# Fuzzy Adaptive Prescribed Performance Control for a Class of Uncertain Nonlinear Systems with Unknown Dead-Zone Inputs

Authors: Wei Xiang; Ning Li; Yeguo Sun
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

This paper proposes a fuzzy adaptive prescribed performance control scheme for a class of uncertain chaotic systems with unknown control gains and unknown dead-zone inputs. Firstly, an error transformation is introduced to transform the original constrained system into an equivalent unconstrained one. Then, based on the error transformation technique and the predefined performance technique, a fuzzy adaptive feedback control method is developed. It is shown that all the signals of the resulting closed-loop system are bounded. Finally, an illustrative example is given to demonstrate the effectiveness and usefulness of the proposed technique.


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## Body

## 1. Introduction

As a hard nonlinearity, dead-zone has been found in many industrial processes. For example, dry friction or stiction is a common source of dead-zone nonlinearities in electromechanical systems, and temperature changes on the surface of these components can produce relevant variations of dead-zone effects [1]. If dead-zone nonlinearities are neglected [2], they will limit the closed-loop system performance and lead to instability. So, these nonlinearities are particularly harmful. Many researchers employed many methods to improve the performances of control systems with dead-zone inputs. The most common approaches are adaptive schemes [3–6], fuzzy systems [7–14], neural networks [15–18], and sliding mode control [19–22]. In order to compensate the negative effects of the dead-zone nonlinearity, an inverse dead-zone as a method is used. But this method leads to a discontinuous control law and requires instantaneous switching, which in practice cannot be accomplished with mechanical actuators. To overcome this limitation, smooth inverses are adopted in [23, 24]. Another method was proposed by Lewis et al. [8] and adopted by Wang et al. [25]. In both works, the dead-zone is treated as a combination of a linear and a saturation function. In [4], Boulkroune and M’Saad proposed fuzzy adaptive observer-based approach to deal with the practical projective synchronization problem for a class of chaotic systems involving dead-zone in the control channel. Wu et al. [5] introduced a smooth inverse of the dead-zone to compensate the effect of the dead-zone in controllers design and proposed an adaptive sliding mode control law to achieve spacecraft attitude tracking problem. By using a four-dimensional energy resource demand supply system, an adaptive neural networks control approach is presented in [17]. The approach not only makes the states of two chaotic systems asymptotic synchronization but also achieved better control performances. In [19], an adaptive fuzzy control scheme is proposed for a class of uncertain multi-input multioutput (MIMO) nonlinear systems with the nonsymmetric control gain matrix and the unknown dead-zone inputs. Boulkroune and M’Saad [20] developed a fuzzy adaptive variable-structure control scheme for a class of uncertain MIMO chaotic systems with both sector nonlinearities and dead-zones. In order to realize the robust compensator, most of the aforementioned control schemes are obtained with the restriction that the control gains are known in advance. However, this assumption does not appear to be realistic in a general case [26]. So it is more advisable to take the effects of the unknown control gains and unknown dead-zone inputs into account for uncertain nonlinear systems.Recently, Bechlioulis and Rovithakis [27] proposed prescribed performance control (PPC) scheme for uncertain nonlinear system. Utilizing a transformation function, the original controlled system is transformed into a new one. If the uniform boundedness of the states of the latter is ensured, we can solve the stability problem for the former. The paper [28] established a control scheme to control unknown pure feedback systems of known high relative degree, exhibiting prescribed performance with respect to trajectory oriented metrics. For nonlinear large-scale systems, Li and Tong [29] employed prescribed performance adaptive fuzzy output-feedback control method to ensure that all the signals in the closed-loop system are bounded. Meanwhile, Li and Tong [30] proposed an adaptive fuzzy output constrained control design approach for MIMO uncertain stochastic nonlinear systems. Sun and Liu [31] presented a fuzzy adaptive control method for MIMO uncertain chaotic systems, which is capable of guaranteeing the prescribed performance bounds. However, the main limitation in [31] is that the effect of both unknown control gains and unknown dead-zone inputs for uncertain nonlinear systems has not been taken into account.To the author’s best knowledge, there are few literatures dealing with the prescribed performance control problem with unknown dead-zone inputs. Inspired by the works in [32], we investigate the tracking control with guaranteed prescribed performance for uncertain nonlinear systems. Compared with related works, there are three main contributions that are worth to be emphasized:(1)
Compared with the results in [31], the uncertain chaotic system with unknown control gain and unknown dead-zone input is considered.(2)
The prescribed performance function (PPF) is incorporated into the control design.(3)
The proposed control law avoids the chattering phenomenon.Motivated by the aforementioned works, this paper focuses on the problem of adaptive fuzzy control for a class of uncertain nonlinear systems with unknown control gains and unknown dead-zone inputs. Based on Lyapunov function, it is proved that all the signals of the closed-loop system are bounded and that the tracking error remains an adjustable neighborhood of the origin with the prescribed performance bounds.The organization of this paper is described as follows. In the next section, system model is derived, and the assumptions are also given. In Section3, the design of the proposed control strategies is discussed. The simulation results are presented to demonstrate the effectiveness of proposed control scheme in Section 4. Conclusion is presented in Section 5.


