By Daizhan Cheng, Hongsheng Qi, Zhiqiang Li
The Boolean community has turn into a robust instrument for describing and simulating mobile networks within which the weather behave in an on–off model. research and regulate of Boolean Networks offers a scientific new method of the research of Boolean regulate networks. the elemental instrument during this strategy is a singular matrix product referred to as the semi-tensor product (STP). utilizing the STP, a logical functionality will be expressed as a traditional discrete-time linear approach. within the gentle of this linear expression, yes significant concerns bearing on Boolean community topology – fastened issues, cycles, temporary instances and basins of attractors – should be simply printed by way of a suite of formulae. This framework renders the state-space method of dynamic regulate structures 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 an eye on difficulties together with controllability, observability, stabilization, disturbance decoupling, identity, optimum regulate, and so on.
The booklet is self-contained, requiring purely wisdom of linear algebra and the fundamentals of the keep watch over concept of linear structures. It starts off with a quick advent to prepositional good judgment and the innovations and homes of the STP and progressing through the (bi)linear expression of Boolean (control) networks to disturbance decoupling and decomposition of Boolean keep an eye on platforms. ultimately multi-valued common sense is taken into account as a extra certain manner of describing genuine networks and stochastic Boolean networks are touched upon. proper numerical calculations are defined in an appendix and a MATLAB® toolbox for the algorithms within the booklet should be downloaded from http://lsc.amss.ac.cn/~dcheng/.
Analysis and keep an eye on of Boolean Networks could be a basic reference for researchers in platforms biology, keep an eye on, platforms technological know-how and physics. The publication was once constructed for a quick direction for graduate scholars and is acceptable for that objective. laptop scientists and logicians can also locate this ebook to be of curiosity.
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Extra info for Analysis and Control of Boolean Networks: A Semi-tensor Product Approach
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] ). 4 Properties of the Semi-tensor Product In this section some fundamental properties of the semi-tensor product of matrices are introduced.
4 Properties of the Semi-tensor Product 45 The following lemma is useful for simplifying some expressions. 1 Let A ∈ Mm×n . Then W[m,q] W[q,n] = Iq ⊗ A. 72) The semi-tensor product has some pseudo-commutative properties. The following are some useful pseudo-commutative properties. Their usefulness will become apparent later. 18 Suppose we are given a matrix A ∈ Mm×n . 1. Let Z ∈ Rt be a column vector. Then AZ T = Z T W[m,t] AW[t,n] = Z T (It ⊗ A). 73) 2. Let Z ∈ Rt be a column vector. Then ZA = W[m,t] AW[t,n] Z = (It ⊗ A)Z.
Arrange it into a matrix by using the ordered multi-index Id(i1 , i2 , . . , is ; n) for columns and the ordered multi-index Id(j1 , j2 , . . , jt ; n) for rows. 27) Mφ = ⎢ . ⎥. ⎢ . ⎥ ⎣ . ⎦ 11···1 · · · c11···n · · · cnn···n cnn···n nn···n nn···n It is the structure matrix of the tensor φ. Now, assume ωi ∈ V ∗ , i = 1, 2, . . , t, and Xj ∈ V , j = 1, 2, . . , s, where ωi are expressed as rows, and Xj are expressed as columns. Then φ(ω1 , . . , ωt , X1 , . . 28) is omitted. Next, we define the power of a matrix.