thesis on fractional order system

Fractional order chaotic systems: history, achievements, applications, and future challenges

• Published: 26 March 2020
• Volume 229 , pages 887–904, ( 2020 )

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• Mohammad Saleh Tavazoei 1  

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Motivated by the importance of study on the complex behaviors, which may be exhibited by fractional order differential equations, this review paper focuses on dynamical fractional order systems exhibiting chaotic behaviors. The review begins with a brief history on the first publications on the above-mentioned subject. Then, the review is continued by investigating the recent progresses relevant to fractional order chaotic systems. Furthermore, a summary on some applications for such systems, which have been reported in the literature, is presented. Finally, the paper is closed by discussing some open problems on the aforementioned research subject. These open problems, as future challenges for further study on fractional order chaotic systems, can specify some direction lines for continuing the research on that subject.

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Tavazoei, M.S. Fractional order chaotic systems: history, achievements, applications, and future challenges. Eur. Phys. J. Spec. Top. 229 , 887–904 (2020). https://doi.org/10.1140/epjst/e2020-900238-8

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DOI : https://doi.org/10.1140/epjst/e2020-900238-8

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Research Article

Controlling and synchronizing a fractional-order chaotic system using stability theory of a time-varying fractional-order system

Roles Conceptualization, Methodology

Affiliation Department of Automation, North China Electric Power University, Baoding, China

Roles Investigation, Software

Affiliation Shenhua Guohua Electric Power Research Institute Corporation, Beijing, China

Roles Formal analysis

* E-mail: [email protected]

Affiliation Department of Cognitive Science, School of Information Science and Engineering, Xiamen University, Xiamen, China

• Yu Huang, 
• Dongfeng Wang, 
• Jinying Zhang, 

• Published: March 30, 2018
• https://doi.org/10.1371/journal.pone.0194112
• Reader Comments

Control and synchronization of fractional-order chaotic systems have attracted wide attention due to their numerous potential applications. To get suitable control method and parameters for fractional-order chaotic systems, the stability analysis of time-varying fractional-order systems should be discussed in the first place. Therefore, this paper analyzes the stability of the time-varying fractional-order systems and presents a stability theorem for the system with the order 0< α <1. This theorem is a sufficient condition which can discriminate the stability of time-varying systems conveniently. Feedback controllers are designed for control and synchronization of the fractional-order Lü chaotic system. The simulation results demonstrate the effectiveness of the proposed theorem.

Citation: Huang Y, Wang D, Zhang J, Guo F (2018) Controlling and synchronizing a fractional-order chaotic system using stability theory of a time-varying fractional-order system. PLoS ONE 13(3): e0194112. https://doi.org/10.1371/journal.pone.0194112

Editor: Jun Ma, Lanzhou University of Technology, CHINA

Received: October 19, 2017; Accepted: February 23, 2018; Published: March 30, 2018

Copyright: © 2018 Huang et al. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the paper and its Supporting Information files.

Funding: This work was supported by the National Key R&D Program of China (2016YFB0600701) and the Fundamental Fund for the Central Universities of China under Grant 2015MS66. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

