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

Authors: Chunzhi Yang; Song Xu; Hui Lv
Journal: Journal of Mathematics
 (2020)

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## Abstract

Based on fuzzy logic systems and prescribed performance technology, this paper investigated the stable tracking problem of a class of nonlinear systems with unknown dead-zone inputs. Firstly, the original system is transformed into an equivalent system. Then, a Nussbaum-type function is added in the predefined performance controller to estimate the unknown control direction, and the boundedness of all signals in the closed-loop system is guaranteed. In addition, tracking errors will always remain within the prescribed performance boundaries. Finally, a comparative simulation example shows the effectiveness of the method proposed in this paper.


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## Body

## 1. Introduction

As is known to all, the dead zone may badly limit the system performance, such as electronic circuits and electric servomotors [1–5]. So, many researchers used different control methods (i.e., backstepping technique, fuzzy systems, neural networks, sliding mode control, and so on) to research the stability of nonlinear systems with unknown control directions [6–9]. For example, based on the terminal sliding mode control, Chen and Ren in [6] proposed an adaptive neural control scheme for permanent magnet synchronous motor (PMSM) servo system with dead-zone input. For uncertain discrete-time systems with nonaffine dead-zone input, Liu and Tong [7] used adaptive neural networks tracking control such that the tracking error converges to a small neighborhood of zero. By using high-dimensional integral Lyapunov–Krasovskii functional theory and direct adaptive control scheme, control of uncertain MIMO systems with time-varying delay and dead-zone input was considered in [8]. In [9], the authors investigated an adaptive backstepping control method for an SISO system with unknown dead-zone input. For eliminating the influence of unknown control directions, one of the commonly applied methods is the Nussbaum-type function technique [10–12]. Liu et al. [10] constructed an adaptive neural controller for discrete-time system, which made all signals of the closed-loop system bounded in the condition of unknown dead-zone input. For a class of MIMO stochastic systems in [11], Li and Tong put forward an adaptive fuzzy output-feedback control method by using Nussbaum gain function and fuzzy logic systems. Xu [12] applied the adaptive neural control scheme to guarantee that the flexible hypersonic flight vehicle system is bounded stable with dead-zone input.However, control methods in aforementioned literature can only guarantee that the tracking error converges to a small residual set, rather than a small residual set with the prescribed performance bounds. For an uncertain nonlinear system [4, 13–15], in order to solve the above problem, the prescribed performance control (PPC) strategy has been proposed. The idea of this method is converting the original system to an equivalent system and ensuring the boundedness of the states of the equivalent system. For example, based on the backstepping technique and the PPC, Wang and Hu [13] proposed an improved PPC for the longitudinal model of an air-breathing hypersonic vehicle (AHV) with unknown dead-zone input nonlinearity. Yang and Chen [14] developed an adaptive neural PPC for near space vehicles (NSVs) with input nonlinearities including saturation and dead zone. It can be seen from the literature [13, 14] that it has certain practical significance to adopt the PPC method in the practical system. For MIMO nonlinear systems with unknown dead-zone inputs, Shi et al. [4] proposed an adaptive fuzzy PPC scheme such that only three parametric adaptive rules were needed. Cao et al. [16] studied a fractional neural network model with impulse amplitude, time-delay time-varying, and reaction diffusion terms. Many other interesting results about PPC of nonlinear systems can be seen in [17–19].Inspired by the work of [4, 20], this paper explores the stability problem that the tracking error meets prescribed performance boundary (PPB) with unknown dead-zone input. Compared with related works, the main contributions of this paper are listed as follows: (1) compared with the work of [14, 17, 19], the introduced error transformation variable zi can be directly differentiated without the participation of the transformation function. (2) The Nussbaum-type function is incorporated into the control method so that the influence of unknown control directions is eliminated. (3) The singular problem in the PPC for nonlinear systems can be avoided by using the proposed method.The structure of this paper is as follows. In Section2, the uncertain system model with unknown dead-zone inputs is raised and provides assumption and lemmas. Section 3 includes the designation of fuzzy adaptive prescribed performance control method. The results of simulation are presented in Section 4. In Section 5, a brief conclusion is given.


