui et al. in [12]. G k = P Gamma k H T k H k P Gamma k H T k R k (B.7) where R is the variance of the measurement noise v k in Equation B.3. The error covariance estimated is updated as follows: P k = I Gamma G k H k )P Gamma k (B.8) B.2.2 The Kalman Filter Algorithm The following ....

C. K. Chui, G. Chen, Kalman Filtering with Real-Time Applications, Springer Verlag publishers, 1987.

....and the propagation of error through this transformation [19] 20] 13] 8] In the 1980 s, researchers began using an algorithm called Kalman Filtering to estimate object state by modeling the system structure. Kalman Filtering is an optimal state estimation process applied to a dynamic system [5], and is described in Section 5. Gennery [9] uses an algorithm very similar to the Kalman Filter to estimate the position and rotation of a rigid object. In [3] Broida and Chellappa employ a Kalman Filter to estimate the 2D rotation and 1D translation of an object from a sequence of 1D images. ....

C. K. Chui and G. Chen. Kalman Filtering with Real-Time Applications. Springer Series in Information Sciences. Springer-Verlag, Berlin Heidelberg, 1987.

....the deterministic elements and one composed of the stochastic elements. Then (1) is the sum of these two systems. The advantage to this approach is that the deterministic system has a well known solution, which may be combined with the solution for the stochastic system to derive the Kalman Filter [3]. The stochastic system is given by x s k 1 = A k x s k G k w k z s k = H k x s k v k ; 2) and the deterministic system is given by x d k 1 = A k x d k B k u k z d k = H k x d k D k u k ; 3) in which x k = x s k x d k , and z k = z s k z d k . To simplify ....

.... x) T (x Gamma x) The well known solution to this problem is given by x k = H T k R Gamma1 k H k ) Gamma1 H T k R Gamma1 k z k ; 4) where R k = V ar(v k ) We may extend this approach to incorporate all of the observed data by exploiting our knowledge of the system dynamics [3]. We define z k as z k = Theta z 0 : z k T : 5) By fairly straightforward manipulations of (2) we obtain z j = C k;j x k e k;j (6) in which 1 The variance of a vector is simply the Covariance matrix of the vector with itself (V ar(v) Cov(v; v) C k;j = 2 6 4 H 0 Phi 0;k ....

C. K. Chui and G. Chen. Kalman Filtering with Real-Time Applications. Springer Series in Information Sciences. Springer-Verlag, Berlin Heidelberg, 1987.

....for jaj 6= 1, s j = A j (A Gamma 1)s 0 r A(A j Gamma 1)s 0 (A Gamma 1)r and that for jaj = 1, s j = s 0 r js 0 r Finally, show that when j 1 we have s j 0 para jaj 1 , s j (1 Gamma 1 A )r ; para jaj 1 : Interpret the asymptotic results obtained above. 6. Chui Chen [25], Problema 2.14) Some typical engineering applications are classified under the designation ARMA (autoregressive moving average) and can be written as: v k = N X i=1 B i v k Gammai M X i=0 A i u k Gammai ; where the matrices B 1 ; BN are n Theta n dimensional, and the ....

Chui, C.K., & G. Chen, 1991: Kalman Filtering with Real--Time Applications, 2nd ed. Springer--Verlag, Vol. 17, 191 pp.

....two disjoint systems, 2 one composed of the deterministic elements of Equation (3.1) see Figure 3. 1(c) and one composed of the stochastic 1 The variance of a vector is simply the covariance matrix of the vector with itself (V ar(v) Cov(v; v) 2 This derivation follows closely that of [84]. 31 v k z 1 k G H A k k w z k x k s k 1 x s Plant Model Measurement Model x k 1 x k v k z k z 1 k G B H k A k k w k u B k Plant Model Measurement Model z 1 B k H A k k u B k k 1 x d k x d z k d Plant Model Measurement Model a) General System Model = b) Stochastic System Model c) ....

