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Stability analysis for discrete-time switched GRNs with persistent dwell-time and time delays
Stability Analysis for Discrete-Time Switched GRNs with Persistent Dwell-Time and Time Delays
Journal Pre-proof
Stability Analysis for Discrete-Time Switched GRNs with Persistent Dwell-Time and Time Delays Tingting Yu, Yue Zhao, Qingshuang Zeng PII: DOI: Reference:
S0016-0032(19)30710-0 https://doi.org/10.1016/j.jfranklin.2019.09.039 FI 4191
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
7 May 2019 14 September 2019 29 September 2019
Please cite this article as: Tingting Yu, Yue Zhao, Qingshuang Zeng, Stability Analysis for DiscreteTime Switched GRNs with Persistent Dwell-Time and Time Delays, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.09.039
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Stability Analysis for Discrete-Time Switched GRNs with Persistent Dwell-Time and Time Delays Tingting Yua , Yue Zhaob,∗, Qingshuang Zenga a Space
Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin, 150001, P.R. China. b Department of Control Science and Engineering, Harbin Institute of Technology, Harbin, 150001, P.R. China.
Abstract In this paper, the problem of exponential stability for discrete-time switched genetic regulatory networks (GRNs) with persistent dwell-time (PDT) switching and time delays is concerned. The novelty of this paper is derived from introducing an extended property of quadratic convex function, and utilizing an improved summation inequality together with an extended reciprocally convex matrix inequality, which is less conservative than the Jensen inequality and the reciprocally convex combination employed in the discrete-time delay systems. In addition, the considered switching regularity is more general than dwell-time (DT) switching and average dwell-time (ADT) switching. Moreover, by introducing the delay partitioning method and the piecewise Lyapunov-Krasovskii functional, a set of sufficient conditions are established in the form of linear matrix inequalities to guarantee the discrete-time PDT switched GRNs with constant time delays and time-varying delays are exponentially stable, respectively. Finally, some numerical examples are exploited to demonstrate the validity and potential of the proposed method. Keywords: Exponential stability, discrete-time switched GRNs, delay partitioning, persistent dwell-time (PDT) switching.
1. Introduction
5
Recently, genetic regulatory networks (GRNs) have drawn considerable attention, because GRNs play an important role in living systems at the molecular level, which can describe the interaction between DNA, RNA and protein. Basically, there are two types of GRNs, the Boolan model [1, 2] and the differential equation model [3]. It is worth noting that time delays are unavoidable in practice and may lead to oscillation and instability of some given system ∗ Corresponding
author Email addresses: [email protected] (Tingting Yu), [email protected] (Yue Zhao), [email protected] (Qingshuang Zeng)
Preprint submitted to Journal of The Franklin Institute
October 14, 2019
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[4, 5, 6, 7, 8]. More significant results on the dynamic analysis of delayed GRNs have been given in the existing literature [9, 10, 11, 12]. As we all known, switched systems are a class of dynamical hybrid systems, which consist of a family of continuous-time (or discrete-time) subsystems and operated by a switching rule. In reality, GRNs often exhibit a special characteristic of network-mode switching, which gives rise to the so-called switched GRNs. In the past decade or so, a number of literatures addressed the analysis and synthesized problems of switched GRNs with Markovian jumping and time delays, which are referred to stability analysis [13, 14, 15], state estimation [16, 17], controller design [18] and so on. In [13], authors researched the global robust stability of Markovian switching uncertain stochastic GRNs with norm bounded parameter uncertainties and unbounded time-varying delays; In [15], Zhu et al. considered the robust stability analysis of Markov jump standard GRNs with parameter uncertainties and mixed time-varying delays; In [16], the problem of event-triggered H∞ state estimation for discrete-time stochastic GRNs with Markovian jumping parameters and time-varying delays was discussed. Very recently, in [14], authors addressed the asymptotic stability of Markovian switching GRNs with leakage and mode-dependent time delays; In [18], the non-uniform sampled-data control was considered to guarantee the passivity of Markov jump GRNs with time-varying delays; The decentralized event-triggered mechanism is proposed in [19] with Markov jump parameters and distributed delays for uncertain delayed GRNs. Motivated by the preceding consideration, the dynamic behaviors of delayed GRNs are subject to Markovian switching parameters. However, due to the high complexity of GRNs, the likelihood of obtaining the specified probability distribution of Markovian jumping is actually questionable [20]. So under the circumstances, two important switching signals, namely DT and ADT were proposed to handle the dynamic behaviors of switched GRNs [21, 22, 23]. However, due to the limitation on switching times of DT and ADT, Hespanha et al. in [24] presented another important switching signal, named PDT switching, which is less conservative and more general than the DT switching and the ADT switching. Up to now, the PDT is seldom addressed since the complexity of their switching rules [25, 26, 27]. In [25], Du et al. designed a fuzzy-parameterdependent fault detection filter for nonlinear systems with PDT switching; In [26], the finite-time output feedback controller is designed for a class of discretetime linear systems with PDT switching. To the best of authors’ knowledge, the exponential stability of discrete-time switched GRNs with PDT switching and time delays has not been considered yet, which constitutes the main motivation of this paper. In this paper, the problem of exponential stability is analyzed for discretetime switched GRNs with PDT switching and time delays. The main contributions of this paper are highlighted as follows: • The switching regularity we considered is PDT switching, which has been proved to be more general than DT switching and ADT switching in [24]. • By introducing an extended property of quadratic convex function (Lemma 2
55
3) and using the improved summation inequality [28] together with the extended reciprocally convex matrix inequality [29], some new exponentially stable criteria are obtained in terms of LMIs for switched GRNs based on the piecewise Lyapunov-Krasovskii functional and delay partitioning method. • Numerical examples are presented to illustrate the effectiveness of the derived results.
60
65
70
Notations: In this paper, In and 0n denote an identity matrix and a zero matrix whose dimensions are n, respectively; Z+ is a set of non-negative integers; Z≥s refers to the set {k ∈ Z+ | k ≥ s}; Rn means that it is the n-dimensional Euclidean space; Rn×m means that it is the set of all n×m real matrices; col{. . . } represents a block-column matrix; diag {. . .} represents a block-diagonal matrix; the notation P > 0 stands for that P is positive definite and real symmetric; λmax (P ) and λmin (P )) denote the maximum eigenvalue and minimum eigenvalue of matrix P , respectively; k·k refers to the Euclidean norm of a vector; sym {A} is used to represent A + AT and the superscript “T ” denotes the transposition of a matrix. For an arbitrary matrix Y and two symmetric matrices X and Z, X Y denotes a symmetric matrix, where ‘∗’ represents the term that is ∗ Z induced by symmetry. 2. System Description and Preliminary Results Consider a class of discrete-time GRNs with time delays of the form [30]: m(k + 1) = Am(k) + Bg(p(k − τ (k))) + J, (1) p(k + 1) = Cp(k) + Dm(k − σ(k)),
75
80
where m(k) = [m1 (k), m2 (k), . . . , mn (k)]T ∈ Rn , mi (k) is the concentration of the ith mRNA at time k, p(k) = [p1 (k), p2 (k), . . . , pn (k)]T ∈ Rn , pi (k) is the concentration of the ith protein at time k. A = diag {a1 , a2 , . . . , an }, C = diag {c1 , c2 , . . . , cn }, ai represents the degradation rates of the mRNA of the ith gene, ci represents the degradation rates of the protein of the ith gene, and satisfy |ai | < 1, |ci | < 1. B = [bij ]n×n ∈ Rn×n is a coupling matrix of the genetic network. D = diag {d1 , d2 , . . . , dn } and di is a positive constant representing the translation rate of the ith gene. g(p(t)) = [g1 (p1 (t)), g2 (p2 (t)), . . . , gn (pn (t))]T and gj (·) represents the feedback regulation of the protein on the transcription in which gj (x) =
85
(x/βj )Hj 1+(x/βj )Hj
is a monotonically increasing function, where βj
being a positive constant and Hj is the Hill coefficient. J = [J1 , J2 , . . . , Jn ]T , Ji represents the basal transcriptional rate of the repressor of gene i. σ(k) and τ (k) are translational and transcriptional delays, respectively. Note that gj (·) is a monotonically increasing function and that it satisfies for ∀x 6= y ∈ R li− ≤
gi (x) − gi (y) ≤ li+ , i = 1, 2, . . . , n, x−y 3
(2)
90
where li− and li+ are known real constants. The delay σ(k) and τ (k) satisfy either (A1) or (A2) as the following form: (A1): Constant time delays: 1 ≤ σ(k) ≡ σ, 1 ≤ τ (k) ≡ τ ; (A2): Time-varying delays: 1 ≤ σ1 ≤ σ(k) ≤ σ2 , 1 ≤ τ1 ≤ τ (k) ≤ τ2 and σ(k) ˙ ≤ σd , τ˙ (k) ≤ τd ,
95
where σ, σ1 , σ2 , τ , τ1 , τ2 , σd and τd are known real constants. Now, we transform the equilibrium point (m∗ , p∗ ) to the origin by the relation xi (k) = mi (k)−m∗i , yi (k) = pi (k)−p∗i and fi (yi (k)) = gi (yi (k)+p∗i )−gi (p∗i ), i = 1, 2, . . . , n. Then, the discrete-time GRNs (1) can be transformed into x(k + 1) = Ax(k) + Bf (y(k − τ (k))), (3) y(k + 1) = Cy(k) + Dx(k − σ(k)), where x(k) = [x1 (k), x2 (k), . . . , xn (k)]T , y(k) = [y1 (k), y2 (k), . . . , yn (k)]T , f (y(k − τ (k))) = [f1 (y(k − τ (k))), f2 (y(k − τ (k))), . . . , fn (y(k − τ (k)))]T . In this paper, we will focus on the exponential stability of discrete-time GRNs with PDT switching and time delays, which are described as follows: x(k + 1) = Aα(k) x(k) + Bα(k) f (y(k − τ (k)), (4) y(k + 1) = Cα(k) y(k) + Dα(k) x(k − σ(k)),
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where the piecewise constant function α(k) : [0, ∞) → N = {1, 2, . . . , N } is the switching signal, and N stands for the number of subsystems. In this paper, we assume that the switching sequence k0 , k1 , . . . , kt , . . . are unknown a priori, but are known instantly. When k ∈ [kt , kt+1 ) means that the α(kt )-th subsystem is active. For α(k) = i ∈ N , {Ai , Bi , Ci , Di } are known real constant matrices with appropriate dimensions. In addition, from (2), we can easily obtain the following condition: li− ≤
fi (y) ≤ li+ , y
∀y 6= 0,
fi (0) = 0,
∀i = 1, 2, . . . , n.
(5)
To this end, we present the following definitions and lemmas which will be required in the next section. 110
115
Definition 1. [24] Consider system (4) and switching instants k0 , k1 , . . . , kt , . . . with k0 = 0. A positive constant value τP DT is said to be the persistent dwell time (PDT), if there exist numerous nonadjacent intervals whose interval length no smaller than τP DT on which α(k) is constant, and consecutive intervals with the above mentioned property are separated by no more than T , where T stands for the period of persistence. The interval defined in the PDT stage is assumed to have the running time (τ -position) of a certain subsystem and a period of persistence (T -position). A 4
certain subsystem is activated in the τ -position, whose running time is at least τP DT . Denote the actual running time of the T -position at the p-stage as T p , p ∈ Z≥1 , then the following equation holds: G(ktp +1 ,ktp+1 ) p
T =
X
Tα(ktp +r ) ≤ T,
r=1
120
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130
(6)
where Tα(ktp +r ) represents for the running time of the subsystem activated at the switching instant ktp +r ∈ [ktp +1 , ktp+1 ), r ∈ Z≥1 , G(ktp +1 , ktp+1 ) represents the switching times within [ktp +1 , ktp+1 ), ktp +1 denotes the next instant switching after ktp at the p-th stage, and ktp+1 stands for the instant switching to the (p + 1)-th stage. Remark 1. As discussed in [24], DT switching and ADT switching can be termed as special cases of PDT switching. It should be noted that PDT switching is more general than ADT switching, since PDT switching has no limitation on the maximum switching times during the period of persistence, while ADT switching strictly limits the upper bound of switching times within an interval of length less than τADT . Definition 2. [30] The discrete-time PDT switched GRNs (4) are said to be exponentially stable under α(k), if the following inequality holds: kz(k)k ≤ ρη (k−k0 ) kz(k0 )kC 1 , ∀k ≥ k0 ,
(7)
where constants ρ ≥ 1, 0 < η < 1, and k z(k0 )kC 1 = z(k) = 135
x(k) y(k)
{kz(k + θ)k , kς(k + θ)k},
sup
− max{σ2 ,τ2 }≤θ≤0
and ς(θ) = z(θ + 1) − z(θ).
