Frank L. Lewis. Optimal Estimation. John Wiley & Sons, Inc., New York, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....can be greatly simpli fied. In this case, only the conditional mean k = E [xklY ] and covariance Px need to be maintained in order to recursively calculate the posterior density p(x lye) which, under these Gaussian assumptions, is itself a Gaussian distribution . It is straightforward to show [2, 10, 23] that this leads to the recursive estimation . prediction of xk) C [y (prediction of y) 6) C. yk ) 7) Px = P ]CPy]C[ 8) While this is a linear recursion, we have not assumed linearity of the model. The optimal terms in this recursion are given by E [f(Xk l, ....

Frank L. Lewis. Optimal Estimation. John Wiley & Sons, Inc., New York, 1986.

....read the derivation of the method in section 3.3 or skip directly to section 3.4 for the explanation of the actual algorithm. Section 3.5 describes an important pitfall in the method and how to compensate for that weakness. 3.2. Definitions The estimator is based upon a discrete Kalman filter [5] and is a modification of the estimator used in [1] Each discrete step i is one millisecond of time, so the estimator updates at 1 kHz. The estimator maintains a state vector x which is a 6 by 1 matrix: xi = rc Pc hc rs Ps hs] T where r, p, and h represent the roll, pitch and heading values at ....

F. Lewis, Optimal Estimation, John Wiley and Sons, USA, 1986.

....conditional density. For Gaussian distributions, the MAP estimate also corresponds to the minimum mean squared error (MMSE) estimator. More generally, as long as the density is unimodal and symmetric around the mean, the MAP estimate provides the Bayes estimate for a broad class of loss functions [17]. Taking MAP as the starting point, allows dual estimation approaches to be divided into two basic classes. The rst, referred to herein as joint estimation methods, attempt to maximize x directly. We can write this optimization problem explicitly as 1 ; w) arg max 1 ;w ....

Frank L. Lewis. Optimal Estimation. John Wiley & Sons, Inc., New York, 1986.

....each application one therefore has to pick the estimator 1 1 Introduction 2 which is found to best trade ooe various properties such as estimation accuracy, ease of implementation, numerical robustness, and computational burden. Up to now the extended Kalman lter (EKF) GKN 74] May82] Lew86] has unquestionably been the dominating state estimation technique. The EKF is based on rst order Taylor approximations of state transition and observation equations about the estimated state trajectory. Application of the lter is therefore contingent upon the assumption that the required ....

....k = g(x k ; w k ) 73) v k and w k are assumed i.i.d. and independent of current and past states, v k (v k ; Q(k) w k ( w k ; R(k) The commonly used state estimation principle for nonlinear systems is brieAEy outlined in the following. In depth treatments of the topic can be found in [Lew86] GKN 74] May82] Ideally, we would like to determine the a priori state and covariance estimates like in the Kalman lter. That is, as the conditional expectations x k = E[x k jY k Gamma1 ] 74) P (k) E Theta (x k Gamma x k ) x k Gamma x k ) T jY k Gamma1 ; 75) where ....

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F. L. Lewis. Optimal Estimation. John Wiley & Sons, New York, NY, 1986.

....parameter, is known to cause resonance problems in closed loop. Here y k and v k are position measurementandvelocity estimate at the kth instance, and T is the sampling period. 1 Alpha beta trackers have also been proposed for optimal estimation of velocity from noisy position measurements (Lewis 1986). An alpha beta tracker is a specialized form of double integrator Kalman filter. In (Glad and Ljung 1984) a Kalman filtering approach has been proposed for the velocity estimation based on position measurements obtained at irregular time instants. Further investigation has been done along this ....

Lewis, F. L. (1986). Optimal Estimation. John Wiley and Sons.

....Kalman ltering (EKF) method. The use of di erent predictive models on short term windows to account for the nonstationarity of speech will be discussed later, in Section 4. 8 Handbook of Neural Networks for Speech Processing Given a linear model f( the well known Kalman lter algorithm [31] optimally combines noisy observations y k at each time step with predictions x k (based on previous observations) to produce the linear least squares estimate of the speech x k . In the linear case with Gaussian statistics, the estimates are the minimum mean square estimates. With no prior ....

....k M 2 3 7 7 7 5 C = 1 0 0 ; B = C T : 8) Because the neural network model is nonlinear, the Kalman lter cannot be applied directly, but requires a linearization of the nonlinear model at the each time step. The resulting algorithm is known as the extended Kalman lter (EKF) [31], and e ectively approximates the nonlinear function with a time varying linear one. Letting 2 v;k and 2 n;k represent the variances of the process noise v k and observation noise n k , respectively, the EKF algorithm is as follows: x k = F [ x k 1 ; w k 1 ] 9) P x;k = AP x;k 1 ....

