ISSN 01464116, Automatic Control and Computer Sciences, 2010, Vol. 44, No. 4, pp. 234–245. © Allerton Press, Inc., 2010. .
Design of DelayRangeDependent Robust Controller for Uncertain Genetic Regulatory Networks with Interval Time Varying Delays1 ChinWen Liao and ChienYu Lu Department of Industrial Education and Technology, National Changhua University of Education, Changhua, Taiwan 500, R. O. C. email:
[email protected] Abstract—This paper investigates the problem of robust stabilization for genetic regulatory networks with interval timevarying delays, which are subject to normbounded timevarying parameter uncer tainties. The time delays including lower and upper bounds of delay are assumed to appear in both the mRNA and protein. The regulatory functions are assumed to be globally Lipschitz continuous. The resulting delayrangedependent robust controller with interval range is designed in terms of improved bounding technique. A sufficient condition for the solvability of the problem is obtained via a linear matrix inequality (LMI). When this LMI is feasible, an explicit expression of a desired state feedback controller is also given. The theory developed in this paper is demonstrated by two numerical exam ples. Key words: delayrangedependent, genetic regulatory networks, interval timevarying delay, linear matrix inequality DOI: 10.3103/S0146411610040073
1
1. INTRODUCTION During the past decades, considerable attention has been devoted to the study of regulatory networks due to the fact that regulatory modeling has played an important role in biological and biomedical sciences [1–4]. Each gene contains some regulatory sequences, called cis elements. Transcription factors and their cofactors as well as other proteins can bind to such elements, and increase or reduce the gene expression levels. This result leads to the change of the corresponding protein levels, which in turn affect other genes’ expression levels. The mechanisms which have evolved to regulate the expression of genes are known as genetic regulatory networks. Theoretical studies on genetic networks may not only contribute to the understanding of the gene expression functions, but also have potential significance on engineering appli cations, such as developing circuits and systems with biotechnological design principles of synthetic genetic regulatory networks and new kinds of integrated circuits like neurochips learnt from biological neural networks [5–6]. Genetic networks are biochemically dynamical systems, and it is natural to model genetic networks by using dynamical system models, which provide a powerful tool for studying gene regulation processes in living organisms. It is known that living organisms respond to external signals thanks to a large variety of genetically precoded responses which is achieved through networks of gene of high connectivity and com plexity. Basically, there are two types of genetic network models, i.e., the Boolean model [7], and the dif ferential equation model [8–10]. In the Boolean model, the activity of each gene is expressed by two states, ON or OFF, which are determined by a Boolean function of its own and other related states. On the other hand, the differential equation model is utilized to describe the concentrations of gene products, such as mRNA and proteins, which are used as the state variables in genetic regulatory networks. As con tinuous values of the gene regulatory systems, using the differential equation approach is more accurate and being able to be a powerful tool for discovering higher order structure of an organism and for gaining deep insights into dynamic behaviors of genetic networks by extracting functional information from
1 The article is published in the original.
234
DESIGN OF DELAYRANGEDEPENDENT ROBUST CONTROLLER
235
observation data. Time delay is an important aspect in the accurate predictions [11]. Time delays are often encountered in biological systems and artificial genetic networks due to slow biochemical reactions such as gene transcription and translation, and protein diffusion between the cytosol and uncleus or the finite switching speed of amplifiers [12]. It has been shown that the existence of time delay is frequently more complicated in a genetic network. Therefore, analysis of timedelay systems has attracted a great deal of attention during the past years. Various types of time delays have been considered and a great number of results have been reported [13–17]. Recently, genetic regulatory networks with time varying have been studied in the literature [8, 18]. For instance, some robust stability results for genetic regulatory networks with different types of parameter uncertainties were proposed in [19–20], respectively. The aforemen tioned results for [20] are delaydependent conditions. This paper deals with the problem of robust controller for uncertain genetic regulatory networks with interval timevarying delays and control input. The time delays are assumed to appear in both mRNA and protein states, and the parameter uncertainties are assumed to be timevarying but normbounded which appear in all the matrices in mRNA and protein states. A delayrangedependent condition with time varying delays in a range for the existence of controllers is proposed and an LMI approach is developed. A controller is sought to guarantee that the resulting system is global asymptotical stability for all admissible uncertainties. An important feature of the results lies in that null equalities in terms of Leibniz⎯Newton formula are not introduced into this paper. Consequently, the advantage of a closed−loop system in this paper is to introduce into several free weighting matrices and reduce the conservatism of the criterion by applying the new bounding technique. Desired controllers can be obtained by the solution to certain LMIs, which can be solved numerically and efficiently by resorting to standard numerical algorithms [21]. Finally, illustrative examples are provided to demonstrate the effectiveness of the proposed method. 2. PRELIMINARIES The activities of a gene is regulated by other genes through the interactions among them, i.e., by tran scription and translation. Consider the following uncertain genetic regulatory network with interval time varying delay m· (t) = –(A + ΔA(t)m(t) + (G0 + ΔG0(t))g(p(t)) + (G1 + ΔG1(t))g(p(t – τp(t))) + I(t), (1) p· (t) = –(C + ΔC(t))p(t) + (D + ΔD(t))m(t – τm(t)),
(2)
where m(t) = [m1(t), m2(t), …, mn(t)]T is the mRNA state vector, p(t) = [p1(t), p2(t), …, pn(t)]T is the protein state vector, A = diag(a1, a2,…, an) > 0 and C = diag(c1, c2,…, cn) > 0, i = 1, 2, …, n are the decay rates of n×n
n×n
mRNA and protein, respectively. D = diag(d1, d2, …, dn) is the translation rate, G 0 and G 1 are the interconnection matrices representing the weighting coefficients of the genes, g(p(t)) = [g1(p1(t)), …, gn(pn(t))]T ∈ Rn is the feedback regulation of the protein on transcription with g(0) = 0, τp(t) and τm(t) are the timevarying delays of the initial translation to emergence of a complete functional protein molecule and the initial transcription to the arrival of the mature mRNA, respectively. τp(t) and τm(t) satisfy τp1 < τp(t) ≤ τp2, τ· p (t) ≤ dp, (3) and τ m1 ≤ τm(t) ≤ τ m2 ,
τ· m (t) ≤ dm,
(4)
where 0 ≤ τp1 < τp2, 0 ≤ τ m1 < τ m2 and dp and dm are known constants. I(t) is the control input vector. ΔA(t), ΔG0(t), ΔG1(t), ΔC(t) and ΔD(t) are normbounded unknown matrices with timevarying parameter uncertainties, which are assumed to be of the form (5) [ΔA(t) ΔG0(t) ΔG1(t) ΔC(t) ΔD(t)] = MF(t)[E1 E20 E21 E3 E4], where M, E1, E20, E21, E3 and E4 are known real constant matrices, and F(⋅): R Rk × l is unknown time varying matrix function satisfying FT(t)F(t) ≤ I. (6) The uncertain matrices ΔA(t), ΔG0(t), ΔG1(t), ΔC(t) and ΔD(t) are said to be admissible if both (5) and (6) hold. I(t) denotes the external control input and satisfies I(t) = Km(t), (7) with K is a gain matrix to be determined. AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 4
2010
236
CHINWEN LIAO, CHIENYU LU
Applying the state feedback controller in (7) to system (1) results in the following closedloop systems m· (t) = –(A – KI + ΔA(t)m(t) + (G0 + ΔG0(t))g (p(t)) + (G1 + ΔG1(t))g(p(t – τp(t))), (8) p· (t) = –(C + ΔC(t))p(t) + (D + ΔD(t))m(t – τm(t)).
