Mathematical Biosciences 250 (2014) 41–53
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Robust H1 observer-based controller for stochastic genetic regulatory networks Hossein Shokouhi-Nejad ⇑, Amir Rikhtehgar-Ghiasi Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran
a r t i c l e
i n f o
Article history: Received 7 September 2013 Received in revised form 22 November 2013 Accepted 5 February 2014 Available online 14 February 2014 Keywords: Genetic regulatory network H1 observer based controller Time-varying delay LMI
a b s t r a c t This study is considered with the robust H1 observer based controller problem for a nonlinear genetic regulatory network (GRN) includes noise and disturbances, delays, and parameter uncertainties. The nonlinear functions describing the feedback regulation are assumed to satisfy the sector-like conditions; the parameter uncertainties are time-varying and unknown but are norm-bounded, and the delays are timevarying. We aim to design robust observer based controller to stabilize the stochastic GRN such that, for all admissible uncertainties, nonlinearities, stochastic perturbations and time varying delays, the dynamics of the GRN and observer are guaranteed to be robustly asymptotically stable in the mean square sense while achieving the prescribed H1 disturbance attenuation level. Based on the Lyapunov method and the stochastic analysis technique, it is shown that if a set of linear matrix inequalities (LMIs) are feasible, the desired observer based controller does exist. Finally, a numerical example is presented to illustrate the effectiveness of the proposed theoretical results. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction In a living cell, there are a complete set of genes but they are not all expressed in every tissue. Genetic regulatory networks (GRNs) are the mechanisms which regulate the expression of genes, where the expression of a gene is regulated by its production. In recent years, GRNs have been widely investigated in the biological and biomedical science. Since dynamical behaviors have been appeared in GRNs, during the past few years, dynamical modeling of genetic regulatory network has become an attractive area and a variety of models have been proposed. Basically, there are two types of genetic network models, the Boolean model [1,2], where the activity of each gene is expressed in one of two states, ON or OFF, and the differential equation model [3–6], where the variables describe the concentrations of gene products, such as mRNAs and proteins. In this paper, differential equation model of GRNs is exploited. Although in normal conditions GRNs can regulate concentrations of mRNA and protein by inherent feedback loops existing in GRNs, sometimes some problems may occur that tend to deteriorate GRNs performance and eventually lead to serious consequences. For instance, inability of regulation by GRN may cause to some fatal disease like cancer [7,8]. Motivated by this, recently
⇑ Corresponding author. Tel.: +98 9133175265. E-mail address: [email protected] (H. Shokouhi-Nejad). http://dx.doi.org/10.1016/j.mbs.2014.02.003 0025-5564/Ó 2014 Elsevier Inc. All rights reserved.
control theory has been developed for attenuating effects of this inability [8–18]. In practice, for identifying genes of interest and developing proper drugs as input control, biologists need to know the exact steady-state values of the GRN states, which are the concentrations of the mRNAs and proteins. Unfortunately, due to the existence of state delay and state-depended noises, the actual GRN measurements are far from the accurate network states. The observer goal is to estimate the network states in order to provide the right amount of proper drugs as artificial input control [8] such that the overall error could asymptotically converge to zero in the mean square sense for a GRN that contains transmission varying delays, intrinsic fluctuations, and parameter uncertainties. Time delays are frequently encountered in a variety of dynamic systems, such as nuclear reactors, chemical engineering systems, biological systems, and population dynamics models. They are usually results in unsatisfactory performances and are frequently a source of instability. In GRNs, time delays are unavoidable due to the slow process of transcription, translation, and translocation so it is very important to consider delays in mathematical GRN models [19–21]. On the other hand, the genetic regulation process always faces to internal noise which is due to the random chemical reactions, and external noise which is originated from environment fluctuations [22–24]. Thus for describing the behavior of internal and external noise in GRNs, stochastic models have been developed. Furthermore, because of the use of an approximate system model for the purpose of simplifying models, it is very likely that the parameters of the model will vary from time to time.
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Therefore, when modeling GRNs, beside stochastic perturbations, parameter uncertainties should also be considered. In recent works, the stabilizing problem of GRNs has received considerable attention [8–10,18]. Note that most existing results are concentrated on the stabilizing problem of GRNs by state feedback control and state feedback requires the availability of all state variables. While in practice, because of the complexity originated from the time delays, intrinsic fluctuation and modeling, there is only partial information about the states of the GRNs. Therefore, to obtain the true states of the networks, we need to estimate them from available measurements. Motivated by the above discussions, we will design two robust estimator based controllers, one without and one with considering H1 criterion, for genetic regulatory networks with internal and external noises and parameter uncertainties as well as time-varying state delays. Using LMI technique, we derive the sufficient conditions for ensuring asymptotic mean square stability of the extended GRN, and then we find the corresponding gains of the controller and observer in terms of the solution of the LMIs. The paper is organized as follows. In Section 2, the uncertain stochastic genetic regulatory networks model with extrinsic noises and parameter perturbations is introduced, some necessary assumptions and lemmas which will be used in the proof of the main results are also given. In Section 3, a robust controller and a robust H1 controller based on observer are designed for stabilizing in the mean square sense of the GRNs .In Section 4, numerical examples are constructed to show the effectiveness of proposed controllers. The conclusions are finally drawn in Section 5.
where b is a positive constant, Hj is the Hill coefficient, and dimension less bounded constant qij is the transcriptional rate of transcription factor j to i. On the other hand, if transcription factor j is a repressor of gene i, then:
g ij ðxpj Þ ¼ qij
0
bH j H
bHj þ xpjj
¼ qij @1
1
H
xpjj H
bHj þ xpjj
A
ð3Þ
The matrix A2 ¼ ½aij 2 Rnn in (1) is such that aij > 0 if transcription factor j is an activator of gene i; aij ¼ 0 if there is no link between gene i and j; and aij < 0 if transcription factor j is a repressor of gene i. In other words, matrix A2 defines the coupling topology, direction, and the transcriptional rate of the GRN Moreover, A1 ; A3 , A4 , B1 , B2 , C 1 , C 2 , G1 and G2 in (1) are known real constant matrices with appropriate dimensions, while DA1 ðtÞ, DA2 ðtÞ, DA3 ðtÞ, DA4 ðtÞ, DB1 ðtÞ, DB2 ðtÞ, DG1 ðtÞ and DG2 ðtÞ are unknown matrices representing time-varying parameter uncertainties. Assumption 1. The parameter uncertainties are assumed to satisfy the following norm-bounded conditions:
2
DA 1 6 4 DA 3 DB 1
3 2 3 DA2 M1 7 6 7 DA4 5 ¼ 4 M2 5FðtÞ½ N1 N2 ; DB2 M3
DG1 M4 ¼ FðtÞN3 DG2 M5
ð4Þ
where Mj ðj ¼ 1; 2; :::; 5Þ and Ni ði ¼ 1; 2; 3Þ are known real constant matrices and FðtÞ is unknown Lebesque-measurable matrix-valued functions subject to the following condition:
2. Model description and some assumptions
F T ðtÞFðtÞ 6 I; 8t
In this Letter, consider the following stochastic nonlinear genetic regulatory network:
Assumption 2 [25]. The function g i ðÞ satisfies the following sector condition:
dxm ðtÞ ¼ ½ðA1 þ DA1 ðtÞÞxm ðtÞ þ ðA2 þ DA2 ðtÞÞgðxp ðt s1 ðtÞÞÞ þ ðB1 þ DB1 Þu1 ðtÞ þ ðG1 þ DG1 ðtÞÞv ðtÞdt
0 6 fi ðxp Þ :¼
þ d1 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞÞdx1 ðtÞ dxp ðtÞ ¼ ½ðA3 þ DA3 ðtÞÞxp ðtÞ þ ðA4 þ DA4 ðtÞÞxm ðt s2 ðtÞÞ þ ðB2 þ DB2 Þu2 ðtÞ þ ðG2 þ DG2 ðtÞÞv ðtÞdt
ð1Þ
which is equivalent to
g T ðxp Þðgðxp Þ Zxp Þ 6 0
ym ðtÞ ¼ C 1 xm ðtÞ
ð6Þ
where Z ¼ diagfz1 ; z2 ; . . . ; zn g > 0 is known.
yp ðtÞ ¼ C 2 xp ðtÞ xp ðtÞ ¼ /p ðtÞ; 8t < 0
xm ðtÞ ¼ /m ðtÞ; n
where xm ðtÞ 2 R , xp ðtÞ 2 Rn are the concentrations of mRNAs and proteins at time t for a GRN with n nodes, respectively; u1 ðtÞ 2 Rq , u2 ðtÞ 2 Rq are the control inputs; ym ðtÞ 2 Rr , yp ðtÞ 2 Rr represent the expression levels of mRNAs and proteins at time t, respectively; /m , /p are initial functions of xm ðtÞ and xp ðtÞ for t < 0, respectively; the time-varying scalars s1 ðtÞ > 0, s2 ðtÞ > 0 denote the feedback regulation delay and the translation delay, respectively, satisfying 1 6 1 and 0 6 s_ 2 ðtÞ 6 d 2 6 1; x1 ðtÞ and x2 ðtÞ are zero0 6 s_ 1 ðtÞ 6 d mean scalar Wiener process (Brownian Motion) on ðX; F ; PÞ with Efxi ðtÞg ¼ 0, Efx2i ðtÞg ¼ t ði ¼ 1; 2Þ, and they are mutually uncorrelated; v ðtÞ 2 RP the disturbance signal that belongs to L2 ½0; 1Þ; and T monotone function gðxp Þ ¼ ½ g 1 ðxp Þ g 2 ðxp Þ . . . g n ðxp Þ 2 Rn represents the feedback regulation of the protein on the transcription, whose elements are generally nonlinear function in the form of P SUM logic: g i ðxp1 ; xp2 ; ::; xpn Þ ¼ nj¼1 g ij ðxpj Þ where g ij is a monotonic function of Hill form. If transcription factor j is an activator of gene i, then: H xpjj Hj
and f i ð0Þ ¼ 0; g i ð0Þ
¼ 0; i ¼ 1; . . . ; n:
þ d2 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞdx2 ðtÞ
g ij ðxpj Þ ¼ qij
g i ðxp Þ 6 zi ; 8xp 2 R; xp – 0; xp
ð5Þ
b þ
H xpjj
ð2Þ
Assumption 3. The noise intensity vectors d1 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞ and d2 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞ in (1) satisfy the following conditions:
trace½dT1 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞd1 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞ 6 xTm ðtÞQ T1 Q 1 xTm ðtÞ þ xTp ðtÞQ T2 Q 2 xTp ðtÞ þ xTm ðt s2 ðtÞÞQ T3 Q 3 xTm ðt s2 ðtÞÞ þ xTm ðt s1 ðtÞÞQ T4 Q 4 xTm ðt s1 ðtÞÞ trace½dT2 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞd2 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞ 6 xTm ðtÞRT1 R1 xTm ðtÞ þ xTp ðtÞRT2 R2 xTp ðtÞ þ xTm ðt s2 ðtÞÞRT3 R3 xTm ðt s2 ðtÞÞ þ xTm ðt s1 ðtÞÞRT4 R4 xTm ðt s1 ðtÞÞ ð7Þ where Q 1 ; Q 2 ; Q 3 ; Q 4 ; R1 ; R2 ; R3 and R4 are the known matrices. The main objective of this Letter is to design a controller on the basis of estimator. Consider the state feedback controller as
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u1 ðtÞ ¼ K 1 ^xm ðtÞ u2 ðtÞ ¼ K 2 ^xp ðtÞ
ð8Þ
said to be robustly asymptotically stable in the mean-square sense if limfkxm ðtÞk2 þ kxp ðtÞk2 g ¼ 0 holds for any initial condition. t!1
where K 1 and K 2 are appropriate dimension matrices to be determined. ^ xm ðtÞ 2 Rn and ^ xp ðtÞ 2 Rn are estimation the of xm ðtÞ and xp ðtÞ, respectively with following dynamics:
^m ðtÞÞ þ B1 u1 ðtÞdt d^xm ðtÞ ¼ ½Am ^xm ðtÞ þ L1 ðym ðtÞ y ^ ^ ^ dxp ðtÞ ¼ ½Ap xp ðtÞ þ L2 ðyp ðtÞ yp ðtÞÞ þ B2 u2 ðtÞdt
ð9Þ
where matrices Am , Ap , L1 and L2 are the observer parameters to be determined. xm ðtÞ, ep ðtÞ ¼ xp ðtÞ ^ xp ðtÞ and combinDefining em ðtÞ ¼ xm ðtÞ ^ ing system (1) and observer (9), yields the following augmented dynamics:
d^xm ðtÞ ¼ ½ðAm þ B1 K 1 Þ^xm ðtÞ þ L1 C 1 em ðtÞdt dem ðtÞ ¼ ½ðA1 þ DA1 ðtÞ L1 C 1 Þem ðtÞ þ ðA2 þ DA2 Þgðxp ðt s1 ðtÞÞÞ þ ðA1 þ DA1 Am þ DB1 K 1 Þ^xm ðtÞ þ ðG1 þ DG1 ðtÞÞv ðtÞdt þ d1 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞdx1 ðtÞ d^xp ðtÞ ¼ ½ðAp þ B2 K 2 Þxp ðtÞ þ L2 C 2 ep ðtÞdt dep ðtÞ ¼ ½ðA3 ðtÞ þ DA3 ðtÞ L2 C 2 Þep ðtÞ þ ðA4 þ DA4 ðtÞÞem ðt s2 ðtÞÞ þ ðA3 ðtÞ þ DA3 ðtÞ Ap þ DB2 ðtÞK 2 Þ^xp ðtÞ þ ðA4 þ DA4 ðtÞÞ^xm ðt s2 ðtÞÞ þ ðG2 þ DG2 ðtÞÞv ðtÞdt þ d2 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞdx2 ðtÞ
xm ðtÞ and Definition 3. For augmented system (10), u1 ðtÞ ¼ K 1 ^ xp ðtÞ are said to be H1 controllers if augmented system u2 ðtÞ ¼ K 2 ^ (10) with v ðtÞ ¼ 0 is mean-square asymptotically stable and also T there exists a l > 0 such that for zðtÞ ¼ eTm ðtÞ eTp ðtÞ
Z t J¼E zT ðsÞzðsÞ l2 v T ðsÞv ðsÞds < 0
ð12Þ
0
holds for all non-zero v ðtÞ 2 L2 ½0; 1Þ under zero initial conditions. Lemma 1 (Schur Complement [29]). Given constant matrices U1 ,
U2 , U3 where U1 ¼ UT1 and U2 ¼ UT2 > 0. Then U1 þ UT3 U1 2 U3 < 0 if T U2 U3 U U 1 3 < 0 or < 0. and only if UT3 U1 U3 U2 Lemma 2 [30]. Let X; Y and H be real matrices of appropriate dimension with HT H 6 I. Then, XHY þ Y T HT X T 6 eXX T þ e1 Y T Y holds for any scalar e > 0. 3. Main results
ð10Þ
In this section, we shall deal with the robust observer based controller design for the augmented system (10) with v ðtÞ ¼ 0 by the following theorem.
