Finite-Time H ‘ State Estimation for Markovian Jump Neural Networks with Time-Varying Delays via an Extended Wirtinger’s Integral Inequality

This study investigates the ﬁnite-time boundedness for Markovian jump neural networks (MJNNs) with time-varying delays. An MJNN consists of a limited number of jumping modes wherein it can jump starting with one mode then onto the next by following a Markovian process with known transition probabilities. By constructing new Lyapunov–Krasovskii functional (LKF) candidates, extended Wirtinger’s, and Wirtinger’s double inequality with multiple integral terms and using activation function conditions, several suﬃcient conditions for Markovian jumping neural networks are derived. Furthermore, delay-dependent adequate conditions on guaranteeing the closed-loop system which are stochastically ﬁnite-time bounded (SFTB) with the prescribed H ∞ performance level are proposed. Linear matrix inequalities are utilized to obtain analysis results. The purpose is to obtain less conservative conditions on ﬁnite-time H ∞ performance for Markovian jump neural networks with time-varying delay. Eventually, simulation examples are provided to illustrate the validity of the addressed method.


Introduction
Due to the great significance of neural networks (NNs) for both practical and theoretical purposes, their dynamics have been explored widely in recent years, such as pattern recognition, signal processing, solving optimization problems, static image processing, associative memories, target tracking, and automatic control. erefore, many research subjects have been studied in a broad spectrum of stability analysis, passivity analysis, control, filtering design, and state estimation and synchronization, concerning to NNs [1][2][3][4][5][6]. In [4], passive filter design for fractional-order quaternionvalued neural networks with neutral delays and external disturbance has been studied. e authors in [6] investigated stability criteria of quaternion-valued neutral-type delayed neural networks. In many implementations of NNs, time delays are inevitable [7] and can lead NNs to instability and oscillation. Hence, the stability analysis with time delays in the NN models under consideration has attracted considerable attention [8][9][10][11][12][13][14].
Due to interconnection failures, sudden environment changes, components, and so on, plenty of structural parameters of neural networks may mutate. In general, there are finite modes in the neural network,s switching or jumping from one mode to another mode by a random form. A Markov chain can be used to describe jumping between different modes of neural networks, and the kinds of systems are called Markovian jump neural networks [15][16][17][18]. Many practical control systems can be modeled as Markovian jump neural networks, such as air intake systems and economic systems [19]. In an MJNN, hopping among operation modes is specified by a Markov process, so it is important to understand the impacts of its stochastic attributes on the stability analysis of delayed MJNNs. Some previous works [15][16][17][18] have discussed certain standard results in relation to MJNN stability analysis. In [20], the authors conducted an asymptotic stability analysis for stochastic and static NNs with time-varying delays that are mode-dependent. e use of linear matrix inequalities (LMIs) has led to important and interesting results concerning various types of NN with MJ parameters [21,22]. e mode-dependent MJNNs with time-varying delays and incomplete transition rates can be found in [23], wherein some LMI-based conditions are proposed to obtain the required results.
In some cases, we are interested in knowing how the modeled system behaves within fixed-and finite-time intervals. In other words, given an initial bounded state, we require the system to remain in a state that is not superior to a particular threshold during a specified time interval. Since this type of stability ensures a faster convergence of the system, it has been widely used in various NNs, such as the MJNNs, and synchronizing neural networks [24]. An important example can be found in controlling the trajectory of a spacecraft between its initial and final locations within a specified time interval. However, because of the lack of other finite-time-bounded operational conditions, it is natural that research interest has shifted to Lyapunov stability in this paper. In addition, based on LMI results, the idea of finitetime boundedness (FTB) has been revisited here. We also studied that finite-time stability involves dynamical systems whose part of the trajectory converges to an equilibrium state in a finite time. Note that the finite-time stability with control frameworks has gained significant attention in recent years [25][26][27][28]. In [29], the authors discussed the finitetime L 2 -gain performance of MJNNs. e design of a finitetime passive controller for uncertain MJ systems is optimized in [28], wherein a robust and fuzzy finite-time passive control is defined along with the finite-time stochastic stability of a nonlinear MJ system. However, finite-time H ∞ state estimation of MJ systems has not been studied much for NNs. is is a primary inspiration for this study. e main contributions of this study are listed as follows: (1) e comprehensive Markovian jump neural networks with state and input constraints are studied.
(2) We have introduced a novel Lyapunov-Krasovskii functional (LKF), including time-varying delays. (3) Wirtinger's double integral inequality, introduced by Park et al. [30], and Wirtinger's integral inequality, extended by Zhang et al. [31], are introduced into the time-derivative of LKF. is time-derivative forms the LMIs which are FTB. ese LMIs deliver more effective outcomes in comparison to previous works.
e numerical examples are also given. (4) To show the real-life application, the four-tank water pumping system and network circuit are considered in this paper in terms of the NN model to show feasibility on a benchmark problem.
Notations are as follows: R n : n-dimensional Euclidean space P > 0: the matrix P is a symmetric matrix min (P): minimum eigenvalue of P max(P): maximum eigenvalue of P I: identity matrix diag · { }: diagonal matrix * : symmetric matrices

