Control of Reaction Systems via Rate Estimation and Feedback Linearization

Julien Léo Billeter, Dominique Bonvin, Diogo Filipe Mateus Rodrigues
Elsevier, 2015
Conference paper

Abstract

Abstract of the conference paper The kinetic identification of chemical reaction systems often represents a time-consuming and complex task. This contribution presents an approach that uses rate estimation and feedback linearization to implement effective control without a kinetic model. The reaction rates are estimated by numerical differentiation of reaction variants. The approach is illustrated in simulation through the temperature control of a continuous stirred-tank reactor. Extended abstract Model identification and controller design are often seen as closely related tasks, since the control law is calculated using the plant model. Previous control approaches based on extensive variables or inventories are examples of this strong dependence on the model [1, 2]. Since the identification of chemical reaction systems can be a time-consuming and complex task, one would ideally like to avoid it as much as possible. The concept of variant and invariant states allows isolating the different rates in chemical reaction systems, thereby facilitating analysis, monitoring and control [3-5]. Using this concept, one can estimate dynamic effects without the need of identifying the corresponding kinetic models. This contribution presents a feedback linearization approach that is based on the estimation of unknown rates, such as the rates of reaction and mass transfer, thus allowing efficient control without the use of kinetic models. Rate estimation uses the numerical differentiation of appropriately transformed extensive variables called rate variants that are invariant with respect to the manipulated variables. A rate variant contains all the information about the corresponding rate and, as such, is decoupled from the other unknown rates. Since it is possible to estimate the unknown rates this way, the controller does not require kinetic information. However, because of the differentiation step, the controller is most effective with frequent and precise measurements of several output variables. Feedback linearization sets a rate of variation for the controlled variables, thereby guaranteeing quick convergence of these variables to their set points. For open chemical reactors, the parameters of the feedback linearization controller are determined by readily available information, such as the reaction stoichiometry, the heats of reaction, the inlet composition or the inlet and outlet flow rates. This novel control strategy is illustrated in simulation for the control of both concentration and temperature in a continuous stirred-tank reactor. [1] Georgakis, Chem. Eng. Sci., 1986, 41, 1471 [2] Farschman et al., AIChE J., 1998, 44, 1841 [3] Asbjørnsen and Fjeld, Chem. Eng. Sci., 1970, 25, 1627 [4] Bhatt et al., Ind. Eng. Chem. Res., 2011, 50, 12960 [5] Srinivasan et al., IFAC Workshop on Thermodynamic Foundations of Mathematical Systems Theory, Lyon, 2013.

Official source

https://infoscience.epfl.ch/record/201512?ln=en

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts

Related publications

Julien Léo Billeter, Dominique Bonvin, Diogo Filipe Mateus Rodrigues
Elsevier, 2015
Conference paper

Abstract

Official source

https://infoscience.epfl.ch/record/201512?ln=en

About this result

Related concepts (20)

A chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the posit

Chemical reaction engineering (reaction engineering or reactor engineering) is a specialty in chemical engineering or industrial chemistry dealing with chemical reactors. Frequently the term relates

The kinetic theory of gases is a simple, historically significant classical model of the thermodynamic behavior of gases, with which many principal concepts of thermodynamics were established. The

Related concepts

Related publications (45)

