# H

Authors: Yantao Wang; Xingming Zhou; Xian Zhang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

This paper is concerned with theH∞ filtering problem for a class of discretetime genetic regulatory networks with random delay and external disturbance. The aim is to design H∞ filter to estimate the true concentrations of mRNAs and proteins based on available measurement data. By introducing an appropriate Lyapunov function, a sufficient condition is derived in terms of linear matrix inequalities (LMIs) which makes the filtering error system stochastically stable with a prescribed H∞ disturbance attenuation level. The filter gains are given by solving the LMIs. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed approach; that is, our approach is available for a smaller H∞ disturbance attenuation level than one in (Liu et al., 2012).


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## Body

## 1. Introduction

Genetic regulatory networks (GRNs) are collections of DNA segments in a cell which interact with each other indirectly through their mRNAs, protein expression products, and other substances. Understanding the nature and functions of various GRNs is very interesting and crucially important for the treatment of many diseases such as cancers [1, 2]. Therefore, in the past decade, the study on GRNs has been put more emphasis by the researchers at interdisciplinary field. Mathematical modeling of GRNs provides a powerful tool for studying gene regulation processes. In general, genetic network models can be classified into two types, that is, the discrete model [3, 4] and the continuous model [5–8]. Usually, a continuous model is described by a (functional) differential equation. Due to slow biochemical reactions such as gene transcription and translation, time delays can play an important role in GRNs, which results that the (functional) differential equation model has been one of the most fashionable GRN models, and a lot of research on analysis and synthesis of GRNs have been recently done based on (functional) differential equation models (see, e.g., [9–15]).The concentrations of gene products, such as mRNAs and proteins, are described as system states in a (functional) differential equation model. In practice, biologists hope to gain actual concentrations of gene products in GRNs. However, due to model errors, external perturbation, time delays, and parameters jump, the steady-state values of GRNs can hardly be obtained. In order to obtain the steady-state values through available measurement data, the design of filter and estimator for (functional) differential equation models of GRNs has been investigated by some scholars (see, e.g., [16–23]). However, due to the requirement for implementing and application of GRNs for computer-based simulation, it is of vital importance to design filter or estimator for delayed discrete-time GRNs (i.e., discretized (functional) differential equation models of GRNs) in today’s digital world, although there are, to the best author’s knowledge, only three results reported at present [24–26]. Zhang et al. [25] is concerned with the set-values filtering for a class of discrete-time GRNs with time-varying parameters, constant time-delay, and bounded external noise. For a class of discrete-time GRNs with random delays described by a Markov chain, Liu et al. [26] designed a filter ensuring that the filtering error system is stochastically stable and has a prescribed H∞ performance. By utilizing the Lyapunov stability theory and stochastic analysis technique, Wang et al. [24] investigated the existing conditions and explicit expressions of H∞ state estimators for a class of stochastic discrete-time GRNs with probabilistic measurement delays described by Bernoulli distributed white sequences. These conditions are given in terms of LMIs and are dependent on the lower and upper bounds of the time-varying delays.It should also be emphasized that for delayed discrete-time GRNs, the stability problem (as the most important properties for any dynamics systems) [27–29], H∞ stabilization problem [30], and passivity problem [31] have been exploited. On the other hand, researchers have been paying attention to the problems of analysis and synthesis for Markovian jump system [32–36] and the filtering problems for some nonlinear systems [37–41].Motivated by the above discussion, in this paper, we will deal with theH∞ filtering problem for a class of discrete-time GRNs with random delay which is described by a Markovian chain. By constructing a novel Lyapunov function different from one in [26], a sufficient LMI condition is first established to ensure the existence of the desired filter. The condition is dependent on the transition probability matrix of the random delay. Then, the explicit expression of the desired filter is shown to ensure the resulting filtering error system to be stochastically stable and have a prescribed H∞ disturbance attenuation level. Moreover, an optimization problem with LMIs constraints is established to design an H∞ filter which ensures an optimal H∞ disturbance attenuation level. Finally, a numerical example is given to show the effectiveness of the proposed approach.


