Synchronisation of two different uncertain fractional-order chaotic systems with unknown parameters using a modified adaptive sliding-mode controller

This article investigates a modified adaptive sliding-mode controller to achieve synchronisation between two different fractional-order chaotic systems with fully unknown parameters. A suitable parameter updating law is designed to tackle the unknown parameters. For constructing the modified adaptive sliding-mode control, a simple sliding surface is designed and the stability of the suggested method is proved using Lyapunov stability theory. Finally, the proposed method is applied to gain chaos synchronisation between two different pairs of fractional-order chaotic systems with uncertain parameters. Numerical simulations are performed to demonstrate the robustness and efficiency of the proposed method.


Introduction
The essence of studying chaotic systems is to understand their structure and behaviour. These systems are deemed important as their study links between science and nature. One way to obtain a perspective glimpse of their complex dynamics is through their chaos control. However, chaos control is almost always an impossible task. This fact is due to the systems' unpredictable behaviour and sensitivity towards initial conditions. Hence, many arbitrary assumptions are necessary to achieve control in practical engineering problems, but this also implies a loss of important information. Systems characterised by a fractional order have attracted much attention in recent years. The ability of fractional-order modelling is to describe, mathematically and physically, phenomena of our real-world more rationally and carefully than the classical integer-order calculus [1][2][3], which has made them particularly well suited to research into nonlinear systems, particularly in physics and biology. Many phenomena show fractional-order dynamics, such as viscoelasticity and the polarisation of the dielectric as well as electrode-electrolyte [4,5]. The synchroni-sation of fractional-order chaotic systems has been documented extensively and has been the centre of interest in recent years, given its potential applicability to diverse fields of physics and engineering sciences such as cryptography, information processing, chemical reactions and biological systems [6].
Various control techniques have been proposed for controlling and synchronising fractional-order chaotic systems; for instance in [7] the authors extended the applications of adaptive control to anti-synchronise different fractional-order chaotic and hyperchaotic dynamical systems. New results on adaptive synchronisation design for chaotic Arneodo system of incommensurate fractional order with unknown parameters based on the Lyapunov stability theory were introduced in [8]. The finite time robust synchronisation problem of a class of uncertain fractional chaotic/hyper-chaotic systems with a novel fractional sliding mode control technique was investigated in [9]. A novel fractional-order fuzzy sliding mode control strategy was developed to realise the deployment of the tethered satellite system (TSS) with input saturation in [10]. A novel robust predictive control strategy was proposed for the synchronisation of fractional-order time-delay chaotic systems in [11]. A new robust optimal control strategy was presented to synchronise a class of fractionalorder chaotic systems with unknown fractional orders and uncertain dynamics, and input nonlinearities were proposed in [12]. A new technique using a recurrent non-singleton type-2 sequential fuzzy neural network for synchronisation of the fractional-order chaotic systems with time-varying delay and uncertain dynamics was presented in [13]. The robust stochastic stabilisation problem for a class of fuzzy Markovian jump systems with time-varying delay and external disturbances via sliding mode control scheme was studied in [14]. The synchronisation issue for a family of time-delayed fractional-order complex dynamical networks (FCDNs) with time delay, unknown bounded uncertainty and disturbance was investigated in [15].
Fortunately, some existing methods for synchronising integer-order systems can be rigorously generalised to fractional-order systems. However, in practical engineering situations, system parameters are often unknown and variable. Therefore, there is much interest in effectively synchronising two fractional-order chaotic systems of undetermined parameters. This is very significant for both theoretical and practical research. Among the aforementioned methods, the adaptive sliding-mode control strategy is an efficient control method for synchronising fractional-order chaotic systems when some or all the system parameters are unknown. The fundamental features of this strategy include its fast response, robustness against perturbations, good transient performance and ease of implementation in real applications. The main contributions of this paper are: a technique for synchronising fractional-order chaotic systems which will serve the purpose of synchronisation of fractional-order as well as integer-order chaotic systems; a modified adaptive sliding-mode synchronisation approach that is relevant to fractional-order chaotic systems with unknown parameters; synchronisation controller and parameter-identification technique are designed based on the Lyapunov stability method to achieve the synchronisation of two different pairs of fractional-order chaotic systems using the proposed method. Computer simulations based on the Adams-Bashforth-Moulton method support the theoretical findings.
This paper is organised as follows: Sect. 2 describes the fractional calculus and its properties. Section 3 presents a step-by-step methodology for the proposed modified adaptive sliding-mode synchronisation controller design of fractional-order chaotic systems with unknown parameters. Two simulation examples are given to demonstrate the effectiveness of the proposed synchronisation schemes in Sects. 4, 5. Finally, a conclusion in Sect. 6 closes the work.

