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Cogn Neurodyn. 2016 April; 10(2): 135–147.
Published online 2016 January 8. doi:  10.1007/s11571-015-9371-z
PMCID: PMC4805683

Delay decomposition approach to filtering analysis of genetic oscillator networks with time-varying delays

Abstract

In this paper, the filtering problem is treated for N coupled genetic oscillator networks with time-varying delays and extrinsic molecular noises. Each individual genetic oscillator is a complex dynamical network that represents the genetic oscillations in terms of complicated biological functions with inner or outer couplings denote the biochemical interactions of mRNAs, proteins and other small molecules. Throughout the paper, first, by constructing appropriate delay decomposition dependent Lyapunov–Krasovskii functional combined with reciprocal convex approach, improved delay-dependent sufficient conditions are obtained to ensure the asymptotic stability of the filtering error system with a prescribed performance. Second, based on the above analysis, the existence of the designed filters are established in terms of linear matrix inequalities with Kronecker product. Finally, numerical examples including a coupled Goodwin oscillator model are inferred to illustrate the effectiveness and less conservatism of the proposed techniques.

Keywords: Coupling delay, Delay decomposition approach, Genetic oscillator networks, filtering, Kronecker product, Linear matrix inequality

Introduction

In living organisms, biological activities are regulated by a network of genes (Jacob and Monod 1961) that interact among themselves to synthesize certain products such as proteins. During these biochemical interaction mechanism, the nonlinear physiological behaviors such as genetic switches (Lin et al. 2015) and genetic oscillations (Stricker et al. 2008) exist predominately in the complex biochemical network which forms the central functions of living cells. To illustrate the oscillatory nature and regulation mechanism of natural genetic oscillators, several mathematical models namely, Goodwin oscillator (Goodwin 1965), repressilator (Fraser and Tiwari 1974), Smolen oscillator (Smolen et al. 1999), circadian oscillator (Im and Taghert 2010) etc., have been constructed and discussed by many researchers. In Goodwin oscillator model, oscillations are induced by a single gene that represses itself. Later, Elowitz and Leibler (2000) proposed a repressilator model by extending the Goodwin oscillator model to a cycle of three genes, in which the oscillations are induced by repressing its successor genes in the cell cycle. In recent years, the robustness analysis of genetic oscillators has received considerable research attention (Amos 2014; O’Brien et al. 2012). An example for a natural genetic oscillator includes a tumor suppressor protein p53 in relation to cancer (Lahav et al. 2004).

It is noted that coupled networks can well describe the dynamical behavior of many real world systems (Gonze 2010; Pastor-Satorras et al. 2003; Rakkiyappan and Sasirekha 2014) including biological oscillators. In this aspect, greater efforts have been made to analyze the synchronization phenomena of coupled genetic oscillators (Li et al. 2007; Lu et al. 2015). Recently, authors in Uriu and Morelli (2014) have analyzed the effect of collective cellular behaviors during embryonic development enhance the synchronization of locally coupled genetic oscillators. In the previous work, it is noted that time delay has not been taken into account in the synchronization analysis of genetic oscillators. However, in genetic networks, time delay plays a crucial role, which is regarded as a source of instability and oscillations (Lakshmanan et al. 2014; Mao 2013; Mathiyalagan et al. 2012) due to the slow process of transcription and translation associated with mRNA and protein respectively. It has been observed that in gene regulator oscillator model (Wang et al. 2014) intercellular delay regulates the collective period of coupled cellular oscillators. By using Lyapunov stability theory and matrix inequality approach, the synchronization criteria for coupled genetic oscillators with delayed coupling has been investigated in Li and Lam (2011). And then, the analysis of exponential synchronization for Markovian jump and switched genetic oscillators with constant and time-varying nonidentical feedback delays have been discussed in Wan et al. (2014); Zhang et al. 2013). Very recently, different from asymptotic and exponential synchronization technique, authors in Alofi et al. (2015) have introduced a new power-rate synchronization technique to handle the unbounded time-varying delays in coupled genetic oscillators.

