# Direct Adaptive Fuzzy Sliding Mode Control with Variable Universe Fuzzy Switching Term for a Class of MIMO Nonlinear Systems

Authors: Guo Haigang; Li Hongxing; Zhao Weijing; Song Zhankui
Journal: Mathematical Problems in Engineering
 (2012)

---

## Abstract

Combining adaptive fuzzy sliding mode control with fuzzy or variable universe fuzzy switching technique, this study develops two novel direct adaptive schemes for a class of MIMO nonlinear systems with uncertainties and external disturbances. The proposed control schemes consist of fuzzy equivalent control terms, fuzzy switching control terms (in scheme one) or variable universe fuzzy switching control terms (in scheme two), and compensation control terms. The compensation control terms are used to relax the assumption on fuzzy approximation error. Based on Lyapunov stability theory, the parameters update laws are adaptively tuned online and the global asymptotic stability of the closed-loop system can be guaranteed. The major contribution of this study is to develop a novel framework for designing direct adaptive fuzzy sliding mode control scheme facing model uncertainties and external disturbances. The derived schemes can effectively solve the chattering problem and the equivalent control calculation in that environment. Simulation results performed on a two-link robotic manipulator demonstrate the feasibility of the proposed control schemes.


---

## Body

## 1. Introduction

Some nonlinear systems, such as robotic manipulator, inverted pendulum, and electrical machines, not only are often highly coupled and time-varying systems, but also suffer from structured and unstructured uncertainties [1, 2]. The control of these systems is an important topic in the field of control. Sliding mode control (SMC) is an effective control scheme to deal with these problems [3–6]. However, this control scheme suffers mainly from two disadvantages. One is the chattering due to discontinuous switching term. The other is the difficulty involved in the calculation of the equivalent control [7]. A thorough knowledge of the plant dynamics is required for this purpose. But in the real world, there are many complex industrial processes whose accurate mathematical models are not available or difficult to formulate.In recent decades, fuzzy control methodology has emerged as a promising way to approach nonlinear control problems since it can incorporate linguistic information from human experts into control strategy [8–13]. Considerable efforts have been done to combine fuzzy system with SMC to overcome the disadvantages of general SMC [1, 14–30]. For example, fuzzy switching technique [15–17] and fuzzy boundary layer technique [18–20] have been developed to eliminate chattering problem. Both the techniques are built on the condition that the equivalent control has already existed. There exist some difficulties for the techniques to obtain a suitable equivalent control if the nominal mathematics model is unknown. In this case, there are generally two kinds of adaptive fuzzy SMC approaches to calculate the equivalent control in the existing literatures: direct [1, 21, 22] and indirect approaches [23–27]. Direct approach is to use fuzzy system directly to approximate the equivalent control term. Indirect approach is first to utilize fuzzy systems to approximate the unknown system functions, then to design the equivalent control based on these estimates. Both the approaches can effectively deal with the calculation of the equivalent control in the presence of model uncertainties and unknown disturbance. But the chattering problem might be encountered no matter what type of adaptive fuzzy SMC.Several indirect schemes which combined adaptive fuzzy SMC with fuzzy switching technique have been reported for SISO nonlinear systems in [28–30]. The proposed schemes can simultaneously overcome the two disadvantages of SMC mentioned above despite of model uncertainties and unknown disturbances. Unfortunately, convergence of the tracking error to zero is guaranteed by assuming that the fuzzy approximation error is very small if not equal to zero and square integrable. This, however, is difficult to show for any given plant [31]. Besides, direct approach may be of more interest not only because of its simple design and easy implementation, but also because it does not need to consider any possible controller singularity problem [32]. However, the constraints on control gain present difficulties for the design of direct adaptive control. Therefore, it will be a challenge for the direct control of MIMO nonlinear systems with model uncertainties and unknown disturbances.Combining adaptive fuzzy SMC with fuzzy or variable universe fuzzy switching technique, this paper proposes two novel direct adaptive control schemes for a class of MIMO nonlinear systems with uncertainties and external disturbances. The difference between them lies in that one scheme employs fuzzy system to estimate the switching control gain and the other uses the variable universe fuzzy system proposed in [33] to do it. Motivated by paper [21], this study relaxes the constraint on the gain matrix and only requires it to be positive definite symmetric besides the inverse of its derivative is bounded by an unknown function. To relax the assumption on fuzzy approximation error [28–30], we append an adaptive compensation term to compensate the effect of fuzzy approximation error [34]. The overall closed-loop systems stability and the online adjustment laws of the updated parameters are built based on Lyapunov stability theory. Lastly, the proposed schemes are utilized to deal with the trajectory tracking problem of robotic manipulators. Simulation results demonstrate that the proposed schemes are effective for a class of MIMO nonlinear systems. The two control schemes cannot only achieve the asymptotical tracking for ideal input signal, but also effectively eliminate the chattering of the general SMC.The rest of this paper is organized as follows. In Sections2 and 3, brief statements about the control system and variable universe fuzzy system are provided, respectively. Section 4 develops two novel direct adaptive fuzzy SMC schemes. Computer simulation results are illustrated in Section 5. Section 6 concludes this paper.


## 2. Problem Statement

Consider the following MIMO nonlinear system [21, 25]:
(2.1)y1(r1)=f1(x)+∑j=1p‍g1j(x)uj+d1,⋮yp(rp)=fp(x)+∑j=1p‍gpj(x)uj+dp,
where x=[y1,…,y1(r1-1),y2,…,y2(r2-1),…,yp,…,yp(rp-1)]T is the state vector which is available for measurement and r=[r1,r2,…,rp]T, u=[u1,u2,…,up]T is the control input vector, y=[y1,y2,…,yp]T is the output vector, fi(x)(i=1,2,…,p) are unknown continuous nonlinear functions, gij(x)(i,j=1,2,…,p) are smooth unknown nonlinear functions, and D(t)=[d1,d2,…,dp]T are unknown external disturbances.Definey=[y1(r1),y2(r2),…,yp(rp)]T, F(x)=[f1(x),f2(x),…,fp(x)]T, and
(2.2)G(x)=[g11(x),…,g1p(x)⋮…,⋮gp1(x),…,gpp(x)].
Then the dynamic system equation (2.1) can be rewritten as
(2.3)y(r)=F(x)+G(x)u+D(t).
Define reference trajectory xd=[xd1,xd2,…,xdp]T and the tracking error
(2.4)e1(t)=xd1(t)-y1(t),⋮ep(t)=xdp(t)-yp(t).
According to [3, 25], for each subsystem, one can define sliding surface as follows:
(2.5)s1(t)=(ddt+λ1)(r1-1)e1(t),λ1>0,⋮sp(t)=(ddt+λp)(rp-1)ep(t),λp>0.
From (2.5), we can see that ei(j)(t)→0, j=0,1,…,ri-1, i=1,2,…,p as si(t)→0. Then, the control duty can be transferred to design the control law such that si(t)→0, i=1,2,…,p.The time derivatives of the sliding surface of each subsystem are(2.6)s˙1(t)=e1(r1)+∑j=1r1-1‍Cr1-1je1(r1-j)λ1(j)=v1-f1(x)-∑j=1p‍g1j(x)uj-d1,⋮s˙p(t)=ep(rp)+∑j=1rp-1‍Crp-1jep(rp-j)λp(j)=vp-fp(x)-∑j=1p‍gpj(x)uj-dp,
where Cnk=n!/k!(n-k)! in which k≤n, and
(2.7)v1=xd1(r1)+∑j=1r1-1‍Cr1-1je1(r1-j)λ1(j),⋮vp=xdp(rp)+∑j=1rp-1‍Crp-1jep(rp-j)λp(j).
Let S=[s1,s2,…,sp]T and v=[v1,v2,…,vp]T. Equation (2.6) can be rewritten as
(2.8)S˙=v-F(x)-G(x)u-D(t).The objective of this paper is to design a control lawu(t) such that the output vector y(t) follows asymptotically the desired trajectory xd(t), with all involved signals in the closed-loop system remaining bounded. In the controller design, the following assumptions are useful for steady analysis and proof.Assumption 2.1.
G
(
x
) is a positive definite and symmetrical matrix.Assumption 2.2.
(
1
/
2
)
∥
(
d
x
/
d
t
)
G
-
1
(
x
)
∥
≤
δ
(
x
), where δ(x) is bounded positive continuous function without knowing its bound.Assumption 2.3.
The ideal trajectoryxdi(t)(i=1,2,…,p) are ri-order derivable, and xdi(t)(i=1,2,…,p) and their j-order derivatives (j=1,2,…,ri-1) are continuous functions of known boundary.Remark 2.4 (see [21]).
Assumption2.1 is useful to the stability analysis and stability proof. There are many physical systems, such as robotic systems and electrical machines, which satisfy the positive definiteness and the symmetry. Assumption 2.2 is not restrictive, since we only assume the existence of δ(x) and not its knowledge. Moreover, there are several physical systems in which the control gain matrix G(x) satisfies the inequality, for example, robotic manipulators, electrical machines, inverted pendulum, and chaotic systems.


