# Robust Output Controllability Analysis and Control Design for Incomplete Boolean Networks with Disturbance Inputs

Authors: Lei Deng; Shihua Fu; Ying Li; Peiyong Zhu
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper addresses the problems of robust-output-controllability and robust optimal output control for incomplete Boolean control networks with disturbance inputs. First, by resorting to the semi-tensor product technique, the system is expressed as an algebraic form, based on which several necessary and sufficient conditions for the robust output controllability are presented. Second, the Mayer-type robust optimal output control issue is studied and an algorithm is established to find a control scheme which can minimize the cost functional regardless of the effect of disturbance inputs. Finally, a numerical example is given to demonstrate the effectiveness of the obtained new results.


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## Body

## 1. Introduction

Boolean network (BN) was firstly introduced by Kauffman for modelling genetic regulatory networks (GRNs) [1]. Since then, it has been used to analyze and simulate cellular networks. A typical BN consists of N nodes, and each can take one of the following two values: 1 or 0, representing the gene is expressed or not, respectively. Furthermore, the state evolution of each node can be determined by Boolean functions. Although BN is a simplified model, it becomes a powerful tool in analysing GRNs. And some significant results were presented [2, 3].Recently, D. Cheng and coauthors have developed an algebraic state space representation (ASSR) of Boolean control networks (BCNs) by using the semitensor product (STP) of matrices [4]. Using the ASSR, many fundamental and landmark results about BCNs, which include but are not limited to controllability and observability [5–8], stability and stabilization [9–11], disturbance decoupling [12, 13], network synchronization [14, 15], output tracking [16], output regulation [17, 18], and optimal control [19], have been obtained in the last few years. The STP is an interesting research topic, and we can refer to [20–23] for some other applications of it.As is well known, output-controllability and optimal output control issues are fundamental concepts in a control theory field. The output controllability describes the ability of an external input to move the output from any initial condition to any final condition in a finite time interval. The optimal output deals with the problem of finding an external input for a given system such that a certain optimality output criterion is achieved. A controllable system is not necessarily output controllable, and an output controllable system is not necessarily controllable. Hence, they are also two important structural properties in modern control theory which reflect the dominant ability of the control inputs over outputs [24, 25]. As a suitable model of GRNs, the study of output controllability (reachability) and optimal control is also meaningful for BCNs, and there are some literatures studying on both of them. Reference [26] discussed the output reachability of BCNs, and several necessary and sufficient conditions were obtained for the verification of output reachability. The output-controllability and optimal output control problems of a state-dependent switched BCN were studied in [27], in which the output controllability criteria were presented and an optimal output control design algorithm for the Mayer-type optimal output problem was established.Robust control is a branch of control theory that deals with uncertainty [28]. It aims to achieve robust performance, such as controllability, in the presence of uncertain parameters or disturbances. It is known that, in practical networks such as GRNs, the disturbances are ubiquitous, which primarily stem from environmental changes, biological uncertainties, and experimental noises [29, 30]. These disturbance inputs may lead to instability of networks or make it difficult to understand the network dynamics. To be consistent with the dynamical behaviours of the practical networks, disturbances should be a characteristic which has to be taken into account when modelling the GRNs by BNs [31]. Thus, two interesting problems with biological significance are robust-output-controllability and robust optimal output control for BCNs with disturbance inputs, which deserve investigation for theoretical developments as well as for practical applications.In order to solve two classical intellectual problems: the wolf-sheep-cabbage puzzle and the missionaries-cannibals problem, the authors of [32] presented a new logical control system (LCS), called the incomplete LCS, in which certain controls can only be applied to certain states. Moreover, in [33], authors modeled a networked evolutionary game with bankruptcy mechanism as an LCS whose certain control-states should be forbidden. These cases show that the investigation of incomplete BCNs is of practical significance. Unfortunately, to the best of authors’ knowledge, there is no result available for robust output controllability and robust optimal output control of incomplete BCNs with disturbance inputs, which motivates the study of this research.This paper investigates the robust-output-controllability and robust optimal output control of incomplete BCNs with disturbance inputs. The main difficulties of extending the results obtained in [27, 34] contain the following two aspects: (1) the dynamics of a BCN with disturbance inputs is much more complicated than a BCN and, in this paper, the disturbance inputs are arbitrary; (2) the existence of control-state avoiding set makes the control law of this paper different from that of the existing results about BCNs since some control-states should be avoided when designing the control sequences. We present several necessary and sufficient conditions for the robust output controllability of incomplete BCNs with disturbance inputs. Moreover, a robust optimal output control design algorithm for the Mayer-type robust optimal output problem is also derived.The remainder of this paper is organized as follows. Section2 provides some preliminary results. In Section 3, we study the robust-output-controllability and robust optimal output control problems of incomplete BCNs with disturbance inputs, and present the main results of this paper. A numerical example is given in Section 4 to illustrate the validity of the obtained new results. Finally, conclusions are drawn in Section 5.