## 2. Preliminaries

Consider the following uncertain nonlinear system, which described as(1)x˙1=f1t,x+d1t+b1ϕ1u1,x˙2=f2t,x+d2t+b2ϕ2u2,…,x˙n=fnt,x+dnt+bnϕnun,where x=x1,x2,…,xnT is the system state vector, fit,x is an unknown smooth nonlinear function, dit is an external disturbance, ϕiui denotes the control input subject to dead-zone type nonlinear, and bi is an unknown control coefficient, i=1,2,…,n. Here, ϕui is described as follows:(2)ϕiui=miui−bir,ifui≥bir,0,ifbil<ui<bir,miui+bil,ifui≤−bil,where mi,bir, and bil are unknown positive constants, i=1,2,…,n. Then, we rewrite (2) as(3)ϕiui=miui+△ui,where(4)△ui=−mibir,ifui≥bir,−miui,ifbil<ui<bir,mibil,ifui≤−bil.From (3), system (1) is described as(5)x˙1=f1t,x+d1t+b1m1u1+b1△u1,x˙2=f2t,x+d2t+b2m2u2+b2△u2,…,x˙n=fnt,x+dnt+bnmnun+bn△un.Designing two adaptive prescribed performance control methods is the control objective of this paper. The goal of this paper is that the tracking errorei satisfies PPB, and all closed-loop system signals are bounded.In order to meet the objective, we proposed the following assumptions.Assumption 1.
The system statex, the reference signal xd, and its derivative x˙d are available and bounded.Assumption 2.
The nonlinear functionfit,x, the control coefficient bi, and the external disturbance dit are unknown but bounded, i=1,2,…,n.Remark 1.
Assumptions1 and 2 are commonly used to address the problem of unknown control direction [21–23]. And owing to the boundedness of bi,Δui, and dit, we know that there exists an unknown positive constant Di∗, such that bi△ui+dit≤Di∗,i=1,2,…,n.
Letb¯i=bimi and Dit=bi△ui+dit, then Dit≤Di∗. The error state is defined as e=e1,e2,…,enT=x1−x1d,x2−x2d,…,xn−xndT, and the following error system is acquired:(6)e˙1=f1t,x+D1t+b¯1u1t−x˙1d,e˙2=f2t,x+D2t+b¯2u2t−x˙2d,…,e˙n=fnt,x+Dnt+b¯nunt−x˙nd.
The following definition and lemmas used in this paper are to overcome the problem of unknown control directions.Definition 1.
A continuous functionN⋅ is called a Nussbaum function, if N⋅ satisfies(7)limt⟶+∞sup1t∫0tNsds=+∞,limt⟶+∞inf1t∫0tNsds=−∞.Remark 2.
Generally, the selection of the Nussbaum functions for the controller design with unknown control direction isξ2cosξ or eξ2cosΠ/2ξ or lnξ+1coslnξ+12 and so on. In this paper, we choose the even Nussbaum function ξ2cosξ to design the PPC controller.Lemma 1.
(see [15]). Let Vt and ξt be smooth functions, Vt≥0 for ∀t∈0,tf, and N⋅ be an even smooth Nussbaum function. If the following inequality holds:(8)Vt≤r0+e−r1t∫0tαxsNξs+1⋅ξ˙se−r1sds,∀t∈0,tf,where r0 and r1 are constants with r1>0, αxt is a time-varying parameter which takes values in the unknown closed intervals I=:l−,l+ with 0∈¯I, then Vt, ξt, and ∫0tαxsNξs+1ξ˙se−r1sds must be bounded on 0,tf.Lemma 2.
(see [4]). Let A∈Rn×n be a symmetric matrix and x∈Rn be a nonzero vector. Let α=xTAx/xTx. Then, there exist eigenvalues of A such that λ1∈−∞,α and λ2∈α,+∞.Remark 3.
In this paper, we will construct the time-varying parameterαt by using Lemma 2. For example, let A=diaga1,a2,…,an and z is a nonzero vector; we can design αt as αt=zTAz/zTz, and then there exist two constants l− and l+ such that(9)l−≤miniai≤αt≤maxiai≤l+.
In this paper, fuzzy logic systems will be used to approximate unknown nonlinear functions. Generally, the output of fuzzy logic system can be expressed as [24–32](10)f^=∑i∈Iwitμixt∑i∈Iμixt,where x=x1t,…,xntT∈C1ℐ,Ω∀t∈ℐ is the input vector. f^:Ω↦ℛ is a Lipschitz continuous mapping, Ω⊆ℛn is a compact subset, and ℛ is a real line. I=∏j=1nℱj, where ℱj consists of Nj fuzzy sets 1≤j≤n. The membership function of the i-th rule is μii∈I, where μi:ℛn↦0,1. wi:ℐ↦ℛ (i∈1,2,…,N), which represents the centroid of the i-th consequent set. Let θt=w1t,…,wNtT and φt=φ1xt,φ2xt,…,φNxtT, where φi (i∈I) is the i-th fuzzy basis function and φi is a continuous mapping, then φi can be defined as(11)φixt=μit∑s∈Iμsxt.
Thus, (10) can be rearranged as(12)f^xt=θTtφt.