Preliminaries and Problem Formulation
Given a probability space (Ω, F, P), where Ω, F, and P represent sample space, σ-algebra of events, and probability measure defined on F, respectively. Let parameter r t , t ≥ 0 be a right continuous Markov chain taking values on (Ω, F, P) a finite set S � 1, 2, . . . , N { } with generator Π � (π ij ) N×N given by where r t ∈ S, Δt > 0, lim Δt⟶0 (o(Δt)/Δ) � 0, and π ij denotes the transition probability from modes i to j satisfying π ij ≥ 0, for i ≠ j, with π ii � − s j�1,j ≠ i π ij , i, j ∈ S. Consider the MJNNs with time-varying delays are as follows: where x(·) � [x 1 (·), x 2 (·), . . . , x n (·)] T ∈ R n is the state vector, y(t) ∈ R p is the output measurement, z(t) ∈ R m denotes the estimated signal, w(t) ∈ R q represents exogenous disturbance belonging to , h 2 (x 2 (t)), . . . , h n (x n (t))] T ∈ R n is a neuron activation function, and J ∈ R n denotes an external input constant vector. A(r t ) > 0 is a diagonal matrix, and B(r t ), B d (r t ), E 1 (r t ), E 2 (r t ), C(r t ), D(r t ), and G(r t ) are connection weight matrices. A time-varying delay is denoted as δ(t), where 0 ≤ δ(t) ≤ δ and _ δ(t) ≤ μ, such that δ and μ are Assumption 1. Each neuron activation function h k (t) (k � 1, 2, . . . , n) is continuous and bounded and satisfies the following condition: where ϱ − k and ϱ + k are constant. en, we define the followings Assumption 2. e external disturbance w(t) fluctuates and satisfies the following inequality: For a MJNN defined as (2), a state estimator is constructed as follows: where x(t) ∈ R n denotes the estimated state and z(t) ∈ R q is the estimated measurement of z(t). en, estimator gain matrix K i is to be constructed.
Definition 1 (stochastically finite-time stable (SFTS) [27]). Given time constant T > 0, an MJNN defined as (7) with w(t) � 0 is SFTS with respect to (c 1 , c 2 , T, R) if there exists a positive matrix R > 0 and scalars c 1 > 0 and c 2 > 0, such that the following inequality holds: Definition 2 (stochastically finite-time boundedness (SFTB) [27]). Given a time constant T > 0, an MJNN defined as (7) is said to be SFTB with respect to (c 1 , c 2 , T, R, d), where there exist R > 0 and scalars c 1 > 0 and c 2 > 0, such that the following inequality holds: Definition 3 (see [32,33]). For T > 0, an MJNN defined as (7) is said to be SFTB with respect to (c 1 , c 2 , T, R, d) and with a prescribed level of noise attenuation c > 0 under a zero initial condition if it holds:

Mathematical Problems in Engineering
Definition 4 (see [34]). A functional V(x(t), r(t), t > 0) � V(x(t), r) is said to be a stochastic positive functional. Its weak infinitesimal operator can be defined as Lemma 1 (see [30]). Let a constant matrix M > 0; the following condition can be defined for all differentiable function ϕ in [a, b] ⟶ R n for scalars a and b with a < b: where Lemma 2 (see [35]). For any constant matrix M > 0, the following inequality holds for all continuously differentiable function φ on [a, b] ⟶ R n×n : where Lemma 3 (see [31]). For a given symmetric matrix W 2 � W T 2 > 0, the following inequality holds for all contin- where

Finite-Time State Estimation.
is section derives the SFTB of the error system in (7).