## 2. System Descriptions and Problem Formulations

Consider the following nonlinear system:(1)x˙1=f1t,x+g1t,xϕ1u1t,x˙2=f2t,x+g2t,xϕ2u2t,⋮x˙n=fnt,x+gnt,xϕnunt,where x=[x1,x2,…,xn]T is the system state vector which is assumed to be available for measurement. u=[u1,u2,…,un]T is the control input vector and f(t,x)=[f1(t,x),f2(t,x),…,fn(t,x)]T is the unknown continuous nonlinear function, g(t,x)=[g1(t,x),g2(t,x),…,gn(t,x)]T is unknown control gain, and ϕi(ui(t)) is the output of the dead-zone nonlinearity, i=1,2,…,n.The dead-zone characteristic can be described as follows:(2)ϕiuit=miuit-b1i,foruit≥b1i0,for-b2i<uit<b1imiuit+b2i,foruit≤-b2i,where mi stand for the right and the left slope, b1i and b2i represent the breakpoints of the input nonlinearity, and the following assumption is given.Assumption 1.
The dead-zone parameters:mi>0, b1i and b2i are unknown bounded constants.Obviously, the output of the dead-zone can be rewritten as(3)ϕiuit=miuit+diuit,where di(ui(t)) can be calculated as (4)diuit=-mib1i,foruit≥b1i-miuit,for-b2i<uit<b1imib2i,foruit≤-b2i.From Assumption 1, there exists an unknown positive constant di∗ such as diuit≤di∗,  i=1,2,…,n.Now, letΦ(u)=[ϕ1(u1(t)),ϕ2(u2(t)),…,ϕn(un(t))]T, D(u)=[d1(u1(t)),d2(u2(t)),…,dn(un(t))]T, and M=diag(m1,m2,…,mn). From (1) and (3), we have (5)x˙=ft,x+diaggt,xΦu,Φu=Mu+Du.So, we can obtain (6)x˙=ft,x+diaggt,xMu+diaggt,xDu.The objective of this paper is to construct a controller for system (1) such that the system state x tracks the reference signal xd∈Rn and all the signals in the closed-loop system remain bounded.To meet the objective, we make the following reasonable assumptions.Assumption 2.
f
(
t
,
x
) and diag(g(t,x)) are unknown but bounded.Assumption 3.
The matrixdiag(g(t,x)) is nonsingular.Assumption 4.
The desired trajectoryxid is a known bounded differentiable function, i=1,2,…,n.Remark 5.
In [19], gi(t,x) is assumed as gi(t,x)≥ιi, and ιi is positive constant, i=1,2,…,n. In this paper, we relax this condition and only suppose diag(g(t,x)) is an unknown nonsingular matrix.Letxd=[x1d,x2d,…,xnd]T, and the tracking error e=x-xd=[e1,e2,…,en]T. So we can obtain the following error dynamical system:(7)e˙=ft,x+diaggt,xMu+diaggt,xDu-x˙d.Usually, we employ sliding mode control scheme [18] to stabilize the error system (7). Firstly, a sliding mode surface is designed as follows: (8)σ=e+∫0tKeτdτ,where K=diag(k1,k2,…,kn), ki>0, i=1,2,…,n. Differentiating σ with time yields (9)σ˙=ft,x+diaggt,xMu+diaggt,xDu-x˙d+Ke.If the nonlinear functions f(t,x) and diag(g(t,x)) and the parameters mi and di∗ are all known, we employ the control law (10)u=-diaggt,xM-1ft,x-x˙d+Ke+Kσ+u¯s,where u¯s=[g1t,xd1∗·sign(σ1),g2t,xd2∗·sign(σ2),…,gnt,xdn∗·sign(σn)]T.Consider the Lyapunov functionV0=(1/2)σTσ. Using (10) yields (11)V˙0=σTσ˙=-σTKσ-∑i=1nσigit,xdi∗-σigit,xdiui.One obtains V˙0≤-σTKσ≤0. We can conclude that σ→0 as t→∞. However, the control law (10) is implemented in cases where f(t,x), diag(g(t,x)), mi, and di∗ are all known. And us may cause chattering phenomena. Meanwhile, the inverse of diag(g(t,x))M cannot be calculated easily.In order to overcome these limitations, we adopt the fuzzy adaptive prescribed performance control scheme to ensure that all the signals are bounded in probability and the system statex(t) can track the given reference xd with the given prescribed performance bounds.