On the other hand, it has been shown that (Li et al. 2007; Li and Li 2009) cellular noises (intrinsic and extrinsic noises) in gene networks affect the dynamics of system both quantitatively and qualitatively. Hence, extrinsic noises resulting from environmental perturbations are unavoidable in modeling genetic oscillator networks (GONs). Based on linear matrix inequality (LMI) approach, the robust synchronization design problem for stochastic genetic oscillators has been discussed in Chen and Hsu (2012) to approximate the nonlinear coupled system. To exhibit more realistic characteristics of GONs, the stochastic synchronous criteria for Markovian jumping GONs with time delays have been derived in Wang et al. (2010). Recently, the authors in Lu et al. (2015) have utilized the drive-response concept, to study the passive synchronization analysis of Markovian jump GONs with external disturbances. However, noise has played a key role in biological systems, only few works have been reported in the literature to minimize the effect of it. Recently, filtering approach has been developed in Revathi et al. (2014), Wang et al. (2008) to estimate the true concentrations of network components in genetic regulatory networks. The objective of filtering is to minimize the norm of the filtering error system from noise inputs to filtering errors. It assumes that the noise inputs are energy-bounded signals rather than Gaussian white noise. Thus, the applicability of the filtering approach is that it does not requires the exact knowledge on the external noise signals. Upto now, the filtering analysis for GONs with extrinsic noises has not yet received much research attention.

Furthermore, in recent years, delay decomposition approach (Hua et al. 2014; Jarina Banu and Balasubramaniam 2014; Lakshmanan et al. 2012; Wu et al. 2011) has been widely used to reduce the conservatism issue of the stability of time delay systems. Despite, these efforts, authors in Cheng et al. (2013), Ge et al. (2014) have utilized delay decomposition approach along with reciprocal convex technique to further reduce the conservatism in terms of LMIs. Motivated by this fact, in this paper, we have made the first attempt to investigate the delay decomposition approach with reciprocal convex lemma for filtering analysis of GONs with time-varying delays and external disturbances.

Inspired by the above works, the main objective of this paper is to investigate the problem of filtering for GONs with extrinsic noises and time-varying delays in network couplings and nonlinear function. By implementing the Lyapunov–Krasovskii functional approach and delay decomposition method combined with reciprocal convex technique, new delay-dependent sufficient stability conditions are derived to guarantee the existence of the designed filters. The main contributions of this paper can be highlighted as follows:

  1. To estimate the true concentrations of network components in GONs, an improved delay-dependent filter design criteria is derived by employing the delay decomposition method for time-varying delays.
  2. The sufficient stability conditions are given in more generalized compact form for the existence of the designed filters.
  3. The applicability of the proposed approach is numerically illustrated by providing the simulation results for a coupled Goodwin oscillator model.

The rest of this paper is organized as follows. “System description and preliminaries” section presents the delay differential equation model describing the N coupled GONs with time-varying delays in nonlinear function and network coupling. Also some preliminaries are given to design the filters against the exogenous disturbances. “Main results” section provides the sufficient conditions in terms of LMIs for the existence of the designed filters. “Numerical examples” section numerically illustrates the conservatism of the proposed delay decomposition approach along with reciprocal convex technique by providing two numerical examples including a Goodwin oscillator model. Finally, “Conclusions” section concludes the paper.

Notation

Throughout this paper, I denotes the identity matrix with compatible dimension. n and n×m denote, respectively, the n-dimensional Euclidean space and the space of all n × m real matrices. L2[0,  ) represents the space of square integrable vector functions over [0,  ). (Q ⊗ R) ∈ ℛmp×nq denotes the Kronecker product of matrices Q ∈ ℛm×n and R ∈ ℛp×q. We use diag{ ·  ·  · } as a block-diagonal matrix. A > 0 ( < 0) means A is a symmetric positive (negative) definite matrix, A-1 denotes the inverse of matrix A. AT denotes the transpose of matrix A and  ∗  denotes the symmetric terms in a symmetric matrix.

System description and preliminaries

The differential equation model of a general delayed GON can be described by the following vector form:

dy(t)dt=Ay(t)+Bf(y(t-τ)),

where y(t) = col{y1(t), y2(t), …, yn(t)} ∈ ℛn represents the concentrations of mRNAs, proteins and chemical complexes; AB are matrices in n×n; f(y(tτ)) = col{f1(y1(tτ)),  f2(y2(tτ)), …, fn(yn(tτ))} ∈ ℛn is a monotonic genetic regulatory function which is usually taken as the Hill form. τ > 0 denotes the translation time delay in the translation process.

It is well known that genetic oscillators in biological networks are tightly coupled between each other in the network dynamics and also external disturbances are inevitable which affects the oscillations of the system. Therefore, the filtering problem is formulated for N coupled GONs by considering time-varying delays and disturbance inputs:

x˙l(t)=Axl(t)+Bf(xl(t-τ(t)))+m=1NwlmGxm(t-τ(t))+D1lνl1(t),yl(t)=Clxl(t)+D2lνl2(t),zl(t)=Hxl(t),xl(t)=φl(t),t[-τ,0],l=1,2,,N,
1

where xl(t) = col{xl1(t), xl2(t), …, xln(t)} ∈ ℛn is the state vector of the lth genetic oscillator; yl(t) ∈ ℛp is the measured output of the lth genetic oscillator and zl(t) ∈ ℛm is the signal to be estimated; νl1(t) and νl2(t) are the extrinsic noises belonging to L2([0,  );ℛq). φl(t) denotes the initial condition of xl(t) defined on the interval [ - τ,  0]. ABClD1lD2lH are known constant matrices with appropriate dimensions.

