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Stability analysis of electrical RLC circuit described by the Caputo-Liouville generalized fractional derivative
Alexandria Engineering Journal (2020) xxx, xxx–xxx
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Stability analysis of electrical RLC circuit described by the Caputo-Liouville generalized fractional derivative Ndolane Sene Laboratoire Lmdan, De´partement de Mathe´matiques de la De´cision, Universite´ Cheikh Anta Diop de Dakar, Faculte´ des Sciences Economiques et Gestion, BP 5683 Dakar Fann, Senegal Received 18 December 2019; revised 2 January 2020; accepted 6 January 2020
KEYWORDS Fractional order derivative; Fractional electrical RLC circuit; Lyapunov direct method
Abstract We consider an electrical RLC circuit in two-dimensional spaces described by a fractional-order derivative. We propose the qualitative properties of the proposed model. We analyze the local asymptotic stability and the global asymptotic stability for the trivial equilibrium point for the electrical RLC circuit. We suggest the solution to the proposed model too. In our investigation, we consider the Caputo-Liouville fractional-order derivative. We use the characteristic matrix for the electrical RLC circuit model to analyze the local asymptotic stability of the trivial equilibrium point. For global asymptotic stability, we use the Lyapunov function method by constructing a Lyapunov function. Ó 2020 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Fractional calculus has experienced considerable advancements in research this last decade. Many new investigations related to the application of fractional calculus in real-world problems appear in the literature, see in [2,6,11,14–16,28,31]. Nowadays, many mathematicians are interested by this new field. This new field is interesting and has many applications. The definition of fractional derivative constitutes the most crucial question in this new field [5,12,16]. How to characterize the fractional derivative used in this new field? What is the importance of a fractional derivative? When and where we can use the fractional derivative. Many questions exist, and these E-mail address: [email protected] Peer review under responsibility of Faculty of Engineering, Alexandria University.
questions have answers nowadays. The responses given for these above questions open many schooles in fractional calculus. But all the existing schools try to develop mathematics and have their definition of the fractional derivative. The important in this new field is not the type of used fractional derivative but scientific novelty contained in fractional calculus papers. Many classical methods find interesting applications in fractional calculus as the homotopy perturbation method [36], Fourier transform, Mellin transforms, and others. In physics, many new diffusion process appear in the fractional calculus: subdiffusion process, super-diffusion process, ballistic-process, hyper-diffusion process and other [21–23,35]. There exist many types of fractional derivatives in fractional calculus: LiouvilleRiemann-derivative [25], Caputo-Liouville-derivative [25], Atangana-Baleanu derivative [1,13,28,26], Caputo-Fabrizio derivative [18], conformable derivative [27], and others [22].
https://doi.org/10.1016/j.aej.2020.01.008 1110-0168 Ó 2020 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: N. Sene, Stability analysis of electrical RLC circuit described by the Caputo-Liouville generalized fractional derivativeRLC circuit – >, Alexandria Eng. J. (2020), https://doi.org/10.1016/j.aej.2020.01.008
2 Fractional derivatives have many advantages comparing them with the classical derivative. The first advantage is the fractional derivatives take into account the memory. The memory effect is a fundamental property for a differential equation. That explain the use the fractional derivatives in modeling the differential equations [3,5]. Another advantage, the fractional derivatives generated many diffusion processes. For example, the superdiffusion, the hyperdiffusion, the ballistic diffusion, and other diffusion processes, see more details in [35]. For discussion, comparison, and review the fractional derivatives with singular kernel and fractional derivatives with no singular kernel see in [5]. Many applications of the fractional derivatives in electronics exist too [4,19,20]. With the fractional derivatives, many new mathematical models appear. And recently, researchers start applying the fractional-order derivatives in electrical circuits. We note many types of fractional electrical circuit recently introduced in the literature, see in [3,4,19,20]. We have the fractional RL circuit modeling introduced in [7,8,32–34]. We have the fractional RC circuit model in [7,8,32–34]. We have the fractional LC circuit introduced in [8,33,34]. Many of the investigations related to the fractional electrical circuits concern the numerical and analytical solutions. In [7], Aguilar et al. study the analytical and the numerical solutions for the fractional RL and RC circuits using the Atangana-Baleanu derivative and bi-order derivatives. In [8], Aguilar et al. propose the solutions of the fractional electrical RL; LC, and RC circuits described by Mittag-Leffler fractional derivative. In [32], Rawdan et al. study the fractional-order RL and LC circuits. They propose a comparison between fractional-order electrical circuits with the conventional electrical circuits. In [9], Aguilar et al. propose modeling the electrical RC and LC circuits in the context of the fractional derivatives. In [10], Aguilar et al. propose work on fractional electrical circuits described by a fractional derivative with regular Kernel. Many other investigations exist in the literature of fractional calculus. The stability analysis is recently introduced in fractional calculus to studies the stability of the fractional differential equations, see in [29]. Many investigations, study the stability analysis of electrical RL; LC; RC circuits [33,34]. The stability of the fractional RLC circuit is not trivial following the proposed model for RLC in the literature. In this present paper, we propose an alternative issue to study the local asymptotic stability and the global asymptotic stability of the trivial solution of the electrical RLC circuit. In this paper, we discuss a new representation of the electrical RLC model. We use the Matignon criterion to study the local asymptotic stability and the Lyapunov direct method to study global asymptotic stability. For novelty, this present work proposes the construction of a new Lyapunov candidate function for the fractional electrical RLC circuit described by the Caputo derivative. We propose solutions to the proposed model, too. The work is structured as follows: In Section 2, we recall the fractional derivatives, integral, stability notions, and properties related to our investigation. In Section 3, we introduce the considered model, prove the existence, and the uniqueness of the solutions. In Section 4, we determine the characteristic matrix of the proposed model, study the local asymptotic stability, and the global asymptotic stability for the fractional RLC circuit. We finish with Section 5 with the conclusions and remarks.
N. Sene 2. Fractional calculus, stability notions and properties We summarize in this section the lemmas and the definitions which are necessaries for our present investigation. We recall the fractional derivative, the fractional integral, the stability notions, the Matignon criterion for local stability, and the Lyapunov condition for the stability global and asymptotic of the trivial equilibrium point of the differential equations. In the following definitions, we give the generalized form of the integral proposed by Riemann and Liouville, and the description of the generalized form of the fractional derivative introduced by Caputo and Liouville. Definition 1 [25]. We choose the following function p : ½0; þ1½!R, conventionally the generalization of the integral proposed by Riemann and Liouville is defined as
gRL I
w;j
p ðtÞ ¼
1 NðwÞ
w1 Z t j t uj du pðuÞ 1j ; u j 0
ð1Þ
where the orders w, and j verify the condition w; j > 0, the function gamma is Nð. . .Þ, for all t > 0, and furthermore 0 < w < 1. Definition 2 [25]. We choose the following function p : ½0; þ1½!R, conventionally the generalization of the derivative proposed by Caputo and Liouville (gLCderivative) is defined as gCL D
w;j
pðtÞ ¼ Iw;j p ðtÞ ¼
1 Nð1 wÞ
Z t 0
tj uj j
w1 p0 ðuÞdu;
ð2Þ
where the orders w, and j verify the condition w; j > 0, the function gamma is Nð. . .Þ, for all t > 0, and furthermore 0 < w < 1. We recall the following definitions and lemmas related to the stability notions and the characterization of the stability in terms of Lyapunov function. We consider the differential equation with gCL-derivative described by the following equation gCL D
w;j
x ¼ pðx; tÞ;
ð3Þ
where the variable x 2 R and the function p is Lipchitz and continous defined as the following form p : Rn Rþ ! Rn . We represent by the variable x the equilibrium point of the fractional equation Eq. (3) and verifying the equation pðx Þ ¼ 0. n
Definition 3 [37]. Consider the equilibrium x of the gCLdifferential Eq. (3) is said stable when, for all n > 0, we can find g ¼ gðnÞ verifying for alll condition kx0 k < g, the state xðtÞ of the gCL-differential Eq. (3) verifies the inequality kxðtÞ x k < n for all t > t0 . In addition, the equilibrium point x is said asymptotically stable when we have stability and the condition lim kxðtÞ x k ¼ 0;
t!þ1
is held.