Related publications

, ,

Many chemical and biochemical industries utilize chemical reaction processes to get desired products from raw materials. Common examples include reaction systems used to manufacture pharmaceutical drugs, vaccines and other chemicals. A chemical reaction system is a complex combination of various rate processes. Apart from the chemical reactions, these systems may include (i) mass transfers between phases, and (ii) heat transfer due to heating and cooling. Also, the reactor can be operated in a continuous mode, which adds additional mass transport due to the inlet and outlet streams. The measured state variables, namely concentrations, temperature and mass, are functions of the underlying reactions, mass transfer between phases, and mass transport due to the inlet and outlet streams. Since these variables are highly coupled and contain effects of all the phenomena, the analysis of reactor performance based on these variables is highly complex. Proper understanding of the reaction system is necessary for process design, control and optimization. The analysis of the chemical reaction systems can be simplified, if a transformation can be made from the measured state variables into alternate variables (named variants) that each describe the dynamic behavior of the reactions, mass and heat transfers, inlets and outlets. The transformed states also include variables that are invariant with respect to time and remain constant during the course of the reaction. A number of applications of the reaction variants and invariants have been studied in the literature [1-7]. Srinivasan et al. [1] have discussed possible applications of reaction and flow variants/invariants for control-related tasks such as model reduction, state accessibility, state reconstruction, and feedback linearizability. Reaction invariants have been used to study the state controllability and observability of continuous stirred-tank reactors [2]. Reaction invariants have also been used to automate the formulation of mole balance equations for the non-reacting part of complex processes (mixing and splitting operations), thereby determining the degrees of freedom for process synthesis [3]. Furthermore, Waller and Makila [4] demonstrated the use of reaction invariants to control pH, assuming that equilibrium reactions are very fast. Gruner et al. [5] showed that, through the use of reaction invariants, the dynamic of reaction-separation processes with fast (equilibrium) reactions resembles the dynamic of corresponding non-reactive systems in a reduced set of transformed variables. Aggarwal et al. [6] considered multi-phase reactors operating at thermodynamic equilibrium and were able to use the concept of reaction invariants, which they called invariant inventories, to reduce the order of dynamic models and to design control strategies accordingly. Recently, Scott and Barton [7] have used reaction invariant relationships to compute bounds for kinetic models of a chemical reaction system operated under batch conditions. While the applications of reaction variants and invariants are clear, the challenge still remains in finding these relationships in the presence of different physical phenomena. Asbjornsen and co-workers [8] introduced a methodology for computing reaction variants and invariants and used it for reactor modeling and control. However, for open reactors, the reaction variants proposed are not fully representative of reactions since they are also affected by the inlet and outlet flows. Friedly [9] proposed to compute the variants of equivalent batch reactions, associating the remainder to transport processes, and used them to describe the dynamics of flow through porous media accompanied by chemical reactions. For open homogeneous reaction systems, Srinivasan et al. [1] developed a nonlinear transformation of the numbers of moles to reaction variants, flow variants, and reaction and flow invariants, thereby separating the effects of reactions and flows. Later, the same authors refined that transformation to make it linear [10] (at the price of losing the one-to-one property) and therefore more easily applicable. They also showed that, for a reactor with an outlet flow, the concept of vessel extent is most useful, as it represents the amount of material associated with a given process (reaction, exchange) that is still in the vessel. Bhatt et al. [11] extended that concept to heterogeneous G-L reaction systems for the case of no reaction and no accumulation in the film, the result being decoupled vessel extents of reaction, mass transfer, inlet and outlet, as well as true invariants, i.e. identically equal to zero. The use of this linear transformation for the task of kinetic identification has been clearly demonstrated [12] and this transformation has been extended to typical measured signals (calorimetry and spectroscopy) [13, 14]. Srinivasan et al. [15] have recently introduced a new transformation that gives the same variants and invariants as the one proposed in [10]. This new transformation, which is conceptually and computationally much simpler, also gives explicit relationships for reaction invariants in terms of the measured state variables. The invariant relationships from the transformation can now directly be used for process monitoring, control and optimization. This transformation also allows computing the reaction invariants in the presence of additional measurements (e.g. flow rates) and can be extended to heterogeneous systems. This contribution will first extend the computation of reaction invariants of homogeneous reaction systems when additional measurements (flow rates) are available. The procedure for writing the invariant relationships will then be adapted to F–F reaction systems with (i) reactions in the bulk of both phases, (ii) unsteady–state mass transfer between phases, and (iii) mass transport due to inlet and outlet streams in both phases. The procedure for writing the reaction invariant relationships will be demonstrated with suitable examples. [1] Srinivasan et al., AIChE J. 44(8), 1858-1867, 1998 [2] Fjeld et al., Chem. Eng. Sci. 29, 1917-1926, 1974 [3] Gadewar et al., Ind. Eng. Chem. Res. 41, 3771-3783, 2002 [4] Waller et al., Ind. Eng. Chem. Process Des. Dev. 20, 1-11, 1981 [5] Gruner et al., AIChE J. 52, 1010-1026, 2006 [6] Aggarwal et al., J. Process Contr. 21, 1390-1406, 2011 [7] Scott et al., Comp. Chem. Eng. 34(5), 717-731, 2010 [8] Asbjornsen et al., Chem. Eng. Sci. 27, 709-717, 1972 [9] Friedly, AIChE J. 37(5), 687-693, 1991 [10] Amrhein et al., AIChE J. 56, 2873, 2010 [11] Bhatt et al., Ind. Eng. Chem. Res. 49, 7704-7717, 2010 [12] Bhatt et al., Chem. Eng. Sci. 83, 24-38, 2012 [13] Srinivasan et al., Chem. Eng. J. 207-208, 785-793, 2012 [14] Billeter et al., Anal. Chim. Acta 767, 21-34, 2013 [15] Srinivasan et al., 1st IFAC Workshop on Thermodynamic Foundations of Mathematical Systems Theory (TFMST), Lyon (France), 2013

2014

Incremental Model Identification of Fluid-Fluid Reaction Systems – Dynamic Accumulation and Reactions in the Diffusion Layer

