# On Robust Synchronization of Drive-Response Boolean Control Networks with Disturbances

Authors: Yuanyuan Li; Jie Zhong; Jianquan Lu; Zhen Wang; Fuad E. Alssadi
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper investigates the robust synchronization of drive-response Boolean control networks (BCNs) with disturbances via semi-tensor product of matrices. Firstly, the definition of robust synchronization is presented for the drive-response BCNs with disturbances. Then, based on the algebraic state space representation of drive-response BCNs, the robustly reachable states/sets are presented to investigate robust synchronization of disturbed BCNs. According to the set of robustly reachable states, some necessary and sufficient criteria are obtained for robust synchronization of drive-response BCNs with disturbances under a given state feedback controller. Finally, an illustrative example is presented to demonstrate the obtained theoretical results.


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## Body

## 1. Introduction

As an efficient mathematical model of biological systems and genetic regulatory networks (GRNs), Boolean network (BN) has attracted much attention as a qualitative tool for analyzing GRNs. BN theory was firstly proposed by Kauffman in 1969 [1] to model a gene as a binary device for studying the behavior of large, randomly constructed networks of these genes. Thanks to Kauffman’s pioneering work [1], investigations on Boolean networks (BNs) have attracted great attention from biologists and systems scientists. Consequently, many fundamental and excellent results have been established for BNs [2]. In a BN, each node is characterized by two possible values, 1 and 0, and its value (1 or 0) indicates its measured abundance (expressed or unexpressed, active or inactive). Meanwhile, the state evolution of each node on network is determined by a series of Boolean rules. In the past few years, some basic problems of BNs have been investigated in [3], including the fixed points, cycles, the basin of attractors, and the transient time. BNs with additional inputs called Boolean control networks (BCNs) were firstly proposed in [2] and could be used to design and analyze therapeutic intervention strategies. For example, in [4], Huang and Inber established a simple Boolean control network (BCN) to simulate the dynamics of signaling system within capillary endothelia cells, where two external inputs represent growth factors and cell shape (spreading). Recently, the controllability of BCNs has been widely investigated by numerous researchers [5–9].Recently, a new matrix product called semi-tensor product (STP) of matrices proposed by Cheng and his colleagues [10, 11] has been successfully used to study Boolean (control) networks and game theory on static games [12, 13], and many excellent results have been obtained [14–23]. Its main idea is to convert a logical function into an algebraic function, and, thus, the dynamic of BN can be uniquely converted into a standard discrete algebraic dynamic [11]. However, it should be noted that one main drawback of the algebraic state expression of BNs is its computational complexity. The algebraic state representation converts a BN with n state variables and m control inputs into a state-control-space of size 2n+m. Thus, any algorithm based on this approach has an exponential time-complexity. Moreover, many problems like determining fixed points and observability of BCNs have already been proved to be NP-hard. Hence, the computational complexity is intrinsic and also independent of the models adopted to describe BNs.Since BN is an efficient model to provide general features of living organism and to well illustrate GRNs, the synchronization problem for BNs (or BCNs) has received considerable research interests in the past few years. The researches on synchronization of BNs (or BCNs) can capture lots of useful information on the evolution of biological systems whose corresponding subsystems influence each other. For example, investigations on synchronized BNs are beneficial to better understand synchronization between two coupled lasers [24]. Hence, there are both theoretical and practical importance for studying the synchronization problem of BNs (or BCNs). Recently, some important results corresponding to complete synchronization for two deterministic BNs have been obtained [25, 26].In the past few years, much attention has been paid on the synchronization phenomenon of collective behavior by large amount of researches [27–34]. Aside from synchronization, the ability of disturbances resisting for synchronized networks has also attracted much attention [35–38]. In real world, the external disturbances are ubiquitous and may also lead the coupled networks to some unexpected behaviors or even break the phenomenon of synchronization. Hence, it is desirable to investigate the ability of coupled networks to resist external disturbances. In order to deal with this situation, it is practically significant to investigate robust synchronization of dynamical networks, which has gained much research attention [39–41]. Gene regulation is an intrinsically noisy process, and it is always subject to intracellular and extracellular disturbances and environment fluctuations [42–44]. For example, a cell apoptosis Boolean model is given to study the antiapoptotic pathway under external disturbance inputs, one may be interested in whether there exists a response system that can achieve synchronization with the apoptosis network without disturbances. Thus, when modeling GRNs, disturbances should be considered, as the states of genes may be subject to instantaneous perturbations and experience abrupt disturbances [45, 46]. These disturbance inputs may prohibit the effectiveness of control strategies in keeping the cellular states of biological systems and GRNs in a desirable set. Thus, it is necessary for us to design some control mechanisms under which some coupled GRNs are robustly synchronized to the external disturbances. Unfortunately, concerning BCNs, to the best of our knowledge, there is no result available corresponding to robust synchronization.In this paper, using STP technique, the robust synchronization problem of drive-response BCNs with disturbances is investigated, and some necessary and sufficient criteria are obtained to check whether a given drive-response BCNs with disturbances can be robustly synchronized. Moreover, a set of robustly reachable states from a given initial state and a union set of robustly reachable states from the initial states setΔ22n are proposed to check robust synchronization. Finally, one numerical example is presented to illustrate the main results.The rest of this paper is structured as follows. In Section2, we present some necessary notations and preliminaries on STP technique. In Section 3, we present our problem formulation and our main results on robust synchronization of drive-response BCNs with disturbances, by resorting to the matrix expression of logical functions and the STP technique. Illustrative example is given to show the validness of our main results in Section 4.


