Extended Kalman Filter

Grimble, M.J., Patton, R.J. and Wise, D.A. (1979) The design of dynamic ship positioning control systems using extended Kalman filtering techniques, IEEE Conference, Oceans 79, CA, San Diego. 

B.M. Quencer, Multicomponent kinetic determinations with the extended Kalman filter. Diss. Abstr. Int. B 54 (1994) 5121-5122. 

M. Gui and S.C. Rutan, Determination of initial concentration of analyte by kinetic detection of the intermediate product in consecutive first-order reactions using an extended Kalman filter. Anal. Chim. Acta, 66 (1994) 1513-1519. 

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. 

Extended Kalman filtering has been a popular method used in the literature to solve the dynamic data reconciliation problem (Muske and Edgar, 1998). As an alternative, the nonlinear dynamic data reconciliation problem with a weighted least squares objective function can be expressed as a moving horizon problem (Liebman et al., 1992), similar to that used for model predictive control discussed earlier. 

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]. 

Equations (8.11) and (8.12) are approximate expressions for propagating the estimate and the error covariance, and in the literature they are referred to as the extended Kalman filter (EKF) propagation equations (Jaswinski, 1970). Other methods for dealing with the same problem are discussed in Gelb (1974) and Anderson and Moore (1979). 

The solution of the minimization problem again simplifies to updating steps of a static Kalman filter. For the linear case, matrices A and C do not depend on x and the covariance matrix of error can be calculated in advance, without having actual measurements. When the problem is nonlinear, these matrices depend on the last available estimate of the state vector, and we have the extended Kalman filter. 

The problem of state-parameter estimation in dynamic systems is considered in terms of decoupling the estimation procedure. By using the extended Kalman filter (EKF) approach, the state-parameter estimation problem is defined and a decoupling procedure developed that has several advantages over the classical approach. 

According to the previous section, in order to deal with the state-parameter estimation problem we have to solve a nonlinear set of filtering equations. The extended Kalman filter leads to the following equations (Ursin, 1980) ... 

Ursin, B. (1980). Asymptotic convergence properties of the extended Kalman filter using filtered state estimates. IEEE Trans. Autom. Control AC-25, 1207-1211. 

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-... 

Since both the on-line dynamic optimization and the model-based control strategy rely on process models, the knowledge of current states and/or model parameters is required. However, in most industrial processes, state variables are not all measurable and some parameters are not known exactly. As a consequence, there is a need for estimating these states and parameters. In this work, two Extended Kalman Filters (EKF) are implemented. The first one is applied to predict the reactant concentration, which will be used for on-line dynamic optimization, from its delayed measurement. The other one is applied to estimate the unknown heat of reaction, which will be used for model-based controller, from the frequently available measurements of temperature. 

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. 

More in general, the use of observers for state estimation in batch reactors has been extensively investigated. In [61], an Extended Kalman Filter (EKF) is adopted... 

V.M. Becerra, P.D. Roberts, and G.W. Griffiths. Applying the extended Kalman filter to systems described by nonlinear differential-algebraic equations. Control Engineering Practice, 9 267-281,2001. 

D.I. Wilson, M. Agarwal, and D.V.T. Rippin. Experiences implementing the extended Kalman filter on an industrial batch reactor. Combustion and Flame, 22 1653-1672, 1998. 

Although there is a close relationship among the various quantitative model-based techniques, observer-based approaches have become very important and diffused, especially within the automatic control community. Luenberger observers [1,45, 53], unknown input observers [44], and Extended Kalman Filters [21] have been mostly used in fault detection and identification for chemical processes and plants. Reviews of several model-based techniques for FD can be found in [8, 13, 35, 50] and, as for the observer-based methods, in [1, 36,44],... 

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. 

Y. Huang, G.V. Reklaitis, and V. Venkatasubramanian. A heuristic extended Kalman filter based estimator for fault identification in a fluid catalytic cracking unit. Industrial Engineering Chemistry Research, 42 3361-3371, 2003. 

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. 

Sorensen and Skogestad (1994) developed control strategies for BREAD processes by repetitive simulation strategy using a simple model in SPEEDUP package. Wilson and Martinez (1997) developed EKF (Extended Kalman Filter) based composition estimator to control BREAD processes. The estimator was found to be quite robust and was able to estimate composition within acceptable accuracy, even in the face of process/model mismatches. Balasubramhanya and Doyle III... 

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... 

The structure of the augmented (extended) Kalman filter is shown in Fig. 3.87, which also presents the schematic methodology for obtaining the exit-computed vector Yk. It can be observed that coupling the process with computation procedures allows parameter identification and control of the process. 

The vector can be calculated either with the normal Kalman Filter (KF) which gives Xk for the discrete equation state (F(Xk, Uk, Vk)) or with the extended Kalman filter (KFE) which gives Pk+i in the calculation system. For this estimation, it is also necessary to obtain the state of the system Xk from the next state Xk+j. This estimation is made by block IT (inversion translator) another IT block gives... 

Using an earlier version of the dynamic model given in Section IV,6, Kiparissides et al (1980b) illustrated the use of such an extended Kalman Filter to infer JV(f), VJit), AJit), and X(f) from measurements taken only on conversion [X(f)J using UV turbidity spectra. Jo and Bankoff (197Q used these filters to track some of the moments of the MWD of PVAc in a solution polymerization process using measurements made on refractive index and viscosity. 

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. 

Keywords Fault Detection and Isolation, Extended Kalman Filter, Dynamic Hybrid Simulation, Object Differential Petri nets. Distance. 

In this paper, the proposed approach to fault detection and isolation is a model-based approach. The first part of this communication focuses on the main fundamental concepts of the simulation library PrODHyS, which allows the simulation of the system reference model of a typical process example. Then, the proposed detection approach is presented. This exploits the extended Kalman Filter, in order to generate a fault indicator. In the last part, this approach is exploited through the simulation of the monitoring of a didactic example. 

The first part eoncerns the generation of the residuals (waved pattern in the Figure 1). In order to obtain an observer of the physical system, a real-time simulation is done in parallel. So, a eomplete state of the system will be available at any time. Thus, it is based on the eomparison between the predicted behavior obtained thanks to the simulation of the reference model (values of state variables) and the real observed behavior (measurements from the process correlated thanks to the Extended Kalman Filter). The main idea is to reconstruct the outputs of the system from the measurement and to use the residuals for fault detection (Mehra and Peschon, 1971, Welch and Bishop, 1995, Simani and Fantuzzi, 2006). A description of the extended Kalman filter ean be found in (Olivier-Maget et al., 2007). Besides the residual is defined aeeording to the following equation ... 

We remind that the thresholds for the detection correspond to the model uncertainties obtained by the adjustment of the Extended Kalman filter. A default of the reactor heating energy feed is introdueed at t = 20 min. This energy feed provides a heat quantity lower than the nominal one. Figure 3 shows the detection stage. It illustrates the evolution of the residuals linked to the liquid composition of water and methanol. From t = 80 min, the values of the both residuals underline the abnormal behavior of the proeess. The diagnosis is launehed at t = 95 min. 

From a Bayesian interpretation, MHE and the extended Kalman filter assume normal or uniform distributions for the prior and the likelihood. Unfortunately, these assumptions are easily violated by nonlinear dynamic systems in which the conditional density is generally asymmetric, potentially multimodal and can vary significantly with time. 

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