An r-Order Finite-Time State Observer for Reaction-Diffusion Genetic Regulatory Networks with Time-Varying Delays

It will be settled out for the open problem of designing an r-order finite-time (F-T) state observer for reaction-diffusion genetic regulatory networks (RDGRNs) with time-varying delays. By assuming the Dirichlet boundary conditions, aiming to estimate the mRNA and protein concentrations via available network measurements. Firstly, sufficient F-T stability conditions for the filtering error system have been investigated via constructing an appropriate Lyapunov–Krasovskii functional (LKF) and using several integral inequalities and (reciprocally) convex technique simultaneously. These conditions are delay-dependent and reaction-diffusion-dependent and can be checked by MATLAB toolbox. Furthermore, a method is proposed to design an r -order F-T state observer, and the explicit expressions of observer gains are given. Finally, a numerical example is presented to illustrate the effectiveness of the proposed method.


Introduction
Recently, due to a great many applications in the real world, genetic regulatory networks have become one of the hot topics in many fields.Much significant results (see [1][2][3][4][5][6][7][8][9][10] and the references therein) have been obtained.Usually, owing to the highly complexity of genetic regulatory networks, spatial homogeneity (i.e., the mRNA and protein concentrations are independent on their space positions) is assumed in the process of modelling genetic regulatory networks.However, this assumption is sometimes unreasonable, for example, the concentrations of proteins in ribosomal gathering are higher than ones of other parts of the cell in the process of translation.So, bringing the diffusing phenomenon into the models of genetic regulatory networks is urgent and necessary, which results in RDGRNs.Generally, the models of genetic regulatory networks are divided into discrete-time models and continuous-time ones [11].A continuous-time model has wide applications in studying the complex features and the nonlinear behaviors of genetic regulatory networks.Moreover, due to the slow processes of transcription and translation, time delays should be considered in the continuous-time models of RDGRNs.It should be emphasized that time delays may lead to poor network performance, even instability.To the best our knowledge, those works in [12][13][14][15][16][17] have researched the problem of stability analysis of delayed RDGRNs.The asymptotic stability analysis of delayed RDGRNs have been involved in [13][14][15] by constructing an appropriate LKF and applying some inequality techniques.In [16], a sufficient condition of F-T stability for delayed RDGRNs has been given by constructing an LKF including quad-slope integrations and applying the Gronwall inequality and Wirtinger-type integral inequality.Related research on uncertain stochastic time-delay RDGRNs and impulsive stochastic time-delay RDGRNs can be found in [12,17], respectively.
Generally speaking, with the change of environment, not all mRNA and protein concentrations are measurable.So, it is important and necessary to estimate the mRNA and protein concentrations.Currently, one of effective approaches to estimate system states is to design observers based on the available measurement.As we all know, the problem of estimating the states of delayed RDGRNs is only studied in [18,19], although some scholars have addressed the diffusion-free case (see [20][21][22] and the references therein).A full-order observer based on available measurement has been designed in [18] by introducing an LKF and employing several integral inequalities, convex approach and Green's second identity.The existence condition and design method of a full-order F-T observer have been given in [19].It is worth emphasizing that all these results are about the fullorder observers, and all approaches proposed in these literatures are not available for designing reduced-order observers.But, the design of a reduced-order observer is necessary, since it can be more easily realized than the full-order one in engineering practise.
The above discussion motivates us to design an r-order (i.e., reduced-order) F-T state observer for delayed RDGRNs.By constructing a novel LKF and employing several integral inequalities and (reciprocally) convex technique to estimate its derivative, a F-T stability criterion in the form of linear matrix inequalities (LMIs) is established for the resulting error system.In addition, we propose a method for designing an r-order F-T state observer for RDGRNs with time-varying delays, and the observer gains are parameterized by the solutions of these LMIs.Furthermore, the method proposed in this paper is explained by a numerical example.
It is worth emphasizing that the method proposed in this paper has the following advantages: (i) The r-order observer is delay-dependent and reaction-diffusion-dependent, which is more practical.
(ii) Compared with the full-order observers, the designed r-order one can save the cost in the engineering.
(iii) For delayed genetic regulatory networks without reaction-diffusion items, the method is still keeping available by removing the corresponding parts of V 1 .
Notation 1.For given n × n matrices X and Y, we say X > Y and X ≥ Y, if X − Y is real symmetric positive definite and semidefinite, respectively.The n × n identity matrix is defined by I n , and the m × n zero matrix by 0 m×n .A T and Sym A stand for the transpose matrix of A and the sum of A and its transpose, respectively.For an arbitrary but fixed positive integer l, we denote by l the set 1, 2, … , l .Let The symbol col