## 2. Some Preliminaries

In this section, we present some preliminaries on semi-tensor product (STP) and matrix expression of logical functions, which will be used in later analysis. The main tool used in this paper is STP of matrices. Using STP of matrices, a Boolean (control) network can be expressed in its equivalent algebraic representation.For statement ease, we firstly present some basic notations:1k=[1,…,1︸k]T, where superscript T presents the transpose of a matrix or a vector; D={0,1} with the usual operations (sum +, product ·, and negation ¬); let δni and Δn={δn1,…,δnn} be the i-th column and the columns’ set of identity matrixes In. When n=2, we simply use Δ=Δ2; let Coli(A) (Rowi(A)) and Col(A) (Row(A)) be the i-th column (row) and the set of columns (rows) of matrix A; given an m×mn matrix A, denote by Blki(A), i=1,…,n, the i-th block of an m×mn matrix A; i.e., A=(Blk1(A),…,Blkn(A)); L=[δni1,…,δnis] is called a logical matrix if Col(L)⊆Δn, which can be simply denoted as L=δn[i1,…,is]. Moreover, here we call i1,…,is the row indexes of columns for matrix L. The set of m×n logical matrices is denoted by Lm×n. Assume that A∈Rm×n is a real matrix (Rm×n denotes the set of real matrices with order m×n), if all the entries of A are positive (nonnegative), that is, Aij>0 (Aij≥0) for any 1≤i≤m,1≤j≤n, then simply denote A>0 (A≥0).Now, we present the definitions of Boolean addition and Boolean product of Boolean matrices. Anm×n matrix A=(aij)m×n is called a Boolean matrix if aij∈D, where Bm×n denotes the set of all m×n Boolean matrices. Then, we define the Boolean addition below: x+By≔x∨y, x,y∈D. Therefore, for two Boolean matrices X,Y∈Bm×n, we can obtain the Boolean addition for X and Y: X+BY=(xij+Byij)m×n, 1≤i≤m,1≤j≤n. Moreover, we can also define the Boolean product of two Boolean matrices A∈Bm×n and B∈Bn×p, denoted by A×BB=(cij)m×p∈Bm×p with cij=∑Bk=1paik∧bkj, 1≤i≤m,1≤j≤p. Particularly, if A∈Bm×m, then A(k)=A(k-1)×BA.Then, we introduce the STP “⋉” between matrices (and in particular, vectors) as follows [10].Definition 1 (see [10]).
Considern×m matrix A and p×q matrix B. The STP of A and B is defined as follows: A⋉B=(A⊗Il/m)(B⊗Il/p), where l=l.c.m.(m,p), denotes the least common multiple of m and p. Here, ⊗ denotes the Kronecker product of matrices.Remark 2.
Ifm=p, then A⋉B=AB, which is the standard matrix product. Thus, the STP of matrices is a generalization of the standard matrix product providing a new way to multiply two matrices with arbitrary dimensions.Definition 3 (see [10]).
Ifσ∈Δ2n, then σ⋉σ=Φ2nσ, where Φ2n is called power-reducing matrix for 2n-valued logical vectors and Φ2n=δ22n[1,2n+2,…,(2n-2)·2n+2n-1,22n].Definition 4 (see [10]).
Anmn×mn matrix W[m,n] is called a swap matrix, if it is constructed by the way: label its columns by (11,12,…,1n,…,m1,…,mn) and similarly label its rows by (11,21,…,m1,…,1n,2n,…,mn). Then its element in the position ((I,J),(i,j)) is assigned as(1)wI,J,i,j=δi,jI,J=1,I=iandJ=j,0,otherwise.When m=n, we denote W[m,n] by W[n] or W[m].Identify Boolean variables 1 and 0 with vectorsδ21 and δ22. That is to say, we consider a Boolean variable X∈D as a vector x∈Δ; thus a Boolean function with n variables f:Dn→D is equivalent with a map f:(Δ2)n→Δ2.Proposition 5 (see [10]).
Letf:(D)n→D be a Boolean function. Then there exists a unique matrix F∈L2×2n such that f(σ1,…,σn)=F⋉σ1⋉⋯⋉σn, for every (σ1,…,σn)∈(Δ2)n, where F is called structure matrix of the logical function f.


## 3. Problem Formulation and Main Results

### 3.1. Problem Formulation

Since disturbance is an important factor when modeling GRNs, here we consider the following drive-response BCNs with disturbances:(2)x1t+1=f1ξ1t,…,ξpt,x1t,…,xnt,⋮xnt+1=fnξ1t,…,ξpt,x1t,…,xnt,y1t+1=g1u1t,…,umt,y1t,…,ynt,⋮ynt+1=gnu1t,…,umt,y1t,…,ynt.Here, xi and yj represent the i-th node and j-th node of drive and response BN, respectively, fi:{1,0}n+p→{1,0}, gj:{1,0}n+m→{1,0} are Boolean functions, i=1,…,n, j=1,…,n. In addition, uα and ξβ are controls and disturbance inputs, α=1,…,m, β=1,…,p. We simply denote by x(t)=(x1(t),…,xn(t))T and y(t)=(y1(t),…,yn(t))T the states of drive and response BN, respectively. We can observe that the state evolution of drive-response BCNs depends on the following initial states x(0)=(x1(0),…,xn(0))T and y(0)=(y1(0),…,yn(0))T. Correspondingly, the dynamic of the state feedback control is in the form of(3)u1t=ψ1x1t,…,xnt,y1t,…,ynt,⋮umt=ψnx1t,…,xnt,y1t,…,ynt.Here, ψi:{1,0}2n→{1,0} (i=1,…,m) are Boolean functions.Now, we present the definition of robust synchronization for drive-response BCNs (2) with disturbances.Definition 6.
Consider the drive-response BCNs (2) with disturbances and the state feedback control (3). Then, system (2) is said to be robustly synchronized if, for any initial states and any disturbance inputs, there exists an integer k such that x(t)=y(t), ∀t≥k.Remark 7.
As we can see from system (2), the disturbances are imposed on the states of drive BN, which implies that the states of drive BN will be affected by the values of disturbances ξi(t) (i=1,…,p). Thus, if there exists a state feedback control in form of (3) such that the response BN can always synchronize with the drive BN regardless of the disturbances, it will be welcome. The main objective of this paper is to establish robust synchronization criteria for the drive-response BCNs (2) with disturbances.


