# Efficient Software Tools for Control and Analysis of Hybrid by Michal Kvasnica

By Michal Kvasnica

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Let the mid-point of the line between the two intersection points xint + tp d and t +t xint + tn d serve as a new interior point, xint := xint + ip 2 in d 7. 3. The algorithm requires an interior point to begin with in step 2. 5)). 4) which are obtained when solving the Chebychev-Ball problem can also be used to initialize the set of non-redundant constraints JN . Obviously, all half-spaces which are ‘touched’ by the ball are non-redundant, provided all duplicate half-spaces have been removed. 3 Efficient Polytope Reduction in Multi-Parametric Programming 33 (a) Half-spaces defining the polytope.

2) for a given parameter x. For any given x, it is possible to obtain the optimizer by solving a standard quadratic programming problem1 . Before going further, we will introduce the following definitions. e. XN = {x ∈ Rn |∃UN ∈ RN , GUN ≤ W + Ex}. 6). 3) limN →∞ XN . 2 (Polytopic/Polyhedral Partition) A collection of polytopic (polyhedral) sets {Pr }R r=1 = {P1 , . , PR } is a polytopic (polyhedral) partition of a polytopic (polyhedral) set Θ if (i) R r=1 Pr = Θ, (ii) (Pr \∂Pr ) ∩ (Pq \∂Pq ) = ∅, ∀r = q, where ∂ denotes the boundary.

Let x(k) denote the measured state at time k and xk denote the predicted state at k steps ahead, given the state x(0). Let uk be the predicted input k steps ahead, given x(0). In this chapter we will give a brief overview of optimal control problems for LTI systems discussed in the literature. 1) are subject to the following constraints x(k) ∈ X ⊆ Rn , u(k) ∈ U ⊆ Rm , k ∈ {0, . . 2) where X and U are polyhedral sets containing the origin in their interior1 . 3a) k=0 The extension to mixed constraints C x x + C u u ≤ C o is straightforward and omitted here.