Assessment and Future Directions of Nonlinear Model Predictive ...

132 T. Alamo et al.In this paper we propose an algorithm that circumvents the huge computationalcomplexity associated to the obtainment of the maximal robust controlinvariant set. Two new algorithms are proposed. The first one provides a convexpolyhedral outer bound of the maximal control invariant set for the piecewiseaffine system. This outer estimation is used, by the second proposed algorithm,to obtain a robust control invariant set for the system (not necessarily the maximalone). The algorithms are based on inner and outer approximations of agiven non-convex set.The paper is organized as follows: Section 2 presents the problem statement.Section 3 presents an algorithm that computes an outer bound of the maximalrobust control invariant set of the piecewise affine system. A procedure to obtaina robust control invariant set is proposed in section 4. An illustrative exampleis given in section 5. The paper draws to a close with a section of conclusions.2 Problem StatementLet us suppose that X is a bounded convex polyhedron. Suppose also that theconvex polyhedra X i , i =1,...,r, with disjoint interiors, form a partition of X.⋃That is, X = r X i .i=1We consider the following piecewise affine system:x + = f(x, u, w) =A i x + B i u + E i w + q i if x ∈ X i (1)where x ∈ R nx is the state vector; x + denotes the successor state; u ∈ U ={ u ∈ R nu : ‖u‖ ∞ ≤ u max } is the control input; w denotes a bounded additiveuncertainty: w ∈ W = { w ∈ R nw : ‖w‖ ∞ ≤ ɛ }.In order to present the results of this paper it is important to refer to thenotion of the one step set [8].Definition 1 (one step set). Given a region Ω, and system (1), the followingsets are defined:Q(Ω) ={ x ∈ X : there is u ∈ U such that f(x, u, w) ∈ Ω,∀w ∈ W }Q i (Ω)={ x ∈ X i : there is u ∈ U such that A i x+B i u+E i w+q i ∈ Ω,∀w ∈ W}The following well-known properties allow us to compute Q(Ω) for a piecewiseaffine system [8]:Property 1. Given a convex polyhedron Ω: Q(Ω) =i =1,...,r are polyhedra.Property 2. If Ω =r⋃i=1 j=1s ⋃j=1r ⋃i=1Q i (Ω), where Q i (Ω),P j and P 1 ,P 2 ,...,P s are convex polyhedra, then Q(Ω) =s⋃Q i (P j ), where Q i (P j ), i =1,...,r, j =1,...,s are convex polyhedra.