### 3.2. Algebraic form of Drive-Response BCNs with Disturbances

In this subsection, we will convert the drive-response BCNs (2) with disturbances and the feedback control (3) into equivalent algebraic forms by applying STP. Using the vector form of Boolean variables and denoting x(t)=⋉i=1nxi(t)∈Δ2n, yt=⋉i=1nyi(t)∈Δ2n, u(t)=⋉i=1mui(t)∈Δ2m, and ξ(t)=⋉i=1pξi(t)∈Δ2p, by Proposition 5, we can convert systems (2) and (3) into the following algebraic forms:(4)xt+1=Fξtxt,yt+1=Gutyt,and(5)ut=Kxtyt,where F∈L2n×2n+p, G∈L2n×2m+n, and K∈L2m×22n are called the state transition matrices of drive BN and response BN and the state feedback gain matrix, respectively.Further, denotez(t)=x(t)⋉y(t)∈Δ22n, and we can combine (4) into the following equation:(6)zt+1=xt+1yt+1=FξtxtGutyt=FI2n+p⊗Gξtxtutyt=FI2n+p⊗GI2p⊗W2m,2nξtutzt≜Lξtutzt,where L=F(I2n+p⊗G)(I2p⊗W[2m,2n])∈L22n×2p+m+2n.Thus, for the drive-response BCNs (4) with disturbances and the feedback control (5), we obtain another equivalent algebraic form as follows:(7)zt+1=Lξtutzt,ut=Kzt,where L∈L22n×2p+m+2n and K∈L2m×22n are the state transition matrices of system (6) and state feedback gain matrix of system (5).Remark 8.
Definingz(t)=x(t)y(t), we transfer the disturbed BCNs (4) and control system (5) into system (7). It has been proved in [10] that, by defining z(t)=x(t)y(t), we can get a bijective mapping (Δ)22n→Δ2n×Δ2n. For example, if z(t)=δ43, we can derive that x(t)=δ22 and y(t)=δ21.