G = [glm]n×n is the matrix describing the inner-coupling between each genetic network node; W = [wlm]N×N is the outer coupling matrix representing the coupling strength and topological structure of the network; wlm is defined as: if there is a link from lth genetic oscillator to the mth genetic oscillator (l ≠ m), then wlm equals to a positive constant, otherwise, wlm = 0; wll=-l=1,lmNwlm. The time-varying delay τ(t) satisfies

0τ(t)τ,τ˙(t)μ<1.
2

Assumption 1

The nonlinear function fi( · ) satisfies the following sector-like condition

0fi(xi)xili,liR,i=1,2,,n

or equivalently fT(x)(f(x) - Lx)  ≤  0, where Ldiag{l1l2, …, ln}  >  0.

To estimate the concentrations of genetic oscillator containing mRNAs, proteins and chemical complexes, the following full-order filter is designed:

x^˙l(t)=AFlx^l(t)+BFlyl(t),z^l(t)=Hx^l(t),x^l(t)=φ^l(t),t[-τ,0],
3

where x^l(t)Rn is the filter state vector of the lth genetic oscillator and AFlBFl are appropriately dimensioned filter matrices to be designed. z^l(t)Rm is the estimate of the zl(t).

By using Kronecker product, systems (1) and (3) can be rewritten in the compact form:

x˙(t)=A~x(t)+B~f(x(t-τ(t)))+(WG)x(t-τ(t))+D1ν1(t),y(t)=Cx(t)+D2ν2(t),z(t)=H~x(t),x(t)=φ(t),t[-τ,0],x^˙(t)=AFx^(t)+BFy(t),z^(t)=H~x^(t),x^(t)=φ^(t),t[-τ,0],

where

x(t)=col{x1(t),,xN(t)},x^(t)=colx^1(t),,x^N(t),y(t)=col{y1(t),,yN(t)},z(t)=colz1(t),,zN(t),A~=IA,B~=IB,C=diagC1,,CN,D1=diag{D11,,D1N},D2=diag{D21,,D2N},H~=IH,f(x(t-τ(t)))=colf(x1(t-τ(t))),,f(xN(t-τ(t))),ν1(t)=col{ν11(t),,νN1(t)},ν2(t)=col{ν21(t),,νN2(t)},AF=diag{AF1,,AFN},BF=diag{BF1,,BFN}.

By defining the state vector e(t)=[xT(t)x^T(t)]T and estimation error z¯(t)=z(t)-z^(t), the augmented filtering error dynamics is obtained as follows:

e˙(t)=A¯1e(t)+A¯2Ke(t-τ(t))+B¯f(Ke(t-τ(t)))+D¯ν¯(t),z¯(t)=H¯e(t),e(t)=φ¯(t),t[-τ,0],
4

where φ¯(t)=[φT(t)φ^T(t)]T, and

A¯1=A~0BFCAF,A¯2=WG0,B¯=B~0,D¯=D100BFD2,H¯=H~-H~,K=I0,ν¯(t)=ν1T(t)ν2T(t)T.

Lemma 2.1

(Park et al. 2011) Letf1f2, …, fN: m → ℝhave positive values in an open subsetDofm. Then, the reciprocally convex combination offioverDsatisfies

min{αiαi>0,iαi=1}i1αifi(t)=ifi(t)+maxgi,j(t)ijgi,j(t)

subject to

gi,j:RmR,gj,i(t)gi,j(t),fi(t)gi,j(t)gi,j(t)fj(t)0.

Definition 2.1

Given a scalar γ > 0, the filtering error system (4) is said to be asymptotically stable with an performance γ, if it is asymptotically stable with ν¯(t)=0 and, under zero initial condition, it satisfies 1Nz¯(t)2γν¯(t)2, for any non-zero ν¯(t)L2[0,).

Main results

In this section, by constructing suitable Lyapunov–Krasovskii functional along with delay-decomposition approach, new set of delay-dependent sufficient stability conditions are derived for the filtering error system (4) with performance γ.

