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 regulate of Boolean Networks offers a scientific new method of the research of Boolean keep watch over networks. the basic device during this procedure is a singular matrix product known as the semi-tensor product (STP). utilizing the STP, a logical functionality could be expressed as a traditional discrete-time linear method. within the mild of this linear expression, sure significant concerns relating Boolean community topology – mounted issues, cycles, brief instances and basins of attractors – may be simply printed through a suite of formulae. This framework renders the state-space method of dynamic keep watch over platforms appropriate to Boolean regulate 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 forth.

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For the remainder of this section we mainly consider k-valued logic. First, we define some unary operators. It is easy to see that there are k k unary operators. We define some which will be useful in the sequel: (1) “negation”, ¬, (2) “rotator”, k , (3) “i-confirmer”, i,k , i = 1, 2, . . , k. 8 1. Let p = i k−1 . Then ¬p = 2. Let p = i k−1 . (k − 1) − i . 35) Then kp = i,k p = i−1 k−1 , 1, i > 0, i = 0. 36) 3. p, p = 0, p = k−i k−1 , k−i k−1 . 11 shows the truth values of these unary operators.

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. Then: B 1. A B and B A have the same characteristic functions. 2. tr(A B) = tr(B A). 3. If A and B are invertible, then A B ∼ B A, where “∼” stands for matrix similarity. 4. , lower triangular, diagonal, orthogonal) matrix. 5.

14 Let A ∈ Mm×n , X ∈ Mn×q , Y ∈ Mp×m . Then Vr (AX) = A Vc (Y A) = A Vr (X), T Vc (Y ). , Rn ). 64) becomes a standard linear mapping. 65), the stacking expression of a matrix polynomial may also be obtained. 2 Let X be a square matrix and p(x) be a polynomial, expressible as p(x) = q(x)x + p0 . Then Vr p(X) = q(X)Vr (X) + p0 Vr (I ). 66) Using linear mappings on matrices, some other useful formulas may be obtained [4]. 15 Let A ∈ Mm×n and B ∈ Mp×q . Then (Ip ⊗ A)W[n,p] = W[m,p] (A ⊗ Ip ), W[m,p] (A ⊗ B)W[q,n] = (B ⊗ A).

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