### 3.3. Main Results on Robust Synchronization

In the following subsection, we will investigate the robust synchronization of the drive-response BCNs (2) with disturbances based on the equivalent algebraic form (7).Consider the drive-response BCNs (2) with disturbances. For a given state feedback control u(t)=Kz(t) with K∈L2m×22n, one can see from Proposition 5 that(8)zt+1=Lξtutzt=LξtKΦ2nzt=LI2p⊗KΦ2nξtzt≜Lξtzt,where L=L(I2p⊗KΦ2n)=[Blk1(L),…,Blk2p(L)]∈L22n×22n+p.Then, in the following sequel, we introduce a set of robustly reachable states from a given initial state at given step.Definition 9.
A stateϖ is said to be a reachable state from the initial state z(0) at the k-th step if there exists a disturbance sequence ξ(0),…,ξ(k-1), such that ϖ=z(k). Then, for all possible disturbance sequences ξ(0),…,ξ(k-1), we can obtain the set of all possible reachable states, which is called the set of robustly reachable states from the initial state z(0) at the k-th step, denoted by RSk(z(0)).According to (8), suppose that ξ(0)=δ2pμ and z(0)=δ22nj, and we can obtain the set of robustly reachable states from z(0) at the first step, which is RS1(z(0))={Colj(Blkμ(L)),μ=1,…,2p}. Then, we can further define the union set of robustly reachable states from the initial states set Δ22n at the k-th step as follows:(9)RSkΔ22n=RSkδ22n1∪RSkδ22n2∪⋯∪RSkδ22n22n.Particularly, when k=0, we denote RS0(Δ22n)=Δ22n.The following proposition gives some important properties for the union setRSk(Δ22n).Proposition 10.
(1) LetRSk(Δ22n) be the union set of robustly reachable states from the initial states set Δ22n at the k-th step, and then we have the following relationship:(10)Δ22n≔RS0Δ22n⊇RS1Δ22n⋯⊇RSkΔ22n⋯.
(2) If there exists an integerk>0 such that RSk+1(Δ22n)=RSk(Δ22n), then we have RSt(Δ22n)=RSk(Δ22n), t≥k.Proof.
( 1 ) Since RS1(Δ22n)⊆Δ22n, we have (11)RSk+1Δ22n=RSkRS1Δ22n⊆RSkΔ22n,k=1,2,….(2) Suppose that there exists an integer k such that RSk+1(Δ22n)=RSk(Δ22n); i.e., RSk(Δ22n)=RS1(RSk(Δ22n)). Then we have(12)RSkΔ22n=RS1RSk+1Δ22n=RSk+2Δ22n=RS2RSk+1Δ22n=RSk+3Δ22n⋮=RSk+τΔ22n,τ=1,2,….This completes the proof.By resorting to the above important properties of the union setRSk(Δ22n), one can derive the following result, which will be an auxiliary result for the robust synchronization.Proposition 11.
Assume thatπ is the smallest positive integer such that RSk+1(Δ22n)=RSk(Δ22n), and then we have 0≤π≤22n-1.Proof.
IfRS1(Δ22n)=Δ22n, then we have RS0(Δ22n)=RS1(Δ22n)=Δ22n, which implies that π=0. Now we claim that 1≤π≤22n-1. To prove this claim, it is sufficient to prove that |RSk(Δ22n)|≤22n-k holds for every 1≤k≤22n-1. We draw the conclusion by induction on k. Consider the case of k=1. Suppose on the contrary that RS1(Δ22n)>22n-1, which implies that |RS1(Δ22n)|=22n. This implies the fact of RS1(Δ22n)=Δ22n, and then we have RSt(Δ22n)=Δ22n,t=1,…. This is a contradiction with the fact that RS0(Δ22n)⊇RS1(Δ22n); i.e., π≥1. Thus, we prove that, for the case of k=1, RS1(Δ22n)≤22n-1.
Now, let1<k≤π and suppose that |RSk(Δ22n)|≤22n-k by induction. According to Proposition 10, we have RSk+1(Δ22n)⊆RSk(Δ22n). This together with the induction hypothesis yields that |RSk+1(Δ22n)|≤|RSk(Δ22n)|≤22n-k. If |RSk+1(Δ22n)|>22n-k-1, then we have |RSk+1(Δ22n)|=|RSk(Δ22n)|=22n-k, which implies that RSk+1(Δ22n)=RSk(Δ22n). This contradicts with the fact of the minimality of π. Thus, one can derive that |RSk(Δ22n)|≤22n-k. This completes the proof.Based on Proposition10, we obtain the following necessary and sufficient condition for robust synchronization of the drive-response BCNs (2) with disturbances.Theorem 12.
Consider the drive-response BCNs (2) with disturbances. Let RSk(Δ22n) be the union set of robustly reachable states from the initial states set Δ22n at the k-th step. Then system (2) can be robustly synchronized if and only if there exists an integer k satisfying 1≤k≤22n-1, such that(13)RSkΔ22n⊆Ξ,where Ξ={δ22n(i-1)2n+i:i=1,…,2n} denotes the synchronized states set.Proof.
Based on Proposition10, if there exists an integer k satisfying 1≤k≤22n-1, such that RSk(Δ22n)⊆Ξ, then, given any integer t≥k, RSt(Δ22n)⊆Ξ. Then, according to the definition of set Ξ, the states of x(t) and y(t) reach synchronization after k steps. The proof of sufficient part can be directly obtained, which is omitted here. This completes the proof.Remark 13.
Ifx(t)=y(t)=δ2nj, then the equality z(t)=x(t)⋉y(t) yields that z(t)=δ22n(j-1)2n+j. Thus, we use the set Ξ={δ22n(i-1)2n+i:i=1,…,2n} to denote the synchronized states set. According to Proposition 10, for the initial states set Δ22n, we can always calculate the union set of robustly reachable states at given step by recurrence method. Moreover, according to Proposition 11, we can conclude that one only needs to calculate the union set of robustly reachable states for finite steps in order to check robust synchronization.Consider a setΠ={δ22nλ1,δ22nλ2,…,δ22nλε}. Note that each state δ22nλj (j=1,…,ε) is a column of the identity matrix I22n, we define the column vector form of the set Π by Boolean summing up of all the states as follows:(14)Πc=…,0,1λ1,0,…,0,1λε,…T∈B2n×1.Here, the λ1,…,λε-th elements of Πc are 1, and the rest are 0. Particularly, when Π=Δ22n, one can derive its corresponding column vector form of Δ22n, denoted by (1,…,1)T∈B2n×1. Thus, for the union set of robustly reachable states from the initial states Δ22n, i.e., RSk(Δ22n), we use RSk(Δ22n) to denote the corresponding column vector form. Then, we have(15)RSkΔ22n=⋃j=122nRSkδ22nj~RSkΔ22n=∑j=122nBRSkδ22nj.Since L in (8) is a 22n×22n+p matrix, we split it into 2p equal blocks as L=[L1,…,L2p], where Lj∈L22n (j=1,…,2p). Then, the following result will provide an algebraic representation for RSk(Δ22n).Theorem 14.
Consider the drive-response BCNs (2) with disturbances. Let RSk(Δ22n) be the union set of robustly reachable states from the initial states set Δ22n at the k-th step. Then, we have the following equation for the column vector form RSk(Δ22n):(16)RSkΔ22n=∑j=122nBColjLˇk,Lˇ=∑j=12pBLj.Proof.
We prove the result by induction. Consider the case ofk=1. Given an initial state z(0)=δ22nj, one can obtain the column vector form of RS1(δ22nj) as follows:(17)RS1δ22nj=L⋉δ2p1δ22nj+B⋯+BL⋉δ2p2pδ22nj=∑i=12pBLiδ22nj=ColjLˇ.Thus, the union set of robustly reachable states from the initial states set Δ22n at the first step is RS1(Δ22n)=∑Bj=122nRS1(δ22nj)=∑Bj=122nColj(Lˇ), which proves the case of k=1.
Then, we assume that (16) holds for the case of k=n and consider the case of k=n+1. Suppose that the reachable states in RSn(Δ22n) are δ22nw1,…,δ22nwr, where r=|RSn(Δ22n)|. Note that the trajectory starting from z(0)∈Δ22n to z(n+1)∈RSn+1(Δ22n) can be decomposed into trajectory starting from z(0) to some z(n)=δ22nwi (i∈{1,…,r}) at the n-th step and the trajectory from the possible states z(n)=δ22nwi to z(n+1)∈RSn+1(Δ22n) at one step. Thus, one can derive that(18)RSn+1Δ22n=RS1RSnΔ22n=RS1δ22nw1+B⋯+BRS1δ22nwr. This together with (17) yields(19)RSn+1Δ22n=∑l=1rB∑j=12pBLjδ22nwl=∑j=12pBLj×B∑l=1rBδ22nwl=Lˇ×BRSnΔ22n.On the other hand, one can obtain that(20)∑i=122nBColiLˇk+1=∑i=122nBColiLˇ×BLˇk=∑i=122nBLˇ×BColiLˇk=Lˇ×B∑i=122nBColiLˇk.Applying the induction hypothesis, by (19), we have ∑Bi=122nColi(Lˇ(n+1))=Lˇ×BRSn(Δ22n)=RSn+1(Δ22n), which proves the case of k=n+1. Thus, according to the mathematical induction, we can conclude that (16) holds for any positive integer k, which completes the proof.LetΞ→=∑j=12nδ22n(j-1)2n+j be the column vector form of the synchronized states set Ξ. Thus, based on Theorem 14, we can obtain the following result for robust synchronization.Corollary 15.
Consider the drive-response BCNs (2) with disturbances. Let Ξ→ be the column vector form of the synchronized states set Ξ. Then system (2) can be robustly synchronized if and only if there exists an integer 1≤k≤22n-1, such that(21)Ξ→-∑j=122nBColiLˇk≥0.


