# Design of Nonfragile State Estimator for Discrete-Time Genetic Regulatory Networks Subject to Randomly Occurring Uncertainties and Time-Varying Delays

Authors: Yanfeng Zhao; Jihong Shen; Dongyan Chen
Journal: Complexity
 (2017)

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## Abstract

We deal with the design problem of nonfragile state estimator for discrete-time genetic regulatory networks (GRNs) with time-varying delays and randomly occurring uncertainties. In particular, the norm-bounded uncertainties enter into the GRNs in random ways in order to reflect the characteristic of the modelling errors, and the so-called randomly occurring uncertainties are characterized by certain mutually independent random variables obeying the Bernoulli distribution. The focus of the paper is on developing a new nonfragile state estimation method to estimate the concentrations of the mRNA and the protein for considered uncertain delayed GRNs, where the randomly occurring estimator gain perturbations are allowed. By constructing a Lyapunov-Krasovskii functional, a delay-dependent criterion is obtained in terms of linear matrix inequalities (LMIs) by properly using the discrete-time Wirtinger-based inequality and reciprocally convex combination approach as well as the free-weighting matrix method. It is shown that the proposed method ensures that the estimation error dynamics is globally asymptotically stable and the desired estimator parameter is designed via the solutions to certain LMIs. Finally, we provide two numerical examples to illustrate the feasibility and validity of the proposed estimation results.


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## Body

## 1. Introduction

Genetic regulatory networks (GRNs), which are biochemically dynamical systems describing highly complicated interactions among DNA, RNA, protein, and metabolic product in the interactive transcriptional and translational processes, have become an attractive research field in the systems biology and biomedical sciences during the past few decades. One of the main objectives in systems biology is to understand how mRNAs and proteins work collectively and interact with each other to perform the complicated biological functions (e.g., the process of transcriptions and translations). In the past few years, a great deal of attention has been devoted to the theoretical analysis and experimental investigation on GRNs. Accordingly, a large amount of important results has been reported in literature making significant contributions for understanding both static and dynamic behaviors of biological systems in detail (see [1–3] and the references therein). Among them, two main classes of models used to characterize the GRNs include discrete models such as Boolean model [4, 5] and continuous models such as differential equation models [6, 7].When modelling GRNs, we should bear in mind the fact that GRNs models are unavoidably affected by modelling uncertainties, including parametric errors, time delays, and stochastic noises. More specifically, the time delays are unavoidable due primarily to the slow process of transcription/translation, translocation, or the finite switching speed of amplifiers. The existence of the time delays could affect the whole system performance and may lead to oscillation, divergence, and instability of the genetic networks. According to the occurrence way of time delays, they can generally be classified into three types: interval time-varying delays [8, 9], distributed delays [10], and random time delays [11, 12]. In addition, gene regulation is an intrinsically noisy process due to intracellular and extracellular noise perturbations, which are derived from random births and deaths of individual molecules and environmental fluctuations. Accordingly, a great deal of effort has been made to deal with the analysis problem of GRNs with time-varying delays as well as stochastic noises and a growing number of approaches have been provided in the literature to examine the dynamical behaviors of the addressed GRNs [2, 3, 13]. On the other hand, the discrete-time version of GRNs receives increasing research interests due to the reason that the continuous-time GRNs are commonly discretized for computer simulation and experimental purposes [14, 15]. In recent years, much effort has been made to tackle the synthesis problems of discrete GRNs with time delays. For example, the robust H∞ control problem has been investigated in [16] for discrete delayed stochastic GRNs. Moreover, the asymptotic stability problems have been investigated in [17–19] for discrete-time uncertain GRNs with time-varying delays and stochastic fluctuations, where some sufficient conditions have been presented to ensure the stability of the addressed GRNs via the linear