(199e) Integrating Data Driven Modeling with First Principles Knowledge

Principles Knowledge

Nikesh Patel, Jake Nease, Siam Aumi, Christopher Ewaschuk, Jie Luo, and

Prashant Mhaskar

Department of Chemical Engineering, McMaster University, Hamilton, Ontario, Canada

E-mail: mhaskar@mcmaster.ca

Batch processes are extremely important for a wide range of manufacturing industries such as chemicals, polymers, specialty glass and ceramics, and steel production. Batch processes carry out a sequence or recipe, which can entail the addition of ingredients and executing processing steps. The recipe can be adjusted based on results from previous batches to maintain and promote quality control.^[1] Additionally, the use of batch, instead of continuous processes, allows for batches that fail to meet the quality standards to be discarded without effecting the end products. The high value of these products means that discarded batches amount to significant revenue losses, making advanced batch process control strategies more attractive as a regulatory mechanism which, in turn, often necessitates a good process model.

First principles models, owing to their ability to capture inherent process dynamics and gains, are often the most sought after modeling approach.^[2-4] These models rely on conservation equations, such as mass, mole, and energy balances. However, they can be high-dimensional and complex, making them difficult to develop and maintain, or use for control design. As a result, data-driven models are often the preferred modeling approach in industry. For batch processes, the availability of historical batch-run data that typically spans a wide range of operating conditions makes data-driven modeling approaches particularly appealing. One gap in using data-driven methods is the explicit incorporation of first-principles knowledge. While multivariate methods, such as partial least squares regression, do allow appending the data matrix columns with calculated variables that capture some first-principles knowledge, they cannot guarantee the final model parameters will capture the desired first principles knowledge.

One popular data-driven modeling approach is subspace identification. Subspace identification involves a two-step procedure where the first step is to use data projection to identify a state trajectory and the second step is to compute the state space matrices. Unlike the majority of system identification approaches, it does not rely on good initial parameters.^[10-12] Subspace identification algorithms have different techniques including: canonical variate analysis,^[13] numerical algorithms for subspace identification of state space models^[14], and multivariate output error state space algorithms.^[15] These subspace identification algorithms can be classified by their use of singular value decomposition (SVD) of matrices under different weightings schemes.

Recent developments of the traditional subspace method have allowed for batch data to be analyzed using the same SVD method.^[16] The key to batch subspace identification is the projection of the batch data utilizing a reordering of the Hankel matrix. It is important to perform the subspace identification using batches with different initial conditions in order to retain a common basis for the states among different batches and capture all the necessary subspace states that result from the different initial conditions. Also, the subspace identification approach has not yet been adapted to allow for the inclusion of first principles based knowledge. This novel addition to the modeling approach is beneficial in that it should prevent the process model from predicting spurious relationships between certain inputs and outputs that can result from model structure and order mismatch and erroneous data.

While some efforts have been made to incorporate first principles knowledge into subspace identification approaches by including constraints, these approaches are expensive computationally and a methodology to address this effectively has not been thoroughly explored.^[17] The approach^[17] proposes a constrained least squares solution, while using a series of weighted constraints to turn the problem into a regular least squares solution. However, it is limited to equality constraints, requiring the exact value of parameters to be known. Additionally, while the system matrices can be solved with improved model predictions, based on simulations, the method requires the system matrices to be solved in an intermediate step rather than being solved simultaneously.^[17]

Motivated by these considerations, this work presents a modified approach to the batch subspace identification.^[18] Compared to the existing subspace identification procedure, this work introduces a novel blending of first-principles based constraints in the subspace identification procedure to generate a constrained model. The integration of first principles knowledge in subspace identification yields the ability to identify models capable of predicting the true process more accurately in comparison to other modeling approaches. This is illustrated in scenarios where the training data contains sensor errors resulting in mismatches between the input and output data that is collected. Our approach is to utilize first-principle knowledge of the process dynamics, in order to add constraints to the subspace identification procedure, without excessively increasing the computational complexity. To achieve this, first a nonlinear optimization problem is used to solve for the system matrices (via state estimates generated using an existing subspace identification technique) with constraints based on first-principles knowledge, hereafter referred to as the constrained model. Then an iterative procedure is used to refine the resulting system matrices and reconcile them with the state trajectories.

References

(1) Lee, K. S.; Lee, J. H. Iterative learning control-based batch process control technique for integrated control of end product properties and transient profiles of process variables. Journal of Process Control 2003, 13, 607-621, Selected Papers from the sixth IFAC Symposium on Bridging Engineering with Science - DYCOPS - 6.

(2) Shi, D.; El-Farra, N. H.; Li, M.; Mhaskar, P.; Christofides, P. D. Predictive control of particle size distribution in particulate processes. Chemical Engineering Science 2006, 61, 268-281.

(3) Bonvin, D. Control and optimization of batch processes. IEEE control systems 2006, 26, 34-45.

(4) Kozub, D. J.; MacGregor, J. F. Feedback control of polymer quality in semi-batch copolymerization reactors. Chemical Engineering Science 1992, 47, 929-942.

(5) Hu, B.; Zhao, Z.; Liang, J. Multi-loop nonlinear internal model controller design under nonlinear dynamic PLS framework using ARX-neural network model. Journal of Process Control 2012, 22, 207-217.

(6) MacGregor, J. F.; Jaeckle, C.; Kiparissides, C.; Koutoudi, M. Process monitoring and diagnosis by multiblock PLS methods. AIChE Journal 40, 826-838.

(7) Söderström, T.; Stoica, P.; Friedlander, B. An indirect prediction error method for system identification. Automatica 1991, 27, 183-188.

(8) Ljung, L. Prediction error estimation methods. Circuits, Systems and Signal Processing 2002, 21, 11-21.

(9) Zhao, Y.; Huang, B.; Su, H.; Chu, J. Prediction error method for identification of LPV models. Journal of process control 2012, 22, 180-193.

(10) Moonen, M.; De Moor, B.; Vandenberghe, L.; Vandewalle, J. On-and off-line identification of linear state-space models. International Journal of Control 1989, 49, 219-232.

(11) Huang, B.; Ding, S. X.; Qin, S. J. Closed-loop subspace identification: an orthogonal projection approach. Journal of process control 2005, 15, 53-66.

(12) Van Overschee, P.; De Moor, B. A unifying theorem for three subspace system identification algorithms. Automatica 1995, 31, 1853-1864.

(13) Larimore, W. E. Statistical optimality and canonical variate analysis system identification. Signal Processing 1996, 52, 131-144, Subspace Methods, Part II: System Identification.

(14) Van Overschee, P.; De Moor, B. N4SID: Subspace algorithms for the identification of combined deterministic-stochastic systems. Automatica 1994, 30, 75-93.

(15) Verhaegen, M.; Dewilde, P. Subspace model identification part 2. Analysis of the elementary output-error state-space model identification algorithm. International journal of control 1992, 56, 1211-1241.

(16) Corbett, B.; Mhaskar, P. Data-driven modeling and quality control of variable duration batch processes with discrete inputs. Industrial & Engineering Chemistry Research 2017, 56, 6962-6980.

(17) Alenany, A.; Shang, H.; Soliman, M.; Ziedan, I. Improved subspace identification with prior information using constrained least squares. IET Control Theory and Applications 2011, 5, 1568-1576(8).

(18) Corbett, B.; Mhaskar, P. Subspace identification for data-driven modeling and quality control of batch processes. AIChE Journal 62, 1581-1601.