# Observer-Based Adaptive Fuzzy Predefined Performance Control of a Class of Nonlinear Pure-Feedback Systems with Input Delay

Authors: Xin Li; Qiang Zhang; Dakuo He
Journal: Complexity
 (2020)

---

## Abstract

This paper presents a problem of observer-based adaptive fuzzy predefined performance control of a class of nonlinear pure-feedback systems with input delay and unknown control direction. Compared with the existing research, a novel predefined performance controller is proposed, which relaxes the assumption that the initial error is known. In addition, it is difficult to design the controllers due to input delay and nonaffine properties of the pure-feedback systems, which can be simplified by Pade approximation. Moreover, dynamic surface control and Nussbaum functions are applied to overcome the calculation explosion caused by repeated differentiations and unknown control direction, respectively. Based on the above methods, an adaptive fuzzy predefined performance controller is proposed, and it is proved that all the signals of a closed-loop system are semiglobally uniformly ultimately bounded (SGUUB). The tracking errors converge within a predefined range, while the observer estimation errors converge within a small zero region. Finally, the simulation results demonstrate the effectiveness of the proposed control system.


---

## Body

## 1. Introduction

In the past years, the adaptive nonlinear systems based on the backstepping method has matured increasingly and received widespread attention in [1, 2]. At an earlier time, there was an unmodeled nonlinear problem in the above system, which greatly limited the application of this technology. To solve the above problems, fuzzy logic systems (FLSs) and neural networks (NNs) were applied extensively to approximate unknown nonlinear function in [3–7]. However, the characteristic of the backstepping method is a class of recursive design procedures coupled with Lyapunov function candidates; hence, the repeated differentiation of virtual controller leads to the complexity explosion problem. Afterwards, the dynamic surface control (DSC) technology was integrated into the backstepping method to solve this problem in [8–10]. In addition, since the unmeasurable state in the application has a great restriction, the state observer was employed to estimate the unmeasured state in [11–15]. Among them, an equivalent output injection sliding mode observer was proposed in [12], which could estimate the status of each follower and its neighbor. And high gain observer was used to estimate the position, course, and speed of the vessel in [13]. In recent years, observer-based adaptive fuzzy control with the DSC technology was investigated in [16–18].It is well known that different from strict-feedback systems, pure-feedback systems have nonaffine structure of the variables, which presents more challenges to the controller design. Fortunately, the mean value theorem was proposed to solve the variables coupling problem of nonaffine structures in [19, 20]. Moreover, pure-feedback systems usually have the problem of unknown input control direction, which could be solved by Nussbaum functions in [21–24]. In addition, the input and output of the control systems have many restrictions, such as input saturation, dead zone, and input delay in [25–33]. It is worth mentioning that an adaptive predictor incorporated with a high-order neural network observer was proposed to obtain the predictions of the future system states in pure-feedback systems in [28], which were applied in the control design to avoid the input delay and nonlinearities. Subsequently, the input delay was solved by Pade approximation technique and intermediate variables in the strict-feedback systems in [29–31], which simplified the controller design. However, to the best of the author’s knowledge, the combination of input delay and pure-feedback systems was rarely considered. Therefore, the controller design of pure-feedback systems with input delay is complicated, which needs to be further developed.On the other side, the predefined control performance is better able to achieve the desired performance, such as overshoot, convergence rate, and convergence accuracy. Therefore, the prescribed performance control was proposed, which can satisfy preset transient and steady-state tracking performance in [34–36]. In particular, an adjustable finite-time prescribed performance function with fast convergence speed was adopted in [37, 38], which ensures real-time adjustment of controller parameters cased by the tracking error. Although the research of the prescribed performance control method is approaching maturity, there was still limitation of unknown initial values. Fortunately, a predefined performance function with time-varying design parameters was proposed to reduce the impact of unknown initial tracking error in [39]. However, it is not applied to the unknown nonlinear pure-feedback systems. In summary, the existing predefined performance control methods are insufficient to deal with a class of nonlinear pure-feedback systems with input delay. Therefore, the controller design for the above conditions needs to be developed.Based on the above discussion, this article presents a method for observer-based adaptive fuzzy predefined performance control of a class of nonlinear pure-feedback systems with unknown control direction and input delay. State observer and FLSs are proposed to solve the problem of approximate unmeasurable state and unknown nonlinear functions, respectively. Compared with the existing literature, the main contributions of this paper are as follows:(1)
In the existing literatures [34, 37], the initial values in predefined performance control are assumed to be known. In order to relax that assumption, a novel predefined performance control method is proposed, which is a variable-parameter scheme independent of the initial error. Therefore, the restriction of the unknown initial error in the predefined performance control is solved.(2)
Compared with [36], the input delay is introduced into the pure-feedback systems. It is difficult to design the controllers due to input delay and nonaffine properties of the pure-feedback systems, which can be simplified by Pade approximation and mean value theorem, respectively.(3)
By combining DSC technology and backstepping method, the issue of complexity explosion caused by repeated differentiations of some intermediate variables is eliminated. And the Nussbaum functions are proposed to solve unknown control direction.The framework of this article is as follows. In Section2, preliminaries and problem formulation are presented. In Section 3, an observer-based adaptive fuzzy predefined performance controller is designed of a class of nonlinear pure-feedback systems with unknown control direction and input delay, and the stability analysis is given. In Section 4, an example simulation is given to verify the feasibility of the proposed method. Finally, the Section 5 is the conclusions and the prospect of the future work.Notations:ℜ denotes the set of real numbers, ℜi denotes the i-dimensional vector space, and R+ is the set of all nonnegative real numbers. ⋅ indicates the Euclidean norm of vectors or matrix. For a matrix X, XT indicates its transpose and X−1 indicates its inverse. For a matrix Q, λminQ stands for the smallest eigenvalue of Q and λmaxQ stands for the largest eigenvalue of Q.


