Computational Intelligence and Neuroscience

Volume 2017, Article ID 9640849, 11 pages

https://doi.org/10.1155/2017/9640849

## Advanced Interval Type-2 Fuzzy Sliding Mode Control for Robot Manipulator

^{1}Republic of Korea Naval Logistics Command, P.O. Box 602, Hyeon-dong, Jinhae-gu, Changwon-si, Gyeongsangnam-do 645-798, Republic of Korea^{2}Department of Computer Engineering, Gachon University, Bokjeong-dong, Sujeong-gu, Seongnam-si, Gyeonggi-do 461-701, Republic of Korea^{3}Department of Electronic Engineering, Incheon National University, Incheon 402-752, Republic of Korea^{4}Department of Digital Electronics, Inha Technical College, 100 Inha-ro, Nam-Gu, Incheon 402-752, Republic of Korea

Correspondence should be addressed to Jong-Wook Park; rk.ca.noehcni@wgnj and Dong W. Kim; rk.ca.ctahni@miknwd

Received 31 October 2016; Revised 14 December 2016; Accepted 9 January 2017; Published 8 February 2017

Academic Editor: Manuel Graña

Copyright © 2017 Ji-Hwan Hwang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, advanced interval type-2 fuzzy sliding mode control (AIT2FSMC) for robot manipulator is proposed. The proposed AIT2FSMC is a combination of interval type-2 fuzzy system and sliding mode control. For resembling a feedback linearization (FL) control law, interval type-2 fuzzy system is designed. For compensating the approximation error between the FL control law and interval type-2 fuzzy system, sliding mode controller is designed, respectively. The tuning algorithms are derived in the sense of Lyapunov stability theorem. Two-link rigid robot manipulator with nonlinearity is used to test and the simulation results are presented to show the effectiveness of the proposed method that can control unknown system well.

#### 1. Introduction

In control engineering, the design of robust controller for a class of uncertain nonlinear multiple-input multiple-output (MIMO) systems remains one of the most challenging tasks. When MIMO systems are nonlinear and uncertain, their control problem becomes more challenging.

Conventional control theory is well suited to applications, where the control inputs can be generated based on analytical model [1, 2]. Sliding mode control (SMC), which is based on the theory of variable structure systems (VSS), has been widely applied to robust control of nonlinear systems [3–5]. SMC performs well in trajectory tracking of some nonlinear systems. The SMC employs a discontinuous control law to drive the state trajectory toward a specified sliding surface and maintain its motion along the sliding surface in the state space. Hung et al. [3] have made a comprehensive survey of the VSS theory. The dynamic performance of the SMC system has been confirmed as an effective robust control approach with respect to system uncertainties and unknown disturbance when the system trajectories belong to predetermined sliding surface [4].

Although the SMC performs well in the nonlinear systems, it suffers from some difficulties. First, due to the highly coupled nonlinear and uncertain dynamics, it is generally difficult or even impossible for many physical systems to obtain accurate mathematical models. Secondly, to operate effectively in the sliding surface, the SMC requires instantaneous change of the control input without sacrificing the robustness against the model uncertainties and external disturbances. The discontinuity in the control action becomes the cause of chattering, which is undesirable in most applications [6]. In the practical implementation, the chattering may cause an unnecessarily large control signal as the system uncertainties are large and may damage system components such as actuators. Thus, the chattering has to be eliminated or alleviated as much as possible. Finally, it is difficult to directly extend the SMC design into a multiple-input multiple-output (MIMO) system, especially when the coupling among the subsystems is unknown.

During the last two decades fuzzy logic system (FLS) has been a dominant topic in intelligent systems research or control community. Because the FLS provide a systematic and efficient framework to incorporate linguistic fuzzy information from human expert, it is particularly suitable for those systems with uncertain or complex dynamics. Owing to universal approximation capability [7] of fuzzy system, many FLS schemes have been developed for handling nonlinear systems, especially in the presence of incomplete knowledge of the system [8, 9].

Some researchers applied fuzzy system to sliding mode control to improve the performance of SMC. The fuzzy sliding mode control (FSMC) forms the equivalent control of SMC. By employing the FLS, the set of linearized mathematical model can be integrated into a global model that is equivalent to the nonlinear system [10, 11].