....a zero mean white noise sequence v k , z k = H k x k v k ; the optimal estimate (in the least squares sense) of x k using the data z k is: x k = H T k R Gamma1 k H k ) Gamma1 H T k R Gamma1 k z k ; 3. 4) where R k is the variance of v k , V ar(v k ) Proof: It is well known (see [84]) that for an over determined set of linear equations, b = Ax, the least squares solution is given by: x = A T WA) Gamma1 A T Wb; where W is a weighting matrix among the terms in x. W may be set to I if equal weighting among the terms in x is desired. If we substitute H k for A and z k ....

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C. K. Chui and G. Chen, Kalman Filtering with Real-Time Applications. Berlin: SpringerVerlag, 1987.

....the Kalman gain K k 1 is also expensive, since it involves n times the solution of a m Theta m matrix vector equation. Fortunately, the computational effort for the measurement update can significantly be reduced in case the measurements are uncorrelated. In that case it can be shown (see e.g. [2, 6]) that the following iterative procedure may be applied (dropping the superscripts k 1) a i = c i P i Gamma1 c T i R ii ) Gamma1 b i = 1 p a i R ii ) Gamma1 K i = a i P i Gamma1 c T i P i = P i Gamma1 Gamma a i P i Gamma1 c T i c i P i Gamma1 x i = x i Gamma1 K i (y i ....

C.K. Chui and G. Chen. Kalman Filtering with real time applications. volume 17 of Springer Series in Information Sciences. Springer Verlag, Berlin, 1987.

....P (n) which describes the dependencies of all weights with each other, based on previous estimations and inputs. Due to the remarkable length of the derivation, only the Global Extended Kalman Filter equations are given below. A detailed mathematical background is presented extensively in [8] and [9] The Global Extended Kalman Filter Equations are: P (n 1) P (n) Gamma K(n) Delta H T (n) Delta P (n) Q(n) K(n) P (n) Delta H(n) Delta h (j(n) Delta S(n) Gamma1 H T (n) Delta P (n) Delta H(n) i Gamma1 b w(n 1) b w(n) K(n) Delta i d(n) Gamma h( ....

C. K. Chui and G. Chen, Kalman Filtering with Real-Time Applications. Springer-- Verlag, 1987.

....of the whole synoptic scale (or macro scale over 1000 km) system would have to be taken into account. This has motivated the present work. 3. An extended Kalman filter for rain cell tracking We develop an extended Kalman filter for an observed system of rain cells. We refer the reader to [1] for a good description of the extended Kalman Filter, a prediction correction filter for a non linear system. The cell motion model is geared towards describing the pattern of rotation near a synoptic scale low pressure. We use a modified form of the image sequence model given above, with the ....

C. K. Chui and G. Chen. Kalman Filtering with Real--Time Applications. Springer Series in Information Sciences. Springer--Verlag, Berlin, second edition, 1991.

....problem is formulated and in section 2.2 the solution is presented. In section 2.3 is described what to do if the system turns out to be nonlinear and in that context the the extended Kalman filter is introduced. A more detailed description of Kalman filters and their applications can be found in [ChC91] and [Hay90] 2.1 The Kalman Filtering Problem Let the n dimensional vector x k denote the state of a discrete time linear stochastic system and let the p dimensional vector y k denote the observed data of the system, where k is the discrete time. The system model, presented in the form of a ....

Chui, C. K. and G. Chen (1991): Kalman Filtering with Real-Time Applications, 2nd Edition, Springer-Verlag, New York.

....Technique Kalman ltering, as pointed out by Lowe [20] is likely to have applications throu ghout Computer Vision as a general method for integrating noisy measurements. The reader is referred to [34] for an introduction of the Kalman lter and its appli cations to 3D computer vision, and to [15, 22, 8] for more theoretical studies. The following subsection collects the Kalman lter formulas for simpler reference. 8.1 Kalman Filter If we denote the state vector by s and denote the measurement vector by x, a linear dynamic system (in discrete time form) can be described by s i 1 = H i s i n ....

