# Robust Controllability and Observability of Boolean Control Networks under Different Disturbances

Authors: Yuanhua Wang; Xiao Zhang; Yaqi Hao
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper investigates robust controllability and observability of Boolean control networks under disturbances. Firstly, under unobservable disturbances, some sufficient conditions are obtained for robust controllability of BCNs. Then an algorithm is proposed to construct the least control sequences which drive the trajectory from a state to a given reachable state. If the disturbances are observable, by defining the order-preserving system, an efficient sufficient condition is obtained for robust controllability of BCNs. Finally, the robust observability problem is converted into an equivalent robust controllability via set controllability and is solved by using the results obtained for set controllability. Some numerical examples are presented to illustrate the obtained results.


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## Body

## 1. Introduction

Boolean network (BN) was first proposed by Kauffman to model gene regularity networks [1]. It has been widely used in some other fields, such as biology and system science [2, 3]. To manipulate BN, external control inputs and outputs have been added to the logical dynamics, which yields Boolean control networks (BCNs) [4].Recently, a new matrix product, called the semi-tensor product (STP) of matrices, has been proposed [5]. Using this method, an algebraic state space representation (ASSR) has been developed for dealing with some classical control problems of logical dynamic systems [6, 7], including controllability and observability [8–11], stability and stabilization [12–15], optimal control [16, 17], and disturbance decoupling [18, 19]. One may refer to [20–27] for some other applications.It is known that the external disturbances are ubiquitous and may lead the network dynamics to some unexpected behaviour. Therefore, when modeling a gene regularity network, disturbances should be considered [28, 29]. In recent years, the disturbance decoupling problem of BCNs has been investigated in [18, 19]. Moreover, the global robust stability and stabilization of BCNs have been studied in [30]. However, to our best knowledge, there is few literatures devoted to the robust controllability and observability of BCNs, though they have obvious importance for BCNs with disturbances. Especially for robust observability problems, they are much harder to be understood and verified, compared with robust controllability problems. A recent work [31] proposes a new method to investigate observability of BCNs by converting it to a problem of set controllability. Under a fundamental framework of [31], we investigate robust controllability and observability problem in our earlier conference paper [32]. Regarding [32], the novelties of this paper consist of the following: (i) we discuss two kinds of control inputs, respectively, for robust controllability under unobservable disturbances: one is a free Boolean sequence, and the other inputs are logical variables satisfying certain input nteworks. For these two cases, we, respectively, give some sufficient conditions for robust controllability. But in [32], we only consider the case that control inputs are free; (ii) we give an actual example to illustrate our theoretical results; please see Example 22. But there is no such example in [32].The main contributions of this paper include the following: (i) For unobservable disturbances, some sufficient conditions for robust controllability and observability of BCNs are obtained, respectively; (ii) based on the definition of order-preserving system, some easily verifiable sufficient conditions for robust controllability and observability under observable disturbances are presented, respectively; (iii) a general algorithm is proposed to design the least control sequences, which performs the required robust controllability or observability of BCNs.The rest of this paper is organized as follows: Section2 presents some preliminaries on the STP of matrices and set controllability of BCNs. Section 3 studies the robust controllability of BCNs under unobservable disturbances and gives an algorithm for designing the least control sequences. Section 4 discusses the robust controllability of BCNs under observable disturbances. The robust observability of BCNs is considered in Section 5. Section 6 is a brief conclusion.Notation. The notations of this paper are fairly standard. Mm×n denotes the set of m×n real matrices. Col(M) (Row(M)) is the set of columns (rows) of M and Coli(M) (Rowi(M)) is the i-th column (row) of M. D≔0,1. Δn≔δni∣i=1,…,n, where δni is the i-th column of the identity matrix In. 1l=(1,1,…,1︸l)T. 1p×q denotes a p×q matrix with all entries equal to 1. A matrix L∈Mm×n is called a logical matrix if Col(L)⊂Δm. Denote by Lm×n the set of m×n logical matrixes. If L=[δni1,δni2,…,δnir], it is briefly denoted as L=δn[i1,i2,…,ir]. Denote by Bm×n the set of m×n Boolean matrices. Let M,N∈Bm×n; then M+BN is the Boolean addition (with respect to +B=∨ and ×B=∧). A matrix C>0 means all the entries are positive; that is, ci,j>0, ∀i,j.