Julien Léo Billeter, Dominique Bonvin, Sriniketh Srinivasan

The identification of kinetic models is an important step for the monitoring, control and optimization of industrial processes. This is particularly the case for highly competitive business sectors such as chemical and pharmaceutical industries, where the current trend of changing markets and strong competition leads to a reduction in the process development costs [1]. Moreover, the PAT initiative of the FDA advocates a better understanding and control of manufacturing processes by the use of modern instrumental technologies and innovative software solutions [2]. Reaction systems can be represented by first-principles kinetic models that describe the time evolution of states – numbers of moles, temperature, volume, pressure – by means of conservation and constitutive equations of differential and algebraic nature. These models are designed to include all kinetic phenomena, whether physical or chemical, involved in the reaction systems. Generally, such kinetic phenomena include the dynamic effects of reactions (stoichiometry and reaction kinetics), transfer of species between phases (mass-transfer rates), and operating conditions (initial conditions as well as inlet and outlet flows). The identification of reaction and mass-transfer rates as well as the estimation of their corresponding rate parameters represents the main challenge in building first-principles models. The task of identification is commonly performed in one step via ‘simultaneous identification’, in which a dynamic model comprising all rate effects is postulated, and the corresponding model parameters are estimated by comparing the measured and modeled concentrations [3]. This procedure is repeated for all combinations of model candidates, and the combination with the best fit is usually selected. The main advantage of this identification method lies in its capability to model complex dynamic effects in a concomitant way and thus to generate enough constraints in the optimization problem so that indirect measurements such as spectroscopic and calorimetric data can be modeled without the use of a calibration step [4, 5]. However, the simultaneous approach can be computationally costly when several candidates are available for each dynamic effect. Furthermore, this method often leads to high parameter correlation with the consequence that any structural mismatch in the modeling of one part of the model can result in errors in all estimated parameters and, in addition, convergence problems can arise from a poor choice of initial guesses [6, 7]. As an alternative, the incremental approach decomposes the identification task into a set of sub-problems of lower complexity [8]. The approach consists in transforming the measured concentrations into decoupled rates or extents, which can then be modeled individually. When needed, prior to the modeling step, the missing or unmeasured states can be reconstructed using the computed rates or extents. In the ‘rated-based incremental identification’ [9], rates are first obtained by differentiation of concentration measurements. Then, postulated rate expressions and rate parameters are estimated one at a time by comparing the measured and modeled rates. However, because of the bias introduced in the differentiation step, the rate parameters estimated by this method are not statistically optimal. That is why, another approach, termed ‘extent-based incremental identification’ [10], that is based on the integral method of parameters estimation has been introduced. In this approach, extents are first computed from measured concentrations, and then postulated rate expressions are integrated individually for each extent and the corresponding rate parameters estimated by comparing the measured and modeled extents. The extent-based identification can also be adapted to analyze calorimetric and spectroscopic data using a calibration step [11, 12]. The transformation to rates or extents reduces the dimensionality of the dynamic model since all redundant states (invariants) can be discarded. More importantly, the remaining states (variants) isolate the effects of the reactions, mass transfers and operating conditions, which can then be analyzed individually [13]. This allows substantially reducing the computational effort, the convergence problems and the correlation between the estimated rate parameters. Recently, the extent-based incremental identification has been extended to fluid-fluid reaction systems undergoing unsteady-state mass transfer and reactions at the interface of the two immiscible phases. This situation is commonly encountered in reaction systems that are limited by diffusion, such as CO2 post-combustion capture and nitration reactions. Such reaction systems can be modeled using the film theory, where the two bulks are separated by a spatially distributed film, located in either of the two phases, in which diffusing species can accumulate and react. In both bulks, the mass balance equations describing the dynamics of chemical species are expressed as ordinary differential equations (ODE) and serve as boundary conditions for the film. The dynamic accumulation in the film is described by Fick’s second law combined with a reaction term, thus leading to partial differential equations (PDE), which can be solved by appropriate spatial discretization and rearrangement in ODEs. The extent-based model identification of fluid-fluid reaction systems with unsteady-state mass transfer and reactions requires a large number of measurements for reconstructing all the states and modeling the dynamics of the film [14]. The difficulty lies in the fact that, with the current state of sensor technology, such measurements can only come from the two homogeneous bulks, which provide information from a well-mixed reactor region and consequently are resolved only in time and not in space. Nevertheless, extents of reaction and extents of mass transfer can be extracted from these bulk measurements. These extents of reaction represent the effect of slow reactions that take place in the bulks of the two phases and can be modeled as before. On the other hand, the extents of mass transfer represent now the combined effect of mass transfer by diffusion through the film and of fast reactions taking place at the interface or in the film. Hence, both the diffusion coefficients and the rate constants of the fast reactions can be estimated by comparing the measured extents of mass transfer and the extents obtained by solving the corresponding PDEs. In the absence of coupling terms in the PDEs due to interactive diffusion and/or reactions, the diffusion coefficients of each species transferring through the film can be estimated incrementally. However, in the case of interactive diffusion and/or reactions, the interdependence of species via the coupling terms of the PDEs calls for a simultaneous identification of the diffusion coefficients and rate constants within the film. This contribution extends the extent-based incremental identification to the analysis of reaction systems with dynamic accumulation and reactions in the film. In particular, the question of whether to use incremental or simultaneous estimation of the diffusion coefficients and rate constants within a diffusion layer will be addressed. [1] Workman et al., Anal. Chem. 83, 4557-4578, 2011 [2] Billeter et al., 100th AIChE Annual Meeting, Philadelphia, 2008 [3] Hsieh et al., Anal. Chem., http://dx.doi.org/10.1021/ac302766m, 2013 [4] Billeter et al., Chemom. Intell. Lab. Syst. 95(2), 170-187, 2009 [5] Zogg et al., Thermochimica Acta 419, 1-17, 2004 [6] Billeter et al., Chemom. Intell. Lab. Syst. 93(2), 120-131, 2008 [7] Billeter et al., Chemom. Intell. Lab. Syst. 98(2), 213-226, 2009 [8] W. Marquardt, Chem. Eng. Res. Des., 83(A6), 561–573, 2005 [9] Brendel et al., Chem. Eng. Sci. 61, 5404-5420, 2006 [10] Bhatt et al., Ind. Eng. Chem. Res. 50, 12960-12974, 2011 [11] Srinivasan et al., Chem. Eng. J. 207-208, 785-793, 2012 [12] Billeter et al., Anal. Chim. Acta 767, 21-34, 2013 [13] Srinivasan et al., IFAC Workshop TFMST, Lyon, 2013 [14] Billeter et al., 104th AIChE Annual Meeting, Pittsburgh, 2012

