# Adaptive Backstepping Fuzzy Control Based on Type-2 Fuzzy System

Authors: Ll Yi-Min; Yue Yang; Li Li
Journal: Journal of Applied Mathematics
 (2012)

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## 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.


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## Body

## 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 Section2, 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 as(2.1)ẋi=fi(x̅i)+Gi(x̅i)xi+11≤i≤n-1,ẋn=fn(x̅n)+Gn(x̅n)un≥2,y=x1,
where x̅i=[x1,x2…xi]T∈Ri is the stable vector and u∈R and y∈R are the input and output of the system, respectively. fi(x̅i), Gi(x̅i) (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 yr(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 setÃ is characterized by a type-2 membership function μÃ(x,u) [22], where x∈X is the primary variable and u∈Jx⊆[0,1] is the secondary variable:(3.1)Ã={((x,u),μÃ(x,u))∣∀x∈X∀u∈Jx⊆[0,1]}.in which 0≤μÃ(x,u)≤1. Ã can also be expressed as follows [22]:
(3.2)Ã=∫x∈X∫u∈JxμÃ(x,u)(x,u)Jx⊆[0,1],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 as(3.3)Ã=∫x∈X∫u∈Jx1(x,u)=∫x∈X[∫u∈Jx1/u]x.
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 as(3.4)uÃ(x)=exp⁡[-12(x-mσ)2],m∈[m1,m2].Uncertainty aboutÃ 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 Ã:(3.5)FOU(Ã)=⋃∀x∈XJx={(x,u):u∈[μ̲Ã(x),μ̅Ã(x)]},where(3.6)μ̲Ã(x)=FOU(Ã)̲∀x∈X,μ̅Ã(x)=FOU(Ã)¯∀x∈X.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 havingn inputs x1∈X1,…,xn∈Xn and one output y∈Y, and assuming there are M rules, the ith rule of the IT2 SMC can be described as(3.7)Rl:IFx1isF̃1l…andxnisF̃nl,THENyisG̃ll=1,…,M.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:(3.8)Ycos⁡(Y1,…,YM,F1,…,FM)=[yl,yr]=∫y1…∫yM∫f1…∫fM1∑i=1Mfiyi/∑i=1Mfi,
where Ycos(x) is an interval output set determined by its left-most point yl and its right-most point yr, and fi∈Fi=⌊f̲i,f̅i⌋. In the meantime, an IT2FLS with singleton fuzzification and meet under minimum or product t-norm f̲i and f̅i can be obtained as(3.9)f̲i=μ̲F̃1i(x1)*⋅⋅⋅*μ̲F̃ni(xn),and(3.10)f̅i=μ̅F̃1i(x1)*⋅⋅⋅*μ̅F̃ni(xn).Also,yi∈Yi and Yi=[yli,yri] is the centroid of the IT2 consequent set G̃i, the centroid of a T2FS, and for any value y∈Ycos, y can be expressed as(3.11)y=∑i=1Mfiyi∑i=1Mfi,
where y is a monotonic increasing function with respect to yi. Also, yl is the minimum associated only with yli, and yr is the maximum associated only with yri. Note that yl and yr depend only on the mixture of f̲i or f̅i 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,
(3.12)yl=∑i=1Mfliyli∑i=1Mfli=∑i=1Myliξli=ylTξl,and(3.13)yr=∑i=1Mfriyri∑i=1Mfri=∑i=1Myriξri=yrTξr,
respectively, where ξli=fli/∑i=1Mfi, ξl=[ξl1,ξl2,…,ξlM], ylT=[yl1,yl2,…,ylM], and ξri=fri/∑i=1Mfi,  ξr=[ξr1,ξr2,…,ξrM],  yrT=[yr1,yr2,…,yrM].In order to computeyl 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 yri  is arranged in ascending order, that is, yr1≤yr2≤···≤yrM.Step 1.