(9)
Remark 1. It is worth noting that the structure of the parameter uncertainties with (5) and (6) has been widely adopted to deal with the robust stability and stabilization problem for uncertain systems in the open literature [19, 22 and 23]. The parameter uncertainties are introduced into the model of the considered genetic regulatory network because of the modeling inaccuracies and/or changes in the environment. If globally robust stabilization is considered, the synthetic genetic regulatory network can eliminate larger parameter variations such that system can work more reliably. In this way, the robust controller problem addressed in this paper is more general and practical than the one investigated in [19–20]. In order to obtain our main results, the functions of the feedback regulation of the protein on transcrip tion in (1) and (2) are assumed to be bounded and satisfy the following assumption. Assumption 1. The functions of the feedback regulation of the protein on transcription gi(⋅), i = 1, 2, …, n, are globally Lipschitz and monotone nondecreasing; that is, there exist constant scalars σi such that for any ζ1, ζ2 ∈ R and ζ1 ≠ ζ2, gi ( ζ1 ) – gi ( ζ2 ) 0 ≤ ≤ σi , ζ1 – ζ2
i = 1, 2, …, n.
(10)
Remark 2. gi(⋅) usually takes the Michaelis–Menten or Hill form. In this paper, gi(⋅) is considered as n gi(p1(t), p2(t), …, pn(t)) = g (pj(t)), which is called SUM logic [24–25] because each transcription j = 1 ij factor acts additively to regulate the i th gene. gij(⋅) is a monotonic function of the Hill form, that is to, if transcription factor j is an activator of gene i, then
∑
H
( p j ( t )/β j ) j ; g ij ( p j ( t ) ) = α ij H 1 + ( p j ( t )/β j ) j if transcription factor j is a repressor of gene i, then H ⎛ ( p j ( t )/β j ) j ⎞ 1 ⎟ , g ij ( p j ( t ) ) = α ij H = α ij ⎜ 1 – H ⎝ 1 + ( p j ( t )/β j ) j 1 + ( p j ( t )/β j ) j⎠
where Hj are the Hill coefficients, βj is a positive constant, and αij is a bounded constant, which is the dimensionless transcriptional rate of transcription factor j to i. Remark 3. There exists an equilibrium point for the genetic regulatory network in (1) and (2). The stabilization problem we address in this paper is to design a state feedback controller in (7) such that the resulting closedloop system (8) and (9) is globally asymptotically robustly stable. 3. MATHEMATICAL FORMULATION OF THE PROPOSED APPROACH This section explores the globally delayrangedependent robust stabilization conditions given in (8) and (9). Specially, for given scalars 0 ≤ τ p1 < τ p2 and 0 ≤ τ m1 < τ m2 , an LMI approach is employed to solve the controller gain matrix in the form of (8) and (9) such that for interval timevarying delays τp(t) and τm(t) satisfying the (3) and (4) are globally asymptotically robust stability. The analysis commences by using the LMI approach to develop some results which are essential to introduce the following Lemma 1 and Lemma 2 [26] for the proof of our main theorem in this section. Lemma 1. Let A, D, S, F, and P be real matrices of appropriate dimensions with P > 0 and F satisfying FT(k)F(k) ≤ I. Then the following statement holds for any ε > 0 and vectors x, y ∈ Rn 2xTDFSy ≤ ε–1xTDDTx + εyTSTSy. For presentation convenience, defining the following new variables
(11)
T η(t) = [mT(t) mT (t – τm(t)) mT (t – τm1) mT(t – τm2) fT(p(t)) fT(p(t – τp(t)) m· (t) pT(t) pT(t – τp(t))
AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 4
2010
DESIGN OF DELAYRANGEDEPENDENT ROBUST CONTROLLER
237
T pT(t – τp1) pT(t – τp2) p· (t)]T,
N = [N1 N2 N3 N4 N5 N6 N7 0 0 0 0 0], S = [S1 S2 S3 S4 S5 S6 S7 0 0 0 0 0], T = [T1 T2 T3 T4 T5 T6 T7 0 0 0 0 0], T