Fractional calculus and its properties
The concept of an integer-order integro-differential operator can be extended by the fractional-order integro-differential operator using a generalisable formulation, that is, where p is the fractional order which could be a complex number, and a, t symbolise the limits of the operation. There are many definitions of the fractional integral and derivative which have been used in the recent literature, precisely, the following three definitions: Grünwald-Letnikov, Riemann-Liouville and Caputo. The current study dealt with the Riemann-Liouville definition [16,17], which was given by where m = p , J is the fractional Riemann-Liouville integral and with 0 < ϕ ≤ 1 and Γ (·) is the gamma function. For r > n ≥ 0, p and q are integers such that 0 ≤ p -1 ≤ r < p and 0 ≤ q -1 ≤ n < q. Then For r, m ≥ 0, there exist integers p and q such that 0 ≤ p -1 ≤ r < p and 0 ≤ q -1 ≤ m < q. Then

Modified adaptive sliding-mode synchronisation of fractional-order chaotic systems with unknown parameters
Consider a drive system of the form where x ∈ R n is the state vector of the drive system, f : R n → R n is a continuous vector function, F : R n → R n×d is a matrix function, and φ ∈ R d is a parameter vector. The controlled response system is given by where y ∈ R n is the state vector of the response system, g : R n → R n is a continuous function, G : R n → R n×k is a matrix function, θ ∈ R k is a parameter vector and U ∈ R n is a controller. The dynamics of the synchronisation errors can be expressed as where e = yx. Our aim is to design an adaptive sliding-mode controller U that makes states of the response system follow those of the drive system asymptotically, so that where · denotes the Euclidean norm. In accordance with the design strategy for adaptive sliding-mode control, we choose an input signal vector as follows: whereφ,θ are the estimated system parameters and K > 0 is a positive gain vector. Substituting a particular expression for U into the error dynamics (8) yields a form that is convenient for the forthcoming stability analysis: where the control input is defined as and s = s(e) is a switching surface that forms the desirable sliding dynamics. The sliding surface can be defined as where C is a positive constant. The following two conditions must be fulfilled on the sliding surface: s(e) = 0 andṡ(e) = 0.
The second condition is indispensable to constrain the state trajectory to remain on the switching surface defined by s(e) = 0. According to sliding-mode control theory, the sliding mode controller is designed as where γ is a positive real number. The resultant error dynamics then follows: The laws for updating parameters can be specified as follows: where λ = sC T . The following theorem introduces the necessary conditions for verifying the stability of the error system in (16). (10) and following the parameter-update laws in (17). The error system is then stabilised at its equilibrium.
Hence, V is positive-definite andV is negative-definite. Then the error system is stable in the sense defined in Lyapunov stability theory [18] and the response system (7) is synchronised with the drive system (6) globally and asymptotically. This completes the proof.
Then (34) reduces tȯ Since both s 2 > 0 and |s| > 0, then, when e = 0,V < 0 and hence V is positive-definite whilė V is negative-definite, so the error system is stable according to the Lyapunov stability theory [18], and the response system (26) is synchronised with the drive system (25) globally and asymptotically. This completes the proof.
For simulations, we use the Adams-Bashforth-Moulton method to solve systems for the fractional order p i = 0.95, i = 1, 2, 3. The uncertain parameters are set to a 1 = 6, b 1 = 2.92,

Synchronisation of fractional-order Chen and Lü chaotic systems using the modified adaptive sliding-mode control method
This section considers the synchronisation behaviour between the fractional-order Chen [22] and Lü systems using the modified adaptive sliding-mode control method. Now let us assume that the fractional-order Chen system is a drive system and the fractional-order Lü system is considered as a response system. In terms of unknown parameters, the drive

Conclusion
In this article, we have proposed a modified adaptive sliding-mode controller to realise chaos synchronisation between two different fractional-order chaotic systems with fully unknown parameters. The Lyapunov stability theory proved the asymptotic stability of the error system at the origin. The design of a suitable adaptive sliding-mode controller ensures target synchronisation. This work provides parameter update laws that estimate the true values of the unknown parameters. This paper also presents two numerical examples of different unknown fractional-order chaotic systems. Simulation results validate the efficiency and performance of the proposed modified adaptive sliding-mode synchronisation strategy. For future study, designing a modified adaptive sliding-mode controller to realise chaos synchronisation in drive-response dynamical networks is worth considering.