Automatica 43 (2007) 105 – 110 www.elsevier.com/locate/automatica Brief paper Parameter convergence in adaptive extremum-seeking control  V. Adetola, M. Guay ∗ Department of Chemical Engineering, Queen’s University, Kingston, Ont., Canada K7L 3N6 Received 26 May 2005; received in revised form 24 April 2006; accepted 28 July 2006 Available online 28 September 2006 Abstract This paper addresses the problem of parameter convergence in adaptive extremum-seeking control design. An alternate version of the popular persistence of excitation condition is proposed for a class of nonlinear systems with parametric uncertainties. The condition is translated to an asymptotic sufficient richness condition on the reference set-point. Since the desired optimal set-point is not known a priori in this type of problem, the proposed method includes a technique for generating perturbation signal that satisfies this condition in closed-loop. This demonstrates its superiority in terms of parameter convergence. The method guarantees parameter convergence with minimal but sufficient level of perturbation. The effectiveness of the proposed method is illustrated with a simulation example.  2006 Elsevier Ltd. All rights reserved. Keywords: Extremum-seeking control; Persistence of excitation; Sufficient richness 1. Introduction Extremum-seeking control (ESC) is a class of adaptive control that deals with regulation to unknown set-points. The controller finds the operating set-points that optimize a per- formance or cost function. The uncertainty associated with the function makes it necessary to use some sort of adap- tation and perturbation to search for the optimal operating conditions. One of the main challenges with ESC and most determinis- tic adaptive control approach is the ability to recover the true unknown values of the parameters. In most approaches, the convergence of parameters to their true values can only be en- sured if the closed-loop trajectories provide sufficient excita- tion for the parameter estimation routine. In standard linear adaptive control approaches, this problem is tractable (Ioannou & Sun, 1996) and can be solved satisfactorily. A dither signal can be introduced momentarily in the control system to achieve the necessary excitation. For nonlinear systems, the problem of determining appropriate excitation conditions remains open.  This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Tor Arne Johansen under the direction of Editor F. Allgöwer. ∗ Corresponding author. Tel.: +1 613 533 2788; fax: +1 613 533 6637. E-mail address: guaym@chee.queensu.ca (M. Guay). 0005-1098/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2006.07.021 Although some limited persistence of excitation (PE) condi- tions have been derived, they remain difficult to apply. Such conditions appear naturally in Guay and Zhang (2003) for the solutions of an adaptive ESC problem. In fact, the fulfillment of such conditions dictates the performance of the optimization routine. This study is focused on adaptive (mode-based) extremum- seeking techniques. In particular, we consider the class of ESC problems introduced in Guay and Zhang (2003) where the structure of the objective function is employed in the design. In contrast to black-box approaches (see Krstic & Wang, 2000 for example), no direct measurement of the objective function is available but must be inferred through the measurements of the state variables and the estimation of model parameters. Ex- amples of this type of problem arise when the objective func- tion reflects profit or cost estimates which are seldom available for measurement. Such function normally involves economic prices such as operating costs and values of products aside from system’s states and unknown parameters. In the previous works in this area (for examples Adetola & Guay, 2006; DeHaan & Guay, 2005; Guay & Zhang, 2003), convergence to the optimum is guaranteed only by assuming the satisfaction of a PE condition. Apart from the fact that it is diffi- cult to choose a signal that satisfies such assumptions, it is nec- essary to select one that achieves a good compromise between