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

Authors: Yan Wang; Ling Liu; Chongxin Liu; Ziwei Zhu; Zhenquan Sun
Journal: Mathematical Problems in Engineering
 (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.


---

## Body

## 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 α 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. Section2 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]).
Theαth-order Caputo fractional derivative of function f(t) is defined as(1)
​
t
0
C
D
t
α
f
t
=
1
Γ
m
-
α
∫
t
0
t
f
m
τ
t
-
τ
α
-
m
+
1
d
τ
,
m
-
1
<
α
<
m
d
m
f
t
d
t
m
,
α
=
m
,where m is the smallest integer number larger than or equal to α, Γ(·) is the Gamma function, and C represents the Caputo definition. In this paper, ​t0CDtα is abbreviated as Dα when t0=0.Properties2 and 3 hold for both the Caputo derivative and Riemann-Liouville derivative.Property 2 (see [2]).
Forα=0, it has (2)​0Dt0ft=ft.Property 3 (see [2]).
The additive index law:(3)​0Dtα​0Dtβft=​0Dtβ​0Dtαft=​0Dtα+βft.Lemma 4 (see [32]).
Letx(t)∈R be a continuous and derivable function. Then, for any t≥t0, it has(4)12​t0CDtαx2t≤xt​t0CDtαxt,∀α∈0,1.Lemma 5 (see [33]).
Letx=0 be an equilibrium point of the following nonautonomous fractional-order system:(5)​0Dtαx=ft,x,where α∈(0,1], the symbol D can denote both the Caputo and Riemann-Liouville fractional operators, and f(t) satisfies the Lipschitz condition. Assume that there exists a Lyapunov function V(t,x(t)) which satisfies(6)α1x≤Vt,xt≤α2x(7)DβVt,xt≤-α3x,where β∈(0,1) and there exists three class K functions αi, i=1,2,3. 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 ueq=Cg/Cb+CguSY, where Cb is the grading capacitance of circuit breaker, Cg is the bus-to-ground capacitance, and note C=Cb+Cg as equivalent capacitance. uSY is the system voltage and it has uSY=Umsin⁡ωt, L is magnetizing inductance, and R is the equivalent resistance corresponding to the system losses. For very high currents, the transformer coil is saturated and the flux-current characteristic ψ-i of the transformer becomes highly nonlinear which is approximated by i=aψ+bψn, where ψ represents the flux of the nonlinear inductance and n is the index of nonlinearity of the curve.Figure 1
The simplified ferroresonance system circuit model.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 as(8)dq1ψdtq1=uRdq2uRdtq2=ωUmCbCcos⁡ωt-uRRC-aψ+bψnC,where q1 is the order of fractional-order magnetizing inductance L and q2 is the order of fractional-order capacitance C. For the general case, it usually has q1≠q2 and q1,q2∈(0,1). Let x1=ψ, x2=uR/ω, and τ=ωt. Then, (8) can be rewritten as(9)dq1x1dτq1=x2dq2x2dτq2=qcos⁡τ-px2-p1x-p2xn,where q=UmCb/ωC, p=1/ωCR, p1=a/ω2C, and p2=b/ω2C. According to most actual 110kV transformer substations in China, set p=1.155×10-2, p1=1.94×10-4, p2=4.99×10-4, and n=11. The simulation in this paper is based on the Adams–Bashforth–Moulton method.Remark 6.
The fractional-order magnetizing inductanceL and fractional-order capacitance C 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, setq1=0.99 and q2=0.98. As the bifurcation diagram with q varying from 0 to 10 shown in Figure 2, the periodic and chaotic states appear alternately. Phase portraits and time series of states of q=5.85 shown in Figures 3–5 indicate that the noncommensurate fractional-order ferroresonance system exhibits chaotic oscillation. For the integer-order case q1=q2=1, the bifurcation diagram with q varying from 0 to 10 is shown in Figure 6 which indicates that the integer-order ferroresonance system exhibits chaotic oscillation with q>0.841. As the phase portraits and time series of states of q=5.85 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.Figure 2
Bifurcation diagram withq.Figure 3
The phase portraits.Figure 4
Time series withx1.Figure 5
Time series withx2.Figure 6
Bifurcation diagram withq.Figure 7
The phase portraits.Figure 8
Time series withx1.Figure 9
Time series withx2.