## 2. Preliminaries and Problem Formulation

### 2.1. System Descriptions and Assumptions

A class of nonlinear pure-feedback systems with input delay is considered as(1)x˙i=fix¯i,xi+1+dit,1≤i≤n−1,x˙n=fnx¯n,ut−δ+dnt,y=x1,where x¯i=x1,…,xiT∈ℜi are the state vector, y∈ℜ is the output, fix¯i,xi+1,fnx¯n,ut−δ are unknown smooth functions, dit means unknown and bounded external disturbance inputs, and δ denotes the input delay, which is an small unknown positive constant caused by network delay. Moreover, the output y is measurable.Because of the coupling between statesxi+1 and ut−δ in smooth functions fix¯i,xi+1 and fnx¯n,ut−δ, which makes the desired control objectives difficult to design, the mean value theorem is used as(2)fix¯i,xi+1=fix¯i,0+gix¯i,xi+1xi+1,fnx¯n,ut−δ=fnx¯n,0+gnx¯n,ut−δut−δ,where gix¯i,xi+1=∂fix¯i,xi+1/∂xi+1|xi+10 and gnx¯n,ut−δ=∂fnx¯n,ut−δ/∂ut−δ|ut−δ0, xi+10 is certain point between zero and xi+1, and ut−δ0 is certain point between zero and ut−δ. Let fix¯i,0=fix¯i, fnx¯n,0=fnx¯n, gix¯i,xi+1=gi, and gnx¯n,ut−δ=gn.Substituting (2) into system (1), one can obtain as(3)x˙i=fix¯i+gixi+1+dit,x˙n=fnx¯n+gnut−δ+dnt,y=x1.To get the actual control inputu by removing the effect of input delay δ, the Pade approximation method and the delay theorem of Laplace transform can be used, which solve the analysis complexity problem caused by time delay, and it follows that(4)ℓut−δ=e−δsℓut≈1−δs/21+δs/2ℓut,where s represents the Laplace variable and ℓut is the Laplace transform of ut.Define the intermediate variablexn+1 as(5)1−δs/21+δs/2ℓut=ℓxn+1−ℓut.By transforming formula (5), one can be given as(6)2ℓut=ℓxn+1t+δs2ℓut.By taking the inverse Laplace transform,(7)x˙n+1=2βu−βxn+1,where β=2/δ is a variable.Remark 1.
Pade approximation has been used in [31]. In this article, since the Pade approximation is applied to solve a class of small time delay problems, e−δs is approximately equal to 1−δs/2/1+δs/2 when the time delay is very small. And the intermediate variable xn+1 is not a real variable of system (1), which can be viewed as an error variable. And this has been verified in the simulation in [31].
By using the above methods, (3) can be further written as(8)x˙i=fix¯i+gixi+1+dit,x˙n=fnx¯n+gnxn+1−gnu+dnt,x˙n+1=−βxn+1+2βu,y=x1.Assumption 1.
(see [9]). The expected signal yd and its derivatives y˙d and y¨d are all known and bounded, which is ϒ=yd,y˙d,y¨d:yd2+y˙d2+y¨d2≤Y¯, where Y¯ is a positive constant.Assumption 2.
(see [20]). The sign of gi is unknown, but gi has the same sign and its public super bound is known, which is 0<gi<g∗.Assumption 3.
The disturbancedi is bounded to a positive constant di∗, that is, di≤di∗.Assumption 4.
(see [27]). There is a known constant si that satisfies fix¯i−fix¯^i≤six¯i−x¯^i, where x¯^i=x^1,x^2,…,x^iT is the estimate of x¯i=x1,x2,…,xiT, and X represents the 2 norm of the vector X.


### 2.2. Fuzzy Logic Systems

Because the nonlinear function is unknown, FLSs is proposed. Build FLSs with the if-then rules.Rq: if x1 is F1q and x2 is F2q and … and xn is Fnq. Then, y is Bq,q=1,2,…,a. Here, x = x1,…,xnT and y are the FLS input and output, respectively. Fuzzy sets Fiq and Bq, associated with the fuzzy functions μFiqxi and μBqy, respectively. a is the rules number. Thus, FLS can be calculated by formula(9)yxt=∑q=1ay˜q∏i=1nμFiqxi∑q=1a∏i=1nμFiqxi,where y˜q=maxy∈RμBqy.Letφq=∏i=1nμFiqxi/∑q=1a∏i=1nμFiqxi and denote θ=y˜1,y˜2,…,y˜aT=θ1,θ2,…,θaT and φTx=φ1x,…,φax; then, FLS can be rewritten as yx=θTφx.Lemma 1.
(see [40]). Let fx be a continuous function defined on a compact set Ω. Then, for any constant ε>0, there exists an FLS such as(10)supx∈Ωfx−θTφx≤ε.
Define the idealized parameter vectorθi∗ as(i)
θ1∗=argminθ1∈Ω1supx1∈U1f^1x1|θ1−f1x1(ii)
θi∗=argminθi∈Ωisupx¯^i∈Uif^ix¯^i|θi−fix¯^i, i=2,…,nHere, Ω1,Ωi,U1,andUi are compact for θ1,θi,x1,andx¯^i, respectively.
By Lemma1, the nonlinear functions can be approximated by the following FLSs:(11)f^1x1|θ1=θ1Tφ1x1,f^ix¯^i|θ^i=θ^iTφix¯^i,i=2,…,n.
The fuzzy minimum approximation errors can be defined asε1x1=f1x1−f^1x1|θ1∗ and εix^i=fix^i−f^ix¯^i|θi∗, where x¯^i are the estimation of the state x¯i.Assumption 5.
The approximation errorεi is bounded, and there is a constant εi∗ that satisfies εi≤εi∗.
From (11), system (8) can be expressed as(12)x˙i=gixi+1+θi∗Tφix¯^i+εix¯^i+di+Δfi,x˙n=gnxn+1−gnu+θn∗Tφnx¯^n+εnx¯^n+dn+Δfn,x˙n+1=−βxn+1+2βu,y=x1,where Δfi=fix¯i−fix^¯i, i=2,…,n.
Rewriting (12) in the following formula,(13)x˙=Ax+F¯∗+ε+d+Δf+Enu,y=E0Tx,where x=x1,x2,…,xn,xn+1T, A=0g10…0000g2…00⋮000…gn−10000…0gn000…0−β, ε=ε1x1,ε2x¯^2,…,εnx¯^n,0T, F¯∗=θ1∗Tφ1x1,θ2∗Tφ2x¯^2,…,θn∗Tφnx¯^n,0T, d=d1,d2,…,dn,0T, Δf=0,Δf2,…,Δfn,0T, E0T=1,0,…,0, and EnT=0,…,0,−gn,2β.