X = [0 0 0 0 0 0 X 1 0 0 0 0 0 ]T, R = [0 R1 0 0 0 0 0 R2 R3 R4 R5 R6], H = [0 H1 0 0 0 0 0 H2 H3 H4 H5 H6] , Γ = [0 Γ1 0 0 0 0 0 Γ2 Γ3 Γ4 Γ5 Γ6], T
T
T
T
T
T T
L = [0 L 1 0 0 0 0 0 L 2 L 3 L 4 L 5 L 6 ] , τp21 = τp2 – τp1 and τm21 = τm2 – τm1. Lemma 2. For real matrix Z11 > 0, Z21 > 0, Z12 > 0, and Z22 > 0, Ni, Si, Ti (i = 1, 2, …, 7), Ri, Hi, Γi, Li (i = 1, 2, …, 6), and X1 with appropriate dimensions, and τp(t) and τm(t) satisfying (3) and (4), then t
∫
–
T T T –1 T T m· ( s )Z 11 m· ( s ) ds ≤ τ m2 η ( t )N Z 11 Nη ( t ) + 2η ( t )N I 1 η ( t ),
(12)
t – τ m2 t – τm ( t )
–
∫
T T T –1 T T m· ( s ) ( Z 11 + Z 21 )m· ( s ) ds ≤ τ m21 η ( t )S ( Z 11 + Z 12 ) Sη ( t ) + 2η ( t )S I 2 η ( t ),
(13)
t – τ m2 t – τ m1
∫
T T T –1 T T m· ( s )Z 21 m· ( s ) ds ≤ τ m21 η ( t )T Z 21 Tη ( t ) + 2η ( t )T I 3 η ( t ),
(14)
T T T –1 T T p· ( s )Z 12 p· ( s ) ds ≤ τ p2 η ( t )R Z 12 Rη ( t ) + 2η ( t )R I 4 η ( t ),
(15)
T T T –1 T T p· ( s ) ( Z 22 + Z 12 )p· ( s ) ds ≤ τ p21 η ( t )H ( Z 12 + Z 22 ) Hη ( t ) + 2η ( t )H I 5 η ( t ),
(16)
–
t – τ m2 t
–
∫
t – τp ( t ) t – τm ( t )
–
∫
t – τ p2 t – τ p1
–
∫
T T T –1 T T p· ( s )Z 22 p· ( s ) ds ≤ τ p21 η ( t )Γ Z 22 Γη ( t ) + 2η ( t )Γ I 6 η ( t ),
t – τ p2
where I1 = [I –I 0 0 0 0 0 0 0 0 0 0],
I2 = [0 I 0 –I 0 0 0 0 0 0 0 0],
I3 = [0 –I I 0 0 0 0 0 0 0 0 0],
I4 = [0 0 0 0 0 0 0 I –I 0 0 0],
I5 = [0 0 0 0 0 0 0 0 I 0 –I 0],
I6 = [0 0 0 0 0 0 0 0 –I I 0 0].
Proof. Since T
–1
T
T
T –1 T –1 I – N Z 11 N Z 11 N N I – N Z 11 N Z 11 0 0 I I
= 0 0 ≥ 0, 0 Z 11
then, one has AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 4
2010
(17)
238
CHINWEN LIAO, CHIENYU LU –1
T
N Z 11 N N N
T
≥ 0.
(18)
Z 11
From (18), it follows that t
∫
0≤
t – τm ( t )
η(t) m· ( s )
T
–1
T
N Z 11 H N N
T
Z 11
t
η ( t ) ds ≤ τ η T ( t )N T Z –1 Nη ( T ) + 2η T ( t )N T m2 11 m· ( s ) t–τ
∫
t
+
m· ( s ) ds
m(t)
t
∫
T T T –1 T T m· ( s )Z 11 m· ( s ) ds = τ m2 η ( t )N Z 11 Nη ( t ) + 2η ( t )N ( m ( t ) – m ( t – τ m ( t ) ) ) +
t – τm ( t )
∫
T m· ( s )Z 11 m· ( s ) ds ,
t – τm ( t )
which rearranged gives t
∫
–1 T m· ( s )Z 11 m· ( s ) ds ≤ τm2ηT(t)NT Z 11 Nη(t) + 2ηT(t)NTI1η(t).
t – τm ( t )
Obviously, (12) holds. The rest of the proof of this Lemma 2 is similar to that of above proof and is omitted here. The following theorem will play an important role in the solvability of the robust genetic regulatory net work stabilization problem. Theorem 1. Under Assumption 1, given scalars 0 ≤ τp1 < τp2, 0 ≤ τ m1 < τ m2 , the equilibrium point of (1) and (2) with interval timevarying delay τp(t) and τm(t) satisfying (3) and (4) is globally asymptotically sta ble if there exist matrices P1 > 0, P2 > 0, Q11 > 0, Q12 > 0, Q21 > 0, Q22 > 0, Q31 > 0, Q32 > 0, Z11 > 0, Z21 > 0, Z12 > 0, Z22 > 0, and diagonal matrices Q4 > 0, Q5 > 0, and a nonsingular X1, and matrices Y, Ni, Si, Ti (i = 1, 2, …, 7), Rj, Hj, Γj, and Lj (j = 1, 2, …, 6) of appropriate dimensions and scalars ε1 > 0 and ε2 > 0 such that the following LMI holds Ω
τ m2 N T
τ m2 N – τ m2 Z 11
τ m21 S
τ m21 T
τ p2 R
τ p21 H
τ p21 Γ
0
0
0
0
0
0
0
XM LM
τ m21 S
T
0
– τ m21 ( Z 11 + Z 21 )
0
0
0
0
0
0
τ m21 T
T
0
0
– τ m21 Z 21
0
0
0
0
0
0
0
0
– τ p2 Z 12
0
0
0
0
T
0
0
0
0
– τ m21 ( Z 22 + Z 12 )
0
0
0
T
0
0
0
0
0
– τ p21 Z 22
0
0
T
M X
T
0
0
0
0
0
0
–ε1 I
0
T
T
0
0
0
0
0
0
0
–ε2 I
τ p2 R
T
τ p21 H τ p21 Γ
M L
< 0,
(19)
where Ω = Ω(i, j),
i, j = 1, 2, …, 12,
3
T
Ω11 = N1 + N 1 +
∑Q
T
i1
T
+ ε1 E 1 E1,
T
Ω12 = – N 1 –T1 + N2 + S 1 ,
Ω13 = N3 +T1,
T
Ω14 = – S 1 + N4,
i=1 T
T
T
Ω15 = N5 – ε1 E 1 E20, Ω16 = N6 – ε1 E 1 E21, Ω17 = –AT X 1 +YT + P1 + N7, Ω18 = 0, Ω19 = 0, Ω110 = 0, T
T
T
T
T
Ω111 = 0, Ω112 = 0, Ω22 = –(1 –dm)Q31 – N2 – N 2 – T2 – T 2 + S2 + S 2 + L1D + DT L 1 + ε2 E 4 E4, AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 4
