Skip to main content

Finite time stability and sliding mode control for uncertain variable fractional order nonlinear systems

Abstract

This paper deals with the finite time stability and control for a class of uncertain variable fractional order nonlinear systems. The variable fractional Lyapunov direct method is developed to provide the basis for the stability proof of the system considered. The sliding mode control method is applied for robust control of uncertain variable fractional order systems; furthermore, the chattering phenomenon is avoided. And the finite time stability of the systems under control law is proved based on the proposed stability criterion. Finally, numerical simulations are proposed and the efficiency of the controller is verified.

Introduction

In recent years, fractional calculus (FC) has been used to describe the natural behavior in many research fields [19]. And the fractional order differential models can better describe some complex dynamic phenomena in many practical engineering problems [1016]. Thus, the theory of FC has been developed rapidly. For example, a definition of variable fractional order operator (VFO) was proposed in order to describe the complex phenomena of the mechanical modeling [17]. FC has a vast of applications in areas of physics and engineering, and it has been applied to chaos control of the fractional order(FO) dynamical systems [18]. And it has become a hot spot in research for the theory analysis and application of control in FO dynamical systems [1921], and there exist many control methods to deal with the control problem of the chaotic systems, such as adaptive control, backstepping method, feedback control method, and \(H_{\infty }\) approach [2225]. Monje et al. [22] detailed fractional order systems and controls by use of fractional calculus in the description and modeling of systems and in a range of control design and practical applications. Aguilar et al. [25] investigated the chaos control for a class of variable-order fractional chaotic systems using robust control strategy. Moreover, sliding mode control (SMC) [2628], which has the advantage of better transient performance, easy realization, rapid response, and insensitivity to external disturbances and so on, is frequently employed, see [2937]. Pisano et al. [29] applied the sliding mode control approaches to stabilize a class of linear uncertain fractional order dynamics and presented two sliding mode control schemes. Jakovljevic et al. [30] dealt with applications of sliding mode based fractional control techniques to address tracking and stabilization control tasks for some classes of nonlinear uncertain fractional order systems.

Aghababa [38] introduced a suitable robust SMC law to realize control in a given finite time for integer-order nonautonomous chaotic systems. By use of the SMC approach, a feedback control has been designed to guarantee asymptotical stability of the chaotic systems in [39]. Combined with a global SMC, Saleh et al. [40] presented a novel adaptive stabilization technique for disturbed chaotic flow. Motivated by the research in SMC of the constant FO chaotic system, many researchers have exploited the VFO operators to investigate the dynamical and control problems [17, 4144]. However, from the mathematical analysis point, it has not been resolved for the problem of the theory to prove the stability of the controller. To the authors’ knowledge, due to the complexity of VFO systems, the results are rare on this topic. In addition, many studies have been devoted to the simulation of the fraction order system in recent years, and a large number of methods have emerged and the theories have gradually improved [4550].

In the present paper, the finite time control is discussed for VFO chaotic systems in the presence of uncertainties and external disturbances. The Lyapunov direct method is extended to the VFO form, and a finite time stability theorem is proposed. Based on the stable results, a VFO sliding mode manifold is designed. And in order to guarantee the finite time reach of the system state trajectories to the above sliding mode manifold, a SMC law is designed in the VFO form. In this paper, our main contribution is to realize the stabilization of variable fractional order uncertain systems in finite time by the SMC approach. Moreover, the theoretical proof is given by use of the variable fractional order Lyapunov theorem. Lastly, simulation results are proposed to display the effectiveness and usefulness of the theoretical analysis.

The organization of this article is presented as follows. The basic definitions of VFO calculus and the basic description of the system are given in Sect. 2. Section 3 is devoted to obtaining the stability of VFO differential system in a finite time and provide the design strategy of the VFOSMC. Section 4 provides the numerical simulations for the viability of the theoretical results.

Preliminaries

The following definitions of VFO operators are adopted in this article.

Definition 2.1

([51])

When the order \(q(t)\) depends on time t, there is an obvious way for accounting for the variation:

$$ I^{q(t)}_{t}x(t)=\frac{1}{\Gamma (q(t))} \int _{t_{0}}^{t}(t-s)^{q(t)-1}x(s)\,ds,\quad 0< q(t)< 1, $$
(1)

provided the integration is defined on \(t\in [t_{0},T]\), and \(\Gamma (\cdot )\) is the gamma function.

Definition 2.2

([51])

The definition of VFO derivatives is as follows:

$$ {}^{C}D_{t}^{q(t)}x(t)= \frac{1}{\Gamma (1-q(t))} \int _{t_{0}}^{t}(t-s)^{-q(t)}x'(s)\,ds,\quad 0< q(t)< 1, $$
(2)

provided the integration is defined on \(t\in [t_{0},T]\), and \(\Gamma (\cdot )\) is the gamma function.

When \(q(t)\) is a constant, Definitions 2.1 and 2.1 are reduced to the Caputo constant fractional order operators.

The n-dimensional uncertain VFO nonlinear dynamical system is described by the following equations:

$$ \textstyle\begin{cases} {}^{C}_{t_{0}}D_{t}^{q(t)}x_{1}(t)=f_{1}(t,X)+\Delta f_{1}(t,X)+d_{1}(t)+u_{1}(t), \\ {}^{C}_{t_{0}}D_{t}^{q(t)}x_{2}(t)=f_{2}(t,X)+\Delta f_{2}(t,X)+d_{2}(t)+u_{2}(t), \\ \ldots \\ {}^{C}_{t_{0}}D_{t}^{q(t)}x_{n}(t)=f_{n}(t,X)+\Delta f_{n}(t,X)+d_{n}(t)+u_{n}(t), \end{cases} $$
(3)

where \(0< q_{1}\leq q(t)\leq q_{2}<1\), \(q_{1}\), \(q_{2}\) are finite constant. \(X(t)=[x_{1}(t),x_{2}(t),\ldots,x_{n}(t)]^{T}\in R^{n}\) is the states vector, \(f_{i}(t,X)\in R\), \(i=1,2,\ldots,n\), denotes a nonlinear function of X and t, \(u_{i}(t)\in R\), \(i=1,2,\ldots,n\), is the control input. \(d_{i}(t)\in R\) represents an unknown model uncertainty, and \(\Delta f_{i}(t,X)\in R\) represents the external disturbances of the system for \(i=1,2,\ldots,n\), which is required to satisfy the following assumptions.

