Abstract
In this article, we study a class of non-linear control systems called Boolean (i.e. over the finite field F 2) monomial control systems (BMCS) defined as mappings such that every component gi is a monomial function in n state variables and in m input variables. These represent a particular class of finite state automata. We successfully apply the theory of Boolean dynamical systems [Colón-Reyes et al., Ann. Comb 8 (2004), pp. 425–439, Delgado-Eckert (2008), Ph.D thesis, http://mediatum2.ub.tum.de/doc/645326/document.pdf], in particular, the graph theoretical notion of ‘loop number’ to investigate controllability issues for BMCS. We found that BMCS containing only one control input are completely controllable, whereas BMCS displaying more than one input variable are harder to control. Additionally, we introduce the principle of loop number assignment, which is in some sense analogous to the well known ‘pole placement’ method for state continuous linear systems. Moreover, we present an algorithm that synthesizes a suitable state feedback controller in order to specify a desired cyclic behaviour of the closed loop system.
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