## 2. Preliminaries

### 2.1. Semi-Tensor Product of Matrices

This subsection is a brief survey on STP of matrices [5].Definition 1.
LetM∈Mm×n, N∈Mp×q, and t=lcmn,p be the least common multiple of n and p. The STP of M and N is defined as(1)M⋉N≔M⊗It/nN⊗It/p∈Mmt/n×qt/p,where ⊗ is the Kronecker product.Whenn=p, the STP becomes the conventional matrix product. Therefore, the STP is a generalization of the conventional matrix product. We can omit the symbol ⋉ without confusion.Proposition 2.
(1) LetX∈Rm be a column and M be a matrix. Then(2)XM=Im⊗MX.
(2) LetX∈Rm and Y∈Rn be two columns. Then W[m,n]⋉X⋉Y=Y⋉X, where W[m,n]∈Mmn×mn is called the swap matrix, which is defined by(3)Wm,n=δmn1,m+1,…,n-1m+1;2,…,n-1m+2;…;m,2m,…,nm.By identifying1~δ21 and 0~δ22, we have Δ~D. Using vector expression, a logical function can be expressed as its algebraic form.Theorem 3.
Letf:D×⋯×D︸n→D be a Boolean function. Then there exists a unique matrix Mf∈L2×2n, called the structure matrix of f, such that in the vector form we have(4)fx1,…,xn=Mf⋉i=1nxi.Theorem 4.
Letu=M⋉i=1nxi and v=N⋉i=1nxi, where xi∈Δ2, i=1,…,n, are logical variables and M∈Lp×2n, N∈Lq×2n are logical matrices. Then(5)uv=M∗N⋉i=1nxi,where ∗ is the Khatri-Lao product.


### 2.2. Set Controllability of BCNs

A BCN is described as(6)x1t+1=f1x1t,…,xnt;u1t,…,umt⋮xnt+1=fnx1t,…,xnt;u1t,…,umt,yjt=hjx1t,…,xnt,j=1,…,p,where xi∈D, i=1,…,n, are state variables, uj∈D, j=1,…,m, are controls, and fi:Dm+n→D, i=1,…,n, and hj:Dn→D, j=1,…,p, are Boolean functions.Definition 5 (see [5]).
System (6) is (1)
controllable fromx0 to xd, if there are T>0 and a sequence of control u(0),…,u(T-1), such that driven by these controls the trajectory can go from x(0)=x0 to x(T)=xd;(2)
controllable atx0, if it is controllable from x0 to any xd∈Dn;(3)
controllable, if it is controllable at anyx0∈Dn.Using Theorems3 and 4, (6) can be converted into its algebraic form as(7)xt+1=Lutxtyt=Hxt,where x(t)=⋉i=1nxi(t), u(t)=⋉j=1muj(t), y(t)=⋉l=1pyl(t), and L∈L2n×2n+m, H∈L2p×2n.Define(8)M≔⋁j=12mLδ2mj,and set(9)C≔⋁i=12nMi,which is called the controllability matrix. Then we have the following result.Theorem 6 (see [33]).
Consider system (6) (by free control sequence). Assume its controllability matrix is C=(ci,j); then system (6) is (1)
controllable from statesδ2nj to δ2ni, if and only if ci,j=1;(2)
controllable atδ2nj, if and only if Colj(C)=12n;(3)
controllable, if and only ifC=12n×2n.Denote byN=1,2,…,2n the set of states for BCNs. Assuming s∈2N, the index column vector of s, denoted by V(s)∈R2n, is defined as(10)Vsi=1,i∈s0,i∉s.Define the set of initial setsP0 and the set of destination sets Pd, respectively, as follows:(11)P0≔s10,s20,…,sα0⊂2N,Pd≔s1d,s2d,…,sβd⊂2N.Using initial sets and destination sets, the set controllability is defined as follows.Definition 7.
Consider system (6) with a set of initial sets P0 and a set of destination sets Pd. System (6) is (1)
set controllable fromsj0∈P0 to sid∈Pd, if there exist x0∈sj0 and xd∈sid, such that xd is controllable from x0;(2)
set controllable atsj0, if, for any sid∈Pd, the system is controllable from sj0 to sid;(3)
set controllable, if it is set controllable at anysj0∈P0.UsingP0 and Pd defined in (11), we define the initial index matrix J0 and the destination index matrix Jd, respectively, as(12)J0≔Vs10Vs20⋯Vsα0∈B2n×α;Jd≔Vs1dVs2d⋯Vsβd∈B2n×β.Then we define a matrix, called the set controllability matrix, as(13)CS≔JdT×BC×BJ0∈Bβ×α.Note that hereafter all the matrix products are assumed to be Boolean product (×B). Hence the symbol ×B is omitted.Definition 8 (see [31]).
Consider system (6) with P0 and Pd as defined in (11). Moreover, the corresponding set controllability matrix is CS=(ci,j), which is defined by (13). Then system (6) is (1)
set controllable fromsj0 to sid, if and only if ci,j=1;(2)
controllable atsj0, if and only if ColjCS=1β;(3)
set controllable, if and only ifCS=1β×α.