### 3.2. Controller Design

In an actual ferroresonance system, it is difficult to obtain accurate values of bus-to-ground capacitanceCg and parameters a and b which results in that parameters q,p,p1, and p2 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:(10)Dαixi=cixi+1+θiTFix1⋯xi,t+fix1⋯xi,tDαnxn=cnu+θnTFnx1⋯xn,t+fnx1⋯xn,ty=x1,i=1,⋯,n-1,where system orders αi∈(0,1) are not all the same. x=[x1,⋯,xn]T is the state vector, u is controller input, and y is the output of system (10). θi∈Rq(i=1,⋯,n-1) is the vector of unknown constant parameters. ci∈R(i=1,⋯,n) is a known constant. Fi(x1⋯xi,t) and fi(x1⋯xi,t) are known smooth nonlinear or linear functions and they are abbreviated as Fi and fi 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:(11)ei=ri-xi,i=1,2,⋯,n.The virtual control laws:(12)r2=1c1Dα1r1-θ^1TF1-f1+k1Dα1-αξe1ri+1=1ciDαiri-θ^iTFi-f1+kiDαi-αξei+ci-1eisignDαi-1-αξei-1signDαi-αξeii=2,⋯,n-1.The parameter update laws:(13)Dαξθ^iT=-Λ·FisignDαi-αξei,i=1,⋯,n.The adaptive control law:(14)u=1cnDαnrn-θ^nTFn-fn+knDαn-αξen+cn-1ensignDαn-1-αξen-1signDαn-αξen.Then the tracking error is stable asymptotically and globally as(15)limt→∞y-r1=0,where αξ is the smallest order of αi. r1 is a smooth reference signal and its (α1,⋯,∑i=1nαi)th-order derivatives are bounded and continuous. ki is a positive constant and Λ=diag(k,k,⋯,k), k>0.Proof.
Step 1. Find the smallest order αξ of noncommensurate fractional-order system (10); let e1=r1-x1, e2=r2-x2. θ~1=θ^1-θ1 is the adaptive parameter tracking error and r2 is the first virtual controller. The first subsystem is given as(16)Dα1x1=c1r2-e2+θ1TF1+f1.Choose the Lyapunov function candidate as(17)V1=Dα1-αξe1+12θ~1TΛ-1θ~1.Note zT=[Dα1-αξe1,θ~1T] and the K-class functions α1 and α2 are selected as α1(z)=min(z,1/2zTΛ-1z), α2(z)=z+1/2zTΛ-1z. Thus, (17) satisfies condition (6). And then, take the αξ order derivative of Lyapunov function (17) and use Lemma 4; we obtain(18)DαξV1=Dα1e1·signDα1-αξe1+Dαξ12θ~1TΛ-1θ~1≤Dα1r1-x1signDα1-αξe1+θ~1TΛ-1Dαξθ^1≤Dα1r1+c1e2-c1r2-θ1TF1-f1×signDα1-αξe1+θ~1TΛ-1Dαξθ^1.Then choose the first virtual control law r2 as(19)r2=1c1Dα1r1-θ^1TF1-f1+k1Dα1-αξe1which leads to (20)DαξV1≤c1e2+θ~1TF1-k1Dα1-αξe1×signDα1-αξe1+θ~1TΛ-1Dαξθ^1≤-k1Dα1-αξe1+c1e2+θ~1TF1×signDα1-αξe1+θ~1TΛ-1Dαξθ^1.Choose the first parameter update law θ^1 as(21)Dαξθ^1=-Λ·F1signDα1-αξe1.Substitute (21) into (20), which leads to(22)DαξV1≤-k1Dα1-αξe1+c1e2signDα1-αξe1.Step 2. For the second subsystem given as(23)Dα2x2=c2r3-e3+θ2TF2+f2,choose the second Lyapunov function candidate which is similar