Please cite this article in press as: N. Sene, Stability analysis of electrical RLC circuit described by the Caputo-Liouville generalized fractional derivativeRLC circuit – >, Alexandria Eng. J. (2020), https://doi.org/10.1016/j.aej.2020.01.008
Stability analysis of electrical RLC circuit
3
We recall the criterion proposed by Matignon for the stability of the fractional differential equation described by the gCLderivative. Lemma 1 [30]. Consider the gCL-differential equation defined by Eq. (3). We said the equilibrium points of (3) are stable locally and asymptotically when all the eigenvalues (hi ) associated with the so-called Jacobian matrix of Eq. (3) evaluated at the equilibrium points verify the condition jarg ðhi Þj P
wp : 2
ð4Þ
Lemma 2 [17]. Consider a vector of differentiable function v 2 Rn . Let’s a positive matrix P 2 Rnn which is a symmetric and a square matrix. Then, with the condition t P 0, the following relationship is held w;j T v Pv 6 2vT PgCL Dw;j v gCL D
w 2 ½0; 1Þ:
ð5Þ
We recall the definitions of the comparison functions, which are also necessaries for our present investigation. For more details see in [37]. Definition 4 [37]. We define the set DP as a class of all functions - : RP0 ! RP0 which are continuous and verifying the conditions -ð0Þ ¼ 0, and -ðsÞ > 0 for all s > 0. We define the set CK as a class of all function which increase and are in the set DP. We define the set CK1 as a class of all function which are in the set CK and are unbounded. Definition 5 [37]. Consider a function # : RP0 RP0 ! RP0 which is continuous its belongs to the class CKL when #ð:; tÞ 2 CK for any t P 0, and #ðs; :Þ is a function which non-increase and converges to zero when its arguments converge to infinity. Let’s recall the definitions of the stability notions which we will use in this next section to prove our main results. Definition 6 [37]. We said the solution x ¼ 0 of the gCLdifferential equation defined by Eq. (3) is globally asymptotically stable when we can find a function #into the set KL verifying for any initial condition g, the relationship represented by kxðt; gÞk 6 #ðknk; tj tj0 Þ;
ð6Þ
is held, with the Euclidean norm k:k. We cite the following Theorem to characterize the asymptotic global stability of the trivial solution of Eq. (3). Theorem 1 [37]. Consider a positive function D : Rþ Rn !R which is continuous exists, and we have the existence of the functions -1 ; -2 ; -3 into the set CK1 and verifying the following relationships 1. -1 ðkxkÞ 6 Dðt; xÞ 6 -2 ðkxkÞ, 2. gCL Dw;j Dðt; xÞ 6 -3 ðkxkÞ, then the solution x ¼ 0 of the gCL-differential Eq. (3) is globally asymptotically stable.
3. Qualitatives properties of the fractional electrical RLC circuit In this paper, we consider the fractional electrical RLC circuit described by the Caputo-Liouville generalized fractional derivative. We describe an alternative representation of the fractional electrical RLC circuit as follows: we choose the state variables as the current through the indicator and the voltage across the Capacitor. Therefore, we consider the following fractional differential equation in two-dimensional space for RLC circuit described by x2 w;j ð7Þ x1 ¼ ; gCL D L r2ð1aÞ Rr1a w;j x2 ¼ ð8Þ x1 x2 ; gCL D C L with the initial boundary conditions given by x1 ð0Þ ¼ x10 and x2 ð0Þ ¼ x20 :
ð9Þ
where r is introduced in the fractional model to preserve the dimensionality of the temporal operator, see in [9] for more information. Note when we combine Eqs. (7) and (8), we recover the classical fractional electrical RLC circuit described by the fractional-order derivative. We have the following procedure. Consider x1 ¼ q. Thus, we write Eq. (7) as the following form x2 w;j q¼ : ð10Þ gCL D L Remplacing Eq. (10) into Eq. (8), we obtain the following relationships r2ð1aÞ Rr1a q LDa;q c q; C L 2ð1aÞ r w;j q Rr1a Dca;q q: q¼ gCL D C gCL D
w;j
q¼
ð11Þ ð12Þ
Finally, we express the fractional electrical RLC circuit described by Caputo-Liouville generalized fractional derivative as the following form LgCL D2w;j q þ
r2ð1aÞ q þ Rr1a gCL Dw;j q ¼ 0: C
ð13Þ
We observe when the order q ¼ 1, we recover the fractional electrical RLC circuit described by Liouville-Caputo derivative expressed as the form LgCL D2w;j q þ Rr1a gCL Dw;j q þ
r2ð1aÞ q ¼ 0: C
ð14Þ
We notice Eq. (13) is not the general form of the electrical RLCcircuit. In Eq. (13), the source term is null. In Eq. (13), when the order a ¼ 1, we recover the conventional electrical RLC circuit defined by the integer-order derivative. In this paper, we consider the fractional equation described by Eqs. (7) and (8). Note that in the literature, getting the analytical solution of the fractional equation represented in Eq. (13) is not trivial. Classically, we can use the Mittag-Leffler function in three dimensional to express it. The graphical representation is not trivial also. The numerical solutions were recently used to approximate the solution to the fractional electrical RLC circuit. The numerical approach is better than the use of the analytical solution. The graphical representation of the analytical solution of Eq. (13) is not trivial. Another inconvenience for the fractional electrical RLC circuit with Eq. (13) is the
Please cite this article in press as: N. Sene, Stability analysis of electrical RLC circuit described by the Caputo-Liouville generalized fractional derivativeRLC circuit – >, Alexandria Eng. J. (2020), https://doi.org/10.1016/j.aej.2020.01.008
4
N. Sene
stability study of the equilibrium point is practically impossible and does not exist in the literature. The difficulty for getting the analytical solution, the above reasons, and the graphical representation of the solution of (13) motives us in this work. We decide to study the stability of the fractional electrical RLC circuit described by the gCL-derivative by utilizing the Eqs. (7) and (8). Let’s establish the existence and the uniqueness of the solution of the fractional electrical RLC circuit described by the gCL-derivative. We begin with the first Eq. (7). Consider the function F 1 ðx1 Þ ¼
