# Adaptive Backstepping Control for a Class of Uncertain Nonaffine Nonlinear Time-Varying Delay Systems with Unknown Dead-Zone Nonlinearity

Authors: Wei-Dong Zhou; Cheng-Yi Liao; Lan Zheng
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

An adaptive backstepping controller is constructed for a class of nonaffine nonlinear time-varying delay systems in strict feedback form with unknown dead zone and unknown control directions. To simplify controller design, nonaffine system is first transformed into an affine system by using mean value theorem and the unknown nonsymmetric dead-zone nonlinearity is treated as a combination of a linear term and a bounded disturbance-like term. Owing to the universal approximation property, fuzzy logic systems (FLSs) are employed to approximate the uncertain nonlinear part in controller design process. By introducing Nussbaum-type function, the a priori knowledge of the control gains signs is not required. By constructing appropriate Lyapunov-Krasovskii functionals, the effect of time-varying delay is compensated. Theoretically, it is proved that this scheme can guarantee that all signals in closed-loop system are semiglobally uniformly ultimately bounded (SUUB) and the tracking error converges to a small neighbourhood of the origin. Finally, the simulation results validate the effectiveness of the proposed scheme.


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## Body

## 1. Introduction

In the past decade, adaptive backstepping design technique has received a great deal of attention since it was pioneered by Kanellakopoulos et al. in 1991 [1]. In [2–4], adaptive backstepping is utilized to construct robust adaptive backstepping controller. The main feature of this approach is that it can handle nonlinear systems without satisfying the matching conditions, but the backstepping design procedure has a shortcoming named explosion of complexity because of the repeated differentiations of virtual controllers. By using dynamic surface control technique, the explosion of complexity shortcoming is overcome [5]. References [6, 7] develop a command filtered backstepping approach which is feasible even when the number of iterations of the backstepping method is large. However, it should be noted that the nonlinear functions are all assumed to be known in the abovementioned methods. Recently, many adaptive backstepping controllers with FLSs or neural networks (NNs) have been developed for nonlinear systems in strict feedback form [8–27]. Owing to the universal approximation property of FLSs or NNs, these control approaches do not require the precise knowledge of system nonlinearities. Nevertheless, the introduced FLSs or NNs may lead to a burdensome computation when the number of the parameters which need to be tuned by online learning laws increases significantly. To handle the inevitable weakness meeting when increasing the number of fuzzy rules or neural network nodes, the optimal weighting vector in FLSs is used as the estimation parameter [8, 9]. In [10, 14, 19, 21, 25, 27], FLSs are utilized to directly approximate the desired control signals instead of the unknown nonlinearities in each backstepping design step. Consequently, the number of parameters needed to be adapted is significantly reduced for only one parameter needed to be estimated online no matter how many fuzzy rules are selected. On the basis of the work in [10], a novel adaptive fuzzy backstepping controller construct method without requirement of the fuzzy basis functions is exploited [22, 23].Dead-zone characteristic is one of the most common actuator nonsmooth nonlinearities encountered in many industrial processes, which can seriously affect the system performance and indeed make the system unstable. Many controller design schemes are developed for systems with unknown dead zone [2, 3, 15–17, 28–36]. Generally, the dead zone is first treated as a combination of a linear and a bounded disturbance-like term, and then the controller that can achieve a good control performance is designed by adopting robust control technique [16, 17, 28–31]. In [32], a novel two-layered fuzzy logic controller which consists of a fuzzy logic-based precompensator and a usual fuzzy PD controller are developed for controlling systems with dead zone. In [33, 34], by introducing a fuzzy logic dead-zone compensator two fuzzy controllers are constructed for motion control system and a DC motor system, respectively. Nevertheless, when there are no suitable rules for the dead-zone nonlinearity, this method may be unfeasible for it depends much on operators or experts experience. In [2, 3, 15, 35, 36], the inverse function of dead zone is utilized to compensate the effect of the dead zone. Using this method, an effective control has been achieved, but the shortcoming that the dead-zone parameters are required to be constants is inevitable. Regrettably, although much progress has been made in the fields of controller design for nonlinear systems with unknown dead zone, nonaffine nonlinear systems with unknown dead zone are seldomly investigated.Time delays frequently occur in practical control systems, such as electrical networks and hydraulic systems. Considering that the existing time delays often cause system instability and performance deterioration, to handle the control problem for systems with time delays is an unavoidable issue. Two main tools Lyapunov-Krasovskii functionals and Lyapunov-Razumikhin functions are usually applied to nonlinear time-delay systems [4, 17–25, 37–39]. In [17–19, 22–24], Lyapunov-Krasovskii functionals are constructed to compensate the unknown time delays. Within these schemes, the condition that the unknown time delays are assumed to be unknown constants is too strict. To solve time-varying delays problem, a novel Lyapunov-Krasovskii functionals are designed on condition that the derivative of time delay functions is less than one [20, 25, 30, 37]. In [4, 21, 38], Lyapunov-Razumikhin lemma-based adaptive backstepping control approaches are proposed for nonlinear systems in which the limitation condition on the derivative of time delay is cancelled. In [17, 24, 25], adaptive fuzzy or neural backstepping controllers are designed for a class of nonlinear time-delay systems with unknown control directions. As control direction, that is, the sign of control gain, decide the direction along which the controller parameters are updated, designing adaptive controllers for these unknown systems with the control direction becoming much more difficult. Nussbaum-type function is utilized to deal with the unknown control direction [17, 24, 25]. A robust adaptive NNs controller is first proposed for a class of nonlinear time-delay systems with unknown dead-zone nonlinearity and unknown control direction [17]. However, in this method, the time delay is supposed to be unknown constants and the NNs introduced to approximate the uncertain nonlinear term may result in complexity computation when the dimension of system increases.Inspired by the preceding discussion, in this paper, a class of nonaffine nonlinear time-varying delay systems with both unknown dead-zone input and completely unknown control direction is investigated and an adaptive fuzzy backstepping control scheme is exploited. The main contributions of this paper can be summarized as follows.(1) Few papers consider nonaffine systems with unknown dead-zone nonlinearity. The difficulty of design controller for nonaffine systems is that the control input appears nonlinear in unknown nonlinear systems. Mean value theorem is used to transform the nonaffine form into an affine form, and then the existing approaches for affine systems can be directly applied [40]. (2) Similar to [10], FLSs are directly employed to approximate the unknown nonlinearities. Considering the norm of the ideal weighting vector in FLSs as the estimation parameter instead of the elements of weighting vector, there is only one parameter that needs to be estimated online in each step. Meanwhile, it should be noted that in this control approach, the basic functions of FLSs do not occur in the control laws and adaptive laws. This improvement can overcome the explosion of complexity caused by repeated differentiations of virtual controllers and the increase of system dimension. (3) The other encountered trouble is how to cope with the unknown time delay terms in system. Compared with [17], the time delay term considered in this paper is time varying and the novel Lyapunov-Krasovskii functionals are employed to stability analysis and synthesis. In particular, here, the reason we use Lyapunov-Krasovskii functionals to construct controller is that this method can provide less conservative and delay-independent results. Using the Lyapunov stability theorem, it is proved that the proposed control schemes can guarantee that all the signals in closed-loop system are bounded and the tracking error is asymptotic convergence. Finally, effectiveness of the developed scheme is demonstrated by the simulation examples.


