Abstract
Markovian jump Hopfield NNs (MJHNNs) have received considerable attention due to their potential for application in various areas. This paper deals with the issue of state estimation concerning a category of MJHNNs with discrete and distributed delays. Both time-invariant and time-variant discrete delay cases are taken into account. The objective is to design full-order state estimators such that the filtering error systems exhibit exponential stability in the mean-square sense. Two sufficient conditions on the mean-square exponential stability of MJHNNs are established utilizing augmented Lyapunov–Krasovskii functionals, the Wirtinger–based integral inequality, the Bessel-Legendre inequality, and the convex combination inequality. Then, linear matrix inequalities-based design methods for the required estimators are developed through eliminating nonlinear coupling terms. The feasibility of these linear matrix inequalities can be readily verified via available Matlab software, thus enabling numerically tractable implementation of the proposed design methods. Finally, two numerical examples with simulations are provided to demonstrate the applicability and less conservatism of the proposed stability criteria and estimators. Lastly, two numerical examples are given to demonstrate the applicability and reduced conservatism of the proposed stability criteria and estimator design methods. Future research could explore further refinement of these analysis and design results, and exporing their extention to more complex neural network models.
1 Introduction
Neural networks (NNs) are composed of numerous interconnected neurons, providing them the ability to process large amounts of data simultaneously. Recently, NNs have been widely utilized in speech recognition [1], tracking control [2], associative memory [3], image restoration [4], and various other fields. As we all know, time delays are inevitable in the above practical applications, which may cause NNs to oscillate or become unstable [5]. This phenomenon has prompted extensive research on the stability analysis of NNs with time delays [6–9]. Notably, [10] designed an output feedback controller for NNs with time-invariant discrete delay to ensure the asymptotic stability of the closed-loop control system. The time delay considered in this study was constant. In cases where the time delay varies over time, [11] utilized a Lyapunov–Krasovskii functional (LKF) and subsequently established a stability criterion based on linear matrix inequalities (LMIs). However, these studies primarily focus on discrete delays, which may oversimplify these scenarios. In NNs, numerous interconnections between neurons form various parallel paths. Due to the varying sizes and complexities of these paths, signal transmission times are distributed within a certain period of time, resulting in distributed delays [12, 13].
On the other hand, in practical scenarios, environmental fluctuations can induce changes in the parameters of NNs. Researchers have recognized the significant advantages of Markovian jump processes in handling random changes in parameters [14–19]. Over the past few decades, many Markovian jump NN models, either in discrete-time form [20–23] or continuous-time form [24–27], have been proposed and studied. Among these models, Markovian jump Hopfield NNs (MJHNNs) have received considerable attention due to their potential for application. Notably, for an MJHNN, the neuron states are often not completely available. Thus, to achieve a given control goal, it is necessary to estimate the neuron states based on available output data, which has led to a growing focus on the topic of state estimation. [28] examined the finite-time state estimation for MJHNNs with discrete delays and presented a discontinuous estimator design method. [29] focused on the design of state estimators for continuous-time MJHNNs with both discrete and distributed delays and proposed a mean-square exponential stability (MSES) criterion and an LMI-based state estimation strategy.
It should be noted that the criterion derived in [29] was based on Jensen’s inequality, and the LKF used therein omitted some items involving time-delay-related integrals, thus leaving room for further improvement. Another discovery is that the discrete delay considered therein was assumed to be time-invariant, which restricts the application scope of the state estimation strategy since the magnitude of delays may vary over time in practice. Inspired by the observations above, we re-examine the state estimation in MJHNNs with discrete and distributed delays. Unlike the assumption of time invariance made in [29], the discrete delay under consideration here is allowed to be time-variant. The primary contributions of this study are as follows:
(1) Establishing MSES criteria for MJHNNs by integrating augmented LKFs, the Wirtinger-based integral inequality (WBII), the Bessel–Legendre inequality (BLI), and the convex combination inequality (CCI). Compared to the criteria proposed in [29], those proposed in this study are less conservative.
(2) Developing design methods for the required estimators to ensure the MSES of the filtering error systems (FESs) by eliminating nonlinear coupling terms. The estimator gain matrices can be easily determined by solving a set of LMIs.
