# Event-Triggered Control for the Stabilization of Probabilistic Boolean Control Networks

Authors: Shiyong Zhu; Jungang Lou; Yang Liu; Yuanyuan Li; Zhen Wang
Journal: Complexity
 (2018)

---

## Abstract

This paper realizes global stabilization for probabilistic Boolean control networks (PBCNs) with event-triggered state feedback control (ETSFC). Via the semitensor product (STP) of matrices, PBCNs with ETSFC are converted into discrete-time algebraic systems, based on which a necessary and sufficient condition is derived for global stabilization of PBCNs. Furthermore, an algorithm is presented to design a class of feasible event-triggered state feedback controllers for global stabilization. Finally, an illustrative example shows the effectiveness of the obtained result.


---

## Body

## 1. Introduction

As a central focus of genomic research, the way cellular systems fail in diseases has attracted much attention [1, 2]. Various mathematical and computational models have been constructed to describe the behavior of gene regulatory networks, such as Bayesian networks [3], differential equation [4], and Boolean networks (BNs) [5]. By right of its simple representation, BNs have attracted much attention, where the state of each gene is described by two levels, either inactive (0 or OFF) or active (1 or ON). The evolution of each node is related to the states of some other nodes, including itself sometimes, and determined by a series of logical functions at each discrete-time point. BNs with control inputs are named by Boolean control networks (BCNs), which are essentially switched systems with switching among different BNs [6].Recently, a new matrix product, named the semitensor product (STP), is presented by Cheng et al. [7]. It converts the logical form of BCNs into the algebraic state space representation (ASSR). Compared with other methods, such as algebra geometry and symbolic dynamics, STP is more convenient and scientific in the study of BCNs. In this area, many interesting researches have been made, such as controllability [8–10], stabilization [11–16], optimal control [17], disturbance decoupling problems [18, 19], and function perturbation [20]. Besides, it has been proved that STP is effective in the study of logical systems [21, 22], game theory [23–25], fault detection [26], nonlinear shift register [27, 28], and so on.However, the determinacy of a Boolean function limits the application of BCNs, due to the existence of the randomness and measuring noise in the real word. Therefore, it is necessary to extend the BCN to the PBCN, which can be regarded as an undetermined system switching between different constituent BCNs in terms of the probabilistic structure. Similar to BCNs, many extended results have been derived for PBCNs, such as controllability [29–31] and stabilization [31–33].The stability and stabilization problems are two fundamental issues in the control of PBCNs. Reference [32] presented a necessary and sufficient condition to judge whether a PBCN can achieve global stabilization with probability one in the finite time by state feedback controllers, and the controller design is also realized. As mentioned in [31], a reachability matrix was defined to study the asymptotic stability and stabilization of PBCNs. But it is worth mentioning that the state feedback controllers here charge too much cost and need a relatively long transfer period sometimes. Synchronously, many interesting controllers were favorable for their unique properties [18, 34, 35]; motivated by which, we aim to design the ETSFC to overcome the cost of control and transfer period. The event-triggered control not only has a wide application in BCNs [18, 36] but also has smart grids [37], multiagent systems [38–43], and so on. For example, [18] first presented two kinds of event-trigger controllers to study the disturbance decoupling problems of BCNs, and some necessary and sufficient conditions were also obtained. In [36], authors used a class of event-triggered controls to realize global convergence for finite evolution and minimized the event set triggering the state at a special case. But the stabilization of PBCNs with ETSFC is still open but meaningful in the realistic world.The main constructions of this paper mainly focus on the following two points.(i)
A series of reachability sets are defined, via which, a necessary and sufficient condition for global stabilization of PBCNs controlled by ETSFC is designed.
(ii)
An algorithm is presented to design a class of event-triggered state feedback controllers to realize global stabilization of PBCNs.The remaining part of this paper is constructed as follows. In Section2, some notations are given and the STP of matrices is introduced. In Section 3, we devote to investigate the necessary and sufficient condition for stabilization of PBCNs with ETSFC and design controllers to realize global stabilization. Besides, an illustrative example is also given. The conclusion is provided in Section 4.Notation 1.
D≔0,1 and Dn=D×⋯×D⏟n. Denote the ith column of matrix A by ColiA and the set of all columns of matrix A by ColA. Δn≔ColIn, where In is an n×n identity matrix. β=lcmw,q is the least common multiple of w and q. Denote the set of m×n logical matrices by Lm×n, where m×n logical matrix B satisfies ColB⊆ColIm. Denote the column vector of length k with all entries equaling 1 and 0 by 1k and 0k, respectively. r=r1,…,rkT is called a k-dimensional probabilistic vector if ri≥0 and Σi=0kri=1, and the set of k-dimensional probabilistic vectors is denoted by Υk. A m×n matrix C is called a probabilistic matrix if ColiC⊆Υm holds for any i=1,2,…,n. The set of all m×n probabilistic matrices is denoted by Υm×n. Define operator “∘” of two probabilistic column vectors E and F as E∘F=p1∧q1,…,pn∧qnT, where pi∧qi=1 if and only if piqi>0, else pi∧qi=0. The Khatri-Rao product of n matrices M^1∗⋯∗M^n is defined as ColjL=⋉i=1nColjM^i.


