Kalman Filter method

Rutan and Carr have recently compared five algorithms with respect to their abilities to deal with outliers within small data sets during calibration. They concluded that the least-median of squares approach or the zero-lag adaptive Kalman filter methods were superior because these two methods generated slope values that were less than 1% in error for small data sets with outlier points. 

There has been a significant amount of work reported on controlling composition during copolymerization reactions. The Kalman filter method is based on a linear approximation of the nonlinear process [55] but has problems with stability and convergence [56-58]. For that reason, numerous nonlinear methods have been developed. Kravaris et al. [59] used temperature tracking as another nonlinear method to control copolymer composition. Model predictive control (MPC) [60-63], as well as nonlinear MPC (NLMPC) [64-67] algorithms have been suggested for control of nonlinear systems. 

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 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. 

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. 

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 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 ( ). 

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. 

Paliwal and Basu, 1987] Paliwal, K. K. and Basu, A. (1987). A speech enhancement method based on Kalman filtering. Proc. IEEE Int. Conf. Acoust., Speech and Signal Proc, pages 177-180. 

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],... 

In addition to keeping the controllers tuned, other methods are available to improve the quality and reliability of process measurements. Overall process balance calculations and the use of predictor/estimator filters (e.g., Kalman filters) can help to improve the quality of measurements. These better-quality measurements are contributing to better control of performance, which will be discussed in more detail in the following subsections. 

The combined methods Various variants 1. Process mathematical models with distributed inputs 2. Capacity to be associated with a Kalman filter 3. Without inequality type constraints The maximum likelihood method... 

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... 

The proposed strategies for stabilization of gas-lifted oil wells are offline methods which are unable to track online dynamic changes of the system. However, system parameters such as flow rate of injected gas and also noise characteristic are not constant with respect to time. An adaptive Linear Quadratic Gaussian (LQG) approach is presented in this paper in which the state estimation is performed using an Adaptive Unscented Kalman Filter (AUKF) to deal with unknown time-varying noise statistics. State-feedback gain is adaptively calculated based on Linear Quadratic Regulator (LQR). Finally, the proposed control scheme is evaluated on a simulation case study. 

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 computational approach described here, based on the combination of the Kalman filter algorithm and iterative optimization by the simulated annealing method, was able to find the optimal alignment of the pure component peaks with respect to the shifted components in the overlapped spectra, and hence, to correctly estimate the contributions of each component in the mixture. The simulated annealing demonstrated superior ability over the other optimization methods, simplex and steepest descent, in yielding more reliable convergences at the expense of not much more computer time, at least for resolving ternary shifted overlapped spectra. 

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]. 

Resolution of overlapping electrochem. peaks with Kalman filtering TITFIT, a comprehensive program, Newton-Gauss-Marquardt method Calcn. using [H+] as independent and [B4] as dependent variable using pocket calculators... 

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. 

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