Notation: In this paper, and denote the set of -dimensional real matrices and -dimensional symmetric positive definite matrices, respectively. and represent the unit matrix and the zero matrix, respectively. We denote as the transpose of the matrix , as the inverse of the matrix , as the column vector with elements , as the diagonal matrix with diagonal elements , as the symmetry term of a symmetric matrix, and as the Euclidean norm. The operator denotes the expectation, and . The notation indicates that the matrix is positive definite (negative definite).
2 Preliminaries
The MJHNN with time-invariant discrete and distributed delays is modeled as follows:where and represent the system state and measurement output, respectively. The positive scalars and denote the time-invariant discrete delay and the distributed delay, respectively. is an initial function defined on . , , , , and are matrix functions of the random jump process . To simplify the notation, we denote as and use a similar notation for the other matrices. In addition, takes values in the set . The transition probability matrix is given bywhere , , and denotes the rate at which the system transitions from mode at time to mode at time , and [30].
In the MJHNN Equation 1, and represent the activation and perturbation functions, respectively. These functions satisfy the following assumptions.
Assumption 1For given matrices and , satisfies
Assumption 2For given matrices and , satisfies
Remark 1Equations 2 and 3 satisfy sector-bounded conditions [31], which have broader applicability than the standard Lipschitz conditions [32, 33] and are widely utilized in the study of NNs [9, 34].The full-order state estimator for the MJHNN Equation 1 is designed as follows:where and represent the state estimate and gain matrices, respectively.Define . Then, we obtain the following FES:whereNext, we introduce the definition of the MSES.
Definition 1The FES Equation 5 has MSES if there exist positive scalars and such thatholds, where is the initial function of .The primary objective of this paper is to establish the MSES criterion for the MJHNN and design the state estimator to ensure that the FES achieves MSES. To facilitate subsequent derivations, we have prepared four lemmas.
Lemma 2[37] (WBII) For any matrix , scalars , and function , the inequalityholds, where
Lemma 4[40] Given matrices , , , and of appropriate dimensions, if there exist scalars such thatholds, then we obtain
3 Stability analysis and state estimation of MJHNNs
In this section, we establish an MSES criterion and design a state estimator for MJHNNs with time-invariant discrete and distributed delays.
It can be deduced from Equations 2, 3, 6 thatandwhereandNext, the MSES criterion for the MJHNN Equation 1 is established.
For given positive scalars and and matrices and , if there exist positive scalars , and and matrices , and , and , and such thathold, for whereand the remaining symbols are defined in Equation 10. Then, the MJHNN Equation 1 achieves MSES.
Proof. DefinewhereThe LKF is constructed as follows:where is defined in Equation 9 andThe infinitesimal generator is defined as is abbreviated as . Then, and have the following first components:The second components are given byand these enable us to deduce thatwhere and are defined in Equation 13. It follows from Equation 14 thatUnder Equation 11, the integral terms in and in can be eliminated to obtainUtilizing Lemma 1, the following inequality holds:Combining Equations 16 and 17 yieldsFor the integral term in , by applying Lemma 2, we can establishwhereThen, it is easy to obtainAdditionally, for the system Equation 1, utilizing the free-weight matrix technique yieldsFrom Equations 7 and 15–20, for any scalars and , we can obtainApplying the Schur complement to Equation 12 yieldswhere . From the definitions of , , and , there exist positive scalars , , , , and such that the following inequalities hold:Combining Equation 21 with Equation 22 results inBy utilizing Dynkin’s formula, for , it follows thatwhereBy changing the order of integration, we can writeSubstituting Equations 24–26 into Equation 23 and combining it with Equation 21, we obtainwhereOne can write the following inequality:Then, we can prove that for any ,holds, whereTherefore, following a similar approach as in [29], system Equation 1 achieves MSES according to Definition 1.
Remark 2[39] considered the BLI and proved that needs to include and to fully benefit from the BLI. Therefore, we consider the state augmentation of and demonstrate its conservative reduction through Example 1.
Remark 3As shown in Equation 17, the term is processed using the BLI, instead of scaling it up by as in [29]. This approach helps further reduce conservatism.Next, the state estimator design method is as follows.
For given positive scalars , , and , and matrices , , , and , there exist positive scalars , , and and matrices , and , and , and , , , and such thathold, for whereThe remaining symbols coincide with those defined in Theorem 1. For the MJHNN Equation 1, the estimator Equation 4 with gain matricesguarantees the MSES of the FES Equation 5.