Computeyr in (3.13) by initially setting frl=(f̅l+f̲l)/2 for l=1,…,M, wheref̅l and f̲l have been precomputed by (3.10) and let yr′=yr.Step 2 :.
FindR(1≤R≤M-1) such that yrR≤yr′≤yrR+1.Step 3.
Computeyr in (3.13) with fri=fi for i≤R and fri=f̅i for i>Rand let yr′′=yr.Step 4.
Ifyr′′≠yr, then go to Step 5; if yr′′=yr, then stop and set yr=yr′′.Step 5.
Setyr′equal to yr′′ and return to Step 2.
The point to separate two sides by numberR can be decided from the above algorithm, one side using lower firing strengths f̲i’s and another side using upper firing strengths f̅i’s. Therefore, yr can be expressed as(3.14)yr=∑i=1Rf̲iyri+∑i=R+1Mf̅iyri∑i=1Rf̲i+∑i=R+1Mf̅i=∑i=1Rq̲riyri+∑i=R+1Mq̅riyri=[Q̲rQ̅r][y̲ry̅r]=ξrTΘr,
where  q̲ri=f̲i/Dr, q̅ri=f̅i/Dr and Dr=(∑i=1Rf̲i+∑i=R+1Mf̅i). In the meantime, we have Q̲r=[q̲r1,q̲r2,…,q̲rR], Q̅r=[q̅r1,q̅r2,…,q̅rR], ξrT=⌊Q̲r,Q̅r⌋, and ΘrT=⌊y̲ry̅r⌋.
The procedure to computeyl is similar to compute yr. In Step 2, it only determines L(1≤L≤M-1), such that ylL≤yl′≤ylL+1. In Step 3, let fli=f̅i for i≤L andfli=f̲i for i>L. Therefore, yl can be expressed as(3.15)yl=∑i=1Lf̲iyli+∑i=L+1Mf̅iyli∑i=1Lf̲i+∑i=L+1Mf̅i=∑i=1Lq̲liyli+∑i=L+1Mq̅liyli=[Q̲lQ̅l][y̲ly̅l]=ξlTΘl,
where q̲li=f̲i/Dl, q̅li=f̅i/Dl and Dl=(∑i=1Lf̲i+∑i=L+1Mf̅i). In the meantime, we have Q̲l=[q̲l1,q̲l2,…,q̲lR],  Q̅l=[q̅l1,q̅l2,…,q̅lR], ξlT=⌊Q̲lQ̅l⌋, and ΘlT=⌊y̲ly̅l⌋.
We defuzzify the interval set by using the average ofyl and yr, hence, the defuzzified crisp output becomes(3.16)Yfuzz2(x)=yl+yr2=12(ξlTΘl+ξrTΘr)=12[ξlTξrT][ΘlΘr]=ξTΘ,
where (1/2)[ξlTξrT]=ξTand [ΘlTΘrT]=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
(3.17)sup⁡x∈Ω|f(x)-ξT(x)Θ|≤ε.


## 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 constantsG̅i and G̲i(1≤i≤n), such that G̲i≤Gi(x̅i)≤G̅i.Assumption 4.2.
There exist positive constantGid(1≤i≤n), such that |Ġi(x̅i)|≤Gid.Assumption 4.3.
Define the optimal parameter vectorsΘi* as
(4.1)Θi*=argmin⁡Θi∈Ωi[sup⁡x̅i∈Ui|fi(x̅i∣Θi)-fi(x̅i)|],
where Ωi, Ui are compact regions for Θi, x̅i, respectively. The fuzzy logic system minimum approximation errors ωi are defined as
(4.2)|ξiTΘi*-fi(x̅i)|≤ωi.T2ABFC method proposed in this paper is summarized in Theorem4.4.Theorem 4.4.
