Online estimation of the internal states is a perquisite for monitoring, control, and fault diagnosis of many engineering processes. A cost effective approach to monitor these variables in real time is to employ model-based state estimation techniques. Dynamic model-based state estimation is a rich and highly active area of research and many novel approaches have emerged over the last few years. In this paper, we review various recent developments in the area of nonlinear state estimators from a Bayesian perspective. In particular, we focus on the constrained state estimation (including systems modeled using differential-algebraic equations), the handling of multi-rate and delayed measurements and recent advances in model parameter estimation. Recent advances on the stability analysis of the estimation error dynamics are also briefly discussed. The review aims to provide an integrated view of important ideas, from the authors' perspective that have driven the research in this area in recent years. © 2012 Elsevier Ltd.

Shankar Narasimhan

Department Of Chemical Engineering

Active area

BAYESIAN PERSPECTIVE

Bayesian state estimation

Constrained state estimation

Cost-effective approach

DELAYED MEASUREMENTS

Differential algebraic equations

ENGINEERING PROCESS

Estimation errors

ESTIMATION TECHNIQUES

INTERNAL STATE

MODEL PARAMETER ESTIMATION

Multi rate

NONLINEAR STATE ESTIMATORS

On-line estimation

Real time

Stability analysis

Estimation

Parameter estimation

State Estimation

IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

Control Engineering Practice

Changes in process parameters and measured/unmeasured disturbances shift the optimal point at which the economic benefits are maximized. Combinations of real time optimization (RTO) techniques, which periodically determine the economically optimum operating point using steady state optimization, and model predictive control (MPC) have been widely employed in the process industry for operating process plants optimally in the face of drifting disturbances and/or parameters. Due to the long wait time between two successive RTO invocations and model inconsistency between the RTO and MPC layers, the conventional RTO schemes can end up operating the plant suboptimally if the process parameters/unmeasured disturbances change significantly during the wait time. Recently proposed frequent RTO approaches attempt to address this difficulty by increasing the frequency of RTO invocation. An online update of the steady state model employed by the RTO layer using dynamic operating data is a major concern in implementation of frequent RTO. Use of a dynamic model based state and parameter estimation for maintenance of the model used in frequent RTO has not received much attention in the literature. In this work, it is proposed to develop a novel integrated frequent RTO and adaptive nonlinear MPC (NMPC) approach for operating a unit operation in a economically optimal manner. A dynamic mechanistic model based simultaneous state and parameter estimation scheme is used as a common link between the RTO and the NMPC components. Estimates of the drifting unmeasured disturbances/parameters generated by the state and parameter estimator are used to update the steady state model used for frequent RTO and the dynamic model used for predictions in NMPC. Use of a single model for carrying out RTO as well as control tasks eliminates difficulties that arise due to the mismatch between models used for RTO and control. The efficacy of the proposed integrated optimizing control scheme is demonstrated by conducting simulation studies on benchmark systems from the literature. Analysis of simulation results reveals that the proposed integrated online optimizing control scheme maintains these systems at their respective economically optimal operating points in the presence of drifting disturbances/parameters. © Copyright © 2019 American Chemical Society.

Industrial and Engineering Chemistry Research

An Integrated Frequent RTO and Adaptive Nonlinear MPC Scheme Based on Simultaneous Bayesian State and Parameter Estimation

Nonlinear state estimation techniques for state dynamics involving additive discrete zero mean white noise have received considerable attention in the past. The probabilistic formulation of the conventional moving horizon estimation (MHE) is also developed under this simplifying assumption. In reality, the unmeasured disturbances affect the states and measurements in much more complex way and thus can not be treated in additive manner. In current work, we formally introduce a probabilistic formulation of MHE which can handle such nonlinear uncertainties in the state dynamics. The efficacy of the proposed MHE has been demonstrated by conducting stochastic simulation studies on a benchmark continuous stirred tank reactor (CSTR) system. It is found that the estimation performance of the proposed MHE formulation is comparable to estimation performance of a sampling based Bayesian estimator (unscented Kalman filter or UKF), which deals with non-additive disturbances systematically, and significantly better than the extended Kalman filter (EKF) that deals with non-additive disturbances through successive linearisation. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