## 3. Variable Universe Fuzzy System

Fuzzy control emerged in decades ago is a promising way to solve nonlinear control problems. It has several excellent properties. For example, it does not require the plant model and can effectively incorporate the semantic knowledge of human experts. Since the universal approximation theorem has been put forward in [10], fuzzy system and fuzzy control evolve faster than before. Specifically, in the control area, fuzzy systems are mainly used as a nonlinear function approximation tool.It should be emphasized that, in this paper, it is assumed that the structure and the membership function parameters of the fuzzy system are properly specified in advance by the designer. This means that the designer decision is needed to determine the structure of the fuzzy system, namely, the pertinent inputs, the number of membership functions for each input, the membership function parameters, and the number of rules.


### 3.1. Fuzzy System

For convenience, we will recall briefly the fuzzy system in the following. LetXi=[-Ei,Ei](i=1,2,…,m) be the universe of input variable zi(i=1,2,…,m) and Y=[-U,U] the universe of output variable uo. 𝔄i={Aij}(j=1,2,…,N) is defined as a fuzzy partition on Xi and 𝔅={Bj}(j=1,2,…,N) a fuzzy partition on Y, where Aij∈ℱ(Xi) and Bj∈ℱ(Y) are termed as the base, and aij and bj are the peak points of Aij and Bj, respectively. 𝔄i and 𝔅 are regarded as linguistic variables so that a group of fuzzy inference rules are formed as follows:
(3.1)Ifz1isA1j,z2isA2j,…,zmisAmj,thenuoisBj(j=1,2,…,N),
where N represents the number of the rules. Singleton fuzzifier, triangle membership function (overlap law is 0.5), product inference engine, and center average defuzzifier are used in this fuzzy system. The derived output of fuzzy system can be written as
(3.2)u^o(z)=ξ(z)Tθ,
where z=[z1,z2,…,zm]T, θ=[u1,u2,…,uN]T is a vector grouping all consequent parameters, and ξ(z)=[ξ1(z),ξ2(z),…,ξN(z)]T is fuzzy basis function vector defined as
(3.3)ξj(z)=∏i=1m‍Aij(zi),j=1,2,…,N.


### 3.2. Variable Universe Fuzzy System

Adaptive fuzzy controller based on fixed universe has limited approximation accuracy according to the interpolation mechanism of fuzzy system [35]. Aiming at the problem, Li first presents the variable universe idea [36]. The core idea of variable universe fuzzy control is that the universes contracts following the decrease of error. Contraction of the universe is equivalent to the increase of the control rules. Therefore, control accuracy is improved.The so-called variable universe means that some universes, for example,Xi and Y, respectively, can change along with changing of variables zi and uo. In this case, the universes are denoted by
(3.4)Xi(zi)=[-αi(zi)Ei,αi(zi)Ei],Y(uo)=[-β(uo)U,β(uo)U],
where αi(zi)(i=1,2,…,m) and β(uo) are, respectively, called contraction-expansion factors of the universes Xi(i=1,2,…,m) and Y. Being relative to the variable universes, the original universes Xi and Y are naturally called initial universes. After the above changes, the output of variable universe fuzzy controller can be written as
(3.5)u^o(β,z)=βζ(z)Tϑ,
where ϑ=[u2,u2,…,uN]T and ζ(z)=[ζ1(z),ζ2(z),…,ζN(z)]T represent the parameter vector of inference consequence and the fuzzy base function vector, respectively, in which ζj(z)=∏i=1mAij(zi/αi(zi)), j=1,2,…,N.Remark 3.1.
Under the framework of variable universe fuzzy control, the parameter which needs to be online adjusted is a scalarβ instead of a parameter vector θ in conventional adaptive fuzzy control scheme. Therefore, variable universe fuzzy control scheme simplifies the design procedure of adaptive fuzzy control. In addition, we hardly need smart expert knowledge in the realm, but need the rough trend of control rules in the design of variable universe fuzzy controller [37]. From this point, variable universe fuzzy control reduces the design difficulty.


## 3.1. Fuzzy System

For convenience, we will recall briefly the fuzzy system in the following. LetXi=[-Ei,Ei](i=1,2,…,m) be the universe of input variable zi(i=1,2,…,m) and Y=[-U,U] the universe of output variable uo. 𝔄i={Aij}(j=1,2,…,N) is defined as a fuzzy partition on Xi and 𝔅={Bj}(j=1,2,…,N) a fuzzy partition on Y, where Aij∈ℱ(Xi) and Bj∈ℱ(Y) are termed as the base, and aij and bj are the peak points of Aij and Bj, respectively. 𝔄i and 𝔅 are regarded as linguistic variables so that a group of fuzzy inference rules are formed as follows:
(3.1)Ifz1isA1j,z2isA2j,…,zmisAmj,thenuoisBj(j=1,2,…,N),
where N represents the number of the rules. Singleton fuzzifier, triangle membership function (overlap law is 0.5), product inference engine, and center average defuzzifier are used in this fuzzy system. The derived output of fuzzy system can be written as
(3.2)u^o(z)=ξ(z)Tθ,
where z=[z1,z2,…,zm]T, θ=[u1,u2,…,uN]T is a vector grouping all consequent parameters, and ξ(z)=[ξ1(z),ξ2(z),…,ξN(z)]T is fuzzy basis function vector defined as
(3.3)ξj(z)=∏i=1m‍Aij(zi),j=1,2,…,N.


## 3.2. Variable Universe Fuzzy System

Adaptive fuzzy controller based on fixed universe has limited approximation accuracy according to the interpolation mechanism of fuzzy system [35]. Aiming at the problem, Li first presents the variable universe idea [36]. The core idea of variable universe fuzzy control is that the universes contracts following the decrease of error. Contraction of the universe is equivalent to the increase of the control rules. Therefore, control accuracy is improved.The so-called variable universe means that some universes, for example,Xi and Y, respectively, can change along with changing of variables zi and uo. In this case, the universes are denoted by
(3.4)Xi(zi)=[-αi(zi)Ei,αi(zi)Ei],Y(uo)=[-β(uo)U,β(uo)U],
where αi(zi)(i=1,2,…,m) and β(uo) are, respectively, called contraction-expansion factors of the universes Xi(i=1,2,…,m) and Y. Being relative to the variable universes, the original universes Xi and Y are naturally called initial universes. After the above changes, the output of variable universe fuzzy controller can be written as
(3.5)u^o(β,z)=βζ(z)Tϑ,
where ϑ=[u2,u2,…,uN]T and ζ(z)=[ζ1(z),ζ2(z),…,ζN(z)]T represent the parameter vector of inference consequence and the fuzzy base function vector, respectively, in which ζj(z)=∏i=1mAij(zi/αi(zi)), j=1,2,…,N.Remark 3.1.
Under the framework of variable universe fuzzy control, the parameter which needs to be online adjusted is a scalarβ instead of a parameter vector θ in conventional adaptive fuzzy control scheme. Therefore, variable universe fuzzy control scheme simplifies the design procedure of adaptive fuzzy control. In addition, we hardly need smart expert knowledge in the realm, but need the rough trend of control rules in the design of variable universe fuzzy controller [37]. From this point, variable universe fuzzy control reduces the design difficulty.