## 2. Preliminaries

### 2.1. STP of Matrices

Definition 1 (see [7]).
Define the STP of matrixA∈Rm×w and B∈Rp×q by
(1)A⋉B=A⊗Iβ/wB⊗Iβ/p,β=lcmw,q,and ⊗ is the tensor (or Kronecker) product.Remark 1.
Ifn=p, A⋉B=A⊗I1B⊗I1=AB. Thus, STP is a generalization of the conventional matrix product. If no confusion arises, the symbol “⋉” can be omitted.Comparing with the general matrix product, the following pseudocommutative properties of STP are presented.Proposition 1 (see [7]).
Multiply a column matrixX∈Rm×1 and any matrix N, then
(2)X⋉N=Im⊗NX.Proposition 2 (see [7]).
Multiply two column matricesX∈Rm×1 and Y∈Rn×1, then
(3)Y⋉X=Wm,n⋉X⋉Y,where Wm,n=In⊗δm1,…,In⊗δmm.Let1~δ21 and 0~δ22, then D~Δ2. Any logical functions with n variables f:Dn→D can be expressed as the equivalent algebraic form by the following lemma.Lemma 1.
Forfx1,x2,…,xn:Dn→D, there exists a unique matrix Mf∈L2×2n, called the structure matrix of function f, such that
(4)fx1,x2,…,xn=Mf⋉i=1nxi,where xi∈Δ2 and ⋉i=1nxi=x1⋉⋯⋉xn∈Δ2n.


### 2.2. Algebraic Representation of PBCNs by ETSFC

A probabilistic Boolean model withn nodes is
(5)x1t+1=f1k1x1t,…,xnt,x2t+1=f2k2x1t,…,xnt,⋮xnt+1=fnknx1t,…,xnt,t≥0,where xi∈D are logic states. At each time step, fij are chosen from a given finite logical function set Fi=fi1,fi2,…,fili with probability pij, where Σj=1lifij=1 holds for any i=1,2,…,n. Ultimately, there are a total of π=Πi=1nli models. If we use the following matrix T to denote the index set of π models:
(6)T=11⋯1111⋯12⋮⋮⋮⋮⋮11⋯1ln11⋯2111⋯22⋮⋮⋮⋮⋮11⋯2ln⋮⋮⋮⋮⋮l1l2⋯ln−1ln,then the λth model can be defined by the λth row of matrix T, denoted by Σλ=f1λ1,f2λ2,…,fnλn, where λ=Σi=1n−1λi−1Πj=in−1lj+1+λn and 1≤λi≤li. Moreover, the probability of the λth model selected is computed to
(7)Pλ=∏i=1npiλi.Using Lemma1, let xt=⋉i=1nxit; we convert the logical form (5) into an algebraic form as
(8)x1t+1=M1xt,x2t+1=M2xt,⋮xnt+1=Mnxt,where Mi are chosen from Mi1,…,Mili with probability pi1,…,pili. Mij are the structure matrices of Boolean functions fik, i=1,…,n, j=1,…,li.DenoteExt as the mathematical expectation of state at time t; the evolution of state expectation is
(9)Exit+1=EMixt=Mi^xt,where Mi^=Σj=1lipijMij and i=1,2,…,n.Further, we multiply equations in (9), which leads to the following equation:
(10)Ext+1=LExt,where L=M^1∗⋯∗M^n∈Υ2n×2n and ∗ is the Khatri-Rao product.Similarly, PBCNs withm inputs can be equivalently described as
(11)x1t+1=f1k1x1t,…,xnt,u1t,…,umt,x2t+1=f2k2x1t,…,xnt,u1t,…,umt,⋮xnt+1=fnknx1t,…,xnt,u1t,…,umt,where xit and ujt∈D are states and inputs, respectively, and functions fji:Dn+m→Dn are logical functions. By Lemma 1, let xt=⋉i=1nxit and ut=⋉i=1nuit; the dynamic of PBCNs (11) can be converted into the following algebraic discrete-time system:
(12)x1t+1=L1k1utxt,x2t+1=L2k2utxt,⋮xnt+1=Lnknutxt,where Lij are the structure matrix of logical function fij. Taking the expected value on both sides of (12), we have
(13)Ex1t+1=L1utExt,Ex2t+1=L2utExt,⋮Exnt+1=LnutExt,where Li==Σj=1lipijLij, i=1,2,…,n.Multiplying (13) together, the algebraic state space representation is as follows:
(14)Ext+1=L~utExt,where ut∈Δ2m is the control input and L~=L1∗⋯∗Ln∈Υ2n×2n+m.The principle of ETSFC is that the controller only works ifxt equals to some designated states, consisting of set Λ, and, in this case, the system switches into (14). Otherwise, the system remains (10) and the controller does not work. The state feedback controller ut is
(15)u1t=k1x1t,…,xnt,u2t=k2x1t,…,xnt,⋮umt=kmx1t,…,xnt,t≥0,where ki:Dn→D are logical functions. Assume Ki is the structure matrix of logical function ki, (15) can be converted into
(16)ut=Kxt,t≥0,where the state feedback gain matrix K=K1∗⋯∗Km∈L2m×2n.Define control signfxt∈Δ2 depending on xt. If fxt=δ21, the system corresponds to dynamic (10), and fxt=δ22 makes the system correspond to PBCN (14). Therefore, the dynamic of PBCNs controlled by ETSFC is presented as
(17)xt+1=L^fxtxt,where L^=LL~ut, ut is the ETSFC which only works when fxt=δ22, that is,
(18)fxt=δ21,xt∈Δ2n\Λ,δ22,xt∈Λ,where set Λ consists of the states triggering the state feedback controller (16).