Lemma 1. [28] For a given symmetric positive definite matrix Θ, integers b ≥ a, any sequence of discrete-time variable x : Z[a, b] → Rn , the following inequality holds: (b − a)
b−1 X i=a
T
η (k)Θη(k) ≥ ≥
θ1 θ2 θ1 θ2
T
T
Θ ∗ Θ ∗
0 θ1 3ρ(a, b)Θ θ2 0 θ1 , 3Θ θ2
where η(k) = x(k + 1) − x(k), θ1 = x(b) − x(a), θ2 = x(b) + x(a) − 2 ρ(a, b) =
b−a+1 b−a−1
for b − a 6= 1, and ρ(a, b) = 1 for b − a = 1. 5
b P
i=a
(8) (9) x(i) b−a+1 ,
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Lemma 2. [29] For a real scalar α ∈ (0, 1), a symmetric matrix Θ ≥ 0, and any matrix S, the following matrix inequality holds: 1 0 Θ + (1 − α)T1 S αΘ ≥ , (10) 1 ∗ Θ + αT2 0 1−α Θ where T1 = Θ − SΘ−1 S T and T2 = Θ − S T Θ−1 S.
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Extending Lemma 1 in [31] to the case where two variables are involved, we present the following lemma which will be used directly in the proof of the main theorem. Lemma 3. For symmetric matrices Z0 , Z1 , Zˆ0 and Zˆ1 , let F (α) = Z0 + αZ1 + α2 Z2 , F (β) = Zˆ0 + β Zˆ1 + β 2 Zˆ2 and F (α, β) = F (α) + F (β) with Z2 ≥ 0 and Zˆ2 ≥ 0. F (α, β) = Z0 + αZ1 + α2 Z2 + Zˆ0 + β Zˆ1 + β 2 Zˆ2 < 0 holds true if the following inequalities hold: F (α1 , β1 ) = Z0 + α1 Z1 + α12 Z2 + Zˆ0 + β1 Zˆ1 + β12 Zˆ2 < 0, F (α2 , β2 ) = Z0 + α2 Z1 + α22 Z2 + Zˆ0 + β2 Zˆ1 + β22 Zˆ2 < 0,
150
where ∀α ∈ [α1 , α2 ], ∀β ∈ [β1 , β2 ]. Proof. Because (α − α1 )(α − α2 ) ≤ 0 and (β − β1 )(β − β2 ) ≤ 0 hold for ∀α ∈ [α1 , α2 ] and ∀β ∈ [β1 , β2 ]. Then, by simple iteration and calculation yields β2 − β α − α1 β − β1 α2 − α + }F (α1 , β1 ) + { + }F (α2 , β2 ) α2 − α1 β2 − β1 α2 − α1 β2 − β1 ≥ Z0 + αZ1 + α2 Z2 + Zˆ0 + β Zˆ1 + β 2 Zˆ2 {
= F (α) + F (β) = F (α, β).
Thus one can draw the conclusion that F (α, β) < 0 from F (α1 , β1 ) < 0 and F (α2 , β2 ) < 0. 155
3. Exponential Stability Analysis 3.1. The Case of a Constant Time Delays In this section, our aim is to derive the new exponential stability criterion for discrete-time switched GRNs (4) with PDT switching and constant time delays (A1) : 1 ≤ σ(t) ≡ σ and 1 ≤ τ (t) ≡ τ .
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Theorem 1. Under the assumption of (A1), for given integers r ≥ 1, d1 ≥ 1, d2 ≥ 1, µ ≥ 1 and a constant 0 < β < 1, where µβ ≥ 1, the discretetime PDT switched GRNs (4) are exponentially stable, if there exist positivedefinite symmetric matrices Pji ∈ Rn×n , Rji ∈ Rn×n , Qji ∈ Rrn×rn , positivedefinite diagonal matrices Π = diag {π1 , π2 , . . . , πn }, V = diag {v1 , v2 , . . . , vn }
6
165
and matrices Mji , Nji , Sji , j = 1, 2, satisfies that for i ∈ N , Ψi =
2 X
WPTj P¯ji WPj +
j=1
+sym
2 X
2 X
¯ ji WQ + WQTj Q j
j=1
˜ PT Pji W ˜R + W j j
j=1
2 X
WRTj Rji WRj +
j=1
2 X
2 X
¯ ji Gj GTj R
j=1
T ˜P WM WSji + WFT Π(L+ + L− )W 2 ji
j=1
˜ PT ΠL+ L− W ˜ P − WFT ΠWF + W T V (L+ + L− )WD −W FD 2 2 T + − T −WD V L L WD − WF D V WF D < 0,
(11)
such that ∀m, n ∈ N , Pjm ≤ µPjn , Qjm ≤ µQjn , Rjm ≤ µRjn , j = 1, 2,
(T + 1) ln µ + (T + τP DT ) ln β < 0,
and ¯ ji = diag {Qji , −Qji } , P¯ji = diag {Pji , Pji } , Q # " −Rji 0 ¯ ji = , R d +1 ∗ − djj −1 3Rji s β d1 +1 In −In 0n×(2r+4)n 0n 0n G1 = , In In 0n×(2r+4)n −2In 0n d1 s β d2 +1 0n×(r+1)n In −In 0n×(r+4)n 0n , G2 = 0n×(r+1)n In In 0n×(r+4)n −2In d2 √ 1 − βIn 0n×(2r+4)n 0n×3n , WP 1 = 0n×(2r+4)n In 0n×3n √ 1 − βIn 0n×(r+4)n 0n×2n 0n×(r+1)n , WP 2 = 0n×(r+1)n 0n×(r+4)n In 0n×2n √ βIrn p 0rn×n 0rn×(r+7)n WQ1 = , 0rn×n β d1 +1 Irn 0rn×(r+7)n √ 0rn×(r+1)n p βIrn 0rn×7n WQ 2 = , 0rn×(r+2)n β d2 +1 Irn 0rn×6n √ βd1 In 0n×3n , WR1 = 0n×(2r+4)n √ βd2 In 0n×2n , WR2 = 0n×(2r+5)n ˜ P = In 0n×(2r+7)n , W ˜ R = p 1 WR , W 1 j j βdj ˜ P = 0n×(r+1)n In 0n×(r+6)n , W 2 WM1i = M1i S1i 0n×(2r+2)n N1i 0n×3n , WM2i = 0n×(r+1)n M2i S2i 0n×(r+2)n N2i 0n×2n , 7
(12) (13)
WS1i = Ai − In 0n×(2r+2)n Bi − In 0n×3n , WS2i = 0n×rn Di Ci − In 0n×(r+3)n − In 0n×2n , WF = 0n×(2r+2)n In 0n×5n , WD = 0n×(2r+1)n In WF D = 0n×(2r+3)n In 0n×4n .
0n×6n
,
Moreover, kz(k)k ≤ ρη (k−k0 ) kz(k0 )kC 1 , where r √ λ2 ≥ 1, η = λ, λ1 = min{λmin (P1i ), λmin (P2i )}, ρ = i∈N λ1 λ2 = max{λmax (P1i ) + λmax (P2i ) + βd1 λmax (Q1i ) + βd2 λmax (Q2i ) i∈N
+
170
d1 (d1 + 1) d2 (d2 + 1) βλmax (R1i ) + βλmax (R2i )}. 2 2
(14)
Proof. By employing the delay-partitioning method to the delays σ = rd1 and τ = rd2 that give r parts. Choose the Lyapunov-Krasovskii function of the following form V (z(k), k) =
3 X
Vj (z(k), k),
(15)
j=1
where V1 (z(k), k) = xT (k)P1i x(k) + y T (k)P2i y(k), k−1 X
V2 (z(k), k) =
β k−l ξ1T (l)Q1i ξ1 (l) +
l=k−d1
V3 (z(k), k) =
−1 X
k−1 X
β k−l ξ2T (l)Q2i ξ2 (l),
l=k−d2 k−1 X
β k−l ς1T (l)R1i ς1 (l) +
−1 X
k−1 X
β k−l ς2T (l)R2i ς2 (l),
s=−d2 l=k+s
s=−d1 l=k+s
and ξ1 (l) = col ξ2 (l) = col
x(l), y(l),
x (l − d1 ) ,
x (l − 2d1 ) , . . . , x (l − rd1 + d1 )
y (l − d2 ) , y (l − 2d2 ) ,
. . . , y (l − rd2 + d2 )
ς1 (l) = x(l + 1) − x(l), ς2 (l) = y(l + 1) − y(l). 175
,
,
For k ∈ [kl , kl+1 ) , we denote the forward difference of Vj (zk , αk ) as ∆Vj (z(k), k) = Vj (z(k + 1), k + 1) − Vj (z(k), k), j = 1, 2, 3, and define the following notations: ψ(k) = col {ξ1 (k), x(k − σ), ξ2 (k), y(k − τ ), f (y(k)), f (y(k − τ )), ς1 (k), ς2 (k), ν1 (k), ν2 (k)} ,
ν1 (k) =
k X
i=k−d1
k X x(i) y(i) , ν2 (k) = . d1 + 1 d2 + 1 i=k−d2
8
Then, we can deduce that ∆V1 (z(k), k) = {2x(k) + ς1 (k)}T P1i ς1 (k) + {2y(k) + ς2 (k)}T P2i ς2 (k), (16) ∆V2 (z(k), k) = −(1 − β)V2 (z(k), k) + β −
2 X j=1
−
ξjT (k)Qji ξj (k)
j=1
β dj +1 ξjT (k − dj ) Qji ξj (k − dj ) ,
∆V3 (z(k), k) = −(1 − β)V3 (z(k), k) + β 2 X
2 X
k−1 X
2 X
(17)
dj ςjT (k)Rji ςj (k)
j=1
β k+1−l ςjT (l)Rji ςj (l).
(18)
j=1 l=k−dj
¯ ji , Gj (j = 1, 2) and applying (8) of According to the definition of matrix R Lemma 1, we can easily obtain − 180
k−1 X
l=k−dj
¯ ji Gj ψ(k). β k+1−l ςjT (l)Rji ςj (l) ≤ ψ T (k)GTj R
(19)
Moreover, for any appropriate matrices Mji , Nji and Sji , j = 1, 2, i ∈ N , we have: T T T 2[xT (k)M1i + ς1T (k)N1i + xT (k − d1 )S1i ]
×{[Ai − I]x(k) − ς1 (k) + Bi f (y(k − τ ))} = 0,
(20)
and T T T 2[y T (k)M2i + ς2T (k)N2i + y T (k − d2 )S2i ]
×{[Ci − I]y(k) − ς2 (k) + Di x(k − σ)} = 0.
185
(21)
From (5), for any scalars πi ≥ 0 and vi ≥ 0, (i = 1, 2, · · · , n), it is easy to get that 2f T (y(k))Π(L+ + L− )y(k) − 2y T (k)ΠL+ L− y(k) −2f T (y(k))Πf (y(k)) ≥ 0, (22) 2f T (y(k − τ ))V (L+ + L− )y(k − τ ) T + − −2y (k − τ )V L L y(k − τ ) −2f T (y(k − τ ))V f (y(k − τ )) ≥ 0, where L+ = diag l1+ , l2+ , . . . , ln+ and L− = diag l1− , l2− , . . . , ln− . Considering inequalities (16)–(22), we have ∆V (z(k), k) + (1 − β)V (z(k), k) ≤ ψ T (t)Ψi ψ(t), 9
where Ψi is established as in (11), which implies that Ψi < 0. Thus it is obviously that V (z(k + 1), k + 1) ≤ βV (z(k), k).
