Journal of Process Control 23 (2013) 1508–1514 Contents lists available at ScienceDirect Journal of Process Control jou r n al hom ep age: www.elsevier.com/locate/jprocont Optimal control of an advection-dominated catalytic fixed-bed reactor with catalyst deactivation I. Aksikas a,∗ , L. Mohammadi b , J.F. Forbes b , Y. Belhamadia c , S. Dubljevic b a Department of Mathematics, Statistics and Physics, Qatar University, Doha, Qatar b Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Canada c Campus Saint-Jean and Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada a r t i c l e i n f o Article history: Received 29 February 2012 Received in revised form 21 September 2013 Accepted 21 September 2013 Available online 31 October 2013 Keywords: Fixed bed reactor Infinite dimensional time-varying system Linear quadratic optimal control Catalyst deactivation a b s t r a c t The paper focuses on the linear-quadratic control problem for a time-varying partial differential equa- tion model of a catalytic fixed-bed reactor. The classical Riccati equation approach, for time-varying infinite-dimensional systems, is extended to cover the two-time scale property of the fixed-bed reac- tor. Dynamical properties of the linearized model are analyzed using the concept of evolution systems. An optimal LQ-feedback is computed via the solution of a matrix Riccati partial differential equation. Numerical simulations are performed to evaluate the closed loop performance of the designed controller on the fixed-bed reactor. The performance of the proposed controller is compared to performance of an infinite dimensional controller formulated by ignoring the catalyst deactivation. Simulation results show that the performance of the proposed controller is better compared to the controller ignoring the catalyst deactivation when the deactivation time is close to the resident time of the reactor. © 2013 Elsevier Ltd. All rights reserved. 1. Introduction Catalytic fixed-bed reactors are the most widely used reactor type and play an important role in chemical industry. The temper- ature and concentrations in this type of reactor vary both with time and space. In order to capture the effects of reaction and trans- port phenomena in the reactor, the reactor model may take the form of a set of partial differential equations (PDEs). These systems are infinite dimensional in nature and are commonly referred to as distributed parameter systems (DPS). The catalyst in a fixed bed reactor deactivates during the operation for a variety of reasons (e.g., poisoning by impurities in feed, formation of coke on catalyst surface, etc). Incorporation of catalyst deactivation in the model of the reactor, results in a time-varying infinite dimensional sys- tem. Optimal operation of catalytic reactors in a chemical plant has important effect on profitability of the plant. Therefore, designing high performance controllers which are able to regulate the reactor temperature and concentrations around the optimal values during the operation time of the reactor is crucial for chemical processes. A common approach for controlling the distributed parame- ter systems is to convert the set of PDEs to a set of ODEs using discretization techniques, for example finite difference or finite ele- ment methods. These discretization methods may result in models ∗ Corresponding author. Tel.: +974 44034615. E-mail addresses: aksikas@qu.edu.qa, aksikas@ualberta.ca (I. Aksikas). that do not capture all dynamic properties of the original system accurately. In order to have accurate finite dimensional models a very fine discretization is needed which results in high order state space systems leading to computationally demanding controllers. In recent years, research on control of DPS has focused on meth- ods that deal with infinite dimensional nature of these systems [1,2]. In the aforementioned work, distributed parameter systems are formulated in a state space form, similar to lumped parameter systems, by introducing a suitable infinite dimensional space set- ting and associated operators, which allows infinite dimensional controllers to be synthesized directly from the infinite dimensional realization of the system [1,3,4]. For diffusion–convection–reaction systems, which are described by parabolic PDEs, Christofides and co-workers studied nonlinear order reduction and control of nonlinear parabolic systems. Dubljevic et al. [5] also used modal decomposition to derive finite-dimensional systems that capture the dominant dynamics of the original PDE which are subsequently used for low dimensional controller design. In the case of first order hyperbolic systems that model convection–reaction type of processes, the eigenvalues of the spa- tial differential operator cluster along vertical or nearly vertical asymptotes in the complex plane [6], which implies that the modal decomposition techniques suitable for parabolic PDEs cannot be used. The optimal control of hyperbolic systems using spectral fac- torization which is based on frequency-domain description of the 0959-1524/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jprocont.2013.09.016