## 4. Adaptive Fuzzy SMC Design

### 4.1. Sliding Mode Control

In this section, we firstly consider that nonlinear function matricesF(x) and G(x) are known. From Assumption 2.1, the matrix G(x) is reversible. Both sides of (2.8) are multiplied by G1(x)=G-1(x), then we obtain that
(4.1)G1(x)S˙=G1(x)v-F1(x)-u-G1(x)D(t),
where F1(x)=G1(x)F(x). The control law is designed as
(4.2)u=G1(x)v-F1(x)-G1(x)D(t)+δ(x)IpS+ηsign(S),
where sign(S)=[sign(s1),sign(s2),…,sign(sp)]T and η=diag(η1,η2,…,ηp) in which ηi>0(i=1,2,…,p), where its aim is to meet the sliding condition. Substituting (4.2) into (4.1), we can obtain
(4.3)G1(x)S˙=-ηsign(S)-δ(x)IpS.We choose a candidate Lyapunov functionV=(1/2)STG1(x)S. For the matrix G(x) being positive definite and symmetrical, G1(x) is also positive definite and symmetrical and satisfies S˙TG1(x)S=STG1(x)S˙. Combining with Assumption 2.2, we have
(4.4)V˙=STG1(x)S˙+12STG˙1(x)S=-ST(ηsign(S)+δ(x)IpS)+12STG˙1(x)S≤-∑i=1p‍ηi|si|.
Therefore, V˙ is negative definite and the control objective can be achieved. However, in engineering practice, nonlinear system matrix F(x) and control gain matrix G(x) are often unknown. Accordingly, the derivative function matrices F1(x) and G1(x) as well as δ(x) are also unknown. Moreover, external disturbance vector D(t) is also unknown. Therefore, the control law (4.2) cannot be implemented. In the following, we employ fuzzy systems to design the control law.


### 4.2. Direct Adaptive Fuzzy SMC with Fuzzy Switching Term (DAFSMC with FSW)

According to the sliding mode control scheme, the control lawu can be decomposed into the equivalent control and the switching control. In the sliding phase, the role of the equivalent control is to force the system dynamics to stay on the sliding surface. In the reaching phase, the switching control is designed to satisfy the sliding mode condition [3, 28, 29].Let(4.5)ueq≜G1(x)v-F1(x)-G1(x)D(t)+δ(x)IpS
and usw≜ηsign(S). Equation (4.2) can rewritten as
(4.6)u=ueq+usw.It is well known that the fuzzy rules used for reasoning are not easy to extract, especially for multi-input(m>2) fuzzy system. Yet, a prominent merit of adaptive fuzzy system is that it does not need initial fuzzy rules and can generate fuzzy rules online based on Lyapunov stability theory [38].So, in what follows, adaptive fuzzy systems as (3.2) are chosen to approximate the equivalent control term ueq. Therefore, we have the following format:
(4.7)ueq=u^eq(Θe*,x)+ω=ΞeT(x)Θe*+ω,
where ω=[ω1,ω2,…,ωp]T and Ξe(x)=[diag(ξe1T(x),ξe2T(x),…,ξepT(x))]T denotes fuzzy approximate error vector and fuzzy base function matrix, respectively.LetΘe*=[θe1*T,θe2*T,…,θep*T]T be the optimal parameter vector, where
(4.8)θei*=argmin∥θei∥≤Mei{supx∈Dx|ueqi-u^eqi(θei,x)|},
in which Mei(i=1,2,…,p) is design constant restraining parameter vector θei(i=1,2,…,p) and Dxis a compact set containing state x, and
(4.9)u^eqi(θei,x)=θeiTξei(x)
denotes the estimate of the equivalent control law ueqi. Due to unknown ideal parameter vector θei*, we estimate θei* by virtue of θei.Remark 4.1.
In this paper, we assume that fuzzy approximation errorsωi(i=1,2,…,p) are bounded for all x∈Dx, that is, |ωi(x)|≤ρi, for all x∈Dx, where ρi is an unknown constant. The knowledge of ρ=diag(ρ1,ρ2,…,ρp) is only required for analysis purpose.Given the approximation errorωi between the equivalent control ueqi and the used fuzzy system u^eqi, the switching control term usw must be modified as (η+ρ)sign(S), that is, uswi=(ηi+ρi)sign(si)(i=1,2,…,p). The objective of this modification is to meet the sliding mode existing condition. In the previous literatures [23, 26], the switching gain matrix (η+ρ) has to be determined in advance. This is difficult when the bound of the approximation error is unknown. Improper switching gain easily causes chattering problem which is undesired in practice.Since the chattering is caused by the switching gain matrix(η+ρ) and the discontinuous function sign(s), let the switching control usw=(η+ρ)sign(s) be replaced by a gain vector K=[k1,k2,…,kp] [17]. Motivated by papers [28–30], in the controller design, we employ fuzzy systems in the form of (3.2) to approximate the gain vector K.According to the switching controluswi in (4.6), we can choose a single-input single-output fuzzy system k^i(si) to approximate the gain ki. Here, the fuzzy system k^i is applied to compensate the system uncertainty to reduce the energy of si and causes the sliding surface to approach zero. It is obvious that the sign of ki is the same as that of si. When |si| is away from zero, |ki| should be chosen a large value to make the system move quickly to the switching surface. When |si| is small, |ki| should be chosen a small value to avoid overshoot. When siis zero, |ki| should be zero. From the above analysis, it is easy to make sis˙i≤0 and guarantee the sliding condition. Briefly, the fuzzy rules can be determined as follows:
(4.10)IfsiisAh,thenkiisBh,h=1,2,…,iL,
where si and ki are the input and the output variables of the fuzzy system, respectively, and iL represents the number of fuzzy rules. Then the estimated switching gain k^i of the ith subsystem can be written as
(4.11)k^i=ξsi(si)Tθsi,
where ξsi(si)=[A1(si/αi(si)),…,AiL(si/αi(si))]T and θsi=[b1,b2,…,biL]T. Let
(4.12)θsi*=argmin∥θsi∥≤Msi{supsi∈Dsi|ki-ξsi(si)Tθsi|},
be the optimal parameter vector, where Msi is design constant restraining parameter vector θsi and Dsi is compact set containing the variable si. The optimal parameter vector θsi* is unknown, so θsi* is estimated by θsi.Further, to cancel the approximation error between the equivalent controlueqi and the used fuzzy system u^eqi, we append a compensation control term uc=[uc1,uc2,…,ucp]T [34]. Therefore, the overall control effort can be modified as
(4.13)ui=u^eqi+k^i+uci=ξeiT(x)θei+ξsiT(si)θsi+uci
or
(4.14)u=u^eq+K^+uc=Ξe(x)TΘe+ΞsT(S)Θs+uc,
where Ξs(S)=[diag(ξs1T(s1),ξs2T(s2),…,ξspT(sp))]T and Θs=[θs1T,θs2T,…,θspT]T. The compensation control term uc will be designed next.Substituting (4.14) into (4.1), we obtain
(4.15)G1(x)S˙=G1(x)v-F1(x)-(ΞeT(x)Θe+ΞsT(S)Θs+uc)-G1(x)D(t)=ω-uc-(ΞeT(x)Θe-ΞeT(x)Θe*)-(ΞsT(S)Θs-ΞsT(S)Θs*)-ΞsT(S)Θs*-δ(x)IpS.
Let Θ~e=Θe-Θe* and Θ~s=Θs-Θs*. Combining the above equation, we have
(4.16)G1(x)S˙=ω-uc-ΞeT(x)Θ~e-ΞsT(S)Θ~s-ΞsT(S)Θs*-δ(x)IpS.To derive the adaptive laws of the parameter vectors, let us consider the Lyapunov candidate function(4.17)V1=12STG1(x)S+12γ1Θ~eTΘ~e+12γ2Θ~sTΘ~s+12γω~Tω~,
where γ1, γ2, γ are positive constants, ω~=ω-ω^ in which the vector ω^=[ω^1,ω^2,…,ω^p]T denotes the estimate of the unknown approximation error vector ω.Theorem 4.2.
For system (2.1), nonlinear function matrices F(x), G(x), and vector D(t) are unknown, and Assumptions 2.1–2.3 hold. The control law is selected as (4.13) or (4.14) in which uci=ω^i with adaptation laws (4.18) as follows:
(4.18)θ˙ei={γ1siξei(x),∥θei∥<Meior∥θei∥=Mei,γ1siθeiTξei(x)≤0,γ1siξei(x)-γ1siθeiθeiT∥θei∥2ξei(x),∥θei∥=Mei,γ1siθeiTξei(x)>0,θ˙si={γ2siξsi(si),∥θsi∥<Msior∥θsi∥=Msi,γ2siθsiTξsi(si)≤0,γ2siξsi(si)-γ2siθsiθsiT∥θsi∥2ξsi(si),∥θsi∥=Msi,γ2siθsiTξsi(si)>0,ω^˙i=γsi,
where Mei and Msi are defined as before. Then one can derive the performance as follows. (1)
The involved signals of the close loop are bounded.(2)
The tracking errors and their derivatives decay to zero asymptotically, in other words, whent→∞, ei(j)→0, j=0,1,…,ri-1, i=1,2,…,p.Proof .
Differentiating (4.17) with respect to t and using (4.16), we have
(4.19)V˙1=STG1(x)S˙+12STG˙1(x)S+1γ1Θ~eTΘ~˙e+1γ2Θ~sTΘ~˙s+1γω~Tω~˙=ST(ω-uc-ΞeT(x)Θ~e-ΞsT(S)Θ~s-ΞsT(S)Θs*-δ(x)IpS)+12STG˙1(x)S+1γ1Θ~eTΘ~˙e+1γ2Θ~sTΘ~˙s+1γω~Tω~˙≤∑i=1p‍siωi-∑i=1p‍siω^i-∑i=1p‍siθ~eiTξei(x)+1γ1∑i=1p‍θ~eiTθ~˙ei+12STG˙1(x)S-δ(x)∥S∥2-∑i=1p‍siθ~siTξsi(si)+1γ2∑i=1p‍θ~siTθ~˙si-∑i=1p‍ηi|si|-1γ∑i=1p‍ω~iTω^˙i≤-1γ1∑i=1p‍θ~eT(γ1siξei(x)-θ˙e)-1γ2∑i=1p‍θ~siT(γ2siξsi(si)-θ~˙si)+1γ∑i=1p‍(γsi(ωi-ω^i)-ω~iTω~˙i)-∑i=1p‍ηi|si|.
Considering the same sign between si and the fuzzy switching term ki, the first inequality in the above derivation is easily established by using the inequality ηi|si|≤siθsi*Tξsi(si)≤(ηi+ρi)|si|. Noticing (4.18), we have
(4.20)V˙1≤-∑i=1p‍ηi|si|<0.
Consequently, all signals in the system are bounded. Obviously, if e(0) is bounded, then e(t) is also bounded for all t. Since the reference trajectory xdis bounded, then the system state x(t) is bounded as well.
To complete the proof and establish asymptotic convergence of the tracking error, we need to prove thatS→0 as t→∞. We rearrange (4.16) as following:
(4.21)S˙=G(x)(ω-ω^-ΞeT(x)Θ~e-ΞsT(S)Θ~s-ΞsT(S)Θs*-δ(x)IpS).
Since G(x) and δ(x) are continuous functions in a compact set Dx, they are bounded. By using the boundness of S, ω^˙∈L∞ and in turn ω^∈L∞. Therefore, S˙∈L∞ holds. Using Barbalat's lemma, S→0(t→∞) holds. This completes the proof.Remark 4.3.
The tracking control using conventional adaptive fuzzy SMC with fuzzy switching control term [28–30] does not tackle the problem of attenuation of the effect of the fuzzy approximation error. We usually have V˙≤Sw-ηSTsign(S). In this case, it can be excepted that Sw should be very small if not equal to zero in [28–30] but here we get V˙≤-∑i=1pηi|si|<0, which improves the stability proof. In order to overcome this restriction, we have proposed ω^ to estimate the fuzzy approximation error ω.Remark 4.4.
In conventional adaptive fuzzy SMC design [24, 39, 40], there is often an assumption that the unknown external disturbance D(t) is bounded by a known positive constant. However, it is often not possible to obtain the bound in complex situations. In order to overcome this restriction, the variable D(t) is incorporated into the equivalent control term ueq and a fuzzy system is used to approximate ueq. Therefore, knowledge of the bound of external disturbance is not necessary in the present paper.


