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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 658424, 27 pages
http://dx.doi.org/10.1155/2012/658424
Research Article

Adaptive Backstepping Fuzzy Control Based on Type-2 Fuzzy System

1Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China
2School of Computer Science Telecommunication Engineering, Zhenjiang, Jiangsu 212013, China

Received 20 August 2011; Revised 28 December 2011; Accepted 29 December 2011

Academic Editor: Alain Miranville

Copyright © 2012 Ll Yi-Min 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

A novel indirect adaptive backstepping control approach based on type-2 fuzzy system is developed for a class of nonlinear systems. This approach adopts type-2 fuzzy system instead of type-1 fuzzy system to approximate the unknown functions. With type-reduction, the type-2 fuzzy system is replaced by the average of two type-1 fuzzy systems. Ultimately, the adaptive laws, by means of backstepping design technique, will be developed to adjust the parameters to attenuate the approximation error and external disturbance. According to stability theorem, it is proved that the proposed Type-2 Adaptive Backstepping Fuzzy Control (T2ABFC) approach can guarantee global stability of closed-loop system and ensure all the signals bounded. Compared with existing Type-1 Adaptive Backstepping Fuzzy Control (T1ABFC), as the advantages of handling numerical and linguistic uncertainties, T2ABFC has the potential to produce better performances in many respects, such as stability and resistance to disturbances. Finally, a biological simulation example is provided to illustrate the feasibility of control scheme proposed in this paper.

1. Introduction

Early results on adaptive control for nonlinear systems are usually obtained based on the assumption that nonlinearities in systems satisfied matching conditions [1, 2]. To control the nonlinear systems with mismatched conditions, backstepping design technique has been developed in [3], and the papers [4, 5] addressed some robust adaptive control results by backstepping design. So far, backstepping approach has become one of the most popular design methods for a series of nonlinear systems. However, in many real plants, not only the nonlinearities in the system are unknown but also prior knowledge of the bounds of these nonlinearities is unavailable. In order to stabilize those nonlinear systems, many Approximator-Based Adaptive Backstepping Control (ABABC) methods have been developed by combining the concepts of adaptive backstepping and several universal approximators, like the Adaptive Backstepping Fuzzy Control (ABFC) [69], Adaptive Backstepping Neural Network Control (ABNNC) [1013], and Adaptive Backstepping Wavelet Control (ABWC) [14, 15].

So far, lots of important results on ABFC have been reported. In [16], direct ABFC scheme has been proposed by combining the modified integral Lyapunov functions and the backstepping technique. In [17], the author introduced the further extended ABFC scheme to the time-delay setting. Recently, in [18], the fuzzy systems are used as feedforward compensators to model some system functions depending on the reference signal. Based on backstepping technique, in [19], adaptive fuzzy controller has been proposed for temperature control in a general class of continuous stirred tank reactors. In [20], the author developed a fuzzy adaptive backstepping design procedure for a class of nonlinear systems with nonlinear uncertainties, unmodeled dynamics, and dynamics disturbances. In [21], a fuzzy adaptive backstepping output feedback control approach is developed for a class of MIMO nonlinear systems with unmeasured states.

However, all the existing ABFC schemes have the common problem that they cannot fully handle or accommodate the uncertainties as they use precise type-1 fuzzy sets. In general, the uncertainty rules will be existed in the following three possible ways [2228]: (i) the words that are used in antecedents and consequents of rules can mean different things to different people; (ii) consequents obtained by polling a group of experts will often be different for the same rule because the experts will not necessarily be in agreement; (iii) noisy training data. So when something is uncertain and the circumstances are fuzzy, we have trouble determining the membership grade even as a crisp number in [0, 1].

To overcome this drawback, we consider using fuzzy sets of type-2 in this paper. The concept of Type-2 Fuzzy Sets (T2FSs) was first introduced in [29] as an extension of the well-known ordinary fuzzy set, the Type-1 Fuzzy Sets (T1FSs). A T2FSs is characterized by a fuzzy membership function; that is, the membership grade for each element is also a fuzzy set in [0, 1]. The membership functions of T2FSs are three-dimensional and include a Footprint of Uncertainty (FOU), which is a new third dimension of T2FSs, and the FOU provides an additional degree of freedom. So compared to Type-1 Fuzzy Logic System (T1FLS), Type-2 Fuzzy Logic System (T2FLS) has many advantages as follows (i) As T2FSs are able to handle the numerical and linguistic uncertainties, T2FLC based on T2FSs will have the potential to produce a better performance than T1FLC. (ii) Using T2FSs to represent the FLC inputs and outputs will also result in the reduction of the FLC rule base compared to using T1FSs. (iii) In a T2FLC each input and output will be represented by a large number of T1FSs, which allows for greater accuracy in capturing the subtle behavior of the user in the environment. (iv) The T2FSs enable us to handle the uncertainty when trying to determine the exact membership functions for the fuzzy sets with the inputs and outputs of the FLC.

The papers [30] by Hagras and [31] by Melin and Castillo were the first two papers on T2FLC. Subsequently, Castillo, Hagras, and Sepulveda presented T2FLC designs, respectively; for details see [3237]. Also, some results on T2 fuzzy sliding-mode controller have been presented in [3840]. Moreover, in recent years, indirect adaptive interval T2 fuzzy control for SISO nonlinear system is proposed in [41] and direct adaptive interval T2 fuzzy control has been developed in [26] for a MIMO nonlinear system. Robust adaptive tracking control of multivariable nonlinear systems based on interval T2 fuzzy approach is developed in [42]. Adaptive control of two-axis motion control system using interval T2 fuzzy neural network is presented in [43]. The author introduced interval T2 fuzzy logic congestion control method for video streaming across IP networks in [44]. And in [45], optimization of interval T2 fuzzy logic controllers for a perturbed autonomous wheeled mobile robot has been developed.

Inspired by all of that, a novel ABFC approach based on T2FSs is drawn in this paper. Compared with traditional T1ABFC, T2ABFC can fully handle or accommodate the uncertainties and achieve higher performances. So, T2ABFC method proposed in this paper succeeds in solving the control problem of a series of nonlinear systems with not only mismatched conditions but also complicated uncertainties.

The rest of this paper is organized as follows. First is the problem formulation, with some preliminaries given in Section 2, and in Section 3, a brief introduction of the interval T2FLS. Indirect adaptive backstepping fuzzy controller design using interval T2FLS is presented in Section 4. In Section 5, a simulation example is provided to illustrate the feasibility of the proposed control scheme. In Section 6, we conclude the work of the paper.

2. Problem Formulation

Consider a class of SISO nonlinear systems described by the differential equations aṡ𝑥𝑖=𝑓𝑖𝑥𝑖+𝐺𝑖𝑥𝑖𝑥𝑖+11𝑖𝑛1,̇𝑥𝑛=𝑓𝑛𝑥𝑛+𝐺𝑛𝑥𝑛𝑢𝑛2,𝑦=𝑥1,(2.1) where 𝑥𝑖=[𝑥1,𝑥2𝑥𝑖]𝑇𝑅𝑖 is the stable vector and 𝑢𝑅 and 𝑦𝑅 are the input and output of the system, respectively. 𝑓𝑖(𝑥𝑖), 𝐺𝑖(𝑥𝑖) (𝑖=1,2𝑛) are unknown smooth nonlinear functions. The control objective of this paper is formulated as follows: for a given bounded reference signal 𝑦𝑟(𝑡)𝐷 with continuous and bounded derivatives up to order 𝜌, where 𝐷𝑅 is a known compact set. Utilize fuzzy logic system and parameters adaptive laws such that(1)all the signals involved in the closed-loop system are ultimately and uniformly bounded,(2)the tracking errors converge to a small neighborhood around zero,(3)the closed-loop system is global stable.

3. Interval Type-2 Fuzzy Logic Systems

In this section, the interval type-2 fuzzy set and the inference of the type-2 fuzzy logic system are presented.

