Adaptive Backstepping Fuzzy Control Based on Type-2 Fuzzy System

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) [6–9], Adaptive Backstepping Neural Network Control (ABNNC) [10–13], 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 [22–28]: (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 [32–37]. Also, some results on T2 fuzzy sliding-mode controller have been presented in [38–40]. 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 as𝑥̇_𝑖=𝑥𝑓̇_𝑖𝑛𝑥_𝑖+𝐺_𝑖𝑥_𝑖𝑥_𝑖+11≤𝑖≤𝑛−1,=𝑓_𝑛𝑥_𝑛+𝑦=𝐺_𝑛𝑥₁𝑥_𝑛𝑢𝑛≥2,,(2.1) where 𝑥_𝑖=[𝑥₁,𝑥₂…𝑥_𝑖]^𝑇∈𝑅^𝑖 is the stable vector and 𝑢∈𝑅 and 𝑦∈𝑅 are the input and output of the system, respectively. 𝑓_𝑖(𝑥_𝑖), 𝐺_𝑖(𝑥_𝑖) (𝑖=1,2…𝑛) are unknown smooth nonlinear functions. The control objective of this paper is formulated as follows: for a given bounded reference signal 𝑦_𝑟(𝑡)∈𝐷 with continuous and bounded derivatives up to order 𝜌, where 𝐷∈𝑅 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 𝜇_𝐴(𝑥,𝑢) [22], where 𝑥∈𝑋 is the primary variable and 𝑢∈𝐽_𝑥⊆[0,1] is the secondary variable:𝐴=(𝑥,𝑢),𝜇_𝐴(𝑥,𝑢)∣∀𝑥∈𝑋∀𝑢∈𝐽_𝑥⊆[0,1].(3.1)in which 0≤𝜇_𝐴(𝑥,𝑢)≤1. 𝐴 can also be expressed as follows [22]: 𝐴=_𝑥∈𝑋_{𝑢∈𝐽𝑥}𝜇_𝐴(𝑥,𝑢)(𝑥,𝑢)𝐽^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, 46–50]. It is described as𝐴=_𝑥∈𝑋_{𝑢∈𝐽𝑥}1(𝑥,𝑢)=_𝑥∈𝑋∫^𝑢∈𝐽_𝑥1/𝑢𝑥.(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 as𝑢_𝐴(𝑥)=exp−12𝑥−𝜎𝑚²,𝑚∈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 [22–24], which is called the FOU of 𝐴:FOU𝐴=_{∀𝑥∈𝑋}𝐽_𝑥=(𝑥,𝑢)∶𝑢∈𝜇_𝐴(𝑥),𝜇_𝐴(𝑥),(3.5)

where𝜇_𝐴(𝑥)=FOU𝐴∀𝑥∈𝑋,𝜇_𝐴(𝑥)=FOU𝐴∀𝑥∈𝑋.(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 𝑥₁∈𝑋₁,…,𝑥_𝑛∈𝑋_𝑛 and one output 𝑦∈𝑌, and assuming there are 𝑀 rules, the 𝑖th rule of the IT2 SMC can be described as𝑅^𝑙∶IF𝑥₁is𝐹^𝑙1…and𝑥_𝑛is𝐹^𝑙𝑛,THEN𝑦is𝐺^𝑙𝑙=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:𝑌_cos𝑌¹,…,𝑌^𝑀,𝐹¹,…,𝐹^𝑀=𝑦_𝑙,𝑦_𝑟=_𝑦1…_𝑦𝑀_𝑓1…_𝑓𝑀1∑^𝑀𝑖=1𝑓^𝑖𝑦^𝑖/∑^𝑀𝑖=1𝑓^𝑖,(3.8) where 𝑌_cos(𝑥) is an interval output set determined by its left-most point 𝑦_𝑙 and its right-most point 𝑦_𝑟, and 𝑓^𝑖∈𝐹^𝑖=⌊𝑓^𝑖,𝑓^𝑖⌋. In the meantime, an IT2FLS with singleton fuzzification and meet under minimum or product 𝑡-norm 𝑓^𝑖 and 𝑓^𝑖 can be obtained as𝑓^𝑖=𝜇𝐹^𝑖1𝑥₁∗⋅⋅⋅∗𝜇𝐹^𝑖𝑛𝑥_𝑛,(3.9)

and𝑓^𝑖=𝜇𝐹^𝑖1𝑥₁∗⋅⋅⋅∗𝜇𝐹^𝑖𝑛𝑥_𝑛.(3.10)

Also, 𝑦^𝑖∈𝑌^𝑖 and 𝑌^𝑖=[𝑦^𝑖𝑙,𝑦^𝑖𝑟] is the centroid of the IT2 consequent set 𝐺^𝑖, the centroid of a T2FS, and for any value 𝑦∈𝑌_cos, 𝑦 can be expressed as𝑦=∑^𝑀𝑖=1𝑓^𝑖𝑦^𝑖∑^𝑀𝑖=1𝑓^𝑖,(3.11) where 𝑦 is a monotonic increasing function with respect to 𝑦^𝑖. Also, 𝑦_𝑙 is the minimum associated only with 𝑦^𝑖𝑙, and 𝑦_𝑟 is the maximum associated only with 𝑦^𝑖𝑟. Note that 𝑦_𝑙 and 𝑦_𝑟 depend only on the mixture of 𝑓^𝑖 or 𝑓^𝑖 values. Therefore, the left-most point 𝑦_𝑙 and the right-most point 𝑦_𝑟 can be expressed as a Fuzzy Basis Function (FBF) expansion, that is, 𝑦_𝑙=∑^𝑀𝑖=1𝑓^𝑖𝑙𝑦^𝑖𝑙∑^𝑀𝑖=1𝑓^𝑖𝑙=^𝑀_𝑖=1𝑦^𝑖𝑙𝜉^𝑖𝑙=𝑦^𝑇𝑙𝜉_𝑙,(3.12)

and𝑦_𝑟=∑^𝑀𝑖=1𝑓^𝑖𝑟𝑦^𝑖𝑟∑^𝑀𝑖=1𝑓^𝑖𝑟=^𝑀_𝑖=1𝑦^𝑖𝑟𝜉^𝑖𝑟=𝑦^𝑇𝑟𝜉_𝑟,(3.13) respectively, where 𝜉^𝑖𝑙=𝑓^𝑖𝑙/∑^𝑀𝑖=1𝑓^𝑖, 𝜉_𝑙=[𝜉^1𝑙,𝜉^2𝑙,…,𝜉^𝑀𝑙], 𝑦^𝑇𝑙=[𝑦^1𝑙,𝑦^2𝑙,…,𝑦^𝑀𝑙], and 𝜉^𝑖𝑟=𝑓^𝑖𝑟/∑^𝑀𝑖=1𝑓^𝑖,  𝜉_𝑟=[𝜉^1𝑟,𝜉^2𝑟,…,𝜉^𝑀𝑟],  𝑦^𝑇𝑟=[𝑦^1𝑟,𝑦^2𝑟,…,𝑦^𝑀𝑟].

