Extended Kalman Filter Estimator Research Articles

Nonlinear control of a multicomponent distillation process coupled with a binary distillation model as an EKF predictor

Abstract The work is devoted to design the globally linearizing control (GLC) strategy for a multicomponent distillation process. The control system is comprised with a nonlinear transformer, a nonlinear closed-loop state estimator [extended Kalman filter (EKF)], and a linear external controller [conventional proportional integral (PI) controller]. The model of a binary distillation column has been used as a state predictor to avoid huge design complexity of the EKF estimator. The binary components are the light key and the heavy key of the multicomponent system. The proposed GLC-EKF (GLC in conjunction with EKF) control algorithm has been compared with the GLC-ROOLE [GLC coupled with reduced-order open-loop estimator (ROOLE)] and the dual-loop PI controller based on set point tracking and disturbance rejection performance. Despite huge process/predictor mismatch, the superiority of the GLC-EKF has been inspected over the GLC-ROOLE control structure.

Composition estimation of multicomponent reactive batch distillation with optimal sensor configuration

An inferential state estimation scheme based on extended Kalman filter (EKF) with optimal selection of sensor locations using principal component analysis (PCA) is presented for composition estimation in multicomponent reactive batch distillation. The properties of PCA are exploited to provide the most sensitive dynamic temperature measurement information of the process to the estimator for accurate estimation of compositions. The state estimator is supported by a simplified dynamic model of reactive batch distillation that includes component balance equations together with thermodynamic relations and reaction kinetics. The performance of the proposed scheme is evaluated by applying it for composition estimation on all trays, reboiler, reflux drum and products of a reactive batch distillation column, in which ethyl acetate is produced through an esterification reaction between acetic acid and ethanol. This quaternary system with azeotropism is highly nonlinear and typically suited for implementation of the proposed scheme. The results demonstrate that the proposed EKF estimation scheme with optimal temperature sensor configuration is effective for inferential estimation of compositions in multicomponent reactive batch distillation.

State Estimator Design for Multicomponent Batch Distillation Columns

In the control of batch distillation columns, one of the problems is the difficulty in monitoring the compositions. This problem can be handled by estimating the compositions from readily available online temperature measurements using a state estimator. In this study, a state estimator that infers the product composition in a multicomponent batch distillation column (MBDC) from the temperature measurements is designed and tested using a batch column simulation. An extended Kalman filter (EKF) is designed as the state estimator and is implemented for performance investigation on the case column with eight trays separating the mixture of cyclo-hexane, n-heptane and toluene. EKF parameters of the diagonal terms of process noise covariance matrix and those of measurement model noise covariance matrix are tuned in the range where the estimator is stable and selected basing on the least IAE score. Although NC-1 temperature measurements is sufficient considering observability criteria, using NC measurements spread through out the column homogeneously improves the performance of EKF estimator. The designed EKF estimator is successfully used in the composition—feedback inferential control of MBDC operated under variable reflux-ratio policy with an acceptable deviation of 0.5–3% from the desired purity level of the products.

Adaptive Blind Equalization of Nonlinear Channels and Chaotic Systems using Coupled EKF and RLS Estimator

A novel approach, which employs coupled extended Kalman filter (EKF) and recursive least squares (RLS) estimator is proposed for adaptive blind equalization of nonlinear channels and chaotic systems. The conventional models of nonlinear FIR Volterra channels, nonlinear IIR channels, and chaotic communication systems are cast into regression models to formulate RLS algorithm to estimate the unknown channel parameters. The state-space representation of the channels have been formulated to develop the EKF to estimate the state from which the input sequence can be recovered. Then, the EKF and RLS estimator are coupled to estimate jointly the channel parameters and the state. The stability problem of the estimator is also addressed. The proposed estimator is corroborated with simulation examples on adaptive blind equalization of nonlinear FIR Volterra channels, nonlinear IIR channels, and chaotic communication systems. Simulation results show that the proposed estimator is effective in recovering the input sequence as well as the channel parameters blindly from the channel output measurement.

Adaptive-Weighting Input-Estimation Approach to Nonlinear Inverse Heat-Conduction Problems

The inverse heat-conduction problem involves surface-heat-flux or heat-source estimation that requires only the temperatures measured at an insulated wall. This problem becomes nonlinear if the thermal properties are temperature-dependent. Innovative adaptive-weighting input-estimation inverse methodology for estimating a time-varying unknown heat source from a nonlinear thermal system is presented. This algorithm includes the extended Kalman filter (EKF) and the recursive least-square estimator. EKF recursively estimates the interior temperature of a body under a system involving noisy measurement and modeling errors. During the EKF estimation procedure, an important regression equation between the observable bias residual innovation and the thermal unknown is provided. Based on this regression model, a recursive least-square estimator weighted by the adaptive weighting factor is proposed to estimate these unknowns, defined as the input. The Kalman tuning parameter is used first to analyze the interactive relationship between measurement noise and modeling-error variance. The superior capabilities of the proposed algorithm are demonstrated through two simulated examples with different types of time-varying heat sources as the unknown inputs. Nomenclature

