Analysis and Control of Boolean Networks: A Semi-tensor by Daizhan Cheng, Hongsheng Qi, Zhiqiang Li

By Daizhan Cheng, Hongsheng Qi, Zhiqiang Li

Research and keep an eye on of Boolean Networks provides a scientific new method of the research of Boolean regulate networks. the basic device during this technique is a singular matrix product referred to as the semi-tensor product (STP). utilizing the STP, a logical functionality may be expressed as a standard discrete-time linear process. within the mild of this linear expression, sure significant matters bearing on Boolean community topology – mounted 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 structures appropriate to Boolean keep an eye on networks. The bilinear-systemic illustration of a Boolean regulate community makes it attainable to enquire easy keep watch over difficulties together with controllability, observability, stabilization, disturbance decoupling and so on.

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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 , . . , ik ; n1 , . . , nk ). Then ω[In1 +···+nt−1 ⊗ W[nt+1 ,nt ] ⊗ Int+2 +···+nk ] is a row vector consisting of the same elements, arranged by the ordered multiindex Id(i1 , . . , it+1 , it , .

53) The following property shows that the semi-tensor product can be expressed by the conventional matrix product plus the Kronecker product. 10 1. If A ∈ Mm×np , B ∈ Mp×q , then A B = A(B ⊗ In ). 54) B = (A ⊗ Ip )B. 55) 2. 10 is a fundamental result. Many properties of the semi-tensor product can be obtained through it. 55) as providing an alternative definition of the semi-tensor product. In fact, the name “semi-tensor product” comes from this proposition. Recall that for A ∈ Mm×n and B ∈ Mp×q , their tensor product satisfies A ⊗ B = (A ⊗ Ip )(In ⊗ B).

2. Let x = {x1 , x2 , . . , x24 }. If we use λ1 , λ2 , λ3 to express the data in the form ai = aλ1 ,λ2 ,λ3 , then under different Id’s they have different arrangements: (a) Using the ordered multi-index Id(λ1 , λ2 , λ3 ; 2, 3, 4), the elements are arranged as x111 x121 x131 .. x112 x122 x132 x113 x123 x133 x114 x124 x134 x231 x232 x233 x234 . (b) Using the ordered multi-index Id(λ1 , λ2 , λ3 ; 3, 2, 4), the elements are arranged as x111 x121 x211 .. x112 x122 x212 x113 x123 x213 x114 x124 x214 x321 x322 x323 x324 .

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