Projective synchronization of different chaotic time-delayed neural networks based on integral sliding mode controller

doi:10.1016/j.amc.2010.05.037

Applied Mathematics and Computation

Volume 217, Issue 1, 1 September 2010, Pages 164-174

https://doi.org/10.1016/j.amc.2010.05.037 Get rights and content

Abstract

In this paper, an integral sliding mode control approach is presented to study the projective synchronization for different chaotic time-delayed neural networks. A sliding mode surface is appropriately constructed and a sliding mode controller is synthesized to guarantee the reachability of the specified sliding surface. The global asymptotic stability of the error dynamical system in the specified switching surface is investigated with the Lyapunov–Krasovskii (L–K) functional method. A delay-dependent sufficient condition is derived and the maximum time-delay value is obtained by means of the linear matrix inequality (LMI) technique. A simulation example is finally exploited to illustrate the feasibility and effectiveness of the proposed approach, verify the conservativeness of L–K method and LMI technique, and exhibit the relationship between the convergence velocity of error system and the gain matrix.

Introduction

Chaos appears in time evolutions of some kinds of nonlinear equations. Chaotic systems intrinsically defy synchronization due to the evolution of a chaotic system sensitive dependence on its initial condition. However, since the pioneer work that Pecora and Carroll introduced a method to realize the synchronization of two identical chaotic systems with different initial conditions [1], many attentions have been attracted on the chaos synchronization over the past decades. The increasing interest in researching the synchronization characteristics of chaotic systems stems from its potential applications in cryptography, secure communication, optimization of nonlinear systems performance, modeling brain activity, and chemical reaction [2], [3], [4], [5], [6], [7]. The phenomena of chaos synchronization in coupled nonlinear systems are being extensively studied in the context of laser dynamics, electronics circuits, chemical and biological systems [8].

Recently, the study of dynamical properties of neural networks appears more and more [9], [10], [11], [12] due to their extensive applications in differential fields such as signal and image processing, combinatorial optimization, pattern recognition and etc. In the electronic implementation of a neural network, time delay will occur in the interaction between the neurons inevitably, and will affect the dynamic behavior of the neural network model. In some particular cases, chaotic and hyperchaotic attractors may be generated by the introduction of delays into neural networks [13], [14], [15]. Therefore, some time-delayed neural networks could be as a model when we study the chaos synchronization.

In practice, the chaotic systems are inevitably subject to some environmental changes, which may render their parameters to be variant. Furthermore, from the point of view of engineering, it is very difficult to keep the two chaotic systems to be identical all the time [16]. It is thus of great significance to study the synchronization problem of nonidentical chaotic neural networks. Obviously, if the neural networks considered are distinct and with time delay, it becomes more complex and challenging. Not any two different chaotic systems can be synchronized. The synchronization can occur only when the response system attractor possesses the same topological characteristic as the drive system in essence, such as Lyapunov exponents and fractal dimensions [17].

Over the past few years, following complete synchronization, several new types of synchronization have been found in interesting chaotic systems, such as anticipatory synchronization [18], lag synchronization [19], phase synchronization [20], antiphase synchronization [21], generalized synchronization [22], and etc. Among all kinds of chaos synchronization, projective synchronization, characterized by a scaling factor that two systems synchronize proportionally, is one of most interesting problems. Projective synchronization is interesting because of its proportionality between the synchronized dynamical states. In application to secure communication, this feature can be used to M-nary digital communication for achieving fast communication [23].

In 1998, projective synchronization phenomenon was first reported and discovered by Gonzalez-Miranda [24]. She observed that when chaotic systems exhibit invariance properties under a special type of continuous transformation, amplification and displacement of the attractor occurs. This degree of amplification or displacement obtained is smoothly dependent on the initial condition. By this definition, complete synchronization is not a special case of projective synchronization. In 1999, Mainieri and Rehacek called this type of synchronization projective synchronization [25], and declared that the two identical systems could be synchronized up to a scaling factor α. Subsequently, some researchers extended the concept of projective synchronization and termed it as generalized projective synchronization [26], [27]. Dibakar Ghosh in Ref. [23] presented a point of view that projective synchronization is not in the category of generalized synchronization because the response system of projective synchronization is not asymptotically stable.