matrix inequality approach.In practice, for the purpose of both the drug design and disease diagnosis, it is worth mentioning that biologists hope to gain actual concentrations of gene products in GRNs. However, due to the existence of the model errors, time delays, and external disturbances, the actual values of GRNs can hardly be obtained and only partial information about the gene states is commonly available in the measurement outputs. As such, the filtering and state estimation problems for complex dynamics systems have been widely investigated; see, for example, [20–26]. To mention a few, the set-values filtering problem has been investigated in [20] for a class of GRNs with time-varying uncertainties and bounded external noise, where new filtering scheme has been given for the addressed GRNs. In [22], the state estimation problem has been studied for a class of discrete-time GRNs with random delays. Moreover, the H∞ state estimation problem has been discussed in [24] for discrete stochastic GRNs with Markovian jumping parameters and time-varying delays. More recently, the sampling-data state estimation algorithm has been provided in [25] for a class of GRNs subject to time-varying delays. On the other hand, the parameter uncertainties may be changed in a random way with certain types and intensity due to various effects, for instance, the changing subsystem interconnections, network-induced random failures, repairs of components and sudden environmental perturbations, and so on. Hence, it is more significant to deal with the effects from the randomly occurring parameter uncertainties onto the networked systems [27–32]. In addition, as discussed in [33], a small or even tiny drift/fluctuation/error of the presented controller/estimator during the parameter implementation may result in unexpected fragility/degradation for whole system performance. In other words, there exist the deviations between the parameters of actually implemented controller/estimator and their expected values, and hence there is a need to design the robust controller/estimator with certain degree of tolerance against the possible deviations. In the past decade, the nonfragile estimation problems have gained much attention with respect to the implementation errors of proposed estimators [34–36]. However, the nonfragile state estimation problem has not been fully studied yet for discrete delayed GRNs, not to mention the case where the randomly occurring parameter uncertainties are also considered. As such, the main purpose of the paper is to provide a robust state estimation method against the mentioned phenomena.Motivated by the aforementioned analysis, in this paper, we aim to investigate the robust nonfragile state estimation problem for a class of discrete GRNs with time-varying delays and randomly occurring uncertainties. The Lyapunov-Krasovskii functional is chosen which utilizes more information about the time delays and a new set of conditions is added to calculate the difference of the designed Lyapunov-Krasovskii functional. Furthermore, as mentioned in [37], it is clearly seen that the Wirtinger-based inequality can reduce the conservatism and provide a more tighter lower delay bound of the summation terms. Therefore, we revisit the nonfragile state estimation problem for discrete GRNs with time-varying delays and randomly occurring uncertainties by using the discrete-time Wirtinger-based inequality and the reciprocally convex combination inequality. A delay-dependent estimation criterion is established in terms of feasibility of a set of LMIs. The main contributions of this paper can be summarized as follows: (1) the discrete Wirtinger-based inequality is utilized for the first time to investigate the nonfragile state estimation problem for GRNs with time-varying delays and randomly occurring uncertainties, where the estimator gain perturbations occur in a random way; (2) by utilizing the discrete-time Wirtinger-based inequality and the reciprocally convex combination approach, the summation terms are addressed and more information of time delays is reflected in the proposed algorithm; and (3) more free-weighting matrices are introduced by adding to new conditions with hope to further reduce the conservativeness induced by time delays. Finally, examples and simulation are given to illustrate the effectiveness and advantages of the proposed main results.Notations. For a matrix A, A>0 denotes that A is a symmetric positive-definite matrix. E[·] stands for the mathematical expectation operator. The superscript T and (-1) represent the transpose and the inverse of the matrix, respectively. I represents an identity matrix with appropriate dimension. For simplicity, ∗ represents the term that is induced by symmetry in symmetric block matrices.


