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

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.

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: 𝑥̇_𝑖(𝑡)=𝑓_𝑖𝐱_𝑖(𝑡)+𝑔_𝑖𝐱_𝑖(𝑡)𝑥_𝑖+1(𝑡)+𝑑_𝑖(𝑡)+ℎ_𝑖𝑥₁𝑡−𝜏₁(𝑡),…,𝑥_𝑖𝑡−𝜏_𝑖(𝑡),𝑥̇_𝑛(𝑡)=𝑓(𝐱(𝑡),𝐷(𝑢(𝑡)))+𝑑_𝑛(𝑡)+ℎ_𝑛𝑥₁𝑡−𝜏₁(𝑡),…,𝑥_𝑛𝑡−𝜏_𝑛(𝑡,𝑦(𝑡))=𝑥₁(𝑡),𝑖=1,…,𝑛−1, (1) where 𝑢∈𝑅 and 𝑦∈𝑅 are the system control input and output, respectively, 𝐱_𝑖(𝑡)=[𝑥₁(𝑡),…,𝑥_𝑖(𝑡)]^𝑇 and 𝐱(𝑡)=[𝑥₁(𝑡),…,𝑥_𝑛(𝑡)]^𝑇 are system states, 𝑓_𝑖(⋅), 𝑔_𝑖(⋅), and 𝑓(⋅) are unknown smooth functions, and 𝑖=1,…,𝑛−1. ℎ_𝑖(⋅) is an unknown smooth function with the unknown time-varying delay terms 𝜏_𝑖(𝑡). There exists a positive constant 𝜏₀ satisfying 0≤𝜏_𝑖(𝑡)≤𝜏₀<+∞, and 𝜑_𝑖(𝑠)=𝑥_𝑖(𝑠), ∀𝑠∈[−𝜏₀,0] is an unknown continuous bounded initial function. 𝑑_𝑖(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 𝑦_𝑑(𝑡) and all signals in closed-loop system are bounded.

Utilizing the mean value theorem [40], function 𝑓(⋅) in (1) can be rewritten as 𝑓(𝐱(𝑡)(𝑢(𝑡(𝐱(𝑡))+(𝐱(𝑡)(𝑢(𝑡,𝐷)))=𝑓,0𝜕𝑓,𝐷)))𝜕𝐷(𝑢(𝑡))_{𝐷(𝑢)=𝜆}𝐷(𝑢(𝑡))=𝑓_𝑛(𝐱(𝑡))+𝑔_𝑛(𝐱(𝑡))𝐷(𝑢(𝑡)), (2) with 𝑓_𝑛(𝐱(𝑡))=𝑓(𝐱(𝑡),0) and 𝑔_𝑛(𝐱(𝑡))=𝜕𝑓(𝐱(𝑡),𝐷(𝑢(𝑡)))/𝜕𝐷(𝑢(𝑡))|_{𝐷(𝑢(𝑡))=𝜆}; 𝜆∈(0,𝐷(𝑢)).

Assumption 1. There exist constants 𝑔 and 𝑔 satisfying 0<𝑔≤|𝑔_𝑖(𝐱_𝑖(𝑡))|≤𝑔.

The nonsymmetric dead-zone input is defined as [17] 𝐷(𝑢(𝑡=𝜑))^𝑟(𝑡)𝑢(𝑡)−𝑚_𝑟,𝑢(𝑡)>𝑚_𝑟,0,−𝑚_𝑙≤𝑢(𝑡)≤𝑚_𝑟,𝜑_𝑙(𝑡)𝑢(𝑡)+𝑚_𝑙,𝑢(𝑡)<−𝑚_𝑙, (3) where 𝜑_𝑟(𝑡) and 𝜑_𝑙(𝑡) are unknown right and left slopes of the dead zone and 𝑚_𝑟 and 𝑚_𝑙 are breakpoints of the dead zone. To deal with dead-zone nonlinearity, the following assumptions are put forward.

Assumption 2. 𝑚_𝑟 and 𝑚_𝑙 are unknown bounded constants. 𝜑_𝑟(𝑡) and 𝜑_𝑙(𝑡) are unknown functions and there are unknown positive constants 𝜑_𝑟0, 𝜑_𝑟1, 𝜑_𝑙0, and 𝜑_𝑙1 satisfying 0<𝜑_𝑟0≤𝜑_𝑟(𝑡)≤𝜑_𝑟1,0<𝜑_𝑙0≤𝜑_𝑙(𝑡)≤𝜑_𝑙1. (4)

Define vectors 𝜼(𝑡) and 𝜿(𝑡) as 𝜼(𝑡)=𝜂_𝑟(𝑡),𝜂_𝑙(𝑡)^𝑇,𝜿(𝑡)=𝜑_𝑟(𝑡),𝜑_𝑙(𝑡)^𝑇, (5) with 𝜂_𝑟(𝑡)=^{1,𝑢(𝑡)≥−𝑚𝑙}_{0,𝑢(𝑡)<−𝑚𝑙} and 𝜂_𝑙(𝑡)=^{1,𝑢(𝑡)≤𝑚𝑟}_{0,𝑢(𝑡)>𝑚𝑟}.

Based on the above analysis, the dead zone can be expressed as 𝐷(𝑢)=𝜼^𝑇(𝑡)𝜿(𝑡)𝑢(𝑡)+𝑑(𝑢(𝑡)), (6) where 𝑑(𝑢(𝑡=))−𝜑_𝑟(𝑡)𝑚_𝑟,𝑢(𝑡)≥𝑚_𝑟,−𝜑_𝑟(𝑡)+𝜑_𝑙(𝑡)𝑢(𝑡),−𝑚_𝑙<𝑢(𝑡)<𝑚_𝑟,𝜑_𝑙(𝑡)𝑚_𝑙,𝑢(𝑡)≤−𝑚_𝑙. (7)

According to Assumption 2, it can be concluded that |𝑑(𝑢(𝑡))|≤𝑑₀ and 𝑑₀ is an unknown positive constant meeting 𝑑₀=(𝜑_𝑟1+𝜑_𝑙1)max{𝑚_𝑟,𝑚_𝑙}. Considering the definition of 𝜼(𝑡) and 𝜿(𝑡), we have 𝜼^𝑇(𝑡)𝜿(𝑡)=𝜑_𝑟(𝑡),𝑢(𝑡)≥𝑚_𝑟,𝜑_𝑟(𝑡)+𝜑_𝑙(𝑡),−𝑚_𝑙<𝑢(𝑡)<𝑚_𝑟,𝜑_𝑙(𝑡),𝑢(𝑡)≤−𝑚_𝑙. (8)

It is easy to obtain that 𝜼^𝑇(𝑡)𝜿(𝑡)∈[min(𝜑_𝑟0,𝜑_𝑙0),𝜑_𝑟1+𝜑_𝑙1]; that is, 𝜼^𝑇(𝑡)𝜿(𝑡) is a positive discontinuous bounded function.

