Assessment and Future Directions of Nonlinear Model Predictive ...

Robust Model Predictive Control for ObstacleAvoidance: Discrete Time CaseSaša V. Raković and David Q. MayneImperial College London, London SW7 2BT, United Kingdomsasa.rakovic@imperial.ac.uk, d.mayne@imperial.ac.uk1 IntroductionThe importance of the obstacle avoidance problem is stressed in [4]. Computationof reachability sets for the obstacle avoidance problem is addressed, for continuoustimesystems in [4, 5] and for discrete-time systems in [12]; further results appear in,for instance [2, 17, 18]. The obstacle avoidance problem is inherently non–convex.Most existing results are developed for the deterministic case when external disturbancesare not present. The main purpose of this paper is to demonstrate thatthe obstacle avoidance problem in the discrete time setup has considerable structureeven when disturbances are present. We extend the robust model predictiveschemes using tubes (sequences of sets of states) [9, 11, 14] to address the robustobstacle avoidance problem and provide a mixed integer programming algorithmfor robust control of constrained linear systems that are required to avoid specifiedobstacles. The resultant robust optimal control problem that is solved on–line hasmarginally increased complexity compared with that required for model predictivecontrol for obstacle avoidance in the deterministic case.This paper is organized as follows. Section 2 discusses the general idea of robustcontrol invariant tubes for the obstacle avoidance problem. Section 3 considersin more detail the case when system being controlled is linear. Section 4 presentsa simple tube controller, establishes its properties and provides an illustrativeexample. Finally, Section 5 gives conclusions and indicates further extensions.Notation: Let N {0, 1, 2,...} and N q {0, 1,...,q} for q ∈ N. Apolyhedronis the (convex) intersection of a finite number of open and/or closedhalf-spaces, a polytope is the closed and bounded polyhedron and a closed(open) polygon is the union of a finite number of polytopes (polyhedra). Giventwo sets U ⊂ R n and V ⊂ R n , the Minkowski set addition is defined byU⊕V {u + v | u ∈U, v ∈V}, the Minkowski/Pontryagin set differenceis: U⊖V {x | x ⊕V ⊆U}. The distance of a point z from a set X is denotedby d(z,X) inf{|z − x| |x ∈ X}.2 Problem FormulationWe consider the following discrete-time, time-invariant system:x + = f(x, u, w) (1)R. Findeisen et al. (Eds.): Assessment and Future Directions, LNCIS 358, pp. 617–627, 2007.springerlink.com c○ Springer-Verlag Berlin Heidelberg 2007