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F. L. Lewis. Optimal Estimation. John Wiley & Sons, Inc., New York, 1986.

....Section 4, we present results of using the UKF for the different areas of nonlinear estimation. 2. The EKF and its Flaws Consider the basic state space estimation framework as in Equations 1 and 2. Given the noisy observation y k , a recursive estimation for x k can be expressed in the form (see [6]) xk = prediction of xk ) Kk [yk (prediction of yk ) 8) This recursion provides the optimal minimum mean squared error (MMSE) estimate for x k assuming the prior estimate x k 1 and current observation y k are Gaussian RandomVariables (GRV) We need not assume linearity of the model. ....

....and then determining the posterior covariance matrices analytically for the linear system. In other words, in the EKF the state distribution is approximated by a GRV which is then propagated analytically through the first order linearization of the nonlinear system. The readers are referred to [6] for the explicit equations. As such, the EKF can be viewed as providing first order approximations to the optimal terms 2 . These approximations, however, can introduce large errors in the true posterior mean and covariance of the transformed (Gaussian) random variable, which may lead to ....

F. L. Lewis. Optimal Estimation. John Wiley & Sons, Inc., New York, 1986.

.... threshold based adaptive decision strategies similar to those used in LSB that can respond to a policy maker s estimate of any trend in damages based on annual observations of the noisy time series Dt ( We calculate this estimate, Dt est ( using a linear, discretetime Kalman filter (Lewis, 1986) a Bayesian estimator that rapidly detects any statistically significant trend in the damage time series. As shown in Figure 4, our adaptive strategies begin with a pre determined abatement rate 1R 1 and can switch to a second period abatement rate 1R 2 in the year, t trig , when either the ....

Lewis, F. L.: 1986, Optimal Estimation, John Wiley & Sons.

.... Since there is one to one correspondence between the observations and the innovation process as stated in the previous section, x(t) must be linearly related to the innovation j(t) associated with the observation z(t) Again for a Gaussian time varying process, the optimal MMSE estimator is linear [53]. Thus, we can express x(tjt) the MMSE estimate, of the state vector x(t) as linear combinations of the innovation sequence, that is, x(tjt) t X k=1 B t (k)j(k) 2.38) CHAPTER 2. BACKGROUND 29 where fB t (k)g is a set of N x dimensional vectors to be determined. According to the ....

....a correction term fj(t)g to the product of the CHAPTER 2. BACKGROUND 30 previous state estimate x(t Gamma 1jt Gamma 1) and the state transition matrix F. Thus is refered to as Kalman gain. 2.5. 3 Kalman Gain In this section, we express the Kalman gain in a convenient form for computation[45, 50, 53]. We rewrite the expression for the Kalman gain, by substituting for x(t Gamma 1) and the innovation j(t) Hffl x (tjt Gamma 1) v(t) as, Efx(t)ffl x (tjt Gamma 1)gHr Gamma1 j (2.46) As ffl x (tjt Gamma 1) and x(t Gamma 1) are orthogonal, Efffl x (tjt Gamma 1) T ffl x (tjt ....

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F.L.Lewis. Optimal Estimation. Wiley-Interscience, 1986.

....the model error. Conversely, if the residual covariance is lower than the known covariance, then W should be increased so that less unmodeled dynamics are added to the assumed system model. The sample measurement covariance can be determined from a recursive relationship given by (see Ref. [11]) ## # RR k k k eeeeR kk kkkk T k = # # # # # # 1 11 1 11 #### (18a) ee k ee kk kk = 11 1 1 ## (18b) The covariance constraint is met when # RR k , after the filter has converged (i.e. the estimate reaches a stochastic steady state so that the effects of ....

Lewis, F.L., Optimal Estimation, John Wiley & Sons, NY, 1986.

....model for the time varying visual field properly handled. Normally, in an exact implementation of a Kalman filter such uncertainty, which is expressed via the incorporation of process noise , can serve the purpose of making the filter stable against modeling errors and responsive to new data [4, 16, 45]. Certainly, a filtering algorithm based on a more systematic approximation of Kalman filter is desirable. This thesis offers such algorithms. Specifically, we first formulate a general Kalman filtering solution for multi frame visual reconstrucition problems and then design various approximation ....