Suppose assumptions are4.1–4.3 tenable, then the fuzzy adaptive output tracking design described by (2.1), control law u=-en-1-knen-ξnT(x̅n)Θ̂n, and parameter adaptive laws Θ̂̇nl=Θ̃̇nl=γnl(enξnlT(x̅n)-cnlΘ̂nl)  Θ̂̇nr=Θ̃̇nr=γnr(enξnrT(x̅n)-cnrΘ̂nr)  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(4.3)e1=x1-yr.
The time derivative of e1 is
(4.4)ė1=ẋ1-ẏr=f1(x̅1)+G1(x̅1)x2-ẏr.
Take x2 as a virtual control, and define
(4.5)α1*=k1e1-1G1(x̅1)(f1(x̅1)-ẏr),
where k1 is a positive constant.
Sincef1(x̅1) and f1(x̅1) 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(x̅1))(f1(x̅1)-ẏr) can be approximated by the following type-2 fuzzy logic system ξ1T(x̅1)Θ1, and we obtain(4.6)α1*=-k1e1-ξ1T(x̅1)Θ1*-ω1.Express x2 as x2=e2+α1 and define
(4.7)α1=-k1e1-ξ1T(x̅1)Θ̂1=-k1e1-(ξ1lT(x̅1)Θ̂1l+ξ1rT(x̅1)Θ̂1r),
then the time derivative of e1 is
(4.8)ė1=f1(x̅1)+G1(x̅1)(e2+α1)-ẏr=G1(x̅1)(e2-k1e1-ξ1T(x̅1)Θ̃1+ω1)=G1(x̅1)(e2-k1e1-ξ1lT(x̅1)Θ̃1l-ξ1rT(x̅1)Θ̃1r+ω1),
where Θ̃1=Θ̂1-Θ1*.
Consider the following Lyapunov function:(4.9)V1=12G1(x̅1)e12+12γ1Θ̃1TΘ̃1=12G1(x̅1)e12+12γ1lΘ̃1lTΘ̃1l+12γ1rΘ̃1rTΘ̃1r,
then the time derivative of V1 is
(4.10)V̇1=e1ė1G1(x̅1)-Ġ1(x̅1)2G12(x̅1)e12+1γ1Θ̃1TΘ̂̇1=e1ė1G1(x̅1)-Ġ1(x̅1)2G12(x̅1)e12+1γ1lΘ̃1lTΘ̂̇1l+1γ1rΘ̃1rTΘ̂̇1r=e1e2-k1e12-e1(Θ̃1lTξ1l+Θ̃1rTξ1r)+e1ω-Ġ1(x̅1)2G12(x̅1)e12+1γ1lΘ̃1lTΘ̂̇1l+1γ1rΘ̃1rTΘ̂̇1r=e1e2-k1e12-Ġ1(x̅1)2G12(x̅1)e12+e1ω1+Θ̃1lT[-e1ξ1lT(x̅1)+1γ1lΘ̂̇1l]+Θ̃1rT[-e1ξ1rT(x̅1)+1γ1rΘ̂̇1r].
Choose the intermediate adaptive laws as
(4.11)Θ̂̇1l=Θ̃̇1l=γ1l(e1ξ1lT(x̅1)-c1lΘ̂1l),Θ̂̇1r=Θ̃̇1r=γ1r(e1ξ1rT(x̅1)-c1rΘ̂1r),
where c1l and c1r are given positive constants.Substituting (4.11) into (4.10) yields
(4.12)V̇1=e1e2+e1ω1-(k1+Ġ1(x̅1)2G12(x̅1))e12-(c1lΘ̃1lTΘ̂1l+c1rΘ̃1rTΘ̂1r),
where k1=k1,0+k1,1,  k1,0>0,k1,1>0,  e1ω1-k1e12≤e1ω1-k1,1e12≤ε12/4k1,1,|ω1|≤|ε1|
We can obtain that(4.13)-c1lΘ̃1lTΘ̂1l=-c1lΘ̃1lT(Θ̃1l+Θ1l*)≤-c1l‖Θ̃1l‖2+c1l‖Θ̃1l‖‖Θ1l*‖≤-c1l‖Θ̃1l‖22+c1l‖Θ1l*‖22,-c1rΘ̃1rTΘ̂1r=-c1rΘ̃1rT(Θ̃1r+Θ1r*)≤-c1r‖Θ̃1r‖2+c1r‖Θ̃1r‖‖Θ1r*‖≤-c1r‖Θ̃1r‖22+c1r‖Θ1r*‖22-(k1,0+Ġ12G12)e12≤-(k1,0-G1d2G12)e12,(k1,0*=k1,0-G1d2G12>0).