## 2. Problem Formulation

Consider the following discrete-time GRN with random delays,n mRNAs, and n proteins [27, 28]:
(1)Mi(k+1)=e-aihMi(k)+ϕi(h)×[∑j=1n‍bijfj(Pj(k-d(k)))+Vi],Pi(k+1)=e-cihPi(k)+φi(h)diMi(k-d(k)),hhhhhhhhhhhhhhhhi=1,2,…,n,
where Mi(k) and Pi(k), respectively, are the concentrations of mRNA and protein of the ith gene; ϕi(h)=(1-e-aih)/ai>0 and φi(h)=(1-e-cih)/ci>0, where h is a given positive real number standing for the uniform discretionary step size; d(k) denotes the random time delay of mRNAs and proteins, and is assumed to be a Markovian chain with state space 𝒩:={1,2,…,d}, and d is a fixed positive integer; ai>0 and ci>0 are the degradation rates of mRNA and protein, respectively; di is the translation rate; Vi=∑j∈Iivij, where vij is a bounded constant denoting the dimensionless transcriptional rate of gene j to i, and Ii is the set of all the repressors of ith gene; bij(i,j=1,2,…,n) are the coupling coefficients satisfying
(2)bij={vij,iftranscriptionfactorjisanactivatorofgenei,0,ifthereisnolinkfromlinknodejtoi,-vij,iftranscriptionfactorjisarepressorofgenei;
the nonlinear function fj(j=1,2,…,n) denotes the feedback regulation of protein in process of transcription. In general, fj is a monotonic function in Hill form; namely, fj(s)=shj/(1+shj)(j=1,2,…,n), where hj is the Hill coefficient. Denote by π:=[πij]n×n the transition probability matrix of d(k), where πij=Prob{d(k+1)=j∣d(k)=i}.Let us rewrite GRN (1) as the following compact matrix form:
(3)M(k+1)=AM(k)+Bf(P(k-d(k)))+V,P(k+1)=CP(k)+DM(k-d(k)),
where
(4)M(k)=[M1(k)M2(k)⋯Mn(k)]T,P(k)=[P1(k)P2(k)⋯Pn(k)]T,f(P(k-d(k)))=[f1(P1(k-d(k)))f2(P2(k-d(k)))⋯fn(Pn(k-d(k)))]T,V=[ϕ1(h)V1ϕ2(h)V2⋯ϕn(h)Vn]T,A=diag⁡(e-a1h,e-a2h,…,e-anh),C=diag⁡(e-c1h,e-c2h,…,e-cnh),D=diag⁡(φ1(h)d1,φ2(h)d2,…,φn(h)dn),B=[ϕi(h)bij]n×n(i=1,2,…,n).Let(M*,P*) be an equilibrium point of GRN (3), where M*=[M1*⋯Mn*]T and P*=[P1*⋯Pn*]T; that is,
(5)M*=AM*+Bf(P*)+V,P*=CP*+DM*.
To simplify the analysis, one can transform the equilibrium point to the origin by the relation xm(k)=M(k)-M* and xp(k)=P(k)-P*. Then the transformed system is changed as follows:
(6)xm(k+1)=Axm(k)+Bg(xp(k-d(k))),xp(k+1)=Cxp(k)+Dxm(k-d(k)),
where g(xp(k))=f(xp(k)+P*)-f(P*). For every i=1,2,…,n, since fi is a monotonic function in Hill form, one can easily obtain that gi is a monotonically increasing function with saturation and satisfies the following inequality:
(7)gi(0)=0,0≤gi(s1)-gi(s2)s1-s2≤li,∀s1,s2∈R,s1≠s2,
where li is a given constant.When we take extracellular perturbations into account, a class of stochastic discrete-time GRN model with random delays is represented as follows:(8)xm(k+1)=Axm(k)+Bg(xp(k-d(k)))+E1w(k),xp(k+1)=Cxp(k)+Dxm(k-d(k))+F1v(k),ym(k)=C1xm(k)+E2w(k),yp(k)=C2xp(k)+F2v(k),zm(k)=G1xm(k),zp(k)=G2xp(k),xm(k)=θm(k),xp(k)=θp(k),hhhhhhlhhhk=-d,-d+1,…,0,
where A,B, C, D, C1, C2, E1, E2, F1, F2, G1, and G2 are constant matrices of appropriate dimension; ym(k):=[ym1(k)⋯ymn(k)]T and yp(k):=[yp1(k)⋯ypn(k)]T denote the expression levels of mRNA and protein, respectively; zm(k):=[zm1(k)⋯zml(k)]T and zp(k):=[zp1(k)⋯zpl(k)]T are the estimated signals; both w(k) and v(k) are exogenous disturbance signals; and θm(k) and θp(k) are the initial conditions of xm(k) and xp(k), respectively.In complex GRNs, only the partial information of the network components can be usually obtained. Therefore, in order to obtain the states of GRNs, we need to estimate them via available measurements [42]. The full order linear filter which need to be designed as the following form:
(9)x^m(k+1)=Afx^m(k)+Bfym(k),x^p(k+1)=Cfx^p(k)+Dfyp(k),z^m(k)=G1fx^m(k)+H1fym(k),z^p(k)=G2fx^p(k)+H2fyp(k),
where x^m(k), x^p(k), z^m(k), and z^p(k) are the estimates of xm(k), xp(k), zm(k), and zp(k), respectively; Af, Bf, Cf, Df∈Rn×n and G1f, G2f, H1f, H2f∈Rl×n are filter parametric matrices to be determined.Set(10)x~m(k)=[xm(k)x^m(k)],x~p(k)=[xp(k)x^p(k)],em(k)=zm(k)-z^m(k),ep(k)=zp(k)-z^p(k).