Using (2) and (6), system (1) can be transformed into the following form: 𝑥̇_𝑖(𝑡)=𝑓_𝑖𝐱_𝑖(𝑡)+𝑔_𝑖𝐱_𝑖(𝑡)𝑥_𝑖+1(𝑡)+𝑑_𝑖(𝑡)+ℎ_𝑖𝑥₁𝑡−𝜏₁(𝑡),…,𝑥_𝑖𝑡−𝜏_𝑖(𝑡),𝑥̇_𝑛(𝑡)=𝑓_𝑛(𝐱)+𝑔_𝑛(𝐱)𝜼^𝑇(𝑡)𝜿(𝑡)𝑢(𝑡)+𝑔_𝑛(𝐱)𝑑(𝑢(𝑡))+ℎ_𝑛𝑥₁𝑡−𝜏₁(𝑡),…,𝑥_𝑛𝑡−𝜏_𝑛(𝑡)+𝑑_𝑛(𝑡),𝑦(𝑡)=𝑥₁(𝑡). (9)

Notation. 𝑥_𝑖, 𝑥_𝑖(𝜏_𝑖), 𝑦, 𝑢, 𝑑_𝑖, 𝑓_𝑖, and 𝑔_𝑖 denote 𝑥_𝑖(𝑡), 𝑥_𝑖(𝑡−𝜏_𝑖(𝑡)), 𝑦(𝑡), 𝑢(𝑡), 𝑑_𝑖(𝑡), 𝑓_𝑖(𝐱_𝑖), and 𝑔_𝑖(𝐱_𝑖), respectively.

Assumption 3. The unknown smooth functions ℎ_𝑖(𝑥₁(𝜏₁),…,𝑥_𝑖(𝜏_𝑖)) satisfy the inequality ℎ_𝑖𝑥₁𝜏₁,…,𝑥_𝑖𝜏_𝑖≤^𝑖_𝑗=1𝐻_𝑖,𝑗𝑥_𝑗𝜏_𝑗, (10) where 𝐻_𝑖,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 𝐻_𝑖,𝑗(⋅) are unknown in this paper.

Assumption 5. The time derivatives of the time-varying delay terms 𝜏_𝑖 are 𝜏̇_𝑖 and satisfy 𝜏̇_𝑖≤𝜏^∗<1 where 𝜏^∗ is an unknown positive constant.

Assumption 6. The external disturbances 𝑑_𝑖 satisfy |𝑑_𝑖|≤𝑑_𝑖, where 𝑑_𝑖 is defined as an unknown positive constant.

Remark 7. The constants 𝑔_𝑖0, 𝑔_𝑖1, 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]: 𝑅^(𝑙)∶If𝑥₁is𝐹^𝑙₁and⋯and𝑥_𝑛is𝐹^𝑙_𝑛,Then𝑦is𝐺^𝑙,𝑙=1,…,𝑀, (11) where 𝐱=[𝑥₁,𝑥₂,…,𝑥_𝑛]^𝑇 and 𝑦 are the FLS inputs and output, respectively. 𝐹^𝑙_𝑖, 𝑖=1,…,𝑛, and 𝐺^𝑙 are fuzzy sets characterized by fuzzy membership functions 𝜇_𝐹𝑙𝑖(𝑥_𝑖) and 𝜇_𝐺𝑙(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]: 𝑦(𝐱)=∑^𝑀_𝑙=1𝑦^𝑙∏^𝑛_𝑖=1𝜇_𝐹𝑙𝑖𝑥_𝑖∑^𝑀_𝑙=1∏^𝑛_𝑖=1𝜇_𝐹𝑙𝑖𝑥_𝑖, (12) where 𝑦^𝑙 is the point at which the membership function 𝜇_𝐺𝑙(𝑦) achieves its maximum value and we assume that 𝜇_𝐺𝑙(𝑦^𝑙)=1. Let 𝜽=[𝑦¹,…,𝑦^𝑀]^𝑇 be a vector grouping all consequent parameters and 𝜉(𝐱)=[𝜉₁(𝐱),…,𝜉_𝑀(𝐱)]^𝑇, where 𝜉_𝑗(𝐱)≜∏^𝑛_𝑖=1𝜇_𝐹^𝑗_𝑖(𝑥_𝑖)/∑^𝑀_𝑙=1(∏^𝑛_𝑖=1𝜇_𝐹𝑙𝑖(𝑥_𝑖)), 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)=𝜽^𝑇𝜉(𝐱). Then according to the universal approximation theorem, any continuous nonlinear function 𝑓(𝐱) can be approximated by the FLS as 𝑓(𝐱)=𝜽^∗_𝑓^𝑇𝜉(𝐱)+𝜀(𝐱), (13) where 𝜽^∗_𝑓 is an optimal parameter satisfying 𝜽^∗_𝑓=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]: lim_𝑘→∞1sup𝑘^𝑘₀𝑁(𝜍)𝑑𝜍=+∞,lim_𝑘→∞1inf𝑘^𝑘₀𝑁(𝜍)𝑑𝜍=−∞. (14)

Functions, such as 𝜍²cos(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∫^𝑡₀[𝑔(𝑥(𝜏))𝑁(𝜍)+1]𝜍𝑒̇^𝑐1𝜏𝑑𝜏 holds with 𝑡∈[0,𝑡_𝑓), 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 ∫^𝑡₀𝑔(𝑥(𝜏))𝑁(𝜍)𝜍𝑑̇𝜏 must be bounded on [0,𝑡_𝑓).

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: 𝜒₁=𝑥₁−𝑦_𝑑𝜒_𝑖=𝑥_𝑖−𝛼_𝑖−1,𝑖=2,…,𝑛, (15) where 𝛼_𝑖−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 𝜗_𝑖=𝜽_𝑖²,𝑖=1,…,𝑛. (16)

For 𝜗_𝑖 are unknown and 𝜗_𝑖 are used to estimate 𝜗_𝑖 with estimation errors 𝜗_𝑖 defined as 𝜗_𝑖=𝜗_𝑖−𝜗_𝑖.

The detailed design procedure is described in the following steps.

Step 1. Consider the Lyapunov-Krasovskii function as 𝑉₁=12𝜒²₁+12𝛾₁𝜗²₁+𝑉_1,0, (17) where 𝑉_1,0=(𝑒^{−𝑟(𝑡−𝜏0)}/1−𝜏^∗)∫^𝑡_{𝑡−𝜏1(𝑡)}𝑒^𝑟𝑠𝐻²_1,1(𝑥₁(𝑠))𝑑𝑠, 𝛾₁ and 𝑟 are design positive parameters.