Definition 1. Itô formula [26]: Consider an n-dimensional stochastic vector process fXðt; xÞg with stochastic differential dXðt; xÞ ¼ Fðt; xÞdt þ Gðt; xÞdW t ðxÞ on t P 0, where W t ðxÞ is an m-dimensional Brownian motion defined on ðX; F ; fF t gt2R ; PÞ. Let V 2 ðRn Rþ ; Rþ Þ, then:
Theorem 1. If there exist five positive scalars k , e1 , e2 , q1 , q2 and 1 , P 2 , P 3 , P 4 , W 1 , W 2 , W 3 , W 4 and matrices positive definite matrices P _ _ 1; K 2 such that the following linear matrix A^m ; A^p ; L^1 ; L^2 ; P1 ; P2 ; K inequalities:
dVðtÞ ¼ LVðtÞdt þ V TX Gdx
ð11Þ
where LVðtÞ ¼ V t þ V Tx F þ 1=2trfGGT V xx g with V t ; V x and V xx denoting the partial derivative on t, the gradient of V and the Hessian matrix of V respectively, and trfg being the sum of the main diagonal entries.
(
2
Y T1 Y T2 6 e1 I 0 6 6 e2 I 4 1I P3 P q 2I P4 P q _ C 1 P3 ¼ P1 C 1
P
C1
0 0 1I q
C2
3
0 7 7 0 7 0 there exists a scalar rðeÞ > 0 such that Efkxm ðtÞk2 g < e and Efkxp ðtÞk2 g < e, 8t > 0 when sup Efk/m ðsÞk2 g < rðeÞ and
^ m þ B1 K ^T þ K 1 þ A T BT þ W 1 P1 ¼ A m 1 1
ð15Þ
sup Efk/p ðsÞk2 g < rðeÞ. Additionally, system (1) with
^ p þ B2 K ^T þ K 2 þ A T BT þ W 2 P2 ¼ A p 2 3
ð16Þ
s60
s60
v ðtÞ 0,
is
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H. Shokouhi-Nejad, A. Rikhtehgar-Ghiasi / Mathematical Biosciences 250 (2014) 41–53
P3 ¼ A1 P 3 þ P 3 AT1 ^L1 C 1 C T1 ^LT1 þ W 3 þ e1 M1 MT1 þ e2 M3 MT3
ð17Þ
P4 ¼ A3 P 4 þ P 4 AT3 ^L2 C 2 C T2 ^LT2 þ W 4 þ e1 M2 MT2 þ e2 M3 MT3
ð18Þ
and
n n
o C2 ¼ diag P 1 RT1 ; P 2 RT2 ; P 3 RT1 ; P 4 RT2 ; P 1 RT3 ; P 2 RT4 ; P 3 RT3 ; P 4 RT4 ; 0
" Y1 ¼ " Y2 ¼
N1 P1 0 1 N1 K 0
0 2 N1 P
3 N1 P
0 3 0 0 N2 P # 0 0 0 0 0 0 0
0 2 N2 K
0 4 N2 P
0 1 N2 P
0
ð19Þ
0 N2 kI 0
#
0
0 0 0 0 0 0 0
ð20Þ
then, using:
1 P 1 ; K 2 ¼ K 2P 1 K1 ¼ K 1 2 ^mP ^pP 1 ; Ap ¼ A 1 Am ¼ A 1 2 L1 ¼ ^L1 P^ 1 ; L2 ¼ ^L2 P^ 1 1
P 3 6 q1 I
ð25Þ
P 4 6 q2 I then, according to Assumption 3, it is easy to see that
o
C1 ¼ diag P 1 Q T1 ; P 2 Q T2 ; P 3 Q T1 ; P 4 Q T2 ; P 1 Q T3 ; P 2 Q T4 ; P 3 Q T3 ; P 4 Q T4 ; 0
If there exist scalars qi ; i ¼ 1; 2 such that the following LMIs hold
ð21Þ
2
for the parameters of the controller and estimator, the augmented system (10) will be robustly asymptotically stable in the mean-square sense when v ðtÞ 0.
h Trace dT1 ð:ÞP 3 d1 ð:Þ 6 kmax ðP 3 Þ xTm ðtÞQ T1 Q 1 xTm ðtÞ þ xTp ðtÞQ T2 Q 2 xTp ðtÞ þxTm ðt s2 ðtÞÞQ T3 Q 3 xTm ðt s2 ðtÞÞi
þxTm ðt s1 ðtÞÞQ T4 Q 4 xTm ðt s1 ðtÞÞ h 6 q1 xTm ðtÞQ T1 Q 1 xTm ðtÞ þ xTp ðtÞQ T2 Q 2 xTp ðtÞ þxTm ðt s2 ðtÞÞQ T3 Q 3 xTm ðt s2 ðtÞÞi
þxTm ðt s1 ðtÞÞQ T4 Q 4 xTm ðt s1 ðtÞÞ trace dhT2 ð:ÞP4 d2 ð:Þ 6 kmax ðP 4 Þ xTm ðtÞRT1 R1 xTm ðtÞ þ xTp ðtÞRT2 R2 xTp ðtÞ þxTm ðt s2 ðtÞÞRT3 R3 xTm ðt s2 ðtÞÞi
þxTm ðt s1 ðtÞÞRT4 R4 xTm ðt s1 ðtÞÞ h 6 q2 xTm ðtÞRT1 R1 xTm ðtÞ þ xTp ðtÞRT2 R2 xTp ðtÞ þxTm ðt s2 ðtÞÞRT3 R3 xTm ðt s2 ðtÞÞi þxTm ðt s1 ðtÞÞRT4 R4 xTm ðt s1 ðtÞÞ
ð26Þ
Considering inequalities (6) and (26) and the facts that 0 6 Proof . Based on the system (10), we construct the following Lyapunov functional:
VðtÞ ¼ ^xTm ðtÞP1 ^xm ðtÞ þ ^xTp ðtÞP2 ^xp ðtÞ þ eTm ðtÞP3 em ðtÞ Z t ^xTm ðsÞP 5 ^xm ðsÞds þ eTp ðtÞP4 ep ðtÞ þ þ þ
Z Z
ts2 ðtÞ
t ts1 ðtÞ
^xTp ðsÞP6 ^xp ðsÞds þ
Z
þ 2eTm ðtÞP3 ðA2 þ DA2 ðtÞÞgðxp ðt s1 ðtÞÞÞ þ 2eTm ðtÞP3 ðA1 þ DA1 ðtÞ Am þ DB1 ðtÞK 1 Þ^xm ðtÞ
eTm ðsÞP7 em ðsÞds
þ eTp ðtÞ½2P 4 ðA3 þ DA3 ðtÞ L2 C 2 Þ þ P8 ep ðtÞ
t ts1 ðtÞ
eTp ðsÞP8 ep ðsÞds
ð22Þ
þ 2eTp ðtÞP4 ðA4 þ DA4 ðtÞÞem ðt s2 ðtÞÞ þ 2eTp ðtÞP4 ðA3 þ DA3 ðtÞ Ap þ DB2 ðtÞK 2 Þ^xp ðtÞ
By Itô formula, given in Definition 1, we can obtain the following stochastic differential along the trajectory of augmented system:
dVðtÞ ¼ LVðtÞdt þ 2 eTm ðtÞP3 d1 ðÞ dx1 ðtÞ þ 2 eTp ðtÞP 4 d2 ðÞ dx2 ðtÞ
ð23Þ
þ 2eTp ðtÞP4 ðA4 þ DA4 ðtÞÞ^xm ðt s2 ðtÞÞ 2 Þ^xT ðt s2 ðtÞÞP5 ^xm ðt s2 ðtÞÞ ð1 d m
1 Þ ^xT ðt s1 ðtÞÞP6 ^xp ðt s1 ðtÞÞ ð1 d p 2 ÞeT ðt s2 ðtÞÞP7 em ðt s2 ðtÞÞ ð1 d m
1 Þ eT ðt s1 ðtÞÞP8 ep ðt s1 ðtÞÞ ð1 d p
where
LVðtÞdt ¼ ^xTm ðtÞ½2P1 ðAm þ B1 K 1 Þ þ P5 ^xm ðtÞ
k g T ðxp ðt s1 ðtÞÞÞðgðxp ðt s1 ðtÞÞÞ Zep ðt s1 ðtÞÞ Z ^xp ðt s1 ðtÞÞÞ h þ q1 eTm ðtÞQ T1 Q 1 eTm ðtÞ þ eTp ðtÞQ T2 Q 2 eTp ðtÞ
þ 2^xTm ðtÞP1 L1 C 1 em ðtÞ þ ^xTp ðtÞ 2P2 ðAp þ B2 K 2 Þ þ P6 ^xp ðtÞ þ 2^xTp ðtÞP2 L2 C 2 ep ðtÞ
þeTm ðt s2 ðtÞÞQ T3 Q 3 eTm ðt s2 ðtÞÞ i þeTm ðt s1 ðtÞÞQ T4 Q 4 eTm ðt s1 ðtÞÞ h þ q1 ^xTm ðtÞQ T1 Q 1 ^xTm ðtÞ þ ^xTp ðtÞQ T2 Q 2 ^xTp ðtÞ
þ eTm ðtÞ½2P3 ðA1 þ DA1 ðtÞ L1 C 1 Þ þ P7 em ðtÞ þ 2eTm ðtÞP3 ðA2 þ DA2 ðtÞÞgðxp ðt s1 ðtÞÞÞ þ 2eTm ðtÞP3 ðA1 þ DA1 ðtÞ Am þ DB1 ðtÞK 1 Þ^xm ðtÞ
þ^xTm ðt s2 ðtÞÞQ T3 Q 3 ^xTm ðt s2 ðtÞÞ i þ^xTm ðt s1 ðtÞÞQ T4 Q 4 ^xTm ðt s1 ðtÞÞ h þ q2 eTm ðtÞRT1 R1 eTm ðtÞ þ eTp ðtÞRT2 R2 eTp ðtÞ
þ eTp ðtÞ½2P4 ðA3 þ DA3 ðtÞ L2 C 2 Þ þ P8 ep ðtÞ þ 2eTp ðtÞP4 ðA4 þ DA4 ðtÞÞem ðt s2 ðtÞÞ þ 2eTp ðtÞP4 ðA3 þ DA3 ðtÞ Ap þ DB2 ðtÞK 2 Þ^xp ðtÞ þ 2eTp ðtÞP4 ðA4
þeTm ðt s2 ðtÞÞRT3 R3 eTm ðt s2 ðtÞÞ i þeTm ðt s1 ðtÞÞRT4 R4 eTm ðt s1 ðtÞÞ h þ q2 ^xTm ðtÞRT1 R1 ^xTm ðtÞ þ ^xTp ðtÞRT2 R2 ^xTp ðtÞ
þ DA4 ðtÞÞ^xm ðt s2 ðtÞÞ ð1 s_ 2 ðtÞÞ^xTm ðt s2 ðtÞÞP5 ^xm ðt s2 ðtÞÞ ð1 s_ 1 ðtÞÞ ^xTp ðt s1 ðtÞÞP6 ^xp ðt s1 ðtÞÞ ð1 s_ 2 ðtÞÞeTm ðt s2 ðtÞÞP7 em ðt s2 ðtÞÞ ð1 s_ 1 ðtÞÞ eTp ðt s1 ðtÞÞP8 ep ðt s1 ðtÞÞ þ trace dT1 ðÞP3 d1 ðÞ þ trace dT2 ðÞP4 d2 ðÞ
LVðtÞdt 6 ^xTm ðtÞ½2P1 ðAm þ B1 K 1 Þ þ P5 ^xm ðtÞ þ 2^xTm ðtÞP1 L1 C 1 em ðtÞ þ ^xTp ðtÞ 2P2 ðAp þ B2 K 2 Þ þ P6 ^xp ðtÞ þ 2^xTp ðtÞP2 L2 C 2 ep ðtÞ þ eTm ðtÞ½2P3 ðA1 þ DA1 ðtÞ L1 C 1 Þ þ P7 em ðtÞ
t
ts2 ðtÞ
s_ 1 ðtÞ 6 d1 6 1, 0 6 s_ 2 ðtÞ 6 d2 6 1 the following inequality is true:
þ^xTm ðt s2 ðtÞÞRT3 R3 ^xTm ðt s2 ðtÞÞ i þ^xTm ðt s1 ðtÞÞRT4 R4 ^xTm ðt s1 ðtÞÞ ð24Þ
ð27Þ
45
H. Shokouhi-Nejad, A. Rikhtehgar-Ghiasi / Mathematical Biosciences 250 (2014) 41–53
2
where
6 6 6 6 6 6 6 6 6 U¼6 6 6 6 6 6 6 6 6 6 4
k>0 On the other hand, it is easy to see that the left hand side of (6) can be represented as: 2 3T 2 32 3 ep ep 0 0 Z=2 6 7 6 76 7 g T ðxp Þðgðxp Þ ðZep þ Z ^xp ÞÞ ¼ 4 ^xp 5 4 0 0 Z=2 54 x^p 5
gðxp Þ
gðxp Þ
I
Z=2 Z=2
ð28Þ Considering (27) and (28) it follows that: T
LVðtÞdt 6 h ðtÞKðtÞhðtÞ
U1
0
U9
0
0
0
0
0
U2
0
U10
0
0
0
0
U3
0
0
0 0
U4
0 A4 P1
0
0 3 A4 P
U5
0
0
0
U6
0
0
U7
0
U8
0
3
7 7 7 7 A2 k 7 7 7 0 7 7 7 0 7 7 P2 Z=2 7 7 7 7 0 7 7 4 Z=2 7 P 5 0
kI
ð35Þ
ð29Þ
where
with
T hðtÞ ¼ ^xTm ðtÞ ^xTp ðtÞ eTm ðtÞ eTp ðtÞ ^xTm ðt s2 ðtÞÞ x^Tp ðt s1 ðtÞÞ eTm ðt s2 ðtÞÞ eTp ðt s1 ðtÞÞ g T ðxp ðt s1 ðtÞÞÞ 2 6 6 6 6 6 6 6 6 6 6 KðtÞ ¼ 6 6 6 6 6 6 6 6 6 4
K1
0
K9
0
0
0
0
0
K2
0
K10
0
0
0
0
K3
0
0
0
0
0
K4
P4 ðA4 þ DA4 ðtÞÞ
0
P 4 ðA4 þ DA4 ðtÞÞ
0
K5
0
0
0
K6
0
0
0
3
7 7 7 7 P3 ðA2 þ DA2 ðtÞÞ 7 7 7 0 7 7 7 0 7 7 7 kðZ=2Þ 7 7 7 0 7 7 7 kðZ=2Þ 5 0
K7
0
K8
ð31Þ
kI
with:
1 1 1 1 P1 1 ¼ P1 ; P2 ¼ P2 ; P3 ¼ P3 ; P4 ¼ P4 ; k ¼ k
K1 ¼ P1 Am þ P1 B1 K 1 þ ATm P1 þ K T1 BT1 P1 þ P5 þ q1 Q T1 Q 1 þ q2 RT1 R1
^ m ¼ Am P ^ p ¼ Ap P 2 1 ¼ K 1 P1 ; K 2 ¼ K 2 P2 ; A 1 ; A K
K2 ¼ P2 Ap þ P2 B2 K 2 þ ATp P þ K T2 BT2 P 2 þ P6 þ q1 Q T2 Q 2 þ q2 RT2 R2
1 ¼ W 1 ; P2 P6 P 2 ¼ W 2 ; P 3 P7 P 3 ¼ W 3 ; P 4 P8 P 4 ¼ W 4 P1 P5 P h i ^ m þ B1 K ^T þ K 1 þ A T BT þ W 1 þ P 1 q Q T Q 1 þ q RT R1 P 1 U1 ¼ A 1 1 2 1 m 1 1 h i ^ p þ B2 K ^T þ K 2 þ A T BT þ W 2 þ P 2 q Q T Q 2 þ q RT R2 P 2 U2 ¼ A 1 2 2 2 p 2 2
