Analysis and Control of Boolean Networks: A Semi-tensor - download pdf or read online

By Daizhan Cheng, Hongsheng Qi, Zhiqiang Li

ISBN-10: 0857290967

ISBN-13: 9780857290960

ISBN-10: 0857290975

ISBN-13: 9780857290977

Research and regulate of Boolean Networks provides a scientific new method of the research of Boolean keep an eye on networks. the elemental instrument during this procedure is a unique matrix product known as the semi-tensor product (STP). utilizing the STP, a logical functionality may be expressed as a traditional discrete-time linear procedure. within the gentle of this linear expression, convinced significant matters referring to Boolean community topology – fastened issues, cycles, temporary instances and basins of attractors – might be simply published by means of a collection of formulae. This framework renders the state-space method of dynamic keep an eye on 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 regulate difficulties together with controllability, observability, stabilization, disturbance decoupling and so on.

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In the following example we give an example of 3-dimensional data. 1 Consider R3 , with its canonical basis {e1 , e2 , e3 }. Any vector X ∈ R3 may then be expressed as X = x1 e1 + x2 e2 + x3 e3 . When the basis is fixed, we simply use X = (x1 , x2 , x3 )T to represent it. From simple vector algebra we know that in R3 there is a cross product, ×, such that for any two vectors X, Y ∈ R3 we D. 1007/978-0-85729-097-7_2, © Springer-Verlag London Limited 2011 19 20 2 have X × Y ∈ R3 , defined as follows: ⎛⎡ e1 X × Y = det⎝⎣ x1 y1 e2 x2 y2 Semi-tensor Product of Matrices ⎤⎞ e3 x3 ⎦⎠ .

3 c13 = 0, 1 c12 = 0, 1 c21 = 0, 2 c12 = 0, 3 c12 = 1, 2 c21 = 0, 2 c23 = 0, 3 c21 = −1, 3 c23 = 0, 2 c32 = 0, 3 c32 = 0, Since the cross product is linear with respect to the coefficients of each vector, the structure constants uniquely determine the cross product. For instance, let X = 3e1 − e3 and Y = 2e2 + 3e3 . Then X × Y = 6e1 × e2 + 9e1 × e3 − 2e3 × e2 − 3e3 × e3 1 2 3 1 2 3 = 6 c12 e1 + c12 e2 + c12 e3 + 9 c13 e1 + c13 e2 + c13 e3 1 2 3 1 2 3 e1 + c32 e2 + c32 e3 − 3 e33 e1 + c33 e2 + c33 e3 − 2 c32 = 2e1 − 9e2 + 6e3 .

Conventionally, φ is called a tensor, where s is called its covariant degree. 22 2 Semi-tensor Product of Matrices It is clear that for a tensor with covariant degree s, its structure constants form a set of s-dimensional data. 2 1. In R3 we define a three linear mapping as φ(X, Y, Z) = X × Y, Z , X, Y, Z ∈ R3 , where ·, · denotes the inner product. Its geometric interpretation is the volume of the parallelogram with X, Y , Z as three adjacent edges [when (X, Y, Z) form a right-hand system, the volume is positive, otherwise, the volume is negative].

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


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