Giving a compact set Ω_𝜅1 as Ω_𝜅1={𝜒₁∣|𝜒₁|<0.2554𝜅₁} with 𝜅₁, a positive design parameter, then a function defined as follows will be employed to design controller 𝑞₁𝜒₁=1,𝜒₁∉Ω_𝜅1,0,𝜒₁∈Ω_𝜅1. (18)

We choose the virtual control law 𝛼₁ and adaptive laws as 𝛼₁=𝑞₁𝜒₁2𝑎²₁𝑁𝜍₁𝜒₁𝜗₁,𝜍₁̇=𝑞₁𝜒₁2𝑎²₁𝜒²₁𝜗₁,̇𝜗₁=𝛾₁2𝑎²₁𝜒²₁−𝜂₁𝜗₁, (19)(20)(21) 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 𝜍₁̇=0; that is, 𝑁(𝜍₁) is bounded. Integrating (21) over [0,𝑡] we get that signal 𝜗₁ is bounded.

Case  2. When |𝜒₁|≥0.2554𝜅₁, the following process is needed.

The time derivative of 𝑉₁ is 𝑉̇₁≤𝜒₁𝑓₁+𝑔₁𝑥₂+ℎ₁𝑥₁𝜏₁+𝑑₁−𝑦̇_𝑑−1𝛾₁𝜗₁̇𝜗₁−𝑟𝑉_1,0+𝑒^{−𝑟(𝑡−𝜏0)}1−𝜏^∗𝑒^𝑟𝑡𝐻²_1,1𝑥₁−1−𝜏̇₁(𝑡)𝑒^{𝑟(𝑡−𝜏1(𝑡))}𝐻²_1,1𝑥₁𝜏₁≤𝜒₁𝑓₁+𝑔₁𝑥₂+ℎ₁𝑥₁𝜏₁+𝑑₁−𝑦̇_𝑑−1𝛾₁𝜗₁̇𝜗₁−𝑟𝑉_1,0+𝐻₁𝑥₁−𝐻²_1,1𝑥₁𝜏₁, (22) with 𝐻₁(𝑥₁)=𝑒^𝑟𝜏0𝐻²_1,1(𝑥₁)/(1−𝜏^∗).

When Assumption 3 holds, we get ℎ₁(𝑥₁(𝜏₁))²≤𝐻²_1,1𝑥₁𝜏₁. (23)

By using Young’s inequality and combining with (23), (22) yields 𝑉̇₁=𝜒₁𝑔₁𝜒₂+𝛼₁+𝜒₁𝑓₁+𝑑₁−𝑦̇_𝑑+14𝜒²₁+ℎ²₁𝑥₁𝜏₁−1𝛾₁𝜗₁̇𝜗₁−𝑟𝑉_1,0+𝐻₁𝑥₁−𝐻²_1,1𝑥₁𝜏₁≤−𝑐₁𝜒²₁+𝜒²₂+𝜒₁𝑔₁𝛼₁+𝜒₁𝑓₁−𝑦̇_𝑑+𝑐₁𝜒₁+12𝜌²𝜒₁+14𝑔²₁𝜒₁+34𝜒₁−12𝜒²₁+12𝜌²𝑑²₁−1𝛾₁𝜗₁̇𝜗₁+𝐻²_1,1𝑥₁𝜏₁−𝑟𝑉_1,0+𝐻₁𝑥₁−𝐻²_1,1𝑥₁𝜏₁=−𝑐₁𝜒²₁−𝑟𝑉_1,0+𝜒²₂+𝜒₁𝑔₁𝛼₁+𝜒₁𝑓₁−1𝛾₁𝜗₁̇𝜗₁+1−16tanh²𝜒₁𝜅₁𝐻₁𝑥₁−12𝜒²₁+12𝜌²𝑑²₁, (24) where 𝑐₁ and 𝜌 are positive constants 𝑓₁=𝑓₁−𝑦̇_𝑑+𝑐₁𝜒₁+12𝜌²𝜒₁+14𝑔²₁𝜒₁+34𝜒₁+16𝜒₁tanh²𝜒₁𝜅₁𝐻₁𝑥₁. (25)

Remark 9. Note that the function 𝐻_𝑖(𝑥_𝑖)/𝜒_𝑖 (𝑖=1,…,𝑛) is not defined at 𝜒_𝑖=0 which leads to the fact that it cannot be approximated by FLSs. To cope with this difficulty, we introduce function (16/𝜒_𝑖)tanh²(𝜒_𝑖/𝜅_𝑖)𝐻_𝑖(𝑥_𝑖) instead of 𝐻_𝑖(𝑥_𝑖)/𝜒_𝑖 according to the effective approach in [19]. The design parameter 𝜅_𝑖 can be adjusted to achieve better performance.

As 𝑓₁ 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 𝑓₁=𝜽₁^∗𝑇𝝃₁𝐙₁+𝛿₁𝐙₁,𝛿₁𝐙₁≤𝜀₁, (26) where 𝛿₁(𝐙₁) stands for approximation error and 𝜀₁ is an unknown constant, 𝐙₁=[𝑥₁,𝑦_𝑑,𝑦̇_𝑑]^𝑇.

Considering the inequality 0<𝝃^𝑇₁(𝐙₁)𝝃₁(𝐙₁)≤1 we obtain 𝜒₁𝑓₁=𝜒₁𝜽₁^∗𝑇𝝃₁𝐙₁+𝜒₁𝛿₁𝐙₁≤12𝑎²₁𝜒²₁𝜗₁+12𝑎²₁+12𝜒²₁+12𝜀²₁. (27)

As |𝜒₁|≥0.2554𝜅₁, we have 1−16tanh²𝜒₁𝜅₁𝐻₁𝑥₁≤0. (28)

Substituting (27) and (28) into (24) results in 𝑉̇₁≤−𝑐₁𝜒²₁−𝑟𝑉_1,0+𝜒²₂+𝜒₁𝑔₁𝛼₁+12𝑎²₁𝜒²₁𝜗₁+12𝑎²₁+12𝜀²₁+12𝜌²𝑑²₁−1𝛾₁𝜗₁̇𝜗₁. (29)

Considering 𝑞(𝜒₁)=1 and using (19)–(21), (29) yields 𝑉̇₁≤−𝑐₁𝜒²₁−𝑟𝑉_1,0+𝜒²₂+𝑔₁𝑁𝜍₁+1𝜍₁̇+12𝑎²₁+12𝜀²₁+12𝜌²𝑑²₁+𝜂₁𝛾₁𝜗₁𝜗₁≤−𝜇₁𝑉₁+𝜒²₂+𝑔₁𝑁𝜍₁+1𝜍₁̇+𝐶₁, (30) with 𝜇₁=min(2𝑐₁,𝜂₁,𝑟) and 𝐶₁=(1/2)𝑎²₁+(1/2)𝜀²₁+(1/2)𝜌²𝑑²₁+(𝜂₁/2𝛾₁)𝜗²₁.