to (17) as(24)V2=V1+Dα2-αξe2+12θ~2TΛ-1θ~2.Similarly, take the αξ order derivative of Lyapunov function (24) and use Lemma 4, which leads to(25)DαξV2=DαξV1+Dα2e2·signDα2-αξe2+Dαξ12θ~2TΛ-1θ~2≤-k1Dα1-αξe1+c1e2signDα1-αξe1+Dα2r2-x2signDα2-αξe2+θ~2TΛ-1Dαξθ^2≤-k1Dα1-αξe1+c1e2signDα1-αξe1+Dα2r2+c2e3-c2r3-θ2TF2-f2×signDα2-αξe2+θ~2TΛ-1Dαξθ^2.Choose the second virtual control law r3 as(26)r3=1c2Dα2r2-θ^2TF2-f2+k2Dα2-αξe2+c1e2signDα1-αξe1signDα2-αξe2.Substituting (26) into (25), we obtain (27)DαξV2≤-k1Dα1-αξe1+c1e2signDα1-αξe1+c2e3+θ~2TF2-k2Dα2-αξe2-c1e2signDα1-αξe1signDα2-αξe2×signDα2-αξe2+θ~2TΛ-1Dαξθ^2≤-k1Dα1-αξe1-k2Dα2-αξe2+c2e3+θ~2TF2signDα2-αξe2+θ~2TΛ-1Dαξθ^2.Choose the second parameter update law θ^2 as(28)Dαξθ^2T=-Λ·F2signDα2-αξe2.Substituting (28) into (27), we obtain(29)DαξV2≤-k1Dα1-αξe1-k2Dα2-αξe2+c2e3signDα2-αξe2.
Step i = ( 2 , ⋯ , n - 1 ). Consider the ith subsystem given as(30)Dαixi=ciri+1-ei+1+θiTFi+fi.Similarly, select the ith Lyapunov function candidate as(31)Vi=Vi-1+Dαi-αξei+12θ~iTΛ-1θ~i.Take the αξ order derivative of Lyapunov function (31) and use Lemma 4, which leads to(32)DαξVi=DαξVi-1+DαieisignDαi-αξei+Dαξ12θ~iTΛ-1θ~i≤-∑j=1i-1kjDαj-αξej+ci-1eisignDαi-1-αξei-1+Dαiri-xisignDαi-αξei+θ~iTΛ-1Dαξθ^i≤-∑j=1i-1kjDαj-αξej+ci-1eisignDαi-1-αξei-1+Dαξri+ciei+1-ciri+1-θiTFi-fi×signDαi-αξei+θ~iTΛ-1Dαξθ^i.Choose the ith virtual control law ri+1 as(33)ri+1=1ciDαiri-θ^iTFi-fi+kiDαi-αξei+ci-1eisignDαi-1-αξei-1signDαi-αξei.Substituting (33) into (32), we obtain(34)DαξVi≤-∑j=1ikjDαj-αξej+ciei+1+θ~iTFisignDαi-αξei+θ~iTΛ-1Dαξθ^i.In a similar way, the ith parameter update law is chosen as(35)Dαξθ^iT=-Λ·FisignDαi-αξei,i=2,⋯,n-1.Substituting (35) into (34), we obtain(36)DαξVi≤-∑j=1ikjDαj-αξej+ciei+1signDαi-αξeii=2,⋯,n-1.
Step n. Finally, the overall Lyapunov function for system (10) is chosen as(37)Vn=Vn-1+Dαn-αξen+12θ~nTΛ-1θ~n.Take the αξ order derivative of Lyapunov function (37) and use Lemma 4, which leads to(38)DαξVn=DαξVn-1+DαnensignDαn-αξen+Dαξ12θ~nTΛ-1θ~n≤-∑j=1n-1kjDαj-αξej+cn-1ensignDαn-1-αξen-1+Dαnrn-xnsignDαn-αξen+θ~nTΛ-1Dαξθ^n≤-∑j=1n-1kjDαj-αξej+cn-1ensignDαn-1-αξen-1+Dαξrn-cnu-θnTFn-fn×signDαn-αξen+θ~nTΛ-1Dαξθ^n.If the parameter update law and adaptive control law are designed as (13) and (14), the negative terms in (38) can be left, which leads to(39)DαξVn≤-∑j=1nkjDαj-αξej.Therefore, according to Lemma 5, all the states of system (10) are globally and asymptotically stable, and tracking errors ei(i=1,⋯,n) will converge to zero asymptotically under the proposed control scheme. The proof is completed.Remark 8.
