Article
Kyungpook Mathematical Journal 2021; 61(2): 353-369
Published online June 30, 2021
Copyright © Kyungpook Mathematical Journal.
Synchronization of Non-integer Chaotic Systems with Uncertainties, Disturbances and Input Non-linearities
Ayub Khan and Nasreen*
Department of Mathematics, Jamia Millia Islamia, New Delhi, India
e-mail : akhan12@jmi.ac.in and nasreen899@gmail.com
Received: March 14, 2019; Revised: November 22, 2020; Accepted: November 25, 2020
Abstract
In this paper, we examine and analyze the concept of different non-integer chaotic systems with external disturbances, uncertainties, and input non-linearities. We consider both drive and response systems with external bounded disturbances and uncertainties. We also consider non-linear control inputs. For synchronization, we introduce the adaptive sliding mode technique, in which we establish the stability of the controlled system by a control which estimates uncertainties and disturbances, and then applies a suitable sliding surface to control them. We use computer simulations to established the efficacy and adeptness of the prospective scheme.
Keywords: chaos synchronization, adaptive sliding mode technique, unknown disturbances, model uncertainties and input non-linearity
1. Introduction
In recent times, non-linear dynamical systems have become a hot topic among researchers. Discovered by Henri Poincaré[26], chaos is a complex phenomenon found in most non-linear dynamical systems, describing the sensitive dependence of the evolution of the system on the initial conditions. Poincaré observed that two neighboring points in state space can very quickly become isolated. The phenomenon of chaos has deep applications in viscoelasticity [14], dielectric polarization, electromagnetic waves[9], diffusion, signal processing, mathematical biology and, of course, chaotic systems. Different procedures have been used to investigate the behavior of the chaotic non-linear systems that surround us. Among these procedures are plotting phase portraits, poincaré sections, or bifurcation diagrams, or finding Lyapunov exponents.
To understand the behaviour of non-linear systems and to stabilize their control, Pecora and Carroll established the idea of synchronization. Under synchronization, trajectories of coupled systems evolve together in a usual pattern. Based on different control techniques such as adaptive backstepping [27], linear and nonlinear feedback synchronization [3], active control [25], sliding mode control [6], adaptive sliding mode technique[12], and time delay feedback[5], researchers have developed many synchronization schemes- schemes such as complete and anti synchronization [10, 21], phase and anti-phase synchronization [20], projective and hybrid function projective synchronization[13, 22], generalised synchronization[28].
To combine the concept of integer order differentiation and n-fold integration, Leibnitz and L'Hospital in 1675 gave the theory of integrals and derivatives of arbitrary order. Systems represented by non-integer differential equations[23] have been studied extensively in recent years. These studies focus on real-life systems and have many multidisciplinary applications. Specifically, it has been seen that non-integer systems, which generalize many well-known integer order systems, have chaotic and hyper-chaotic behavior. Some such systems are Lorenz systems[8], Chen systems, Rössler systems[15], Liu-systems[6], Genesio-Tesi systems[7], Chua systems[29], complex t-system [19] and complex Lu-systems[24].
In this manuscript, we synchronize two different non-integer chaotic systems. We treat a non-integer chaotic Liu-system[6] as drive a system, and a chaotic Genesio-Tesi system[7] as response a system. We do so with model uncertainties, and external bounded disturbances, but also with non-linear input. The considertion of these elements together seems to be novel in the literature. As uncertainties and disturbances introduce a dreadful change in chaotic systems, dynamics, and synchronization reduce this. Researchers have introduced various schemes [4, 12] to examine the synchronization of chaotic systems with various disturbances and uncertainties. Generally the sliding mode control technique is an efficient approach for dealing with uncertainties and disturbances.
In practice when we encounter a controller in a real-life systems, physical limitations cause some non-linearity in the control inputs. It has been shown that non-linear input can cause a severe decay in the system performance. If the controller is poorly designed, then system failure becomes worse. Therefore, non-linear input effects must be taken into consideration when evaluating and implementing a control scheme for chaotic systems. Researchers have designed various techniques to synchronize integer-order chaotic systems with non-linear inputs [2, 16]. It seems, however, synchronization among non-integer chaotic systems in the presence of external disturbances, model uncertainties and non-linear input has not been discovered. In our paper, we investigate the synchronization under these perturbations. We introduce an adaptive sliding mode control scheme to synchronize the considered systems. We estimated the disturbances and uncertainties through a adaptive control rule and we chose a suitable sliding surface to counter their effect. We design the appropriate controllers using known control techniques and Lyapunov stability theory.
As motivated above, we summarize here the main aspects of the this paper.
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1. We propose a novel synchronization scheme for non-integer chaotic systems.
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2. We design non-linear control inputs for non-integer chaotic systems.
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3. We compare our proposed methodology with the previously published literature. We consider disturbances, uncertainties, and non-linear input. Even with this, our methodology yields better results than the previously publishedliterature.
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4. We ilustrate an application in secure communication in Section 5.
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5. We use numerical simulations to validate and visualize our results.
2. Preliminaries and Problem Formulation
The fractional order derivative can be defined in various forms[23] such as Riemann-Lioville's derivative, Grünwald Letnikov's derivative, Caputo's derivative. Here we have taken Caputo's derivative defined as
where
Consider the drive system of
for
Consider the response system of
for
Assumption 1. The trajectories of non-integer chaotic systems are bounded so here we have assumed that the model uncertainties
Consequently we have
for
Assumption 2. It is assumed that if external disturbances
Thus for
where
In order to achieve synchronization, here we define synchronization error as
for
To achieve synchronization, we have to establish that the error system (2.3) is stable. For that our aim is to design control laws for any two non-integer chaotic systems with model uncertainties, external disturbances, and non-linear input to established that it is stable asymptotically:
To minimize the error, we choose the suitable sliding surface which is as follows:
where
To discuss the error system (2.3) at the chosen sliding surface (2.4), it is necessary that it should satisfy the following condition for
The derivative of (2.4) yields the following equation
Then, by considering the necessary condition
Hence, the system (2.3) is asymptotically stable which shows that the slave system (2.2) can be tackled by the master system (2.1) by constructing the appropriate control inputs.
