# Linear Model Predictive Control for a Coupled CSTR and Axial Dispersion Tubular Reactor with Recycle

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

#### 2.1. System Representation

#### 2.2. Open-Loop Stability

## 3. Discrete Representation

#### 3.1. Discrete Operators

#### 3.2. Discrete Adjoint Operators

## 4. Luenberger Observer Design

## 5. Model Predictive Control for Linear Coupled ODE-PDE System

#### 5.1. Optimization Problem

#### 5.2. Terminal State Penalty Operator

#### 5.3. Stability Constraint

## 6. Simulation Results

#### 6.1. Observer Design and Open-Loop Response

#### 6.2. MPC Implementation

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Ray, W. Advanced Process Control; McGraw-Hill: New York, NY, USA, 1981; p. 1984. [Google Scholar]
- Muske, K.R.; Rawlings, J.B. Model predictive control with linear models. AIChE J.
**1993**, 39, 262–287. [Google Scholar] [CrossRef] - Rawlings, J.B. Tutorial overview of model predictive control. IEEE Control Syst. Mag.
**2000**, 20, 38–52. [Google Scholar] - Eaton, J.W.; Rawlings, J.B. Model-Predictive Control of Chemical Processes. Chem. Eng. Sci.
**1992**, 47, 705–720. [Google Scholar] [CrossRef] - Richalet, J.; Rault, A.; Testud, J.; Papon, J. Model predictive heuristic control: Applications to industrial processes. Automatica
**1978**, 14, 413–428. [Google Scholar] [CrossRef] - Shang, H.; Forbes, J.F.; Guay, M. Model Predictive Control for Quasilinear Hyperbolic Distributed Parameter Systems. Ind. Eng. Chem. Res.
**2004**, 43, 2140–2149. [Google Scholar] [CrossRef] - Armaou, A.; Christofides, P. Dynamic optimization of dissipative PDE systems using nonlinear order reduction. Chem. Eng. Sci.
**2002**, 57, 5083–5114. [Google Scholar] [CrossRef] - Krstic, M.; Smyshlyaev, A. Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays. Syst. Control Lett.
**2008**, 57, 750–758. [Google Scholar] [CrossRef] - Oh, M.; Pantelides, C. A Modelling and Simulation Language for Combined Lumped and Distributed Parameter Systems. Comput. Chem. Eng.
**1996**. [Google Scholar] [CrossRef] - Mohammadi, L.; Aksikas, I.; Dubljevic, S.; Forbes, J.F. Optimal boundary control of coupled parabolic PDE–ODE systems using infinite-dimensional representation. J. Process Control
**2015**, 33, 102–111. [Google Scholar] [CrossRef] - Moghadam, A.A.; Aksikas, I.; Dubljevic, S.; Forbes, J.F. Boundary optimal (LQ) control of coupled hyperbolic PDEs and ODEs. Automatica
**2013**, 49, 526–533. [Google Scholar] [CrossRef] - Susto, G.A.; Krstic, M. Control of PDE–ODE cascades with Neumann interconnections. J. Frankl. Inst.
**2010**, 347, 284–314. [Google Scholar] [CrossRef] - Hasan, A.; Aamo, O.M.; Krstic, M. Boundary observer design for hyperbolic PDE–ODE cascade systems. Automatica
**2016**, 68, 75–86. [Google Scholar] [CrossRef] - Tang, S.; Xie, C. State and output feedback boundary control for a coupled PDE–ODE system. Syst. Control Lett.
**2011**, 60, 540–545. [Google Scholar] [CrossRef] - Krstic, M.; Smyshlyaev, A. Boundary Control of Pdes: A Course on Backstepping Designs; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 2008. [Google Scholar]
- Meglio, F.D.; Argomedo, F.B.; Hu, L.; Krstic, M. Stabilization of coupled linear heterodirectional hyperbolic PDE–ODE systems. Automatica
**2018**, 87, 281–289. [Google Scholar] [CrossRef] [Green Version] - Mayne, D.; Rawlings, J.; Rao, C.; Scokaert, P. Constrained model predictive control: Stability and optimality. Automatica
**2000**, 36, 789–814. [Google Scholar] [CrossRef] - Chen, H.; Allgöwer, F. Nonlinear Model Predictive Control Schemes with Guaranteed Stability. In Nonlinear Model Based Process Control; Springer Netherlands: Dordrecht, The Netherlands, 1998. [Google Scholar] [CrossRef]
- García, C.E.; Prett, D.M.; Morari, M. Model predictive control: Theory and practice—A survey. Automatica
**1989**, 25, 335–348. [Google Scholar] [CrossRef] - Ito, K.; Kunisch, K. Receding horizon optimal control for infinite dimensional systems. ESAIM Control. Optim. Calc. Var.
**2002**, 8, 741–760. [Google Scholar] [CrossRef] [Green Version] - Dubljevic, S.; El-Farra, N.H.; Mhaskar, P.; Christofides, P.D. Predictive control of parabolic PDEs with state and control constraints. Int. J. Robust Nonlinear Control
**2006**, 16, 749–772. [Google Scholar] [CrossRef] - Liu, L.; Huang, B.; Dubljevic, S. Model predictive control of axial dispersion chemical reactor. J. Process Control
**2014**, 24, 1671–1690. [Google Scholar] [CrossRef] - Dubljevic, S.; Christofides, P.D. Predictive control of parabolic PDEs with boundary control actuation. Chem. Eng. Sci.
**2006**, 61, 6239–6248. [Google Scholar] [CrossRef] - Bonis, I.; Xie, W.; Theodoropoulos, C. A linear model predictive control algorithm for nonlinear large-scale distributed parameter systems. AIChE J.
**2012**, 58, 801–811. [Google Scholar] [CrossRef] - Ai, L.; San, Y. Model Predictive Control for Nonlinear Distributed Parameter Systems based on LS-SVM. Asian J. Control
**2013**, 15, 1407–1416. [Google Scholar] [CrossRef] - Kazantzis, N.; Kravaris, C. Energy-predictive control: A new synthesis approach for nonlinear process control. Chem. Eng. Sci.
**1999**, 54, 1697–1709. [Google Scholar] [CrossRef] - Åström, K.J.; Wittenmark, B. Computer-Controlled Systems: Theory and Design, 2nd ed.; Prentice-Hall, Inc.: Upper Saddle River, NJ, USA, 1990. [Google Scholar]
- Havu, V.; Malinen, J. The Cayley Transform as a Time Discretization Scheme. Numer. Funct. Anal. Optim.
**2007**, 28, 825–851. [Google Scholar] [CrossRef] [Green Version] - Hairer, E.; Lubich, C.; Wanner, G. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 2006. [Google Scholar] [CrossRef] [Green Version]
- Mayne, D.Q. Model predictive control: Recent developments and future promise. Automatica
**2014**, 50, 2967–2986. [Google Scholar] [CrossRef] - Rawlings, J.; Mayne, D.; Diehl, M. Model Predictive Control: Theory, Computation, and Design; Nob Hill Publishing: Santa Barbara, CA, USA, 2017. [Google Scholar]
- Scokaert, P.O.M.; Mayne, D.Q.; Rawlings, J.B. Suboptimal model predictive control (feasibility implies stability). IEEE Trans. Autom. Control
**1999**. [Google Scholar] [CrossRef] [Green Version] - Fogler, H.S. Elements of Chemical Reaction Engineering; Prentice-Hall: Upper Saddle River, NJ, USA, 2005. [Google Scholar]
- Ozorio Cassol, G.; Dubljevic, S. Discrete Output Regulator Design for the linearized Saint-Venant-Exner model. Unpublished work.
- Xu, Q.; Dubljevic, S. Linear Model Predictive Control for Transport-Reaction Processes. AIChE J.
**2017**, 63. [Google Scholar] [CrossRef] - Curtain, R.; Zwart, H. An Introduction to Infinite Dimensional Linear Systems Theory; Springer: Berlin, Germany, 1995. [Google Scholar]