Multiplying (30) by 𝑒^𝜇1𝑡 and then integrating over [0,𝑡], we get 𝑉₁≤𝛽₁+𝑒^{−𝜇1𝑡𝑡}₀𝑔₁𝑁𝜍₁+1𝜍₁̇𝑒^𝜇1𝜏𝑑𝜏+𝑒^{−𝜇1𝑡𝑡}₀𝜒²₂𝑒^𝜇1𝜏𝑑𝜏, (31) where 𝛽₁=(𝐶₁/𝜇₁)+(𝑉₁(0)−(𝐶₁/𝜇₁))𝑒^−𝜇1𝑡.

Noting that there is an extra term 𝑒^−𝜇1𝑡∫^𝑡₀𝜒²₂𝑒^𝜇1𝜏𝑑𝜏 within (31), we suppose that 𝜒₂ can be regulated as bounded; then we have 𝑒^{−𝜇1𝑡𝑡}₀𝜒²₂𝑒^𝜇1𝑠𝑑𝑠≤𝑒^−𝜇1𝑡sup_{𝑠∈[0,𝑡]}𝜒²₂(𝑠)^𝑡₀𝑒^𝜇1𝑠≤1𝑑𝑠𝜇₁sup_{[]𝑠∈0,𝑡}𝜒²₂(𝑠). (32)

As the boundedness of the extra term 𝑒^−𝜇1𝑡∫^𝑡₀𝜒²₂𝑒^𝜇1𝜏𝑑𝜏 is obtained from (32), directly applying Lemma 8, we get the conclusion that signals 𝑉₁(𝑡), 𝜍₁(𝑡), and ∫^𝑡₀𝑔₁𝑁(𝜍₁)𝜍₁̇𝑑𝜏 hence 𝜒₁, 𝑁(𝜍₁), 𝜗₁, and 𝜗₁ are all bounded on [0,𝑡_𝑓). 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 Step k).

Step k (2≤𝑘≤𝑛−1). Considering that steps 2≤𝑘≤𝑛−1 have a similar procedure, Step k is presented as follows.
Choose the following Lyapunov-Krasovskii function: 𝑉_𝑘=12𝜒²_𝑘+12𝛾_𝑘𝜗²_𝑘+𝑉_𝑘,0, (33) with 𝛾_𝑘 being a design positive parameter, 𝑉_𝑘,0=𝑒^{−𝑟(𝑡−𝜏0)}1−𝜏^∗𝑘𝑖_𝑖=2𝑗=1^𝑡_{𝑡−𝜏𝑗(𝑡)}𝑒^𝑟𝑠𝐻²_𝑖,𝑗𝑥_𝑗(𝑠)+1𝑑𝑠2^𝑡_{𝑡−𝜏1(𝑡)}𝑒^𝑟𝑠𝐻²_1,1𝑥₁(𝑠)𝑑𝑠. (34)

Similar to Step 1, the virtual control law 𝛼_𝑘 and adaptive laws are designed as 𝛼_𝑘=𝑞_𝑘𝜒_𝑘2𝑎²_𝑘𝑁𝜍_𝑘𝜒_𝑘𝜗_𝑘,𝜍_𝑘̇=𝑞_𝑘𝜒_𝑘2𝑎²_𝑘𝜒²_𝑘𝜗_𝑘,̇𝜗_𝑘=𝛾_𝑘2𝑎²_𝑘𝜒²_𝑘−𝜂_𝑘𝜗_𝑘, (35)(36)(37) with 𝑎_𝑘 and 𝜂_𝑘 being design positive parameters; the function 𝑞_𝑘(𝜒_𝑘) is defined as 𝑞_𝑘𝜒_𝑘=1,𝜒_𝑘∉Ω_𝜅𝑘0,𝜒_𝑘∈Ω_𝜅𝑘, (38) where Ω_𝜅𝑘∶={𝜒_𝑘∣|𝜒_𝑘|<0.2554𝜅_𝑘} stands for a compact set and 𝜅_𝑘 is a design positive parameter which decides the size of convergence region.

Case  1. In this case, we suppose that |𝜒_𝑘|<0.2554𝜅_𝑘. It is obvious that the tracking error 𝜒_𝑘 is bounded. From (35)–(37), when selecting bounded initial values, we achieve the boundedness of 𝛼_𝑘, 𝜍_𝑘, and 𝑁(𝜍_𝑘). After integrating (37) over [0,𝑡], we conclude that signal 𝜗_𝑘 is bounded.

Case  2. In case 2, the tracking error satisfies |𝜒_𝑘|≥0.2554𝜅_𝑘.

The time derivative of 𝑉_𝑘,0 is 𝑉̇_𝑘,0=−𝑟𝑉_𝑘,0+𝑒^{−𝑟(𝑡−𝜏0)}1−𝜏^∗×^𝑘𝑖_𝑖=2𝑗=1𝑒^𝑟𝑡𝐻²_𝑖,𝑗𝑥_𝑗−1−𝜏̇_𝑗𝑒^{𝑟(𝑡−𝜏𝑗(𝑡))}𝐻²_𝑖,𝑗𝑥_𝑗𝜏_𝑗+12𝑒^𝑟𝑡𝐻²_1,1𝑥₁−121−𝜏̇_𝑗𝑒^{𝑟(𝑡−𝜏𝑗(𝑡))}𝐻²_1,1𝑥₁𝜏₁≤−𝑟𝑉_𝑘,0+𝐻_𝑘𝐱_𝑘−^𝑘𝑖_𝑖=2𝑗=1𝐻²_𝑖,𝑗𝑥_𝑗𝜏_𝑗−12𝐻²_1,1𝑥₁𝜏₁, (39) where 𝐻_𝑘(𝐱_𝑘)=(𝑒^𝑟𝜏0/(1−𝜏^∗)){∑^𝑘_𝑖=2∑^𝑖_𝑗=1𝐻²_𝑖,𝑗(𝑥_𝑗)+(1/2)𝐻²_1,1(𝑥₁)}.

