Combining adaptive fuzzy sliding mode control with fuzzy or variable universe fuzzy switching technique, this study develops two novel direct adaptive schemes for a class of MIMO nonlinear systems with uncertainties and external disturbances. The proposed control schemes consist of fuzzy equivalent control terms, fuzzy switching control terms (in scheme one) or variable universe fuzzy switching control terms (in scheme two), and compensation control terms. The compensation control terms are used to relax the assumption on fuzzy approximation error. Based on Lyapunov stability theory, the parameters update laws are adaptively tuned online and the global asymptotic stability of the closedloop system can be guaranteed. The major contribution of this study is to develop a novel framework for designing direct adaptive fuzzy sliding mode control scheme facing model uncertainties and external disturbances. The derived schemes can effectively solve the chattering problem and the equivalent control calculation in that environment. Simulation results performed on a twolink robotic manipulator demonstrate the feasibility of the proposed control schemes.
Some nonlinear systems, such as robotic manipulator, inverted pendulum, and electrical machines, not only are often highly coupled and timevarying systems, but also suffer from structured and unstructured uncertainties [
In recent decades, fuzzy control methodology has emerged as a promising way to approach nonlinear control problems since it can incorporate linguistic information from human experts into control strategy [
Several indirect schemes which combined adaptive fuzzy SMC with fuzzy switching technique have been reported for SISO nonlinear systems in [
Combining adaptive fuzzy SMC with fuzzy or variable universe fuzzy switching technique, this paper proposes two novel direct adaptive control schemes for a class of MIMO nonlinear systems with uncertainties and external disturbances. The difference between them lies in that one scheme employs fuzzy system to estimate the switching control gain and the other uses the variable universe fuzzy system proposed in [
The rest of this paper is organized as follows. In Sections
Consider the following MIMO nonlinear system [
Define
The time derivatives of the sliding surface of each subsystem are
The objective of this paper is to design a control law
The ideal trajectory
Assumption
Fuzzy control emerged in decades ago is a promising way to solve nonlinear control problems. It has several excellent properties. For example, it does not require the plant model and can effectively incorporate the semantic knowledge of human experts. Since the universal approximation theorem has been put forward in [
It should be emphasized that, in this paper, it is assumed that the structure and the membership function parameters of the fuzzy system are properly specified in advance by the designer. This means that the designer decision is needed to determine the structure of the fuzzy system, namely, the pertinent inputs, the number of membership functions for each input, the membership function parameters, and the number of rules.
For convenience, we will recall briefly the fuzzy system in the following. Let
Adaptive fuzzy controller based on fixed universe has limited approximation accuracy according to the interpolation mechanism of fuzzy system [
The socalled variable universe means that some universes, for example,
Under the framework of variable universe fuzzy control, the parameter which needs to be online adjusted is a scalar
In this section, we firstly consider that nonlinear function matrices
We choose a candidate Lyapunov function
According to the sliding mode control scheme, the control law
Let
It is well known that the fuzzy rules used for reasoning are not easy to extract, especially for multiinput
So, in what follows, adaptive fuzzy systems as (
Let
In this paper, we assume that fuzzy approximation errors
Given the approximation error
Since the chattering is caused by the switching gain matrix
According to the switching control
Further, to cancel the approximation error between the equivalent control
Substituting (
To derive the adaptive laws of the parameter vectors, let us consider the Lyapunov candidate function
For system (
The involved signals of the close loop are bounded.
The tracking errors and their derivatives decay to zero asymptotically, in other words, when
Differentiating (
To complete the proof and establish asymptotic convergence of the tracking error, we need to prove that
The tracking control using conventional adaptive fuzzy SMC with fuzzy switching control term [
In conventional adaptive fuzzy SMC design [
As stated in Section
Similar to the analysis in Section
Let
Substituting (
To derive the adaptive law of the parameter vectors, we consider the Lyapunov candidate function as
For system (
The involved signals of the close loop are bounded.
The tracking errors and their derivatives decay to zero asymptotically, in other words, when
Differentiating (
To summarize the above analysis, the stepbystep procedures for the two direct adaptive fuzzy SMCs are proposed as follows.
Design Procedure:
Select proper positive coefficients
Specify design constant vectors
Define
Construct the fuzzy rule bases for the fuzzy system
Construct the fuzzy systems
Construct the control law (
Use the adaptive laws (
In this section, we test the proposed control schemes on the trajectory tracking control of the twolink rigid robot manipulators moving a horizontal plane. The equations of motion of the manipulators can be expressed in matrix form as follows [
In the simulation, the following parameter values of the plant are used:
The object is to design control law
Fuzzy rule list of variable universe fuzzy switching control.
Input 
PB  PM  PS  ZE  NS  NM  NB 
Output 
PB  PM  PS  ZE  NS  NM  NB 
In this paper, we consider that the nonlinear function matrices
In the whole simulation, the design parameters used are chosen as follows:
To verify the robust stability of the proposed schemes, external disturbances are chosen as square wave signal
The performance indices in 20 seconds.
Controller  rod1  rod2  