Theorem 3.1

For given scalarsτμand positive integerd, the filtering error system (4) is asymptotically stable with anperformanceγfor time-varying delayτ(t)satisfying (2), if there exist positive definite matricesP,Q~,R~,R~h,S~hand for any matricesU~h(h=1,2,,d)with appropriate dimensions such that the following LMIs hold

Ξ+ΠhΨ1NΣΓ-γ2I0-I<0,
5
S~hU~hS~h0,
6

where

Ξ=sym(Δ1TPΔs)+Δ1T(KTQ~K+KTL~TR~L~K+KTR~1K)Δ1-(1-μ){Δd+2TQ~Δd+2+Δd+3TR~Δd+3}+q=2dΔqTR~qΔq-q=1dΔq+1TR~qΔq+1+q=1d(τ/d)2ΔsTKTS~qKΔs,Πh=-Δd+2-Δh+1Δh-Δd+2TS~hU~hS~hΔd+2-Δh+1Δh-Δd+2-p=1,phd(Δp+1-Δp+2)T×S~p(Δp+1-Δp+2),Γ=q=1d(τ/d)2D¯TKTS~qKD¯,Σ=H¯0n,(d+2)nT,Ψ=Ψ1,d+4T0n,dnΨd+2,d+4TΨd+3,d+4TT,Ψ1,d+4=PD¯+q=1d(τ/d)2A¯1TKTS~qKD¯,Ψd+2,d+4=q=1d(τ/d)2A¯2TKTS~qKD¯,Ψd+3,d+4=q=1d(τ/d)2B¯TKTS~qKD¯,Δk=0n,(k-1)nIn0n,(d-k+3)n,Δs=A¯10n,dnA¯2B¯,k=1,2,,d+3,h=1,2,,d.

Proof

Choosing the Lyapunov–Krasovskii functional for the system (4) as

V(t,e(t))=n=14Vn(t,e(t)),
7

where

V1(t,e(t))=eT(t)Pe(t),V2(t,e(t))=t-τ(t)teT(s)KTQ~Ke(s)ds+t-τ(t)tfT(Ke(s))R~f(Ke(s))ds,V3(t,e(t))=h=1dt-hδt-(h-1)δeT(s)KTR~hKe(s)ds,V4(t,e(t))=δh=1d-hδ-(h-1)δt+θte˙T(s)KTS~hKe˙(s)dsdθ,

with P=diag{IP1,IP2},Q~=IQ,R~=IR,R~h=IRh,S~h=ISh and d ≥ 1 is the number of divisions of the interval [ - τ, 0]; δτ/d is the length of the each subinterval and h varies from 1, 2, …, d.

Taking the time derivative of V(te(t)) along the trajectory of the system (4), we have

V˙1(t,e(t))=2eT(t)PA¯1e(t)+A¯2Ke(t-τ(t))+B¯f(Ke(t-τ(t)))+D¯ν¯(t),
8
V˙2(t,e(t))=eT(t)KTQ~Ke(t)-(1-τ˙(t))eT(t-τ(t))KTQ~Ke(t-τ(t))+fT(Ke(t))R~f(Ke(t))-(1-τ˙(t))fT(Ke(t-τ(t)))R~f(Ke(t-τ(t)))eT(t)KTQ~Ke(t)-(1-μ)eT(t-τ(t))KTQ~Ke(t-τ(t))+eT(t)KTL~TR~L~Ke(t)-(1-μ)fT(Ke(t-τ(t)))R~f(Ke(t-τ(t))),
9
V˙3(t,e(t))=eT(t)KTR~1Ke(t)-eT(t-τ/d)KTR~1Ke(t-τ/d)+eT(t-τ/d)KT×R~2Ke(t-τ/d)-eT(t-2τ/d)KTR~2Ke(t-2τ/d)+eT(t-2τ/d)KT×R~3Ke(t-2τ/d)-eT(t-3τ/d)KTR~3Ke(t-3τ/d)++eT(t-(d-1)τ/d)KTR~(d-1)Ke(t-(d-1)τ/d)-eT(t-τ)KTR~dKe(t-τ),V˙4(t,e(t))=(τ/d)2e˙T(t)KT(S~1+S~2++S~d)Ke˙(t)-τ/dt-τ/dte˙T(s)KTS~1Ke˙(s)ds-τ/dt-2τ/dt-τ/de˙T(s)KTS~2Ke˙(s)ds-τ/dt-3τ/dt-2τ/de˙T(s)KTS~3Ke˙(s)ds--τ/dt-τt-(d-1)τ/de˙T(s)KTS~dKe˙(s)ds.
10