### 4.3. Direct Adaptive Fuzzy SMC with Variable Universe Fuzzy Switching Term (DAFSMC with VUFSW)

As stated in Section3.2, variable universe fuzzy system possesses high static precision by virtue of contraction-expansion factor. In this subsection, to accelerate response speed and improve the control accuracy, we consider an adaptive fuzzy SMC with VUFSW.Similar to the analysis in Section4.2, the variables si and ki are still taken as the input and the output variables of the variable universe fuzzy system. Then the variable universe fuzzy switching control law k^i of the ith subsystem can be written as
(4.22)k^i=βiζsi(si)Tθsi,
where ζsi(si)=[A1(si/αi(si)),…,AiL(si/αi(si))]T and θsi=[b1,b2,…,biL]T.Letβ*=diag(β1*,β2*,…,βp*) be the optimal parameter vector, where
(4.23)βi*=argmin|βi|≤Mi{supsi∈Dsi|ki-βiζsi(si)Tθsi|},
in which Mi is the design constant. Similarly, since the ideal parameter βi* is unknown, βi is employed to estimate βi*. Therefore, the overall control effort can be modified as
(4.24)ui=u^eqi+k^i+uci=ξeiT(x)θei+βiζsiT(si)θsi+ω^i
or
(4.25)u=u^eq+K^+uc=ΞeT(x)Θe+βΞsT(S)Θs+ω^,
where Ξs(S)=[diag(ζs1T(s1),ζs2T(s2),…,ζspT(sp))]T and Θs=[θs1T,θs2T,…,θspT]T.Substituting (4.25) into (4.1), we obtain
(4.26)G1(x)S˙=G1(x)v-F1(x)-(ΞeTΘe+βΞsT(S)Θs+ω^)-G1(x)D(t)=(ω-ω^)-(ΞeTΘe-ΞeTΘe*)-(βΞsT(S)Θs-β*ΞsT(S)Θs)-β*ΞsT(S)Θs-δ(x)IpS.
Let Θ~e=Θe-Θe* and β~=β-β*. Then
(4.27)G1(x)S˙=ω~-ΞeT(x)Θ~e-β~ΞsT(S)Θs-β*ΞsT(S)Θs-δ(x)IpS.To derive the adaptive law of the parameter vectors, we consider the Lyapunov candidate function as(4.28)V2=12STG1(x)S+12γ1Θ~eTΘ~e+12γ3β~Tβ~T+12γω~Tω~,
where γ3 is a positive constant.Theorem 4.5.
For system (2.1), nonlinear function matrices F(x), G(x), and vector D(t) are unknown, and Assumptions 2.1–2.3 hold. The control law is selected as (4.24) or (4.25), with adaptation laws (4.29) as follows:
(4.29)θ˙ei={γ1siξei(x),∥θei∥<Meior∥θei∥=Mei,γ1siθeiTξei(x)≤0,γ1siξei(x)-γ1siθeiθeiT∥θei∥2ξei(x),∥θei∥=Mei,γ1siθeiTξei(x)>0,β˙i={γ3siθsiTζsi(si),|βi|<Mior|βi|=Mi,γ3siθsiTζsi(si)≤0,0,|βi|=Mi,γ3siθsiTζsi(si)>0,ω^˙=γsi,
where Mei and Mi are defined as before. Then one can derive the performance as follows.(1)
The involved signals of the close loop are bounded.(2)
The tracking errors and their derivatives decay to zero asymptotically, in other words, whent→∞, ei(j)→0, j=0,1,…,ri-1, i=1,2,…,p.Proof.
Differentiating (4.28) with respect to t and using (4.27), we have
(4.30)V˙2=STG1(x)S˙+12STG˙1(x)S+1γ1Θ~eTΘ~˙e+1γ3β~Tβ~˙+1γω~Tω~˙=ST(ω~-ΞeT(x)Θ~e-β~ΞsT(S)Θs-β*ΞsT(S)Θs-δ(x)IpS)+12STG˙1(x)S+1γ1Θ~eTΘ~˙e+1γ3β~Tβ~˙+1γω~Tω~˙≤∑i=1p‍siω~i-1γ∑i=1p‍ω~iTω^˙i-∑i=1p‍siθ~eiTξei(x)+1γ1∑i=1p‍θ~eiTθ˙ei+12STG˙1(x)S-δ(x)∥S∥2-∑i=1p‍siβ~iθsiTζsi(si)+1γ3∑i=1p‍β~iβ˙i-∑i=1p‍ηi|si|≤1γ∑i=1p‍ω~i(γsi-ω^˙i)-1γ1∑i=1p‍θ~eiT(γ1siξei(x)-θ˙ei)-1γ3∑i=1p‍β~i(γ3siθsiTζsi(si)-β˙i)-∑i=1p‍ηi|si|.
Noticing that (4.29), we have
(4.31)V˙2≤-∑i=1p‍ηi|si|<0.
Therefore, all signals in the system are bounded. In order to show the boundedness of S˙, we rearrange (4.27) as follows:
(4.32)S˙=G(x)(ω~-ΞeT(x)Θ~e-β~ΞsT(S)Θs-β*ΞsT(S)Θs-δ(x)IpS).
Similarly, we can derive S˙∈L∞.To summarize the above analysis, the step-by-step procedures for the two direct adaptive fuzzy SMCs are proposed as follows.Design Procedure:Step 1.
Select proper positive coefficientsλ1,λ2,…,λp and learning coefficients γ1,γ2,γ3, and γ.Step 2.
Specify design constant vectorsMe=[Me1,Me2,…,Mep]T, Ms=[Ms1,Ms2,…,Msp]T and M=[M1,M2,…,Mp]T.Step 3.
Definemi fuzzy sets Fi for variable xi to achieve an uniform coverage of the universe of discourse. Select the initial parameter vector θei(0)=0_mi×1(i=1,2,…,p).Step 4.
Construct the fuzzy rule bases for the fuzzy systemuswi. Define iL fuzzy sets Ai for variable si to achieve a uniform coverage of the universe of discourse. Select the initial parameter vectors θsi(0)iL×1(i=1,2,…,p) for DAFSMC with FSW and the initial values βi(0)(i=1,2,…,p) for DAFSMC with VUFSW.Step 5.
Construct the fuzzy systemsu^eqi in (4.9), and k^i in (4.11) or (4.22).Step 6.
Construct the control law (4.13) or (4.14) with the adaptive laws in (4.18) or construct the control law (4.24) or (4.25) with the adaptive laws in (4.29).Step 7.
Use the adaptive laws (4.18) or (4.29) to adjust the parameters θei, θsi or βi, and ω^i.


## 4.1. Sliding Mode Control

In this section, we firstly consider that nonlinear function matricesF(x) and G(x) are known. From Assumption 2.1, the matrix G(x) is reversible. Both sides of (2.8) are multiplied by G1(x)=G-1(x), then we obtain that
(4.1)G1(x)S˙=G1(x)v-F1(x)-u-G1(x)D(t),
where F1(x)=G1(x)F(x). The control law is designed as
(4.2)u=G1(x)v-F1(x)-G1(x)D(t)+δ(x)IpS+ηsign(S),
where sign(S)=[sign(s1),sign(s2),…,sign(sp)]T and η=diag(η1,η2,…,ηp) in which ηi>0(i=1,2,…,p), where its aim is to meet the sliding condition. Substituting (4.2) into (4.1), we can obtain
(4.3)G1(x)S˙=-ηsign(S)-δ(x)IpS.We choose a candidate Lyapunov functionV=(1/2)STG1(x)S. For the matrix G(x) being positive definite and symmetrical, G1(x) is also positive definite and symmetrical and satisfies S˙TG1(x)S=STG1(x)S˙. Combining with Assumption 2.2, we have
(4.4)V˙=STG1(x)S˙+12STG˙1(x)S=-ST(ηsign(S)+δ(x)IpS)+12STG˙1(x)S≤-∑i=1p‍ηi|si|.
Therefore, V˙ is negative definite and the control objective can be achieved. However, in engineering practice, nonlinear system matrix F(x) and control gain matrix G(x) are often unknown. Accordingly, the derivative function matrices F1(x) and G1(x) as well as δ(x) are also unknown. Moreover, external disturbance vector D(t) is also unknown. Therefore, the control law (4.2) cannot be implemented. In the following, we employ fuzzy systems to design the control law.


