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Robust control of constrained max‐plus‐linear systems
Author(s) -
Necoara Ion,
De Schutter Bart,
van den Boom Ton J. J.,
Hellendoorn Hans
Publication year - 2008
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.1309
Subject(s) - bounded function , polytope , mathematics , linear programming , linear system , convexity , sequence (biology) , parametric statistics , nonlinear system , mathematical optimization , control theory (sociology) , state (computer science) , optimal control , bellman equation , function (biology) , robust control , control (management) , computer science , algorithm , discrete mathematics , mathematical analysis , statistics , physics , quantum mechanics , artificial intelligence , evolutionary biology , biology , financial economics , economics , genetics
Max‐plus‐linear (MPL) systems are a class of nonlinear systems that can be described by models that are ‘linear’ in the max‐plus algebra. We provide here solutions to the three types of finite‐horizon min–max control problems for uncertain MPL systems, depending on the nature of the control input over which we optimize: open‐loop input sequences, disturbances feedback policies, and state feedback policies. We assume that the uncertainty lies in a bounded polytope and that the closed‐loop input and state sequence should satisfy a given set of linear inequality constraints for all admissible disturbance realizations. Despite the fact that the controlled system is nonlinear, we provide sufficient conditions that allow one to preserve convexity of the optimal value function and its domain. As a consequence, the min–max control problems can be either recast as a linear program or solved via N parametric linear programs, where N is the prediction horizon. In some particular cases of the uncertainty description (e.g. interval matrices), by employing results from dynamic programming, we show that a min–max control problem can be recast as a deterministic optimal control problem. Copyright © 2008 John Wiley & Sons, Ltd.