Vinay A. Bavdekar

Constrained dual ensemble Kalman filter for state and parameter estimation

The performance of a state estimator is dependent on the accuracy of the process model used. Since processes undergo various changes as time progresses, it is essential to adapt the model parameters to reflect the change in process conditions and maintain the accuracy of the model predictions. In several cases, it may be necessary to account for the physical bounds on the states and parameters while computing their estimates. In this work, a constrained dual ensemble Kalman filter (C-EnKF) for state and parameter estimation is proposed to construct the state and parameter estimates that are consistent with their physical limits. The efficacy of the proposed dual C-EnKF is demonstrated on two simulation case studies. The results obtained demonstrate that the proposed approach tracks parameter changes with reasonable accuracy, while maintaining the state and parameter estimates within their physical limits.

Maximum likelihood estimation of noise covariance matrices for state estimation of autonomous hybrid systems

Abstract A critical aspect of developing Bayesian state estimators for hybrid systems, that involve a combination of continuous and discrete state variables, is to have a reasonably accurate characterization of the stochastic disturbances affecting their dynamics. Recently, Bavdekar et al. (2011) have proposed a maximum likelihood (ML) based framework for estimation of the noise covariance matrices from operating input–output data when an EKF is used for state estimation. In this work, the ML framework is extended to estimation of the noise covariance matrices associated with autonomous hybrid systems, and, to a wider class of recursive Bayesian filters. Under the assumption that the innovations generated by an estimator form a white noise sequence, the proposed ML framework computes the noise covariance matrices such that they maximize the log-likelihood function of the estimator innovations. The efficacy of the proposed scheme is demonstrated through the simulation and experimental studies on the benchmark three-tank system.

Evaluation of Adaptive Extended Kalman Filter Algorithms for State Estimation in Presence of Model-Plant Mismatch

Abstract The occurrence of model-plant mismatch is a common problem in dynamic model based applications such as state estimation. The use of an inaccurate model results in biased estimates of the states. Hence, conventional state estimation algorithms are modified in various ways to compensate for model-plant mismatch. In this work, the performance of four adaptive state estimation algorithms is compared in the presence of a model plant mismatch arising due to random drifts in parameter values. The comparison is carried out through simulations on a benchmark non-isothermal CSTR problem. Simulation results demonstrate that online re-identification of the parameters susceptible to drift or change is the most effective approach to minimize the effect of model-plant mismatch on the state estimates.

Identification of Noise Covariances for State Estimation of Autonomous Hybrid Systems

Abstract The occurrence of dynamic systems that involve both continuous and discrete state variables is becoming increasingly common in process industries, especially those that manufacture multiple products. To operate such processes efficiently, it is essential to monitor and tightly control the state variables associated with them. Recently, Prakash et al. (2010b) have proposed the use of derivative free state estimators, namely the unscented Kalman filter and ensemble Kalman filter, for estimating continuous and discrete states associated with autonomous hybrid systems. A critical aspect of developing such Bayesian state estimators is to have a reasonably accurate characterisation of such unmeasured disturbances and noise. In this work, it is proposed to identify noise covariances associated with autonomous hybrid systems from the operating data. The state estimators used for this purpose are UKF and EnKF. The problem of estimating the noise covariance matrices is formulated as a constrained optimisation problem, in which a suitable objective function of the innovation sequence is minimised. The efficacy of the proposed covariance estimation scheme is demonstrated by simulating the benchmark three-tank system from the literature.

An Ensemble Kalman Filter for Systems Governed by Differential Algebraic Equations (DAEs)

Abstract Many process systems can be realistically described by a set of nonlinear differential algebraic equations (DAEs). To carry out state estimation of these systems, the conventional sequential Bayesian estimation schemes have to be modified to accommodate nonlinear algebraic constraints. In this work, we present a modified formulation of the Ensemble Kalman Filter for state estimation of systems described by DAEs. The proposed formulation can utilize measurements obtained either from, the differential or algebraic states. The efficacy of the proposed EnKF formulation is demonstrated by simulating two benchmark examples from the literature. The simulation results indicate that the proposed EnKF algorithm can efficiently track both the differential and the algebraic states with reasonable accuracy.

Moving Window Ensemble Kalman Filter for Delayed and Multi-Rate Measurements

Abstract Irregularly sampled measurements with variable time delays are a common scenario in various applications in the process industry. The existing ensemble Kalman filter algorithms do not take into account the presence of such delayed measurements to generate improved estimates of the states. In this work, a moving window ensemble Kalman filter (EnKF) formulation is proposed to make use of the delayed and non-uniformly sampled measurements to generate better estimates of the states. In practice, it may also become necessary to account for the physical bounds on the states. A constrained version of the moving window EnKF is also developed to yield state estimates that are consistent with the bounds. The efficacy of the unconstrained moving window EnKF is demonstrated by application on a simulated continuous fermenter problem, while the efficacy of the constrained moving window EnKF is demonstrated by simulation of a benchmark gas-phase batch reactor system.

Identification of Noise Covariances for State Estimation of a Continuous Fermenter using Extended EM Algorithm

Abstract Despite developments in sensor technology, monitoring a biological process using regular sensor measurements is often difficult. Development of Bayesian state observers, such as extended Kalman filter(EKF), is an attractive alternative for soft-sensing of such complex systems. The performance of EKF is dependent on the accurate characterisation of the uncertainties in the state dynamics and in the measurements. In this work, an extended expectation maximisation (EM) algorithm is developed for estimation of the state and measurement noise covariances for the EKF using irregularly sampled multi-rate measurements. The efficacy of the proposed approach is demonstrated on a benchmark continuous fermenter system. The simulation results reveal that the proposed approach generates fairly accurate estimates of the noise covariances.

SCALING AND DISCRIMINATION ISSUES IN MONITORING, FAULT DETECTION AND DIAGNOSIS

Abstract The performance of data based monitoring algorithms is crucially dependent on the ability to discriminate between patterns of normal and fault data. In this paper, we analyze discriminatory properties of PCA, FDA and nonlinear scaled version of PCA algorithm proposed by (Ding et al. , 2002). We demonstrate improved discriminatory performance of the nonlinearly scaled PCA over traditional algorithms like PCA and FDA. The scaling and discrimination issues have been analyzed for each of the above algorithms using normal and fault data generated from the bench-marked Tennessee Eastman (TE) problem. The TE problem is used to highlight the superiority of the nonlinear scaled PCA (SPCA) over PCA and FDA.

A Comparison of Moving Horizon and Bayesian State Estimators with an Application to a pH Process

Abstract The moving horizon estimator (MHE) formulation utilizes a window of measurements to compute the estimates of the states in that particular window. This approach leads to smoothing of the state estimates included in the window, since future information is used to compute the same. However, the effect of smoothing, in the MHE algorithm, on the state estimates has not been studied in the literature. In this work the performance of the MHE is compared with recursive Bayesian state estimators (such as UKF, EnKF) to study the effect of the moving window of the past data on the quality of state estimates, via an application on a benchmark pH simulation case study. The simulations are carried out for two scenarios–the ideal case and the case with a parametric model-plant mismatch. The results obtained indicate that the use of MHE results in improved state estimates when compared to the recursive Bayesian state estimators, but does not help compensate for model-plant-mismatch.