## 2. Model Description and Preliminaries

In this paper, we consider the following discrete GRNs with time-varying delays,n mRNA, and n proteins:(1)Mk+1=AMk+BfPk-τk+L,Pk+1=CPk+DMk-δk,where M(k)=[M1(k),M2(k),…,Mn(k)]T, P(k)=[P1(k),P2(k),…,Pn(k)]T∈Rn are the concentrations of mRNA and protein at instant k, respectively. A=diag⁡a1,a2,…,an and C=diag⁡c1,c2,…,cn are real constant diagonal matrices representing the decay rates of mRNA and protein with entries ai<1 and ci<1, D=diag⁡d1,d2,…,dn, di is the translation rate, and B=(bij)n×n is the coupling matrix of the genetic networks. L=[l1,l2,…,ln]T stands for the basal rates of degradation. τk>0 and δk>0 are the time-varying delays denoting the feedback regulation delay and the translation delay satisfying 0<τm≤τk≤τM and 0<δm≤δk≤δM. In addition, the nonlinear function f(k)=[f1(k),f2(k),…,fn(k)]T represents the feedback regulation of the protein on the transcription. It is a monotonic function in the Hill form; that is, fj(x)=(x/βj)Hj/(1+(x/βj)Hj), x∈R, where Hj is the Hill coefficient and βj is a positive constant.Let[M∗T,P∗T]=[M1∗,M2∗,…,Mn∗,P1∗,P2∗,…,Pn∗] be an equilibrium point of system (1). Then, it is easy to obtain that(2)M∗=AM∗+BfP∗+L,P∗=CP∗+DM∗.Subsequently, we can shift the equilibrium point [M∗T,P∗T] of system (1) to the origin point via the transformations x(k)=M(k)-M∗ and y(k)=P(k)-P∗. Then, system (1) can be converted into the following form:(3)xk+1=Axk+Bgyk-τk,yk+1=Cyk+Dxk-δk. with gyk=fyk+P∗-fP∗.As mentioned above, the GRNs are often the large-scale networks and it should be noted that system parameter uncertainties may be subject to random changes in real circumstances due to some factors such as repairs of components and sudden environmental disturbances. Thus, they may occur in a probabilistic way with various types and intensities. By considering the randomly occurring uncertainties, system (3) becomes(4)xk+1=A+α1kΔAkxk+B+β2kΔBkgyk-τk,yk+1=C+β1kΔCkyk+D+α2kΔDkxk-δk,where ΔA(k), ΔB(k), ΔC(k), and ΔD(k) denote the parameter uncertainties satisfying the following condition:(5)ΔAkΔBkΔCkΔDk=E1FkM1M2M3M4,where E1, M1, M2, M3, and M4 are known constant matrices and F(k) is an unknown time-varying matrix satisfying FTkFk≤I. The stochastic variables αik and βi(k)(i=1,2) are used to characterize the randomly occurring uncertainties and are mutually independent Bernoulli-distributed white noise sequences satisfying(6)Probαik=1=αi,Probαik=0=1-αi,Probβik=1=βi,Probβik=0=1-βi, where αi and βi∈[0,1](i=1,2) are known constants.The state components of the GRNs are usually not completely accessible. Consequently, it is necessary to make use of the available output information and design a state estimator to estimate the state vector of the addressed GRNs. For this case, assume that the measurement outputs are given as follows:(7)zxk=W1xk,zyk=W2yk,where zx(k) and zy(k) are the measurements of model (4); W1 and W2 are the known matrices.The main objective of this paper is to estimate the concentrations of mRNA and protein in (4) from the available network outputs in (7). In the sequel, we design the following nonfragile state