2010
DESIGN OF DELAYRANGEDEPENDENT ROBUST CONTROLLER T
Ω15 = –N3 – T3 + S3 + T 2 ,
T
Ω24 = –N4 – T3 –S2 + S4 + T 2 , T
T
239
Ω25 = –N5 – T5 + S5, Ω26 = –N6 – T6 + S6, T
T
T
T
T
Ω27 = –N7 – T7 + S7, Ω22 = –L1C + R 1 + DT L 2 – ε2 E 4 E3, Ω29 = – R 1 – T 1 + H 1 + DT L 3 , T
T
T
Ω23 = T 1 + DT L 4 ,
T
T
T
Ω211 = – H 1 + DT L 5 , Ω212 = –L1 + DT L 6 , Ω33 = –Q11 + T3 + T 3 ,
T
Ω34 = – S 3 + T4,
Ω35 = T5, Ω36 = T6, Ω37 = T7, Ω38 = 0, Ω39 = 0, Ω310 = 0, Ω311 = 0, Ω312 = 0, T
Ω44 = – S 4 – S4 – Q21, Ω45 = –S5, Ω46 = –S6, Ω47 = –S7, Ω48 = 0, Ω49 = 0, Ω410 = 0, Ω411 = 0, T
T
T
Ω412 = 0, Ω55 = – Q 4 – Q4 + ε1 E 20 E20,
T
Ω56 = ε1 E 20 E21,
T
Ω57 = G 0 X 1 ,
Ω58 = Q4Σ, Ω59 = 0,
T
T
T
Ω510 = 0, Ω511 = 0, Ω512 = 0, Ω66 = –Q5Σ–1 –(Q5Σ–1 )T + ε1 E 21 E21, Ω67 = G 1 X 1 ,
Ω68 = 0, Ω69 = Q5,
T
Ω610 = 0, Ω611 = 0, Ω612 = 0, Ω77 = –X1 – X 1 + τm2Z11 + τm21Z21, Ω78 = 0, Ω79 = 0, Ω710 = 0, 3
T
T
Ω711 = 0, Ω712 = 0, Ω88 = –L2C – CT L 2 + R2 + R 2 +
∑Q
T
i2
+ ε2 E 3 E3,
i=1 T Ω89 = –CT L 3
–
T Γ2
–
T R2
+
T H2
T Ω810 = –CT L 4
+ R3,
T
T
+ R4, Ω881 = –CT L 5 – H 2 + R5,
T
T
T
T
Ω812 = –L2 + CT L 6 + R6 + P2, Ω99 = –(1 –dp)Q32 –Γ3 – Γ 3 – R3 – R 3 + H3 + H 3 , T
T
T
Ω910 = –Γ4 + Γ 2 + H4 – R4, Ω911 = –Γ5 – R5 – H 3 + H5, Ω912 = –L3 – Γ6 – R6 + H6, Ω1010 = –Q12 + Γ4 + Γ 4 , T
Ω1011 = Γ5 – H 4 ,
T
Ω1012 = Γ6,
Ω1111 = –Q22 – H5 – H 5 ,
Ω1112 = –L5 – H6,
T
Ω1212 = –L6 – L 6 + τp2Z12 + τp21Z22, in which Σ = diag(σ1, σ2, …, σn), σi > 0, i = 1, 2, …, n are given in Assumption 1. In this case, an appropriate delayrangedependent global robust stabilizing state feedback –1 controller K can be chosen as K = X 1 Y. Proof. Define the following Lyapunov⎯Krasovskii functional candidate for system (8) and (9) as t
2
T
T
V ( x ( t ), y ( t ) ) = m ( t )P 1 m ( t ) + p ( t )P 2 p ( t ) +
∑ ∫
t
m ( s )Q i1 m ( s ) ds +
i = 1 t – τ mt 0
+
∫ ∫
T m· ( s )Z 11 m· ( s ) ds dθ +
– τ m2 t + θ
∫
t – τp ( t )
0 T
p ( s )Q 32 p ( s ) ds +
t
2
T m· ( s )Z 21 m· ( s ) ds dθ +
– τ m2 t + θ
t
+
∫ ∫
T
m ( s )Q 31 m ( s ) ds
t – τm ( t )
– τ m1 t
t
∫
T
∑∫
T
p ( s )Q i2 p ( s ) ds
(20)
i = 1 t – τ pi – τ p1 t
t
∫ ∫ p· ( s )Z T
12 p ( s ) ds dθ
·
– τ p2 t + θ
∫ ∫ p· ( s )Z T
+
22 p ( s ) ds dθ.
·
– τ p2 t + θ
For any matrices Lj (j = 1, 2, …, 6), a nonsingular matrix X1 of appropriate dimensions, and diagonal matrices Q4 > 0, Q5 > 0 the following null equations hold T 2η ( t )X [ – ( A – KI + ΔA ( t ) )m ( t ) + ( G 0 + ΔG 0 ( t ) )g ( p ( t ) ) + ( G 1 + ΔG 1 ( t ) )g ( p ( t – τ p ( t ) ) ) – m· ( t ) ] = 0, (21) T 2η ( t )L [ – ( C + ΔC ( t ) )p ( t ) + ( D + ΔD ( t ) )m ( t – τ m ( t ) ) – p· ( t ) ] = 0,
AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 4
2010
(22)
240
CHINWEN LIAO, CHIENYU LU T
T
2g ( p ( t ) )Q 4 g ( p ( t ) ) – 2g ( p ( t ) )Q 4 g ( p ( t ) ) = 0,
(23)
T
(24)
T
2g ( p ( t – τ p ( t ) ) )Q 5 p ( – τ p ( t ) ) – 2g ( p ( t – τ p ( t ) ) )Q 5 p ( t – τ p ( t ) ) = 0. Differentiating V(m(t), p(t)) along the solutions of (8) and (9) gives
T T V· ( m ( t ), p ( t ) ) = 2m ( t )P 1 m· ( t ) + 2p ( t )P 2 p· ( t ) +
2
∑ m ( t )Q T
T
i1 m ( t )
– m ( t – τ mi )Q i1 m ( t – τ mi )
i=1 T
T
T
+ m ( t )Q 31 m ( t ) – ( 1 – τ· m ( t ) )m ( t – τ m ( t ) )Q 31 m ( t – τ m ( t ) ) + τ m2 m· ( t )Z 11 m· ( t ) t – τ m1
t
∫
–
T T m· ( s )Z 11 m· ( s ) ds + τ m21 m· ( t )Z 21 m· ( t ) –
t – τ m2
∫
T m· ( s )Z 21 m· ( s ) ds
(25)
t – τ m2
2
+
∑ p ( t )Q T
i2 p ( t )
T
T
T
– p ( t – τ pi )Q i2 p ( t – τ pi ) + p ( t )Q 32 p ( t ) – ( 1 – τ· p ( t ) )p ( t – τ p ( t ) )Q 32 p ( t – τ p ( t ) )
i=1
t – τ p1
t
∫
T + τ p2 p· ( t )Z 12 p· ( t ) –
∫
T T p· ( s )Z 12 p· ( s ) ds + τ p21 p· ( t )Z 22 p· ( t ) –
t – τ p2
T p· ( s )Z 21 p· ( s ) ds.
