(552g) Multi-Rate Reduced-Order Observer Design with Application to Monitoring of Gas-Phase Polyethylene Reactors

State observers find many applications in the areas of process monitoring, feedback control, and fault detection. Many available observer design methods tackle the state estimation problem by using a continuous-time model with continuous outputs, and this works well under fast sampling rates [1-4]. In polymerization processes, however, it is inevitable to use measurements with large sampling periods (e.g., gel permeation chromatography, gas chromatography, melt index deviceâ?¦) in order to monitor product quality. This poses a significant challenge on observer design. To deal with the sampled measurements, one possible approach is to first develop a discrete-time description of the dynamics, with discretization time step equal to the sampling period. But this would not account for the inter-sample behavior and a part of the information is lost at the moment when the continuous-time model is being discretized.

A full-order observer based on an available continuous-time observer design, coupled with inter-sample output predictor, was proposed recently [5]. During the time period in between two consecutive output measurements, an inter-sample predictor is designed to obtain an estimate of the rate of change of the output, in order to be able to continuously apply a correction on the most recent measurement during this period. This predictor is re-initialized once the most recent measurement is available. It is proved in [5] that the sampled-data implementation inherits the stability, performance and robustness properties of the continuous-time observer as long as the sampling period is not too large.

The present paper is motivated by the same idea of inter-sample predictor, but the development will be in a reduced-order observer framework, to eliminate the redundancy of the full-order formulation, where the sampled states are estimated twice, in the observer and in the inter-sample predictor. In the proposed formulation, a continuous-time reduced-order observer will be coupled with an inter-sample predictor for the slow-sampled measurements. Moreover, the present work will provide a generalization in the sense of handling multi-rate output measurements, allowing the possibility of both continuous and slow-sampled measurements to be processed. Stability analysis of the error dynamics of the overall multi-rate reduced-order sampled-data observer will be performed in the linear case. Explicit conditions that allow the estimation of the maximum allowable sampling period for stability will be derived by using Lyapunovâ??s second method [6].

Moreover, in the present work, we apply the proposed observer design framework to a gas-phase polyethylene reactor to monitor the number of active catalyst sites. Polyethylene is the most popular synthetic commodity polymer. A large fraction of polyethylene is produced in gas-phase reactors using Ziegler-Natta catalysts [7]. A dynamic process model that also includes the dynamics of the product properties (i.e., instantaneous/cumulative melt index and density) was derived in [8, 9]. Gas compositions in the bleed stream leaving from the top of the reactor are measured by on-line gas chromatography in a discrete manner. Cumulative melt index and density are measured off-line in a quality control lab, and these are directly related to molecular weight and copolymer composition respectively. The only continuous measurement is the temperature of the reactor. Using the proposed design framework of multi-rate reduced-order observer, we are able to obtain a continuous estimate of the unmeasured state, with a convergence rate that is comparable to the case of continuous measurements. In addition, by using the inter-sample predictor, we are able to estimate the outputs in between consecutive samples as well.

References:

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[9] K. B. Mcauley, D. A. Macdonald, and P. J. Mclellan, "Effects of Operating-Conditions on Stability of Gas-Phase Polyethylene Reactors," AIChE Journal, vol. 41, pp. 868-879, Apr 1995.