A Control Theory for Boolean Monomial Dynamical Systems

Abstract

Recently criteria for determining when a certain type of nonlinear discrete dynamical system is a fixed point system have been developed. This theory can be used to determine if certain events modeled by those systems reach a steady state. In this work we formalize the idea of a "stabilizable" discrete dynamical system. We present necessary and sufficient conditions for a Boolean monomial dynamical control system to be stabilizable in terms of properties of the dependency graph associated with the system. We use the equivalence of periodicity of the dependency graph and loop numbers to develop a new O(n 2 log n) algorithm for determining the loop numbers of the strongly connected components of the dependency graph, and hence a new O(n 2 log n) algorithm for determining when a Boolean monomial dynamical system is a fixed point system. Finally, we show how this result can be used to determine if a Boolean monomial dynamical control system is stabilizable in time O(n 2 log n).

On persistent excitations for the identification of switched linear dynamical systems over finite fields

“…The line D intersects with all the hyperplanes at N distinct points under the condition that it is not parallel with any hyperplanes. In other words, the equation of degree N p N (µv + w 0 ) = 0 (17) admits N distinct integer roots {µ s } N s=1 under the constraint p N (v) = 0 (or equivalently v / ∈ S s ). Therefore, the intersection of D with all of the hyperplanes are given by…”

“…Apart from modeling, control issues have also been addressed for discrete systems in the literature. The issues of controllability, reachability and state feedback control have been investigated using different approaches such as the theory of polynomial dynamical systems [15], graph theory [16,17], max-plus algebra [18,19], finite field theory [14,20], discrete abstraction and state transition graphs [5], linear modular systems [21] to name a few. Observability for DDSs has also been the subject of a number of research works.…”

Matrix representations of vectorial Boolean functions and eigenanalysis

“…Vectorial Boolean functions [1] play an important role in cryptography as nonlinear components of symmetric algorithms [2]. They are also used in control theory to model discrete dynamical systems [3]. This paper aims at giving a unified overview on various representations of vectorial Boolean functions.…”