### 2.1. Prescribed Performance

According to [14, 15], the prescribed performance is achieved by ensuring that tracking error e=[e1,e2,…,en]T evolves strictly within predefined decaying bounds as follows: (12)-δiminμit<eit<δimaxμit,t≥0,i=1,2,…,n,where δimin and δimax are design constants and the performance functions μi(t) are bounded and strictly positive decreasing smooth functions and limt→∞μi(t)=μi∞>0. Choosing the performance function μi(t) and the constants δimin and δimax appropriately determines the performance bounds of the error ei, i=1,2,…,n.To represent (11) by an equality form, we employ an error transformation as (13)eit=μitsizi,i=1,2,…,n,where zi is the transformed error, and si(·) is smooth, strictly increasing function, satisfying the following condition: (14)-δimin≤sizi≤δimax,limzi→-∞sizi=-δimin,limzi→+∞sizi=δimax.Note thatsi(zi) are strictly increasing functions; we have (15)zi=si-1eitμit,i=1,2,…,n.Differentiating (15) with respect to time yields (16)z˙i=∂si-1∂eit/μit1μitfit,x+git,xmiui+git,xdiui-x˙di-eitμ˙itμit.Let (17)ri=∂si-1∂eit/μit1μit>0,hi=-x˙di-eitμ˙itμit.Then (16) can be rewritten as (18)z˙i=rifit,x+git,xmiui+git,xdiui+hi,i=1,2,…,n.Let z=[z1,z2,…,zn]T,  H=[h1,h2,…,hn]T; then (18) can be written into the following form: (19)z˙=diagrft,x+diaggt,xMu+diaggt,xDu+H,where diag(r)=diag(r1,r2,…,rn).Remark 6.
Usually, we choosesi(zi)=tanh⁡(zi)=(δimaxezi-δimine-zi)/(ezi+e-zi). So, we can calculate that ri=(1/(λi+δimin)+1/(δimax-λi))/2μ>0, λi=ei(t)/μi(t).
Iffi(t,x),  gi(t,x),  mi, and di∗ are all known, the following controller (20)u=-diaggt,xM-1ft,x+H+Kz+us,where us=[signz1g1t,xd1∗, signz2g2t,xd2∗,…,signzngnt,xdn∗]T, and K=diag(k1,k2,…,kn) with ki>0 (i=1,2,…,n), can meet the control objective. Indeed, consider the Lyapunov function V1=(1/2)zTz. Using (19) and (20) yields V˙1=zTz˙≤0. According to the above inequality, V1 is always negative, which implies that zi∈L∞. Then, according to the properties of function si(zi), we know that -δimin<si(zi)<δimax, which implies -δiminμi(t)<ei(t)<δimaxμi(t). Then, one can conclude that tracking control of system (2) is achieved.
However, the termus is discontinuous, which may cause chattering phenomena, and inverse matrix (diag(g(t,x))M)-1 cannot be calculated easily. So, motivated by [11, 12], we modify controller (20) as follows: (21)u=-zzTdiagrft,x+zTdiagrH+zTdiagrKz+z¯Tdiagru¯szTdiagrdiaggt,xMz,where z¯=[z1,z2,…,zn]T, u¯s=[g1t,xd1∗,g2t,xd2∗,…,gnt,xdn∗]T. The control objective can also be realized.Remark 7.
In controller (21), we do not need to calculate the matrix (diag(g(t,x))M)-1.Note thatf(t,x), diag(g(t,x)), M, and u¯s are unknown in (21); we need to use fuzzy logic system to approximate the nonlinear unknown functions.