x2 r2ð1aÞ Rr1a and F 2 ðx2 Þ ¼ x1 x2 : L C L
x1 ¼ F 1 ðx1 Þ;
ð16Þ
gCL D
w;j
x2 ¼ F 1 ðx2 Þ:
ð17Þ
We can easily observe the function F 1 is locally Lipchitz and satisfies the relationship ð18Þ
We can easily observe the function F 2 is locally Lipchitz and satisfies the relationship ð19Þ
1a
where the constant k ¼ RrL . We define Picard’s operators using the following procedure. Applying the generalized fractional integral to Eqs. (16) and (17), we have that x1 ðtÞ ¼ x10 þ gRL Iw;j F 1 ðx1 Þ;
ð20Þ
x2 ðtÞ ¼ x20 þ gRL Iw;j F 2 ðx2 Þ:
ð21Þ
operators are defined by the following
P1 x ¼ x10 þ gRL Iw;j F 1 ðx1 ;Þ
ð22Þ
P2 x ¼ x20 þ gRL Iw;j F 2 ðx2 :Þ
ð23Þ
Utilizing the first condition Eq. (9), the first Picard’s operator can be redefined as the following form P1 x ¼ x10 :
ð24Þ
The contraction of P1 is trivial. The contraction proves the existence and the uniqueness of the solution of the gCLdifferential equation defined by Eq. (7). We first determine the operator P2 is well definite. For that, we evaluate the following relation kP2 x x20 k ¼ 6 6 6
kgRL Iw;j F 2 ðx2 Þk; w;j kF 2 ðx2 Þk; gRL I kF 2 ðx2 ÞkgRL Iw;j ð1Þ;
ð25Þ
q1a Tq a ð Þ m; CðaÞ q
where kF 2 ðx2 Þk 6 m. Thus the operator P2 is well definite. We can see the operator P2 defines a contraction. We have the following proof. kP2 u P2 vk ¼ 6 6
kgRL Iw;j ½F 2 u F 2 vk; w;j k½F 2 u F 2 vk; gRL I k½F 2 u F 2 vkgRL I
w;j
ð1Þ:
Under which we conclude using the Banach fixed Theorem, the fractional differential Eq. (8) has a unique solution.
In this section, we consider the trivial equilibrium point. We study its Local stability and its global asymptotic stability. For the local stability, we use the characteristic matrix of the fractional differential equation defined by Eqs. (7) and (8). For the global asymptotic stability, we use the Lyapunov direct method by constructing a new Lyapunov candidate function. It is not trivial and constitutes the novelty of this paper. We begin by the local stability of the trivial equilibrium 2ð1aÞ 1a point. For simplification d ¼ L1 ; l ¼ r C and b ¼ RrL . We write Eqs. (7) and (8) as follows
w;j
The Picards relationships
The operator P2 is a contraction under the condition defined by the relationship q a q1a T 1 < : ð28Þ Cða þ 1Þ q k
4. Stability analysis of the fractional electrical RLC circuit model
gCL D
kF 2 ðuÞ F 2 ðvÞk 6 kku vk:
ð27Þ
ð15Þ
Thus, we rewrite Eqs. (7) and (8) as the following forms
kF 1 ðuÞ F 1 ðvÞk ¼ 0:
Recalling Eq. (19), it follows the relationships a q1a Tq kP2 u P2 vk 6 kku vk: CðaÞ q
ð26Þ
gCL D
w;j
x1 ¼ dx2 ;
ð29Þ
gCL D
w;j
x2 ¼ lx1 þ bx2 :
ð30Þ
After the application of the Laplace transform to Eqs. (29) and (30), we obtain the following equations sa x1 d x2 ¼ sa1 x10 ;
ð31Þ
l x1 þ ðsa bÞ x2 ¼ sa1 x20 :
ð32Þ
Using Eqs. (31) and (32), the characteristic matrix is expressed in the following form a d s : ð33Þ D¼ l sa b We use Eq. (33) to get the eigenvalues of the characteristic matrix. For local stability of x1 ; x2 ¼ ð0; 0Þ, we give the following Theorem. Note that the trivial equilibrium point satisfies the equations Dca;q x1 ¼ 0 and Dca;q x1 ¼ 0. Theorem 2. The trivial equilibrium of the fractional electrical RLC circuit represented by Eqs. (7) and (8) is locally stable if the following condition is held R2 P 4LC:
ð34Þ
In this above Theorem, we use the Characteristic matrix to investigate the local stability. The characteristic polynomial of the matrix D is obtained by calculating det ðDÞ. From which the eigenvalues are obtained by solving the equation det ðDÞ ¼ 0. Proof. We first calculate the determinant of the characteristic matrix given by the following relationship det ðDÞ ¼ ¼
sa ðsa bÞ ld; s2a bsa ld:
ð35Þ
Please cite this article in press as: N. Sene, Stability analysis of electrical RLC circuit described by the Caputo-Liouville generalized fractional derivativeRLC circuit – >, Alexandria Eng. J. (2020), https://doi.org/10.1016/j.aej.2020.01.008
Stability analysis of electrical RLC circuit
5
Let’s k ¼ sa , thus the characteristic polynomial is given by the following expression det ðDÞ ¼ k2 bk ld:
ð36Þ
From which the following values represent the eigenvalues of the matrix D pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b b2 þ 4ld b þ b2 þ 4ld and k2 ¼ : ð37Þ k1 ¼ 2 2 Note that by the simple calculations, we observe all the eigenvalues are negatives and satisfy the Matignon criterion in Eq. (4) if the following relationship is held b2 þ 4ld P 0; R2 4 LC P 0; L2
ð38Þ
R2 P 4LC:
In Theorem 2, the stability of the trivial point is local. The contribution of this work is to prove we can have better. In other words, we can get the global asymptotic stability by constructing a Lyapunov candidate function. Note that in stability analysis, proving the global asymptotic stability using the Lyapunov candidate functions is not trivial. Because in many cases finding the Lyapunov functions are not trivial and impossible in many cases. Recently, researchers make many advances in the construction of these functions. Notably, in the biological models, see in [24]. This present work proposes a new alternative to analyze the global asymptotic stability of the electrical RLC circuit described by fractional-order derivative using a suitable Lyapunov function. The second part of this section is to prove the global asymptotic stability of the equilibrium point of the fractional electrical RLC circuit (7)-(8) by using the Lyapunov direct method. We propose the following Theorem. h Theorem 3. The trivial equilibrium point of the fractional electrical RLC circuit represented by Eqs. (7) and (8) is globally asymptotically stable when the following condition is held 1 r1a P : 2
ð39Þ
Proof. Let’s the Lyapunov candidate function defined by VðxÞ ¼ xT Px where x ¼ ðx1 ; x2 Þ and the positive definite matrix P is expressed as P¼
bL 2lL R 1
1 2 R
! :
ð40Þ
Using Eq. (5) and applying the gCL-derivative along the trajectories, it yields that
4l 2 x1 x2 þ x22 ; V ¼ gCL Dw;j xT Px 6 xT PgCL Dw;j x ¼ 2b R L 4l 4b þ 2lx21 þ 2bx1 x2 þ x1 x2 þ x22 ; R R
2 4b 2 þ x2 ; ¼ 2lx21 þ L R
T 2 4b 2 w;j w;j 2 þ x: V ¼ gCL D x Px 6 2lx1 þ ð41Þ gCL D L R 2 gCL D