K3 ¼ P3 ðA1 þ DA1 ðtÞÞ þ ðA1 þ DA1 ðtÞÞT P3 P3 L1 C 1 C T1 LT1 P3 þ P7 þ q1 Q T1 Q 1 þ q2 RT1 R1
K4 ¼ P4 ðA3 þ DA3 ðtÞÞ þ ðA3 þ DA3 ðtÞÞT P4 P4 L2 C 2 C T2 LT2 P4 T 1Q 2Q 2
þ P8 þ q
U3 ¼ A1 P 3 þ P 3 AT1 L1 C 1 P 3 P 3 C T1 LT1 þ W 3
T 2 R2 R2
h i 3 þ P3 q1 Q T1 Q 1 þ q2 RT1 R1 P
þq
K5 ¼ ð1 d2 ÞP5 þ q1 Q T3 Q 3 þ q2 RT3 R3
U4 ¼ A3 P 4 þ P 4 AT3 L2 C 2 P 4 P 4 C T2 LT2 þ W 4
K6 ¼ ð1 d1 ÞP6 þ q1 Q T4 Q 4 þ q2 RT4 R4 K7 ¼ ð1 d2 ÞP7 þ q1 Q T3 Q 3 þ q2 RT3 R3 K8 ¼ ð1 d1 ÞP8 þ q1 Q T4 Q 4 þ q2 RT4 R4 K9 ¼ P1 L1 C 1 þ ðA1 þ DA1 ðtÞÞT P3 ATm P3 þ K T1 DBT1 ðtÞP3 K10 ¼ P2 L2 C 2 þ ðA3 þ DA3 ðtÞÞT P4 ATp P4 þ K T2 DBT2 ðtÞP4 ð32Þ
ð33Þ
in which
1
h
3
T 1Q 4Q 4
2 3
i 4 þ q2 RT4 R4 P
^T U9 ¼ L1 C 1 P 3 þ P 1 AT1 A m ^T U10 ¼ L2 C 2 P 4 þ P2 AT3 A m
we have
LVðtÞdt 6 U þ DUðtÞ
h i 4 þ P4 q1 Q T2 Q 2 þ q2 RT2 R2 P h i U5 ¼ ð1 d2 ÞW 1 þ P 1 q1 Q T3 Q 3 þ q2 RT3 R3 P 1 h i U6 ¼ ð1 d1 ÞW 2 þ P 2 q1 Q T4 Q 4 þ q2 RT4 R4 P 2 h i U7 ¼ ð1 d2 ÞW 3 þ P 3 q Q T Q 3 þ q RT R3 P 3
U8 ¼ ð1 d1 ÞW 4 þ P 4 q
Then, pre- and post- multiply the two sides of (31) by
n o 1 1 1 1 1 1 1 1 Diag P1 1 ; P2 ; P3 ; P4 ; P1 ; P2 ; P3 ; P4 ; k I
ð30Þ
ð36Þ
ð34Þ and
46
H. Shokouhi-Nejad, A. Rikhtehgar-Ghiasi / Mathematical Biosciences 250 (2014) 41–53
2
1 DAT ðtÞ þ K T DBT ðtÞ 0 0 P 1 1 1
6 6 6 6 6 6 6 6 6 6 6 DUðtÞ ¼ 6 6 6 6 6 6 6 6 6 6 6 4
0
0
0
0
0
2 DAT ðtÞ þ K T DBT ðtÞ P 2 2 3
0
0
0
0
0
0
0
0
0
DA1 ðtÞP3 þ P 3 DAT1 ðtÞ
0
0
DA3 ðtÞP 4 þ P4 DAT3 ðtÞ
DA4 ðtÞP 1
3 0 DA4 ðtÞP
0
0
0
0
0
0
0
0
0
0
0
Above equation is equal to
ð38Þ
where
¼ FðtÞ
0
0
FðtÞ
"
0 0 M T1 0 0
" X2 ¼
0
0 0 M T3 0 0
0
ð39Þ
0
0 0 0 0 0
M T2
0 0 0 0 0
0
0 0 0 0 0
M T3
0 0 0 0 0
#T ð40Þ
#T ð41Þ
Using Lemma 2, it is clear that:
ð42Þ
therefore
þ e1 Y T Y 1 þ e1 Y T Y 2 LVðtÞ 6 U 1 2 1 2
6 LVðtÞ 6 6 4
U1
0
U9
0
0
0
0
0
U2
U10
0
0
0
0
0 3 U
0
0
0
0
0
4 U
A4 P 1
0
A4 P3
0
U5
0
0
0
U6
0
0
U7
0
U8
0
3
7 7 7 A2 kI 7 7 7 0 7 7 7 0 7 7 P2 Z=2 7 7 7 0 7 7 P4 Z=2 5 kI
7 0 7 5 e2 I
e1 I
ð46Þ
Since EfdVðtÞg ¼ EfLVðtÞdtg, in order to show that the augmented system (10) is asymptotically stable in mean square sense and the system (1) with v ðtÞ ¼ 0 is asymptotically stabilizable by observer-based control (8), it is enough to show that the right hand side of (46) is negative. Since the right hand side of (46) is not satisfied LMI conditions yet, we use schur complement again and yield to (13). Furthermore, making the changes in the ^mP ^p P 1P 1 , K 2 ¼ K 2P 1 , Am ¼ A 1 , Ap ¼ A 1 , variables as K 1 ¼ K 1 2 1 2 1 ^ 1 and L2 ¼ ^ ^ L1 ¼ ^ controller and observer gains can be L1 P L P 2 2 1 obtained. h
In this section, H1 observer based controller for the GRN (1) with exogenous disturbance is proposed. According to the following theorem, system (1) is mean-square robustly asymptotically stable with the attenuation level of l.
2
0
W Y T1
6 6 6 6 6 6 6 6 6 6 4
Y T2
Y T3
1 C
2 C
e1 I
0
0
0
0
e2 I
0
0
0
e3 I
0
0
1I q
0
2I q
3 P q 1I P 4 P q 2I P _
with
4 ¼ U4 þ e1 M 2 M T þ e2 M 3 MT U 2 3
3
k; e1 ; e2 ; Theorem 2. For given scalar l, if there exist positive scalars i ; W i ; i ¼ 1; :::; 4 and e3 ; q 1 ; q_2 and positive definite matrices P _ 1; K 2 such that following LMIs. matrices P 1 ; P 2 ; K
ð44Þ
3 ¼ U3 þ e1 M 1 M T þ e2 M 3 MT U 1 3
Y T2
ð43Þ
where
6 6 6 6 6 6 6 6 ¼6 U 6 6 6 6 6 6 6 4
ð37Þ
3.1. H1 controller
T T 1 T DUðtÞ 6 e1 X 1 X T1 þ e1 1 Y 1 Y 1 þ e2 X 2 X 2 þ e2 Y 2 Y 2
2
Y T1
U
and
X1 ¼
2
FðtÞ
7 7 7 7 7 DA2 ðtÞ kI 7 7 7 7 0 7 7 7 0 7 7 7 7 0 7 7 7 0 7 7 7 0 5 0 0
By Schur complement, inequality (43) is equivalent to
T T T T T T DUðtÞ ¼ X 1 FðtÞY 1 þ Y 1 F ðtÞX 1 þ X 2 FðtÞY 2 þ Y 2 F ðtÞX 2
3
0
C 1 P 3 ¼ P1 C 1 _
ð45Þ
C 2 P 4 ¼ P2 C 2 holds where:
3 7 7 7 7 7 7 0
ð65Þ
Now, per and post multiply the two side of (64) by
n o 1 1 1 1 1 1 1 1 Diag P1 1 ; P 2 ; P 3 ; P 4 ; P 1 ; P 2 ; P 3 ; P 4 ; k I; I therefore, the condition of (65) is changed to
8t > 0
H4 þ DH4 ðtÞ 6 0;
ð66Þ
where H4 is a constant matrix and DH4 ðtÞ is a time-varying matrix given by:
T z ðsÞzðsÞ l2 v T ðsÞv ðsÞ þ LVðsÞ ds
0
Z t _T ¼E h ðsÞH2 ðsÞhðsÞ ds
ð61Þ
0
where
2
K1
0
K9
0
0
0
0
0
K2
0
K10
0
0
0
0
K3 þ I
0
0
0
0
0
0
P4 ðA4 þ DA4 ðtÞÞ
0
6 6 6 6 6 6 6 6 6 6 6 H2 ðtÞ ¼ 6 6 6 6 6 6 6 6 6 6 6 4
K4 þ I P4 ðA4 þ DA4 ðtÞÞ
3
0
7 7 7 7 P3 ðA2 þ DA2 ðtÞÞ P3 ðG1 þ DG1 ðtÞÞ 7 7 7 0 P4 ðG2 þ DG2 ðtÞÞ 7 7 7 0 0 7 7 7 0 0 7 7 7 0 0 7 7 7 0 0 7 7 7 0 0 5 0
0
K5
0
0
0
K6
0
0
K7
0
K8
2
U9
0
0
0
0
0
0
0
0
U10
0
0
0
0
0
0
A2 kI
0 7 7 7 G1 7 7 G2 7 7 7 0 7 7 0 7 7 7 0 7 7 0 7 7 7 0 5
Using (6), (28), and (61), one can write:
JðtÞ 6 JðtÞ E
¼E
0
Z
t
8 > > > > > > > :
0
2
t
ep 6 6 k6 x^p 4 gðxp Þ
3 9 ep 0 0 Z=2 > > > = 6 76 7 > 6 76 ^ 7 0 Z=2 76 xp 7ds 6 0 4 54 5 > > > > ; gðxp Þ Z=2 Z=2 I
3T 2 7 7 7 5
32
h_T ðsÞH3 ðsÞhðsÞds
0
ð63Þ
U1 0 6 U2 6
6 6 6 6 6 6 6 H4 ¼ 6 6 6 6 6 6 6 6 6 4
ð62Þ
l2 I
U3 þ I
0
0
0
0
U4 þ I A4 P 1 0 A4 P 3 U5 0 0 U6 0 U7
0
0
0
0
0 0
U8
2 Z=2 P 0 4 Z=2 P
kI
3
l2 I ð67Þ
where
2
K1 6 6 6 6 6 6 6 6 6 H3 ðtÞ ¼ 6 6 6 6 6 6 6 6 6 4
0
K2
K9 0 0 0 0 K10 0 0 K3 þ I 0 0 0 K4 þ I P4 ðA4 þ DA4 ðtÞÞ 0 K5 0 K6
0
0
0
0
0
0
P4 ðA4 þ DA4 ðtÞÞ
0
0
0
0
0
K7
0 K8
0
0
3
7 0 0 7 7 P3 ðA2 þ DA2 ðtÞÞ P3 ðG1 þ DG1 ðtÞÞ 7 7 0 P4 ðG2 þ DG2 ðtÞÞ 7 7 7 7 0 0 7 7 kðZ=2Þ 0 7 7 7 0 0 7 7 kðZ=2Þ 0 7 7 5 kI 0
l2 I
ð64Þ
49
H. Shokouhi-Nejad, A. Rikhtehgar-Ghiasi / Mathematical Biosciences 250 (2014) 41–53
1 DAT ðtÞ þ K T DBT ðtÞ 0 0 P 1 1 1
6 6 6 6 6 6 6 6 6 6 DH4 ðtÞ ¼ 6 6 6 6 6 6 6 6 6 4
0
0
0
0
0
0 0
0
0
T DBT ðtÞ P 2 DAT3 ðtÞ þ K 2 2
0
0
0
0
DA1 ðtÞP 3 þ P 3 DAT1 ðtÞ
0
0
0
0
0 DA2 ðtÞ kI
DA3 ðtÞP 4 þ
4 DAT ðtÞ P 3
DA4 ðtÞP 1 0
It is not difficult to see that:
3
0 0 M T1 0 0
" 2 ¼ X
0
"
0 0 0 0 0 0
M T2
0 0 0 0 0 0
0
0 0 0 0 0 0
M T3
0 0 0 0 0 0
0
0 0 0 0 0 0
0 0 M T3 0 0
3 ¼ X
0
0
0 0 MT4 0 0
MT5
0
0
0
0
0
0 0
0 0
0 0
0
0
0
0
)
ð73Þ
#T
In this section, in order to show the effectiveness of the approaches in Section 3, a numerical example is provided. Consider the dynamic of the synthetic oscillatory network for Escherichia coli as follow [31]:
#T
8 dm ðtÞ a < dti ¼ mi ðtÞ þ 1þp n
i ¼ lacl; tetR; cl
: dpi ðtÞ
j ¼ cl; lacl; tetR
j
dt
3
ð70Þ
then
T e1 i Yi Yi
¼ bðpi mi Þ
6 6 6 6 6 6 6 6 6 H¼6 6 6 6 6 6 6 6 6 4
3 0 0 0:02 7 6 7 6 0 A2 ¼ 6 0:2 0 7; 5 4 0 0:2 0
A1 ¼ diagf5; 2; 0:05g; 3
U1
0
U9
0
0
0
0
0
0
0
U2
U10
0
0
0
0
0
0 ^3 U
A2 kI
0
0 3 A4 P
0
0 1 A4 P
0
0 ^4 U
0
0
U5
0
0
0
0
U6
0
0
P 2 Z=2
0
0
0 7 7 7 G1 7 7 G2 7 7 7 0 7 7 0 7 7 7 0 7 7 0 7 7 7 0 5
U7
U8 P 4 Z=2
kI
l2 I
^ 3 ¼ U3 þ I þ e1 M 1 M T þ e2 M3 MT þ e3 M 4 M T U 1 3 4 ^ 4 ¼ U4 þ I þ e1 M 2 M T þ e2 M3 MT þ e3 M 5 M T U 2
3
A3 ¼ diagf3:5; 3:5; 3:5g; B1 ¼ diagf0:1; 0:2; 0:2g;
A4 ¼ diagf1:2; 3:1; 0:03g
B2 ¼ diagf0:3; 0:2; 0:3g;
C 1 ¼ diagf0:1; 0:24; 1:7g;
C 2 ¼ diagf1:4; 0:2; 1:22g
s1 ðtÞ ¼ 0:2 þ 0:1sinð0:3tÞ; s2 ðtÞ ¼ 0:6 þ 0:1 sinð0:7tÞ; g i ðxp Þ ¼ x2p =ð1 þ x2p Þ; i ¼ 1;2;:::; n: Z ¼ diagf0:6; 0:6; 0:6g;
It is easy to check that the maximal value of the derivative of g i ðxp Þ is less than zi ¼ 0:6. Without loss of generality, the noise intensify vectors are set as
5
ð72Þ Using Schur Complement twice on right hand side of (71), and observing (47), it can be inferred that the right-hand of (71) is negative definite, and therefore H4 þ DH4 ðtÞ < 0, 8t > 0. h
ð74Þ
2
ð71Þ
where
!
