Design of Nonfragile State Estimator for Discrete-Time Genetic Regulatory Networks Subject to Randomly Occurring Uncertainties and Time-Varying Delays
We deal with the design problem of nonfragile state estimator for discrete-time genetic regulatory networks (GRNs) with time-varying delays and randomly occurring uncertainties. In particular, the norm-bounded uncertainties enter into the GRNs in random ways in order to reflect the characteristic of the modelling errors, and the so-called randomly occurring uncertainties are characterized by certain mutually independent random variables obeying the Bernoulli distribution. The focus of the paper is on developing a new nonfragile state estimation method to estimate the concentrations of the mRNA and the protein for considered uncertain delayed GRNs, where the randomly occurring estimator gain perturbations are allowed. By constructing a Lyapunov-Krasovskii functional, a delay-dependent criterion is obtained in terms of linear matrix inequalities (LMIs) by properly using the discrete-time Wirtinger-based inequality and reciprocally convex combination approach as well as the free-weighting matrix method. It is shown that the proposed method ensures that the estimation error dynamics is globally asymptotically stable and the desired estimator parameter is designed via the solutions to certain LMIs. Finally, we provide two numerical examples to illustrate the feasibility and validity of the proposed estimation results.
Genetic regulatory networks (GRNs), which are biochemically dynamical systems describing highly complicated interactions among DNA, RNA, protein, and metabolic product in the interactive transcriptional and translational processes, have become an attractive research field in the systems biology and biomedical sciences during the past few decades. One of the main objectives in systems biology is to understand how mRNAs and proteins work collectively and interact with each other to perform the complicated biological functions (e.g., the process of transcriptions and translations). In the past few years, a great deal of attention has been devoted to the theoretical analysis and experimental investigation on GRNs. Accordingly, a large amount of important results has been reported in literature making significant contributions for understanding both static and dynamic behaviors of biological systems in detail (see [1–3] and the references therein). Among them, two main classes of models used to characterize the GRNs include discrete models such as Boolean model [4, 5] and continuous models such as differential equation models [6, 7].
When modelling GRNs, we should bear in mind the fact that GRNs models are unavoidably affected by modelling uncertainties, including parametric errors, time delays, and stochastic noises. More specifically, the time delays are unavoidable due primarily to the slow process of transcription/translation, translocation, or the finite switching speed of amplifiers. The existence of the time delays could affect the whole system performance and may lead to oscillation, divergence, and instability of the genetic networks. According to the occurrence way of time delays, they can generally be classified into three types: interval time-varying delays [8, 9], distributed delays , and random time delays [11, 12]. In addition, gene regulation is an intrinsically noisy process due to intracellular and extracellular noise perturbations, which are derived from random births and deaths of individual molecules and environmental fluctuations. Accordingly, a great deal of effort has been made to deal with the analysis problem of GRNs with time-varying delays as well as stochastic noises and a growing number of approaches have been provided in the literature to examine the dynamical behaviors of the addressed GRNs [2, 3, 13]. On the other hand, the discrete-time version of GRNs receives increasing research interests due to the reason that the continuous-time GRNs are commonly discretized for computer simulation and experimental purposes [14, 15]. In recent years, much effort has been made to tackle the synthesis problems of discrete GRNs with time delays. For example, the robust control problem has been investigated in  for discrete delayed stochastic GRNs. Moreover, the asymptotic stability problems have been investigated in [17–19] for discrete-time uncertain GRNs with time-varying delays and stochastic fluctuations, where some sufficient conditions have been presented to ensure the stability of the addressed GRNs via the linear matrix inequality approach.