## 3.1. Problem Formulation

Since disturbance is an important factor when modeling GRNs, here we consider the following drive-response BCNs with disturbances:(2)x1t+1=f1ξ1t,…,ξpt,x1t,…,xnt,⋮xnt+1=fnξ1t,…,ξpt,x1t,…,xnt,y1t+1=g1u1t,…,umt,y1t,…,ynt,⋮ynt+1=gnu1t,…,umt,y1t,…,ynt.Here, xi and yj represent the i-th node and j-th node of drive and response BN, respectively, fi:{1,0}n+p→{1,0}, gj:{1,0}n+m→{1,0} are Boolean functions, i=1,…,n, j=1,…,n. In addition, uα and ξβ are controls and disturbance inputs, α=1,…,m, β=1,…,p. We simply denote by x(t)=(x1(t),…,xn(t))T and y(t)=(y1(t),…,yn(t))T the states of drive and response BN, respectively. We can observe that the state evolution of drive-response BCNs depends on the following initial states x(0)=(x1(0),…,xn(0))T and y(0)=(y1(0),…,yn(0))T. Correspondingly, the dynamic of the state feedback control is in the form of(3)u1t=ψ1x1t,…,xnt,y1t,…,ynt,⋮umt=ψnx1t,…,xnt,y1t,…,ynt.Here, ψi:{1,0}2n→{1,0} (i=1,…,m) are Boolean functions.Now, we present the definition of robust synchronization for drive-response BCNs (2) with disturbances.Definition 6.
Consider the drive-response BCNs (2) with disturbances and the state feedback control (3). Then, system (2) is said to be robustly synchronized if, for any initial states and any disturbance inputs, there exists an integer k such that x(t)=y(t), ∀t≥k.Remark 7.
As we can see from system (2), the disturbances are imposed on the states of drive BN, which implies that the states of drive BN will be affected by the values of disturbances ξi(t) (i=1,…,p). Thus, if there exists a state feedback control in form of (3) such that the response BN can always synchronize with the drive BN regardless of the disturbances, it will be welcome. The main objective of this paper is to establish robust synchronization criteria for the drive-response BCNs (2) with disturbances.


## 3.2. Algebraic form of Drive-Response BCNs with Disturbances

In this subsection, we will convert the drive-response BCNs (2) with disturbances and the feedback control (3) into equivalent algebraic forms by applying STP. Using the vector form of Boolean variables and denoting x(t)=⋉i=1nxi(t)∈Δ2n, yt=⋉i=1nyi(t)∈Δ2n, u(t)=⋉i=1mui(t)∈Δ2m, and ξ(t)=⋉i=1pξi(t)∈Δ2p, by Proposition 5, we can convert systems (2) and (3) into the following algebraic forms:(4)xt+1=Fξtxt,yt+1=Gutyt,and(5)ut=Kxtyt,where F∈L2n×2n+p, G∈L2n×2m+n, and K∈L2m×22n are called the state transition matrices of drive BN and response BN and the state feedback gain matrix, respectively.Further, denotez(t)=x(t)⋉y(t)∈Δ22n, and we can combine (4) into the following equation:(6)zt+1=xt+1yt+1=FξtxtGutyt=FI2n+p⊗Gξtxtutyt=FI2n+p⊗GI2p⊗W2m,2nξtutzt≜Lξtutzt,where L=F(I2n+p⊗G)(I2p⊗W[2m,2n])∈L22n×2p+m+2n.Thus, for the drive-response BCNs (4) with disturbances and the feedback control (5), we obtain another equivalent algebraic form as follows:(7)zt+1=Lξtutzt,ut=Kzt,where L∈L22n×2p+m+2n and K∈L2m×22n are the state transition matrices of system (6) and state feedback gain matrix of system (5).Remark 8.
Definingz(t)=x(t)y(t), we transfer the disturbed BCNs (4) and control system (5) into system (7). It has been proved in [10] that, by defining z(t)=x(t)y(t), we can get a bijective mapping (Δ)22n→Δ2n×Δ2n. For example, if z(t)=δ43, we can derive that x(t)=δ22 and y(t)=δ21.