## 2.1. Semi-Tensor Product of Matrices

This subsection is a brief survey on STP of matrices [5].Definition 1.
LetM∈Mm×n, N∈Mp×q, and t=lcmn,p be the least common multiple of n and p. The STP of M and N is defined as(1)M⋉N≔M⊗It/nN⊗It/p∈Mmt/n×qt/p,where ⊗ is the Kronecker product.Whenn=p, the STP becomes the conventional matrix product. Therefore, the STP is a generalization of the conventional matrix product. We can omit the symbol ⋉ without confusion.Proposition 2.
(1) LetX∈Rm be a column and M be a matrix. Then(2)XM=Im⊗MX.
(2) LetX∈Rm and Y∈Rn be two columns. Then W[m,n]⋉X⋉Y=Y⋉X, where W[m,n]∈Mmn×mn is called the swap matrix, which is defined by(3)Wm,n=δmn1,m+1,…,n-1m+1;2,…,n-1m+2;…;m,2m,…,nm.By identifying1~δ21 and 0~δ22, we have Δ~D. Using vector expression, a logical function can be expressed as its algebraic form.Theorem 3.
Letf:D×⋯×D︸n→D be a Boolean function. Then there exists a unique matrix Mf∈L2×2n, called the structure matrix of f, such that in the vector form we have(4)fx1,…,xn=Mf⋉i=1nxi.Theorem 4.
Letu=M⋉i=1nxi and v=N⋉i=1nxi, where xi∈Δ2, i=1,…,n, are logical variables and M∈Lp×2n, N∈Lq×2n are logical matrices. Then(5)uv=M∗N⋉i=1nxi,where ∗ is the Khatri-Lao product.


## 2.2. Set Controllability of BCNs

A BCN is described as(6)x1t+1=f1x1t,…,xnt;u1t,…,umt⋮xnt+1=fnx1t,…,xnt;u1t,…,umt,yjt=hjx1t,…,xnt,j=1,…,p,where xi∈D, i=1,…,n, are state variables, uj∈D, j=1,…,m, are controls, and fi:Dm+n→D, i=1,…,n, and hj:Dn→D, j=1,…,p, are Boolean functions.Definition 5 (see [5]).
System (6) is (1)
controllable fromx0 to xd, if there are T>0 and a sequence of control u(0),…,u(T-1), such that driven by these controls the trajectory can go from x(0)=x0 to x(T)=xd;(2)
controllable atx0, if it is controllable from x0 to any xd∈Dn;(3)
controllable, if it is controllable at anyx0∈Dn.Using Theorems3 and 4, (6) can be converted into its algebraic form as(7)xt+1=Lutxtyt=Hxt,where x(t)=⋉i=1nxi(t), u(t)=⋉j=1muj(t), y(t)=⋉l=1pyl(t), and L∈L2n×2n+m, H∈L2p×2n.Define(8)M≔⋁j=12mLδ2mj,and set(9)C≔⋁i=12nMi,which is called the controllability matrix. Then we have the following result.Theorem 6 (see [33]).
Consider system (6) (by free control sequence). Assume its controllability matrix is C=(ci,j); then system (6) is (1)
controllable from statesδ2nj to δ2ni, if and only if ci,j=1;(2)
controllable atδ2nj, if and only if Colj(C)=12n;(3)
controllable, if and only ifC=12n×2n.Denote byN=1,2,…,2n the set of states for BCNs. Assuming s∈2N, the index column vector of s, denoted by V(s)∈R2n, is defined as(10)Vsi=1,i∈s0,i∉s.Define the set of initial setsP0 and the set of destination sets Pd, respectively, as follows:(11)P0≔s10,s20,…,sα0⊂2N,Pd≔s1d,s2d,…,sβd⊂2N.Using initial sets and destination sets, the set controllability is defined as follows.Definition 7.
Consider system (6) with a set of initial sets P0 and a set of destination sets Pd. System (6) is (1)
set controllable fromsj0∈P0 to sid∈Pd, if there exist x0∈sj0 and xd∈sid, such that xd is controllable from x0;(2)
set controllable atsj0, if, for any sid∈Pd, the system is controllable from sj0 to sid;(3)
set controllable, if it is set controllable at anysj0∈P0.UsingP0 and Pd defined in (11), we define the initial index matrix J0 and the destination index matrix Jd, respectively, as(12)J0≔Vs10Vs20⋯Vsα0∈B2n×α;Jd≔Vs1dVs2d⋯Vsβd∈B2n×β.Then we define a matrix, called the set controllability matrix, as(13)CS≔JdT×BC×BJ0∈Bβ×α.Note that hereafter all the matrix products are assumed to be Boolean product (×B). Hence the symbol ×B is omitted.Definition 8 (see [31]).
Consider system (6) with P0 and Pd as defined in (11). Moreover, the corresponding set controllability matrix is CS=(ci,j), which is defined by (13). Then system (6) is (1)
set controllable fromsj0 to sid, if and only if ci,j=1;(2)
controllable atsj0, if and only if ColjCS=1β;(3)
set controllable, if and only ifCS=1β×α.


