Model predictive control of a non-isothermal axial dispersion tubular reactor with recycle

https://doi.org/10.1016/j.compchemeng.2020.107159Get rights and content

Highlights

  • Model predictive control of a non-isothermal axial dispersion tubular reactor with recycle.

  • The proposed controller is designed to manipulate the system to operate at an unstable condition despite physical limitations and input disturbance.

  • The design of the controller accounts for different mass and energy Peclet numbers in the axial dispersion tubular reactor.

  • The discrete version of the model is based on the application of energy-preserving Cayley-Tustin time discretization scheme.

Abstract

This paper addresses the model predictive output controller design for a non-isothermal axial dispersion tubular reactor accounting for energy and mass transport in a recycle stream. The underlying transport-reaction process is characterized by different mass and energy Peclet numbers. Open-loop analysis reveals unstable operating conditions based on the reactor’s parameters. The discrete model of the system is obtained by considering a Crank-Nicolson type of discretization method without any model reduction and/or spatial approximation for the system of coupled parabolic PDEs. The proposed predictive controller accounts for asymptotic close-loop system stabilization and/or naturally present input and state constraints with the rejection of possible disturbances arising from reactor operations. To account for the output controller design, a discrete observer is developed to reconstruct the infinite dimensional states in the predictive control realization. Finally, the controller’s performance is assessed via simulation studies, implying proper state stabilization and constraints satisfaction with input disturbance rejection.

Introduction

Many transport-reaction processes present in petrochemical, biochemical and pharmaceutical unit operations belong to distributed parameter system (DPS) models. Within a finite spatial domain, the mathematical formulation of the mentioned processes, arising from the first-principle modeling, usually takes the form of a set of parabolic partial differential equations (PDEs) in which the intrinsic feature of reaction, convection and diffusion phenomena can be captured. The non-isothermal axial dispersion reactors have attracted great attention since their models account for a large number of reactor realizations in industry (see Varma and Aris, 1977 for a survey). In particular, the dynamical properties and control of axial dispersion tubular reactors, where mass and thermal phenomena are taking place, has been the objective of numerous studies over the years (see Hlavacek, Hofmann, 1970, Hlavacek, Hofmann, 1970, Cohen, Poore, 1974, Georgakis, Aris, Amundson, 1977, Bokovi, Krsti, 2002), as several complexities are observed in the dynamic description and in the practical realization of the reactor operation (Marwaha and Luss, 2003). In addition, when a dispersive chemical tubular reactor is considered with a recycle of energy and mass flow, the model analysis and subsequent controller design become more challenging and require careful consideration since competitive effects of mass and heat transfer are present in the recycle, which may induce instability in apparently stable reactor operations (Luss and Amundson, 1967).

The salient feature of the axial dispersion reactor model is the mathematical description given by parabolic partial differential equations (PDEs), which can admit a variety of boundary modeling realizations to account for the physically meaningful setting present in the industry. Among several modeling realizations, the Danckwerts’ boundary conditions (Danckwerts, 1953) reflect the physically relevant inlet flux transport and zero flux conditions at the reactor outlet. In general, the reactor models are numerically solved by the application of standard (high-order) and reduced-order PDE to ordinary differential equation (ODE) including finite difference discretization schemes (Badillo-Hernandez et al., 2019), methods relying on dynamically moving the discretization mesh to minimize the discretization error (Liu and Jacobsen, 2004), or by applying a variety of spectral methods, such as proper orthogonal decomposition (POD), to study oscillatory reactors’ regimes (Bizon et al., 2008). Along with the modeling efforts, the controller designs and associated spatial discretization techniques are proposed for PDEs to obtain sets of ordinary differential equations (ODEs), and then the reduced models (if possible) are utilized for the synthesis of finite dimensional controllers (see Balas, 1979, Ray, 1981, Curtain, 1982, Antoniades, Christofides, 2000). The significant drawback of this approach, notably when it comes to parabolic PDEs, is the order of the discretization used (in the case of finite difference methods applied to approximate the spatial derivatives) and/or number of modes that must be considered to reach the desired order of model approximation, which may lead to high dimensional controllers that are difficult to be implemented. Consequently, the regulator design for dissipative PDEs has been the objective of many studies over the years. Different design methodologies were considered, such as approximate inertial manifolds, dynamic optimization, robust control, and linear quadratic control in the frequency domain (e.g. Christofides, Daoutidis, 1997, Armaou, Christofides, 2002, Aksikas, Moghadam, Forbes, 2017 to cite a few). In the same vein, there are several contributions focused on the stability and dynamical properties of the axial dispersion tubular reactors with multiple steady states (Jensen, Ray, 1982, Dochain, 2018, Bildea, Dimian, Cruz, Iedema, 2004).