From (4.12), (4.13), it follows that
(4.14)V̇1≤e1e2-k1,0*e12+ε124k1,1-c1l‖Θ̃1l‖22-c1r‖Θ̃1r‖22+c1l‖Θ1l*‖22+c1r‖Θ1r*‖22.Step 2.
Differentiatinge2 yields
(4.15)ė2=ẋ2-α̇1=f2(x̅2)+G2(x̅2)x3-α̇1.
Take x3 as a virtual control, and define
(4.16)α2*=-e1-k2e2-1G2(x̅2)(f2(x̅2)-α̇1),
where k2 is a positive constant.
From (4.7), we obtain
(4.17)α̇1=∂α1∂x1(f1+G1x2)+∂α1∂yrẏr+∂α1∂Θ̂1l[γ1l(e1lξ1l-c1lΘ̂1l)]+∂α1∂Θ̂1r[γ1r(e1ξ1r-c1rΘ̂1r)].
Sincef2(x̅2) and G2(x̅2) are unknown, ideal controller is not available in practice, so we can assume that the unknown function -(1/G2(x̅2))(f2(x̅2)-α̇1)can be approximated by the following type-2 fuzzy logic system ξ2T(x̅2)Θ2, and obtain
(4.18)α2=-e1-k2e2-ξ2T(x̅2)Θ̂2.
Express x3 as x3=e3+α2 and the time derivative of e2 is
(4.19)ė2=f2(x̅2)+G2(x̅2)(e3+α2)-α̇1=G2(x̅2)(e3-e1-k2e2-ξ2T(x̅2)Θ̃2+ω2),
where Θ̃2=Θ̂2-Θ2*.
Consider the following Lyapunov function(4.20)V2=V1+12G2(x̅2)e22+12γ2Θ̃2TΘ̃2=V1+12G2(x̅2)e22+12γ2lΘ̃2lTΘ̃2l+12γ2rΘ̃2rTΘ̃2r,
then the time derivative of V2 is
(4.21)V̇2=V̇1-e1e2+e2e3-k2e22-Ġ2(x̅2)2G22(x̅2)e22+e2ω2-Θ̃2lT[e2ξ2lT(x̅2)-1γ2lΘ̂̇2l]-Θ̃2rT[e2ξ2rT(x̅2)-1γ2rΘ̂̇2r].
Choose the intermediate adaptive laws as
(4.22)Θ̂̇2l=Θ̃̇2l=γ2l(e2ξ2lT(x̅2)-c2lΘ̂2l),Θ̂̇2r=Θ̃̇2r=γ2r(e2ξ2rT(x̅2)-c2rΘ̂2r).
Substituting (4.22) into (4.21) yields
(4.23)V̇2≤e2e-k1,0*e12-k2,0*e22+ε124k1,1+ε224k2,1-c1l‖Θ̃1l‖22-c1r‖Θ̃1r‖22-c2l‖Θ̃2l‖22-c2r‖Θ̃2r‖22+c1l‖Θ1l*‖22+c1r‖Θ1r*‖22+c2l‖Θ2l*‖22+c2r‖Θ2r*‖22.Stepi(3≤i≤n-1)
A similar procedure is employed recursively at each step. By defining(4.24)ei=xi-αi-1
the time derivative of ei is
(4.25)ėi=ẋi-α̇i-1=fi(x̅i)+Gi(x̅i)xi+1-α̇i-1.