## 2. Problem Formulation and Preliminaries

Consider the following SISO nonaffine nonlinear time-varying delay system:(1)x˙i(t)=fi(x-i(t))+gi(x-i(t))xi+1(t)+di(t)+hi(x1(t-τ1(t)),…,xi(t-τi(t))),x˙n(t)=f(x(t),D(u(t)))+dn(t)+hn(x1(t-τ1(t)),…,xn(t-τn(t))),y(t)=x1(t),i=1,…,n-1,
where u∈R and y∈R are the system control input and output, respectively, x-i(t)=[x1(t),…,xi(t)]T and x(t)=[x1(t),…,xn(t)]T are system states, fi(·), gi(·), and f(·) are unknown smooth functions, and i=1,…,n-1. hi(·) is an unknown smooth function with the unknown time-varying delay terms τi(t). There exists a positive constant τ0 satisfying 0≤τi(t)≤τ0<+∞, and φi(s)=xi(s), ∀s∈[-τ0,0] is an unknown continuous bounded initial function. di(t) denotes the unknown external disturbance, i=1,…,n. D(u(t)) is the unknown dead-zone input.The control objective is to design an adaptive backstepping controller for system (1) such that the system output y(t) tracks the desired trajectory yd(t) and all signals in closed-loop system are bounded.Utilizing the mean value theorem [40], function f(·) in (1) can be rewritten as
(2)f(x(t),D(u(t)))=f(x(t),0)+∂f(x(t),D(u(t)))∂D(u(t))|D(u)=λD(u(t))=fn(x(t))+gn(x(t))D(u(t)),
with fn(x(t))=f(x(t),0) and gn(x(t))=∂f(x(t),D(u(t)))/∂D(u(t))|D(u(t))=λ; λ∈(0,D(u)).Assumption 1.
There exist constantsg_ and g- satisfying 0<g_≤|gi(x-i(t))|≤g-.The nonsymmetric dead-zone input is defined as [17]
(3)D(u(t))={φr(t)(u(t)-mr),u(t)>mr,0,-ml≤u(t)≤mr,φl(t)(u(t)+ml),u(t)<-ml,
where φr(t) and φl(t) are unknown right and left slopes of the dead zone and mr and ml are breakpoints of the dead zone. To deal with dead-zone nonlinearity, the following assumptions are put forward.Assumption 2.
m
r and ml are unknown bounded constants. φr(t) and φl(t) are unknown functions and there are unknown positive constants φr0, φr1, φl0, and φl1 satisfying
(4)0<φr0≤φr(t)≤φr1,0<φl0≤φl(t)≤φl1.Define vectorsη(t) and κ(t) as
(5)η(t)=[ηr(t),ηl(t)]T,κ(t)=[φr(t),φl(t)]T,
with ηr(t)={1,u(t)≥-ml0,u(t)<-ml and ηl(t)={1,u(t)≤mr0,u(t)>mr.Based on the above analysis, the dead zone can be expressed as(6)D(u)=ηT(t)κ(t)u(t)+d(u(t)),
where
(7)d(u(t))={-φr(t)mr,u(t)≥mr,-(φr(t)+φl(t))u(t),-ml<u(t)<mr,φl(t)ml,u(t)≤-ml.According to Assumption2, it can be concluded that |d(u(t))|≤d0 and d0 is an unknown positive constant meeting d0=(φr1+φl1)max⁡{mr,ml}. Considering the definition of η(t) and κ(t), we have
(8)ηT(t)κ(t)={φr(t),u(t)≥mr,φr(t)+φl(t),-ml<u(t)<mr,φl(t),u(t)≤-ml.It is easy to obtain thatηT(t)κ(t)∈[min⁡(φr0,φl0),φr1+φl1]; that is, ηT(t)κ(t) is a positive discontinuous bounded function.Using (2) and (6), system (1) can be transformed into the following form:
(9)x˙i(t)=fi(x-i(t))+gi(x-i(t))xi+1(t)+di(t)+hi(x1(t-τ1(t)),…,xi(t-τi(t))),x˙n(t)=fn(x)+gn(x)ηT(t)κ(t)u(t)+gn(x)d(u(t))+hn(x1(t-τ1(t)),…,xn(t-τn(t)))+dn(t),y(t)=x1(t).Notation.
x
i, xi(τi), y, u, di, fi, and gi denote xi(t), xi(t-τi(t)), y(t), u(t), di(t), fi(x-i), and gi(x-i), respectively.Assumption 3.