## 3. Robust Controllability of BCNs under Unobservable Disturbances

Consider the following disturbed BCN:(14)x1t+1=f1x1t,…,xnt;u1t,…,umt;ξ1t,…,ξqt⋮xnt+1=fnx1t,…,xnt;u1t,…,umt;ξ1t,…,ξqt,with(15)yjt=hjx1t,…,xnt,j=1,…,p,where xi(t),uj(t),ξk(t),ys(t)∈D are states, controls, disturbances, and outputs, respectively, i=1,…,n, j=1,…,m, k=1,…,q, s=1,…,p. Using vector expression with x(t)=⋉i=1nxi(t), u(t)=⋉j=1muj(t), ξ(t)=⋉k=1qξk(t), and y(t)=⋉s=1pys(t), the algebraic forms of (14) and (15) are expressed as(16)xt+1=Futξtxtwith(17)yt=Hxt,where F∈L2n×2n+m+q, H∈L2p×2n.First, the robust controllability of disturbed BCNs is defined as follows.Definition 9.
System (14) is (1)
robust controllable fromx0 to xd, if there exist T>0 and a control sequence ut∣t=0,1,…,T-1, such that, under any ξ(t), t=0,1,…,T-1, the state can be driven from x(0)=x0 to x(T)=xd;(2)
robust controllable atx0, if it is robust controllable from x0 to any xd∈Dn;(3)
robust controllable, if it is robust controllable at anyx0∈Dn.In this paper, we consider two kinds of control inputs.(i)
The control is a free Boolean sequence.(ii)
The control inputs are logical variables satisfying certain input networks, such as(18)uit+1=giu1t,…,umt,i=1,…,m.


### 3.1. Robust Controllability with a Free Control Sequence

For the algebraic form (16) of system (14), define(19)M≔⋁j=12m⋀k=12qFδ2mjδ2qk,and(20)CR≔⋁i=12nMi,which is called the robust controllability matrix; then we have the following result.Theorem 10.
Consider system (14) under unobservable disturbances. Assume CR=(ci,jR); then system (14) is(1)
robust controllable fromδ2nj to δ2ni, if ci,jR=1;(2)
robust controllable atδ2nj, if Colj(CR)=12n;(3)
robust controllable, ifCR=12n×2n.Proof.
It is easy to see that the system is controllable fromδ2nj to δ2ni; then there exists a finite control sequence u(0),u(1),⋯,u(T-1) such that(21)δ2nr0=δ2nj→u0δ2nr1→u1δ2nr2→u1⋯→uT-1δ2nrT=δ2ni.Because of the construction of matrix M, particularly (Fδ2mjδ2q1)⋀Fδ2mjδ2q2⋀⋯⋀Fδ2mjδ2q2q, at any step t no matter which disturbance happens, we can always find the corresponding u(t) to realize the corresponding one step transfer in (21). Hence, system (14) is controllable from δ2nj to δ2ni. (The detailed argument is similar to the proof of Theorem 6.)Since robust controllability for any disturbance is a strong requirement, next we consider robust controllability under a constrained set of disturbance inputs. First, a constrained set for disturbance inputs is denoted byQd=δ2qω1,δ2qω2,…,δ2qωτ, where τ is the cardinality of set Qd. Define the robust controllability matrix under the constrained disturbances set Qd as(22)CdR≔⋁i=12nMdi,where(23)Md≔⋁j=12m⋀k=1τFδ2mjδ2qωk.Then we have the following result.Corollary 11.
Consider system (14) under the constrained disturbances set Qd. Assume CdR=(cdR(i,j)); then system (14) is(1)
robust controllable fromδ2nj to δ2ni, if cdR(i,j)=1;(2)
robust controllable atδ2nj, if Colj(CdR)=12n;(3)
robust controllable, ifCdR=12n×2n.Remark 12.
The main advantage of Theorem10 is when the iterative steps increase, the corresponding matrices, which are defined in (19) and (20), do not increase their dimensions. Hence it is easily verifiable and computable. The main disadvantage is that this result is not necessary; refer to the following example.Example 13.
Consider the following BCN with unobservable disturbance:(24)xt+1=Futξtxt,where x(t)=x1(t)⋉x2(t)∈Δ4, u(t),ξ(t)∈Δ2, and F=δ4[2,3,3,1,3,2,2,4,1,4,4,4,4,4,4,2]. Assuming x0=δ41 and xd=δ44, then the state trajectory can be presented in Figure 1. It is easy to see that, under unobservable disturbance, system (24) is robust controllable from δ41 to δ44. However, by immediate calculation, we have(25)M=⋁j=12⋀k=12Fδ2jδ2k=0000000000000110.It follows that(26)CR≔⋁i=12nMi=0000000000000110.Obviously, (CR)4,1=0. Hence we conclude that Theorem 10 is only sufficient but not necessary.Figure 1
State trajectory under unobservable disturbance.


### 3.2. Robust Controllability with Input Networks

Using the STP method, the input networks (18) can be expressed in its algebraic form as(27)ut+1=Gut,where G∈L2m×2m.Putting (16) and (27) together, we have(28)xt+1=Futξtxtut+1=Gut.Set w(t)=u(t)x(t); then (28) can be converted into(29)wt+1=Φξtwt,where Φ=[G(12qT⊗I2m⊗12nT)]∗(FW[2q,2m]).Similar to the argument in [34], we construct the set of initial sets P0 and the set of destination sets Pd as(30)P0=Pd=si∣i=1,…,2n,where si=uδ2ni∣u∈Δ2m=δ2m1δ2ni,δ2m2δ2ni,…,δ2m2mδ2ni.Then it is easy to see that eachsi corresponds to a unique state δ2ni. Hence system (29) is set controllable from sj to si, which is equivalent to the controllability from state δ2nj to δ2ni of system (14).Theorem 14.
System (14) with inputs (18) is robust controllable, if and only if the extended system (29) is robust set controllable with respect to P0 and Pd, which are defined in (30).DefineMg≔⋀k=12qΦδ2qk, and CgR≔⋁i=12n+mMg(i). Using (12), we have the robust set controllable matrix as(31)CsgR≔JdT×BCgR×BJ0∈B2n×2n.Similar to Theorem 10, we have an easily verifiable sufficient condition, which is shown as follows.Theorem 15.
Consider system (14) with inputs (18). Assuming CsgR=(ci,j), then system (14) is (1)
robust controllable fromδ2nj to δ2ni, if ci,j=1;(2)
robust controllable atδ2nj, if Colj(CsgR)=12n;(3)
robust controllable, ifCsgR=12n×2n.