In order to compute 𝑦_𝑙 and 𝑦_𝑟, 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 𝑦_𝑟. Without loosing of generality, assume 𝑦^𝑖𝑟 is arranged in ascending order, that is, 𝑦^1𝑟≤𝑦^2𝑟≤⋅⋅⋅≤𝑦^𝑀𝑟.

Step 1. Compute𝑦_𝑟 in (3.13) by initially setting 𝑓^𝑙𝑟=(𝑓^𝑙+𝑓^𝑙)/2 for 𝑙=1,…,𝑀, where𝑓^𝑙 and 𝑓^𝑙 have been precomputed by (3.10) and let 𝑦^𝑟=𝑦_𝑟.

Step 2 :. Find 𝑅(1≤𝑅≤𝑀−1) such that 𝑦^𝑅𝑟≤𝑦^𝑟≤𝑦^𝑟𝑅+1.

Step 3. Compute 𝑦_𝑟 in (3.13) with 𝑓^𝑖𝑟=𝑓^𝑖 for 𝑖≤𝑅 and 𝑓^𝑖𝑟=𝑓^𝑖 for 𝑖>𝑅and let 𝑦^𝑟=𝑦_𝑟.

Step 4. If 𝑦^𝑟≠𝑦_𝑟, then go to Step 5; if 𝑦^𝑟=𝑦_𝑟, then stop and set 𝑦_𝑟=𝑦^𝑟.

Step 5. Set 𝑦^𝑟equal to 𝑦^𝑟 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 𝑓^𝑖’s and another side using upper firing strengths 𝑓^𝑖’s. Therefore, 𝑦_𝑟 can be expressed as𝑦_𝑟=∑^𝑅𝑖=1𝑓^𝑖𝑦^𝑖𝑟+∑^{𝑀𝑖=𝑅+1}𝑓^𝑖𝑦^𝑖𝑟∑^𝑅𝑖=1𝑓^𝑖+∑^{𝑀𝑖=𝑅+1}𝑓^𝑖=^𝑅_𝑖=1𝑞^𝑖𝑟𝑦^𝑖𝑟+^𝑀_𝑖=_𝑅+1𝑞^𝑖𝑟𝑦^𝑖𝑟=𝑄_𝑟𝑄_𝑟⎡⎢⎢⎣𝑦^𝑟𝑦_𝑟⎤⎥⎥⎦=𝜉_𝑇𝑟Θ_𝑟,(3.14) where 𝑞^𝑖𝑟=𝑓^𝑖/𝐷_𝑟, 𝑞^𝑖𝑟=𝑓^𝑖/𝐷_𝑟 and 𝐷_𝑟=(∑^𝑅𝑖=1𝑓^𝑖+∑^{𝑀𝑖=𝑅+1}𝑓^𝑖). In the meantime, we have 𝑄_𝑟=[𝑞^1𝑟,𝑞^2𝑟,…,𝑞^𝑅𝑟], 𝑄_𝑟=[𝑞^1𝑟,𝑞^2𝑟,…,𝑞^𝑅𝑟], 𝜉^𝑇𝑟=⌊𝑄_𝑟,𝑄_𝑟⌋, and Θ^𝑇𝑟=⌊𝑦_𝑟𝑦_𝑟⌋.
The procedure to compute 𝑦_𝑙 is similar to compute 𝑦_𝑟. In Step 2, it only determines 𝐿(1≤𝐿≤𝑀−1), such that 𝑦^𝐿𝑙≤𝑦^𝑙≤𝑦^𝑙𝐿+1. In Step 3, let 𝑓^𝑖𝑙=𝑓^𝑖 for 𝑖≤𝐿 and𝑓^𝑖𝑙=𝑓^𝑖 for 𝑖>𝐿. Therefore, 𝑦_𝑙 can be expressed as𝑦_𝑙=∑^𝐿𝑖=1𝑓^𝑖𝑦^𝑖𝑙+∑^{𝑀𝑖=𝐿+1}𝑓^𝑖𝑦^𝑖𝑙∑^𝐿𝑖=1𝑓^𝑖+∑^{𝑀𝑖=𝐿+1}𝑓^𝑖=^𝐿_𝑖=1𝑞^𝑖𝑙𝑦^𝑖𝑙+^𝑀_𝑖=_𝐿+1𝑞^𝑖𝑙𝑦^𝑖𝑙=𝑄_𝑙𝑄_𝑙⎡⎢⎢⎣𝑦^𝑙𝑦_𝑙⎤⎥⎥⎦=𝜉_𝑇𝑙Θ_𝑙,(3.15) where 𝑞^𝑖𝑙=𝑓^𝑖/𝐷_𝑙, 𝑞^𝑖𝑙=𝑓^𝑖/𝐷_𝑙 and 𝐷_𝑙=(∑^𝐿𝑖=1𝑓^𝑖+∑^{𝑀𝑖=𝐿+1}𝑓^𝑖). In the meantime, we have 𝑄_𝑙=[𝑞^1𝑙,𝑞^2𝑙,…,𝑞^𝑅𝑙], 𝑄_𝑙=[𝑞^1𝑙,𝑞^2𝑙,…,𝑞^𝑅𝑙], 𝜉^𝑇𝑙=⌊𝑄_𝑙𝑄_𝑙⌋, and Θ^𝑇𝑙=⌊𝑦_𝑙𝑦_𝑙⌋.
We defuzzify the interval set by using the average of 𝑦_𝑙 and 𝑦_𝑟, hence, the defuzzified crisp output becomes𝑌_fuzz2(𝑥)=𝑦^𝑙+𝑦_𝑟2=21𝜉^𝑇𝑙Θ_𝑙+𝜉^𝑇𝑟Θ_𝑟=12𝜉^𝑇𝑙𝜉^𝑇𝑟⎡⎢⎢⎣Θ^𝑙Θ_𝑟⎤⎥⎥⎦=𝜉_𝑇Θ,(3.16) where (1/2)[𝜉^𝑇𝑙𝜉^𝑇𝑟]=𝜉^𝑇and [Θ^𝑇𝑙Θ^𝑇𝑟]=Θ^𝑇.