## 3. Synchronization Controller Design

### 3.1. Prescribed Performance

The steady-state performance ofei can be preserved by setting the performance constraint condition as(13)−λ2μt<ei<λ1μt,i=1,2,…,n,where λ1andλ2 are positive constants, which can regulate the error boundaries. μt is denoted by(14)μt=μ0−μ∞e−λ¯t+μ∞,where λ¯>0 and μ0,μ∞>0. Obviously, μt is a decreasing function and μ0=μ0, limt⟶+∞μt=μ∞.We abbreviateμt to μ and give an error transformation as(15)zi=λ1eiλ1μ−eilei+λ2eiλ2μ+ei1−lei,where lei satisfies(16)lei=1,ei≥0,0,ei<0.Obviously, if we choose the initial conditionei0 as −λ2μ0<ei0<λ1μ0 and zi is bounded, then the error state ei will satisfy the inequality that −λ2μ<ei<λ1μ. Because when ei approaches −λ2μ0 or λ1μ0, zi will approach infinity, which contradicts the boundedness of zi. So, here, the work for us is to prove that the error transformation variable zi is bounded, zii=1,2,…,n.Remark 4.
The proposed transformation variablezi in this paper can be directly expressed by ei and parameters λ1 and λ2, see (15), while the traditional transformation variable zi in [4, 14] needs to rely on the transformation function tanhzi=ei/μ to transform. Therefore, the proposed error transformation variable zi is simpler and is more convenient for the design of the controller ui.