## 2.1. STP of Matrices

Definition 1 (see [7]).
Define the STP of matrixA∈Rm×w and B∈Rp×q by
(1)A⋉B=A⊗Iβ/wB⊗Iβ/p,β=lcmw,q,and ⊗ is the tensor (or Kronecker) product.Remark 1.
Ifn=p, A⋉B=A⊗I1B⊗I1=AB. Thus, STP is a generalization of the conventional matrix product. If no confusion arises, the symbol “⋉” can be omitted.Comparing with the general matrix product, the following pseudocommutative properties of STP are presented.Proposition 1 (see [7]).
Multiply a column matrixX∈Rm×1 and any matrix N, then
(2)X⋉N=Im⊗NX.Proposition 2 (see [7]).
Multiply two column matricesX∈Rm×1 and Y∈Rn×1, then
(3)Y⋉X=Wm,n⋉X⋉Y,where Wm,n=In⊗δm1,…,In⊗δmm.Let1~δ21 and 0~δ22, then D~Δ2. Any logical functions with n variables f:Dn→D can be expressed as the equivalent algebraic form by the following lemma.Lemma 1.
Forfx1,x2,…,xn:Dn→D, there exists a unique matrix Mf∈L2×2n, called the structure matrix of function f, such that
(4)fx1,x2,…,xn=Mf⋉i=1nxi,where xi∈Δ2 and ⋉i=1nxi=x1⋉⋯⋉xn∈Δ2n.