estimator:(8)x^k+1=Ax^k+Bgy^k-τk+K1+γ1kΔK1kzxk-W1x^k,y^k+1=Cy^k+Dx^k-δk+K2+γ2kΔK2kzyk-W2y^k,where x^(k), y^(k) are the estimations of x(k) and y(k), and the nonfragile estimator gain matrices K1, K2 are to be determined. The real-valued matrices ΔK1(k) and ΔK2(k) represent possible estimator gain fluctuations. It is assumed that ΔK1(k) and ΔK2(k) have the following structure:(9)ΔK1kΔK2k=E2FkM5M6,where E2, M5, and M6 are the known constant matrices and F(k) is the unknown time-varying matrix satisfying FT(k)F(k)≤I. The stochastic variables γi(k)(i=1,2) represent the perturbations in the estimator gain matrices and are mutually independent Bernoulli-distributed white noise sequences taking on values of 0 or 1 with(10)Probγik=1=γi,Probγik=0=1-γi,where γi∈[0,1](i=1,2) are known constants.Letting the estimation error bex~(k)=x(k)-x^(k) and y~(k)=y(k)-y^(k), the estimation error dynamics can be described as follows:(11)x~k+1=A-K1W1x~k+Bgy~k-τk+α1kΔAkxk-γ1kΔK1kW1x~k+β2kΔBkgyk-τk,y~k+1=C-K2W2y~k+Dx~k-δk+β1kΔCkyk-γ2kΔK2kW2y~k+α2kΔDkxk-δk,where g(y~(k-τ(k)))=g(y(k-τ(k)))-g(y^(k-τ(k))). Initial conditions for the uncertain GRNs (4) and state estimator system (8) are assumed to be (x(k),y(k))=(ϕ(k),ψ(k)) and (x^(k),y^(k))=(ϕ^(k),ψ^(k)) on [-τ,0], where τ=max{τM,δM}.To facilitate further derivations, set(12)exk=xkx~k,eyk=yky~k.Combining (4) with (11), we obtain the augmented estimation error dynamics:(13)exk+1=A¯1exk+B¯1geyk-τk+E¯1ϱ1k+E¯2ϱ2k,eyk+1=C¯1eyk+D¯1exk-δk+E¯1ϱ3k+E¯2ϱ4k,ϱik=F¯kςik,i=1,2,3,4,ς1k=α1kM¯1exk+β2kM¯2geyk-τk,ς2k=γ1kM¯5exk,ς3k=β1kM¯3eyk+α2kM¯4exk-δk,ς4k=γ2kM¯6eyk,where(14)A¯1=A00A-K1W1,C¯1=C00C-K2W2,B¯1=B00B,D¯1=D00D,E¯1=E100E1,E¯2=E200E2,F¯k=Fk00Fk,M¯1=M10M10,M¯2=M20M20,M¯3=M30M30,M¯4=M40M40,M¯5=000-M5W1,M¯6=000-M6W2.Assumption 1.
The nonlinear functiongi(·)(i=1,2,…,n) is a monotonically increasing function and satisfies the following condition:(15)0≤gixixi≤li,xi≠0,gi0=0,which is equivalent to(16)gTxgx-Lx≤0,where L=diag⁡l1,l2,…,ln>0 and li(i=1,2,…,n) are known constants.Definition 2.
System (8) is said to be a robust asymptotic state estimator of the GRNs (4) if estimation error system (13) is globally robustly asymptotically stable in the mean square; that is,(17)limk→+∞⁡Ex~k2+y~k2=0.To end of this section, we introduce the following lemmas which will be frequently used in the subsequent developments.Lemma 3 ([38] (Schur complement)).
Given constant matricesS11, S12, and S22 with appropriate dimensions, where S11=S11T and S22=S22T<0, then S11-S12S22-1S12T<0 if and only if(18)S=S11S12S12TS22<0.Lemma 4 ([37] (discrete-time Wirtinger-based inequality)).
For a given positive-definite matrixR and three nonnegative integers a,b,k satisfying a≤b≤k, denote(19)χk,a,b=1b-a2∑j=k-bk-a-1xj+xk-a-xk-b,a<b2xk-a,a=b.Then, one has(20)-b-a∑j=k-bk-a-1ηTiRηi≤-Ω0Ω1TR003RΩ0Ω1,where (21)ηk=xk+1-xk,Ω0=xk-a-xk-b,Ω1=xk-a+xk-b-χk,a,b.Lemma 5 (see [39]).
For given positive integersn,m, a scalar α∈(0,1), an n×n-matrix R>0, and two n×m-matrices S1,S2, for all vector ξ∈Rm, the function Θ(α,R) is given by(22)Θα,R=1αξTS1TRS1ξ+11-αξTS2TRS2ξ. If there is a matrix X∈Rn×n such that RX∗R>0, then the following inequality holds:(23)min⁡Θα,R≥S1ξS2ξTRX∗RS1ξS2ξ.