### 3.2. Control Design and Stability Analysis

The time derivative ofzi is(17)zi.=μΠiei.−μ˙Πiei,where Πi=λ12/λ1μ−ei2lei+λ22/λ2μ+ei21−lei. So, we obtain the following error transformation dynamic system.Due to the fact thatfit,x is unknown, we need to employ fuzzy logic system to approximate fit,x. So, the approximate of function fit,x can be expressed as f^ix,θ^fi=θ^fiTφfix,i=1,2,…,n, where θ^fi is the parameter vector and the fuzzy function vector is φfix. For fit,x, there is an optimal parameter vector θfi∗ such that(18)fit,x=θfi∗Tφfix+εfix,where θfi∗=argminθ^fisupfit,x−θ^fiTφfix and εfix is the approximation error, which is bounded as εfix≤εfi∗, where εfi∗ is a positive constant, i=1,2,…,n.LetD¯it=εfix+Dit, then D¯t≤εfi∗+Di∗≜D¯i∗. The error transformation dynamic system is rewritten as follows:(19)z˙1=μΠ1f1t,x+D1t+b¯1u1t−x˙1d−μ˙Π1e1,z˙2=μΠ2f2t,x+D2t+b¯2u2t−x˙2d−μ˙Π2e2,…,z˙n=μΠnfnt,x+Dnt+b¯nunt−x˙nd−μ˙Πnen.Next, we design the controllerui to guarantee the boundedness of zi. Let z=z1,z2,…,znT,Π=Π1,Π2,…,ΠnT, and another controller as follows:(20)ui=ziNξu¯μΠi,u¯=Δ2Δz+σ,ξ˙=z2u¯,whereΔ=Δ1+Δ2, Δ1=μθ^Π∑i=1nφfix2, Δ2=μΠD^+∑i=1nμ˙Πiei+1/2z, θ∗≜∑i=1nθfi∗2, D∗=∑i=1nD¯i∗, θ^ and D^ are the estimates of θ∗ and D∗, respectively, and the parameters adaptive laws are given as(21)θ^˙=μΠ∑i=1nφfix2⋅z−γ1θ^,D^˙=μΠz−γ2D^,where γ1andγ2 are positive constants.Now, consider the Lyapunov function:(22)V=12zTz+12θ˜2+12D˜2,where θ˜=θ∗−θ^andD˜=D∗−D^. By differentiating V, one achieves(23)V˙=−12zTz+zTz˙+z2+θ˜θ˜˙+D˜D˜˙=−12zTz+∑i=1nziμΠiθfi∗Tφfix+ziμΠiD¯it−μ˙ziΠiei−zix˙idμΠi+zi22+∑i=1nziμΠib¯iui+θ˜θ˜˙+D˜D˜˙≤−12zTz+∑i=1nziμΠiθfi∗φfix+ziμΠiD¯it+μ˙ziΠiei+zix˙idμΠi+zi22+∑i=1nziμΠib¯iui+θ˜θ˜˙+D˜D˜˙≤−12zTz+μzΠ⋅∑i=1nθfi∗φfix+zμΠ∑i=1nD¯i∗︸D∗=D˜+D^+z⋅∑i=1nμ˙Πiei+z∑i=1nx˙idμΠi+12z2+∑i=1nziμΠib¯iui+θ˜θ˜˙+D˜D˜˙.According to Cauchy–Schwarz inequality,(24)∑i=1nθfi∗φfix≤∑i=1nθfi∗2⋅∑i=1nφfix2=θ∗∑i=1nφfix2=θ˜+θ^∑i=1nφfix2.Considering its adaptive law (21), it is concluded that(25)θ˜θ˜˙=−θ˜θ˜˙=−θ˜μΠ∑i=1nφfix2⋅z−γ1θ^=−θ˜μΠ∑i=1nφfix2⋅z+γ1θ˜θ^=−θ˜μΠ∑i=1nφfix2⋅z+γ1θ˜θ∗−θ˜≤−γ12θ˜2−θ˜μΠ∑i=1nφfix2⋅z+γ12θ∗2.Similarly,(26)D˜D˜˙≤−γ22D˜2−D˜μΠz+γ22D∗2.So, one gets(27)V˙≤−γ¯¯V+zμΠθ^∑i=1nφfix2+μΠD^+∑i=1nμ˙Πiei+∑i=1nx˙idμΠi+z2+∑i=1nzib¯ziNξu¯+R¯¯0,where γ¯¯=min1,γ1,γ2 and R¯¯0=γ1/2θ∗2+γ2/2D∗2. For zi≠0, we define αt=∑i=1nzib¯izi/∑i=1nzi2=zTUz/zTz, where U=diagb¯1,b¯2,…,b¯n. By using Lemma 2, there are two constants l+ and l− such that l−≤minib¯i≤αt≤maxibi≤l+. So,(28)∑i=1nzib¯zi=z2αt.Obviously, ifzi=0, (28) also holds. By substituting uit in (20) into (27), it yields(29)V˙≤−γ¯¯V+αtNξ+1ξ˙+R¯¯¯0,where R¯¯¯0=R¯¯0+σ. Integrating (29) over 0,t, one obtains(30)V≤V0+R¯¯¯0γ¯¯+∫0t1+αsNξξ˙eγ¯¯s−tds.According to Lemma2, Vt is bounded into 0,tf. So, zi is bounded for any given positive constant tf, and then the boundedness of zi is true for tf=+∞ (see [20]).Through the above analysis and discussion, the following theorem is obtained.Theorem 1.
Consider the uncertain nonlinear system (1) with dead-zone inputs and unknown control directions. The proposed control method (20) can guarantee the boundedness of all the signals in the closed-loop system. Furthermore, for any ei0 satisfies −λ2μ0<ei0<λ1μ0, the error ei will satisfy the prescribed performance (13).