Our next step is to design the appropriate control inputs in order to stabilize the error system and attain synchronization on the chosen sliding surface
The control inputs are designed as follows
where
The adaptive laws are chosen as
Theorem 2.1. For the error system (2.3) with control laws (2.8) and adaptive laws (2.9), if the following condition is fulfilled:
then the synchronization error converges to
The derivative of
By substituting the value of
Using adaptive laws (2.9) and substituting the values of
Using Assumption (2)
Substituting
Using equations (2.8), (2.9) and (2.10) and simplifying, we get the following inequality:
Integrating the above equation from
Since
Remark 2.2.([1, 17]) The signum function behaves as a rigid switcher in the prospective control law and it can cause chattering. Therefore, we modify the controller to prevent chattering:
where
Remark 2.3. After substituting the controller (2.11) into
Using condition (2.10) and
Using Theorem 2.1 together with the Barbalat Lemma, we obtain
3. Illustration Example
To check the applicability and efficacy of the proposed control scheme, we consider the following two different non-integer chaotic systems:
Drive system[6]
where
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Figure 1. Phase Portraits of fractional order Liu chaotic system for
α=0.95 (a)u2-u1 axis (b)u2-u3 axis (c)u3-u1 axis (d)u1-u2-u3 axis.
The uncertainties and disturbances for the drive system are taken as
for
Response system[7]
where
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Figure 2. Phase Portraits of fractional order Genesio-Tesi chaotic system for
α=0.95 (a)v1-v2 axis (b)v2-v3 axis (c)v1-v3 axis (d)v1-v2-v3 axis.
The uncertainties and disturbances for response system are taken as
for
The non-linear control inputs are taken as
The error dynamical system can be written as
Choosing a suitable sliding surface (2.4) using control law (2.11) and adaptive law (2.9), we take
Using Theorem 2.1 and (2.11), we get the following control laws.
For the systems (3.1) and (3.2) take
Figures 1 and 2 show the phase portraits of system (3.1) and (3.2). Figure 3 shows the synchronized state trajectories of system (3.1) and (3.2). Figure 4 shows the synchronized error and that the sliding surface converges to zero at approximately time
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Figure 3. Synchronized state trajectories which are synchronized at time t=2 unit(approx.)(a)
u1-v1 (b)u2-v2 (c)u3-v3
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Figure 4. (a) Synchronization error (b) sliding surface converging to zero at time t=2 unit(approx.).
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Figure 5. Estimated values of (a)uncertainties bounds (b)disturbances bounds.
4. Comparison of the Proposed Scheme with the Previous Published Literature
1. First of all, we compared our synchronization result to the integer-order Liu system and Genesio-Tesi system for the same set of parameter values and initial conditions. For
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Figure 6. Synchronization Error for integer order Liu system and Genesio-Tesi system.
2. In [6], the author adopted the active control and sliding mode control methods to analyse the complete synchronization between two identical non-integer Genesio Tesi systems with parameter values
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Figure 7. ynchronization Error for two identical non-integer Genesio-Tesi system.
3. In [18], the complete synchronization between two identical non-integer Genesio-Tesi systems with fifth order non-linearity based on the adaptive control method was studied with parameter values
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Figure 8. Synchronization Error for two identical non-integer Genesio-Tesi system with fifth order non-linearity.
4. In Section 4 of [25], the author anti-synchronized two identical non-integer Genesio-Tesi systems using the active control method with parameter values
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Figure 9. Synchronization Error for two identical non-integer Genesio-Tesi system.
5. In Section 5 of [25], the author anti-synchronized a non-integer Genesio-Tesi system and a Qi system using the active control method with parameter values
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Figure 10. synchronization Error for non-integer Genesio-Tesi system and Qi-system.
Our methodology significantly beats published liturature in these settings.
5. Application to Secure Communication
Non-integer, chaotic systems have applications in many fields, such as physics, chemical science, and secure communication.
In this manuscript, we show an application in secure communication.
Here we take a simple additive encryption masking scheme, to validate our proposed scheme.
The information input signal is selected as
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Figure 11. Additive encryption masking scheme (a) Information input signal(
IS ) (b) Chaotic encrypted signal(CS ) (c) Decrypted signal(DS ) (d)Error between original signal and decrypted signal.
6. Conclusion
In this paper, a robust adaptive sliding mode technique has been used to achieve synchronization between two different fractional-order chaotic systems with model uncertainties, external disturbances, and non-linear inputs. Synchronization of non-integer chaotic systems in the presence of uncertainties, disturbances and non-linear control inputs has not been examined in the prior literature. We synchronized non-integer chaotic Liu-system and Genesio-Tesi systems. We chose a suitable sliding surface and estimated the bounded uncertainties and disturbances using update laws to achieve the desired synchronization and reduce the consequence of external uncertainties and disturbances and non-linear input. Then, using the considered control scheme and Lyapunov stability theory, we designed appropriate controllers. Although we have taken non-integer chaotic systems with uncertainties, disturbances, and non-linearities, we get better synchronization results. This scheme should perform a significant role to enhance security in communications. Computational methods were used to evaluate the efficiency of the considered scheme.
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