## 2. Preliminaries

Firstly, we list some notations which will be used throughout this paper.(i)
D≔{1,0} and Dn≔D×⋯×D︸n.(ii)
δni: the i-th column of the identity matrix In.(iii)
Δn≔{δni∣i=1,2,⋯,n}, and for compactness, Δ≔Δ2. By identifying 1~δ21 and 0~δ22, where “~” denotes two equivalent forms of the same object, the logical variable in D then takes value from Δ.(iv)
Col(A)(Row(A)) is the set of columns (rows) of A. Coli(A)(Rowi(A)) is the i-th column (row) of A.(v)
Mm×n: the set of m×n real matrices. A matrix A∈Mm×n is called a logical matrix, if the columns of A are elements of Δm. Denote by Lm×n the set of m×n logical matrices.(vi)
IfA∈Lm×n, by definition it can be expressed as A=δmi1,δmi2,⋯,δmin, and its shorthand is A=δmi1i2⋯in.(vii)
A matrixB∈Mm×n is called a Boolean matrix, if its entries bij∈D, for every i,j. Denote by Bm×n the set of m×n Boolean matrices.(viii)
A matrixL∈Mm×n is called an incomplete logical matrix, if (1)ColL⊂Δm⋃0m,where0m is an m dimensional zero vector. Identifying 0m with δm0, an incomplete logical matrix L can briefly expressed as L=δmi1i2⋯in, where certain ir could be 0. Denote by Lm×nI the set of m×n incomplete logical matrices.(ix)
ForA∈Mm×r,B∈Mn×r, (2)A∗B=Col1A⋉Col1B,⋯,ColrA⋉ColrBis the Khatri-Rao product of A and B.Some necessary definitions are given as follows.Definition 1 (see [4]).
ForA∈Mm×n and B∈Mp×q, the STP, denoted by A⋉B, is defined as follows: (3)A⋉B≔A⊗Is/nB⊗Is/p,where s=l.c.m.(n,p), denoting the least common multiple of n and p, and ⊗ denotes the Kronecker product.It is noted that the STP is a generalization of the ordinary matrix product. In this paper, we simply call it “product" and omit the symbol “⋉" if no confusion arises.Definition 2.
The intersection of Boolean matrices is defined as(4)A∧B=aij∧bij∈Bm×n,∀A,B∈Bm×n.Definition 3.
LetA=(aij)∈Bm×n be a matrix; then we call A>0 if and only if aij>0, ∀i,j.Finally, we give some conclusions which will be used in the remainder of the paper.Lemma 4 (see [4]).
Any logical functionf(x1,⋯,xn) with logical arguments x1,⋯,xn∈Δ can be expressed in a multilinear form as (5)fx1,⋯,xn=Mfx1⋯xn, where Mf∈L2×2n is unique, called the structure matrix of logical function f.Consider a disturbed BCN as follows:(6)x1t+1=f1Ξt,Xt,Ut,x2t+1=f2Ξt,Xt,Ut,⋮xnt+1=fnΞt,Xt,Ut;yjt=hjXt,j=1,⋯,p,where X(t)=(x1(t),x2(t),⋯,xn(t))∈Dn denotes the state variable, U(t)=(u1(t),⋯,um(t))∈Dm denotes the control input, Ξ(t)=(ξ1(t),⋯,ξr(t))∈Dr denotes the disturbance input, Y(t)=(y1(t),y2(t),⋯,yp(t))∈Dp denotes the output variable, and fi:Dn+m+r↦D, i=1,⋯,n, and hj:Dn↦D,j=1,⋯,p are Boolean functions. Denote the output trajectory of the system (6) by Y(t;X(0),Ut-1,Ξt-1), where X(0)∈Dn, Ut-1≔{U(0),⋯,U(t-1)}⊆Dm, and Ξt-1≔{Ξ(0),⋯,Ξ(t-1)}⊆Dr represent the initial state, the control sequence, and the sequence of disturbance inputs, respectively.In order to convert the system (6) into an algebraic form, we define x(t)=⋉i=1nxi(t), u(t)=⋉i=1mui(t),ξ(t)=⋉i=1rξi(t), and y(t)=⋉i=1pyi(t). Assume that the structure matrices for fi, hj are Fi∈L2×2n+m+r, Hj∈L2×2n, respectively, then the system (6) can be converted into the following algebraic form:(7)xt+1=Lξtutxt,yt=Hxt,where L=∗i=1nFi∈L2n×2n+m+r is called the state transition matrix, H=∗j=1nHj∈L2p×2n, and ∗ is the Khatri-Rao product.


## 3. Main Results

In this section, we study the robust-output-controllability and robust optimal output control of disturbed incomplete BCNs and present the main results of this paper.


### 3.1. An Algebraic Form for Disturbed Incomplete BCNs

First, we give the concept of disturbed incomplete BCNs.Definition 5.
Consider the disturbed BCN (7). Suppose certain controls are not applicable to certain states. Precisely, there exists a set of pairs(8)Θ≔δ2mαi,δ2nβi∣i=1,⋯,s⊂Δ2m×Δ2n,such that the control δ2mαi is not applicable to δ2nβi. Θ is called the control-state avoiding set. Supposing that Θ≠∅, then the disturbed BCN is called the disturbed incomplete BCN.Consider the disturbed incomplete BCN (7) with control-state avoiding set Θ and split L into 2r blocks as L≔[L1,L2,⋯,L2r], where Lk∈L2n×2n+m,k=1,2,⋯,2r. For u(t)=δ2mαi,x(t)=δ2nβi, a straightforward computation shows that u(t)⋉x(t)=δ2mαi⋉δ2nβi=δ2m+nai, where ai=(αi-1)2n+βi. Thus, when ξ(t)=δ2rk,u(t)=δ2mαi and x(t)=δ2nβi, we have (9)xt+1=Lξtutxt=Lkutxt=Lkδ2m+nai=ColaiLk,where ai=(αi-1)2n+βi and k=1,2,⋯,2r. If (δ2mαi,δ2nβi)∈Θ, we let Colai(Lk)=δ2n0. In this case, the corresponding x(t+1) does not exist. We also denote it briefly as x(t+1)=δ2n0. Then, it is easy to obtain the following result.Proposition 6.
A disturbed incomplete BCN with control-state avoiding setΘ as in (8) has its algebraic expression as in (7) except that (10)ColaiLk=δ2n0, where ai=(αi-1)2n+βi,i=1,2,⋯,s, and k=1,2,⋯,2r.We denote the new blocks obtained from Proposition6 by LkΘ, that is,(11)ColaiLkΘ=δ2n0,ai=αi-12n+βi;ColaiLkΘ=ColaiLk,otherwise.Let IΘ be the matrix obtained from an identity matrix I2n+m×2n+m substituting the columns with indexes a1,⋯,as by δ2n+m0; then LkΘ can be calculated by LkΘ=LkIΘ.Construct a new matrix as follows:(12)LΘ=L1Θ,L2Θ,⋯,L2rΘ,and then it is easy to see that LΘ is the state transition matrix of a disturbed incomplete BCN with control-state avoiding set Θ. In fact, LΘ can be further calculated as(13)LΘ=L1Θ,L2Θ,⋯,L2rΘ=L1IΘ,L2IΘ,⋯,L2rIΘ=L⋉δ2r1⋉IΘ,L⋉δ2r2⋉IΘ,⋯,L⋉δ2r2r⋉IΘ=L⋉δ2r1⋉IΘ,δ2r2⋉IΘ,⋯,δ2r2r⋉IΘ=LI2r⊗IΘ.Based on (7) and above analysis, the following result is obvious.Proposition 7.
Consider a disturbed incomplete BCN with control-state avoiding setΘ. It can be expressed as an algebraic form:(14)xt+1=LΘξtutxt,yt=Hxt,where LΘ∈L2n×2n+m+rI is given in (12).Proof.
We only need to prove that the state trajectories of the disturbed incomplete logical system (14) are equivalent to that of the system (7) with control-state avoiding set Θ.
DenoteW={δ2m+nai=δ2mαi⋉δ2nβi∣(δ2mαi,δ2nβi)∈Θ}, where ai=(αi-1)2n+βi. It is obvious that W and Θ are equivalent. For any ξ(t)=δ2rk,k=1,⋯,2r, we first prove the case of u(t)x(t)=δ2m+nai∈W. A simple calculation shows that(15)xt+1=Lξtutxt=Lkutxt=ColaiLk=δ2n0=ColaiLkΘ=LkΘutxt=LΘξtutxt.
Whenu(t)x(t)=δ2m+nai∉W, we have(16)xt+1=Lξtutxt=Lkutxt=ColaiLk=ColaiLkΘ=LkΘutxt=LΘξtutxt. Hence, in both cases, the state trajectories of the disturbed incomplete logical system (14) are equivalent to that of the system (7) with control-state avoiding set W. Since W and Θ re equivalent, this completes the proof.


