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

Fan, Xiaofei; Wang, Yantao; Wu, Ligang; Zhang, Xian

doi:https://doi.org/10.1155/2018/7257083

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 7257083 | https://doi.org/10.1155/2018/7257083

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

Xiaofei Fan,^1,2Yantao Wang,^1,3Ligang Wu,⁴and Xian Zhang^1,3

Academic Editor: Mohamed Boutayeb

Received27 Feb 2018

Accepted02 Jul 2018

Published20 Sept 2018

Abstract

It will be settled out for the open problem of designing an -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 -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.

1. 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 [1–10] 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 [12–17] have researched the problem of stability analysis of delayed RDGRNs. The asymptotic stability analysis of delayed RDGRNs have been involved in [13–15] 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 [20–22] 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 full-order 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 -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 -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 -order observer is delay-dependent and reaction-diffusion-dependent, which is more practical.(ii)Compared with the full-order observers, the designed -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 .(iv)The method can also extend to some other time-delay models, including Markov jump neural networks [23–25] and stochastic delayed systems [26].

Notation 1. For given matrices and , we say and , if is real symmetric positive definite and semidefinite, respectively. The identity matrix is defined by , and the zero matrix by . and stand for the transpose matrix of and the sum of and its transpose, respectively. For an arbitrary but fixed positive integer , we denote by the set . Let with , . The set of all functions having the continuous second derivatives is defined by . and represent the norms on and are defined by and The symbol refers to .

2. Problem Formulation

Consider the following delayed RDGRN [15]: where

, , and stand for the concentrations of mRNAs and proteins, respectively; , , and are the rate constants; and denote the diagonal diffusion rate matrices; represents the coupling matrix with elements defined as in [15]; is the Hill function, is the sum of dimensionless transcriptional rates which repress gene , and are delays subject to where , , , and are nonnegative real numbers.

The expression of indicates that for some common scalar .

Assume that is the unique equilibrium solution of (3). Set

Then the delayed RDGRN (3) turns into where

Next, it is assumed that the initial conditions and the Dirichlet boundary conditions of (8) are as follows: where , and and are functions in . Furthermore, let the network outputs be with the full-row-rank constant matrices and . Here, and represent the expression levels of mRNAs and proteins at time , respectively.

This paper aims at designing an -order F-T state observer for the delayed RDGRN (8), which is described as:

Here, and are the -order observer states, and , , , and are the observer gains.

Remark 1. Clearly, when the observer (12) is of full-order. So, our method is also available to establish full-order 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 , , and , if We say that system (14) is F-T stable with respect to positive scalars , , and , if so, it is its trivial solution.
To achieve our aim, one requires to seek -order observer gains , , , and such that system (14) is F-T stable with respect to positive scalars , , and .

3. Preliminaries

The following two lemmas are needed to design an -order F-T state observer.

Lemma 1 (Jensen’s inequality) [27]. For given scalars , an integral function and a matrix , there holds the following inequality:

Lemma 2 (Wirtinger-type integral inequalities) [28]. For given scalars , a function which is derivative and a matrix , there hold the following inequalitieswhere

4. Design Method of Observer

In this section, a method to design an -order F-T state observer for the delayed RDGRN (8) is proposed, that is, determine the observer gains , and such that the error system (14) is F-T stable. For this end, we define where , and 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).

Theorem 1. For given scalars , and satisfying (5) and positive constants , and , system (14) is F-T stable with respect to , and T, if there exist matrices , and diagonal matrices and and matrices , and of appropriate dimensions, such that the following inequalities are feasible for and κ∈{0,}: where , and , , , and are defined previously.