Then the time derivative of 𝜒_𝑘 is 𝜒̇_𝑘=𝑓_𝑘+𝑔_𝑘𝑥_𝑘+1+ℎ_𝑘𝑥₁𝜏₁,…,𝑥_𝑘𝜏_𝑘+𝑑_𝑘−𝛼̇_𝑘−1, (40) with 𝛼̇_𝑘−1=^𝑘−1_𝑖=1𝜕𝛼_𝑘−1𝜕𝜍_𝑖12𝑎²_𝑖𝑒²_𝑖𝜗_𝑖+𝜕𝛼_𝑘−1𝜕𝑥_𝑖𝑓_𝑖+𝑔_𝑖𝑥_𝑖+1+ℎ_𝑖𝑥₁𝜏₁,…,𝑥_𝑖𝜏_𝑖+𝑑_𝑖(𝑡)+𝜕𝛼_𝑘−1𝜕𝜗_𝑖𝛾_𝑖2𝑎²_𝑖𝑒²_𝑖−𝛾_𝑖𝜗_𝑖+𝜕𝛼_𝑘−1𝜕𝑦_𝑑𝑦̇_𝑑. (41)

Remark 10. Here 𝛼_𝑘−1 is the function of 𝛼_𝑘−2. When 𝑘=2, we have 𝛼_𝑘−2=𝑦_𝑑; that is, 𝛼_𝑘−1 is the function of 𝐱_𝑘−1, 𝑦_𝑑, 𝝇_𝑘−1, and 𝜗_𝑘−1, where 𝝇_𝑘−1=[𝜍₁,…,𝜍_𝑘−1]^𝑇 and 𝜗_𝑘−1=[𝜗₁,…,𝜗_𝑘−1]^𝑇. Then we derive the time derivative of 𝛼_𝑘−1 in (41).

Using (40) and (41), the derivative of 𝑉_𝑘 is 𝑉̇_𝑘=𝜒_𝑘𝑓_𝑘+𝑔_𝑘𝑥_𝑘+1+ℎ_𝑘𝑥₁𝜏₁,…,𝑥_𝑘𝜏_𝑘+𝑑_𝑘−𝛼̇_𝑘−1−1𝛾_𝑘𝜗_𝑘̇𝜗_𝑘+𝑉̇_𝑘,0=𝜒_𝑘𝑓_𝑘+𝑔_𝑘𝑥_𝑘+1+ℎ_𝑘𝑥₁𝜏₁,…,𝑥_𝑘𝜏_𝑘+𝑑_𝑘−^𝑘−1_𝑖=1𝜕𝛼_𝑘−1𝜕𝜍_𝑖𝜍_𝑖̇+𝜕𝛼_𝑘−1𝜕𝑥_𝑖𝑥̇_𝑖+𝜕𝛼_𝑘−1𝜕𝜗_𝑖̇𝜗_𝑖−𝜕𝛼_𝑘−1𝜕𝑦_𝑑𝑦̇_𝑑−1𝛾_𝑘𝜗_𝑘̇𝜗_𝑘+𝑉̇_𝑘,0. (42)

Owing to Assumption 3, we get ℎ_𝑖(𝑥₁(𝜏₁),…,𝑥_𝑖(𝜏_𝑖))²≤2^𝑖_𝑗=1𝐻²_𝑖,𝑗𝑥_𝑗𝜏_𝑗,2≤𝑖≤𝑛. (43)

Utilizing Young’s inequality (42) yields 𝑉̇_𝑘≤−𝑐_𝑘𝜒²_𝑘+𝜒_𝑘𝑔_𝑘𝛼_𝑘+𝜒²_𝑘+1+𝜒_𝑘𝑓_𝑘−^𝑘−1_𝑖=1𝜕𝛼_𝑘−1𝜕𝜍_𝑖12𝑎²_𝑖𝜒²_𝑖𝜗_𝑖+𝜕𝛼_𝑘−1𝜕𝑥_𝑖𝑓_𝑖+𝑔_𝑖𝑥_𝑖+1+𝜕𝛼_𝑘−1𝜕𝜗_𝑖𝛾_𝑖2𝑎²_𝑖𝜒²_𝑖−𝛾_𝑖𝜗_𝑖−𝜕𝛼_𝑘−1𝜕𝑦_𝑑𝑦̇_𝑑+12𝜌²+12^𝑘−1_𝑖=1𝜕𝛼_𝑘−1𝜕𝑥_𝑖²𝜒_𝑘+12𝜌²+𝑐_𝑘+14𝑔²_𝑘+1𝜒_𝑘−12𝜒²_𝑘+^𝑘𝑖_𝑖=2𝑗=1𝐻²_𝑖,𝑗𝑥_𝑗𝜏_𝑗+12𝐻²_1,1𝑥₁𝜏₁+12^𝑘_𝑖=1𝜌²𝑑²_𝑖−1𝛾_𝑘𝜗_𝑘̇𝜗_𝑘+𝑉̇_𝑘,0. (44)

Substituting (39) into (44) results in 𝑉̇_𝑘≤−𝑐_𝑘𝜒²_𝑘−𝑟𝑉_𝑘,0+𝜒_𝑘𝑔_𝑘𝛼_𝑘+𝜒²_𝑘+1+𝜒_𝑘𝑓_𝑘−12𝜒²_𝑘+12^𝑘_𝑖=1𝜌²𝑑²_𝑖−1𝛾_𝑘𝜗_𝑘̇𝜗_𝑘+1−16tanh²𝜒_𝑘𝜅_𝑘𝐻_𝑘𝐱_𝑘, (45) with 𝑓_𝑘=𝑓_𝑘−^𝑘−1_𝑖=1𝜕𝛼_𝑘−1𝜕𝜍_𝑖12𝑎²_𝑖𝜒²_𝑖𝜗_𝑖+𝜕𝛼_𝑘−1𝜕𝑥_𝑖𝑓_𝑖+𝑔_𝑖𝑥_𝑖+1+𝜕𝛼_𝑘−1𝜕𝜗_𝑖𝛾_𝑖2𝑎²_𝑖𝜒²_𝑖−𝛾_𝑖𝜗_𝑖−𝜕𝛼_𝑘−1𝜕𝑦_𝑑𝑦̇_𝑑+12𝜌²+12^𝑘−1_𝑖=1𝜕𝛼_𝑘−1𝜕𝑥_𝑖²𝜒_𝑘+12𝜌²+𝑐_𝑘+14𝑔²_𝑘+1𝜒_𝑘+16𝜒_𝑘tanh²𝜒_𝑘𝜅_𝑘𝐻_𝑘𝐱_𝑘. (46)

Similarly, 𝑓_𝑘 can be approximated by FLSs to an arbitrary given accuracy as 𝑓_𝑘=𝜽_𝑘^∗𝑇𝝃_𝑘𝐙_𝑘+𝛿_𝑘𝐙_𝑘,𝛿_𝑘𝐙_𝑘≤𝜀_𝑘, (47) where 𝐙_𝑘=[𝐱_𝑘,𝑦_𝑑,𝑦̇_𝑑,𝝇_𝑘−1,𝜗_𝑘−1]^𝑇, 𝛿_𝑘(𝐙_𝑘) represents approximation error, and 𝜀_𝑘 is an unknown positive constant.