IAE (rad)  ITAE (rad · s)  ISV (N^{2})  IAE (rad)  ITAE (rad · s)  ISV (N^{2})  
DAFSMC with CSW  0.8485  1.5820 

0.4191  0.3299  704.3263 
DAFSMC with FSW  2.3991  12.8742 

1.1504  5.3901  253.8947 
DAFSMC with VUFSW  0.3923  0.3867 

0.2852  0.3180  437.4319 
The performance indices in 100 seconds.
Controller  rod1  rod2  

IAE (rad)  ITAE (rad · s)  ISV (N^{2})  IAE (rad)  ITAE (rad · s)  ISV (N^{2})  
DAFSMC with CSW  1.4397  33.2521 

0.4722  6.6922 

DAFSMC with FSW  3.9996  97.6183 

2.1688  63.2731 

DAFSMC with VUFSW  0.5073  7.2268 

0.3844  6.1505 

Angular trajectory tracking curves of robotic manipulators.
Tracking curve of
Tracking curve of
Angular trajectory tracking error curves of robotic manipulators.
Tracking error of
Tracking error of
Sliding mode dynamic evolution curves.
Sliding surface
Sliding surface
Control input torques of robotic manipulators.
Control effort of
Control effort of
As shown in Figures
Both IAE and ITAE are used as evaluating error performance, while the criterion ISV shows energy consumption. It is well known that there is a tradeoff between error performance and energy consumption, that is, when IAE and ITAE are improved, ISV becomes worse, and vice versa. Conservative control input is often required to guarantee the stability of the control system in DAFSMC with CSW scheme. Therefore, DAFSMC with CSW expends relatively more energy to achieve the tracking task than DAFSMC with FSW. It also implies that the indices IAE and ITAE of DAFSMC with FSW become worse. But DAFSMC with VUFSW simultaneously improves IAE and ITAE as well as ISV as compared to the others as stated in Tables
As already stated in Section
Compared with [
A novel framework is developed to design a direct adaptive fuzzy SMC for a class of MIMO nonlinear systems with model uncertainties and unknown disturbances. Combining adaptive fuzzy SMC with fuzzy or variable universe fuzzy switching technique, this study proposes two novel direct adaptive fuzzy SMC schemes. The derived schemes effectively overcome the two disadvantages of general SMC. Besides, the constraint on the control gain matrix and the fuzzy approximation error are relaxed. Future works will focus on the extension of the framework to more general MIMO nonlinear systems such as the continuoustime or the discretetime nonaffine nonlinear systems.
This work was partly supported by NSFC project (no. 61074044), Basic Scientific and Technological Frontier project of Henan province (no. 092300410178), and Scientific Research key project of Henan provincial Education Department (no. 12B110007) and Specialized Research Fund for the Doctoral Program of Higher Education (no.20090041110003).