### 2.3. Nussbaum-Type Function

A continuous functionNμ is called the Nussbaum function if it has the following properties:(14)limm⟶∞sup1m∫0mNμdμ=∞,limm⟶∞inf1m∫0mNμdμ=−∞,where m is the integral upper boundary. For instance, the frequently used continuous Nussbaum-type functions contain μ2cosμ,μ2sinμ,eμ2cosμ, and so on. In this work, the continuous Nussbaum-type function Nμ=μ2cosμ is utilized.Lemma 2.
(see [41, 42]). Smooth functions V⋅ and μ⋅ are defined on 0,tf, where Vt≥0∀∈0,tf and Nμ is a Nussbaum-type function. If the following inequality holds(15)Vt≤c±∫0t∑l=1ngNμl+1μ˙ldτ,where g is a nonzero constant and c represents appropriate constant, then Vt, μt, and ∫0t∑l=1ngNμl+1μ˙ldτ must be bounded on 0,tf.Remark 2.
The parametersgi and gn are time-varying parameters and their signs are unknown. If the control direction changes rapidly, it is difficult to effectively guarantee the stability of the closed-loop system under the self-adaptive condition. Therefore, compared with the control direction which is assumed to be known, Assumption2 is more flexible in the application. In addition, similar to [20], this paper uses the Nussbaum functions to solve the control direction problem, which relaxes the prior knowledge.


### 2.4. Fuzzy State Observer Design

To estimate the unmeasurable states of the system, the corresponding fuzzy observer is designed as(16)x^˙i=gix^i+1+θiTφix^¯i+kiy−x^1,x^˙n=gnx^n+1−gnu+θnTφnx^¯n+kny−x^1,x^˙n+1=−βx^n+1+2βu,y^=x^1.Rewriting (16) in the following formula,(17)x^˙=A0x^+Ky+F¯+Enu,y^=E0Tx^,where x^=x^1,x^2,…,x^n,x^n+1T, A0=−k1g10…000−k20g2…000⋮−kn−100…0gn−10−kn00…00gn000…00−β, K=k1,…,kn,0T, and F¯=θ1Tφ1x1,θ2Tφ2x¯^2,…,θnTφnx¯^n,0T.The observer gain matrixK is given such that A0 is a Hurwitz matrix. Therefore, for any chosen positive definite matrix Q=QT>0, there is a positive definite matrix P=PT>0 that satisfies(18)A0TP+PA0=−Q.The observer errors can be obtained as(19)e=x−x^=e1,…,enT.From (12), (16), and (17), the observer error is(20)e˙=A0e+d+ε+Δf+Θ˜,where θ˜i=θi∗−θi and Θ˜=θ1˜Tφ1x1,…,θn˜Tφnx¯^nT.Consider the Lyapunov function candidate as(21)V0=eTPe.The time derivative ofV0 with (20) is(22)V˙0=e˙TPe+eTPe˙,=eTA0T+dT+εT+ΔfT+Θ˜TPe+eTPA0e+d+ε+Δf+Θ˜,=eTA0TP+PA0e+2eTPd+ε+Δf+Θ˜,=−eTQe+2eTPd+ε+Δf+Θ˜.By using Young’s inequality and Assumptions3–5, the inequalities can be obtained as(23)2eTPε+2eTPd≤2e2+P2ε∗2+P2d∗2,(24)2eTPΘ˜≤e2+P2∑l=1nθl˜Tθl˜,(25)2eTPΔf≤e2+P2Δf2≤e2+P2∑j=2nLj2e2=r0e2,where r0=1+P2∑j=2nLj2.Substituting (23)–(25) into (22) yields(26)V˙0≤−eTQe+r0+3e2+P2ε∗2+P2d∗2+P2∑l=1nθl˜Tθl˜.


### 2.5. Tracking Error Transformation

A new variable parameter independent of the initial error is proposed, which satisfies the expected variable-parameter scheme tracking performance constraints. Then, the tracking error is defined asE1=y−yd. Predefined performance control constraint will be obtained as inequality holds for all t≥0:(27)−ιdtρt<E1t<ιutρt,where smooth function ιdt and ιut satisfies the follow properties. 1ιdt>0, ιut>0 and strictly decreasing. 2limt⟶0ιdt=+∞; limt⟶∞ιdt=C1,C1∈R+; limt⟶0ιut=+∞; and limt⟶∞ιut=C2,C2∈R+.In this article,ιdt and ιut can be chosen as(28)ι˙dt=−λdιdt+hd,ι˙ut=−λuιut+hu,where λd, λu, hd, and hu are positive constants.Andρt is an appropriate performance boundary function and is defined as ρt=ρ0−ρ∞e−λt+ρ∞, which satisfies the following. 1ρt is positive and strictly decreasing. 2limt⟶∞ρt=ρ∞>0.Integrating equality (28) over 0,t, we have(29)ιdt=ιd0−hdλde−λdt+hdλd.In the same way,(30)ιut=ιu0−huλue−λut+huλu.By the above analysis,ιdt and ιut converge exponentially to constants hd/λd and hu/λu, and the convergence rate can be improved by adding λd and λu. When ιdt and ιut converge to constant values, inequality constraint (27) is degenerated as follows:(31)−hdλdρt<E1t<huλuρt.According to (31), when the system is stable, the upper bound of the steady-state error is maxhd/λd,hu/huρ∞, and the error convergence speed and the maximum overpass can be adjusted by the coefficient λd, λu, hd, hu, and ρt.The inequality constraint is transformed into equality constant, and the error transformation functionϕz,ιd,ιu is defined as(32)E1t=ρtϕz,ιd,ιu,where z is transform error, and the continuous function ϕz,ιdown,ιup satisfies the following properties. 1ϕz,ιd,ιu is smooth and strictly increasing; 2−ιdt<ϕz,ιd,ιu<ιut; 3limz⟶−∞ϕz,ιd,ιu=−ιdt; limz⟶+∞ϕz,ιd,ιu=ιut.By the properties of functionϕz,ιd,ιu, the inverse transformation is(33)z1=ϕ−1E1tρt,ιd,ιu.The error transformation function is defined as(34)ϕz,ιd,ιu=ιuez−ιde−zez+e−z.Therefore, by (32), the boundless of the error transformation function z1 results in the prescribed performance of the tracking error E1.Differentiating (33), it yields(35)z1.t=∂ϕ−1∂E1/ρ∂E1/ρ∂t=∂ϕ−1∂E1/ρE1.−E1ρ˙ρ.Combining (35) and (12), one can obtain as(36)z1.=ϖg1tx2+θ1∗Tφ1x1+ε1x1+d1+D,where ϖ=∂ϕ−1/∂E1/ρ1/ρ>0 and D=−∂ϕ−1/∂E1/ρ1/ρyd.+E1ρ˙/ρ.By derivation,(37)z1.=12logE1ρ+ιu–12logιd–E1ρ,ϖ=12ρ1E1/ρ+ιd−1E1/ρ−ιu.Remark 3.
Note that the functionsιd and ιu are set to solve the problem of unknown initial error E0. In the predefined performance control of [34], the initial error is assumed to be known, but in the actual control, the exact value of the initial error is often not obtained, which limits the use of the predefined performance control. The reason why the initial values of ιdt and ιut are set to infinity is mainly to guarantee E0∈−ιd,ιu; however, ιd0 and ιu0 just need to be set to a sufficiently large constant in the actual design. The facts justify this treatment.