As the fuzzy basis function 𝝃_𝑘(𝐙_𝑘) satisfies 0<𝝃^𝑇_𝑘(𝐙_𝑘)𝝃_𝑘(𝐙_𝑘)≤1, we get the following inequality: 𝜒_𝑘𝑓_𝑘=𝜒_𝑘𝜽_𝑘^∗𝑇𝝃_𝑘𝐙_𝑘+𝜒_𝑘𝛿_𝑘𝐙_𝑘≤12𝑎²_𝑘𝜒²_𝑘𝜗_𝑘+12𝑎²_𝑘+12𝜒²_𝑘+12𝜀²_𝑘. (48)

Substituting (48) into (45), we obtain 𝑉̇_𝑘≤−𝑐_𝑘𝜒²_𝑘−𝑟𝑉_𝑘,0+𝜒_𝑘𝑔_𝑘𝛼_𝑘+𝜒²_𝑘+1−1𝛾_𝑘𝜗_𝑘̇𝜗_𝑘+12𝑎²_𝑘𝜒²_𝑘𝜗_𝑘+12𝑎²_𝑘+12𝜀²_𝑘+12^𝑘_𝑖=1𝜌²𝑑²_𝑖+1−16tanh²𝜒_𝑘𝜅_𝑘𝐻_𝑘𝐱_𝑘. (49)

As |𝜒_𝑘|≥0.2554𝜅_𝑘, we can derive 1−16tanh²𝜒_𝑘𝜅_𝑘𝐻_𝑘𝐱_𝑘≤0. (50)

Applying (35)–(37) and (50), (49) produces 𝑉̇_𝑘≤−𝜇_𝑘𝑉_𝑘+𝜒²_𝑘+1+𝑔_𝑘𝑁𝜍_𝑘+1𝜍_𝑘̇+𝐶_𝑘, (51) where 𝜇_𝑘=min(2𝑐_𝑘,𝑟,𝜂_𝑘) and 𝐶_𝑘=(1/2)𝑎²_𝑘+(1/2)𝜀²_𝑘+(1/2)∑^𝑘_𝑖=1𝜌²𝑑²_𝑖+(𝜂_𝑘/2𝛾_𝑘)𝜗²_𝑘.

Multiplying (51) by 𝑒^𝜇𝑘𝑡 results in 𝑑𝑉𝑑𝑡^𝑘𝑒^𝜇𝑘𝑡≤𝑒^𝜇𝑘𝑡𝜒²_𝑘+1+𝑔_𝑘𝑁𝜍_𝑘+1𝜍_𝑘̇𝑒^𝜇𝑘𝑡+𝐶_𝑘𝑒^𝜇𝑘𝑡. (52)

Integrating (52) over [0, t], we obtain 𝑉_𝑘≤𝛽_𝑘+𝑒^{−𝜇𝑘𝑡𝑡}₀𝑔_𝑘𝑁𝜍_𝑘+1𝜍_𝑘̇𝑒^𝜇𝑘𝜏𝑑𝜏+𝑒^{−𝜇𝑘𝑡𝑡}₀𝜒²_𝑘+1𝑒^𝜇𝑘𝜏𝑑𝜏, (53) with 𝛽_𝑘=(𝐶_𝑘/𝜇_𝑘)+(𝑉_𝑘(0)−(𝐶_𝑘/𝜇_𝑘))𝑒^{−𝜇𝑘𝑡}.

Remark 11. The discussion of (53) is similar to the analysis of (31). If 𝜒_𝑘+1 can be regulated as bounded, by utilizing Lemma 8, the boundedness of signals 𝑉_𝑘(𝑡), 𝜍_𝑘(𝑡), and ∫^𝑡₀𝑔_𝑘𝑁(𝜍_𝑘)𝜍_𝑘̇𝑑𝜏 is achieved. Thus, we can guarantee that signals 𝜒_𝑘, 𝑁(𝜍_𝑘), 𝜗_𝑘, and 𝜗_𝑘 are all bounded on [0,𝑡_𝑓). The effect of the extra term 𝑒^{−𝜇𝑘𝑡}∫^𝑡₀𝑒²_𝑘+1𝑒^𝜇𝑘𝜏𝑑𝜏 will be handled in the next step.

Step n. Consider Lyapunov-Krasovskii function as follows: 𝑉_𝑛=12𝜒²_𝑛+12𝛾_𝑛𝜗²_𝑛+𝑉_𝑛,0, (54) where 𝛾_𝑛 is a design positive parameter, and 𝑉_𝑛,0=𝑒^{−𝑟(𝑡−𝜏0)}1−𝜏^∗𝑛𝑖_𝑖=2𝑗=1^𝑡_{𝑡−𝜏𝑗(𝑡)}𝑒^𝑟𝑠𝐻²_𝑖,𝑗𝑥_𝑗(𝑠)+1𝑑𝑠2^𝑡_{𝑡−𝜏1(𝑡)}𝑒^𝑟𝑠𝐻²_1,1𝑥₁(𝑠)𝑑𝑠. (55)

We choose the following actual control input 𝑢 and adaptive laws: 𝑞𝑢=^𝑛𝜒_𝑛2𝑎²_𝑛𝑁𝜍_𝑛𝜒_𝑛𝜗_𝑛,𝜍_𝑛̇=𝑞_𝑛𝜒_𝑛2𝑎²_𝑛𝜒²_𝑛𝜗_𝑛,̇𝜗_𝑛=𝛾_𝑛2𝑎²_𝑛𝜒²_𝑛−𝜂_𝑛𝜗_𝑛, (56)(57)(58) where 𝑎_𝑛 and 𝜂_𝑛 are design positive parameters. The function 𝑞_𝑛(𝜒_𝑛) is defined as 𝑞_𝑛𝜒_𝑛=1,𝜒_𝑛∉Ω_𝜅𝑛,0,𝜒_𝑛∈Ω_𝜅𝑛, (59) with Ω_𝜅𝑛∶={𝜒_𝑛∣|𝜒_𝑛|<0.2554𝜅_𝑛} denoting a compact set and 𝜅_𝑛 is a design positive parameter.

Similarly, we analyze the 𝑛th-subsystem from two cases.

Case  1. In Case 1, 𝜒_𝑛 satisfies |𝜒_𝑛|<0.2554𝜅_𝑛. As 𝜅_𝑛 is a positive design parameter; we obtain that 𝜒_𝑛 is bounded. In addition, we can conclude that the signals 𝑢, 𝜍_𝑛, 𝑁(𝜍_𝑛), 𝜗_𝑛, and 𝜗_𝑛 are bounded.

Case  2. We suppose that |𝜒_𝑛|≥0.2554𝜅_𝑛 in this case.