### 2.2. Fuzzy Logic Systems

The basic configuration of a fuzzy logic system consists of a fuzzifier, some fuzzy IF-THEN rules, a fuzzy inference engine, and a defuzzifier. The fuzzy inference engine uses the fuzzy IF-THEN rules to perform a mapping from an input vectorx=[x1,x2,…,xn]T∈Rn to an output α(x)∈R. The ith fuzzy rule is written asRulei: ifx1isF1iand…andxnisFni  then αxisαi,where F1i,F2i,…, and Fni are fuzzy sets and αi is the fuzzy singleton for the output in the ith rule. By using the singleton fuzzifier, product inference, and the center-average defuzzifier, the output of the fuzzy system can be expressed as follows: (22)αx=∑j=1Nαj∏i=1nμFijxi∑j=1N∏i=1nμFijxi=θTψx,where μFij(xi) is the degree of membership of xi to Fij, N is the number of fuzzy rules, θ=[α1,…,αN]T is the adjustable parameter vector, and ψ(x)=[p1(x),p2(x),…,pN(x)]T, where (23)pjx=∏i=1nμFijxi∑j=1N∏i=1nμFijxiis the fuzzy basis function. It is assumed that fuzzy basis functions are selected so that there is always at least one active rule.


## 2.1. Prescribed Performance

According to [14, 15], the prescribed performance is achieved by ensuring that tracking error e=[e1,e2,…,en]T evolves strictly within predefined decaying bounds as follows: (12)-δiminμit<eit<δimaxμit,t≥0,i=1,2,…,n,where δimin and δimax are design constants and the performance functions μi(t) are bounded and strictly positive decreasing smooth functions and limt→∞μi(t)=μi∞>0. Choosing the performance function μi(t) and the constants δimin and δimax appropriately determines the performance bounds of the error ei, i=1,2,…,n.To represent (11) by an equality form, we employ an error transformation as (13)eit=μitsizi,i=1,2,…,n,where zi is the transformed error, and si(·) is smooth, strictly increasing function, satisfying the following condition: (14)-δimin≤sizi≤δimax,limzi→-∞sizi=-δimin,limzi→+∞sizi=δimax.Note thatsi(zi) are strictly increasing functions; we have (15)zi=si-1eitμit,i=1,2,…,n.Differentiating (15) with respect to time yields (16)z˙i=∂si-1∂eit/μit1μitfit,x+git,xmiui+git,xdiui-x˙di-eitμ˙itμit.Let (17)ri=∂si-1∂eit/μit1μit>0,hi=-x˙di-eitμ˙itμit.Then (16) can be rewritten as (18)z˙i=rifit,x+git,xmiui+git,xdiui+hi,i=1,2,…,n.Let z=[z1,z2,…,zn]T,  H=[h1,h2,…,hn]T; then (18) can be written into the following form: (19)z˙=diagrft,x+diaggt,xMu+diaggt,xDu+H,where diag(r)=diag(r1,r2,…,rn).Remark 6.
Usually, we choosesi(zi)=tanh⁡(zi)=(δimaxezi-δimine-zi)/(ezi+e-zi). So, we can calculate that ri=(1/(λi+δimin)+1/(δimax-λi))/2μ>0, λi=ei(t)/μi(t).
Iffi(t,x),  gi(t,x),  mi, and di∗ are all known, the following controller (20)u=-diaggt,xM-1ft,x+H+Kz+us,where us=[signz1g1t,xd1∗, signz2g2t,xd2∗,…,signzngnt,xdn∗]T, and K=diag(k1,k2,…,kn) with ki>0 (i=1,2,…,n), can meet the control objective. Indeed, consider the Lyapunov function V1=(1/2)zTz. Using (19) and (20) yields V˙1=zTz˙≤0. According to the above inequality, V1 is always negative, which implies that zi∈L∞. Then, according to the properties of function si(zi), we know that -δimin<si(zi)<δimax, which implies -δiminμi(t)<ei(t)<δimaxμi(t). Then, one can conclude that tracking control of system (2) is achieved.
However, the termus is discontinuous, which may cause chattering phenomena, and inverse matrix (diag(g(t,x))M)-1 cannot be calculated easily. So, motivated by [11, 12], we modify controller (20) as follows: (21)u=-zzTdiagrft,x+zTdiagrH+zTdiagrKz+z¯Tdiagru¯szTdiagrdiaggt,xMz,where z¯=[z1,z2,…,zn]T, u¯s=[g1t,xd1∗,g2t,xd2∗,…,gnt,xdn∗]T. The control objective can also be realized.Remark 7.
In controller (21), we do not need to calculate the matrix (diag(g(t,x))M)-1.Note thatf(t,x), diag(g(t,x)), M, and u¯s are unknown in (21); we need to use fuzzy logic system to approximate the nonlinear unknown functions.