**Figure 2.**Eigenvalues distribution for unstable coupled ODE-PDE system when there is no diffusion within the tubular reactor.

**Figure 3.**Effects of diffusivity within the dispersive reactor on the eigenvalues placement for unstable coupled ODE-PDE system.

**Figure 4.**Shifting the unstable eigenvalue of the observer for different values of the observer gain.

**Figure 6.**(

**a**) Evolution of the discrete error dynamics with the value ${L}_{c}=5$ for observer gain; (

**b**) The estimated state profile evolution (${x}_{I}(\zeta ,k)$) through dispersive tubular reactor constructed on the basis of discrete-time coupled ODE-PDE system in an open-loop condition; (

**c**) Dynamics reconstruction of the scalar variable within the CSTR in an open-loop condition.

**Figure 7.**(

**I**) (a), (b) and (c) demonstrate the comparison between input profiles under model predictive control law: with and without input/state constraints, with constraints and with disturbance for $20\le t\le 25$; (

**II**) (d), (e) and (f) denote reconstructed dynamics of the scalar variable within the CSTR under model predictive law: with and without input/state constraints, with constraints and with disturbance for $20\le t\le 25$; (

**III**) Evolution of the stabilized spatial profile for the tubular reactor with all constraints.

Parameters | Values |
---|---|

v | $1.8$ |

F | 1 |

D | $0.35$ |

${a}_{1}$ | $-0.25$ |

$\psi $ | $-1$ |

${a}_{2}$ | 1 |

R | $0.5$ |

${u}^{min}$ | $-0.09$ |

${u}^{max}$ | 0 |

${{x}_{F}}^{min}$ | 0 |

${{x}_{F}}^{max}$ | $0.65$ |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Khatibi, S.; Ozorio Cassol, G.; Dubljevic, S.
Linear Model Predictive Control for a Coupled CSTR and Axial Dispersion Tubular Reactor with Recycle. *Mathematics* **2020**, *8*, 711.
https://doi.org/10.3390/math8050711

**AMA Style**

Khatibi S, Ozorio Cassol G, Dubljevic S.
Linear Model Predictive Control for a Coupled CSTR and Axial Dispersion Tubular Reactor with Recycle. *Mathematics*. 2020; 8(5):711.
https://doi.org/10.3390/math8050711

**Chicago/Turabian Style**

Khatibi, Seyedhamidreza, Guilherme Ozorio Cassol, and Stevan Dubljevic.
2020. "Linear Model Predictive Control for a Coupled CSTR and Axial Dispersion Tubular Reactor with Recycle" *Mathematics* 8, no. 5: 711.
https://doi.org/10.3390/math8050711