Formally a type-2 fuzzy set 𝐴 is characterized by a type-2 membership function 𝜇𝐴(𝑥,𝑢) [22], where 𝑥𝑋 is the primary variable and 𝑢𝐽𝑥[0,1] is the secondary variable:(𝐴=𝑥,𝑢),𝜇𝐴(𝑥,𝑢)𝑥𝑋𝑢𝐽𝑥[]0,1.(3.1)in which 0𝜇𝐴(𝑥,𝑢)1. 𝐴 can also be expressed as follows [22]: 𝐴=𝑥𝑋𝑢𝐽𝑥𝜇𝐴(𝑥,𝑢)𝐽(𝑥,𝑢)𝑥[]0,1,(3.2)

Due to the facts that an Interval Type-2 Fuzzy Logic Control (IT2FLC) is computationally far less intensive than a general T2FLC and thus better suited for real-time computation in embedded computational artifacts, our learning and adaptation technique use an IT2FLC (using interval T2FSs to represent the inputs and outputs). The IT2FSs, currently the most widely used kind of T2FSs, are characterized by IT2 membership functions in which the secondary membership grades are equal to 1. The theoretic background of IT2 FLS can be seen in [23, 4650]. It is described as𝐴=𝑥𝑋𝑢𝐽𝑥1=(𝑥,𝑢)𝑥𝑋𝑢𝐽𝑥1/𝑢𝑥.(3.3) Also, a Gaussian primary membership function with uncertain mean and fixed standard deviation having an interval type-2 secondary membership function can be called an interval type-2 Gaussian membership function. Consider the case of a Gaussian primary membership function having an uncertain mean in [𝑚1,𝑚2] and a fixed standard deviation 𝜎. It can be expressed as𝑢𝐴1(𝑥)=exp2𝑥𝑚𝜎2𝑚,𝑚1,𝑚2.(3.4)

Uncertainty about 𝐴 can be expressed by the union of all the primary memberships, and is bounded by an upper membership function and a lower membership function [2224], which is called the FOU of 𝐴:𝐴=FOU𝑥𝑋𝐽𝑥=𝜇(𝑥,𝑢)𝑢𝐴(𝑥),𝜇𝐴(𝑥),(3.5)

where𝜇𝐴𝐴(𝑥)=FOU𝑥𝑋,𝜇𝐴(𝑥)=𝐴FOU𝑥𝑋.(3.6)

The concept of FOU, associated with the concepts of lower and upper membership functions, models the uncertainties in the shape and position of the T1FSs.

The distinction between T1 and T2 rules is associated with the nature of the membership functions; the structure of the rules remains exactly the same in the T1 case, but all the sets involved are T2 now. We can consider a T2FLS having 𝑛 inputs 𝑥1𝑋1,,𝑥𝑛𝑋𝑛 and one output 𝑦𝑌, and assuming there are 𝑀 rules, the 𝑖th rule of the IT2 SMC can be described as𝑅𝑙IF𝑥1𝐹is𝑙1and𝑥𝑛𝐹is𝑙𝑛𝐺,THEN𝑦is𝑙𝑙=1,,𝑀.(3.7)

Since the output of the inference-engine is a T2FS, it must be type-reduced and the defuzzifier is used to generate a crisp output. The type-reducer is an extension of T1 defuzzifier obtained by applying the extension principle [29]. There are many kinds of type-reduction methods [23, 24, 51, 52], such as the centroid, center of sets, center of sums, and height type-reduction, and these are elaborated upon in [25]. As in [23], the most commonly used type-reduction method is the center of sets type-reducer, as it has reasonable computational complexity that lies between the computationally expensive centroid type-reduction and the simple height and modified height type-reduction which have a problem when only one rule fires [25]. The type-reduced set using the center of sets type-reduction can be expressed as follows:𝑌cos𝑌1,,𝑌𝑀,𝐹1,,𝐹𝑀=𝑦𝑙,𝑦𝑟=𝑦1𝑦𝑀𝑓1𝑓𝑀1𝑀𝑖=1𝑓𝑖𝑦𝑖/𝑀𝑖=1𝑓𝑖,(3.8) where 𝑌cos(𝑥) is an interval output set determined by its left-most point 𝑦𝑙 and its right-most point 𝑦𝑟, and 𝑓𝑖𝐹𝑖=𝑓𝑖,𝑓𝑖. In the meantime, an IT2FLS with singleton fuzzification and meet under minimum or product 𝑡-norm 𝑓𝑖 and 𝑓𝑖 can be obtained as𝑓𝑖=𝜇𝐹𝑖1𝑥1𝜇𝐹𝑖𝑛𝑥𝑛,(3.9)

and𝑓𝑖=𝜇𝐹𝑖1𝑥1𝜇𝐹𝑖𝑛𝑥𝑛.(3.10)

Also, 𝑦𝑖𝑌𝑖 and 𝑌𝑖=[𝑦𝑖𝑙,𝑦𝑖𝑟] is the centroid of the IT2 consequent set 𝐺𝑖, the centroid of a T2FS, and for any value 𝑦𝑌cos, 𝑦 can be expressed as𝑦=𝑀𝑖=1𝑓𝑖𝑦𝑖𝑀𝑖=1𝑓𝑖,(3.11) where 𝑦 is a monotonic increasing function with respect to 𝑦𝑖. Also, 𝑦𝑙 is the minimum associated only with 𝑦𝑖𝑙, and 𝑦𝑟 is the maximum associated only with 𝑦𝑖𝑟. Note that 𝑦𝑙 and 𝑦𝑟 depend only on the mixture of 𝑓𝑖 or 𝑓𝑖 values. Therefore, the left-most point 𝑦𝑙 and the right-most point 𝑦𝑟 can be expressed as a Fuzzy Basis Function (FBF) expansion, that is, 𝑦𝑙=𝑀𝑖=1𝑓𝑖𝑙𝑦𝑖𝑙𝑀𝑖=1𝑓𝑖𝑙=𝑀𝑖=1𝑦𝑖𝑙𝜉𝑖𝑙=𝑦𝑇𝑙𝜉𝑙,(3.12)

and𝑦𝑟=𝑀𝑖=1𝑓𝑖𝑟𝑦𝑖𝑟𝑀𝑖=1𝑓𝑖𝑟=𝑀𝑖=1𝑦𝑖𝑟𝜉𝑖𝑟=𝑦𝑇𝑟𝜉𝑟,(3.13) respectively, where 𝜉𝑖𝑙=𝑓𝑖𝑙/𝑀𝑖=1𝑓𝑖, 𝜉𝑙=[𝜉1𝑙,𝜉2𝑙,,𝜉𝑀𝑙], 𝑦𝑇𝑙=[𝑦1𝑙,𝑦2𝑙,,𝑦𝑀𝑙], and 𝜉𝑖𝑟=𝑓𝑖𝑟/𝑀𝑖=1𝑓𝑖,  𝜉𝑟=[𝜉1𝑟,𝜉2𝑟,,𝜉𝑀𝑟],  𝑦𝑇𝑟=[𝑦1𝑟,𝑦2𝑟,,𝑦𝑀𝑟].

In order to compute 𝑦𝑙 and 𝑦𝑟, the Karnik-Mendel iterative procedure is needed [25, 51]. It has been shown in [25, 26, 49, 51]. For illustrative purposes, we briefly provide the computation procedure for 𝑦𝑟. Without loosing of generality, assume 𝑦𝑖𝑟 is arranged in ascending order, that is, 𝑦1𝑟𝑦2𝑟𝑦𝑀𝑟.

Step 1. Compute𝑦𝑟 in (3.13) by initially setting 𝑓𝑙𝑟=(𝑓𝑙+𝑓𝑙)/2 for 𝑙=1,,𝑀, where𝑓𝑙 and 𝑓𝑙 have been precomputed by (3.10) and let 𝑦𝑟=𝑦𝑟.