2013

Concept of Variants and Invariants for Reaction Systems, with Application to Estimation, Control and Optimization

Diogo Filipe Mateus Rodrigues

The concept of reaction variants and invariants for lumped reaction systems has been known for several decades. Its applications encompass model identification, data reconciliation, state estimation and control using kinetic models. In this thesis, the concept of variants and invariants is extended to distributed reaction systems and used to develop new applications to estimation, control and optimization. The thesis starts by reviewing the material and heat balances and the concept of variants and invariants for several lumped reaction systems. Different definitions of variants and invariants, in particular the vessel extents, are presented for the case of homogeneous reaction systems, and transformations to variants and invariants are obtained. The extension to systems with heat balance and mass transfer is also reviewed. The concept of extents is generalized to distributed reaction systems, which include many processes involving reactions and described by partial differential equations. The concept of extents and the transformation to extents are detailed for various configurations of tubular reactors and reactive separation columns, as well as for a more generic framework that is independent of the configuration. New developments of the extent-based incremental approach for model identification are presented. The approach, which compares experimental and modeled extents, results in maximum-likelihood parameter estimation if the experimental extents are uncorrelated and the modeled extents are unbiased. Furthermore, the identification problem can be reformulated as a convex optimization problem that is solved efficiently to global optimality. The estimation of unknown rates without the knowledge or the identification of the rate models is described. This method exploits the fact that the variants computed from the available measurements allow isolating the different rates. Upon using a Savitzky-Golay filter for differentiation of variants, one can show that the resulting rate estimator is optimal and obtain the error and variance of the rate estimates. The use of variants and invariants for reactor control is also considered. Firstly, offset-free control via feedback linearization is implemented using kinetic models. Then, it is shown how rate estimation can be used for control via feedback linearization without kinetic models. By designing an outer-loop feedback controller, the expected values of the controlled variables converge exponentially to their setpoints. This thesis presents an approach to speed up steady-state optimization, which takes advantage of rate estimation without rate models to speed up the estimation of steady state for imperfectly known dynamic systems with fast and slow states. Since one can use feedback control to speed up convergence of the fast part, rate estimation allows estimating the steady state of the slow part during transient operation. The application to dynamic optimization is also shown. Adjoint-free optimal control laws are computed for all the types of arcs in the solution. In the case of reactors, the concept of extents allows the symbolic computation of optimal control laws in a systematic way. A parsimonious input parameterization is presented, which approximates the optimal inputs well with few parameters. For each arc sequence, the optimal parameter values are computed via numerical optimization. The theoretical results are illustrated by simulated examples of reaction systems.

EPFL2018