190
(23)
Suppose α(ktp ) = i, α(ktp +1 + T p ) = j and consider an arbitrary switching occurs within T p , then we can get from inequality (12) and (23) that Vj (z(ktp +1 + T p ), ktp +1 + T p ) ≤ µVl (z(ktp +1 + T p ), ktp +1 + T p )
≤ µβ Tl Vl (z(ktp +1 + T p − Tl ), ktp +1 + T p − Tl ) ≤ ···
≤ µG(ktp ,ktp +1 +T ) β Tl +Tm +···+Tn β τP DT Vi (z(ktp ), ktp ),
195
p
(24)
where l, m, . . . , n represent for all the possible indices of subsystems which are being switched within T p , with corresponding running time Tl , Tm , . . . , Tn , respectively. p p According to µ ≥ 1 and µβ ≥ 1, we have µT +1 β T ≤ µT +1 β T . It is noted that (13) implies µT +1 β T +τP DT < 1. Let kt1 = k0 and λ = µT +1 β T +τP DT , then as from (24), it follows that Vα(ktp ) (z(ktp ), ktp ) ≤ λVα(ktp−1 ) (z(ktp−1 ), ktp−1 ) ≤ · · · ≤ λp−1 Vα(kt1 ) (z(kt1 ), kt1 ) ≤ λp−1 Vα(k0 ) (z(k0 ), k0 ).
(25)
Notice from (15) that Vα(ktp ) (z(ktp ), ktp ) ≥ λ1 kz(ktp )k2 , Vα(k0 ) (z(k0 ), k0 ) ≤ λ2 kz(k0 )k2C 1 ,
200
(26)
where λ1 and λ2 are defined in (14). Following from (25) and (26), we can deduce that 1 λ2 p−1 1 p−1 kz(k)k2 ≤ Vα(ktp ) (z(ktp ), ktp ) ≤ λ Vα(k0 ) (z(k0 ), k0 ) ≤ λ kz(k0 )k2C 1 . λ1 λ1 λ1 q √ Furthermore, denoting p − 1 = ktp − k0 , ktp = k, ρ = λλ21 and η = λ, we have kz(k)k ≤ ρη (k−k0 ) kz(k0 )kC 1 . Therefore, from Definition 2, one can get that the discrete-time PDT switched GRNs in (4) are exponentially stable.
205
210
Remark 2. The analysis of stability for switched GRNs has been widely investigated, but most of these results are based on ADT switching. One innovation of Theorem 1 consists in the generality of PDT switching when compared to ADT switching [32, 21]. Another one is that Theorem 1 rest with the improved summation inequality, which has been proved to be less conservative than the usual Jensen inequality in [28]. Hence, a new set of exponentially stable criteria has been proposed with PDT switching in Theorem 1, which is more flexible. 10
215
220
225
3.2. The Case of a Time-Varying Delays In this case, we propose a new exponential stability criterion for discretetime switched GRNs (4) with PDT switching and time-varying delay (A2): 1 < σ1 ≤ σ(k) ≤ σ2 , 1 < τ1 ≤ τ (k) ≤ τ2 and σ(k) ˙ ≤ σd , τ˙ (k) ≤ τd . We set σ12 = σ2 − σ1 , τ12 = τ2 − τ1 , then partition the time delays σ(k) and τ (k) into two parts: constant parts σ1 , τ1 and time-varying parts υˆ(k), υ˜(k), i.e., σ(k) = σ1 + υˆ(k), τ (k) = τ1 + υ˜(k), where 0 ≤ υˆ(k) ≤ σ12 , 0 ≤ υ˜(k) ≤ τ12 . Then, we set σ1 = rd1 and τ1 = rd2 , by using the delay partitioning approach, one can get: Theorem 2. Under the assumption of (A2), for given integers r ≥ 1, d1 ≥ 1, d2 ≥ 1, µ ≥ 1 and a constant 0 < β < 1, where µβ ≥ 1, the discrete-time PDT switched GRNs (4) are exponentially stable, if there exist positive-definite symmetric matrices Pji ∈ Rn×n , Rjki ∈ Rn×n , Qj1i ∈ Rrn×rn , Qj2i ∈ Rn×n , Lji ∈ Rn×n , positive-definite diagonal matrices Π = diag {π1 , π2 , . . . , πn }, H = diag {h1 , h2 , . . . , hn } and matrices Mji , Nji , Sji ∈ Rn×n , Tji ∈ R2n×2n , j, k = 1, 2 such that for i ∈ N , Φ10i GT11 T1i GT21 T2i ¯ 12i 02n×2n < 0, −R Φ1i = ∗ (27) ¯ 22i ∗ ∗ −R Φ2i and ∀m, n ∈ N ,
Φ20i = ∗ ∗
T GT12 T1i ¯ 12i −R ∗
T GT22 T2i 02n×2n < 0, ¯ 22i −R
(28)
Pjm ≤ µPjn , Rjkm ≤ µRjkn , Qjkm ≤ µQjkn , Ljm ≤ µLjn , j, k = 1, 2,
(T + 1) ln µ + (T + τP DT ) ln β < 0,
and Φj0i = Φ0i − Σ1ji − Σ2ji , j = 1, 2, 230
where Φ0i =
2 X
WPTj P¯ji WPj +
j=1
+
2 X
2 X 2 X
¯ jki WQ + WQTjk Q jk
k=1 j=1
2 X 2 X
WRTjk Rjki WRjk
k=1 j=1
¯ ji WL + GT3 R ¯ 11i G3 + GT4 R ¯ 21i G4 WLTj L j
j=1
+sym
2 nX j=1
˜ T Pji W ˜R + W Pj j1
2 X
T ˜P WM WSji + WFT Π(L+ + L− )W 2 ji
j=1
˜ T ΠL+ L− W ˜ P − W T ΠWF + W T H(L+ + L− )WD −W P2 F FD 2 o T + − T −WD HL L WD − WF D HWF D , 11
¯ ji = diag { Lji , −Lji } , j = 1, 2, P¯ji = diag {Pji , Pji } , L " # −Rj1i 0 0 ¯ j1i = ¯ j2i = Rj2i R , R , j = 1, 2, dj +1 ∗ − dj −1 3Rj1i ∗ 3Rj2i
¯ jki = diag {Qjki , −Qjki } , j = 1, 2, k = 1, 2, Q √ 1 − βIn 0n×(2r+8)n 0n×7n , WP1 = 0n×(2r+8)n In 0n×7n √ 0n×(r+3)n 1 − βIn 0n×(r+6)n 0n×6n WP2 = , 0n×(r+3)n 0n×(r+6)n In 0n×6n q q ˜ R , j = 1, 2, WR = βσ 2 W ˜ WRj1 = βd2j W j1 12 12 R11 , q 2 W ˜R , W ˜ P = In 0n×(2r+15)n , WR22 = βτ12 21 1 ˜ P = 0n×(r+3)n In 0n×(r+12)n , W 2 ˜ R = 0n×(2r+8)n In 0n×7n , W 11 ˜ R = 0n×(2r+9)n In 0n×6n , W 21 √ βIrn p 0rn×n 0rn×(r+15)n , WQ11 = β d1 +1 Irn 0rn×(r+15)n 0rn×n √ βIn 0n×(r+2)n 0n×(r+13)n p WQ12 = , 0n×(r+2)n β σ2 +1 In 0n×(r+13)n √ 0rn×(r+3)n p βIrn 0rn×13n WQ21 = , 0rn×(r+4)n β d2 +1 Irn 0rn×12n √ 0n×(r+12)n 0n×(r+3)n p βIn WQ22 = , 0n×(2r+5)n β τ2 +1 In 0n×10n p β(σ12 + 1)In p 0n×(r+1)n 0n×(r+14)n , WL1 = 0n×(r+1)n β σ2 +1 In 0n×(r+14)n p β(τ12 + 1)In 0n×(r+12)n 0n×(r+3)n p WL2 = , 0n×(2r+4)n β τ2 +1 In 0n×11n WM1i = M1i S1i 0n×(2r+6)n N1i 0n×7n , WM2i = 0n×(r+3)n M2i S2i 0n×(r+4)n N2i 0n×6n , WS1i = Ai − In 0n×(2r+6)n Bi −In 0n×7n , WS2i = 0n×(r+1)n Di 0n Ci − In 0n×(r+5)n − In 0n×6n , WF = 0n×(2r+6)n In 0n×9n , WD = 0n×(2r+4)n In 0n×11n , WF D = 0n×(2r+7)n In 0n×8n , p In −In 0n×(2r+8)n 0n 0n×5n , G13 = β d1 +1 In In 0n×(2r+8)n −2In 0n×5n 12
GT23
GT11
GT12
G21 G22 Rj1T i Σjli
p d +1 2 = β
0(r+3)n×n In −In 0(r+6)n×n 0n 04n×n
0(r+3)n×n In In 0(r+6)n×n −2In 04n×n
0rn×n In −In
0rn×n In In
,
p = β σ2 +1 0(r+10)n×n 0(r+10)n×n , 0n −2In 03n×n 03n×n 0(r+1)n×n 0(r+1)n×n In In p −In In = β σ2 +1 0(r+10)n×n 0(r+10)n×n , 0n −2In 02n×n 02n×n p 0n×(2r+3)n In −In 0n×9n = β τ2 +1 0n×(2r+3)n In In 0n×9n p 0n×(2r+4)n In −In 0n×9n β τ2 +1 = 0n×(2r+4)n In In 0n×9n ¯ j2i ¯ j2i Tji R Tji 2R = ¯ j2i ¯ j2i , Rj2T i = ∗ 2R ∗ R T Gj1 Gj1 = RjlT i , j, l = 1, 2, Gj2 Gj2
0n −2In 0n −2In ,
0n 0n ,
,
Proof. Choose a Lyapunov-Krasovskii function of the following form: W (z(k), k) =
4 X
Wj (z(k), k),
(29)
j=1
where W1 (z(k), k) = xT (k)P1i x(k) + y T (k)P2i y(k), W2 (z(k), k) =
k−1 X
β k−l xT (l)Q12i x(l) +
l=k−σ2
+
2 X
k−1 X
β k−l y T (l)Q22i y(l)
l=k−τ2 k−1 X
β k−l ζjT (l)Qj1i ζj (l),
j=1 l=k−dj
W3 (z(k), k) =
2 X
−1 X
k−1 X
dj β k−l ςjT (l)Rj1i ςj (l)
j=1 s=−dj l=k+s
13
+σ12
−σ 1 −1 k−1 X X
β k−l ς1T (l)R12i ς1 (l)
s=−σ2 l=k+s
+τ12
−τ 1 −1 k−1 X X
β k−l ς2T (l)R22i ς2 (l),
s=−τ2 l=k+s
W4 (z(k), k) =
−σ 1 +1 X
k−1 X
β k−l xT (l)L1i x(l)
s=−σ2 +1 l=k−1+s
+
−τ 1 +1 X
k−1 X
β k−l y T (l)L2i y(l),
s=−τ2 +1 l=k−1+s 235
and ζ1 (l) = col ζ2 (l) = col
x(l), y(l),
x (l − d1 ) , x (l − 2d1 ) , . . . , x (l − rd1 + d1 )
y (l − d2 ) , y (l − 2d2 ) , . . . , y (l − rd2 + d2 )
,
.