Step 2 :. Find 𝑅(1𝑅𝑀1) such that 𝑦𝑅𝑟𝑦𝑟𝑦𝑟𝑅+1.

Step 3. Compute 𝑦𝑟 in (3.13) with 𝑓𝑖𝑟=𝑓𝑖 for 𝑖𝑅 and 𝑓𝑖𝑟=𝑓𝑖 for 𝑖>𝑅and let 𝑦𝑟=𝑦𝑟.

Step 4. If 𝑦𝑟𝑦𝑟, then go to Step 5; if 𝑦𝑟=𝑦𝑟, then stop and set 𝑦𝑟=𝑦𝑟.

Step 5. Set 𝑦𝑟equal to 𝑦𝑟 and return to Step 2.
The point to separate two sides by number 𝑅 can be decided from the above algorithm, one side using lower firing strengths 𝑓𝑖’s and another side using upper firing strengths 𝑓𝑖’s. Therefore, 𝑦𝑟 can be expressed as𝑦𝑟=𝑅𝑖=1𝑓𝑖𝑦𝑖𝑟+𝑀𝑖=𝑅+1𝑓𝑖𝑦𝑖𝑟𝑅𝑖=1𝑓𝑖+𝑀𝑖=𝑅+1𝑓𝑖=𝑅𝑖=1𝑞𝑖𝑟𝑦𝑖𝑟+𝑀𝑖=𝑅+1𝑞𝑖𝑟𝑦𝑖𝑟=𝑄𝑟𝑄𝑟𝑦𝑟𝑦𝑟=𝜉𝑇𝑟Θ𝑟,(3.14) where 𝑞𝑖𝑟=𝑓𝑖/𝐷𝑟, 𝑞𝑖𝑟=𝑓𝑖/𝐷𝑟 and 𝐷𝑟=(𝑅𝑖=1𝑓𝑖+𝑀𝑖=𝑅+1𝑓𝑖). In the meantime, we have 𝑄𝑟=[𝑞1𝑟,𝑞2𝑟,,𝑞𝑅𝑟], 𝑄𝑟=[𝑞1𝑟,𝑞2𝑟,,𝑞𝑅𝑟], 𝜉𝑇𝑟=𝑄𝑟,𝑄𝑟, and Θ𝑇𝑟=𝑦𝑟𝑦𝑟.
The procedure to compute 𝑦𝑙 is similar to compute 𝑦𝑟. In Step 2, it only determines 𝐿(1𝐿𝑀1), such that 𝑦𝐿𝑙𝑦𝑙𝑦𝑙𝐿+1. In Step 3, let 𝑓𝑖𝑙=𝑓𝑖 for 𝑖𝐿 and𝑓𝑖𝑙=𝑓𝑖 for 𝑖>𝐿. Therefore, 𝑦𝑙 can be expressed as𝑦𝑙=𝐿𝑖=1𝑓𝑖𝑦𝑖𝑙+𝑀𝑖=𝐿+1𝑓𝑖𝑦𝑖𝑙𝐿𝑖=1𝑓𝑖+𝑀𝑖=𝐿+1𝑓𝑖=𝐿𝑖=1𝑞𝑖𝑙𝑦𝑖𝑙+𝑀𝑖=𝐿+1𝑞𝑖𝑙𝑦𝑖𝑙=𝑄𝑙𝑄𝑙𝑦𝑙𝑦𝑙=𝜉𝑇𝑙Θ𝑙,(3.15) where 𝑞𝑖𝑙=𝑓𝑖/𝐷𝑙, 𝑞𝑖𝑙=𝑓𝑖/𝐷𝑙 and 𝐷𝑙=(𝐿𝑖=1𝑓𝑖+𝑀𝑖=𝐿+1𝑓𝑖). In the meantime, we have 𝑄𝑙=[𝑞1𝑙,𝑞2𝑙,,𝑞𝑅𝑙], 𝑄𝑙=[𝑞1𝑙,𝑞2𝑙,,𝑞𝑅𝑙], 𝜉𝑇𝑙=𝑄𝑙𝑄𝑙, and Θ𝑇𝑙=𝑦𝑙𝑦𝑙.
We defuzzify the interval set by using the average of 𝑦𝑙 and 𝑦𝑟, hence, the defuzzified crisp output becomes𝑌fuzz2𝑦(𝑥)=𝑙+𝑦𝑟2=12𝜉𝑇𝑙Θ𝑙+𝜉𝑇𝑟Θ𝑟=12𝜉𝑇𝑙𝜉𝑇𝑟Θ𝑙Θ𝑟=𝜉𝑇Θ,(3.16) where (1/2)[𝜉𝑇𝑙𝜉𝑇𝑟]=𝜉𝑇and [Θ𝑇𝑙Θ𝑇𝑟]=Θ𝑇.

Lemma 3.1. (Wang [53]). Let 𝑓(𝑥) be a continuous function defined on a compact set Ω. Then for any constant 𝜀>0, there exists a fuzzy logic system (3.16) such as sup𝑥Ω||𝑓(𝑥)𝜉𝑇||(𝑥)Θ𝜀.(3.17)

4. Adaptive Backstepping Fuzzy Controller Design Using IT2FLS

In this section, our objective is to use IT2FLS to approximate the nonlinear functions. With type-reduction, the IT2FLS is replaced by the average of two T1FLSs. Ultimately, the adaptive laws, by means of backstepping design technique, will be developed to adjust the parameters to attenuate the approximation error and external disturbance.

To begin with, some assumptions are given as follows.

Assumption 4.1. There exist positive constants 𝐺𝑖 and 𝐺𝑖(1𝑖𝑛), such that 𝐺𝑖𝐺𝑖(𝑥𝑖)𝐺𝑖.

Assumption 4.2. There exist positive constant 𝐺𝑑𝑖(1𝑖𝑛), such that |̇𝐺𝑖(𝑥𝑖)|𝐺𝑑𝑖.

Assumption 4.3. Define the optimal parameter vectors Θ𝑖 as Θ𝑖=argminΘ𝑖Ω𝑖sup𝑥𝑖𝑈𝑖||𝑓𝑖𝑥𝑖Θ𝑖𝑓𝑖𝑥𝑖||,(4.1) where Ω𝑖, 𝑈𝑖 are compact regions for Θ𝑖, 𝑥𝑖, respectively. The fuzzy logic system minimum approximation errors 𝜔𝑖 are defined as ||𝜉𝑇𝑖Θ𝑖𝑓𝑖𝑥𝑖||𝜔𝑖.(4.2)

T2ABFC method proposed in this paper is summarized in Theorem 4.4.

Theorem 4.4. Suppose assumptions are 4.14.3 tenable, then the fuzzy adaptive output tracking design described by (2.1), control law 𝑢=𝑒𝑛1𝑘𝑛𝑒𝑛𝜉𝑇𝑛(𝑥𝑛)Θ𝑛, and parameter adaptive laws ̇Θ𝑛𝑙=̇Θ𝑛𝑙=𝛾𝑛𝑙(𝑒𝑛𝜉𝑇𝑛𝑙(𝑥𝑛)𝑐𝑛𝑙Θ𝑛𝑙)  ̇Θ𝑛𝑟=̇Θ𝑛𝑟=𝛾𝑛𝑟(𝑒𝑛𝜉𝑇𝑛𝑟(𝑥𝑛)𝑐𝑛𝑟Θ𝑛𝑟) based on T2FLS guarantees that closed-loop system is globally uniform ultimately bounded, output tracking error converges to a small neighborhood of the origin and resistant to extern disturbance. (𝑘𝑛, 𝛾𝑛𝑙, 𝛾𝑛𝑟, 𝑐𝑛𝑙, 𝑐𝑛𝑟 are all design parameters).

The detailed design and certification procedures are described in the following steps.