## 2.1. System Descriptions and Assumptions

A class of nonlinear pure-feedback systems with input delay is considered as(1)x˙i=fix¯i,xi+1+dit,1≤i≤n−1,x˙n=fnx¯n,ut−δ+dnt,y=x1,where x¯i=x1,…,xiT∈ℜi are the state vector, y∈ℜ is the output, fix¯i,xi+1,fnx¯n,ut−δ are unknown smooth functions, dit means unknown and bounded external disturbance inputs, and δ denotes the input delay, which is an small unknown positive constant caused by network delay. Moreover, the output y is measurable.Because of the coupling between statesxi+1 and ut−δ in smooth functions fix¯i,xi+1 and fnx¯n,ut−δ, which makes the desired control objectives difficult to design, the mean value theorem is used as(2)fix¯i,xi+1=fix¯i,0+gix¯i,xi+1xi+1,fnx¯n,ut−δ=fnx¯n,0+gnx¯n,ut−δut−δ,where gix¯i,xi+1=∂fix¯i,xi+1/∂xi+1|xi+10 and gnx¯n,ut−δ=∂fnx¯n,ut−δ/∂ut−δ|ut−δ0, xi+10 is certain point between zero and xi+1, and ut−δ0 is certain point between zero and ut−δ. Let fix¯i,0=fix¯i, fnx¯n,0=fnx¯n, gix¯i,xi+1=gi, and gnx¯n,ut−δ=gn.Substituting (2) into system (1), one can obtain as(3)x˙i=fix¯i+gixi+1+dit,x˙n=fnx¯n+gnut−δ+dnt,y=x1.To get the actual control inputu by removing the effect of input delay δ, the Pade approximation method and the delay theorem of Laplace transform can be used, which solve the analysis complexity problem caused by time delay, and it follows that(4)ℓut−δ=e−δsℓut≈1−δs/21+δs/2ℓut,where s represents the Laplace variable and ℓut is the Laplace transform of ut.Define the intermediate variablexn+1 as(5)1−δs/21+δs/2ℓut=ℓxn+1−ℓut.By transforming formula (5), one can be given as(6)2ℓut=ℓxn+1t+δs2ℓut.By taking the inverse Laplace transform,(7)x˙n+1=2βu−βxn+1,where β=2/δ is a variable.Remark 1.
Pade approximation has been used in [31]. In this article, since the Pade approximation is applied to solve a class of small time delay problems, e−δs is approximately equal to 1−δs/2/1+δs/2 when the time delay is very small. And the intermediate variable xn+1 is not a real variable of system (1), which can be viewed as an error variable. And this has been verified in the simulation in [31].
By using the above methods, (3) can be further written as(8)x˙i=fix¯i+gixi+1+dit,x˙n=fnx¯n+gnxn+1−gnu+dnt,x˙n+1=−βxn+1+2βu,y=x1.Assumption 1.
(see [9]). The expected signal yd and its derivatives y˙d and y¨d are all known and bounded, which is ϒ=yd,y˙d,y¨d:yd2+y˙d2+y¨d2≤Y¯, where Y¯ is a positive constant.Assumption 2.
(see [20]). The sign of gi is unknown, but gi has the same sign and its public super bound is known, which is 0<gi<g∗.Assumption 3.
The disturbancedi is bounded to a positive constant di∗, that is, di≤di∗.Assumption 4.
(see [27]). There is a known constant si that satisfies fix¯i−fix¯^i≤six¯i−x¯^i, where x¯^i=x^1,x^2,…,x^iT is the estimate of x¯i=x1,x2,…,xiT, and X represents the 2 norm of the vector X.


## 2.2. Fuzzy Logic Systems

Because the nonlinear function is unknown, FLSs is proposed. Build FLSs with the if-then rules.Rq: if x1 is F1q and x2 is F2q and … and xn is Fnq. Then, y is Bq,q=1,2,…,a. Here, x = x1,…,xnT and y are the FLS input and output, respectively. Fuzzy sets Fiq and Bq, associated with the fuzzy functions μFiqxi and μBqy, respectively. a is the rules number. Thus, FLS can be calculated by formula(9)yxt=∑q=1ay˜q∏i=1nμFiqxi∑q=1a∏i=1nμFiqxi,where y˜q=maxy∈RμBqy.Letφq=∏i=1nμFiqxi/∑q=1a∏i=1nμFiqxi and denote θ=y˜1,y˜2,…,y˜aT=θ1,θ2,…,θaT and φTx=φ1x,…,φax; then, FLS can be rewritten as yx=θTφx.Lemma 1.
(see [40]). Let fx be a continuous function defined on a compact set Ω. Then, for any constant ε>0, there exists an FLS such as(10)supx∈Ωfx−θTφx≤ε.
Define the idealized parameter vectorθi∗ as(i)
θ1∗=argminθ1∈Ω1supx1∈U1f^1x1|θ1−f1x1(ii)
θi∗=argminθi∈Ωisupx¯^i∈Uif^ix¯^i|θi−fix¯^i, i=2,…,nHere, Ω1,Ωi,U1,andUi are compact for θ1,θi,x1,andx¯^i, respectively.
By Lemma1, the nonlinear functions can be approximated by the following FLSs:(11)f^1x1|θ1=θ1Tφ1x1,f^ix¯^i|θ^i=θ^iTφix¯^i,i=2,…,n.
The fuzzy minimum approximation errors can be defined asε1x1=f1x1−f^1x1|θ1∗ and εix^i=fix^i−f^ix¯^i|θi∗, where x¯^i are the estimation of the state x¯i.Assumption 5.
The approximation errorεi is bounded, and there is a constant εi∗ that satisfies εi≤εi∗.
From (11), system (8) can be expressed as(12)x˙i=gixi+1+θi∗Tφix¯^i+εix¯^i+di+Δfi,x˙n=gnxn+1−gnu+θn∗Tφnx¯^n+εnx¯^n+dn+Δfn,x˙n+1=−βxn+1+2βu,y=x1,where Δfi=fix¯i−fix^¯i, i=2,…,n.
Rewriting (12) in the following formula,(13)x˙=Ax+F¯∗+ε+d+Δf+Enu,y=E0Tx,where x=x1,x2,…,xn,xn+1T, A=0g10…0000g2…00⋮000…gn−10000…0gn000…0−β, ε=ε1x1,ε2x¯^2,…,εnx¯^n,0T, F¯∗=θ1∗Tφ1x1,θ2∗Tφ2x¯^2,…,θn∗Tφnx¯^n,0T, d=d1,d2,…,dn,0T, Δf=0,Δf2,…,Δfn,0T, E0T=1,0,…,0, and EnT=0,…,0,−gn,2β.