t – τ p2
Using Assumption 1 and noting that Q4 > 0 and Q5 > 0 are diagonal matrices, one has T
T
2g ( p ( t ) )Q 4 g ( p ( t ) ) ≤ 2g ( p ( t ) )Q 4 Σp ( t ), T
T
(26)
–1
(27)
T m· ( s )Z 11 m· ( s ) ds,
(28)
T m· ( s )Z 21 m· ( s ) ds,
(29)
– 2g ( p ( t – τ p ( t ) ) )Q 5 p ( t – τ p ( t ) ) ≤ – 2g ( p ( t – τ p ( t ) ) )Q 5 Σ g ( p ( t – τ p ( t ) ) ), where Σ = diag(σ1, ω2, …, σn). On the other hand, the following equations are also true t – τm ( t )
t
–
∫
∫
T
m· ( s )Z 11 m· ( s ) ds = –
t – τ m2
t – τ m2
t – τ m1
–
∫
m· ( s )Z 21 m· ( s ) ds = –
∫
t – τ p2
t – τm ( t )
∫
t T p· ( s )Z 12 p· ( s ) ds –
t – τ p2
t – τ p1
∫
t – τ p2
∫
T p· ( s )Z 11 p· ( s ) ds,
(30)
T p· ( s )Z 21 p· ( s ) ds.
(31)
t – τp ( t )
t – τp ( t ) T p· ( s )Z 22 p· ( s ) ds = –
∫
m· ( s )Z 21 m· ( s ) ds –
t – τp ( t ) T p· ( s )Z 12 p· ( s ) ds = –
t – τ p2
–
t – τ m1 T
t – τ m2
t
∫
∫
∫
t – τm ( t )
t – τm ( t ) T
t – τ m2
–
t T m· ( s )Z 11 m· ( s ) ds –
t – τ p1 T p· ( s )Z 22 p· ( s ) ds –
∫
t – τp ( t )
AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 4
2010
DESIGN OF DELAYRANGEDEPENDENT ROBUST CONTROLLER
241
Combining (21)–(24) and using (26)–(31) results in T T V· ( m ( t ), p ( t ) ) ≤ 2m ( t )P 1 m· ( t ) + 2p ( t )P 2 p· ( t ) +
2
∑ m ( t )Q T
i1 m ( t )
T
– m ( t – τ mi )Q i1 m ( t – τ mi )
i=1 T
T
T
+ m ( t )Q 31 m ( t ) – ( 1 – d m ( t ) )m ( t – τ m ( t ) )Q 31 m ( t – τ m ( t ) ) + τ m2 m· ( t )Z 11 m· ( t ) t – τm ( t )
∫
–
t – τm ( t )
t
∫
T m· ( s )Z 11 m· ( s ) ds –
t – τ m2
∫
T T m· ( s )Z 11 m· ( s ) ds + τ m21 m· ( t )Z 21 m· ( t ) –
t – τm ( t )
t – τ m2
t – τ m1
∫
–
T m· ( s )Z 21 m· ( s ) ds
2
T m· ( s )Z 21 m· ( s ) ds +
t – τm ( t )
∑ p ( t )Q T
i2 p ( t )
T
T
– p ( t – τ pi )Q i2 p ( t – τ pi ) + p ( t )Q 32 p ( t )
i=1
(32)
t – τp ( t ) T T – ( 1 – d p ( t ) )p ( t – τ p ( t ) )Q 32 p ( t – τ p ( t ) ) + τ p2 p· ( t )Z 12 p· ( t ) –
∫
T p· ( s )Z 12 p· ( s ) ds
1 – τ p2 t – τp ( t )
t
–
∫
∫
T T p· ( s )Z 11 p· ( s ) ds + τ p21 p· ( t )Z 22 p· ( t ) –
1 – τp ( t )
t – τ p1 T p· ( s )Z 22 p· ( s ) ds –
t – τ p2
∫
T p· ( s )Z 21 p· ( s ) ds
t – τp ( t )
T
+ 2η ( t )X [ – ( A – KI + ΔA ( t ) )m ( t ) + ( G 0 + ΔG 0 ( t ) )g ( p ( t ) ) + ( G 1 + ΔG 1 ( t ) )g ( p ( t – τ p ( t ) ) ) – m· ( t ) ] T T + 2η ( t )L [ – ( C + ΔC ( t ) )p ( t ) + ( D + ΔD ( t ) )m ( t – τ m ( t ) ) – p· ( t ) ] + 2g ( p ( t ) )Q 4 Σ ( p ( t ) ) T
T
–1
T
– 2g ( p ( t ) )Q 4 g ( p ( t ) ) + 2 g ( p ( t – τ p ( t ) ) )Q 5 p ( – τ p ( t ) ) – 2g ( p ( t – τ p ( t ) ) )Q 5 Σ g ( p ( t – τ p ( t ) ) ). Employing Lemma 1, Lemma 2 and rearranging, (32) can be written as T T –1 T –1 T –1 T –1 V· ≤ η ( t ){Ω + τ m2 N Z 11 N + τ m21 S ( Z 11 + Z 21 ) S + τ m21 T Z 21 T + τ p2 R Z 12 R –1
T
+ τ p21 H ( Z 22 + Z 12 ) H + τ p21 Γ –1
T
–1 Z 22 Γ
+
–1 T T ε 1 XMM X
–1
+
–1 T T ε 2 LMM L }η ( t ).