Step 1. Define the tracking error for the system as 𝑒1=𝑥1𝑦𝑟.(4.3) The time derivative of 𝑒1 is ̇𝑒1=̇𝑥1̇𝑦𝑟=𝑓1𝑥1+𝐺1𝑥1𝑥2̇𝑦𝑟.(4.4) Take 𝑥2 as a virtual control, and define 𝛼1=𝑘1𝑒11𝐺1𝑥1𝑓1𝑥1̇𝑦𝑟,(4.5) where 𝑘1 is a positive constant.
Since 𝑓1(𝑥1) and 𝑓1(𝑥1) are unknown, ideal controller is not available in practice. By Lemma 3.1, fuzzy logic systems are universal approximators, so we can assume that the unknown function (1/𝐺1(𝑥1))(𝑓1(𝑥1)̇𝑦𝑟) can be approximated by the following type-2 fuzzy logic system 𝜉𝑇1(𝑥1)Θ1, and we obtain𝛼1=𝑘1𝑒1𝜉𝑇1𝑥1Θ1𝜔1.(4.6)Express 𝑥2 as 𝑥2=𝑒2+𝛼1 and define 𝛼1=𝑘1𝑒1𝜉𝑇1𝑥1Θ1=𝑘1𝑒1𝜉𝑇1𝑙𝑥1Θ1𝑙+𝜉𝑇1𝑟𝑥1Θ1𝑟,(4.7) then the time derivative of 𝑒1 is ̇𝑒1=𝑓1𝑥1+𝐺1𝑥1𝑒2+𝛼1̇𝑦𝑟=𝐺1𝑥1𝑒2𝑘1𝑒1𝜉𝑇1𝑥1Θ1+𝜔1=𝐺1𝑥1𝑒2𝑘1𝑒1𝜉𝑇1𝑙𝑥1Θ1𝑙𝜉𝑇1𝑟𝑥1Θ1𝑟+𝜔1,(4.8) where Θ1=Θ1Θ1.
Consider the following Lyapunov function:𝑉1=12𝐺1𝑥1𝑒21+12𝛾1Θ𝑇1Θ1=12𝐺1𝑥1𝑒21+12𝛾1𝑙Θ𝑇1𝑙Θ1𝑙+12𝛾1𝑟Θ𝑇1𝑟Θ1𝑟,(4.9) then the time derivative of 𝑉1 is ̇𝑉1=𝑒1̇𝑒1𝐺1𝑥1̇𝐺1𝑥12𝐺21𝑥1𝑒21+1𝛾1Θ𝑇1̇Θ1=𝑒1̇𝑒1𝐺1𝑥1̇𝐺1𝑥12𝐺21𝑥1𝑒21+1𝛾1𝑙Θ𝑇1𝑙̇Θ1𝑙+1𝛾1𝑟Θ𝑇1𝑟̇Θ1𝑟=𝑒1𝑒2𝑘1𝑒21𝑒1Θ𝑇1𝑙𝜉1𝑙+Θ𝑇1𝑟𝜉1𝑟+𝑒1̇𝐺𝜔1𝑥12𝐺21𝑥1𝑒21+1𝛾1𝑙Θ𝑇1𝑙̇Θ1𝑙+1𝛾1𝑟Θ𝑇1𝑟̇Θ1𝑟=𝑒1𝑒2𝑘1𝑒21̇𝐺1𝑥12𝐺21𝑥1𝑒21+𝑒1𝜔1+Θ𝑇1𝑙𝑒1𝜉𝑇1𝑙𝑥1+1𝛾1𝑙̇Θ1𝑙+Θ𝑇1𝑟𝑒1𝜉𝑇1𝑟𝑥1+1𝛾1𝑟̇Θ1𝑟.(4.10) Choose the intermediate adaptive laws as ̇Θ1𝑙=̇Θ1𝑙=𝛾1𝑙𝑒1𝜉𝑇1𝑙𝑥1𝑐1𝑙Θ1𝑙,̇Θ1𝑟=̇Θ1𝑟=𝛾1𝑟𝑒1𝜉𝑇1𝑟𝑥1𝑐1𝑟Θ1𝑟,(4.11) where 𝑐1𝑙 and 𝑐1𝑟 are given positive constants.Substituting (4.11) into (4.10) yields ̇𝑉1=𝑒1𝑒2+𝑒1𝜔1𝑘1+̇𝐺1𝑥12𝐺21𝑥1𝑒21𝑐1𝑙Θ𝑇1𝑙Θ1𝑙+𝑐1𝑟Θ𝑇1𝑟Θ1𝑟,(4.12) where 𝑘1=𝑘1,0+𝑘1,1,  𝑘1,0>0,𝑘1,1>0,  𝑒1𝜔1𝑘1𝑒21𝑒1𝜔1𝑘1,1𝑒21𝜀21/4𝑘1,1,|𝜔1||𝜀1|
We can obtain that 𝑐1𝑙Θ𝑇1𝑙Θ1𝑙=𝑐1𝑙Θ𝑇1𝑙Θ1𝑙+Θ1𝑙𝑐1𝑙Θ1𝑙2+𝑐1𝑙Θ1𝑙Θ1𝑙𝑐1𝑙Θ1𝑙22+𝑐1𝑙Θ1𝑙22,𝑐1𝑟Θ𝑇1𝑟Θ1𝑟=𝑐1𝑟Θ𝑇1𝑟Θ1𝑟+Θ1𝑟𝑐1𝑟Θ1𝑟2+𝑐1𝑟Θ1𝑟Θ1𝑟𝑐1𝑟Θ1𝑟22+𝑐1𝑟Θ1𝑟22𝑘1,0+̇𝐺12𝐺21𝑒21𝑘1,0𝐺𝑑12𝐺21𝑒21,𝑘1,0=𝑘1,0𝐺𝑑12𝐺21.>0(4.13)
From (4.12), (4.13), it follows that ̇𝑉1𝑒1𝑒2𝑘1,0𝑒21+𝜀214𝑘1,1𝑐1𝑙Θ1𝑙22𝑐1𝑟Θ1𝑟22+𝑐1𝑙Θ1𝑙22+𝑐1𝑟Θ1𝑟22.(4.14)

