Fractional-Order Adaptive Backstepping Control of a Noncommensurate Fractional-Order Ferroresonance System

Wang, Yan; Liu, Ling; Liu, Chongxin; Zhu, Ziwei; Sun, Zhenquan

doi:https://doi.org/10.1155/2018/8091757

Mathematical Problems in Engineering

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Mathematical Theories and Applications for Nonlinear Control Systems

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Research Article | Open Access

Volume 2018 | Article ID 8091757 | https://doi.org/10.1155/2018/8091757

Fractional-Order Adaptive Backstepping Control of a Noncommensurate Fractional-Order Ferroresonance System

Yan Wang,¹Ling Liu,¹Chongxin Liu,¹Ziwei Zhu,¹and Zhenquan Sun²

Academic Editor: S. Hassan Hosseinnia

Received30 Jul 2018

Revised03 Oct 2018

Accepted29 Oct 2018

Published07 Nov 2018

Abstract

In this paper, fractional calculus is applied to establish a novel fractional-order ferroresonance model with fractional-order magnetizing inductance and capacitance. Some basic dynamic behaviors of this fractional-order ferroresonance system are investigated. And then, considering noncommensurate orders of inductance and capacitance and unknown parameters in an actual ferroresonance system, this paper presents a novel fractional-order adaptive backstepping control strategy for a class of noncommensurate fractional-order systems with multiple unknown parameters. The virtual control laws and parameter update laws are designed in each step. Thereafter, a novel fractional-order adaptive controller is designed in terms of the fractional Lyapunov stability theorem. The proposed control strategy requires only one control input and can force the output of the chaotic system to track the reference signal asymptotically. Finally, the proposed method is applied to a noncommensurate fractional-order ferroresonance system with multiple unknown parameters. Numerical simulation confirms the effectiveness of the proposed method. In addition, the proposed control strategy also applies to commensurate fractional-order systems with unknown parameters.

1. Introduction

Fractional calculus is a generalization of ordinary integration and differentiation to arbitrary order [1]. In recent years, some scholars pointed out that there exist a large number of fractal phenomena in nature [2], which should not be modeled and analyzed by the classical calculus but fractional calculus. Two main outstanding advantages of fractional calculus have drawn great attention of scholars. The first advantage is that fractional calculus has unlimited memory and can take into account the previous responses up to the present time. Fractional calculus can provide clearer and more accurate description than the classical calculus methods without memory which is only a particular case. Furthermore, with the help of fractional-order calculus, we can obtain more accurate descriptions of the physical phenomena, such as electrical circuit [3, 4], power system [5, 6], signal processing [7], secure communication [8], bioengineering [9], and image encryption [10]. Second, the fractional-order can enhance the flexibility of parameters and reveal various unusual characteristics of the fractional-order system and fractional-order controllers which remain concealed in the integer-order method. It has been confirmed that the controller based on fractional calculus leads to better closed-loop performance by improving transient and steady-state responses than integer-order approaches [11, 12]. One of the most striking applications of fractional calculus is fractional-order controllers. In recent years, some excellent fractional-order controllers have been investigated, such as fractional-order PID control [13], fractional-order sliding mode control [5], fractional fuzzy control [14], and fractional-order backstepping control [15].

As a complex nonlinear phenomenon in a power system, ferroresonance can result in chaotic oscillation and push the power system to instability, which may cause overvoltage and overcurrent, voltage collapse, and even large-scale blackout [16–18]. At present, the trend of establishing a large-scale power grid, which may cause more uncertainties and disturbances in the power system, results in some ferroresonance incidents. Thus, it is necessary to suppress the chaotic oscillation caused by ferroresonance and many scholars have studied some excellent methods [19–21] to eliminate ferroresonance. In fact, the actual inductance and capacitance modeled by fractional calculus are more accurate than the classical integer method [3, 4]. In addition, the fractional-order model can provide clear description of real systems with unmodeled dynamics, uncertainties, and noise, which the integer-order model fails to do. Many real dynamical circuits, like RLC circuit, resonance circuits with ultracapacitors, fractional-order Chua’s circuit, and so on, have been investigated by the fractional-order model [22–26]. However, to the best of our knowledge, there are almost no reports about the fractional-order ferroresonance system, especially the more general noncommensurate case. Thus, it is necessary to investigate, study, and suppress ferroresonance by the novel and interesting fractional-order method.