In the past 15 years, many techniques for chaos synchronization have been developed. These include periodic parametric perturbation [28], drive-response synchronization [29], adaptive control [30], sliding mode (or variable structure) control [31], [32], [33], backstepping control [34], fuzzy control [35], impulsive control [36], and many others. The sliding mode control theory introduced by Utkin [37] in 1977 provides an efficient way to the robust control problem. Its main advantages are fast response, good transient performance and robustness to variations of system parameters.

In this paper, an integral sliding mode control approach will be developed to address the projective synchronization problem of different chaotic time-delayed neural networks. It overcomes some limitations of the previous work where projective synchronization has been investigated only in chaotic systems without time delay [25], [26], [27], [33], [34] or in delayed chaotic systems always with identical structure and same parameters [38], [39]. Firstly, an integral sliding surface is properly constructed and an integral sliding mode controller is designed to guarantee the reachability of the specified sliding surface. Then, by employing the L–K function and LMI technique, the delay-dependent condition is derived under which the resulting error system is globally asymptotically stable in the specified switching surface. Generally speaking, delay-dependent conditions are less conservative than the delay-independent ones, especially when the size of delay is small. Moreover, in engineering practice, information on the delay range is generally available [40]. The main contributions of this study are to propose an integral sliding mode controller to deal with the projective synchronization of different drive-response systems with delay, obtain a delay-dependent LMI criterion for the stability of error system by constructing an appropriate Lyapunov–Krasovskii function, verify the conservativeness of the maximum time-delay value solved by L–K method and LMI technique [40], [41], [42], [43] and show the variation of convergence velocity of the error dynamical system with the gain matrix in the simulation.

Notation

Throughout the paper, for any real symmetric matrices A and B, the notation A ⩾ B or A > B means that matrix A − B is positive semi-definite or positive definite, and A ⩾ 0 or A > 0 means that matrix A is positive semi-definite or positive definite. A^T and A⁻¹ represent the transpose and the inverse of matrixA respectively.

Section snippets

Problem description and preliminaries

A class of delayed chaotic neural networks is described by the following time-delay differential equations: ${\dot{x}}_{i} (t) = - c_{i} x_{i} (t) + \sum_{j = 1}^{n} a_{ij} g_{j} (x_{j} (t)) + \sum_{j = 1}^{n} b_{ij} g_{j} (x_{j} (t - τ)) + H_{i},$ where n ⩾ 2 denotes the number of neurons in the network, x_i(t) denotes the state variable associated with the ith neuron, τ ⩾ 0 denotes the transmission time delay, c_i > 0 denotes the self connection of neurons, a_ij indicates the strength of the neuron interconnections within the network, b_ij indicates the strength of the neuron

Sliding surface and equivalent control law design

It can be seen clearly from (8) that the dynamic behavior of the error system depends on the chaotic states x(t) and x(t − τ) of the drive system (2) because of the difference between the drive and response systems and the existence of scaling factor α in projective synchronization. Therefore, projective synchronization between different chaotic neural networks (2), (4) cannot be achieved only by utilizing output feedback control. An integral sliding mode control approach will be proposed to