Assumption 1

Suppose that the unknown model uncertainty \(\Delta f_{i}(t,X)\) for \(i=1, 2,\ldots , n\) is differentiable and satisfies

$$\begin{aligned} \bigl\vert D^{q(t)}\Delta f_{i}(t,X) \bigr\vert \leq M_{i}^{\Delta f}, \end{aligned}$$

where \(M_{i}^{\Delta f}>0\) is a constant which is known for \(i=1, 2,\ldots , n\).

Assumption 2

Suppose that the external disturbance \(d_{i}(t)\) is differentiable for \(i=1, 2,\ldots , n\),

$$\begin{aligned} \bigl\vert D^{q(t)}d_{i}(t) \bigr\vert \leq M_{i}^{d}, \end{aligned}$$

where \(M_{i}^{d}>0\) is a constant which is known for \(i=1, 2,\ldots , n\).

Remark 1

From the applied points of view, the uncertain terms and external disturbances are always bounded, a designed control input always has a finite magnitude. Thus, the above assumption is realistic and not restricting.

Main results

Stability analysis of the VFO system

This part is to derive some criteria of stability for the VFO differential systems.

Definition 3.1

The constant \(x_{0}\) is an equilibrium point of the VFO system

$$ \textstyle\begin{cases} {}^{C}_{t_{0}}D_{t}^{q(t)}x(t)=f(t,x(t)),\quad t\in (t_{0},T], \\ x(t_{0})=x_{0}, \end{cases} $$
(4)

where \(0< q_{1}\leq q(t,x(t))\leq q_{2}<1\) if \(f(t,x_{0})=0\).

The following theorem is an extended Lyapunov direct method into a VFO form, which provides the basis of the asymptotic stability analysis for the VFO system.

Theorem 3.2

Suppose that \(x=0\) is an equilibrium point of VFO system (4) and \(D\subset R^{n}\) is a domain containing the origin. If there exists a continuously differential function \(V(t,x(t)):[0,\infty )\times D\rightarrow R\) such that, for arbitrary positive constants \(\alpha _{1}\), \(\alpha _{2}\), \(\alpha _{3}\), a, b, the following inequality holds:

$$ \textstyle\begin{cases} \alpha _{1} \Vert x \Vert ^{a}\leq V(t,x(t))\leq \alpha _{2} \Vert x \Vert ^{ab}, \\ D^{q(t)}V(t,x(t))\leq -\alpha _{3} \Vert x \Vert ^{ab}, \end{cases} $$
(5)

where \(x\in D\), \(0< q_{1}\leq q(t)\leq q_{2}<1\), \(t\in [0,\infty )\).

Then the equilibrium point of system (4) is asymptotically stable.

Proof

Divide the interval \([0,\infty )\) into the subintervals \([t_{k},t_{k+1}]\), \(k=1,2,\ldots ,n,\ldots \) , which requires \(\lim_{k\rightarrow \infty }t_{k}=\infty \). Denote \(\chi _{k}=t_{k+1}-t_{k}\) with \(\inf_{k}\chi _{k}>0\) and \(0<\sup_{k}\chi _{k}<1\) for \(k=1,2,\ldots ,n,\ldots \) , then the following inequality is obtained:

$$ \chi _{k}^{-q(t)}\leq \textstyle\begin{cases} (\frac{1}{\chi _{k}})^{q_{2}},&0< \chi _{k}< 1, \\ (\frac{1}{\chi _{k}})^{q_{1}},&1\leq \chi _{k}, \end{cases} $$
(6)

with \(\widehat{\chi _{k}}=max\{(\frac{1}{\chi _{k}})^{q_{2}},( \frac{1}{\chi _{k}})^{q_{1}}\}\). According to the property of \(\Gamma (t)\) on \((0,1]\), we have \(\Gamma (1-q_{1})\leq \Gamma (1-q(t))\leq \Gamma (1-q_{2})\), which gives that, for \(t\in [t_{k},t_{k+1})\), \(k=1,\ldots ,n,\ldots \) , together with Definition 2.1 and Definition 2.2

$$\begin{aligned} {}^{C}_{t_{k}}D^{q(t)}_{t}V \bigl(t,x(t)\bigr)&= \int _{t_{k}}^{t} \frac{(t-s)^{-q(t)}}{\Gamma (1-q(t))}V' \bigl(s,x(s)\bigr)\,ds \\ &\geq \frac{1}{\Gamma (1-q_{1})} \int _{t_{k}}^{t}(t-s)^{-q(t)}V' \bigl(s,x(s)\bigr)\,ds \\ &\geq \frac{\widehat{\chi _{k}}}{\Gamma (1-q_{1})} \int _{t_{k}}^{t}\biggl( \frac{t-s}{\chi _{k}} \biggr)^{-q_{2}}V'\bigl(s,x(s)\bigr)\,ds \\ &=\frac{1}{H_{k}}{{}^{C}_{t_{k}}D^{q_{2}}_{t}}V \bigl(t,x(t)\bigr), \end{aligned}$$
(7)

where \(H_{k}= \frac{\Gamma (1-q_{1})}{\chi _{k}^{q_{2}}\widehat{\chi _{k}}\Gamma (1-q_{2})}>0\). Thus, we obtain

$$ {{}^{C}_{t_{k}}D_{t}^{q_{2}}}V\bigl(t,x(t) \bigr)\leq H_{k} {{}^{C}_{t_{k}}D_{t}^{q(t)}}V \bigl(t,x(t)\bigr),\quad t\in [t_{k},t_{k+1}), k=1,\ldots ,n, $$

then we get

$$ {{}^{C}_{0}D_{t}^{q_{2}}}V \bigl(t,x(t)\bigr)\leq H {{}^{C}_{0}D_{t}^{q(t)}}V \bigl(t,x(t)\bigr)\quad \text{for } t\in [0,\infty ), $$
(8)

where \(H=H_{i}\) for \(t\in [t_{i},t_{i+1})\). From inequalities (5) and (8), it is obtained that

$$\begin{aligned} {{}^{C}_{0}D_{t}^{q_{2}}}V \bigl(t,x(t)\bigr) \leq & H {{}^{C}_{0}D_{t}^{q(t)}}V \bigl(t,x(t)\bigr) \\ \leq & -H\alpha _{3}\alpha _{2}^{-1}V \bigl(t,x(t)\bigr). \end{aligned}$$
(9)

