(383e) Handling Delayed and Irregular Measurements in Batch Subspace Model Identification Framework

Batch processes form an integral part of the manufacturing industry, either to directly produce specialty chemicals, or as lead up to continuous operation (in the form of process startups). Traditional approach for control of batch processes has been to use open loop, recipe based policy. In this approach, a predefined input trajectory is applied to each batch [1]. The approach assumes that the desired product quality can be obtained by repeating historically successful input profiles. Although these approaches are easy to implement and do not require a model for the process, they are incapable of rejecting disturbances that affect the process. This has motivated the use of feedback control strategies.

A classical closed loop control approach in batch process is trajectory tracking. In this approach, a predefined set-point trajectory for a measured process variable, often obtained from past successful batches, is tracked. PI controllers are often used for implementing trajectory tracking control. When implementing trajectory tracking approaches (with PI or otherwise) even with perfect tracking, the desired quality may not be obtained, as the relationship between the measured/tracked variable and final quality may change significantly with changes in the process conditions.

These challenges are addressed by control strategies that are cognizant of the causal relationship not only between the manipulated and the (online) measured variables, but also between the manipulated and the final quality variable. A popular model based control approach, model predictive control (MPC), has increasingly been studied for the control of batch processes [2-5]. The reasons for popularity of these control schemes are twofold: first, the feedback controller can counter the model uncertainties that are associated with model simplifications and measurement errors, the other being their ability to handle the ever present input/output constraints. In implementing these formulations, where possible, good first principles models are preferred due to the ability to predict process behavior beyond the data set used to estimate the model parameters. However, first principles models are difficult to develop as it is often impractical to estimate the parameters through experiments, particularly because of the associated cost. Further, these first principles models, for instance, in case of distributed parameter systems such as batch particulate processes, pose computational challenges for direct use in predictive control formulation.

These issues have motivated the use of simpler, often linear, models derived from past batches database. A variety of approaches for development of data-driven models have been proposed. One excellent approach is partial least squares (PLS), which models the process in a projected latent space [6]. These models are essentially time-varying linear models, linearized around mean past trajectories, and therefore require the batches to be of same length, or to recognize an appropriate alignment variable. To account for these limitations, a multi-model approach was proposed in [5]. These models were based on the 'current measurements' of the process instead of the 'time'. These developments were followed by contributions in the area of integration of these data-driven models with the advanced control formulations [4-5]. More recently a subspace identification based batch control approach was proposed in [7] where a LTI state-space model of the batch process is estimated. Subspace identification methods, in general, are a class of model identification methods that are non-iterative in nature and models a process as a state-space linear time invariant system, identifying the system matrices along with the order of the system. These methods utilize efficient matrix factorization methods such as singular value decomposition (SVD) and QR factorization for their implementation. This is in contrast to classical system identification techniques, which are iterative optimization based algorithms that minimize the prediction errors (PEM) using efficient methods such as maximum likelihood estimation (MLE) and expectation maximization (EM) algorithm. These methods offer well established theoretical properties and have been widely studied [8-11]. However, most of the existing classical and subspace identification approaches focus on continuous operation. The subspace identification approach in [7] was different from conventional subspace identification approach for continuous process in the way Hankel matrices are constructed to utilize data from multiple batches. This formulation will henceforth be referred to as batch subspace identification.

The existing formulations of batch subspace identification method have, however, dealt generally with all process variables being measured at a fixed rate [7,12,13]. Recently, [14] discussed the formulation for a multi rate process, to handle the case where some of the measurements are available at different rate than others. However, these results do not explicitly consider delayed measurements in the batch subspace identification formulation. Further, the formulation of batch subspace identification needs to be generalized for the case of measurements arriving at a unsteady rate during the batch, that is, the rate at which variables are measured is not same during the entire batch duration.

Motivated by these considerations, this work presents a subspace identification based state-space modeling and control approach for batch process with delayed and infrequent measurements. The key idea is to adapt the recently developed batch subspace identification approach to explicitly handle delayed and irregular measurements. The ideas proposed in this work will be illustrated through a particulate batch process.

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