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This paper presents a robust adaptive fuzzy sliding mode control method for a class of uncertain nonlinear systems. The fractional order calculus is employed in the parameter updating stage. The underlying stability analysis as well as parameter update law design is carried out by Lyapunov based technique. In the simulation, two examples including a comparison with the traditional integer order counterpart are given to show the effectiveness of the proposed method. The main contribution of this paper consists in the control performance is better for the fractional order updating law than that of traditional integer order.

Fuzzy systems have been applied to many control problems effectively because they do not need accurate mathematical model of the controlled system and they can also cooperate with human expert knowledge. It is well known that fuzzy systems and neural networks can uniformly approximate any nonlinear continuous function over some compact set [

Parameter tuning in adaptive control systems is an important part of the overall mechanism alleviating the influence associated with the changes in the parameters and uncertainties of the systems. Many studies can be found in the past two decades, and the domain of adaptation has become a blend of techniques of dynamical systems theory, optimization, soft computing, and heuristics. Now, tuning of parameters based on some set of observations has been facilitated [

A common feature of all these control methods and the cited research is the fact that the differentiation and integration are performed in integer order. Up to now, with the development of complex engineering applications and natural science, fractional calculus as well as fractional differential equation theory and their applications begin to attract more and more attention from physicists to engineers [

In the stability analysis of fractional order systems, the Lyapunov function

As a result of the discussion above, the absence of methods designed by fractional differintegration in robust control is visible. The objective of this paper is to fill this gap to the extent that covers the following aspects:

Consider the following MIMO nonlinear dynamic system which can be described by

If we denote

The objective of this paper is to construct a control input

The desired trajectory signal

The control gain matrix

Assumption

Let us define the tracking error as

The fuzzy logic system that employs singleton fuzzification, sum-product inference, and center-of-sets defuzzification, as shown in Figure

Structure of a fuzzy inference system.

Suppose that the functions

Then we can construct the following ideal controller

Consider system (

Substituting the ideal control input (

Multiplying

Let us define the following Lyapunov function:

Its derivative with respect to time can be given by

By using Assumption

From (

Let

In the rest of this section, we assume that the target output of the fuzzy system is known such that the approximation error is available for parameter updating process. Let

Let the approximation error of the fuzzy system be

Now we are ready to give the following results.

Suppose the following boundedness conditions hold:

Then the approximation error will converge to zero within some finite time satisfying

Noting that

Let us define

According to the Leibniz rule of the fractional differentiation, we have

Then we have

Substituting (

Since

Now, let us prove that first hitting to the switching surface happens in some finite time

Applying the fractional integration operator

Multiplying

After some straightforward manipulators, we can obtain

And this ends the proof of Theorem

The assumptions (

The boundedness condition (

In this paper, we will prove the boundedness conditions and establish a fundamental lemma. This lemma is established for stability analysis of fractional order systems, especially for Mittag-Leffler stability [

Let

Since

On the other hand, because

It is known that the Gamma function is nonzero everywhere along the real line, and there is in fact no complex number

Since

From above discussion, we can obtain the following inequality:

Since the nonlinear functions

Let us denote

Noting that there are

From above discussions, now we are ready to give the following theorem.

Consider system (

From Theorems

Two nonlinear systems are utilized to show the effectiveness of the proposed hybrid control scheme.

Firstly, let us use the following MIMO system:

System (

The fuzzy systems have

Membership functions in fuzzy system.

The design parameters are chosen as

Figure

Simulation results for Example 1: (a) tracking errors:

Time response of the fuzzy system parameters:

In this Section, a 3D saturated multiscroll chaotic system will be used. The comparison between our control method and the control method proposed in [

A 3D saturated multiscroll chaotic system can be described by [

3D multiscroll attractor: (a)

The initial values are chosen as

The design parameters are chosen as

Figure

Comparison between our control method and the control method proposed in [

This paper proposes a fractional order integration method for updating the parameters of fuzzy systems. It is shown that the proposed controller is applicable to MIMO nonlinear systems. According to the results in this paper, the fractional order updating law outperforms the updating mechanisms exploiting integer-order operators. To demonstrate the effectiveness of fractional order operators in the fuzzy system parameters updating, this paper investigates a wide range of applications from the domain of adaptive control. Specifically, the adaptive fuzzy sliding mode control method is focused on in this paper.

The authors declare that there is no conflict of interests regarding the publication of this paper.

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