w;j
From Eq. (41), we observe Theorem 1 is satisfyed. Thus the trivial solution of the electrical RLC circuit is global asymptotic stability if and only if the following relationship is held 2 L 2 L
þ 4b 6 0; R 1a
4rL P 0; r1a P 12 :
ð42Þ
We can observe when the fractional order a ¼ 1, Eq. (41) is expressed as the following form 2 2 Da;q V ¼ Da;q xT Px 6 x21 x22 : C L
ð43Þ
Thus using Eq. (43), the trivial solution of fractional electrical RLC circuit is globally asymptotically stable. This new investigation related to the Lyapunov candidate function for the electrical RLC circuit described by a fractional-order derivative opens a new door in stability analysis. We can extend this present investigation in the following direction. Probably, we can construct a Lyapunov candidate function, which avoids condition (39) in the stability criterion for global asymptotic stability. h 5. Solutions procedures for the fractional RLC circuit In this section, we propose the solution of the fractional electrical RLC circuit when we consider the Eqs. (7) and (8). The procedure is classical. We use an algebraic procedure in this section. We consider x ¼ ðx1 ; x2 Þ, we write the fractional differential Eqs. (7) and (8) as the following form gCL D
w;j
x ¼ Ax;
where the matrix d 0 A¼ : l b
ð44Þ
ð45Þ
With the algebraic manipulations, we write the matrix A as follows A ¼ P1 DP, where we express the matrix P, and the diagonal matrix D as ! db d 0 0 l P¼ and D ¼ : ð46Þ 0 b 1 1 Using the change variable y ¼ Px, we write Eq. (44) as the following form gCL D
w;j
y ¼ Dy:
ð47Þ
We express the solution of the fractional differential Eq. (44) using the Laplace transform as the following form j w ! t ; ð48Þ y1 ðtÞ ¼ y10 Ew d j j w ! t ð49Þ y2 ðtÞ ¼ y20 Ew b j With the change variable y ¼ Px, we express the solutions of the fractional differential Eq. (44) as the following form
Please cite this article in press as: N. Sene, Stability analysis of electrical RLC circuit described by the Caputo-Liouville generalized fractional derivativeRLC circuit – >, Alexandria Eng. J. (2020), https://doi.org/10.1016/j.aej.2020.01.008
6
N. Sene
Fig. 1
States variables x2 in times.
Fig. 2
States variables x2 in times.
Fig. 3
States variables x1 in times.
Please cite this article in press as: N. Sene, Stability analysis of electrical RLC circuit described by the Caputo-Liouville generalized fractional derivativeRLC circuit – >, Alexandria Eng. J. (2020), https://doi.org/10.1016/j.aej.2020.01.008
Stability analysis of electrical RLC circuit
7
Fig. 4
j w ! l t y10 Ew d ; db j j w ! j w ! l t t y Ew d þ y20 Ew b : x2 ðtÞ ¼ b d 10 j j x1 ðtÞ ¼
States variables x1 in times.
ð50Þ ð51Þ
Using the initial conditions, the solutions of the fractional differential Eq. (44) take the following form j w ; x1 ðtÞ ¼ x10 Ew d tj h i j j w j w w x2 ðtÞ ¼ x10 Ew b tj Ew d tj þ x20 Ew b tj : Let’s R ¼ 50X; L ¼ 100H, and the initial conditions x10 ¼ x20 ¼ 0:5. In Fig. 1, we depict the solutions x2 of the fractional electrical RLC circuit with w ¼ 1, and the order j varies and satisfies the condition j 1. In Fig. 2, we depict the solutions x2 of the fractional electrical RLC circuit with w ¼ 1, and the order j varie and satisfies the condition j P 1. In Fig. 3, we depict the solutions x1 of the fractional electrical RLC circuit with w ¼ 1, and the order j varie and satisfies the condition j 1. In Fig. 4, we depict the solution x1 of the fractional electrical RLC circuit with w ¼ 1 and the order j varie and satisfies the condition j P 1.
6. Conclusion In this paper, we have discussed the local asymptotic stability and the global asymptotic stability for the trivial equilibrium point of the fractional electrical RLC circuit described by gCL-derivative. We have proposed a new criterion for local asymptotic stability, which depends on the Resistance and the current across the indicator. We give the condition for the global asymptotic stability which depends on the fractional-order derivative and the fractional parameter time r. We hope this present work will open new doors in stability analysis for the fractional electrical circuits.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] T. Abdeljawad, D. Baleanu, On fractional derivatives with generalized Mittag-Leffler kernels, Adv. Diff. Equat. (2018:) 468. [2] K.A. Abro, J.F. Gomez-Aguilar, A comparison of heat and mass transfer on a Walter’s-B fluid via Caputo-Fabrizio versus Atangana-Baleanu fractional derivatives using the Fox-H function, Eur. Phys. J. Plus 134 (3) (2019) 1–10. [3] J.F.G. Aguilar, Fundamental solutions to electrical circuits of non-integer order via fractional derivatives with and without singular kernels, Eur. Phys. J. Plus 135 (5) (2018) 197. [4] J.F.G. Aguilar, Behavior characteristics of a cap-resistor, memcapacitor, and a memristor from the response obtained of RC and RL electrical circuits described by fractional differential equations, Turk. J. Elec. Eng. Comp. Sci. 24 (2016) 1421–1433. [5] J.F. Gomez-Aguilar, A. Atangana, New insight in fractional differentiation: power, exponential decay and Mittag-Leffler laws and applications, Eur. Phys. J. Plus 132(1) (2017) 1–23. [6] J.F.G. Aguilar, D. Baleanu, Fractional Transmission Line with Losses, Z. Naturforsch. 69a (2014) 539–546. [7] J.F. Gomez-Aguilar, R.F. Escobar-Jimenez, V.H. OlivaresPeregrino, M.A. Taneco-Hernandez, G.V. Guerrero-Ramirez, Electrical circuits RC and RL involving fractional operators with bi-order, Adv. Mech. Eng. 9 (6) (2017) 1–10. [8] J.F. Gomez-Aguilar, A. Atangana, V.F. Morales-Delgado, Electrical circuits RC, LC, and RL described by AtanganaBaleanu fractional derivatives, Int. J. Circ. Theor. Appl. 45 (11) (2017) 1514–1533. [9] J.F. Gomez-Aguilar, R.G. Juan, R.H.J. Roberto, G.C. Manuel, Fractional RC and LC Electrical Circuits, Ingen, Investig. Tecno. 15 (2) (2014) 311–319. [10] J.F. Gomez-Aguilar, T. Cordova-Fraga, J.E. EscalanteMartinez, C. Calderon-Ramon, R.F. Escobar-Jime´nez, Electrical circuits described by a fractional derivative with regular Kernel, Revist. Mex. Fis. 62 (2) (2016) 144–154.