### 3.3. Control Design

For system (14) with any disturbance, assume x0 can be driven to xd by a proper free control sequence. Since the trajectory from x0 to xd is in general not unique, in the following sequel, we try to find the shortest one. We propose the following “back stepping" algorithm.Algorithm 16.
Setx0=δ2nj and xd=δ2ni. Assume the (i,j)-th element of robust controllability matrix, ci,jR=1.(i)
Step 1. Define (at most) k-step controllability matrix as CkR≔⋁l=1kM(l). Find smallest k, such that CkRi,j=1.(ii)
Step 2. If k=1, then C1R=M. Find α0 such that Fδ2mα0δ2qri,j=1, ∀r. Set u(0)=δ2mα0; stop. Else, find one s, such that Mi,s=1 and CkRs,j=1.(iii)
Step 3. Find αk-1 such that Fδ2mαk-1δ2qri,s=1 (note that αk-1 is not unique, but there is at least one such block). Set u(k-1)=δ2mαk-1.(iv)
Step 4. Set k=k-1 and replace i by s. Go back to Step 2.The following proposition is an immediate consequence of Algorithm16.Proposition 17.
The control sequence generated by Algorithm16 is a least sequence of controls, which can drive the trajectory from x0 to xd. Moreover, the corresponding trajectory is {x(0)=x0,x(1),…,x(s)=xd}, which is also produced by the algorithm.It is easily verified that Algorithm16 can also be applied to the case with the constrained disturbances set.Example 18.
Consider the following BCN with unobservable disturbance:(32)x1t+1=x1t↔x2t∨utx2t+1=¬x1t∧ξt.Setting x(t)=x1(t)⋉x2(t), we can get its algebraic form as(33)xt+1=Futξtxt,where F=δ4[2,2,1,1,2,2,2,2,2,4,3,1,2,4,4, 2]. It is easy to calculate that(34)M=⋁j=12⋀k=12Fδ2jδ2k=0000110000000100.It follows that(35)CR≔⋁i=14Mi=0000110000001100.According to Theorem 10, since (CR)4,1=1, we can see that system (32) is robust controllable from x0=δ41 to xd=δ44. What is the least control sequence for driving δ41 to δ44?
Using Algorithm16, we can find the smallest k=2, such that (C2R)4,1=1. It is easy to find that M4,2=1 and (C1R)2,1=1. Choose α1=2, such that Fδ2α1δ2r4,2=1, r=1,2. Set u(1)=δ22.
According to Step 4 in Algorithm16, set k=k-1=1, and replace i=4 by s=2. Then we have C1R=M. Find α0=1 (or 2), such that Fδ2α0δ2r2,1=1, r=1,2. Set u(0)=δ21(orδ22).
Hence the least control sequence isu0=δ21orδ22,u1=δ22, which can drive δ41 to δ44, and the trajectory is {x(0)=δ41,x(1)=δ42,x(2)=δ44}.
For system (32) with constrained disturbance set Qd=δ21, using (22) and (23), we have(36)Md=⋁j=12Fδ2jδ21=0011110000100100,and(37)CdR≔⋁i=14Mdi=1111111100101111.According to Corollary 11, we conclude that system (32) with Qd=δ21 is robust controllable at x0=δ43.