–1
(33)
–1
If Ω + τm2NT Z 11 N + τm21ST(Z11 + Z21)–1Sτm21TT Z 21 T + τp2RT Z 12 R + τp21ΓT Z 22 Γ + τp21HT(Z22 + Z12)–1H + –1
–1
ε 1 XMM TXT + ε 2 LMM TLT< 0, –1 which is equivalent to (17) by Schur complement and change of variable such that K = X 1 Y, then V· < 0, which ensures the globally robustly asymptotically stable in (8) and (9) for any delays τp(t) and τm(t) satis fying (3) and (4). This completes the proof of Theorem 1. Remark 4. Based on the Lemma 2, Theorem 1 proposes a delayrangedependent criterion such that for all admissible uncertainties, the state feedback controller design for the uncertain genetic regulatory network can be approached by solving an LMI. These free weighting matrices Ni, Si, Ti (i = 1, 2, …, 7) and Ri, Hi, Γi, Li (i =1, 2, …, 6) are introduced into the LMI condition (19) in Theorem 1. Even for τm1 = 0 and τp1 = 0, the result in Theorem 1 may lead to the delaydependent stability criteria. In fact, if Z21 = β1I and Z12 = β2I, with β1 > 0 and β2 > 0, being sufficient small scalars, Ti = 0, i = 1, 2, …, 7 and Rj = 0, j = 1, 2, …, 6, Theorem 1 results in the delay dependent criterion. That is to say, Z21, Z12, Ti = 0 (i = 1, 2, …, 7) and Rj = 0 (j = 1, 2, …, 6) may provide some extra freedom in the selections of them in Theorem 1. It should be noted that these free weighting variables are not required to be symmetric. The purpose of introduction of these free weighting variables is to reduce conservatism in the existing results. The results in [19, 20] fail to deal with this above case when Theorem 1 in this paper is without state feedback controller. Thus Theorem 1 present here is more general and practical than the existing results in [19, 20]. Remark 5. Estimating the upper bound of the derivative of Lyapunov functional, some useful terms are T T –τ ignored, for example, the derivatives of 0–τ t m· (s)Z11 m· (s)dsdθ, –τm1 t m· (s)Z21 m· (s)dsdθ,
∫ ∫ m2
∫ ∫
T 0 t p· (s)Z12 p· (s)dsdθ, – τ p2 t + θ
and
– τ p1
∫ ∫
t
– τ p2 t + θ
∫ ∫
t+θ
m2
t+θ
T T p· (s)Z22 p· (s)dsdθ, in (20) are often estimated as τm2 m· (t)Z11 m· (t) –
AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 4
2010
242
CHINWEN LIAO, CHIENYU LU
∫
T t m· (s)Z11 m· (s)ds, 1 – τ m2
and
∫
1 – τ m1
1 – τ m2
T τm21 m· (t)Z21 m· (t) –
T τp21 p· (t)Z22 p· (t) T m· (s)Z21 m· (s)ds, –
–
∫
∫
1 – τ p1 1 – τ p2
∫
1 – τ m1
1 – τ m2
T T m· (s)Z21 m· (s)ds, τp2 p· (t)Z12 p· (t) –
T p· (s)Z21 p· (s)ds,
T t p· (s)Z12 p· (s)ds, 1 – τ p2
and –
these
∫
1 – τ p1 1 – τ p2
terms
–
∫
∫
T t p· Z12 p· (s)ds 1 – τ p2
T t m· (s)Z11 m· (s)ds, 1 – τ m2
–
T p· (s)Z21 p· (s)ds are often ignored, which may
bring conservativeness. In Theorem 1, all these terms are conserved, which can reduce conservativeness due to introduce into the free weighting matrices by applying Lemma 2 (improved bounding technique). In this sense, Therorem1 produces a larger range of the timevarying delays to keep the stabilization of sys tems. Also, the derivative of timevarying delay is generally less than 1. The condition may have limited use. Therefore, Theorem 1 in this paper applies Lemma 2 (improved bounding technique) such that the above condition can be removed with the LMI approach being used. Remark 6. In deriving the globally robust stabilization in theorem 1, no model transformation has been performed. That is to say, a common approach is to transform the original system to another one by employing the NewtonLeibniz formula. Since, the conditions of upper bounds are not used in our deri vation. Based on Lemma 2 (improved bounding technique), free weighting matrices are introduced into Theorem 1. This feature has the potential to enable us to obtain lessconservative results. Two illustrative examples are now presented to demonstrate the usefulness of the proposed approach. 4. EXAMPLES Example 1. Consider the synthetic oscillatory network of transcriptional regulators for Escherichia coli [24]. The repressilator is a cyclic negativefeedback loop comprising three repressor genes (lacl, tetR, and cl) and their promoters. The kinetics of the system are described by six coupled firstorder differential equations as follows ε m· i = – m i + , n 1 + pj p· i = –β(pi – mi), where i = lacl, tetR, cl; j = cl, lacl, tetR, each of theses six molecular species participates in transcription, translation, and degradation reactions. mi and pi are the concentrations of the three mRNA and repressor protein, and β > 0 is the ratio of the protein decay rate to mRNA decay rate. Considering the transcrip tional time delay, the above equation can rearrange into vector form in (1) and (2) with adjust some param eters, where A = diag{2, 2, 2}, C = diag{2.5, 2.5, 2.5}, D = diag{0.8, 0.8, 0.8}, 0 2.5 – 2.5 G0 = – 2.5 0 0 , 0 2.5 0 0.02 0.02 0 E1 = – 0.1 0.2 0.1 , 0.2 0 0.1