## 2.2. Fuzzy Logic Systems

The basic configuration of a fuzzy logic system consists of a fuzzifier, some fuzzy IF-THEN rules, a fuzzy inference engine, and a defuzzifier. The fuzzy inference engine uses the fuzzy IF-THEN rules to perform a mapping from an input vectorx=[x1,x2,…,xn]T∈Rn to an output α(x)∈R. The ith fuzzy rule is written asRulei: ifx1isF1iand…andxnisFni  then αxisαi,where F1i,F2i,…, and Fni are fuzzy sets and αi is the fuzzy singleton for the output in the ith rule. By using the singleton fuzzifier, product inference, and the center-average defuzzifier, the output of the fuzzy system can be expressed as follows: (22)αx=∑j=1Nαj∏i=1nμFijxi∑j=1N∏i=1nμFijxi=θTψx,where μFij(xi) is the degree of membership of xi to Fij, N is the number of fuzzy rules, θ=[α1,…,αN]T is the adjustable parameter vector, and ψ(x)=[p1(x),p2(x),…,pN(x)]T, where (23)pjx=∏i=1nμFijxi∑j=1N∏i=1nμFijxiis the fuzzy basis function. It is assumed that fuzzy basis functions are selected so that there is always at least one active rule.


## 3. Main Results

Letdiag(g¯(t,x))=diag(g(t,x))M=diag(g1(t,x)m1,g2(t,x)m2,…,gn(t,x)mn). By applying the introduced fuzzy systems, approximation of functions fi(t,x), gi(t,x)mi, and git,xdi∗(ui) can be expressed as follows: (24)f^ix,θfi=θfiTψfix,g¯^ix,θgi=θgiTψgix,g^ix,θgi=θgiTψgix,i=1,2,…,n.Optimal parameters θfi∗, θgi∗, and θgi∗ can be defined such that (25)θfi∗=argminθfisup⁡fit,x-f^ix,θfi,θgi∗=argminθgisup⁡git,xmi-g¯^ix,θgi,θgi∗=argminθgisup⁡git,xdi∗-g^ix,θgi,i=1,2,…,n.Define the parameter estimation errors and the fuzzy approximation errors as follows:(26)θ~fi=θfi∗-θfi,θ~gi=θgi∗-θgi,θ~gi=θgi∗-θgi,εfix=fit,x-fix,θfi∗,εgix=git,xmi-gix,θgi∗,εgix=git,xdi∗-gix,θgi∗,i=1,2,…,n.Assumption 8.
ε
f
i
(
x
), εgi(x), and ε|gi|(x) are all bounded, respectively.Denotef^(x,θf∗) = [f^1(x,θf1∗),f^2(x,θf2∗),…,f^n(x,θfn∗)]T, diag(g¯^(x,θg∗)) = diag(g¯^1(x,θg1∗),g¯^2(x,θg2∗),…,g¯^n(x,θgn∗)), and u-^s∗ = [g^1(x,θ|g1|∗),g^2(x,θ|g2|∗),…,g^n(x,θ|gn|∗)]T.The controller can be constructed as(27)u=zu0,where (28)u0=zTdiagrf^x,θf+zTdiagrH+zTdiagrKz+z¯Tdiagru¯^s+ur-zTdiagrdiagg¯^x,θgz+μzTdiagrz+ur2,where μ=diag(g¯^(x,θg))+ϵ, ϵ is a small positive constant, ur is a compensated controller, which will be designed later, and K=diag(k1,k2,…,kn), ki>0,  i=1,2,…,n.Substituting (27) and (28) into zTz˙ yields (29)zTz˙=-zTdiagrKz+zTdiagrft,x-f^x,θf+zTdiagrdiagg¯t,x-diagg¯^x,θgzu0+μzTdiagrzu0+ur2u0-ur+zTdiagrdiaggt,xDu-z¯Tdiagru¯^s.Note thatzTdiag(r)diag(g(t,x))D(u)≤z¯Tdiag(r)u¯s. So, (21) can be written as follows: (30)zTz˙=-∑i=1nrikizi2+∑i=1nriziθ~fiTψfix+∑i=1nrizi2u0θ~giTψfix+∑i=1nriziθ~giTψgix+∑i=1nrizigit,xdiui-git,xdi∗-ur+u0ur2+Λ,where Λ=∑i=1nriziεfi+∑i=1nrizi2u0εgi+∑i=1nri|zi|ε|gi|+∑i=1nμrizi2u0.According to Assumption8, there exist unknown positive constants εf, εg, and ε|g| such that(31)maxiεfi≤εf,maxiεgi≤εg,maxiεgi≤εg.So, we have (32)Λ≤εf+εg∑i=1nrizi+εg+μ∑i=1nrizi2u0.Let εf+ε|g|=ε1,  εg=ε2. To generate the approximations f(t,x), diag(g¯(t,x)),  u¯s, ε1, and ε2 online, we choose the following adaptation laws: (33)θ˙fi=κfiriziψfix,θ˙gi=κgirizi2u0ψgix,θ˙gi=κgiriziψgix,ε^˙1=κε1∑i=1nrizi,ε^˙2=κε2∑i=1nrizi2u0,where ε^1, ε^2 are the estimates of ε1 and ε2, respectively. κfi,  κgi,  κ|gi|,  κε1, and κε2 are positive constants, i=1,2,…,n. And ur is designed as (34)u˙r=-u0ur+1-Πurur2+ν2,ν˙=-Πνur2+ν2,where Π=ε^1∑i=1nri|zi|+(ε^2+μ)∑i=1nrizi2|u0|.So, we obtain the following theorem.Theorem 9 (consider system (19)).
Suppose that Assumptions1–8 are satisfied. Then controller (27) with the adaptation law given by (33) can guarantee all signals in the closed-loop system are bounded in probability, and the tracking error e(t) remains in a neighborhood of the origin within the prescribed performance bounds for all t≥0.Proof.
Consider a Lyapunov function asV=V2+V3, where (35)V2=12zTz+∑i=1n1κfiθ~fiTθ~fi+∑i=1n1κgiθ~giTθ~gi+∑i=1n1κgiθ~giTθ~gi,V3=12ur2+ν2+1κε1ε~12+1κε2ε~22,where ε~1=ε1-ε^1, ε~2=ε2-ε^2.
The time derivative ofV2 is given by (36)V˙2=zTz˙-∑i=1n1κfiθ~fiTθ˙fi+1κgiθ~giTθ˙gi+1κgiθ~giTθ˙gi.Substituting (33) into (36), we have (37)V˙2=-∑i=1nrikizi2+∑i=1nrizigit,xdiui-git,xdi∗-ur+u0ur2+Λ≤-∑i=1nrikizi2-ur+u0ur2+Λ.Notice that (38)Λ≤ε1∑i=1nrizi+ε2+μ∑i=1nrizi2u0=ε~1∑i=1nrizi+ε~2∑i=1nrizi2u0+Π.Substituting (38) into (37), one can obtain (39)V˙2≤-∑i=1nrikizi2-ur+u0ur2+ε~1∑i=1nrizi+ε~2∑i=1nrizi2u0+Π.
The time derivative ofV3 is (40)V˙3=uru˙r+νν˙-1κε1ε~1ε^˙1-1κε2ε~2ε^˙2.Substituting (34) into (40), one gets (41)V˙3=-ur2u0+ur-ε~1∑i=1nrizi-ε~2∑i=1nrizi2u0-Π.Combining (40) and (41) gives (42)V˙t≤-∑i=1nrikizi2.Therefore, according to Lyapunov theorem, V(t) is always negative, so, V(t) is uniformly ultimately bounded; thus, the transformed error zi is bounded; that is, zi∈L∞. Then, according to the properties of function si(zi), we know that -δimin<si(zi)<δimax. Then, one can conclude that tracking control of system (7) with prescribed error performance (4) is achieved. This completes the proof.Remark 10.
Compared with the results in [15], the unknown dead-zone inputs are considered in the paper. Meanwhile, the control law (27) can avoid the singular problem.


