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alt="images"/> and images images. The first‐order Taylor expansion of the vector F around images can be expressed as follows:

      (1.20)equation

      (1.21)equation

      where matrices images, and images, images, images and the scalars images, images correspond to the images and images inequality and equality constraints of the sets images and images, respectively. This problem serves as the basis that will be discussed in Part I, where its solution properties and solution strategies among other things are in focus. Part II then focusses on the application of such problems to optimal control, as the use of parameters enables the formulation of explicit model predictive control problems.

      Multi‐parametric programming is intimately related to the properties and operations applicable to polytopes. In the following, some basic definitions on polytopes are stated, which are used throughout the book.

      Definition 1.9

      A function images, where images is a polytope, is called piecewise affine if it is possible to partition images into disjoint polytopes, called critical regions, images and

      (1.22)equation

      Remark 1.2 The definition of piecewise quadratic is analogous.

      The set images is called a images‐dimensional polytope if it satisfies

      where images is finite.

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.

       A polytope is called bounded if and only if there exists a finite and such for all .

       A polytope, which is closed and bounded, is called compact.

       Let be an ‐dimensional polytope. Then, a subset of a polytope is called a face of if it can be represented as(1.24) for some inequality , which holds for all . The faces of polytopes of dimension , 1, and 0 are referred to as facets, edges, and vertices, respectively.

       Two polytopes and are called disjoint if . Similarly, two polytopes and are called overlapping if . Lastly, two polytopes and are called adjacent or neighboring if is a ‐dimensional polytope.

       Let and be two adjacent polytopes. Then the facet‐to‐facet property is said to hold if is a facet of both and (see Figure 1.2 for an illustration).

       Let be an ‐dimensional polytope. Then, there exists a series of vertices such that(1.25)

       Eq. (1.23) is referred to the halfspace (or H) representation, while Eq. (1.25) denotes the vertex (or V) representation. The process of moving from the halfspace to the vertex representation is referred to as vertex enumeration.

       The Chebyshev center of a polytope is given as the largest Euclidean ball that lies in a polytope [2]. It can be determined by solving the following linear programming (LP) problem:(1.26) where the solution denotes the radius of the largest Euclidean ball. Based on the solution of problem (1.26), the following conclusions can be drawn:– Problem (1.26) is infeasible: The polytope is empty.– : The polytope is lower‐dimensional.– : The polytope is full‐dimensional.

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