State estimation from plant measurements plays a critical role in all advanced monitoring and control technologies because the performance of an advanced feedback control system is directly affected by the quality of the state estimates and most complex product properties are not measurable but must be inferred from other measurements combined with nonlinear property models. Moving horizon estimation (MHE) is an optimization-based method that has been shown to provide accurate state estimates that are robust to significant process disturbances and model errors (1; 2; 3). MHE naturally handles nonlinear models and incorporates hard state constraints. To provide high-quality state estimates, MHE requires a dynamic model of the process, which includes the statistics of the stochastic disturbances affecting the states and measurements. Most chemical processes are characterized by both nonlinearity in the dynamics and significant levels of process and sensor noise. If these noises are modeled as zero-mean Gaussian sequences, then their covariances are required to specify their statistics. These statistics are usually unknown, but can be estimated from routine process operating data. For nonlinear systems, the linear time-varying autocovariance least-squares (ALS) technique was developed to estimate these covariances. Despite the use of linearizations, this approach has accurately estimated the covariances of nonlinear industrial examples (4; 5).

The goal of this presentation is to apply a previously proposed design method for nonlinear state estimation (5) to build and validate state estimators for weakly observable systems, with focus on industrial polymerization processes. Two process examples are of particular interest in this work: (i) a gas-phase ethylene copolymerization process model taken from the literature (6; 7; 8); and (ii) an industrial propylene polymerization process from ExxonMobil Chemical Company. Specifically, the proposed design method consists of the following steps: nonlinear process model selection, stochastic disturbance model selection, covariance identification from operating data, determination of the subset of the state to be estimated online from the measurements, and estimator selection and implementation. This task is challenging because polymerization process models, in addition to containing many unobservable and weakly observable modes, are nonlinear and large dimensional (around 50 states and 20 measurements); the product properties of most interest are complex nonlinear functions of the process state; the stochastic disturbance structure is unknown a priori; the most informative laboratory measurements are characterized by missing values and significant time delays. Moreover, industrial polymerization processes are routinely operated over a wide range of conditions, producing many different grades with significantly different properties, and the models need to provide accurate predictions throughout a grade transition, and not just within a particular grade.

For the modeling task, we use fully nonlinear stochastic models in discrete time obtained by combining available information, such as a deterministic set of nonlinear differential equations describing the physical principles of the process and a routine set of operating data that provide a typical sample of the measurement and process disturbances affecting the system. Integrating disturbance models are used to provide offset free control of the properties of interest, while maintaining the complexity low enough so that the disturbance statistics can be determined from the available measurements. For the estimation of these statistics, specifically the covariances of the process (Q) and measurement (R) noises, case studies of the two process examples mentioned above with different data sets are considered. For the ethylene copolymerization example, simulated data sets are generated using the literature model and an assumed Q and R. For the propylene polymerization example, industrial data sets are provided by ExxonMobil Chemical Company. Also for this industrial example, the stochastic structure of the disturbance model is identified from data by determining the minimum number of independent disturbances affecting the states, using the ALS-SDP technique (9). These estimated covariances correspond to the Q and R weights used in the MHE formulation and are used to specify the noise statistics of the state estimator.

As indicated by preliminary results on the copolymerization example (10), physical models for polymerization may be overly complex considering the available measurements; they may contain many unobservable and weakly observable modes. Overly complex structures lead to ill-conditioned or singular ALS problems for disturbance variance estimation. Ill-conditioning leads to unrealistic data demands for reliable estimates. In general, overly complex disturbance models for weakly observable system models must be avoided. To reduce or eliminate this ill-conditioning problem that may also plague the state estimation step, we design a reduced-order extended Kalman Filter to estimate only the strongly observable system states. This reduced filter is used to perform the ALS estimation of noise covariances. One example of such a filter is the Schmidt-Kalman filter (SKF) that was originally developed for navigation systems to improve numerical stability and reduce computational complexity of Kalman filters (11; 12), and later used to tackle weakly observable systems (13; 14). The general idea of this technique is to remove weakly observable states in the Kalman filter gain calculation, producing a filter that does not estimate the removed state variables, but still keeps track of the influences these states have on the gain applied to the other states. Preliminary results on the implementation of this technique to the copolymerization example show that better conditioned state estimation and covariance estimation problems are obtained. In this presentation, we further explore this approach by implementing it on both process examples cited above and by comparing it with MHE using weights defined by the ALS covariances. Also, we show that high-quality state estimates can be obtained after the specification of the noise statistics of these estimators by ALS. The main contributions of this work are in the development of optimization-based moving horizon estimators for weakly observable systems, nonlinear estimation using physical models, nonlinear covariance estimation from data, and building low complexity disturbance models for nonlinear systems.

References

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