(767f) On the State and Output Sensitivity of First-Principles Models

First-principle models serve for a wide range of purposes in today's process industry. Not only are they used to understand the underlying dynamics of complex physical processes, but also to monitor, optimize, predict, and make relevant decisions of economic and environmental impact. Generally, these process models are a collection of differential equations based on momentum, material and energy balances. They also contain algebraic relationships that represent kinetic, transport, and thermodynamic phenomena. The resulting description of such models can be as complex and detailed as required to the expense of the introduction of several latent variables and a larger parametrization. Analogously, prior knowledge is frequently not sufficient to fully derive appropriate mathematical models and their equations involve unknown parameters that might affect the system's behavior. The identification of these parameters is therefore important because one or several of them might affect the performance of the model-based operation support system. Under the assumption of no under-modeling, the model parameters could be seen as scaling variables that allow to tune the physical relations describing the phenomena in physical systems. Adjusting these parameters will allow to capture the current process behavior. Furthermore, the development of process simulation models is time-consuming and expensive; therefore, a strategy to perform model adjustment is necessary to support the effort and investment put in building these models.

A relevant application area is the estimation of parameters in reaction system models. These parameters play a major role in the synthesis and implementation of model-based technology. Moreover, a nice feature of reaction systems is that they can be recast in the form of a linear parameter-varying (LPV) using a linear transformation that preserves the physical meaning of the system states [1]. This linear representation allows to analyze the parametric sensitivity in a simpler way. Although, sensitivity analysis has mostly been limited to constant parameters, the analysis can also be extended to time-varying parameters. Following this idea, it is well-known that time-varying parameters might be unidentifiable from “frozen” linearized time-invariant (LTI) realizations of the process. However, they may be fully identifiable from the dynamic LPV representation [2]. This is especially important if the parameters are needed for control [3].

In this work we focus on linear (parameter-varying) state space representation of chemical processes to assess the parameter sensitivity problem using state and output sensitivities [4]. The analysis is extended to the case of parameter-varying systems, allowing us to deal with the identifiability problem using well-known system-theoretic tools to derive conditions when the model is unidentifiable. The state and output sensitivity representation also allows for a convenient way to monitor parameter sensitivity online, so more reliable parameter estimates can be computed. A simple compartmental model and a CSTR examples are shown to illustrate the sensitivity analysis results.

Acknowledgments: This work has been done within the INSPEC project with the support of the Institute for Sustainable Process Technology (ISPT).

[1] Amrhein, M., Bhatt, N., Srinivasan, B., & Bonvin, D. (2010). “Extents of reaction and flow for homogeneous reaction systems with inlet and outlet streams”. AIChE journal, 56(11), 2873-2886.

[2]Mohammadpour, J., & Scherer, C. W. (Eds.). (2012). “Control of linear parameter varying systems with applications”. Springer Science & Business Media.

[3] Marquez-Ruiz, A., Mendez-Blanco, C. S., & Özkan, L. (2018). “Control of homogeneous reaction systems using extent-based LPV models”. IFAC-PapersOnLine, 51(18), 548-553.

[4] Stigter, Johannes D., and Karel J. Keesman. “Optimal parametric sensitivity control of a fed-batch reactor.” Automatica 40.8 (2004): 1459-1464.