Then a nonnegative function \(\overline{G(t)}\) exists such that

$$ {{}^{C}_{0}D_{t}^{q_{2}}}V \bigl(t,x(t)\bigr)+\overline{G(t)}=-\alpha _{3}\alpha _{2}^{-1} HV\bigl(t,x(t)\bigr). $$
(10)

Applying the Laplace transform to (10) with \(V(0)=V(0,x(0))\), we derive that

$$ s^{q_{2}}V(s)-V(0)s^{q_{2}-1}+\overline{G(s)}=-\alpha _{3}\alpha _{2}^{-1} HV(s), $$

where \(V(s)=L\{V(t,x(t))\}\) and \(G(s)=L\{D(t)\}\). After some manipulations, it is obtained that

$$ V(s)= \frac{V(0)s^{q_{2}-1}-G(s)}{s^{q_{2}}+H\alpha _{3}\alpha _{2}^{-1}}. $$

By the inverse Laplace transform, one can get

$$\begin{aligned} V(t)&=V(0)E_{q_{2}}\bigl(-\alpha _{3}\alpha _{2}^{-1}Ht^{q_{2}}\bigr)- \int _{0}^{t}(t-s)^{q_{2}-1}E_{q_{2},q_{2}} \bigl[-H \alpha _{3}\alpha _{2}^{-1}(t-s)^{q_{2}} \bigr]G(s)\,ds \\ &\leq V(0)E_{q_{2}}\bigl(-\alpha _{3}\alpha _{2}^{-1}Ht^{q_{2}}\bigr). \end{aligned}$$

In terms of inequality (5), it implies that

$$ \Vert x \Vert \leq \bigl[V(0)\alpha _{1}^{-1}E_{q_{2}} \bigl(-H\alpha _{3}\alpha _{2}^{-1}t^{q_{2}} \bigr)\bigr]^{ \frac{1}{a}}. $$

Thus, the proof is completed. □

The coming definition and theorems are concerned with finite time stability of the systems.

Definition 3.3

Assume that D is some open connected set, \(W(Y,t)\) is a function of variables Y, t. Then a function \(Y(t)\), \(t_{0}\leq t < T\), \(T>t_{0}\) is called a solution of the differential inequality

$$\begin{aligned} {}^{C}_{t_{0}}D^{q}Y(t)\leq W \bigl(Y(t),t\bigr) \end{aligned}$$
(11)

on \([t_{0}, T)\) if \(Y(t)\), and its fractional order derivative satisfies inequality (11) on \([t_{0}, T)\).

Theorem 3.4

Suppose that \(W(X,t)\) is continuous on \(D\in R^{2}\), which is an open connected set, and \(X(t)\) is a solution of the following initial value problem:

$$\begin{aligned} {}^{C}_{t_{0}}D^{q}X(t)=\lambda X(t), X(t_{0})=x_{0}, \end{aligned}$$

on \([t_{0}, T]\), where λ is a known constant. If \(Y(t)\) is a solution of inequality (11) on \([t_{0}, T]\) with \(Y(t_{0})\leq X(t_{0})\), then \(Y(t)\leq X(t)\) for \(t_{0}\leq t \leq T\).

Proof

Set \(P(t)=Y(t)-X(t)\), taking the FO derivative on time yields

$$\begin{aligned} {}^{C}_{t_{0}}D^{q}P(t)&={{}^{C}_{t_{0}}D^{q}} \bigl(Y(t)-X(t)\bigr) \\ &={{}^{C}_{t_{0}}D^{q}}Y(t)-{{}^{C}_{t_{0}}D^{q}} X(t) \\ &\leq \lambda Y(t)-\lambda X(t) \\ &=\lambda P(t), \end{aligned}$$

which combined with \(Y(t_{0})\leq X(t_{0})\) gives \(P(t_{0})\leq 0\). The following is to validate the inequality holds:

$$ P(t)\leq 0,\quad \forall t\in [t_{0},T). $$
(12)

By the contradiction method, if there exist \(t_{1}, t_{2} \in (t_{0},T)\), \(t_{1}< t_{2}\) satisfying

$$ \textstyle\begin{cases} P(t)< 0,&t\in (t_{0},t_{1}), \\ P(t)>0,&t\in (t_{1},t_{2}], \\ P(t_{1})=0,&t=t_{1}. \end{cases} $$

Suppose that \(\lambda >0\), applying the fractional operator \(I^{q}\) to the following inequality:

$$\begin{aligned} {}^{C}_{t_{0}}D^{q}P(t)\leq \lambda P(t), \end{aligned}$$

then it is obtained that

$$\begin{aligned} P(t_{1})-P(t_{0})\leq \frac{1}{\Gamma (q)} \int _{t_{0}}^{t_{1}}(t_{1}-s)^{q-1} \lambda P(s)\,ds. \end{aligned}$$

Since \(P(t)<0\) for \(t\in (t_{0},t_{1})\), then \(-P(t_{0})<0\), which is a contradiction. Assume \(\lambda <0\), following a similar approach, we have

$$\begin{aligned} P(t_{2})-P(t_{1})\leq \frac{1}{\Gamma (q)} \int _{t_{1}}^{t_{2}}(t_{2}-s)^{q-1} \lambda P(s)\,ds< 0, \end{aligned}$$

then \(P(t_{2})<0\), which means that inequality (12) holds. □

Theorem 3.5

Assume that \(V(t)\) is a continuous and positive definite function which satisfies

$$ {}^{C}_{t_{0}}D^{q_{2}}V(t)\leq -\alpha V(t) $$
(13)

for \(t\geq t_{0}\), where α is a positive constant. Then the following inequality can be got:

$$ V(t)\leq V(t_{0})E_{q_{2}}\bigl(-\alpha t^{q_{2}}\bigr) $$
(14)

for \(t_{0}\leq t< t^{*}\) with \(t^{*}=(\frac{\Gamma (q_{2}+1)}{\alpha })^{{q_{2}^{-1}}}\), where \(E_{q_{2}}(t)\) is a Mittag-Leffler function which is denoted by \(E_{q_{2}}(t)=\sum_{k=0}^{\infty } \frac{t^{k}}{\Gamma ({q_{2}}k+1)}\). Moreover, \(V(t_{0})\geq 0\) with any given \(t_{0}\), and \(V(t)=0\) for \(t\geq t^{*}\).