## 3. Main Results

In this section, we first consider that there are no parameter uncertainties in GRNs (4). Also, the estimator can be selected as (8); then augmented error system (13) can be rewritten as(24)exk+1=A¯1exk+B¯1geyk-τk+E¯2ϱ2k,eyk+1=C¯1eyk+D¯1exk-δk+E¯2ϱ4k,ϱik=F¯kςik,i=2,4,ς2k=γ1kM¯5exk,ς4k=γ2kM¯6eyk.Our main aim is to design the nonfragile state estimator to guarantee that estimation error system (24) is globally asymptotically stable. In other words, by designing a proper Lyapunov-Krasovskii functional, together with the discrete-time Wirtinger-based inequality, reciprocally convex combination approach, and the free-weighting matrices method, we are interested in looking for the estimator gain matrices K1 and K2 such that the error dynamics governed by (24) is globally asymptotically stable. Further, the result is extended to handle the nonfragile state estimation problem for GRNs (4) with randomly occurring uncertainties.Theorem 6.
For given positive scalarsδm, δM, τm, and τM, estimation error system (24) is globally asymptotically stable, if there exist scalars εi>0(i=2,4), matrices P1>0, P2>0, R>0, Zj>0(j=1,2,3,4), and Ql>0(l=1,2,…,6), positive diagonal matrices Λ1, Λ2, and Λ3, matrices Yi=Yi1Yi2∗Yi3(i=1,2), and block diagonal matrices N1, N2, N3, N4, X1, and X3, such that the following LMIs hold:(25)Σ=Ξ0ε2Ψ2Tε4Ψ4T∗-ε2I2n0∗∗-ε4I2n<0,where (26)Ψ2=γ1M¯50n×19n,Ψ4=0n×4nγ2M¯60n×15n,Ξ0=∑i=19Ξi,Ξ1=2e1P1e9T+e9P1e9T+2e5P2e10T+e10P2e10T,Ξ2=e1Q1+1+δ1Q2+Q3e1T-e3Q1e3T-e2Q2e2T-e4Q3e4T,Ξ3=e5Q4+1+τ1Q5+Q6e5T-e7Q4e7T-e6Q5e6T-e8Q6e8T,Ξ4=e111+τ1Re11T-e12Re12T,Ξ5=e9δ1Z1+δmZ2e9T+e10τ1Z3+τmZ4e10T-1δmρ1Z~2ρ1T-1δ1ρ7Θ1ρ7T-1τmρ4Z~4ρ4T-1τ1ρ8Θ2ρ8T,Ξ6=2e5LΛ1e11T-2e11Λ1e11T+2e6LΛ2e12T-2e12Λ2e12T+2e5-e6LΛ3e11-e12T-2e11-e12Λ3e11-e12T,Ξ7=-e19ε2I2ne19T-e20ε4I2ne20T,Ξ8=2e9N1B¯1e12T+2e9N1E¯2e19T-2e9N1e9T+2e1N2B¯1e12T+2e1N2E¯2e19T-2e1N2e9T+2e10N3D¯1e2T+2e10N3E¯2e20T-2e10N3e10T+2e5N4D¯1e2T+2e5N4E¯2e20T-2e5N4e10T,Ξ9=2e9N1A~1-X1W¯1e1T+2e1N2A~1-X1W¯1e1T+2e10N3C~1-X3W¯2e5T+2e5N4C~1-X3W¯2e5T,ei=02n×i-12nI2n×2n02n×20-i2nT,i=1,2,…,20,ρ1=e1-e33e1+e3-e13,ρ2=e2-e43e2+e4-e14,ρ3=e3-e23e3+e2-e15,ρ4=e5-e73e5+e7-e16,ρ5=e6-e83e6+e8-e17,ρ6=e7-e63e7+e6-e18,ρ7=ρ2ρ3,ρ8=ρ5ρ6,Z~i=Zi00Zi,i=1,2,3,4,Θ1=Z~1Y1∗Z~1,Θ2=Z~3Y2∗Z~3,A~1=A-I00A-I,C~1=C-I00C-I,W¯1=000-W1,W¯2=000-W2.And the remaining terms are zero. Then, the estimator gain matrices are determined by(27)K1=N12-1X1,K2=N32-1X3.Proof.
Set(28)η1k=exk+1-exk,η2k=eyk+1-eyk,δ1=δM-δm,τ1=τM-τm.We construct the following Lyapunov-Krasovskii functional for discrete-time GRNs (24):(29)Vk=∑i=15Vik,where(30)V1k=exTkP1exk+eyTkP2eyk,V2k=∑i=k-δmk-1exTiQ1exi+∑i=k-δMk-1exTiQ3exi+∑i=k-δkk-1exTiQ2exi+∑j=-δM+1-δm∑i=k+jk-1exTiQ2exi,V3k=∑i=k-τmk-1eyTiQ4eyi+∑i=k-τMk-1eyTiQ6eyi+∑i=k-τkk-1eyTiQ5eyi+∑j=-τM+1-τm∑i=k+jk-1eyTiQ5eyi,V4k=∑i=k-τkk-1gTeyiRgeyi+∑j=-τM+1-τm∑i=k+jk-1gTeyiRgeyi,V5k=∑j=-δM-δm-1∑i=k+jk-1η1TiZ1η1i+∑j=-δm-1∑i=k+jk-1η1TiZ2η1i+∑j=-τM-τm-1∑i=k+jk-1η2TiZ3η2i+∑j=-τm-1∑i=k+jk-1η2TiZ4η2i.For convenience, define the following notations: (31)χ1k=χk,0,δm,χ2k=χk,δk,δM,χ3k=χk,δm,δk,χ4k=χk,0,τm,χ5k=χk,τk,τM,χ6k=χk,τm,τk,ξTk=exTk,exTk-δk,exTk-δm,exTk-δM,eyTk,eyTk-τk,eyTk-τm,eyTk-τM,η1Tk,η2Tk,gTeyk,gTeyk-τk,χ1Tk,χ2Tk,χ3Tk,χ4Tk,χ5Tk,χ6Tk,ϱ2Tk,ϱ4Tk.Calculate the difference of V(k) by defining ΔV(k)=V(k+1)-V(k) along the solutions of (24):(32)ΔV1k=V1k+1-V1k=exTk+1P1exk+1-exTkP1exk+eyTk+1P2eyk+1-eyTkP2eyk=η1k+exkTP1η1k+exk-exTkP1exk+η2k+eykTP2η2k+eyk-eyTkP2eyk=ξTkΞ1ξk,ΔV2k=exTkQ1exk-exTk-δmQ1exk-δm+exTkQ3exk-exTk-δMQ3exk-δM+δ1exTkQ2exk+∑i=k+1-δk+1kexTiQ2exi-∑i=k-δkk-1exTiQ2exi-∑j=-δM+1-δmexTk+jQ2exk+j≤exTkQ1exk+1+δ1exTkQ2exk+exTkQ3exk-exTk-δmQ1exk-δm-exTk-δkQ2exk-δk-exTk-δMQ3exk-δM=ξTkΞ2ξk.In the same way, we obtain(33)ΔV3k≤eyTkQ4eyk+1+τ1eyTkQ5eyk+eyTkQ6eyk-eyTk-τmQ4eyk-τm-eyTk-τkQ5eyk-τk-eyTk-τMQ6eyk-τM=ξTkΞ3ξk,(34)ΔV4k=τ1gTeykRgeyk+∑i=k+1-τk+1kgTeyiRgeyi-∑i=k-τkk-1gTeyiRgeyi-∑j=-τM+1-τMgTeyk+jRgeyk+j≤1+τ1gTeykRgeyk-gTeyk-τkRgeyk-τk=ξTkΞ4ξk,(35)ΔV5k=δ1η1TkZ1η1k-∑i=k-δMk-δm-1η1TiZ1η1i+δmη1TkZ2η1k-∑i=k-δmk-1η1TiZ2η1i+τ1η2TkZ3η2k-∑i=k-τMk-τm-1η2TiZ3η2i+τmη2TkZ4η2k-∑i=k-τmk-1η2TiZ4η2i.By Lemma 4, it follows that(36)-∑i=k-δmk-1η1TiZ2η1i≤-1δmexk-exk-δmexk+exk-δm-χ1kTZ2003Z2exk-exk-δmexk+exk-δm-χ1k=-1δmξTkρ1Z~2ρ1Tξk.Next, it follows from Lemmas 4 and 5 that(37)-∑i=k-δMk-δm-1η1TiZ1η1i=-∑i=k-δMk-δk-1η1TiZ1η1i-∑i=k-δkk-δm-1η1TiZ1η1i≤-1δM-δmξTkρ2Z~1ρ2Tξk-δk-δmδM-δm1δM-δkξTkρ2Z~1ρ2Tξk-1δM-δmξTkρ3Z~1ρ3Tξk-δM-δkδM-δm1δk-δmξTkρ3Z~1ρ3Tξk=-1δM-δmδM-δmδM-δkξTkρ2Z~1ρ2Tξk+δM-δmδk-δmξTkρ3Z~1ρ3Tξk≤-1δM-δmξTkρ7Θ1ρ7Tξk.Based on (35)–(37), we have(38)ΔV5k≤η1Tkδ1Z1+δmZ2η1k+η2Tkτ1Z3+τmZ4η2k-ξTk1δmρ1Z~2ρ1T+1δ1ρ7Θ1ρ7T+1τmρ4Z~4ρ4T+1τ1ρ8Θ2ρ8Tξk=ξTkΞ5ξk.
From Assumption1, for diagonal matrices Λ1, Λ2, and Λ3>0 and a known matrix L=diag⁡{l1,l2,…,ln}>0, it is not difficult to see that the inequalities(39)2gTeykΛ1geyk-2eyTkLΛ1geyk≤0,(40)2gTeyk-τkΛ2geyk-τk-2eyTk-τkLΛ2geyk-τk≤0,(41)2geyk-geyk-τkTΛ3geyk-geyk-τk-2eyk-eyk-τkTLΛ3geyk-eyk-τk≤0hold. In addition, since η1(k)=ex(k+1)-ex(k), η2(k)=ey(k+1)-ey(k), by introducing relaxation matrices N1=diag⁡{N11,N12}, N2=diag⁡{N21,N12}, N3=diag⁡{N31,N32}, and N4=diag⁡{N41,N32} with appropriate dimensions, we obtain(42)2η1TkN1+exTkN2A¯1-I2nexk+B¯1geyk-τk+E¯2ϱ2k-η1k=0,(43)2η2TkN3+eyTkN4C¯1-I2neyk+D¯1exk-δk+E¯2ϱ4k-η2k=0.From FT(k)F(k)≤I, we have(44)ϱiTkϱik≤ζiTkζik,i=2,4.Then, there exist positive scalars ε2 and ε4 satisfying(45)εiξTkΨiTkΨikξk-ϱiTkϱik≥0.Now, combining (32)–(45) and taking mathematical expectation, one can obtain the following inequality:(46)EΔVk≤EξTk∑i=19Ξi+ε2Ψ2TΨ2+ε4Ψ4TΨ4ξk<0.Using the Schur complement Lemma, it can be shown that the considered uncertain system (24) is globally asymptotically stable in mean square sense, which completes the proof of the theorem.Remark 7.
It is worthwhile to mention that two constraint conditions (i.e., (39)-(40)) on feedback regulatory function were used on dealing with the state estimation problem for discrete GRNs with time-varying delays. In this paper, we add inequality (41) with hope to reduce the conservativeness introduced by time delays.Now, we are in a position to deal with the nonfragile estimator design for GRNs (4) with time-varying delays and randomly occurring uncertainties.Theorem 8.
For given positive scalarsδm, δM, τm, and τM, system (8) is said to be a globally robustly asymptotic state estimator of GRNs (4), if there exist scalars εi>0(i=1,2,3,4), matrices P1>0, P2>0, R>0, Zj>0(j=1,2,3,4), and Ql>0(l=1,2,…,6), positive diagonal matrices Λ1, Λ2, and Λ2, matrices Yi=Yi1Yi2∗Yi3(i=1,2), and block diagonal matrices N1, N2, N3, N4, X1, and X3 with appropriate dimensions, such that the following LMIs hold:(47)Σ~=∑i=111Ξiε1Ψ1Tε2Ψ2Tε3Ψ3Tε4Ψ4T∗-ε1I000∗∗-ε2I00∗∗∗-ε3I0∗∗∗∗-ε4I<0,with (48)Ψ1=α1M¯10n×10nβ2M¯20n×10n,Ψ2=γ1M¯50n×21n,Ψ3=0n×nα2M¯40n×2nβ1M¯30n×17n,Ψ4=0n×4nγ2M¯60n×17n,Ξ10=2e9N1+e1N2E¯1e21T+2e10N3+e5N4E¯1e22T,Ξ11=-e21ε1I2ne21T-e22ε3I2ne22T,and the other parameters are defined as in Theorem 6. Moreover, the estimation gain matrices are determined by (49)K1=N12-1X1,K2=N32-1X3.Proof.
