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 watch over of Boolean Networks offers a scientific new method of the research of Boolean regulate networks. the basic device during this strategy is a singular matrix product known as the semi-tensor product (STP). utilizing the STP, a logical functionality should be expressed as a standard discrete-time linear procedure. within the mild of this linear expression, convinced significant matters bearing on Boolean community topology – fastened issues, cycles, temporary occasions and basins of attractors – might be simply published through a suite of formulae. This framework renders the state-space method of dynamic regulate platforms appropriate to Boolean keep an eye on networks. The bilinear-systemic illustration of a Boolean keep watch over community makes it attainable to enquire simple keep an eye on difficulties together with controllability, observability, stabilization, disturbance decoupling and so on.

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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). 56) to form the product. 10. 11 Let A and B be matrices with proper dimensions such that A is well defined.

1 Multiple-Dimensional Data Roughly speaking, linear algebra mainly concerns two kinds of objects: vectors and matrices. An n-dimensional vector is expressed as X = (x1 , x2 , . . , xn ). Its elements are labeled by one index, i, where xi is the ith element of X. For an m × n matrix ⎤ ⎡ a11 a12 · · · a1n ⎥ ⎢a ⎢ 21 a22 · · · a2n ⎥ ⎥, ⎢ A=⎢ . ⎥ . ⎦ ⎣ . am1 am2 · · · amn elements are labeled by two indices, i and j , where ai,j is the element of A located in the ith row and j th column. In this way, it is easy to connect the dimension of a set of data with the number of indices.

For instance, in a chemical factory the stove temperature may be classified as “very high”, “high”, “average”, “low”, or “very low”. Here, we could use “true” or “1” for the first case, and “false” or “0” for the fifth case. But what of the intermediate cases? 25” for “low”. In Beijing, the television broadcast uses “clean”, “mildly polluted”, and “severely polluted” to describe the air quality. 5”, and “1”) to classify statement 2. For wealth, “below the poverty line”, “low income”, “middle class”, “high income”, etc.

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