Abstract
This paper is concerned with the filtering problem for a class of discretetime genetic regulatory networks with random delay and external disturbance. The aim is to design filter to estimate the true concentrations of mRNAs and proteins based on available measurement data. By introducing an appropriate Lyapunov function, a sufficient condition is derived in terms of linear matrix inequalities (LMIs) which makes the filtering error system stochastically stable with a prescribed disturbance attenuation level. The filter gains are given by solving the LMIs. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed approach; that is, our approach is available for a smaller disturbance attenuation level than one in (Liu et al., 2012).
1. Introduction
Genetic regulatory networks (GRNs) are collections of DNA segments in a cell which interact with each other indirectly through their mRNAs, protein expression products, and other substances. Understanding the nature and functions of various GRNs is very interesting and crucially important for the treatment of many diseases such as cancers [1, 2]. Therefore, in the past decade, the study on GRNs has been put more emphasis by the researchers at interdisciplinary field. Mathematical modeling of GRNs provides a powerful tool for studying gene regulation processes. In general, genetic network models can be classified into two types, that is, the discrete model [3, 4] and the continuous model [5–8]. Usually, a continuous model is described by a (functional) differential equation. Due to slow biochemical reactions such as gene transcription and translation, time delays can play an important role in GRNs, which results that the (functional) differential equation model has been one of the most fashionable GRN models, and a lot of research on analysis and synthesis of GRNs have been recently done based on (functional) differential equation models (see, e.g., [9–15]).
The concentrations of gene products, such as mRNAs and proteins, are described as system states in a (functional) differential equation model. In practice, biologists hope to gain actual concentrations of gene products in GRNs. However, due to model errors, external perturbation, time delays, and parameters jump, the steady-state values of GRNs can hardly be obtained. In order to obtain the steady-state values through available measurement data, the design of filter and estimator for (functional) differential equation models of GRNs has been investigated by some scholars (see, e.g., [16–23]). However, due to the requirement for implementing and application of GRNs for computer-based simulation, it is of vital importance to design filter or estimator for delayed discrete-time GRNs (i.e., discretized (functional) differential equation models of GRNs) in today’s digital world, although there are, to the best author’s knowledge, only three results reported at present [24–26]. Zhang et al. [25] is concerned with the set-values filtering for a class of discrete-time GRNs with time-varying parameters, constant time-delay, and bounded external noise. For a class of discrete-time GRNs with random delays described by a Markov chain, Liu et al. [26] designed a filter ensuring that the filtering error system is stochastically stable and has a prescribed performance. By utilizing the Lyapunov stability theory and stochastic analysis technique, Wang et al. [24] investigated the existing conditions and explicit expressions of state estimators for a class of stochastic discrete-time GRNs with probabilistic measurement delays described by Bernoulli distributed white sequences. These conditions are given in terms of LMIs and are dependent on the lower and upper bounds of the time-varying delays.
It should also be emphasized that for delayed discrete-time GRNs, the stability problem (as the most important properties for any dynamics systems) [27–29], stabilization problem [30], and passivity problem [31] have been exploited. On the other hand, researchers have been paying attention to the problems of analysis and synthesis for Markovian jump system [32–36] and the filtering problems for some nonlinear systems [37–41].
Motivated by the above discussion, in this paper, we will deal with the filtering problem for a class of discrete-time GRNs with random delay which is described by a Markovian chain. By constructing a novel Lyapunov function different from one in [26], a sufficient LMI condition is first established to ensure the existence of the desired filter. The condition is dependent on the transition probability matrix of the random delay. Then, the explicit expression of the desired filter is shown to ensure the resulting filtering error system to be stochastically stable and have a prescribed disturbance attenuation level. Moreover, an optimization problem with LMIs constraints is established to design an filter which ensures an optimal disturbance attenuation level. Finally, a numerical example is given to show the effectiveness of the proposed approach.