## 2.2. Algebraic Representation of PBCNs by ETSFC

A probabilistic Boolean model withn nodes is
(5)x1t+1=f1k1x1t,…,xnt,x2t+1=f2k2x1t,…,xnt,⋮xnt+1=fnknx1t,…,xnt,t≥0,where xi∈D are logic states. At each time step, fij are chosen from a given finite logical function set Fi=fi1,fi2,…,fili with probability pij, where Σj=1lifij=1 holds for any i=1,2,…,n. Ultimately, there are a total of π=Πi=1nli models. If we use the following matrix T to denote the index set of π models:
(6)T=11⋯1111⋯12⋮⋮⋮⋮⋮11⋯1ln11⋯2111⋯22⋮⋮⋮⋮⋮11⋯2ln⋮⋮⋮⋮⋮l1l2⋯ln−1ln,then the λth model can be defined by the λth row of matrix T, denoted by Σλ=f1λ1,f2λ2,…,fnλn, where λ=Σi=1n−1λi−1Πj=in−1lj+1+λn and 1≤λi≤li. Moreover, the probability of the λth model selected is computed to
(7)Pλ=∏i=1npiλi.Using Lemma1, let xt=⋉i=1nxit; we convert the logical form (5) into an algebraic form as
(8)x1t+1=M1xt,x2t+1=M2xt,⋮xnt+1=Mnxt,where Mi are chosen from Mi1,…,Mili with probability pi1,…,pili. Mij are the structure matrices of Boolean functions fik, i=1,…,n, j=1,…,li.DenoteExt as the mathematical expectation of state at time t; the evolution of state expectation is
(9)Exit+1=EMixt=Mi^xt,where Mi^=Σj=1lipijMij and i=1,2,…,n.Further, we multiply equations in (9), which leads to the following equation:
(10)Ext+1=LExt,where L=M^1∗⋯∗M^n∈Υ2n×2n and ∗ is the Khatri-Rao product.Similarly, PBCNs withm inputs can be equivalently described as
(11)x1t+1=f1k1x1t,…,xnt,u1t,…,umt,x2t+1=f2k2x1t,…,xnt,u1t,…,umt,⋮xnt+1=fnknx1t,…,xnt,u1t,…,umt,where xit and ujt∈D are states and inputs, respectively, and functions fji:Dn+m→Dn are logical functions. By Lemma 1, let xt=⋉i=1nxit and ut=⋉i=1nuit; the dynamic of PBCNs (11) can be converted into the following algebraic discrete-time system:
(12)x1t+1=L1k1utxt,x2t+1=L2k2utxt,⋮xnt+1=Lnknutxt,where Lij are the structure matrix of logical function fij. Taking the expected value on both sides of (12), we have
(13)Ex1t+1=L1utExt,Ex2t+1=L2utExt,⋮Exnt+1=LnutExt,where Li==Σj=1lipijLij, i=1,2,…,n.Multiplying (13) together, the algebraic state space representation is as follows:
(14)Ext+1=L~utExt,where ut∈Δ2m is the control input and L~=L1∗⋯∗Ln∈Υ2n×2n+m.The principle of ETSFC is that the controller only works ifxt equals to some designated states, consisting of set Λ, and, in this case, the system switches into (14). Otherwise, the system remains (10) and the controller does not work. The state feedback controller ut is
(15)u1t=k1x1t,…,xnt,u2t=k2x1t,…,xnt,⋮umt=kmx1t,…,xnt,t≥0,where ki:Dn→D are logical functions. Assume Ki is the structure matrix of logical function ki, (15) can be converted into
(16)ut=Kxt,t≥0,where the state feedback gain matrix K=K1∗⋯∗Km∈L2m×2n.Define control signfxt∈Δ2 depending on xt. If fxt=δ21, the system corresponds to dynamic (10), and fxt=δ22 makes the system correspond to PBCN (14). Therefore, the dynamic of PBCNs controlled by ETSFC is presented as
(17)xt+1=L^fxtxt,where L^=LL~ut, ut is the ETSFC which only works when fxt=δ22, that is,
(18)fxt=δ21,xt∈Δ2n\Λ,δ22,xt∈Λ,where set Λ consists of the states triggering the state feedback controller (16).