On-line states and parameter identification of acetone–butanol–ethanol fermentation process

In the present work, the use of extended Kalman filter (EKF) to infer the outlet compositions and growth rate of the acetone–butanol–ethanol fermentation process from outlet gas CO 2 measurement has been discussed. In order to tune the filter ( Q matrix), different methods consist of trial and error; Loan, Mont Carlo simulations and continuous case have been investigated for process noise covariance matrix. The reliability of the developed estimator is discussed with respect to disturbance on CO 2 concentration. The performance of the estimator has been evaluated by comparison with other filters such as Wiener and linearized Kalman filter as well as simulation and experimental results. The results show, EKF estimator provides good agreement with the composition values measured off-line while Q is adjusting by Loan method.

EKF and ANFIS Estimator Design in Multicomponent Batch Distillation Columns

In the control of batch distillation columns, one of the problems is the difficulty of monitoring the compositions. This problem can be handled by estimating the compositions from readily available online temperature measurements using a state estimator. In this study, an extended Kalman Filter (EKF) and an Adaptive Neuro-Fuzzy Inference System (ANFIS) state estimators that infer the product composition in a multi component batch distillation column (MBDC) from the temperature measurements are designed and tested using a batch column simulation. The designed EKF and ANFIS estimators are successfully used in the composition – feedback inferential control of MBDC operated under variable reflux-ratio policy with an acceptable deviation from the desired purity level of the products

Sensorless IPMSM Drive with EKF Estimation of Speed and Rotor Position

In this paper, a new sensorless Interior Permanent Magnet Synchronous Motor (IPMSM) drive method with Extended Kalman Filtering (EKF) estimation of speed and rotor position is proposed. This method is on the basis of the sensorless Surface Permanent Magnet Synchronous Motor (SPMSM) drive [1] that can be applied to motors with no salient pole and is developed so as to be able to be applied to motors with salient pole such as IPMSM. Noise and vibration in mechanical plant becomes larger if rotor position estimation error occurs. Furthermore, the rotor position estimation error causes noise and vibration. The feature of the method is that rotor position estimation error does not occur even if motor current or motor voltage change. Therefore, the method can be applied with arbitrary motor voltage waveform and, consequently, the method is suitable for sensorless IPMSM drive. Numerical simulations are illustrated to verify the effect of the method.

STAP for GPS receiver synchronization

A space-time adaptive processing (STAP) algorithm for delay tracking and acquisition of the GPS signature sequence with interference rejection capability is developed. The interference can consist of both broadband and narrowband jammers, and is mitigated in two steps. The narrowband jammers are modelled as vector autoregressive (VAR) processes and rejected by temporal whitening. The spatial nulling is implicitly achieved by estimating a sample covariance matrix and feeding its inverse into the extended Kalman filter (EKF). The EKF estimates of the code delay and the fading channel are used for a t-test for acquisition detection. Computer simulations demonstrate robust performance of the algorithm in severe jamming, and also show that the algorithm outperforms the conventional delay-locked loop (DLL).

Feedback control of the Rayleigh–Bénard convection by means of mode reduction

AbstractAn optimal feedback control is synthesized for the Rayleigh–Bénard convection by means of empirical reduction of modes. The Boussinesq equation is reduced to a minimal set of ordinary differential equations by using the Karhunen–Loève Galerkin procedure. The state feedback control synthesis, that drives the intensity of convection to a preset trajectory by adjusting heat flux at the bottom of the system, is constructed using this low‐dimensional dynamic model by first performing an extended Kalman filter estimate of the velocity and temperature fields and then developing the optimal feedback law by means of the linear regulator theory. The present technique allows for the practical implementation of modern control concepts to the natural convection and is found to yield satisfactory results. Copyright © 2002 John Wiley & Sons, Ltd.

3-D head pose recovery for interactive virtual reality avatars

This paper discusses a tracking method allowing real-time recovery of the three-dimensional (3-D) position and orientation of a moving head. The described method uses a wireframe model of the head, a feature-based matching algorithm, and an extended Kalman filter estimator. The resulting motion tracking system works in a realistic environment without makeup on the face, with uncalibrated camera, and unknown lighting conditions and background.

Feedback control of natural convection

An applicable method is developed for the identification and feedback control of natural convection. The Boussinesq equation is reduced to a small set of ordinary differential equations by means of the Karhunen–Loève Galerkin procedure [Int. J. Heat Mass Transfer 39 (1996) 3311]. Based on this low-dimensional dynamic model, a feedback control synthesis is constructed by first performing an extended Kalman filter estimate of the velocity and temperature fields to treat the measurement errors and then developing the optimal feedback law by means of the linear quadratic regulator theory. The present method allows for the practical implementation of modern control concepts to many flow systems including natural convection.