2. Problem Formulation
Consider the following discrete-time GRN with random delays, mRNAs, and proteins [27, 28]: where and , respectively, are the concentrations of mRNA and protein of the th gene; and , where is a given positive real number standing for the uniform discretionary step size; denotes the random time delay of mRNAs and proteins, and is assumed to be a Markovian chain with state space , and is a fixed positive integer; and are the degradation rates of mRNA and protein, respectively; is the translation rate; , where is a bounded constant denoting the dimensionless transcriptional rate of gene to , and is the set of all the repressors of th gene; are the coupling coefficients satisfying the nonlinear function denotes the feedback regulation of protein in process of transcription. In general, is a monotonic function in Hill form; namely, , where is the Hill coefficient. Denote by the transition probability matrix of , where .
Let us rewrite GRN (1) as the following compact matrix form: where
Let be an equilibrium point of GRN (3), where and ; that is, To simplify the analysis, one can transform the equilibrium point to the origin by the relation and . Then the transformed system is changed as follows: where . For every , since is a monotonic function in Hill form, one can easily obtain that is a monotonically increasing function with saturation and satisfies the following inequality: where is a given constant.
When we take extracellular perturbations into account, a class of stochastic discrete-time GRN model with random delays is represented as follows: where , , , , , , , , , , and are constant matrices of appropriate dimension; and denote the expression levels of mRNA and protein, respectively; and are the estimated signals; both and are exogenous disturbance signals; and and are the initial conditions of and , respectively.
In complex GRNs, only the partial information of the network components can be usually obtained. Therefore, in order to obtain the states of GRNs, we need to estimate them via available measurements [42]. The full order linear filter which need to be designed as the following form: where , , , and are the estimates of , , , and , respectively; , , , and , , , are filter parametric matrices to be determined.
Set Then the filtering error system can be expressed as where For convenience, for a nonnegative integer we define
Definition 1 (see [26]). The delay is said to be the random delay described by a Markovian chain if it is bound by , and is a Markovian chain with state space and transition probability matrix .
Definition 2 (see [26]). When and , the filtering error system (11) is said to be stochastically stable, if for every initial condition and initial mode , where represents the mathematical expectation operator.
Definition 3. For a given constant , the filtering error system (11) is said to be stochastically stable with disturbance attenuation level if it is stochastically stable with and , and under the zero initial conditions it satisfies the following inequality: for all nonzero , , and initial mode .
The objective of this paper is to design a filter of form (9) such that the filtering error system (11) is stochastically stable with disturbance attenuation level . In order to realize the aim, we first introduce the following lemma.
Lemma 4 (see [43]). For symmetric matrices and , the matrix inequality holds, if and only if there is a matrix such that
3. Stability Analysis and Filter Design
The stability analysis for the filtering error system (11) with and is presented by the following theorem.
Theorem 5. The filtering error system (11) with and is stochastically stable, if there exist matrices , , , and such that the following matrix inequalities (18) and (19) hold for all : where
Proof. Choose an appropriate Lyapunov function for the filtering error system (11) with and as follows:
with
where and . By taking the forward difference of the function along with the solution of system (11), one can obtain that
Additionally, it can be verified that
Similarly, the following inequalities (25) can be derived:
In view of (7), we can conclude that
Then, it follows from (26) that
Now, combining (23)–(25) and (27) results in
where , and is defined as in (18).
Due to (18), formula (28) results in
where denotes the minimal eigenvalue of . Since
we obtain
by taking the conditional expectation and summing from to on both sides of (29). Consequently, by Definition 2, one can conclude from the above inequality that the filtering error system (11) is stochastically stable, and the proof is thus completed.
Remark 6. It is worth noting that the filtering problem for (8) has been studied in [26], but the obtained results in [26] are not dependent on the transition probability matrix of the random delay described by a Markovian chain. In order to reduce the conservatism and give the explicit expression of the desired filter, in the above theorem we have constituted intensive studying of the filtering problem for (8) and have investigated a result dependent on the transition probability matrix of the random delay described by a Markovian chain.