C.K. Chui and G. Chen. Kalman Filtering with Real-Time Applications. Sprin ger Ser. Info. Sci., Vol. 17. Springer, Berlin, Heidelberg, 1987.

.... lter minimizes the following cost function F i ( s i ) E[ s i Gamma s i ) T M( s i Gamma s i ) where M is an arbitrary, positive semide nite matrix. The optimal estimate s i of the state vector s i is easily understood to be a least squares estimate of s i with the properties that [3]: 1. the transformation that yields s i from [x T 0 Delta Delta Delta x T i ] T is linear, 2. s i is unbiased in the sense that E[ s i ] E[s i ] 3. it yields a minimum variance estimate with the inverse of covariance matrix of measurement as the optimal weight. By inspecting the ....

....the measurement noise process are uncorrelated and that they are all Gaussian white noise sequences. These assumptions are adequate in solving the problems addressed in this monograph. In the case that noise processes are correlated or they are not white (i.e. colored) the reader is referred to [3] for the derivation of the Kalman lter equations. The numerical unstability of Kalman lter implementation is well known. Several techniques are developed to overcome those problems, such as square root ltering and U D factorization. See [13] for a thorough discussion. There exist many other ....

C.K. Chui and G. Chen. Kalman Filtering with Real-Time Applications. Springer Ser. Info. Sci., Vol. 17. Springer, Berlin, Heidelberg, 1987.

....as accurate as that given by the best method possible from a data set of size fflk. The efficiency depends on how well the functions in the prior are modelled by M on average. It increases towards unity as M becomes larger. The actual algorithm is simply a standard Kalman filter algorithm [1] with a particular method for setting the initial mean and variance. 1 In one dimensional cases it is usually called a random process. 1. Use prior knowledge to select a mean hfi 0 = b f 0 , and covariance kernel D f; f T E 0 = V 0 . 2. Define model space M by selecting basis Phi. ....

C. K. Chui and G. Chen. Kalman Filtering with Real-Time Applications. Springer Series in Information Sciences. Springer-Verlag, Berlin, 1987.

....than the values themselves, as proposed by Ashok and Ben Akiva (1993) Estimated and predicted deviations are finally added to an updated historical OD matrix to get estimated and predicted OD matrices. The solution to the model formulation is given by a square root Kalman Filtering algorithm (Chui and Chen, 1987). The input to the OD estimation and prediction model for an estimation time interval consists primarily of the following matrices: An updated OD matrix of drivers departing in that interval. A number of assignment matrices, equal to the number OD flows of prior intervals that contribute to ....

Chui, C.K., and G. Chen (1987). Kalman Filtering with Real-Time Applications. Springer Verlag.

....parameters. For each iteration, this regressor algorithm requires (N 11) multiplications which is O(N ) 2.2. IIR Kalman Output of the IIR Kalman regressor is an estimate of the actual system output obtained by using the Kalman filter on the following state space model of the original system [3]: w(n 1) Aw(n) Bx(n) 8) d(n) Cw(n 1) v(n) 9) where C = Theta 1 0 Delta Delta Delta 0 . Since the actual parameters are unknown in Eqn. 8) we can form A and B matrices by using the estimated parameters at time n. Then, we get w(n 1) A(n)w(n) B(n)x(n) u(n) ....

....The proposed algorithms are compared with two gradient descent type IIR adaptation algorithms, CRA [6] and BRLE [2] Furthermore, we provided comparison results with the extended Kalman filter (EKF) algorithm which is given in Table 3. Note that the EKF algorithm is an O(N 2 ) algorithm [3, 4]. 3.1. Simulation Example 1 The system to be identified is chosen as in [2] H(z) 1 1 Gamma1:7z Gamma1 0:7225z Gamma2 . The input is a unit variance white Gaussian process. The output noise, v(n) is chosen as white Gaussian. oe v is varied to investigate the sensitivity of the ....