## 2.3. Nussbaum-Type Function

A continuous functionNμ is called the Nussbaum function if it has the following properties:(14)limm⟶∞sup1m∫0mNμdμ=∞,limm⟶∞inf1m∫0mNμdμ=−∞,where m is the integral upper boundary. For instance, the frequently used continuous Nussbaum-type functions contain μ2cosμ,μ2sinμ,eμ2cosμ, and so on. In this work, the continuous Nussbaum-type function Nμ=μ2cosμ is utilized.Lemma 2.
(see [41, 42]). Smooth functions V⋅ and μ⋅ are defined on 0,tf, where Vt≥0∀∈0,tf and Nμ is a Nussbaum-type function. If the following inequality holds(15)Vt≤c±∫0t∑l=1ngNμl+1μ˙ldτ,where g is a nonzero constant and c represents appropriate constant, then Vt, μt, and ∫0t∑l=1ngNμl+1μ˙ldτ must be bounded on 0,tf.Remark 2.
The parametersgi and gn are time-varying parameters and their signs are unknown. If the control direction changes rapidly, it is difficult to effectively guarantee the stability of the closed-loop system under the self-adaptive condition. Therefore, compared with the control direction which is assumed to be known, Assumption2 is more flexible in the application. In addition, similar to [20], this paper uses the Nussbaum functions to solve the control direction problem, which relaxes the prior knowledge.


## 2.4. Fuzzy State Observer Design

To estimate the unmeasurable states of the system, the corresponding fuzzy observer is designed as(16)x^˙i=gix^i+1+θiTφix^¯i+kiy−x^1,x^˙n=gnx^n+1−gnu+θnTφnx^¯n+kny−x^1,x^˙n+1=−βx^n+1+2βu,y^=x^1.Rewriting (16) in the following formula,(17)x^˙=A0x^+Ky+F¯+Enu,y^=E0Tx^,where x^=x^1,x^2,…,x^n,x^n+1T, A0=−k1g10…000−k20g2…000⋮−kn−100…0gn−10−kn00…00gn000…00−β, K=k1,…,kn,0T, and F¯=θ1Tφ1x1,θ2Tφ2x¯^2,…,θnTφnx¯^n,0T.The observer gain matrixK is given such that A0 is a Hurwitz matrix. Therefore, for any chosen positive definite matrix Q=QT>0, there is a positive definite matrix P=PT>0 that satisfies(18)A0TP+PA0=−Q.The observer errors can be obtained as(19)e=x−x^=e1,…,enT.From (12), (16), and (17), the observer error is(20)e˙=A0e+d+ε+Δf+Θ˜,where θ˜i=θi∗−θi and Θ˜=θ1˜Tφ1x1,…,θn˜Tφnx¯^nT.Consider the Lyapunov function candidate as(21)V0=eTPe.The time derivative ofV0 with (20) is(22)V˙0=e˙TPe+eTPe˙,=eTA0T+dT+εT+ΔfT+Θ˜TPe+eTPA0e+d+ε+Δf+Θ˜,=eTA0TP+PA0e+2eTPd+ε+Δf+Θ˜,=−eTQe+2eTPd+ε+Δf+Θ˜.By using Young’s inequality and Assumptions3–5, the inequalities can be obtained as(23)2eTPε+2eTPd≤2e2+P2ε∗2+P2d∗2,(24)2eTPΘ˜≤e2+P2∑l=1nθl˜Tθl˜,(25)2eTPΔf≤e2+P2Δf2≤e2+P2∑j=2nLj2e2=r0e2,where r0=1+P2∑j=2nLj2.Substituting (23)–(25) into (22) yields(26)V˙0≤−eTQe+r0+3e2+P2ε∗2+P2d∗2+P2∑l=1nθl˜Tθl˜.