Considering the stochastic and delayed characters of their interactions, the above equation can rearrange into genetic regulatory network (1). It is assumed that the third mRNA, cl, cannot work as well as other mRNAs and proteins. The parameters of the system are specified as follow:
i¼1
2
ð68Þ
4. Simulation results
0 0 0 0 0 0
3 X
0
#T
1X T þ e1 Y T Y 1 þ e2 X 2X T þ e1 Y T Y 2 þ e3 X 3X T DH4 ðtÞ 6 e1 X 1 1 1 2 2 2 3 1 T þ e Y Y3
þ H4 þ DH4 ðtÞ 6 H
0
¼ l2 xml_add>subject to (47) with l
then, considering Lemma 2, we have:
3
0
7 7 7 7 DG1 ðtÞ 7 7 DG2 ðtÞ 7 7 7 0 7 7 0 7 7 7 0 7 7 0 7 7 7 0 5 0
ð69Þ
3
1; Y 2; Y 3 are given in (51)–(53) and: where Y
"
0 DA4 ðtÞP 3
3
Corollary 1. To reduce the effects of the disturbance vector, the following linear objective minimization problem can be used
1 FðtÞ Y 1 þ Y T FðtÞ X T þ X 2 FðtÞ Y 2 þ Y T FðtÞ X T DH4 ðtÞ ¼ X 1 1 2 2 3 FðtÞ Y 3 þ Y T FðtÞ X T þX
1 ¼ X
0
)
2
R1 ðÞ ¼ Q 1 xm ðtÞ þ Q 2 xp ðtÞ þ Q 3 xm ðt s2 ðtÞÞ þ Q 4 xp ðt s1 ðtÞÞ R2 ðÞ ¼ R1 xm ðtÞ þ R2 xp ðtÞ þ R3 xm ðt s2 ðtÞÞ þ R4 xp ðt s1 ðtÞÞ For simplify, we just let
50
H. Shokouhi-Nejad, A. Rikhtehgar-Ghiasi / Mathematical Biosciences 250 (2014) 41–53
2 Q 1 ¼ R1 ¼ 0;
0:1 0
6 Q 2 ¼ R2 ¼ 4 0
0:2
0:1
0 3 0:2 0 0:1 6 7 Q 3 ¼ R3 ¼ 4 0 0:1 0 5 0:1 0 0:15
0
3
2
7 5;
6 M 3 ¼ 4 0:18
0:1 0:2
2
G2 ¼ diagf0:5; 0:7; 0:2g;
6
4
4
2
2
0
-2
-6
-6
4
6
8
10
12
14
16
18
-8
20
0
4
2
6
8
0.25
0.2
0.2
0.15
0.15
Protein 2
mRNA 2
0.25
0.1
10
0.05
0
0
-0.05
-0.05
2
4
6
8
10
12
14
16
18
-0.1
20
0
2
4
6
8
35
15
30
10
25
5
20
Protein 3
mRAN 3
20
0
10
-10
5
-15
0
2
4
6
8
10
12
Time (Sec.)
16
18
20
10
12
14
16
18
20
15
-5
0
14
Time (Sec.)
Time (Sec.)
-20
12
0.1
0.05
0
0:2
Time (Sec.)
Time (Sec.)
-0.1
6 M 4 4 0:18
-2
-4
2
7 0:11 5;
0:2
0:53 0:16
0
-4
0
2
0:6
3
7 0:11 5;
By Theorem 2 and Corollary 1, using YALMIP toolbox in MATLAB, the feasible solutions of (73) can be obtained. Here, we have
M 2 ¼ diagf0:33; 0:29; 0:31g
6
-8
3
0:21 0:17 0:3 0:21 0:17 0:3 3 0:4 0:2 0:1 6 7 M 5 ¼ 4 0:1 0:3 0:2 5; N 1 ¼ diagf0:8; 0:55; 0:21g; 0:1 0:2 0:3 2 3 0:4 0:6 0:3 6 7 N2 ¼ diagf0:1; 0:3; 0:9g; N3 ¼ 4 0:3 0:55 0:67 5 0:21 0:3 0:2
Protein 1
mRNA 1
M 1 ¼ diagf0:43; 0:3; 0:4g;
0:6
2
and Q 4 ¼ R4 ¼ 0. Then, we choose the following additional matrices:
G1 ¼ diagf0:9; 0:3; 0:3g;
0:53 0:16
14
16
18
20
-5 0
2
4
6
8
10
12
Time (Sec.)
Fig. 1. Concentrations of the mRNA (left column) and protein (right column) with u(t) = 0.
14
16
18
20
51
H. Shokouhi-Nejad, A. Rikhtehgar-Ghiasi / Mathematical Biosciences 250 (2014) 41–53
2
3
2
0:0062 0:0007 0:0002 6 7 0:0006 5; P1 ¼ 4 0:0007 0:0096 0:0002 0:0006 0:0281 2 3 0:0063 0:0001 0:0006 6 7 P2 ¼ 4 0:0001 0:0056 0:0013 5; 2
0:0000
0:0158 0:0035 0:0031 6 7 0:0059 5; P5 ¼ 4 0:0035 0:0351 0:0031 0:0059 0:2789 2 3 0:0126 0:0001 0:0018 6 7 P6 ¼ 4 0:0001 0:0105 0:0040 5;
0:0159 3
0:0006 0:0013 0:0050
2
0:0001
6 7 P3 ¼ 4 0:0000 0:0059 0:0001 5; 2
0:0001 0:0001 0:0071 0:0020
0:0001 0:0002
3
0:0116
6 P7 ¼ 4 0:0000 3
2
6 7 P4 ¼ 4 0:0001 0:0027 0:0002 5; 0:0002 0:0002 0:0067
0:0146 3
0:0018 0:0040
0:0008 0:0015
0:0008
0:0000 0:0174
7 0:0007 5;
0:0007
0:0397
0:0001 0:0002
3
6 7 P8 ¼ 4 0:0001 0:0027 0:0004 5 0:0002 0:0004 0:0029
0.2
0.2 0.15
0.15
Protein 1
mRNA 1
0.1 0.05 0
0.1
0.05
-0.05 0 -0.1 -0.15
0
1
2
3
4
5
6
7
-0.05
8
0
1
2
3
0.2
0.2
0.15
0.15
0.1
0.05
0
-0.05
4
5
6
7
8
6
7
8
Time (Sec.)
Protein 2
mRNA 2
Time (Sec.)
0.1
0.05
0
0
1
2
3
4
5
6
7
-0.05
8
0
1
2
3
Time (Sec.)
4
5
Time (Sec.)
0.25
0.2
0.2 0.15
Protein 3
mRAN 3
0.15 0.1 0.05
0.1
0.05
0 0 -0.05 -0.1
0
1
2
3
4
5
Time (Sec.)
6
7
8
-0.05
0
1
2
3
4
5
6
7
8
Time (Sec.)
Fig. 2. The true values (solid line) and the estimates (dashed line) of the concentrations of the mRNA (left column) and protein (right column) with proposed controller.
52
H. Shokouhi-Nejad, A. Rikhtehgar-Ghiasi / Mathematical Biosciences 250 (2014) 41–53
2
198:6215
6 P^ 1 ¼ 4 2:5206
0:4376
0:1797
3
and with the following parameter variations:
7 0:2759 5; 51:9375 13:8428 141:1335 2 3 506:0167 85:4435 18:3874 6 7 P^ 2 ¼ 4 1:7437 369:1918 2:0288 5; 13:9632 75:4923 149:3526 l ¼ l2 ¼ 0:0052; e1 ¼ 350:6; e2 ¼ 581:7;
2
169:4216
6 Q ðtÞ ¼ 4
e3 ¼ 1210; k ¼ 0:0051; q1 ¼ 0:0084; q2 ¼ 0:0078 Therefore, the controller and observer parameters can be obtained as follow:
2
0:3536 6 K 1 ¼ K 1 P 1 ¼ 4 0:0504 1:5456 2 6:8727 2 P2 ¼ 6 K2 ¼ K 4 0:0114 2
0:3558
0:1238
L2 ¼ ^L2 P^ 1 2
3
7 1:3069 5; 2:9332 28:1318 3 0:5027 3:5175 7 2:2984 2:7642 5; 3:5151
0:6849
9:3482 3
0
0
0
0:2 þ 0:2 sinðtÞ
0
0
0
0:3 þ 0:3 cosðtÞ
3 7 5
The states of the system (1) with uðtÞ ¼ 0 is shown in Fig. 1, so it is not mean-square asymptotically stable. From Corollary 1, we know that the system (1) is mean-square robustly asymptotically stabilizable. Using Corollary 1, the states of the true values and the estimates of the concentrations of the mRNA and protein are shown in Fig. 2 where the left and the right columns are devoted to the concentrations of mRNA and protein, respectively. Moreover, Fig. 3 shows the ratio of the energy of the T zðtÞ ¼ eTm ðtÞ eTp ðtÞ to the energy of the disturbance v ðtÞ where initial conditions are set to zero. It is clear that this ratio is less than ¼ 0:0052 for all t. the optimal value l 5. Conclusions
1:5752 0:2132 4:4390
^ m P1 ¼ 6 0:7170 Am ¼ A 4 0:0837 0:7424 1:6636 2 8:5101 0:3412 ^ p P2 ¼ 6 Ap ¼ A 3:7512 4 0:4447
L1 ¼ ^L1 P^ 1 1
0:6078
0:1 þ 0:1 sinðtÞ
7 4:1725 5; 20:0061 3 2:1243 7 1:8262 5;
0:5013 0:1765 4:1780 2 3 26:2158 1:4243 0:7272 6 7 7:1360 0:6318 5; ¼ 4 2:8021 9:0989 3:8621 3:0297 2 3 1:1724 1:1748 0:7140 6 7 ¼ 4 0:1062 3:6399 0:5689 5 0:0989 0:9398 1:8201
To simulate this example, we run system (1) for the one second with zero exogenous disturbances, and thereafter, parameter variT ations are considered with disturbance v ðtÞ ¼ ½ et et et uðtÞ where uðtÞ ¼ 1 when t P 0, and uðtÞ ¼ 0 for t < 0. We run the simulation with the following initial conditions:
/m ðtÞ ¼ ½ 0:2 0:2 0:2 T ;
/p ðtÞ ¼ ½ 0:2 0:2 0:2 T ;
In this research, the robust controller on the basis of observer for a model of uncertain time delay stochastic genetic regulatory networks has been investigated. Based on Lyapunov stability theory, LMI conditions, which could be easily solved by using the Matlab LMI control toolbox, have been proposed to design observer based controller such that, for all admissible nonlinearities and time delays, the overall error dynamic is robustly asymptotically stable in the mean square sense and a prescribed H1 disturbance attention level is guaranteed. A numerical example is finally presented to illustrate the application and effectiveness of the obtained results. It should be pointed out that in the GRN model (1), we did not consider any constraints on the delays, and thus, the designed controller is delay-independent which naturally involves conservatism. To reduce conservatism, one can exert delay bounds as design constraints, which yields a delay-dependent controller design. To design such a delay-dependent filter, one should change the Lyapunov function in an appropriate way. In addition, there are some new methods in delay-dependent controller design such as free-weighting matrices and delay-partitioning approaches which
^xm ð0Þ ¼ ½ 0:1 0:1 0:1 T ;
^xp ð0Þ ¼ ½ 0:1 0:1 0:1 T
can be employed to further reduce conservatism of the design. ∫ z(t)Tz(t) / ∫ v(t)Tv(t)
-3
3.5
x 10
References
3
Magnitude
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
6
7
Time (Sec.) Fig. 3. The ratio of the energy of z(t) to the energy of v(t).
8
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H. Shokouhi-Nejad, A. Rikhtehgar-Ghiasi / Mathematical Biosciences 250 (2014) 41–53 [9] W. Pan, Z. Wang, H. Gao, Y. Li, M. Du, Robust H1 feedback control for uncertain stochastic delayed genetic regulatory networks with additive and multiplicative noise, Int. J. Robust Nonlinear Control 20 (18) (2010) 2093. [10] C.W. Liao, C.Y. Lu, Design of delay-range-dependent robust controller for uncertain genetic regulatory networks with interval time-varying delays, Autom. Control Comput. Sci. 44 (4) (2010) 234. [11] M. Mohammadian, A.H. Abolmasoumi, H.R. Momeni, H1 mode-independent filter design for Markovian jump genetic regulatory networks with timevarying delays, Neurocomputing 87 (2012) 10. [12] W. Wanga, S. Zhong, F. Liu, New delay-dependent stability criteria for uncertain genetic regulatory networks with time-varying delays, Neurocomputing 93 (2012) 19. [13] Y. Sun, G. Feng, J. Cao, Robust stochastic stability analysis of genetic regulatory networks with disturbance attenuation, Neurocomputing 79 (2012) 39. [14] W. Wanga, S. Zhong, Stochastic stability analysis of uncertain genetic regulatory networks with mixed time-varying delays, Neurocomputing 82 (2012) 143. [15] A. Liu, L. Yu, W. Zhang, B. Chen, H1 filtering for discrete-time genetic regulatory networks with random delays, Math. Biosci. 239 (2012) 97. [16] W. Wang, S. Zhong, F. Liu, Robust filtering of uncertain stochastic genetic regulatory networks with time-varying delays, Chaos, Solitons Fractals 45 (2012) 915. [17] S.M. Mousavi, V. Majd, Robust filtering of extended stochastic genetic regulatory networks with parameter uncertainties, disturbances, and timevarying delays, Neurocomputing 74 (2011) 2123. [18] Y. He, J. Zeng, M. Wu, C. Zhang, Robust stabilization and H1 controllers design for stochastic genetic regulatory networks with time-varying delays and structured uncertainties, Math. Biosci. 236 (2012) 53. [19] H. Wu, X. Liao, S. Guo, W. Feng, Z. Wang, Stochastic stability for uncertain genetic regulatory networks with interval time-varying delay, Neurocomputing 72 (2009) 3263.
