The Kalman filter

In the design of state observers in section 8.4.3, it was assumed that the measurements y = Cx were noise free. In practice, this is not usually the case and therefore the observed state vector x may also be contaminated with noise. 

In Sections 41.2 and 41.3 we applied a recursive procedure to estimate the model parameters of time-invariant systems. After each new measurement, the model parameters were updated. The updating procedure for time-variant systems consists of two steps. In the first step the system state j - 1) at time /), is extrapolated to the state x(y) at time by applying the system equation (eq. (41.15)) in Table 41.10). At time tj a new measurement is carried out and the result is used to 

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. 

The algorithm is initialized in the same way as for a time-invariant system. The sequence of the estimations is as follows  

Extrapolate the estimate of the system state to x(2ll) and the associated uncertainty P(211) 

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Other chemometrics methods to improve caUbration have been advanced. The method of partial least squares has been usehil in multicomponent cahbration (48—51). In this approach the concentrations are related to latent variables in the block of observed instmment responses. Thus PLS regression can solve the colinearity problem and provide all of the advantages discussed earlier. Principal components analysis coupled with multiple regression, often called Principal Component Regression (PCR), is another cahbration approach that has been compared and contrasted to PLS (52—54). Cahbration problems can also be approached using the Kalman filter as discussed (43). 

The Kalman filter single variable estimation problem... 

The Kalman filter is a eomplementary form of the Weiner filter. Let be a measurement of a parameter x and let its varianee Pa be given by... 

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. 

If the forward velocity of the ship is the state variable u, a best estimate of which is given by the Kalman filter, the gain scheduling controller can be expressed as... 

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... 

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. 

An important property of a Kalman filter is that during the measurement and estimation process, regions of the measurement range can be identified where the model is invalid. This allows us to take steps to avoid these measurements affecting the accuracy of the estimated parameters. Such a filter is called the adaptive Kalman fdter. An increasing number of applications of the Kalman filter... 

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. 

Before we introduce the Kalman filter, we reformulate the least-squares algorithm discussed in Chapter 8 in a recursive way. By way of illustration, we consider a simple straight line model which is estimated by recursive regression. Firstly, the measurement model has to be specified, which describes the relationship between the independent variable x, e.g., the concentrations of a series of standard solutions, and the dependent variable, y, the measured response. If we assume a straight line model, any response is described by ... 

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... 

The recursive property of the Kalman filter allows the detection of such model deviations, and offers the possibility of disregarding the measurements in the region where the model is invalid. This filter is the so-called adaptive Kalman filter. 

The above example illustrates the self adaptive capacity of the Kalman filter. The large interferences introduced at the wavelengths 26 and 28 10 cm have not really influenced the end result. At wavelengths 26 and 28 10 cm , the innovation is large due to the interfered. At 30 10 cm the innovation is high because the concentration estimates obtained in the foregoing step are poor. However, the observation at 30 10 cm is unaffected by which the concentration estimates are restored within the true value. In contrast, the OLS estimates obtained for the above example are inaccurate (j , = 0.148 and JCj = 0.217) demonstrating the sensitivity of OLS for model errors. 

One of the earliest applications of the Kalman filter in analytical chemistry was multicomponent analysis by UV-Vis spectrometry of time and wavelength independent concentrations, which was discussed by several authors [7-10]. Initially, the spectral range was scanned in the upward and downward mode, but later on... 

T.F. Brown and S.D. Brown, Resolution of overlapped electrochemical peaks with the use of the Kalman filter. Anal. Chem., 53 (1981) 1410-1417. 

S.C. Rutan and S.D. Brown, Pulsed photoacoustic spectroscopy and spectral deconvolution with the Kalman filter for determination of metal complexation parameters. Anal. Chem., 55 (1983) 1707-1710. 

S.C. Rutan, E. Bouveresse, K.N. Andrew, P.J. Worsfold and D.L. Massart, Correction for drift in multivariate systems using the Kalman filter. Chemom. Intell. Lab. Syst., 35 (1996) 199-211. 

C.A. Scolari and S.D. Brown, Multicomponent determination in flow-injection systems with square-wave voltammetric detection using the Kalman filter. Anal. Chim. Acta, 178 (1985) 239-246. 

The previous method supposes complete knowledge of the system and depends on the measurement quality of instruments (errors, availability), leading to severe effects on the accuracy of the on-line estimates. Therefore, a good noise filtration algorithm (like the Kalman filter or derivative) should be employed to improve the reliability of the estimated values before their use. 

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 most popular of these estimators is the Kalman Filter. Use of this estimator requires a level of knowledge in statistics and stochastic control that is... 

The proposed technique is based on an extension to time-varying systems of Wiener s optimal filtering method (l-3). The estimation of the corrected chromato gram is optimal in the sense of minimizing the estimation error variance. A test for verifying the results is proposed, which is based on a comparison between the "innovations" sequence and its corresponding expected standard deviation. The technique is tested on both synthetic and experimental examples, and compared with an available recursive algorithm based on the Kalman filter ( ). 

Note that the proposed check must be perfomed after having obtained the estimation of u. In contrast, in the Kalman filter technique (jt), the corresponding values of e and Eg may be recursively calculated along with the input estimate. 

The proposed technique is numerically "robust", and its results are comparable to those obtained through a recursive method based on the Kalman filter ( L). It should be noted that because the present technique utilizes all of the information simultaneously, the results have been compared to those of the optimal smoother estimates in (1 ), which are "better" than the true filtered estimates. 

In this work, an Extended Kalman Filter (EKF), an extension of the Kalman Filter, is designed to reconstruct the current state variables from the delayed state measurement. The advantage of the EKF is that it requires information only from the previous sampling time and allows prior knowledge of a system via process models to be used for the estimation. The algorithm of the EKF can be seen in Appendix A. 

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