## 3.3. Main Results on Robust Synchronization

In the following subsection, we will investigate the robust synchronization of the drive-response BCNs (2) with disturbances based on the equivalent algebraic form (7).Consider the drive-response BCNs (2) with disturbances. For a given state feedback control u(t)=Kz(t) with K∈L2m×22n, one can see from Proposition 5 that(8)zt+1=Lξtutzt=LξtKΦ2nzt=LI2p⊗KΦ2nξtzt≜Lξtzt,where L=L(I2p⊗KΦ2n)=[Blk1(L),…,Blk2p(L)]∈L22n×22n+p.Then, in the following sequel, we introduce a set of robustly reachable states from a given initial state at given step.Definition 9.
A stateϖ is said to be a reachable state from the initial state z(0) at the k-th step if there exists a disturbance sequence ξ(0),…,ξ(k-1), such that ϖ=z(k). Then, for all possible disturbance sequences ξ(0),…,ξ(k-1), we can obtain the set of all possible reachable states, which is called the set of robustly reachable states from the initial state z(0) at the k-th step, denoted by RSk(z(0)).According to (8), suppose that ξ(0)=δ2pμ and z(0)=δ22nj, and we can obtain the set of robustly reachable states from z(0) at the first step, which is RS1(z(0))={Colj(Blkμ(L)),μ=1,…,2p}. Then, we can further define the union set of robustly reachable states from the initial states set Δ22n at the k-th step as follows:(9)RSkΔ22n=RSkδ22n1∪RSkδ22n2∪⋯∪RSkδ22n22n.Particularly, when k=0, we denote RS0(Δ22n)=Δ22n.The following proposition gives some important properties for the union setRSk(Δ22n).Proposition 10.
(1) LetRSk(Δ22n) be the union set of robustly reachable states from the initial states set Δ22n at the k-th step, and then we have the following relationship:(10)Δ22n≔RS0Δ22n⊇RS1Δ22n⋯⊇RSkΔ22n⋯.
(2) If there exists an integerk>0 such that RSk+1(Δ22n)=RSk(Δ22n), then we have RSt(Δ22n)=RSk(Δ22n), t≥k.Proof.
( 1 ) Since RS1(Δ22n)⊆Δ22n, we have (11)RSk+1Δ22n=RSkRS1Δ22n⊆RSkΔ22n,k=1,2,….(2) Suppose that there exists an integer k such that RSk+1(Δ22n)=RSk(Δ22n); i.e., RSk(Δ22n)=RS1(RSk(Δ22n)). Then we have(12)RSkΔ22n=RS1RSk+1Δ22n=RSk+2Δ22n=RS2RSk+1Δ22n=RSk+3Δ22n⋮=RSk+τΔ22n,τ=1,2,….This completes the proof.By resorting to the above important properties of the union setRSk(Δ22n), one can derive the following result, which will be an auxiliary result for the robust synchronization.Proposition 11.
Assume thatπ is the smallest positive integer such that RSk+1(Δ22n)=RSk(Δ22n), and then we have 0≤π≤22n-1.Proof.
IfRS1(Δ22n)=Δ22n, then we have RS0(Δ22n)=RS1(Δ22n)=Δ22n, which implies that π=0. Now we claim that 1≤π≤22n-1. To prove this claim, it is sufficient to prove that |RSk(Δ22n)|≤22n-k holds for every 1≤k≤22n-1. We draw the conclusion by induction on k. Consider the case of k=1. Suppose on the contrary that RS1(Δ22n)>22n-1, which implies that |RS1(Δ22n)|=22n. This implies the fact of RS1(Δ22n)=Δ22n, and then we have RSt(Δ22n)=Δ22n,t=1,…. This is a contradiction with the fact that RS0(Δ22n)⊇RS1(Δ22n); i.e., π≥1. Thus, we prove that, for the case of k=1, RS1(Δ22n)≤22n-1.
Now, let1<k≤π and suppose that |RSk(Δ22n)|≤22n-k by induction. According to Proposition 10, we have RSk+1(Δ22n)⊆RSk(Δ22n). This together with the induction hypothesis yields that |RSk+1(Δ22n)|≤|RSk(Δ22n)|≤22n-k. If |RSk+1(Δ22n)|>22n-k-1, then we have |RSk+1(Δ22n)|=|RSk(Δ22n)|=22n-k, which implies that RSk+1(Δ22n)=RSk(Δ22n). This contradicts with the fact of the minimality of π. Thus, one can derive that |RSk(Δ22n)|≤22n-k. This completes the proof.Based on Proposition10, we obtain the following necessary and sufficient condition for robust synchronization of the drive-response BCNs (2) with disturbances.Theorem 12.
Consider the drive-response BCNs (2) with disturbances. Let RSk(Δ22n) be the union set of robustly reachable states from the initial states set Δ22n at the k-th step. Then system (2) can be robustly synchronized if and only if there exists an integer k satisfying 1≤k≤22n-1, such that(13)RSkΔ22n⊆Ξ,where Ξ={δ22n(i-1)2n+i:i=1,…,2n} denotes the synchronized states set.Proof.
