The Kalman Filter Equations

This method is frequently used for filtering, smoothing and identifying parameters in the case of a dynamic time process. It has been developed taking into account the following conditions (i) acceptance of the gaussian distribution of the disturbances and exits of the variables of the process (ii) there is a local linear dependence between the exit vector and the state vector in the mathematical model of the process. 

The Kalman filter problem. Considering the relations (3.258) and (3.259)) we can write the following discrete-time system  

The dimensions of the state vector X and of the observation vector (exit vector) are N and M respectively. This short introduction is completed by assuming that Rk is positive (Rk 0). 

The problem considered here is the estimation of the state vector X (which contains the unknown parameters) from the observations of the vectors = [yo yi.yk ] Because the collection of variables Y = (yoYi - -yk) jointly gaussian, we can estimate X by maximizing the likelihood of conditional probability distributions p(Xk/Yk), which are given by the values of conditional variables. Moreover, we can also search the estimate X, which minimizes the mean square error k = Xk — Xk. In both cases (maximum likelihood or least squares), the optimal estimate for the jointly gaussian variables is the conditional mean and the error in the estimate is the conventional covariance. 

In what follows, we will develop the conditional mean and covariance for the couple Xk and Yk. This is followed by a description of the Kalman filter and a rapid and practical method for a recursive or iterative calculation of the conditional mean and covariance for the random variable vector 

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The script file kalfild.m solves, in forward-time, the discrete solution of the Kalman filter equations, using equations (9.74), (9.75) and (9.76) in a recursive process. The MATLAB command Iqed gives the same result. 

These two relations are the basis for other important developments of the Kalman filter equations. Concerning the problem considered above, the calculation of the minimum mean square error can be carried out either ... 

Here z(y) is the minimum mean square estimate of z for a given observation of y and this is used in Rel. (3.278) by means of Rel. (3.268). This new random vector, V, has several interesting properties, which are important for the development of the Kalman filter equations ... 

The Kalman Filter Equations are here obtained from the formulation of the Kalman filter with the purpose of finding a recursive estimation procedure for the solution of a problem (estimation of state vector). Before detailing the procedure, we have to introduce other new notations. The minimum square estimate of for the given observations Yj= [yo>yi --yil defined by xy. Furthermore represents the error covariance associated with... 

Combining relations (3.287), (3.288) and (3.289) results in a set of recursive equations, which are called the Kalman filter equations ... 

To be operational, the Kalman filter equations must be completed with the starting conditions Xq/ i and CCo/ i, which correspond to k = 0 in relations (3.290) and (3.291). These conditions are obtained from the statistical starting of the initial state vector ... 

This follows from the rewriting XjX,= Xt.i Xt.i+ x, x,, X,V = XnV-i + and the application of the matrix inversion lemma. This leads to the Kalman filter equations... 

The purpose of this paper is to show the application of Kalman Filter theory to the problem of extracting the motion parameters from a sequence of images of a rigid body which is moving in a three dimensional space. The objective is to apply the Kalman Filter equations to the system of a moving object using the state variable approach. The system is described by a set of equations that specify the position and orientation of the body in space and how these coordinates are changing in time. 

The Kalman Filter equations for this system are shown in this section. The discrete Kalman Filter was developed for linear systems that have the following model ... 

The general form of the Kalman filter usually eontains a diserete model of the system together with a set of reeursive equations that eontinuously update the Kalman gain matrix K and the system eovarianee matrix P. 

Equations (9.71)-(9.76) are illustrated in Figure 9.7 whieh shows the bloek diagram of the Kalman filter. 

The plant deseribed in Example 9.8 by equations (9.185) and (9.186) is to be eontrolled by a Linear Quadratie Gaussian (LQG) eontrol seheme that eonsists of a LQ Regulator eombined with the Kalman filter designed in Example 9.8. The... 

In this chapter we discuss the principles of the Kalman filter with reference to a few examples from analytical chemistry. The discussion is divided into three parts. First, recursive regression is applied to estimate the parameters of a measurement equation without considering a systems equation. In the second part a systems equation is introduced making it necessary to extend the recursive regression to a Kalman filter, and finally the adaptive Kalman filter is discussed. In the concluding section, the features of the Kalman filter are demonstrated on a few applications. 

Equations (41.15) and (41.19) for the extrapolation and update of system states form the so-called state-space model. The solution of the state-space model has been derived by Kalman and is known as the Kalman filter. Assumptions are that the measurement noise v(j) and the system noise w(/) are random and independent, normally distributed, white and uncorrelated. This leads to the general formulation of a Kalman filter given in Table 41.10. Equations (41.15) and (41.19) account for the time dependence of the system. Eq. (41.15) is the system equation which tells us how the system behaves in time (here in j units). Equation (41.16) expresses how the uncertainty in the system state grows as a function of time (here in j units) if no observations would be made. Q(j - 1) is the variance-covariance matrix of the system noise which contains the variance of w. 

Equation (41.11) represents the (deterministic) system equation which describes how the concentrations vary in time. In order to estimate the concentrations of the two compounds as a function of time during the reaction, the absorbance of the mixture is measured as a function of wavelength and time. Let us suppose that the pure spectra (absorptivities) of the compounds A and B are known and that at a time t the spectrometer is set at a wavelength giving the absorptivities h (0- The system and measurement equations can now be solved by the Kalman filter given in Table 41.10. By way of illustration we work out a simplified example of a reaction with a true reaction rate constant equal to A , = 0.1 min and an initial concentration a , (0) = 1. The concentrations are spectrophotometrically measured every 5 minutes and at the start of the reaction after 1 minute. Each time a new measurement is performed, the last estimate of the concentration A is updated. By substituting that concentration in the system equation xff) = JC (0)exp(-A i/) we obtain an update of the reaction rate k. With this new value the concentration of A is extrapolated to the point in time that a new measurement is made. The results for three cycles of the Kalman filter are given in Table 41.11 and in Fig. 41.7. The... 