### 3.2. Robust Output Controllability Analysis of Disturbed Incomplete BCNs

Now, we give definitions of the robust output controllability for disturbed incomplete BCNs.Definition 8.
Consider the disturbed incomplete BCN (14).(i)
yf∈Δ2p is said to be s-robust-output-reachable from the initial state x(0)∈Δ2n, if one can find a control sequence us-1⊆Δ2m such that for arbitrary disturbance inputs ξs-1⊆Δ2r, the output at time t=s satisfies y(s;x(0),us-1,ξs-1)=yf, where us-1≔{u(0),⋯,u(s-1)} and ξs-1≔{ξ(0),⋯,ξ(s-1)}. The s-robust-output-reachable set of the initial state x(0) is denoted by Rs(x(0)).(ii)
The system (14) is said to be s-robust-output-controllable at the initial state x(0)∈Δ2n, if for any output state yf∈Δ2p, one can find a control sequence us-1⊆Δ2m such that, for arbitrary disturbance inputs ξs-1⊆Δ2r, the output at time t=s satisfies y(s;x(0),us-1,ξs-1)=yf. The system (14) is said to be s-robust-output-controllable, if the system is s-robust-output-controllable at any x(0)∈Δ2n.(iii)
The system (14) is said to be robust-output-controllable at the initial state x(0)∈Δ2n, if for any output state yf∈Δ2p, one can find a positive integer T and a control sequence uT-1⊆Δ2m such that, for arbitrary disturbance inputs ξT-1⊆Δ2r, the output at time t=T is y(T;x(0),uT-1,ξT-1)=yf. The system (14) is said to be robust-output-controllable, if the system is robust-output-controllable at any x(0)∈Δ2n.To obtain necessary and sufficient conditions for the robust output controllability of disturbed incomplete BCNs, we first consider the robust reachability of disturbed incomplete BCNs.Consider the disturbed incomplete BCN (14), and denote by(17)L~Θ=L1Θ∧L2Θ∧⋯∧L2rΘ.SplitL~Θ into 2m square blocks as(18)L~Θ=L~1Θ,L~2Θ,⋯,L~2mΘ,whereL~jΘ∈L2n×2nI,j=1,2,⋯,2m. Then we have the following result.Theorem 9.
Consider the disturbed incomplete BCN (14). Let xf=δ2nq and x(0)=δ2nr be given. Then,
(i)xf is robust reachable from x(0) at the s-th step, if and only if(19)Msq,r>0,
whereM=∑j=12m‍(L~jΘ) and (Ms)q,r denotes the (q,r)th element of Ms;
(ii)xf is robust reachable from x(0), if and only if(20)∑s=12m+n+rMsq,r>0.Proof.
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(i) (Necessity). We prove it by induction on s. Consider the case s=1. It is assumed that x(1)=δ2nq can be robust reachable from x(0)=δ2nr at the first step; then one can find at least one control u(0)=δ2mj such that, for any disturbance input ξ(0)=δ2rk, the following equation will hold:(21)δ2nq=LkΘδ2mjδ2nr,∀k=1,2,⋯,2r.
Multiplying (21) from the left by (δ2nq)T yields(22)1=δ2nqTLkΘδ2mjδ2nr,∀k=1,2,⋯,2r.
For each blockLkΘ∈L2n×2n+mI, split it into 2m equal blocks as(23)LkΘ=Lk1Θ,Lk2Θ,⋯,Lk2mΘ,where LkΘ is given in (12). Then according to (22), we have(24)L1jΘq,r=L2jΘq,r=⋯=L2rjΘq,r=1.Since L~Θ=L1Θ∧L2Θ∧⋯∧L2rΘ, one knows that(25)L~Θδ2mj=L~jΘ=L1Θ∧L2Θ∧⋯∧L2rΘδ2mj=L1Θδ2mj∧L2Θδ2mj∧⋯∧L2rΘδ2mj=L1jΘ∧L2jΘ∧⋯∧L2rjΘ.
Combining (24) and (25), we have (26)L~jΘq,r=1.
Thus,(27)Mq,r=L~1Θ+L~2Θ+⋯+L~2mΘq,r≥L~jΘq,r=1.
This proves the situation fors=1.
For the induction step, we assume that, for anyl≤s, if x(l)=δ2nq can be robust reachable from x(0)=δ2nr at the lth step, then (Ml)q,r>0. In the following, we need to prove that (19) holds for l=s+1. Assuming that x(s+1)=δ2nq can be robust reachable from x(0)=δ2nr at the (s+1)th step, then there must exist a path from δ2nr to δ2nk0 at the sth step and a path from δ2nk0 to δ2nq in one step. Applying the induction hypothesis, we have(28)Msk0,r>0,Mq,k0>0. Therefore,(29)Ms+1q,r=MMsq,r=∑k=12nMq,kMsk,r≥Mq,k0Msk0,r>0.The proof of necessity is completed.