0 – 2.5 – 2.5 G1 = – 2.5 0 2.5 , 2.5 2.5 0
0.02 – 0.02 0 E20 = 0.1 – 0.2 0.01 , 0.2 0 0.1
M=
0.1 0 0.1 0 0.2 0.1 , 0.2 0.1 0.3
0.02 – 0.02 0 E21 = 0.1 – 0.2 0 , 0.2 0 0.1
0.01 0.02 0.01 0.01 0.02 0.01 2 2 E3 = – 0.1 0.3 0.1 , E4 = – 0.1 0.3 0.1 , gi(pi) = p i /(1 + p i ), where the Hill coefficient n = 2, 0.1 0 0.2 0.01 0.2 0.1 α = 2.5. In this example, we assume the functions of the feedback regulation of the protein on the tran scription satisfy Assumption 1 with σ1 = 0.075, σ2 = 0.518, σ3 = 0.426. Using the Matlab LMI Control Toolbox to solve the LMI (20) for all interval timevarying delays satisfying τp(t) = 4 + 0.4sin(tπ/2)(i.e. the lower bound τp1 = 3.6 and the upper bound τp2 = 4.4) and τm(t) = 2 + 0.5sin(tπ/2) (i.e. the lower bound τm1 = 1.5 and the upper bound τm2 = 2.5). The purpose is the design of a state feedback controller such that AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 4
2010
DESIGN OF DELAYRANGEDEPENDENT ROBUST CONTROLLER
243
the resulting closedloop system is globally robustly stable for estimation of the deviation of the perturbed trajectory from the equilibrium point. The feasible solutions for LMI (16) can be found as follows – 1.0825 – 0.0039 – 0.0011 0.0434 – 0.0024 – 0.0008 Y = – 0.0040 – 1.0861 0.0033 , X1 = – 0.0023 0.0418 00021 , the stabilization problem is solvable, – 0.0011 0.0033 – 1.0930 – 0.0007 0.0021 0.0373 –1
and a desired controller gain is given by K = X 1 Y as – 25.0304 – 1.5073 – 0.4827 K = – 1.4520 – 26.1432 1.5293 . – 0.4175 1.5321 – 29.3981 That is to say, globally robust stabilization is considered, the synthetic genetic regulatory network can eliminate larger parameter variations such that system can work more reliably. Example 2. Consider the uncertain genetic regulatory networks with the following parameters A = diag{2, 3, 4, 5, 2}, C = diag{5, 4, 5, 4.5, 4}, D = diag{0.3, 0.2, 0.4, 0.1, 0.1}, 0 2.5 – 2.5 0 – 2.5 – 2.5 0 0 2.5 0 G0 = 0 2.5 0 – 2.5 0 , 2.5 – 2.5 0 0 – 2.5 0 0 2.5 – 2.5 0
0 2.5 – 2.5 0 – 2.5 – 2.5 0 – 2.5 2.5 0 G1 = 0 2.5 0 – 2.5 0 , 2.5 – 2.5 0 0 – 2.5 – 2.5 0 2.5 – 2.5 0
0.1 0.01 M = 0.1 0.2 0.03
0.02 – 0.1 E1 = 0 0.2 – 0.3
0.3 – 0.1 E20 = 0 0.1 – 0.2
0.01 0.2 0.1 0.03 0.02
– 0.1 0 0.1 – 0.1 0.4
0.1 0.2 0.03 0.01 0.01
0 0.1 0.1 – 0.4 0.3
0.2 0.1 0.01 0.2 0.02
0.1 – 0.1 – 0.4 0.2 – 0.04
0.03 0.02 0.01 , 0.02 0.4 – 0.2 0.4 0.3 , – 0.04 0.02
– 0.1 0.2 0.1 – 0.3 0.4
0 0.1 0.1 – 0.4 0.3
0.2 – 0.3 – 0.4 0.2 0.01
0.1 0 0 0 0 0.1 E21 = 0 0.1 0.1 – 0.1 – 0.1 – 0.3 – 0.2 0.4 0.1
– 0.3 0.4 0.3 , 0.01 0.02
– 0.1 – 0.1 – 0.3 0.2 – 0.3
– 0.2 0.4 0.1 , – 0.3 0.02
0.01 – 0.02 0.01 – 0.3 0.1 0.2 – 0.1 0.3 0.1 – 0.4 – 0.02 0.3 0.1 0.1 0.4 – 0.1 0.3 0.1 – 0.2 0.3 2 2 E3 = 0.01 0.1 0.2 0.4 0.03 , E4 = 0.3 0.1 0.1 0.3 0.2 , gi(pi) = p i /(1 + p i ), the func – 0.3 0.1 0.4 002 0.03 0.1 – 0.2 0.3 0.1 – 0.3 0.1 0.4 0.03 0.03 0.2 – 0.4 0.3 0.2 – 0.3 0.2 tions of the feedback regulation of the protein on the transcription in this example are assumed to satisfy Assumption 1 with σ1 = 0.034, σ2 = 0.429, σ3 = 0.508, σ4 = 0.214, and σ5 = 0.372. By the Matlab LMI Control Toolbox, it can be verified that Theorem 1 in this paper is feasible solution for all interval time varying delays satisfying τp(t) = 2 + 2sin(tπ/2) (i.e. the lower bound τp1 = 0 and the upper bound τp2 = 4) AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 4
2010
244
CHINWEN LIAO, CHIENYU LU
and τm(t) = 3 + 3sin(tπ/2) (i.e. the lower bound τm1 = 0 and the upper bound τm2 = 6), the desired con –1
troller gain K = X 1 Y is obtained as – 42.8640 – 10.9737 –1 K = X 1 Y = – 8.4395 – 4.4847 – 1.9700 – 0.4751 – 0.0110 Y = – 0.0124 – 0.0004 – 0.0018
– 0.0083 – 0.4570 0.0138 – 0.0134 0.0054
– 0.0089 0.0134 – 0.4370 – 0.0138 0.0025
0.0001 – 0.0109 – 0.0110 – 0.4229 – 0.0126
– 8.8869 – 37.4681 6.2298 – 5.6187 4.3194
– 7.6729 7.0741 – 31.6372 – 6.1207 – 0.9066
– 0.0015 0.0056 0.0017 , – 0.0166 – 0.4732
– 3.0332 – 5.6272 – 4.7166 – 31.9353 – 11.9046
0.0127 – 0.0041 X1 = – 0.0035 – 0.0001 – 0.0010
– 1.7023 3.3145 – 1.7810 , where – 10.0702 – 39.0314
– 0.0035 0.0146 0.0035 – 0.0029 0.0027
– 0.0036 0.0044 0.0159 – 0.0031 0.0014
0.0002 – 0.0032 – 0.0024 0.0156 – 0.0053
– 0.0007 0.0019 0.0003 . – 0.0037 0.0137