Case 1: If 0 ≤ τ(t) < τ/d, then

V˙4(t,e(t))=(τ/d)2e˙T(t)KT(S~1+S~2++S~d)Ke˙(t)-τ/dt-τ/dt-τ(t)e˙T(s)KTS~1Ke˙(s)ds-τ/dt-τ(t)te˙T(s)KTS~1Ke˙(s)ds-τ/dt-2τ/dt-τ/de˙T(s)KTS~2Ke˙(s)ds-τ/dt-3τ/dt-2τ/de˙T(s)KTS~3Ke˙(s)ds--τ/dt-τt-(d-1)τ/de˙T(s)KTS~dKe˙(s)ds(τ/d)2e˙T(t)KT(S~1+S~2++S~d)Ke˙(t)-τ/dτ/d-τ(t)[eT(t-τ(t))KT-eT(t-τ/d)KT]S~1[Ke(t-τ(t))-Ke(t-τ/d)]-τ/dτ-τ(t)[eT(t)KT-eT(t-τ(t))KT]S~1[Ke(t)-Ke(t-τ(t))]-τ/dt-2τ/dt-τ/de˙T(s)KTS~2Ke˙(s)ds-τ/dt-3τ/dt-2τ/de˙T(s)KTS~3Ke˙(s)ds--τ/dt-τt-(d-1)τ/de˙T(s)KTS~dKe˙(s)ds

By Lemma 2.1, we have

V˙4(t,e(t))(τ/d)2e˙T(t)KT(S~1+S~2++S~d)Ke˙(t)-Ke(t-τ(t))-Ke(t-τ/d)Ke(t)-Ke(t-τ(t))TS~1U~1S~1×Ke(t-τ(t))-Ke(t-τ/d)Ke(t)-Ke(t-τ(t))-[eT(t-τ/d)KT-eT(t-2τ/d)KT]S~2[Ke(t-τ/d)-Ke(t-2τ/d)]-[eT(t-2τ/d)KT-eT(t-3τ/d)KT]S~3[Ke(t-2τ/d)-Ke(t-3τ/d)]--[eT(t-(d-1)τ/d)KT-eT(t-τ)KT]S~d×Ke(t-(d-1)τ/d)-Ke(t-τ).
11

Case 2: If τ/d ≤ τ(t) < 2τ/d, then

V˙4(t,e(t))(τ/d)2e˙T(t)KT(S~1+S~2++S~d)Ke˙(t)-[eT(t)KT-eT(t-τ/d)KT]×S~1[Ke(t)-Ke(t-τ/d)]-Ke(t-τ(t))-Ke(t-2τ/d)Ke(t-τ/d)-Ke(t-τ(t))T×S~2U~2S~2Ke(t-τ(t))-Ke(t-2τ/d)Ke(t-τ/d)-Ke(t-τ(t))-[eT(t-2τ/d)KT-eT(t-3τ/d)KT]S~3[Ke(t-2τ/d)-Ke(t-3τ/d)]--[eT(t-(d-1)τ/d)KT-eT(t-τ)KT]S~d×[Ke(t-(d-1)τ/d)-Ke(t-τ)].
12

Case 3: If 2τ/d ≤ τ(t) < 3τ/d, then

V˙4(t,e(t))(τ/d)2e˙T(t)KT(S~1+S~2++S~d)Ke˙(t)-[eT(t)KT-eT(t-τ/d)KT]×S~1[Ke(t)-Ke(t-τ/d)]-[eT(t-τ/d)KT-eT(t-2τ/d)KT]×S~2[Ke(t-τ/d)-Ke(t-2τ/d)]-Ke(t-τ(t))-Ke(t-3τ/d)Ke(t-2τ/d)-Ke(t-τ(t))T×S~3U~3S~3Ke(t-τ(t))-Ke(t-3τ/d)Ke(t-2τ/d)-Ke(t-τ(t))--[eT(t-(d-1)τ/d)KT-eT(t-τ)KT]S~d[Ke(t-(d-1)τ/d)-Ke(t-τ)].
13

 ⋮ 

Case d: If (d - 1)τ/d ≤ τ(t) ≤ τ, then

V˙4(t,e(t))(τ/d)2e˙T(t)KT(S~1+S~2++S~d)Ke˙(t)-[eT(t)KT-eT(t-τ/d)KT]×S~1[Ke(t)-Ke(t-τ/d)]-[eT(t-τ/d)KT-eT(t-2τ/d)KT]×S~2[Ke(t-τ/d)-Ke(t-2τ/d)]-[eT(t-2τ/d)KT-eT(t-3τ/d)KT]×S~3[Ke(t-2τ/d)-Ke(t-3τ/d)]--Ke(t-τ(t))-Ke(t-τ)Ke(t-(d-1)τ/d)-Ke(t-τ(t))TS~hU~hS~h×Ke(t-τ(t))-Ke(t-τ)Ke(t-(d-1)τ/d)-Ke(t-τ(t)).
14