(Sufficiency). We still prove it by induction on s. From the proof of necessity, one can easily see that if (M)q,r>0, then (21) must hold, which implies that control δ2mj can steer system (14) from δ2mr to δ2mq in one step under arbitrary disturbance inputs. This proves (19) for s=1.
We assume that, for anyl≤s, if (Ml)q,r>0, then δ2mq can be robust reachable from δ2mr at the lth step. By (29), one can obviously see that if (Ms+1)q,r>0, then there exists at least a positive integer k0 such that (Ms)k0,r>0 and (M)q,k0>0. Based on the induction hypothesis, δ2nk0 can be robust reachable from δ2nr at the sth step and δ2nq can be robust reachable from δ2nk0 in one step. Thus, δ2nq can be robust reachable from δ2nr at the (s+1)th step.
The result (ii) can be easily obtained from (i).From Theorem9 and Definition 8, one can obtain the following robust output controllability results.Theorem 10.
Consider the disturbed incomplete BCN (14).
(i)yf=δ2pl is s-robust-output-reachable from the initial state x(0)=δ2nr, if and only if(30)HMsl,r>0;(ii) The system (14) is s-robust-output-controllable at the initial state x(0)=δ2nr, if and only if(31)ColrHMs>0;(iii) The system (14) is s-robust-output-controllable, if and only if(32)HMs>0;(iv) The system (14) is robust-output-controllable at the initial state x(0)=δ2nr, if and only if(33)∑s=12m+n+rColrHMs>0;(v) The system (14) is robust-output-controllable, if and only if(34)∑s=12m+n+rHMs>0.Proof.
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(i) (Sufficiency). Assuming that (30) holds, and we need to prove that there exists a control sequence us-1≔{u(t):t=0,⋯,s-1}⊆Δ2m such that, for arbitrary disturbance inputs ξs-1≔{ξ(t):t=0,⋯,s-1}⊆Δ2r, the output satisfies y(s;δ2nr,us-1,ξs-1)=δ2pl. Since(35)HMsl,r=∑q=12nHl,qMsq,r>0,there must exist an integer 1≤q0≤2n such that (H)l,q0=1 and (Ms)q0,r>0.
Based on Theorem9, δ2nq0 is robust reachable from δ2nr at the sth step. In other words, we can find a control sequence us-1⊆Δ2m such that (36)xs;δ2nr,us-1,ξs-1=δ2nq0,∀ξs-1⊆Δ2p. Correspondingly,(37)ys;δ2nr,us-1,ξs-1=Hδ2nq0=Colq0H=δ2pl,∀ξs-1⊆Δ2p,which infers that δ2pl is s-robust-output-reachable from x(0)=δ2nr.
(Necessity). Assuming that δ2pl is s-robust-output-reachable from δ2nr, then there exists a sequence of control signals us-1 under which y(s;δ2nr,us-1,ξs-1)=δ2pl regardless of the effect of disturbance inputs ξs-1. Since (38)ys;δ2nr,us-1,ξs-1=Hxs;δ2nr,us-1,ξs-1, letting x(s;δ2nr,us-1,ξs-1)=δ2nq0, it is easy to check that Colq0(H)=δ2pl and δ2nq0 is robust reachable from δ2nr at the sth step. In other words, (H)l,q0=1 and (Ms)q0,r>0. Hence, (39)HMsl,r=∑i=12nHl,iMsi,r≥Hl,q0Msq0,r>0, which means that (30) holds.
The results (ii) and (iii) can be derived from (i) and Definition8 easily. Thus, we omit them.
Next, we prove result (iv).
(Sufficiency). Assuming that (33) holds, then for any l∈{1,2,⋯,2p}, there exists an integer 1≤s≤2n+m+r such that (HMs)l,r>0. Based on result (i), yf=δ2pl is s-robust-output-reachable from the initial state x(0). For the arbitrariness of l, we know system (14) is robust-output-controllable at the initial state x(0)=δ2nr.
(Necessity). Suppose that the system (14) is robust-output-controllable at the initial state x(0)=δ2nr. Then for any output yf=δ2pl, there exists an integer 1≤sl≤2m+n+r such that, for arbitrary disturbance inputs, the inequality (HMsl)l,r>0 holds, ∀l∈{1,2,⋯,2p}. Thus, (40)∑s=12m+n+rHMsl,r≥HMsll,r>0,∀l∈1,2,⋯,2p, which means that (33) holds.
The result (v) follows from (i) and Definition8, and we omit it. The proof is completed.Corollary 11.
By Theorem10, for a given initial state x(0)=δ2nr and an integer s>0, one can successively obtain that(41)Rsx0=δ2pl:HMsl,r>0.


