(149d) Model Predictive Control of an Axial Dispersion Tubular Reactor with Recycle: A Distributed Parameter System with State Delay
AIChE Annual Meeting
2023
2023 AIChE Annual Meeting
Computing and Systems Technology Division
Interactive Session: Systems and Process Control
Tuesday, November 7, 2023 - 3:30pm to 5:00pm
In many petrochemical, biochemical, and pharmaceutical unit operations, transport-reaction processes are described by distributed parameter system (DPS) models. The use of DPS models has enabled the capture of the intrinsic features of these operations, such as reactions, convection, and diffusion, within a finite spatial domain. [1] Various types of delays have also been addressed using DPS models in the same field. Therefore, the corresponding systems are typically formulated using partial differential equations (PDEs), which make them belong to the class of infinite-dimensional systems. [2]
DPS models may be addressed using either the “Early Lumping†or the “Late Lumping†method. In the early lumping approach, the original infinite-dimensional system is reduced to a finite-dimensional system. [1] However, such an approximation might result in an unwanted mismatch between the behavior of the original system and its finite-dimensional approximation. [3] Another approach to address DPS models is to use the “Late Lumping†method, treating the original system as an infinite-dimensional system and then implementing numerical methods to obtain the solutions. While using the late lumping method produces more accurate models to predict the behavior of the original system, controlling the DPS becomes mathematically challenging. [1]
Axial dispersion tubular reactors are one type of distributed parameter systems in the field of chemical and process engineering that have received a significant amount of attention from academia and industry due to their prevalence and practical complexity. Numerous works have been done on the modeling and the control of these reactors as distributed parameter systems, including the control of an axial dispersion tubular reactor with an instantaneous recycle. [4] Another important class of chemical and process engineering problems that classify as infinite-dimensional systems are those that show some aspect of delay. [5] The notion of delay in these systems may be equivalently represented by an infinite-dimensional pure transport setup. [2] Although delay systems that refer to input-output delays are well-known in the field of chemical engineering and process control, [2,6] less attention has been drawn to those systems with state delays. This may be due to the fact that such systems are assumed to be uncommon in the field of chemical and process systems engineering.
Acknowledging tubular reactors equipped with recycle as one of the most common industrial setups, it can be shown that the notion of state delay may need to be used to model such a widely used chemical engineering setup, as it takes time for the recycled flow to re-enter the reactor inlet. This is shown in the attached Figure. Thus, in this work, the behavior of a tubular reactor with recycle is modeled using an infinite-dimensional transient system of PDEs describing reaction, transport, diffusion, and state delay; addressed by the Late Lumping approach. Knowing that most modern control strategies focus on the discrete-time setting, a Cayley-Tustin transformation is used to convert the original continuous-time infinite-dimensional system to its discrete-time representation. Hence, neither spatial discretization nor model order reduction is done in the scope of this work. A model-predictive control structure has also been proposed to ensure the stability of the controlled system, as well as its ability to appropriately track given set points within a finite time horizon.
DPS models may be addressed using either the “Early Lumping†or the “Late Lumping†method. In the early lumping approach, the original infinite-dimensional system is reduced to a finite-dimensional system. [1] However, such an approximation might result in an unwanted mismatch between the behavior of the original system and its finite-dimensional approximation. [3] Another approach to address DPS models is to use the “Late Lumping†method, treating the original system as an infinite-dimensional system and then implementing numerical methods to obtain the solutions. While using the late lumping method produces more accurate models to predict the behavior of the original system, controlling the DPS becomes mathematically challenging. [1]
Axial dispersion tubular reactors are one type of distributed parameter systems in the field of chemical and process engineering that have received a significant amount of attention from academia and industry due to their prevalence and practical complexity. Numerous works have been done on the modeling and the control of these reactors as distributed parameter systems, including the control of an axial dispersion tubular reactor with an instantaneous recycle. [4] Another important class of chemical and process engineering problems that classify as infinite-dimensional systems are those that show some aspect of delay. [5] The notion of delay in these systems may be equivalently represented by an infinite-dimensional pure transport setup. [2] Although delay systems that refer to input-output delays are well-known in the field of chemical engineering and process control, [2,6] less attention has been drawn to those systems with state delays. This may be due to the fact that such systems are assumed to be uncommon in the field of chemical and process systems engineering.
Acknowledging tubular reactors equipped with recycle as one of the most common industrial setups, it can be shown that the notion of state delay may need to be used to model such a widely used chemical engineering setup, as it takes time for the recycled flow to re-enter the reactor inlet. This is shown in the attached Figure. Thus, in this work, the behavior of a tubular reactor with recycle is modeled using an infinite-dimensional transient system of PDEs describing reaction, transport, diffusion, and state delay; addressed by the Late Lumping approach. Knowing that most modern control strategies focus on the discrete-time setting, a Cayley-Tustin transformation is used to convert the original continuous-time infinite-dimensional system to its discrete-time representation. Hence, neither spatial discretization nor model order reduction is done in the scope of this work. A model-predictive control structure has also been proposed to ensure the stability of the controlled system, as well as its ability to appropriately track given set points within a finite time horizon.
[1] Ray, W. Advanced process control. (1982): 505.
[2] Curtain, R. and Zwart, H. Introduction to infinite-dimensional systems theory: a state-space approach. Vol. 71. Springer Nature, 2020.
[3] Alizadeh Moghadam A., Aksikas I., Dubljevic S., Forbes J. Boundary optimal (LQ) control of coupled hyperbolic PDEs and ODEs. Automatica. 2013;49(2):526-33.
[4] Khatibi, S., Ozorio Cassol G. and Dubljevic S. Model predictive control of a non-isothermal axial dispersion tubular reactor with recycle. Computers & Chemical Engineering 145 (2021): 107159.
[5] Chen C. Linear System Theory and Design. 4th ed. Oxford Series in Electrical and Computer Engineering. Cary, NC: Oxford University Press; 2012.
[6] Ozorio Cassol G., Ni D. and Dubljevic S. Heat exchanger system boundary regulation. AIChE Journal. 2019;65(8):e16623.