1. Introduction

Fractional-order calculus which extends the descriptive abilities of integer-order calculus can be traced to the work of Leibniz and Hospital in 1695. The integer-order calculus depends only on the local characteristics of a function’s, but fractional-order calculus accumulates all information of the function in a certain time, which is also called memory property. Mathematical models based on fractional-order calculus can describe the dynamic behavior of an actual system accurately in many areas, thereby it is necessary to facilitate the improvement of its design and control stability for fractional-order dynamic systems [ 1 ]. Recently, fractional-order chaotic control and synchronization have attracted increasing attention. In [ 2 ], Razminia A et al . synchronized a unidirectional coupling structure for the two fractional order chaotic systems using a sliding mode control methodology. In [ 3 ], Wu GC et al . presented a nonlinear coupling method to study the master-slave synchronization for the fractional differential equation. In [ 4 ], Golmankhaneh AK et al . have presented the chaos synchronization of two identical and nonidentical fractional orders of a new chaotic system by using active control. In [ 5 ], Jajarmi A et al . used a linear state feedback controller together with an active control technique in order to control a hyperchaotic financial system. In [ 6 ], a Lyapunov approach is adopted for deriving the parameter adaptation laws and proving the stability of the generalized projective synchronization (GPS) of two incommensurate fractional-order chaotic closed-loop systems. A linear feedback controller is proposed to achieve synchronisation of a fractional-order system with uncertainties and disturbance and guarantees the bounded state error for any bounded interference infinite time [ 7 ]. In [ 8 ], a simple but practical method to synchronize almost all familiar fractional-order chaotic systems which are including the commensurate system and incommensurate case, autonomous system, and the nonautonomous case has been put forward, and sufficient conditions are derived to guarantee synchronization of these systems. In [ 9 ], Shao SY et al . studies the fractional-order disturbance observer (FODO)-based adaptive sliding mode synchronization control for a class of fractional-order chaotic systems with unknown bounded disturbances. In [ 10 ], Soukkou A et al . proposed a fractional-order prediction-based feedback control scheme (Fo-PbFC) to stabilize the unstable equilibrium points and to synchronize the fractional-order chaotic systems (FoCS). In [ 11 ], Nourian et al . estimated the unknown coefficients of the system and demonstrated the stabilization of the synchronizer system by using the adaptive rule and a proper Lyapunov candidate function. In [ 12 ], Maheri et al . put forward a robust adaptive nonlinear feedback controller scheme to realize the synchronization of two different fractional-order chaotic systems in the condition of fully unknown parameters, external disturbance and uncertainties. In [ 13 ], Zhou et al . designed an adaptive controller to synchronize two entirely different fractional-order chaotic systems with uncertain parameters. Combining with appropriate parameter estimation laws. In [ 14 ], Yang proposed a single-state proportional feedback method to synchronize two identical generalized Lorenz systems. Used Lyapunov stability theory and a fractional-order differential inequality. In [ 15 ], Zhang et al . developed a modified adaptive control scheme and adaptive parameter laws to robustly synchronize coupled with fractional-order chaotic systems without certain parameters and perturbations. In [ 16 ], Xiang et al . investigated a robust synchronization for a class of systems with external disturbances.

In addition, many scholars have made great contributions in the field of the control and stability of time-varying fractional order systems. In [ 17 ], Aguila-Camacho N et al . put forward a new lemma for the Caputo fractional derivatives which has been proved to be useful in order to find the fractional-order extension of Lyapunov functions and can be used to demonstrate the stability of many fractional order systems including nonlinear and time-varying. In [ 18 ], Bao HB et al . put forward sufficient conditions which ensure the drive–response systems to achieve adaptive synchronization of fractional-order memristor-based neural networks with time-varying delay. In [ 19 ], the authors dealt with the fractional-order neural networks with impulsive effects and time-varying delay, and established several sufficient conditions guaranteeing the global Mittag–Leffler stability of the equilibrium point of the neural networks.

However, the most basic control and synchronization problem of chaotic systems are that of stability. Stability is a precondition for normal operation of systems and the main factor of system designs. A Lyapunov direct method is a core issue in integer-order stability theory, which is also a basic stability theorem for control systems.

It has been proven that the Lyapunov direct method is a relatively complete theoretical for integer-order systems both in theoretical study and practical application. As the transfer function of fractional-order systems is usually not a rational function of complex variable s , the stability analysis of fractional-order systems is far more complicated than that of integer-order systems. Many scholars have carried out extensive research on time-invariant fractional-order systems and made considerable achievements. For fractional-order LTI systems, in [ 20 ], Semary et al . discussed their physical and non-physical transfer functions, stability, poles, time domain, frequency domain, their relationships for different fractional-order differential equations and other basic concepts. In [ 21 ], Wang et al . used the argument principle of complex analysis to deduce two stability criteria for linear time-invariant fractional-order systems, which can determine system stability without utilizing characteristic roots. They also used Laplace transform and residue theorem to discuss the internal and external stability conditions of a linear time-invariant fractional-order system [ 22 ]. Pakzad put forward a practical analytical model to discuss the stability robustness of a class of linear time-invariant fractional-order systems with single and multiple commensurate delays of retarded type, against delay uncertainties [ 23 ].