Proof. DefinewhereThe LKF is constructed as follows:where is defined in Equation 9 andAlong similar lines to Theorem 1, by combining Equations 5 and 8, we can obtain . Then, under Equation 30, one haswhere are defined in Equation 29 andIn light of Lemma 4, Equations 27 and 28 can ensure the MSES of the FES Equation 5.
4 Extension to the time-variant discrete-delay situation
Consider the MJHNN with time-variant discrete and distributed delays as follows:where represents a time-variant discrete delay and satisfiesThen, the estimator and the FES for the MJHNN Equation 31 are considered as follows:We can provide the condition of the MSES for the MJHNN Equation 31 as follows.
For given positive scalars , , and and matrices and , there exist positive scalars and and matrices , , , and , , , , and , and , , and and such thathold, for and , whereand the remaining symbols coincide with those defined in theorem 1. Then, the MJHNN Equation 31 achieves MSES.
Proof. DefinewhereThe LKF is constructed as follows:where is defined in Equation 9 andBy the definition of in Equation 14, along the MJHNN Equation 31, we havewhere includes the following components:The initial components are , while the subsequent components demand two distinct expressions for , depending on and . It can be attained thatSubstituting Equation 38 into Equation 37, we obtainTherefore, we obtain andThe components of are as follows:and it is easy to obtainThen, we deduce from Equation 35 thatHence, leveraging the matrices and defined in Equation 35, we can derive the following equation for matrices and :One can derive from Equations 36–41 thatIt follows from Equation 33 thatBy applying Lemma 1 to the integral term in , one can obtainandUtilizing Lemma 2, we can deduce from Equation 45 thatleading toAccording to Lemma 2, can be computed as follows:Similar to Theorem 1, Equation 20 can be written asFrom Equation 7 and Equations 42–49, for any scalars and , we can obtainwhere are defined in Equation 35. Using a similar approach to Theorem 1, we can derive that Equations 34 and 35 can ensure the MSES of the MJHNN Equation 31.
Then, based on Theorem 2, we can establish the following estimator design method.
For given positive scalars , , , and and matrices , , , and , there exist positive scalars , , and and matrices , , , and , , , , and , and , , and , , , and such thathold, for and , whereand the remaining symbols coincide with those defined in theorem 3. For the MJHNN Equation 31, the estimator Equation 32 with gain matricesguarantees the MSES of the FES Equation 32.
5 Numerical example
Example 1Consider MJHNNs Equation 5 and Equation 32 withBy solving LMIs in theorems 1 and 3 in this paper and Theorem 2 in [29], we can obtain the maximum admissible upper bound for different values of and , as shown in Table 1, which clearly shows the effectiveness of theorems 1 and 3 in this paper. Compared to Theorem 2 in [29], theorem 1 in this study is less conservative.The maximum admissible upper bound for various and is presented in Table 2. It can be observed that for the same and , in theorem 1 is larger than in Theorem 2 of [29].In addition, for different combinations of , , and , solving Theorem 3 yields the maximum admissible upper bound , as shown in Table 3. As increases, decreases under the corresponding and conditions.
TABLE 1
| Method | ||||
|---|---|---|---|---|
| Theorem 2 in [29] | 0.5 | 2.0222 | 1.7123 | 1.2863 |
| Theorem 1 | 2.0777 | 1.7473 | 1.3058 | |
| Theorem 3 | 1.9167 | 1.5911 | 1.1169 | |
| ( = 0.45, = 0.55) | ||||
| Theorem 2 in [29] | 1.0 | 1.9535 | 1.6741 | 1.2739 |
| Theorem 1 | 2.0453 | 1.7347 | 1.2983 | |
| Theorem 3 | 1.8992 | 1.5776 | 1.1081 | |
| ( = 0.95, = 1.05) | ||||
| Theorem 2 in [29] | 1.5 | 1.9161 | 1.6462 | 1.2675 |
| Theorem 1 | 2.0094 | 1.7137 | 1.2882 | |
| Theorem 3 | 1.8743 | 1.5585 | 1.0979 | |
| ( = 1.45, = 1.55) | ||||
| Theorem 2 in [29] | 2.0 | 1.9015 | 1.6358 | 1.2659 |
| Theorem 1 | 1.9795 | 1.6908 | 1.2800 | |
| Theorem 3 | 1.8423 | 1.5402 | 1.0952 | |
| ( = 1.95, = 2.05) | ||||
Maximum allowable delay upper bound for various and .