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N. Sene [25] J. Fahd, T. Abdeljawad, A modified Laplace transform for certain generalized fractional operators, Res. Nonl. Anal. 2 (2018) 88–98. [26] J. Hristov, On the Atangana-Baleanu derivative and its relation to the fading memory concept: the diffusion equation formulation, in: J. Gomez, L. Torres, R. Escobar (Eds.), Fractional Derivatives with Mittag-Leffler Kernel, Studies in Systems, Decision and Control, 194, Springer, Cham, 2019, pp. 175–193. [27] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014) 65–70. [28] H. Khan, F. Jarad, T. Abdeljawad, A. Khan, A singular ABCfractional differential equation with p-Laplacian operator, Chaos, Soli. Fract. 129 (2019) 56–61. [29] K. Liu, J. Wang, Y. Zhou, D. O’Regan, Hyers-Ulam stability and existence of solutions for fractional differential equations with Mittag-Leffler kernel, Chaos, Sol. Fract. 132 (2020) 109534. [30] D. Matignon, Stability results for fractional differential equations with applications to control processing, Comput. Engin. Syst. Appl. 2 (1996) 963–968. [31] K.M. Owolabi, Numerical analysis and pattern formation process for space fractional superdiffusive systems, Disct. Contin. Dyn. Syst. Ser. S 12 (3) (2019) 543–566. [32] A.G. Radwan, K.N. Salama, Fractional-order RC and RL circuits, Circuits Syst. Signal Process 31 (2012) 1901–1915. [33] N. Sene, J.F. Gomez-Aguilar, Analytical solutions of electrical circuits considering certain generalized fractional derivatives, Eur. Phys. J. Plus 134 (2019) 260. [34] N. Sene, Fractional input stability for electrical circuits described by the Riemann-Liouville and the Caputo fractional derivatives, AIMS Math. 4 (1) (2019) 147–165. [35] N. Sene, K. Abdelmalek, Analysis of the fractional diffusion equations describ e d by Atangana-Baleanu-Caputo fractional derivative, Chaos Soli. Fract. 127 (2019) 158–164. [36] N. Sene, Homotopy perturbation q-laplace transform method and its application to the fractional diffusion equation and the fractional diffusion-reaction equation, Fractal Fract. 3 (2019) 14. [37] N. Sene, Stability analysis of the generalized fractional differential equations with and without exogenous inputs, J. Nonlinear Sci. Appl. 12 (2019) 562–572.
Please cite this article in press as: N. Sene, Stability analysis of electrical RLC circuit described by the Caputo-Liouville generalized fractional derivativeRLC circuit – >, Alexandria Eng. J. (2020), https://doi.org/10.1016/j.aej.2020.01.008
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ORIGINAL ARTICLE
Stability analysis of electrical RLC circuit described by the Caputo-Liouville generalized fractional derivative Ndolane Sene Laboratoire Lmdan, De´partement de Mathe´matiques de la De´cision, Universite´ Cheikh Anta Diop de Dakar, Faculte´ des Sciences Economiques et Gestion, BP 5683 Dakar Fann, Senegal Received 18 December 2019; revised 2 January 2020; accepted 6 January 2020
KEYWORDS Fractional order derivative; Fractional electrical RLC circuit; Lyapunov direct method
Abstract We consider an electrical RLC circuit in two-dimensional spaces described by a fractional-order derivative. We propose the qualitative properties of the proposed model. We analyze the local asymptotic stability and the global asymptotic stability for the trivial equilibrium point for the electrical RLC circuit. We suggest the solution to the proposed model too. In our investigation, we consider the Caputo-Liouville fractional-order derivative. We use the characteristic matrix for the electrical RLC circuit model to analyze the local asymptotic stability of the trivial equilibrium point. For global asymptotic stability, we use the Lyapunov function method by constructing a Lyapunov function. Ó 2020 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Fractional calculus has experienced considerable advancements in research this last decade. Many new investigations related to the application of fractional calculus in real-world problems appear in the literature, see in [2,6,11,14–16,28,31]. Nowadays, many mathematicians are interested by this new field. This new field is interesting and has many applications. The definition of fractional derivative constitutes the most crucial question in this new field [5,12,16]. How to characterize the fractional derivative used in this new field? What is the importance of a fractional derivative? When and where we can use the fractional derivative. Many questions exist, and these E-mail address: [email protected] Peer review under responsibility of Faculty of Engineering, Alexandria University.
questions have answers nowadays. The responses given for these above questions open many schooles in fractional calculus. But all the existing schools try to develop mathematics and have their definition of the fractional derivative. The important in this new field is not the type of used fractional derivative but scientific novelty contained in fractional calculus papers. Many classical methods find interesting applications in fractional calculus as the homotopy perturbation method [36], Fourier transform, Mellin transforms, and others. In physics, many new diffusion process appear in the fractional calculus: subdiffusion process, super-diffusion process, ballistic-process, hyper-diffusion process and other [21–23,35]. There exist many types of fractional derivatives in fractional calculus: LiouvilleRiemann-derivative [25], Caputo-Liouville-derivative [25], Atangana-Baleanu derivative [1,13,28,26], Caputo-Fabrizio derivative [18], conformable derivative [27], and others [22].
https://doi.org/10.1016/j.aej.2020.01.008 1110-0168 Ó 2020 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: N. Sene, Stability analysis of electrical RLC circuit described by the Caputo-Liouville generalized fractional derivativeRLC circuit – >, Alexandria Eng. J. (2020), https://doi.org/10.1016/j.aej.2020.01.008
2 Fractional derivatives have many advantages comparing them with the classical derivative. The first advantage is the fractional derivatives take into account the memory. The memory effect is a fundamental property for a differential equation. That explain the use the fractional derivatives in modeling the differential equations [3,5]. Another advantage, the fractional derivatives generated many diffusion processes. For example, the superdiffusion, the hyperdiffusion, the ballistic diffusion, and other diffusion processes, see more details in [35]. For discussion, comparison, and review the fractional derivatives with singular kernel and fractional derivatives with no singular kernel see in [5]. Many applications of the fractional derivatives in electronics exist too [4,19,20]. With the fractional derivatives, many new mathematical models appear. And recently, researchers start applying the fractional-order derivatives in electrical circuits. We note many types of fractional electrical circuit recently introduced in the literature, see in [3,4,19,20]. We have the fractional RL circuit modeling introduced in [7,8,32–34]. We have the fractional RC circuit model in [7,8,32–34]. We have the fractional LC circuit introduced in [8,33,34]. Many of the investigations related to the fractional electrical circuits concern the numerical and analytical solutions. In [7], Aguilar et al. study the analytical and the numerical solutions for the fractional RL and RC circuits using the Atangana-Baleanu derivative and bi-order derivatives. In [8], Aguilar et al. propose the solutions of the fractional electrical RL; LC, and RC circuits described by Mittag-Leffler fractional derivative. In [32], Rawdan et al. study the fractional-order RL and LC circuits. They propose a comparison between fractional-order electrical circuits with the conventional electrical circuits. In [9], Aguilar et al. propose modeling the electrical RC and LC circuits in the context of the fractional derivatives. In [10], Aguilar et al. propose work on fractional electrical circuits described by a fractional derivative with regular Kernel. Many other investigations exist in the literature of fractional calculus. The stability analysis is recently introduced in fractional calculus to studies the stability of the fractional differential equations, see in [29]. Many investigations, study the stability analysis of electrical RL; LC; RC circuits [33,34]. The stability of the fractional RLC circuit is not trivial following the proposed model for RLC in the literature. In this present paper, we propose an alternative issue to study the local asymptotic stability and the global asymptotic stability of the trivial solution of the electrical RLC circuit. In this paper, we discuss a new representation of the electrical RLC model. We use the Matignon criterion to study the local asymptotic stability and the Lyapunov direct method to study global asymptotic stability. For novelty, this present work proposes the construction of a new Lyapunov candidate function for the fractional electrical RLC circuit described by the Caputo derivative. We propose solutions to the proposed model, too. The work is structured as follows: In Section 2, we recall the fractional derivatives, integral, stability notions, and properties related to our investigation. In Section 3, we introduce the considered model, prove the existence, and the uniqueness of the solutions. In Section 4, we determine the characteristic matrix of the proposed model, study the local asymptotic stability, and the global asymptotic stability for the fractional RLC circuit. We finish with Section 5 with the conclusions and remarks.
