Finite-time and fixed-time synchronization control of fuzzy Cohen-Grossberg neural networks

doi:10.1016/j.fss.2019.12.002

Fuzzy Sets and Systems

Volume 394, 1 September 2020, Pages 87-109

https://doi.org/10.1016/j.fss.2019.12.002 Get rights and content

Abstract

This paper aims to investigate the finite-time and fixed-time robust synchronization of fuzzy Cohen-Grossberg neural networks with discontinuous activations. To deal with the discontinuous property, the framework of Filippov solution is invoked to solve the inexistence of the classical solutions. For the purpose of achieving fixed-time synchronization, we turn to investigating the fixed-time stability problem of the error system between the drive-response systems. By the functional differential inclusions theory, inequality technique and the non-smooth Lyapunov-Krasovskii functional and designing a simple discontinuous state-feedback control law for the response neural system, some sufficient algebraic criteria are derived to achieve synchronization within a fixed time, and the settling time is given. Moreover, based on the fixed-time robust synchronization and under the same basic assumptions, we further complete the finite-time synchronization of the addressed drive-response systems by designing a simple switching adaptive controller, and the upper bound of the settling time is also estimated. It should be regarded as the first time to study the finite-time synchronization of fuzzy Cohen-Grossberg neural networks with discontinuous activations based on the adaptive control. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed synchronization strategies.

Introduction

In 1996, Yang and Yang [36] put forward the fuzzy cellular neural networks based on the traditional cellular neural network in order to cope with the uncertainty or vagueness in human cognitive processes and modeling neural systems more practical in real world problems such as image processing and pattern recognition problems. After that, fuzzy cellular neural networks with delays were extensively considered by researchers; for instance, see [5], [15], [21], [22], [23], [24], [26], [27], [39], [40]. For example, Jia [15] studied the finite-time stability for a class of fuzzy cellular neural networks with multi-proportional delays by applying the finite-time stability theory and differential inequality techniques; Li et al. [21] considered the global asymptotic/exponential stability of fuzzy cellular neural networks with time delays by using Lyapunov-Krasovskii functionals, M-matrix theory and linear matrix inequality approach; Based on the Lyapunov-Krasovskii functional, stochastic analysis theory and the Itô's formula as well as the Dynkin formula, Zhu and Li [39] established new results on exponential and almost sure exponential stability of stochastic fuzzy delayed Cohen-Grossberg neural networks.

Synchronization, which means the dynamical signals of chaotic coupled system achieve an identical behavior with time moving. In reality, it is significant to consider the synchronization of its different potential applications including biological systems, intelligent control, secure communication, and image protection. During the past several years, various kinds of synchronization control of fuzzy neural networks have been proposed, see, [1], [7], [31], [33], [38], [37] and the references therein. For example, Yu et al. [37] studied the exponential lag synchronization for delayed fuzzy cellular neural networks via periodically intermittent control by utilizing inequality technique, Lyapunov functional theory and the analysis method; By using Lyapunov-Krasovskii functional and linear matrix inequality method, Ratnavelu et al. [31] considered the synchronization of fuzzy bidirectional associative memory neural networks with various time delays; Abdurahman et al. [1] investigated the finite-time synchronization for fuzzy cellular neural networks with time-varying delays by applying the finite-time stability theory, inequality technique and the analysis method.

The finite-time synchronization possesses a critical issue that the settling time is dependent on the initial conditions. But, the initial conditions of real world practical models can hardly be estimated. In 2012, Polyakov [29] investigated a nonlinear feedback design law for the fixed-time stabilization of linear control systems. Polyakov pointed out that the settling time of fixed-time synchronization is independent of the initial synchronization errors and the upper bound of the settling time can be estimated in advance. Based on these, the fixed time synchronization is more applicable than finite time synchronization. Compared with many finite-time synchronization problems on the fuzzy neural networks, the research of fixed-time synchronization is still in a primitive stage, and few research has been investigated on the fixed-time synchronization of fuzzy neural networks, see [41], [42].