....of algorithms in the subsequent chapters, we briefly present here some fundamental facts relevant to our visual reconstruction problems as well as establish some notation that will be used frequently in the sequel. For derivations and other details, introductory texts on estimation theory (e.g. [16, 45, 74]) should be consulted. The maximum likelihood (ML) estimation problem deals with estimation of an unknown vector x based on an observed sample of a random vector y whose distribution is parameterized by x. Specifically, let P y ( Deltajx) be the probability density function for such a ....

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F. L. Lewis. Optimal Estimation. John Wiley & Sons, New York, 1986.

....where E f Deltag represents expected value and x k = x k Gamma x k is the estimation error. Let the Kalman filter have the following form x Gamma k 1 = A k x k B k u k (2a) x k 1 = x Gamma k 1 K k 1 (y k 1 Gamma C k 1 x Gamma k 1 ) 2b) where x 0 = x 0 . It can be shown [1, 2] that with this initial condition, the estimation error has zero mean. The motivation for Equations (2) is as follows. Equation (2a) predicts the state estimate x Gamma k 1 based on the plant dynamics. The predicted estimate is then corrected by Equation (2b) based on the difference between ....

....corrected by Equation (2b) based on the difference between actual measurement y k 1 and the predicted measurement C k 1 x k 1 . The amount of correction is determined by the Kalman gain matrix K k 1 , which is chosen such that the estimation error covariance is minimized. It can be shown (cf. [1, 2]) that the optimal K k 1 is given by K k 1 = P k 1 C T k 1 R Gamma1 k 1 = P Gamma k 1 C T k 1 (C k 1 P Gamma k 1 C T k 1 R k 1 ) Gamma1 ; 3) where P Gamma k 1 is the covariance of predicted estimate x Gamma k 1 . We see from Equation (3) that the Kalman gain matrix K k 1 ....

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F.L. Lewis. Optimal Estimation. John Wiley and Sons, Inc., New York, 1986.

.... the open loop system, and it becomes I(u) E[J(u) For this motion control problem (event move robot) the algorithms cost is identified with the optimal value of I: C = I(u ) e(0) T P e(0) N X k=1 tr(P DCvD T ) 14) where P is the solution of a discrete algebraic Ricatti equation (Lewis, 1986), and N the number of samples in the trajectory. A lower bound for the Reliability can be obtained based on a method described by McInroy and Saridis (1994) when the specifications are quadratic in the tracking error e(kTs ) e(kTs ) T Qse(kTs ) ffl; k = 1; N; Qs 0 (15) where Qs is ....

Lewis, F. L. (1986). Optimal Estimation. John Wiley and Sons.

....parallelism is not as straightforward as detecting local point like features. It would be possible to use lines in the reconstruction process and to help segment the generated surfaces but that subject is not pursued here. 4. 3 3 D Reconstruction Method Incremental optimal (Kalman) filtering [Lewis, 1986] is used to reconstruct 3 D feature positions from their 2 D positions in the image sequence. This is fairly straightforward because the tracking arm provides a good estimate for the camera position for each image. For each feature, its position in each successive camera image defines a ray from ....

....3 Theta3 (z current Gamma x old ) 4.21) where P old 3 Theta3 and Pnew 3 Theta3 are the old and new (updated) covariance matrices for a feature, respectively, and ( Gamma1 represents matrix inversion. The common factor K 3 Theta3 in equations 4.20 and 4. 21 is referred to as the Kalman gain [Lewis, 1986, page 70] x old and xnew are the old and updated feature position estimates, respectively. z is the estimated mean of the measurement seen by the camera. This value is estimated to be at the same depth from the camera as the previous estimate of a feature s position but along the current ray. ....

Lewis, F. L. (1986). Optimal Estimation (with an introduction to stochastic control theory). John Wiley & Sons.

....a short version of chapter 6 of [1] That chapter is included with the CD ROM version of this paper. 3 Approach The frequency domain analysis draws upon linear systems theory, spectral analysis, and the Fourier and Z transforms. This section provides a brief overview; for details please see [3] [9] [12] 14] 15] 16] Functions and signals are often defined in the time domain. A function f(t) returns its value based upon the time t. However, it is possible to represent the same function in the frequency domain with a different set of basis functions. Converting representations is performed ....

....because it has an efficient recursive formulation, suitable for computer implementation. Efficiency is important because the filter must operate in real time to be of any use to the head motion prediction problem. This section outlines the basic operation of the filter; for details please see [3] [9]. Since the inputs are assumed to arrive at discrete, evenly spaced intervals, the type of filter used is the Discrete Kalman filter. Figure 4 shows the high level operation of the Kalman filter. The Kalman filter maintains two matrices, X and P. X is an N by 1 matrix that holds the state ....