Fulltext

IFAC-PapersOnLine

Batch and moving Horizon estimation for systems subjected to non-additive stochastic disturbances

Online estimation of internal states and parameters is often required for process monitoring, control and fault diagnosis. The conventional approach to estimate drifting parameters is to artificially model them as a random walk process and estimate them simultaneously with the states. However, tuning of the random walk model is not a trivial exercise. Recently, Valluru et al. (2017) have developed a moving window based state and parameter estimator which assumes that the parameters change slowly and remain constant within the window. Also, in another development, a moving window based recursive filter, receding horizon nonlinear Kalman (RNK) filter has been proposed by Rengaswamy et al. (2013). In this work, a novel simultaneous state and parameter estimator is proposed by combining the window based parameter variation model with RNK filter formulation. The performance of the RNK based estimator is demonstrated by conducting simulation studies on the benchmark quadruple tank system and a CSTR system. The efficacy of RNK based estimator is compared with that of the conventional simultaneous EKF approach and Moving Horizon Estimator (MHE) based state and parameter approach. Analysis of the simulation results reveals that the proposed state and parameter estimation scheme is able to generate better estimation performance than that of the simultaneous EKF and closer to that of the MHE based parameter estimator with less computational efforts. © 2018

Simultaneous State and Parameter Estimation using Receding-horizon Nonlinear Kalman Filter

Nonlinear state estimation is a pre-requisite for advanced process control and fault diagnosis tasks. In literature, various recursive nonlinear filtering techniques have been proposed and used for state estimation of nonlinear systems. Over the last few years, Moving Horizon Estimation (MHE) is increasingly being used for state estimation of nonlinear systems. Moving horizon estimation works with a window of data and hence requires additional online computation compared to recursive nonlinear filters. However, MHE performs both smoothing and filtering and thus has the potential to obtain more accurate state estimates as compared to the recursive filters. Most of the available comparisons of MHE with recursive filters are based on simulation case studies where the true states and parameters, as well as noise processes are exactly known. In this work, we apply MHE to a three-Tank experimental setup and compare its performance with various nonlinear filters available in literature such as Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF) and Gaussian Sum EKF (GSEKF). We present some of the challenges in experimental implementation of state estimation approaches. These include presence of unknown disturbances, non-whiteness of noise signals as well as lack of accurate measurements. We then discuss the approach followed for obtaining model parameters and noise characterization to make the models amenable for filter implementation. It is found that EKF, GSEKF and UKF perform as well as MHE as far as accuracy is concerned, but require significantly lower computational efforts. © 2017 IEEE.

2017 6th International Conference on Computer Applications in Electrical Engineering - Recent Advances, CERA 2017

Nonlinear state estimation for three tank experimental setup: A comparative evaluation