Problem Formulation
Consider the following delayed RDGRN [15]: where , and p i t, x stand for the concentrations of mRNAs and proteins, respectively; a i , c i , and b i are the rate constants; D k > 0 and D * k > denote the diagonal diffusion rate matrices; W represents the coupling matrix with elements defined as in [15]; g j s → s H /1 + s H is the Hill function, q i is the sum of dimensionless transcriptional rates which repress gene i, κ t and ρ t are delays subject to where ρ, κ, μ ρ , and μ κ are nonnegative real numbers.

Complexity
The expression of g i indicates that for some common scalar ξ i > 0.
Assume that m * x , p * x is the unique equilibrium solution of (3).Set Then the delayed RDGRN (3) turns into Next, it is assumed that the initial conditions and the Dirichlet boundary conditions of ( 8) are as follows: where d = max κ, ρ , and ψ t, x and ψ * t, x are functions in C 2 −d, 0 × Ω, ℝ n .Furthermore, let the network outputs be with the full-row-rank constant matrices N m and N p .
Here, z m t, x and z p t, x represent the expression levels of mRNAs and proteins at time t, respectively.This paper aims at designing an r-order F-T state observer for the delayed RDGRN (8), which is described as: Here, m t, x and p t, x are the r-order observer states, and Â, Ĉ, Mm , and Mp are the observer gains.
Remark 1.Clearly, when r = n the observer ( 12) is of fullorder.So, our method is also available to establish fullorder observers for the delayed RDGRN (8).

Define the augmented vectors
According to (8), (11), and (12), one can obtain the resulting error system as follows: where Definition 1 (see [12]).The trivial solution of system ( 14) is called F-T stable with respect to positive scalars c 1 , c 2 , and T, if We say that system (14) is F-T stable with respect to positive scalars c 1 , c 2 , and T, if so, it is its trivial solution.To achieve our aim, one requires to seek r-order observer gains Â, Ĉ, Mm , and Mp such that system ( 14) is F-T stable with respect to positive scalars c 1 , c 2 , and T.

Preliminaries
The following two lemmas are needed to design an r-order F-T state observer.
Lemma 1 (Jensen's inequality) [27].For given scalars a < b, an integral function χ a, b → ℝ n and a matrix M T = M > 0, there holds the following inequality: Lemma 2 (Wirtinger-type integral inequalities) [28].For given scalars a < b, a function χ a, b → ℝ n which is derivative and a matrix M T = M > 0, there hold the following inequalities where

