By Daizhan Cheng, Hongsheng Qi, Zhiqiang Li
The Boolean community has turn into a robust device for describing and simulating mobile networks during which the weather behave in an on–off style. research and keep watch over of Boolean Networks offers a scientific new method of the research of Boolean keep watch over networks. the elemental device during this process is a singular matrix product referred to as the semi-tensor product (STP). utilizing the STP, a logical functionality could be expressed as a standard discrete-time linear method. within the mild of this linear expression, definite significant concerns touching on Boolean community topology – fastened issues, cycles, temporary instances and basins of attractors – should be simply published by means of a collection of formulae. This framework renders the state-space method of dynamic keep an eye on platforms acceptable to Boolean keep watch over networks. The bilinear-systemic illustration of a Boolean keep an eye on community makes it attainable to enquire simple keep watch over difficulties together with controllability, observability, stabilization, disturbance decoupling, id, optimum keep an eye on, and so forth.
The publication is self-contained, requiring simply wisdom of linear algebra and the fundamentals of the regulate thought of linear platforms. It starts off with a quick creation to prepositional good judgment and the ideas and houses of the STP and progressing through the (bi)linear expression of Boolean (control) networks to disturbance decoupling and decomposition of Boolean regulate structures. eventually multi-valued good judgment is taken into account as a extra detailed method of describing genuine networks and stochastic Boolean networks are touched upon. suitable numerical calculations are defined in an appendix and a MATLAB® toolbox for the algorithms within the e-book may be downloaded from http://lsc.amss.ac.cn/~dcheng/.
Analysis and regulate of Boolean Networks may be a basic reference for researchers in platforms biology, keep an eye on, platforms technology and physics. The e-book used to be constructed for a brief direction for graduate scholars and is appropriate for that function. laptop scientists and logicians can also locate this publication to be of curiosity.
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Extra resources for Analysis and Control of Boolean Networks: A Semi-tensor Product Approach
Nk ). W[m,n] can be constructed in an alternative way which is convenient in some applications. Denoting by δni the ith column of the identity matrix In , we have the following. 7 W[m,n] = δn1 1 δm ··· δnn 1 δm ··· δn1 m δm ··· δnn m . δm For convenience, we provide two more forms of swap matrix: ⎡ T⎤ Im ⊗ δn1 ⎥ ⎢ .. W[m,n] = ⎣ ⎦ . 49) Im ⊗ δnn T and, similarly, 1 m . , . . 50) The following factorization properties reflect the blockwise permutation property of the swap matrix. 51) W[pq,r] = (W[p,r] ⊗ Iq )(Ip ⊗ W[q,r] ) = (W[q,r] ⊗ Ip )(Iq ⊗ W[p,r] ).
This allows a very useful generalization of the previous proposition, which we now state as a theorem. 2 1. ,ik ) be a column vector with its elements arranged by the ordered multi-index Id(i1 , . . , ik ; n1 , . . , nk ). Then [In1 +···+nt−1 ⊗ W[nt ,nt+1 ] ⊗ Int+2 +···+nk ]X is a column vector consisting of the same elements, arranged by the ordered multi-index Id(i1 , . . , it+1 , it , . . , tk ; n1 , . . , nt+1 , nt , . . , nk ). 2. ,ik ) be a row vector with its elements arranged by the ordered multi-index Id(i1 , .
Nk ), if the elements are labeled by i1 , . . , ik and arranged as follows: Let it , t = 1, . . , k, run from 1 to nt with the order that t = k first, then t = k − 1, and so on, until t = 1. ,βk if and only if there exists 1 ≤ j ≤ k such that αi = βi , i = 1, . . , j − 1, αj < βj . If the numbers n1 , . . , nk of i1 , . . , ik are equal, we may use Id(i1 , . . , ik ; n) := Id(i1 , . . , ik ; n, . . , n). If ni are obviously known, the expression of Id can be simplified as Id(i1 , .
Analysis and Control of Boolean Networks: A Semi-tensor Product Approach by Daizhan Cheng, Hongsheng Qi, Zhiqiang Li
Categories: System Theory