Automatica 45 (2009) 1542–1548 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Brief paper LQ control design of a class of hyperbolic PDE systems: Application to fixed-bed reactor ✩ Ilyasse Aksikas a,∗ , Adrian Fuxman a , J. Fraser Forbes a , Joseph J. Winkin b a Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB, Canada, T6G 2G6 b Department of Mathematics, University of Namur, Belgium article info Article history: Received 12 April 2007 Received in revised form 28 December 2008 Accepted 18 February 2009 Available online 20 March 2009 Keywords: Linear-quadratic regulator Hyperbolic partial differential equations Infinite-dimensional systems Fixed-bed reactors Closed-loop stability abstract A general linear controller design method for a class of hyperbolic linear partial differential equation (PDEs) systems is presented. This is achieved by using an infinite-dimensional Hilbert state-space description with infinite-dimensional (distributed) input and output. A state LQ-feedback operator is computed via the solution of a matrix Riccati differential equation in the space variable. The proposed method is applied to a fixed-bed chemical reactor control problem, where one elementary reaction takes place. An optimal controller is designed for linearized fixed-bed reactor model, it is applied to the original nonlinear model and the resulting closed-loop stability is analyzed. Numerical simulations are performed to show the performance of the designed controller. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction Many unit operations in chemical plants include transport pro- cesses that can best be described by partial differential equa- tions (PDEs): see Christofides (2001), Dochain (1994) and Laabissi, Achhab, Winkin, and Dochain (2001). When diffusive transport is negligible and convective transport is dominant, processes can be described by first-order hyperbolic PDEs. The class of such processes includes tubular reactors, in which the medium is not homogeneous (e.g., fixed-bed reactors). Significant research efforts have been focused on the develop- ment of control methods for PDE systems. In order to respect the distributed nature of these systems, many approaches have been developed to control infinite-dimensional systems, e.g. linear- quadratic methods (see Bensoussan, Da Prato, Delfour, and Mit- ter (2007) and Curtain and Zwart (1995)), sliding mode control techniques (see Orlov and Utkin (1987)) and flatness-based control approach (see Lynch and Rudolph (2000)). ✩ The material in this paper was partially presented at European Control Conference, Kos, Greece, 2007. This paper was recommended for publication in revised form by Associate Editor Nicolas Petit under the direction of Editor Miroslav Krstic. ∗ Corresponding author. Tel.: +1 780 492 6238; fax: +1 780 492 2881. E-mail addresses: aksikas@ulaberta.ca (I. Aksikas), afuxman@ualberta.ca (A. Fuxman), fraser.forbes@ualberta.ca (J.F. Forbes), joseph.winkin@fundp.ac.be (J.J. Winkin). Linear-Quadratic (LQ) optimal control plays a crucial role in the control literature. It is known that the solution of the LQ optimal control problem for infinite-dimensional systems can be obtained by two popular methods. One is based on the solution of an algebraic operator Riccati equation for state-space model (see e.g. Curtain and Zwart (1995)). This method was recently applied to a particular class of hyperbolic PDEs: see Aksikas, Winkin, and Dochain (2008). The second alternative, called spectral factorization, derives the feedback operator via the solution of an operator Diophantine equation: see Callier and Winkin (1990, 1992). This method was recently applied to a nonisothermal plug flow reactor in order to regulate the temperature and the reactant concentration in the reactor: see Aksikas, Winkin, and Dochain (2007). In Aksikas et al. (2008), it was assumed that all the coefficients in the convective transport term are identical in all equations. In this paper, the work (Aksikas et al., 2008) is extended to a more general class of hyperbolic PDEs by using an infinite-dimensional Hilbert state-space description with infinite- dimensional (distributed) input and output and the new approach is applied to a fixed-bed reactor model (Aksikas & Forbes, 2007). The aim of this paper is to design a spatially distributed control so that the distributed state x(z , t ) is driven in an optimal fashion towards the equilibrium spatial profile of the state variables i.e. x(z , t ) → x ss . The contributions of this paper can be summarized as follows. Section 2 describes both the hyperbolic PDEs that we are interested in and their infinite-dimensional state-space formulation. In designing an optimal controller, an important exponential stability 0005-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2009.02.017