Step 2. Differentiating 𝑒2 yields ̇𝑒2=̇𝑥2̇𝛼1=𝑓2𝑥2+𝐺2𝑥2𝑥3̇𝛼1.(4.15) Take 𝑥3 as a virtual control, and define 𝛼2=𝑒1𝑘2𝑒21𝐺2𝑥2𝑓2𝑥2̇𝛼1,(4.16) where 𝑘2 is a positive constant.
From (4.7), we obtain ̇𝛼1=𝜕𝛼1𝜕𝑥1𝑓1+𝐺1𝑥2+𝜕𝛼1𝜕𝑦𝑟̇𝑦𝑟+𝜕𝛼1𝜕Θ1𝑙𝛾1𝑙𝑒1𝑙𝜉1𝑙𝑐1𝑙Θ1𝑙+𝜕𝛼1𝜕Θ1𝑟𝛾1𝑟𝑒1𝜉1𝑟𝑐1𝑟Θ1𝑟.(4.17)
Since 𝑓2(𝑥2) and 𝐺2(𝑥2) are unknown, ideal controller is not available in practice, so we can assume that the unknown function (1/𝐺2(𝑥2))(𝑓2(𝑥2)̇𝛼1)can be approximated by the following type-2 fuzzy logic system 𝜉𝑇2(𝑥2)Θ2, and obtain 𝛼2=𝑒1𝑘2𝑒2𝜉𝑇2𝑥2Θ2.(4.18) Express 𝑥3 as 𝑥3=𝑒3+𝛼2 and the time derivative of 𝑒2 is ̇𝑒2=𝑓2𝑥2+𝐺2𝑥2𝑒3+𝛼2̇𝛼1=𝐺2𝑥2𝑒3𝑒1𝑘2𝑒2𝜉𝑇2𝑥2Θ2+𝜔2,(4.19) where Θ2=Θ2Θ2.
Consider the following Lyapunov function𝑉2=𝑉1+12𝐺2𝑥2𝑒22+12𝛾2Θ𝑇2Θ2=𝑉1+12𝐺2𝑥2𝑒22+12𝛾2𝑙Θ𝑇2𝑙Θ2𝑙+12𝛾2𝑟Θ𝑇2𝑟Θ2𝑟,(4.20) then the time derivative of 𝑉2 is ̇𝑉2=̇𝑉1𝑒1𝑒2+𝑒2𝑒3𝑘2𝑒22̇𝐺2𝑥22𝐺22𝑥2𝑒22+𝑒2𝜔2Θ𝑇2𝑙𝑒2𝜉𝑇2𝑙𝑥21𝛾2𝑙̇Θ2𝑙Θ𝑇2𝑟𝑒2𝜉𝑇2𝑟𝑥21𝛾2𝑟̇Θ2𝑟.(4.21) Choose the intermediate adaptive laws as ̇Θ2𝑙=̇Θ2𝑙=𝛾2𝑙𝑒2𝜉𝑇2𝑙𝑥2𝑐2𝑙Θ2𝑙,̇Θ2𝑟=̇Θ2𝑟=𝛾2𝑟𝑒2𝜉𝑇2𝑟𝑥2𝑐2𝑟Θ2𝑟.(4.22) Substituting (4.22) into (4.21) yields ̇𝑉2𝑒2𝑒𝑘1,0𝑒21𝑘2,0𝑒22+𝜀214𝑘1,1+𝜀224𝑘2,1𝑐1𝑙Θ1𝑙22𝑐1𝑟Θ1𝑟22𝑐2𝑙Θ2𝑙22𝑐2𝑟Θ2𝑟22+𝑐1𝑙Θ1𝑙22+𝑐1𝑟Θ1𝑟22+𝑐2𝑙Θ2𝑙22+𝑐2𝑟Θ2𝑟22.(4.23)

Step i (3𝑖𝑛1)
A similar procedure is employed recursively at each step. By defining 𝑒𝑖=𝑥𝑖𝛼𝑖1(4.24) the time derivative of 𝑒𝑖 is ̇𝑒𝑖=̇𝑥𝑖̇𝛼𝑖1=𝑓𝑖𝑥𝑖+𝐺𝑖𝑥𝑖𝑥𝑖+1̇𝛼𝑖1.(4.25) Take 𝑥𝑖+1 as a virtual control, and define 𝛼𝑖=𝑒𝑖1𝑘𝑖𝑒𝑖1𝐺𝑖𝑥𝑖𝑓𝑖𝑥𝑖̇𝛼𝑖1,(4.26) where 𝑘𝑖 is a positive constant: ̇𝛼𝑖1=𝑖1𝑘=1𝜕𝛼𝑖1𝜕𝑥𝑘𝐺𝑘𝑥𝑘+1+𝑓𝑘+𝜕𝛼𝑖1𝜕𝑦𝑟̇𝑦𝑟+𝑖1𝑘=1𝜕𝛼𝑖1𝜕Θ𝑘𝑙𝛾𝑘𝑙𝑒𝑘𝜉𝑇𝑘𝑙𝑐𝑘𝑙Θ𝑘𝑙+𝑖1𝑘=1𝜕𝛼𝑖1𝜕Θ𝑘𝑟𝛾𝑘𝑟𝑒𝑘𝜉𝑇𝑘𝑟𝑐𝑘𝑟Θ𝑘𝑟.(4.27) The unknown function (1/𝐺𝑖(𝑥𝑖))(𝑓𝑖(𝑥𝑖)̇𝛼𝑖1) can be approximated by the following type-2 fuzzy logic system 𝜉𝑇𝑖(𝑥𝑖)Θ𝑖, and we can obtain 𝛼𝑖=𝑒𝑖1𝑘𝑖𝑒𝑖𝜉𝑇𝑖𝑥𝑖Θ𝑖.(4.28) Express 𝑥𝑖+1 as 𝑥𝑖+1=𝑒𝑖+1+𝛼𝑖 and the time derivative of 𝑒𝑖 is ̇𝑒𝑖=𝑓𝑖𝑥𝑖+𝐺𝑖𝑥𝑖𝑒𝑖+1+𝛼𝑖̇𝛼𝑖1=𝐺𝑖𝑥𝑖𝑒𝑖+1𝑒𝑖1𝑘𝑖𝑒𝑖𝜉𝑇𝑖𝑥𝑖Θ𝑖+𝜔𝑖,(4.29) where Θ𝑖=Θ𝑖Θ𝑖.
Consider the following Lyapunov function:𝑉𝑖=𝑉𝑖1+12𝐺𝑖𝑥𝑖𝑒2𝑖+12𝛾𝑖Θ𝑇𝑖Θ𝑖=𝑉𝑖1+12𝐺𝑖𝑥𝑖𝑒2𝑖+12𝛾𝑖𝑙Θ𝑇𝑖𝑙Θ𝑖𝑙+12𝛾𝑖𝑟Θ𝑇𝑖𝑟Θ𝑖𝑟,(4.30) then the time derivative of 𝑉𝑖 is ̇𝑉𝑖=̇𝑉𝑖1𝑒𝑖1𝑒𝑖+𝑒𝑖𝑒𝑖+1𝑘𝑖𝑒2𝑖̇𝐺𝑖𝑥𝑖2𝐺2𝑖𝑥𝑖𝑒2𝑖+𝑒𝑖𝜔𝑖Θ𝑇𝑖𝑙𝑒𝑖𝜉𝑇𝑖𝑙𝑥𝑖1𝛾𝑖𝑙̇Θ𝑖𝑙Θ𝑇𝑖𝑟𝑒𝑖𝜉𝑇𝑖𝑟𝑥𝑖1𝛾𝑖𝑟̇Θ𝑖𝑟.(4.31) Choose the intermediate adaptive laws as ̇Θ𝑖𝑙=̇Θ𝑖𝑙=𝛾𝑖𝑙𝑒𝑖𝜉𝑇𝑖𝑙𝑥𝑖𝑐𝑖𝑙Θ𝑖𝑙,̇Θ𝑖𝑟=̇Θ𝑖𝑟=𝛾𝑖𝑟𝑒𝑖𝜉𝑇𝑖𝑟𝑥𝑖𝑐𝑖𝑟Θ𝑖𝑟.(4.32) Substituting (4.32) into (4.31) yields ̇𝑉𝑖𝑒𝑖𝑒𝑖+1𝑖𝑘=1𝑘𝑘,0𝑒2𝑘+𝑖𝑘=1𝜀2𝑘4𝑘𝑘,1𝑖𝑘=1𝑐𝑘𝑙Θ𝑘𝑙22𝑖𝑘=1𝑐𝑘𝑟Θ𝑘𝑟22+𝑖𝑘=1𝑐𝑘𝑙Θ𝑘𝑙22+𝑖𝑘=1𝑐𝑘𝑟Θ𝑘𝑟22.(4.33)