Based on Proposition10, if there exists an integer k satisfying 1≤k≤22n-1, such that RSk(Δ22n)⊆Ξ, then, given any integer t≥k, RSt(Δ22n)⊆Ξ. Then, according to the definition of set Ξ, the states of x(t) and y(t) reach synchronization after k steps. The proof of sufficient part can be directly obtained, which is omitted here. This completes the proof.Remark 13.
Ifx(t)=y(t)=δ2nj, then the equality z(t)=x(t)⋉y(t) yields that z(t)=δ22n(j-1)2n+j. Thus, we use the set Ξ={δ22n(i-1)2n+i:i=1,…,2n} to denote the synchronized states set. According to Proposition 10, for the initial states set Δ22n, we can always calculate the union set of robustly reachable states at given step by recurrence method. Moreover, according to Proposition 11, we can conclude that one only needs to calculate the union set of robustly reachable states for finite steps in order to check robust synchronization.Consider a setΠ={δ22nλ1,δ22nλ2,…,δ22nλε}. Note that each state δ22nλj (j=1,…,ε) is a column of the identity matrix I22n, we define the column vector form of the set Π by Boolean summing up of all the states as follows:(14)Πc=…,0,1λ1,0,…,0,1λε,…T∈B2n×1.Here, the λ1,…,λε-th elements of Πc are 1, and the rest are 0. Particularly, when Π=Δ22n, one can derive its corresponding column vector form of Δ22n, denoted by (1,…,1)T∈B2n×1. Thus, for the union set of robustly reachable states from the initial states Δ22n, i.e., RSk(Δ22n), we use RSk(Δ22n) to denote the corresponding column vector form. Then, we have(15)RSkΔ22n=⋃j=122nRSkδ22nj~RSkΔ22n=∑j=122nBRSkδ22nj.Since L in (8) is a 22n×22n+p matrix, we split it into 2p equal blocks as L=[L1,…,L2p], where Lj∈L22n (j=1,…,2p). Then, the following result will provide an algebraic representation for RSk(Δ22n).Theorem 14.
Consider the drive-response BCNs (2) with disturbances. Let RSk(Δ22n) be the union set of robustly reachable states from the initial states set Δ22n at the k-th step. Then, we have the following equation for the column vector form RSk(Δ22n):(16)RSkΔ22n=∑j=122nBColjLˇk,Lˇ=∑j=12pBLj.Proof.
We prove the result by induction. Consider the case ofk=1. Given an initial state z(0)=δ22nj, one can obtain the column vector form of RS1(δ22nj) as follows:(17)RS1δ22nj=L⋉δ2p1δ22nj+B⋯+BL⋉δ2p2pδ22nj=∑i=12pBLiδ22nj=ColjLˇ.Thus, the union set of robustly reachable states from the initial states set Δ22n at the first step is RS1(Δ22n)=∑Bj=122nRS1(δ22nj)=∑Bj=122nColj(Lˇ), which proves the case of k=1.
Then, we assume that (16) holds for the case of k=n and consider the case of k=n+1. Suppose that the reachable states in RSn(Δ22n) are δ22nw1,…,δ22nwr, where r=|RSn(Δ22n)|. Note that the trajectory starting from z(0)∈Δ22n to z(n+1)∈RSn+1(Δ22n) can be decomposed into trajectory starting from z(0) to some z(n)=δ22nwi (i∈{1,…,r}) at the n-th step and the trajectory from the possible states z(n)=δ22nwi to z(n+1)∈RSn+1(Δ22n) at one step. Thus, one can derive that(18)RSn+1Δ22n=RS1RSnΔ22n=RS1δ22nw1+B⋯+BRS1δ22nwr. This together with (17) yields(19)RSn+1Δ22n=∑l=1rB∑j=12pBLjδ22nwl=∑j=12pBLj×B∑l=1rBδ22nwl=Lˇ×BRSnΔ22n.On the other hand, one can obtain that(20)∑i=122nBColiLˇk+1=∑i=122nBColiLˇ×BLˇk=∑i=122nBLˇ×BColiLˇk=Lˇ×B∑i=122nBColiLˇk.Applying the induction hypothesis, by (19), we have ∑Bi=122nColi(Lˇ(n+1))=Lˇ×BRSn(Δ22n)=RSn+1(Δ22n), which proves the case of k=n+1. Thus, according to the mathematical induction, we can conclude that (16) holds for any positive integer k, which completes the proof.LetΞ→=∑j=12nδ22n(j-1)2n+j be the column vector form of the synchronized states set Ξ. Thus, based on Theorem 14, we can obtain the following result for robust synchronization.Corollary 15.
Consider the drive-response BCNs (2) with disturbances. Let Ξ→ be the column vector form of the synchronized states set Ξ. Then system (2) can be robustly synchronized if and only if there exists an integer 1≤k≤22n-1, such that(21)Ξ→-∑j=122nBColiLˇk≥0.