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 sup_𝑥∈Ω||𝑓(𝑥)−𝜉^𝑇(𝑥)Θ||≤𝜀.(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 𝐺_𝑖 and 𝐺_𝑖(1≤𝑖≤𝑛), such that 𝐺_𝑖≤𝐺_𝑖(𝑥_𝑖)≤𝐺_𝑖.

Assumption 4.2. There exist positive constant 𝐺^𝑑𝑖(1≤𝑖≤𝑛), such that |𝐺̇_𝑖(𝑥_𝑖)|≤𝐺^𝑑𝑖.

Assumption 4.3. Define the optimal parameter vectors Θ^∗𝑖 as Θ^∗𝑖=argmin_{Θ𝑖∈Ω𝑖}sup_{𝑥𝑖∈𝑈𝑖}||𝑓_𝑖𝑥_𝑖∣Θ_𝑖−𝑓_𝑖𝑥_𝑖||,(4.1) where Ω_𝑖, 𝑈_𝑖 are compact regions for Θ_𝑖, 𝑥_𝑖, respectively. The fuzzy logic system minimum approximation errors 𝜔_𝑖 are defined as ||𝜉^𝑇𝑖Θ^∗𝑖−𝑓_𝑖𝑥_𝑖||≤𝜔_𝑖.(4.2)

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

Theorem 4.4. Suppose assumptions are 4.1–4.3 tenable, then the fuzzy adaptive output tracking design described by (2.1), control law 𝑢=−𝑒_𝑛−1−𝑘_𝑛𝑒_𝑛−𝜉^𝑇𝑛(𝑥_𝑛)Θ_𝑛, and parameter adaptive laws Θ̇_𝑛𝑙=Θ̇_𝑛𝑙=𝛾_𝑛𝑙(𝑒_𝑛𝜉^𝑇𝑛𝑙(𝑥_𝑛)−𝑐_𝑛𝑙Θ_𝑛𝑙)  Θ̇_𝑛𝑟=Θ̇_𝑛𝑟=𝛾_𝑛𝑟(𝑒_𝑛𝜉^𝑇𝑛𝑟(𝑥_𝑛)−𝑐_𝑛𝑟Θ_𝑛𝑟) 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. (𝑘_𝑛, 𝛾_𝑛𝑙, 𝛾_𝑛𝑟, 𝑐_𝑛𝑙, 𝑐_𝑛𝑟 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=𝑥₁−𝑦_𝑟.(4.3) The time derivative of 𝑒₁ is 𝑒₁̇=𝑥̇₁−𝑦̇_𝑟=𝑓₁𝑥₁+𝐺₁𝑥₁𝑥₂−𝑦̇_𝑟.(4.4) Take 𝑥₂ as a virtual control, and define 𝛼^∗1=𝑘₁𝑒₁−1𝐺₁𝑥₁𝑓¹𝑥₁−𝑦̇_𝑟,(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)−𝑦̇_𝑟) can be approximated by the following type-2 fuzzy logic system 𝜉^𝑇1(𝑥₁)Θ₁, and we obtain𝛼^∗1=−𝑘₁𝑒₁−𝜉^𝑇1𝑥₁Θ^∗1−𝜔₁.(4.6)Express 𝑥₂ as 𝑥₂=𝑒₂+𝛼₁ and define 𝛼₁=−𝑘₁𝑒₁−𝜉^𝑇1𝑥₁Θ₁=−𝑘₁𝑒₁−𝜉^𝑇1𝑙𝑥₁Θ_1𝑙+𝜉^𝑇1𝑟𝑥₁Θ_1𝑟,(4.7) then the time derivative of 𝑒₁ is 𝑒₁̇=𝑓₁𝑥₁+𝐺₁𝑥₁𝑒₂+𝛼₁−𝑦̇_𝑟==𝐺𝐺¹₁𝑥₁𝑒₂−𝑘₁𝑒₁−𝜉^𝑇1𝑥₁Θ₁+𝜔₁𝑥₁𝑒₂−𝑘₁𝑒₁−𝜉^𝑇1𝑙𝑥₁Θ_1𝑙−𝜉^𝑇1𝑟𝑥₁Θ_1𝑟+𝜔₁,(4.8) where Θ₁=Θ₁−Θ^∗1.
Consider the following Lyapunov function:𝑉₁=12𝐺₁𝑥₁𝑒²¹+12𝛾₁Θ^𝑇1Θ₁=12𝐺₁𝑥₁𝑒²¹+12𝛾_1𝑙Θ^𝑇1𝑙Θ_1𝑙+12𝛾_1𝑟Θ^𝑇1𝑟Θ_1𝑟,(4.9) then the time derivative of 𝑉₁ is 𝑉₁̇=𝑒¹𝑒₁̇𝐺₁𝑥₁−𝐺̇¹𝑥₁2𝐺²¹𝑥₁𝑒²¹+1𝛾₁Θ^𝑇1Θ̇₁=𝑒¹𝑒₁̇𝐺₁𝑥₁−𝐺̇¹𝑥₁2𝐺²¹𝑥₁𝑒²¹+1𝛾_1𝑙Θ^𝑇1𝑙Θ̇_1𝑙+1𝛾_1𝑟Θ^𝑇1𝑟Θ̇_1𝑟=𝑒₁𝑒₂−𝑘₁𝑒²¹−𝑒₁Θ^𝑇1𝑙𝜉_1𝑙+Θ^𝑇1𝑟𝜉_1𝑟+𝑒¹𝜔−𝐺̇¹𝑥₁2𝐺²¹𝑥₁𝑒²¹+1𝛾_1𝑙Θ^𝑇1𝑙Θ̇_1𝑙+1𝛾_1𝑟Θ^𝑇1𝑟Θ̇_1𝑟=𝑒₁𝑒₂−𝑘₁𝑒²¹−𝐺̇¹𝑥₁2𝐺²¹𝑥₁𝑒²¹+𝑒₁𝜔₁+Θ^𝑇1𝑙−𝑒¹𝜉^𝑇1𝑙𝑥₁+1𝛾_1𝑙Θ̇_1𝑙+Θ^𝑇1𝑟−𝑒¹𝜉^𝑇1𝑟𝑥₁+1𝛾_1𝑟Θ̇_1𝑟.(4.10) Choose the intermediate adaptive laws as Θ̇_1𝑙Θ̇_1𝑟=Θ̇_1𝑙=𝛾_1𝑙𝑒¹𝜉^𝑇1𝑙𝑥₁−𝑐_1𝑙Θ_1𝑙,=Θ̇_1𝑟=𝛾_1𝑟𝑒¹𝜉^𝑇1𝑟𝑥₁−𝑐_1𝑟Θ_1𝑟,(4.11) where 𝑐_1𝑙 and 𝑐_1𝑟 are given positive constants.Substituting (4.11) into (4.10) yields 𝑉₁̇=𝑒₁𝑒₂+𝑒₁𝜔₁−𝑘₁+𝐺̇¹𝑥₁2𝐺²¹𝑥₁𝑒²¹−𝑐_1𝑙Θ^𝑇1𝑙Θ_1𝑙+𝑐_1𝑟Θ^𝑇1𝑟Θ_1𝑟,(4.12) where 𝑘₁=𝑘_1,0+𝑘_1,1,  𝑘_1,0>0,𝑘_1,1>0,  𝑒₁𝜔₁−𝑘₁𝑒²¹≤𝑒₁𝜔₁−𝑘_1,1𝑒²¹≤𝜀²¹/4𝑘_1,1,|𝜔₁|≤|𝜀₁|
We can obtain that −𝑐_1𝑙Θ^𝑇1𝑙Θ_1𝑙=−𝑐_1𝑙Θ^𝑇1𝑙Θ^1𝑙+Θ^∗1𝑙≤−𝑐^1𝑙‖‖Θ_1𝑙‖‖²+𝑐_1𝑙‖‖Θ_1𝑙‖‖‖‖Θ^∗1𝑙‖‖≤−𝑐^1𝑙‖‖Θ_1𝑙‖‖²2+𝑐^1𝑙‖‖Θ^∗1𝑙‖‖²−𝑐_1𝑟2,Θ^𝑇1𝑟Θ_1𝑟=−𝑐_1𝑟Θ^𝑇1𝑟Θ^1𝑟+Θ^∗1𝑟≤−𝑐^1𝑟‖‖Θ_1𝑟‖‖²+𝑐_1𝑟‖‖Θ_1𝑟‖‖‖‖Θ^∗1𝑟‖‖≤−𝑐^1𝑟‖‖Θ_1𝑟‖‖²2+𝑐^1𝑟‖‖Θ^∗1𝑟‖‖²−𝑘_1,02+𝐺̇¹2𝐺²¹𝑒²¹≤−𝑘_1,0−𝐺^𝑑12𝐺²¹𝑒²¹,𝑘^∗1,0=𝑘_1,0−𝐺^𝑑12𝐺²¹>0.(4.13)
From (4.12), (4.13), it follows that 𝑉₁̇≤𝑒₁𝑒₂−𝑘^∗1,0𝑒²¹+𝜀²¹4𝑘_1,1−𝑐^1𝑙‖‖Θ_1𝑙‖‖²2−𝑐^1𝑟‖‖Θ_1𝑟‖‖²2+𝑐^1𝑙‖‖Θ^∗1𝑙‖‖²2+𝑐^1𝑟‖‖Θ^∗1𝑟‖‖²2.(4.14)