Considering the Eqs. (8)–(11) for Case 1, we have

V˙(t,e(t))ζ(t)ν¯(t)TΞ+Π1ΨΓζ(t)ν¯(t)=ηT(t)Ξ+Π1ΨΓη(t),
15

where

ηT(t)=[ζT(t)ν¯T(t)],ζT(t)=[eT(t)eT(t-τ/d)KTeT(t-2τ/d)KTeT(t-3τ/d)KTeT(t-(d-1)τ/d)KTeT(t-τ)KTeT(t-τ(t))KTfT(Ke(t-τ(t)))],Π1=-Δd+2-Δ2Δ1-Δd+2TS~1U~1S~1Δd+2-Δ2Δ1-Δd+2-p=2,p1d(Δp+1-Δp+2)TS~p×(Δp+1-Δp+2).

Now, we show that for Case 1, the filtering error system (4) with ν¯(t)=0, is asymptotically stable.

If ν¯(t)=0, from (15), we obtain

V˙(t,e(t))ζT(t)(Ξ+Π1)ζ(t).

From (5), we can conclude that ΞΠ1 < 0, which implies V˙(t,e(t))<0. Thus, the filtering error system (4) with ν¯(t)=0 is asymptotically stable.

For performance analysis of filtering error system (4) with non-zero ν¯(t), we define the following performance index

JT=0T[1Nz¯T(t)z¯(t)-γ2ν¯T(t)ν¯(t)+V˙(t,e(t))]dt-0TV˙(t,e(t))dt0T[ηT(t)Ωη(t)]dt,

where Ω=(Ξ+Π1)+1NΣΣTΨΓ-γ2I.

By Schur complement, for h=1, (5) is equivalent to Ω < 0 and consequently, if (5) and (6) hold, then JT < 0, ∀ T > 0. That is

0[1Nz¯T(t)z¯(t)-γ2ν¯T(t)ν¯(t)]dt<0,

which means that 1Nz¯(t)2γν¯(t)2 for any nonzero ν¯(t). Therefore, the filtering error system (4) is asymptotically stable with performance γ. Hence the theorem holds for Case 1. Similarly, Theorem 3.1 holds for all other cases by considering Eqs. (12), (13), (14) for Case 2, Case 3, ..., Case d. This completes the proof.

Remark 1

It has been noted that, in the proof process of Theorem 3.1, by employing a more general delay decomposition approach and reciprocal convex technique, better conservative results are obtained in terms of LMIs without using any free-weighting matrix method and model transformation. The conservative reduction increases with increase in number of delay intervals.

Based on Theorem 3.1, the following theorem presents a solution to H filter design for the filtering error system (4).

Theorem 3.2

For given scalarsτμand a positive integerd, the filtering error system (4) is asymptotically stable with anperformanceγfor time-varying delayτ(t)satisfying (2), if there exist positive definite matricesP1P2QRRhSh, matricesAFBFandUh (h = 1, 2, …, d)with appropriate dimensions such that the following LMIs hold