On the other hand, from the previous literature review, many results on the stability and synchronization of the fuzzy neural networks were based on the assumption that the activation functions of are continuous, Lipschitz continuous or even smooth. However, in practice, the neuron activation functions possess jump discontinuities with respect to states. In recent years, many researchers have devoted themselves to investigating the neural network systems with discontinuous activation functions since the pioneering contribution of Forti and Nistri [11]. See, to name a few, [2], [6], [8], [9], [12], [18], [19], [20]. For example, Abdujelil et al. [2] investigated the general decay synchronization of memristor-based Cohen-Grossberg neural networks with mixed time-delays and discontinuous activations based on the concept of Filippov solution, theory of differential inclusion, Lyapunov-Krasovskii functionals and employing useful inequality techniques. By utilizing the discontinuous state feedback control method, Duan et al. [9] further considered the finite-time synchronization of delayed fuzzy cellular neural networks with discontinuous activations.

However, to the best of our knowledge, there has been few literature to study the finite-time synchronization problem of fuzzy neural networks with discontinuous activations by designing the adaptive controller, although the adaptive synchronization schemes have been widely applied to considerable neural network systems, such as reaction-diffusion neural networks, fuzzy cellular neural networks and so on. See, to name a few, [27], [32].

Based on the above analysis, in this paper, our works have mainly focused on developing new finite-time and fixed-time robust synchronization control criterion for a class of fuzzy Cohen-Grossberg neural networks with discontinuous activations. In order to handle this problem easily, first of all, we have to deal with the discontinuous right sides by the framework of Filippov solutions. Then, by using functional differential inclusions theory and constructing an appropriate Lyapunov-Krasovskii functional, the robust fixed-time synchronization is achieved by designing a simple discontinuous state-feedback control law, and the finite-time synchronization of the addressed drive-response systems can be achieved by designing a simple switching adaptive controller. Several remarks are provided to show the novelty and importance of the theory results, which summarize and extend some existing works. Finally, numerical examples and corresponding simulations have been investigated to verify the correctness of the main theorems.

The highlights and major contributions of this paper are reflected in the subsequent key aspects:

•
(1) We focus on the study of fuzzy Cohen-Grossberg neural networks with discontinuous neuron activations and time-varying delays, which can extend some previous results to the discontinuous case, such as [1], [7], [15], [21], [22], [26], [28], [33], [39], [40]. Moreover, many other Cohen-Grossberg drive-response models and fuzzy cellular drive-response models with delays are also the special cases of our considered model, such as [9].
•
(2) Since fixed-time synchronization is more practical than the asymptotic synchronization and the finite-time synchronization. We first attempt to address the fixed-time synchronization control problem for the proposed models.
•
(3) By designing a switching state-feedback control law for the response neural system, new verifiable algebraic criteria are given to guarantee that the response system can robustly synchronize with the drive system in a fixed time. Moreover, the settling time is also given and can be easily estimated.
•
(4) Adaptive control sometimes take more advantages than the state-feedback control in the synchronization control. By designing a simple switching adaptive, we investigate the finite-time synchronization of the addressed drive-response systems. Moreover, the upper bounds of the settling time are also estimated. It is the first time to study the finite-time synchronization of fuzzy Cohen-Grossberg neural networks with discontinuous activations with adaptive control.

The remainder part of this paper is organized as follows. System description and some preliminaries are presented in Section 2. Two control design schemes are proposed and employed to ensure the finite-time and fixed-time robust synchronization in Section 3. In Section 4, numerical examples are given to illustrate the effectiveness of the obtained results. Finally, conclusions are drawn in Section 5.

Section snippets

System description

In this paper, we consider the following delayed fuzzy Cohen-Grossberg neural networks with discontinuous activations: ${\dot{x}}_{i} (t) = d_{i} (x_{i} (t)) [- c_{i} (t, x_{i} (t)) + \sum_{j = 1}^{n} a_{i j} (t) f (x_{j} (t)) + \sum_{j = 1}^{n} b_{i j} ν_{j} + ⋀_{j = 1}^{n} T_{i j} ν_{j} + ⋀_{j = 1}^{n} α_{i j} (t) f_{j} (x_{j} (t - τ_{j} (t))) + ⋁_{j = 1}^{n} β_{i j} (t) f_{j} (x_{j} (t - τ_{j} (t))) + ⋁_{j = 1}^{n} S_{i j} ν_{j} + I_{i} (t)], i = 1, 2, . . ., n,$ with initial conditions $x_{i 0} (θ) = ϕ_{i} (θ), θ \in [- τ, 0],$ where $i, j = 1, 2, . . . n$ , $n \geq 2$ is the number of neurons in the network, $x_{i} (t)$ denotes the state of the ith unit at time t; $d_{i} (x_{i} (t))$ denotes the amplification function, $c_{i} (t, x_{i} (t))$ is