Step 2. Differentiating 𝑒₂ yields 𝑒₂̇=𝑥̇₂−𝛼̇₁=𝑓₂𝑥₂+𝐺₂𝑥₂𝑥₃−𝛼̇₁.(4.15) Take 𝑥₃ as a virtual control, and define 𝛼^∗2=−𝑒₁−𝑘₂𝑒₂−1𝐺₂𝑥₂𝑓²𝑥₂−𝛼̇₁,(4.16) where 𝑘₂ is a positive constant.
From (4.7), we obtain 𝛼̇₁=𝜕𝛼¹𝜕𝑥₁𝑓₁+𝐺₁𝑥₂+𝜕𝛼¹𝜕𝑦_𝑟𝑦̇_𝑟+𝜕𝛼¹𝜕Θ_1𝑙𝛾^1𝑙𝑒^1𝑙𝜉_1𝑙−𝑐_1𝑙Θ_1𝑙+𝜕𝛼¹𝜕Θ_1𝑟𝛾^1𝑟𝑒¹𝜉_1𝑟−𝑐_1𝑟Θ_1𝑟.(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)−𝛼̇₁)can be approximated by the following type-2 fuzzy logic system 𝜉^𝑇2(𝑥₂)Θ₂, and obtain 𝛼₂=−𝑒₁−𝑘₂𝑒₂−𝜉^𝑇2𝑥₂Θ₂.(4.18) Express 𝑥₃ as 𝑥₃=𝑒₃+𝛼₂ and the time derivative of 𝑒₂ is 𝑒₂̇=𝑓₂𝑥₂+𝐺₂𝑥₂𝑒₃+𝛼₂−𝛼̇₁=𝐺₂𝑥₂𝑒₃−𝑒₁−𝑘₂𝑒₂−𝜉^𝑇2𝑥₂Θ₂+𝜔₂,(4.19) where Θ₂=Θ₂−Θ^∗2.
Consider the following Lyapunov function𝑉₂=𝑉₁+12𝐺₂𝑥₂𝑒²²+12𝛾₂Θ^𝑇2Θ₂=𝑉₁+12𝐺₂𝑥₂𝑒²²+12𝛾_2𝑙Θ^𝑇2𝑙Θ_2𝑙+12𝛾_2𝑟Θ^𝑇2𝑟Θ_2𝑟,(4.20) then the time derivative of 𝑉₂ is 𝑉₂̇=𝑉₁̇−𝑒₁𝑒₂+𝑒₂𝑒₃−𝑘₂𝑒²²−𝐺̇²𝑥₂2𝐺²²𝑥₂𝑒²²+𝑒₂𝜔₂−Θ^𝑇2𝑙𝑒²𝜉^𝑇2𝑙𝑥₂−1𝛾_2𝑙Θ̇_2𝑙−Θ^𝑇2𝑟𝑒²𝜉^𝑇2𝑟𝑥₂−1𝛾_2𝑟Θ̇_2𝑟.(4.21) Choose the intermediate adaptive laws as Θ̇_2𝑙Θ̇_2𝑟=Θ̇_2𝑙=𝛾_2𝑙𝑒²𝜉^𝑇2𝑙𝑥₂−𝑐_2𝑙Θ_2𝑙,=Θ̇_2𝑟=𝛾_2𝑟𝑒²𝜉^𝑇2𝑟𝑥₂−𝑐_2𝑟Θ_2𝑟.(4.22) Substituting (4.22) into (4.21) yields 𝑉₂̇≤𝑒₂𝑒−𝑘^∗1,0𝑒²¹−𝑘^∗2,0𝑒²²+𝜀²¹4𝑘_1,1+𝜀²²4𝑘_2,1−𝑐^1𝑙‖‖Θ_1𝑙‖‖²2−𝑐^1𝑟‖‖Θ_1𝑟‖‖²2−𝑐^2𝑙‖‖Θ_2𝑙‖‖²−𝑐_2𝑟2‖‖Θ_2𝑟‖‖²2+𝑐^1𝑙‖‖Θ^∗1𝑙‖‖²2+𝑐^1𝑟‖‖Θ^∗1𝑟‖‖²2+𝑐^2𝑙‖‖Θ^∗2𝑙‖‖²2+𝑐^2𝑟‖‖Θ^∗2𝑟‖‖²2.(4.23)

