2022 Annual Meeting

(363d) Selection of Self-Optimizing Controlled Variables Using Principal Component Analysis

Authors

Kumar, S., Indian Institute of Technology Tirupati
In the era of data science, efficient use of process data to improve the profitability of a chemical plant using machine learning methods is increasingly becoming important. Self-optimizing control provides a pragmatic approach to convert economic goals into control goals by selecting an additional set of controlled variables corresponding to the unconstrained degrees of freedom such that the effect of disturbances and measurement noise are mitigated. These set of controlled variables are kept at a constant setpoint strategy in a typical feedback loop to improve the economic performance. In general, these Self Optimizing Controlled (SOC) variables are often chosen to be linear combinations of measurements. There exist several approaches for the selection of self-optimizing controlled variables viz minimum singular value rule, minimum loss method, null space method, or exact local method. All the mentioned methods are model based approaches for the selection of SOC variables. To the best of our knowledge, there are only a limited number of works on the data-based approach to the selection of controlled variables. In the work of Johannes [1], a Partial Least Squares (PLS) is used to obtain self optimizing controlled variables in which the cost function of an optimal operation problem is expressed as a quadratic function in terms of process measurements, and a linear process model is assumed. It was shown that the steady state gain matrix and quadratic cost function can be used to easily compute the linear combination of measurements as SOC variables by the use of optimality equations. Very recently, Luppi [2] developed a principal component analysis based selection of controlled variables using only the steady state gain matrix. However, the resulting controlled variables do not possess self-optimizing properties. Hence, in this work, the focus is to develop a data-driven approach to determine the SOC variables using principal component analysis.

In this work, we propose two main approaches that are based on principal component analysis to identify SOC variables. In the first approach, we follow the philosophy of null space approach, and obtain the sensitivity matrix numerically. This is useful in the case when the disturbances are measured as the sensitivity matrix is a function of disturbances. In this case, we show that the singular value decomposition of the optimal sensitivity matrix is used to obtain the optimal linear combination of measurements as SOC variables. It should be noted that the number of SOC variables can be obtained heuristically using the percentage variance captured. In the second approach, we follow the philosophy of exact local method. In this case, we define the adjusted data matrix as the sum of output measurements (written in terms of deviation variables) and economic sensitivity of manipulated variables, and show that the left null space of the adjusted data matrix yields the SOC variables. Also, this method does not assume that disturbances are measured, and thus it is practically convenient. Following the first approach, we perform singular value decomposition on the adjusted data matrix to obtain the SOC variables. In a similar fashion, the number of SOC variables are selected heuristically. To demonstrate the efficacy of proposed data-driven approaches, a benchmark case study of continuous stirred tank reactor is used to compare it with existing model-based approaches presented in literature.

References

[1] Jäschke, J. and Skogestad, S., “Using process data for finding self-optimizing controlled variables,” IFAC Proceedings Volumes, 46 (32), pp. 451-456 (2013).

[2] Luppi, P. A., et al., “Optimal Measurement Selection and Principal Component Analysis-Based Combination as Controlled Variables,” Industrial & Engineering Chemistry Research, 60 (1), pp. 457-472 (2021).