Discrete-time Mittag–Leffler state estimation for fractional-order quaternion memristive neural networks

Abstract

This article investigates the state estimation of fractional-order memristive systems with discrete-time terms. By considering discrete fractional calculus, we propose a novel and efficient criterion for ensuring the global Mittag–Leffler stability of the estimation error system. Additionally, by utilizing a functional that incorporates a discrete fractional sum element, we derive the stability condition for the concerned system. It is noteworthy that the proposed approach integrates a vector optimization method, which enhances the understanding of how to construct a meaningful convex closure formed by quaternions. Finally, numerical simulations are conducted to validate the theoretical results.

Introduction

In¹, the physicist L.O. Chua pointed out that a fourth fundamental element should exist, and he termed this hypothetical entity the “memristor”². Its potential discovery could mark a significant leap forward in the development of increasingly powerful circuitry. Beyond its intrinsic scientific importance, the real excitement lies in the practical implementation of memristive systems, which represent a specific instance of a broader category of nonlinear dynamic devices.

Recent advancements in memristive neural networks have shifted focus toward understanding their dynamic behaviors^3–11. For example, the impact of Lvy noise on these systems was explored in³, while the stability of memristive systems was analyzed by using a sliding mode controller in⁴. Additionally, studies in⁵ and⁶ delved into fractional-order memristive models. Notably, quaternion neural networks have garnered significant attention. The findings in⁹ demonstrated that by applying fundamental quaternion algorithms, Lagrange exponential stability conditions for quaternion memristive models can be established.

Simultaneously, quaternion-valued systems have gained increasing attention among scholars, leading to significant progress in the modeling of high-dimensional (three- and four-dimensional) data^12–14. Consequently, a substantial portion of research has been dedicated to this area^15–23. Among these studies, the existence of equilibrium points was verified in¹⁶, while local stability conditions were also examined, highlighting the system’s high storage capacity. In¹⁷, a convex optimization method was introduced for analyzing neural networks with quaternion-valued parameters. Furthermore,²¹ concentrated on quaternion-valued models in both continuous-time and discrete-time scenarios. These developments reveal the quaternion-valued system is a highly promising research direction with considerable potential.

For a system to effectively perform specific tasks, particularly for quaternion-valued memristive systems (which are relatively large-scale models), it is often crucial to obtain complete information about its states. However, this has proven to be an extremely challenging task for several reasons, including the sheer size of the system, the physical limitations of the devices involved, and the limited availability of measurement resources. Accordingly, it is quite necessary to estimate the system states through accessible estimations, and thus the state estimation issues have become the focus of research^24–29. For example, via an event-triggered estimator, the memristive system with discrete terms was considered in²⁴, non-fragile estimation was discussed in^25,26, the results in²⁷ addressed the exponential estimation problem for a class of quaternion memristive model.

A major issue in the dynamic of the quaternion memristive model is the existence of convex closures. When simply following the convex closure described in the earlier results, it is found that the convex closure herein is made up by quaternion parameters, which leaves us facing the following dilemma: how to ensure the convex closure is meaningful. Over the course of multiple training, a vector order method is proposed, which resolves this issue as it allows us to compare the size of quaternions.

Based on the aforementioned characterization of the quaternion memristive system, we will proceed to construct a series of conclusions that resolves the Mittag–Leffler state estimation issues and meets all requirements. We provide a rigorous analysis of our method and then give some empirical results. This entails the following challenges: (i) The Mittag–Leffler stability is proposed for the quaternion-valued discrete-time memristive system; (ii) In^19,20, several researchers have utilized the decomposition method to study the dynamic behaviors of quaternion-valued neural networks. However, since quaternion-valued neural networks inherently represent a four-dimensional system, the decomposition approach may compromise the integrity of entire system in practical applications. In contrast to these studies, our work directly investigates quaternion-valued memristive neural networks without decomposing it into several subsystems, preserving the system’s full dimensionality and rigor; (iii) Fractional-order operators are introduced to a system with discrete-time terms, which consider the overall function information.

The remainder of this article is as follows. In “Preliminaries” section, some preliminary knowledge is introduced, including necessary hypotheses, definitions and lemmas. The main theorems are stated and proven in “Main results” section. In “Numerical simulation” section, we demonstrate the effectiveness of main results by means of numerical experiments. Finally, conclusions are drawn in “Conclusion” section.

Preliminaries

Notations: Set , where i, j, k represent imaginary units whose operations satisfy Hamilton rules, i.e. The conjugate of m is , the modulus of is . Besides, for , set be the modulus of , and , be the norm of , where is the conjugate transpose of . For a function , . Let , .

Set

, is called t to the m rising³⁰. Besides, set

where , and , .

Definition 1

³¹ The nabla discrete fractional sum is

where , , .

Definition 2

³² The Riemann-Liouville fractional difference is

where , , . For convenience, is short for in the following.

Definition 3

³³ For , , and with the real part of is positive, the nabla discrete Mittag–Leffler function with two parameters is defined as

Lemma 1

Let be a function, then

holds, where , .