Numerical simulations

Assume that the following delayed chaotic Hopfield neural networks [13] is the drive system: $[\begin{matrix} {\dot{x}}_{1} (t) \\ {\dot{x}}_{2} (t) \end{matrix}] = - C [\begin{matrix} x_{1} (t) \\ x_{2} (t) \end{matrix}] + A [\begin{matrix} g_{1} (x_{1} (t)) \\ g_{2} (x_{2} (t)) \end{matrix}] + B [\begin{matrix} g_{1} (x_{1} (t - τ)) \\ g_{2} (x_{2} (t - τ)) \end{matrix}],$ where g_j(x_j(t)) = tanh (x_j(t)), j = 1, 2, and the following delayed chaotic cellular neural networks [14] is the response system: $[\begin{matrix} {\dot{z}}_{1} (t) \\ {\dot{z}}_{2} (t) \end{matrix}] = - D [\begin{matrix} z_{1} (t) \\ z_{2} (t) \end{matrix}] + P [\begin{matrix} f_{1} (z_{1} (t)) \\ f_{2} (z_{2} (t)) \end{matrix}] + Q [\begin{matrix} f_{1} (z_{1} (t - τ)) \\ f_{2} (z_{2} (t - τ)) \end{matrix}] + u (t),$ where f_j(z_j(t)) = 0.5(∣z_j(t) + 1∣ − ∣z_j(t) − 1∣), j = 1, 2.

For numerical simulation, we choose the fourth-order Runge–Kutta method to solve the

Conclusions

In the fields of secure communication, projective synchronization can be used to extend binary digital to M-nary digital communication for achieving fast communication, so the research on projective synchronization has important theoretical significance and improved value. Furthermore, projective synchronization has been applied in the research of secure communication due to the unpredictability of the scaling factors. In this paper, the projective synchronization problem has been studied for

Acknowledgements

This research is supported by the NNSF of China under Grant No. 10532050, the National Outstanding Young Funds of China under Grant No. 10625211 and the Program of Shanghai Subject Chief Scientist No. 08XD14044.

References (46)

S. Boccaletti et al.
The synchronization of chaotic systems
Phys. Rep.
(2002)
P. Balasubramaniam et al.
Global asymptotic stability of stochastic recurrent neural networks with multiple discrete delays and unbounded distributed delays
Appl. Math. Comput.
(2008)
L.Q. Zhou et al.
Global exponential periodicity and stability of cellular neural networks with variable and distributed delays
Appl. Math. Comput.
(2008)
H. Huang et al.
On global asymptotic stability of recurrent neural networks with time-varying delays
Appl. Math. Comput.
(2003)
Z. Chen et al.
Discrete analogue of high-order periodic Cohen–Grossberg neural networks with delay
Appl. Math. Comput.
(2009)
H.T. Lu
Chaotic attractors in delayed neural networks
Phys. Lett. A
(2002)
P. Yu et al.
Study of double Hopf bifurcation and chaos for an oscillator with time-delayed feedback
Commun. Nonlinear Sci. Numer. Simul.
(2002)
H. Huang et al.
Synchronization of nonidentical chaotic neural networks with time delays
Neural Networks
(2009)
C.Y. Chee et al.
Control of the formation of projective synchronization in lower-dimensional discrete-time systems
Phys. Lett. A
(2003)
Q. Wang et al.
Generalized Q–S (lag, anticipated and complete) synchronization in modified Chua’s circuit and Hindmarsh–Rose systems
Appl. Math. Comput.
(2006)

D.X. Wang et al.

Lag synchronizing chaotic system based on a single controller

Commun. Nonlinear Sci. Numer. Simul.

(2008)

D. Ghosh et al.

On the various kinds of synchronization in delayed Duffing–Van der Pol system

Commun. Nonlinear Sci. Numer. Simul.

(2008)

J.P. Yan et al.

Generalized projective synchronization of a unified chaotic system

Chaos Solitons Fract.

(2005)

G.H. Li

Generalized projective synchronization of two chaotic systems by using active control

Chaos Solitons Fract.

(2006)

V.V. Astakhov et al.

Synchronization of chaotic oscillators by periodic parametric perturbations

Phys. D

(1997)

X.S. Yang et al.

A note on mathematical aspects of drive–response type synchronization

Chaos Solitons Fract.

(1999)

M. Roopaei et al.

Unknown nonlinear chaotic gyros synchronization using adaptive fuzzy sliding mode control with unknown dead-zone input

Commun. Nonlinear Sci. Numer. Simul.

