We give a state-feedback control method for fractional-order nonlinear systems subject to input saturation. First, a sufficient condition is derived for the asymptotical stability of a class of fractional-order nonlinear systems. Then based on Gronwall-Bellman lemma and a sector bounded condition of the saturation function, a linear state-feed back controller is designed. Finally, two simulation examples are presented to show the validity of the proposed method.

The idea of fractional calculus has been proposed since the development of the integer order calculus, with the initial works of Leibniz and L’Hospital. In spite of being such an old topic, the developments in this field were rather slow. However, in the last two decades, fractional differential equations have been utilized more and more to model various physical phenomena. In fact, recent developments of fractional calculus are dominated by modern examples of applications in differential and integral equations, physics, fluid mechanics, signal processing, mathematical biology, viscoelasticity, electrochemistry, and many others. And we can refer to [

As a result of the growing applications, the study of stability of fractional differential equations has attracted much attention. The earliest discussion on stability of fractional differential equations (FDEs) can be traced back to Matignon [

On the other hand, because fractional differential operators are nonlocal and have weakly singular kernels, some methods in handling integer-order systems cannot be simply extended to fractional-order systems. To the best of our knowledge, there are only several results on the stability of fractional-order nonlinear systems. For example, the definition of Mittag-Leffler stability and the fractional Lyapunov direct method to discuss the stability of fractional-order nonlinear dynamic systems are proposed in [

All real world technical systems are subjected to input constraints. In many engineered systems, input saturation does exist due to a limited size of sensors, actuators, and some interfacing devices. The existence of input saturation may decrease the control performance or cause oscillations and even lead to instability of the system [

Though significant research efforts have been put to the fractional-order time-invariant systems, fractional-order systems subject to input saturation have rarely been investigated in the literature. Here, with the help of the Laplace transform, Mittag-Leffler function, and Gronwall inequality, a state-feedback controller is designed for a class of fractional-order nonlinear systems subject to input saturation. Compared with [

The Caputo definition of fractional-order derivatives can be written as [

The Laplace transform of Caputo fractional derivative can be given as follows:

Throughout this paper, the following definition and lemmas will be used.

The Mittag-Leffler function with two parameters can be written as

The Laplace transform of Mittag-Leffler function can be given as

If

If

In this section, a sufficient condition is given for the asymptotical stability and stabilization for a class of fractional-order nonlinear systems with input saturation.

Let us consider the following fractional-order nonlinear system:

The nonlinear function

The equilibrium point of system (

The constant

In this paper, let the equilibrium point be

For convenience, we state all theorems for the case when the equilibrium point is

Assumption

Now we are ready to give the following results.

Consider the fractional dynamic system (

By taking the Laplace transform on system (

Taking the Laplace inverse transform on (

According to Lemma

Based on the properties of Assumption

Then (

By using Lemma

Since

In [

If the nonlinear function

Now let us consider the following controlled fractional-order chaotic system subject to input saturation with

In this paper, we construct the state-feedback control input

A memoryless nonlinearity

Based on Definition

Let

Then the closed-loop system (

Consider system (

By taking the Laplace transform on system (

Taking the Laplace inverse transform on (

According to Lemma

Noting that

From Definition

Let

By using Theorem

In this section, two fractional-order nonlinear (chaotic) systems are utilized to show the effectiveness of the proposed control method.

Consider the following.

The fractional-order Genesio-Tesi chaotic system can be written as [

And we can easily know that

In Figure

Chaotic behavior of fractional-order Genesio-Tesi system.

The control gain matrices are chosen as

From the above discussion, we know that

Let

Simulation results for fractional-order Genesio-Tesi system. (a), (b), and (c): time response of

Consider the following.

The so-called

Apparently, Assumption

Simulation results for fractional-order

In the simulation, the control gain matrices are chosen as

From the above discussion, we know that

Let

State trajectories of fractional-order

Time responses of control inputs subject to input saturation.

We investigate control problem for a class of fractional-order nonlinear systems subject to input saturation by means of linear state-feedback control. Based on the sector bounded condition of the saturation function, the Mittag-Leffler function, and the Laplace transform technique, a sufficient condition is given for the stabilization of such systems. It is shown that linear state-feedback controller can be used to control the fractional-order nonlinear systems. Two simulation studies are given to confirm the effectiveness of the proposed method.

The author declares that there is no conflict of interests regarding the publication of this paper.

This work is supported by National Natural Science Foundation of China (Grant no. 61001086) and Fundamental Research Funds for the Central Universities (Grant no. ZYGX2011X004).