Step i (3≤𝑖≤𝑛−1)
A similar procedure is employed recursively at each step. By defining 𝑒_𝑖=𝑥_𝑖−𝛼_𝑖−1(4.24) the time derivative of 𝑒_𝑖 is 𝑒_𝑖̇=𝑥̇_𝑖−𝛼̇_𝑖−1=𝑓_𝑖𝑥_𝑖+𝐺_𝑖𝑥_𝑖𝑥_𝑖+1−𝛼̇_𝑖−1.(4.25) Take 𝑥_𝑖+1 as a virtual control, and define 𝛼^∗𝑖=−𝑒_𝑖−1−𝑘_𝑖𝑒_𝑖−1𝐺_𝑖𝑥_𝑖𝑓^𝑖𝑥_𝑖−𝛼̇_𝑖−1,(4.26) where 𝑘_𝑖 is a positive constant: 𝛼̇_𝑖−1=^𝑖−1_𝑘=1𝜕𝛼_𝑖−1𝜕𝑥_𝑘𝐺_𝑘𝑥_𝑘+1+𝑓_𝑘+𝜕𝛼^𝑖−1𝜕𝑦_𝑟𝑦̇_𝑟+^𝑖−1_𝑘=1𝜕𝛼_𝑖−1𝜕Θ_𝑘𝑙𝛾^𝑘𝑙𝑒^𝑘𝜉^𝑇𝑘𝑙−𝑐_𝑘𝑙Θ_𝑘𝑙+_𝑖−1_𝑘=1𝜕𝛼_𝑖−1𝜕Θ_𝑘𝑟𝛾^𝑘𝑟𝑒^𝑘𝜉^𝑇𝑘𝑟−𝑐_𝑘𝑟Θ_𝑘𝑟.(4.27) The unknown function −(1/𝐺_𝑖(𝑥_𝑖))(𝑓_𝑖(𝑥_𝑖)−𝛼̇_𝑖−1) can be approximated by the following type-2 fuzzy logic system 𝜉^𝑇𝑖(𝑥_𝑖)Θ_𝑖, and we can obtain 𝛼_𝑖=−𝑒_𝑖−1−𝑘_𝑖𝑒_𝑖−𝜉^𝑇𝑖𝑥_𝑖Θ_𝑖.(4.28) Express 𝑥_𝑖+1 as 𝑥_𝑖+1=𝑒_𝑖+1+𝛼_𝑖 and the time derivative of 𝑒_𝑖 is 𝑒_𝑖̇=𝑓_𝑖𝑥_𝑖+𝐺_𝑖𝑥_𝑖𝑒_𝑖+1+𝛼_𝑖−𝛼̇_𝑖−1=𝐺_𝑖𝑥_𝑖𝑒_𝑖+1−𝑒_𝑖−1−𝑘_𝑖𝑒_𝑖−𝜉^𝑇𝑖𝑥_𝑖Θ_𝑖+𝜔_𝑖,(4.29) where Θ_𝑖=Θ_𝑖−Θ^∗𝑖.
Consider the following Lyapunov function:𝑉_𝑖=𝑉_𝑖−1+12𝐺_𝑖𝑥_𝑖𝑒^2𝑖+12𝛾_𝑖Θ^𝑇𝑖Θ_𝑖=𝑉_𝑖−1+12𝐺_𝑖𝑥_𝑖𝑒^2𝑖+12𝛾_𝑖𝑙Θ^𝑇𝑖𝑙Θ_𝑖𝑙+12𝛾_𝑖𝑟Θ^𝑇𝑖𝑟Θ_𝑖𝑟,(4.30) then the time derivative of 𝑉_𝑖 is 𝑉_𝑖̇=𝑉̇_𝑖−1−𝑒_𝑖−1𝑒_𝑖+𝑒_𝑖𝑒_𝑖+1−𝑘_𝑖𝑒^2𝑖−𝐺̇^𝑖𝑥_𝑖2𝐺^2𝑖𝑥_𝑖𝑒^2𝑖+𝑒_𝑖𝜔_𝑖−Θ^𝑇𝑖𝑙𝑒^𝑖𝜉^𝑇𝑖𝑙𝑥_𝑖−1𝛾_𝑖𝑙Θ̇_𝑖𝑙−Θ^𝑇𝑖𝑟𝑒^𝑖𝜉^𝑇𝑖𝑟𝑥_𝑖−1𝛾_𝑖𝑟Θ̇_𝑖𝑟.(4.31) Choose the intermediate adaptive laws as Θ̇_𝑖𝑙Θ̇_𝑖𝑟=Θ̇_𝑖𝑙=𝛾_𝑖𝑙𝑒^𝑖𝜉^𝑇𝑖𝑙𝑥_𝑖−𝑐_𝑖𝑙Θ_𝑖𝑙,=Θ̇_𝑖𝑟=𝛾_𝑖𝑟𝑒^𝑖𝜉^𝑇𝑖𝑟𝑥_𝑖−𝑐_𝑖𝑟Θ_𝑖𝑟.(4.32) Substituting (4.32) into (4.31) yields 𝑉_𝑖̇≤𝑒_𝑖𝑒_𝑖+1−^𝑖_𝑘=1𝑘^∗𝑘,0𝑒^2𝑘+^𝑖_𝑘=1𝜀^2𝑘4𝑘_𝑘,1−^𝑖_𝑘=1𝑐_𝑘𝑙‖‖Θ_𝑘𝑙‖‖²2−^𝑖_𝑘=1𝑐_𝑘𝑟‖‖Θ_𝑘𝑟‖‖²+_𝑖2_𝑘=1𝑐_𝑘𝑙‖‖Θ^∗𝑘𝑙‖‖²2+^𝑖_𝑘=1𝑐_𝑘𝑟‖‖Θ^∗𝑘𝑟‖‖²2.(4.33)