## 2.5. Tracking Error Transformation

A new variable parameter independent of the initial error is proposed, which satisfies the expected variable-parameter scheme tracking performance constraints. Then, the tracking error is defined asE1=y−yd. Predefined performance control constraint will be obtained as inequality holds for all t≥0:(27)−ιdtρt<E1t<ιutρt,where smooth function ιdt and ιut satisfies the follow properties. 1ιdt>0, ιut>0 and strictly decreasing. 2limt⟶0ιdt=+∞; limt⟶∞ιdt=C1,C1∈R+; limt⟶0ιut=+∞; and limt⟶∞ιut=C2,C2∈R+.In this article,ιdt and ιut can be chosen as(28)ι˙dt=−λdιdt+hd,ι˙ut=−λuιut+hu,where λd, λu, hd, and hu are positive constants.Andρt is an appropriate performance boundary function and is defined as ρt=ρ0−ρ∞e−λt+ρ∞, which satisfies the following. 1ρt is positive and strictly decreasing. 2limt⟶∞ρt=ρ∞>0.Integrating equality (28) over 0,t, we have(29)ιdt=ιd0−hdλde−λdt+hdλd.In the same way,(30)ιut=ιu0−huλue−λut+huλu.By the above analysis,ιdt and ιut converge exponentially to constants hd/λd and hu/λu, and the convergence rate can be improved by adding λd and λu. When ιdt and ιut converge to constant values, inequality constraint (27) is degenerated as follows:(31)−hdλdρt<E1t<huλuρt.According to (31), when the system is stable, the upper bound of the steady-state error is maxhd/λd,hu/huρ∞, and the error convergence speed and the maximum overpass can be adjusted by the coefficient λd, λu, hd, hu, and ρt.The inequality constraint is transformed into equality constant, and the error transformation functionϕz,ιd,ιu is defined as(32)E1t=ρtϕz,ιd,ιu,where z is transform error, and the continuous function ϕz,ιdown,ιup satisfies the following properties. 1ϕz,ιd,ιu is smooth and strictly increasing; 2−ιdt<ϕz,ιd,ιu<ιut; 3limz⟶−∞ϕz,ιd,ιu=−ιdt; limz⟶+∞ϕz,ιd,ιu=ιut.By the properties of functionϕz,ιd,ιu, the inverse transformation is(33)z1=ϕ−1E1tρt,ιd,ιu.The error transformation function is defined as(34)ϕz,ιd,ιu=ιuez−ιde−zez+e−z.Therefore, by (32), the boundless of the error transformation function z1 results in the prescribed performance of the tracking error E1.Differentiating (33), it yields(35)z1.t=∂ϕ−1∂E1/ρ∂E1/ρ∂t=∂ϕ−1∂E1/ρE1.−E1ρ˙ρ.Combining (35) and (12), one can obtain as(36)z1.=ϖg1tx2+θ1∗Tφ1x1+ε1x1+d1+D,where ϖ=∂ϕ−1/∂E1/ρ1/ρ>0 and D=−∂ϕ−1/∂E1/ρ1/ρyd.+E1ρ˙/ρ.By derivation,(37)z1.=12logE1ρ+ιu–12logιd–E1ρ,ϖ=12ρ1E1/ρ+ιd−1E1/ρ−ιu.Remark 3.
Note that the functionsιd and ιu are set to solve the problem of unknown initial error E0. In the predefined performance control of [34], the initial error is assumed to be known, but in the actual control, the exact value of the initial error is often not obtained, which limits the use of the predefined performance control. The reason why the initial values of ιdt and ιut are set to infinity is mainly to guarantee E0∈−ιd,ιu; however, ιd0 and ιu0 just need to be set to a sufficiently large constant in the actual design. The facts justify this treatment.