The time derivative of 𝑉_𝑛 is 𝑉̇_𝑛=𝜒_𝑛𝑓_𝑛+𝑔_𝑛𝜼^𝑇𝜿𝑢+𝑔_𝑛𝑑(𝑢)+ℎ_𝑛𝑥₁𝜏₁,…,𝑥_𝑛𝜏_𝑛+𝑑_𝑛−𝛼̇_𝑛−1−1𝛾_𝑛𝜗_𝑛̇𝜗_𝑛+𝑉̇_𝑛,0. (60)

From the definition of 𝛼_𝑛−1, we obtain 𝛼̇_𝑛−1=^𝑛−1_𝑖=1𝜕𝛼_𝑛−1𝜕𝜍_𝑖12𝑎²_𝑖𝜒²_𝑖𝜗_𝑖+𝜕𝛼_𝑛−1𝜕𝑥_𝑖𝑓_𝑖+𝑔_𝑖𝑥_𝑖+1+ℎ_𝑖𝑥₁𝜏₁,…,𝑥_𝑖𝜏_𝑖+𝑑_𝑖+𝜕𝛼_𝑛−1𝜕𝜗_𝑖𝛾_𝑖2𝑎²_𝑖𝜒²_𝑖−𝛾_𝑖𝜗_𝑖+𝜕𝛼_𝑛−1𝜕𝑦_𝑑𝑦̇_𝑑. (61)

Applying Young’s inequality and (43), (60) can be rewritten as 𝑉̇_𝑛≤−𝑐_𝑛𝜒²_𝑛+𝜒_𝑛𝑔_𝑛𝜼^𝑇𝜿𝑢+𝜒_𝑛𝑓_𝑛−^𝑛−1_𝑖=1𝜕𝛼_𝑛−1𝜕𝜍_𝑖12𝑎²_𝑖𝜒²_𝑖𝜗_𝑖+𝜕𝛼_𝑛−1𝜕𝑥_𝑖𝑓_𝑖+𝑔_𝑖𝑥_𝑖+1+𝜕𝛼_𝑛−1𝜕𝜗_𝑖𝛾_𝑖2𝑎²_𝑖𝜒²_𝑖−𝛾_𝑖𝜗_𝑖−𝜕𝛼_𝑛−1𝜕𝑦_𝑑𝑦̇_𝑑+12𝜌²+12^𝑛−1_𝑖=1𝜕𝛼_𝑛−1𝜕𝑥_𝑖²𝜒_𝑛+12𝜌²+𝑐_𝑛+1𝜒_𝑛−12𝜒²_𝑛+^𝑛𝑖_𝑖=2𝑗=1𝐻²_𝑖,𝑗𝑥_𝑗𝜏_𝑗+12𝐻²_1,1𝑥₁𝜏₁+12^𝑛_𝑖=1𝜌²𝑑²_𝑖−1𝛾_𝑛𝜗_𝑛̇𝜗_𝑛+𝑉̇_𝑛,0. (62)

The time derivative of 𝑉_𝑛,0 is 𝑉̇_𝑛,0=−𝑟𝑉_𝑛,0+𝑒^{−𝑟(𝑡−𝜏0)}1−𝜏^∗×12𝑒^𝑟𝑡𝐻²_1,1𝑥₁−121−𝜏̇₁𝑒^{𝑟(𝑡−𝜏1(𝑡))}𝐻²_1,1𝑥₁𝜏₁+^𝑛𝑖_𝑖=2𝑗=1𝑒^𝑟𝑡𝐻²_𝑖,𝑗𝑥_𝑗−1−𝜏̇_𝑗𝑒^{𝑟(𝑡−𝜏𝑗(𝑡))}𝐻²_𝑖,𝑗𝑥_𝑗𝜏_𝑗≤−𝑟𝑉_𝑛,0+𝐻_𝑛𝐱_𝑛−^𝑛𝑖_𝑖=2𝑗=1𝐻²_𝑖,𝑗𝑥_𝑗𝜏_𝑗−12𝐻²_1,1𝑥₁𝜏₁, (63) where 𝐻_𝑛(𝐱_𝑛)=(𝑒^𝑟𝜏0/(1−𝜏^∗)){∑^𝑛_𝑖=2∑^𝑖_𝑗=1𝐻²_𝑖,𝑗(𝑥_𝑗)+(1/2)𝐻²_1,1(𝑥₁)}.

By utilizing (63), (62) yields 𝑉̇_𝑛≤−𝑐_𝑛𝜒²_𝑛−𝑟𝑉_𝑛,0+𝜒_𝑛𝑔_𝑛𝜼^𝑇𝜿𝑢+𝜒_𝑛𝑓_𝑛−12𝜒²_𝑛+12^𝑛_𝑖=1𝜌²𝑑²_𝑖−1𝛾_𝑛𝜗_𝑛̇𝜗_𝑛+1−16tanh²𝜒_𝑛𝜅_𝑛𝐻_𝑛𝐱_𝑛, (64) with 𝑓_𝑛=𝑓_𝑛−^𝑛−1_𝑖=1𝜕𝛼_𝑛−1𝜕𝜍_𝑖12𝑎²_𝑖𝜒²_𝑖𝜗_𝑖+𝜕𝛼_𝑛−1𝜕𝑥_𝑖𝑓_𝑖+𝑔_𝑖𝑥_𝑖+1+𝜕𝛼_𝑛−1𝜕𝜗_𝑖𝛾_𝑖2𝑎²_𝑖𝜒²_𝑖−𝛾_𝑖𝜗_𝑖−𝜕𝛼_𝑛−1𝜕𝑦_𝑑𝑦̇_𝑑+12𝜌²+12^𝑛−1_𝑖=1𝜕𝛼_𝑛−1𝜕𝑥_𝑖²𝜒_𝑛+12𝜌²+𝑐_𝑛+1𝜒_𝑛+16𝜒_𝑛tanh²𝜒_𝑛𝜅_𝑛𝐻_𝑛𝐱_𝑛. (65)

By using FLSs, function 𝑓_𝑛 can be approximated as 𝑓_𝑛=𝜽_𝑛^∗𝑇𝝃_𝑛𝐙_𝑛+𝛿_𝑛𝐙_𝑛,𝛿_𝑛𝐙_𝑛≤𝜀_𝑛, (66) where 𝐙_𝑛=[𝐱_𝑛,𝑦_𝑑,𝑦̇_𝑑,𝝇_𝑛−1,𝜗_𝑛−1]^𝑇 with 𝝇_𝑛−1=[𝜍₁,…,𝜍_𝑛−1]^𝑇 and 𝜗_𝑛−1=[𝜗₁,…,𝜗_𝑛−1]^𝑇, 𝛿_𝑛(𝐙_𝑛) expresses the approximation error, and 𝜀_𝑛 is an unknown positive constant.

