State estimation from plant measurements plays an important role in advanced process control. For example, the performance of a closed-loop control system is directly affected by the quality of the current state estimates. The challenge of state estimation is to determine good estimates in the face of noisy and incomplete output measurements. Moving Horizon Estimator (MHE) is an optimization based state estimator that is robust to process disturbances and model errors. Moreover, MHE naturally handles nonlinear models and incorporates hard state constraints, such as non-negative concentrations and pressures, in its formulation [1-4]. For optimal performance, this state estimator requires knowledge about the noise statistics affecting the states and measurements. 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 process operating data [5-7]. The Autocovariance Least-Squares (ALS) technique has proven to effectively estimate the covariances of system disturbances from data for linear [5,6] and nonlinear systems [7], using both simulated and real process data. This technique uses routine process operating data and thus does not require input-output testing to be applied to the system.

This presentation introduces a design method for all of the steps of nonlinear state estimation, from nonlinear stochastic modeling to estimator implementation. The effectiveness of the developed method is illustrated with an industrial polymerization process from ExxonMobil Chemical Company.

First, the modeling task is performed by showing that a nonlinear stochastic model can be well represented by a nonlinear deterministic model, obtained from first principles, with an added noise component that can be estimated from operating data containing the measurement noise. Then, the covariances of the process (Q) and measurement (R) noises are estimated using the time-varying ALS technique formulated for nonlinear models [7]. For this estimation, two cases using different data sets are considered: 1) simulated data generated using a published model of an industrial gas-phase ethylene copolymerization process [8-10] and an assumed Q and R; 2) real industrial data of the same process provided by ExxonMobil Chemical Company. For the second case, the stochastic structure of the disturbance model is also identified from data, by determining the minimum number of independent disturbances affecting the states, using the ALS-SDP technique [5,6]. These estimated covariances correspond to the Q and R weights used in the MHE formulation [1] and are used to specify the noise statistics of the state estimator. Next, MHE is implemented in both scenarios (real and simulated data) to show that better state estimates can be obtained after the specification of its noise statistics. Comparisons of the implemented estimator are made to other commonly used estimation techniques, such as Extended Kalman Filter (EKF) and a typical industrial estimator (heuristic KF). Finally, the ability of MHE to deal with infrequent asynchronous lab measurements is analyzed and it is demonstrated that the covariance estimation of the measurement noises using ALS scales this covariance based on the frequencies of the measurements.

References:

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[2] Douglas G. Robertson, Jay H. Lee, and James B. Rawlings. A moving horizon-based approach for least-squares state estimation. AIChE J., 42(8):2209-2224, 1996.

[3] Douglas G. Robertson and Jay H. Lee. On the use of constraints in least squares estimation and control. Automatica, 38(7):1113-1123, 2002.

[4] Christopher V. Rao, James B. Rawlings, and David Q. Mayne. Constrained state estimation for nonlinear discrete-time systems: Stability and moving horizon approximations. IEEE Trans. Auto. Cont., 48(2):246-258, 2003.

[5] Murali R. Rajamani and James B. Rawlings. Estimation of the disturbance structure from data using semidefinite programming and optimal weighting. Technical Report 2007-02, TWMCC, Dept. of Chem. and Bio. Eng., University of Wisconsin-Madison (Available at http://jbrwww.che.wisc.edu/tech-reports.html), 2007.

[6] Murali R. Rajamani and James B. Rawlings. Estimation of the disturbance structure from data using semidefinite programming and optimal weighting. Accepted for publication in Automatica, 2007.

[7] Murali R. Rajamani. Data-based Techniques to Improve State Estimation in Model Predictive Control. PhD thesis, Dept. of Chem. and Bio. Eng., University of Wisconsin-Madison (Available at http://jbrwww.che.wisc.edu/theses.html), 2007.

[8] K. B. McAuley, J. F. MacGregor, and A. E. Hamielec. A kinetic model for industrial gas-phase ethylene copolymerization. AICHE J., 36(6):837-850, 1990.

[9] S. A. Dadebo, M. L. Bell, P. J. McLellan, and K. B. McAuley. Temperature control of industrial gas phase polyethylene reactors. J. Proc. Cont., 7(2):83-95, 1997.

[10] Adiwinata Gani, Prashant Mhaskar, and Panagiotis D. Christofides. Fault-tolerant control of a polyethylene reactor. J. Proc. Cont., 17(5):439-451, 2007.