## 3. Control Design and Stability Analysis

In this section, by utilizing adaptive backstepping technique and Lyapunov stability theorem, an observer-based adaptive fuzzy predefined performance decentralized control approach is developed.The coordinate changes are as follows:(38)E1=y−yd,Ei=x^i−α¯ii=2,…,n−1,En=x^n−α¯n+x^n+1gnβ,ξi=α¯i−αi−1,where αi is the virtual control law and α¯i and ξi are the filter signal and filter error of the first-order filter, respectively. Define a first-order filter as(39)ζiα¯˙i+α¯i=αi−1,α¯i0=αi−10,i=2,…,n,where ζi is the design constant.Step 1.
By substituting (38) into (36), one can obtain as(40)z˙1=ϖg1E2+ξ2+α1+e2+θ1∗φ1x1+ε1x1+d1+D.
Consider the Lyapunov function candidate as(41)V1=V0+12z12+12υ1θ1˜Tθ1˜,where υ1>0 is a design constant. The time derivative of V1 and substitute (40) and (26) into (41) as(42)V˙1=V˙0+z1z˙1+1υ1θ1˜Tθ˜.1≤−eTQe+r0+3e2+P2ε∗2+P2d∗2+P2∑l=1nθl˜Tθl˜+θ1∗φ1x1+z1ϖg1E2+ξ2+α1+e2+ε1x1+d1+D+1υ1θ1˜Tθ˜.1.
By using Young’s inequality,(43)z1ϖg1E2+ξ2≤ϖ2z12+12g1∗2E22+ξ22,(44)z1ϖg1e2≤12ϖ2z12+12g1∗2e2,(45)z1ϖε1≤12ϖ2z12+12ε12,(46)z1ϖd1≤12ϖ2z12+12d12.
From (43)–(46), one can obtain as(47)V˙1≤−λminQ−r0−3−12g1∗2e2+P2ε∗2+P2d∗2+P2∑l=1nθl˜Tθl˜+z1ϖ52ϖz1+g1α1+θ1Tφ1x1+D+12ε1∗2+12d1∗2+12g1∗2E22+ξ22+1υ1θ1˜Tυ1φ1x1z1ϖ−θ.1.
Select the virtual control functionα1 and the adaptive laws θ1 as(48)α1=Nμ1ω1z1+52ϖz1+θ1Tφ1x1+Dϖ,(49)μ˙1=z1ϖω1z1+52ϖz1+θ1Tφ1x1+Dϖ,(50)θ.1=υ1φ1x1z1ϖ−σθ1,where ω1>0 and σ>0 are design constants.
Substituting (48)–(50) into (47) yields(51)V˙1≤−r1e2+P2ε∗2+P2d∗2+P2∑l=1nθl˜Tθl˜−ω1ϖz12+συ1θ1˜Tθ1+12g1∗2E22+ξ22+gNμ1+1μ˙1+12ε1∗2+12d1∗2,where r1=λminQ−r0−3−1/2g1∗2.
By using Young’s inequality,(52)συ1θ˜1Tθ1=συ1θ˜1T−θ˜1+θ1∗≤−σ2υ1θ˜1Tθ˜1+σ2υ1θ1∗Tθ1∗.
Substitute (52) into (51) as follows:(53)V˙1≤−r1e2+P2∑l=1nθl˜Tθl˜−ω1ϖz12−σ2υ1θ˜1Tθ˜1+12g1∗2E22+ξ22+gNμ1+1μ˙1+M1,where M1=P2ε∗2+P2d∗2+1/2ε1∗2+1/2d1∗2+σ/2υ1θ1∗Tθ1∗.Step 2.
(ii=2,…,n−1). The time derivative of Ei is(54)E˙i=x^˙i−α¯i.=gix^i+1+θi∗Tφix¯^i+εi+kie1−α¯˙i=giEi+1+ξi+1+αi+θi∗Tφix¯^i+εi+kie1−α¯˙i.
Consider the Lyapunov function candidate as(55)Vi=Vi−1+12Ei2+12υiθ˜iTθ˜i+12ξi2,where υi>0 is a design constant. The derivative of Vi with time is(56)V˙i=V˙i−1+EiE˙i+1υiθ˜iTθ˜.i+ξiξ˙i≤−r1e2+P2∑l=1nθ˜lTθ˜l−ω1ϖz12−∑l=1i−1σ2υlθ˜lTθ˜l+12∑l=1i−1gl∗2El+12+ξl+12+Mi−1−∑l=2i−1ωlEl2+EigiEi+1+ξi+1+αi+θi∗Tφix¯^i+εi+kie1−α¯˙i+1υiθ˜iTθ˜.i+ξiξ˙i+∑l=1i−1gNμl+1μ˙l,where Mi−1=Mi−2+1/2εi−1∗2+σ/2υi−1θi−1∗Tθi−1∗.
By using Young’s inequality,(57)EigiEi+1+ξi+1≤Ei2+12gi∗2Ei+12+ξi+12,(58)Eiεi≤12Ei2+12εi∗2.
From (57) and (58), one can obtain as(59)Vi.≤−r1e2+P2∑l=1nθ˜lTθ˜l−ω1ϖz12−∑l=1i−1σ2υlθ˜lTθ˜l+12∑l=1i−1gl∗2El+12+ξl+12+Mi−1−∑l=2i−1ωlEl2+Eigiαi+θiTφix¯i^+32Ei+kie1−α¯˙i+12gi∗2Ei+12+ξi+12+12εi∗2+1υiθ˜iTυiEiφix¯^i−θ.i+ξiξ˙i+∑l=1i−1gNμl+1μ˙l.
Select the virtual control functionαi and the adaptive laws θi as(60)αi=NμiωiEi+32Ei+θiTφix¯^i+kie1−α¯˙i,(61)μ˙i=EiωiEi+32Ei+θiTφix¯^i+kie1−α¯˙i,(62)θ.i=υiEiφix¯^i−σθi.
Substituting (60)–(62) into (59) yields(63)V˙i≤−r1e2+P2∑l=1nθl˜Tθl˜−ω1ϖz12−∑l=1i−1σ2υlθ˜lTθ˜l+12∑l=1igl∗2El+12+ξl+12+Mi+∑l=1iξlξ˙l−∑l=2iωlEl2+∑l=1igNμl+1μ˙l+Mi−1+12εi∗2+συiθi˜Tθi.
According to the definition ofα1,α2, and αi, one can get α˙i=ηiE1,…,Ei,Ei+1,θ1,…,θi,yd,y˙d,y¨d,ξ2,…,ξi, where ηi is a continuous function. Given that any ψ, Ci=E1,…,Ei,Ei+1,θ1,…,θi,yd,y˙d,y¨d,ξ2,…,ξiT:ϑ<ψ is a prefixed compact set, where the compact set Ci can be made larger as needed. Therefore, the maximum value of the continuous function ηi is Bi+1 on Ci∗Δ0 based on Assumption 1 of the compact set ϒ and the compact set Ci. It is obvious that(64)ξ˙i=−ξiχi+ηi−1E1,…,Ei,θ1,…,θi−1,yd,y˙d,y¨d,ξ2,…,ξi−1≤−ξiχi+Bi.
Utilising Young’s inequality,(65)συiθ˜iTθi=συiθ˜iT−θ˜i+θi∗≤−σ2υiθ˜iTθ˜i+σ2υiθi∗Tθi∗,(66)ξiξ˙i≤ξi−ξiχi+Bi≤−ξi2χi+ξiBi≤−1χi−Bi22πiξi2+πi2,where πi is the design constant.
Substitute (65) and (66) into (63) as follows:(67)V˙i≤−r1e2+P2∑l=1nθl˜Tθl˜−ω1ϖz12−∑l=1iσ2υlθ˜lTθ˜l+12∑l=1igl∗2El+12+ξl+12−∑l=1i1χi−Bi22πiξi2−∑l=2iωlEl2+∑l=1igNμl+1μ˙l+Mi,where Mi=Mi−1+1/2εi∗2+σ/2υiθi∗Tθi∗+πi/2.Step 3.
n. The time derivative of En is(68)E˙n=x^˙n−α¯˙n+x^˙n+1gnβ=gnx^n+1−u+θn∗Tφnx¯^n+εn+kne1−α¯˙n+gnβ−βx^n+1+2βu=gnu+θn∗Tφnx¯^n+εn+kne1−α¯˙n.
Consider the Lyapunov function candidate as(69)Vn=Vn−1+12En2+12υnθ˜nTθ˜n+12ξn2,where υn>0 is a design constant. The derivative of Vn with time along with (68) is(70)V˙n=V˙n−1+EnE˙n+1υnθ˜nTθ˜.n+ξnξ˙n=V˙n−1+Engnu+θn∗Tφnx¯^n+εn+kne1−α¯˙n+1υnθ˜nTθ˜.n+ξnξ˙n.
By using Young’s inequality,(71)Enεn≤12En2+12εn∗2.
From (71), one can obtain(72)V˙n≤V˙n−1+Engnu+θnTφnx¯^n+12En+kne1−α¯˙n+12εn∗2+1υnθ˜nTυnEnφnx¯^n−θ.n+ξnξ˙n.
Select the actual control inputu and the adaptive laws θn as(73)u=NμnωnEn+12En+θnTφnx¯^n+kne1−α¯˙n,(74)μ˙n=EnωnEn+12En+θnTφnx¯^n+kne1−α¯˙n,(75)θ.n=υnEnφnx¯^n−σθn.
Substituting (73)–(75) into (72) yields(76)V˙n≤−r1e2+P2∑l=1nθ˜lTθ˜l−ω1ϖz12−∑l=1n−1σ2υlθ˜lTθ˜l+12∑l=1n−1gl∗2El+12+ξl+12+∑l=1nξlξ˙l−∑l=2nωlEl2+∑l=1ngNμl+1μ˙l+Mn−1+12εn∗2+συnθ˜nTθn.
Combining Young’s inequality and inequality (64), one can obtain(77)συnθ˜nTθn=συnθ˜nT−θ˜n+θn∗≤−σ2υnθ˜nTθ˜n+σ2υnθn∗Tθn∗,(78)ξnξ˙n≤−ξn2χn+ξnBn≤−1χn−Bn22πnξn2+πn2.
Substitute (77) and (78) into (76) as follows(79)V˙n≤−r1e2+P2∑l=1nθ˜lTθ˜l−ω1ϖz12−∑l=1nσ2υlθ˜lTθ˜l+12∑l=1n−1gl∗2El+12+ξl+12+Mn−∑l=1n1χn−Bn22πnξn2−∑l=2nωlEl2+∑l=1ngNμl+1μ˙l,where Mn=Mn−1+1/2εn∗2+πn/2+σ/2υlθl∗Tθl∗.The following theorem is summarized by the above controller design and stability analysis.Theorem 1.
Consider a class of nonlinear pure-feedback systems (1), with uncertain functions, unmeasurable states, and input delay, and the state observer is designed as (16). Under Assumptions1–5, the adaptive laws are designed as (48), (60), and (73). The virtual control functions are chosen as (50) and (62). And the actual control input function (75) can guarantee all the signals in the closed-loop system are SGUUB. Meanwhile, the tracking error does not deviate from the prescribed performance bound (31), and the observer errors converge within a small zero region.Proof.
Let(80)c¯=min2r1λminP,2υlσ2υl−P2,2ϖω1,2ωk,21χn−Bn22πn,k=2,…,n;l=1,2,…,n,where c¯ will be positive by choosing appropriate parameters.Then, (79) can be finally expressed as(81)V˙n≤−c¯Vn+∑l=1ngNμl+1μ˙l+12∑l=1n−1gl∗2El+12+ξl+12+Mn,where Mn=P2ε∗2+P2d∗2+∑l=1n1/2εl∗2+∑l=inπl/2+∑l=1nσ/2υlθl∗Tθl∗ is bounded.Multiplying both sides of (81) by e−c¯t generates(82)dVne−c¯tdt≤∑l=1ngNμl+1μ˙le−c¯t+Mne−c¯t+12∑l=1n−1gl∗2El+12+ξl+12e−c¯t.Integrating the above inequality over0,tf and then multiplying both sides by e−c¯t yields(83)Vn≤V0e−c¯t+Mn+1c¯1−e−c¯t+e−c¯t∫0t∑l=1ngNμl+1μ˙le−c¯τdτ+e−c¯t∫0t12∑l=1n−1gl∗2El+12+ξl+12e−c¯τdτ.Based on Lemma 2,Vn and μ can be proved to be bounded. In addition, since El+1 and ξl+1 are semiglobal and ultimately uniformly bounded, e−c¯t∫0t1/2∑l=1n−1gl∗2El+12+ξl+12e−c¯τdτ is bounded. Therefore, the stability of the whole closed-loop system is demonstrated. Furthermore, since z1 can be proved to be uniformly bounded, the prescribed performance tracking control is achieved.


