Workshop D3 "Groebner Bases in Control Theory and Signal Processing" of the Special Semester on Groebner Bases and Related Methods
Sprache des Tagungstitel:
Englisch
Original Kurzfassung:
In systems, containing parameters, it often happens, that some structural properties (like the controllability) hold only for the generic case (i.e. for almost all values of parameters). It means, that there might exist some parameter constellations, such that a generically controllable system, specialized at these constellations becomes non--controllable. We provide an algorithmic way to detect such and similar phenomena, which we call "genericity violation", not restricting ourselves to the generically controllable systems. We assume during the computations, that the parameters $\{p_i\}$ are algebraically independent, and hence, we allowe any operations with the elements of $\K(p_1,\dots,p_n)$, in particular we allow divisions by any polynomial involving $p_i$. There exist several variations of Comprehensive Gr\"obner bases (going back to V.~Weispfennig). The result of a typical C. G. B is a tree, which consists of different Gr\"obner bases together with the constraints on parameters, which imply the particular form of the attached Gr\"obner basis. Algorithmically, this approach is complicated, complex, and aimed at parametric Gr\"obner bases in their full generality. In addition, a reasonable implementation of this method is hardly available. We propose an approach, which works in the situation, where we expect that there are several parameter constellations, leading to the bases, different from the generic Gr\"obner basis. On the other side, it seems reasonable to require, that a sequence of substitutions of parameters (chosen randomly) leads us to results with the same leading submodule. We show the implementation of the methods above in library "control.lib" for the computer algebra system SINGULAR. We also demonstrate the impact of genericity on some interesting examples, e.~g. on the "two pendula" example from the book of Poldermann--Willems, and comment on the other examples. We discuss the general algorithm and investigate its separate algebraic ..........