The unknown smooth functionshi(x1(τ1),…,xi(τi)) satisfy the inequality
(10)|hi(x1(τ1),…,xi(τi))|≤∑j=1iHi,j(xj(τj)),
where Hi,j(·) are unknown positive smooth functions.Remark 4.
Compared with [17] in which the bounding functions are required to be known, it should be emphasized that the nonlinear functions Hi,j(·) are unknown in this paper.Assumption 5.
The time derivatives of the time-varying delay termsτi are τ˙i and satisfy τ˙i≤τ*<1 where τ* is an unknown positive constant.Assumption 6.
The external disturbancesdi satisfy |di|≤d-i, where d-i is defined as an unknown positive constant.Remark 7.
The constantsgi0, gi1, τ0, τ*, and d-i are only required for analytical purposes and their values are not necessarily known in control laws and adaptive laws.Before we derive our results, the FLSs and Nussbaum-type function should be introduced.The FLS has a basic configuration which contains fuzzifier, fuzzy rule base, fuzzy reference engine, and defuzzifier, such four components. The fuzzy rule base is composed of a series If-Then inference rules in the following form [41]:
(11)R(l):Ifx1isF1land⋯andxnisFnl,hhhhhhThenyisGl,l=1,…,M,
where x=[x1,x2,…,xn]T and y are the FLS inputs and output, respectively. Fil, i=1,…,n, and Gl are fuzzy sets characterized by fuzzy membership functions μFil(xi) and μGl(y), respectively, and M is the number of fuzzy rule. The final output of the fuzzy system can be expressed by using the singleton fuzzifier, product inference engine, and center-average defuzzifier as follows [41]:
(12)y(x)=∑l=1My-l(∏i=1nμFil(xi))∑l=1M(∏i=1nμFil(xi)),
where y-l is the point at which the membership function μGl(y) achieves its maximum value and we assume that μGl(y-l)=1. Let θ=[y-1,…,y-M]T be a vector grouping all consequent parameters and ξ(x)=[ξ1(x),…,ξM(x)]T, where ξj(x)≜∏i=1nμFij(xi)/∑l=1M(∏i=1nμFil(xi)), j=1,2,…,M is the vector of fuzzy basis function. Then, using the conception of fuzzy basis functions [41], the output of the fuzzy logic system can be formulated as y(x)=θTξ(x). Then according to the universal approximation theorem, any continuous nonlinear function f(x) can be approximated by the FLS as
(13)f(x)=θf*Tξ(x)+ε(x),
where θf* is an optimal parameter satisfying θf*=argmin⁡θf⁡(sup⁡x∈x⁡|f^(x)-f(x)|) and ε(x) is the minimum approximation error satisfying |ε(x)|≤b0 (b0 is a positive constant).Nussbaum-type function is successfully applied to cope with the problem caused by unknown control direction [17, 24, 25, 27]. A function which has the following properties is called Nussbaum function [42]:
(14)lim⁡k→∞sup⁡1k∫0kN(ς)dς=+∞,lim⁡k→∞inf⁡1k∫0kN(ς)dς=-∞.Functions, such asς2cos⁡(ς), ς2sin(ς) and exp⁡(ς2)cos⁡((π/2)ς), are commonly used as Nussbaum functions for nonlinear systems with unknown control direction. In this paper, the Nussbaum function N(ς)=ς2cos⁡(ς) is employed.Lemma 8 (see [17]).
V
(
t
) and ς(t) are smooth functions with V(t)≥0. If the inequality V(t)≤c0+e-c1t∫0t[g(x(τ))N(ς)+1]ς˙ec1τdτ holds with t∈[0,tf), c0 is a suitable constant, N(ς)=ς2cos⁡(ς), c1 is a positive constant, and g(x) is a time-varying parameter which takes values in unknown closed intervals I=[g_,g-] and 0∉I, then V(t), ς(t), and ∫0tg(x(τ))N(ς)ς˙dτ must be bounded on [0,tf).


## 3. Controller Design and Stability Analysis

In this section, an adaptive fuzzy control scheme is presented by using backstepping technique combined with Lyapunov-Krasovskii functionals and Nussbaum type functions. The backstepping design is based on the following change of coordinates:(15)χ1=x1-ydχi=xi-αi-1,i=2,…,n,
where αi-1 is a virtual control which should be designed for the corresponding (i-1)th subsystem. In general, the design procedure contains n steps. FLSs are employed to approximate the unknown nonlinear term. Then, let us define unknown constants satisfying
(16)ϑi=∥θi∥2,i=1,…,n.Forϑi are unknown and ϑ^i are used to estimate ϑi with estimation errors ϑ~i defined as ϑ~i=ϑi-ϑ^i.The detailed design procedure is described in the following steps.Step 1.