## 4. Illustrative Example

In this section, we present an illustrative example to demonstrate the effectiveness of our obtained results.Consider the following drive-response BCNs with one disturbance input, and each BN has two nodes:(22a)x1t+1=ξt∧x1t∧x2tx2t+1=ξt∧x1t∧x-2t,and(22b)y1t+1=u1t∧y1t∨u¯1t∧u2t∧y1t∨u¯2t∧y1t∧y2t∨y¯1t∧y¯2ty2t+1=u1t∧y2t∨u¯1t∧u¯2t∧y2t∨u¯2t∧y¯2t,where x1,x2∈D and y1,y2∈D are two state variables of drive BN and response BN respectively; u1,u2∈D are two control inputs and ξ∈D is an external disturbance. Here, we use x¯ to substitute the symbol ¬x, in order to simplify the logical functions. For example, if x=1 (x=0), we have x¯=0 (x¯=1).Suppose that the state feedback control is given in the following form:(23)u1t=x1t∨x¯1t∧x2t∨x¯2t∧y¯1t∧y¯2tu2t=x1t∨x¯1t∧x¯2t∧y¯1t∧y¯2t∨x¯1t∧x2t.Then, using the vector forms of Boolean variables and denoting x(t)=⋉i=12xi(t)∈Δ4, y(t)=⋉i=12yi(t)∈Δ4, and u(t)=⋉i=12ui(t)∈Δ4, we can obtain the following algebraic forms of the drive-response BCNs (22a) and (22b) with disturbance and the state feedback control (23):(24)xt+1=Fξtxt,yt+1=Gutyt,ut=Kxtyt,where F=δ4[2,3,4,4,3,3,4,4], G=δ4[1,2,3,4,1,2,3,4,1,2,3,4,2,3,4,1], and K=δ4[1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,1].Then, denotez(t)=x(t)⋉y(t)∈Δ16, and we can obtain the following equivalent algebraic form of the drive-response BCNs (24) with disturbance:(25)zt+1=Lξtutzt,where L=F(I2n+p⊗G)(I2p⊗W[2m,2n])∈L16×27. We present the row indexes of columns in matrix L as follows: (26)L=δ165,6,7,8,9,10,11,12,13,14,15,16,13,14,15,16,5,6,7,8,9,11,12,13,14,15,16,13,14,15,16,5,6,7,8,9,10,11,12,13,14,15,16,13,14,15,16,5,6,7,8,9,10,11,12,13,14,15,16,13,14,15,16,6,7,8,5,10,11,12,9,14,15,16,13,14,15,16,13,9,10,11,12,9,10,11,12,13,14,15,16,13,14,15,16,9,10,11,12,9,10,11,12,13,14,15,16,13,14,15,16,9,10,11,12,9,10,11,12,13,14,15,16,13,14,15,16,10,11,12,9,10,11,12,9,14,15,16,13,14,15,16,13.Figure1 shows the row indexes of columns in matrix L.Figure 1
The row indexes of each column of matrixL obtained in system (25). Each point corresponds to the row index of each column, which implies the position of element 1.Thus, under the state feedback controlu(t)=Kx(t)y(t), we obtain the following system: z(t+1)=Lξ(t)z(t), where L=L(I2⊗KΦ4)∈L16×32. A calculation yields (27)L=δ165,6,7,8,9,10,11,12,13,14,15,16,14,15,16,16,9,10,11,12,9,10,11,12,13,14,15,16,14,15,16,16. Our objective is to check whether the drive-response BCNs (22a) and (22b) with disturbances can be robustly synchronized. As discussed in Remark 13, if x(t)=y(t)=δ4j (1≤j≤4) then the equality z(t)=x(t)⋉y(t) yields that z(t)=δ16(j-1)4+j. Thus, we can obtain the synchronized states set Ξ={δ16i:i=1,6,11,16} and the unsynchronized states set Ξ¯={δ16i:i=2,3,4,5,7,8,9,10,12,13,14,15}. For example, if z(t)=δ162, one can obtain that x(t)≠y(t), since x(t)=δ41 and y(t)=δ42. A calculation yields (28)Lˇ=δ165+δ169,δ166+δ1610,δ167+δ1611,δ168+δ1612,δ169,δ1610,δ1611,δ1612,δ1613,δ1614,δ1615,δ1616,δ1614,δ1615,δ1616,δ1616. Thus, according to Corollary 15, we can obtain that(29)∑j=116BColjLˇ6=δ1616≤Ξ→=∑j=1,6,11,16Bδ16j,which implies that the drive-response BCNs (22a) and (22b) with disturbance can be robustly synchronized. The set of trajectories starting from the initial states set Δ16 converging to synchronized states set Ξ={δ16i:i=1,6,11,16} from time t=0 to time t=6 can be given as follows: (30)z10=δ161z20=δ162⋮z160=δ1616→ξ0⋯→ξ4z155=δ1615z165=δ1616→ξ5z166=δ1616⊆Ξ.As we can see from the above trajectories, at time t=4, there exist two states δ1614 and δ1615 in RS4(Δ16) that belong to the unsynchronized states set Ξ¯. At time t=5, there still exists one state δ1615 in RS5(Δ16) belonging to Ξ¯. Thus, the drive-response BCNs (22a) and (22b) with disturbance can not be robustly synchronized at time t=5. However, at time t=6, there is only one state δ1616 in RS6(Δ16), which belongs to the synchronized states set Ξ. Thus, the drive-response BCNs (22a) and (22b) with disturbance is robustly synchronized at time t=6. Moreover, the row indexes of the column vector RSk(Δ16) versus time k=1,2,…,6 are plotted in Figure 2. Thus, the validness of Corollary 15 has been well illustrated by this example.Figure 2
The whole row indexes of the column vectors obtained inRSk(Δ16) (k=1,2,…,6). Each point corresponds to the row index of each column, which implies the position of element 1.


## 5. Conclusion

In this paper, we have investigated robust synchronization of drive-response BCNs with disturbances. Based on the fundamental properties of STP, we firstly have obtained the algebraic representations of drive-response BCNs with disturbances. By defining a set of robustly reachable states from a given initial state at a given time step, we have obtained the set of all possible reachable states for any given disturbance sequences from the global initial states set after given steps, which is called the union set of robustly reachable states from the global initial states set. Then, based on the defined union sets, several necessary and sufficient criteria have been obtained for robust synchronization of disturbed BCNs. Moreover, the obtained results also have been verified that one only needs to calculate finite steps to check whether the drive-response BCNs with disturbances can be robustly synchronized.



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