(2010)

N. Cai et al.

Modified projective synchronization of chaotic systems with disturbances via active sliding mode control

Commun. Nonlinear Sci. Numer. Simul.

(2010)

G.H. Li et al.

Generalized projective synchronization between two different chaotic systems using active backstepping control

Phys. Lett. A

(2006)

Y.Q. Yang et al.

Exponential lag synchronization of a class of chaotic delayed neural networks with impulsive effects

Phys. A

(2007)

C.F. Feng et al.

Generalized projective synchronization in time-delayed chaotic systems

Chaos Solitons Fract.

(2008)

E. Fridman

Effects of small delays on stability of singularly perturbed systems

Automatica

(2002)

E. Fridman et al.

H-infinity control of linear state-delay descriptor systems: an LMI approach

Linear Algebra Appl.

(2002)

Cited by (87)

Saturated and asymmetric saturated control for projective synchronization of inertial neural networks with delays and discontinuous activations through matrix measure method
2023, ISA Transactions
The projective synchronization work presented in this article is focused on a class of nonlinear discontinuous coupled inertial neural networks with mixed time-varying delays and a cluster topological structure. The synchronization problem for discontinuous coupled inertial neural networks with clustering topology is examined in consideration with the mismatched parameters and the mutual influence among various clusters. To determine the required conditions for network convergence under the influence of an extensive range of impulses, the matrix measure technique and the average impulsive intervals are employed. To illustrate the effectiveness of the theoretical findings, graphical representation of varied impulsive ranges for multiple cases are provided using numerical simulations.
Adaptive neural networks-based integral sliding mode control for T-S fuzzy model of delayed nonlinear systems
2023, Applied Mathematics and Computation
This paper presents a method for designing adaptive neural networks (NNs) based on integral sliding mode control (ISMC) for delayed Takagi-Sugeno fuzzy systems with external disturbances. To do this, a fuzzy sliding surface is constructed with the information of the delayed states, and an adaptive radial basis function (RBF) NNs-based fuzzy ISMC is designed. Here, the RBF NN function is utilized to approximate the nonlinear perturbation term. The resulting stability criteria are derived based on suitable Lyapunov functionals with the help of generalized free-weight matrix-based inequality. The resulting conditions ensure the asymptotic stability with $H_{\infty}$ attenuation level. Moreover, the designed sliding motion achieved the reachability condition under the adaptive RBF NNs based on the sliding mode controller. To end with, a design and comparative examples are given to show the success of the nominated control scheme and less conservatism of the derived sufficient conditions.
H<inf>∞</inf> synchronization of persistent dwell-time switched neural networks based on an observer-based sliding mode scheme
2021, Nonlinear Analysis: Hybrid Systems
This paper addresses the issue of $H_{\infty}$ synchronization of switched neural networks via an observer-based sliding mode control scheme. With regard to the switched neural networks, the persistent dwell-time switching law is introduced to govern the switchings among subsystems, which is a more general switching law. Furthermore, in view of the difficulty in the measurement of most disturbances in various applications, a disturbance observer is constructed. Subsequently, an integral sliding surface and a sliding mode control law are constructed based on the disturbance observer states to guarantee the reachability of the sliding surface. Then, based on the Lyapunov stability and sliding mode control theories, sufficient conditions are derived to ensure that the sliding mode dynamics are globally uniformly asymptotically stable with a prescribed $H_{\infty}$ performance. Finally, the validity of the proposed method is verified by a numerical example.
Long-life control of tumor growth via synchronizing to a less severe case
2021, Biomedical Signal Processing and Control
