VOLUME 85, AUGUST 2007 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING 447 INTRODUCTION R eal-time optimization has seen a resurgence of interest in the recent years. The traditional approach is the model- based repeated optimization where the model is adapted using the available measurements and numerical optimization is performed on the updated model (Marlin and Hrymak, 1996; Zhang et al., 2002). An alternative approach to real-time optimi- zation known as the “extremum seeking” allows treating the optimization problem as a control problem with the advantages related to sensitivity reduction and disturbance rejection. The optimization problem becomes one of regulating the norm of the gradient at zero. The crucial point in extremum-seeking methods is the computation of this gradient for which several techniques have been proposed. Firstly, the system is perturbed using an external excitation signal in order to numerically compute the gradient (Blackman, 1962; Krstic and Wang, 2000). Also, the excitation Dependence of the Error in the Optimal Solution of Perturbation-Based Extremum Seeking Methods on the Excitation Frequency Moncef Chioua 1* , Bala Srinivasan 1 , Martin Guay 2 and Michel Perrier 1 1. Département de génie chimique, École Polytechnique Montréal, Montréal, QC, Canada 2. Department of Chemical Engineering, Queen’s University, Kingston, ON, Canada can be generate internally by sliding mode control as in Yaodong et al. (2003). Alternately, an adapted model of the system is used for analytical evaluation of the gradient (Guay and Zhang, 2003). In addition, the gradient can be computed using finite difference between the outputs of multiple units running in parallel as in Srinivasan (2007). These class of methods have been successively applied in simulation to the on-line optimiza- tion of a variety of (bio-)chemical processes (Guay et al., 2004; Wang et al., 1999). This paper deals with the first form of gradient estimation, i.e., extremum-seeking methods based on perturbations. The renewed popularity of perturbation-based methods (Blackman, 1962) is mainly due to the publication of Krstic and Wang (2000) where a formal proof of convergence has been established. In perturbation-based extremum-seeking methods, an excitation signal is added to the input, and the gradient, computed from the correlation between the input and output variations, is forced to zero. The main drawback of the method is that the speed of convergence, which is linked to the dither frequency, is slow due to the low value of dither frequency typically chosen. Increasing the excitation frequency may cause instability, but that could be corrected by phase compensation. In this paper, it is shown that an additional problem exists, i.e., the distance between the optimum and solution reached by the perturbation method is proportional to the square of the frequency of excitation and does not go to zero even when the amplitude of the excitation goes to zero. However, for Wiener/Hammerstein approximations, the error will indeed go to zero with the excitation amplitude. Simulation results on a distributed reaction system are used to illustrate the concepts presented in this work. Dans les méthodes de recherche des extrêmes basée sur les perturbations, un signal d’excitation est ajouté à l’entrée, et le gradient, calculé à partir de la corrélation entre les variations d’entrée et de sortie, est forcé vers zéro. Le principal inconvénient de la méthode est que la vitesse de convergence, qui est liée à la fréquence du signal de superposition, est lente à cause de la faible valeur de fréquence du signal de superposition typiquement choisie. Le fait d’accroître la fréquence d’excitation peut entraîner de l’instabilité, mais cela pourrait être corrigée par la compensation de phase. Dans cet article, on montre qu’un problème supplémentaire existe, à savoir que la distance entre l’optimum et la solution obtenue par la méthode de perturbation est proportionnelle au carré de la fréquence d’excitation et ne tend pas vers zéro même quand l’amplitude de l’excitation tend vers zéro. Toutefois, pour les approximations de Wiener/Hammerstein, l’erreur tendra vers zéro avec l’amplitude de l’excitation. Les résultats des simulations sur un système de réaction distribué permettent d’illustrer les concepts présentés dans ce travail. Keywords: real-time optimization, extremum seeking, perturbations method, reactor control, chemical reactors * Author to whom correspondence may be addressed. E-mail address: moncef.chioua@polymtl.ca