H ∞ Filtering for Discrete-Time Genetic Regulatory Networks with Random Delay Described by a Markovian Chain

and Applied Analysis 3 can easily obtain that g i is a monotonically increasing function with saturation and satisfies the following inequality: g i (0) = 0, 0 ≤ g i (s 1 ) − g i (s 2 ) s 1 − s 2 ≤ l i , ∀s 1 , s 2 ∈ R, s 1 ̸ = s 2 , (7) where l i 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: x m (k + 1) = Ax m (k) + Bg (x p (k − d (k))) + E 1 w (k) , x p (k + 1) = Cx p (k) + Dx m (k − d (k)) + F 1 V (k) , y m (k) = C 1 x m (k) + E 2 w (k) , y p (k) = C 2 x p (k) + F 2 V (k) , z m (k) = G 1 x m (k) , z p (k) = G 2 x p (k) , x m (k) = θ m (k) , x p (k) = θ p (k) , k = −d, −d + 1, . . . , 0, (8) where A, B, C, D, C 1 , C 2 , E 1 , E 2 , F 1 , F 2 , G 1 , and G 2 are constant matrices of appropriate dimension; y m (k) := [y m1 (k) ⋅ ⋅ ⋅ y mn (k)] T and y p (k) := [y p1 (k) ⋅ ⋅ ⋅ y pn (k)] T denote the expression levels of mRNA and protein, respectively; z m (k) := [z m1 (k) ⋅ ⋅ ⋅ z ml (k)] T and z p (k) := [z p1 (k) ⋅ ⋅ ⋅ z pl (k)] T are the estimated signals; bothw(k) and V(k) are exogenous disturbance signals; and θ m (k) and θ p (k) are the initial conditions of x m (k) and x p (k), 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: x m (k + 1) = A f x m (k) + B f y m (k) , x p (k + 1) = C f x p (k) + D f y p (k) , ?̂? m (k) = G 1f x m (k) + H 1f y m (k) , ?̂? p (k) = G 2f x p (k) + H 2f y p (k) , (9) where x m (k), x p (k), ?̂? m (k), and ?̂? p (k) are the estimates of x m (k),x p (k), z m (k), and z p (k), respectively;A f ,B f ,C f ,D f ∈ R n×n and G 1f , G 2f , H 1f , H 2f ∈ R l×n are filter parametric matrices to be determined. Set x m (k) = [ x m (k) x m (k) ] , x p (k) = [ x p (k) x p (k) ] , e m (k) = z m (k) − ?̂? m (k) , e p (k) = z p (k) − ?̂? p (k) . (10) Then the filtering error system can be expressed as x m (k + 1) = Ax m (k) + Bg (Z 1 x p (k − d (k))) + Ew (k) , x p (k + 1) = Cx p (k) + DZ 1 x m (k − d (k)) + FV (k) , e m (k) = G 1f x m (k) + H 1f w (k) , e p (k) = G 2f x p (k) + H 2f V (k) , x m (k) = ̃ θ m (k) , x p (k) = ̃ θ p (k) , k = −d, −d + 1, . . . , 0, (11) where ̃ θ m (k) = [ θ m (k) 0 ] , ̃ θ p (k) = [ θ p (k) 0 ] , A = [ A 0 B f C 1 A f ] , B = [ B 0 ] , C = [ C 0 D f C 2 C f ] , D = [ D 0 ] , E = [ E 1 B f E 2 ] , F = [ F 1 D f F 2 ] , G 1f = [G 1 − H 1f C 1 −G 1f ] , G 2f = [G 2 − H 2f C 2 −G 2f ] , H 1f = −H 1f E 2 , H 2f = −H 2f F 2 , Z 1 = [I 0] . (12) For convenience, for a nonnegative integer k we define Θ k = {x m (k) , x m (k − 1) , . . . , x m (k − d) , x p (k) , x p (k − 1) , . . . , x p (k − d)} . (13) Definition 1 (see [26]). Thedelayd(k) is said to be the random delay described by a Markovian chain if it is bound by 1 ≤ d(k) ≤ d, and {d(k) ∈ N, k = 0, 1, 2, . . .} is a Markovian chain with state spaceN and transition probability matrix π. Definition 2 (see [26]). When w(k) = 0 and V(k) = 0, the filtering error system (11) is said to be stochastically stable, if


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][6][7][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][10][11][12][13][14][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][17][18][19][20][21][22][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][25][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 where   () and   (), respectively, are the concentrations of mRNA and protein of the th gene; where Let ( * ,  * ) be an equilibrium point of GRN (3), where 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  = 1, 2, . . ., , 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 , , , ,  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 x (), x (), ẑ (), and ẑ () are the estimates of   (),   (),   (), and   (), respectively;   ,   ,   ,   ∈  × and  1 ,  2 ,  1 ,  2 ∈  × 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 1 ≤ () ≤ , and {() ∈ N,  = 0, 1, 2, . ..} is a Markovian chain with state space N and transition probability matrix .
Definition 3.For a given constant  > 0, the filtering error system ( 11) is said to be stochastically stable with  ∞ disturbance attenuation level  if it is stochastically stable with () = 0 and V() = 0, and under the zero initial conditions it satisfies the following inequality: for all nonzero (), V() ∈  2 [0, +∞), and initial mode (0).
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  > 0 and  > 0, the matrix inequality holds, if and only if there is a matrix  such that

Stability Analysis and 𝐻 ∞ Filter Design
The stability analysis for the filtering error system (11) with () = 0 and V() = 0 is presented by the following theorem.
Due to (18), formula (28) results in where  min denotes the minimal eigenvalue of −Ω.Since we obtain by taking the conditional expectation {⋅ | Θ 0 , (0)} and summing from  = 0 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.
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 () = [  ()   () V  ()]  , 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

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: 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  () = V () = { sin (0.3) ,  ≤ 20, 0,  > 20. (50) 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 (52) This verifies that the  ∞ disturbance attenuation level is below the given upper bound.

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.
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