DESIGN OF ROBUST PREDICTIVE CONTROL LAWS USING SET MEMBERSHIP IDENTIFIED MODELS M. Canale, L. Fagiano, and M.C. Signorile ABSTRACT This paper investigates the robust design of nonlinear model predictive control (NMPC) laws that employ approximated models, derived directly from process input-output data. In particular, a nonlinear set membership (NSM) identification technique is used to obtain a system model and a bound of the related uncertainty. The latter is used to carry out a robust control design, via a min-max formulation of the optimal control problem underlying the NMPC methodology. A numerical example with a nonlinear oscillator shows the effectiveness of the proposed approach. Key Words: Predictive control, robust stability, nonlinear control. I. INTRODUCTION Nonlinear model predictive control (MPC, see e.g. [14]), also referred to as receding horizon control, is a control technique in which the current control move is computed by solving on-line a constrained finite horizon optimal control problem (FHOCP). In each sampling period, a measure or estimate of the system state is used as the initial condition for the FHOCP and, according to a receding horizon (RH) strat- egy, only the first element of the solution sequence is applied to the system. Then, the procedure is repeated in the following sampling period, when a new measure of the state is available. The model employed in the FHOCP is typically a “physical” model (i.e. derived from physical laws) or a nonlinear para- metric function (e.g. a neural network), whose parameters are identified by using measured process data. In regard to the robustness analysis and robust design of NMPC, much progress has been made but many questions, such as uncertainty description and efficiency of the on-line computation, remain open. References [17] and [1] depicted the development of robust MPC, describing the several solu- tions proposed during the years. Using the contraction prin- ciple, [19] derived some necessary and sufficient conditions for robust stability, but the results could be conservative and difficult to verify. Reference [7] assured robust stability through the computation of some weights. Unfortunately, the existence of such weights was only a sufficient condition and consequently could be restrictive. Reference [11] introduced a procedure to guarantee robust stability providing a non- restrictive result, which may turn out to be unsuitable for on-line computation because of its complexity. Reference [20] proposed to achieve robust stability by enforcing a robust state contraction constraint through the optimization of a quadratic problem of medium size. The problem of designing predictive controllers in the presence of unmodeled dynamics was studied by [4] and [12]. More recently, [13] carried out a regional input-to-state stability (ISS) analysis of NMPC, [10] derived a suboptimal NMPC law with ISS guarantees, and [18] presented a robust NMPC scheme in the presence of state-dependent uncertainties and additive bounded perturba- tions. The concept of ISS has also been successfully exploited in [5] where a min-max MPC design approach has been introduced for the case of nonlinear time varying systems in the presence of delays. Although several methods, like those described above, face the problem of robust stability, it has to be noted that, to the best of the authors’ knowledge, in the nonlinear case there is no rigorous procedure to obtain a suitable description of the uncertainty associated with the employed model. This issue hampers the possibility of performing a systematic robust- ness analysis or a synthesis procedure to derive robust NMPC control laws. In fact, in most practical cases only a model of the system to be controlled is available, without any uncer- tainty description and/or estimate. Basically, this issue is due to the difficulty of evaluating model uncertainty when non- linear parametric models, either “physical” or “black–box”, are employed. Indeed, the parameters of such models are usually identified from system input/output data. With such a Manuscript received June 1, 2011; revised December 30, 2011; accepted May 3, 2012. The authors are with Dipartimento di Automatica e Informatica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy (e-mail: {massimo.canale, lorenzo.fagiano,maria.signorile}@polito.it). Lorenzo Fagiano is also with Department of Mechanical Engineering, University of California, Santa Barbara—CA, USA. M. Canale is the corresponding author. This research has received funding from European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement n. PIOF-GA-2009-252284— Marie Curie project “Innovative Control, Identification and Estimation Methodologies for Sustainable Energy Technologies.” Asian Journal of Control, Vol. 15, No. 3, pp. 1–9, May 2013 Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.560 © 2012 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society