The proof of this theorem follows from Theorem6 directly. According to (13), for any matrices N1, N2, N3, and N4, we have(50)2η1TkN1+exTkN2A¯1-I2nexk+B¯1geyk-τk+E¯1ϱ1k+E¯2ϱ2k-η1k=0,(51)2η2TkN3+eyTkN4C¯1-I2neyk+D¯1exk-δk+E¯1ϱ3k+E¯2ϱ4k-η2k=0.Similarly, we get(52)ϱiTkϱik≤ζiTkζik,i=1,2,3,4.Then, for positive scalars εi, we have(53)εiξ~TkΨiTkΨikξ~k-ϱiTkϱik≥0,i=1,2,3,4,with (54)ξ~Tk=exTk,exTk-δk,exTk-δm,exTk-δM,eyTk,eyTk-τk,eyTk-τm,eyTk-τM,η1Tk,η2Tk,gTeyk,gTeyk-τk,χ1Tk,χ2Tk,χ3Tk,χ4Tk,χ5Tk,χ6Tk,ϱ2Tk,ϱ4Tk,ϱ1Tk,ϱ3Tk.Combining the above and taking the mathematical expectation, one has(55)EΔVk≤Eξ~TkΣ~ξ~k.By using the Schur complement Lemma, it is easy to conclude that the proof of this theorem is complete.Remark 9.
In this paper, a nonfragile state estimation algorithm has been presented for discrete-time GRNs with time-varying delays and randomly occurring uncertainties. It is worth mentioning that it is the first time to use the discrete Wirtinger-based inequality and the reciprocally convex inequality to handle the state estimation problem for discrete-time delayed GRNs subject to randomly occurring uncertainties and estimator gain perturbations. In particular, the reciprocally convex approach, the discrete Wirtinger-based inequality, and the free-weighting matrix method are fully utilized in order to reduce the conservativeness of the developed estimation criterion. Besides, by introducing two new zero equations (42) and (43), the free-weighting matrices N1, N2, N3, and N4 have been introduced in the main results with hope to further reduce the possible conservativeness induced by time-varying delays, which could be utilized to deal with related state estimation problems for complex systems with time delays.Remark 10.
It should be noted that the presented method in this paper is based on LMIs. The standard LMI system has a polynomial-time complexity bounded byO(MN), where M is the total row size of the LMI system and N is the total number of scalar decision variables. For example, let us look at the LMIs condition for the addressed GRNs (as described in Theorem 6); we have M=44n and N=2+13n(n+1)/2+2(n(n+1)+n2)+3n+8n2+2n2=(1/2)(41n2+23n+4). Therefore, the computational complexity of the LMI-based state estimation criterion can be represented by O(902n3+506n2+88n).Remark 11.
It is worthwhile to point out that our major focus is on the nonfragile state estimation problem for discrete-time GRNs subject to time-varying delays and randomly occurring uncertainties. Due to such a complicated system, we have decided to carry out research on the feasibility problem with satisfactory performance requirements for addressed GRNs; that is, we made great efforts to propose a new state estimation method against the above-mentioned phenomena. Nevertheless, in case that the convergence precision of the proposed state estimation method becomes a concern, some additional conditions can be provided, which constitutes one of our future research topics.