Proof

Consider the initial value problem of the FO differential systems:

$$ {}^{C}_{t_{0}}D^{q_{2}}X(t)=-\alpha X(t),\quad X(t_{0})=V(t_{0}). $$

Then this initial value problem has a unique solution as follows:

$$ X(t)= X(t_{0})E_{q_{2}}\bigl(-\alpha t^{q_{2}}\bigr) \quad \text{for } t_{0}\leq t. $$

Therefore, on the basis of Theorem 3.4, one obtains that

$$ V(t)\leq X(t)= V(t_{0})E_{q_{2}}\bigl(-\alpha t^{q_{2}} \bigr) \quad \text{for } t_{0} \leq t< t^{*}, $$

where \(t^{*}=(\frac{\Gamma (q_{2}+1)}{\alpha })^{{{q_{2}}^{-1}}}\) and \(V(t)=0\) for \(\forall t\geq t^{*}\). □

Finite time control of the VFO system by SMC approach

For VFO differential system (3), the VFO sliding mode is proposed as follows:

$$ s_{i}(t)={{}^{C}_{0}D^{q(t)}}x_{i}+ \beta _{i}x_{i}+\bar{\beta _{i}}sgn(x_{i}) \vert x_{i} \vert ^{q(t)}, $$
(15)

where \(\beta _{i}>0\), \(\bar{\beta _{i}}>0\) for \(i=1,2,\ldots , n\).

Remark 2

The representation of the sliding mode is related to the variable fractional order operator, and it is focused on the VFO systems. When the VFO parameter is a constant, it can be used to deal with the constant FO systems.

If the states of the system reach the sliding mode surface, then it is obtained that

$$ s_{i}(t)=0,\quad i=1,2,\ldots ,n. $$

Thus,

$$ {}^{C}_{0}D^{q(t)}x_{i}=- \beta _{i}x_{i}-\bar{\beta _{i}}sgn(x_{i}) \vert x_{i} \vert ^{q(t)}. $$
(16)

Lemma 3.6

Assume that \(x(t)\) is a continuous differential function, then the following inequality holds for any time instant \(t\geq 0\):

$$ \frac{1}{2} {{}^{C}_{0}D_{t}^{q(t)}}x^{2}(t) \leq x(t){{}^{C}_{0}D_{t}^{q(t)}}x(t),\quad 0< q(t)< 1. $$

Theorem 3.7

Consider the VFO sliding mode dynamics (16), then its state trajectories converge to zero asymptotically in a finite time.

Proof

Choose the Lyapunov functional as

$$ V_{1}(t)=\sum_{i=1}^{n}x_{i}^{2}(t), $$
(17)

and apply VFO derivative \(q(t)\) to Lyapunov function (17) with respect to time. We obtain that along with Lemma 3.6 by subsisting equation (16) of \({{}^{C}_{0}D^{q(t)}_{t}}x_{i}(t)\),

$$\begin{aligned} {{}^{C}_{0}D^{q(t)}_{t}}V_{1}(t) &=\sum_{i=1}^{n}{{}^{C}_{0}D^{q(t)}_{t}}x_{i}^{2}(t) \\ &\leq \sum_{i=1}^{n}x_{i}(t){{}^{C}_{0}D^{q(t)}_{t}}x_{i}(t) \\ &=\sum_{i=1}^{n}x_{i}(t) \bigl[-\beta _{i}x_{i}(t)- \bar{\beta _{i}}sgn(x_{i}) \vert x_{i} \vert ^{q(t)}\bigr] \\ &\leq -aV_{1}(t), \end{aligned}$$

where \(a=\min \{\beta _{i}, \bar{\beta _{i}}\}\). Then, according to Theorem 3.4 and Theorem 3.5, the state variables \(x_{i}\), \(i=1,2,\ldots ,n\), asymptotically tend to zero in a finite time.

The following step is to design a robust sliding control law based on the sliding mode approach. Then the state trajectories of the VFO system are forced to the sliding mode surface in a finite time. Subsequently, the control law is given as follows:

$$\begin{aligned} u_{i}(t)=u^{i}_{eq}+u^{i}_{sw} \end{aligned}$$
(18)

with

$$ \textstyle\begin{cases} u^{i}_{eq}=-f_{i}(t,X)-\bar{\beta _{i}}sgn(x_{i}) \vert x_{i} \vert ^{q(t)}- \beta _{i}x_{i}, \\ {}^{C}_{0}D^{q(t)}u^{i}_{sw}=-(M_{i}^{\Delta f}+M_{i}^{d})sgn(s_{i})- \xi ^{i}_{1}s_{i}-\xi ^{i}_{2}sgn(s_{i}), \end{cases} $$
(19)

where \(u^{i}_{eq}\) is the equivalent control, and \(u^{i}_{sw}\) is the reaching law with \(\bar{\beta _{i}}>0\), \(\beta _{i}>0\), \(\xi ^{i}_{1}>0\), \(\xi ^{i}_{2}>0\), \(i=1,2, \ldots ,n\). □

The next theorem ensures that system trajectories (3) converge to the sliding mode surface under the controller.

Theorem 3.8

Consider VFO system (3). If controller (18) is applied to system (3) with \(\bar{\beta _{i}}>0\), \(\beta _{i}>0\), \(\xi ^{i}_{1}>0\), \(\xi ^{i}_{2}>0\), then the states of system (3) are driven to reach to the sliding mode surface (15) asymptotically from the initial conditions in the finite time and stay on it forever.