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Contents lists available at ScienceDirect
Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs
Robust H1 observer-based controller for stochastic genetic regulatory networks Hossein Shokouhi-Nejad ⇑, Amir Rikhtehgar-Ghiasi Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran
a r t i c l e
i n f o
Article history: Received 7 September 2013 Received in revised form 22 November 2013 Accepted 5 February 2014 Available online 14 February 2014 Keywords: Genetic regulatory network H1 observer based controller Time-varying delay LMI
a b s t r a c t This study is considered with the robust H1 observer based controller problem for a nonlinear genetic regulatory network (GRN) includes noise and disturbances, delays, and parameter uncertainties. The nonlinear functions describing the feedback regulation are assumed to satisfy the sector-like conditions; the parameter uncertainties are time-varying and unknown but are norm-bounded, and the delays are timevarying. We aim to design robust observer based controller to stabilize the stochastic GRN such that, for all admissible uncertainties, nonlinearities, stochastic perturbations and time varying delays, the dynamics of the GRN and observer are guaranteed to be robustly asymptotically stable in the mean square sense while achieving the prescribed H1 disturbance attenuation level. Based on the Lyapunov method and the stochastic analysis technique, it is shown that if a set of linear matrix inequalities (LMIs) are feasible, the desired observer based controller does exist. Finally, a numerical example is presented to illustrate the effectiveness of the proposed theoretical results. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction In a living cell, there are a complete set of genes but they are not all expressed in every tissue. Genetic regulatory networks (GRNs) are the mechanisms which regulate the expression of genes, where the expression of a gene is regulated by its production. In recent years, GRNs have been widely investigated in the biological and biomedical science. Since dynamical behaviors have been appeared in GRNs, during the past few years, dynamical modeling of genetic regulatory network has become an attractive area and a variety of models have been proposed. Basically, there are two types of genetic network models, the Boolean model [1,2], where the activity of each gene is expressed in one of two states, ON or OFF, and the differential equation model [3–6], where the variables describe the concentrations of gene products, such as mRNAs and proteins. In this paper, differential equation model of GRNs is exploited. Although in normal conditions GRNs can regulate concentrations of mRNA and protein by inherent feedback loops existing in GRNs, sometimes some problems may occur that tend to deteriorate GRNs performance and eventually lead to serious consequences. For instance, inability of regulation by GRN may cause to some fatal disease like cancer [7,8]. Motivated by this, recently
⇑ Corresponding author. Tel.: +98 9133175265. E-mail address: [email protected] (H. Shokouhi-Nejad). http://dx.doi.org/10.1016/j.mbs.2014.02.003 0025-5564/Ó 2014 Elsevier Inc. All rights reserved.
control theory has been developed for attenuating effects of this inability [8–18]. In practice, for identifying genes of interest and developing proper drugs as input control, biologists need to know the exact steady-state values of the GRN states, which are the concentrations of the mRNAs and proteins. Unfortunately, due to the existence of state delay and state-depended noises, the actual GRN measurements are far from the accurate network states. The observer goal is to estimate the network states in order to provide the right amount of proper drugs as artificial input control [8] such that the overall error could asymptotically converge to zero in the mean square sense for a GRN that contains transmission varying delays, intrinsic fluctuations, and parameter uncertainties. Time delays are frequently encountered in a variety of dynamic systems, such as nuclear reactors, chemical engineering systems, biological systems, and population dynamics models. They are usually results in unsatisfactory performances and are frequently a source of instability. In GRNs, time delays are unavoidable due to the slow process of transcription, translation, and translocation so it is very important to consider delays in mathematical GRN models [19–21]. On the other hand, the genetic regulation process always faces to internal noise which is due to the random chemical reactions, and external noise which is originated from environment fluctuations [22–24]. Thus for describing the behavior of internal and external noise in GRNs, stochastic models have been developed. Furthermore, because of the use of an approximate system model for the purpose of simplifying models, it is very likely that the parameters of the model will vary from time to time.
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H. Shokouhi-Nejad, A. Rikhtehgar-Ghiasi / Mathematical Biosciences 250 (2014) 41–53
Therefore, when modeling GRNs, beside stochastic perturbations, parameter uncertainties should also be considered. In recent works, the stabilizing problem of GRNs has received considerable attention [8–10,18]. Note that most existing results are concentrated on the stabilizing problem of GRNs by state feedback control and state feedback requires the availability of all state variables. While in practice, because of the complexity originated from the time delays, intrinsic fluctuation and modeling, there is only partial information about the states of the GRNs. Therefore, to obtain the true states of the networks, we need to estimate them from available measurements. Motivated by the above discussions, we will design two robust estimator based controllers, one without and one with considering H1 criterion, for genetic regulatory networks with internal and external noises and parameter uncertainties as well as time-varying state delays. Using LMI technique, we derive the sufficient conditions for ensuring asymptotic mean square stability of the extended GRN, and then we find the corresponding gains of the controller and observer in terms of the solution of the LMIs. The paper is organized as follows. In Section 2, the uncertain stochastic genetic regulatory networks model with extrinsic noises and parameter perturbations is introduced, some necessary assumptions and lemmas which will be used in the proof of the main results are also given. In Section 3, a robust controller and a robust H1 controller based on observer are designed for stabilizing in the mean square sense of the GRNs .In Section 4, numerical examples are constructed to show the effectiveness of proposed controllers. The conclusions are finally drawn in Section 5.
where b is a positive constant, Hj is the Hill coefficient, and dimension less bounded constant qij is the transcriptional rate of transcription factor j to i. On the other hand, if transcription factor j is a repressor of gene i, then:
g ij ðxpj Þ ¼ qij
0
bH j H
bHj þ xpjj
¼ qij @1
1
H
xpjj H
bHj þ xpjj
A
ð3Þ
The matrix A2 ¼ ½aij 2 Rnn in (1) is such that aij > 0 if transcription factor j is an activator of gene i; aij ¼ 0 if there is no link between gene i and j; and aij < 0 if transcription factor j is a repressor of gene i. In other words, matrix A2 defines the coupling topology, direction, and the transcriptional rate of the GRN Moreover, A1 ; A3 , A4 , B1 , B2 , C 1 , C 2 , G1 and G2 in (1) are known real constant matrices with appropriate dimensions, while DA1 ðtÞ, DA2 ðtÞ, DA3 ðtÞ, DA4 ðtÞ, DB1 ðtÞ, DB2 ðtÞ, DG1 ðtÞ and DG2 ðtÞ are unknown matrices representing time-varying parameter uncertainties. Assumption 1. The parameter uncertainties are assumed to satisfy the following norm-bounded conditions:
2
DA 1 6 4 DA 3 DB 1
3 2 3 DA2 M1 7 6 7 DA4 5 ¼ 4 M2 5FðtÞ½ N1 N2 ; DB2 M3
DG1 M4 ¼ FðtÞN3 DG2 M5
ð4Þ
where Mj ðj ¼ 1; 2; :::; 5Þ and Ni ði ¼ 1; 2; 3Þ are known real constant matrices and FðtÞ is unknown Lebesque-measurable matrix-valued functions subject to the following condition:
2. Model description and some assumptions
F T ðtÞFðtÞ 6 I; 8t
In this Letter, consider the following stochastic nonlinear genetic regulatory network:
Assumption 2 [25]. The function g i ðÞ satisfies the following sector condition:
dxm ðtÞ ¼ ½ðA1 þ DA1 ðtÞÞxm ðtÞ þ ðA2 þ DA2 ðtÞÞgðxp ðt s1 ðtÞÞÞ þ ðB1 þ DB1 Þu1 ðtÞ þ ðG1 þ DG1 ðtÞÞv ðtÞdt
0 6 fi ðxp Þ :¼
þ d1 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞÞdx1 ðtÞ dxp ðtÞ ¼ ½ðA3 þ DA3 ðtÞÞxp ðtÞ þ ðA4 þ DA4 ðtÞÞxm ðt s2 ðtÞÞ þ ðB2 þ DB2 Þu2 ðtÞ þ ðG2 þ DG2 ðtÞÞv ðtÞdt
ð1Þ
which is equivalent to
g T ðxp Þðgðxp Þ Zxp Þ 6 0
ym ðtÞ ¼ C 1 xm ðtÞ
ð6Þ
where Z ¼ diagfz1 ; z2 ; . . . ; zn g > 0 is known.
yp ðtÞ ¼ C 2 xp ðtÞ xp ðtÞ ¼ /p ðtÞ; 8t < 0
xm ðtÞ ¼ /m ðtÞ; n
where xm ðtÞ 2 R , xp ðtÞ 2 Rn are the concentrations of mRNAs and proteins at time t for a GRN with n nodes, respectively; u1 ðtÞ 2 Rq , u2 ðtÞ 2 Rq are the control inputs; ym ðtÞ 2 Rr , yp ðtÞ 2 Rr represent the expression levels of mRNAs and proteins at time t, respectively; /m , /p are initial functions of xm ðtÞ and xp ðtÞ for t < 0, respectively; the time-varying scalars s1 ðtÞ > 0, s2 ðtÞ > 0 denote the feedback regulation delay and the translation delay, respectively, satisfying 1 6 1 and 0 6 s_ 2 ðtÞ 6 d 2 6 1; x1 ðtÞ and x2 ðtÞ are zero0 6 s_ 1 ðtÞ 6 d mean scalar Wiener process (Brownian Motion) on ðX; F ; PÞ with Efxi ðtÞg ¼ 0, Efx2i ðtÞg ¼ t ði ¼ 1; 2Þ, and they are mutually uncorrelated; v ðtÞ 2 RP the disturbance signal that belongs to L2 ½0; 1Þ; and T monotone function gðxp Þ ¼ ½ g 1 ðxp Þ g 2 ðxp Þ . . . g n ðxp Þ 2 Rn represents the feedback regulation of the protein on the transcription, whose elements are generally nonlinear function in the form of P SUM logic: g i ðxp1 ; xp2 ; ::; xpn Þ ¼ nj¼1 g ij ðxpj Þ where g ij is a monotonic function of Hill form. If transcription factor j is an activator of gene i, then: H xpjj Hj
and f i ð0Þ ¼ 0; g i ð0Þ
¼ 0; i ¼ 1; . . . ; n:
þ d2 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞdx2 ðtÞ
g ij ðxpj Þ ¼ qij
g i ðxp Þ 6 zi ; 8xp 2 R; xp – 0; xp
ð5Þ
b þ
H xpjj
ð2Þ
Assumption 3. The noise intensity vectors d1 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞ and d2 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞ in (1) satisfy the following conditions:
trace½dT1 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞd1 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞ 6 xTm ðtÞQ T1 Q 1 xTm ðtÞ þ xTp ðtÞQ T2 Q 2 xTp ðtÞ þ xTm ðt s2 ðtÞÞQ T3 Q 3 xTm ðt s2 ðtÞÞ þ xTm ðt s1 ðtÞÞQ T4 Q 4 xTm ðt s1 ðtÞÞ trace½dT2 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞd2 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞ 6 xTm ðtÞRT1 R1 xTm ðtÞ þ xTp ðtÞRT2 R2 xTp ðtÞ þ xTm ðt s2 ðtÞÞRT3 R3 xTm ðt s2 ðtÞÞ þ xTm ðt s1 ðtÞÞRT4 R4 xTm ðt s1 ðtÞÞ ð7Þ where Q 1 ; Q 2 ; Q 3 ; Q 4 ; R1 ; R2 ; R3 and R4 are the known matrices. The main objective of this Letter is to design a controller on the basis of estimator. Consider the state feedback controller as
43
H. Shokouhi-Nejad, A. Rikhtehgar-Ghiasi / Mathematical Biosciences 250 (2014) 41–53
u1 ðtÞ ¼ K 1 ^xm ðtÞ u2 ðtÞ ¼ K 2 ^xp ðtÞ
ð8Þ
said to be robustly asymptotically stable in the mean-square sense if limfkxm ðtÞk2 þ kxp ðtÞk2 g ¼ 0 holds for any initial condition. t!1
where K 1 and K 2 are appropriate dimension matrices to be determined. ^ xm ðtÞ 2 Rn and ^ xp ðtÞ 2 Rn are estimation the of xm ðtÞ and xp ðtÞ, respectively with following dynamics:
^m ðtÞÞ þ B1 u1 ðtÞdt d^xm ðtÞ ¼ ½Am ^xm ðtÞ þ L1 ðym ðtÞ y ^ ^ ^ dxp ðtÞ ¼ ½Ap xp ðtÞ þ L2 ðyp ðtÞ yp ðtÞÞ þ B2 u2 ðtÞdt
ð9Þ
where matrices Am , Ap , L1 and L2 are the observer parameters to be determined. xm ðtÞ, ep ðtÞ ¼ xp ðtÞ ^ xp ðtÞ and combinDefining em ðtÞ ¼ xm ðtÞ ^ ing system (1) and observer (9), yields the following augmented dynamics:
d^xm ðtÞ ¼ ½ðAm þ B1 K 1 Þ^xm ðtÞ þ L1 C 1 em ðtÞdt dem ðtÞ ¼ ½ðA1 þ DA1 ðtÞ L1 C 1 Þem ðtÞ þ ðA2 þ DA2 Þgðxp ðt s1 ðtÞÞÞ þ ðA1 þ DA1 Am þ DB1 K 1 Þ^xm ðtÞ þ ðG1 þ DG1 ðtÞÞv ðtÞdt þ d1 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞdx1 ðtÞ d^xp ðtÞ ¼ ½ðAp þ B2 K 2 Þxp ðtÞ þ L2 C 2 ep ðtÞdt dep ðtÞ ¼ ½ðA3 ðtÞ þ DA3 ðtÞ L2 C 2 Þep ðtÞ þ ðA4 þ DA4 ðtÞÞem ðt s2 ðtÞÞ þ ðA3 ðtÞ þ DA3 ðtÞ Ap þ DB2 ðtÞK 2 Þ^xp ðtÞ þ ðA4 þ DA4 ðtÞÞ^xm ðt s2 ðtÞÞ þ ðG2 þ DG2 ðtÞÞv ðtÞdt þ d2 ðxm ðtÞ; xp ðtÞ; xm ðt s2 ðtÞÞ; xp ðt s1 ðtÞÞÞdx2 ðtÞ
xm ðtÞ and Definition 3. For augmented system (10), u1 ðtÞ ¼ K 1 ^ xp ðtÞ are said to be H1 controllers if augmented system u2 ðtÞ ¼ K 2 ^ (10) with v ðtÞ ¼ 0 is mean-square asymptotically stable and also T there exists a l > 0 such that for zðtÞ ¼ eTm ðtÞ eTp ðtÞ
Z t J¼E zT ðsÞzðsÞ l2 v T ðsÞv ðsÞds < 0
ð12Þ
0
holds for all non-zero v ðtÞ 2 L2 ½0; 1Þ under zero initial conditions. Lemma 1 (Schur Complement [29]). Given constant matrices U1 ,
U2 , U3 where U1 ¼ UT1 and U2 ¼ UT2 > 0. Then U1 þ UT3 U1 2 U3 < 0 if T U2 U3 U U 1 3 < 0 or < 0. and only if UT3 U1 U3 U2 Lemma 2 [30]. Let X; Y and H be real matrices of appropriate dimension with HT H 6 I. Then, XHY þ Y T HT X T 6 eXX T þ e1 Y T Y holds for any scalar e > 0. 3. Main results
ð10Þ
In this section, we shall deal with the robust observer based controller design for the augmented system (10) with v ðtÞ ¼ 0 by the following theorem.