Ξ¯+Π¯hΨ¯(1)Ψ¯(2)1NΣ¯Γ¯00-γ2I0-I<0,
16
IShIUhISh0,
17

where

Ξ¯=sym(Δ¯1T(IP1)Δ¯s1)+sym(Δ¯2T(IP2)Δ¯s2)+Δ¯1T[IQ+ILTRL+IR1]Δ¯1-(1-μ)Δ¯d+3T(IQ)Δ¯d+3+Δ¯d+4T(IR)Δ¯d+4+q=2dΔ¯q+1T(IRq)Δ¯q+1-q=1dΔ¯q+2T(IRq)Δ¯q+2+q=1d(τ/d)2Δ¯s1T(ISq)Δ¯s1,Π¯1=-Δ¯d+3-Δ¯3Δ¯1-Δ¯d+3TIS1IU1IS1Δ¯d+3-Δ¯3Δ¯1-Δ¯d+3-p=2,p1d(Δ¯p+1-Δ¯p+2)T(ISp)(Δ¯p+1-Δ¯p+2),Π¯h=-Δ¯d+3-Δ¯h+2Δ¯h+1-Δ¯d+3TIShIUhIShΔ¯d+3-Δ¯h+2Δ¯h+1-Δ¯d+3-(Δ¯1-Δ¯3)T(IS1)(Δ¯1-Δ¯3)-p=2,phd(Δ¯p+1-Δ¯p+2)T(ISp)(Δ¯p+1-Δ¯p+2),Γ¯=q=1d(τ/d)2D1T(ISq)D1-γ2I,Ψ¯(1)=Ψ¯1,d+5(1)T0n,(d+1)nΨ¯d+2,d+5(1)TΨ¯d+3,d+5(1)TT,Ψ¯(2)=0nΨ¯2,d+6(2)T0n,(d+2)nT,Σ¯=IH-(IH)0n,(d+2)nT,Ψ¯1,d+5(1)=(IP1)D1+q=1d(τ/d)2(IATSq)D1,Ψ¯2,d+6(2)=[(IP2)BF]D2,Ψ¯d+2,d+5(1)=q=1d(τ/d)2(WTGTSq)D1,Ψ¯d+3,d+5(1)=q=1d(τ/d)2(IBTSq)D1,Δ¯k=0n,(k-1)nIn0n,(d-k+4)n,Δ¯s1=IA0n,(d+1)nWGIB,Δ¯s2=BFCAF0n,(d+2)n,k=1,2,,d+4,h=2,3,,d.

Further, the parameters of the desired filters are choosen as

AF=(IP2)-1A¯F,BF=(IP2)-1B¯F,
18

where A¯F=diag{A¯F1,A¯F2,,A¯FN},B¯F=diag{B¯F1,B¯F2,,B¯FN}.

Proof

Substituting (4) and the values of P,Q~,R~,R~h,S~h and U~h into (5) and (6), we obtain (16) and (17), in which the parameters are defined as (IP2)AF=A¯F,(IP2)BF=B¯F. Hence, if (16), (17) hold, the filter gain matrices are given by (18). This completes the proof.

Numerical examples

In this section, the applicability of the proposed delay decomposition approach for filtering design is demonstrated by considering two numerical examples with simulation results for the coupled GONs (1).

Example 1

Consider the coupled GONs (1) with n = 2,  N = 3 and the parameters are defined as

A=-1.40.20.5-1.7,B=-0.80.4-0.20.6,G=1001,C1=0.90.50.20.1,C2=0.10.60.20.3,C3=0.20.60.20.3,D11=0.60.20.10.3,D12=0.50.20.20.1,D13=0.40.20.40.1,D21=0.30.20.10.3,D22=-0.50.10.20.3,D23=10.10.40.5,H=0.50.20.20.3.

The coupling matrix is assumed to be

W=0.6-10.50.50.5-10.50.50.5-1,

where κ = 0.6 is called the coupling strength of the genetic oscillators in the symmetric networks. The nonlinear regulation function can be taken as f(xl(t-τ(t)))=xl(t-τ(t))2(1+xl(t-τ(t))2) for l = 1,  2,  3 and it can be easily verified that L = 0.65I. For the time-varying delay τ(t) = 1.2 + 0.1cos(t) with τ = 1.3,  μ = 0.1 and decomposing the delay interval into two parts as [0,  τ/2) ⋃ [τ/2,  τ], the filter matrices designed in (3) are obtained by solving the LMIs in Theorem 3.2 for γ = 0.8.

AF1=-0.5156-7.10447.0894-0.5068,AF2=-0.5159-4.65764.6422-0.5071,AF3=-0.5166-5.26355.2472-0.5072,BF1=0.03170.01770.00830.0022,BF2=0.01890.01160.04490.0391,BF3=0.01420.01110.01200.0049.

By choosing the initial conditions x1(t)=x^1(t)=[0.6,0.6]T,x2(t)=x^2(t)=[0.8,0.2]T,x3(t)=x^3(t)=[0.4,0.8]T and the disturbance inputs νl1(t) = sin(t)e-2tνl2(t) = sin(t)e-4t, the simulation results for Example 1 are shown in Fig. 1 and and2.2. Figure Figure11 shows the state responses of three coupled GONs and its estimates. Figure Figure22 shows the estimation errors z¯l(t) of three coupled GONs. The simulation results demonstrate that the designed filters are feasible and effective for the considered GONs. Furthermore, the maximum allowable upper bound (MAUB) values of τ for various values of μ are obtained in Table Table11 by increasing the number of divisions of the delay interval d.