N. Sene 2. Fractional calculus, stability notions and properties We summarize in this section the lemmas and the definitions which are necessaries for our present investigation. We recall the fractional derivative, the fractional integral, the stability notions, the Matignon criterion for local stability, and the Lyapunov condition for the stability global and asymptotic of the trivial equilibrium point of the differential equations. In the following definitions, we give the generalized form of the integral proposed by Riemann and Liouville, and the description of the generalized form of the fractional derivative introduced by Caputo and Liouville. Definition 1 [25]. We choose the following function p : ½0; þ1½!R, conventionally the generalization of the integral proposed by Riemann and Liouville is defined as
gRL I
w;j
p ðtÞ ¼
1 NðwÞ
w1 Z t j t uj du pðuÞ 1j ; u j 0
ð1Þ
where the orders w, and j verify the condition w; j > 0, the function gamma is Nð. . .Þ, for all t > 0, and furthermore 0 < w < 1. Definition 2 [25]. We choose the following function p : ½0; þ1½!R, conventionally the generalization of the derivative proposed by Caputo and Liouville (gLCderivative) is defined as gCL D
w;j
pðtÞ ¼ Iw;j p ðtÞ ¼
1 Nð1 wÞ
Z t 0
tj uj j
w1 p0 ðuÞdu;
ð2Þ
where the orders w, and j verify the condition w; j > 0, the function gamma is Nð. . .Þ, for all t > 0, and furthermore 0 < w < 1. We recall the following definitions and lemmas related to the stability notions and the characterization of the stability in terms of Lyapunov function. We consider the differential equation with gCL-derivative described by the following equation gCL D
w;j
x ¼ pðx; tÞ;
ð3Þ
where the variable x 2 R and the function p is Lipchitz and continous defined as the following form p : Rn Rþ ! Rn . We represent by the variable x the equilibrium point of the fractional equation Eq. (3) and verifying the equation pðx Þ ¼ 0. n
Definition 3 [37]. Consider the equilibrium x of the gCLdifferential Eq. (3) is said stable when, for all n > 0, we can find g ¼ gðnÞ verifying for alll condition kx0 k < g, the state xðtÞ of the gCL-differential Eq. (3) verifies the inequality kxðtÞ x k < n for all t > t0 . In addition, the equilibrium point x is said asymptotically stable when we have stability and the condition lim kxðtÞ x k ¼ 0;
t!þ1
is held.
Please cite this article in press as: N. Sene, Stability analysis of electrical RLC circuit described by the Caputo-Liouville generalized fractional derivativeRLC circuit – >, Alexandria Eng. J. (2020), https://doi.org/10.1016/j.aej.2020.01.008
Stability analysis of electrical RLC circuit
3
We recall the criterion proposed by Matignon for the stability of the fractional differential equation described by the gCLderivative. Lemma 1 [30]. Consider the gCL-differential equation defined by Eq. (3). We said the equilibrium points of (3) are stable locally and asymptotically when all the eigenvalues (hi ) associated with the so-called Jacobian matrix of Eq. (3) evaluated at the equilibrium points verify the condition jarg ðhi Þj P
wp : 2
ð4Þ
Lemma 2 [17]. Consider a vector of differentiable function v 2 Rn . Let’s a positive matrix P 2 Rnn which is a symmetric and a square matrix. Then, with the condition t P 0, the following relationship is held w;j T v Pv 6 2vT PgCL Dw;j v gCL D
w 2 ½0; 1Þ:
ð5Þ
We recall the definitions of the comparison functions, which are also necessaries for our present investigation. For more details see in [37]. Definition 4 [37]. We define the set DP as a class of all functions - : RP0 ! RP0 which are continuous and verifying the conditions -ð0Þ ¼ 0, and -ðsÞ > 0 for all s > 0. We define the set CK as a class of all function which increase and are in the set DP. We define the set CK1 as a class of all function which are in the set CK and are unbounded. Definition 5 [37]. Consider a function # : RP0 RP0 ! RP0 which is continuous its belongs to the class CKL when #ð:; tÞ 2 CK for any t P 0, and #ðs; :Þ is a function which non-increase and converges to zero when its arguments converge to infinity. Let’s recall the definitions of the stability notions which we will use in this next section to prove our main results. Definition 6 [37]. We said the solution x ¼ 0 of the gCLdifferential equation defined by Eq. (3) is globally asymptotically stable when we can find a function #into the set KL verifying for any initial condition g, the relationship represented by kxðt; gÞk 6 #ðknk; tj tj0 Þ;
ð6Þ
is held, with the Euclidean norm k:k. We cite the following Theorem to characterize the asymptotic global stability of the trivial solution of Eq. (3). Theorem 1 [37]. Consider a positive function D : Rþ Rn !R which is continuous exists, and we have the existence of the functions -1 ; -2 ; -3 into the set CK1 and verifying the following relationships 1. -1 ðkxkÞ 6 Dðt; xÞ 6 -2 ðkxkÞ, 2. gCL Dw;j Dðt; xÞ 6 -3 ðkxkÞ, then the solution x ¼ 0 of the gCL-differential Eq. (3) is globally asymptotically stable.