In practice, for the purpose of both the drug design and disease diagnosis, it is worth mentioning that biologists hope to gain actual concentrations of gene products in GRNs. However, due to the existence of the model errors, time delays, and external disturbances, the actual values of GRNs can hardly be obtained and only partial information about the gene states is commonly available in the measurement outputs. As such, the filtering and state estimation problems for complex dynamics systems have been widely investigated; see, for example, [20–26]. To mention a few, the set-values filtering problem has been investigated in  for a class of GRNs with time-varying uncertainties and bounded external noise, where new filtering scheme has been given for the addressed GRNs. In , the state estimation problem has been studied for a class of discrete-time GRNs with random delays. Moreover, the state estimation problem has been discussed in  for discrete stochastic GRNs with Markovian jumping parameters and time-varying delays. More recently, the sampling-data state estimation algorithm has been provided in  for a class of GRNs subject to time-varying delays. On the other hand, the parameter uncertainties may be changed in a random way with certain types and intensity due to various effects, for instance, the changing subsystem interconnections, network-induced random failures, repairs of components and sudden environmental perturbations, and so on. Hence, it is more significant to deal with the effects from the randomly occurring parameter uncertainties onto the networked systems [27–32]. In addition, as discussed in , a small or even tiny drift/fluctuation/error of the presented controller/estimator during the parameter implementation may result in unexpected fragility/degradation for whole system performance. In other words, there exist the deviations between the parameters of actually implemented controller/estimator and their expected values, and hence there is a need to design the robust controller/estimator with certain degree of tolerance against the possible deviations. In the past decade, the nonfragile estimation problems have gained much attention with respect to the implementation errors of proposed estimators [34–36]. However, the nonfragile state estimation problem has not been fully studied yet for discrete delayed GRNs, not to mention the case where the randomly occurring parameter uncertainties are also considered. As such, the main purpose of the paper is to provide a robust state estimation method against the mentioned phenomena.
Motivated by the aforementioned analysis, in this paper, we aim to investigate the robust nonfragile state estimation problem for a class of discrete GRNs with time-varying delays and randomly occurring uncertainties. The Lyapunov-Krasovskii functional is chosen which utilizes more information about the time delays and a new set of conditions is added to calculate the difference of the designed Lyapunov-Krasovskii functional. Furthermore, as mentioned in , it is clearly seen that the Wirtinger-based inequality can reduce the conservatism and provide a more tighter lower delay bound of the summation terms. Therefore, we revisit the nonfragile state estimation problem for discrete GRNs with time-varying delays and randomly occurring uncertainties by using the discrete-time Wirtinger-based inequality and the reciprocally convex combination inequality. A delay-dependent estimation criterion is established in terms of feasibility of a set of LMIs. The main contributions of this paper can be summarized as follows: the discrete Wirtinger-based inequality is utilized for the first time to investigate the nonfragile state estimation problem for GRNs with time-varying delays and randomly occurring uncertainties, where the estimator gain perturbations occur in a random way; by utilizing the discrete-time Wirtinger-based inequality and the reciprocally convex combination approach, the summation terms are addressed and more information of time delays is reflected in the proposed algorithm; and more free-weighting matrices are introduced by adding to new conditions with hope to further reduce the conservativeness induced by time delays. Finally, examples and simulation are given to illustrate the effectiveness and advantages of the proposed main results.
Notations. For a matrix , denotes that is a symmetric positive-definite matrix. stands for the mathematical expectation operator. The superscript and represent the transpose and the inverse of the matrix, respectively. represents an identity matrix with appropriate dimension. For simplicity, represents the term that is induced by symmetry in symmetric block matrices.
2. Model Description and Preliminaries
In this paper, we consider the following discrete GRNs with time-varying delays, mRNA, and proteins:where , are the concentrations of mRNA and protein at instant , respectively. and are real constant diagonal matrices representing the decay rates of mRNA and protein with entries and , , is the translation rate, and is the coupling matrix of the genetic networks. stands for the basal rates of degradation. and are the time-varying delays denoting the feedback regulation delay and the translation delay satisfying and . In addition, the nonlinear function represents the feedback regulation of the protein on the transcription. It is a monotonic function in the Hill form; that is, , , where is the Hill coefficient and is a positive constant.
Let be an equilibrium point of system (1). Then, it is easy to obtain thatSubsequently, we can shift the equilibrium point of system (1) to the origin point via the transformations and . Then, system (1) can be converted into the following form: with .
As mentioned above, the GRNs are often the large-scale networks and it should be noted that system parameter uncertainties may be subject to random changes in real circumstances due to some factors such as repairs of components and sudden environmental disturbances. Thus, they may occur in a probabilistic way with various types and intensities. By considering the randomly occurring uncertainties, system (3) becomeswhere , , , and denote the parameter uncertainties satisfying the following condition:where , , , , and are known constant matrices and is an unknown time-varying matrix satisfying . The stochastic variables and are used to characterize the randomly occurring uncertainties and are mutually independent Bernoulli-distributed white noise sequences satisfying where and are known constants.
The state components of the GRNs are usually not completely accessible. Consequently, it is necessary to make use of the available output information and design a state estimator to estimate the state vector of the addressed GRNs. For this case, assume that the measurement outputs are given as follows:where and are the measurements of model (4); and are the known matrices.