All the above stability analyses are for time-invariant fractional-order systems. However, the above results are not widely used due to various reasons. For example, the eigenvalue criterion cannot be applied in time-varying fractional-order systems [ 24 ]. Therefore, this paper analyzes the stability of the time-varying fractional-order systems and presents a stability theorem for the system with the order 0< α <1. This theorem is a sufficient condition which can discriminate the stability of time-varying systems conveniently. Feedback controllers are designed for control and synchronization of the fractional-order Lü chaotic system.

The rest of the paper is organized as follows. Section 2 analyzes the development status and the stability of fractional-order systems. Section 3 presents a stability theorem for these systems with the order 0< α <1. Feedback controllers for fractional-order Lü chaotic system’s control and synchronization are designed on the basis of previous stability theorem in Section 4. Finally, the conclusion is drawn according to the present study in Section 5.

2. Development status of fractional-order system and stability

2.1 definition of fractional-order calculus.

Nowadays, many different definitions of fractional-order calculation were presented, in [ 25 ]. The most common definition, with αϵ(0,1), is shown as Eq 1 and was proposed by M. Caputo in 1967. Eq 1 is important for integral transformation because the initial value expressions generated in integral transformation are all in the form of integer-order derivatives, which can be effectively applied in practice.

Where x ( t ) is a function with an arbitrary integer order; the fractional order meets 0< α <1; t 0 I t α is a fractional-order integral with order α of function x ( t ) between [ t 0 , t ]; Γ (·); denotes the gamma function.

2.2 Development of stability analysis of fractional-order system

Theorem 1 When 0<α<1, x∊R n , A ∊R nxn , the fractional-order system t 0 D α t x(t) = A x(t), t≥t 0 is asymptotically stable if and only if all the characteristic values of matrix A satisfy |arg(eig( A ))|>απ/2. Furthermore, the system is stable if and only if all the characteristic values of matrix A satisfy |arg(eig( A ))|≥απ/2, which can be found in [ 21 ].

Theorem 1 is the existing stability criterion of a linear time-invariant fractional-order system with 0<α<1. This theorem is suitable only for a linear time-invariant fractional-order system, but it is often misused [ 26 ]. For time-invariant fractional-order nonlinear systems, if all the eigenvalues of the Jacobi matrix at equilibrium are stable, then the equilibrium is called stable equilibrium point. However, Theorem 1 is not suitable for time-varying fractional-order systems.

Definition 2 A continuous function β : [0, t )→[0,∞) is said to belong to class-k if it is strictly increasing and β (0) = 0.

3. Stability of time-varying fractional-order systems

3.1. fractional-order system stability analysis.

For the system (6), we present stability Theorem 3 after introducing Lemma 1 as follows:

As H(t) is a real symmetric matrix, whose eigenvalues are real numbers. Let λ min and λ max be the respective minimum and maximum eigenvalues of H(t) . We thus obtain the following asymptotic stability sufficiency with Theorem 3.

Theorem 3 The sufficient condition for asymptotic stability of the fractional-order system (6) with equilibrium point x e =0 is that the maximum eigenvalue of H(t) satisfies λ max < 0.

Considering the theorem’s condition λ max < 0, we can easily obtain the conclusion because V(x , t) satisfies the condition (1) of Theorem 2, and V α (x,t) satisfies the condition (2) of Theorem 2. Hence, according to Theorem 2, the time-varying fractional-order system (6) with equilibrium point x e is asymptotically stable.

3.2 Examples of fractional-order system stability analysis

The stability analysis of two typical systems is given to demonstrate the effectiveness of the proposed stability theory.

All the eigenvalues are negative, we can conclude that system (9) is stable from Theorem 3.