Then the filtering error system can be expressed as
(11)x~m(k+1)=A-x~m(k)+B-g(Z1x~p(k-d(k)))+E-w(k),x~p(k+1)=C-x~p(k)+D-Z1x~m(k-d(k))+F-v(k),em(k)=G-1fx~m(k)+H-1fw(k),ep(k)=G-2fx~p(k)+H-2fv(k),x~m(k)=θ~m(k),x~p(k)=θ~p(k),hhhhhhhhhhhlk=-d,-d+1,…,0,
where
(12)θ~m(k)=[θm(k)0],θ~p(k)=[θp(k)0],A-=[A0BfC1Af],B-=[B0],C-=[C0DfC2Cf],D-=[D0],E-=[E1BfE2],F-=[F1DfF2],G-1f=[G1-H1fC1-G1f],G-2f=[G2-H2fC2-G2f],H-1f=-H1fE2,H-2f=-H2fF2,Z1=[I0].
For convenience, for a nonnegative integer k we define
(13)Θk={x~m(k),x~m(k-1),…,x~m(k-d),x~p(k),x~p(k-1),…,x~p(k-d)}.Definition 1 (see [26]).
The delayd(k) is said to be the random delay described by a Markovian chain if it is bound by 1≤d(k)≤d, and {d(k)∈𝒩,k=0,1,2,…} is a Markovian chain with state space 𝒩 and transition probability matrix π.Definition 2 (see [26]).
Whenw(k)=0 and v(k)=0, the filtering error system (11) is said to be stochastically stable, if
(14)∑k=0∞‍E{∥x~m(k)∥2+∥x~p(k)∥2∣Θ0,d(0)}<∞
for every initial condition Θ0 and initial mode d(0), where E{·} represents the mathematical expectation operator.Definition 3.
For a given constantγ>0, the filtering error system (11) is said to be stochastically stable with H∞ disturbance attenuation level γ if it is stochastically stable with w(k)=0 and v(k)=0, and under the zero initial conditions it satisfies the following inequality:
(15)∑k=0∞‍E{[em(k)ep(k)]T[em(k)ep(k)]∣Θ0,d(0)}<γ2∑k=0∞‍[w(k)v(k)]T[w(k)v(k)]
for all nonzero w(k), v(k)∈l2[0,+∞), and initial mode d(0).The objective of this paper is to design a filter of form (9) such that the filtering error system (11) is stochastically stable with H∞ disturbance attenuation level γ. In order to realize the aim, we first introduce the following lemma.Lemma 4 (see [43]).
For symmetric matricesP>0 and Q>0, the matrix inequality
(16)[-P-1A*-Q]<0
holds, if and only if there is a matrix R such that
(17)[P-R-RTRTA*-Q]<0.


## 3. Stability Analysis andH∞ Filter Design

The stability analysis for the filtering error system (11) with w(k)=0 and v(k)=0 is presented by the following theorem.Theorem 5.