3. Qualitatives properties of the fractional electrical RLC circuit In this paper, we consider the fractional electrical RLC circuit described by the Caputo-Liouville generalized fractional derivative. We describe an alternative representation of the fractional electrical RLC circuit as follows: we choose the state variables as the current through the indicator and the voltage across the Capacitor. Therefore, we consider the following fractional differential equation in two-dimensional space for RLC circuit described by x2 w;j ð7Þ x1 ¼ ; gCL D L r2ð1aÞ Rr1a w;j x2 ¼ ð8Þ x1 x2 ; gCL D C L with the initial boundary conditions given by x1 ð0Þ ¼ x10 and x2 ð0Þ ¼ x20 :
ð9Þ
where r is introduced in the fractional model to preserve the dimensionality of the temporal operator, see in [9] for more information. Note when we combine Eqs. (7) and (8), we recover the classical fractional electrical RLC circuit described by the fractional-order derivative. We have the following procedure. Consider x1 ¼ q. Thus, we write Eq. (7) as the following form x2 w;j q¼ : ð10Þ gCL D L Remplacing Eq. (10) into Eq. (8), we obtain the following relationships r2ð1aÞ Rr1a q LDa;q c q; C L 2ð1aÞ r w;j q Rr1a Dca;q q: q¼ gCL D C gCL D
w;j
q¼
ð11Þ ð12Þ
Finally, we express the fractional electrical RLC circuit described by Caputo-Liouville generalized fractional derivative as the following form LgCL D2w;j q þ
r2ð1aÞ q þ Rr1a gCL Dw;j q ¼ 0: C
ð13Þ
We observe when the order q ¼ 1, we recover the fractional electrical RLC circuit described by Liouville-Caputo derivative expressed as the form LgCL D2w;j q þ Rr1a gCL Dw;j q þ
r2ð1aÞ q ¼ 0: C
ð14Þ
We notice Eq. (13) is not the general form of the electrical RLCcircuit. In Eq. (13), the source term is null. In Eq. (13), when the order a ¼ 1, we recover the conventional electrical RLC circuit defined by the integer-order derivative. In this paper, we consider the fractional equation described by Eqs. (7) and (8). Note that in the literature, getting the analytical solution of the fractional equation represented in Eq. (13) is not trivial. Classically, we can use the Mittag-Leffler function in three dimensional to express it. The graphical representation is not trivial also. The numerical solutions were recently used to approximate the solution to the fractional electrical RLC circuit. The numerical approach is better than the use of the analytical solution. The graphical representation of the analytical solution of Eq. (13) is not trivial. Another inconvenience for the fractional electrical RLC circuit with Eq. (13) is the
Please cite this article in press as: N. Sene, Stability analysis of electrical RLC circuit described by the Caputo-Liouville generalized fractional derivativeRLC circuit – >, Alexandria Eng. J. (2020), https://doi.org/10.1016/j.aej.2020.01.008
4
N. Sene
stability study of the equilibrium point is practically impossible and does not exist in the literature. The difficulty for getting the analytical solution, the above reasons, and the graphical representation of the solution of (13) motives us in this work. We decide to study the stability of the fractional electrical RLC circuit described by the gCL-derivative by utilizing the Eqs. (7) and (8). Let’s establish the existence and the uniqueness of the solution of the fractional electrical RLC circuit described by the gCL-derivative. We begin with the first Eq. (7). Consider the function F 1 ðx1 Þ ¼
x2 r2ð1aÞ Rr1a and F 2 ðx2 Þ ¼ x1 x2 : L C L
x1 ¼ F 1 ðx1 Þ;
ð16Þ
gCL D
w;j
x2 ¼ F 1 ðx2 Þ:
ð17Þ
We can easily observe the function F 1 is locally Lipchitz and satisfies the relationship ð18Þ
We can easily observe the function F 2 is locally Lipchitz and satisfies the relationship ð19Þ
1a
where the constant k ¼ RrL . We define Picard’s operators using the following procedure. Applying the generalized fractional integral to Eqs. (16) and (17), we have that x1 ðtÞ ¼ x10 þ gRL Iw;j F 1 ðx1 Þ;
ð20Þ
x2 ðtÞ ¼ x20 þ gRL Iw;j F 2 ðx2 Þ:
ð21Þ
operators are defined by the following
P1 x ¼ x10 þ gRL Iw;j F 1 ðx1 ;Þ
ð22Þ
P2 x ¼ x20 þ gRL Iw;j F 2 ðx2 :Þ
ð23Þ
Utilizing the first condition Eq. (9), the first Picard’s operator can be redefined as the following form P1 x ¼ x10 :
ð24Þ
The contraction of P1 is trivial. The contraction proves the existence and the uniqueness of the solution of the gCLdifferential equation defined by Eq. (7). We first determine the operator P2 is well definite. For that, we evaluate the following relation kP2 x x20 k ¼ 6 6 6
kgRL Iw;j F 2 ðx2 Þk; w;j kF 2 ðx2 Þk; gRL I kF 2 ðx2 ÞkgRL Iw;j ð1Þ;
ð25Þ
q1a Tq a ð Þ m; CðaÞ q
where kF 2 ðx2 Þk 6 m. Thus the operator P2 is well definite. We can see the operator P2 defines a contraction. We have the following proof. kP2 u P2 vk ¼ 6 6
kgRL Iw;j ½F 2 u F 2 vk; w;j k½F 2 u F 2 vk; gRL I k½F 2 u F 2 vkgRL I
w;j
ð1Þ:
Under which we conclude using the Banach fixed Theorem, the fractional differential Eq. (8) has a unique solution.
In this section, we consider the trivial equilibrium point. We study its Local stability and its global asymptotic stability. For the local stability, we use the characteristic matrix of the fractional differential equation defined by Eqs. (7) and (8). For the global asymptotic stability, we use the Lyapunov direct method by constructing a new Lyapunov candidate function. It is not trivial and constitutes the novelty of this paper. We begin by the local stability of the trivial equilibrium 2ð1aÞ 1a point. For simplification d ¼ L1 ; l ¼ r C and b ¼ RrL . We write Eqs. (7) and (8) as follows
w;j
The Picards relationships
The operator P2 is a contraction under the condition defined by the relationship q a q1a T 1 < : ð28Þ Cða þ 1Þ q k
4. Stability analysis of the fractional electrical RLC circuit model
gCL D
kF 2 ðuÞ F 2 ðvÞk 6 kku vk:
ð27Þ
ð15Þ
Thus, we rewrite Eqs. (7) and (8) as the following forms
kF 1 ðuÞ F 1 ðvÞk ¼ 0:
Recalling Eq. (19), it follows the relationships a q1a Tq kP2 u P2 vk 6 kku vk: CðaÞ q
ð26Þ
gCL D
w;j
x1 ¼ dx2 ;
ð29Þ
gCL D
w;j
x2 ¼ lx1 þ bx2 :
ð30Þ
After the application of the Laplace transform to Eqs. (29) and (30), we obtain the following equations sa x1 d x2 ¼ sa1 x10 ;
ð31Þ
l x1 þ ðsa bÞ x2 ¼ sa1 x20 :
ð32Þ
Using Eqs. (31) and (32), the characteristic matrix is expressed in the following form a d s : ð33Þ D¼ l sa b We use Eq. (33) to get the eigenvalues of the characteristic matrix. For local stability of x1 ; x2 ¼ ð0; 0Þ, we give the following Theorem. Note that the trivial equilibrium point satisfies the equations Dca;q x1 ¼ 0 and Dca;q x1 ¼ 0. Theorem 2. The trivial equilibrium of the fractional electrical RLC circuit represented by Eqs. (7) and (8) is locally stable if the following condition is held R2 P 4LC:
ð34Þ
In this above Theorem, we use the Characteristic matrix to investigate the local stability. The characteristic polynomial of the matrix D is obtained by calculating det ðDÞ. From which the eigenvalues are obtained by solving the equation det ðDÞ ¼ 0. Proof. We first calculate the determinant of the characteristic matrix given by the following relationship det ðDÞ ¼ ¼