The unsmooth Lyapunov functionV=x1 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 Vi=Vi-1+Dαi-αξei+1/2θ~iTΛ-1θ~i. The term Dαi-αξei is used to solve the problem brought by the noncommensurate orders, and the term 1/2θ~iTΛ-1θ~i can simplify the forms of parameter update laws effectively.Remark 9.
When system orders satisfyαi=α(i=1,⋯,n), system (10) becomes a strict-feedback commensurate fractional-order system with unknown parameters. It can be described as(40)Dαxi=cixi+1+θiTFix1⋯xi,t+fix1⋯xi,t.Dαxn=cnu+θnTFnx1⋯xn,t+fnx1⋯xn,ty=x1,i=1,⋯,n-1For system (40), according to Property 2, adaptive control law (14), and parameter update laws, (13) can be rewritten as(41)u=1cnDαrn-θ^nTFn-fn+knen+cn-1ensignen-1signen(42)Dαθ^iT=-Λ·Fisignei,i=1,⋯,n.For system (40), in each step, the Lyapunov function candidate is selected as(43)Vi=Vi-1+ei+12θ~iTΛ-1θ~i,i=2,⋯,n.Similar to the noncommensurate case, consider system (40) with adaptive control law (41) and parameter update law (42), and take the α order derivative of the overall Lyapunov function Vn. Finally, it concludes as(44)DαVn≤-∑j=1nkjej.For the commensurate fractional-order system with unknown parameters, according to Lemma 5, all the states are stable globally and tracking errors ei(i=1,⋯,n) 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:
( 1 ) More common conditions for the strict-feedback system are considered, including the noncommensurate orders, nonautonomous system, and multiple unknown parameters
( 2 ) A novel Lyapunov function candidate Vi=Vi-1+Dαi-αξei+1/2θ~iTΛ-1θ~i for the noncommensurate fractional-order system with unknown parameters is presented and the proposed Lyapunov function also applies to the commensurate fractional-order system
( 3 ) Compared with existing methods, the proposed adaptive control strategy requires only one controller input which reduces the complexity and eases the implementation


## 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 ueq=Cg/Cb+CguSY, where Cb is the grading capacitance of circuit breaker, Cg is the bus-to-ground capacitance, and note C=Cb+Cg as equivalent capacitance. uSY is the system voltage and it has uSY=Umsin⁡ωt, L is magnetizing inductance, and R is the equivalent resistance corresponding to the system losses. For very high currents, the transformer coil is saturated and the flux-current characteristic ψ-i of the transformer becomes highly nonlinear which is approximated by i=aψ+bψn, where ψ represents the flux of the nonlinear inductance and n is the index of nonlinearity of the curve.Figure 1
The simplified ferroresonance system circuit model.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 as(8)dq1ψdtq1=uRdq2uRdtq2=ωUmCbCcos⁡ωt-uRRC-aψ+bψnC,where q1 is the order of fractional-order magnetizing inductance L and q2 is the order of fractional-order capacitance C. For the general case, it usually has q1≠q2 and q1,q2∈(0,1). Let x1=ψ, x2=uR/ω, and τ=ωt. Then, (8) can be rewritten as(9)dq1x1dτq1=x2dq2x2dτq2=qcos⁡τ-px2-p1x-p2xn,where q=UmCb/ωC, p=1/ωCR, p1=a/ω2C, and p2=b/ω2C. According to most actual 110kV transformer substations in China, set p=1.155×10-2, p1=1.94×10-4, p2=4.99×10-4, and n=11. The simulation in this paper is based on the Adams–Bashforth–Moulton method.Remark 6.
The fractional-order magnetizing inductanceL and fractional-order capacitance C 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, setq1=0.99 and q2=0.98. As the bifurcation diagram with q varying from 0 to 10 shown in Figure 2, the periodic and chaotic states appear alternately. Phase portraits and time series of states of q=5.85 shown in Figures 3–5 indicate that the noncommensurate fractional-order ferroresonance system exhibits chaotic oscillation. For the integer-order case q1=q2=1, the bifurcation diagram with q varying from 0 to 10 is shown in Figure 6 which indicates that the integer-order ferroresonance system exhibits chaotic oscillation with q>0.841. As the phase portraits and time series of states of q=5.85 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.Figure 2
Bifurcation diagram withq.Figure 3
The phase portraits.Figure 4
Time series withx1.Figure 5
Time series withx2.Figure 6
Bifurcation diagram withq.Figure 7
The phase portraits.Figure 8
Time series withx1.Figure 9
Time series withx2.