The filtering error system (11) with w(k)=0 and v(k)=0 is stochastically stable, if there exist matrices ς:=diag⁡(ς1,ς2,…,ςn)>0, μ:=diag⁡(μ1,μ2,…,μn)>0, PiT(r)=Pi(r)>0(i=1,2,…,6;r=1,2,…,d), and PjT=Pj>0(j=2,3,5,6) such that the following matrix inequalities (18) and (19) hold for all r∈𝒩:
(18)Ω:=Ω~+Ω^<0,(19)P-j(r)<Pj,j=2,3,5,6,
where
(20)Ω^=Λ1TP-1(r)Λ1+Λ2T(dP-3(r)+d2+d2P3)Λ2+Λ3TP-4(r)Λ3,Λ1=[A-00B-00],Λ2=[A--I00B-00],Λ3=[0C-D-Z1000],Ω~=[Ω110Ω13000*-P4(r)00-Z1TςL-Z1TCTμL**Ω3300-Z1TDTμL***Ω44Ω450****Ω55Ω56*****Ω66],Ω11=(d-1)P2+P-2(r)-P1(r)-Ω13,Ω13=1rP3(r)+1rP3,Ω33=-P2(r)-Ω13,Ω44=-P5(r)-Ω45,Ω45=1rP6(r)+1rP6,Ω55=(d-1)P5+P-5(r)-Ω56-Ω45-ς,Ω56=-dP-6(r)-(d2+d)P62,Ω66=-Ω56-μ,L=diag⁡(-l12,-l22,…,-ln2),P-i(r)=∑s=1d‍πrsPi(s),i=1,2,…,6.Proof.
Choose an appropriate Lyapunov functionV(Θk,k,d(k)) for the filtering error system (11) with w(k)=0 and v(k)=0 as follows:
(21)V(Θk,k,d(k))=∑i=13‍(Vm,i(Θk,k,d(k))+Vp,i(Θk,k,d(k)))
with
(22)Vm,1(Θk,k,d(k))=x~mT(k)P1(d(k))x~m(k),Vp,1(Θk,k,d(k))=x~pT(k)P4(d(k))x~p(k),Vm,2(Θk,k,d(k))=∑i=k-d(k)k-1‍x~mT(i)P2(d(k))x~m(i)+∑j=-d+1-1‍∑i=k+jk-1‍x~mT(i)P2x~m(i),Vp,2(Θk,k,d(k))=∑i=k-d(k)k-1‍gT(Z1x~p(i))P5(d(k))g(Z1x~p(i))+∑j=-d+1-1‍∑i=k+jk-1‍gT(Z1x~p(i))P5g(Z1x~p(i)),Vm,3(Θk,k,d(k))=∑j=-d(k)-1‍∑i=k+jk-1‍ηT(i)P3(d(k))η(i)+∑j=-d-1‍∑l=j-1‍∑i=k+lk-1‍ηT(i)P3η(i),Vp,3(Θk,k,d(k))=∑j=-d(k)-1‍∑i=k+jk-1‍ζT(i)P6(d(k))ζ(i)+∑j=-d-1‍∑l=j-1‍∑i=k+lk-1‍ζT(i)P6ζ(i),
where η(k)=x~m(k+1)-x~m(k) and ζ(k)=g(Z1x~p(k+1))-g(Z1x~p(k)). By taking the forward difference of the function Vm,1(Θk,k,d(k)) along with the solution of system (11), one can obtain that
(23)E{Vm,1(Θk+1,k+1,d(k+1))∣Θk,d(k)=r}-Vm,1(Θk,k,r)=∑s=1d‍πrsx~mT(k+1)P1(s)x~m(k+1)-x~mT(k)P1(r)x~m(k)=x~mT(k+1)P-1(r)x~m(k+1)-x~mT(k)P1(r)x~m(k).
Additionally, it can be verified that
(24)E{Vm,2(Θk+1,k+1,d(k+1))∣Θk,d(k)=r}-Vm,2(Θk,k,r)=∑s=1d‍πrs∑i=k+1-sk‍x~mT(i)P2(s)x~m(i)-∑i=k-rk-1‍x~mT(i)P2(r)x~m(i)+∑j=-d+1-1‍∑i=k+1+jk‍x~mT(i)P2x~m(i)-∑j=-d+1-1‍∑i=k+jk-1‍x~mT(i)P2x~m(i)=x~mT(k)P-2(r)x~m(k)-x~mT(k-r)P2(r)x~m(k-r)+∑i=k+1-sk-1‍x~mT(i)P-2(r)x~m(i)-∑i=k+1-rk-1‍x~mT(i)P2(r)x~m(i)+∑j=-d+1-1‍x~mT(k)P2x~m(k)-∑j=k+1-dk-1‍x~mT(j)P2x~m(j)≤x~mT(k)P-2(r)x~m(k)-x~mT(k-r)P2(r)x~m(k-r)+(d-1)x~mT(k)P2x~m(k)+∑i=k+1-dk-1‍x~mT(i)P-2(r)x~m(i)-∑i=k+1-dk-1‍x~mT(i)P2x~m(i)≤x~mT(k)[(d-1)P2+P-2(r)]x~mT(k)-x~mT(k-r)P2(r)x~mT(k-r),E{Vm,3(Θk+1,k+1,d(k+1))∣Θk,d(k)=r}-Vm,3(Θk,k,r)=∑s=1d‍πrs∑j=-s-1‍∑i=k+1+jk‍ηT(i)P3(s)η(i)-∑j=-r-1‍∑i=k+jk-1‍ηT(i)P3(r)η(i)+∑j=-d-1‍∑l=j-1‍[∑i=k+1+lk‍ηT(i)P3η(i)-∑i=k+lk-1‍ηT(i)P3η(i)]≤∑j=-d-1‍ηT(k)P-3(r)η(k)-∑j=-r-1‍ηT(k+j)P3(r)η(k+j)+∑j=-d-1‍∑i=k+1+jk-1‍ηT(i)P-3(r)η(i)+d2+d2ηT(k)P3η(k)-∑j=-d-1‍∑i=k+1+jk-1‍ηT(i)P3η(i)-∑j=-d-1‍ηT(k+j)P3η(k+j)≤ηT(k)[dP-3(r)+d2+d2P3]η(k)-∑j=-r-1‍ηT(k+j)1rP3(r)∑j=-r-1‍η(k+j)+∑j=-d-1‍∑i=k+1+jk-1‍ηT(i)P-3(r)η(i)-∑j=-d-1‍∑i=k+1+jk-1‍ηT(i)P3η(i)-∑j=-r-1‍ηT(k+j)1rP3∑j=-r-1‍η(k+j)≤ηT(k)[dP-3(r)+d2+d2P3]η(k)-∑j=-r-1‍ηT(k+j)1r(P3(r)+P3)∑j=-r-1‍η(k+j).
Similarly, the following inequalities (25) can be derived:
(25)E{Vp,1(Θk+1,k+1,d(k+1))∣Θk,d(k)=r}-Vp,1(Θk,k,r)=∑s=1d‍πrsx~pT(k+1)P4(s)x~p(k+1)-x~pT(k)P4(r)x~p(k)=x~pT(k+1)P-4(r)x~p(k+1)-x~pT(k)P4(r)x~p(k),E{Vp,2(Θk+1,k+1,d(k+1))∣Θk,d(k)=r}-Vp,2(Θk,k,r)=∑s=1d‍πrs∑i=k+1-sk‍gT(Z1x~p(i))P5(s)g(Z1x~p(i))-∑i=k-rk-1‍gT(Z1x~p(i))P5(r)g(Z1x~p(i))+∑j=-d+1-1‍∑i=k+1+jk‍gT(Z1x~p(i))P5g(Z1x~p(i))-∑j=-d+1-1‍∑i=k+jk-1‍gT(Z1x~p(i))P5g(Z1x~p(i))=gT(Z1x~p(k))P-5(r)g(Z1x~p(k))-gT(Z1x~p(k-r))P5(r)g(Z1x~p(k-r))+∑s=1d‍πrs∑i=k+1-sk-1‍gT(Z1x~p(i))P5(s)g(Z1x~p(i))-∑i=k+1-rk-1‍gT(Z1x~p(i))P5(r)g(Z1x~p(i))+∑j=-d+1-1‍[gT(Z1x~p(k))P5g(Z1x~p(k))-gT(Z1x~p(k+j))P5g(Z1x~p(k+j))]≤gT(Z1x~p(k))[(d-1)P5+P-5(r)]g(Z1x~p(k))-gT(Z1x~p(k-r))P5(r)g(Z1x~p(k-r))+∑i=k+1-dk-1‍gT(Z1x~p(i))P-5(r)g(Z1x~p(i))-∑i=k+1-dk-1‍gT(Z1x~p(i))P5g(Z1x~p(i))≤gT(Z1x~p(k))[(d-1)P5+P-5(r)]g(Z1x~p(k))-gT(Z1x~p(k-r))P5(r)g(Z1x~p(k-r)),E{Vp,3(Θk+1,k+1,d(k+1))∣Θk,d(k)=r}-Vp,3(Θk,k,r)=∑s=1d‍πrs∑j=-s-1‍∑i=k+1+jk‍ζT(i)P6(s)ζ(i)-∑j=-r-1‍∑i=k+jk-1‍ζT(i)P6(r)ζ(i)+∑j=-d-1‍∑l=j-1‍[∑i=k+1+lk‍ζT(i)P6ζ(i)-∑i=k+lk-1‍ζT(i)P6ζ(i)]≤∑s=1d‍πrs∑j=-d-1‍ζT(k)P6(s)ζ(k)-∑j=-r-1‍ζT(k+j)P6(r)ζ(k+j)+∑s=1d‍πrs∑j=-d-1‍∑i=k+1+jk-1‍ζT(i)P6(s)ζ(i)-∑j=-r-1‍∑i=k+1+jk-1‍ζT(i)P6(r)ζ(i)+d2+d2ζT(k)P6ζ(k)-∑j=-d-1‍∑i=k+1+jk-1‍ζT(i)P6ζ(i)-∑j=-d-1‍ζT(k+j)P6ζ(k+j)≤ζT(k)Ω56ζ(k)-∑j=-r-1‍ζT(k+j)Ω45∑j=-r-1‍ζ(k+j).