Proof

The Lyapunov function is defined as

$$ V_{2}(s_{i})=\frac{1}{2}\sum _{i=1}^{n}s_{i}^{2}(t). $$
(20)

Calculating the variable fractional order derivative \(q(t)\) for \(V_{2}(s_{i})\) with respect to time, it is obtained from Lemma 3.6 that

$$\begin{aligned} {}^{C}_{0}D^{q(t)}_{t}V_{2}(s_{i})& \leq \sum_{i=1}^{n}s_{i} {{}^{C}_{0}D^{q(t)}_{t}}s_{i}. \end{aligned}$$
(21)

According to the sliding mode (15), it is rewritten under controller (18) as follows:

$$\begin{aligned} {}^{C}_{0}D^{q(t)}_{t}V_{2}(s_{i})& \leq \sum_{i=1}^{n}s_{i} \bigl[{{}^{C}_{0}D^{q(t)}_{t}} \bigl({{}^{C}_{0}D^{q(t)}_{t}}x_{i}+ \beta _{i}x_{i}+\bar{\beta _{i}}sgn(x_{i}) \vert x_{i} \vert ^{q(t)}\bigr)\bigr] \\ &=\sum_{i=1}^{n}s_{i} \bigl[{{}^{C}_{0}D^{q(t)}_{t}} \bigl(f_{i}(t,X)+ \Delta f_{i}(t,X)+d_{i}(t)+u_{i}(t)+ \beta _{i}x_{i}+\bar{\beta _{i}}sgn(x_{i}) \vert x_{i} \vert ^{q(t)}\bigr)\bigr] \\ &=\sum_{i=1}^{n}s_{i} \bigl[{{}^{C}_{0}D^{q(t)}_{t}} \bigl(f_{i}(t,X)+ \Delta f_{i}(t,X)+d_{i}(t)-f_{i}(t,X)- \beta _{i}x_{i}- \bar{\beta _{i}}sgn(x_{i}) \vert x_{i} \vert ^{q(t)} \\ & \quad {}+u^{i}_{sw}+\beta _{i}x_{i}+ \bar{\beta _{i}}sgn(x_{i}) \vert x_{i} \vert ^{q(t)}\bigr)\bigr] \\ &=\sum_{i=1}^{n}s_{i} \bigl[{{}^{C}_{0}D^{q(t)}_{t}}\Delta f_{i}(t,X)+{{}^{C}_{0}D^{q(t)}_{t}}d_{i}(t)+{{}^{C}_{0}D^{q(t)}_{t}}u^{i}_{sw} \bigr] \\ &\leq \sum_{i=1}^{n} \bigl[M_{i}^{\Delta } \vert s_{i} \vert +M_{i}^{d} \vert s_{i} \vert -M_{i}^{ \Delta } \vert s_{i} \vert -M_{i}^{d} \vert s_{i} \vert -\xi ^{i}_{1}s^{2}_{i}-\xi ^{i}_{2} \vert s_{i} \vert \bigr] \\ &=-\sum_{i=1}^{n}\xi ^{i}_{1}s^{2}_{i}-\sum _{i=1}^{n} \xi ^{i}_{2} \vert s_{i} \vert \\ &\leq -\xi V_{2}(s_{i}), \end{aligned}$$

where \(\xi =\min \{\xi ^{1}_{1},\xi ^{2}_{1},\ldots,\xi ^{n}_{1}\}\) is a constant.

From Theorem 3.4 and Theorem 3.5, we obtain that the state trajectories of VFO system (3) will be driven to \(s_{i}(t)=0\), \(i=1,2,\ldots ,n\), as \(t\rightarrow \infty \) in a finite time and stay on it forever. Combined with Theorem 3.7 and Theorem 3.8, the trajectories will converge to zero asymptotically in a finite time. □

Numerical simulation

The simulation results are presented to validate our theoretical results in this section.

Control of the VFO brushless motor system by SMC

The VFO brushless motor system is stated as follows:

$$ \textstyle\begin{cases} {}^{C}_{0}D^{q(t)}x_{1}=-0.875x_{1}+x_{2}x_{3}+\Delta f_{1}(X,t)+d_{1}(t)+u_{1}(t), \\ {}^{C}_{0}D^{q(t)}x_{2}=-x_{2}+55x_{3}-x_{1}x_{3}+\Delta f_{2}(X,t)+d_{2}(t)+u_{2}(t), \\ {}^{C}_{0}D^{q(t)}x_{3}=4(x_{2}-x_{3})+\Delta f_{3}(X,t)+d_{3}(t)+u_{3}(t), \end{cases} $$
(22)

where \(\Delta f_{i}(X,t)\), \(i=1,2,3\), and \(d_{i}(t)\), \(i=1,2,3\), denote the perturbation and uncertainty terms of the system, respectively,

$$ \textstyle\begin{cases} \Delta f_{1}(X,t)+d_{1}(t)=-0.15\cdot sin(2t)+0.2\cdot cos(3t)x_{1}, \\ \Delta f_{2}(X,t)+d_{2}(t)=0.2\cdot sin(3t)+0.25\cdot sin(4t)x_{2}, \\ \Delta f_{3}(X,t)+d_{3}(t)=-0.25\cdot cos(4t)+0.3\cdot sin(2t)x_{3}. \end{cases} $$
(23)

Under the initial conditions

$$ x_{1}(0)=10,\qquad x_{2}(0)=-5,\qquad x_{3}(0)=5, $$

system (22) is chaotic behavior with the following VFO:

$$ q(t)= \textstyle\begin{cases} 0.96+0.002t/T,&t\in [0,T], \\ 0.96,&t>T. \end{cases} $$

The chaotic trajectories of uncontrolled system (22) are illustrated in Fig. 1. From the figure, we can find that the system presents chaotic behavior under the initial value condition.