For k ∈ [kl , kl+1 ) , we denote the forward difference of Wj (zk , αk ) as ∆Wj (z(k), k) = Wj (z(k+1), k+1)−Wj (z(k), k), j = 1, 2, 3, 4, and define the following notations: φ(k) = col {ζ1 (k), x(k − σ1 ), x(k − σ(k)), x(k − σ2 ), ζ2 (k), y(k − τ1 ), y(k − τ (k)), y(k − τ2 ), f (y(k)), f (y(k − τ (k))), ς1 (k), ς2 (k),
ν1 (k), ν2 (k), ν3 (k), ν4 (k), ν5 (k), ν6 (k)} , k−σ X1
ν3 (k) =
i=k−σ(k)
ν5 (k) =
k−τ X1
k−σ(k) X x(i) x(i) , ν4 (k) = , σ(k) − σ1 + 1 σ2 − σ(k) + 1 i=k−σ2
k−τ (k)
i=k−τ (k)
X y(i) y(i) , ν6 (k) = . τ (k) − τ1 + 1 τ2 − τ (k) + 1 i=k−τ2
Then we have ∆W1 (z(k), k) = {2x(k) + ς1 (k)}T P1i ς1 (k) + {2y(k) + ς2 (k)}T P2i ς2 (k), ∆W2 (z(k), k) =
2 X j=1
βζjT (k)Qj1i ζj (k) −
2 X j=1
(30)
β dj +1 ζjT (k − dj ) Qj1i ζj (k − dj )
−(1 − β)W2 (z(k), k) − β σ2 +1 xT (k − σ2 ) Q12i x (k − σ2 ) −β τ2 +1 y T (k − τ2 ) Q22i y (k − τ2 )
+βxT (k)Q12i x(k) + βy T (k)Q22i y(k), ∆W3 (z(k), k) =
2 X j=1
βd2j ςjT (k)Rj1i ςj (k) −
2 X
k−σ 1 −1 X l=k−σ2
(31)
dj β k+1−l ςjT (l)Rj1i ςj (l)
j=1 l=k−dj
−(1 − β)W3 (z(k), k) − σ12 14
k−1 X
β k+1−l ς1T (l)R12i ς1 (l)
2 T 2 T +βσ12 ς1 (k)R12i ς1 (k) + βτ12 ς2 (k)R22i ς2 (k)
−τ12
k−τ 1 −1 X
β k+1−l ς2T (l)R22i ς2 (l),
(32)
l=k−τ2
∆W4 (z(k), k) ≤ β(σ12 + 1)xT (k)L1i x(k) + β(τ12 + 1)y T (k)L2i y(k)
−(1 − β)W4 (z(k), k) − β σ2 +l xT (k − σ(k)) L1i x (k − σ(k)) −β τ2 +l y T (k − τ (k)) L2i y (k − τ (k)) .
240
(33)
¯ j1i and Gj3 (j = 1, 2), Applying (8) of Lemma 1 and the definition of matrices R we can easily get −dj
k−1 X
l=k−dj
¯ j1i Gj3 φ(k). β k+1−l ςjT (l)Rj1i ςj (l) ≤ φT (k)GTj3 R
(34)
Using (9) of Lemma 1 and (10) of Lemma 2 yields −σ12
k−σ 1 −1 X
β k+1−l ς1T (l)R12i ς1 (l) ≤ −φT (k)Σ1i φ(k),
(35)
−τ12
k−τ 1 −1 X
β k+1−l ς2T (l)R22i ς2 (l) ≤ −φT (k)Σ2i φ(k),
(36)
l=k−σ2
and
l=k−τ2
where Σji =
Gj1 Gj2
+γj2 GTj1 γ1j = (−1)j
245
Tji Gj1 ¯ j2i R Gj2 −1 T T ¯ −1 ¯ ¯ ¯ j2i − Tji Rj2i − Tji Rj2i Tji Gj1 + γj1 GTj2 R Rj2i Tji Gj2 ,
T
¯ j2i R ∗
σj − σ(k) τj − τ (k) , γ2j = (−1)j , j = 1, 2. σ12 τ12
Moreover, for any matrices Mji , Nji and Sji , j = 1, 2, the following equations are true: T T T 2[xT (k)M1i + ς1T (k)N1i + xT (k − d1 )S1i ]
× {[Ai − I]x(k) − ς1 (k) + Bi f (y(k − τ (k)))} = 0,
(37)
and T T T 2[y T (k)M2i + ς2T (k)N2i + y T (k − d2 )S2i ]
× {[Ci − I]y(k) − ς2 (k) + Di x(k − σ(k))} = 0.
15
(38)
From (5), for any scalar πi ≥ 0 and hi ≥ 0, we can get that for i = 1, 2, · · · , n 2f T (y(k))Π(L+ + L− )y(k) − 2y T (k)ΠL+ L− y(k) −2f T (y(k))Πf (y(k)) ≥ 0, (39) 2f T (y(k − τ (k)))H(L+ + L− )y(k − τ (k)) T + − −2y (k − τ (k))HL L y(k − τ (k)) −2f T (y(k − τ (k)))Hf (y(k − τ (k))) ≥ 0. Considering (30)–(39), we obtain
∆W (z(k), k) + (1 − β)W (z(k), k) ≤ φT (k)Φi φ(k), where Φi = Φ0i − Σ1i − Σ2i . 250
It follows from Lemma 3 that Φi < 0 if the following two inequalities hold: Φ1i = Φ0i − Σ11i − Σ21i + Φ2i = Φ0i − Σ12i − Σ22i +
2 X
¯ −1 T T Gj1 < 0, GTj1 Tji R j2i ji
j=1
2 X
T ¯ −1 GTj2 Tji Rj2i Tji Gj2 < 0,
j=1
which are equivalent to (27) and (28), respectively, based on the Schur complement. Hence, if (27) and (28) hold, then we can obtain ∆W (z(k), k) + (1 − β)W (z(k), k) ≤ φT (k)Φi φ(k) < 0. Based on the similar method in Theorem 1, one can easily accomplish the proof. 255
260
265
Remark 3. The Lyapunov-Krasovskii function used in the proof of Theorem 2 inspired by but different from [33]. By applying the delay-partitioning idea, we first partition the time delays σ(k) and τ (k) into two parts: the constant parts and time-varying parts, then divide the constant parts into r parts. Based on the previous discussion and due to the results we derived not only depend on the delay bounds, but also depend on the delay partitioning, we construct that Lyapunov-Krasovskii function. Remark 4. It is worth to point out that extending the problem of exponential stability to the switched GRNs with PDT switching and time-varying delays is meaningful, which can make the issue under consideration more general. Moreover, by introducing Lemma 3 and utilizing an extended reciprocally convex matrix inequality [29] and an improved summation inequality [28], which has been proved to be less conservative than the popular reciprocally convex combination lemma (RCCL) and the traditional Jensen inequality, some new exponentially stable criteria are derived. 16
270
4. Illustrative Examples In this section, two numerical examples are presented to illustrate the effectiveness of the theoretical results derived above.
275
Example 1. When N = 2, consider the discrete-time PDT switched GRNs (4) with constant time delays and the following parameters: Subsystem #1 0.1 0 0.4 0 A1 = , C1 = , 0 0.1 0 0.4 0.3 0 0.25 −0.5 D1 = , B1 = , 0 0.3 0.2 0 Subsystem #2 A2 = D2 =
280
285
290
0.6 0 −0.1 0
0.2 0 , C2 = , 0 0.2 0 −0.2 0 , B2 = . 0.1 0.2 0.4
0 0.6
Let us choose the nonlinear regulatory functions as gj (x) = x2 /(1 + x2 ). Through the simple calculation, one has L− = diag{0, 0}, L+ = diag{0.65, 0.65}. Consider µ = 1.25, β = 0.81 and r = 2, then derived by Theorem 1, we can get the maximum allowable delay σ = 4 for the fixed delay τ = 6, such that the discrete-time PDT switched GRNs (4) are exponentially stable. Furthermore, by using the Matlab LMI toolbox and solving the inequalities of (11)–(13) in Theorem 1, it is easy to get the feasible solutions. Here we only list the matrix variables P11 , P21 , R11 and R21 as follows: 100.5717 −17.7231 162.0385 −5.3798 P11 = , P21 = , −17.7231 183.6755 −5.3798 153.7282 6.6601 −8.6280 22.7325 −1.4570 R11 = , R21 = . −8.6280 50.8056 −1.4570 24.4833 Using Theorem 1, one can find that the discrete-time PDT switched GRNs (4) are exponentially stable under the above conditions. Moreover, for given T = 3, the admissible PDT can be computed as τP DT = 2. The actual running time of the PDT switching is described in Table 1, and the evolutions of the system modes satisfying PDT switching are generated randomly as shown in Fig. 1, where ’1’ and ’2’ represent the first and second subsystem, respectively. Then consider (14) yield λ1 = 932.3251, λ2 = 1883.712, η = 0.9227 and ρ = 1.4214, thus kz(k)k ≤ 1.4214 × 0.9227k−k0 kz(k0 )kC 1 .
17
Table 1: Running time of τ -position and T -position
τ -position T -position
0-2 s 2-5 s
5-8 s 8-9 s
9-14 s 14-16 s
16-19 s 19-20 s
20-25 s 25-27 s
27-29 s 29-30 s
Modes
2
1
0
5
10
15
20
25
30
Time in samples
Figure 1: PDT switching signal
295
300
Remark 5. It is worth mentioning that under the above conditions, when choosing r = 2, we can not get the available solutions in [21]. Moreover, in [32], authors investigated exponential stability for continuous-time GRNs with timevarying delays in the light of ADT switching. However, in this paper, we focus on the exponential stability for discrete-time PDT switching GRNs with constant delays and time-varying delays, respectively. Hence, from the above, one can see that the method we derived is less conservative than that in [21] and [32] to a certain extent.
Example 2. When N = 2, we consider the discrete-time PDT switched GRNs (4) with time-varying delays and the following parameters: Subsystem #1 0 0 −0.5 0 0 , A1 = 0.4I3 , C1 = D1 = 0.3I3 , B1 = −0.5 0 −0.5 0 18
Subsystem #2
0 A2 = 0.4I3 , C2 = 0.2I3 , D2 = 0.08I3 , B2 = −0.6 0
305
310
315
0 0 −0.6
−0.6 0 , 0
and gj (x) = x2 /(1 + x2 ), L− = diag{0, 0, 0}, L+ = diag{0.65, 0.65, 0.65}. For kπ given τ (k) = 4 + 2 cos( kπ 2 ), σ(k) = 5 + sin( 2 ), it is easy to get the following parameters: τ1 = 2, τd = 2, σ1 = 4, σd = 1. Consider β = 0.95, µ = 1.2, τ2 = 6 and r = 2, according to Theorem 2 and using the Matlab LMI toolbox, the corresponding admissible upper bounds of σ2 is σ2 = 6.4661. Let the initial T T . Then and y(0) = 0.3 0.2 0.5 conditions be x(0) = 0.2 0.2 0.3 the state trajectories of subsystem # 1 and subsystem # 2 are demonstrate in Figs. 2 and 3, respectively, which can be seen that both the subsystem # 1 and subsystem # 2 are exponentially stable. Figs. 4 exhibits the state trajectories of the whole switched GRNs (4), which proved that the whole switched GRNs is exponentially stable under the PDT switching. 0.3
0.6 m1 m2 m3 protein concentration
mRNA concentration
0.2
0.1
0
−0.1
−0.2
p1 p2 p3
0.5 0.4 0.3 0.2 0.1 0
0
5
10 t/sec.
15
−0.1
20
0
5
10 t/sec.
15
20
Figure 2: mRNA and Protein concentrations mj , pj (j = 1, 2, 3) of subsystem 1
5. Conclusion
320
325
In this paper, the problem of exponential stability has been investigated for discrete-time switched GRNs with PDT switching and time delays. Combined with an extended property of quadratic convex function (Lemma 3), an improved summation inequality (Lemma 1) and an extended reciprocally convex matrix inequality (Lemma 2), we derived some new criteria for discrete-time switched GRNs to ensure the switched GRNs with constant delays and time-varying delays are exponential stable, respectively, by employing the PDT method, piecewise Lyapunov-Krasovskii functional and delay partitioning method. The achieved results are provided in the forms of LMIs which can be easily solved 19
0.3
0.6 m1 m2 m3
protein concentration
mRNA concentration
0.2
0.1
0
−0.1
−0.2
p1 p2 p3
0.5 0.4 0.3
0.02
0.2
−0.02
0
10.5
0.1
11
11.5
0 0
5
10 t/sec.
15
−0.1
20
0
5
10 t/sec.
15
20
Figure 3: mRNA and Protein concentrations mj , pj (j = 1, 2, 3) of subsystem 2 0.3
0.6 m1 m2 m3
0.1
0
−0.1
−0.2
p1 p2 p3
0.5 protein concentration
mRNA concentration
0.2
0.4 0.3 0.2 0.1 0
0
5
10 t/sec.
15
−0.1
20
0
5
10 t/sec.
15
20
Figure 4: mRNA and Protein concentrations mj , pj (j = 1, 2, 3) of the whole switched GRNs (4)
330
by using MATLAB toolbox. At last, numerical examples have been given to illustrate the usefulness of our proposed results. In future research, we can extend the main results in this paper to some more intricate systems and actual systems, such as include both stable and unstable subsystems. Furthermore, the problem of controller and filter design for switched GRNs with PDT switching can be explored. Acknowledgments
335
This work was supported in part by the National Natural Science Foundation of China (61673130, 61703120, 41772377 and 61703123), and the Self-Planned Task of State Key Laboratory of Robotics and System (HIT)(SKLRS201806B).
