Tham khảo tài liệu 'discrete time systems part 17', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Discrete Time Systems with Event-Based Dynamics Recent Developments in Analysis and Synthesis Methods 469 Fig. 1. Strongly connected dependency graph Gf Vf Ef nf with loop number LGf Vf 6 of a 24-dimensional Boolean monomial dynamical system f G MF24 F2 . Circles blue demarcate each of the six loop equivalence classes. Essentially the dependency graph is a closed path of length 6. and because of 2 in the previous theorem clearly ã IJ Nt b Given one loop equivalence class a G Vg the set of all the t loop equivalence classes can be ordered in the following manner a a 1 u N1 b .fi j u Nj b .dl t-1 u Nt-1 b bGữị bGữị bGữị For any c G u Nt-1 b it must hold N1 c G ãị ifN1 c n ãj 0 with j i then ãị ãj . Thus the graph G can be visualized as see Fig. 1 ai ai 1 ai j a i j 1 mod t . ai t-1 a i t mod t Due to the fact a IJ Nt b V a G Vg we can conclude that the claims of the previous lemma still bGa- hold if the sequence lengths m and m are replaced by the more general lengths Kt m and K t in where K Ả G N. Boolean monomial control systems Control theoretic questions studied We start this section with the formal definition of a time invariant monomial control system over a finite field. Using the results stated in the previous section we provide a very compact nomenclature for such systems. After further elucidations and in particular after providing the formal definition of a monomial feedback controller we clearly state the main control theoretic problem to be studied in Section of this chapter. Definition 54. Let Fq be a finite field n G N a natural number and m G No a nonnegative integer. A mapping g Fn X Fm Fn is called time invariant monomial control system over Fq if for 470 Discrete Time Systems every i 1 . n there are two tuples Ai1 . Ain Ef and Bi1 . Bim 6 Eq1 such that gi x u xAM 1. V x u Ff X Fq Remark 55. In the case m 0 we have Fq F0 the set containing the empty tuple and nm n0 n n Li tits n q X R q R q X R q R q X R q . Xit o 4- Í Ler worts g t-s a ĩ I
