Journal of Process Control 28 (2015) 9–16 Contents lists available at ScienceDirect Journal of Process Control j ourna l ho me pa ge: www.elsevier.com/locate/jprocont Identification of switched ARX models via convex optimization and expectation maximization András Hartmann a,c,∗ , João M. Lemos a , Rafael S. Costa a,c , João Xavier b , Susana Vinga c a INESC-ID, Instituto Superior Técnico, Universidade de Lisboa, R Alves Redol 9, 1000-029 Lisboa, Portugal b ISR, Instituto Superior Técnico, Universidade de Lisboa, Av Rovisco Pais 1, 1049-001 Lisboa, Portugal c IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal a r t i c l e i n f o Article history: Received 18 October 2014 Received in revised form 12 February 2015 Accepted 12 February 2015 Available online 10 March 2015 Keywords: Parameter identification Time-varying parameter Hybrid systems Convex optimization Expectation maximization a b s t r a c t This article addresses the problem of parameter identification for Switched affine AutoRegressive models with eXogenous inputs (SARX). The system includes continuous domain states that depend on discrete time-varying parameters. The identification of such systems typically results in non-convex problems that could be tackled as a mixed integer program. However, in this case, the computational complexity would be intractable in many practical applications. Another approach involves heuristics in order to deliver approximate solutions. This article proposes a three-step method based on solving a regularized convex optimization problem, followed by a clustering step, yielding a partial solution to the problem. When substituted back into the original problem, the partial solution renders it convex. Finally, this convex problem is solved in the third step, yielding an approximate solution. It is found that each step significantly improves the parameter estimation results on the systems considered. A beneficial property of the method is that it relies upon only one scalar tuning parameter, to which the final results are not highly sensitive. The performance of the algorithm is compared with other methods on a simulated system, and illustrated in an experimental biological dataset of diauxic bacterial growth. © 2015 Elsevier Ltd. All rights reserved. 1. Introduction One limitation of many mathematical models describing dynamic systems is that the parameters are assumed to be invariant during the observation period. This premise does however not nec- essary hold in many cases, such as certain applications in economics [7], computer vision [29], systems biology [9] or longitudinal clini- cal datasets [15]. In the last decade, much attention has been given to dynamical modeling using Linear Parameter Varying (LPV) and Abbreviations: MSE, Mean Squared Error; RMSE, Root Mean Squared Error; SON, Sum Of Norm; ARX, affine AutoRegressive model with eXogenous inputs; SARX, Switched affine AutoRegressive model with eXogenous inputs; PWARX, PieceWise affine AutoRegressive models with eXogenous input; LPV, Linear Parameter Varying; QP, Quadratic Program; EM, Expectation Maximization; RANSAC, RANdom SAmple Consensus; LASSO, Least Absolute Shrinkage and Selection Operator; PRBS, Pseudo- Random Binary Sequence. ∗ Corresponding author at: IDMEC – Instituto de Engenharia Mecânica, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal. Tel.: +351 218 419 502; fax: +351 218 498 097. E-mail addresses: andras.hartmann@tecnico.ulisboa.pt (A. Hartmann), jlml@inesc-id.pt (J.M. Lemos), rafael.s.costa@tecnico.ulisboa.pt (R.S. Costa), jxavier@isr.ist.utl.pt (J. Xavier), susanavinga@tecnico.ulisboa.pt (S. Vinga). closely related linear hybrid systems. The similarity between LPV and hybrid systems is that the system parameters may be subject to abrupt changes that are represented by different linear submodels. The main difference is that LPV systems do not limit the number of submodels like hybrid systems. In this sense hybrid systems are considered to be a special case of LPVs [26]. In the rest of this article we focus on the identification of a type of hybrid systems, for more information about the definition and identification of LPV systems, the interested reader is referred to the comprehensive book of [27] and references therein. Hybrid models comprise both continuous and discrete states (parameters) [16]. Typically the dynamics can be modeled with continuous state evolution and the transitions between submodels being represented by the changes of discrete states. Various classes of hybrid systems were proposed [22], with different definitions of discrete and continuous states and their corresponding interac- tions. The equivalences between some of the classes (under mild conditions) were established by Heemels et al. [11,12], Weiland et al. [32]. In the literature, a great emphasis has been placed on hybrid system identification; for a comprehensive overview see Paoletti et al. [22], Lygeros [16] and references therein. Many contributions focus on PieceWise affine AutoRegressive models with eXogenous http://dx.doi.org/10.1016/j.jprocont.2015.02.003 0959-1524/© 2015 Elsevier Ltd. All rights reserved.