Take xi+1 as a virtual control, and define
(4.26)αi*=-ei-1-kiei-1Gi(x̅i)(fi(x̅i)-α̇i-1),
where ki is a positive constant:
(4.27)α̇i-1=∑k=1i-1∂αi-1∂xk(Gkxk+1+fk)+∂αi-1∂yrẏr+∑k=1i-1∂αi-1∂Θ̂kl[γkl(ekξklT-cklΘ̂kl)]+∑k=1i-1∂αi-1∂Θ̂kr[γkr(ekξkrT-ckrΘ̂kr)].
The unknown function -(1/Gi(x̅i))(fi(x̅i)-α̇i-1) can be approximated by the following type-2 fuzzy logic system ξiT(x̅i)Θi, and we can obtain
(4.28)αi=-ei-1-kiei-ξiT(x̅i)Θ̂i.
Express xi+1 as xi+1=ei+1+αi and the time derivative of ei is
(4.29)ėi=fi(x̅i)+Gi(x̅i)(ei+1+αi)-α̇i-1=Gi(x̅i)(ei+1-ei-1-kiei-ξiT(x̅i)Θ̃i+ωi),
where Θ̃i=Θ̂i-Θi*.
Consider the following Lyapunov function:(4.30)Vi=Vi-1+12Gi(x̅i)ei2+12γiΘ̃iTΘ̃i=Vi-1+12Gi(x̅i)ei2+12γilΘ̃ilTΘ̃il+12γirΘ̃irTΘ̃ir,
then the time derivative of  Vi is
(4.31)V̇i=V̇i-1-ei-1ei+eiei+1-kiei2-Ġi(x̅i)2Gi2(x̅i)ei2+eiωi-Θ̃ilT[eiξilT(x̅i)-1γilΘ̂̇il]-Θ̃irT[eiξirT(x̅i)-1γirΘ̂̇ir].
Choose the intermediate adaptive laws as
(4.32)Θ̂̇il=Θ̃̇il=γil(eiξilT(x̅i)-cilΘ̂il),Θ̂̇ir=Θ̃̇ir=γir(eiξirT(x̅i)-cirΘ̂ir).
Substituting (4.32) into (4.31) yields
(4.33)V̇i≤eiei+1-∑k=1ikk,0*ek2+∑k=1iεk24kk,1-∑k=1ickl‖Θ̃kl‖22-∑k=1ickr‖Θ̃kr‖22+∑k=1ickl‖Θkl*‖22+∑k=1ickr‖Θkr*‖22.Stepn
In the final design step, the actual control inputu will appears. Defining
(4.34)en=xn-αn-1,
the time derivative of en is
(4.35)ėn=ẋn-α̇n-1=fn(x̅n)+Gn(x̅n)u-α̇n-1.
Define the actual control input u as
(4.36)u=-en-1-knen-ξnT(x̅n)Θ̂n,
where kn is a positive constant, Θ̃i=Θ̂i-Θi*.Choose the whole Lyapunov function as
(4.37)V=Vn-1+12Gn(x̅n)en2+12γnΘ̃nTΘ̃n=Vn-1+12Gn(x̅n)en2+12γnlΘ̃nlTΘ̃nl+12γnrΘ̃nrTΘ̃nr,
then the time derivative of V is
(4.38)V̇=V̇n-1-en-1en-knen2-Ġn(x̅n)2Gn2(x̅n)en2+enωn-Θ̃nlT[enξnlT(x̅n)-1γnlΘ̂̇nl]-Θ̃nrT[enξnrT(x̅n)-1γnrΘ̂̇nr].
Choose the actual adaptive laws as
(4.39)Θ̂̇nl=Θ̃̇nl=γnl(enξnlT(x̅n)-cnlΘ̂nl),Θ̂̇nr=Θ̃̇nr=γnr(enξnrT(x̅n)-cnrΘ̂nr).