## 3.1. Prescribed Performance

The steady-state performance ofei can be preserved by setting the performance constraint condition as(13)−λ2μt<ei<λ1μt,i=1,2,…,n,where λ1andλ2 are positive constants, which can regulate the error boundaries. μt is denoted by(14)μt=μ0−μ∞e−λ¯t+μ∞,where λ¯>0 and μ0,μ∞>0. Obviously, μt is a decreasing function and μ0=μ0, limt⟶+∞μt=μ∞.We abbreviateμt to μ and give an error transformation as(15)zi=λ1eiλ1μ−eilei+λ2eiλ2μ+ei1−lei,where lei satisfies(16)lei=1,ei≥0,0,ei<0.Obviously, if we choose the initial conditionei0 as −λ2μ0<ei0<λ1μ0 and zi is bounded, then the error state ei will satisfy the inequality that −λ2μ<ei<λ1μ. Because when ei approaches −λ2μ0 or λ1μ0, zi will approach infinity, which contradicts the boundedness of zi. So, here, the work for us is to prove that the error transformation variable zi is bounded, zii=1,2,…,n.Remark 4.
The proposed transformation variablezi in this paper can be directly expressed by ei and parameters λ1 and λ2, see (15), while the traditional transformation variable zi in [4, 14] needs to rely on the transformation function tanhzi=ei/μ to transform. Therefore, the proposed error transformation variable zi is simpler and is more convenient for the design of the controller ui.