Simultaneous occurrence of gross errors (outliers/biases/drifts) in the measured signals, and drifting disturbances/parameter variations affecting the system dynamics can lead to biased state estimates, and, in turn, can lead to deterioration in the performance of model-based monitoring and control schemes. In this work, robust recursive and moving window based Bayesian state and parameter estimators are developed that are robust w.r.t. gross errors in the measurements and can simultaneously estimate non-additive unmeasured disturbance/parameter variations. Using Bayes’ rule, the update step of Kalman filter (KF) is recast as an optimization problem. The optimization is then modified by replacing the likelihood term in the objective function with cost function defined by an M-estimator. The M-estimators considered in this work are Huber's Fair function and Hampel's redescending estimator. The reformulated KF is then used as a basis for reformulating extended Kalman filter (EKF). This re-formulated EKF is then used for developing robust simultaneous state and parameter estimation schemes. In particular, a robust version of recently proposed moving window based state and parameter estimator [1] has been developed. The resulting formulation can be viewed as a hybrid approach, in which the gross errors in the measurements are dealt with in a passive manner, with an active elimination of model plant mismatch by estimating unmeasured disturbance/parameter variations simultaneously. The efficacy of the proposed robust state and parameter estimators is demonstrated by conducting simulation studies and experimental studies. Analysis of the simulation and experimental results reveal that the proposed robust recursive and moving window based state and parameter estimators significantly reduce or completely nullify the effect of gross errors on the state estimates while simultaneously estimating drifting unmeasured disturbances/parameters. The simulation study also underscores the importance of simultaneous estimation of unmeasured disturbances/parameters while achieving robustness using the M-estimators. Moreover, Hampel's redescending estimator is found to be a better choice of M-estimator than the popular Huber's Fair function, as the redescending estimator can completely nullify the effect of gross errors on the state and parameter estimates. © 2018 Elsevier Ltd

Journal of Process Control

Development of robust extended Kalman filter and moving window estimator for simultaneous state and parameter/disturbance estimation

Nonlinear constrained state estimation is an important task in performance monitoring, online optimization and control. There has been recent interest in developing estimators based on the idea of unscented transformation for constrained nonlinear systems. One of these approaches is the unscented recursive nonlinear dynamic data reconciliation (URNDDR) method. The URNDDR approach follows the traditional predictor-corrector framework. Constraints are handled in the prediction step through a projection algorithm and in the correction step through an optimization formulation. It has been shown that URNDDR produces very accurate estimates at the cost of computational expense. However, there are two issues that need to be addressed in the URNDDR framework: (i) URNDDR approach was primarily developed to handle bound constraints and needs to be enhanced to handle general nonlinear equality and inequality constraints, and (ii) computational concerns in the application of the URNDDR approach needs to be addressed. In this paper, a new estimation technique named constrained unscented recursive estimator (CURE) is proposed, which eliminates these disadvantages of URNDDR, while providing estimates with almost the same accuracy. © 2012 Elsevier Ltd. All rights reserved.

Constrained unscented recursive estimator for nonlinear dynamic systems

State estimation is an important problem in process operations. For linear dynamical systems, Kalman Filter (KF) results in optimal estimates. Chemical engineering problems are characterized by nonlinear models and constraints on the states. Nonlinearities in these models are handled effectively by the Extended Kalman Filter (EKF), whereas constraints pose more serious problems. Several constrained estimation problems where the EKF approach fails have been reported in the literature. To address this issue, receding horizon approaches such as the Moving Horizon Estimation (MHE) have been proposed. The MHE approach has been shown to provide the most reliable estimates in several example problems; albeit at a high computational price. Unlike the KF, the MHE formulation does not use an explicit predictor-corrector approach. In this paper, we study the following questions in nonlinear constrained state estimation: (i) can the EKF be extended to include a receding horizon in a simple intuitive fashion? (ii) are there any performance gains over an EKF due to a receding horizon? and, (iii) are there any computational gains over the standard MHE through such an extension? A Receding Nonlinear Kalman (RNK) Filter formulation is proposed to answer these questions. The RNK formulation follows a predictor-corrector approach and uses linearization of the state space model for covariance calculation much like the EKF approach. We demonstrate through examples that inclusion of a receding horizon improves performance over the standard EKF approach. We also discuss the computational properties of RNK in comparison with MHE. © 2011 Elsevier B.V.

