Identification of Bouc–Wen type models using multi-objective optimization algorithms Gilberto A. Ortiz ⇑ , Diego A. Alvarez, Daniel Bedoya-Ruíz Universidad Nacional de Colombia, 170004 Manizales, Colombia article info Article history: Received 23 April 2012 Accepted 15 October 2012 Keywords: Bouc–Wen–Baber–Noori model Hysteresis Multi-objective optimization System identification NSGA-II abstract Most of the published literature concerned with the parameter estimation of the Bouc–Wen model of hysteresis via evolutionary algorithms uses a single objective function (the mean square error between the known displacements and the estimated ones) and considers the original Bouc–Wen model of hyster- esis (without degradation and pinching) in the identification process. In this paper, a novel method for the identification of the parameters of the Bouc–Wen–Baber–Noori (BWBN) model of hysteresis is pre- sented. The methodology is based on a multi-objective evolutionary optimization algorithm called NSGA-II [39]; therefore, a set of objective functions is employed instead of the traditional single objective function. The proposed methodology identifies the structural system and allows the observation of multi- modality of the BWBN model of hysteresis. The performance of the algorithm is evaluated using simu- lated and real data. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction The Bouc–Wen (BW) model has been widely employed in struc- tural engineering for the analytical description of smooth hysteretic behavior given its capability to describe several patterns that match the response of a wide class of hysteretic systems. The model was introduced by Bouc [1,2] and later generalized by Wen [3], with the advantage of computational simplicity, because only one auxil- iary nonlinear differential equation is needed to describe the hyster- etic behavior. In addition, closed-form expressions are available which ease the usage of the model in nonlinear random vibration analysis by the equivalent linearization method (see for instance [4,5]). This model has been extended to describe various character- istics of hysteretic behavior, like degradation of stiffness and strength [6–8], pinching effect (see e.g. [9,10]), biaxial hysteresis [11], asymmetry of the peak restoring force [12,13], among others, leading to a broader class of models, henceforth named BW-type models. Bouc–Wen type models have been used in an ample range of applications such as vibration of steel structures [14,15], concrete structures [16], wood joints [9,17] or base isolation devices for buildings [18]. It has been also used to model magnetorheological dampers [19], piezoelectric elements [20], and soil dynamics [21], among others. When a BW-type model is employed in practical applications, the first step to perform is the identification of the model param- eters, that is, given a set of experimental input/output data, the model parameters must be tuned so that the output of the model matches as good as possible the experimental data. In conse- quence, system identification techniques are usually employed in order to perform this task. Once the BW-type model parame- ters have been found, the resulting model is regarded as a ‘‘good’’ approximation of the true experimental hysteresis when the dis- similarities between the experimental data and the output of the mathematical model are small enough. Thereafter, the fitted model is employed either to predict the behavior of the physical hysteretic element under different excitations or for control pur- poses. Take into account that this is not the end of the road; the important stage of model validation comes after that when the model is usually tested against unknown excitations and the re- sults are assessed against the response of the true system (see for example [22]). Several methods have been proposed for the identification of the parameters of the BW-type models using experimental input and output data. The procedures suggested in the literature to tune the parameters of the BW-type models can be classified into two major groups as (see also a recent survey on BW model parameter identification by Ismail et al. [23]):  methods based on the minimization of a loss function, using for example least square estimation (see e.g. [24]), Gauss–Newton methods like the Levenberg–Marquardt algorithm (see e.g. [25]), evolutionary algorithms (see e.g. [26,27]), differential evolution (see e.g. [28,29]), particle swarm optimization (see e.g. [30,31]), the generalized reduced gradient method (see e.g. [25]), etc.; in this case, the error difference between the time histories is minimized. 0045-7949/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruc.2012.10.016 ⇑ Corresponding author. Tel.: +57 68879300x50256. E-mail address: gialorga@gmail.com (G.A. Ortiz). Computers and Structures 114–115 (2013) 121–132 Contents lists available at SciVerse ScienceDirect Computers and Structures journal homepage: www.elsevier.com/locate/compstruc