The eigenvalues of H ( t ) can be calculated: λ 1,2 = -2 b ±2a Taking a = 0.25, b = 0.5 (λ 1 = -0.5,λ 2 = -1.5),this system is stable according to Theorem 3. And if we take a = 1, b = 0.5(λ 1 = 1,λ 2 = -3), it is unable to determine the stability of this system from Theorem 3.

These examples show that Theorem 3 can discriminate the stability of time-varying fractional-order systems accurately. However, it is worth noting that this theorem is only a sufficient condition rather than a sufficient and necessary condition.

4. Control and synchronization of fractional-order Lü chaotic system

With the global boom of complex network research [ 27 ], chaotic systems as a part of complex networks are being widely applied [ 28 ]. The robust control and synchronization of chaotic systems have been gaining increasing attention. However, because of the lack of a stability analysis method for fractional-order systems, no systematic solution exists for the control and synchronization of a fractional-order chaotic system. With the use of the time-varying fractional-order stability theorem proposed in this paper, two controllers are designed for the fractional-order Lü chaotic system’s tracking control and synchronization.

4.1 Tracking control of fractional-order Lü chaotic system

By solving det ( λI − H ( t )) = 0, we obtain the eigenvalues of H(t) .Combining the root locus analysis, we determine that all the eigenvalues of H(t) are less than 0 if k <−29.7. According to Theorem 3, the controlled Lü system (13) is uniformly asymptotically stable.

Fig 1 . shows the fractional-order Lü system (13) controlled to the zero equilibrium point with k = -35.

• PPT PowerPoint slide
• PNG larger image
• TIFF original image

The motion curves of each state of the fractional-order Lü chaotic system when control action is added at t = 10s.

https://doi.org/10.1371/journal.pone.0194112.g001

The solid lines in Fig 1 . shows the motion curves of each state of the fractional-order Lü chaotic system when control action is added at t = 10s. Clearly, the system gradually converges to equilibrium point S 0 = (0,0,0) after the control action is added. The above design shows that we can easily obtain a feedback control parameter k to make the system stable using H(t) -based Theorem 3. Given the time-varying state x 1 contained in A(t) , obtaining a feedback control parameter k using A(t) -based Theorem 1 is difficult.

4.2 Synchronization of fractional-order Lü chaotic system

In accordance with the design process of the tracking controller in Section 4.1, the same feedback coefficient k can guarantee the stability of the synchronization systems. The synchronization results of the fractional-order Lü system when k = -35 are shown in Fig 2 .

The motion curves of each state of the fractional-order Lü chaotic driving system and response system when the control action is added at t = 10 s.

https://doi.org/10.1371/journal.pone.0194112.g002

The solid lines in Fig 2 show the motion curves of each state of the fractional-order Lü chaotic driving system and response system when the control action is added at t = 10 s. Clearly, the response curves tend to the driving curves after the control action is added. The error curves in sub-figures d)–e) of Fig 2 show the quickness and effectiveness of the method.

The above design shows that we can easily obtain a feedback control parameter k to make the system stable according to the proposed method. However, the time-varying state x 1 is contained in A(t) , so it is difficult to obtain a feedback control parameter k based on Theorem 1.

5. Conclusion

A sufficient stability theorem for time-varying fractional-order systems is proposed because the existing stability determination methods for fractional-order systems are complicated and difficult to apply. On the basis of the proposed theorem, a feedback controller for the fractional-order Lü chaotic system is designed for tracking control and synchronization. Simulation results demonstrate the effectiveness of the method.

Supporting information

https://doi.org/10.1371/journal.pone.0194112.s001

Acknowledgments

This work was supported by the National Key R&D Program of China (2016YFB0600701) and the Fundamental Fund for the Central Universities of China under Grant 2015MS66. The authors would like to give our heartfelt thanks to anonymous reviewers for the constructive suggestions in improving the paper.