Computer Aided Chemical Engineering

Receding Nonlinear Kalman (RNK) Filter for Nonlinear Constrained State Estimation

This work deals with state estimation in the presence of delayed and infrequent measurements. While most measurements (referred to as secondary measurements) are available frequently and instantaneously, there might be a delay associated with acquiring other measurements (primary measurements) due to long analysis times involved. The primary measurements are usually sampled at irregular intervals and the exact delay is also unknown. The traditional fixed-lag smoothing algorithm, which has been applied for a variety of chemical processes systems, can be computationally inefficient for such situations and alternate methods to handle delays are necessary. In this paper, we analyze several existing methods to incorporate measurement delays and reinterpret their results under a common unified framework (for Extended Kalman Filter). Extensions to handle time-varying and uncertain delays, as well as out of sequence measurement arrival are also presented. Simulation studies on a linear distillation column and a nonlinear polymerization reactor are used to compare the performance of these methods based on RMSE values and computation times. A large scale nonlinear reactive distillation column example is also used to illustrate the practicality of the suggested method. © 2010 Elsevier Ltd. All rights reserved.

Incorporating delayed and infrequent measurements in Extended Kalman Filter based nonlinear state estimation

Kalman filter and its variants have been used for state estimation of systems described by ordinary differential equation (ODE) models. While state and parameter estimation of ODE systems has been studied extensively, differential algebraic equation (DAE) systems have received much less attention. However, most realistic chemical engineering processes are modelled as DAE systems and hence state and parameter estimation of DAE systems is a significant problem. Becerra et al. (2001) proposed an extension of the extended kalman filter (EKF) for estimating the states of a system described by nonlinear differential-algebraic equations (DAE). One limitation of this approach is that it only utilizes measurements of the differential states, and is therefore not applicable to processes in which algebraic states are measured. In this paper, we address the state estimation of constrained nonlinear DAE systems. The novel aspects of this work are: (i) development of a modified EKF approach that can utilize measurements of both algebraic and differential states, (ii) development of a recursive approach for the inclusion of constraints, and (iii) development of approaches that utilize unscented sampling in state and parameter estimation of nonlinear DAE systems; this has not been attempted before. The utility of these estimators is demonstrated using electrochemical and reactive distillation processes. © 2010 Elsevier Ltd.

Chemical Engineering Science

Recursive state estimation techniques for nonlinear differential algebraic systems

There is growing realization that on-line model maintenance is the key to realizing long term benefits of a predictive control scheme. In this work, a novel intelligent nonlinear state estimation strategy is proposed, which keeps diagnosing the root cause(s) of the plant model mismatch by isolating the subset of active faults (abrupt changes in parameters/disturbances, biases in sensors/actuators, actuator/sensor failures) and auto-corrects the model on-line so as to accommodate the isolated faults/failures. To carry out the task of fault diagnosis in multivariate nonlinear time varying systems, we propose a nonlinear version of the generalized likelihood ratio (GLR) based fault diagnosis and identification (FDI) scheme (NL-GLR). An active fault tolerant NMPC (FTNMPC) scheme is developed that makes use of the fault/failure location and magnitude estimates generated by NL-GLR to correct the state estimator and prediction model used in NMPC formulation. This facilitates application of the fault tolerant scheme to nonlinear and time varying processes including batch and semi-batch processes. The advantages of the proposed intelligent state estimation and FTNMPC schemes are demonstrated by conducting simulation studies on a benchmark CSTR system, which exhibits input multiplicity and change in the sign of steady state gain, and a fed batch bioreactor, which exhibits strongly nonlinear dynamics. By simulating a regulatory control problem associated with an unstable nonlinear system given by Chen and Allgower [H. Chen, F. Allgower, A quasi infinite horizon nonlinear model predictive control scheme with guaranteed stability, Automatica 34(10) (1998) 1205-1217], we also demonstrate that the proposed intelligent state estimation strategy can be used to maintain asymptotic closed loop stability in the face of abrupt changes in model parameters. Analysis of the simulation results reveals that the proposed approach provides a comprehensive method for treating both faults (biases/drifts in sensors/actuators/model parameters) and failures (sensor/ actuator failures) under the unified framework of fault tolerant nonlinear predictive control. © 2008 Elsevier Ltd. All rights reserved.

Journal	Control Engineering Practice
ISSN	09670661
Open Access	No