Finite‐time observability of probabilistic Boolean control networks

Abstract

In this paper, the finite-time observability (FO) of probabilistic Boolean control networks (PBCNs) based on set reachability and parallel extension is investigated. Under algebraic state space representation of PBCNs, finite-time set reachability (FSR), finite-time single input sequence set reachability (FSSSR), finite-time arbitrary input sequence set reachability (FASSR), as well as finite-time output feedback set reachability (FOSR) are addressed, where some efficient criteria are proposed. In the second part of this paper, several types of FO are provided, which can be converted to the corresponding FSR based on the parallel extension technique. Consequently, the obtained results for FSR are utilized to deal with FO. Finally, two examples including a biological network are given to demonstrate the theoretical results.

Further results for synchronization of two coupled Boolean networks

“…, then, an algebraic form for (8a) and (8b) is obtained, v(t + 1) = Fv(t), w(t + 1) = Gv(t)w(t), where F = 𝛿 8 [4,7,5,8,3,6,4,7], G = 𝛿 8 [3,3,3,3,3,4,3,4, … ] ∈  8×64 , the detailed information for matrix G is omitted here. Let 𝜉(t) = v(t)w(t) ∈ Δ 64 , then we get…”

“…Then, an algebraic form for (10a) and (10b) is obtained, v(t + 1) = Fv(t), w(t + 1) = Gv(t)w(t), where F = 𝛿 8 [3,7,8,8,1,5,6,6], G = 𝛿 8 [5, 1, 6, 2, 7, 3, 8, 4, … ] ∈  8×64 , the detailed information for matrix G is omitted here. Then, we get 64 and 𝛿 62 64 of (11) before and after {1, 2}-perturbation 𝜉(t + 1) = L𝜉(t), (11) where L = 𝛿 64 [21, 17, 22, 18, 23, 19, 24, 20,…”

“…To overcome this difficulty, Cheng et al derived the equivalent algebraic form of BNs by using the semi-tensor product (STP) of matrices in 2010 [4]. Then, the researches on BNs are further developed, includ-ing controllability, observability, optimal control, synchronization, output tracking, and state estimation [5][6][7][8][9][10][11][12][13][14][15][16].…”

Semi‐tensor product‐based algebra‐logic mixed representation and fault diagnosis for a class of gate networks

“…Besides, based on the Boolean function of gate networks, an algebraic representation, that is, a linear form of the product of a matrix and a vector, was proposed through the technology of semi-tensor product of matrices, which was an original theory first proposed by Cheng and Qi in 2009 [22]. Thereafter, extensive superior work has been done on the applications of logic systems based on their algebraic representations, such as state estimation [23,24], detectability, and observability [25][26][27] as well as control of Boolean networks [28][29][30][31][32], synchronization of Boolean networks [33,34], games [35][36][37], fault detection of digital circuits [38][39][40], and transformation of two feedback shift registers [41].…”

Reconstructibility analysis and set‐observer design for Boolean control networks

“…Thus, the logic-based problems can be transformed into algebraic problems. Based on STP method, some theoretical problems of BCNs, such as controllability and observability [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], reconstructibility or observer design [17][18][19][20][21][22][23][24][25][26][27], stability and stabilization [28][29][30][31][32][33] and optimal control [34][35][36][37][38][39], have been discussed.…”