By Daizhan Cheng, Hongsheng Qi, Zhiqiang Li

ISBN-10: 0857290967

ISBN-13: 9780857290960

ISBN-10: 0857290975

ISBN-13: 9780857290977

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 method is a singular matrix product referred to as the semi-tensor product (STP). utilizing the STP, a logical functionality might be expressed as a traditional discrete-time linear process. within the mild of this linear expression, sure significant matters bearing on Boolean community topology – mounted issues, cycles, brief 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 keep an eye on structures acceptable to Boolean keep an eye on networks. The bilinear-systemic illustration of a Boolean keep watch over community makes it attainable to enquire easy regulate difficulties together with controllability, observability, stabilization, disturbance decoupling and so on.

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Research and regulate of Boolean Networks provides a scientific new method of the research of Boolean keep an eye on networks. the elemental software during this procedure is a unique 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 method.

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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. 74) 3. Let X ∈ Rm be a row vector. Then T XT A = Vr (A) X. 75) 4. Let Y ∈ Rn be a row vector. Then AY = Y T Vc (A). 76) 5. Let X ∈ Rm be a column vector and Y ∈ Rn a row vector. Then XY = Y W[m,n] X. 19 Let A ∈ Mm×n and B ∈ Ms×t . 18 Assume ⎡ a A = 11 a21 a12 , a22 where m = n = 2, s = 3 and t = 2.

Xm S m and take their sum. i Step 2: Multiply m i=1 xi S by Y (which is a standard inner product). It is easy to check that this algorithm produces the same result. Now, in the first step it seems that we have (S 1 · · · S n ) × X. This calculation motivates a new algorithm, which is defined as follows. 5 Let T be an np-dimensional row vector and X a p-dimensional column vector. Split T into p equal blocks, named T 1 , . . , T p , which are 1 × n matrices. Define a left semi-tensor product, denoted by , as p T X= T i xi ∈ R n .

17 1. Let X = (x11 , x12 , x13 , x21 , x22 , x23 ). That is, {xij } is arranged by the ordered multi-index Id(i, j ; 2, 3). A straightforward computation shows ⎡ 1 ⎢0 ⎢ ⎢0 Y = W[23] X = ⎢ ⎢0 ⎢ ⎣0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 ⎤ ⎡ ⎤ ⎤⎡ x11 0 x11 ⎥ ⎢ ⎥ ⎢ 0⎥ ⎥ ⎢ x12 ⎥ ⎢ x21 ⎥ ⎢ x13 ⎥ ⎢ x12 ⎥ 0⎥ ⎥=⎢ ⎥. ⎥⎢ ⎥ ⎢ ⎥ ⎢ 0⎥ ⎥ ⎢ x21 ⎥ ⎢ x22 ⎥ 0 ⎦ ⎣ x22 ⎦ ⎣ x13 ⎦ x23 x23 1 That is, Y is the rearrangement of the elements xij in the order of Id(j, i; 3, 2). 2. Let X = (x1 , x2 , . . , xm )T ∈ Rm , Y = (y1 , y2 , .

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

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