Discovering suitable methods for treatment of diseases is a fundamental question in bioengineering. Researches show that chaos in tumor behavior can result in increasing tumor size and cell density. From this viewpoint, in this paper a novel idea is introduced for treatment of cancers. While researches try to remove the tumor, we can change the chaotic manner to non-chaotic one via a suitable control method to decrease the tumor size. In other words, the chaotic behavior of tumor can be transformed into a behavior which is not dangerous for cancer patients, so they can live longer with less aggressive cancer. An identical non-chaotic system is considered as a less severe case and being a master system to perform this idea. Then, control laws are designed such that the chaotic model (slave system) follows the master system. This strategy permits the patients to live with a tumor for years, theoretically. Also, this treatment is a new theoretical paradigm for transforming malignant tumors into a benign tumor. Three reliable tumor growth models with chaotic behavior are presented in the literature: tumor-immune-vascular, metastasis tumor growth and delayed tumor-immune models are investigated to show the efficiency of the presented method. Stability of the control law is demonstrated by using Lyapunov stability theory for each case. Finally, simulation results are offered for the tumor models with unknown parameters. They demonstrate the proposed method's effectiveness and show that this approach can be made for many cancerous models.
Sliding mode control for finite-time and fixed-time synchronization of delayed complex-valued recurrent neural networks with discontinuous activation functions and nonidentical parameters
2021, European Journal of Control
Citation Excerpt :
As a result, one can get desired benefits such as robustness, tracking ability and insensitivity to external disturbance. In comparison with the existing works of [19,23,46,53], the fixed-time sliding mode control approach (SMC) of this letter is superior to them from the following aspects: 1) Based on the idea given in [46], the constructed integral sliding mode surface is extended. 2) The fixed-time synchronization between drive-response systems (1) and (2) is assured under the fixed-time SMC, but the synchronization in fixed-time cannot be obtained by these SMC in [19,23,46,53].
This paper considers the finite-time and fixed-time synchronization of delayed complex-valued recurrent neural networks (CVRNNs) with discontinuous activation functions and nonidentical parameters via sliding mode control. Firstly, we design a sliding surface involving integral structure and a discontinuous control. Secondly, by constructing Lyapunov functional and using the differential inequality technique, some sufficient conditions are derived to guarantee the finite-time and fixed-time synchronization of delayed complex-valued recurrent neural networks with discontinuous activation functions and nonidentical parameters. Finally, two simulation examples are shown to illustrate the proposed methods.
Synchronization of a novel model for memristive neural networks via sliding mode control
2020, ISA Transactions
In this paper, a novel memristive neural networks model is developed. In the new model, the states of memristors are related to the initial resistance of the memristors and the amount of charge flowing through them in a specific direction, which embodies the memory characteristic of memristors. As a consequence, parameters in the model vary continuously and cannot be determined by the states of neurons. Existing results on synchronization of memristive neural networks are useless to this model. To investigate the synchronization of the new model, the main difficulty is how to deal with the time-varying parameter mismatches between the drive and response networks. Since the error is unbounded and only utilizing output feedback control is not enough, a sliding mode controller is designed. An integral sliding surface is designed for the desired sliding motion, and a feasible control law is proposed. Moreover, an example is given to demonstrate the novelty of our model and to illustrate the effectiveness of the sliding mode controller.

View all citing articles on Scopus

View full text

Projective synchronization of different chaotic time-delayed neural networks based on integral sliding mode controller

Abstract

Introduction

Section snippets

Problem description and preliminaries

Sliding surface and equivalent control law design

Numerical simulations

Conclusions

Acknowledgements

Phys. Rep.

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Comput.

Phys. Lett. A

Commun. Nonlinear Sci. Numer. Simul.

Neural Networks

Phys. Lett. A

Appl. Math. Comput.

Commun. Nonlinear Sci. Numer. Simul.

Commun. Nonlinear Sci. Numer. Simul.

Chaos Solitons Fract.

Chaos Solitons Fract.

Phys. D

Chaos Solitons Fract.

Commun. Nonlinear Sci. Numer. Simul.

Commun. Nonlinear Sci. Numer. Simul.

Phys. Lett. A

Phys. A

Chaos Solitons Fract.

Automatica

Linear Algebra Appl.