## 3.2. Control Design and Stability Analysis

The time derivative ofzi is(17)zi.=μΠiei.−μ˙Πiei,where Πi=λ12/λ1μ−ei2lei+λ22/λ2μ+ei21−lei. So, we obtain the following error transformation dynamic system.Due to the fact thatfit,x is unknown, we need to employ fuzzy logic system to approximate fit,x. So, the approximate of function fit,x can be expressed as f^ix,θ^fi=θ^fiTφfix,i=1,2,…,n, where θ^fi is the parameter vector and the fuzzy function vector is φfix. For fit,x, there is an optimal parameter vector θfi∗ such that(18)fit,x=θfi∗Tφfix+εfix,where θfi∗=argminθ^fisupfit,x−θ^fiTφfix and εfix is the approximation error, which is bounded as εfix≤εfi∗, where εfi∗ is a positive constant, i=1,2,…,n.LetD¯it=εfix+Dit, then D¯t≤εfi∗+Di∗≜D¯i∗. The error transformation dynamic system is rewritten as follows:(19)z˙1=μΠ1f1t,x+D1t+b¯1u1t−x˙1d−μ˙Π1e1,z˙2=μΠ2f2t,x+D2t+b¯2u2t−x˙2d−μ˙Π2e2,…,z˙n=μΠnfnt,x+Dnt+b¯nunt−x˙nd−μ˙Πnen.Next, we design the controllerui to guarantee the boundedness of zi. Let z=z1,z2,…,znT,Π=Π1,Π2,…,ΠnT, and another controller as follows:(20)ui=ziNξu¯μΠi,u¯=Δ2Δz+σ,ξ˙=z2u¯,whereΔ=Δ1+Δ2, Δ1=μθ^Π∑i=1nφfix2, Δ2=μΠD^+∑i=1nμ˙Πiei+1/2z, θ∗≜∑i=1nθfi∗2, D∗=∑i=1nD¯i∗, θ^ and D^ are the estimates of θ∗ and D∗, respectively, and the parameters adaptive laws are given as(21)θ^˙=μΠ∑i=1nφfix2⋅z−γ1θ^,D^˙=μΠz−γ2D^,where γ1andγ2 are positive constants.Now, consider the Lyapunov function:(22)V=12zTz+12θ˜2+12D˜2,where θ˜=θ∗−θ^andD˜=D∗−D^. By differentiating V, one achieves(23)V˙=−12zTz+zTz˙+z2+θ˜θ˜˙+D˜D˜˙=−12zTz+∑i=1nziμΠiθfi∗Tφfix+ziμΠiD¯it−μ˙ziΠiei−zix˙idμΠi+zi22+∑i=1nziμΠib¯iui+θ˜θ˜˙+D˜D˜˙≤−12zTz+∑i=1nziμΠiθfi∗φfix+ziμΠiD¯it+μ˙ziΠiei+zix˙idμΠi+zi22+∑i=1nziμΠib¯iui+θ˜θ˜˙+D˜D˜˙≤−12zTz+μzΠ⋅∑i=1nθfi∗φfix+zμΠ∑i=1nD¯i∗︸D∗=D˜+D^+z⋅∑i=1nμ˙Πiei+z∑i=1nx˙idμΠi+12z2+∑i=1nziμΠib¯iui+θ˜θ˜˙+D˜D˜˙.According to Cauchy–Schwarz inequality,(24)∑i=1nθfi∗φfix≤∑i=1nθfi∗2⋅∑i=1nφfix2=θ∗∑i=1nφfix2=θ˜+θ^∑i=1nφfix2.Considering its adaptive law (21), it is concluded that(25)θ˜θ˜˙=−θ˜θ˜˙=−θ˜μΠ∑i=1nφfix2⋅z−γ1θ^=−θ˜μΠ∑i=1nφfix2⋅z+γ1θ˜θ^=−θ˜μΠ∑i=1nφfix2⋅z+γ1θ˜θ∗−θ˜≤−γ12θ˜2−θ˜μΠ∑i=1nφfix2⋅z+γ12θ∗2.Similarly,(26)D˜D˜˙≤−γ22D˜2−D˜μΠz+γ22D∗2.So, one gets(27)V˙≤−γ¯¯V+zμΠθ^∑i=1nφfix2+μΠD^+∑i=1nμ˙Πiei+∑i=1nx˙idμΠi+z2+∑i=1nzib¯ziNξu¯+R¯¯0,where γ¯¯=min1,γ1,γ2 and R¯¯0=γ1/2θ∗2+γ2/2D∗2. For zi≠0, we define αt=∑i=1nzib¯izi/∑i=1nzi2=zTUz/zTz, where U=diagb¯1,b¯2,…,b¯n. By using Lemma 2, there are two constants l+ and l− such that l−≤minib¯i≤αt≤maxibi≤l+. So,(28)∑i=1nzib¯zi=z2αt.Obviously, ifzi=0, (28) also holds. By substituting uit in (20) into (27), it yields(29)V˙≤−γ¯¯V+αtNξ+1ξ˙+R¯¯¯0,where R¯¯¯0=R¯¯0+σ. Integrating (29) over 0,t, one obtains(30)V≤V0+R¯¯¯0γ¯¯+∫0t1+αsNξξ˙eγ¯¯s−tds.According to Lemma2, Vt is bounded into 0,tf. So, zi is bounded for any given positive constant tf, and then the boundedness of zi is true for tf=+∞ (see [20]).Through the above analysis and discussion, the following theorem is obtained.Theorem 1.
Consider the uncertain nonlinear system (1) with dead-zone inputs and unknown control directions. The proposed control method (20) can guarantee the boundedness of all the signals in the closed-loop system. Furthermore, for any ei0 satisfies −λ2μ0<ei0<λ1μ0, the error ei will satisfy the prescribed performance (13).


