Integrating Data-Driven Modeling with First-Principles Knowledge

Abstract

This Article addresses the problem of integrating subspace-based model identification with first-principles modeling for handling scenarios where the subspace model identifies spurious relationships between inputs and outputs. The key motivation is to suitably synergize the two approaches while retaining the simplicity of subspace-based model identification. In the proposed methodology, as is done with traditional subspace identification, state trajectories that best describe the input−output data are first computed (which implicitly correspond to an underlying linear time invariant model). In computing the system matrices using the state trajectories, constraints derived from first-principles understanding are incorporated into the optimization problem. To reconcile the resulting mismatch between the state trajectories and the system matrices, an iterative process is utilized. First, the system matrices computed from the optimization problem are utilized to re-estimate the state trajectories (this time utilizing a state estimator and the input and output trajectories). The state trajectories are, in turn, utilized to resolve the system matrices using the input−output data. The process is repeated until convergence occurs between successive state trajectories, thus yielding state trajectories and "consistent" system matrices. The efficacy of the proposed approach is shown via simulations using a nonlinear process example.

Subspace Based Model Identification for an Industrial Bioreactor: Handling Infrequent Sampling Using Missing Data Algorithms

“…While this would be possible in principle (and reasonable for the process considered in [18]), in the present instance, this would lead to each of the batches having very sparse measurements-thereby comprising mostly missing data. In this case, the recently developed missing data approach [24] would not be directly applicable.…”

“…As a result of these considerations, a missing data subspace modeling approach using PCA and PLS steps was recently developed [24]. Specifically, the addition of PCA and PLS steps to the subspace identification approach allows for the missing observations to be accounted for.…”

“…The second step replaces SVD with PCA, to handle missing observations, to identify the states of the system whereupon traditional subspace approaches can be utilized. The approach in [24], however, does not directly handle discrete additions (of glucose), and thus is not directly applicable. Another recent result [27] that explicitly handles discrete additions is not applicable due to two reasons--the first is that the results in [27] do not handle missing data, and the second is that a direction application of the approach in [27] coupled with the missing data approach of Patel et al [24] would lead to batches with almost no data, in turn making the approach inapplicable.…”

“…The approach in [24], however, does not directly handle discrete additions (of glucose), and thus is not directly applicable. Another recent result [27] that explicitly handles discrete additions is not applicable due to two reasons--the first is that the results in [27] do not handle missing data, and the second is that a direction application of the approach in [27] coupled with the missing data approach of Patel et al [24] would lead to batches with almost no data, in turn making the approach inapplicable. Finally, while the results in [24,27] provide a modeling framework, the resultant subspace models do not necessarily provide insight into the process dynamics and could be improved by augmenting with tools to enable easier access to the practitioner.…”

“…Motivated by the above considerations, the present manuscript adapts the missing data approach of Patel et al [24] to specifically handle the discrete addition nature of the Sartorius Bioreactor along with the missing data in the metabolite measurements and develops a data driven dynamic model that also predicts variables that can be much better interpreted by the practitioner. The approach is designed to handle batch data with variable batch length without the need for batch alignment techniques.…”