IFAC PapersOnLine 51-15 (2018) 84–89 ScienceDirect Available online at www.sciencedirect.com 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2018.09.095 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. 1. INTRODUCTION Uncertainty is one of the main motivations for feedback control, where controller synthesis is based only on the available knowledge about the process to be controlled. For this purpose, mathematical models are built in order to represent the dynamics of the system, usually condensed as parametric or non-parametric models, where the existing mismatch between the model and the true process is com- monly referred to as the modelling error. The modelling error is a consequence of the interplay existing between control and identification, in the sense that the quality of the model is a compromise between having a simple model for control design and having a complex model to avoid uncertainty (Antoulas, 2005). In modeled-based control design, the closed-loop perfor- mance is subjected to the size of the modelling error, and then the control algorithm must be robustly designed in order to ensure closed-loop stability and performance. Previous works (see, e.g., Francis (1987), Zhou et al. (1996), and Goodwin et al. (2000)) have studied this topic under the name of robust control, and its solution assumes spectral properties of the modelling error. Indeed, fundamental limitations on the closed-loop performance are posed in terms of the largest gain of the modelling error (or a weighted version of it), also known as its (linear)  2 (or L 2 for continuous-time systems) gain van der Schaft (2017). The  2 -gain of a system provides a measure of how large can the output become for a given input. Apart from being useful in analysis of disturbance rejection properties, the computation of the  2 -gain can be used for estab- lishing robust stability in the presence of norm bounded uncertainties. For nonlinear systems, the control design commonly follows from the linearization of the system. As  This work was partially supported by the Swedish Research Council under contracts 2015-04393 and 2016-06079. discussed in Johansson (1999), the robustness of piecewise linear systems can be analyzed in terms of the  2 /L 2 -gain of the system. Along this work, we refer as  2 -gain to the conventional linear  2 -gain. The problem of approximating its nonlinear version has been studied, e.g., in Dower and Kellet (2008). The problem of finding the  2 -gain of a system can be seen as a input design problem (Barenthin, 2008), where the input signal is designed so that the norm gain is maximum. The difficulty of this approach lies on finding the right parametrization of the input signal. For linear and time-invariant (LTI) systems, it is well known that the input signal can be parametrized in terms of the frequency (Fairman, 1998). Then, the problem of finding the optimal input signal is equivalent to find the peak of the frequency response of the system, commonly known as the H ∞ norm of the system (Francis, 1987). In M¨ uller et al. (2017), the  2 -gain of an LTI system was estimated by restricting the input signal to be a multi-sine signal, parametrized by the amplitude of each frequency. A multi- armed bandit approach was then employed to iteratively update these parameters. For LTI systems, other methods such as the parametric gradient descent method (Bar- enthin, 2008), and the ones presented in Bruinsma and Steinbuch (1990), Belur and Praagman (2011) can be employed. However, for general nonlinear systems, the unknown shape of the input signal maximizing the norm gain in the output increases the difficulty of finding the  2 -gain. In other words, the parametrization of the input signal is not trivial for a general nonlinear system. This issue describes the main obstacle when trying to extend the exposed traditional methods for LTI systems to nonlinear systems. One way to circumvent the task of parametrizing the input signal is to use an iterative input design (Hjalmarsson, 2002) approach that not requires parametrization. In this Keywords:  2 -gain, Input and excitation design, non-linear system identification, identification for control. Abstract: In this work the problem of computing the maximum gain of non-linear systems, also known as its  2 -gain, from input-output data is studied. From an input design perspective, this problem reduces to find an optimal input sequence, of bounded norm, maximizing the norm gain of the output, where our target estimation corresponds to the ratio of these quantities. The novelty of this approach lies on the fact that the input signal is a realization of a stationary process with finite memory whose range is a finite set of values. Based on recent developents on input design for nonlinear systems, our approach leads to a linear program whose optimal cost gives an approximation of the  2 -gain of the system. An illustrative example shows how well the algorithm performs compared to other methods approximating this quantity. * Automatic Control Department, KTH Royal Institute of Technology, Stockholm, Sweden (e-mails: {mimr2,crro}@ kth.se). Matias I. M¨ uller * Cristian R. Rojas * A Markov Chain Approach to Compute the  2 -gain of Nonlinear Systems 