Theorem 1.
For scalars c 1 , c 2 , T, d, δ, μ, and α, an MJNN defined as (7) where P i , Q s , and W s are symmetric positive definite (PD), and U t > 0 and U u > 0 are diagonal, such that the following inequality holds: where 4 Mathematical Problems in Engineering In addition, the desired control gain matrices can be calculated by Proof. Construct LKF for an MJNN defined as (7): where By differentiating the above LKF to obtain its time derivatives along with the trajectory of the MJNN defined as (7), we obtain Mathematical Problems in Engineering Utilizing Lemma 2, we obtain By applying Lemma 3, the following inequality can be written as By applying Lemma 1, we obtain − δ where is can be written algebraically as f(e(t)) ≤ 0, f(e(t − δ(t))) en, the following inequality holds for any positive matrices U t � diag u 1 , u 2 , . . . , u n and U u � diag u 1 , u 2 , . . . , u n }: For convenience, consider a matrix N with appropriate dimension, and the following zero equality holds: erefore, from (22)-(34), given that α > 0, we obtain where We can write this as LV(e(t)) ≤ αV(x(t)) + αw T (t)Xw(t).
Multiplying both sides of (37) by e − αt and then integrating from 0 to t, where t ∈ [0, T], we obtain Conversely, Mathematical Problems in Engineering 7 where Note that en, from (18), we obtain Based on Definition 2, an MJNN defined as (7) is SFTB. □ Corollary 1. Given scalars c 1 , c 2 , T, δ, μ, and α, an MJNN defined as (7) where P i , Q s , and W s are symmetric PD, and U t > 0 and U u > 0 are diagonal such that the following inequality holds: where ψ ij is defined in eorem 1.
Proof. It can be proved in a similar way as eorem 1. e proof is omitted for brevity. where Proof. In a similar way to the proof in eorem 1, we obtain It can be deduced from (47) and (50) that

Mathematical Problems in Engineering
By integrating (51) from 0 to T, we obtain Subsequently, the following inequality is obtained: Hence, we conclude that the MJNN defined as (7) is SFTB.
□ Remark 1. Consider the following error system from the MJNN defined as (7), with w(t) � 0 and without MJ parameters:

_ e(t) � − (A + KC)e(t) − KDe(x(t − δ(t))) + Bf(e(t)) + B d f(e(t − δ(t))).
(54) Corollary 2. Given scalars δ and μ, the error system (54) with w(t) � 0 is said to be stable if there exist feasible matrices where P i , Q s , and W s are symmetric PD, and U t > 0 and U u > 0 are diagonal, such that the following inequality holds: where Proof. Following similar ideas as in the proof of eorem 1. e proof is omitted for brevity. □ Remark 2. Consider a NN from the MJNN defined as (7) with C � 0, D � 0, w(t) � 0, and no MJ parameters:

Corollary 3. Given scalars δ and μ, the error system (57) with w(t) � 0 is said to be stable if there exist feasible matrices
where P i , Q s , and W s are symmetric PD, and U t > 0 and U u > 0 are diagonal, such that the following inequality holds:

Mathematical Problems in Engineering
where Proof. It can be proved in a similar way to eorem 1. e proof is omitted for brevity. □ Remark 3. e stability analysis of time-delay systems can be classified into two categories, i.e., delay-dependent stability criteria and delay-independent ones. Also, it is well known that delay-dependent stability criteria, which use the information on the size of time delays, are less conservative than delay-independent ones. us, more attention has been paid to the derivation of delay-dependent stability criteria for time-delay systems.

Remark 4.
It is important to note that some pioneering works have been done on finite-time H ∞ state estimation for Markovian jump neural networks based on interval timevarying delay with simple LKF techniques. In [29], the authors studied finite-time boundedness for Markovian jump neural networks with L 2 gain analysis. Authors in [26] formulated finite-time stabilization of uncertain neural networks. Exponential state estimation problem has been designed for Markovian jumping neural networks in [18]. e model consider in this present study is more practical than that proposed by [18,26,29], whereas in this paper, we consider finite-time H ∞ state estimation problem with the combination of Markovian jump neural networks' interval time-varying delay model, which is another advantage. However, the authors in [18,26,29] used some simple techniques in LKFs to solve the stability problems to those articles. A new LKF with double and triple integral terms and utilizing extended Writnger's integral inequality (EWII) techniques has been proposed for the stochastically finitetime bounded analysis of Markovian jump system in this paper. Consider that some less conservative results can occur in our method and can be provided in the numerical example section with real-life examples. Hence, the results presented in this paper are essentially new.
Remark 5. Typically, finite time stability with H ∞ control, state estimation approach, and interval time-varying delay is not simply applied to Markovian jump neural networks. Some research publications have handled such issues [17,18,26,29]. As it is, the author utilized some elementary LKFs to deal with the stability problems in those articles. Novel LKF with EWII has been proposed; in addition, the developed stochastic stability criteria tested for feasibility of the benchmark problem to explore the real-world application in this paper. However, the desired control was completely studied for the considered neural network model with the real-world application problem (e.g., four-tank pumping system and network circuit), which is the principle commitment and inspiration of our work.