Here 4 is the target state vector at time index k and Wg contains two random variables which describe the unknown process error, which is assumed to be a Gaussian random variable with expectation zero and covariance matrix Q. In addition to the target dynamic model, a measurement equation is needed to implement the Kalman filter. This measurement equation maps the state vector t. to the measurement domain. In the next section different measurement equations are considered to handle various types of association strategies. 

The target state vector tk measured by the multilateration procedure can be considered directly as a target plot input of the association process. In this case, the input of the Kalman filter describes the same parameters that the internal state vector does. It is characteristic for the plot-to-track association procedure that the measurement equation contains directly the target state vector tk which is influenced by noise ftsk only ... 

The respective Kalman filter equations for the position correction and prediction steps can now be formulated based on equations (18) and (19), (20) or (21) accordingly for the different mentioned association schemes. Since the measurement equation is nonlinear in case of range-velocity-to-track or frequency-to-track association, the Extended Kalman filter is used for this particular application [16]. 

It should be noted that the solution of the minimization problem simplifies to the updating step of a Kalman filter. In fact, if instead of applying the matrix inversion lemma to Eq. (8.19) to produce Eq. (8.20), the inversion is performed on the estimate equation (8.18), the well-known form of the Kaman filter equations is obtained. 

When a model state is described by nonlinear equations, the extended Kalman filter has been applied using the well-known Kalman filter equations for the linearization of equations. If the state vector is enlarged with the parameter vector (P]j is used because it corresponds to the discrete version of the state model) and if it is considered to be constant or varying slowly, then it is possible to transform the problem of parameters estimation into a problem of state estimation. The P i i = P]j + njj with n]j white noise correction represents the model suggested for... 

Zf. Furthermore, it is assumed to be statistically independent to F. The essential steps of the Kalman filter algorithm are to predict and filter at each time step with the data set P// = yi, yz, , yw - When the measurements up to the nth time step T> = yi, y2,..., yn are available, the predicting procedure is applied to estimate y +i by using the conditional PDF p(X +i T>n, C), which is multi-variate Gaussian for linear systems. By using Equation (2.181), the predicted state vector at the (n + l)th time step can be estimated from the filtered state at the nth time step ... 

With this TVAR vector model, the predicting and filtering steps for the PM lo concentrations can be performed with the Kalman filter. The essential steps of the Kalman filter are to predict and filter the measured PMio concentration alternately. When the measured PMio concentrations up to the (n - 1 )th day are available, the predicting procedure is applied to give the one-step-ahead prediction of the PMio concentration. By using Equations (2.204) and (2.205), the predicted state vector on the nth day can be estimated from the filtered state vector on the (n - l)th day ... 

Corticosteroids Blue Tetrazolium Use of the Kalman filter algorithm to resolve mixtures of cortisone and hydrocortisone with a pseudo-first order rate constant ratio as low as 1.8 Comparison with logarithmic-extrapolation and proportional-equations methods... 

Generally speaking, the Kalman filter is a set of recursive equations used to regenerate estimates of the system state after every transition the system makes (i.e., at the begmning of each period, in our terms). Assume an initial state vector estimate X and an initial error covariance matrix Then, at the beginning of every period t, the estimate Xt is produced by ... 

Another successful measurement feedback/state estimation method is the EKF [37], The Kalman filter predicts the process states and makes correction to the prediction once the measurement becomes available. The general form of the EKF equation is... 

The Kalman filter algorithm is initialized using an initial state estimate vector X o i and its error covariance matrix P[oi i], both assumed known. Hereafter, it propagates by computing state estimates X i +in recursively from the following equation ... 

The Kalman filter is applied to estimate the system states, and, subsequently, the response of the top floor deiipi is estimated according to Eq. 28, where the term lePiip] is disregarded. The noise covariance matrices Q, R, and S are calculated from Eq. 9, for Cp and Rm given by the following equations ... 

The paper is divided into four sections. Section I shows a review of related papers. Section II deals with the set of equations that describes the system of the body in three dimensional motion. Section III develops the application of the Kalman Filter theory to this problem and Section IV describes the experiment and some results. Conclusions are shown at the end of the paper. 

Figure 1 shows a block diagram for the perturbed state of a robot, e, subject to both the process noise w and measurement noise y. The actually measured perturbed state is denoted as z. The Kalman filter is the best linear estimator in the sense that it produces unbiased, minimum variance estimates (Kalman and Bucy, 1961 Brown, 1983). Let (t) be the estimated perturbed state and 6eg(t) be the residual which is the difference between the true measured perturbed state, z(t), and the estimated perturbed state based on 6a (t), here denoted as (t). It has already been shown (Lewis, 1986) that Cx satisfies a differential equation which can be schematically represented by the block diagram shown in Fig. 2 where K(t) is a Kalman filter gain. K(t) is to be calculated according to the equation... 

At the first eonferenee of the International Federation of Automatie Control (IFAC), Kalman (1960) introdueed the dual eoneept of eontrollability and observability. At the same time Kalman demonstrated that when the system dynamie equations are linear and the performanee eriterion is quadratie (LQ eontrol), then the mathematieal problem has an explieit solution whieh provides an optimal eontrol law. Also Kalman and Buey (1961) developed the idea of an optimal filter (Kalman filter) whieh, when eombined with an optimal eontroller, produeed linear-quadratie-Gaussian (LQG) eontrol. 

The reeursive equations (9.74)-(9.76) that ealeulate the Kalman gain matrix and eovarianee matrix for a Kalman filter are similar to equations (9.29) and (9.30) that... 

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