Proof

. Let , then according to Lemma 1 in³⁴, when , it holds that

1

Set , where . Then, it follows from (1) that

On the other hand,

thus,

holds, if .

Now, set , i.e., . Hence, can be also deemed as a complex-valued function, the proof is accomplished.

Lemma 2

Let be a decrescent scalar function defined on . If there exists a constant satisfying

then it holds that

Proof

. Considering that is equivalent to . Then, by virtue of Lemma 6 in³⁵, the above conclusion is correct.

Consider the following model

2

where , , denotes the state, , are the connection weights, represents the time delay, is the external input, and signifies the activation function.

For , is supposed to satisfy the following condition

where .

The connection weights are defined as

where the switching jump .

Remark 1

As is known, the memristive connections are switching between two distinct values, which indicates that the value of the connection at a fixed time must be in , where , . Therefore, it is necessary to determine which one is bigger between and . Based on the comparing principle for quaternions (vector ordering approach in³⁶), the values of , , and can be readily obtained.

Then, system (2) could be expressed as below

3

Obviously, there exists and satisfying

4

The measurement of (2) is

5

where signifies the measurement output, and are constants.

Now, the estimator is given by

6

with

7

and , is the estimator control gain. Hence,

8

Similarly, there exists and such that

9

Define , the error system would be described as

10

In recent years, the quaternion-valued system has demonstrated exceptional performance in single-image dehazing³⁷ and associative memory³⁸. This insight will serve as our foundation to construct some efficient results for the estimation of memristive systems. In order to arrive at this objective, we shall give the following definition.

Definition 4

System (10) is referred as globally Mittag–Leffler stable, if there have , satisfying

where , and .

Main results

Theorem 1

Let be given, the error system (10) is globally Mittag–Leffler stable, if there exist scalars , , such that

hold.

Proof

. Consider a candidate function as below

11

Before moving on, the following tight estimation is necessary

12

Thus, the derivative of along with (10) can be enlarged as

13

It then follows directly from Lemma 2.5 in³⁴ that

14

Analogously, it can be decuced that

15

and

16

where , with .

Applying (14)–(16) to (13), it will modify to

17

Based on the Razumikhin condition

one has

18

Therefore, it follows from Lemma 2 that

It is apparent to observe from (11) that

19

Hence,

20

then based on Definition 3 (where ), one can conclude that (10) is globally Mittag–Leffler stable.

Remark 2

It is worth pointing out that we can attain Mittag–Leffler stability condition via constructing an appropriate function in Theorem 1. We will now proceed to investigate the stability conclusion by establishing a brand new function, and the correspongding results are presented as follows.

Theorem 2

Let be specified, the error system (10) can achieve the stable performance, if there exist scalars , , satisfying

Proof

Construct the following candidate function

21

Following a similar idea as mentioned above, the derivative of along with (10) can be calculated as

which further implies that (10) is globally asymptotically stable.

Remark 3

As a side note, by choosing different estimation controllers and auxiliary functions, we can obtain various stability conclusions. For instance, if (7) is taken as

then the Mittag–Leffler quasi-state estimation result can be derived.

Numerical simulation

Example 1

Considering a two-dimensional system (2), in which the parameters are described as

Besides, , , , the activation function is , which implies that . Additionally, we suppose , , , , , , . Then, one can make a simple calculation according to the following two criteria

thus , and it is obvious that , which means

On the other hand, let , one has

Simulation resuls are illustrated in Figs. 1234, which confirm the stability trend of the original system according to Theorem 1. Preliminary experiments are in line with our theoretical findings.

Figure 1.

Transient behaviors of system state , state estimation and error state .

Figure 2.

Transient behaviors of system state , state estimation and error state .

Figure 3.

Transient behaviors of system state , state estimation and error state .

Figure 4.

Transient behaviors of system state , state estimation and error state .

Conclusion

State estimation of memristive systems with fractional derivative and discrete-time term presents a significant challenge. In this work, we demonstrate that by employing discrete fractional calculus and a functional incorporating a discrete fractional sum element, the state estimation problem for such systems can be effectively addressed. The proposed results are not only applicable to the study of memristive systems but also offer a more efficient approach for exploring systems with quaternion elements. Importantly, the proposed method integrates the advantages of vector optimization techniques, providing deeper insights into achieving a meaningful convex closure. A numerical example is included to validate the effectiveness of state estimator in determining the system’s actual states.

Author contributions

Qun Huang conceived the manuscript. All authors contributed to manuscript preparation, and revised the manuscript.

Funding

This study was supported by the Natural Science Foundation of Nantong Municipality under Grant No. JC2024008 and the Science and Technology Research Program of Chongqing Municipal Education Commission under Grant No. KJZD-K202201202.

Data availability

In this paper, numerical simulation is used to carry out the experiment, and all the data have been presented in the manuscript.

Declarations

Competing interests

The authors declare no competing interests.