As an extension of the well-known ordinary fuzzy set (type-1 fuzzy sets), the concept of type-2 fuzzy sets (T2FS) was first introduced by Zadeh [12]. The sets are fuzzy sets whose membership grades themselves are type-1 fuzzy sets. They are very useful in circumstances where it is difficult to determine an exact membership function for a fuzzy set. They are useful for incorporating uncertainties [13].

In this paper, we propose a novel advanced interval type-2 fuzzy sliding mode control (AIT2FSMC) for a class of uncertain nonlinear MIMO systems. To inherit the strength of these two methods, we combine IT2FLS and SMC into one methodology. The AIT2FSMC system is comprised of a fuzzy control design and a hitting control design. For resembling a feedback linearization (FL) control law, IT2 fuzzy system is designed. For compensating the approximation error between the FL control law and IT2 fuzzy system, sliding mode controller is designed, respectively.

The tuning algorithms are derived in the sense of Lyapunov stability theorem. The two-link robot manipulator is used to test the proposed method and the simulation results show the AIT2FSMC can control the unknown system well.

The organization of this paper is as follows. Problem formulation and notation are presented in Section 2. In Section 3, IT2FLS is briefly introduced. Section 4 describes the design process and the stability analysis of AIT2FSMC. In Section 5, the simulation results are presented to show the effectiveness of the proposed control for a two-link robot manipulator. Finally, conclusions are given in Section 6.

#### 2. Notation and Problem Formulation

In this section, we present the problem formulation for a class of MIMO nonlinear dynamic systems. Consider the following class of MIMO nonlinear dynamic systems: where is the fully measurable state vector and , is the control input vector, is the output vector, and , are continuous nonlinear functions, and , are continuous nonlinear functions.

Let us denote

Then, system (1) can be rewritten in the following compact form:

The control problem is to design a control law which assures that the system tracks a -dimensional desired vector , which belongs to a class of continuous functions on . In this paper, we make the following assumption.

*Assumption 1. *The matrix is positive definite; then there exists , such that , with being an identity matrix. In the following may be known or not.

Although this assumption restricts the considered class of MIMO nonlinear systems, many physical systems, such as robotic systems [5], fulfill the above property.

*Assumption 2. *The desired trajectory , , is a known bounded function of time with bounded known derivatives, and is assumed to be -times differentiable.

Let us define the tracking error asand the sliding surfaces as

The time derivatives of the sliding surfaces can be written aswhere are given as follows:where

Denote

Then, (6) can be written in the compact form

If the nonlinear functions and are known, one can use a sliding mode controller. When the closed loop system is in the sliding mode, it satisfies , and then the traditional sliding mode control law is obtained by the following equation: where is an equivalent control law and is a hitting control law and with for . Using (10) and (11), we can obtain the following equation:

Multiplying to (12) gives

Let us consider the following Lyapunov function candidate:whose time derivative is given by

With (13), (15) can be reexpressed aswhich implies that as Therefore, and all its derivatives up to converge to zero [5].

According to the above analysis, the control law (11) is easily obtained if the nonlinear functions and are known. However, in this paper, these nonlinear functions are assumed to be unknown, so the above design method cannot be applied directly.

#### 3. Interval Type-2 Fuzzy Logic System

The theory and design of interval type-2 fuzzy logic systems (FLS) are presented well in [13–15]. The brief description of the interval type-2 FLS is depicted here. Detailed descriptions can be found in [13–15]. In particular, refer to [13, 15] for more notations and calculations of type-2 fuzzy logic equations.

A T2FS in the universal set is denoted as which is characterized by a type-2 membership function in (17). can be referred to as a secondary membership function (MF) or also referred to as secondary set, which is a type-1 set in . In (17) is a secondary grade, which is the amplitude of a secondary MF; that is, . The domain of a secondary MF is called the primary membership of . In (17), is the primary membership of , where for ; is a fuzzy set in , rather than a crisp point in .

When , , then the secondary MFs are interval sets such that in (17) can be called an interval type-2 MF [13]. Therefore, T2FS can be rewritten as

Also, a Gaussian primary MF with uncertain mean and fixed standard deviation having an interval type-2 secondary MF can be called an interval type-2 Gaussian MF. A 2D interval type-2 Gaussian MF with an uncertain mean in and a fixed standard deviation is shown in Figure 1. It can be expressed as