Fixed-time synchronization under discontinuous state-feedback controller

In order to achieve the fixed-time synchronization, the following discontinuous control law is designed for the response neural networks $u_{i} (t) = - s i g n (ε_{i} (t)) (λ_{i} + ζ_{i} | ε_{i} (t) | + ϱ_{i} | ε_{i} (t - τ_{i} (t)) | + κ_{i} | ε_{i} (t) |^{δ} + σ_{i} | ε_{i} (t) |^{θ}),$ where $ε_{i} (t) = y_{i} (t) - x_{i} (t)$ , $δ > 1$ , $0 \leq θ < 1$ , $λ_{i}$ , $ζ_{i}$ , $ϱ_{i}$ , $κ_{i}$ , $σ_{i}$ are the parameters to be designed later, $i = 1, 2, . . ., n$ , $j = 1, 2, . . ., n$ .

Theorem 3.1

Suppose that the conditions (H1)-(H4) hold, then the response system (2.4) can robustly synchronize with the drive system (2.3) in a fixed time based on the controller

Numerical examples and simulations

In this section, numerical examples are dedicated to showing the effectiveness of the proposed criteria.

Conclusion

In this paper, we have dealt with the finite-time and fixed-time robust synchronization of fuzzy Cohen-Grossberg neural networks with discontinuous activations. In order to achieving fixed-time synchronization of the proposed drive-response systems, we turn to investigating the fixed-time stability problem of the error system between the drive-response systems. We present a novel discontinuous state-feedback control inputs to the response neural system. Then, under the concept of Filippov

Authors' contributions

All authors read and approved the manuscript.

Acknowledgements

The authors thank the anonymous reviewers for their insightful suggestions which improved this work significantly. This work was jointly supported by the National Natural Science Foundation of China (61773217), Hunan Provincial Science and Technology Project Foundation (2019RS1033), Scientific Research Fund of Hunan Provincial Education Department(18A013), Hunan Normal University National Outstanding Youth Cultivation Project (XP1180101), Construct Program of the Key Discipline in Hunan Province

References (42)

A. Abdurahman et al.
Finite-time synchronization for fuzzy cellular neural networks with time-varying delays
Fuzzy Sets Syst.
(2016)
J.Y. Chen et al.
Asymptotic stability of delayed fractional-order fuzzy neural networks with impulse effects
J. Franklin Inst.
(2018)
Y. Cao et al.
Robust passivity analysis for uncertain neural networks with leakage delay and additive time-varying delays by using general activation function
Math. Comput. Simul.
(2019)
W. Ding et al.
Synchronization of delayed fuzzy cellular neural networks based on adaptive control
Phys. Lett. A
(2008)
X.S. Ding et al.
Robust fixed-time synchronization for uncertain complex-valued neural networks with discontinuous activation functions
Neural Netw.
(2017)
L. Duan et al.
Finite-time synchronization of delayed fuzzy cellular neural networks with discontinuous activations
Fuzzy Sets Syst.
(2019)
M. Forti et al.
Generalized Lyaponov approach for convergence of neural networks with discontinuous or non-Lipschitz activations
Physica D
(2006)
C. Hu et al.
Fixed-time stability of dynamical systems and fixed-time synchronization of coupled discontinuous neural networks
Neural Netw.
(2017)
R.W. Jia
Finite-time stability of a class of fuzzy cellular neural networks with multi-proportional delays
Fuzzy Sets Syst.
(2017)
C.Z. Li et al.
Exponential stability of fuzzy Cohen-Grossberg neural networks with time delays and impulsive effects
Commun. Nonlinear Sci. Numer. Simul.
(2010)

X.D. Li et al.