Step n
In the final design step, the actual control input 𝑢 will appears. Defining 𝑒𝑛=𝑥𝑛𝛼𝑛1,(4.34) the time derivative of 𝑒𝑛 is ̇𝑒𝑛=̇𝑥𝑛̇𝛼𝑛1=𝑓𝑛𝑥𝑛+𝐺𝑛𝑥𝑛𝑢̇𝛼𝑛1.(4.35) Define the actual control input 𝑢 as 𝑢=𝑒𝑛1𝑘𝑛𝑒𝑛𝜉𝑇𝑛𝑥𝑛Θ𝑛,(4.36) where 𝑘𝑛 is a positive constant, Θ𝑖=Θ𝑖Θ𝑖.Choose the whole Lyapunov function as 𝑉=𝑉𝑛1+12𝐺𝑛𝑥𝑛𝑒2𝑛+12𝛾𝑛Θ𝑇𝑛Θ𝑛=𝑉𝑛1+12𝐺𝑛𝑥𝑛𝑒2𝑛+12𝛾𝑛𝑙Θ𝑇𝑛𝑙Θ𝑛𝑙+12𝛾𝑛𝑟Θ𝑇𝑛𝑟Θ𝑛𝑟,(4.37) then the time derivative of 𝑉 is ̇̇𝑉𝑉=𝑛1𝑒𝑛1𝑒𝑛𝑘𝑛𝑒2𝑛̇𝐺𝑛𝑥𝑛2𝐺2𝑛𝑥𝑛𝑒2𝑛+𝑒𝑛𝜔𝑛Θ𝑇𝑛𝑙𝑒𝑛𝜉𝑇𝑛𝑙𝑥𝑛1𝛾𝑛𝑙̇Θ𝑛𝑙Θ𝑇𝑛𝑟𝑒𝑛𝜉𝑇𝑛𝑟𝑥𝑛1𝛾𝑛𝑟̇Θ𝑛𝑟.(4.38) Choose the actual adaptive laws as ̇Θ𝑛𝑙=̇Θ𝑛𝑙=𝛾𝑛𝑙𝑒𝑛𝜉𝑇𝑛𝑙𝑥𝑛𝑐𝑛𝑙Θ𝑛𝑙,̇Θ𝑛𝑟=̇Θ𝑛𝑟=𝛾𝑛𝑟𝑒𝑛𝜉𝑇𝑛𝑟𝑥𝑛𝑐𝑛𝑟Θ𝑛𝑟.(4.39) Substituting (4.39) into (4.38) yields ̇𝑉𝑛𝑘=1𝑘𝑘,0𝑒2𝑘+𝑛𝑘=1𝜀2𝑘4𝑘𝑘,1𝑛𝑘=1𝑐𝑘𝑙Θ𝑘𝑙22𝑛𝑘=1𝑐𝑘𝑟Θ𝑘𝑟22+𝑛𝑘=1𝑐𝑘𝑙Θ𝑘𝑙22+𝑛𝑘=1𝑐𝑘𝑟Θ𝑘𝑟22.(4.40)𝑘𝑘,0 is chosen such that 𝑘𝑘,0>(𝜌/2𝐺𝑘), then 𝑘𝑘,0>𝜌2𝐺𝑘+̇𝐺𝑘2𝐺2𝑘(𝑘=1,,𝑛),(4.41) where 𝜌 is a positive constant and 𝛿𝑛𝑘=1𝜀2𝑘4𝑘𝑘,1+𝑛𝑘=1𝑐𝑘𝑙Θ𝑘𝑙22+𝑛𝑘=1𝑐𝑘𝑟Θ𝑘𝑟22.(4.42) Then (4.40) becomes ̇𝑉𝑛𝑘=1𝑘𝑘,0𝑒2𝑘+𝛿𝑛𝑘=1𝑐𝑘𝑙Θ𝑘𝑙22𝑛𝑘=1𝑐𝑘𝑟Θ𝑘𝑟22<𝑛𝑘=1𝜌2𝐺𝑘𝑒2𝑘+𝛿𝑛𝑘=1𝜌Θ𝑇𝑘𝑙Θ𝑘𝑙2𝑛𝑘=1𝜌Θ𝑇𝑘𝑟Θ𝑘𝑟2<𝜌𝑛𝑘=112𝐺𝑘𝑒2𝑘+𝑛𝑘=1Θ𝑇𝑘𝑙Θ𝑘𝑙2+𝑛𝑘=1Θ𝑇𝑘𝑟Θ𝑘𝑟2+𝛿<𝜌𝑉+𝛿.(4.43) From (4.43) we have 𝑡𝑉(𝑡)𝑉0𝑒𝜌(𝑡𝑡0)+𝛿𝜌.(4.44) It can be shown that the signals 𝑥(𝑡), 𝑒(𝑡), Θ𝑙(𝑡), Θ𝑟(𝑡), and 𝑢(𝑡) are globally uniformly ultimately bounded and that |𝑦𝑛(𝑡)𝑦𝑟(𝑡)|2𝑉(𝑡)𝑒(𝜌/2)(𝑡𝑡0)+2𝛿/𝜌. In order to achieve the tracking error convergences to a small neighborhood around zero, the parameters 𝜌 and 𝛿 should be chosen appropriately, then it is possible to make 2𝛿/𝜌 as small as desired. Denote 𝜙>2𝛿/𝜌. Since as 𝑡, 𝑒(𝛿/2)(𝑡𝑡0)0, therefore, it follows that there exists 𝑇, when 𝑡𝑇, |𝑦(𝑡)𝑦𝑟(𝑡)|𝜙.

5. Simulation

In this section, we provide a biological simulation example to illustrate the feasibility of the control scheme proposed in this paper.

In recent years, interest in adaptive control systems has increased rapidly along with interest and progress in control topics. The adaptive control has a variety of specific meanings, but it often implies that the system is capable of accommodating unpredictable environmental changes, whether these changes arise within the system or external to it.

Adaptation is a fundamental characteristic of living organisms such as prey-predator systems and many other biological models since these systems attempt to maintain physiological equilibrium in the midst changing environmental conditions.

Background Knowledge
The Salton Sea, which is located in the southeast desert of California, came into the limelight due to deaths of fish and fish catching birds on a massive scale. Recently, Chattopadhyay and Bairagi [54] proposed and analyzed an eco-epidemiological model on Salton Sea.We assume that there are two populations.

(1)The prey population, Tilapia fish, whose population density is denoted by 𝑁, which is the number of Tilapia fish per unit designated area.(2)In the absence of bacterial infection, the fish population grows according to a logistic law with carrying capacity 𝐾(𝐾𝑅+), with an intrinsic birth rate constant 𝑟(𝑟𝑅+), such that𝑑𝑁𝑁𝑑𝑡=𝑟𝑁1𝐾.(5.1)(3)In the presence of bacterial infection, the total fish population 𝑁 is divided into two classes, namely, susceptible fish population, denoted by 𝑆, and infected fish population, denoted by 𝐼. Therefore, at any time 𝑡, the total density of prey (i.e., fish) population is𝑁(𝑡)=𝑆(𝑡)+𝐼(𝑡).(5.2)(4)Only the susceptible fish population 𝑆 is capable of reproducing with the logistic law, and the infected fish population 𝐼 dies before having the capacity of reproduction. However, the infected fish 𝐼 still contributes with 𝑆 to population growth towards the carrying capacity.(5)Liu et al. [55] concluded that the bilinear mass action incidence rate due to saturation or multiple exposures before infection could lead to a nonlinear incidence rate as 𝜆𝑆𝑝𝐼𝑞 with 𝑝 and 𝑞 near 1 and without a periodic forcing term, which have much wider range of dynamical behaviors in comparison to bilinear incidence rate 𝜆𝑆𝐼. Here, 𝜆𝑅+ is the force of infection or rate of transmission. Therefore, the evolution equation for the susceptible fish population 𝑆 can be written as 𝑑𝑆𝑑𝑡=𝑟𝑆1𝑆+𝐼𝐾𝜆𝑆𝐼.(5.3)(6)The predator population, Pelican birds, whose population density is denoted by 𝑃, which is the number of birds per unit designated area.(7)It is assumed that Pelicans cannot distinguish the infected and healthy fish. They consume the fish that are readily available. Since the prey population is infected by a disease, infected preys are weakened and become easier to predate, while susceptible (healthy) preys easily escape predation. Considering this fact, it is assumed that the Pelicans mostly consume the infected fish only. The natural death rate of infected prey (not due to predation) is denoted by 𝜇(𝜇𝑅+). 𝑑 is the total death of predator population (including natural death and death due to predation of infected prey). 𝑚 is the search rate, 𝜃 is the conversion factor, and 𝑎 is the half saturation coefficient.(8)The system appears to exhibit a chaotic behavior for a range of parametric values. The range of the system parameters for which the subsystems converge to limit cycles is determined. In Figure 1, typical chaotic attractor for the model system is obtained for the parameter values as see [54]:𝑟=22𝐾=400𝜆=0.06𝜇=3.4𝑚=15.5𝑎=15𝑑=8.3𝜃=10.0.(5.4)

fig1
Figure 1: Typical chaotic attractor for the model system.