20
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Journal Pre-proof
Stability Analysis for Discrete-Time Switched GRNs with Persistent Dwell-Time and Time Delays Tingting Yu, Yue Zhao, Qingshuang Zeng PII: DOI: Reference:
S0016-0032(19)30710-0 https://doi.org/10.1016/j.jfranklin.2019.09.039 FI 4191
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
7 May 2019 14 September 2019 29 September 2019
Please cite this article as: Tingting Yu, Yue Zhao, Qingshuang Zeng, Stability Analysis for DiscreteTime Switched GRNs with Persistent Dwell-Time and Time Delays, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.09.039
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Stability Analysis for Discrete-Time Switched GRNs with Persistent Dwell-Time and Time Delays Tingting Yua , Yue Zhaob,∗, Qingshuang Zenga a Space
Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin, 150001, P.R. China. b Department of Control Science and Engineering, Harbin Institute of Technology, Harbin, 150001, P.R. China.
Abstract In this paper, the problem of exponential stability for discrete-time switched genetic regulatory networks (GRNs) with persistent dwell-time (PDT) switching and time delays is concerned. The novelty of this paper is derived from introducing an extended property of quadratic convex function, and utilizing an improved summation inequality together with an extended reciprocally convex matrix inequality, which is less conservative than the Jensen inequality and the reciprocally convex combination employed in the discrete-time delay systems. In addition, the considered switching regularity is more general than dwell-time (DT) switching and average dwell-time (ADT) switching. Moreover, by introducing the delay partitioning method and the piecewise Lyapunov-Krasovskii functional, a set of sufficient conditions are established in the form of linear matrix inequalities to guarantee the discrete-time PDT switched GRNs with constant time delays and time-varying delays are exponentially stable, respectively. Finally, some numerical examples are exploited to demonstrate the validity and potential of the proposed method. Keywords: Exponential stability, discrete-time switched GRNs, delay partitioning, persistent dwell-time (PDT) switching.
1. Introduction
5
Recently, genetic regulatory networks (GRNs) have drawn considerable attention, because GRNs play an important role in living systems at the molecular level, which can describe the interaction between DNA, RNA and protein. Basically, there are two types of GRNs, the Boolan model [1, 2] and the differential equation model [3]. It is worth noting that time delays are unavoidable in practice and may lead to oscillation and instability of some given system ∗ Corresponding
author Email addresses: [email protected] (Tingting Yu), [email protected] (Yue Zhao), [email protected] (Qingshuang Zeng)
Preprint submitted to Journal of The Franklin Institute
October 14, 2019
10
15
20
25
30
35
40
45
50
[4, 5, 6, 7, 8]. More significant results on the dynamic analysis of delayed GRNs have been given in the existing literature [9, 10, 11, 12]. As we all known, switched systems are a class of dynamical hybrid systems, which consist of a family of continuous-time (or discrete-time) subsystems and operated by a switching rule. In reality, GRNs often exhibit a special characteristic of network-mode switching, which gives rise to the so-called switched GRNs. In the past decade or so, a number of literatures addressed the analysis and synthesized problems of switched GRNs with Markovian jumping and time delays, which are referred to stability analysis [13, 14, 15], state estimation [16, 17], controller design [18] and so on. In [13], authors researched the global robust stability of Markovian switching uncertain stochastic GRNs with norm bounded parameter uncertainties and unbounded time-varying delays; In [15], Zhu et al. considered the robust stability analysis of Markov jump standard GRNs with parameter uncertainties and mixed time-varying delays; In [16], the problem of event-triggered H∞ state estimation for discrete-time stochastic GRNs with Markovian jumping parameters and time-varying delays was discussed. Very recently, in [14], authors addressed the asymptotic stability of Markovian switching GRNs with leakage and mode-dependent time delays; In [18], the non-uniform sampled-data control was considered to guarantee the passivity of Markov jump GRNs with time-varying delays; The decentralized event-triggered mechanism is proposed in [19] with Markov jump parameters and distributed delays for uncertain delayed GRNs. Motivated by the preceding consideration, the dynamic behaviors of delayed GRNs are subject to Markovian switching parameters. However, due to the high complexity of GRNs, the likelihood of obtaining the specified probability distribution of Markovian jumping is actually questionable [20]. So under the circumstances, two important switching signals, namely DT and ADT were proposed to handle the dynamic behaviors of switched GRNs [21, 22, 23]. However, due to the limitation on switching times of DT and ADT, Hespanha et al. in [24] presented another important switching signal, named PDT switching, which is less conservative and more general than the DT switching and the ADT switching. Up to now, the PDT is seldom addressed since the complexity of their switching rules [25, 26, 27]. In [25], Du et al. designed a fuzzy-parameterdependent fault detection filter for nonlinear systems with PDT switching; In [26], the finite-time output feedback controller is designed for a class of discretetime linear systems with PDT switching. To the best of authors’ knowledge, the exponential stability of discrete-time switched GRNs with PDT switching and time delays has not been considered yet, which constitutes the main motivation of this paper. In this paper, the problem of exponential stability is analyzed for discretetime switched GRNs with PDT switching and time delays. The main contributions of this paper are highlighted as follows: • The switching regularity we considered is PDT switching, which has been proved to be more general than DT switching and ADT switching in [24]. • By introducing an extended property of quadratic convex function (Lemma 2
55
3) and using the improved summation inequality [28] together with the extended reciprocally convex matrix inequality [29], some new exponentially stable criteria are obtained in terms of LMIs for switched GRNs based on the piecewise Lyapunov-Krasovskii functional and delay partitioning method. • Numerical examples are presented to illustrate the effectiveness of the derived results.
60
65
70
Notations: In this paper, In and 0n denote an identity matrix and a zero matrix whose dimensions are n, respectively; Z+ is a set of non-negative integers; Z≥s refers to the set {k ∈ Z+ | k ≥ s}; Rn means that it is the n-dimensional Euclidean space; Rn×m means that it is the set of all n×m real matrices; col{. . . } represents a block-column matrix; diag {. . .} represents a block-diagonal matrix; the notation P > 0 stands for that P is positive definite and real symmetric; λmax (P ) and λmin (P )) denote the maximum eigenvalue and minimum eigenvalue of matrix P , respectively; k·k refers to the Euclidean norm of a vector; sym {A} is used to represent A + AT and the superscript “T ” denotes the transposition of a matrix. For an arbitrary matrix Y and two symmetric matrices X and Z, X Y denotes a symmetric matrix, where ‘∗’ represents the term that is ∗ Z induced by symmetry. 2. System Description and Preliminary Results Consider a class of discrete-time GRNs with time delays of the form [30]: m(k + 1) = Am(k) + Bg(p(k − τ (k))) + J, (1) p(k + 1) = Cp(k) + Dm(k − σ(k)),
75
80
where m(k) = [m1 (k), m2 (k), . . . , mn (k)]T ∈ Rn , mi (k) is the concentration of the ith mRNA at time k, p(k) = [p1 (k), p2 (k), . . . , pn (k)]T ∈ Rn , pi (k) is the concentration of the ith protein at time k. A = diag {a1 , a2 , . . . , an }, C = diag {c1 , c2 , . . . , cn }, ai represents the degradation rates of the mRNA of the ith gene, ci represents the degradation rates of the protein of the ith gene, and satisfy |ai | < 1, |ci | < 1. B = [bij ]n×n ∈ Rn×n is a coupling matrix of the genetic network. D = diag {d1 , d2 , . . . , dn } and di is a positive constant representing the translation rate of the ith gene. g(p(t)) = [g1 (p1 (t)), g2 (p2 (t)), . . . , gn (pn (t))]T and gj (·) represents the feedback regulation of the protein on the transcription in which gj (x) =
85
(x/βj )Hj 1+(x/βj )Hj
is a monotonically increasing function, where βj
being a positive constant and Hj is the Hill coefficient. J = [J1 , J2 , . . . , Jn ]T , Ji represents the basal transcriptional rate of the repressor of gene i. σ(k) and τ (k) are translational and transcriptional delays, respectively. Note that gj (·) is a monotonically increasing function and that it satisfies for ∀x 6= y ∈ R li− ≤
gi (x) − gi (y) ≤ li+ , i = 1, 2, . . . , n, x−y 3
(2)
90
where li− and li+ are known real constants. The delay σ(k) and τ (k) satisfy either (A1) or (A2) as the following form: (A1): Constant time delays: 1 ≤ σ(k) ≡ σ, 1 ≤ τ (k) ≡ τ ; (A2): Time-varying delays: 1 ≤ σ1 ≤ σ(k) ≤ σ2 , 1 ≤ τ1 ≤ τ (k) ≤ τ2 and σ(k) ˙ ≤ σd , τ˙ (k) ≤ τd ,
95
where σ, σ1 , σ2 , τ , τ1 , τ2 , σd and τd are known real constants. Now, we transform the equilibrium point (m∗ , p∗ ) to the origin by the relation xi (k) = mi (k)−m∗i , yi (k) = pi (k)−p∗i and fi (yi (k)) = gi (yi (k)+p∗i )−gi (p∗i ), i = 1, 2, . . . , n. Then, the discrete-time GRNs (1) can be transformed into x(k + 1) = Ax(k) + Bf (y(k − τ (k))), (3) y(k + 1) = Cy(k) + Dx(k − σ(k)), where x(k) = [x1 (k), x2 (k), . . . , xn (k)]T , y(k) = [y1 (k), y2 (k), . . . , yn (k)]T , f (y(k − τ (k))) = [f1 (y(k − τ (k))), f2 (y(k − τ (k))), . . . , fn (y(k − τ (k)))]T . In this paper, we will focus on the exponential stability of discrete-time GRNs with PDT switching and time delays, which are described as follows: x(k + 1) = Aα(k) x(k) + Bα(k) f (y(k − τ (k)), (4) y(k + 1) = Cα(k) y(k) + Dα(k) x(k − σ(k)),
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where the piecewise constant function α(k) : [0, ∞) → N = {1, 2, . . . , N } is the switching signal, and N stands for the number of subsystems. In this paper, we assume that the switching sequence k0 , k1 , . . . , kt , . . . are unknown a priori, but are known instantly. When k ∈ [kt , kt+1 ) means that the α(kt )-th subsystem is active. For α(k) = i ∈ N , {Ai , Bi , Ci , Di } are known real constant matrices with appropriate dimensions. In addition, from (2), we can easily obtain the following condition: li− ≤
fi (y) ≤ li+ , y
∀y 6= 0,
fi (0) = 0,
∀i = 1, 2, . . . , n.
(5)
To this end, we present the following definitions and lemmas which will be required in the next section. 110
115
Definition 1. [24] Consider system (4) and switching instants k0 , k1 , . . . , kt , . . . with k0 = 0. A positive constant value τP DT is said to be the persistent dwell time (PDT), if there exist numerous nonadjacent intervals whose interval length no smaller than τP DT on which α(k) is constant, and consecutive intervals with the above mentioned property are separated by no more than T , where T stands for the period of persistence. The interval defined in the PDT stage is assumed to have the running time (τ -position) of a certain subsystem and a period of persistence (T -position). A 4
certain subsystem is activated in the τ -position, whose running time is at least τP DT . Denote the actual running time of the T -position at the p-stage as T p , p ∈ Z≥1 , then the following equation holds: G(ktp +1 ,ktp+1 ) p
T =
X
Tα(ktp +r ) ≤ T,
r=1
120
125
130
(6)
where Tα(ktp +r ) represents for the running time of the subsystem activated at the switching instant ktp +r ∈ [ktp +1 , ktp+1 ), r ∈ Z≥1 , G(ktp +1 , ktp+1 ) represents the switching times within [ktp +1 , ktp+1 ), ktp +1 denotes the next instant switching after ktp at the p-th stage, and ktp+1 stands for the instant switching to the (p + 1)-th stage. Remark 1. As discussed in [24], DT switching and ADT switching can be termed as special cases of PDT switching. It should be noted that PDT switching is more general than ADT switching, since PDT switching has no limitation on the maximum switching times during the period of persistence, while ADT switching strictly limits the upper bound of switching times within an interval of length less than τADT . Definition 2. [30] The discrete-time PDT switched GRNs (4) are said to be exponentially stable under α(k), if the following inequality holds: kz(k)k ≤ ρη (k−k0 ) kz(k0 )kC 1 , ∀k ≥ k0 ,
(7)
where constants ρ ≥ 1, 0 < η < 1, and k z(k0 )kC 1 = z(k) = 135
x(k) y(k)
{kz(k + θ)k , kς(k + θ)k},
sup
− max{σ2 ,τ2 }≤θ≤0
and ς(θ) = z(θ + 1) − z(θ).