### 3.3. Robust Optimal Output Control of Disturbed Incomplete Boolean Control Networks

In this subsection, we investigate the Mayer-type robust optimal output control problem of disturbed incomplete Boolean control networks and present a new design procedure for the problem.Consider a disturbed incomplete BCN with the algebraic state representation (14). The Mayer-type robust optimal output control problem can be described as follows: find a control sequence us-1≔{u(t):t=0,⋯,s-1} such that, for arbitrary disturbance inputs ξs-1≔{ξ(t):t=0,⋯,s-1}, the cost functional(42)Jus-1,ξs-1;x0=λTysis minimized under the given initial state x(0), where λ=λ1⋯λ2pT∈M2p×1 is a given constant vector and s≥1 is a fixed termination time.Minimizing the cost functionalJ is equal to find out the minimum value of J under the following constraint:(43)ys∈Rsx0=δ2pq1,⋯,δ2pqα,And, meanwhile, we need to design a control sequence {u(0),u(1),⋯,u(s-1)} which steers x(0) to the optimal terminal output y∗(s) regardless of the effect of disturbance inputs, where Rs(x(0)) can be computed by (41).It is noted that wheny(s)=δ2pqi, J(us-1,ξs-1;x(0))=λTy(s)=λTδ2pqi=Colqi(λT)=λqi. Thus, to find out the minimum value of J, we just need to compute the minimum value of λq,q=q1,⋯,qα.According to the above discussion, we obtain the following algorithm to find a sequence of controlsu(0),⋯,u(s-1) such that, for any disturbance inputs ξ(0),⋯,ξ(s-1), the cost functional J is minimized at the fixed termination time s.Algorithm 12.
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Step 1. Compute the s-robust-output-reachable set with the initial state x(0)=δ2nr by (41), denoted by Rs(x(0))={δ2pq1,⋯,δ2pqα};
Step 2. Compute the optimal value J∗ =min{λq:q=q1,⋯,qα}≔λqw∗;
Step 3. Let y(s)=δ2pqw∗. Find an integer 1≤q≤2n such that (H)qw∗,q=1 and (Ms)q,r>0. Let x(s)=δ2nq;
Step 4. Find two integers 1≤β≤2n and 1≤j≤2m such that (M)q,β>0,(Ms-1)β,r>0 and (L~jΘ)q,β>0. Let x(s-1)=δ2nβ and u(s-1)=δ2mj;
Step 5. If s-1=1, find j′ such that (L~j′Θ)β,r>0 and let u(0)=δ2mj′, stop. Otherwise, replace s by s-1, and replace q by β. Go back to step 4.Proposition 13.
The control sequence{u(0),⋯,u(s-1)} generated by Algorithm 12 can minimize the cost functional (42) at the fixed termination time s regardless of the effect of disturbance inputs.Proof.
Sincey(s)=δ2pqw∗ can minimize the cost functional J, we just need to show that the control sequence {u(0),⋯,u(s-1)} can steer the state x(0) to the objective output y∗(s) under arbitrary disturbance inputs.
Notice thaty(s)∈Rs(x(0)); by Theorem 10, we can find an integer q such that (H)qw∗,q=1 and (Ms)q,r>0. Let x(s)=δ2nq. By (23), one knows that for arbitrary disturbance input ξ(s-1)=δ2rk,k=1,2,⋯,2r, it follows the result (44)xs=LΘξs-1us-1xs-1=LkΘus-1xs-1=LkΘδ2mjδ2nβ=ColβLkjΘ=δ2nq.To ensure that δ2nq can be reached from δ2nβ in one step with the control u(s-1)=δ2mj for arbitrary disturbance input, the following results should be kept:(45)δ2nq=L1Θδ2mjδ2nβ=ColβL1jΘ,δ2nq=L2Θδ2mjδ2nβ=ColβL2jΘ,⋮δ2nq=L2rΘδ2mjδ2nβ=ColβL2rjΘ.Combining (25) and (45), we obtain Colβ(L~jΘ)=δ2nq.
In the same way, we can find integersj′ and β′ such that (46)Mβ,β′>0,Ms-2β′,r>0,L~j′Θβ,β′>0, which implies that δ2nβ can be reached from δ2nβ′ in one step with the control u(s-2)=δ2mj′ for arbitrary disturbance.
Do the same procedure, we can finally obtain the control sequence{u(0),⋯,u(s-1)} which steers the state x(0) to the objective output y∗(s) regardless of the effect of disturbance inputs.


## 3.1. An Algebraic Form for Disturbed Incomplete BCNs

First, we give the concept of disturbed incomplete BCNs.Definition 5.
Consider the disturbed BCN (7). Suppose certain controls are not applicable to certain states. Precisely, there exists a set of pairs(8)Θ≔δ2mαi,δ2nβi∣i=1,⋯,s⊂Δ2m×Δ2n,such that the control δ2mαi is not applicable to δ2nβi. Θ is called the control-state avoiding set. Supposing that Θ≠∅, then the disturbed BCN is called the disturbed incomplete BCN.Consider the disturbed incomplete BCN (7) with control-state avoiding set Θ and split L into 2r blocks as L≔[L1,L2,⋯,L2r], where Lk∈L2n×2n+m,k=1,2,⋯,2r. For u(t)=δ2mαi,x(t)=δ2nβi, a straightforward computation shows that u(t)⋉x(t)=δ2mαi⋉δ2nβi=δ2m+nai, where ai=(αi-1)2n+βi. Thus, when ξ(t)=δ2rk,u(t)=δ2mαi and x(t)=δ2nβi, we have (9)xt+1=Lξtutxt=Lkutxt=Lkδ2m+nai=ColaiLk,where ai=(αi-1)2n+βi and k=1,2,⋯,2r. If (δ2mαi,δ2nβi)∈Θ, we let Colai(Lk)=δ2n0. In this case, the corresponding x(t+1) does not exist. We also denote it briefly as x(t+1)=δ2n0. Then, it is easy to obtain the following result.Proposition 6.
A disturbed incomplete BCN with control-state avoiding setΘ as in (8) has its algebraic expression as in (7) except that (10)ColaiLk=δ2n0, where ai=(αi-1)2n+βi,i=1,2,⋯,s, and k=1,2,⋯,2r.We denote the new blocks obtained from Proposition6 by LkΘ, that is,(11)ColaiLkΘ=δ2n0,ai=αi-12n+βi;ColaiLkΘ=ColaiLk,otherwise.Let IΘ be the matrix obtained from an identity matrix I2n+m×2n+m substituting the columns with indexes a1,⋯,as by δ2n+m0; then LkΘ can be calculated by LkΘ=LkIΘ.Construct a new matrix as follows:(12)LΘ=L1Θ,L2Θ,⋯,L2rΘ,and then it is easy to see that LΘ is the state transition matrix of a disturbed incomplete BCN with control-state avoiding set Θ. In fact, LΘ can be further calculated as(13)LΘ=L1Θ,L2Θ,⋯,L2rΘ=L1IΘ,L2IΘ,⋯,L2rIΘ=L⋉δ2r1⋉IΘ,L⋉δ2r2⋉IΘ,⋯,L⋉δ2r2r⋉IΘ=L⋉δ2r1⋉IΘ,δ2r2⋉IΘ,⋯,δ2r2r⋉IΘ=LI2r⊗IΘ.Based on (7) and above analysis, the following result is obvious.Proposition 7.
Consider a disturbed incomplete BCN with control-state avoiding setΘ. It can be expressed as an algebraic form:(14)xt+1=LΘξtutxt,yt=Hxt,where LΘ∈L2n×2n+m+rI is given in (12).Proof.
We only need to prove that the state trajectories of the disturbed incomplete logical system (14) are equivalent to that of the system (7) with control-state avoiding set Θ.
DenoteW={δ2m+nai=δ2mαi⋉δ2nβi∣(δ2mαi,δ2nβi)∈Θ}, where ai=(αi-1)2n+βi. It is obvious that W and Θ are equivalent. For any ξ(t)=δ2rk,k=1,⋯,2r, we first prove the case of u(t)x(t)=δ2m+nai∈W. A simple calculation shows that(15)xt+1=Lξtutxt=Lkutxt=ColaiLk=δ2n0=ColaiLkΘ=LkΘutxt=LΘξtutxt.
Whenu(t)x(t)=δ2m+nai∉W, we have(16)xt+1=Lξtutxt=Lkutxt=ColaiLk=ColaiLkΘ=LkΘutxt=LΘξtutxt. Hence, in both cases, the state trajectories of the disturbed incomplete logical system (14) are equivalent to that of the system (7) with control-state avoiding set W. Since W and Θ re equivalent, this completes the proof.