## 3. Main Results

In this section, the event-triggered stabilization of PBCNs is investigated.Definition 2.
PBCN (14) is said to be globally stabilized to state s∗ with probability one, if for any initial state x0∈Δ2n, there is a positive integer U and a control sequence u such that Pxt;x0,u=s∗=1, ∀t≥U.Then, for a nonempty setA⊆Δ2n, let R0A=A, and RkA can be expressed by induction as follows:
(19)RkA=δ2njColjL∘12n−∑a∈⋃i=0k−1RiAa=02n.Furthermore, defineRA as
(20)RA=⋃i=12nRiA.Remark 2.
RkA can also be represented as follows:
(21)RkA=δ2njPxt+1∈⋃i=0k−1RiA,xt=δ2nj=1.Assumings∗=δ2nr and the basins of s∗ by Rδ2nr, for i≥1, a series of sets are defined as follows:
(22)Si′=δ2nj∈Δ2nΩi−1thereexistsaδ2mvjsuchthatColjBlkvjL~∘12n−∑a∈Ωi−1a=02n,Si=RSi′∪Ωi−1Si′,where S0=δ2nr, S~0=Rδ2nr, Ω0=S0∪S0′, S~i=Si′∪Si, and Ωi=⋃k=0iS~k.For the above sets, we have some useful properties below.Proposition 3 (see [32]).
One has(i)
Ω1=δ2nr, then Ωt=δ2nr, ∀t≥1,
(ii)
ifΩt−1=Ωt, for some t≥1, then Ωj=Ωt, ∀j≥t.Theorem 1.
System (17) is globally stabilized to a designated state s∗=δ2nr with probability one, if and only if, the following conditions are satisfied:
(i)
ColrL=δ2nr or there exists an integer 1≤vr≤2m satisfying ColrBlkvrL~=δ2nr.
(ii)
There exists an integer0≤N≤2n−1 such that ΩN=Δ2n.Proof 1.
Sufficiency: condition (i) shows thatPxt+1=δ2nr∣xt=δ2nr=1 holds by the designed controller. According to condition (ii) and the constructing procedure of Si′ and Si, for any initial state x0=δ2nj∈Δ2n\δ2nr, it claims that x0∈S~0 or x0∈⋃i=1NSi′ or x0∈⋃i=1NSi. Therefore, there are three cases.
Case 1:
Ifx0=δ2nj∈S~0, there exists a positive integer kj satisfying δ2nj∈Rkjδ2nr, which implies that Pxkj=δ2nr∣x0=x0=1.
Case 2:
Ifx0=δ2nj∈Sij′, we can find a control u=δ2mvj such that L~ux0=L~⋉δ2mvj⋉δ2nj=ColjBlkvjL~∈Ωi−1.
Case 3:
Ifx0=δ2nj∈Sij, there exists Tj′ such that PxTj′∈Sij′∪Ωi−1∣x0=x0=1. The case in Sij′ has been discussed before. Alternatively, the state in Ωi will enter Ωi−1.
In a word, every statex0∈Δ2n will arrive at δ2nj with probability one.
Necessity: system (17) is globally stabilizable to s∗=δ2nr with probability one; δ2nr should be fixed.
Assume that condition (ii) does not hold; any state inΔ2n\ΩN cannot be steered into δ2nr via the intermittent control, which is in contradiction with Definition 2.
In the following, we aim to prove that the smallest integerN∗ satisfying condition (ii) is no more than 2n−1. It is enough to show that
(23)∣Ωτ∣≥τ+1,where 0≤τ≤N. We will show it by induction. When τ=1, if S0~∪S1~<2, that is, S0~∪S1~=1, thus S0~∪S1~=δ2nr. By Proposition 3, it holds that ΩN=δ2nr, which is a contradiction with Definition 2.
Suppose thatΩτ−1≥τ, for some 0≤τ≤N. It can be immediately obtained that
(24)Ωτ≥Ωτ−1≥τ.
IfΩτ≤τ+1, then
(25)Ωτ=Ωτ−1=τ,which means that Ωτ=Ωτ−1. Proposition 3 implies that Ωτ−1=ΩN=Δ2n, which is a contradiction to the minimality of integer N.
Moreover, it also holds thatΩN=Δ2n=2n≥N+1, that is to say, N≤2n−1.
Assume that conditions (i) and (ii) in Theorem1 hold, then the event-triggered state feedback controllers can be designed by the following algorithm.Algorithm 1
Step 1: InitializeΛ=∅, if ColrL=δ2nr, then Λ=⋃i=1NSi′; otherwise, ColrK=δ2mvr satisfying ColrBlkvrL~=δ2nr and Λ=δ2nr∪⋃i=1NSi′. Then go to Step 2.
Step 2: For every stateδ2nj in Δ2n\δ2nr, find a unique zj such that δ2nj∈Szj′. Then, let ColjK=δ2mvj satisfying ColjBlkvjL~∘12n−∑a∈Ωzj−1a=02n. Then go to Step 3.
Step 3: For remainderδ2nj∈Δ2n\Λ, let ColjK=δ2nvj be a random vector in Δ2m. Return set Λ and matrix K, where Λ is the event-set-triggering control (15) and matrix K is the state feedback gain matrix.Proof 2.