## 4.2. Direct Adaptive Fuzzy SMC with Fuzzy Switching Term (DAFSMC with FSW)

According to the sliding mode control scheme, the control lawu can be decomposed into the equivalent control and the switching control. In the sliding phase, the role of the equivalent control is to force the system dynamics to stay on the sliding surface. In the reaching phase, the switching control is designed to satisfy the sliding mode condition [3, 28, 29].Let(4.5)ueq≜G1(x)v-F1(x)-G1(x)D(t)+δ(x)IpS
and usw≜ηsign(S). Equation (4.2) can rewritten as
(4.6)u=ueq+usw.It is well known that the fuzzy rules used for reasoning are not easy to extract, especially for multi-input(m>2) fuzzy system. Yet, a prominent merit of adaptive fuzzy system is that it does not need initial fuzzy rules and can generate fuzzy rules online based on Lyapunov stability theory [38].So, in what follows, adaptive fuzzy systems as (3.2) are chosen to approximate the equivalent control term ueq. Therefore, we have the following format:
(4.7)ueq=u^eq(Θe*,x)+ω=ΞeT(x)Θe*+ω,
where ω=[ω1,ω2,…,ωp]T and Ξe(x)=[diag(ξe1T(x),ξe2T(x),…,ξepT(x))]T denotes fuzzy approximate error vector and fuzzy base function matrix, respectively.LetΘe*=[θe1*T,θe2*T,…,θep*T]T be the optimal parameter vector, where
(4.8)θei*=argmin∥θei∥≤Mei{supx∈Dx|ueqi-u^eqi(θei,x)|},
in which Mei(i=1,2,…,p) is design constant restraining parameter vector θei(i=1,2,…,p) and Dxis a compact set containing state x, and
(4.9)u^eqi(θei,x)=θeiTξei(x)
denotes the estimate of the equivalent control law ueqi. Due to unknown ideal parameter vector θei*, we estimate θei* by virtue of θei.Remark 4.1.
In this paper, we assume that fuzzy approximation errorsωi(i=1,2,…,p) are bounded for all x∈Dx, that is, |ωi(x)|≤ρi, for all x∈Dx, where ρi is an unknown constant. The knowledge of ρ=diag(ρ1,ρ2,…,ρp) is only required for analysis purpose.Given the approximation errorωi between the equivalent control ueqi and the used fuzzy system u^eqi, the switching control term usw must be modified as (η+ρ)sign(S), that is, uswi=(ηi+ρi)sign(si)(i=1,2,…,p). The objective of this modification is to meet the sliding mode existing condition. In the previous literatures [23, 26], the switching gain matrix (η+ρ) has to be determined in advance. This is difficult when the bound of the approximation error is unknown. Improper switching gain easily causes chattering problem which is undesired in practice.Since the chattering is caused by the switching gain matrix(η+ρ) and the discontinuous function sign(s), let the switching control usw=(η+ρ)sign(s) be replaced by a gain vector K=[k1,k2,…,kp] [17]. Motivated by papers [28–30], in the controller design, we employ fuzzy systems in the form of (3.2) to approximate the gain vector K.According to the switching controluswi in (4.6), we can choose a single-input single-output fuzzy system k^i(si) to approximate the gain ki. Here, the fuzzy system k^i is applied to compensate the system uncertainty to reduce the energy of si and causes the sliding surface to approach zero. It is obvious that the sign of ki is the same as that of si. When |si| is away from zero, |ki| should be chosen a large value to make the system move quickly to the switching surface. When |si| is small, |ki| should be chosen a small value to avoid overshoot. When siis zero, |ki| should be zero. From the above analysis, it is easy to make sis˙i≤0 and guarantee the sliding condition. Briefly, the fuzzy rules can be determined as follows:
(4.10)IfsiisAh,thenkiisBh,h=1,2,…,iL,
where si and ki are the input and the output variables of the fuzzy system, respectively, and iL represents the number of fuzzy rules. Then the estimated switching gain k^i of the ith subsystem can be written as
(4.11)k^i=ξsi(si)Tθsi,
where ξsi(si)=[A1(si/αi(si)),…,AiL(si/αi(si))]T and θsi=[b1,b2,…,biL]T. Let
(4.12)θsi*=argmin∥θsi∥≤Msi{supsi∈Dsi|ki-ξsi(si)Tθsi|},
be the optimal parameter vector, where Msi is design constant restraining parameter vector θsi and Dsi is compact set containing the variable si. The optimal parameter vector θsi* is unknown, so θsi* is estimated by θsi.Further, to cancel the approximation error between the equivalent controlueqi and the used fuzzy system u^eqi, we append a compensation control term uc=[uc1,uc2,…,ucp]T [34]. Therefore, the overall control effort can be modified as
(4.13)ui=u^eqi+k^i+uci=ξeiT(x)θei+ξsiT(si)θsi+uci
or
(4.14)u=u^eq+K^+uc=Ξe(x)TΘe+ΞsT(S)Θs+uc,
where Ξs(S)=[diag(ξs1T(s1),ξs2T(s2),…,ξspT(sp))]T and Θs=[θs1T,θs2T,…,θspT]T. The compensation control term uc will be designed next.Substituting (4.14) into (4.1), we obtain
(4.15)G1(x)S˙=G1(x)v-F1(x)-(ΞeT(x)Θe+ΞsT(S)Θs+uc)-G1(x)D(t)=ω-uc-(ΞeT(x)Θe-ΞeT(x)Θe*)-(ΞsT(S)Θs-ΞsT(S)Θs*)-ΞsT(S)Θs*-δ(x)IpS.
Let Θ~e=Θe-Θe* and Θ~s=Θs-Θs*. Combining the above equation, we have
(4.16)G1(x)S˙=ω-uc-ΞeT(x)Θ~e-ΞsT(S)Θ~s-ΞsT(S)Θs*-δ(x)IpS.To derive the adaptive laws of the parameter vectors, let us consider the Lyapunov candidate function(4.17)V1=12STG1(x)S+12γ1Θ~eTΘ~e+12γ2Θ~sTΘ~s+12γω~Tω~,
where γ1, γ2, γ are positive constants, ω~=ω-ω^ in which the vector ω^=[ω^1,ω^2,…,ω^p]T denotes the estimate of the unknown approximation error vector ω.Theorem 4.2.
For system (2.1), nonlinear function matrices F(x), G(x), and vector D(t) are unknown, and Assumptions 2.1–2.3 hold. The control law is selected as (4.13) or (4.14) in which uci=ω^i with adaptation laws (4.18) as follows:
(4.18)θ˙ei={γ1siξei(x),∥θei∥<Meior∥θei∥=Mei,γ1siθeiTξei(x)≤0,γ1siξei(x)-γ1siθeiθeiT∥θei∥2ξei(x),∥θei∥=Mei,γ1siθeiTξei(x)>0,θ˙si={γ2siξsi(si),∥θsi∥<Msior∥θsi∥=Msi,γ2siθsiTξsi(si)≤0,γ2siξsi(si)-γ2siθsiθsiT∥θsi∥2ξsi(si),∥θsi∥=Msi,γ2siθsiTξsi(si)>0,ω^˙i=γsi,
where Mei and Msi are defined as before. Then one can derive the performance as follows. (1)
The involved signals of the close loop are bounded.(2)
The tracking errors and their derivatives decay to zero asymptotically, in other words, whent→∞, ei(j)→0, j=0,1,…,ri-1, i=1,2,…,p.Proof .
Differentiating (4.17) with respect to t and using (4.16), we have
(4.19)V˙1=STG1(x)S˙+12STG˙1(x)S+1γ1Θ~eTΘ~˙e+1γ2Θ~sTΘ~˙s+1γω~Tω~˙=ST(ω-uc-ΞeT(x)Θ~e-ΞsT(S)Θ~s-ΞsT(S)Θs*-δ(x)IpS)+12STG˙1(x)S+1γ1Θ~eTΘ~˙e+1γ2Θ~sTΘ~˙s+1γω~Tω~˙≤∑i=1p‍siωi-∑i=1p‍siω^i-∑i=1p‍siθ~eiTξei(x)+1γ1∑i=1p‍θ~eiTθ~˙ei+12STG˙1(x)S-δ(x)∥S∥2-∑i=1p‍siθ~siTξsi(si)+1γ2∑i=1p‍θ~siTθ~˙si-∑i=1p‍ηi|si|-1γ∑i=1p‍ω~iTω^˙i≤-1γ1∑i=1p‍θ~eT(γ1siξei(x)-θ˙e)-1γ2∑i=1p‍θ~siT(γ2siξsi(si)-θ~˙si)+1γ∑i=1p‍(γsi(ωi-ω^i)-ω~iTω~˙i)-∑i=1p‍ηi|si|.
Considering the same sign between si and the fuzzy switching term ki, the first inequality in the above derivation is easily established by using the inequality ηi|si|≤siθsi*Tξsi(si)≤(ηi+ρi)|si|. Noticing (4.18), we have
(4.20)V˙1≤-∑i=1p‍ηi|si|<0.
Consequently, all signals in the system are bounded. Obviously, if e(0) is bounded, then e(t) is also bounded for all t. Since the reference trajectory xdis bounded, then the system state x(t) is bounded as well.
To complete the proof and establish asymptotic convergence of the tracking error, we need to prove thatS→0 as t→∞. We rearrange (4.16) as following:
(4.21)S˙=G(x)(ω-ω^-ΞeT(x)Θ~e-ΞsT(S)Θ~s-ΞsT(S)Θs*-δ(x)IpS).
Since G(x) and δ(x) are continuous functions in a compact set Dx, they are bounded. By using the boundness of S, ω^˙∈L∞ and in turn ω^∈L∞. Therefore, S˙∈L∞ holds. Using Barbalat's lemma, S→0(t→∞) holds. This completes the proof.Remark 4.3.
The tracking control using conventional adaptive fuzzy SMC with fuzzy switching control term [28–30] does not tackle the problem of attenuation of the effect of the fuzzy approximation error. We usually have V˙≤Sw-ηSTsign(S). In this case, it can be excepted that Sw should be very small if not equal to zero in [28–30] but here we get V˙≤-∑i=1pηi|si|<0, which improves the stability proof. In order to overcome this restriction, we have proposed ω^ to estimate the fuzzy approximation error ω.Remark 4.4.
In conventional adaptive fuzzy SMC design [24, 39, 40], there is often an assumption that the unknown external disturbance D(t) is bounded by a known positive constant. However, it is often not possible to obtain the bound in complex situations. In order to overcome this restriction, the variable D(t) is incorporated into the equivalent control term ueq and a fuzzy system is used to approximate ueq. Therefore, knowledge of the bound of external disturbance is not necessary in the present paper.