Design Method of Observer
In this section, a method to design an r-order F-T state observer for the delayed RDGRN ( 8) is proposed, that is, determine the observer gains Â, Ĉ, Mm , and Mp such that the error system ( 14) is F-T stable.For this end, we define Complexity where L k , D k , and D * k are the same with previous ones.Now we can provide an approach to design an r-order F-T state observer for the delayed RDGRN (8).
In addition, based on a feasible solution of ( 21), (22), and ( 23), an r-order F-T state observer can be represented by (12) with the following gains: Proof 1. Choose the following LKF functional: V t, e m , e p = 〠 6 i=1 V i t, e m , e p , 25 where Then, calculating the derivatives of V i t, e m , e p i ∈ 6 along on the solution of system (14), one can obtain that Using Dirichlet boundary conditions, one can derive that This, together with the so-called Wirtinger's inequality [29], implies that Secondly, in view of ( 21), the reciprocally convex technique [30] and Lemma 2, it follows that −ρ The combination of ( 30) and ( 42), ( 43), (44), and (45) gives Combining (31), (50), and (51), we have Fourthly, it is obvious that By Lemmas 1 and 2, one can obtain respectively.Similarly, By (32) and ( 53), ( 54), ( 55), (56), and (57), it is obtained that Finally, from (6) and the relationship between f i and g i , it is clear to say that f 0 = 0 and that is, for given diagonal matrices Z 1 > 0 and Z 2 > 0, we get The combination of ( 22), ( 28), ( 29), ( 41), ( 46), ( 52), (39), and (60) yields For any t ∈ 0, T , integrating the two sides of (62) from 0 to t, we derive V t, e m , e p ≤ V 0, e m 0, x , e p 0, x + t 0 αV s, e m , e p ds 63 By means of the so-called Gronwall inequality, one can obtain V T, e m , e p ≤ e αT V 0, e m 0, x , e p 0, x 64 According to the expression of V t, e m , e p , it apparently will come V 0, e m 0, x , e p 0, x  14) is F-T stable with respect to c 1 , c 2 , and T. The proof is completed.
Following the proof of Theorem 1, one can easily derive the following conclusion which gives a method to design an r-order observer for the delayed RDGRN (8).
Remark 2. Different from [18,19], this paper gives an r -order F-T state observer (12) for the delayed RDGRN (8).It should be mentioned that, compared with the fullorder observer, the reduced-order one is more practical.Particularly, a reduced-order observer can save the cost in the engineering applications.
11 Complexity Remark 3. In the proof of Theorem 1, we employ the socalled Wirtinger's inequality to obtain the equalities (18,19), which claims the necessity of Dirichlet boundary conditions.However, by employing the technique used in [14,15], one can deal with the cases of Robin boundary conditions and Neumann boundary conditions, which will make the LMI conditions corresponding to ones in Theorems 1 and 2 more conservative.Remark 4. For delayed genetic regulatory networks without reaction-diffusion items, the method proposed in Theorems 1 and 2 is still keeping available by removing the corresponding parts of V 1 t, e m , e p .Remark 5.For delayed genetic regulatory networks without reaction-diffusion items, Zhang et al. [22] proposed a method to design full-and reduced-order state observers.It should be pointed that this method cannot be applied to delayed RDGRNs, since the equivalent decompositions of output matrices are required.Remark 6.Several techniques used in this paper may be available for some other time-delay models: (1) Wirtinger-type integral inequality, instead of Jensen's inequality, is applied to estimate some integral items in the derivative of LKFs; (2) The convex technique and reciprocally convex technique are organically combined; (3) The coefficients, 1/ρ t and 1/ρ − ρ t , are introduced into the augmented vector η t, x .

An Illustrative Example
In this section, we will give a numerical example to verify the availability of the proposed method to design the r-order F-T state observer.

Conclusions
The design problem of r-order F-T state observer of RDGRNs with time-varying delays has been researched under Dirichlet boundary conditions.Utilizing available measurement outputs, we proposed a method to design r-order F-T observer which can be used to estimate the mRNA and protein concentrations.Sufficient F-T stability conditions for error system have been investigated by constructing an appropriate LKF and employing several integral inequalities and (reciprocally) convex technique.Thereby, the concrete expression of r-order F-T state observer is given.A numerical example is presented to illustrate the validity of the proposed method.It is worth emphasizing that the reduced-order observer problem of delayed RDGRNs is studied at the first time.
In literature, the problem of full-order state estimation for complex systems have been addressed (see, for example, [23][24][25]).However, all approaches proposed in these 13 Complexity literature are not available for designing reduced-order observers.Therefore, extending the method presented in this paper to the other system models will be left for future work.

References
[1] X. Zhang, L. Wu, and J. Zou, "Globally asymptotic stability analysis for genetic regulatory networks with mixed delays: an M-matrix-based approach," IEEE/ACM Transactions on

Figure 1 :
Figure 1: The real trajectories of mRNA concentration ( m t, x 2 ).

Figure 4 :
Figure 4: The estimation protein concentrations by the 1-order observer ( p t, x 2 ).

Figure 5 :
Figure 5: The mRNA concentrations of error system ( e m t, x 2 ).

Figure 6 :
Figure 6: The protein concentrations of error system ( e p t, x 2 ).

Figure 2 :
Figure 2: The real trajectories of protein concentration ( p t, x 2 ).