Consider the Lyapunov-Krasovskii function as(17)V1=12χ12+12γ1ϑ~12+V1,0,
where V1,0=(e-r(t-τ0)/1-τ*)∫t-τ1(t)tersH1,12(x1(s))ds, γ1 and r are design positive parameters.Giving a compact setΩκ1 as Ωκ1={χ1∣|χ1|<0.2554κ1} with κ1, a positive design parameter, then a function defined as follows will be employed to design controller
(18)q1(χ1)={1,χ1∉Ωκ1,0,χ1∈Ωκ1.We choose the virtual control lawα1 and adaptive laws as
(19)α1=q1(χ1)2a12N(ς1)χ1ϑ^1,(20)ς˙1=q1(χ1)2a12χ12ϑ^1,(21)ϑ^˙1=γ12a12χ12-η1ϑ^1,
where a1 and η1 are design positive parameters.Case  1. In this case, we have |χ1|<0.2554κ1. Apparently, the tracking error χ1 is bounded. According to (18)–(20), we can conclude that when we select bounded initial values, α1 is bounded and ς˙1=0; that is, N(ς1) is bounded. Integrating (21) over [0,t] we get that signal ϑ^1 is bounded.Case  2. When |χ1|≥0.2554κ1, the following process is needed.The time derivative ofV1 is
(22)V˙1≤χ1(f1+g1x2+h1(x1(τ1))+d1-y˙d)-1γ1ϑ~1ϑ^˙1-rV1,0+e-r(t-τ0)1-τ*[ertH1,12(x1)+e-r(t-τ0)1-τ*=-(1-τ˙1(t))er(t-τ1(t))H1,12(x1(τ1))]≤χ1(f1+g1x2+h1(x1(τ1))+d1-y˙d)-1γ1ϑ~1ϑ^˙1-rV1,0+H1(x1)-H1,12(x1(τ1)),
with H1(x1)=erτ0H1,12(x1)/(1-τ*).When Assumption3 holds, we get
(23)|h1(x1(τ1))|2≤H1,12(x1(τ1)).By using Young’s inequality and combining with (23), (22) yields
(24)V˙1=χ1g1(χ2+α1)+χ1(f1+d1-y˙d)+14χ12+h12(x1(τ1))-1γ1ϑ~1ϑ^˙1-rV1,0+H1(x1)-H1,12(x1(τ1))≤-c1χ12+χ22+χ1g1α1+χ1(f1-y˙d+c1χ1+12ρ2χ1+14g12χ1+34χ1)-12χ12+12ρ2d-12-1γ1ϑ~1ϑ^˙1+H1,12(x1(τ1))-rV1,0+H1(x1)-H1,12(x1(τ1))=-c1χ12-rV1,0+χ22+χ1g1α1+χ1f1--1γ1ϑ~1ϑ^˙1+(1-16tanh2(χ1κ1))H1(x1)-12χ12+12ρ2d-12,
where c1 and ρ are positive constants
(25)f1-=f1-y˙d+c1χ1+12ρ2χ1+14g12χ1+34χ1+16χ1tanh2(χ1κ1)H1(x1).Remark 9.
Note that the functionHi(xi)/χi (i=1,…,n) is not defined at χi=0 which leads to the fact that it cannot be approximated by FLSs. To cope with this difficulty, we introduce function (16/χi)tanh2(χi/κi)Hi(xi) instead of Hi(xi)/χi according to the effective approach in [19]. The design parameter κi can be adjusted to achieve better performance.Asf1- consists of unknown nonlinear functions f1, g1, and H1(x1), f1- cannot be directly used to construct controller. According to the universal approximation property of FLSs, f1- can be rewritten as
(26)f1-=θ1*Tξ1(Z1)+δ1(Z1),|δ1(Z1)|≤ε1,
where δ1(Z1) stands for approximation error and ε1 is an unknown constant, Z1=[x1,yd,y˙d]T.Considering the inequality0<ξ1T(Z1)ξ1(Z1)≤1 we obtain
(27)χ1f1-=χ1θ1*Tξ1(Z1)+χ1δ1(Z1)≤12a12χ12ϑ1+12a12+12χ12+12ε12.As|χ1|≥0.2554κ1, we have
(28)(1-16tanh2(χ1κ1))H1(x1)≤0.Substituting (27) and (28) into (24) results in
(29)V˙1≤-c1χ12-rV1,0+χ22+χ1g1α1+12a12χ12ϑ1+12a12+12ε12+12ρ2d-12-1γ1ϑ~1ϑ^˙1.Consideringq(χ1)=1 and using (19)–(21), (29) yields
(30)V˙1≤-c1χ12-rV1,0+χ22+[g1N(ς1)+1]ς˙1+12a12+12ε12+12ρ2d-12+η1γ1ϑ~1ϑ^1≤-μ1V1+χ22+[g1N(ς1)+1]ς˙1+C1,
with μ1=min⁡(2c1,η1,r) and C1=(1/2)a12+(1/2)ε12+(1/2)ρ2d-12+(η1/2γ1)ϑ12.Multiplying (30) by eμ1t and then integrating over [0,t], we get
(31)V1≤β1+e-μ1t∫0t[g1N(ς1)+1]ς˙1eμ1τdτ+e-μ1t∫0tχ22eμ1τdτ,
where β1=(C1/μ1)+(V1(0)-(C1/μ1))e-μ1t.Noting that there is an extra terme-μ1t∫0tχ22eμ1τdτ within (31), we suppose that χ2 can be regulated as bounded; then we have
(32)e-μ1t∫0tχ22eμ1sds≤e-μ1tsup⁡s∈[0,t](χ22(s))∫0teμ1sds≤1μ1sup⁡s∈[0,t](χ22(s)).As the boundedness of the extra terme-μ1t∫0tχ22eμ1τdτ is obtained from (32), directly applying Lemma 8, we get the conclusion that signals V1(t), ς1(t), and ∫0tg1N(ς1)ς˙1dτ hence χ1, N(ς1), ϑ^1, and ϑ~1 are all bounded on [0,tf). Consequently, if χ2 is bounded, we can get the conclusion that all signals in Step 1 are bounded. In addition, the boundedness of χ2 will be proved in step 2 (see Stepk).Step k (2≤k≤n-1).
Considering that steps2≤k≤n-1 have a similar procedure, Stepk is presented as follows.