Although the stabilization of the system is investigated in the aforementioned studies, the issue of input and state constraints, which are naturally present in the process, is generally not considered for the transport-reaction systems with recycle streams. Hence, within the optimal framework, the model predictive control (MPC), or the so-called online receding horizon control, is introduced by control practitioners to compute the required manipulated variable that optimizes the open-loop performance objective subjected to constraints (Muske and Rawlings, 1993). A significant number of contributions have been focused on properties of MPC controllers for parabolic PDEs, including spatial discretization methods, constraints validation with system performance for a class of the Riesz spectral systems with separable spectrums, piecewise predictive feedback control, and data-based modeling using repeated online linearization (Dufour, Tour, Blanc, Laurent, 2003, Dubljevic, El-Farra, Mhaskar, Christofides, 2006, Bonis, Xie, Theodoropoulos, 2012).

For the implementation of digital controllers, the discrete version of the overall system is mainly required for control realizations. Over the years, for the conversion of the models or controllers into a discrete time setting, classical methods such as explicit or implicit Euler and Runge-Kutta methods are usually taken into account. It is proven from linear system theory that by increasing the sampling period, the accuracy of the discretization can be degraded, which leads to the mapping of stable continuous system into an unstable discrete one (Aström, Wittenmark, 1990, Kazantzis, Kravaris, 1999). This issue becomes more prominent when it comes to distributed parameter systems, given that the infinite dimensional state-space control realization needs to be accounted for in the controller design. Motivated by the aforementioned issues, this work considers a robust and accurate transformation of a continuous linear infinite dimensional system to a discrete one by the application of the Cayley-Tustin time discretization technique (Crank-Nicolson midpoint integration rule), in which the system properties and intrinsic energy of the DPS model are preserved (i.e., Hamiltonian preserving) (V.Havu, Malinen, 2007, Xu, Dubljevic, 2017). In addition, the practical realizations usually account for the output controller designs (Xie et al., 2019) due to the fact that infinite dimensional system states cannot be directly measured. Hence, the discrete observer developed in this work is based on the Cayley-Tustin method, which does not account for any model reduction or/and spatial approximation, as the discretization of the underlying operators is the usual procedure found in the literature to reconstruct the state variables given by transport-reaction processes (e.g. Dochain, 2001, Dochain, 2000, Mohd Ali, Ha Hoang, Hussain, Dochain, 2015, Alonso, Kevrekidis, Banga, Frouzakis, 2004, Bitzer, Zeitz, 2002).

In this manuscript, the claimed novelty is the extension of linear MPC designs for the finite-dimensional system (Mayne, 2014, Rawlings, Mayne, Diehl, 2017) to the the infinite-dimensional one. Particularly, the transport-reaction system described by a system of coupled parabolic PDEs is considered, which might represent a non-isothermal axial dispersion tubular reactor with mass and thermal recycle flow. In addition, the proposed findings provide an insight into the dynamical properties of the system since the model of interest considers different mass and heat Peclet numbers in the spatial operators, accounting for the distinction between heat and mass transport phenomena. Moreover, different values of mass and thermal Peclet numbers lead to increased complexity in the model’s representation and make it difficult to find the analytic solution for the eigenvalues and corresponding eigenfunctions. Due to the fact that one cannot realize the measurement of mass and temperature along the reactor, a discrete observer for the system of parabolic PDEs is proposed. The discrete-time observer design accounts for the available output measurement taken at the exit of the reactor (considered to be the reactor temperature) and reconstructs the system’s states. Finally, the controller design provides optimal stabilization of the system with the inclusion of state and input constraints, as well as input disturbance rejection in the control law.

