(687b) A Subspace Identification Based Dynamic Soft Sensor Approach for Digester Control

Kraft pulping is the commonly applied chemical pulping process that normally utilizes a (Kamyr) continuous digester. The most important quality variable in pulping is the Kappa number, which represents the residual lignin content of pulp, and it is desired to minimize the variations of the Kappa number of the pulp product. The continuous digester control problem is very challenging [1-6], and one of the reasons is that the main controlled variable, Kappa number, is measured infrequently and not in real time.

In this work, based on the formulation of the subspace identification [7-9], we derive a dynamic PLS model as the soft sensor to predict the Kappa number using process inputs and secondary measurements. By reducing the future horizon to one, the correlation between the future input and future disturbance can be eliminated. Therefore the developed method works well for closed-loop data. Because the developed PLS model is a reduced higher-order ARX model of the process, the soft sensor is not a static one but a dynamic one which captures the process dynamics; In addition, the model can be updated easily on line to capture any process changes.

Starting with a state-space representation of a linear system:

where y^s and y^c are secondary and primary variables that are measured at different frequencies; process innovation e(k) is assumed to be stationary white noise sequence. By representing the system described by Eqns (1) to (3) in its predictor form, we can obtain the extended state space model of the system, which is also the foundation for the subspace identification methods,

where subscript f denotes the future horizon, p denotes the past horizon, and . The rest of the notations are the standard notation used in subspace identification (e.g. see [7]), and the superscript ?*? indicates these matrices are constructed with predictor matrices instead of system matrices.  Notice that many subspace identification methods do not apply to closed-loop data because the future input matrix contained in  are correlated to future disturbance matrix  due to feedback control because the future horizon is always larger than the system order. This limitation can be avoided by setting f = 1 and Eqn.(4) becomes the higher-order ARX model of the process,

Based on Eqn. (5), a linear regression between the current primary output and the past inputs and secondary outputs can capture the process dynamics because the future innovation e^c(k) is independent of the past inputs and outputs. In this work, we apply partial least squares to capture the correlation between augmented input and output variables under closed-loop operation. PLS also naturally removes the noise and reduces the order of the ARX model.

In subspace identification formulation, the past horizon is usually much larger than the process order in order for the contribution from the initial state to be negligible. Consequently, a large number of regressor variables are included in the soft sensor while only a few of them contribute significantly to the prediction of the primary variable. Therefore, it is desirable to select most related variables to build the soft sensor. In this work, how to select process variables to be included in the soft sensor is also discussed.

Finally, the developed method is evaluated using both simulated digester process and industrial data. We used the fundamental model developed in [10] to simulate a single vessel digester, and the industrial data was provided by MeadWestvaco Corporation.

Reference

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