## 3.1. Robust Controllability with a Free Control Sequence

For the algebraic form (16) of system (14), define(19)M≔⋁j=12m⋀k=12qFδ2mjδ2qk,and(20)CR≔⋁i=12nMi,which is called the robust controllability matrix; then we have the following result.Theorem 10.
Consider system (14) under unobservable disturbances. Assume CR=(ci,jR); then system (14) is(1)
robust controllable fromδ2nj to δ2ni, if ci,jR=1;(2)
robust controllable atδ2nj, if Colj(CR)=12n;(3)
robust controllable, ifCR=12n×2n.Proof.
It is easy to see that the system is controllable fromδ2nj to δ2ni; then there exists a finite control sequence u(0),u(1),⋯,u(T-1) such that(21)δ2nr0=δ2nj→u0δ2nr1→u1δ2nr2→u1⋯→uT-1δ2nrT=δ2ni.Because of the construction of matrix M, particularly (Fδ2mjδ2q1)⋀Fδ2mjδ2q2⋀⋯⋀Fδ2mjδ2q2q, at any step t no matter which disturbance happens, we can always find the corresponding u(t) to realize the corresponding one step transfer in (21). Hence, system (14) is controllable from δ2nj to δ2ni. (The detailed argument is similar to the proof of Theorem 6.)Since robust controllability for any disturbance is a strong requirement, next we consider robust controllability under a constrained set of disturbance inputs. First, a constrained set for disturbance inputs is denoted byQd=δ2qω1,δ2qω2,…,δ2qωτ, where τ is the cardinality of set Qd. Define the robust controllability matrix under the constrained disturbances set Qd as(22)CdR≔⋁i=12nMdi,where(23)Md≔⋁j=12m⋀k=1τFδ2mjδ2qωk.Then we have the following result.Corollary 11.
Consider system (14) under the constrained disturbances set Qd. Assume CdR=(cdR(i,j)); then system (14) is(1)
robust controllable fromδ2nj to δ2ni, if cdR(i,j)=1;(2)
robust controllable atδ2nj, if Colj(CdR)=12n;(3)
robust controllable, ifCdR=12n×2n.Remark 12.
The main advantage of Theorem10 is when the iterative steps increase, the corresponding matrices, which are defined in (19) and (20), do not increase their dimensions. Hence it is easily verifiable and computable. The main disadvantage is that this result is not necessary; refer to the following example.Example 13.
Consider the following BCN with unobservable disturbance:(24)xt+1=Futξtxt,where x(t)=x1(t)⋉x2(t)∈Δ4, u(t),ξ(t)∈Δ2, and F=δ4[2,3,3,1,3,2,2,4,1,4,4,4,4,4,4,2]. Assuming x0=δ41 and xd=δ44, then the state trajectory can be presented in Figure 1. It is easy to see that, under unobservable disturbance, system (24) is robust controllable from δ41 to δ44. However, by immediate calculation, we have(25)M=⋁j=12⋀k=12Fδ2jδ2k=0000000000000110.It follows that(26)CR≔⋁i=12nMi=0000000000000110.Obviously, (CR)4,1=0. Hence we conclude that Theorem 10 is only sufficient but not necessary.Figure 1
State trajectory under unobservable disturbance.


## 3.2. Robust Controllability with Input Networks

Using the STP method, the input networks (18) can be expressed in its algebraic form as(27)ut+1=Gut,where G∈L2m×2m.Putting (16) and (27) together, we have(28)xt+1=Futξtxtut+1=Gut.Set w(t)=u(t)x(t); then (28) can be converted into(29)wt+1=Φξtwt,where Φ=[G(12qT⊗I2m⊗12nT)]∗(FW[2q,2m]).Similar to the argument in [34], we construct the set of initial sets P0 and the set of destination sets Pd as(30)P0=Pd=si∣i=1,…,2n,where si=uδ2ni∣u∈Δ2m=δ2m1δ2ni,δ2m2δ2ni,…,δ2m2mδ2ni.Then it is easy to see that eachsi corresponds to a unique state δ2ni. Hence system (29) is set controllable from sj to si, which is equivalent to the controllability from state δ2nj to δ2ni of system (14).Theorem 14.
System (14) with inputs (18) is robust controllable, if and only if the extended system (29) is robust set controllable with respect to P0 and Pd, which are defined in (30).DefineMg≔⋀k=12qΦδ2qk, and CgR≔⋁i=12n+mMg(i). Using (12), we have the robust set controllable matrix as(31)CsgR≔JdT×BCgR×BJ0∈B2n×2n.Similar to Theorem 10, we have an easily verifiable sufficient condition, which is shown as follows.Theorem 15.
Consider system (14) with inputs (18). Assuming CsgR=(ci,j), then system (14) is (1)
robust controllable fromδ2nj to δ2ni, if ci,j=1;(2)
robust controllable atδ2nj, if Colj(CsgR)=12n;(3)
robust controllable, ifCsgR=12n×2n.