## 3.2. Robust Output Controllability Analysis of Disturbed Incomplete BCNs

Now, we give definitions of the robust output controllability for disturbed incomplete BCNs.Definition 8.
Consider the disturbed incomplete BCN (14).(i)
yf∈Δ2p is said to be s-robust-output-reachable from the initial state x(0)∈Δ2n, if one can find a control sequence us-1⊆Δ2m such that for arbitrary disturbance inputs ξs-1⊆Δ2r, the output at time t=s satisfies y(s;x(0),us-1,ξs-1)=yf, where us-1≔{u(0),⋯,u(s-1)} and ξs-1≔{ξ(0),⋯,ξ(s-1)}. The s-robust-output-reachable set of the initial state x(0) is denoted by Rs(x(0)).(ii)
The system (14) is said to be s-robust-output-controllable at the initial state x(0)∈Δ2n, if for any output state yf∈Δ2p, one can find a control sequence us-1⊆Δ2m such that, for arbitrary disturbance inputs ξs-1⊆Δ2r, the output at time t=s satisfies y(s;x(0),us-1,ξs-1)=yf. The system (14) is said to be s-robust-output-controllable, if the system is s-robust-output-controllable at any x(0)∈Δ2n.(iii)
The system (14) is said to be robust-output-controllable at the initial state x(0)∈Δ2n, if for any output state yf∈Δ2p, one can find a positive integer T and a control sequence uT-1⊆Δ2m such that, for arbitrary disturbance inputs ξT-1⊆Δ2r, the output at time t=T is y(T;x(0),uT-1,ξT-1)=yf. The system (14) is said to be robust-output-controllable, if the system is robust-output-controllable at any x(0)∈Δ2n.To obtain necessary and sufficient conditions for the robust output controllability of disturbed incomplete BCNs, we first consider the robust reachability of disturbed incomplete BCNs.Consider the disturbed incomplete BCN (14), and denote by(17)L~Θ=L1Θ∧L2Θ∧⋯∧L2rΘ.SplitL~Θ into 2m square blocks as(18)L~Θ=L~1Θ,L~2Θ,⋯,L~2mΘ,whereL~jΘ∈L2n×2nI,j=1,2,⋯,2m. Then we have the following result.Theorem 9.
Consider the disturbed incomplete BCN (14). Let xf=δ2nq and x(0)=δ2nr be given. Then,
(i)xf is robust reachable from x(0) at the s-th step, if and only if(19)Msq,r>0,
whereM=∑j=12m‍(L~jΘ) and (Ms)q,r denotes the (q,r)th element of Ms;
(ii)xf is robust reachable from x(0), if and only if(20)∑s=12m+n+rMsq,r>0.Proof.
​​
(i) (Necessity). We prove it by induction on s. Consider the case s=1. It is assumed that x(1)=δ2nq can be robust reachable from x(0)=δ2nr at the first step; then one can find at least one control u(0)=δ2mj such that, for any disturbance input ξ(0)=δ2rk, the following equation will hold:(21)δ2nq=LkΘδ2mjδ2nr,∀k=1,2,⋯,2r.
Multiplying (21) from the left by (δ2nq)T yields(22)1=δ2nqTLkΘδ2mjδ2nr,∀k=1,2,⋯,2r.
For each blockLkΘ∈L2n×2n+mI, split it into 2m equal blocks as(23)LkΘ=Lk1Θ,Lk2Θ,⋯,Lk2mΘ,where LkΘ is given in (12). Then according to (22), we have(24)L1jΘq,r=L2jΘq,r=⋯=L2rjΘq,r=1.Since L~Θ=L1Θ∧L2Θ∧⋯∧L2rΘ, one knows that(25)L~Θδ2mj=L~jΘ=L1Θ∧L2Θ∧⋯∧L2rΘδ2mj=L1Θδ2mj∧L2Θδ2mj∧⋯∧L2rΘδ2mj=L1jΘ∧L2jΘ∧⋯∧L2rjΘ.
Combining (24) and (25), we have (26)L~jΘq,r=1.
Thus,(27)Mq,r=L~1Θ+L~2Θ+⋯+L~2mΘq,r≥L~jΘq,r=1.
This proves the situation fors=1.
For the induction step, we assume that, for anyl≤s, if x(l)=δ2nq can be robust reachable from x(0)=δ2nr at the lth step, then (Ml)q,r>0. In the following, we need to prove that (19) holds for l=s+1. Assuming that x(s+1)=δ2nq can be robust reachable from x(0)=δ2nr at the (s+1)th step, then there must exist a path from δ2nr to δ2nk0 at the sth step and a path from δ2nk0 to δ2nq in one step. Applying the induction hypothesis, we have(28)Msk0,r>0,Mq,k0>0. Therefore,(29)Ms+1q,r=MMsq,r=∑k=12nMq,kMsk,r≥Mq,k0Msk0,r>0.The proof of necessity is completed.