Definition 1. Itô formula [26]: Consider an n-dimensional stochastic vector process fXðt; xÞg with stochastic differential dXðt; xÞ ¼ Fðt; xÞdt þ Gðt; xÞdW t ðxÞ on t P 0, where W t ðxÞ is an m-dimensional Brownian motion defined on ðX; F ; fF t gt2R ; PÞ. Let V 2 ðRn Rþ ; Rþ Þ, then:
Theorem 1. If there exist five positive scalars k , e1 , e2 , q1 , q2 and 1 , P 2 , P 3 , P 4 , W 1 , W 2 , W 3 , W 4 and matrices positive definite matrices P _ _ 1; K 2 such that the following linear matrix A^m ; A^p ; L^1 ; L^2 ; P1 ; P2 ; K inequalities:
dVðtÞ ¼ LVðtÞdt þ V TX Gdx
ð11Þ
where LVðtÞ ¼ V t þ V Tx F þ 1=2trfGGT V xx g with V t ; V x and V xx denoting the partial derivative on t, the gradient of V and the Hessian matrix of V respectively, and trfg being the sum of the main diagonal entries.
(
2
Y T1 Y T2 6 e1 I 0 6 6 e2 I 4 1I P3 P q 2I P4 P q _ C 1 P3 ¼ P1 C 1
P
C1
0 0 1I q
C2
3
0 7 7 0 7 0 there exists a scalar rðeÞ > 0 such that Efkxm ðtÞk2 g < e and Efkxp ðtÞk2 g < e, 8t > 0 when sup Efk/m ðsÞk2 g < rðeÞ and
^ m þ B1 K ^T þ K 1 þ A T BT þ W 1 P1 ¼ A m 1 1
ð15Þ
sup Efk/p ðsÞk2 g < rðeÞ. Additionally, system (1) with
^ p þ B2 K ^T þ K 2 þ A T BT þ W 2 P2 ¼ A p 2 3
ð16Þ
s60
s60
v ðtÞ 0,
is
44
H. Shokouhi-Nejad, A. Rikhtehgar-Ghiasi / Mathematical Biosciences 250 (2014) 41–53
P3 ¼ A1 P 3 þ P 3 AT1 ^L1 C 1 C T1 ^LT1 þ W 3 þ e1 M1 MT1 þ e2 M3 MT3
ð17Þ
P4 ¼ A3 P 4 þ P 4 AT3 ^L2 C 2 C T2 ^LT2 þ W 4 þ e1 M2 MT2 þ e2 M3 MT3
ð18Þ
and
n n
o C2 ¼ diag P 1 RT1 ; P 2 RT2 ; P 3 RT1 ; P 4 RT2 ; P 1 RT3 ; P 2 RT4 ; P 3 RT3 ; P 4 RT4 ; 0
" Y1 ¼ " Y2 ¼
N1 P1 0 1 N1 K 0
0 2 N1 P
3 N1 P
0 3 0 0 N2 P # 0 0 0 0 0 0 0
0 2 N2 K
0 4 N2 P
0 1 N2 P
0
ð19Þ
0 N2 kI 0
#
0
0 0 0 0 0 0 0
ð20Þ
then, using:
1 P 1 ; K 2 ¼ K 2P 1 K1 ¼ K 1 2 ^mP ^pP 1 ; Ap ¼ A 1 Am ¼ A 1 2 L1 ¼ ^L1 P^ 1 ; L2 ¼ ^L2 P^ 1 1
P 3 6 q1 I
ð25Þ
P 4 6 q2 I then, according to Assumption 3, it is easy to see that
o
C1 ¼ diag P 1 Q T1 ; P 2 Q T2 ; P 3 Q T1 ; P 4 Q T2 ; P 1 Q T3 ; P 2 Q T4 ; P 3 Q T3 ; P 4 Q T4 ; 0
If there exist scalars qi ; i ¼ 1; 2 such that the following LMIs hold
ð21Þ
2
for the parameters of the controller and estimator, the augmented system (10) will be robustly asymptotically stable in the mean-square sense when v ðtÞ 0.
h Trace dT1 ð:ÞP 3 d1 ð:Þ 6 kmax ðP 3 Þ xTm ðtÞQ T1 Q 1 xTm ðtÞ þ xTp ðtÞQ T2 Q 2 xTp ðtÞ þxTm ðt s2 ðtÞÞQ T3 Q 3 xTm ðt s2 ðtÞÞi
þxTm ðt s1 ðtÞÞQ T4 Q 4 xTm ðt s1 ðtÞÞ h 6 q1 xTm ðtÞQ T1 Q 1 xTm ðtÞ þ xTp ðtÞQ T2 Q 2 xTp ðtÞ þxTm ðt s2 ðtÞÞQ T3 Q 3 xTm ðt s2 ðtÞÞi
þxTm ðt s1 ðtÞÞQ T4 Q 4 xTm ðt s1 ðtÞÞ trace dhT2 ð:ÞP4 d2 ð:Þ 6 kmax ðP 4 Þ xTm ðtÞRT1 R1 xTm ðtÞ þ xTp ðtÞRT2 R2 xTp ðtÞ þxTm ðt s2 ðtÞÞRT3 R3 xTm ðt s2 ðtÞÞi
þxTm ðt s1 ðtÞÞRT4 R4 xTm ðt s1 ðtÞÞ h 6 q2 xTm ðtÞRT1 R1 xTm ðtÞ þ xTp ðtÞRT2 R2 xTp ðtÞ þxTm ðt s2 ðtÞÞRT3 R3 xTm ðt s2 ðtÞÞi þxTm ðt s1 ðtÞÞRT4 R4 xTm ðt s1 ðtÞÞ
ð26Þ
Considering inequalities (6) and (26) and the facts that 0 6 Proof . Based on the system (10), we construct the following Lyapunov functional:
VðtÞ ¼ ^xTm ðtÞP1 ^xm ðtÞ þ ^xTp ðtÞP2 ^xp ðtÞ þ eTm ðtÞP3 em ðtÞ Z t ^xTm ðsÞP 5 ^xm ðsÞds þ eTp ðtÞP4 ep ðtÞ þ þ þ
Z Z
ts2 ðtÞ
t ts1 ðtÞ
^xTp ðsÞP6 ^xp ðsÞds þ
Z
þ 2eTm ðtÞP3 ðA2 þ DA2 ðtÞÞgðxp ðt s1 ðtÞÞÞ þ 2eTm ðtÞP3 ðA1 þ DA1 ðtÞ Am þ DB1 ðtÞK 1 Þ^xm ðtÞ
eTm ðsÞP7 em ðsÞds
þ eTp ðtÞ½2P 4 ðA3 þ DA3 ðtÞ L2 C 2 Þ þ P8 ep ðtÞ
t ts1 ðtÞ
eTp ðsÞP8 ep ðsÞds
ð22Þ
þ 2eTp ðtÞP4 ðA4 þ DA4 ðtÞÞem ðt s2 ðtÞÞ þ 2eTp ðtÞP4 ðA3 þ DA3 ðtÞ Ap þ DB2 ðtÞK 2 Þ^xp ðtÞ
By Itô formula, given in Definition 1, we can obtain the following stochastic differential along the trajectory of augmented system:
dVðtÞ ¼ LVðtÞdt þ 2 eTm ðtÞP3 d1 ðÞ dx1 ðtÞ þ 2 eTp ðtÞP 4 d2 ðÞ dx2 ðtÞ
ð23Þ
þ 2eTp ðtÞP4 ðA4 þ DA4 ðtÞÞ^xm ðt s2 ðtÞÞ 2 Þ^xT ðt s2 ðtÞÞP5 ^xm ðt s2 ðtÞÞ ð1 d m
1 Þ ^xT ðt s1 ðtÞÞP6 ^xp ðt s1 ðtÞÞ ð1 d p 2 ÞeT ðt s2 ðtÞÞP7 em ðt s2 ðtÞÞ ð1 d m
1 Þ eT ðt s1 ðtÞÞP8 ep ðt s1 ðtÞÞ ð1 d p
where
LVðtÞdt ¼ ^xTm ðtÞ½2P1 ðAm þ B1 K 1 Þ þ P5 ^xm ðtÞ
k g T ðxp ðt s1 ðtÞÞÞðgðxp ðt s1 ðtÞÞÞ Zep ðt s1 ðtÞÞ Z ^xp ðt s1 ðtÞÞÞ h þ q1 eTm ðtÞQ T1 Q 1 eTm ðtÞ þ eTp ðtÞQ T2 Q 2 eTp ðtÞ
þ 2^xTm ðtÞP1 L1 C 1 em ðtÞ þ ^xTp ðtÞ 2P2 ðAp þ B2 K 2 Þ þ P6 ^xp ðtÞ þ 2^xTp ðtÞP2 L2 C 2 ep ðtÞ
þeTm ðt s2 ðtÞÞQ T3 Q 3 eTm ðt s2 ðtÞÞ i þeTm ðt s1 ðtÞÞQ T4 Q 4 eTm ðt s1 ðtÞÞ h þ q1 ^xTm ðtÞQ T1 Q 1 ^xTm ðtÞ þ ^xTp ðtÞQ T2 Q 2 ^xTp ðtÞ
þ eTm ðtÞ½2P3 ðA1 þ DA1 ðtÞ L1 C 1 Þ þ P7 em ðtÞ þ 2eTm ðtÞP3 ðA2 þ DA2 ðtÞÞgðxp ðt s1 ðtÞÞÞ þ 2eTm ðtÞP3 ðA1 þ DA1 ðtÞ Am þ DB1 ðtÞK 1 Þ^xm ðtÞ
þ^xTm ðt s2 ðtÞÞQ T3 Q 3 ^xTm ðt s2 ðtÞÞ i þ^xTm ðt s1 ðtÞÞQ T4 Q 4 ^xTm ðt s1 ðtÞÞ h þ q2 eTm ðtÞRT1 R1 eTm ðtÞ þ eTp ðtÞRT2 R2 eTp ðtÞ
þ eTp ðtÞ½2P4 ðA3 þ DA3 ðtÞ L2 C 2 Þ þ P8 ep ðtÞ þ 2eTp ðtÞP4 ðA4 þ DA4 ðtÞÞem ðt s2 ðtÞÞ þ 2eTp ðtÞP4 ðA3 þ DA3 ðtÞ Ap þ DB2 ðtÞK 2 Þ^xp ðtÞ þ 2eTp ðtÞP4 ðA4
þeTm ðt s2 ðtÞÞRT3 R3 eTm ðt s2 ðtÞÞ i þeTm ðt s1 ðtÞÞRT4 R4 eTm ðt s1 ðtÞÞ h þ q2 ^xTm ðtÞRT1 R1 ^xTm ðtÞ þ ^xTp ðtÞRT2 R2 ^xTp ðtÞ
þ DA4 ðtÞÞ^xm ðt s2 ðtÞÞ ð1 s_ 2 ðtÞÞ^xTm ðt s2 ðtÞÞP5 ^xm ðt s2 ðtÞÞ ð1 s_ 1 ðtÞÞ ^xTp ðt s1 ðtÞÞP6 ^xp ðt s1 ðtÞÞ ð1 s_ 2 ðtÞÞeTm ðt s2 ðtÞÞP7 em ðt s2 ðtÞÞ ð1 s_ 1 ðtÞÞ eTp ðt s1 ðtÞÞP8 ep ðt s1 ðtÞÞ þ trace dT1 ðÞP3 d1 ðÞ þ trace dT2 ðÞP4 d2 ðÞ
LVðtÞdt 6 ^xTm ðtÞ½2P1 ðAm þ B1 K 1 Þ þ P5 ^xm ðtÞ þ 2^xTm ðtÞP1 L1 C 1 em ðtÞ þ ^xTp ðtÞ 2P2 ðAp þ B2 K 2 Þ þ P6 ^xp ðtÞ þ 2^xTp ðtÞP2 L2 C 2 ep ðtÞ þ eTm ðtÞ½2P3 ðA1 þ DA1 ðtÞ L1 C 1 Þ þ P7 em ðtÞ
t
ts2 ðtÞ
s_ 1 ðtÞ 6 d1 6 1, 0 6 s_ 2 ðtÞ 6 d2 6 1 the following inequality is true:
þ^xTm ðt s2 ðtÞÞRT3 R3 ^xTm ðt s2 ðtÞÞ i þ^xTm ðt s1 ðtÞÞRT4 R4 ^xTm ðt s1 ðtÞÞ ð24Þ
ð27Þ
45
H. Shokouhi-Nejad, A. Rikhtehgar-Ghiasi / Mathematical Biosciences 250 (2014) 41–53
2
where
6 6 6 6 6 6 6 6 6 U¼6 6 6 6 6 6 6 6 6 6 4
k>0 On the other hand, it is easy to see that the left hand side of (6) can be represented as: 2 3T 2 32 3 ep ep 0 0 Z=2 6 7 6 76 7 g T ðxp Þðgðxp Þ ðZep þ Z ^xp ÞÞ ¼ 4 ^xp 5 4 0 0 Z=2 54 x^p 5
gðxp Þ
gðxp Þ
I
Z=2 Z=2
ð28Þ Considering (27) and (28) it follows that: T
LVðtÞdt 6 h ðtÞKðtÞhðtÞ
U1
0
U9
0
0
0
0
0
U2
0
U10
0
0
0
0
U3
0
0
0 0
U4
0 A4 P1
0
0 3 A4 P
U5
0
0
0
U6
0
0
U7
0
U8
0
3
7 7 7 7 A2 k 7 7 7 0 7 7 7 0 7 7 P2 Z=2 7 7 7 7 0 7 7 4 Z=2 7 P 5 0
kI
ð35Þ
ð29Þ
where
with
T hðtÞ ¼ ^xTm ðtÞ ^xTp ðtÞ eTm ðtÞ eTp ðtÞ ^xTm ðt s2 ðtÞÞ x^Tp ðt s1 ðtÞÞ eTm ðt s2 ðtÞÞ eTp ðt s1 ðtÞÞ g T ðxp ðt s1 ðtÞÞÞ 2 6 6 6 6 6 6 6 6 6 6 KðtÞ ¼ 6 6 6 6 6 6 6 6 6 4
K1
0
K9
0
0
0
0
0
K2
0
K10
0
0
0
0
K3
0
0
0
0
0
K4
P4 ðA4 þ DA4 ðtÞÞ
0
P 4 ðA4 þ DA4 ðtÞÞ
0
K5
0
0
0
K6
0
0
0
3
7 7 7 7 P3 ðA2 þ DA2 ðtÞÞ 7 7 7 0 7 7 7 0 7 7 7 kðZ=2Þ 7 7 7 0 7 7 7 kðZ=2Þ 5 0
K7
0
K8
ð31Þ
kI
with:
1 1 1 1 P1 1 ¼ P1 ; P2 ¼ P2 ; P3 ¼ P3 ; P4 ¼ P4 ; k ¼ k
K1 ¼ P1 Am þ P1 B1 K 1 þ ATm P1 þ K T1 BT1 P1 þ P5 þ q1 Q T1 Q 1 þ q2 RT1 R1
^ m ¼ Am P ^ p ¼ Ap P 2 1 ¼ K 1 P1 ; K 2 ¼ K 2 P2 ; A 1 ; A K
K2 ¼ P2 Ap þ P2 B2 K 2 þ ATp P þ K T2 BT2 P 2 þ P6 þ q1 Q T2 Q 2 þ q2 RT2 R2
1 ¼ W 1 ; P2 P6 P 2 ¼ W 2 ; P 3 P7 P 3 ¼ W 3 ; P 4 P8 P 4 ¼ W 4 P1 P5 P h i ^ m þ B1 K ^T þ K 1 þ A T BT þ W 1 þ P 1 q Q T Q 1 þ q RT R1 P 1 U1 ¼ A 1 1 2 1 m 1 1 h i ^ p þ B2 K ^T þ K 2 þ A T BT þ W 2 þ P 2 q Q T Q 2 þ q RT R2 P 2 U2 ¼ A 1 2 2 2 p 2 2
K3 ¼ P3 ðA1 þ DA1 ðtÞÞ þ ðA1 þ DA1 ðtÞÞT P3 P3 L1 C 1 C T1 LT1 P3 þ P7 þ q1 Q T1 Q 1 þ q2 RT1 R1