In view of (7), we can conclude that
(26)gi(s)[gi(s)-lis]≤0,∀s∈R,i=1,2,…,n.
Then, it follows from (26) that
(27)-gT(Z1x~p(k))ςg(Z1x~p(k))-2x~pT(k)Z1TςLg(Z1x~p(k))≥0,-gT(Z1x~p(k+1))μg(Z1x~p(k+1))-2x~pT(k+1)Z1TμLg(Z1x~p(k+1))≥0.
Now, combining (23)–(25) and (27) results in
(28)E{V(Θk+1,k+1,d(k+1))∣Θk,d(k)=r}-V(Θk,k,r)≤ξT(k)Ωξ(k),
where ξT(k)=[x~mT(k)x~pT(k)x~mT(k-r)gT(Z1x~p(k-r))gT(Z1x~p(k))gT(Z1x~p(k+1))], and Ω is defined as in (18).
Due to (18), formula (28) results in
(29)E{V(Θk+1,k+1,d(k+1))∣Θk,d(k)=r}≤V(Θk,k,r)-λmin⁡{x~mT(k)x~m(k)+x~pT(k)x~p(k)},
where λmin⁡ denotes the minimal eigenvalue of -Ω. Since
(30)E{E{V(Θk+1,k+1,d(k+1))∣Θk,d(k)}∣Θ0,d(0)}=E{V(Θk+1,k+1,d(k+1))∣Θ0,d(0)},
we obtain
(31)E{∥x~m(k)∥2+∥x~p(k)∥2∣Θ0,d(0)}≤λmin⁡-1V(Θ0,0,d(0))<∞.
by taking the conditional expectation E{·∣Θ0,d(0)} and summing from k=0 to +∞ on both sides of (29). Consequently, by Definition 2, one can conclude from the above inequality that the filtering error system (11) is stochastically stable, and the proof is thus completed.Remark 6.
It is worth noting that theH∞ filtering problem for (8) has been studied in [26], but the obtained results in [26] are not dependent on the transition probability matrix of the random delay described by a Markovian chain. In order to reduce the conservatism and give the explicit expression of the desired filter, in the above theorem we have constituted intensive studying of the H∞ filtering problem for (8) and have investigated a result dependent on the transition probability matrix of the random delay described by a Markovian chain.Remark 7.
The novel Lyapunov functional in this paper is selected to be of (21). Since in (21) we have not only chosen the triple summation term but also considered sufficiently the information of the random delay described by a Markovian chain, the conservatism might be reduced than one in [26], which will be illustrated through a numerical example in Section 4.Theorem5 does not give a design procedure for the desired filter. Based on Theorem 5, the following theorem offers an approach to design a H∞ filter for GRN (8) such that the filtering error system (11) is stochastically stable with H∞ disturbance attenuation level γ.Theorem 8.