Lemma 1. [28] For a given symmetric positive definite matrix Θ, integers b ≥ a, any sequence of discrete-time variable x : Z[a, b] → Rn , the following inequality holds: (b − a)
b−1 X i=a
T
η (k)Θη(k) ≥ ≥
θ1 θ2 θ1 θ2
T
T
Θ ∗ Θ ∗
0 θ1 3ρ(a, b)Θ θ2 0 θ1 , 3Θ θ2
where η(k) = x(k + 1) − x(k), θ1 = x(b) − x(a), θ2 = x(b) + x(a) − 2 ρ(a, b) =
b−a+1 b−a−1
for b − a 6= 1, and ρ(a, b) = 1 for b − a = 1. 5
b P
i=a
(8) (9) x(i) b−a+1 ,
140
Lemma 2. [29] For a real scalar α ∈ (0, 1), a symmetric matrix Θ ≥ 0, and any matrix S, the following matrix inequality holds: 1 0 Θ + (1 − α)T1 S αΘ ≥ , (10) 1 ∗ Θ + αT2 0 1−α Θ where T1 = Θ − SΘ−1 S T and T2 = Θ − S T Θ−1 S.
145
Extending Lemma 1 in [31] to the case where two variables are involved, we present the following lemma which will be used directly in the proof of the main theorem. Lemma 3. For symmetric matrices Z0 , Z1 , Zˆ0 and Zˆ1 , let F (α) = Z0 + αZ1 + α2 Z2 , F (β) = Zˆ0 + β Zˆ1 + β 2 Zˆ2 and F (α, β) = F (α) + F (β) with Z2 ≥ 0 and Zˆ2 ≥ 0. F (α, β) = Z0 + αZ1 + α2 Z2 + Zˆ0 + β Zˆ1 + β 2 Zˆ2 < 0 holds true if the following inequalities hold: F (α1 , β1 ) = Z0 + α1 Z1 + α12 Z2 + Zˆ0 + β1 Zˆ1 + β12 Zˆ2 < 0, F (α2 , β2 ) = Z0 + α2 Z1 + α22 Z2 + Zˆ0 + β2 Zˆ1 + β22 Zˆ2 < 0,
150
where ∀α ∈ [α1 , α2 ], ∀β ∈ [β1 , β2 ]. Proof. Because (α − α1 )(α − α2 ) ≤ 0 and (β − β1 )(β − β2 ) ≤ 0 hold for ∀α ∈ [α1 , α2 ] and ∀β ∈ [β1 , β2 ]. Then, by simple iteration and calculation yields β2 − β α − α1 β − β1 α2 − α + }F (α1 , β1 ) + { + }F (α2 , β2 ) α2 − α1 β2 − β1 α2 − α1 β2 − β1 ≥ Z0 + αZ1 + α2 Z2 + Zˆ0 + β Zˆ1 + β 2 Zˆ2 {
= F (α) + F (β) = F (α, β).
Thus one can draw the conclusion that F (α, β) < 0 from F (α1 , β1 ) < 0 and F (α2 , β2 ) < 0. 155
3. Exponential Stability Analysis 3.1. The Case of a Constant Time Delays In this section, our aim is to derive the new exponential stability criterion for discrete-time switched GRNs (4) with PDT switching and constant time delays (A1) : 1 ≤ σ(t) ≡ σ and 1 ≤ τ (t) ≡ τ .
160
Theorem 1. Under the assumption of (A1), for given integers r ≥ 1, d1 ≥ 1, d2 ≥ 1, µ ≥ 1 and a constant 0 < β < 1, where µβ ≥ 1, the discretetime PDT switched GRNs (4) are exponentially stable, if there exist positivedefinite symmetric matrices Pji ∈ Rn×n , Rji ∈ Rn×n , Qji ∈ Rrn×rn , positivedefinite diagonal matrices Π = diag {π1 , π2 , . . . , πn }, V = diag {v1 , v2 , . . . , vn }
6
165
and matrices Mji , Nji , Sji , j = 1, 2, satisfies that for i ∈ N , Ψi =
2 X
WPTj P¯ji WPj +
j=1
+sym
2 X
2 X
¯ ji WQ + WQTj Q j
j=1
˜ PT Pji W ˜R + W j j
j=1
2 X
WRTj Rji WRj +
j=1
2 X
2 X
¯ ji Gj GTj R
j=1
T ˜P WM WSji + WFT Π(L+ + L− )W 2 ji
j=1
˜ PT ΠL+ L− W ˜ P − WFT ΠWF + W T V (L+ + L− )WD −W FD 2 2 T + − T −WD V L L WD − WF D V WF D < 0,
(11)
such that ∀m, n ∈ N , Pjm ≤ µPjn , Qjm ≤ µQjn , Rjm ≤ µRjn , j = 1, 2,
(T + 1) ln µ + (T + τP DT ) ln β < 0,
and ¯ ji = diag {Qji , −Qji } , P¯ji = diag {Pji , Pji } , Q # " −Rji 0 ¯ ji = , R d +1 ∗ − djj −1 3Rji s β d1 +1 In −In 0n×(2r+4)n 0n 0n G1 = , In In 0n×(2r+4)n −2In 0n d1 s β d2 +1 0n×(r+1)n In −In 0n×(r+4)n 0n , G2 = 0n×(r+1)n In In 0n×(r+4)n −2In d2 √ 1 − βIn 0n×(2r+4)n 0n×3n , WP 1 = 0n×(2r+4)n In 0n×3n √ 1 − βIn 0n×(r+4)n 0n×2n 0n×(r+1)n , WP 2 = 0n×(r+1)n 0n×(r+4)n In 0n×2n √ βIrn p 0rn×n 0rn×(r+7)n WQ1 = , 0rn×n β d1 +1 Irn 0rn×(r+7)n √ 0rn×(r+1)n p βIrn 0rn×7n WQ 2 = , 0rn×(r+2)n β d2 +1 Irn 0rn×6n √ βd1 In 0n×3n , WR1 = 0n×(2r+4)n √ βd2 In 0n×2n , WR2 = 0n×(2r+5)n ˜ P = In 0n×(2r+7)n , W ˜ R = p 1 WR , W 1 j j βdj ˜ P = 0n×(r+1)n In 0n×(r+6)n , W 2 WM1i = M1i S1i 0n×(2r+2)n N1i 0n×3n , WM2i = 0n×(r+1)n M2i S2i 0n×(r+2)n N2i 0n×2n , 7
(12) (13)
WS1i = Ai − In 0n×(2r+2)n Bi − In 0n×3n , WS2i = 0n×rn Di Ci − In 0n×(r+3)n − In 0n×2n , WF = 0n×(2r+2)n In 0n×5n , WD = 0n×(2r+1)n In WF D = 0n×(2r+3)n In 0n×4n .
0n×6n
,
Moreover, kz(k)k ≤ ρη (k−k0 ) kz(k0 )kC 1 , where r √ λ2 ≥ 1, η = λ, λ1 = min{λmin (P1i ), λmin (P2i )}, ρ = i∈N λ1 λ2 = max{λmax (P1i ) + λmax (P2i ) + βd1 λmax (Q1i ) + βd2 λmax (Q2i ) i∈N
+
170
d1 (d1 + 1) d2 (d2 + 1) βλmax (R1i ) + βλmax (R2i )}. 2 2
(14)
Proof. By employing the delay-partitioning method to the delays σ = rd1 and τ = rd2 that give r parts. Choose the Lyapunov-Krasovskii function of the following form V (z(k), k) =
3 X
Vj (z(k), k),
(15)
j=1
where V1 (z(k), k) = xT (k)P1i x(k) + y T (k)P2i y(k), k−1 X
V2 (z(k), k) =
β k−l ξ1T (l)Q1i ξ1 (l) +
l=k−d1
V3 (z(k), k) =
−1 X
k−1 X
β k−l ξ2T (l)Q2i ξ2 (l),
l=k−d2 k−1 X
β k−l ς1T (l)R1i ς1 (l) +
−1 X
k−1 X
β k−l ς2T (l)R2i ς2 (l),
s=−d2 l=k+s
s=−d1 l=k+s
and ξ1 (l) = col ξ2 (l) = col
x(l), y(l),
x (l − d1 ) ,
x (l − 2d1 ) , . . . , x (l − rd1 + d1 )
y (l − d2 ) , y (l − 2d2 ) ,
. . . , y (l − rd2 + d2 )
ς1 (l) = x(l + 1) − x(l), ς2 (l) = y(l + 1) − y(l). 175
,
,
For k ∈ [kl , kl+1 ) , we denote the forward difference of Vj (zk , αk ) as ∆Vj (z(k), k) = Vj (z(k + 1), k + 1) − Vj (z(k), k), j = 1, 2, 3, and define the following notations: ψ(k) = col {ξ1 (k), x(k − σ), ξ2 (k), y(k − τ ), f (y(k)), f (y(k − τ )), ς1 (k), ς2 (k), ν1 (k), ν2 (k)} ,
ν1 (k) =
k X
i=k−d1
k X x(i) y(i) , ν2 (k) = . d1 + 1 d2 + 1 i=k−d2
8
Then, we can deduce that ∆V1 (z(k), k) = {2x(k) + ς1 (k)}T P1i ς1 (k) + {2y(k) + ς2 (k)}T P2i ς2 (k), (16) ∆V2 (z(k), k) = −(1 − β)V2 (z(k), k) + β −
2 X j=1
−
ξjT (k)Qji ξj (k)
j=1
β dj +1 ξjT (k − dj ) Qji ξj (k − dj ) ,
∆V3 (z(k), k) = −(1 − β)V3 (z(k), k) + β 2 X
2 X
k−1 X
2 X
(17)
dj ςjT (k)Rji ςj (k)
j=1
β k+1−l ςjT (l)Rji ςj (l).
(18)
j=1 l=k−dj
¯ ji , Gj (j = 1, 2) and applying (8) of According to the definition of matrix R Lemma 1, we can easily obtain − 180
k−1 X
l=k−dj
¯ ji Gj ψ(k). β k+1−l ςjT (l)Rji ςj (l) ≤ ψ T (k)GTj R
(19)
Moreover, for any appropriate matrices Mji , Nji and Sji , j = 1, 2, i ∈ N , we have: T T T 2[xT (k)M1i + ς1T (k)N1i + xT (k − d1 )S1i ]
×{[Ai − I]x(k) − ς1 (k) + Bi f (y(k − τ ))} = 0,
(20)
and T T T 2[y T (k)M2i + ς2T (k)N2i + y T (k − d2 )S2i ]
×{[Ci − I]y(k) − ς2 (k) + Di x(k − σ)} = 0.
185
(21)
From (5), for any scalars πi ≥ 0 and vi ≥ 0, (i = 1, 2, · · · , n), it is easy to get that 2f T (y(k))Π(L+ + L− )y(k) − 2y T (k)ΠL+ L− y(k) −2f T (y(k))Πf (y(k)) ≥ 0, (22) 2f T (y(k − τ ))V (L+ + L− )y(k − τ ) T + − −2y (k − τ )V L L y(k − τ ) −2f T (y(k − τ ))V f (y(k − τ )) ≥ 0, where L+ = diag l1+ , l2+ , . . . , ln+ and L− = diag l1− , l2− , . . . , ln− . Considering inequalities (16)–(22), we have ∆V (z(k), k) + (1 − β)V (z(k), k) ≤ ψ T (t)Ψi ψ(t), 9
where Ψi is established as in (11), which implies that Ψi < 0. Thus it is obviously that V (z(k + 1), k + 1) ≤ βV (z(k), k).