Figure 1
figure1

The chaotic trajectories of uncontrolled system (22)

According to the sliding mode surface (15), the following sliding mode surfaces in this simulation are utilized:

$$ s_{i}(t)={{}^{C}_{0}D^{q(t)}}x_{i}+ \beta _{i}x_{i}+\bar{\beta _{i}}sgn(x_{i}) \vert x_{i} \vert ^{q(t)},\quad i=1,2,3. $$
(24)

Subsequently, the controller is designed according to (18) in order to stabilize the chaotic system

$$ \textstyle\begin{cases} u_{1}(t)=0.875x_{1}-x_{2}x_{3}-\bar{\beta }sgn(x_{1}) \vert x_{1} \vert ^{q(t)}- \beta x_{1}+u^{1}_{sw}, \\ {}^{C}_{0}D^{q(t)}u^{1}_{sw}=-(M_{1}^{\Delta f}+M_{1}^{d})sgn(s_{1})- \xi ^{(1)}_{1}s_{1}-\xi ^{(1)}_{2}sgn(s_{1}), \\ u_{2}(t)=x_{2}-55x_{3}+x_{1}x_{3}-\bar{\beta }sgn(x_{2}) \vert x_{2} \vert ^{q(t)}- \beta x_{2}+u^{2}_{sw}, \\ {}^{C}_{0}D^{q(t)}u^{2}_{sw}=-(M_{2}^{\Delta f}+M_{2}^{d})sgn(s_{2})- \xi ^{(2)}_{1}s_{2}-\xi ^{(2)}_{2}sgn(s_{2}), \\ u_{3}(t)=-4(x_{2}-x_{3})-\bar{\beta }sgn(x_{3}) \vert x_{3} \vert ^{q(t)}-\beta x_{3}+u^{3}_{sw}, \\ {}^{C}_{0}D^{q(t)}u^{3}_{sw}=-(M_{3}^{\Delta f}+M_{3}^{d})sgn(s_{3})- \xi ^{(3)}_{1}s_{3}-\xi ^{(3)}_{2}sgn(s_{3}), \end{cases} $$
(25)

the constant parameters are

$$ \textstyle\begin{cases} \beta _{1}=8,\qquad \bar{\beta _{1}}=6,\qquad \beta _{2}=4,\qquad \bar{\beta _{2}}=5,\qquad \beta _{3}=3,\qquad \bar{\beta _{3}}=2, \\ M_{i}^{\Delta f}=M_{i}^{d}=0.05, \qquad \xi ^{(i)}_{1}=0.5,\qquad \xi ^{(i)}_{2}=0.4,\quad i=1,2,3. \end{cases} $$
(26)

Then the trajectories of the system under the controller are depicted in Fig. 2, which shows that the trajectories of the system can be stabilized to the origin. The time responses are showed in Fig. 3 and Fig. 4 for the control inputs (25) and the sliding mode surfaces (24), which show that the time responses have been driven to the origin in a finite time. It is concluded that the state variables converge to the origin in a finite time. Moreover, the chaotic behavior is suppressed.

Figure 2
figure2

The state trajectories of system (22) under controller (25)

Figure 3
figure3

The history of control input (25)

Figure 4
figure4

The time response of the sliding mode surface (24)

Control of the VFO electrostatic transducer by SMC approach

Consider the VFO uncertain nonlinear system

$$ \textstyle\begin{cases} {}^{C}_{0}D^{q(t)}x_{1}=x_{2}+u_{1}(t), \\ {}^{C}_{0}D^{q(t)}x_{2}=x_{3}+0.15cos(2t)+u_{2}(t), \\ {}^{C}_{0}D^{q(t)}x_{3}=-(6.8-0.2sin(t))x_{1}-3.92x_{2}-x_{3}+(1+0.3*cos(0.5t))x^{2}_{1}(t) \\ \hphantom{{}^{C}_{0}D^{q(t)}x_{3}=}{}+1.2*cos(3t)+u_{3}(t), \end{cases} $$
(27)

with the model uncertainty terms of the system as follows:

$$ \textstyle\begin{cases} \Delta f_{1}(X,t)+d_{1}(t)=0, \\ \Delta f_{2}(X,t)+d_{2}(t)=0.15cos(2t), \\ \Delta f_{3}(X,t)+d_{3}(t)=1.2cos(3t). \end{cases} $$
(28)

By (15), the sliding mode is designed as

$$ s_{i}(t)=D^{q(t)}x_{i}+\beta _{i}x_{i}+\bar{\beta _{i}}sgn(x_{i}) \vert x_{i} \vert ^{q(t)},\quad i=1,2,3, $$
(29)

in terms of (18), the controller is proposed as

$$ \textstyle\begin{cases} u_{1}(t)=-x_{2}-\bar{\beta }sgn(x_{1}) \vert x_{1} \vert ^{q(t)}-\beta x_{1}+u^{sw}_{1}, \\ {}^{C}_{0}D^{q(t)}u^{sw}_{1}=-(M_{1}^{\Delta f}+M_{1}^{d})sgn(s_{1})- \xi ^{(1)}_{1}s_{1}-\xi ^{(1)}_{2}sgn(s_{1}), \\ u_{2}(t)=-x_{3}-\bar{\beta }sgn(x_{2}) \vert x_{2} \vert ^{q(t)}-\beta x_{2}+u^{sw}_{2}, \\ {}^{C}_{0}D^{q(t)}u^{sw}_{2}=-(M_{2}^{\Delta f}+M_{2}^{d})sgn(s_{2})- \xi ^{(2)}_{1}s_{2}-\xi ^{(2)}_{2}sgn(s_{2}), \\ u_{3}(t)=(6.8-0.2sin(t))x_{1}+3.92x_{2}+x_{3}-(1+0.3cos(0.5t))x^{2}_{1}- \bar{\beta }sgn(x_{3}) \vert x_{3} \vert ^{q(t)} \\ \hphantom{u_{3}(t)=}{}-\beta x_{3}+u^{3}_{sw}, \\ {}^{C}_{0}D^{q(t)}u^{3}_{sw}=-(M_{3}^{\Delta f}+M_{3}^{d})sgn(s_{3})- \xi ^{(3)}_{1}s_{3}-\xi ^{(3)}_{2}sgn(s_{3}), \end{cases} $$
(30)

the parameters satisfy

$$ \beta _{i}= \bar{\beta _{i}}=2, \qquad M_{i}^{\Delta f}=M_{i}^{d}=1,\qquad \xi ^{(i)}_{1}=4,\qquad \xi ^{(i)}_{2}=5, \quad i=1,2,3,4. $$
(31)

Under the initial value

$$ x_{1}(0)=1,\qquad x_{2}(0)=1,\qquad x_{3}(0)=1, $$

system (27) shows chaotic phenomena, which is indicated by Fig. 5. By using the proposed SMC (30), the state trajectories of system (27) are described by Fig. 6. From the figure, we can see that the system converges to zero quickly. Figure 7 and Fig. 8 demonstrate that the sliding mode surface (29) responses converge to zero and controller (30) can stabilize system (27) effectively in a finite time. Therefore, the control inputs give a good performance in practice.