K4 ¼ P4 ðA3 þ DA3 ðtÞÞ þ ðA3 þ DA3 ðtÞÞT P4 P4 L2 C 2 C T2 LT2 P4 T 1Q 2Q 2
þ P8 þ q
U3 ¼ A1 P 3 þ P 3 AT1 L1 C 1 P 3 P 3 C T1 LT1 þ W 3
T 2 R2 R2
h i 3 þ P3 q1 Q T1 Q 1 þ q2 RT1 R1 P
þq
K5 ¼ ð1 d2 ÞP5 þ q1 Q T3 Q 3 þ q2 RT3 R3
U4 ¼ A3 P 4 þ P 4 AT3 L2 C 2 P 4 P 4 C T2 LT2 þ W 4
K6 ¼ ð1 d1 ÞP6 þ q1 Q T4 Q 4 þ q2 RT4 R4 K7 ¼ ð1 d2 ÞP7 þ q1 Q T3 Q 3 þ q2 RT3 R3 K8 ¼ ð1 d1 ÞP8 þ q1 Q T4 Q 4 þ q2 RT4 R4 K9 ¼ P1 L1 C 1 þ ðA1 þ DA1 ðtÞÞT P3 ATm P3 þ K T1 DBT1 ðtÞP3 K10 ¼ P2 L2 C 2 þ ðA3 þ DA3 ðtÞÞT P4 ATp P4 þ K T2 DBT2 ðtÞP4 ð32Þ
ð33Þ
in which
1
h
3
T 1Q 4Q 4
2 3
i 4 þ q2 RT4 R4 P
^T U9 ¼ L1 C 1 P 3 þ P 1 AT1 A m ^T U10 ¼ L2 C 2 P 4 þ P2 AT3 A m
we have
LVðtÞdt 6 U þ DUðtÞ
h i 4 þ P4 q1 Q T2 Q 2 þ q2 RT2 R2 P h i U5 ¼ ð1 d2 ÞW 1 þ P 1 q1 Q T3 Q 3 þ q2 RT3 R3 P 1 h i U6 ¼ ð1 d1 ÞW 2 þ P 2 q1 Q T4 Q 4 þ q2 RT4 R4 P 2 h i U7 ¼ ð1 d2 ÞW 3 þ P 3 q Q T Q 3 þ q RT R3 P 3
U8 ¼ ð1 d1 ÞW 4 þ P 4 q
Then, pre- and post- multiply the two sides of (31) by
n o 1 1 1 1 1 1 1 1 Diag P1 1 ; P2 ; P3 ; P4 ; P1 ; P2 ; P3 ; P4 ; k I
ð30Þ
ð36Þ
ð34Þ and
46
H. Shokouhi-Nejad, A. Rikhtehgar-Ghiasi / Mathematical Biosciences 250 (2014) 41–53
2
1 DAT ðtÞ þ K T DBT ðtÞ 0 0 P 1 1 1
6 6 6 6 6 6 6 6 6 6 6 DUðtÞ ¼ 6 6 6 6 6 6 6 6 6 6 6 4
0
0
0
0
0
2 DAT ðtÞ þ K T DBT ðtÞ P 2 2 3
0
0
0
0
0
0
0
0
0
DA1 ðtÞP3 þ P 3 DAT1 ðtÞ
0
0
DA3 ðtÞP 4 þ P4 DAT3 ðtÞ
DA4 ðtÞP 1
3 0 DA4 ðtÞP
0
0
0
0
0
0
0
0
0
0
0
Above equation is equal to
ð38Þ
where
¼ FðtÞ
0
0
FðtÞ
"
0 0 M T1 0 0
" X2 ¼
0
0 0 M T3 0 0
0
ð39Þ
0
0 0 0 0 0
M T2
0 0 0 0 0
0
0 0 0 0 0
M T3
0 0 0 0 0
#T ð40Þ
#T ð41Þ
Using Lemma 2, it is clear that:
ð42Þ
therefore
þ e1 Y T Y 1 þ e1 Y T Y 2 LVðtÞ 6 U 1 2 1 2
6 LVðtÞ 6 6 4
U1
0
U9
0
0
0
0
0
U2
U10
0
0
0
0
0 3 U
0
0
0
0
0
4 U
A4 P 1
0
A4 P3
0
U5
0
0
0
U6
0
0
U7
0
U8
0
3
7 7 7 A2 kI 7 7 7 0 7 7 7 0 7 7 P2 Z=2 7 7 7 0 7 7 P4 Z=2 5 kI
7 0 7 5 e2 I
e1 I
ð46Þ
Since EfdVðtÞg ¼ EfLVðtÞdtg, in order to show that the augmented system (10) is asymptotically stable in mean square sense and the system (1) with v ðtÞ ¼ 0 is asymptotically stabilizable by observer-based control (8), it is enough to show that the right hand side of (46) is negative. Since the right hand side of (46) is not satisfied LMI conditions yet, we use schur complement again and yield to (13). Furthermore, making the changes in the ^mP ^p P 1P 1 , K 2 ¼ K 2P 1 , Am ¼ A 1 , Ap ¼ A 1 , variables as K 1 ¼ K 1 2 1 2 1 ^ 1 and L2 ¼ ^ ^ L1 ¼ ^ controller and observer gains can be L1 P L P 2 2 1 obtained. h
In this section, H1 observer based controller for the GRN (1) with exogenous disturbance is proposed. According to the following theorem, system (1) is mean-square robustly asymptotically stable with the attenuation level of l.
2
0
W Y T1
6 6 6 6 6 6 6 6 6 6 4
Y T2
Y T3
1 C
2 C
e1 I
0
0
0
0
e2 I
0
0
0
e3 I
0
0
1I q
0
2I q
3 P q 1I P 4 P q 2I P _
with
4 ¼ U4 þ e1 M 2 M T þ e2 M 3 MT U 2 3
3
k; e1 ; e2 ; Theorem 2. For given scalar l, if there exist positive scalars i ; W i ; i ¼ 1; :::; 4 and e3 ; q 1 ; q_2 and positive definite matrices P _ 1; K 2 such that following LMIs. matrices P 1 ; P 2 ; K
ð44Þ
3 ¼ U3 þ e1 M 1 M T þ e2 M 3 MT U 1 3
Y T2
ð43Þ
where
6 6 6 6 6 6 6 6 ¼6 U 6 6 6 6 6 6 6 4
ð37Þ
3.1. H1 controller
T T 1 T DUðtÞ 6 e1 X 1 X T1 þ e1 1 Y 1 Y 1 þ e2 X 2 X 2 þ e2 Y 2 Y 2
2
Y T1
U
and
X1 ¼
2
FðtÞ
7 7 7 7 7 DA2 ðtÞ kI 7 7 7 7 0 7 7 7 0 7 7 7 7 0 7 7 7 0 7 7 7 0 5 0 0
By Schur complement, inequality (43) is equivalent to
T T T T T T DUðtÞ ¼ X 1 FðtÞY 1 þ Y 1 F ðtÞX 1 þ X 2 FðtÞY 2 þ Y 2 F ðtÞX 2
3
0
C 1 P 3 ¼ P1 C 1 _
ð45Þ
C 2 P 4 ¼ P2 C 2 holds where:
3 7 7 7 7 7 7 0
ð65Þ
Now, per and post multiply the two side of (64) by
n o 1 1 1 1 1 1 1 1 Diag P1 1 ; P 2 ; P 3 ; P 4 ; P 1 ; P 2 ; P 3 ; P 4 ; k I; I therefore, the condition of (65) is changed to
8t > 0
H4 þ DH4 ðtÞ 6 0;
ð66Þ
where H4 is a constant matrix and DH4 ðtÞ is a time-varying matrix given by:
T z ðsÞzðsÞ l2 v T ðsÞv ðsÞ þ LVðsÞ ds
0
Z t _T ¼E h ðsÞH2 ðsÞhðsÞ ds
ð61Þ
0
where
2
K1
0
K9
0
0
0
0
0
K2
0
K10
0
0
0
0
K3 þ I
0
0
0
0
0
0
P4 ðA4 þ DA4 ðtÞÞ
0
6 6 6 6 6 6 6 6 6 6 6 H2 ðtÞ ¼ 6 6 6 6 6 6 6 6 6 6 6 4
K4 þ I P4 ðA4 þ DA4 ðtÞÞ
3
0
7 7 7 7 P3 ðA2 þ DA2 ðtÞÞ P3 ðG1 þ DG1 ðtÞÞ 7 7 7 0 P4 ðG2 þ DG2 ðtÞÞ 7 7 7 0 0 7 7 7 0 0 7 7 7 0 0 7 7 7 0 0 7 7 7 0 0 5 0
0
K5
0
0
0
K6
0
0
K7
0
K8
2
U9
0
0
0
0
0
0
0
0
U10
0
0
0
0
0
0
A2 kI
0 7 7 7 G1 7 7 G2 7 7 7 0 7 7 0 7 7 7 0 7 7 0 7 7 7 0 5
Using (6), (28), and (61), one can write:
JðtÞ 6 JðtÞ E
¼E
0
Z
t
8 > > > > > > > :
0
2
t
ep 6 6 k6 x^p 4 gðxp Þ
3 9 ep 0 0 Z=2 > > > = 6 76 7 > 6 76 ^ 7 0 Z=2 76 xp 7ds 6 0 4 54 5 > > > > ; gðxp Þ Z=2 Z=2 I
3T 2 7 7 7 5
32
h_T ðsÞH3 ðsÞhðsÞds
0
ð63Þ
U1 0 6 U2 6
6 6 6 6 6 6 6 H4 ¼ 6 6 6 6 6 6 6 6 6 4
ð62Þ
l2 I
U3 þ I
0
0
0
0
U4 þ I A4 P 1 0 A4 P 3 U5 0 0 U6 0 U7
0
0
0
0
0 0
U8
2 Z=2 P 0 4 Z=2 P
kI
3
l2 I ð67Þ
where
2
K1 6 6 6 6 6 6 6 6 6 H3 ðtÞ ¼ 6 6 6 6 6 6 6 6 6 4
0
K2
K9 0 0 0 0 K10 0 0 K3 þ I 0 0 0 K4 þ I P4 ðA4 þ DA4 ðtÞÞ 0 K5 0 K6
0
0
0
0
0
0
P4 ðA4 þ DA4 ðtÞÞ
0
0
0
0
0
K7
0 K8
0
0
3
7 0 0 7 7 P3 ðA2 þ DA2 ðtÞÞ P3 ðG1 þ DG1 ðtÞÞ 7 7 0 P4 ðG2 þ DG2 ðtÞÞ 7 7 7 7 0 0 7 7 kðZ=2Þ 0 7 7 7 0 0 7 7 kðZ=2Þ 0 7 7 5 kI 0
l2 I
ð64Þ
49
H. Shokouhi-Nejad, A. Rikhtehgar-Ghiasi / Mathematical Biosciences 250 (2014) 41–53
1 DAT ðtÞ þ K T DBT ðtÞ 0 0 P 1 1 1
6 6 6 6 6 6 6 6 6 6 DH4 ðtÞ ¼ 6 6 6 6 6 6 6 6 6 4
0
0
0
0
0
0 0
0
0
T DBT ðtÞ P 2 DAT3 ðtÞ þ K 2 2
0
0
0
0
DA1 ðtÞP 3 þ P 3 DAT1 ðtÞ
0
0
0
0
0 DA2 ðtÞ kI
DA3 ðtÞP 4 þ
4 DAT ðtÞ P 3
DA4 ðtÞP 1 0
It is not difficult to see that:
3
0 0 M T1 0 0
" 2 ¼ X
0
"
0 0 0 0 0 0
M T2
0 0 0 0 0 0
0
0 0 0 0 0 0
M T3
0 0 0 0 0 0
0
0 0 0 0 0 0
0 0 M T3 0 0
3 ¼ X
0
0
0 0 MT4 0 0
MT5
0
0
0
0
0
0 0
0 0
0 0
0
0
0
0
)
ð73Þ
#T
In this section, in order to show the effectiveness of the approaches in Section 3, a numerical example is provided. Consider the dynamic of the synthetic oscillatory network for Escherichia coli as follow [31]:
#T
8 dm ðtÞ a < dti ¼ mi ðtÞ þ 1þp n
i ¼ lacl; tetR; cl
: dpi ðtÞ
j ¼ cl; lacl; tetR
j
dt
3
ð70Þ
then
T e1 i Yi Yi
¼ bðpi mi Þ
6 6 6 6 6 6 6 6 6 H¼6 6 6 6 6 6 6 6 6 4
3 0 0 0:02 7 6 7 6 0 A2 ¼ 6 0:2 0 7; 5 4 0 0:2 0
A1 ¼ diagf5; 2; 0:05g; 3
U1
0
U9
0
0
0
0
0
0
0
U2
U10
0
0
0
0
0
0 ^3 U
A2 kI
0
0 3 A4 P
0
0 1 A4 P
0
0 ^4 U
0
0
U5
0
0
0
0
U6
0
0
P 2 Z=2
0
0
0 7 7 7 G1 7 7 G2 7 7 7 0 7 7 0 7 7 7 0 7 7 0 7 7 7 0 5
U7
U8 P 4 Z=2
kI
l2 I
^ 3 ¼ U3 þ I þ e1 M 1 M T þ e2 M3 MT þ e3 M 4 M T U 1 3 4 ^ 4 ¼ U4 þ I þ e1 M 2 M T þ e2 M3 MT þ e3 M 5 M T U 2
3
A3 ¼ diagf3:5; 3:5; 3:5g; B1 ¼ diagf0:1; 0:2; 0:2g;
A4 ¼ diagf1:2; 3:1; 0:03g
B2 ¼ diagf0:3; 0:2; 0:3g;
C 1 ¼ diagf0:1; 0:24; 1:7g;
C 2 ¼ diagf1:4; 0:2; 1:22g
s1 ðtÞ ¼ 0:2 þ 0:1sinð0:3tÞ; s2 ðtÞ ¼ 0:6 þ 0:1 sinð0:7tÞ; g i ðxp Þ ¼ x2p =ð1 þ x2p Þ; i ¼ 1;2;:::; n: Z ¼ diagf0:6; 0:6; 0:6g;
It is easy to check that the maximal value of the derivative of g i ðxp Þ is less than zi ¼ 0:6. Without loss of generality, the noise intensify vectors are set as
5
ð72Þ Using Schur Complement twice on right hand side of (71), and observing (47), it can be inferred that the right-hand of (71) is negative definite, and therefore H4 þ DH4 ðtÞ < 0, 8t > 0. h
ð74Þ
2
ð71Þ
where
!