## 3.3. Control Design

For system (14) with any disturbance, assume x0 can be driven to xd by a proper free control sequence. Since the trajectory from x0 to xd is in general not unique, in the following sequel, we try to find the shortest one. We propose the following “back stepping" algorithm.Algorithm 16.
Setx0=δ2nj and xd=δ2ni. Assume the (i,j)-th element of robust controllability matrix, ci,jR=1.(i)
Step 1. Define (at most) k-step controllability matrix as CkR≔⋁l=1kM(l). Find smallest k, such that CkRi,j=1.(ii)
Step 2. If k=1, then C1R=M. Find α0 such that Fδ2mα0δ2qri,j=1, ∀r. Set u(0)=δ2mα0; stop. Else, find one s, such that Mi,s=1 and CkRs,j=1.(iii)
Step 3. Find αk-1 such that Fδ2mαk-1δ2qri,s=1 (note that αk-1 is not unique, but there is at least one such block). Set u(k-1)=δ2mαk-1.(iv)
Step 4. Set k=k-1 and replace i by s. Go back to Step 2.The following proposition is an immediate consequence of Algorithm16.Proposition 17.
The control sequence generated by Algorithm16 is a least sequence of controls, which can drive the trajectory from x0 to xd. Moreover, the corresponding trajectory is {x(0)=x0,x(1),…,x(s)=xd}, which is also produced by the algorithm.It is easily verified that Algorithm16 can also be applied to the case with the constrained disturbances set.Example 18.
Consider the following BCN with unobservable disturbance:(32)x1t+1=x1t↔x2t∨utx2t+1=¬x1t∧ξt.Setting x(t)=x1(t)⋉x2(t), we can get its algebraic form as(33)xt+1=Futξtxt,where F=δ4[2,2,1,1,2,2,2,2,2,4,3,1,2,4,4, 2]. It is easy to calculate that(34)M=⋁j=12⋀k=12Fδ2jδ2k=0000110000000100.It follows that(35)CR≔⋁i=14Mi=0000110000001100.According to Theorem 10, since (CR)4,1=1, we can see that system (32) is robust controllable from x0=δ41 to xd=δ44. What is the least control sequence for driving δ41 to δ44?
Using Algorithm16, we can find the smallest k=2, such that (C2R)4,1=1. It is easy to find that M4,2=1 and (C1R)2,1=1. Choose α1=2, such that Fδ2α1δ2r4,2=1, r=1,2. Set u(1)=δ22.
According to Step 4 in Algorithm16, set k=k-1=1, and replace i=4 by s=2. Then we have C1R=M. Find α0=1 (or 2), such that Fδ2α0δ2r2,1=1, r=1,2. Set u(0)=δ21(orδ22).
Hence the least control sequence isu0=δ21orδ22,u1=δ22, which can drive δ41 to δ44, and the trajectory is {x(0)=δ41,x(1)=δ42,x(2)=δ44}.
For system (32) with constrained disturbance set Qd=δ21, using (22) and (23), we have(36)Md=⋁j=12Fδ2jδ21=0011110000100100,and(37)CdR≔⋁i=14Mdi=1111111100101111.According to Corollary 11, we conclude that system (32) with Qd=δ21 is robust controllable at x0=δ43.


## 4. Robust Controllability of BCNs under Observable Disturbances

Assume the disturbanceξ(t)=δ2qr at time t; that is, the disturbance is dependent on time t, where r∈1,2,…,2q; then the dynamic model at time t is(38)xt+1=F~rutxt,where F~(r)=FW[2q,2m]δ2qr.Now we consider the overall robust controllability of system (14) by a free control sequence. It is easily extended to the case with networked inputs (18).We give a necessary condition as follows, which is obvious, because the constant disturbance is legal.Lemma 19.
Assuming system (14) is robust controllable, then each subsystem (38) is robust controllable.Then we consider the robust controllability problem under the following assumption:(A1) Each subsystem (38) with F~(r) is controllable, r=1,…,2q.For each pair of statesxd∈Δ2n and x∈Δ2n, we define the distance between them under model r.(39)dxdrx=minkstatexis  driven  toxdwithinksteps.It is noted that, under assumption (A1), the distance dxdr(x) is well defined for any xd and any x. If we only consider the reachability to xd, then assumption (A1) can be replaced by the following assumption.(A2) Each subsystem with F~(r) is reachable to the fixed xd, r=1,…,2q.Next we define the order-preserving of system (14).Definition 20.
System (14) is (1)
order-preserving atxd, if for any x,y∈Δ2n, and any 1≤r,s≤2q, we have(40)dxdrx>dxdry⇔dxdsx>dxdsy.(2)
order-preserving, if it is order-preserving at anyxd∈Δ2n.Theorem 21.
Consider system (14) with observable disturbances. (1)
Assume (A2) is satisfied and system (14) is order-preserving at the fixed xd. Then it is robust reachable to xd; i.e., it is robust controllable from any x0 to xd.(2)
Assume (A1) is satisfied and system (14) is order-preserving at any xd. Then it is robust controllable.Proof.
We only prove for the fixed statexd case. The proof for general case is exactly the same. If we know the model at time t is r, then we can design the control u(t) to drive x(t) to x(t+1), where(41)dxdrxt+1=dxdrxt-1.Since the number of all possible models is 2q<∞, after finite times, at least one model should be reached as many times as we wish. Because of the order-preserving, it is easy to see that at each time the distance is decreasing at least one. Hence the distance dxdr(x(t)) will eventually reach zero. That is, system (14) will reach xd.Example 22 (see [35]).
Consider a lac operon regulatory network model(42)x1t+1=¬u1t∧x3t∨u2t,x2t+1=x1t,x3t+1=¬u1t∧x2t∧u2t∨x3t∧¬x2t,where x1,x2, and x3 are state variables, which denote the mRNA, the LacZ polypeptide, and the internal lactose, respectively, and u1,u2 are control inputs, which represent the external lactose and the external glucose, respectively. The variable equal to 1 (or 0) means the corresponding molecular has a high (or low) concentration. In order to investigate the robust controllability of the model, we introduce an artificial disturbance ξ in x2(t+1)=x1(t), which becomes x2(t+1)=x1(t)∨ξ(t).
Using the vector form of logical variables and settingx(t)=x1(t)⋉x2(t)⋉x3(t), u(t)=u1(t)⋉u2(t), by using the STP, we have the following algebraic form:(43)xt+1=Fξtutxt,where(44)F=δ86,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,1,1,1,2,1,1,1,2,1,1,1,2,3,3,3,4,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,2,6,1,6,2,6,1,6,2,6,1,6,4,8,3,8.Assume the disturbance is observable at time t; then we have(45)xt+1=Fδ2rutxt=Mrutxt,where(46)M1=δ86,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,1,1,1,2,1,1,1,2,1,1,1,2,3,3,3,4,M2=δ86,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,2,6,1,6,2,6,1,6,2,6,1,6,4,8,3,8.According to (8) and (9), we calculate that(47)C1=⋁i=18M1~i=1111111111111111111111111111111100000000111111110000000011111111,C2=⋁i=18M2~i=0010001010101010000000100000100000000000111111110000000011111111,where Mp~=⋁j=14Mpδ4j, p=1,2.
It is obvious that both subsystems are not controllable. Hence system (43) does not satisfy Assumption (A1). From the above matrices C1 and C2, it is easy to see that both subsystems are all reachable to xd=δ88. Using (39), for r=1 and r=2, respectively, we can figure out that dxdr(δ8i)=2, i=1,2,3,4, and dxdr(δ8j)=1, j=5,6,7. Obviously, this system is order-preserving at xd=δ88. According to Theorem 21, we conclude that system (43) is robust reachable to δ88; i.e., it is robust controllable from any x0 to δ88.