TABLE 2
| Method | ||||||
|---|---|---|---|---|---|---|
| Theorem 2 in [29] | 1.97 | 0.8471 | 1.67 | 1.0571 | 1.27 | 1.2389 |
| Theorem 1 | 2.1964 | 2.5566 | 3.7560 | |||
| Theorem 2 in [29] | 1.98 | 0.7668 | 1.68 | 0.9221 | 1.28 | 0.7310 |
| Theorem 1 | 1.9912 | 2.2648 | 2.0045 | |||
| Theorem 2 in [29] | 1.99 | 0.6949 | 1.69 | 0.7948 | 1.29 | 0.3709 |
| Theorem 1 | 1.8080 | 2.0199 | 1.4085 | |||
| Theorem 2 in [29] | 2.00 | 0.6298 | 1.70 | 0.6673 | 1.30 | 0.1085 |
| Theorem 1 | 1.6431 | 1.7966 | 0.9128 |
Maximum allowable delay upper bound for various and .
TABLE 3
| 0.5 | 0.5 | 1.7005 | 1.5293 | 1.2861 |
| 1.0 | 2.1101 | 1.9703 | 1.7635 | |
| 1.5 | 2.5509 | 2.4244 | 2.2457 | |
| 1.0 | 0.5 | 1.5133 | 1.2680 | 0.8427 |
| 1.0 | 1.9077 | 1.7082 | 1.3149 | |
| 1.5 | 2.3342 | 2.1532 | 1.7882 | |
Maximum allowable delay upper bound for various , , and in Theorem 3.
Example 2Consider MJHNNs Equation 5 and Equation 32 withNext, we design state estimators to ensure the MSES of FESs with discrete and distributed delays, considering cases of time-invariant and time-variant discrete delays.
Case 1Time-invariant discrete delay.For the FES Equation 5, with , , and , solving the LMIs in Theorem 2 yields the estimator Equation 4 with gain matrices
Case 2Time-variant discrete delay.For the FES Equation 32, with , , , and , solving the LMIs in Theorem 4 yields the estimator Equation 32 with gain matricesIn the simulations, we set , , andThe simulations of the state estimators are shown in Figures 1–4. For the MJHNN with time-invariant discrete and distributed delays, Figure 1 shows the state trajectories and their corresponding estimations, while Figure 2 shows the trajectories of the FES. Figures 3, 4 show the corresponding trajectories under time-variant discrete delay scenarios. The proposed state estimator design methods are effective for MJHNNs with discrete and distributed delays in both cases of time-invariant and time-variant discrete delays.
FIGURE 1
FIGURE 2
FIGURE 3
FIGURE 4
6 Conclusion
This paper has investigated the state estimation for MJHNNs with discrete and distributed delays. Specifically, both time-invariant and time-variant discrete delay cases are considered. Two conditions for the MSES of MJHNNs have been proposed utilizing augmented LKFs, the WBII, the BLI, and the CCI. The LMIs-based design methods for the required estimators have been developed by eliminating nonlinear coupling terms. Lastly, two numerical examples are given to demonstrate the applicability and reduced conservatism of the proposed stability criteria and estimator design methods. Future research could explore further refinement of these analysis and design results, and exporing their extention to more complex neural network models.
Statements
Data availability statement
The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.
Author contributions
LG: Investigation, Methodology, Writing–review and editing. WH: Writing–original draft.
Funding
The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
Hopfield neural networks, Markovian jump, mean-square exponential stability, state estimation, time delays
Citation
Guo L and Huang W (2024) State estimation for Markovian jump Hopfield neural networks with mixed time delays. Front. Phys. 12:1447788. doi: 10.3389/fphy.2024.1447788
Received
12 June 2024
Accepted
26 August 2024
Published
12 September 2024
Volume
12 - 2024
Edited by
Shiping Wen, University of Technology Sydney, Australia
Reviewed by
Guici Chen, Wuhan University of Science and Technology, China
Guodong Zhang, South-Central University for Nationalities, China
Updates
Copyright
© 2024 Guo and Huang.
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*Correspondence: Lili Guo, guolili@ahut.edu.cn
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.