## 4. Simulation Example

Consider a class of nonlinear pure-feedback systems with unavailable states, unknown control direction, and input delay:(84)x˙1=f1x1,x2+d1t,x˙2=f2x¯2,ut−δ+d2t,y=x1,where x1 and x2 are the system states and u and y are the system input and output, respectively. The smooth functions are used as f1x1,x2=x1+x1x2 and fx¯2,ut−δ=2+sinx1x21+ut−δ, and the external disturbances in this simulation are given as d1t=0.02cost and d2t=0.03sint. The input delay is chosen as δ=0.01, and the reference signal is given as yd=sint.The parameters in control functions and adaptive laws are given asω1=0.1, ω2=40, υ1=υ2=5, and σ1=σ2=20. Parameter in a first-order filter is ζ2=0.05. As for the state observer, the observer gains are selected as K=k1,k2T=10,100T. In predefined performance controls, ιu0=ιd0=30, λd=λu=0.8, λ=2, hu=hd=0.4, and ρ∞=0.4. The initial value of the system states are x10=0.3 and x20=0.5, and the initial values for the other parameters are zero.The simulation results are shown in Figures1–7, where the red and blue lines represent the approach proposed in this article, and the black line represents the removal of the variable-parameter predefined performance. Figure 1 shows the output trajectories of y and the expected output signal yd. The output tracking error and the prescribed performance boundaries are shown in Figures 2, and Figures 3 and 5 illustrate the trajectories of system states x1 and x2 and their estimates x^1 and x^2, respectively. The Tracking errors e1 and e2 of state x1 and x2 are shown in Figures 4, and Figures 6 and 7 show the trajectories of the actual control input u. It can be seen from the figure, compared to removing variable-parameter predefined performance, the controller in this paper is obviously better in both the initial oscillation frequency and the tracking effect.Figure 1
Trajectories ofy and yd.Figure 2
Tracking errorE1 with prescribed performance ιuρ and −ιdρ.Figure 3
Trajectory ofx1 and x^1.Figure 4
Tracking errore1.Figure 5
Trajectory ofx2 and x^2.Figure 6
Tracking errore2.Figure 7
The trajectory of actual control inputu.Remark 4.
Compared with the existing literature, an adaptive fuzzy predefined performance controller is proposed in this paper, which makes the tracking error convergence in the preset range better. It can be seen from Figure2 that different from the literatures [34, 37], the novel predefined performance control is a variable-parameter scheme, which relaxes the limitations of known initial error in predefined performance. In addition, from the simulation data, it can be seen that the simultaneous consideration of input delay and pure-feedback system brings great difficulty to controller design. The control method proposed in this paper has excellent control performance, which is shown in Figure 1.


## 5. Conclusion

In this paper, an observer-based adaptive fuzzy predefined performance controller has been introduced of a class of nonlinear pure-feedback systems with unknown control direction and input delay. A novel predefined performance with variable-parameter scheme has been investigated, which solved the problem of unknown initial error. In order to overcome system complexity caused by input delay and pure-feedback systems, the Pade approximation and mean value theorem has been employed, respectively. In addition, Nussbaum functions have been used to deal with the unknown control direction and a first-order filter has applied to approximate repeated differentiations problem of the virtual controllers. State observer and FLSs have been proposed to estimate the unmeasured states and approximate the unknown nonlinear functions, respectively. Therefore, it has been proved that stability of the entire closed-loop system is SGUUB in limitation of the predefined performance control. The tracking errors have converged within a predefined range, while the observer estimation errors have converged within a small zero region. Finally, the simulation results have verified the effectiveness of the proposed control method. In the future research, an observer-based adaptive fuzzy predefined performance controller will be considered in multiagent systems.



---

*Source: 8859328-2020-12-10.xml*