Numerical Examples
is section shows our results through some numerical examples on MJNNs with 2 operation modes to demonstrate the effectiveness of the proposed approach. (62) us, the system is SFTB with the external disturbance c � 0.90.

Example 1. Consider an MJNNs with MJ parameters (i � 2):
To demonstrate the capability of the proposed approach, we show the effectiveness of the theoretical results, as shown in Figures 1-4 . Figure 1 demonstrates the MJ mode r t . Figures 2 and 3 show the behaviors of the error system and state estimation of the error system, respectively. Figure 4 illustrates that the state x(t) of the system converges to zero. Furthermore, the superiority of our theoretical results is demonstrated through the simulation result of x T (t)Rx(t) in Figures 5 and 6. erefore, the proposed MJNN (7) is STFB.      System state x (t) Figure 4: State trajectories of the system. is example shows a comparison of the conservativeness in the stability condition concerning the results in [36][37][38]. e maximum allowable delay bound (MADB) of δ for various μ can be calculated using the MATLAB LMI toolbox. e MADBs of δ for some values of μ in Example 2 are summarized in Table 1. We found that the outcomes of our proposed method produced better results than the previous research [36][37][38].
Example 3. Consider the NN (57) with the following parameters: For some values of μ, the MADBs of δ are obtained and summarized in Table 2. We compare these results with those of previous studies [36,39,40]. As shown in Table 2, the MADB are larger than those obtained from [36,39,40]. It shows the superiority that the proposed stability criterion is less conservative than the previous works.
Remark 6. We calculated upper bounds with different delta, and they are listed in Tables 1 and 2. We provide comparisons with the results obtained in previous studies to show the improvements obtained by our proposed method.

Example 4.
e NNs have similar characteristics to the neurons in a biological organism, leading to the nervous system. e NNs can represent not only the nervous systems with neurons but also the engineering systems such as the four-tank water pumping system, as shown in Figure 7. e four-tank water pumping system is equipped with 2 water  Mathematical Problems in Engineering 13 pumps and 4 interconnected tanks with two valves. Voltage ] 1 and ] 2 are two input processes of two supplying pumps. e four-tank water pumping system can be modeled as a neural network model. Previous studies in [41][42][43] suggested the state-space equations of this four-tank system which is an application of the neural networks. State feedback controller modeled as follows: where Another control problem of our interests is obtained by adding transport delays δ(t) through delaying the inlet of incoming water into the tanks. Hence, the proposed approach has been used to study this problem here. Timevarying transport delays between valves and tanks have also been considered in the previous works, but they have not been considered the following aspects. For simplicity, it was assumed that δ 1 � 0, δ 2 � 0, and δ 3 � δ(t) (since δ(t) ≤ δ). In this example, the control input u(t) indicates the amount of water pumped. erefore, it is naturally a nonlinear function and can be written as follows: e four-tank system (65) can be rewritten to the form of system (57) with K � 1 as follows: 1} with δ � 6.5 and μ � 0.5. Using MATLAB LMI toolbox and solving the inequalities in Corollary 2, we are able to obtain feasible solution, which lead to a conclusion that FTPS (68) is stable.

Y1 Y2
Pump U1 h1 h2 Pump U2 h4 h3 Tank 1   Tank 3  Tank 4 Tank 2 Figure 7: Schematic representation of the four-tank water pumping system. e product R i C i � δ i , i � 1, 2, . . . , n, is called as the time constant of the i th neuron. An identical time constant for each neuron would require, that is, C i � C and R i � R, for all i. In this case, every individual value for δ i would have to be chosen in a way that compensates for C i and R i . It is important to note that the time constant δ i describes the convergence of the neural state e i of the i th neuron. Because of the high-level gain of the transfer function, the output V i might be saturated very fast. us, even if the state e i is still far from reaching its equilibrium point, the output V i might already be saturated, and by observing only V i , it might appear as if the circuit had converged in merely a fraction of the time constant δ i .

Conclusion
Herein, we studied the SFTB of MJNNs with time-varying delays. Using an LKF with Wirtinger's integral inequalities, a sufficient condition was derived such that the MJNNs were SFTB and satisfied a prescribed level of H ∞ disturbance attenuation in a finite-time interval. We illustrated the effectiveness of our main results with five numerical examples. We also compared to show that our results are less conservative than some existing ones. Future works focus on the discrete versions of these inequalities and their applications. [45][46][47].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper. x 1 (t) x 2 (t) Figure 9: Circuit diagram for delayed NN.  16 Mathematical Problems in Engineering