Lewis, Frank L. Optimal Estimation. John Wiley & Sons, 1986.

.... system, and it becomes I(u) E[J(u) For this motion control problem (event move robot) we identify the algorithms cost with the optimal value of I: C = I(u ) e(0) T P e(0) N X k=1 tr(P DC v D T ) 10) where P is the solution of a discrete algebraic Ricatti equation (Lewis, 1986) [6], and N the number of samples in the trajectory. A lower bound for the Reliability can be obtained based on a method described by McInroy and Saridis (1990) 11] when the specifications are quadratic in the tracking error e(kT s ) e(kT s ) T Q s e(kT s ) ffl; k = 1; N; Q s 0 (11) ....

F. L. Lewis. Optimal Estimation. John Wiley and Sons, 1986.

....variances oe 2 v and oe 2 n , respectively. In practice, the noise variances must be estimated directly from the noisy data (see Section 3.0.2) 2.1. Extended Kalman Filter For a linear model with known parameters, the Kalman filter (KF) algorithm can be readily used to estimate the states [8]. At each time step, the filter computes the linear least squares estimate x(k) and prediction x Gamma (k) as well as their error covariances, Px (k) and P Gamma x (k) In the linear case with Gaussian statistics, the estimates are the minimum mean square estimates. With no prior ....

....covariances, Px (k) and P Gamma x (k) In the linear case with Gaussian statistics, the estimates are the minimum mean square estimates. With no prior informationon x, they reduce to the maximum likelihood estimates. A smoothing version of the KF (known as the forwardbackward Kalman filter (FB) [8]) is achieved by combining the standard KF with a backwards information filter. Effectively, an inverse covariance is propagated backwards in time to form backwards state estimates that are combined with the forward state estimates. In the case of a known linear model, this produces the maximum ....

F. L. Lewis. Optimal Estimation. John Wiley & Sons, Inc., New York, 1986.

....and oe 2 n , respectively. Methods for estimating the noise variances directly from the noisy data are described later in this paper. Extended Kalman Filter State Estimation For a linear model with known parameters, the Kalman filter (KF) algorithm can be readily used to estimate the states [9]. At each time step, the filter computes the linear least squares estimate x(k) and prediction x Gamma (k) as well as their error covariances, PX (k) and P Gamma X (k) In the linear case with Gaussian statistics, the estimates are the minimum mean square estimates. With no prior ....

F. Lewis. Optimal Estimation. John Wiley & Sons, Inc. New York, 1986.

....and C = B T . If the model is linear, then f(x(k) takes the form w T x(k) and F [x(k) can be written as Ax(k) where A is in controllable canonical form. If the model is linear, and the parameters w are known, the Kalman filter (KF) algorithm can be readily used to estimate the states (see Lewis, 1986). At each time step, the filter computes the linear least squares estimate x(k) and prediction x Gamma (k) as well as their error covariances, P x (k) and P Gamma x (k) In the linear case with Gaussian statistics, the estimates are the minimum mean square estimates. With no prior ....

....is a batch method that gives a smoothed estimate given all data. Hence, only the estimates x(N ) at the final time step will match. An exact equivalence for all time is achieved by combining the Kalman filter with a backwards information filter to produce a forward backward (FB) smoothing filter (Lewis, 1986). 2 Effectively, an inverse covariance is propagated backwards in time to form backwards state estimates that are combined with the forward estimates. When the data set is large, the FB filter offers significant computational advantages over the batch form. When the model is nonlinear, the ....

F. Lewis. Optimal Estimation John Wiley & Sons, Inc. New York. 1986.

....this filter requires tuning effort for each operational condition, and also causes severe resonance problems when the velocity dependent control algorithms (e.g. a PD control) are used. Alpha beta trackers have also been proposed for optimal estimation of velocity from noisy position information [17]. However, an alpha beta tracker is a specialized form of double integrator Kalman filter. In [11] a Kalman filtering approach has been proposed for the velocity estimation based on position measurements obtained at irregular time instants. Here the problem is formulated as a state estimation ....

F.L. Lewis, "Optimal Estimation," John Wiley & Sons, 1986.

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Frank L. Lewis. Optimal Estimation. John Wiley & Sons, Inc., New York, 1986.

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F. L. Lewis. Optimal Estimation. John Wiley & Sons, Inc., New York, 1986.

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Frank L. Lewis. Optimal Estimation. John Wiley & Sons, Inc., New York, 1986.

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F. L. Lewis, Optimal Estimation. John Wiley, New York, 1986.

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F. Lewis. Optimal Estimation John Wiley & Sons, Inc. New York. 1986.

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