Within the aforementioned fractional-order control method, the backstepping control strategy, which can simplify the controller, is an effective control technique for the systems with strict-feedback structure. Various excellent fractional-order backstepping strategies have been investigated and proposed in recent studies. In [15], a fractional-order backstepping controller to realize the stabilization of a fractional-order chaotic system was proposed via fractional Lyapunov functions. In [27], an adaptive backstepping controller to stabilize fractional-order Chua’s circuit was designed. Ref. [28] promoted the adaptive backstepping technique to the system which does not have strict-feedback structure and avoided singularity effectively in the proposed controller. In [14], an adaptive fuzzy backstepping controller was designed for the fractional-order system with unknown external disturbances. In [29], an adaptive backstepping controller was designed for the noncommensurate fractional-order system via the fractional-order Lyapunov indirect method. In [12], a finite time fractional-order adaptive backstepping controller was applied to robotic manipulators with uncertainties and external disturbances.

In fact, the orders of actual inductance and capacitance of the ferroresonance system are not all the same which means that the fractional-order ferroresonance system is a noncommensurate fractional-order system. Nevertheless, most of the aforementioned fractional-order backstepping methods only apply to the commensurate fractional-order systems. Though the fractional-order extension of the Lyapunov direct method proposed in [30] has been used in some present works, designing an excellent controller for noncommensurate fractional-order systems is still an open and challenging problem [31]. To the best of our knowledge, there are almost no reports about noncommensurate fractional-order backstepping control for noncommensurate fractional-order via the fractional-order Lyapunov direct method. Furthermore, considering the uncertainties of a real ferroresonance system in practice, investigation of fractional-order adaptive backstepping control for noncommensurate fractional-order systems with unknown parameters is necessary.

Motivated by the above considerations, we establish a novel noncommensurate fractional-order ferroresonance model. And then, to suppress undesirable chaotic behaviors of the proposed system, a novel fractional-order adaptive controller is designed for the noncommensurate fractional-order system with unknown parameters via the fractional-order Lyapunov direct method. Virtual control laws and parameter update laws are designed in terms of the backstepping procedures. The stability of the system under control is guaranteed by the fractional Lyapunov stability theorem. Finally, chaotic behaviors of the noncommensurate and commensurate fractional-order ferroresonance systems with multiple unknown parameters are eliminated, which illustrates the effectiveness and feasibility of the proposed method.

The rest of this paper is organized as follows. Section 2 contains preliminary knowledge throughout this paper. Dynamic analysis and the adaptive controller of the fractional-order ferroresonance system are presented in Section 3. Section 4 contains the simulation results. Finally, the conclusion is drawn in Section 5.

2. Preliminary

In the development of fractional calculus theory, many kinds of definitions were proposed. At present, the widely accepted fractional definitions are the Riemann-Liouville definition and Caputo definition. The initial value conditions of the Caputo fractional-order differential equations are clearer than the Riemann-Liouville definition. Thus, in this paper, the Caputo fractional calculus definition is employed.

Definition 1 (see [11]). The th-order Caputo fractional derivative of function is defined aswhere is the smallest integer number larger than or equal to , is the Gamma function, and represents the Caputo definition. In this paper, is abbreviated as when .

Properties 2 and 3 hold for both the Caputo derivative and Riemann-Liouville derivative.

Property 2 (see [2]). For , it has

Property 3 (see [2]). The additive index law:

Lemma 4 (see [32]). Let be a continuous and derivable function. Then, for any , it has

Lemma 5 (see [33]). Let be an equilibrium point of the following nonautonomous fractional-order system:where , the symbol can denote both the Caputo and Riemann-Liouville fractional operators, and satisfies the Lipschitz condition. Assume that there exists a Lyapunov function which satisfieswhere and there exists three class K functions , . Then system (5) is Mittag-Leffler stable, asymptotically.

3. Main Results

3.1. Dynamic Analysis

A classic two-order nonautonomous ferroresonance chaotic circuit model [17, 18] is shown in Figure 1. The circuit is derived by a sinusoidal voltage source , where is the grading capacitance of circuit breaker, is the bus-to-ground capacitance, and note as equivalent capacitance. is the system voltage and it has , is magnetizing inductance, and is the equivalent resistance corresponding to the system losses. For very high currents, the transformer coil is saturated and the flux-current characteristic of the transformer becomes highly nonlinear which is approximated by , where represents the flux of the nonlinear inductance and is the index of nonlinearity of the curve.