Estimation of systemic vascular bed parameters for artificial heart control

An extended Kalman filter estimator for the identification of systemic circulation model parameters during cardiac ejection and cardiac filling is described. The estimator has been developed for use in the control of a cardiac ventricular assist device. A lumped element circuit with a time-varying capacitor was used to represent the systemic circulation and the left ventricle. Since the haemodynamic variables that are measurable in patients with impaired cardiac function vary dramatically as the patients move through different levels of care, the estimator was designed so that it can be used with different sets of blood pressure and flow measurements. Preliminary evaluation of the performance of the estimator using data from a computer simulation and from a patient during open-heart surgery is presented. The robustness of the estimator to variations in parameter initialization is also described.

A digital DS spread-spectrum receiver with joint channel and Doppler shift estimation

A digital spread-spectrum receiver design is presented for communication over multipath channels with severe Doppler shifts. The characteristics of the underwater channel relevant to spread-spectrum system design are discussed, and a channel model for short-range communications (less than 10 km) is defined. The receiver considered uses a digital coherent RAKE combiner, coupled with an extended Kalman filter (EKF)-based estimator for channel parameters and pseudonoise code delay. Receiver performance is evaluated by computing average bit-error rate (BER) versus iterations of the EKF joint estimator, using both fixed and time-varying channels. It is shown that the BER obtained using the EKF joint estimator closely tracks the optimum BER obtained when the channel, delay, and Doppler parameters are known exactly. Finally, the Cramer-Rao lower bound for time-invariant joint channel, delay, and Doppler estimation is derived, and compared with the ensemble averaged mean-squared error of the EKF estimator. >

Application of Identification to an scp Fermentation Process

A linear estimator to identify the parameters and state of a continuous single cell protein fermentation process is derived, using a dynamic mathematical model based on process stoichiometric properties. Observability effects on the process are discussed. Linear estimation results are compared with results for the extended Kalman filter (EKF). Linear smoothing filters are derived for estimating the nonlinear rate of substrate and oxygen consumption. The linear and EKF estimators tracked state variations equally well. The advantages of the linear estimator are better convergence properties and lower implementation requirements.

Identification of photosynthesis-light models for aquatic systems I. Theory and simulations

The paper describes a general procedure which uses a noisy series of diel oxygen concentration measurements to test the adequacy of non-linear photosynthesis-light models containing uncertain parameters. The extended Kalman filter (EKF), used with an oxygen mass belance equation and observed stream oxygen concentrations, provides serial estimates of the state (oxygen concentration) and the model parameters. Objective criteria of model adequacy are based on the statistical behavior of the differences between measurements and estimates of the state (the innovations sequence). While the expected behavior of the innovations can be derived from theoretical considerations for linear systems, no such theory exists for non-linear systems. Simulated data were used to demonstrate thatthe theoretical criteria are applicable to non-linear photosynthesis-light models. Five mass balance equations containing photosynthesis-light relationships were each used with five different levels of system and parameter noise to produce 25 time series of oxygen concentration. The procedure was applied repetitively in attempts to identify adequate models for each of these data sets. In 1250 tests of model adequacy there were no Type I errors (failure to accept the correct model as adequate) and the incidence of Type II errors (accepting as adequate an incorrect model) was less than 3%. These results show the EKF estimation procedure to be reliable and robust and suggest that it should be useful in examining the photosynthesis-light ( P-I) relationship in streams.

Modelling algal behaviour in the river thames

Forecasting the movement and growth of algae in river systems as particularly important for operational managers responsible for the distribution and supply of potable water. Algae affect the taste and smell of water and pose considerable filtration problems at water treatment plants. In a collaborative study with the Thames Water Authority, algal models have been developed for the River Thames. The non-linear processes controlling algal growth are examined using a generalized sensitivity analysis technique and the dominant parameters controlling system behaviour are identified. The extended Kalman filter (EKF) is then used to estimate these important parameters. The technique of using generalized sensitivity analysis prior to EKF estimation is suggested as a pragmatic approach to the problem of identifying the subset of physically, chemically or biologically meaningful parameters controlling system behaviour in mechanistic models.

A sequential failure detection approach and the identification of failure parameters

This paper is concerned with the problem of a failure diagnosis for a discrete-time system with parametric failure, in which the occurrence time and mode of parametric failure cannot be estimated in advance. The failure diagnosis system which is proposed consists of three parts : (i) a normal mode filter, (ii) a detector for anomaly states, and (iii) an adaptive extended Kalman filter. The normal mode filter is called the optimal Kalman filter and transports the information of its innovation sequence to the detector. The detector which is based on the SPRT approach detects anomaly states affected by the parametric failure. The adaptive extended Kalman filter estimates simultaneously system parameters and the states under the failure mode. The adaptive procedure is directed by increasing the calculated covariance on the basis of hypothesis tests for the estimation errors of unknown parameters. Numerical results for a simple plant model illustrate the effectiveness of the proposed failure diagnosis system.