Substituting (4.39) into (4.38) yields
(4.40)V̇≤-∑k=1nkk,0*ek2+∑k=1nεk24kk,1-∑k=1nckl‖Θ̃kl‖22-∑k=1nckr‖Θ̃kr‖22+∑k=1nckl‖Θkl*‖22+∑k=1nckr‖Θkr*‖22.kk,0* is chosen such that kk,0*>(ρ/2Gk), then
(4.41)kk,0>(ρ2Gk)+(Ġk2Gk2)(k=1,…,n),
where ρ is a positive constant and
(4.42)δ≤∑k=1nεk24kk,1+∑k=1nckl‖Θkl*‖22+∑k=1nckr‖Θkr*‖22.
Then (4.40) becomes
(4.43)V̇≤-∑k=1nkk,0*ek2+δ-∑k=1nckl‖Θ̃kl‖22-∑k=1nckr‖Θ̃kr‖22<-∑k=1nρ2Gkek2+δ-∑k=1nρΘ̃klTΘ̃kl2-∑k=1nρΘ̃krTΘ̃kr2<-ρ(∑k=1n12Gkek2+∑k=1nΘ̃klTΘ̃kl2+∑k=1nΘ̃krTΘ̃kr2)+δ<-ρV+δ.
From (4.43) we have
(4.44)V(t)≤V(t0)e-ρ(t-t0)+δρ.
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)(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)-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 byN, 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 capacityK(K∈R+), with an intrinsic birth rate constant r(r∈R+), such that(5.1)dNdt=rN(1-NK).(3)
In the presence of bacterial infection, the total fish populationN 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 is(5.2)N(t)=S(t)+I(t).(4)
Only the susceptible fish populationS 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, 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
(5.3)dSdt=rS(1-S+IK)-λSI.(6)
The predator population, Pelican birds, whose population density is denoted byP, 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 byμ(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 Figure1, typical chaotic attractor for the model system is obtained for the parameter values as see [54]:(5.4)r=22K=400λ=0.06μ=3.4m=15.5a=15d=8.3θ=10.0.Typical chaotic attractor for the model system.
(a)(b)(c)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 substitutingS,I,P by x1,x2,x3, respectively, we can write down the following differential equations:(5.5)dx1dt=rx1(1-x1K)-(rK+λ)x1x2,dx2dt=λx1x2-μx2-mx2x2+ax3,dx3dt=θx2x3x2+a-dx3+u.
The backstepping design algorithm for type-2 adaptive fuzzy control is proposed as follows.Step 1.