Practically, the populations of prey and predator are supposed to be stable or constant, so controller will be used to achieve the intended target. In this model, we add a controller to the third differential equation, that is to say, we control the population of Tilapia fish to be constant by changing the population of Pelican birds in some ways.

From the above assumptions and substituting 𝑆,𝐼,𝑃 by 𝑥1,𝑥2,𝑥3, respectively, we can write down the following differential equations:𝑑𝑥1𝑑𝑡=𝑟𝑥1𝑥11𝐾𝑟𝐾𝑥+𝜆1𝑥2,𝑑𝑥2𝑑𝑡=𝜆𝑥1𝑥2𝜇𝑥2𝑚𝑥2𝑥2𝑥+𝑎3,𝑑𝑥3=𝑑𝑡𝜃𝑥2𝑥3𝑥2+𝑎𝑑𝑥3+𝑢.(5.5) The backstepping design algorithm for type-2 adaptive fuzzy control is proposed as follows.

Step 1. Define the type-2 membership functions as 𝜇1𝑀1𝑥1=𝑒(𝑥150(1±10%)/10)2,𝜇2𝑀1𝑥1=𝑒(𝑥160(1±10%)/10)2,𝜇3𝑀1𝑥1=𝑒(𝑥170(1±10%)/10)2,𝜇1𝑀2𝑥2=𝑒(𝑥250(1±10%)/10)2,𝜇2𝑀2𝑥2=𝑒(𝑥260(1±10%)/10)2,𝜇3𝑀2𝑥2,=𝑒(𝑥270(1±10%)/10)2,𝜇1𝑀3𝑥3=𝑒(𝑥320(1±10%)/10)2,𝜇2𝑀3𝑥3=𝑒(𝑥330(1±10%)/10)2,𝜇3𝑀3𝑥3=𝑒(𝑥340(1±10%)/10)2,(5.6) then the fuzzy basis functions are obtained as𝜉1𝑙=𝜇1𝑀1𝐴,𝜇2𝑀1𝐴,𝜇3𝑀1𝐴,𝐴=𝜇1𝑀1+𝜇2𝑀1+𝜇3𝑀1,𝜉1𝑟=𝜇1𝑀1𝐵,𝜇2𝑀1𝐵,𝜇3𝑀1𝐵,𝐵=𝜇1𝑀1+𝜇2𝑀1+𝜇3𝑀1,𝜉2𝑙=𝜇1𝑀1𝜇1𝑀2𝐶,𝜇1𝑀1𝜇2𝑀2𝐶,𝜇1𝑀1𝜇3𝑀2𝐶,𝜇2𝑀1𝜇1𝑀2𝐶,𝜇2𝑀1𝜇2𝑀2𝐶,𝜇2𝑀1𝜇3𝑀2𝐶,𝜇3𝑀1𝜇1𝑀2𝐶,𝜇3𝑀1𝜇2𝑀2𝐶,𝜇3𝑀1𝜇3𝑀2𝐶,𝐶=𝜇1𝑀1𝜇1𝑀2+𝜇1𝑀1𝜇2𝑀2+𝜇1𝑀1𝜇3𝑀2+𝜇2𝑀1𝜇1𝑀2+𝜇2𝑀1𝜇2𝑀2+𝜇2𝑀1𝜇3𝑀2+𝜇3𝑀1𝜇1𝑀2+𝜇3𝑀1𝜇2𝑀2+𝜇3𝑀1𝜇3𝑀2,𝜉1𝑟=𝜇1𝑀1𝜇1𝑀2𝐷,𝜇1𝑀1𝜇2𝑀2𝐷,𝜇1𝑀1𝜇3𝑀2𝐷,𝜇2𝑀1𝜇1𝑀2𝐷,𝜇2𝑀1𝜇2𝑀2𝐷,𝜇2𝑀1𝜇3𝑀2𝐷,𝜇3𝑀1𝜇1𝑀2𝐷,𝜇3𝑀1𝜇2𝑀2𝐷,𝜇3𝑀1𝜇3𝑀2𝐷,𝐷=𝜇1𝑀1𝜇1𝑀2+𝜇1𝑀1𝜇2𝑀2+𝜇1𝑀1𝜇3𝑀2+𝜇2𝑀1𝜇1𝑀2+𝜇2𝑀1𝜇2𝑀2+𝜇2𝑀1𝜇3𝑀2+𝜇3𝑀1𝜇1𝑀2+𝜇3𝑀1𝜇2𝑀2+𝜇3𝑀1𝜇3𝑀2,𝜉3𝑙=𝜇1𝑀2𝜇1𝑀3𝐸,𝜇1𝑀2𝜇2𝑀3𝐸,𝜇1𝑀2𝜇3𝑀3𝐸,𝜇2𝑀2𝜇1𝑀3𝐸,𝜇2𝑀2𝜇2𝑀3𝐸,𝜇2𝑀2𝜇3𝑀3𝐸,𝜇3𝑀2𝜇1𝑀3𝐸,𝜇3𝑀2𝜇2𝑀3𝐸,𝜇3𝑀2𝜇3𝑀3𝐸,𝐸=𝜇1𝑀2𝜇1𝑀3+𝜇1𝑀2𝜇2𝑀3+𝜇1𝑀2𝜇3𝑀3+𝜇2𝑀2𝜇1𝑀3+𝜇2𝑀2𝜇2𝑀3+𝜇2𝑀2𝜇3𝑀3+𝜇3𝑀2𝜇1𝑀3+𝜇3𝑀2𝜇2𝑀3+𝜇3𝑀2𝜇3𝑀3,𝜉3𝑟=𝜇1𝑀2𝜇1𝑀3𝐹,𝜇1𝑀2𝜇2𝑀3𝐹,𝜇1𝑀2𝜇3𝑀3𝐹,𝜇2𝑀2𝜇1𝑀3𝐹,𝜇2𝑀2𝜇2𝑀3𝐹,𝜇2𝑀2𝜇3𝑀3𝐹,𝜇3𝑀2𝜇1𝑀3𝐹,𝜇3𝑀2𝜇2𝑀3𝐹,𝜇3𝑀2𝜇3𝑀3𝐹,𝐹=𝜇1𝑀2𝜇1𝑀3+𝜇1𝑀2𝜇2𝑀3+𝜇1𝑀2𝜇3𝑀3+𝜇2𝑀2𝜇1𝑀3+𝜇2𝑀2𝜇2𝑀3+𝜇2𝑀2𝜇3𝑀3+𝜇3𝑀2𝜇1𝑀3+𝜇3𝑀2𝜇2𝑀3+𝜇3𝑀2𝜇3𝑀3,𝜉1𝑙=𝜉11𝑙,𝜉21𝑙,𝜉31𝑙,𝑇𝜉1𝑟=𝜉11𝑟,𝜉21𝑟,𝜉31𝑟𝑇,𝜉2𝑙=𝜉12𝑙,𝜉22𝑙𝜉92𝑙,𝑇𝜉2𝑟=𝜉12𝑟,𝜉22𝑟𝜉92𝑟𝑇,𝜉3𝑙=𝜉13𝑙,𝜉23𝑙𝜉93𝑙,𝑇𝜉3𝑟=𝜉13𝑟,𝜉23𝑟𝜉93𝑟𝑇.(5.7)