Figure 5
figure5

The chaotic trajectories of uncontrolled system (27)

Figure 6
figure6

The state trajectories of the system under controller (30)

Figure 7
figure7

The time response of the sliding mode surface (29)

Figure 8
figure8

The history of the control input (30)

Conclusion

This paper investigates a control problem of the VFO nonlinear system. A robust controller is proposed to stabilize the system in the present with uncertainty and external disturbance. By applying the sliding mode control to the system, a VFO derivative sliding mode surface is designed. And then, a control law has been designed which is free of chattering signal for a kind of VFO system. According to the proposed stability criteria, the finite time stability of the controlled systems has been proved. Lastly, numerical results are provided to illustrate the validity and efficiency of the proposed FO controllers. Furthermore, the proposed results motivate the development of theoretical and practical tools for implementing the proposed controllers to the fractional model.

Availability of data and materials

This paper is a theoretical work, and it is not based on any data.

References

  1. 1.

    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

    MATH  Book  Google Scholar 

  2. 2.

    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  3. 3.

    Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20, 763–769 (2016)

    Article  Google Scholar 

  4. 4.

    Atangana, A., Koca, I.: Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order. Chaos Solitons Fractals 89, 447–454 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Li, C., Zhang, F.: A survey on the stability of fractional differential equations. Eur. Phys. J. Spec. Top. 193(1), 27–47 (2011)

    Article  Google Scholar 

  6. 6.

    Li, C., Li, Z.: Asymptotic behaviours of solution to Caputo–Hadamard fractional partial differential equation with fractional Laplacian. Int. J. Comput. Math. 98, 305–339 (2021)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Jiang, J., Cao, D., Chen, H., Zhao, K.: The vibration transmissibility of a single degree of freedom oscillator with nonlinear fractional order damping. Int. J. Syst. Sci. 48(11), 2379–2393 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Jiang, J., Guirao, J.L.G., Chen, H., Cao, D.: The boundary control strategy for a fractional wave equation with external disturbances. Chaos Solitons Fractals 121, 92–97 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Gao, W., Veeresha, P., Baskonus, H.M., Prakasha, D., Kumar, P.: A new study of unreported cases of 2019-nCOV epidemic outbreaks. Chaos Solitons Fractals 138, 109929 (2020)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Shiri, B., Wu, G.C., Baleanu, D.: Collocation methods for terminal value problems of tempered fractional differential equations. Appl. Numer. Math. 156, 385–395 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Ilhan, E., Kymaz, I.: A generalization of truncated m-fractional derivative and applications to fractional differential equations. Appl. Math. Nonlinear Sci. 5(1), 171–188 (2020)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Baleanu, D., Jajarmi, A., Sajjadi, S.S., Asad, J.H.: The fractional features of a harmonic oscillator with position-dependent mass. Commun. Theor. Phys. 72(5), 055002 (2020)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Veeresha, P., Prakasha, D.G., Baskonus, H.M., Yel, G.: An efficient analytical approach for fractional Lakshmanan-Porsezian-Daniel model. Math. Methods Appl. Sci. 43(7), 4136–4155 (2020)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Al-Refai, M.: Maximum principles for nonlinear fractional differential equations in reliable space. Fundam. Inform. 6(2), 95–99 (2020)

    Google Scholar 

  15. 15.

    Sweilam, N.H., Hasan, M.M.A.: Efficient method for fractional Levy-Feller advection-dispersion equation using Jacobi polynomials. Prog. Fract. Differ. Appl. 6(2), 115–128 (2020)

    Article  Google Scholar 

  16. 16.

    Zhang, Y., Cattani, C., Yang, X.J.: Local fractional homotopy perturbation method for solving non-homogeneous heat conduction equations in fractal domains. Entropy 17(10), 6753–6764 (2015)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Coimbra, C.F.: Mechanics with variable-order differential operators. Ann. Phys. 12(11–12), 692–703 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Effati, S., Nik, H.S., Jajarmi, A.: Hyperchaos control of the hyperchaotic Chen system by optimal control design. Nonlinear Dyn. 73(1–2), 499–508 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Jajarmi, A., Baleanu, D.: On the fractional optimal control problems with a general derivative operator. Asian J. Control 40(2) (2019). https://doi.org/10.1002/asjc.2282

    Article  Google Scholar 

  20. 20.

    Jajarmi, A., Hajipour, M.: An efficient recursive shooting method for the optimal control of time-varying systems with state time-delay. Appl. Math. Model. 40(4), 2756–2769 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Jajarmi, A., Pariz, N., Effati, S., Kamyad, A.V.: Infinite horizon optimal control for nonlinear interconnected large-scale dynamical systems with an application to optimal attitude control. Asian J. Control 14(5), 1239–1250 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., Feliu, V.: Fractional-Order Systems and Controls: Fundamentals and Applications. Springer, New York (2010)

    MATH  Book  Google Scholar 

  23. 23.

    Chen, D., Zhang, R., Ma, X., Liu, S.: Chaotic synchronization and anti-synchronization for a novel class of multiple chaotic systems via a sliding mode control scheme. Nonlinear Dyn. 69(1–2), 35–55 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Azar, A.T., Vaidyanathan, S., Ouannas, A.: Fractional Order Control and Synchronization of Chaotic Systems, vol. 688. Springer, Berlin (2017)

    MATH  Book  Google Scholar 

  25. 25.

    Zuñiga-Aguilar, C., Gómez-Aguilar, J., Escobar-Jiménez, R., Romero-Ugalde, H.: Robust control for fractional variable-order chaotic systems with non-singular kernel. Eur. Phys. J. Plus 133(1), 1–13 (2018)

    MATH  Article  Google Scholar 

  26. 26.

    Edwards, C., Spurgeon, S.: Sliding Mode Control: Theory and Applications. CRC Press, Florida (1998)

    MATH  Book  Google Scholar 

  27. 27.

    Dadras, S., Momeni, H.R.: Control of a fractional-order economical system via sliding mode. Phys. A, Stat. Mech. Appl. 389(12), 2434–2442 (2010)

    Article  Google Scholar 

  28. 28.