## 4. Simulation Studies

Consider the Chen chaotic system as a simulation example:(31)x˙1=35x2−35x1︸f1x+d1t+b1ϕ1u1,x˙2=−x1x3−7x1+28x2︸f2x+d2t+b2ϕ2u2,x˙3=−3x3+x1x2︸f3x+d3t+b3ϕ3u3,where d1t=d2t=d3t=2sin2t, b1=2,b2=−2,b3=2, and ϕiui is described as(32)ϕiui=3ui−2,ui≥2,0,−2<ui<2,3ui+2,ui≤−2,i=1,2,3. One reference signal xd=sintsintsintT. Choose the fuzzy membership function as.Firstly, we use the method in [15] to control Chen chaotic system (31), and controller is designed as follows:(33)ui=ξi2cosξiu¯i,u¯i=η^fiTφfix+ρ^tanheiεi+ciei,ξ˙i=eiu¯i,η^˙fi=eiφfix−γ1iη^fi,ρ^˙i=eitanheiε−γ2iρ^i,i=1,2,3,where η^fiTφfix is the estimate of fix and fix=ηfi∗Tφfix+εfix, where εfix is the bounded approximation error. Let dit+εfix+x˙id≤ρi, where ρ^i is the estimate of ρi. The parameters are ε=0.5, γi1=3,γi2=2, i=1,2,3. The initial values are x10=2, x20=1, x30=−2, ξ10=0.01, ξ20=0.1, ξ30=0.1, ρ^10=0.05,ρ^20=−0.3,ρ^30=0.4, η^f10=η^f20=η^f30=0, μt=3e−2.5t+0.1. By using control scheme (33), the trajectories of x1, x2, and x3 are shown in Figures 1–3. The corresponding error trajectories are shown in Figure 4. It can be seen that, in the initial control stage, e1 and e2 will have a jump beyond the prescribed performance boundary (PPB)−μt,μt and then remain within the BBP −μt,μt, while e3, on the contrary, is within the PPB −μt,μt at the initial stage and stays outside the PPB −μt,μt after 3 seconds.Figure 1
x1 and x1d with control scheme (33).Figure 2
x2 and x2d with control scheme (33).Figure 3
x3 and x3d with control scheme (33).Figure 4
e1, e2, and x3 with control scheme (33).In order to compare with the control scheme (33), we employ our proposed control scheme (20). The parameters are λ1=λ2=1 and σ=0.5, and the initial values are D¯^1=0.2,D¯^2=0.3,andD¯^3=0.4, Other parameters and initial values in this example are same as those of the control scheme (33). The simulation results are shown in Figures 5–8. The states x1,x2, and x3 can track the reference signal x1d, x2d, and x3d quickly after a transient response process in Figures 5–7. From Figure 8, errors e1,e2, and e3 are not beyond the PPB −μt,μt. In Figure 9, two control effects of error state e are compared directly. So, these results verify that the proposed control scheme (20) has a better control effect than control scheme (33).Figure 5
x1 and x1d with proposed control scheme (20).Figure 6
x2 and x2d with proposed control scheme (20).Figure 7
x3 and x3d with proposed control scheme (20).Figure 8
e1, e2, and x3 with proposed control scheme (20).Figure 9
e with control schemes (20) and (33).


## 5. Conclusions

The problem of robust fuzzy adaptive prescribed performance control for uncertain nonlinear system is investigated. The proposed control approach is to make sure that all signals in closed-loop system are bounded and tracking errors converge to a small neighborhood of zero with the prescribed performance bounds. The Nussbaum-type function is incorporated into two control methods such that the influence of unknown control directions is eliminated. It is unnecessary to use the inverse operation in the proposed methods. Simulation results verify the effectiveness and robustness of the proposed method. How to effectively estimate the unknown function of the nonlinear system is the future research direction.



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*Source: 8849831-2020-12-23.xml*