For given a scalarγ>0 and a positive integer d, if for each r∈𝒩, there exist matrices PiT(r)=Pi(r)>0(i=1,2,…,6), PjT=Pj>0(j=2,3,5,6),
(32)Rk:=[Rk1Rk2Rk3Rk2]T,det⁡Rk2≠0,k=1,2,ς:=diag⁡(ς1,ς2,…,ςn)>0, μ:=diag⁡(μ1,μ2,…,μn)>0, A-f, B-f, C-f, D-f, G1f, H1f, G2f, and H2f, such that the following LMIs (34) and (35) hold, then the filtering error system (11) is stochastically stable with H∞ disturbance attenuation level γ. Moreover, the required filter is given by (9) with
(33)Af=R12-1A-f,Bf=R12-1B-f,Cf=R22-1C-f,Df=R22-1D-f,(34)Υ:=[Υ110000Υ16*Υ22000Υ26**Υ3300Υ36***-I0Υ46****-IΥ56*****Υ66]<0,(35)P-j(r)<Pj,j=2,3,5,6,
where
(36)Υ11=P-1(r)-R1-R1T,Υ22=dP-3(r)+d2+d2P3-R1-R1T,Υ33=P-4(r)-R2-R2T,P-i(r)=∑s=1d‍πrsPi(s),i=1,2,…,6,Υ16=R1TΨ1+(Z1+Z2)T(B-fΨ2+A-fΨ3),Υ26=R1TΨ4+(Z1+Z2)T(B-fΨ2+A-fΨ3),Υ36=R2TΨ5+(Z1+Z2)T(D-fΨ6+C-fΨ7),Z1=[I0],Z2=[0I],Ψ1=[AZ100B00E1000000000],Ψ2=[C1Z100000E20],Ψ3=[Z20000000],Ψ4=[(A-I)Z100B00E10-Z20000000],Ψ5=[0CZ1DZ10000F100000000],Ψ6=[0C2Z100000F2],Ψ7=[0Z2000000],Υ46=[G-1f00000H-1f0],Υ56=[0G-2f00000H-2f],Υ66=[Ω~0Φ2*-γ2I0**-γ2I],Φ2=[00000-F1TμL]T,
and L, G-1f, G-2f, H-1f, and H-2f are defined as previously.Proof.
A
f, Bf, Cf, and Df are defined as in (33). Then it is easy to verify that Υ16=R1TΛ-1, Υ26=R1TΛ-2, and Υ36=R2TΛ-3, where
(37)Λ-1=[Λ1E-0],Λ-2=[Λ2E-0],Λ-3=[Λ30F-],
and Λ1, Λ2, Λ3, E-, and F- are defined as previously. This, together with (34) and Lemma 4, implies that(38)[-P-1-1(r)0000Λ-1*-(dP-3(r)+d2+d2P3)-1000Λ-2**-P-4-1(r)00Λ-3***-I0Υ46****-IΥ56*****Υ66]<0.Due to the Schur complement lemma, inequality (38) is equal to
(39)Φ+Φ-<0,
where
(40)Φ-=[0000-γ2I000-γ2I]+Υ46TΥ46+Υ56TΥ56,Φ=[Ω~0Φ2000Φ2T00]+Λ-1TP-1(r)Λ-1+Λ-2T(dP-3(r)+d2+d2P3)Λ-2+Λ-3TP-4(r)Λ-3.
Thus
(41)Λ:=Υ66+Λ-1TP-1(r)Λ-1+Λ-2T(dP-3(r)+d2+d2P3)Λ-2+Λ-3TP-4(r)Λ-3<0.
Noting that Ω is a submatrix of Λ, we can conclude that Ω<0. By Theorem 5, the filtering error system (11) with w(k)=0 and v(k)=0 is stochastically stable.
Choose the same Lyapunov function as in (21) for the filtering error system (11) and employ the similar approach in the proof of Theorem 5, one has
(42)ΔVk≔E{V(Θk+1,k+1,d(k+1))∣Θk,d(k)=r}-V(Θk,k,r)≤E{δT(k)Φδ(k)},
where δ(k)=[ξT(k)wT(k)vT(k)]T, and ξ(k) is defined as previously. To deal with the H∞ performance, the following performance function is considered
(43)JK:=∑k=0K‍E{[em(k)ep(k)]T[em(k)ep(k)]-γ2[w(k)v(k)]T[w(k)v(k)]∣Θ0,d(0)}.
Due to the zero initial condition and
(44)E{E{V(Θk+1,k+1,d(k+1))∣Θk,d(k)}∣Θ0,d(0)}=E{V(Θk+1,k+1,d(k+1))∣Θ0,d(0)},
it is easy to see from (39) and (42) that
(45)JK=∑k=0K‍E{[em(k)ep(k)]T[em(k)ep(k)]-γ2[w(k)v(k)]T[w(k)v(k)]+ΔVk∣Θ0,d(0)}-∑k=0K‍E{ΔVk∣Θ0,d(0)}=∑k=0K‍E{[em(k)ep(k)]T[em(k)ep(k)]-γ2[w(k)v(k)]T[w(k)v(k)]+ΔVk∣Θ0,d(0)}-E{V(ΘK+1,K+1,d(K+1))∣Θ0,d(0)}+V(Θ0,0,d(0))≤∑k=0K‍E{emT(k)em(k)+epT(k)ep(k)-γ2wT(k)w(k)-γ2vT(k)v(k)+ΔVk∣Θ0,d(0)}≤∑k=0K‍E{δT(k)(Φ+Φ-)δ(k)∣Θ0,d(0)}<0.
