The hurwitz stability of the multi-affine polynomial families

In this thesis, the robust stability of the multilineer polynomials family is considered. The results which are known in the interval polynomials family and polytope of polynomials are given. Counter examples are provided and it is determined that this results do not valid in the multilineer polynomials family. According to edge theorem; if all edges are stable in the polytope of polynomials, polytope of polynomials is stable too. The edge theorem does not valid in the any polytope of matrices. A particular polytope of matrices in which the Edge Theorem is valid are investigated. The Mapping Theorem which leads to a sufficient condition for robust stability of multilineer polynomial family is considered. Also it is shown that the Mapping Theorem leads to a necessary and sufficient condition for robust stability of particular case according to character of value set. A method which can be efficiently for test of stability in the multilineer polynomials family whose number of uncertain parameters is low is noticed. This method can be use for control whether a system of polynomial equation has a solution on a box or not. If the multilineer polynomials family is stable, the method results in the finite step. The several examples related with theorems are investigated. Keywords: Robust Stable, Multilineer Polynomials Family, Polytope of Matrices