The manuscript is organized as follows: Section 2 addresses the model description of the axial dispersion tubular reactor with recycle. In Section 3, the linearized system is derived and the system is defined in an appropriate abstract Hilbert space. This is followed, in Section 4, by the linear system stability analysis and obtaining the relevant eigenfunctions and adjoint eigenfunctions of the system. In Section 5, the time discretization of the overall system is accomplished by the Cayley-Tustin technique. Then, the discrete observer design is provided in Section 6, while the optimization problem for coupled unstable parabolic PDEs is presented in Section 7. Finally, the performance of the proposed controller is demonstrated with a simulation study in Section 8.

Section snippets

Model representation

The chemical process shown in Fig. 1 represents a non-isothermal tubular reactor involving convection, molecular diffusion with macroscopic back mixing (dispersion) (Levenspiel, 1999), and a first-order irreversible reaction AB, where the reaction is considered to be exothermic. After passing through a separator, the unreacted component A is recycled and fed back to the tubular reactor. The dynamics of the system can be directly deduced from energy and mass balances on a slice with

System linearization

Consider the following deviation variables:[x1(ζ,t)=m1(ζ,t)m1ss(ζ)x2(ζ,t)=m2(ζ,t)m2ss(ζ)]we assume the cooling jacket temperature (Tc(t)) as the manipulated input variable and the reactor outlet temperature as the measured output of the system, which is required to reconstruct the states in the subsequent observer design. Then, the new input and output can be defined as follows:[u(t)=Tw(t)Twssy(t)=m2(ζ=1,t)m2ss(ζ=1)]

The reaction rate is linearized around its steady-state as it depends on

Linear system stability analysis (PeTPem)

Based on the linear system representation, it is possible to analyze the internal stability by solving the eigenvalue problem associated with the system, see Eq. (11). In particular, the stability assessment is performed with different settings of the Peclet numbers, which is studied less often in the literature (see Hastir et al., 2020), as the constant value for Peclet numbers (PeT=Pem) implies the assumption of same transport properties of the mass and heat flow in the axial dispersion

Discrete time representation

In this subsection, for the mentioned coupled convection-diffusion parabolic PDEs, mapping the continuous time setting to discrete one is considered by the application of the Cayley-Tustin time discretization method. Let us consider Δt as the sampling time, then by applying Crank-Nicolson time discretization to Eq. (12), one can derive the following:x(jΔt)x((j1)Δt)ΔtAx(jΔt)+x((j1)Δt)2+Bu(jΔt),j1y(jΔt)Cx(kΔt)+x((k1)Δt)2+Du(kΔt),j1Next, we consider u(jΔt)/Δt as the approximation of u(jΔt),

Observer design for coupled parabolic PDEs

To address the issue of having access to all the state variables, the discrete output observer is considered in this work. The Luenberger observer is one of the practical and easy to realize observers, which is considered in a discrete modern controller realizations. However, the discrete operators cannot be directly used in calculating the observer gain in discrete-time setting. Therefore, the design is performed first in the continuous time setting. Then, the discrete observer gain is

Model predictive control for unstable coupled parabolic PDEs

This section addresses the design of the proposed model predictive controller for the coupled parabolic PDEs. The discrete-time model dynamics with eigenfunctions of the system are used to find the solution of the optimization problem.

Simulation results

This section is dedicated to the implementation of the model predictive controller for the axial dispersion tubular reactor with recycle, see the scheme in Fig. 4. The particular choice of parameters given by Table 1 leads to three equilibria for the system as shown in Fig. 2. The outer two profiles are stable, while the middle one is unstable and will be considered in the simulation study. Based on the model predictive control designed in Section 7, our objective is to stabilize the system

Conclusion

In this contribution, the design of a model predictive controller and discrete observer were investigated for an axial dispersion tubular reactor with recycle. The discrete version of the overall system is provided by the application of the Cayley-Tustin time discretization on the linearization of a coupled nonlinear convection-diffusion-reaction PDEs system. An unstable operating condition with different energy and mass Peclet numbers is considered for the tubular reactor. The discrete

CRediT authorship contribution statement

Seyedhamidreza Khatibi: Software, Writing - original draft, Writing - review & editing, Methodology. Guilherme Ozorio Cassol: Writing - review & editing, Methodology. Stevan Dubljevic: Writing - review & editing, Methodology, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

The support provided by CAPES - 88881.128514/2016-01 (Brazil) for Guilherme Ozorio Cassol is gratefully acknowledged.

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