sa ðsa bÞ ld; s2a bsa ld:
ð35Þ
Please cite this article in press as: N. Sene, Stability analysis of electrical RLC circuit described by the Caputo-Liouville generalized fractional derivativeRLC circuit – >, Alexandria Eng. J. (2020), https://doi.org/10.1016/j.aej.2020.01.008
Stability analysis of electrical RLC circuit
5
Let’s k ¼ sa , thus the characteristic polynomial is given by the following expression det ðDÞ ¼ k2 bk ld:
ð36Þ
From which the following values represent the eigenvalues of the matrix D pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b b2 þ 4ld b þ b2 þ 4ld and k2 ¼ : ð37Þ k1 ¼ 2 2 Note that by the simple calculations, we observe all the eigenvalues are negatives and satisfy the Matignon criterion in Eq. (4) if the following relationship is held b2 þ 4ld P 0; R2 4 LC P 0; L2
ð38Þ
R2 P 4LC:
In Theorem 2, the stability of the trivial point is local. The contribution of this work is to prove we can have better. In other words, we can get the global asymptotic stability by constructing a Lyapunov candidate function. Note that in stability analysis, proving the global asymptotic stability using the Lyapunov candidate functions is not trivial. Because in many cases finding the Lyapunov functions are not trivial and impossible in many cases. Recently, researchers make many advances in the construction of these functions. Notably, in the biological models, see in [24]. This present work proposes a new alternative to analyze the global asymptotic stability of the electrical RLC circuit described by fractional-order derivative using a suitable Lyapunov function. The second part of this section is to prove the global asymptotic stability of the equilibrium point of the fractional electrical RLC circuit (7)-(8) by using the Lyapunov direct method. We propose the following Theorem. h Theorem 3. The trivial equilibrium point of the fractional electrical RLC circuit represented by Eqs. (7) and (8) is globally asymptotically stable when the following condition is held 1 r1a P : 2
ð39Þ
Proof. Let’s the Lyapunov candidate function defined by VðxÞ ¼ xT Px where x ¼ ðx1 ; x2 Þ and the positive definite matrix P is expressed as P¼
bL 2lL R 1
1 2 R
! :
ð40Þ
Using Eq. (5) and applying the gCL-derivative along the trajectories, it yields that
4l 2 x1 x2 þ x22 ; V ¼ gCL Dw;j xT Px 6 xT PgCL Dw;j x ¼ 2b R L 4l 4b þ 2lx21 þ 2bx1 x2 þ x1 x2 þ x22 ; R R
2 4b 2 þ x2 ; ¼ 2lx21 þ L R
T 2 4b 2 w;j w;j 2 þ x: V ¼ gCL D x Px 6 2lx1 þ ð41Þ gCL D L R 2 gCL D
w;j
From Eq. (41), we observe Theorem 1 is satisfyed. Thus the trivial solution of the electrical RLC circuit is global asymptotic stability if and only if the following relationship is held 2 L 2 L
þ 4b 6 0; R 1a
4rL P 0; r1a P 12 :
ð42Þ
We can observe when the fractional order a ¼ 1, Eq. (41) is expressed as the following form 2 2 Da;q V ¼ Da;q xT Px 6 x21 x22 : C L
ð43Þ
Thus using Eq. (43), the trivial solution of fractional electrical RLC circuit is globally asymptotically stable. This new investigation related to the Lyapunov candidate function for the electrical RLC circuit described by a fractional-order derivative opens a new door in stability analysis. We can extend this present investigation in the following direction. Probably, we can construct a Lyapunov candidate function, which avoids condition (39) in the stability criterion for global asymptotic stability. h 5. Solutions procedures for the fractional RLC circuit In this section, we propose the solution of the fractional electrical RLC circuit when we consider the Eqs. (7) and (8). The procedure is classical. We use an algebraic procedure in this section. We consider x ¼ ðx1 ; x2 Þ, we write the fractional differential Eqs. (7) and (8) as the following form gCL D
w;j
x ¼ Ax;
where the matrix d 0 A¼ : l b
ð44Þ
ð45Þ
With the algebraic manipulations, we write the matrix A as follows A ¼ P1 DP, where we express the matrix P, and the diagonal matrix D as ! db d 0 0 l P¼ and D ¼ : ð46Þ 0 b 1 1 Using the change variable y ¼ Px, we write Eq. (44) as the following form gCL D
w;j
y ¼ Dy:
ð47Þ
We express the solution of the fractional differential Eq. (44) using the Laplace transform as the following form j w ! t ; ð48Þ y1 ðtÞ ¼ y10 Ew d j j w ! t ð49Þ y2 ðtÞ ¼ y20 Ew b j With the change variable y ¼ Px, we express the solutions of the fractional differential Eq. (44) as the following form
Please cite this article in press as: N. Sene, Stability analysis of electrical RLC circuit described by the Caputo-Liouville generalized fractional derivativeRLC circuit – >, Alexandria Eng. J. (2020), https://doi.org/10.1016/j.aej.2020.01.008
6
N. Sene
Fig. 1
States variables x2 in times.
Fig. 2
States variables x2 in times.
Fig. 3
States variables x1 in times.
Please cite this article in press as: N. Sene, Stability analysis of electrical RLC circuit described by the Caputo-Liouville generalized fractional derivativeRLC circuit – >, Alexandria Eng. J. (2020), https://doi.org/10.1016/j.aej.2020.01.008
Stability analysis of electrical RLC circuit
7
Fig. 4
j w ! l t y10 Ew d ; db j j w ! j w ! l t t y Ew d þ y20 Ew b : x2 ðtÞ ¼ b d 10 j j x1 ðtÞ ¼
States variables x1 in times.
ð50Þ ð51Þ
Using the initial conditions, the solutions of the fractional differential Eq. (44) take the following form j w ; x1 ðtÞ ¼ x10 Ew d tj h i j j w j w w x2 ðtÞ ¼ x10 Ew b tj Ew d tj þ x20 Ew b tj : Let’s R ¼ 50X; L ¼ 100H, and the initial conditions x10 ¼ x20 ¼ 0:5. In Fig. 1, we depict the solutions x2 of the fractional electrical RLC circuit with w ¼ 1, and the order j varies and satisfies the condition j 1. In Fig. 2, we depict the solutions x2 of the fractional electrical RLC circuit with w ¼ 1, and the order j varie and satisfies the condition j P 1. In Fig. 3, we depict the solutions x1 of the fractional electrical RLC circuit with w ¼ 1, and the order j varie and satisfies the condition j 1. In Fig. 4, we depict the solution x1 of the fractional electrical RLC circuit with w ¼ 1 and the order j varie and satisfies the condition j P 1.
6. Conclusion In this paper, we have discussed the local asymptotic stability and the global asymptotic stability for the trivial equilibrium point of the fractional electrical RLC circuit described by gCL-derivative. We have proposed a new criterion for local asymptotic stability, which depends on the Resistance and the current across the indicator. We give the condition for the global asymptotic stability which depends on the fractional-order derivative and the fractional parameter time r. We hope this present work will open new doors in stability analysis for the fractional electrical circuits.
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Please cite this article in press as: N. Sene, Stability analysis of electrical RLC circuit described by the Caputo-Liouville generalized fractional derivativeRLC circuit – >, Alexandria Eng. J. (2020), https://doi.org/10.1016/j.aej.2020.01.008