## 5. Robust Observability of BCNs

First, we give the definition of robust observability for disturbed BCNs.Definition 23.
System (14)-(15) under unobservable disturbances is robust observable, if for any two initial states x0≠z0, and under any disturbance ξt∣t=0,1,⋯, there exists a control sequence ut∣t=0,1,⋯, such that the corresponding output sequences are distinct; that is,(48)yt∣ut,x0,t≠y~t∣ut,z0,t.As suggested in [31], we consider two kinds of state pairs. A pair (x,z)∈Δ2n×Δ2n is y-indistinguishable if Hx=Hz. Otherwise, (x,z) is called y-distinguishable.Next, we split the product state spaceΔ2n×Δ2n into a partition of three components as(49)D≔zx∣z=x,Θ≔zx∣z≠xandHz=Hx,Ξ≔zx∣Hz≠Hx.Using algebraic form (16), we construct a dual system as(50)zt+1=Futξtztxt+1=Futξtxt.Set w(t)=z(t)x(t); we have(51)zt+1=FI2m+q+n⊗12nTutξtztxtxt+1=FI2m+q⊗12nTutξtztxt.Then (51) can be expressed as a new system(52)wt+1=Lutξtwt,where L=[F(I2m+q+n⊗12nT)]∗[F(I2m+q⊗12nT)]∈L22n×22n+m+q.Now the observability problem of system (14)-(15) can be converted into a set controllability problem of the extended system (52). Construct the set of initial sets and the set of destination sets as follows:(53)P0≔⋃zx∈Θzxand(54)Pd≔Ξ.Note that (53) means each state pair zx in Θ is an element of P0, while (54) means Pd has only one element, which is Ξ. Then we have the following result.Theorem 24.
Consider system (14)-(15) under unobservable disturbances with P0 and Pd as defined in (53) and (54), respectively. The system is robust observable, if system (52) is robust set controllable. Precisely speaking, the conditions in Theorem 10 are satisfied.Example 25.
Recall Example18. Assuming that the output is(55)yt=x1t∨x2t,we have its algebraic form as(56)yt=Hxt,where H=δ2[1,1,1,2].
It is easy to figure out that(57)Θ=δ41,δ42,δ41,δ43,δ42,δ43~δ162,δ163,δ167=θ1,θ2,θ3,and(58)Ξ=δ41,δ44,δ42,δ44,δ43,δ44~δ164,δ168,δ1612.Set w(t)=z(t)x(t); we have a new system as(59)wt+1=Lutξtwt,where L=δ16[6,6,5,5,⋯,6,8,8,6]∈L16×64.
We calculate(60)CR≔⋁i=116Mi∈B16×16,where M=⋁j=12⋀k=12Lδ2jδ2k.
UsingP0=θ1,θ2,θ3 and Pd=Ξ, we have(61)Jd=∑δ16i∈Ξδ16i,J0=δ162,3,7.It follows that(62)CSR≔JdT×BCR×BJ0=1,0,0.According to Theorem 24, since Colj(CSR)=1, we conclude that system (32) with output (55) is robust observable at s10=θ1.Similarly, we can obtain an easily verifiable sufficient condition for the robust observability of BCNs under observable disturbances.Theorem 26.
System (14)-(15) with observable disturbances is robust observable, if system (52) is robust set ontrollable from P0 to Pd. Precisely speaking, the conditions in Theorem 21 are satisfied.


## 6. Conclusion

Robust controllability and observability of disturbed BCNs have been investigated in this paper. Some sufficient conditions for robust controllability under unobservable and observable disturbances have been obtained, respectively. By constructing an extended system for a given BCN, an observability problem can be converted to a set controllability problem of its extended system. Then some corresponding conditions for robust observability are obtained for two cases, respectively. A universal algorithm is presented to construct the least control sequences to realize the required robust controllability.The results in this paper can easily be extended to mix-valued logical systems.



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