    Utkin, V.: Sliding Modes in Control and Optimization. Springer, New York (2013)

    Google Scholar 

  29. 29.

    Pisano, A., Rapaić, M.R., Jeličić, Z.D., Usai, E.: Sliding mode control approaches to the robust regulation of linear multivariable fractional-order dynamics. Int. J. Robust Nonlinear Control 20(18), 2045–2056 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Jakovljević, B., Pisano, A., Rapaić, M.R., Usai, E.: On the sliding-mode control of fractional-order nonlinear uncertain dynamics. Int. J. Robust Nonlinear Control 26, 782–798 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Aghababa, M.P.: A novel terminal sliding mode controller for a class of non-autonomous fractional-order systems. Nonlinear Dyn. 73(1–2), 679–688 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Yin, C., Chen, Y.Q., Zhong, S.: Fractional-order sliding mode based extremum seeking control of a class of nonlinear systems. Automatica 50(12), 3173–3181 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Chen, D., Zhang, R., Sprott, J.C., Chen, H.: Synchronization between integer-order chaotic systems and a class of fractional order chaotic systems via sliding mode control. Chaos, Interdiscip. J. Nonlinear Sci. 22(2), 023130 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Yin, C., Dadras, S., Zhong, S., Chen, Y.: Control of a novel class of fractional-order chaotic systems via adaptive sliding mode control approach. Appl. Math. Model. 37(4), 2469–2483 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Vinagre, B.M., Petrá, I., Podlubny, I.: Using fractional order adjustment rules and fractional order reference models in model-reference adaptive control. Sci. World J. 29(1–4), 269–279 (2002)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Razminia, A., Baleanu, D.: Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics 23(7), 873–879 (2013)

    Article  Google Scholar 

  37. 37.

    Yin, C., Cheng, Y., Chen, Y.Q., Stark, B., Zhong, S.: Adaptive fractional-order switching-type control method design for 3D fractional order nonlinear systems. Nonlinear Dyn. 82(1–2), 39–52 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Aghababa, M.P.: A fractional sliding mode for finite-time control scheme with application to stabilization of electrostatic and electromechanical transducers. Appl. Math. Model. 39(20), 6103–6113 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Hua, W., Han, Z.Z., Xie, Q.Y., Wei, Z.: Sliding mode control for chaotic systems based on LMI. Commun. Nonlinear Sci. Numer. Simul. 14(4), 1410–1417 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    Mobayen, S., Ma, J., Pujol-Vazquez, G., Acho, L., Zhu, Q.: Adaptive finite-time stabilization of chaotic flow with a single unstable node using a nonlinear function-based global sliding mode. Iran. J. Sci. Technol. Trans. Electr. Eng. 43, 339–347 (2019)

    Article  Google Scholar 

  41. 41.

    Soon, C.M., Coimbra, C.F.M., Kobayashi, M.H.: The variable viscoelasticity oscillator. Ann. Phys. 14(6), 378–389 (2005)

    MATH  Article  Google Scholar 

  42. 42.

    Diaz, G., Coimbra, C.F.M.: Nonlinear dynamics and control of a variable order oscillator with application to the Van der Pol equation. Nonlinear Dyn. 56(1–2), 145–157 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Jiang, J., Chen, H., Guirao, J.L., Cao, D.: Existence of the solution and stability for a class of variable fractional order differential systems. Chaos Solitons Fractals 128, 269–274 (2019)

    MathSciNet  Article  Google Scholar 

  44. 44.

    Jiang, J., Cao, D., Chen, H.: Sliding mode control for a class of variable-order fractional chaotic systems. J. Franklin Inst. 357(15), 10127–10158 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Baleanu, D., Ghanbari, B., Asad, J.H., Jajarmi, A., Pirouz, H.M.: Planar system-masses in an equilateral triangle: numerical study within fractional calculus. Comput. Model. Eng. Sci. 124(3), 953–968 (2020)

    Google Scholar 

  46. 46.

    Jajarmi, A., Baleanu, D.: A new iterative method for the numerical solution of high-order non-linear fractional boundary value problems. Front. Phys. 8, 220 (2020)

    Article  Google Scholar 

  47. 47.

    Odibat, Z., Baleanu, D.: Numerical simulation of initial value problems with generalized Caputo-type fractional derivatives. Appl. Numer. Math. 156, 94–105 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    Veeresha, P., Baskonus, H.M., Prakasha, D., Gao, W., Yel, G.: Regarding new numerical solution of fractional schistosomiasis disease arising in biological phenomena. Chaos Solitons Fractals 133, 109661 (2020)

    MathSciNet  Article  Google Scholar 

  49. 49.

    Yokuş, A., Gülbahar, S.: Numerical solutions with linearization techniques of the fractional Harry Dym equation. Appl. Math. Nonlinear Sci. 4(1), 35–42 (2019)

    MathSciNet  Article  Google Scholar 

  50. 50.

    Gao, W., Veeresha, P., Prakasha, D., Senel, B., Baskonus, H.M.: Iterative method applied to the fractional nonlinear systems arising in thermoelasticity with Mittag-Leffler kernel. Fractals 28(08), 2040040 (2020)

    Article  Google Scholar 

  51. 51.

    Lorenzo, C.F., Hartley, T.T.: Variable order and distributed order fractional operators. Nonlinear Dyn. 29(1–4), 57–98 (2002)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgements

Not applicable.

Funding

This paper has been supported by National Natural Science Foundation of China (No.12002194), Ministerio de Ciencia, Innovación y Universidades (No. PGC2018-097198-B-I00) and Fundación Séneca de la Región de Murcia (No.20783/PI/18).

Author information

Affiliations

Authors

Contributions

All authors have worked in research in an equal way to obtain the results of this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Juan L. G. Guirao.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jiang, J., Li, H., Zhao, K. et al. Finite time stability and sliding mode control for uncertain variable fractional order nonlinear systems. Adv Differ Equ 2021, 127 (2021). https://doi.org/10.1186/s13662-021-03286-z

Download citation

Keywords

  • Uncertain variable fractional order system
  • Finite time stability
  • Robust control
  • Sliding mode control method
\