We only need to prove that system (17) is globally stabilized to δ2nr under the controller designed above. For the desired state δ2nr, if ColrL=δ2nr, along the trajectory of (10), xt;x0=δ2nr, for any t≥0, only if x0=δ2nr, that is, δ2nr∈Λ; else, find a δ2nvr such that ColrBlkvrL~=δ2nr. Thus, system (17) can stay at the desired state δ2nr with probability one.
For any initial stateδ2nj∈Δ2n\δ2nr, we split it into three cases: x0∈S~0 or x0∈⋃i=1NSi′ or x0∈⋃i=1NSi.
Case 1:
Forx0∈S~0, according to the proof of Theorem 1, it will reach δ2nr.
Case 2:
Forx0=δ2nj∈⋃i=1NSi′, there exists a zj such that δ2nj∈Szj′. Because δ2nj∈Λ, it will trigger the controller (16), under which, ColjBlkvjL~∈Ωzj−1.
Case 3:
Forx0=δ2nj∈⋃i=1NSi, there exists an integer ij such that for δ2nij, we claim that it will enter into the set Sij′∪Ωij−1 and the state in Sij will enter into Ωij−1, which has been discussed in Case 2.
By mathematical induction, system (17) will stabilize to δ2nr.Remark 3.
IfColrL=δ2nr, the controller (16) will be triggered forever and we can only cut down the cost on transient routes.Remark 4.
Whenl1=l2=⋯=ln=1, PBCN (14) deduces a conventional BCN. The obtained results also can be applied to the event-triggered stabilization of BCNs. Thus, our result can be regarded as a generalization of that in [36] to some extent.Example 1.
Consider the following probabilistic Boolean network:(26)Xt+1=f1Xt,Yt,Zt,Yt+1=f2Xt,Yt,Zt,Zt+1=f3Xt,Yt,Zt,
wheref1∈f11,f12 with probabilistic 1/2, 1/2, respectively, and f2∈f12,f22 with probabilistic 2/3, 1/3, respectively.
(27)f11X,Y,Z=X∧Z∨Y∧¬Z∨¬Y∧Z,f12X,Y,Z=X∧Y∨¬X∧¬Y∧Z,f21X,Y,Z=X∧Y∧Z∨X∧¬Y∧¬Z∨¬X∧Y∧¬Z,f22X,Y,Z=X∧¬Y∧¬Z,f3X,Y,Z=1.
Then we can obtain(28)L=23000013000000000013121200161000000000000101300000000000121201160100000000.
Assume the control system is given as(29)Xt+1=u1t∧f1Xt,Yt,Zt,Yt+1=u2t∧f2Xt,Yt,Zt,Zt+1=f3Xt,Yt,Zt,
Similarly, we obtainL~ as follows:
(30)L~=00000000000000000000000000000000000000001121200010000000000000000000000000000102300000000000000000011111111012120113010000000000000000.
In a word, the dynamics of this system can be described as(31)xt+1=LL~utfxtxt,
Next, we will design the ETSFC to stabilize system (31) to δ87. By Algorithm 1, Ωi, i=0,1,…, can be calculated as follows.
Step 1:
SinceCol7L≠δ87, then δ87∈Λ and there exists u=δ21 such that Col7Blk1L~=δ87.
Step 2:
By the dynamicxt+1=Lxt, we can calculate R1δ87=δ85,δ88, R2δ87=δ84, and Rkδ87=δ84,δ85,δ88,k≥3, so Rδ87=δ84,δ85,δ87,δ88.
Step 3:
LetΩ0=S~0=Rδ87, then Δ2n\Ω0=δ81,δ82,δ83,δ86. Next, we consider dynamic xt+1=L~utxt. For δ81, there exists u=δ21 such that Col1Blk1L~∈Ω0. For δ82, there exists u=δ21 such that Col2Blk1L~∈Ω0. For δ83, there exists u=δ21 such that Col3Blk1L~∈Ω0. For δ86, there exists u=δ22 such that Col6Blk2L~∈Ω0.
SoS1′=δ81,δ82,δ83,δ86 and Ω1=Δ8. Thus, the PBCN (31) is globally stabilized to δ87 via ETSFC by Theorem 1. In terms of Algorithm 1, set Λ can be expressed by Λ=δ81,δ82,δ83,δ86,δ87 and one of the feasible state feedback matrices K can be δ21,1,1,2,2,2,1,1.


## 4. Conclusion

In this technique, the global stabilization of PBCNs has been investigated by ETSFC. Under the framework of STP, the algebraic representation of PBCNs has been obtained from the logical form. A necessary and sufficient condition has been derived based on reachability sets. Finally, we have designed a class of feasible event-triggered state feedback controls to realize global stabilization. A numerical example has shown the effectiveness of our results.



---

*Source: 9259348-2018-09-13.xml*