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Design and implementation of fractional order PID controllers for process control systems

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• Edet, Emmanuel
• Strathclyde Thesis Copyright
• University of Strathclyde
• Doctoral (Postgraduate)
• Doctor of Philosophy (PhD)
• Department of Electronic and Electrical Engineering.
• New methods of designing Fractional Order PID (FOPID) controllers are developed for regulating product's purity in distillation columns. It is widely known that poorly tuned PID controllers lead to bad quality of products and accompanied reduction in profit margin in the process industry. In distillation columns, tight composition control of products with 98% purity level is not achievable with simple pressure controllers only due to sensitivity to disturbances. Therefore, FOPID controllers are proposed to solve these multivariable impurity problems. FOPID controllers have extra tuning parameters that can counteract effects of time delays in distillation columns if properly tuned. This property is exploited as a tool for improved performance. Original contributions of this thesis include the development of three new design methods for multivariable FOPID controllers and results are analysed using inverse maximum singular value of relevant sensitivity functions to assess robust stability. Several conventional PID controller design methods are also reviewed for the purpose of comparison. Thereafter, a decentralised FOPID control system is developed for multivariable systems based on plant's critical frequency point and results show improved performance over conventional PID controllers. The proposed critical frequency point method provides very easy-to-use tuning rules similar to Cohen-Coontuning rule for integer-order PID controllers. In addition, a new decentralised multivariable FOPID controller is also proposed based on Internal Model Control(IMC) method but settings are tuned using Biggest Log-magnitude Technique (BLT). This IMC-FOPID control design scheme overcomes the need for critical frequency point experiments. Another contribution of this thesis is the development of a novel discrete Fractional Order Predictive PI (FOPPI) control design scheme suitable for linear multivariable systems. Comparative study of these methods is presented.Simulation results prove that both top and bottoms products' purity of 98% are achievable with improved disturbance rejection when using the proposed FOPPI controller. In comparison, simpler FOPID control design schemes developed in continuous time are found to meet design objectives but at the expense of having a more conservative control action.
• Katebi, Reza, 1954-
• Doctoral thesis
• 10.48730/t7mn-2t58
• 9912681991802996

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In recent time, the application of fractional derivatives has become quite apparent in modeling mechanical and electrical properties of real materials. Fractional integrals and derivatives has found wide application in the control of dynamical systems, when the controlled system or/and the controller is described by a set of fractional order differential equations. In the present work a fractional order system has been represented by a higher integer order system, which is further approximated by second order plus time delay (SOPTD) model. The approximation to a SOPTD model is carried out by the minimization of the two norm of the actual and approximated system. Further, the effectiveness of a fractional order controller in meeting a set of frequency domain specifications is determined based on the frequency response of an integer order PID and a fractional order PID (FOPID) controller, designed for the approximated SOPTD model. The advent of fuzzy logic has led to greater flexibility in designing controllers for systems with time varying and nonlinear characteristics by exploiting the system observations in a linguistic manner. In this regard, a fractional order fuzzy PID controller has been developed based on the minimization different optimal control based integral performance indices. The indices have been minimized using genetic algorithms. Simulation results show that the fuzzy fractional order PID controller is able to outperform the classical PID, fuzzy PID and FOPID controllers.

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Modeling a Dynamic System Using Fractional Order Calculus

Fractional calculus is the integration and differentiation to an arbitrary or fractional order. The techniques of fractional calculus are not commonly taught in engineering curricula since physical laws are expressed in integer order notation. Dr. Richard Magin (2006) notes how engineers occasionally encounter dynamic systems in which the integer order methods do not properly model the physical characteristics and lead to numerous mathematical operations. In the following study, the application of fractional order calculus to approximate the angular position of the disk oscillating in a Newtonian fluid was experimentally validated. The proposed experimental study was conducted to model the nonlinear response of an oscillating system using fractional order calculus. The integer and fractional order mathematical models solved the differential equation of motion specific to the experiment. The experimental results were compared to the integer order and the fractional order analytical solutions. The fractional order mathematical model in this study approximated the nonlinear response of the designed system by using the Bagley and Torvik fractional derivative. The analytical results of the experiment indicate that either the integer or fractional order methods can be used to approximate the angular position of the disk oscillating in the homogeneous solution. The following research was in collaboration with Dr. Richard Mark French, Dr. Garcia Bravo, and Rajarshi Choudhuri, and the experimental design was derived from the previous experiments conducted in 2018.

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