Let k→∞; it is concluded from Definition 3 that the filtering error system (11) is stochastically stable with H∞ disturbance attenuation level γ.
The proof is thus completed.Remark 9.
What can be seen from Theorem8 is that the scalar γ can be calculated as an optimization variable to obtain the minimum H∞ disturbance attenuation level. To be more specific, the minimal H∞ disturbance attenuation level can be obtained by solving the following convex optimization problem:
(46)min⁡s.t.(34)-(35)β,β=γ2.
Note that if there exists a solution β* to the problem (46), then the minimal H∞ disturbance attenuation level is β*.


## 4. Illustrative Example

In this section we illustrate the effectiveness of the proposed approach by testing the following numerical example which has been used in [26].Consider GRN (8) with the following parameters:
(47)A=[0.36790000.36790000.3679],B=[00-0.126-0.126000-0.1260],E1=[0.30.50],F1=[0.60.40.2],C=[0.36790000.60650000.3679],D=[0.63210000.39350000.6321],E2=[0.50.40.2],F2=[0.20.60.3],G2=G1=C2=C1=[0.30000.20000.3].
The regulation function is taken as gi(x)=x2/(1+x2)(i=1,2,3). It is easy to know that the derivative of gi(x) is less than l=0.65, which shows L=diag⁡(-0.325,-0.325,-0.325). Suppose the bound of the time delay is d=3: then d(k)∈𝒩={1,2,3}. The transition probability matrix Π is given by
(48)Π=[0.30.50.20.40.30.30.20.50.3].
By solving the optimization problem (46), it can be obtained that the optimal disturbance attenuation level γ* is 0.2289, which is better than one (i.e., 1.5046) in [26]. And the corresponding filter gain matrices are as follows:
(49)Af=[0.30330.0362-0.0085-0.11720.06750.0442-0.0196-0.02320.3032],Bf=[-1.36571.0166-0.11680.2405-1.8421-0.09690.05210.4326-1.7666],Cf=[0.0604-0.0945-0.0015-0.21210.43250.0640-0.10540.15550.1184],Df=[0.4505-2.1899-0.27790.0936-1.85830.3270-0.42671.2589-2.8255],G1f=[-0.0965-0.18090.0340-0.0772-0.14470.0272-0.0386-0.07240.0136],G2f=[0.0413-0.0725-0.00970.1240-0.2175-0.02920.0620-0.1087-0.0146],H1f=[0.6784-0.90470.1134-0.25730.27620.0907-0.1286-0.36191.0454],H2f=[1.1304-0.3575-0.03200.3912-0.0724-0.09600.1956-0.53620.9520].In the following simulation setup, the noise signal is chosen as(50)w(k)=v(k)={sin(0.3k),k≤20,0,k>20.Let the filtering error system run by random sequenced(k), the trajectories and their estimations of the mRNAs and proteins are shown in Figures 1 and 2, where the solid line and dotted line describe the state trajectories and estimations of mRNAs and proteins, respectively. The filtering errors are shown in Figures 3 and 4. It can be seen from Figures 3 and 4 that the filtering error converges to zero in the absence of disturbances.Figure 1
Trajectories and estimations of mRNAs.Figure 2
Trajectories and estimations of proteins.Figure 3
Estimation error of mRNAs.Figure 4
Estimation error of proteins.Next, we illustrate theH∞ performance of the filtering error system (11). By direct computation, we have
(51)∑k=060‍[w(k)v(k)]T[w(k)v(k)]=20.9454.
For values of 1000 random sequences of d(k), we obtain by MATLAB that the maximum of ∑k=060‍[em(k)ep(k)]T[em(k)ep(k)] is 0.1647, and hence the maximum disturbance attenuation level is
(52)∑k=060[em(k)ep(k)]T[em(k)ep(k)]∑k=060[w(k)v(k)]T[w(k)v(k)]=0.164720.9454=0.0887<γ*.
This verifies that the H∞ disturbance attenuation level is below the given upper bound.


## 5. Conclusion

In this paper, we investigate the filtering problem on a class of discrete-time GRNs with random delays. The filtering error system is established as a Markovian switched system and the random delay is described as a Markovian chain. By introducing an appropriate Lyapunov function, sufficient conditions for concerned problems are derived in terms of LMIs. The designed filter guarantees that the filtering error system is stochastically stable withH∞ disturbance attenuation level. Finally, the effectiveness and performance of the obtained results are demonstrated by a numerical example.



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*Source: 257971-2014-03-02.xml*