5. CONCLUSIONS In this paper, the problem of robust stabilization for uncertain genetic regulatory network with interval timevarying delay has been studied. Via improved bounding technique, a delayrangedependent condi tion for the solvability of this problem, which free weighting matrices are introduced into the sufficient condition to reduce the conservatism, has been established such that the desired controller guarantees asymptotic robust stabilization of the system. Two illustrative examples have been presented to demon strate the effectiveness of the proposed approach. It is expected that the synthetic genetic regulatory net work can widely be applied in filtering larger environmental extrinsic disturbances and suffering more intrinsic fluctuation such that system can work more reliably. REFERENCES 1. Elowitz, M.B. and Leibler, S., A synthetic oscillatory network of transcriptional regulators, Nature, 2000, vol. 403, pp. 335–338. 2. Becskei, A. and Serrano, L., Engineering stability in gene networks by autoregulation, Nature, 200, vol. 405, pp. 590–593. 3. Hasty, J., McMillen, D., Isaacs, F., and Collins, J., Computational studies of gene regulatory networks: in numero molecular biology, Nat. Rev. Genet., 2001, vol. 2, pp. 268–279. 4. Kobayashi, T., Chen, L., and Aihara, K., Modeling genetic switches with positive feedback loops, J. Theor. Biol., 2000, vol. 221, pp. 379–399. 5. Gardner, T.S., Cantor, C.R., and Collins, J.J., Construction of a genetic toggle switch in Escherichia coli, Nature, 2000, vol. 403, pp. 339–342. 6. McAdams, H. and Shapiro, L., Circuit simulation of genetic networks, Science, 1995, vol. 269, pp. 650–656. 7. Huang, S., Gene expression profiling, genetic networks, and cellular states: an integrating concept for tumor igenesis and drug discovery, Molecular Medicine J., 1999, vol. 77, pp. 469–480. 8. Li, C., Chen, L., and Aihara, K., Stability of genetic networks with SUM regulatory logic: Lure systems and LMI approach, IEEE Trans. Circuits and Systems–I: Regular Papers, 2006, vol. 53, pp. 2451–2458. 9. Chen, T., He, H.L., Church, G.M., Modeling gene expression with differential equations, Pacific Symposium of Biocomputing, 1999, pp. 29–40. 10. Grammaticos, B., Carstea, A.S., and Ramani, A., On the dynamics of a gene regulatory network, J. Phys. A, Math. Gen., 2006, vol. 39, pp. 2965–2971. 11. Tian, T., Burrage, K., Burrage, P.M., and Carletti M., Stochastic delay differential equations for genetic regu latory networks, J. Computational and Applied Mathematics, 2007, vol. 205, pp. 696–707. 12. Mahaffy, J.M. and Pao, C.V., Models of genetic control by repression with time delays and spatial effects, J. Math. Biol., 1984, vol. 20, pp. 39–57. 13. Monk, N.A.M., Oscillatory express of Hes1, p53, and NF–kB driven by transcriptional time delays, Curr. Biol., 2003, vol. 13, pp. 1409–1413. 14. Chen, L. and Aihara, K., Stability of genetic regulatory networks with time delay, IEEE Trans. CAS–I, 2002, vol. 49, pp. 602–608. AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 4
2010
DESIGN OF DELAYRANGEDEPENDENT ROBUST CONTROLLER
245
15. Bratsun, D., Volfson, D., Tsimring, L.S. and Hasty, J., Delay–induced stochastic oscillations in gene regula tion, Proc. Natl. Acd. Sci. USA, 2005, vol. 102, pp. 14593–14598. 16. Smolen, P., Baxter, D.A., and Byrne, J.H., A reduced model clarifies the role of feedback loops and time delays in the Drosophila circadian oscillator, Biophy. J., 2002, vol. 83, pp. 2349–2359. 17. Zhu, Y., Song, K., Li, P., and Wang, H., Feedback control of the genetic networks with time delay, Proceedings of the 5th World Congress on Intelligent Control and Automation, June 15–19, 2004, Hangzhou, P. R. China, 2004, pp. 5549–5553. 18. Feng, W., Zhang, W., Xu, G., and Wu, H., A note on the robust stability of genetic regulatory networks with time–varying delays, CIS 2008, 2008, pp. 279–283. 19. Ren, F. and Cao, J., Asymptotic and robust stability of genetic regulatory networks with time–varying delays, Neurocomputing, 2008, vol. 71, pp. 834–842. 20. Wang, Z., Gao, H., Cao, J., and Liu, X., On delayed genetic regulatory networks with polytopic uncertainties: robust stability analysis, IEEE Trans. Nanobioscience, 2008, vol. 7, pp. 154–163. 21. Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V., Linear matrix inequalities in system and control theory, Philadelphia: SIAM, 1994. 22. Lu, C.Y., Tsai, H.H., Su, T.J., Tsai Jason, S.H., and Liao, C.W., A delay–dependent approach to passivity anal ysis for uncertain neural networks with time–varying delay, Neural Process Lett., 2008, vol. 27, pp. 237–246. 23. Lu, C.Y., Su, T.J., and Tsai Jason, S. H., On the robust stabilization of uncertain stochastic time–delay sys tems–an LMI–based approach, J. Franklin Institute, 2005, vol. 342, pp. 473–487. 24. Kalir, S., Mangan, S., and Alon, U., A coherent feed–forward loop with a SUM input function prolongs fla gella expression in Escherichia coli, Molo. Syst. Biol., 2005, DOI: 10.1038/msb4100010. 25. Yuh, C.H., Bolouri, H., and Davidson, E.H., Genomic cis–regulatory logic: experimental and computational analysis of a sea urchin gene, Science,1998, vol. 279, pp. 1896–1902. 26. Huang, H., Feng, G., and Cao, J., Robust state estimation for uncertain neural networks with time–varying delay, IEEE Trans. Neural Networks, 2008, vol. 19, pp. 1329–1339.
AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 4
2010