## 4. Numerical Simulations

In this section, we present two numerical examples to illustrate the effectiveness and correctness of the proposed state estimation scheme.Example 1.
Consider three-node GRNs with the following parameters:(56)A=D=0.3I,C=0.1I,Fk=diag⁡sink,cosk,0,B=0.5×00-1-1010-10,W1=0.30.500.20.10.15,W2=0.40.1-0.5-0.20.10.3,E2=0.20.300.100.200.30,M5=-0.3-0.05-0.40.06-0.10.02,M6=0.10.030.020.10.030.1.The nonlinear genetic regulatory function g(s) is taken to be as usual Hill function. That is, gs=s2/1+s2. Then, it is easy to see that (57)L=diag⁡0.65,0.65,0.65.This means that Assumption 1 holds. Moreover, assume δm=1, δM=5, τm=2, τM=4, γ1=0.3, and γ2=0.7. By using the LMIs control Toolbox in Matlab to solve the LMIs in Theorem 6, we can obtain feasible solutions. Due to space consideration, only the following few parameters are listed: ε2=1.1713; ε4=1.0898; the nonfragile state estimator can be designed with the estimator gain matrices described by(58)K1=0.0968-0.1305-0.13210.27280.0909-0.2096,K2=-0.21670.12960.50650.8443-0.07590.1388.
The numerical simulation affirms our theoretical results. Figure1 plots the trajectories of mRNA concentrations, its estimates, and the estimation errors with initial conditions ϕ(k)=[0.20.20.4]T and ϕ^(k)=[0.80.50.9]T. Figure 2 depicts the trajectories of protein concentrations, its estimates, and the estimation errors with initial conditions ψ(k)=[0.10.50.3]T and ψ^(k)=[0.50.80.3]T. It follows from Figures 1 and 2 that the estimation error between the original GRNs and its estimation approaches zero asymptotically. In addition, in order to illustrate the advantage of the main results, we can set E2=0. Then, error system (24) is equivalent to system (6) in [27]. Subsequently, it is not difficult to check that when the feedback regulation delay τ(k) and the translation delay δ(k) satisfy 2≤τ(k)≤5 and 2≤δ(k)≤9, respectively, LMI (25) in Theorem 6 of our paper is feasible while LMI condition (7) in [27] is infeasible. Therefore, it is concluded that our results are less conservative than those in [27]. The major reason is that we have made additional efforts to propose new results with more information of time delays by taking the advantage of the discrete-time Wirtinger-based inequality, the reciprocally convex combination approach, and the free-weight matrix technique. On the other hand, it should be mentioned that, compared to [27] without consideration of estimator gain perturbations, the developed state estimation approach is more applicable.Figure 1
mRNA concentrations, its estimates, and the estimation error of Example1.Figure 2
Protein concentrations, its estimates, and the estimation error of Example1.Example 2.
To demonstrate the effectiveness of Theorem8, we consider the following parameters as in Example 1, with(59)E1=0.200.10.2-0.1000.20.3,E2=0.10-0.20.20.100.100.3,M1=0.20.05-0.30.2-0.100.20.30.6,M2=0.10.300.20.10.20.100.3,M3=-0.10-0.0600.10.20.30.2-0.2,M4=0.20.10.2-0.50.20.1-0.30.20,M5=-0.10.020.050.10.10.02,M6=0.20.030.30.1-0.10.Let the delaysδ(k) and τ(k) vary randomly within the intervals [2,5]. The mathematical expectations of Bernoulli-distributed white sequences can be taken as α1=0.5, α2=0.4, β1=0.3, β2=0.2, γ1=0.4, and γ2=0.3. By solving the LMIs in Theorem 8 with the help of Matlab LMI toolbox, we can obtain feasible solutions and scalars ε1=0.1385, ε2=0.6089, ε3=0.0910, and ε4=0.5866. According to Theorem 8, the nonfragile state estimator with randomly occurring uncertainties can be designed with the estimator gain matrices defined as follows:(60)K1=0.1198-0.2160-0.27080.40290.1412-0.3462,K2=0.08420.07780.32870.57490.01430.1117.The simulation results are shown in Figures 3 and 4. The actual state responses x(k) and y(k), their estimates x^(k) and y^(k), and the estimation error are depicted in Figures 3 and 4. The initial conditions are taken as ϕ(k)=[0.50.30.6]T, ϕ^(k)=[0.30.80.2]T, ψ(k)=[0.70.30.4]T, and ψ^(k)=[0.20.90.8]T. It is also observed from Figures 3 and 4 that the estimation error between the original GRNs and the estimator system approaches zero asymptotically. It should be noticed that when we do not consider the nonfragility of estimator gain perturbations, the admissibility of upper bounds of time delays (under different lower bounds of time delays) guaranteeing the globally asymptotical stability of the resulting estimation error system for the addressed GRNs can be obtained and listed in Table 1 by using Theorem2 in [27] and Theorem 8 in this paper, respectively, which further illustrate the advantages and less conservatism of the newly proposed result.Table 1
Admissible upper boundδM with different δm, when τm=2 and τM=5.
δ
m
2
4
6
8
10
[[27], Theorem2]
6
8
10
12
14
Theorem8
7
9
11
13
15Figure 3
mRNA concentrations, its estimates, and the estimation error of Example2.Figure 4
Protein concentrations, its estimates, and the estimation error of Example2.


## 5. Conclusion

In this paper, we have investigated the nonfragile state estimator design for a class of the discrete-time GRNs with interval time-varying delays and randomly occurring uncertainties. The norm-bounded uncertainties enter into the GRNs in random ways, and such randomly occurring uncertainties have been characterized by certain mutually uncorrelated Bernoulli-distributed white noise sequences. Under these circumstances, the state estimator has been designed to estimate the true concentration of the mRNA and the protein from available measurements. By constructing a Lyapunov-Krasovskii functional, new delay-dependent estimation criterion has been obtained in terms of LMIs by using the discrete-time Wirtinger-based inequality and the reciprocally convex approach. A new constraint condition on feedback regulatory function has been used with hope to reduce the conservativeness of the estimation criterion. Then, the desired state estimation algorithm has been provided in view of the solutions to LMIs, which can ensure that the estimation error dynamics are globally asymptotically stable. Finally, numerical simulations have been given to illustrate the applicability and usefulness of the developed theoretical results. It is worth mentioning that the proposed method in this paper can be extended to address the nonfragile state estimation problems for Markovian jump systems as in [40, 41] with different types of time delays and time-varying systems in [42–44] with unreliable measurements, which constitute interesting topic for future research.



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