Automatica 45 (2009) 2234–2243Contents lists available at ScienceDirectAutomaticajournal homepage: www.elsevier.com/locate/automaticaGeneralized bang–bang control for feedforward constrained regulation $Luca Consolini, Aurelio Piazzi ∗Dipartimento di Ingegneria dell’Informazione, Università di Parma, Viale G.P. Usberti 181A, 43124 Parma, Italya r t i c l e i n f oArticle history:Received 7 August 2008Received in revised form28 February 2009Accepted 29 June 2009Available online 7 August 2009Keywords:Feedforward controlGeneralized bang–bang controlSet-point constrained regulationInput–output constraintsMinimum-time controlLinear programmingLinear systemsa b s t r a c tIn the behavioral framework for continuous-time linear scalar systems, simple sufficient conditions forthe solution of the minimum-time rest-to-rest feedforward constrained control problem are provided.The investigation of the time-optimal input–output pair reveals that the input or the output saturateson the assigned constraints at all times except for a set of zero measure. The resulting optimal input iscomposed of sequences of bang–bang functions and linear combinations of the modes associated to thezero dynamics. This signal behavior constitutes a generalized bang–bang control that can be fruitfullyexploited for feedforward constrained regulation. Using discretization, an arbitrarily good approximationof the optimal generalized bang–bang control is found by solving a sequence of linear programmingproblems. Numerical examples are included.© 2009 Elsevier Ltd. All rights reserved.1. IntroductionRecently in the control engineering literature, it has been em-phasized that to achieve high performances in real applications,due attention has to be paid to the constraints which all the plantvariables must comply with. In particular, the main approaches tocontrol system design with input and output constraints are thefollowing:• Antiwindup and override feedback schemes. This is the stan-dardapproachinthepracticalindustrialcontext;see,forexam-ple, the recent book of Glattfelder and Schaufelberger (2003).• Model predictive control. In the receding horizon strategy,input constraints as well as output ones can be naturally con-sidered in designing the feedback controller; see, for instance,Maciejowski (2002).Inthispaperweaddressthesubjectofcontrollingacontinuous-time scalar linear system with input and output constraints bysetting a purely feedforward regulation problem to be solved inminimum-time. We assume that the system is stable and want to$The material in this paper was partially presented at 2006 IEEE Conference onDecisionandControl,SanDiego(California,USA),13–15December2006.ThispaperwasrecommendedforpublicationinrevisedformbyAssociateEditorMarioSznaierunder the direction of Editor Roberto Tempo.∗Corresponding author. Tel.: +39 0521 905733; fax: +39 0521 905723.E-mail addresses: lucac@ce.unipr.it (L. Consolini), aurelio.piazzi@unipr.it(A. Piazzi).find a minimum-time feedforward input that brings the systemfrom a current rest condition to a new desired rest condition whilesatisfying at all times given amplitude constraints on the input andthe output. In such a way, we can naturally deal with both actuatorlimitations and overshooting and undershooting requirements.It is well known that the minimum-time feedforward controlwith input constraints only is given by the so-called bang–bangcontrol, i.e. the input signal switches between its extreme allowedvalues (Lewis & Syrmos, 1995). In a behavioral setting, this papershows that in the presence of both input and output constraintsthe minimum-time input–output pair enjoys the property thatthe saturation of the input or the output signal occurs almosteverywhere. Therefore, the optimal feedforward input is given bya sequence of bang–bang functions and linear combinations of thesystem zero modes. This type of optimal control can be viewedas a generalized bang–bang control. For the actual computation ofthis time-optimal control, the proposed idea is to discretize thecontinuous-time system and to solve the resulting discrete-timeproblem by means of linear programming. In fact, in the discrete-time case, input and output constraints can be represented aslinear inequalities and the minimum number of steps needed fora rest-to-rest transition can be found with a sequence of linearfeasibility tests.The idea of using linear programming for solving a minimum-time problem for linear discrete-time systems subject to ampli-tude input constraints dates back to Zadeh (1962). Subsequently,various contributions have appeared by focusing on some im-provements for this discrete-time problem (Bashein, 1971; Kim &Engell, 1994; Scott, 1986). In this paper, we prove that the optimal0005-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2009.06.030