Step n
In the final design step, the actual control input 𝑢 will appears. Defining 𝑒_𝑛=𝑥_𝑛−𝛼_𝑛−1,(4.34) the time derivative of 𝑒_𝑛 is 𝑒_𝑛̇=𝑥̇_𝑛−𝛼̇_𝑛−1=𝑓_𝑛𝑥_𝑛+𝐺_𝑛𝑥_𝑛𝑢−𝛼̇_𝑛−1.(4.35) Define the actual control input 𝑢 as 𝑢=−𝑒_𝑛−1−𝑘_𝑛𝑒_𝑛−𝜉^𝑇𝑛𝑥_𝑛Θ_𝑛,(4.36) where 𝑘_𝑛 is a positive constant, Θ_𝑖=Θ_𝑖−Θ^∗𝑖.Choose the whole Lyapunov function as 𝑉=𝑉_𝑛−1+12𝐺_𝑛𝑥_𝑛𝑒^2𝑛+12𝛾_𝑛Θ^𝑇𝑛Θ_𝑛=𝑉_𝑛−1+12𝐺_𝑛𝑥_𝑛𝑒^2𝑛+12𝛾_𝑛𝑙Θ^𝑇𝑛𝑙Θ_𝑛𝑙+12𝛾_𝑛𝑟Θ^𝑇𝑛𝑟Θ_𝑛𝑟,(4.37) then the time derivative of 𝑉 is 𝑉̇=𝑉̇_𝑛−1−𝑒_𝑛−1𝑒_𝑛−𝑘_𝑛𝑒^2𝑛−𝐺̇^𝑛𝑥_𝑛2𝐺^2𝑛𝑥_𝑛𝑒^2𝑛+𝑒_𝑛𝜔_𝑛−Θ^𝑇𝑛𝑙𝑒^𝑛𝜉^𝑇𝑛𝑙𝑥_𝑛−1𝛾_𝑛𝑙Θ̇_𝑛𝑙−Θ^𝑇𝑛𝑟𝑒^𝑛𝜉^𝑇𝑛𝑟𝑥_𝑛−1𝛾_𝑛𝑟Θ̇_𝑛𝑟.(4.38) Choose the actual adaptive laws as Θ̇_𝑛𝑙Θ̇_𝑛𝑟=Θ̇_𝑛𝑙=𝛾_𝑛𝑙𝑒^𝑛𝜉^𝑇𝑛𝑙𝑥_𝑛−𝑐_𝑛𝑙Θ_𝑛𝑙,=Θ̇_𝑛𝑟=𝛾_𝑛𝑟𝑒^𝑛𝜉^𝑇𝑛𝑟𝑥_𝑛−𝑐_𝑛𝑟Θ_𝑛𝑟.(4.39) Substituting (4.39) into (4.38) yields 𝑉̇≤−^𝑛_𝑘=1𝑘^∗𝑘,0𝑒^2𝑘+^𝑛_𝑘=1𝜀^2𝑘4𝑘_𝑘,1−^𝑛_𝑘=1𝑐_𝑘𝑙‖‖Θ_𝑘𝑙‖‖²2−^𝑛_𝑘=1𝑐_𝑘𝑟‖‖Θ_𝑘𝑟‖‖²+_𝑛2_𝑘=1𝑐_𝑘𝑙‖‖Θ^∗𝑘𝑙‖‖²2+^𝑛_𝑘=1𝑐_𝑘𝑟‖‖Θ^∗𝑘𝑟‖‖²2.(4.40)𝑘^∗𝑘,0 is chosen such that 𝑘^∗𝑘,0>(𝜌/2𝐺_𝑘), then 𝑘_𝑘,0>𝜌2𝐺_𝑘+2𝐺𝐺̇^𝑘2𝑘(𝑘=1,…,𝑛),(4.41) where 𝜌 is a positive constant and 𝛿≤^𝑛_𝑘=1𝜀^2𝑘4𝑘_𝑘,1+^𝑛_𝑘=1𝑐_𝑘𝑙‖‖Θ^∗𝑘𝑙‖‖²2+^𝑛_𝑘=1𝑐_𝑘𝑟‖‖Θ^∗𝑘𝑟‖‖²2.(4.42) Then (4.40) becomes 𝑉̇≤−^𝑛_𝑘=1𝑘^∗𝑘,0𝑒^2𝑘+𝛿−^𝑛_𝑘=1𝑐_𝑘𝑙‖‖Θ_𝑘𝑙‖‖²2−^𝑛_𝑘=1𝑐_𝑘𝑟‖‖Θ_𝑘𝑟‖‖²<−_𝑛2_𝑘=1𝜌2𝐺_𝑘𝑒^2𝑘+𝛿−^𝑛_𝑘=1𝜌Θ^𝑇𝑘𝑙Θ_𝑘𝑙2−^𝑛_𝑘=1𝜌Θ^𝑇𝑘𝑟Θ_𝑘𝑟<−𝜌_𝑛2_𝑘=112𝐺_𝑘𝑒^2𝑘+^𝑛_𝑘=1Θ^𝑇𝑘𝑙Θ_𝑘𝑙2+^𝑛_𝑘=1Θ^𝑇𝑘𝑟Θ_𝑘𝑟<−𝜌𝑉+𝛿.2+𝛿(4.43) From (4.43) we have 𝑉(𝑡)≤𝑉𝑡₀𝑒^{−𝜌(𝑡−𝑡0)}+𝛿𝜌.(4.44) It can be shown that the signals 𝑥(𝑡), 𝑒(𝑡), Θ_𝑙(𝑡), Θ_𝑟(𝑡), and 𝑢(𝑡) are globally uniformly ultimately bounded and that |𝑦_𝑛(𝑡)−𝑦_𝑟(𝑡)|≤√2𝑉(𝑡)𝑒^{(𝜌/2)(𝑡−𝑡0)}+√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)(𝑡−𝑡0)}→0, therefore, it follows that there exists 𝑇, when 𝑡≥𝑇, |𝑦(𝑡)−𝑦_𝑟(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(𝐾∈𝑅₊), with an intrinsic birth rate constant 𝑟(𝑟∈𝑅₊), such that𝑑𝑁𝑑𝑡=𝑟𝑁1−𝑁𝐾.(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 𝜆𝑆^𝑝𝐼^𝑞 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, 𝜆∈𝑅₊ is the force of infection or rate of transmission. Therefore, the evolution equation for the susceptible fish population 𝑆 can be written as 𝑑𝑆𝑑𝑡=𝑟𝑆1−𝑆𝐾+𝐼−𝜆𝑆𝐼.(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(𝜇∈𝑅₊). 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)