(Sufficiency). We still prove it by induction on s. From the proof of necessity, one can easily see that if (M)q,r>0, then (21) must hold, which implies that control δ2mj can steer system (14) from δ2mr to δ2mq in one step under arbitrary disturbance inputs. This proves (19) for s=1.
We assume that, for anyl≤s, if (Ml)q,r>0, then δ2mq can be robust reachable from δ2mr at the lth step. By (29), one can obviously see that if (Ms+1)q,r>0, then there exists at least a positive integer k0 such that (Ms)k0,r>0 and (M)q,k0>0. Based on the induction hypothesis, δ2nk0 can be robust reachable from δ2nr at the sth step and δ2nq can be robust reachable from δ2nk0 in one step. Thus, δ2nq can be robust reachable from δ2nr at the (s+1)th step.
The result (ii) can be easily obtained from (i).From Theorem9 and Definition 8, one can obtain the following robust output controllability results.Theorem 10.
Consider the disturbed incomplete BCN (14).
(i)yf=δ2pl is s-robust-output-reachable from the initial state x(0)=δ2nr, if and only if(30)HMsl,r>0;(ii) The system (14) is s-robust-output-controllable at the initial state x(0)=δ2nr, if and only if(31)ColrHMs>0;(iii) The system (14) is s-robust-output-controllable, if and only if(32)HMs>0;(iv) The system (14) is robust-output-controllable at the initial state x(0)=δ2nr, if and only if(33)∑s=12m+n+rColrHMs>0;(v) The system (14) is robust-output-controllable, if and only if(34)∑s=12m+n+rHMs>0.Proof.
​​
(i) (Sufficiency). Assuming that (30) holds, and we need to prove that there exists a control sequence us-1≔{u(t):t=0,⋯,s-1}⊆Δ2m such that, for arbitrary disturbance inputs ξs-1≔{ξ(t):t=0,⋯,s-1}⊆Δ2r, the output satisfies y(s;δ2nr,us-1,ξs-1)=δ2pl. Since(35)HMsl,r=∑q=12nHl,qMsq,r>0,there must exist an integer 1≤q0≤2n such that (H)l,q0=1 and (Ms)q0,r>0.
Based on Theorem9, δ2nq0 is robust reachable from δ2nr at the sth step. In other words, we can find a control sequence us-1⊆Δ2m such that (36)xs;δ2nr,us-1,ξs-1=δ2nq0,∀ξs-1⊆Δ2p. Correspondingly,(37)ys;δ2nr,us-1,ξs-1=Hδ2nq0=Colq0H=δ2pl,∀ξs-1⊆Δ2p,which infers that δ2pl is s-robust-output-reachable from x(0)=δ2nr.
(Necessity). Assuming that δ2pl is s-robust-output-reachable from δ2nr, then there exists a sequence of control signals us-1 under which y(s;δ2nr,us-1,ξs-1)=δ2pl regardless of the effect of disturbance inputs ξs-1. Since (38)ys;δ2nr,us-1,ξs-1=Hxs;δ2nr,us-1,ξs-1, letting x(s;δ2nr,us-1,ξs-1)=δ2nq0, it is easy to check that Colq0(H)=δ2pl and δ2nq0 is robust reachable from δ2nr at the sth step. In other words, (H)l,q0=1 and (Ms)q0,r>0. Hence, (39)HMsl,r=∑i=12nHl,iMsi,r≥Hl,q0Msq0,r>0, which means that (30) holds.
The results (ii) and (iii) can be derived from (i) and Definition8 easily. Thus, we omit them.
Next, we prove result (iv).
(Sufficiency). Assuming that (33) holds, then for any l∈{1,2,⋯,2p}, there exists an integer 1≤s≤2n+m+r such that (HMs)l,r>0. Based on result (i), yf=δ2pl is s-robust-output-reachable from the initial state x(0). For the arbitrariness of l, we know system (14) is robust-output-controllable at the initial state x(0)=δ2nr.
(Necessity). Suppose that the system (14) is robust-output-controllable at the initial state x(0)=δ2nr. Then for any output yf=δ2pl, there exists an integer 1≤sl≤2m+n+r such that, for arbitrary disturbance inputs, the inequality (HMsl)l,r>0 holds, ∀l∈{1,2,⋯,2p}. Thus, (40)∑s=12m+n+rHMsl,r≥HMsll,r>0,∀l∈1,2,⋯,2p, which means that (33) holds.
The result (v) follows from (i) and Definition8, and we omit it. The proof is completed.Corollary 11.
By Theorem10, for a given initial state x(0)=δ2nr and an integer s>0, one can successively obtain that(41)Rsx0=δ2pl:HMsl,r>0.