Choose the following Lyapunov-Krasovskii function:(33)Vk=12χk2+12γkϑ~k2+Vk,0,
with γk being a design positive parameter,
(34)Vk,0=e-r(t-τ0)1-τ*{∑i=2k∑j=1i∫t-τj(t)tersHi,j2(xj(s))dse-r(t-τ0)1-τ*w+12∫t-τ1(t)tersH1,12(x1(s))ds∑i=2k∑j=1i∫t-τj(t)tersHi,j2(xj(s))ds}.Similar to Step1, the virtual control law αk and adaptive laws are designed as
(35)αk=qk(χk)2ak2N(ςk)χkϑ^k,(36)ς˙k=qk(χk)2ak2χk2ϑ^k,(37)ϑ^˙k=γk2ak2χk2-ηkϑ^k,
with ak and ηk being design positive parameters; the function qk(χk) is defined as
(38)qk(χk)={1,χk∉Ωκk0,χk∈Ωκk,
where Ωκk∶={χk∣|χk|<0.2554κk} stands for a compact set and κk is a design positive parameter which decides the size of convergence region.Case  1. In this case, we suppose that |χk|<0.2554κk. It is obvious that the tracking error χk is bounded. From (35)–(37), when selecting bounded initial values, we achieve the boundedness of αk, ςk, and N(ςk). After integrating (37) over [0,t], we conclude that signal ϑ^k is bounded.Case  2. In case 2, the tracking error satisfies |χk|≥0.2554κk.The time derivative ofVk,0 is
(39)V˙k,0=-rVk,0+e-r(t-τ0)1-τ*×{∑i=2k∑j=1i(ertHi,j2(xj)-(1-τ˙j)er(t-τj(t))Hi,j2(xj(τj)))+12ertH1,12(x1)-12(1-τ˙j)er(t-τj(t))H1,12(x1(τ1))∑i=2k∑j=1i(ertHi,j2(xj)-(1-τ˙j)er(t-τj(t))Hi,j2(xj(τj)))}≤-rVk,0+Hk(x-k)-∑i=2k∑j=1iHi,j2(xj(τj))-12H1,12(x1(τ1)),
where Hk(x-k)=(erτ0/(1-τ*)){∑i=2k∑j=1iHi,j2(xj)+(1/2)H1,12(x1)}.Then the time derivative ofχk is
(40)χ˙k=fk+gkxk+1+hk(x1(τ1),…,xk(τk))+dk-α˙k-1,
with
(41)α˙k-1=∑i=1k-1[∂αk-1∂ςi(12ai2ei2ϑ^i)w+∂αk-1∂xi(fi+gixi+1+hi(x1(τ1),…,xi(τi))w+∂αk-1∂xie+di(t))+∂αk-1∂ϑ^i(γi2ai2ei2-γiϑ^i)]+∂αk-1∂ydy˙d.Remark 10.
Hereαk-1 is the function of αk-2. When k=2, we have αk-2=yd; that is, αk-1 is the function of xk-1, yd, ςk-1, and ϑ^-k-1, where ςk-1=[ς1,…,ςk-1]T and ϑ^-k-1=[ϑ^1,…,ϑ^k-1]T. Then we derive the time derivative of αk-1 in (41).Using (40) and (41), the derivative of Vk is
(42)V˙k=χk(fk+gkxk+1+hk(x1(τ1),…,xk(τk))=χkw+dk-α˙k-1)-1γkϑ~kϑ^˙k+V˙k,0=χk{∑i=1k-1[∂αk-1∂ςiς˙ix˙i+]fk+gkxk+1+hk(x1(τ1),…,xk(τk))=weχk+dk-∑i=1k-1[∂αk-1∂ςiς˙i+∂αk-1∂xix˙i+∂αk-1∂ϑ^iϑ^˙i]=weχk-∂αk-1∂ydy˙d∑i=1k-1[∂αk-1ϑ^iϑ^˙i]}-1γkϑ~kϑ^˙k+V˙k,0.Owing to Assumption3, we get
(43)|hi(x1(τ1),…,xi(τi))|2≤2∑j=1iHi,j2(xj(τj)),hhhhhhhh2≤i≤n.Utilizing Young’s inequality (42) yields
(44)V˙k≤-ckχk2+χkgkαk+χk+12+χk{fk-∑i=1k-1[∂αk-1∂ςi(12ai2χi2ϑ^i)+∂αk-1∂xi(fi+gixi+1)+χk++χk++∂αk-1∂ϑ^i(γi2ai2χi2-γiϑ^i)]-∂αk-1∂ydy˙d+χkw+(12ρ2+12)∑i=1k-1(∂αk-1∂xi)2χk+χkw+(12ρ2+ck+14gk2+1)χk∑i=1k-1[∂αk-1∂ςi(12ai2χi2ϑ^i)+∂αk-1∂xi(fi+gixi+1)}-12χk2+∑i=2k∑j=1iHi,j2(xj(τj))+12H1,12(x1(τ1))+12∑i=1kρ2d-i2-1γkϑ~kϑ^˙k+V˙k,0.Substituting (39) into (44) results in
(45)V˙k≤-ckχk2-rVk,0+χkgkαk+χk+12+χkf-k-12χk2+12∑i=1kρ2d-i2-1γkϑ~kϑ^˙k+(1-16tanh2(χkκk))Hk(x-k),
with
(46)fk-=fk-∑i=1k-1[∂αk-1∂ςi(12ai2χi2ϑ^i)+∂αk-1∂xi(fi+gixi+1)fk-fk-+∂αk-1∂ϑ^i(γi2ai2χi2-γiϑ^i)]-∂αk-1∂ydy˙d+(12ρ2+12)∑i=1k-1(∂αk-1∂xi)2χk+(12ρ2+ck+14gk2+1)χk+16χktanh2(χkκk)Hk(x-k).Similarly,fk- can be approximated by FLSs to an arbitrary given accuracy as
(47)fk-=θk*Tξk(Zk)+δk(Zk),|δk(Zk)|≤εk,
where Zk=[xk,yd,y˙d,ςk-1,ϑ^-k-1]T, δk(Zk) represents approximation error, and εk is an unknown positive constant.As the fuzzy basis functionξk(Zk) satisfies 0<ξkT(Zk)ξk(Zk)≤1, we get the following inequality:
(48)χkfk-=χkθk*Tξk(Zk)+χkδk(Zk)≤12ak2χk2ϑk+12ak2+12χk2+12εk2.Substituting (48) into (45), we obtain