With the help of fractional calculus, losses of actual magnetizing inductance and capacitance can be described more accurately [3, 4]. Therefore, we try to investigate some dynamic behaviors of the ferroresonance system with noncommensurate fractional-order magnetizing inductance and capacitance. The system shown in Figure 1 can be described aswhere is the order of fractional-order magnetizing inductance and is the order of fractional-order capacitance . For the general case, it usually has and . Let , , and . Then, (8) can be rewritten aswhere , , , and . According to most actual 110kV transformer substations in China, set , , , and . The simulation in this paper is based on the Adams–Bashforth–Moulton method.

Remark 6. The fractional-order magnetizing inductance and fractional-order capacitance are fractance devices which can be approximately equivalent to the tree or chain structure of ideal inductance, capacitance, and resistance [34–36]. Due to the resistance of the equivalent tree or chain structure, impedance and capacitance of fractance devices contain both real and imaginary parts. The real parts will increase the loss of the fractional-order system. Furthermore, the fractance devices will result in more complicated dynamic behaviors than the integer-order model.
To investigate dynamic behaviors with different orders of the proposed system, set and . As the bifurcation diagram with varying from 0 to 10 shown in Figure 2, the periodic and chaotic states appear alternately. Phase portraits and time series of states of shown in Figures 3–5 indicate that the noncommensurate fractional-order ferroresonance system exhibits chaotic oscillation. For the integer-order case , the bifurcation diagram with varying from 0 to 10 is shown in Figure 6 which indicates that the integer-order ferroresonance system exhibits chaotic oscillation with . As the phase portraits and time series of states of shown in Figures 7–9, the system also exhibits chaotic oscillation. Although there exists some difference of bifurcation behaviors between Figures 2 and 6, serious overvoltage occurs in both the integer-order and noncommensurate fractional-order case.

3.2. Controller Design

In an actual ferroresonance system, it is difficult to obtain accurate values of bus-to-ground capacitance and parameters and which results in that parameters , and are actually unknown parameters. To suppress undesirable chaotic behaviors in the power system and design an effective controller for such noncommensurate fractional-order systems with multiple unknown parameters, let us consider the following strict-feedback noncommensurate fractional-order system with unknown parameters:where system orders are not all the same. is the state vector, is controller input, and is the output of system (10). is the vector of unknown constant parameters. is a known constant. and are known smooth nonlinear or linear functions and they are abbreviated as and in this paper.

Obviously, noncommensurate fractional-order ferroresonance system (8) is a particular case of the strict-feedback noncommensurate fractional-order system with unknown parameters (10). To design an excellent adaptive backstepping controller for system (10), in each step, a virtual controller and parameter update law are designed systematically until the last equation which contains the controller.

The result is stated by the following theorem.

Theorem 7. For system (10), let the following designs:
The tracking error variables:The virtual control laws:The parameter update laws:The adaptive control law:Then the tracking error is stable asymptotically and globally aswhere is the smallest order of . is a smooth reference signal and its th-order derivatives are bounded and continuous. is a positive constant and , .

Proof. Step 1. Find the smallest order of noncommensurate fractional-order system (10); let , . is the adaptive parameter tracking error and is the first virtual controller. The first subsystem is given asChoose the Lyapunov function candidate asNote and the K-class functions and are selected as , . Thus, (17) satisfies condition (6). And then, take the order derivative of Lyapunov function (17) and use Lemma 4; we obtainThen choose the first virtual control law aswhich leads to Choose the first parameter update law asSubstitute (21) into (20), which leads toStep 2. For the second subsystem given aschoose the second Lyapunov function candidate which is similar to (17) asSimilarly, take the order derivative of Lyapunov function (24) and use Lemma 4, which leads toChoose the second virtual control law asSubstituting (26) into (25), we obtain Choose the second parameter update law asSubstituting (28) into (27), we obtainStep . Consider the th subsystem given asSimilarly, select the th Lyapunov function candidate asTake the order derivative of Lyapunov function (31) and use Lemma 4, which leads toChoose the th virtual control law asSubstituting (33) into (32), we obtainIn a similar way, the th parameter update law is chosen asSubstituting (35) into (34), we obtainStep n. Finally, the overall Lyapunov function for system (10) is chosen asTake the order derivative of Lyapunov function (37) and use Lemma 4, which leads toIf the parameter update law and adaptive control law are designed as (13) and (14), the negative terms in (38) can be left, which leads toTherefore, according to Lemma 5, all the states of system (10) are globally and asymptotically stable, and tracking errors will converge to zero asymptotically under the proposed control scheme. The proof is completed.