Step 2. We assume that there exist some language rules of unknown functions 1, 2, and 3, respectively.
The unknown function 1:1(𝑟/𝐾)+𝜆𝑟𝑥1𝑥11𝐾̇𝑦𝑟.(5.8)
The unknown function 2:𝑥2+𝑎𝑚𝑥2𝜆𝑥1𝑥2𝜇𝑥2̇𝛼1.(5.9)
The unknown function 3:𝜃𝑥2𝑥3𝑥2+𝑎̇𝛼2𝜃𝑥2𝑥3𝑥2+𝑎̇𝛼2.(5.10)
The fuzzy rules of the Unknown Function 1 (UF1) are as follows.
R1: if 𝑥1 is Small (S), then UF1 is Powerful (P).R2: if 𝑥1 is Medium (M), then UF1 is Weak (W).R3: if 𝑥1 is Large (L), then UF1 is Modest (M).
The fuzzy rules of the Unknown Function 2 (UF2) are as follows.
R1: if 𝑥1 is S, 𝑥2 is S, then UF2 is W.R2: if 𝑥1 is S, 𝑥2 is M, then UF2 is M.R3: if 𝑥1 is S, 𝑥2 is L, then UF2 is P.R4: if 𝑥1 is M, 𝑥2 is S, then UF2 is W.R5: if 𝑥1 is M, 𝑥2 is M, then UF2 is M.R6: if 𝑥1 is M, 𝑥2 is L, then UF2 is P.R7: if 𝑥1 is L, 𝑥2 is S, then UF2 is P.R8: if 𝑥1 is L, 𝑥2 is M, then UF2 is M.R9: if 𝑥1 is L, 𝑥2 is L, then UF2 is W.
The fuzzy rules of the Unknown Function 3 (UF3) are as follows.
R1: if 𝑥2 is S, 𝑥3 is S, then UF3 is M.R2: if 𝑥2 is S, 𝑥3 is M, then UF3 is W.R3: if 𝑥2 is S, 𝑥3 is L, then UF3 is P.R4: if 𝑥2 is M, 𝑥3 is S, then UF3 is W.R5: if 𝑥2 is M, 𝑥3 is M, then UF3 is M.R6: if 𝑥2 is M, 𝑥3 is L, then UF3 is P.R7: if 𝑥2 is L, 𝑥3 is S, then UF3 is M.R8: if 𝑥2 is L, 𝑥3 is M, then UF3 is W.R9: if 𝑥2 is L, 𝑥3 is L, then UF3 is P.Step 3. Specify positive design parameters as follows: 𝑘1=𝑘2=𝑘3=0.5,𝛾1𝑙=𝛾2𝑙=𝛾3𝑙=𝛾1𝑟=𝛾2𝑟=𝛾3𝑟𝑐=1,1𝑙=𝑐2𝑙=𝑐3𝑙=0.5,𝑐1𝑟=𝑐2𝑟=𝑐3𝑟=0.6.(5.11)
The first intermediate controller is chosen as𝛼1=𝑘1𝑒1𝜉𝑇1𝑙𝑥1Θ1𝑙+𝜉𝑇1𝑟𝑥1Θ1𝑟.(5.12)
The first adaptive laws are designed aṡΘ1𝑙=̇Θ1𝑙=𝛾1𝑙𝑒1𝜉𝑇1𝑙𝑥1𝑐1𝑙Θ1𝑙,̇Θ1𝑟=̇Θ1𝑟=𝛾1𝑟𝑒1𝜉𝑇1𝑟𝑥1𝑐1𝑟Θ1𝑟.(5.13)
The second intermediate controller is chosen as𝛼2=𝑒1𝑘2𝑒2𝜉𝑇2𝑙𝑥2Θ2𝑙+𝜉𝑇2𝑟𝑥2Θ2𝑟.(5.14)
The second adaptive laws are designed aṡΘ2𝑙=̇Θ2𝑙=𝛾2𝑙𝑒2𝜉𝑇2𝑙𝑥2𝑐2𝑙Θ2𝑙,̇Θ2𝑟=̇Θ2𝑟=𝛾2𝑟𝑒2𝜉𝑇2𝑟𝑥2𝑐2𝑟Θ2𝑟.(5.15)
The actual controller is chosen as𝑢=𝑒2𝑘3𝑒3𝜉𝑇3𝑙𝑥3Θ3𝑙+𝜉𝑇3𝑟𝑥3Θ3𝑟.(5.16)
The actual adaptive laws are designed aṡΘ3𝑙=̇Θ3𝑙=𝛾3𝑙𝑒3𝜉𝑇3𝑙𝑥3𝑐3𝑙Θ3𝑙,̇Θ3𝑟=̇Θ3𝑟=𝛾3𝑟𝑒3𝜉𝑇3𝑟𝑥3𝑐3𝑟Θ3𝑟.(5.17) The reference output is specified as 𝑦𝑟(𝑡)=93.
The performance of T2ABFC is compared by traditional T1ABFC in state variable outputs, controller trajectory, tracking error, and resistance to disturbances. The simulation results are shown in Figures 111.(1)Figures 2 and 3 show the state variable output with T2ABFC and T1ABFC, respectively. We can obtain that the T2ABFC has a higher performance in terms of response speed than T1ABFC.(2)Figure 4 shows the trajectories of controller based on T2ABFC and T1ABFC, respectively.(3)Figure 5 shows the trajectories of tracking error based on T2ABFC and T1ABFC, respectively. By comparing, the superiority of T2ABFC in tracking performance is obvious. In order to show the property of resistance to disturbances, training data is corrupted by a random noise ±0.05𝑥 and ±0.5𝑥, that is, 𝑥 is replaced by (1 ± random(0.05)) 𝑥 and (1± random (0.5)) 𝑥.(4)Figures 6 and 7 show the responses of state variables 𝑥1 with disturbances based on T2ABFC and T1ABFC, respectively.(5)Figures 8 and 9 show the responses of state variables 𝑥2 with disturbances based on T2ABFC and T1ABFC, respectively.(6)Figures 10 amd 11 show the responses of state variables 𝑥3 with disturbances based on T2ABFC and T1ABFC respectively. It is obvious that the system based on T2ABFC has the better property of resistance to disturbances.
From all the outputs of simulation, we can obtain that T2ABFC method proposed in this paper guaranteeS the higher tracking performance and resistance to external disturbances than traditional T1ABFC method.

fig2
Figure 2: State variable outputs with T2ABFC.
fig3
Figure 3: State variable outputs with T1ABFC.
658424.fig.004
Figure 4: Trajectories of controller.
658424.fig.005
Figure 5: Trajectories of tracking error.
658424.fig.006
Figure 6: Trajectories of the state variable 𝑥1 by T2ABFC with external disturbances.
658424.fig.007
Figure 7: Trajectories of the state variable 𝑥1 by T1ABFC with external disturbances.
658424.fig.008
Figure 8: Trajectories of the state variable 𝑥2 by T2ABFC with external disturbances.
658424.fig.009
Figure 9: Trajectories of the state variable 𝑥2 by T1ABFC with external disturbances.
658424.fig.0010
Figure 10: Trajectories of the state variable 𝑥3 by T2ABFC with external disturbances.
658424.fig.0011
Figure 11: Trajectories of the state variable 𝑥3 by T1ABFC with external disturbances.

6. Conclusion

In this paper, we solve the globally stable adaptive backstepping fuzzy control problem for a class of nonlinear systems and T2ABFC is recommended in this approach. From the simulation results, the main conclusions can be drawn.

(1) T2ABFC guarantees the outputs of the closed-loop system follow the reference signal, and all the signals in the closed-loop system are uniform ultimately bounded.

(2) Compared by traditional T1ABFC, T2ABFC has higher performances, in terms of stability, response speed, and resistance to external disturbances.

Acknowledgment

Work supported by National Natural Science Foundation of China (11072090) and project of advanced talents of Jiangsu University (10JDG093).

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