## 3.2. Controller Design

In an actual ferroresonance system, it is difficult to obtain accurate values of bus-to-ground capacitanceCg and parameters a and b which results in that parameters q,p,p1, and p2 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:(10)Dαixi=cixi+1+θiTFix1⋯xi,t+fix1⋯xi,tDαnxn=cnu+θnTFnx1⋯xn,t+fnx1⋯xn,ty=x1,i=1,⋯,n-1,where system orders αi∈(0,1) are not all the same. x=[x1,⋯,xn]T is the state vector, u is controller input, and y is the output of system (10). θi∈Rq(i=1,⋯,n-1) is the vector of unknown constant parameters. ci∈R(i=1,⋯,n) is a known constant. Fi(x1⋯xi,t) and fi(x1⋯xi,t) are known smooth nonlinear or linear functions and they are abbreviated as Fi and fi 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:(11)ei=ri-xi,i=1,2,⋯,n.The virtual control laws:(12)r2=1c1Dα1r1-θ^1TF1-f1+k1Dα1-αξe1ri+1=1ciDαiri-θ^iTFi-f1+kiDαi-αξei+ci-1eisignDαi-1-αξei-1signDαi-αξeii=2,⋯,n-1.The parameter update laws:(13)Dαξθ^iT=-Λ·FisignDαi-αξei,i=1,⋯,n.The adaptive control law:(14)u=1cnDαnrn-θ^nTFn-fn+knDαn-αξen+cn-1ensignDαn-1-αξen-1signDαn-αξen.Then the tracking error is stable asymptotically and globally as(15)limt→∞y-r1=0,where αξ is the smallest order of αi. r1 is a smooth reference signal and its (α1,⋯,∑i=1nαi)th-order derivatives are bounded and continuous. ki is a positive constant and Λ=diag(k,k,⋯,k), k>0.Proof.
Step 1. Find the smallest order αξ of noncommensurate fractional-order system (10); let e1=r1-x1, e2=r2-x2. θ~1=θ^1-θ1 is the adaptive parameter tracking error and r2 is the first virtual controller. The first subsystem is given as(16)Dα1x1=c1r2-e2+θ1TF1+f1.Choose the Lyapunov function candidate as(17)V1=Dα1-αξe1+12θ~1TΛ-1θ~1.Note zT=[Dα1-αξe1,θ~1T] and the K-class functions α1 and α2 are selected as α1(z)=min(z,1/2zTΛ-1z), α2(z)=z+1/2zTΛ-1z. Thus, (17) satisfies condition (6). And then, take the αξ order derivative of Lyapunov function (17) and use Lemma 4; we obtain(18)DαξV1=Dα1e1·signDα1-αξe1+Dαξ12θ~1TΛ-1θ~1≤Dα1r1-x1signDα1-αξe1+θ~1TΛ-1Dαξθ^1≤Dα1r1+c1e2-c1r2-θ1TF1-f1×signDα1-αξe1+θ~1TΛ-1Dαξθ^1.Then choose the first virtual control law r2 as(19)r2=1c1Dα1r1-θ^1TF1-f1+k1Dα1-αξe1which leads to (20)DαξV1≤c1e2+θ~1TF1-k1Dα1-αξe1×signDα1-αξe1+θ~1TΛ-1Dαξθ^1≤-k1Dα1-αξe1+c1e2+θ~1TF1×signDα1-αξe1+θ~1TΛ-1Dαξθ^1.Choose the first parameter update law θ^1 as(21)Dαξθ^1=-Λ·F1signDα1-αξe1.Substitute (21) into (20), which leads to(22)DαξV1≤-k1Dα1-αξe1+c1e2signDα1-αξe1.Step 2. For the second subsystem given as(23)Dα2x2=c2r3-e3+θ2TF2+f2,choose the second Lyapunov function candidate which is similar