Existence and global stability analysis of equilibrium of fuzzy cellular neural networks with time delay in the leakage term under impulsive perturbations

J. Franklin Inst.

(2011)

Y.K. Li et al.

Existence and global exponential stability of equilibrium for discrete-time fuzzy BAM neural networks with variable delays and impulses

Fuzzy Sets Syst.

(2013)

Y.F. Liu et al.

Multiple μ-stability and multiperiodicity of delayed memristor-based fuzzy cellular neural networks with nonmonotonic activation functions

Math. Comput. Simul.

(2019)

X.Y. Liu et al.

Finite-time and fixed-time bipartite consensus of multi-agent systems under a unified discontinuous control protocol

J. Franklin Inst.

(2019)

F.R. Meng et al.

Periodicity of Cohen-Grossberg-type fuzzy neural networks with impulses and time-varying delays

Neurocomputing

(2019)

P. Mani et al.

Adaptive control for fractional order induced chaotic fuzzy cellular neural networks and its application to image encryption

Inf. Sci.

(2019)

A. Polyakov et al.

Finite-time and fixed-time stabilization: implicit Lyapunov function approach

Automatica

(2015)

K. Ratnavelu et al.

Synchronization of fuzzy bidirectional associative memory neural networks with various time delays

Appl. Math. Comput.

(2015)

R.Q. Tang et al.

Finite-time cluster synchronization for a class of fuzzy cellular neural networks via non-chattering quantized controllers

Neural Netw.

(2019)

Y. Wan et al.

Robust fixed-time synchronization of delayed Cohen-Grossberg neural networks

Neural Netw.

(2016)

J. Yu et al.

Exponential lag synchronization for delayed fuzzy cellular neural networks via periodically intermittent control

Math. Comput. Simul.

(2012)

Cited by (72)