(a)

(b)

(c)

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=𝑟𝑥¹1−𝑥𝐾¹−𝐾𝑟+𝜆𝑥¹𝑑𝑥₂𝑥₂,𝑑𝑡=𝜆𝑥¹𝑥₂−𝜇𝑥₂−𝑚𝑥²𝑥₂+𝑎𝑥³𝑑𝑥₃,𝑑𝑡=𝜃𝑥²𝑥₃𝑥₂+𝑎−𝑑𝑥³+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 𝜇𝜇^1𝑀_11𝑀2𝑥₁=𝑒^{−(𝑥1−50∗(1±10%)/10)2},𝜇^2𝑀₁𝑥₁=𝑒^{−(𝑥1−60∗(1±10%)/10)2},𝜇^3𝑀₁𝑥₁=𝑒^{−(𝑥1−70∗(1±10%)/10)2},𝜇_1𝑀3𝑥₂=𝑒^{−(𝑥2−50∗(1±10%)/10)2},𝜇^2𝑀₂𝑥₂=𝑒^{−(𝑥2−60∗(1±10%)/10)2},𝜇^3𝑀₂𝑥₂,=𝑒^{−(𝑥2−70∗(1±10%)/10)2},𝑥₃=𝑒^{−(𝑥3−20∗(1±10%)/10)2},𝜇^2𝑀₃𝑥₃=𝑒^{−(𝑥3−30∗(1±10%)/10)2},𝜇^3𝑀₃𝑥₃=𝑒^{−(𝑥3−40∗(1±10%)/10)2},(5.6) then the fuzzy basis functions are obtained as：𝜉_1𝑙=⎡⎢⎢⎣𝜇^1𝑀1𝐴,𝜇𝐴^2𝑀1,𝜇^3𝑀₁𝐴⎤⎥⎥⎦,𝐴=𝜇^1𝑀₁𝜉_1𝑟+𝜇^2𝑀₁+𝜇^3𝑀₁,=⎡⎢⎢⎣𝜇^1𝑀₁𝐵,𝜇^2𝑀₁𝐵,𝜇𝐵^3𝑀1⎤⎥⎥⎦,𝐵=𝜉_2𝑙𝜇^1𝑀₁+𝜇^2𝑀₁+𝜇^3𝑀₁,=⎡⎢⎢⎣𝜇^1𝑀1∗𝜇^1𝑀₂𝐶,𝜇^1𝑀1∗𝜇^2𝑀₂𝐶,𝜇^1𝑀1∗𝜇^3𝑀₂𝐶,𝜇^2𝑀1∗𝜇^1𝑀₂𝐶,𝜇^2𝑀1∗𝜇^2𝑀₂𝐶,𝜇^2𝑀1∗𝜇^3𝑀₂𝐶,𝜇^3𝑀₁∗𝜇^1𝑀₂𝐶,𝜇^3𝑀₁∗𝜇^2𝑀₂𝐶,𝜇^3𝑀₁∗𝜇^3𝑀₂𝐶=𝜇_1𝑀1𝐶⎤⎥⎥⎦,∗𝜇^1𝑀₂+𝜇^1𝑀₁∗𝜇^2𝑀₂+𝜇^1𝑀₁∗𝜇^3𝑀₂+𝜇^2𝑀₁∗𝜇^1𝑀₂+𝜇^2𝑀₁∗𝜇^2𝑀₂+𝜇^2𝑀₁∗𝜇^3𝑀₂𝜉_1𝑟+𝜇^3𝑀₁∗𝜇^1𝑀₂+𝜇^3𝑀₁∗𝜇^2𝑀₂+𝜇^3𝑀₁∗𝜇^3𝑀₂,=⎡⎢⎢⎣𝜇^1𝑀₁∗𝜇^1𝑀₂𝐷,𝜇^1𝑀₁∗𝜇^2𝑀₂𝐷,𝜇^1𝑀₁∗𝜇^3𝑀₂𝐷,𝜇^2𝑀₁∗𝜇^1𝑀₂𝐷,𝜇^2𝑀₁∗𝜇^2𝑀₂𝐷,𝜇^2𝑀₁∗𝜇^3𝑀₂𝜇_3𝑀1𝐷,∗𝜇^1𝑀₂𝐷,𝜇^3𝑀1∗𝜇^2𝑀₂𝐷,𝜇^3𝑀1∗𝜇^3𝑀₂𝐷=𝐷⎤⎥⎥⎦,𝜇^1𝑀₁∗𝜇^1𝑀₂+𝜇^1𝑀₁∗𝜇^2𝑀₂+𝜇^1𝑀₁∗𝜇^3𝑀₂+𝜇^2𝑀₁∗𝜇^1𝑀₂+𝜇^2𝑀₁∗𝜇^2𝑀₂+𝜇^2𝑀₁∗𝜇^3𝑀₂𝜉_3𝑙+𝜇^3𝑀₁∗𝜇^1𝑀₂+𝜇^3𝑀₁∗𝜇^2𝑀₂+𝜇^3𝑀₁∗𝜇^3𝑀₂,=⎡⎢⎢⎣𝜇^1𝑀2∗𝜇^1𝑀₃𝐸,𝜇^1𝑀2∗𝜇^2𝑀₃𝐸,𝜇^1𝑀2∗𝜇^3𝑀₃𝐸,𝜇^2𝑀2∗𝜇^1𝑀₃𝐸,𝜇^2𝑀2∗𝜇^2𝑀₃𝐸,𝜇^2𝑀2∗𝜇^3𝑀₃𝐸,𝜇^3𝑀₂∗𝜇^1𝑀₃𝐸,𝜇^3𝑀₂∗𝜇^2𝑀₃𝐸,𝜇^3𝑀₂∗𝜇^3𝑀₃𝐸=𝜇_1𝑀2𝐸⎤⎥⎥⎦,∗𝜇^1𝑀₃+𝜇^1𝑀₂∗𝜇^2𝑀₃+𝜇^1𝑀₂∗𝜇^3𝑀₃+𝜇^2𝑀₂∗𝜇^1𝑀₃+𝜇^2𝑀₂∗𝜇^2𝑀₃+𝜇^2𝑀₂∗𝜇^3𝑀₃𝜉_3𝑟+𝜇^3𝑀₂∗𝜇^1𝑀₃+𝜇^3𝑀₂∗𝜇^2𝑀₃+𝜇^3𝑀₂∗𝜇^3𝑀₃,=⎡⎢⎢⎣𝜇^1𝑀₂∗𝜇^1𝑀₃𝐹,𝜇^1𝑀₂∗𝜇^2𝑀₃𝐹,𝜇^1𝑀₂∗𝜇^3𝑀₃𝐹,𝜇^2𝑀₂∗𝜇^1𝑀₃𝐹,𝜇^2𝑀₂∗𝜇^2𝑀₃𝐹,𝜇^2𝑀₂∗𝜇^3𝑀₃𝜇_3𝑀2𝐹,∗𝜇^1𝑀₃𝐹,𝜇^3𝑀2∗𝜇^2𝑀₃𝐹,𝜇^3𝑀2∗𝜇^3𝑀₃𝐹=𝐹⎤⎥⎥⎦,𝜇^1𝑀₂∗𝜇^1𝑀₃+𝜇^1𝑀₂∗𝜇^2𝑀₃+𝜇^1𝑀₂∗𝜇^3𝑀₃+𝜇^2𝑀₂∗𝜇^1𝑀₃+𝜇^2𝑀₂∗𝜇^2𝑀₃+𝜇^2𝑀₂∗𝜇^3𝑀₃𝜉_1𝑙+𝜇^3𝑀₂∗𝜇^1𝑀₃+𝜇^3𝑀₂∗𝜇^2𝑀₃+𝜇^3𝑀₂∗𝜇^3𝑀₃,𝜉_2𝑙=𝜉^11𝑙,𝜉^21𝑙,𝜉^31𝑙,^𝑇𝜉_1𝑟=𝜉^11𝑟,𝜉^21𝑟,𝜉^31𝑟^𝑇,𝜉_3𝑙=𝜉^12𝑙,𝜉^22𝑙…𝜉^92𝑙,^𝑇𝜉_2𝑟=𝜉^12𝑟,𝜉^22𝑟…𝜉^92𝑟^𝑇,=𝜉^13𝑙,𝜉^23𝑙…𝜉^93𝑙,^𝑇𝜉_3𝑟=𝜉^13𝑟,𝜉^23𝑟…𝜉^93𝑟^𝑇.(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)+𝜆𝑟𝑥¹1−𝑥𝐾¹−𝑦̇^𝑟.(5.8)