Define the type-2 membership functions as(5.6)μM11(x1)=e-(x1-50*(1±10%)/10)2,μM12(x1)=e-(x1-60*(1±10%)/10)2,μM13(x1)=e-(x1-70*(1±10%)/10)2,μM21(x2)=e-(x2-50*(1±10%)/10)2,μM22(x2)=e-(x2-60*(1±10%)/10)2,μM23(x2),=e-(x2-70*(1±10%)/10)2,μM31(x3)=e-(x3-20*(1±10%)/10)2,μM32(x3)=e-(x3-30*(1±10%)/10)2,μM33(x3)=e-(x3-40*(1±10%)/10)2,
then the fuzzy basis functions are obtained as：(5.7)ξ1l=[μ̲M11A,μ̲M12A,μ̅M13A],A=μ̲M11+μ̲M12+μ̅M13,ξ1r=[μ̅M11B,μ̅M12B,μ̲M13B],B=μ̅M11+μ̅M12+μ̲M13,ξ2l=[μ̲M11*μ̲M21C,μ̲M11*μ̲M22C,μ̲M11*μ̅M23C,μ̲M12*μ̲M21C,μ̲M12*μ̲M22C,μ̲M12*μ̅M23C,μ̅M13*μ̲M21C,μ̅M13*μ̲M22C,μ̅M13*μ̅M23C],C=μ̲M11*μ̲M21+μ̲M11*μ̲M22+μ̲M11*μ̅M23+μ̲M12*μ̲M21+μ̲M12*μ̲M22+μ̲M12*μ̅M23+μ̅M13*μ̲M21+μ̅M13*μ̲M22+μ̅M13*μ̅M23,ξ1r=[μ̅M11*μ̅M21D,μ̅M11*μ̅M22D,μ̅M11*μ̲M23D,μ̅M12*μ̅M21D,μ̅M12*μ̅M22D,μ̅M12*μ̲M23D,μ̲M13*μ̅M21D,μ̲M13*μ̅M22D,μ̲M13*μ̲M23D],D=μ̅M11*μ̅M21+μ̅M11*μ̅M22+μ̅M11*μ̲M23+μ̅M12*μ̅M21+μ̅M12*μ̅M22+μ̅M12*μ̲M23+μ̲M13*μ̅M21+μ̲M13*μ̅M22+μ̲M13*μ̲M23,ξ3l=[μ̲M21*μ̲M31E,μ̲M21*μ̲M32E,μ̲M21*μ̅M33E,μ̲M22*μ̲M31E,μ̲M22*μ̲M32E,μ̲M22*μ̅M33E,μ̅M23*μ̲M31E,μ̅M23*μ̲M32E,μ̅M23*μ̅M33E],E=μ̲M21*μ̲M31+μ̲M21*μ̲M32+μ̲M21*μ̅M33+μ̲M22*μ̲M31+μ̲M22*μ̲M32+μ̲M22*μ̅M33+μ̅M23*μ̲M31+μ̅M23*μ̲M32+μ̅M23*μ̅M33,ξ3r=[μ̅M21*μ̅M31F,μ̅M21*μ̅M32F,μ̅M21*μ̲M33F,μ̅M22*μ̅M31F,μ̅M22*μ̅M32F,μ̅M22*μ̲M33F,μ̲M23*μ̅M31F,μ̲M23*μ̅M32F,μ̲M23*μ̲M33F],F=μ̅M21*μ̅M31+μ̅M21*μ̅M32+μ̅M21*μ̲M33+μ̅M22*μ̅M31+μ̅M22*μ̅M32+μ̅M22*μ̲M33+μ̲M23*μ̅M31+μ̲M23*μ̅M32+μ̲M23*μ̲M33,ξ1l=[ξ1l1,ξ1l2,ξ1l3],Tξ1r=[ξ1r1,ξ1r2,ξ1r3]T,ξ2l=[ξ2l1,ξ2l2…ξ2l9],Tξ2r=[ξ2r1,ξ2r2…ξ2r9]T,ξ3l=[ξ3l1,ξ3l2…ξ3l9],Tξ3r=[ξ3r1,ξ3r2…ξ3r9]T.Step 2.
We assume that there exist some language rules of unknown functions 1, 2, and 3, respectively.
The unknown function 1:(5.8)1(r/K)+λ[rx1(1-x1K)-ẏr].
The unknown function 2:(5.9)x2+amx2(λx1x2-μx2-α̇1).
The unknown function 3:(5.10)θx2x3x2+a-α̇2θx2x3x2+a-α̇2.
The fuzzy rules of the Unknown Function 1 (UF1) are as follows.
R1:
ifx1 is Small (S), then UF1 is Powerful (P).R2:
ifx1 is Medium (M), then UF1 is Weak (W).R3:
ifx1 is Large (L), then UF1 is Modest (M).
The fuzzy rules of the Unknown Function 2 (UF2) are as follows.