In addition, based on a feasible solution of (21), (22), and (23), an -order F-T state observer can be represented by (12) with the following gains:

Proof 1. Choose the following LKF functional: where

Then, calculating the derivatives of along on the solution of system (14), one can obtain that

Firstly, it follows from Green formula that

Using Dirichlet boundary conditions, one can derive that

This, together with the so-called Wirtinger’s inequality [29], implies that

In a similar way,

From (14) we get and

By means of Dirichlet boundary conditions, Green formula and [15], Lemma 4, it yields that

Similarly,

Combining (27) and (35), (36), (37), (38), (39), and (40), we get

Secondly, in view of (21), the reciprocally convex technique [30] and Lemma 2, it follows that

Similarly,

Again using Lemma 2, one can obtain and

The combination of (30) and (42), (43), (44), and (45) gives

Thirdly, it is clear that

It follows from Lemma 2 that and

By (47), it implies that

Similarly,

Combining (31), (50), and (51), we have

Fourthly, it is obvious that

By Lemmas 1 and 2, one can obtain and respectively. Similarly, and

By (32) and (53), (54), (55), (56), and (57), it is obtained that

Finally, from (6) and the relationship between and , it is clear to say that and that is, for given diagonal matrices and , we get

The combination of (22), (28), (29), (41), (46), (52), (39), and (60) yields

Since depends affinely upon and , one can derive from (22) that for any and . Utilizing (61), we can get

For any , integrating the two sides of (62) from 0 to , we derive

By means of the so-called Gronwall inequality, one can obtain

According to the expression of , it apparently will come

Combination of (64) and (65) derives

In addition, when , one can easily derive that where is the minimum eigenvalue of . It can be seen from (66) and (67) that

By Definition 1 and (23), under Dirichlet boundary conditions, system (14) is F-T stable with respect to , , and . The proof is completed.

Following the proof of Theorem 1, one can easily derive the following conclusion which gives a method to design an -order observer for the delayed RDGRN (8).

Corollary 1. For given scalars , , , and subject to (5), system (14) is asymptotically stable under Dirichlet boundary conditions, if there are matrices , , , and , diagonal matrices and , and matrices , , , , and of appropriate dimensions, such that LMIs (21) and (22) with are feasible for and . In addition, the desired -order state observer is described by (12) and (24).

Since inequality (23) is not an LMI, the Toolbox YALMIP of MATLAB is not applicable. Now we will do the following transformation:

Theorem 2. For given scalars , , , and satisfying (5) and positive constants , and , system (14) is F-T stable with respect to , , and under Dirichlet boundary conditions, if there are real numbers , , , , and matrices , , , , and , diagonal matrices and , and matrices , , , , and of appropriate dimensions, such that LMIs (21), (22) and the following (69), (70), (71), (72), (73), (74), and (75) are feasible for and :

In addition, the desired -order F-T state observer is described by (12) and (24).

Proof 2. It follows from (69), (70), (71) and (72) that and

This, together with (74) and (75), implies that which shows that (23) holds. By Theorem 1, we complete the proof.

Finally, we make several remarks on the method proposed in this paper.

Remark 2. Different from [18, 19], this paper gives an -order F-T state observer (12) for the delayed RDGRN (8). It should be mentioned that, compared with the full-order observer, the reduced-order one is more practical. Particularly, a reduced-order observer can save the cost in the engineering applications.

Remark 3. In the proof of Theorem 1, we employ the so-called 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 .

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, and , are introduced into the augmented vector .

5. An Illustrative Example

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

Example 1. Consider the delayed RDGRN (8), where , , , ,and

In order to save space, we only design the -order F-T state observer. Let , , , and . The LMIs in Theorem 2 is solved by means of the MATLAB’s toolbox. The solution matrices are listed as follows:

Furthermore, we can obtain the corresponding observer gains as follows:

When the initial function and for , the state responses of RDGRN (8), -order observer (12), and error system (14) are presented in Figures 1–6. From which, it is seen that our approach is effective.

6. Conclusions

The design problem of -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 -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 -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, [23–25]). However, all approaches proposed in these 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.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request. All data are included within the manuscript.

Conflicts of Interest

The authors declare that there is no conflict of interests regarding the publication of this article.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Grant nos. 61174126, 11371006, 61222301, and 11501182), and the Fundamental Research Funds for the Central Universities (Grant no. HIT.BRETIV.201303).

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Copyright © 2018 Xiaofei Fan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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