## 4.3. Direct Adaptive Fuzzy SMC with Variable Universe Fuzzy Switching Term (DAFSMC with VUFSW)

As stated in Section3.2, variable universe fuzzy system possesses high static precision by virtue of contraction-expansion factor. In this subsection, to accelerate response speed and improve the control accuracy, we consider an adaptive fuzzy SMC with VUFSW.Similar to the analysis in Section4.2, the variables si and ki are still taken as the input and the output variables of the variable universe fuzzy system. Then the variable universe fuzzy switching control law k^i of the ith subsystem can be written as
(4.22)k^i=βiζsi(si)Tθsi,
where ζsi(si)=[A1(si/αi(si)),…,AiL(si/αi(si))]T and θsi=[b1,b2,…,biL]T.Letβ*=diag(β1*,β2*,…,βp*) be the optimal parameter vector, where
(4.23)βi*=argmin|βi|≤Mi{supsi∈Dsi|ki-βiζsi(si)Tθsi|},
in which Mi is the design constant. Similarly, since the ideal parameter βi* is unknown, βi is employed to estimate βi*. Therefore, the overall control effort can be modified as
(4.24)ui=u^eqi+k^i+uci=ξeiT(x)θei+βiζsiT(si)θsi+ω^i
or
(4.25)u=u^eq+K^+uc=ΞeT(x)Θe+βΞsT(S)Θs+ω^,
where Ξs(S)=[diag(ζs1T(s1),ζs2T(s2),…,ζspT(sp))]T and Θs=[θs1T,θs2T,…,θspT]T.Substituting (4.25) into (4.1), we obtain
(4.26)G1(x)S˙=G1(x)v-F1(x)-(ΞeTΘe+βΞsT(S)Θs+ω^)-G1(x)D(t)=(ω-ω^)-(ΞeTΘe-ΞeTΘe*)-(βΞsT(S)Θs-β*ΞsT(S)Θs)-β*ΞsT(S)Θs-δ(x)IpS.
Let Θ~e=Θe-Θe* and β~=β-β*. Then
(4.27)G1(x)S˙=ω~-ΞeT(x)Θ~e-β~ΞsT(S)Θs-β*ΞsT(S)Θs-δ(x)IpS.To derive the adaptive law of the parameter vectors, we consider the Lyapunov candidate function as(4.28)V2=12STG1(x)S+12γ1Θ~eTΘ~e+12γ3β~Tβ~T+12γω~Tω~,
where γ3 is a positive constant.Theorem 4.5.
For system (2.1), nonlinear function matrices F(x), G(x), and vector D(t) are unknown, and Assumptions 2.1–2.3 hold. The control law is selected as (4.24) or (4.25), with adaptation laws (4.29) as follows:
(4.29)θ˙ei={γ1siξei(x),∥θei∥<Meior∥θei∥=Mei,γ1siθeiTξei(x)≤0,γ1siξei(x)-γ1siθeiθeiT∥θei∥2ξei(x),∥θei∥=Mei,γ1siθeiTξei(x)>0,β˙i={γ3siθsiTζsi(si),|βi|<Mior|βi|=Mi,γ3siθsiTζsi(si)≤0,0,|βi|=Mi,γ3siθsiTζsi(si)>0,ω^˙=γsi,
where Mei and Mi are defined as before. Then one can derive the performance as follows.(1)
The involved signals of the close loop are bounded.(2)
The tracking errors and their derivatives decay to zero asymptotically, in other words, whent→∞, ei(j)→0, j=0,1,…,ri-1, i=1,2,…,p.Proof.
Differentiating (4.28) with respect to t and using (4.27), we have
(4.30)V˙2=STG1(x)S˙+12STG˙1(x)S+1γ1Θ~eTΘ~˙e+1γ3β~Tβ~˙+1γω~Tω~˙=ST(ω~-ΞeT(x)Θ~e-β~ΞsT(S)Θs-β*ΞsT(S)Θs-δ(x)IpS)+12STG˙1(x)S+1γ1Θ~eTΘ~˙e+1γ3β~Tβ~˙+1γω~Tω~˙≤∑i=1p‍siω~i-1γ∑i=1p‍ω~iTω^˙i-∑i=1p‍siθ~eiTξei(x)+1γ1∑i=1p‍θ~eiTθ˙ei+12STG˙1(x)S-δ(x)∥S∥2-∑i=1p‍siβ~iθsiTζsi(si)+1γ3∑i=1p‍β~iβ˙i-∑i=1p‍ηi|si|≤1γ∑i=1p‍ω~i(γsi-ω^˙i)-1γ1∑i=1p‍θ~eiT(γ1siξei(x)-θ˙ei)-1γ3∑i=1p‍β~i(γ3siθsiTζsi(si)-β˙i)-∑i=1p‍ηi|si|.
Noticing that (4.29), we have
(4.31)V˙2≤-∑i=1p‍ηi|si|<0.
Therefore, all signals in the system are bounded. In order to show the boundedness of S˙, we rearrange (4.27) as follows:
(4.32)S˙=G(x)(ω~-ΞeT(x)Θ~e-β~ΞsT(S)Θs-β*ΞsT(S)Θs-δ(x)IpS).
Similarly, we can derive S˙∈L∞.To summarize the above analysis, the step-by-step procedures for the two direct adaptive fuzzy SMCs are proposed as follows.Design Procedure:Step 1.
Select proper positive coefficientsλ1,λ2,…,λp and learning coefficients γ1,γ2,γ3, and γ.Step 2.
Specify design constant vectorsMe=[Me1,Me2,…,Mep]T, Ms=[Ms1,Ms2,…,Msp]T and M=[M1,M2,…,Mp]T.Step 3.
Definemi fuzzy sets Fi for variable xi to achieve an uniform coverage of the universe of discourse. Select the initial parameter vector θei(0)=0_mi×1(i=1,2,…,p).Step 4.
Construct the fuzzy rule bases for the fuzzy systemuswi. Define iL fuzzy sets Ai for variable si to achieve a uniform coverage of the universe of discourse. Select the initial parameter vectors θsi(0)iL×1(i=1,2,…,p) for DAFSMC with FSW and the initial values βi(0)(i=1,2,…,p) for DAFSMC with VUFSW.Step 5.
Construct the fuzzy systemsu^eqi in (4.9), and k^i in (4.11) or (4.22).Step 6.
Construct the control law (4.13) or (4.14) with the adaptive laws in (4.18) or construct the control law (4.24) or (4.25) with the adaptive laws in (4.29).Step 7.
Use the adaptive laws (4.18) or (4.29) to adjust the parameters θei, θsi or βi, and ω^i.