Fig. 1
The state trajectories xl(t) of three coupled GONs and its estimates x^l(t) for Example 1
Fig. 2
The estimation errors z¯l(t) of three coupled GONs for Example 1
Table 1
MAUB τ for different values of μ for Example 1

Remark 2

From Table Table1,1, it can be verified that, the maximum value of the upper bound τ is obtained by dividing the delay interval into multiple equidistant subintervals. The improvement in the results are observed when d ≥ 2 compared to the case of d = 1. This shows the effectiveness of utilizing delay decomposition approach to get possible conservative delay-dependent stability results for filtering analysis of GONs.

Example 2

An example for a mathematical model of biochemical oscillator has been discussed in (Goodwin 1965) based on a negative feedback loop. The mechanism of the single cell oscillatory behavior of genetic regulatory network is described by a set of differential equations (Alofi et al. 2015; Li and Lam 2011):

dx1(t)dt=-b1x1(t)+11+x3H,dx2(t)dt=b2(x1(t)-x2(t)),dx3(t)dt=b3(x2(t)-x3(t)),
19

where x1(t),  x2(t) and x3(t) represent the concentrations of mRNA, protein and inhibitor, respectively. b1b2b3 are positive rate constants and H represents the Hill coefficient.

The single cell oscillator model described in (19) can be represented in the form of (1) by considering three coupled Goodwin oscillators with time-varying delays in the coupling term and disturbance inputs. Then, the matrices associated with system (1) can be defined as

A=-b100b2-b200b3-b3,B=00-1000000,f(xl(t-τ(t)))=00xl3(t-τ(t))H(1+xl3(t-τ(t))H),

where b1b2b3 are selected as 1.44, 0.75, 0.86 and H = 2. The rest of the matrices in (1) are assumed to be

G=100010001,C1=0.210.130.020.20.180.220.040.30.21,C2=0.010.30.020.20.130.20.40.30.1,C3=0.110.30.020.20.130.020.40.30.1,D1l=0.20.10.20.30.20.10.30.20.1,D21=-0.30.70.40.30.20.30.10.50.2,D22=D23=0.30.70.40.30.20.30.10.50.2,H=0.40.20.30.40.20.50.10.40.3,l=1,2,3,

and the time-varying delay τ(t) is assumed to be 0.22 + 0.08cos(t) with τ = 0.3, μ = 0.08, d = 2 and the coupling matrix is assumed to be the same as defined in Example 1. With the above parameters, for γ = 1, the desired filter matrices are obtained as

AF1=-0.51394.5668-0.1619-4.5844-0.5108-0.70020.13360.6757-0.5178,AF2=-0.5143-3.18546.17923.1675-0.5102-1.4217-6.20901.3961-0.5197,AF3=-0.51662.9921-6.9470-3.0101-0.5078-3.40356.91173.3801-0.5230,BF1=0.02760.01550.01880.04270.02540.05830.04290.03490.0424,BF2=0.01720.02390.02310.01320.03370.07560.06620.02130.0263,BF3=0.04620.01920.01290.2031-0.0860-0.1883-0.02740.08250.1643.

Figures Figures3,3, ,44 and and55 illustrate the simulation results for Example 2 with random initial conditions. In particular, Fig. 3 shows the time evolution of mRNAs concentration xl1(t), its estimates x^l1(t) and estimation errors z¯l1(t). Figures Figures44 and and55 depict the concentrations of proteins and inhibitors, xl2(t),  xl3(t), its estimates x^l2(t),x^l3(t) and estimation errors z¯l2(t),z¯l3(t) for all coupled genetic oscillators l = 1, 2, 3.

Fig. 3
mRNA concentrations of all coupled oscillators, its estimates and estimation errors for Example 2
Fig. 4
Protein concentrations of all coupled oscillators, its estimates and estimation errors for Example 2
Fig. 5
Inhibitor concentrations of all coupled oscillators, its estimates and estimation errors for Example 2

Conclusions

The problem of filtering for GONs with external disturbances and time-varying delays in nonlinear function as well as in coupling nodes has been investigated. In order to guarantee the existence of the filters, suitable Lyapunov–Krasovskii functional has employed combined with Kronecker product and delay decomposition approach. Further, less conservative sufficient stability conditions have been derived in terms of LMIs by using reciprocal convex combination lemma. Then the desired filters have been obtained by solving the appropriate LMIs by using Matlab LMI toolbox. Numerical examples have been provided to illustrate the practical importance and reduced conservatism of the proposed approach. In particular, a simple genetic network containing Goodwin oscillators has also been discussed.

Acknowledgments

This work was supported by UGC-BSR—Research fellowship in Mathematical Sciences—2012–2013, Govt. of India, New Delhi.

Contributor Information

V. M. Revathi, moc.liamg@csag.ihtaver.

P. Balasubramaniam, moc.liamg@urgulab.

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