(49)V˙k≤-ckχk2-rVk,0+χkgkαk+χk+12-1γkϑ~kϑ^˙k+12ak2χk2ϑk+12ak2+12εk2+12∑i=1kρ2d-i2+(1-16tanh2(χkκk))Hk(x-k).As|χk|≥0.2554κk, we can derive
(50)(1-16tanh2(χkκk))Hk(x-k)≤0.Applying (35)–(37) and (50), (49) produces
(51)V˙k≤-μkVk+χk+12+[gkN(ςk)+1]ς˙k+Ck,
where μk=min⁡(2ck,r,ηk) and Ck=(1/2)ak2+(1/2)εk2+(1/2)∑i=1kρ2d-i2+(ηk/2γk)ϑk2.Multiplying (51) by eμkt results in
(52)ddt(Vkeμkt)≤eμktχk+12+[gkN(ςk)+1]ς˙keμkt+Ckeμkt.Integrating (52) over [0,t], we obtain
(53)Vk≤βk+e-μkt∫0t[gkN(ςk)+1]ς˙keμkτdτ+e-μkt∫0tχk+12eμkτdτ,
with βk=(Ck/μk)+(Vk(0)-(Ck/μk))e-μkt.Remark 11.
The discussion of (53) is similar to the analysis of (31). If χk+1 can be regulated as bounded, by utilizing Lemma 8, the boundedness of signals Vk(t), ςk(t), and ∫0tgkN(ςk)ς˙kdτ is achieved. Thus, we can guarantee that signals χk, N(ςk), ϑ^k, and ϑ~k are all bounded on [0,tf). The effect of the extra term e-μkt∫0tek+12eμkτdτ will be handled in the next step.Step n.
Consider Lyapunov-Krasovskii function as follows:(54)Vn=12χn2+12γnϑ~n2+Vn,0,
where γn is a design positive parameter, and
(55)Vn,0=e-r(t-τ0)1-τ*{∑i=2n∑j=1i∫t-τj(t)tersHi,j2(xj(s))dse-r(t-τ0)1-τ*h+12∫t-τ1(t)tersH1,12(x1(s))ds∑i=2n∑j=1i∫t-τj(t)tHi,j2(xj)}.We choose the following actual control inputu and adaptive laws:
(56)u=qn(χn)2an2N(ςn)χnϑ^n,(57)ς˙n=qn(χn)2an2χn2ϑ^n,(58)ϑ^˙n=γn2an2χn2-ηnϑ^n,
where an and ηn are design positive parameters. The function qn(χn) is defined as
(59)qn(χn)={1,χn∉Ωκn,0,χn∈Ωκn,
with Ωκn∶={χn∣|χn|<0.2554κn} denoting a compact set and κn is a design positive parameter.Similarly, we analyze thenth-subsystem from two cases.Case  1. In Case 1, χn satisfies |χn|<0.2554κn. As κn is a positive design parameter; we obtain that χn is bounded. In addition, we can conclude that the signals u, ςn, N(ςn), ϑ^n, and ϑ~n are bounded.Case  2. We suppose that |χn|≥0.2554κn in this case.The time derivative ofVn is
(60)V˙n=χn(fn+gnηTκu+gnd(u)+hn(x1(τ1),…,xn(τn))χne+dn-α˙n-1(fn+gnηTκu+gnd(u)+hn(x1(τ1),…,xn(τn)))-1γnϑ~nϑ^˙n+V˙n,0.From the definition ofαn-1, we obtain
(61)α˙n-1=∑i=1n-1[∂αn-1∂ςi(12ai2χi2ϑ^i)=w+∂αn-1∂xi(fi+gixi+1+hi(x1(τ1),…,xi(τi))+di)=w+∂αn-1∂ϑ^i(γi2ai2χi2-γiϑ^i)]+∂αn-1∂ydy˙d.Applying Young’s inequality and (43), (60) can be rewritten as
(62)V˙n≤-cnχn2+χngnηTκu+χn{fn-∑i=1n-1[∂αn-1∂ςi(12ai2χi2ϑ^i)+∂αn-1∂xi(fi+gixi+1)+χn{fn∑i=1n-1[∂∂ςi+∂αn-1∂ϑ^i(γi2ai2χi2-γiϑ^i)]-∂αn-1∂ydy˙d+χnew+(12ρ2+12)∑i=1n-1(∂αn-1∂xi)2χn+χnew+(12ρ2+cn+1)χn∑i=1n-1[∂αn-1∂ςi∂αn-1∂xi}-12χn2+∑i=2n∑j=1iHi,j2(xj(τj))+12H1,12(x1(τ1))+12∑i=1nρ2d-i2-1γnϑ~nϑ^˙n+V˙n,0.The time derivative ofVn,0 is
(63)V˙n,0=-rVn,0+e-r(t-τ0)1-τ*×{∑j=1iHi,j2-er(t-τj(t))12ertH1,12(x1)-12(1-τ˙1)er(t-τ1(t))H1,12(x1(τ1))+e-r1τ*+∑i=2n∑j=1i(ertHi,j2(xj)-(1-τ˙j)er(t-τj(t))Hi,j2(xj(τj)))}≤-rVn,0+Hn(x-n)-∑i=2n∑j=1iHi,j2(xj(τj))-12H1,12(x1(τ1)),
where Hn(x-n)=(erτ0/(1-τ*)){∑i=2n∑j=1iHi,j2(xj)+(1/2)H1,12(x1)}.By utilizing (63), (62) yields
(64)V˙n≤-cnχn2-rVn,0+χngnηTκu+χnfn--12χn2+12∑i=1nρ2d-i2-1γnϑ~nϑ^˙n+(1-16tanh2(χnκn))Hn(x-n),
with
(65)fn-=fn-∑i=1n-1[∂αn-1∂ςi(12ai2χi2ϑ^i)+∂αn-1∂xi(fi+gixi+1)fn-fn-+∂αn-1∂ϑ^i(γi2ai2χi2-γiϑ^i)]-∂αn-1∂ydy˙d+(12ρ2+12)∑i=1n-1(∂αn-1∂xi)2χn+(12ρ2+cn+1)χn+16χntanh2(χnκn)Hn(x-n).By using FLSs, functionfn- can be approximated as
(66)fn-=θn*Tξn(Zn)+δn(Zn),|δn(Zn)|≤εn,