R1:
ifx1 is S, x2 is S, then UF2 is W.R2:
ifx1 is S, x2 is M, then UF2 is M.R3:
ifx1 is S, x2 is L, then UF2 is P.R4:
ifx1 is M, x2 is S, then UF2 is W.R5:
ifx1 is M, x2 is M, then UF2 is M.R6:
ifx1 is M, x2 is L, then UF2 is P.R7:
ifx1 is L, x2 is S, then UF2 is P.R8:
ifx1 is L, x2 is M, then UF2 is M.R9:
ifx1 is L, x2 is L, then UF2 is W.
The fuzzy rules of the Unknown Function 3 (UF3) are as follows.
R1:
ifx2 is S, x3 is S, then UF3 is M.R2:
ifx2 is S, x3 is M, then UF3 is W.R3:
ifx2 is S, x3 is L, then UF3 is P.R4:
ifx2 is M, x3 is S, then UF3 is W.R5:
ifx2 is M, x3 is M, then UF3 is M.R6:
ifx2 is M, x3 is L, then UF3 is P.R7:
ifx2 is L, x3 is S, then UF3 is M.R8:
ifx2 is L, x3 is M, then UF3 is W.R9:
ifx2 is L, x3 is L, then UF3 is P.Step 3.
Specify positive design parameters as follows:(5.11)k1=k2=k3=0.5,γ1l=γ2l=γ3l=γ1r=γ2r=γ3r=1,c1l=c2l=c3l=0.5,c1r=c2r=c3r=0.6.
The first intermediate controller is chosen as(5.12)α1=-k1e1-(ξ1lT(x̅1)Θ̂1l+ξ1rT(x̅1)Θ̂1r).
The first adaptive laws are designed as(5.13)Θ̂̇1l=Θ̃̇1l=γ1l(e1ξ1lT(x̅1)-c1lΘ̂1l),Θ̂̇1r=Θ̃̇1r=γ1r(e1ξ1rT(x̅1)-c1rΘ̂1r).
The second intermediate controller is chosen as(5.14)α2=-e1-k2e2-(ξ2lT(x̅2)Θ̂2l+ξ2rT(x̅2)Θ̂2r).
The second adaptive laws are designed as(5.15)Θ̂̇2l=Θ̃̇2l=γ2l(e2ξ2lT(x̅2)-c2lΘ̂2l),Θ̂̇2r=Θ̃̇2r=γ2r(e2ξ2rT(x̅2)-c2rΘ̂2r).
The actual controller is chosen as(5.16)u=-e2-k3e3-(ξ3lT(x̅3)Θ̂3l+ξ3rT(x̅3)Θ̂3r).
The actual adaptive laws are designed as(5.17)Θ̂̇3l=Θ̃̇3l=γ3l(e3ξ3lT(x̅3)-c3lΘ̂3l),Θ̂̇3r=Θ̃̇3r=γ3r(e3ξ3rT(x̅3)-c3rΘ̂3r).
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 1–11.(1)
Figures2 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)
Figure4 shows the trajectories of controller based on T2ABFC and T1ABFC, respectively.(3)
Figure5 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)
Figures6 and 7 show the responses of state variables x1 with disturbances based on T2ABFC and T1ABFC, respectively.(5)
Figures8 and 9 show the responses of state variables x2 with disturbances based on T2ABFC and T1ABFC, respectively.(6)
Figures10 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.State variable outputs with T2ABFC.
(a)(b)(c)State variable outputs with T1ABFC.
(a)(b)(c)Figure 4
Trajectories of controller.Figure 5
Trajectories of tracking error.Figure 6
Trajectories of the state variablex1 by T2ABFC with external disturbances.Figure 7
Trajectories of the state variablex1 by T1ABFC with external disturbances.Figure 8
Trajectories of the state variablex2 by T2ABFC with external disturbances.Figure 9
Trajectories of the state variablex2 by T1ABFC with external disturbances.Figure 10
Trajectories of the state variablex3 by T2ABFC with external disturbances.Figure 11
Trajectories of the state variablex3 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.



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*Source: 658424-2012-03-26.xml*