190
(23)
Suppose α(ktp ) = i, α(ktp +1 + T p ) = j and consider an arbitrary switching occurs within T p , then we can get from inequality (12) and (23) that Vj (z(ktp +1 + T p ), ktp +1 + T p ) ≤ µVl (z(ktp +1 + T p ), ktp +1 + T p )
≤ µβ Tl Vl (z(ktp +1 + T p − Tl ), ktp +1 + T p − Tl ) ≤ ···
≤ µG(ktp ,ktp +1 +T ) β Tl +Tm +···+Tn β τP DT Vi (z(ktp ), ktp ),
195
p
(24)
where l, m, . . . , n represent for all the possible indices of subsystems which are being switched within T p , with corresponding running time Tl , Tm , . . . , Tn , respectively. p p According to µ ≥ 1 and µβ ≥ 1, we have µT +1 β T ≤ µT +1 β T . It is noted that (13) implies µT +1 β T +τP DT < 1. Let kt1 = k0 and λ = µT +1 β T +τP DT , then as from (24), it follows that Vα(ktp ) (z(ktp ), ktp ) ≤ λVα(ktp−1 ) (z(ktp−1 ), ktp−1 ) ≤ · · · ≤ λp−1 Vα(kt1 ) (z(kt1 ), kt1 ) ≤ λp−1 Vα(k0 ) (z(k0 ), k0 ).
(25)
Notice from (15) that Vα(ktp ) (z(ktp ), ktp ) ≥ λ1 kz(ktp )k2 , Vα(k0 ) (z(k0 ), k0 ) ≤ λ2 kz(k0 )k2C 1 ,
200
(26)
where λ1 and λ2 are defined in (14). Following from (25) and (26), we can deduce that 1 λ2 p−1 1 p−1 kz(k)k2 ≤ Vα(ktp ) (z(ktp ), ktp ) ≤ λ Vα(k0 ) (z(k0 ), k0 ) ≤ λ kz(k0 )k2C 1 . λ1 λ1 λ1 q √ Furthermore, denoting p − 1 = ktp − k0 , ktp = k, ρ = λλ21 and η = λ, we have kz(k)k ≤ ρη (k−k0 ) kz(k0 )kC 1 . Therefore, from Definition 2, one can get that the discrete-time PDT switched GRNs in (4) are exponentially stable.
205
210
Remark 2. The analysis of stability for switched GRNs has been widely investigated, but most of these results are based on ADT switching. One innovation of Theorem 1 consists in the generality of PDT switching when compared to ADT switching [32, 21]. Another one is that Theorem 1 rest with the improved summation inequality, which has been proved to be less conservative than the usual Jensen inequality in [28]. Hence, a new set of exponentially stable criteria has been proposed with PDT switching in Theorem 1, which is more flexible. 10
215
220
225
3.2. The Case of a Time-Varying Delays In this case, we propose a new exponential stability criterion for discretetime switched GRNs (4) with PDT switching and time-varying delay (A2): 1 < σ1 ≤ σ(k) ≤ σ2 , 1 < τ1 ≤ τ (k) ≤ τ2 and σ(k) ˙ ≤ σd , τ˙ (k) ≤ τd . We set σ12 = σ2 − σ1 , τ12 = τ2 − τ1 , then partition the time delays σ(k) and τ (k) into two parts: constant parts σ1 , τ1 and time-varying parts υˆ(k), υ˜(k), i.e., σ(k) = σ1 + υˆ(k), τ (k) = τ1 + υ˜(k), where 0 ≤ υˆ(k) ≤ σ12 , 0 ≤ υ˜(k) ≤ τ12 . Then, we set σ1 = rd1 and τ1 = rd2 , by using the delay partitioning approach, one can get: Theorem 2. Under the assumption of (A2), for given integers r ≥ 1, d1 ≥ 1, d2 ≥ 1, µ ≥ 1 and a constant 0 < β < 1, where µβ ≥ 1, the discrete-time PDT switched GRNs (4) are exponentially stable, if there exist positive-definite symmetric matrices Pji ∈ Rn×n , Rjki ∈ Rn×n , Qj1i ∈ Rrn×rn , Qj2i ∈ Rn×n , Lji ∈ Rn×n , positive-definite diagonal matrices Π = diag {π1 , π2 , . . . , πn }, H = diag {h1 , h2 , . . . , hn } and matrices Mji , Nji , Sji ∈ Rn×n , Tji ∈ R2n×2n , j, k = 1, 2 such that for i ∈ N , Φ10i GT11 T1i GT21 T2i ¯ 12i 02n×2n < 0, −R Φ1i = ∗ (27) ¯ 22i ∗ ∗ −R Φ2i and ∀m, n ∈ N ,
Φ20i = ∗ ∗
T GT12 T1i ¯ 12i −R ∗
T GT22 T2i 02n×2n < 0, ¯ 22i −R
(28)
Pjm ≤ µPjn , Rjkm ≤ µRjkn , Qjkm ≤ µQjkn , Ljm ≤ µLjn , j, k = 1, 2,
(T + 1) ln µ + (T + τP DT ) ln β < 0,
and Φj0i = Φ0i − Σ1ji − Σ2ji , j = 1, 2, 230
where Φ0i =
2 X
WPTj P¯ji WPj +
j=1
+
2 X
2 X 2 X
¯ jki WQ + WQTjk Q jk
k=1 j=1
2 X 2 X
WRTjk Rjki WRjk
k=1 j=1
¯ ji WL + GT3 R ¯ 11i G3 + GT4 R ¯ 21i G4 WLTj L j
j=1
+sym
2 nX j=1
˜ T Pji W ˜R + W Pj j1
2 X
T ˜P WM WSji + WFT Π(L+ + L− )W 2 ji
j=1
˜ T ΠL+ L− W ˜ P − W T ΠWF + W T H(L+ + L− )WD −W P2 F FD 2 o T + − T −WD HL L WD − WF D HWF D , 11
¯ ji = diag { Lji , −Lji } , j = 1, 2, P¯ji = diag {Pji , Pji } , L " # −Rj1i 0 0 ¯ j1i = ¯ j2i = Rj2i R , R , j = 1, 2, dj +1 ∗ − dj −1 3Rj1i ∗ 3Rj2i
¯ jki = diag {Qjki , −Qjki } , j = 1, 2, k = 1, 2, Q √ 1 − βIn 0n×(2r+8)n 0n×7n , WP1 = 0n×(2r+8)n In 0n×7n √ 0n×(r+3)n 1 − βIn 0n×(r+6)n 0n×6n WP2 = , 0n×(r+3)n 0n×(r+6)n In 0n×6n q q ˜ R , j = 1, 2, WR = βσ 2 W ˜ WRj1 = βd2j W j1 12 12 R11 , q 2 W ˜R , W ˜ P = In 0n×(2r+15)n , WR22 = βτ12 21 1 ˜ P = 0n×(r+3)n In 0n×(r+12)n , W 2 ˜ R = 0n×(2r+8)n In 0n×7n , W 11 ˜ R = 0n×(2r+9)n In 0n×6n , W 21 √ βIrn p 0rn×n 0rn×(r+15)n , WQ11 = β d1 +1 Irn 0rn×(r+15)n 0rn×n √ βIn 0n×(r+2)n 0n×(r+13)n p WQ12 = , 0n×(r+2)n β σ2 +1 In 0n×(r+13)n √ 0rn×(r+3)n p βIrn 0rn×13n WQ21 = , 0rn×(r+4)n β d2 +1 Irn 0rn×12n √ 0n×(r+12)n 0n×(r+3)n p βIn WQ22 = , 0n×(2r+5)n β τ2 +1 In 0n×10n p β(σ12 + 1)In p 0n×(r+1)n 0n×(r+14)n , WL1 = 0n×(r+1)n β σ2 +1 In 0n×(r+14)n p β(τ12 + 1)In 0n×(r+12)n 0n×(r+3)n p WL2 = , 0n×(2r+4)n β τ2 +1 In 0n×11n WM1i = M1i S1i 0n×(2r+6)n N1i 0n×7n , WM2i = 0n×(r+3)n M2i S2i 0n×(r+4)n N2i 0n×6n , WS1i = Ai − In 0n×(2r+6)n Bi −In 0n×7n , WS2i = 0n×(r+1)n Di 0n Ci − In 0n×(r+5)n − In 0n×6n , WF = 0n×(2r+6)n In 0n×9n , WD = 0n×(2r+4)n In 0n×11n , WF D = 0n×(2r+7)n In 0n×8n , p In −In 0n×(2r+8)n 0n 0n×5n , G13 = β d1 +1 In In 0n×(2r+8)n −2In 0n×5n 12
GT23
GT11
GT12
G21 G22 Rj1T i Σjli
p d +1 2 = β
0(r+3)n×n In −In 0(r+6)n×n 0n 04n×n
0(r+3)n×n In In 0(r+6)n×n −2In 04n×n
0rn×n In −In
0rn×n In In
,
p = β σ2 +1 0(r+10)n×n 0(r+10)n×n , 0n −2In 03n×n 03n×n 0(r+1)n×n 0(r+1)n×n In In p −In In = β σ2 +1 0(r+10)n×n 0(r+10)n×n , 0n −2In 02n×n 02n×n p 0n×(2r+3)n In −In 0n×9n = β τ2 +1 0n×(2r+3)n In In 0n×9n p 0n×(2r+4)n In −In 0n×9n β τ2 +1 = 0n×(2r+4)n In In 0n×9n ¯ j2i ¯ j2i Tji R Tji 2R = ¯ j2i ¯ j2i , Rj2T i = ∗ 2R ∗ R T Gj1 Gj1 = RjlT i , j, l = 1, 2, Gj2 Gj2
0n −2In 0n −2In ,
0n 0n ,
,
Proof. Choose a Lyapunov-Krasovskii function of the following form: W (z(k), k) =
4 X
Wj (z(k), k),
(29)
j=1
where W1 (z(k), k) = xT (k)P1i x(k) + y T (k)P2i y(k), W2 (z(k), k) =
k−1 X
β k−l xT (l)Q12i x(l) +
l=k−σ2
+
2 X
k−1 X
β k−l y T (l)Q22i y(l)
l=k−τ2 k−1 X
β k−l ζjT (l)Qj1i ζj (l),
j=1 l=k−dj
W3 (z(k), k) =
2 X
−1 X
k−1 X
dj β k−l ςjT (l)Rj1i ςj (l)
j=1 s=−dj l=k+s
13
+σ12
−σ 1 −1 k−1 X X
β k−l ς1T (l)R12i ς1 (l)
s=−σ2 l=k+s
+τ12
−τ 1 −1 k−1 X X
β k−l ς2T (l)R22i ς2 (l),
s=−τ2 l=k+s
W4 (z(k), k) =
−σ 1 +1 X
k−1 X
β k−l xT (l)L1i x(l)
s=−σ2 +1 l=k−1+s
+
−τ 1 +1 X
k−1 X
β k−l y T (l)L2i y(l),
s=−τ2 +1 l=k−1+s 235
and ζ1 (l) = col ζ2 (l) = col
x(l), y(l),
x (l − d1 ) , x (l − 2d1 ) , . . . , x (l − rd1 + d1 )
y (l − d2 ) , y (l − 2d2 ) , . . . , y (l − rd2 + d2 )
,
.
For k ∈ [kl , kl+1 ) , we denote the forward difference of Wj (zk , αk ) as ∆Wj (z(k), k) = Wj (z(k+1), k+1)−Wj (z(k), k), j = 1, 2, 3, 4, and define the following notations: φ(k) = col {ζ1 (k), x(k − σ1 ), x(k − σ(k)), x(k − σ2 ), ζ2 (k), y(k − τ1 ), y(k − τ (k)), y(k − τ2 ), f (y(k)), f (y(k − τ (k))), ς1 (k), ς2 (k),
ν1 (k), ν2 (k), ν3 (k), ν4 (k), ν5 (k), ν6 (k)} , k−σ X1
ν3 (k) =
i=k−σ(k)
ν5 (k) =
k−τ X1
k−σ(k) X x(i) x(i) , ν4 (k) = , σ(k) − σ1 + 1 σ2 − σ(k) + 1 i=k−σ2
k−τ (k)
i=k−τ (k)
X y(i) y(i) , ν6 (k) = . τ (k) − τ1 + 1 τ2 − τ (k) + 1 i=k−τ2
Then we have ∆W1 (z(k), k) = {2x(k) + ς1 (k)}T P1i ς1 (k) + {2y(k) + ς2 (k)}T P2i ς2 (k), ∆W2 (z(k), k) =
2 X j=1
βζjT (k)Qj1i ζj (k) −
2 X j=1
(30)
β dj +1 ζjT (k − dj ) Qj1i ζj (k − dj )
−(1 − β)W2 (z(k), k) − β σ2 +1 xT (k − σ2 ) Q12i x (k − σ2 ) −β τ2 +1 y T (k − τ2 ) Q22i y (k − τ2 )
+βxT (k)Q12i x(k) + βy T (k)Q22i y(k), ∆W3 (z(k), k) =
2 X j=1
βd2j ςjT (k)Rj1i ςj (k) −
2 X
k−σ 1 −1 X l=k−σ2
(31)
dj β k+1−l ςjT (l)Rj1i ςj (l)
j=1 l=k−dj
−(1 − β)W3 (z(k), k) − σ12 14
k−1 X
β k+1−l ς1T (l)R12i ς1 (l)
2 T 2 T +βσ12 ς1 (k)R12i ς1 (k) + βτ12 ς2 (k)R22i ς2 (k)
−τ12
k−τ 1 −1 X
β k+1−l ς2T (l)R22i ς2 (l),
(32)
l=k−τ2
∆W4 (z(k), k) ≤ β(σ12 + 1)xT (k)L1i x(k) + β(τ12 + 1)y T (k)L2i y(k)
−(1 − β)W4 (z(k), k) − β σ2 +l xT (k − σ(k)) L1i x (k − σ(k)) −β τ2 +l y T (k − τ (k)) L2i y (k − τ (k)) .
