International Journal of Control Vol. 82, No. 7, July 2009, 1219–1228 Optimal control design for time-varying catalytic reactors: a Riccati equation-based approach Ilyasse Aksikas * , J. Fraser Forbes and Youssef Belhamadia Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB, Canada (Received 27 June 2008; final version received 20 September 2008) The linear quadratic (LQ) optimal control problem is studied for a partial differential equation model of a time- varying catalytic reactor. First, the dynamical properties of the linearised model are studied. Next, an LQ-control feedback is computed by using the corresponding operator Riccati differential equation, whose solution can be obtained via a related matrix Riccati partial differential equation. Finally, the designed controller is applied to the non-linear reactor system and tested numerically. Keywords: catalytic reactors; evolution systems; time-varying infinite-dimensional systems; LQ-optimal control; Riccati equation 1. Introduction The dynamics of non-isothermal plug flow reactors are usually described by non-linear partial differential equations derived from mass and energy balances (see, e.g. Winkin, Dochain, and Ligarius 1998; Christofides 2001; Laabissi, Achhab, Winkin, and Dochain 2001; Aksikas 2005; Aksikas, Winkin, and Dochain 2007) and references therein. The main source of non-linearities in the dynamics of a (bio)-chemical reaction are often due to the kinetic terms in the model equations. Industrial catalytic reactors are most often modelled as plug flow reactors and play an important role in many industrial processes (e.g. methanol, ammonia, sulphuric acid, nitric acid and other petrochemicals). Commonly, the conversion within a catalytic reactor decreases with time, as catalytic deactivation or catalyst decay occurs. As this catalytic deactivation occurs, the chemical kinetics change. In Aksikas et al. (2007), the linear-quadratic (LQ) control problem was studied for a partial differential equation model of a non-isothermal plug flow reactor. This was achieved, under the assumption of time- invariant reaction, by using a non-linear time-invariant infinite-dimensional Hilbert state-space description and the spectral factorisation approach, which is based on the frequency domain description of the system (e.g. Callier and Winkin 1992). The objective of this article is to study the same problem in the case of time-varying reaction. This is motivated by the fact that time-varying rates of reaction arise from loss of catalyst activity, which is an important issue in catalytic reactors, and the literature provides several models for catalyst deactivation (Lie and Himmelblau 2000). For the purpose of this article, we will adopt a simple exponential decay model form. As one of the central themes in modern control theory, the Riccati equation approach will be adopted here in order to solve the LQ problem. Numerous research works are concerned with the Riccati equation for time-varying infinite-dimensional systems (e.g. Da Prato and Ichikawa 1990; Pandolfi 1992; Jacob 1995; Phat 2001; Bensoussan, Da Prato, Delfour, and Mitter 2007). The idea in this article is to study the Riccati equation in the case of the catalytic reactor, which requires an extensive analysis of its dynamical proper- ties on the basis of evolution systems theory (Pazy 1983). The article is organised as follows. In x 2, we recall some basic results on evolution systems and LQ control problem for infinite-dimensional time-varying systems. Section 3 describes both the dynamics of the time-varying catalytic reactor that we are interested in, its steady state profile and its linearised model around this profile. In designing an LQ-controller, some useful results on the dynamical properties of the linearised model are established in x 4. The optimal control design problem is the subject of x 5. An LQ-control feedback is computed by using the corresponding operator Riccati differential equation, whose solution can be obtained via a related matrix Riccati partial differential equation. Finally, the controller is applied to the non- linear closed-loop system and tested numerically in x 6. *Corresponding author. Email: aksikas@ualberta.ca. ISSN 0020–7179 print/ISSN 1366–5820 online ß 2009 Taylor & Francis DOI: 10.1080/00207170802492381 http://www.informaworld.com