where Zn=[x-n,yd,y˙d,ςn-1,ϑ^-n-1]T with ςn-1=[ς1,…,ςn-1]T and ϑ^-n-1=[ϑ^1,…,ϑ^n-1]T, δn(Zn) expresses the approximation error, and εn is an unknown positive constant.Similarly, we can derive the following inequality:(67)χnfn-=χnθn*Tξn(Zn)+χnδn(Zn)≤12an2χn2ϑn+12an2+12χn2+12εn2.Considering|χn|≥0.2554κn, we get
(68)(1-16tanh2(χnκn))Hn(x-n)≤0.Applying (56)–(58), (67), and (68), (64) results in
(69)V˙n≤-μnVn+[gnηTκN(ςn)+1]ς˙n+Cn,
where Cn=(1/2)an2+(1/2)εn2+(1/2)∑i=1nρ2d-i2+(1/2)ρ2d02+(ηn/2γn)ϑn2 and μn=min⁡(2cn,r,ηn).Multiplying (69) by eμnt and then integrating it over [0,t], we have
(70)Vn≤βn+e-μnt∫0t[gnηTκN(ςn)+1]ς˙neμnτdτ,
with βn=(Cn/μn)+(Vn(0)-(Cn/μn))e-μnt.Considering functionηT(t)κ(t) satisfies ηT(t)κ(t)∈[min⁡(φr0,φl0),φr1+φl1], and Assumption 2 holds, we can derive |gnηT(t)κ(t)|∈[gn0min⁡(φr0,φl0),(φr1+φl1)gn1]. Noting (70), applying Lemma 8, we can conclude that signals Vn(t), ςn(t), and ∫0tgnηTκN(ςn)ς˙ndτ are bounded. Hence, χn, u, N(ςn), ϑ^n, and ϑ~n are SUUB on [0,tf).The main result is summarized in the following theorem.Theorem 12.
Consider nonaffine nonlinear time-varying delay system (1), when Assumptions 1–6 hold, by applying the control law (56), virtual control laws (19), and (35) and adaptive laws (20), (21), (36), (37), (57), and (58); then with bounded initial conditions, it is guaranteed that all the signals in closed-loop system are SUUB and the tracking error eventually converges to a small neighbourhood of the origin.Proof.
Owing to the previous analysis, we get the conclusion that the term∫0t[gnηTκN(ςn)+1]ς˙ndτ is bounded.
Noting (70), we suppose that the upper bound λn satisfies
(71)∫0t|[gnηTκN(ςn)+1]ς˙n|eμn(τ-t)dτ≤λn.
From (54), (70), and (71), we have
(72)|χn|≤2(βn+λn),|ϑ~n|≤2γn(βn+λn).
Thus, we can conclude the boundedness of the signalsχn, ϑ^n, and ϑ~n.
In the rest of the steps fromn-1 to 1, we acquire
(73)Vi≤βi+e-μit∫0t[giN(ςi)+1]ς˙ieμiτdτ+e-μit∫0tχi+12eμiτdτ,i=1,…,n-1.
As the boundedness ofχi+1 is guaranteed in step i+1, we define an upper bound λi as
(74)∫0t|[giN(ςi)+1]ς˙i|eμi(τ-t)dτ+∫0tχi+12eμi(τ-t)dτ≤λi.
Investigating the definition ofVi(t) and combing (73) and (74), we obtain
(75)|χi|≤2(βi+λi),(76)|ϑ~i|≤2γi(βi+λi).
Hence, signalsχi, ϑ^i, and ϑ~i are bounded.
According to the whole abovementioned analysis, the boundedness of all signals in closed-loop system is proved. The tracking errorχ1 converges to a small neighbourhood of the origin by selecting appropriate design parameters.Remark 13.
According to the above analysis, we know that tracking error depends onai, ϑi, εi, λi, γi, ηi, r, ci, and di-. As ϑi, εi, λi, and di- are unknown, a concrete estimation of the tracking error is impossible. From inequality (75), it is clear that by reducing ai and ηi, meanwhile increasing ci, γi, and r, the tracking error will be diminished. Simultaneously, it is worth pointing out that the parameters ϑi, εi, λi, and di- are not used in the control law and adaptive laws design, which are employed for stability analysis.


## 4. Simulation

In this section, two simulation examples are employed to validate the effectiveness of the proposed adaptive fuzzy tracking control approach. The desired tracking trajectory isyd=sin(t)+cos⁡(0.5t). The dead-zone D(u) is defined as
(77)D(u)={(1-0.3sin(u))(u-0.5),u>0.5,0,-0.25≤u≤0.5,(0.8-0.2cos⁡(u))(u+0.25),u<-0.25.Example 1.