Remark 8. The unsmooth Lyapunov function is selected to analyze the stability of fractional-order systems in recent literature [5, 12, 37]. In this paper, the unsmooth Lyapunov function candidate is selected as . The term is used to solve the problem brought by the noncommensurate orders, and the term can simplify the forms of parameter update laws effectively.

Remark 9. When system orders satisfy , system (10) becomes a strict-feedback commensurate fractional-order system with unknown parameters. It can be described asFor system (40), according to Property 2, adaptive control law (14), and parameter update laws, (13) can be rewritten asFor system (40), in each step, the Lyapunov function candidate is selected asSimilar to the noncommensurate case, consider system (40) with adaptive control law (41) and parameter update law (42), and take the order derivative of the overall Lyapunov function . Finally, it concludes asFor the commensurate fractional-order system with unknown parameters, according to Lemma 5, all the states are stable globally and tracking errors will converge to zero asymptotically under the proposed control method.

Remark 10. To illustrate the advantages of the proposed control method, the main contributions of our work are presented here as follows:
More common conditions for the strict-feedback system are considered, including the noncommensurate orders, nonautonomous system, and multiple unknown parameters
A novel Lyapunov function candidate for the noncommensurate fractional-order system with unknown parameters is presented and the proposed Lyapunov function also applies to the commensurate fractional-order system
Compared with existing methods, the proposed adaptive control strategy requires only one controller input which reduces the complexity and eases the implementation

4. Simulation Results

In this section, the proposed control strategy is applied to both the noncommensurate and commensurate fractional-order ferroresonance system with unknown parameters. The system is described aswhere , and are unknown parameters and is the controller. Applying the control strategy proposed in Section 3, adaptive control law is selected asThe parameter update laws are selected aswhere is the smallest order between and , is the reference signal, is the virtual control law, and and are the tracking errors. The constants are selected as , , and . The initial condition is selected as. Case 1: Tracking performance of the noncommensurate fractional-order ferroresonance system of and under control is shown in Figures 10 and 11. Case 2: Tracking performance of the commensurate fractional-order ferroresonance system of under control is shown in Figures 12 and 13.

From Figures 10–13, it can be seen that the state can track the reference signal within 2 s. Under the proposed control method, the system voltage and flux are restored to the normal level gradually. In addition, the proposed control method applies to both the noncommensurate case and commensurate fractional-order ferroresonance system with multiple unknown parameters by just one control input.

5. Conclusions

In this paper, fractional calculus is applied to investigate the dynamic behaviors of a ferroresonance system with noncommensurate fractional-order magnetizing inductance and capacitance. The simulations illustrate that chaotic oscillation could occur in both the integer-order and noncommensurate fractional-order case. Considering the unknown parameters of a real system and the different orders of actual magnetizing inductance and capacitance, a novel fractional-order adaptive backstepping control method is proposed to suppress chaotic behaviors of noncommensurate fractional-order strict-feedback chaotic systems with unknown parameters via the fractional-order Lyapunov direct method. Parameter update laws and virtual control laws are designed in each step. And then, a novel adaptive backstepping controller is designed to force the system output to track the reference signal. Asymptotic stability of the fractional-order chaotic system under control is guaranteed by the fractional Lyapunov stability theorem. Compared with existing methods, the proposed control scheme not only applies to the noncommensurate case, but also requires just one control input which can simplify implementation in practice. Finally, the proposed control strategy is applied to the noncommensurate fractional-order ferroresonance system with unknown parameters. Simulation results demonstrate the effectiveness of the proposed method. In addition, it is worth noting that the proposed method also applies to the commensurate fractional-order ferroresonance system with unknown parameters.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

There are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This project was supported by the National Natural Science Foundation of China (Grant No. 51877162) and the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (Grant No. 51521065).

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Copyright © 2018 Yan Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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