Adaptive Kalman filtering

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. 

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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. 

Non-adaptive Kalman filter with interference at 26 and 28 10 3 cm-> (see Table 41.5 for the starting conditions)... 

J. Chen and S.C. Rutan, Identification and quantification of overlapped peaks in liquid chromatography with UV diode array detection using an adaptive Kalman filter. Anal. Chim. Acta, 335(1996) 1-10. 

S.C. Rutan, Adaptive Kalman filtering. Anal. Chem., 63 (1991) 1103A-1109A. 

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. 

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. 

S. C. Rutan and P. W. Carr, Anal. Chim. Acta, 215, 131 (1989). Comparison of Robust Regression Methods Based on Least-Median and Adaptive Kalman Filtering Approaches Applied to Linear Calibration Data. 

Mehra, R. K. On the identification of variances and adaptive Kalman filtering, IEEE Transactions On Automatic Control, Apr. 1970, AC-15, pp. 175-184. 

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... 

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. 

No degree of sophistication in the control system (be it adaptive control, Kalman filters, expert systems, etc.) will work if you do not know how your process works. Many people have tried to use complex controllers to overcome ignorance about the process fundamentals, and they have failed Learn how the process works before you start designing its control system. 

M.A. Beyer, W. Grote, and G. Reinig. Adaptive exact linearization control of batch polymerization reactors using a Sigma-Point Kalman filter. Journal of Process Control, 18 663-675, 2008. 

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. 

In the recent years Simulated Moving Bed (SMB) technology has become more and more attractive for complex separation tasks. To ensure the compliance with product specifications, a robust control is required. In this work a new optimization bas adaptive control strategy for the SMB is proposed A linearized reduced order model, which accounts for the periodic nature of the SMB process is used for online optimization and control purposes. Concentration measurements at the raffinate and extract outlets are used as the feedback information together with a periodic Kalman filter to remove model errors and to handle disturbances. The state estimate from the periodic Kalman filter is then used for the prediction of the outlet concentrations over a pre-defined time horizon. Predicted outlet concentrations constitute the basis for the calculation of the optimal input adjustments, which maximize the productivity and minimize the desorbent consumption subject to constraints on product purities. 

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. 

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. 

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]. 

The process of filtering the obtained conversion data with the aid of a polyma-ization model, the so-called Kalman filtering [1] or the model reference adaptive control [2], makes it possible to control the process accurately and to obtain the desired microstructuie (see Qiapter 7). 

To summarize, we propose a so-called MMSE forecast adaptive base-stock policy. This policy employs the Kalman filter technique to calculate minimum mean square error (MMSE) forecasts of future demands at the beginning of each period. A fixed safety stock 7 set at the beginning of the planning horizon, is then added to the MMSE forecast to form the target level /3t for this period. Then, the following rule is applied if the current inventory position is lower than the target level, an order is placed to fill this gap otherwise, no order is placed. The advantage of our policy is that it is intuitive and easily implementable. But, not less importantly, it can be tailored for use in information-rich supply chains, for which the characterization of optimal policies is virtually impossible. 

An example of the adaptive SoC method using extended Kalman filter was studied by Plett [11-13]. Different cell models were investigated by the author and it was shown that the model with five state variables produces the best results [12]. By comparison, the model in Figure 15.11 has four states in total, including SoC and three additional ones (voltages of the capacitors). The results of SoC estimation showed that accuracy of 1% is possible for a presented specific test procedure. Moreover, the method was able to adapt... 

Chapter 12 considers the combination of optimal control with state and parameter estimation. The separation principle is developed, which states that the design of a control problem with measurement and model uncertainty can be treated by first performing a Kalman filter estimate of the states and then developing the optimal control law based upon the estimated states. For linear regulator problems, the problem is known as the linear quadratic Gaussian (LQG) problem. The inclusion of model parameter identification results in adaptive control algorithms. 

In this paper the feed preparation process of the copper flash smelter at Outokumpu Harjavalta plant is studied from a control theoretic perspective. The aim of the study is to identify a dynamic process model from experimental data and to compare different model structures. A sampling campaign was arranged to provide data for identification. The process was modelled with an adaptive ARX model and with a blending tanks model where the process units were modelled as first-order systems. A Kalman filter was used to estimate the process state. The Kalman filter was the most efficient algorithm for predicting the process output and it has been successfully used online to the process. 

Figure 3. One hour forward prediction of arsenic content in the feed mixture ( ) analysis, (—) time-invariant ARX, ( ) adaptive ARK, (...) Kalman filter applied to the blending tanks model...

Wan EA, Van Der Merwe R (2000) The unscented Kalman filter for nonlinear estimation. Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing Communication, and Control Symposium, 153-158... 

The Intel 8251, a Multibus example, a protocol adapter example, a block transfer example, a fifth-order digital elliptic wave filter example, a Kalman filter example, the BTL310, an ADPCM, the Risc-1, the MCS6502, and the IBM System/370. 

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