Consider the following nonlinear time-delay system:(78)x˙1(t)=x12(t)+x1(t)x2(t)+0.5x1(t-τ1(t))+0.5cos⁡(t),x˙2(t)=-x1(t)+D(u(t))|D(u(t))|+0.1-2x12(t)x2(t),+x12(t-τ1(t))x2(t-τ2(t))+0.2sin(t),y=x1,
where τ1(t)=0.2(1+cos⁡(t)) and τ2(t)=0.3(1+sin(t)).
According to Theorem12, the control laws and the adaptive laws are chosen as
(79)α1=q1(χ1)2a12N(ς1)χ1ϑ^1,u=q2(χ2)2a22N(ς2)χ2ϑ^2,ς˙1=q1(χ1)2a12χ12ϑ^1,ϑ^˙1=γ12a12χ12-η1ϑ^1,ς˙2=q2(χ2)2a22χ22ϑ^2,ϑ^˙2=γ22a22χ22-η2ϑ^2.
In the simulation, the design parameters are selected asa1=0.5, a2=0.5, γ1=10, γ2=20, η1=0.1, η2=0.1, κ1=0.1, κ2=0.1, and r=1. The initial values are chosen as x1(ω)=0.1 and x2(ω)=0 for ω≤0, ς1(0)=0.01, ς2(0)=0.01, ϑ^1(0)=0.01, and ϑ^2(0)=0.01. The simulation results are shown in Figures 1–5, respectively.Figure 1
Trajectories of system outputy and reference signal yd.Figure 2
Trajectory of the system statex2.Figure 3
Control inputu.Figure 4
The curve of adaptive parametersϑ^1 and ϑ^2.Figure 5
The curve of adaptive parametersς1 and ς2.From Figure1, it can be seen that good tracking performance is achieved. The response curve of state variable is shown in Figure 2. Figure 3 depicts the trajectory of the control input. We can conclude that the control input is bounded. Figures 4 and 5 display the adaptive parameters ϑ^1, ϑ^2, and ς1, ς2, respectively.Example 2.
To further demonstrate the feasibility of the controller, we present the following nonlinear system:(80)x˙1(t)=16x1(t)x2(t)+2x1(t-τ1(t))-0.5sin⁡(x1)cos⁡⁡(t),x˙2(t)=-x1(t)-x2(t)+(x1(t)+x2(t))x3(t)+x1(t-τ1(t))x2(t-τ2(t))-0.5sin(10t),x˙3(t)=10(x2(t)-x33(t)+0.143x3(t))+sin(5t)+x1(t-τ1(t))+x3(t-τ3(t))+D3(u(t))+(1+x22(t))D(u(t))+sin(0.1D(u(t))),y(t)=x1(t),
where τ1(t)=0.2(1+cos⁡(t)), τ2(t)=0.3(1+sin(t)), and τ3(t)=0.1(0.5+0.2cos⁡(t)).
Similar to Example1, the control laws and the adaptive laws are chosen as
(81)α1=q1(χ1)2a12N(ς1)χ1ϑ^1,α2=q2(χ2)2a22N(ς2)χ2ϑ^2,u=q3(χ3)2a32N(ς3)χ3ϑ^3,ς˙1=q1(χ1)2a12χ12ϑ^1,ϑ^˙1=γ12a12χ12-η1ϑ^1,ς˙2=q2(χ2)2a22χ22ϑ^2,ϑ^˙2=γ22a22χ22-η2ϑ^2,ς˙3=q3(χ3)2a32χ32ϑ^3,ϑ^˙3=γ32a32χ32-η3ϑ^3.
In this example, we choose the design parameters asa1=0.2, a2=0.2, a3=0.2, γ1=20, γ2=20, γ3=20, η1=0.1, η2=0.1, η3=0.1, κ1=0.1, κ2=0.1, and κ3=0.1. The initial values are set to be x1(ω)=0.1, x2(ω)=0, and x3(ω)=0 for ω≤0, ς1(0)=0.01, ς2(0)=0.01, ς3(0)=0.01, ϑ^1(0)=0.01, ϑ^2(0)=0.01, and ϑ^3(0)=0.01. The simulation results are shown in Figures 6–11, respectively.Figure 6
Trajectories of system outputy and reference signal yd.Figure 7
Trajectory of the system statex2.Figure 8
Trajectory of the system statex3.Figure 9
Control inputu.Figure 10
The curve of adaptive parametersϑ^1, ϑ^2, and ϑ^3.Figure 11
The curve of adaptive parametersς1, ς2, and ς3.From Figure6, it can be concluded that a good tracking performance is obtained. Figures 7 and 8 show the trajectory of state variables x2 and x3, respectively. Figure 9 depicts the curve of the control input signal. Figures 10 and 11 display the adaptive parameters ϑ^1, ϑ^2, ϑ^3, and ς1, ς2, ς3, respectively.From the simulation results, it is seen that fairly good tracking performances are achieved; meanwhile, all the other signals in closed-loop system are bounded.


## 5. Conclusions

In this paper, an adaptive fuzzy backstepping control scheme is presented for a class of nonaffine nonlinear time-delay systems with unknown control direction and unknown dead-zone input nonlinearity. By choosing appropriate Lyapunov-Krasovskii functionals, the adaptive fuzzy controller is designed based on backstepping technique and FLSs. The proposed controller guarantees that all the signals in the closed-loop system are bounded and the tracking error eventually converges to a small neighbourhood of the origin. In addition, the number of the parameters which need to be tuned online is significantly reduced. This makes our scheme easily realized in practice. The simulation results illustrate the effectiveness and feasibility of the proposed approach.



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