C. Chui and G. Chen, Kalman Filtering with RealTime Applications. Berlin: Springer-Verlag, 1991.

....unrealistic and better models for its correlation structure should be investigated. In that sense, it could be worth exploring a first order autoregressive model for the system noise. This approach can be handled through the Kalman filter theory with colored noise; see for example chapter 5 of Chui and Chen (1990). Even though the computations would be more complex than those for the independent state noise case, the incorporation of an autoregressive structure for the wind error could eventually improve the performance of the Kalman filter. Our AR(1) model is a model of the autocorrelation of the ....

Chui, C.K., and G. Chen, 1990: Kalman Filtering with Real-Time Applications. Springer-Verlag, New York, 195pp.

....by computing O(N 2 ) multiplications. Also, a robust way of updating the required covariance matrices is provided. CHAPTER 1. INTRODUCTION 3 In chapter 5, we provide extensive comparison results between the approaches investigated in this work and earlier proposed approaches to IIR adaptation [2, 3, 12, 13]. In chapter 6, we provide the conclusions of our work and address potential areas for future work. Chapter 2 IIR System Model and Proposed Adaptive IIR Filter Structure 2.1 IIR System Model As shown in Fig. 2.1, in a typical adaptive filtering application, input, x(n) and noisy output, d(n) ....

....these two stages of the adaptation can be combined into one in an augmented state space description of the system. This approach has been proposed for combined state estimation and tracking of slowly varying system parameters once a close initial estimate to the system parameters is available [12, 13]. In this chapter, we provide the augmented state space description corresponding to IIR adaptive filtering and then derive the corresponding extended Kalman algorithm for the estimation of the augmented state. Also, we use a robust method, which is presented in the Appendix A, for the choice of ....

[Article contains additional citation context not shown here]

C. Chui and G Chen. Kalman Filtering with Real-Time Applications. Springer-Verlag, Berlin, 1991.

....For each iteration, this regression algorithm requires (N 11) multiplications which is O(N ) 2.2. IIR Kalman Output of the IIR Kalman regressor is an estimate of the actual system output obtained by using the Kalman filter on the following state space model of the original system [3]: w(n 1) Aw(n) Bx(n) 11) d(n) Cw(n 1) v(n) 12) where C = Theta 1 0 Delta Delta Delta 0 . Since the actual parameters are unknown in Eqn. 11) we can form A and B matrices by using the estimated parameters at time n. Then, we get w(n 1) A(n)w(n) B(n)x(n) u(n) ....

C. Chui and G. Chen, Kalman Filtering with Real-Time Applications. Berlin: Springer-Verlag, 1991.

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C.K. Chui, G. Chen,"Kalman Filtering with Real-Time Applications," Springer-Verlag, 1987.

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C. K. Chui and G. Chen, Kalman filtering : with real-time applications. Information Sciences, New York: Springer-Verlag, 1991.

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C.K. Chui and G.Chen, Kalman Filtering with real-time applications, 2nd ed., Springer-Verlag, 1991.

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C. K. Chui and G. Chen. Kalman Filtering with Real Time Applications. Springer-Verlag, third edition, 1999.

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C. K. Chui and G. Chen. Kalman Filtering with Real-Time Applications. Springer, 1999.

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Chui, C. K. and G. Chen. Kalman filtering with real time applications. Springer, BerlinHeidelberg -New York, 1987.

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C.K. Chui and G. Chen, Kalman Filtering with Real-Time Applications, Springer Series in Information Sciences No. 17, 2nd edn., T.S. Huang (ed.), Springer, Berlin, 1991.

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C. K. Chui and G. Chen, Kalman Filtering with Real-Time Applications.Springer-- Verlag, 1987.

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Chui, CK and Chen, G. 1987. Kalman filtering with real time applications. Springer Verlag.

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C.K. Chui and G. Chen, Kalman Filtering: With Real Time Applications, 2nd Ed., New York: SpringerVerlag, 1991.

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C.K. Chui and G. Chen, Kalman filtering with real-time applications, Springer-Verlag, New York, 1991.

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