This paper deals with the problem of pseudo-state sliding mode control of fractional SISO nonlinear systems with model inaccuracies. Firstly, a stable fractional sliding mode surface is constructed based on the Routh-Hurwitz conditions for fractional differential equations. Secondly, a sliding mode control law is designed using the theory of Mittag-Leffler stability. Further, we utilize the control methodology to synchronize two fractional chaotic systems, which serves as an example of verifying the viability and effectiveness of the proposed technique.

Fractional calculus has a long history of three hundred years, over which a firm theoretical foundation has been established. In the past few decades, with deep understanding of the power of fractional calculus and rapid development of computer technology, an enormous number of interesting and novel applications have emerged in physics, chemistry, engineering, finance, and other sciences [

Several pioneering attempts to develop fractional control methodologies have been made, such as TID controller [

Very recently, by applying fractional calculus to advanced nonlinear control theory, several fractional nonlinear control schemes have been proposed, such as fractional sliding mode control, fractional adaptive control, and fractional optimal control. Exactly, to design fractional sliding mode controls, various fractional sliding surfaces have been constructed in [

Motivated by the above contributions, this paper proposes a sliding mode control design for fractional SISO nonlinear systems in the presence of model inaccuracies. By constructing a stable fractional sliding mode surface on the basis of Routh-Hurwitz conditions, a sliding mode control law is designed. Further, stability analysis is performed using Mittag-Leffler stability theory. Comparing this with methods in the previous papers, we utilize the fractional derivative of the sliding mode surface instead of first-order derivative, to obtain the equivalent control law. Moreover, to carry out the stability analysis of the closed-loop fractional nonlinear system, we use the fractional derivative of the Lyapunov function candidate in terms of Theorem 2 in [

The rest of the paper is organized as follows. Section

Fractional calculus is a generalization of integration and differentiation to noninteger order fundamental operator

The three most frequently used definitions for fractional calculus are the Grünwald-Letnikov definition, the Riemann-Liouville definition, and the Caputo definition [

The Grünwald-Letnikov derivative definition of order

The Riemann-Liouville derivative definition of order

However, applied problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial conditions, which contains

The Caputo definition of fractional derivative can be written as

In the rest of the paper, we use the Caputo approach to describe the fractional systems and the Grünwald-Letnikov approach to propose numerical simulations. To simplify the notation, we denote the fractional-order derivative of order

Consider fractional SISO nonlinear systems

The estimation error on

The desired tracking signal is denoted as

In this paper, our goal is to design a suitable fractional sliding mode controller to make the output

Now we are ready to give the design steps.

Firstly, the tracking error is defined as

Then, the tracking error vector is defined as

A fractional sliding surface is proposed as

The fractional sliding surface (

Taking its

Substituting the second equation of system (

In terms of Assumption

Bounds of

Design the control law as

To ensure the stability of the fractional system (

Under Assumptions

Consider the following candidate Lyapunov function:

Taking its

Substituting (

Substituting the control law (

Substituting the estimation error (

In this section, we apply the fractional sliding mode control method proposed in Section

The fractional Arneodo system is represented as

The fractional Genesio-Tesi system is described as

Both of the fractional Arneodo system (

In terms of (

The control law is designed as

Initial conditions for the above two systems are, respectively, chosen as

Synchronization of the fractional Arneodo system and the fractional Genesio-Tesi system with the control input (

Synchronization errors with the control input (

Time history of fractional sliding mode with the control input (

Time history of the control input (

From the simulation results, we see that synchronization performance is excellent but is obtained at the price of high control chattering. It can be eliminated by replacing the discontinuous switching control law

Numerical simulations with the modified control law (

Synchronization of the fractional Arneodo system and the fractional Genesio-Tesi system with the control input (

Synchronization errors with the control input (

Time history of fractional sliding mode with the control input (

Time history of the control input (

From the above simulation results, one can easily see that the fractional Arneodo system and the fractional Genesio-Tesi system can be effectively synchronized via the proposed sliding mode control technique. Furthermore, the control chattering caused by the discontinuous control law (

In this paper, we have investigated the pseudo-state sliding control design for fractional SISO nonlinear systems with model inaccuracies. A stable fractional sliding mode surface has been constructed based on the Routh-Hurwitz conditions for fractional differential equations. Then, a sliding mode control law is designed using the Mittag-Leffler stability theorem. Finally, numerical simulations of synchronization of the fractional Arneodo system and the fractional Genesio-Tesi system have been performed to demonstrate the effectiveness of the proposed control technique.

As for the future perspectives, our research activities will be on,

designing adaptive sliding control to deal with parametric uncertainties in

generalizing the method to fractional MIMO nonlinear systems,

generalizing the method to incommensurate nonlinear systems.

The authors do not have any possible conflict of interests.