to (17) as(24)V2=V1+Dα2-αξe2+12θ~2TΛ-1θ~2.Similarly, take the αξ order derivative of Lyapunov function (24) and use Lemma 4, which leads to(25)DαξV2=DαξV1+Dα2e2·signDα2-αξe2+Dαξ12θ~2TΛ-1θ~2≤-k1Dα1-αξe1+c1e2signDα1-αξe1+Dα2r2-x2signDα2-αξe2+θ~2TΛ-1Dαξθ^2≤-k1Dα1-αξe1+c1e2signDα1-αξe1+Dα2r2+c2e3-c2r3-θ2TF2-f2×signDα2-αξe2+θ~2TΛ-1Dαξθ^2.Choose the second virtual control law r3 as(26)r3=1c2Dα2r2-θ^2TF2-f2+k2Dα2-αξe2+c1e2signDα1-αξe1signDα2-αξe2.Substituting (26) into (25), we obtain (27)DαξV2≤-k1Dα1-αξe1+c1e2signDα1-αξe1+c2e3+θ~2TF2-k2Dα2-αξe2-c1e2signDα1-αξe1signDα2-αξe2×signDα2-αξe2+θ~2TΛ-1Dαξθ^2≤-k1Dα1-αξe1-k2Dα2-αξe2+c2e3+θ~2TF2signDα2-αξe2+θ~2TΛ-1Dαξθ^2.Choose the second parameter update law θ^2 as(28)Dαξθ^2T=-Λ·F2signDα2-αξe2.Substituting (28) into (27), we obtain(29)DαξV2≤-k1Dα1-αξe1-k2Dα2-αξe2+c2e3signDα2-αξe2.
Step i = ( 2 , ⋯ , n - 1 ). Consider the ith subsystem given as(30)Dαixi=ciri+1-ei+1+θiTFi+fi.Similarly, select the ith Lyapunov function candidate as(31)Vi=Vi-1+Dαi-αξei+12θ~iTΛ-1θ~i.Take the αξ order derivative of Lyapunov function (31) and use Lemma 4, which leads to(32)DαξVi=DαξVi-1+DαieisignDαi-αξei+Dαξ12θ~iTΛ-1θ~i≤-∑j=1i-1kjDαj-αξej+ci-1eisignDαi-1-αξei-1+Dαiri-xisignDαi-αξei+θ~iTΛ-1Dαξθ^i≤-∑j=1i-1kjDαj-αξej+ci-1eisignDαi-1-αξei-1+Dαξri+ciei+1-ciri+1-θiTFi-fi×signDαi-αξei+θ~iTΛ-1Dαξθ^i.Choose the ith virtual control law ri+1 as(33)ri+1=1ciDαiri-θ^iTFi-fi+kiDαi-αξei+ci-1eisignDαi-1-αξei-1signDαi-αξei.Substituting (33) into (32), we obtain(34)DαξVi≤-∑j=1ikjDαj-αξej+ciei+1+θ~iTFisignDαi-αξei+θ~iTΛ-1Dαξθ^i.In a similar way, the ith parameter update law is chosen as(35)Dαξθ^iT=-Λ·FisignDαi-αξei,i=2,⋯,n-1.Substituting (35) into (34), we obtain(36)DαξVi≤-∑j=1ikjDαj-αξej+ciei+1signDαi-αξeii=2,⋯,n-1.
Step n. Finally, the overall Lyapunov function for system (10) is chosen as(37)Vn=Vn-1+Dαn-αξen+12θ~nTΛ-1θ~n.Take the αξ order derivative of Lyapunov function (37) and use Lemma 4, which leads to(38)DαξVn=DαξVn-1+DαnensignDαn-αξen+Dαξ12θ~nTΛ-1θ~n≤-∑j=1n-1kjDαj-αξej+cn-1ensignDαn-1-αξen-1+Dαnrn-xnsignDαn-αξen+θ~nTΛ-1Dαξθ^n≤-∑j=1n-1kjDαj-αξej+cn-1ensignDαn-1-αξen-1+Dαξrn-cnu-θnTFn-fn×signDαn-αξen+θ~nTΛ-1Dαξθ^n.If the parameter update law and adaptive control law are designed as (13) and (14), the negative terms in (38) can be left, which leads to(39)DαξVn≤-∑j=1nkjDαj-αξej.Therefore, according to Lemma 5, all the states of system (10) are globally and asymptotically stable, and tracking errors ei(i=1,⋯,n) will converge to zero asymptotically under the proposed control scheme. The proof is completed.Remark 8.