Considering the stochastic and delayed characters of their interactions, the above equation can rearrange into genetic regulatory network (1). It is assumed that the third mRNA, cl, cannot work as well as other mRNAs and proteins. The parameters of the system are specified as follow:
i¼1
2
ð68Þ
4. Simulation results
0 0 0 0 0 0
3 X
0
#T
1X T þ e1 Y T Y 1 þ e2 X 2X T þ e1 Y T Y 2 þ e3 X 3X T DH4 ðtÞ 6 e1 X 1 1 1 2 2 2 3 1 T þ e Y Y3
þ H4 þ DH4 ðtÞ 6 H
0
¼ l2 xml_add>subject to (47) with l
then, considering Lemma 2, we have:
3
0
7 7 7 7 DG1 ðtÞ 7 7 DG2 ðtÞ 7 7 7 0 7 7 0 7 7 7 0 7 7 0 7 7 7 0 5 0
ð69Þ
3
1; Y 2; Y 3 are given in (51)–(53) and: where Y
"
0 DA4 ðtÞP 3
3
Corollary 1. To reduce the effects of the disturbance vector, the following linear objective minimization problem can be used
1 FðtÞ Y 1 þ Y T FðtÞ X T þ X 2 FðtÞ Y 2 þ Y T FðtÞ X T DH4 ðtÞ ¼ X 1 1 2 2 3 FðtÞ Y 3 þ Y T FðtÞ X T þX
1 ¼ X
0
)
2
R1 ðÞ ¼ Q 1 xm ðtÞ þ Q 2 xp ðtÞ þ Q 3 xm ðt s2 ðtÞÞ þ Q 4 xp ðt s1 ðtÞÞ R2 ðÞ ¼ R1 xm ðtÞ þ R2 xp ðtÞ þ R3 xm ðt s2 ðtÞÞ þ R4 xp ðt s1 ðtÞÞ For simplify, we just let
50
H. Shokouhi-Nejad, A. Rikhtehgar-Ghiasi / Mathematical Biosciences 250 (2014) 41–53
2 Q 1 ¼ R1 ¼ 0;
0:1 0
6 Q 2 ¼ R2 ¼ 4 0
0:2
0:1
0 3 0:2 0 0:1 6 7 Q 3 ¼ R3 ¼ 4 0 0:1 0 5 0:1 0 0:15
0
3
2
7 5;
6 M 3 ¼ 4 0:18
0:1 0:2
2
G2 ¼ diagf0:5; 0:7; 0:2g;
6
4
4
2
2
0
-2
-6
-6
4
6
8
10
12
14
16
18
-8
20
0
4
2
6
8
0.25
0.2
0.2
0.15
0.15
Protein 2
mRNA 2
0.25
0.1
10
0.05
0
0
-0.05
-0.05
2
4
6
8
10
12
14
16
18
-0.1
20
0
2
4
6
8
35
15
30
10
25
5
20
Protein 3
mRAN 3
20
0
10
-10
5
-15
0
2
4
6
8
10
12
Time (Sec.)
16
18
20
10
12
14
16
18
20
15
-5
0
14
Time (Sec.)
Time (Sec.)
-20
12
0.1
0.05
0
0:2
Time (Sec.)
Time (Sec.)
-0.1
6 M 4 4 0:18
-2
-4
2
7 0:11 5;
0:2
0:53 0:16
0
-4
0
2
0:6
3
7 0:11 5;
By Theorem 2 and Corollary 1, using YALMIP toolbox in MATLAB, the feasible solutions of (73) can be obtained. Here, we have
M 2 ¼ diagf0:33; 0:29; 0:31g
6
-8
3
0:21 0:17 0:3 0:21 0:17 0:3 3 0:4 0:2 0:1 6 7 M 5 ¼ 4 0:1 0:3 0:2 5; N 1 ¼ diagf0:8; 0:55; 0:21g; 0:1 0:2 0:3 2 3 0:4 0:6 0:3 6 7 N2 ¼ diagf0:1; 0:3; 0:9g; N3 ¼ 4 0:3 0:55 0:67 5 0:21 0:3 0:2
Protein 1
mRNA 1
M 1 ¼ diagf0:43; 0:3; 0:4g;
0:6
2
and Q 4 ¼ R4 ¼ 0. Then, we choose the following additional matrices:
G1 ¼ diagf0:9; 0:3; 0:3g;
0:53 0:16
14
16
18
20
-5 0
2
4
6
8
10
12
Time (Sec.)
Fig. 1. Concentrations of the mRNA (left column) and protein (right column) with u(t) = 0.
14
16
18
20
51
H. Shokouhi-Nejad, A. Rikhtehgar-Ghiasi / Mathematical Biosciences 250 (2014) 41–53
2
3
2
0:0062 0:0007 0:0002 6 7 0:0006 5; P1 ¼ 4 0:0007 0:0096 0:0002 0:0006 0:0281 2 3 0:0063 0:0001 0:0006 6 7 P2 ¼ 4 0:0001 0:0056 0:0013 5; 2
0:0000
0:0158 0:0035 0:0031 6 7 0:0059 5; P5 ¼ 4 0:0035 0:0351 0:0031 0:0059 0:2789 2 3 0:0126 0:0001 0:0018 6 7 P6 ¼ 4 0:0001 0:0105 0:0040 5;
0:0159 3
0:0006 0:0013 0:0050
2
0:0001
6 7 P3 ¼ 4 0:0000 0:0059 0:0001 5; 2
0:0001 0:0001 0:0071 0:0020
0:0001 0:0002
3
0:0116
6 P7 ¼ 4 0:0000 3
2
6 7 P4 ¼ 4 0:0001 0:0027 0:0002 5; 0:0002 0:0002 0:0067
0:0146 3
0:0018 0:0040
0:0008 0:0015
0:0008
0:0000 0:0174
7 0:0007 5;
0:0007
0:0397
0:0001 0:0002
3
6 7 P8 ¼ 4 0:0001 0:0027 0:0004 5 0:0002 0:0004 0:0029
0.2
0.2 0.15
0.15
Protein 1
mRNA 1
0.1 0.05 0
0.1
0.05
-0.05 0 -0.1 -0.15
0
1
2
3
4
5
6
7
-0.05
8
0
1
2
3
0.2
0.2
0.15
0.15
0.1
0.05
0
-0.05
4
5
6
7
8
6
7
8
Time (Sec.)
Protein 2
mRNA 2
Time (Sec.)
0.1
0.05
0
0
1
2
3
4
5
6
7
-0.05
8
0
1
2
3
Time (Sec.)
4
5
Time (Sec.)
0.25
0.2
0.2 0.15
Protein 3
mRAN 3
0.15 0.1 0.05
0.1
0.05
0 0 -0.05 -0.1
0
1
2
3
4
5
Time (Sec.)
6
7
8
-0.05
0
1
2
3
4
5
6
7
8
Time (Sec.)
Fig. 2. The true values (solid line) and the estimates (dashed line) of the concentrations of the mRNA (left column) and protein (right column) with proposed controller.
52
H. Shokouhi-Nejad, A. Rikhtehgar-Ghiasi / Mathematical Biosciences 250 (2014) 41–53
2
198:6215
6 P^ 1 ¼ 4 2:5206
0:4376
0:1797
3
and with the following parameter variations:
7 0:2759 5; 51:9375 13:8428 141:1335 2 3 506:0167 85:4435 18:3874 6 7 P^ 2 ¼ 4 1:7437 369:1918 2:0288 5; 13:9632 75:4923 149:3526 l ¼ l2 ¼ 0:0052; e1 ¼ 350:6; e2 ¼ 581:7;
2
169:4216
6 Q ðtÞ ¼ 4
e3 ¼ 1210; k ¼ 0:0051; q1 ¼ 0:0084; q2 ¼ 0:0078 Therefore, the controller and observer parameters can be obtained as follow:
2
0:3536 6 K 1 ¼ K 1 P 1 ¼ 4 0:0504 1:5456 2 6:8727 2 P2 ¼ 6 K2 ¼ K 4 0:0114 2
0:3558
0:1238
L2 ¼ ^L2 P^ 1 2
3
7 1:3069 5; 2:9332 28:1318 3 0:5027 3:5175 7 2:2984 2:7642 5; 3:5151
0:6849
9:3482 3
0
0
0
0:2 þ 0:2 sinðtÞ
0
0
0
0:3 þ 0:3 cosðtÞ
3 7 5
The states of the system (1) with uðtÞ ¼ 0 is shown in Fig. 1, so it is not mean-square asymptotically stable. From Corollary 1, we know that the system (1) is mean-square robustly asymptotically stabilizable. Using Corollary 1, the states of the true values and the estimates of the concentrations of the mRNA and protein are shown in Fig. 2 where the left and the right columns are devoted to the concentrations of mRNA and protein, respectively. Moreover, Fig. 3 shows the ratio of the energy of the T zðtÞ ¼ eTm ðtÞ eTp ðtÞ to the energy of the disturbance v ðtÞ where initial conditions are set to zero. It is clear that this ratio is less than ¼ 0:0052 for all t. the optimal value l 5. Conclusions
1:5752 0:2132 4:4390
^ m P1 ¼ 6 0:7170 Am ¼ A 4 0:0837 0:7424 1:6636 2 8:5101 0:3412 ^ p P2 ¼ 6 Ap ¼ A 3:7512 4 0:4447
L1 ¼ ^L1 P^ 1 1
0:6078
0:1 þ 0:1 sinðtÞ
7 4:1725 5; 20:0061 3 2:1243 7 1:8262 5;
0:5013 0:1765 4:1780 2 3 26:2158 1:4243 0:7272 6 7 7:1360 0:6318 5; ¼ 4 2:8021 9:0989 3:8621 3:0297 2 3 1:1724 1:1748 0:7140 6 7 ¼ 4 0:1062 3:6399 0:5689 5 0:0989 0:9398 1:8201
To simulate this example, we run system (1) for the one second with zero exogenous disturbances, and thereafter, parameter variT ations are considered with disturbance v ðtÞ ¼ ½ et et et uðtÞ where uðtÞ ¼ 1 when t P 0, and uðtÞ ¼ 0 for t < 0. We run the simulation with the following initial conditions:
/m ðtÞ ¼ ½ 0:2 0:2 0:2 T ;
/p ðtÞ ¼ ½ 0:2 0:2 0:2 T ;
In this research, the robust controller on the basis of observer for a model of uncertain time delay stochastic genetic regulatory networks has been investigated. Based on Lyapunov stability theory, LMI conditions, which could be easily solved by using the Matlab LMI control toolbox, have been proposed to design observer based controller such that, for all admissible nonlinearities and time delays, the overall error dynamic is robustly asymptotically stable in the mean square sense and a prescribed H1 disturbance attention level is guaranteed. A numerical example is finally presented to illustrate the application and effectiveness of the obtained results. It should be pointed out that in the GRN model (1), we did not consider any constraints on the delays, and thus, the designed controller is delay-independent which naturally involves conservatism. To reduce conservatism, one can exert delay bounds as design constraints, which yields a delay-dependent controller design. To design such a delay-dependent filter, one should change the Lyapunov function in an appropriate way. In addition, there are some new methods in delay-dependent controller design such as free-weighting matrices and delay-partitioning approaches which
^xm ð0Þ ¼ ½ 0:1 0:1 0:1 T ;
^xp ð0Þ ¼ ½ 0:1 0:1 0:1 T
can be employed to further reduce conservatism of the design. ∫ z(t)Tz(t) / ∫ v(t)Tv(t)
-3
3.5
x 10
References
3
Magnitude
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
6
7
Time (Sec.) Fig. 3. The ratio of the energy of z(t) to the energy of v(t).
8
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