The unknown function 2:𝑥₂+𝑎𝑚𝑥₂𝜆𝑥₁𝑥₂−𝜇𝑥₂−𝛼̇₁.(5.9)
The unknown function 3:𝜃𝑥₂𝑥₃𝑥₂+𝑎−𝛼̇²𝜃𝑥₂𝑥₃𝑥₂+𝑎−𝛼̇².(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=𝑘₂𝑐=𝑘₃=0.5,𝛾_1𝑙=𝛾_2𝑙=𝛾_3𝑙=𝛾_1𝑟=𝛾_2𝑟=𝛾_3𝑟=1,_1𝑙=𝑐_2𝑙=𝑐_3𝑙=0.5,𝑐_1𝑟=𝑐_2𝑟=𝑐_3𝑟=0.6.(5.11)
The first intermediate controller is chosen as𝛼₁=−𝑘₁𝑒₁−𝜉^𝑇1𝑙𝑥₁Θ_1𝑙+𝜉^𝑇1𝑟𝑥₁Θ_1𝑟.(5.12)
The first adaptive laws are designed asΘ̇_1𝑙Θ̇_1𝑟=Θ̇_1𝑙=𝛾_1𝑙𝑒¹𝜉^𝑇1𝑙𝑥₁−𝑐_1𝑙Θ_1𝑙,=Θ̇_1𝑟=𝛾_1𝑟𝑒¹𝜉^𝑇1𝑟𝑥₁−𝑐_1𝑟Θ_1𝑟.(5.13)
The second intermediate controller is chosen as𝛼₂=−𝑒₁−𝑘₂𝑒₂−𝜉^𝑇2𝑙𝑥₂Θ_2𝑙+𝜉^𝑇2𝑟𝑥₂Θ_2𝑟.(5.14)
The second adaptive laws are designed asΘ̇_2𝑙Θ̇_2𝑟=Θ̇_2𝑙=𝛾_2𝑙𝑒²𝜉^𝑇2𝑙𝑥₂−𝑐_2𝑙Θ_2𝑙,=Θ̇_2𝑟=𝛾_2𝑟𝑒²𝜉^𝑇2𝑟𝑥₂−𝑐_2𝑟Θ_2𝑟.(5.15)
The actual controller is chosen as𝑢=−𝑒₂−𝑘₃𝑒₃−𝜉^𝑇3𝑙𝑥₃Θ_3𝑙+𝜉^𝑇3𝑟𝑥₃Θ_3𝑟.(5.16)
The actual adaptive laws are designed asΘ̇_3𝑙Θ̇_3𝑟=Θ̇_3𝑙=𝛾_3𝑙𝑒³𝜉^𝑇3𝑙𝑥₃−𝑐_3𝑙Θ_3𝑙,=Θ̇_3𝑟=𝛾_3𝑟𝑒³𝜉^𝑇3𝑟𝑥₃−𝑐_3𝑟Θ_3𝑟.(5.17) The reference output is specified as 𝑦_𝑟(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 1–11.(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.

(a)

(b)

(c)

Figure 2 

State variable outputs with T2ABFC.

(a)

(b)

(c)

Figure 3 

State variable outputs with T1ABFC.

Figure 4 

Trajectories of controller.

Figure 5 

Trajectories of tracking error.

Figure 6 

Trajectories of the state variable 𝑥1 by T2ABFC with external disturbances.

Figure 7 

Trajectories of the state variable 𝑥1 by T1ABFC with external disturbances.

Figure 8 

Trajectories of the state variable 𝑥2 by T2ABFC with external disturbances.

Figure 9 

Trajectories of the state variable 𝑥2 by T1ABFC with external disturbances.

Figure 10 

Trajectories of the state variable 𝑥3 by T2ABFC with external disturbances.

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).