The unsmooth Lyapunov functionV=x1 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 Vi=Vi-1+Dαi-αξei+1/2θ~iTΛ-1θ~i. The term Dαi-αξei is used to solve the problem brought by the noncommensurate orders, and the term 1/2θ~iTΛ-1θ~i can simplify the forms of parameter update laws effectively.Remark 9.
When system orders satisfyαi=α(i=1,⋯,n), system (10) becomes a strict-feedback commensurate fractional-order system with unknown parameters. It can be described as(40)Dαxi=cixi+1+θiTFix1⋯xi,t+fix1⋯xi,t.Dαxn=cnu+θnTFnx1⋯xn,t+fnx1⋯xn,ty=x1,i=1,⋯,n-1For system (40), according to Property 2, adaptive control law (14), and parameter update laws, (13) can be rewritten as(41)u=1cnDαrn-θ^nTFn-fn+knen+cn-1ensignen-1signen(42)Dαθ^iT=-Λ·Fisignei,i=1,⋯,n.For system (40), in each step, the Lyapunov function candidate is selected as(43)Vi=Vi-1+ei+12θ~iTΛ-1θ~i,i=2,⋯,n.Similar to the noncommensurate case, consider system (40) with adaptive control law (41) and parameter update law (42), and take the α order derivative of the overall Lyapunov function Vn. Finally, it concludes as(44)DαVn≤-∑j=1nkjej.For the commensurate fractional-order system with unknown parameters, according to Lemma 5, all the states are stable globally and tracking errors ei(i=1,⋯,n) 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:
( 1 ) More common conditions for the strict-feedback system are considered, including the noncommensurate orders, nonautonomous system, and multiple unknown parameters
( 2 ) A novel Lyapunov function candidate Vi=Vi-1+Dαi-αξei+1/2θ~iTΛ-1θ~i for the noncommensurate fractional-order system with unknown parameters is presented and the proposed Lyapunov function also applies to the commensurate fractional-order system
( 3 ) 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 as(45)dq1x1dτq1=x2dq2x2dτq2=qcos⁡τ-px2-p1x-p2x11+u,where p,p1,p2, and q are unknown parameters and u is the controller. Applying the control strategy proposed in Section 3, adaptive control law is selected as(46)u=Dq2r2+k2Dq2-δe2+e2signDq1-δe1signDq2-δe2-p^x2+p^1x1+p^2x111+q^cos⁡τ.The parameter update laws are selected as(47)Dδp^=-kx2signDq2-δe2Dδp^1=-kx2signDq2-δe2Dδp^2=-kx111signDq2-δe2Dδq^=-kcos⁡τsignDq2-δe2,where δ is the smallest order between q1 and q2, r1=cos⁡t is the reference signal, r2=Dq1r1+k1Dq1-δe1 is the virtual control law, and e1=r1-x1 and e2=r2-x2 are the tracking errors. The constants are selected as k1=1.5, k2=8, and k=3. The initial condition is selected as(x1(0),x2(0))=(0,0).Case 1: Tracking performance of the noncommensurate fractional-order ferroresonance system of q1=0.99 and q2=0.98 under control is shown in Figures 10 and 11.Case 2: Tracking performance of the commensurate fractional-order ferroresonance system of q1=q2=0.99 under control is shown in Figures 12 and 13.Figure 10
The tracking performancex1-r1 of q1=0.99 and q2=0.98.Figure 11
The tracking performancex2-r2 of q1=0.99 and q2=0.98.Figure 12
The tracking performancex1-r1 of q1=q2=0.99.Figure 13
The tracking performancex2-r2 of q1=q2=0.99.From Figures10–13, it can be seen that the state x1 can track the reference signal r1=cos⁡t 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.



---

*Source: 8091757-2018-11-07.xml*