Fixed/predefined-time projective synchronization for a class of fuzzy inertial discontinuous neural networks with distributed delays
2024, Fuzzy Sets and Systems
In this article, fixed-time projective synchronization (FTPS) and predefined-time projective synchronization (PTPS) for a class of discontinuous fuzzy inertial neural networks (FINNs) with distributed delays are considered. To this end, regarding differential inclusion theories and the Lyapunov stability method, more precise and improved setting times of FTPS are systematically constructed. Furthermore, a more general and novel predefined-time stability lemma is established to make the synchronized time more flexible and can be an accurate value in advance. Meanwhile, some effective feedback controllers and new criteria are established to guarantee the proposed FINNs achieve FTPS and PTPS, respectively. Finally, the corresponding numerical simulations and image encryption application are offered to confirm the obtained results.
Fixed-time synchronization of delayed inertial Cohen–Grossberg neural networks with desynchronizing impulses
2024, Communications in Nonlinear Science and Numerical Simulation
This article focuses on achieving fixed-time (FXT) synchronization of inertial Cohen–Grossberg neural networks (ICGNNs) in the presence of time-varying delays and desynchronizing impulsive effects. Firstly, a new lemma is presented to achieve FXT stability of the impulsive systems with destabilizing impulses. The settling-time function is shown to depend on both the parameters of impulsive sequence and continuous-time subsystems. Furthermore, based on the proposed lemma, sufficient conditions are established to achieve FXT synchronization of ICGNNS with desynchronizing impulses by designing a unified controller. Two numerical examples are taken into consideration to verify the efficacy of the proposed theoretical results.
Time-synchronized control for dynamic positioning system
2024, Ocean Engineering
This paper investigates a time-synchronized control for a class of dynamic positioning systems subjected to the influence of unknown disturbances. In general, it is worth noting that the methodology of time-synchronized control achieves the important outcome of simultaneous convergence of the multiple state-elements in the closed loop dynamic system. Despite this very significant property, the virtues of this methodology have not been completely explored in the application of marine system control design. Recent developments have demonstrated that time-synchronized control has further essential advantages in saving energy consumption, optimizing and shortening the output trajectory, due to the inherent properties of ratio persistence, all of which are critical in marine applications. In this work, to articulate and design the appropriate fixed-time-synchronized control, the methodology of a singularity-free sliding mode is proposed. The different conditions in control design and input-to-state stability and analytical conditions pertaining to singularity avoidance are investigated. A robust time-synchronized controller is developed and incorporated with a disturbance observer in a super-twisting structure for marine applications. With successful fixed-time convergence results, other related properties of time-synchronized stability are further discussed for the dynamic positioning system. Detailed comparative simulation cases are conducted to demonstrate the superiority of proposed controller. The simulation carried out on the disturbed system demonstrates the effectiveness of the developed method.
The synchronization of K-valued Fuzzy cognitive maps
2024, Fuzzy Sets and Systems
Fuzzy cognitive maps (FCMs) with the abilities of knowledge representation and causal reasoning have been widely applied in the control-related fields of multi-agent systems (MAS), such as the mobile robots and unmanned aerial vehicles (UAVs). From the perspective of system applications, the cooperative control of group behaviors is a meaningful topic in this community, where the synchronization of agents is one of fundamental issues. However, how to achieve the synchronization of FCM-based control mechanism is still an open problem. In this article, two finite state fuzzy cognitive map systems with unidirectional coupling in the drive-response configuration are introduced. By utilizing matrix semi-tensor product (STP), the coupling dynamical systems are transformed into algebraic forms, based on which the necessary and sufficient conditions for complete synchronization are presented. The corresponding theoretical proofs are given and quantitative analysis are implemented. Finally, several experiments are given to prove the effectiveness of the proposed methods.
Hyperbolic function-based fixed/preassigned-time stability of nonlinear systems and synchronization of delayed fuzzy Cohen–Grossberg neural networks
2024, Neurocomputing
In this article, the fixed-time (FIT) and preassigned-time (PRT) stability of nonlinear models and the FIT/PRT synchronization of discontinuous delayed fuzzy Cohen–Grossberg neural networks (DFCGNNs) are explored, respectively. Firstly, by means of hyperbolic-cosine function and Gudermannian function, a novel FIT stability theorem is established by utilizing Lyapunov method. Then, based on the developed stability theorem, several new discontinuous control schemes with hyperbolic-cosine function are developed to discuss the FIT synchronization of DFCGNNs. Furthermore, the PRT synchronization is also explored by slightly modifying FIT control schemes, where the synchronization time is predetermined. Eventually, two numerical examples are provided to validate the theoretical research.
Synchronization analysis of nabla fractional-order fuzzy neural networks with time delays via nonlinear feedback control
2024, Fuzzy Sets and Systems
This paper is committed to explore synchronization issues for a class of nabla fractional-order fuzzy neural networks with time delays. First, we establish two useful nabla fractional inequalities by feat of the theory of nabla fractional calculus and $N$ -transform together with nabla convolution. Next, we design two distinctive controllers including nonlinear delay feedback controller and hybrid nonlinear delay controller, some easily verified quasi-synchronization (QS) criteria and complete synchronization (CS) criteria are derived by virtue of newly proposed inequalities therein and some inequality techniques. Finally, the correctness of theoretical results is indicated finally through numerical simulations. What is more important, theoretical results therein can offer some new avenues about QS and CS analysis for nabla fractional dynamical networks.

View all citing articles on Scopus

View full text

Finite-time and fixed-time synchronization control of fuzzy Cohen-Grossberg neural networks

Abstract

Introduction

Section snippets

System description

Fixed-time synchronization under discontinuous state-feedback controller

Numerical examples and simulations

Conclusion

Authors' contributions

Acknowledgements

Fuzzy Sets Syst.

J. Franklin Inst.

Math. Comput. Simul.

Phys. Lett. A

Neural Netw.

Fuzzy Sets Syst.

Physica D

Neural Netw.

Fuzzy Sets Syst.

Commun. Nonlinear Sci. Numer. Simul.

J. Franklin Inst.

Fuzzy Sets Syst.

Math. Comput. Simul.

J. Franklin Inst.

Neurocomputing

Inf. Sci.

Automatica

Appl. Math. Comput.

Neural Netw.

Neural Netw.

Math. Comput. Simul.