Kalman filters applicability

Kalman Filter Application to Tridimensional Rigid Body Motion Parameter Estimation from a Sequence of Images... 

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. 

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... 

D. Wienke, T. Vijn and L. Buydens, Quality self-monitoring of intelligent analyzers and sensor based on an extended Kalman filter an application to graphite furnace atomic absorption spectroscopy. Anal. Chem., 66 (1994) 841-849. 

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]. 

In this chapter different aspects of data processing and reconciliation in a dynamic environment were briefly discussed. Application of the least square formulation in a recursive way was shown to lead to the classical Kalman filter formulation. A simpler situation, assuming quasi-steady-state behavior of the process, allows application of these ideas to practical problems, without the need of a complete dynamic model of the process. 

Quantile probability plots (QQ-plots) are useful data structure analysis tools originally proposed by Wilk and Gnanadesikan (1968). By means of probability plots they provide a clear summarization and palatable description of data. A variety of application instances have been shown by Gnanadesikan (1977). Durovic and Kovacevic (1995) have successfully implemented QQ-plots, combining them with some ideas from robust statistics (e.g., Huber, 1981) to make a robust Kalman filter. 

In order to perform the on-line optimization strategy, the knowledge of current state variables and/or parameters in the process models is required. Due to the fact that some of these variables cannot be known exactly or sometime can be measured with time delay, it is essential to include an on-line estimator to estimate these process variables using available process measurements as well. The sequence of an estimation and optimization procedure is known as an estimation-optimization task [6], As in several estimation techniques, an Extended Kalman Filter (EKF) has become increasingly popular because it is relatively easy to implement. It has been found that the EKF can be applied to a number of chemical process applications with great success. Once the estimate of unknown process variables is deter-... 

The literature focused on model-based FD presents a few applications of observers to chemical plants. In [10] an unknown input observer is adopted for a CSTR, while in [7] and [21] an Extended Kalman Filter is used in [9] and [28] Extended Kalman Filters are used for a distillation column and a CSTR, respectively in [45] a generalized Luenberger observer is presented in [24] a geometric approach for a class of nonlinear systems is presented and applied to a polymerization process in [38] a robust observer is used for sensor faults detection and isolation in chemical batch reactors, while in [37] the robust approach is compared with an adaptive observer for actuator fault diagnosis. 

Y. Chetouani, N. Mouhab, J.M. Cosmao, and L. Estel. Application of extended Kalman filtering to chemical reactor fault detection. Chemical Engineering Communications, 189(9) 1222-1241,2002. 

R. Li and J.H. Olson. Fault detection and diagnosis in a closed-loop nonlinear distillation process application of extended Kalman filter. Industrial Engineering Chemical Research, 30(5) 898-908, 1991. 

The Kalman filter has its origin in the need for rapid on-line curve fitting. In some situations, such as chemical kinetics, it is desirable to calculate a model whilst the reaction is taking place rather than wait until the end. In on-line applications such as process control, it may be useful to see a smoothed curve as the process is taking place, in real time, rather than later. The general philosophy is diat, as something evolves widi time, more information becomes available so the model can be refined. As each successive sample is recorded, the model improves. It is possible to predict die response from information provided at previous sample times and see how this differs from die observed response, so changing the model. 

R. Pachter, R. B. Altman, and O. Jardetzky, /. Magn. Reson., 89, 578 (1990). The Dependence of a Protein Solution Structure on the Quality of the Input NMR Data. Application of the Double-Iterated Kalman Filter Technique to Oxytocin. 

Often, we do not know all parameters of the model or we want to reduce the complexity of modeling. Therefore, in real application, the exact value of R is not known a priori. If the actual process and measurement noises are not zero-mean white noises, the residual in the unscented Kalman filter will also not be a white noise. If this happened, the Kalman filter would diverge or at best converge to a large bound. To prevent the filter from divergence, we use adaptive version of UKF as follows. 

In this paper, we present a method for the fault detection and isolation based on the residual generation. The main idea is to reconstruct the outputs of the system from the measurement using the extended Kalman filter. The estimations are compared to the values of the reference model and so, deviations are interpreted as possible faults. The reference model is simulated by the dynamic hybrid simulator, PrODHyS. The use of this method is illustrated through an application in the field of chemical process. 

The applicability of the Kalman filter requires an accurate knowledge of the response of each component and an efficient procedure for background removal. Background subtraction has recently been treated with cubic splines polynomials(5,6] as smoothing interpolators between peak valleys and this has proved to be efficient for baseline resolution particularly for very low signal-to-noise ratios [7]. 

The essential requisite to the use of the Kalman filter is the linearity of the model i.e. the overall response must be a linear combination of the component signals. Because of this constrain, the alignment of each component signal with respect to the overall signal is critical. In other words, the correct application of the Kalman filter to the resolution of overlapped responses requires coincidence of the signals on the axis of the independent variable in the scans of each pure component and of the mixture, at least in the spectral window of the composite signal chosen for the deconvolution[8,9]. 

Other methods for parameter adaptation are known. The use of a Kahnan filter is the most popular one. The basis of such a filter is the battery model shown in Fig. 8.14. The Kalman filter takes the statistical knowledge of the parameter and the measurement into account. Applications are described in Refs. [16] and [17]. 

Hashemi and Epstein (1982) linearized the set of ordinary differential equations (ODEs) resulting from the application of the method of moments on an MSMPR crystallizer model and used singular value decomposition to define controllability and observability indices. These indices aid in selecting measurements and manipulated and control variables. Myerson et al. (1987) suggested the manipulation of the feed flow rate and the crystallizer temperature according to a nonlinear optimal stochastic control scheme with a nonlinear Kalman filter for state estimation. 

Application of the adaptive Kalman filter [73,75] allows the iterative calculation of the data of a third component the other two are known. Using the law of conservation of mass, the unknown concentration can be calculated from the known concentrations of the other two. 

Equation (24) is fully equivalent to Eq. (23). This type of constrained inversion is popular (see [19]) in applications of satellite remote sensing for retrieving vertical profiles of atmospheric properties (pressure, temperature, gaseous concentrations, etc.). Equation (24) is also widely used in engineering (e.g. see textbook [30]) and other applications [33], such as assimilation of geophysical parameters [34], where Eq. (24) is known as a Kalman filter" named after the author [10] who originated the technique. 

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