Remark 7. The novel Lyapunov functional in this paper is selected to be of (21). Since in (21) we have not only chosen the triple summation term but also considered sufficiently the information of the random delay described by a Markovian chain, the conservatism might be reduced than one in [26], which will be illustrated through a numerical example in Section 4.
Theorem 5 does not give a design procedure for the desired filter. Based on Theorem 5, the following theorem offers an approach to design a filter for GRN (8) such that the filtering error system (11) is stochastically stable with disturbance attenuation level .
Theorem 8. For given a scalar and a positive integer , if for each , there exist matrices , , , , , , , , , , , and , such that the following LMIs (34) and (35) hold, then the filtering error system (11) is stochastically stable with disturbance attenuation level . Moreover, the required filter is given by (9) with where and , , , , and are defined as previously.
Proof. , , , and are defined as in (33). Then it is easy to verify that , , and , where
and , , , , and are defined as previously. This, together with (34) and Lemma 4, implies thatDue to the Schur complement lemma, inequality (38) is equal to
where
Thus
Noting that is a submatrix of , we can conclude that . By Theorem 5, the filtering error system (11) with and is stochastically stable.
Choose the same Lyapunov function as in (21) for the filtering error system (11) and employ the similar approach in the proof of Theorem 5, one has
where , and is defined as previously. To deal with the performance, the following performance function is considered
Due to the zero initial condition and
it is easy to see from (39) and (42) that
Let ; it is concluded from Definition 3 that the filtering error system (11) is stochastically stable with disturbance attenuation level .
The proof is thus completed.
Remark 9. What can be seen from Theorem 8 is that the scalar can be calculated as an optimization variable to obtain the minimum disturbance attenuation level. To be more specific, the minimal disturbance attenuation level can be obtained by solving the following convex optimization problem: Note that if there exists a solution to the problem (46), then the minimal disturbance attenuation level is .
4. Illustrative Example
In this section we illustrate the effectiveness of the proposed approach by testing the following numerical example which has been used in [26].
Consider GRN (8) with the following parameters: The regulation function is taken as . It is easy to know that the derivative of is less than , which shows . Suppose the bound of the time delay is : then . The transition probability matrix is given by By solving the optimization problem (46), it can be obtained that the optimal disturbance attenuation level is 0.2289, which is better than one (i.e., 1.5046) in [26]. And the corresponding filter gain matrices are as follows:
In the following simulation setup, the noise signal is chosen as
Let the filtering error system run by random sequence , the trajectories and their estimations of the mRNAs and proteins are shown in Figures 1 and 2, where the solid line and dotted line describe the state trajectories and estimations of mRNAs and proteins, respectively. The filtering errors are shown in Figures 3 and 4. It can be seen from Figures 3 and 4 that the filtering error converges to zero in the absence of disturbances.
Next, we illustrate the performance of the filtering error system (11). By direct computation, we have For values of 1000 random sequences of , we obtain by MATLAB that the maximum of is , and hence the maximum disturbance attenuation level is This verifies that the disturbance attenuation level is below the given upper bound.
5. Conclusion
In this paper, we investigate the filtering problem on a class of discrete-time GRNs with random delays. The filtering error system is established as a Markovian switched system and the random delay is described as a Markovian chain. By introducing an appropriate Lyapunov function, sufficient conditions for concerned problems are derived in terms of LMIs. The designed filter guarantees that the filtering error system is stochastically stable with disturbance attenuation level. Finally, the effectiveness and performance of the obtained results are demonstrated by a numerical example.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grant no. 11371006, the fund of Heilongjiang Province Innovation Team Support Plan under Grant no. 2012TD007, the fund of Heilongjiang University Innovation Team Support Plan under Grant no. Hdtd2010-03, the Fund of Key Laboratory of Electronics Engineering, College of Heilongjiang Province, (Heilongjiang University), China, the fund of Heilongjiang Education Committee, and the Heilongjiang University Innovation Fund for Graduates. The authors thank the anonymous referees for their helpful comments and suggestions which improve greatly this note.