## 5. Simulation Results and Analysis

In this section, we test the proposed control schemes on the trajectory tracking control of the two-link rigid robot manipulators moving a horizontal plane. The equations of motion of the manipulators can be expressed in matrix form as follows [21, 25]:
(5.1)(q¨1q¨2)=-(M11,M12M21,M22)-1(-hq˙2,-h(q˙1+q˙2)hq˙1,0)(q˙1q˙2)+(M11,M12M21,M22)-1(u1u2)+(d1d2),
where
(5.2)M11=a1+2a3cosq2+2a4sinq2,M22=a2,M21=M12=a2+a3cosq2+a4sinq2,h=a3sinq2-a4cosq2,
in which
(5.3)a1=I1+m1lc12+Ie+melce2+mel12,a2=Ie+melce2,a3=mel1lcecosδe,a4=mel1lcesinδe.In the simulation, the following parameter values of the plant are used:m1=1, me=2, l1=1, lc1=0.5, lce=0.6, I1=0.12, Ie=0.25, δe=30o. Let y=[q1,q2]T, u=[u1,u2]T, x=[q1,q˙1,q2,q˙2]T,
(5.4)F(x)=(f1(x)f2(x))=-(M11,M12M21,M22)-1(-hq˙2,-h(q˙1+q˙2)hq˙1,0)(q˙1q˙2),G(x)=(g11,g12g21,g22)=(M11,M12M21,M22)-1,D(t)=(d1d2).
Then, the robotic manipulators dynamics given by (5.1) can be expressed as
(5.5)y¨=F(x)+G(x)u+D(t).The object is to design control lawu to force the system output q1 and q2 to track the desired trajectories yd1=sint and yd2=sint, respectively. The initial state is selected as x0=[1,0,1,0]T. The fuzzy system k^i(i=1,2) in (4.9) have x=[q1,q˙1,q2,q˙2]T as inputs. For each input variable, we define three triangular membership functions uniformly distributed on the interval [-1,1]. si and ki are the input and the output of the fuzzy system u^swi(i=1,2) in (4.11) and (4.22), respectively. We define seven triangle membership functions uniformly distributed on the input domain [-2,2] and the output domain [-1,1], respectively. The used fuzzy rules are showed in Table 1, in which the fuzzy labels used in this study are negative big (NB), negative medium (NM), negative small (NS), zero (ZE), positive small (PS), positive medium (PM), and positive big (PB).Table 1
Fuzzy rule list of variable universe fuzzy switching control.
Inputsi
PB
PM
PS
ZE
NS
NM
NB
Outputki
PB
PM
PS
ZE
NS
NM
NBIn this paper, we consider that the nonlinear function matricesF(x) and G(x) are assumed to be completely unknown, that is, the design of the proposed controller does not require the knowledge of the system's model. Moreover, the external disturbance D(t) is also unknown. In fact, these functions are only required for simulation purpose. To make a fair comparison, in the simulation, we consider the direct adaptive fuzzy SMC proposed by A. Boulkroune et al. in [21] where a classical switching control term is employed to eliminate the fuzzy approximation error (hereafter referred to as a DAFSMC with CSW).In the whole simulation, the design parameters used are chosen as follows:λ1=λ2=5, γ1=1, γ2=8, γ3=8, γ=1, Me1=2, Me2=1, Ms1=50, Ms2=20, M1=M2=2, η1=13, η2=6 and the contraction-expansion factors αi=1-λexp(-ksi2)(i=1,2) with λ=0.95, k=1. The initial conditions of the online adjustable parameter vectors are selected as θei=[0_]81×1(i=1,2), θsi=[-3,-2,-1,0,1,2,3]T(i=1,2), and β(0)=diag(5,1).To verify the robust stability of the proposed schemes, external disturbances are chosen as square wave signald1=d2=square(2πt) [21]. The response curves, the tracking error curves, and sliding mode dynamic evolution curves under the aforementioned three control schemes are illustrated in Figures 1, 2, and 3, respectively. Their associated control efforts are illustrated in Figure 4. For evaluating numerically their tracking performance, the integral of the absolute error (IAE), the integral of the time multiplied by the absolute error (ITAE), and the integral of square value (ISV) of the control input are also considered because mere visual observation of response curve is not always enough to make a sound comparison. The corresponding tracking performance indices in first 20 seconds and in first 100 seconds are tabulated in Tables 2 and 3, respectively.Table 2
The performance indices in 20 seconds.
Controller
rod1
rod2
IAE (rad)
ITAE (rad · s)
ISV (N2)
IAE (rad)
ITAE (rad · s)
ISV (N2)
DAFSMC with CSW
0.8485
1.5820
3.3838
e
+
03
0.4191
0.3299
704.3263
DAFSMC with FSW
2.3991
12.8742
1.9102
e
+
03
1.1504
5.3901
253.8947
DAFSMC with VUFSW
0.3923
0.3867
2.9613
e
+
03
0.2852
0.3180
437.4319Table 3
The performance indices in 100 seconds.
Controller
rod1
rod2
IAE (rad)
ITAE (rad · s)
ISV (N2)
IAE (rad)
ITAE (rad · s)
ISV (N2)
DAFSMC with CSW
1.4397
33.2521
1.6863
e
+
04
0.4722
6.6922
3.5550
e
+
03
DAFSMC with FSW
3.9996
97.6183
9.0121
e
+
03
2.1688
63.2731
1.2830
e
+
03
DAFSMC with VUFSW
0.5073
7.2268
8.6772
e
+
03
0.3844
6.1505
1.2590
e
+
03Angular trajectory tracking curves of robotic manipulators.
(a)
Tracking curve ofq1
(b)
Tracking curve ofq2Angular trajectory tracking error curves of robotic manipulators.
(a)
Tracking error ofq1
(b)
Tracking error ofq2Sliding mode dynamic evolution curves.
(a)
Sliding surfaces1
(b)
Sliding surfaces2Control input torques of robotic manipulators.
(a)
Control effort ofq1
(b)
Control effort ofq2As shown in Figures1–4, the proposed two schemes can effectively achieve the trajectory tracking of the joint angles despite of the system uncertainties and external disturbances. Furthermore, they can effectively alleviate the chattering, which is the main disadvantage of general SMC. It can also be seen from Figures 1–3, the performance specifications of DAFSMC with VUFSW are much better than those of DAFSMC with CSW and DAFSMC with FSW in terms of reaching time, steady precision.Both IAE and ITAE are used as evaluating error performance, while the criterion ISV shows energy consumption. It is well known that there is a trade-off between error performance and energy consumption, that is, when IAE and ITAE are improved, ISV becomes worse, and vice versa. Conservative control input is often required to guarantee the stability of the control system in DAFSMC with CSW scheme. Therefore, DAFSMC with CSW expends relatively more energy to achieve the tracking task than DAFSMC with FSW. It also implies that the indices IAE and ITAE of DAFSMC with FSW become worse. But DAFSMC with VUFSW simultaneously improves IAE and ITAE as well as ISV as compared to the others as stated in Tables2 and 3. It is undeniable that DAFSMC with VUFSW expends more energy than DAFSMC with FSW in the initial stage due to its fast response speed. However, thanks to the high precision of the variable universe fuzzy control, its energy consumption reduces quickly in the steady state.Remark 5.1.
As already stated in Section1, in the presence of model uncertainties and external disturbances, the two disadvantages of general SMC cannot be overcome just using fuzzy switching technique or adaptive fuzzy SMC. Combining adaptive fuzzy SMC with fuzzy or variable universe fuzzy switching technique, a novel framework for designing direct adaptive fuzzy SMC is developed for a class of MIMO nonlinear systems in this study. The derived schemes can effectively overcome these disadvantages of the general SMC.Remark 5.2.
Compared with [22, 24, 28–30, 38, 41], our proposed control schemes have several advantages as follows. (i) The proposed schemes can simultaneously solve the chattering and the calculation of the equivalent control in the presence of model uncertainties and unknown disturbances. (ii) The constraint on the control gain matrix is relaxed. In this study, the gain matrix is just required to be positive definite symmetric and the inverse of its derivative is bounded by an unknown function. (iii) An adaptive compensation term is appended to remove the assumption on fuzzy approximation error in the stability proof. (iv) The derived DAFSMC with VUFSW achieves better performances than the others in terms of response speed, steady accuracy, IAE, ITAE, and ISV.


## 6. Conclusion

A novel framework is developed to design a direct adaptive fuzzy SMC for a class of MIMO nonlinear systems with model uncertainties and unknown disturbances. Combining adaptive fuzzy SMC with fuzzy or variable universe fuzzy switching technique, this study proposes two novel direct adaptive fuzzy SMC schemes. The derived schemes effectively overcome the two disadvantages of general SMC. Besides, the constraint on the control gain matrix and the fuzzy approximation error are relaxed. Future works will focus on the extension of the framework to more general MIMO nonlinear systems such as the continuous-time or the discrete-time nonaffine nonlinear systems.



---

*Source: 543039-2012-12-18.xml*