## 3.3. Robust Optimal Output Control of Disturbed Incomplete Boolean Control Networks

In this subsection, we investigate the Mayer-type robust optimal output control problem of disturbed incomplete Boolean control networks and present a new design procedure for the problem.Consider a disturbed incomplete BCN with the algebraic state representation (14). The Mayer-type robust optimal output control problem can be described as follows: find a control sequence us-1≔{u(t):t=0,⋯,s-1} such that, for arbitrary disturbance inputs ξs-1≔{ξ(t):t=0,⋯,s-1}, the cost functional(42)Jus-1,ξs-1;x0=λTysis minimized under the given initial state x(0), where λ=λ1⋯λ2pT∈M2p×1 is a given constant vector and s≥1 is a fixed termination time.Minimizing the cost functionalJ is equal to find out the minimum value of J under the following constraint:(43)ys∈Rsx0=δ2pq1,⋯,δ2pqα,And, meanwhile, we need to design a control sequence {u(0),u(1),⋯,u(s-1)} which steers x(0) to the optimal terminal output y∗(s) regardless of the effect of disturbance inputs, where Rs(x(0)) can be computed by (41).It is noted that wheny(s)=δ2pqi, J(us-1,ξs-1;x(0))=λTy(s)=λTδ2pqi=Colqi(λT)=λqi. Thus, to find out the minimum value of J, we just need to compute the minimum value of λq,q=q1,⋯,qα.According to the above discussion, we obtain the following algorithm to find a sequence of controlsu(0),⋯,u(s-1) such that, for any disturbance inputs ξ(0),⋯,ξ(s-1), the cost functional J is minimized at the fixed termination time s.Algorithm 12.
​​
Step 1. Compute the s-robust-output-reachable set with the initial state x(0)=δ2nr by (41), denoted by Rs(x(0))={δ2pq1,⋯,δ2pqα};
Step 2. Compute the optimal value J∗ =min{λq:q=q1,⋯,qα}≔λqw∗;
Step 3. Let y(s)=δ2pqw∗. Find an integer 1≤q≤2n such that (H)qw∗,q=1 and (Ms)q,r>0. Let x(s)=δ2nq;
Step 4. Find two integers 1≤β≤2n and 1≤j≤2m such that (M)q,β>0,(Ms-1)β,r>0 and (L~jΘ)q,β>0. Let x(s-1)=δ2nβ and u(s-1)=δ2mj;
Step 5. If s-1=1, find j′ such that (L~j′Θ)β,r>0 and let u(0)=δ2mj′, stop. Otherwise, replace s by s-1, and replace q by β. Go back to step 4.Proposition 13.
The control sequence{u(0),⋯,u(s-1)} generated by Algorithm 12 can minimize the cost functional (42) at the fixed termination time s regardless of the effect of disturbance inputs.Proof.
Sincey(s)=δ2pqw∗ can minimize the cost functional J, we just need to show that the control sequence {u(0),⋯,u(s-1)} can steer the state x(0) to the objective output y∗(s) under arbitrary disturbance inputs.
Notice thaty(s)∈Rs(x(0)); by Theorem 10, we can find an integer q such that (H)qw∗,q=1 and (Ms)q,r>0. Let x(s)=δ2nq. By (23), one knows that for arbitrary disturbance input ξ(s-1)=δ2rk,k=1,2,⋯,2r, it follows the result (44)xs=LΘξs-1us-1xs-1=LkΘus-1xs-1=LkΘδ2mjδ2nβ=ColβLkjΘ=δ2nq.To ensure that δ2nq can be reached from δ2nβ in one step with the control u(s-1)=δ2mj for arbitrary disturbance input, the following results should be kept:(45)δ2nq=L1Θδ2mjδ2nβ=ColβL1jΘ,δ2nq=L2Θδ2mjδ2nβ=ColβL2jΘ,⋮δ2nq=L2rΘδ2mjδ2nβ=ColβL2rjΘ.Combining (25) and (45), we obtain Colβ(L~jΘ)=δ2nq.
In the same way, we can find integersj′ and β′ such that (46)Mβ,β′>0,Ms-2β′,r>0,L~j′Θβ,β′>0, which implies that δ2nβ can be reached from δ2nβ′ in one step with the control u(s-2)=δ2mj′ for arbitrary disturbance.
Do the same procedure, we can finally obtain the control sequence{u(0),⋯,u(s-1)} which steers the state x(0) to the objective output y∗(s) regardless of the effect of disturbance inputs.


## 4. An Illustrative Example

In this section, we give an example to demonstrate the efficiency of the results obtained in this paper.Consider an incomplete BCN with one disturbance input whose dynamic equations are given as follows:(47)x1t+1=¬u1t∧x2t∨x3t,x2t+1=¬u1t∧u2t∧x1t∧ξtx3t+1=¬u1t∧u2t∨u3t∧x1tIn this example, the measured outputs are assumed as(48)y1t=x1t∧x2t∨x3t,y2t=x1t∧x2t∨¬x1t∧¬x2t.Setx(t)=⋉i=13xi(t),u(t)=⋉i=13ui(t) and y(t)=⋉i=12yi(t). Assume the control-state avoiding set Θ≔{(δ81,δ82),(δ81,δ85),(δ85,δ86),(δ86,δ84),(δ87,δ84)}. Then we can express the disturbed incomplete BCN (47) in its algebraic form as(49)xt+1=LΘξtutxt,yt=Hxt,where(50)LΘ=δ880880888888888888888888888888888111530371110333733304448444844488088088888888888888888888888888833373037333033373330444844484448∈L23×23+3+1I,and H=δ4[11242413].Skipping a tedious calculation, we can obtain(51)HM3=33313332222122226662666522212222,where M=∑j=123‍L~jΘ and L~Θ=L1Θ∧L2Θ can be calculated by (12) and (17). From Theorem 10, it is concluded that the system (49) is 3-robust-output-controllable. It is also robust-output-controllable.( 2 ) Consider the Mayer-type robust optimal output control problem of the disturbed incomplete Boolean control network (49). The constant vector λT=[3142], and we aim to determine a control strategy such that, for arbitrary disturbance input, the cost functional(52)Jus-1,ξs-1;x0=λTys,is minimized under the initial value x(0)=δ84 and s=3.Now, we follow Algorithm12 step by step as follows:Step 1. Compute the 3-robust-output-reachable set with the initial value x(0)=δ84, and obtain that Rs(x(0))=δ4[1,2,3,4];Step 2. J∗=min{λq,q=1,2,3,4}=1, and qw∗=2;Step 3. Let y(3)=δ42, and find q=3 such that (H)2,3=1 and (M3)3,4>0. So x(3)=δ83;Step 4. We can find β=7,j=6 such that (M)3,7>0,(M2)7,4>0, and (L~6Θ)3,7>0. So x(2)=δ87 and u(2)=δ86;Step 5. We can find β′=8,j′=5,j′′=2 such that (M)7,8>0,(M)8,4>0, and (L~5Θ)7,8>0,(L~2Θ)8,4>0. So x(1)=δ88,u(1)=δ85 and u(0)=δ82.At last, we obtain an optimal control{u(0),u(1),u(2)}={δ82,δ85,δ86} to minimize the cost functional J regardless of the effect of disturbance inputs.From another aspect, for the initial valuex(0)=δ84, under the control {u(0),u(1),u(2)}={δ82,δ85,δ86}, the state trajectories of the system (49) are x(1)=δ88,x(2)=δ87,x(3)=δ83. At this time, y(3)=δ42, J(us-1,ξs-1;x(0))=λTδ42=1. Hence, the control {u(0),u(1),u(2)}={δ82,δ85,δ86} is an optimal control.


## 5. Conclusions

In this paper, we have investigated the robust-output-controllability and robust optimal output control problems of incomplete BCNs with disturbance inputs. We have proposed several necessary and sufficient conditions for the robust output controllability based on the algebraic representation of the system. Then, we have discussed the Mayer-type robust optimal output control issue and presented an algorithm to find a control scheme which can minimize the cost functional regardless of the effect of disturbance inputs. Finally, an illustrative example has been given to support the results. It should be pointed out that event-triggered control has attracted a great deal of attention from scholars in the last two decades. Hence, future works can study the following issues:(1) output-controllability of higher-order incomplete BCNs via event-triggered control and (2) event-triggered control for robust optimal output control of incomplete BCNs with disturbance inputs.



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