## 4. Numerical Simulations

In this section, the Genesio chaotic system [33] is also used to illustrate the effectiveness of the proposed control scheme. The uncertain Genesio chaotic system is described: (43)x˙1=x2+d1t,x︸f1t,x+g1t,xϕ1u1t,x˙2=x3+d2t,x︸f2t,x+g2t,xϕ2u2t,x˙3=-6x1-2.92x2-1.2x3+x12+d3t,x︸f3t,x+g3t,xϕ3u3t,where d1(t,x)=2+2sin⁡(2t), d2(t,x)=3-cos⁡(x2), d3(t,x)=0.2+3sin⁡(4t), g1(t,x)=2-cos⁡(t), g2(t,x)=2-cos⁡(2t), g3(t,x)=2-sin⁡(x2), mi=2, b1i=1.5, b2i=-7, i=1,2,3.Firstly, we employ the sliding mode control scheme (see [32]) to control system (43). We assume that the desired trajectory is xd=[x1d,x2d,x3d]=[sin⁡t,sin⁡t,sin⁡(t)]T. Let e1=x1-x1d, e2=x2-x2d, e3=x3-x3d, and diag(g(t,x))≥δ; δ is a positive constant. So, the error dynamic system can be rewritten as follows:(44)e˙1=f1t,x-x˙1d+g1t,xϕ1u1t,e˙2=f2t,x-x˙2d+g2t,xϕ1u2t,e˙3=f3t,x-x˙3d+gnt,xϕ1u3t.The sliding surface s=[s1,s2,s3]T is designed as follows: (45)s1=e1+∫0te1τdτ,s2=e2+∫0te2τdτ,s3=e3+∫0te3τdτ.Let f¯(t,x)=f(t,x)+diag(g(t,x))D(u), diag(g¯(t,x))=diag(g(t,x))M. And the control scheme for error system (24) is designed as (46)ui=ui1+ui2,ui1=-g¯^ix,θg¯iϵ+g¯^ix,θg¯i2-1ei-xid+f¯^ix,θf¯i+kisignsi,ui2=-1δε1+ε2ui1+ϵ+g¯^ix,θg¯i2-1ei-xid+f¯^ix,θf¯i+kisignsisignsi,θ˙f¯i=κf¯iψf¯isi,θ˙g¯i=κg¯iψg¯iui1si,i=1,2,3,where ϵ is a small positive constant. The initial values of the chaotic system are [x1(0),x2(0),x3(0)]T=[-1,-2,1]T. The design parameters are chosen as follows: κf¯i=κg¯i=3,  i=1,2,3., k1=k2=k3=2, ϵ=0.01, δ=1, The initial conditions for the adaptive parameters are selected as θf¯i=θg¯i=0.01,  i=1,2,3. By using the sliding mode control scheme (46), the simulation result is shown in Figure 1.Figure 1
Time responsese1, e2, and e3 of system (44) by using the control scheme (46).From Figure1, the error states are beyond the preset boundary y(t) in the previous stage, where y1(t)=3.17e-1.17t+0.2.The transformation functions aresi(zi)=(2/π)arctan⁡(zi), i=1,2,3. We define three membership functions uniformly distributed on the interval [-2,2]. δimin=δimax=1. Firstly, according to the proposed control scheme (27), we give a block diagram (see Figure 2).Figure 2
A block diagram for the proposed control scheme (27).The simulation result is shown in Figure3 by using the control scheme (27). In order to improve the control effect, we modify y1(t) as y2(t)=3.17e-1.17t+0.05. Compared with Figure 3, the tracking errors are improved in Figure 4. From Figure 5, we can see that the chatter phenomenon is eliminated.Figure 3
Time responsese1,  e2, and e3 of system (44) by using the present control scheme (27) with y1(t).Figure 4
Time responsese1,  e2, and e3 of system (44) by using the present control scheme (27) with y2(t).Figure 5
Time responsesϕ1(u1),  ϕ2(u2), and ϕ3(u3) of system (44) by using the present control scheme (27).From the simulation results in Figures4 and 5, we know that the proposed control scheme can guarantee that all the error states are bounded. Moreover, the tracking errors can remain within the prescribed performance bounds all the time without showing chatter phenomenon. So, the proposed control scheme in this paper can achieve the objective.


## 5. Conclusions

In this paper, a fuzzy adaptive prescribed performance control scheme has been developed for a class of uncertain nonlinear systems with unknown control gains and unknown dead-zone inputs. By using fuzzy logic systems and the prescribed performance technique, the stability of the closed-loop system has been improved. Simulation results have shown the effectiveness of the proposed scheme.



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*Source: 4386515-2017-03-26.xml*