Explicit Description of AE Solution Sets for Parametric Linear Systems

Consider linear systems whose input data are linear functions of uncertain parameters varying within given intervals. We are interested in an explicit description of the so-called AE parametric solution sets (where all universally quantified parameters precede all existentially quantified ones) by a set of inequalities not involving the parameters. This work presents how to obtain explicit description of AE parametric solution sets by combining a modified Fourier–Motzkin type elimination of existentially quantified parameters with the elimination of the universally quantified parameters. Some necessary (and sufficient) conditions for existence of nonempty AE parametric solution sets are discussed, as well as some properties of the parametric AE solution sets, e.g., shape of the solution set and some inclusion relations. Explicit descriptions of particular classes of AE parametric solution sets (tolerable, controllable, any two-dimensional) are given. Numerical examples illustrate the solution sets and their properties.