240
(33)
¯ j1i and Gj3 (j = 1, 2), Applying (8) of Lemma 1 and the definition of matrices R we can easily get −dj
k−1 X
l=k−dj
¯ j1i Gj3 φ(k). β k+1−l ςjT (l)Rj1i ςj (l) ≤ φT (k)GTj3 R
(34)
Using (9) of Lemma 1 and (10) of Lemma 2 yields −σ12
k−σ 1 −1 X
β k+1−l ς1T (l)R12i ς1 (l) ≤ −φT (k)Σ1i φ(k),
(35)
−τ12
k−τ 1 −1 X
β k+1−l ς2T (l)R22i ς2 (l) ≤ −φT (k)Σ2i φ(k),
(36)
l=k−σ2
and
l=k−τ2
where Σji =
Gj1 Gj2
+γj2 GTj1 γ1j = (−1)j
245
Tji Gj1 ¯ j2i R Gj2 −1 T T ¯ −1 ¯ ¯ ¯ j2i − Tji Rj2i − Tji Rj2i Tji Gj1 + γj1 GTj2 R Rj2i Tji Gj2 ,
T
¯ j2i R ∗
σj − σ(k) τj − τ (k) , γ2j = (−1)j , j = 1, 2. σ12 τ12
Moreover, for any matrices Mji , Nji and Sji , j = 1, 2, the following equations are true: T T T 2[xT (k)M1i + ς1T (k)N1i + xT (k − d1 )S1i ]
× {[Ai − I]x(k) − ς1 (k) + Bi f (y(k − τ (k)))} = 0,
(37)
and T T T 2[y T (k)M2i + ς2T (k)N2i + y T (k − d2 )S2i ]
× {[Ci − I]y(k) − ς2 (k) + Di x(k − σ(k))} = 0.
15
(38)
From (5), for any scalar πi ≥ 0 and hi ≥ 0, we can get that for i = 1, 2, · · · , n 2f T (y(k))Π(L+ + L− )y(k) − 2y T (k)ΠL+ L− y(k) −2f T (y(k))Πf (y(k)) ≥ 0, (39) 2f T (y(k − τ (k)))H(L+ + L− )y(k − τ (k)) T + − −2y (k − τ (k))HL L y(k − τ (k)) −2f T (y(k − τ (k)))Hf (y(k − τ (k))) ≥ 0. Considering (30)–(39), we obtain
∆W (z(k), k) + (1 − β)W (z(k), k) ≤ φT (k)Φi φ(k), where Φi = Φ0i − Σ1i − Σ2i . 250
It follows from Lemma 3 that Φi < 0 if the following two inequalities hold: Φ1i = Φ0i − Σ11i − Σ21i + Φ2i = Φ0i − Σ12i − Σ22i +
2 X
¯ −1 T T Gj1 < 0, GTj1 Tji R j2i ji
j=1
2 X
T ¯ −1 GTj2 Tji Rj2i Tji Gj2 < 0,
j=1
which are equivalent to (27) and (28), respectively, based on the Schur complement. Hence, if (27) and (28) hold, then we can obtain ∆W (z(k), k) + (1 − β)W (z(k), k) ≤ φT (k)Φi φ(k) < 0. Based on the similar method in Theorem 1, one can easily accomplish the proof. 255
260
265
Remark 3. The Lyapunov-Krasovskii function used in the proof of Theorem 2 inspired by but different from [33]. By applying the delay-partitioning idea, we first partition the time delays σ(k) and τ (k) into two parts: the constant parts and time-varying parts, then divide the constant parts into r parts. Based on the previous discussion and due to the results we derived not only depend on the delay bounds, but also depend on the delay partitioning, we construct that Lyapunov-Krasovskii function. Remark 4. It is worth to point out that extending the problem of exponential stability to the switched GRNs with PDT switching and time-varying delays is meaningful, which can make the issue under consideration more general. Moreover, by introducing Lemma 3 and utilizing an extended reciprocally convex matrix inequality [29] and an improved summation inequality [28], which has been proved to be less conservative than the popular reciprocally convex combination lemma (RCCL) and the traditional Jensen inequality, some new exponentially stable criteria are derived. 16
270
4. Illustrative Examples In this section, two numerical examples are presented to illustrate the effectiveness of the theoretical results derived above.
275
Example 1. When N = 2, consider the discrete-time PDT switched GRNs (4) with constant time delays and the following parameters: Subsystem #1 0.1 0 0.4 0 A1 = , C1 = , 0 0.1 0 0.4 0.3 0 0.25 −0.5 D1 = , B1 = , 0 0.3 0.2 0 Subsystem #2 A2 = D2 =
280
285
290
0.6 0 −0.1 0
0.2 0 , C2 = , 0 0.2 0 −0.2 0 , B2 = . 0.1 0.2 0.4
0 0.6
Let us choose the nonlinear regulatory functions as gj (x) = x2 /(1 + x2 ). Through the simple calculation, one has L− = diag{0, 0}, L+ = diag{0.65, 0.65}. Consider µ = 1.25, β = 0.81 and r = 2, then derived by Theorem 1, we can get the maximum allowable delay σ = 4 for the fixed delay τ = 6, such that the discrete-time PDT switched GRNs (4) are exponentially stable. Furthermore, by using the Matlab LMI toolbox and solving the inequalities of (11)–(13) in Theorem 1, it is easy to get the feasible solutions. Here we only list the matrix variables P11 , P21 , R11 and R21 as follows: 100.5717 −17.7231 162.0385 −5.3798 P11 = , P21 = , −17.7231 183.6755 −5.3798 153.7282 6.6601 −8.6280 22.7325 −1.4570 R11 = , R21 = . −8.6280 50.8056 −1.4570 24.4833 Using Theorem 1, one can find that the discrete-time PDT switched GRNs (4) are exponentially stable under the above conditions. Moreover, for given T = 3, the admissible PDT can be computed as τP DT = 2. The actual running time of the PDT switching is described in Table 1, and the evolutions of the system modes satisfying PDT switching are generated randomly as shown in Fig. 1, where ’1’ and ’2’ represent the first and second subsystem, respectively. Then consider (14) yield λ1 = 932.3251, λ2 = 1883.712, η = 0.9227 and ρ = 1.4214, thus kz(k)k ≤ 1.4214 × 0.9227k−k0 kz(k0 )kC 1 .
17
Table 1: Running time of τ -position and T -position
τ -position T -position
0-2 s 2-5 s
5-8 s 8-9 s
9-14 s 14-16 s
16-19 s 19-20 s
20-25 s 25-27 s
27-29 s 29-30 s
Modes
2
1
0
5
10
15
20
25
30
Time in samples
Figure 1: PDT switching signal
295
300
Remark 5. It is worth mentioning that under the above conditions, when choosing r = 2, we can not get the available solutions in [21]. Moreover, in [32], authors investigated exponential stability for continuous-time GRNs with timevarying delays in the light of ADT switching. However, in this paper, we focus on the exponential stability for discrete-time PDT switching GRNs with constant delays and time-varying delays, respectively. Hence, from the above, one can see that the method we derived is less conservative than that in [21] and [32] to a certain extent.
Example 2. When N = 2, we consider the discrete-time PDT switched GRNs (4) with time-varying delays and the following parameters: Subsystem #1 0 0 −0.5 0 0 , A1 = 0.4I3 , C1 = D1 = 0.3I3 , B1 = −0.5 0 −0.5 0 18
Subsystem #2
0 A2 = 0.4I3 , C2 = 0.2I3 , D2 = 0.08I3 , B2 = −0.6 0
305
310
315
0 0 −0.6
−0.6 0 , 0
and gj (x) = x2 /(1 + x2 ), L− = diag{0, 0, 0}, L+ = diag{0.65, 0.65, 0.65}. For kπ given τ (k) = 4 + 2 cos( kπ 2 ), σ(k) = 5 + sin( 2 ), it is easy to get the following parameters: τ1 = 2, τd = 2, σ1 = 4, σd = 1. Consider β = 0.95, µ = 1.2, τ2 = 6 and r = 2, according to Theorem 2 and using the Matlab LMI toolbox, the corresponding admissible upper bounds of σ2 is σ2 = 6.4661. Let the initial T T . Then and y(0) = 0.3 0.2 0.5 conditions be x(0) = 0.2 0.2 0.3 the state trajectories of subsystem # 1 and subsystem # 2 are demonstrate in Figs. 2 and 3, respectively, which can be seen that both the subsystem # 1 and subsystem # 2 are exponentially stable. Figs. 4 exhibits the state trajectories of the whole switched GRNs (4), which proved that the whole switched GRNs is exponentially stable under the PDT switching. 0.3
0.6 m1 m2 m3 protein concentration
mRNA concentration
0.2
0.1
0
−0.1
−0.2
p1 p2 p3
0.5 0.4 0.3 0.2 0.1 0
0
5
10 t/sec.
15
−0.1
20
0
5
10 t/sec.
15
20
Figure 2: mRNA and Protein concentrations mj , pj (j = 1, 2, 3) of subsystem 1
5. Conclusion
320
325
In this paper, the problem of exponential stability has been investigated for discrete-time switched GRNs with PDT switching and time delays. Combined with an extended property of quadratic convex function (Lemma 3), an improved summation inequality (Lemma 1) and an extended reciprocally convex matrix inequality (Lemma 2), we derived some new criteria for discrete-time switched GRNs to ensure the switched GRNs with constant delays and time-varying delays are exponential stable, respectively, by employing the PDT method, piecewise Lyapunov-Krasovskii functional and delay partitioning method. The achieved results are provided in the forms of LMIs which can be easily solved 19
0.3
0.6 m1 m2 m3
protein concentration
mRNA concentration
0.2
0.1
0
−0.1
−0.2
p1 p2 p3
0.5 0.4 0.3
0.02
0.2
−0.02
0
10.5
0.1
11
11.5
0 0
5
10 t/sec.
15
−0.1
20
0
5
10 t/sec.
15
20
Figure 3: mRNA and Protein concentrations mj , pj (j = 1, 2, 3) of subsystem 2 0.3
0.6 m1 m2 m3
0.1
0
−0.1
−0.2
p1 p2 p3
0.5 protein concentration
mRNA concentration
0.2
0.4 0.3 0.2 0.1 0
0
5
10 t/sec.
15
−0.1
20
0
5
10 t/sec.
15
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Figure 4: mRNA and Protein concentrations mj , pj (j = 1, 2, 3) of the whole switched GRNs (4)
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by using MATLAB toolbox. At last, numerical examples have been given to illustrate the usefulness of our proposed results. In future research, we can extend the main results in this paper to some more intricate systems and actual systems, such as include both stable and unstable subsystems. Furthermore, the problem of controller and filter design for switched GRNs with PDT switching can be explored. Acknowledgments
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This work was supported in part by the National Natural Science Foundation of China (61673130, 61703120, 41772377 and 61703123), and the Self-Planned Task of State Key Laboratory of Robotics and System (HIT)(SKLRS201806B).
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