The main objective of this paper is to estimate the concentrations of mRNA and protein in (4) from the available network outputs in (7). In the sequel, we design the following nonfragile state estimator:where , are the estimations of and , and the nonfragile estimator gain matrices , are to be determined. The real-valued matrices and represent possible estimator gain fluctuations. It is assumed that and have the following structure:where , , and are the known constant matrices and is the unknown time-varying matrix satisfying . The stochastic variables represent the perturbations in the estimator gain matrices and are mutually independent Bernoulli-distributed white noise sequences taking on values of or withwhere are known constants.
Letting the estimation error be and , the estimation error dynamics can be described as follows:where . Initial conditions for the uncertain GRNs (4) and state estimator system (8) are assumed to be and on , where .
To facilitate further derivations, setCombining (4) with (11), we obtain the augmented estimation error dynamics:where
Assumption 1. The nonlinear function is a monotonically increasing function and satisfies the following condition:which is equivalent towhere and are known constants.
Definition 2. System (8) is said to be a robust asymptotic state estimator of the GRNs (4) if estimation error system (13) is globally robustly asymptotically stable in the mean square; that is,
To end of this section, we introduce the following lemmas which will be frequently used in the subsequent developments.
Lemma 3 ( (Schur complement)). Given constant matrices , , and with appropriate dimensions, where and , then if and only if
Lemma 4 ( (discrete-time Wirtinger-based inequality)). For a given positive-definite matrix and three nonnegative integers satisfying , denoteThen, one haswhere
Lemma 5 (see ). For given positive integers , a scalar , an -matrix , and two -matrices , for all vector , the function is given by If there is a matrix such that , then the following inequality holds:
3. Main Results
In this section, we first consider that there are no parameter uncertainties in GRNs (4). Also, the estimator can be selected as (8); then augmented error system (13) can be rewritten asOur main aim is to design the nonfragile state estimator to guarantee that estimation error system (24) is globally asymptotically stable. In other words, by designing a proper Lyapunov-Krasovskii functional, together with the discrete-time Wirtinger-based inequality, reciprocally convex combination approach, and the free-weighting matrices method, we are interested in looking for the estimator gain matrices and such that the error dynamics governed by (24) is globally asymptotically stable. Further, the result is extended to handle the nonfragile state estimation problem for GRNs (4) with randomly occurring uncertainties.
Theorem 6. For given positive scalars , , , and , estimation error system (24) is globally asymptotically stable, if there exist scalars , matrices , , , , and , positive diagonal matrices , , and , matrices , and block diagonal matrices , , , , , and , such that the following LMIs hold:where And the remaining terms are zero. Then, the estimator gain matrices are determined by
Proof. SetWe construct the following Lyapunov-Krasovskii functional for discrete-time GRNs (24):whereFor convenience, define the following notations: Calculate the difference of by defining along the solutions of (24):In the same way, we obtainBy Lemma 4, it follows thatNext, it follows from Lemmas 4 and 5 thatBased on (35)–(37), we haveFrom Assumption 1, for diagonal matrices , , and and a known matrix , it is not difficult to see that the inequalitieshold. In addition, since , , by introducing relaxation matrices , , , and with appropriate dimensions, we obtainFrom , we haveThen, there exist positive scalars and satisfyingNow, combining (32)–(45) and taking mathematical expectation, one can obtain the following inequality:Using the Schur complement Lemma, it can be shown that the considered uncertain system (24) is globally asymptotically stable in mean square sense, which completes the proof of the theorem.
Remark 7. It is worthwhile to mention that two constraint conditions (i.e., (39)-(40)) on feedback regulatory function were used on dealing with the state estimation problem for discrete GRNs with time-varying delays. In this paper, we add inequality (41) with hope to reduce the conservativeness introduced by time delays.
Now, we are in a position to deal with the nonfragile estimator design for GRNs (4) with time-varying delays and randomly occurring uncertainties.
Theorem 8. For given positive scalars , , , and , system (8) is said to be a globally robustly asymptotic state estimator of GRNs (4), if there exist scalars , matrices , , , , and , positive diagonal matrices , , and , matrices , and block diagonal matrices , , , , , and with appropriate dimensions, such that the following LMIs hold:with and the other parameters are defined as in Theorem 6. Moreover, the estimation gain matrices are determined by
Proof. The proof of this theorem follows from Theorem 6 directly. According to (13), for any matrices , , , and , we haveSimilarly, we getThen, for positive scalars , we havewith