Similarly, we can derive the following inequality: 𝜒_𝑛𝑓_𝑛=𝜒_𝑛𝜽_𝑛^∗𝑇𝝃_𝑛𝐙_𝑛+𝜒_𝑛𝛿_𝑛𝐙_𝑛≤12𝑎²_𝑛𝜒²_𝑛𝜗_𝑛+12𝑎²_𝑛+12𝜒²_𝑛+12𝜀²_𝑛. (67)

Considering |𝜒_𝑛|≥0.2554𝜅_𝑛, we get 1−16tanh²𝜒_𝑛𝜅_𝑛𝐻_𝑛𝐱_𝑛≤0. (68)

Applying (56)–(58), (67), and (68), (64) results in 𝑉̇_𝑛≤−𝜇_𝑛𝑉_𝑛+𝑔_𝑛𝜼^𝑇𝜿𝑁𝜍_𝑛+1𝜍_𝑛̇+𝐶_𝑛, (69) where 𝐶_𝑛=(1/2)𝑎²_𝑛+(1/2)𝜀²_𝑛+(1/2)∑^𝑛_𝑖=1𝜌²𝑑²_𝑖+(1/2)𝜌²𝑑²₀+(𝜂_𝑛/2𝛾_𝑛)𝜗²_𝑛 and 𝜇_𝑛=min(2𝑐_𝑛,𝑟,𝜂_𝑛).

Multiplying (69) by 𝑒^𝜇𝑛𝑡 and then integrating it over [0,𝑡], we have 𝑉_𝑛≤𝛽_𝑛+𝑒^{−𝜇𝑛𝑡𝑡}₀𝑔_𝑛𝜼^𝑇𝜿𝑁𝜍_𝑛+1𝜍_𝑛̇𝑒^𝜇𝑛𝜏𝑑𝜏, (70) with 𝛽_𝑛=(𝐶_𝑛/𝜇_𝑛)+(𝑉_𝑛(0)−(𝐶_𝑛/𝜇_𝑛))𝑒^{−𝜇𝑛𝑡}.

Considering function 𝜼^𝑇(𝑡)𝜿(𝑡) satisfies 𝜼^𝑇(𝑡)𝜿(𝑡)∈[min(𝜑_𝑟0,𝜑_𝑙0),𝜑_𝑟1+𝜑_𝑙1], and Assumption 2 holds, we can derive |𝑔_𝑛𝜼^𝑇(𝑡)𝜿(𝑡)|∈[𝑔_𝑛0min(𝜑_𝑟0,𝜑_𝑙0),(𝜑_𝑟1+𝜑_𝑙1)𝑔_𝑛1]. Noting (70), applying Lemma 8, we can conclude that signals 𝑉_𝑛(𝑡), 𝜍_𝑛(𝑡), and ∫^𝑡₀𝑔_𝑛𝜼^𝑇𝜿𝑁(𝜍_𝑛)𝜍_𝑛̇𝑑𝜏 are bounded. Hence, 𝜒_𝑛, 𝑢, 𝑁(𝜍_𝑛), 𝜗_𝑛, and 𝜗_𝑛 are SUUB on [0,𝑡_𝑓).

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 ∫^𝑡₀[𝑔_𝑛𝜼^𝑇𝜿𝑁(𝜍_𝑛)+1]𝜍_𝑛̇𝑑𝜏 is bounded.
Noting (70), we suppose that the upper bound 𝜆_𝑛 satisfies ^𝑡₀𝑔_𝑛𝜼^𝑇𝜿𝑁𝜍_𝑛+1𝜍_𝑛̇𝑒^{𝜇𝑛(𝜏−𝑡)}𝑑𝜏≤𝜆_𝑛. (71)
From (54), (70), and (71), we have 𝜒_𝑛≤2𝛽_𝑛+𝜆_𝑛,𝜗_𝑛≤2𝛾_𝑛𝛽_𝑛+𝜆_𝑛. (72)
Thus, we can conclude the boundedness of the signals 𝜒_𝑛, 𝜗_𝑛, and 𝜗_𝑛.
In the rest of the steps from 𝑛−1 to 1, we acquire 𝑉_𝑖≤𝛽_𝑖+𝑒^{−𝜇𝑖𝑡𝑡}₀𝑔_𝑖𝑁𝜍_𝑖+1𝜍_𝑖̇𝑒^𝜇𝑖𝜏𝑑𝜏+𝑒^{−𝜇𝑖𝑡𝑡}₀𝜒²_𝑖+1𝑒^𝜇𝑖𝜏𝑑𝜏,𝑖=1,…,𝑛−1. (73)
As the boundedness of 𝜒_𝑖+1 is guaranteed in step 𝑖+1, we define an upper bound 𝜆_𝑖 as ^𝑡₀𝑔_𝑖𝑁𝜍_𝑖+1𝜍_𝑖̇𝑒^{𝜇𝑖(𝜏−𝑡)}𝑑𝜏+^𝑡₀𝜒²_𝑖+1𝑒^{𝜇𝑖(𝜏−𝑡)}𝑑𝜏≤𝜆_𝑖. (74)
Investigating the definition of 𝑉_𝑖(𝑡) and combing (73) and (74), we obtain 𝜒_𝑖≤2𝛽_𝑖+𝜆_𝑖,𝜗_𝑖≤2𝛾_𝑖𝛽_𝑖+𝜆_𝑖. (75)(76)
Hence, signals 𝜒_𝑖, 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 error 𝜒₁ 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 on 𝑎_𝑖, 𝜗_𝑖, 𝜀_𝑖, 𝜆_𝑖, 𝛾_𝑖, 𝜂_𝑖, 𝑟, 𝑐_𝑖, and 𝑑_𝑖. As 𝜗_𝑖, 𝜀_𝑖, 𝜆_𝑖, and 𝑑_𝑖 are unknown, a concrete estimation of the tracking error is impossible. From inequality (75), it is clear that by reducing 𝑎_𝑖 and 𝜂_𝑖, meanwhile increasing 𝑐_𝑖, 𝛾_𝑖, and 𝑟, the tracking error will be diminished. Simultaneously, it is worth pointing out that the parameters 𝜗_𝑖, 𝜀_𝑖, 𝜆_𝑖, and 𝑑_𝑖 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 is 𝑦_𝑑=sin(𝑡)+cos(0.5𝑡). The dead-zone 𝐷(𝑢) is defined as 𝐷(𝑢)=((𝑢)((𝑢)1−0.3sin))(𝑢−0.5,𝑢>0.5,0,−0.25≤𝑢≤0.5,0.8−0.2cos))(𝑢+0.25,𝑢<−0.25. (77)

Example 1. Consider the following nonlinear time-delay system: 𝑥̇₁(𝑡)=𝑥²₁(𝑡)+𝑥₁(𝑡)𝑥₂(𝑡)+0.5𝑥₁𝑡−𝜏₁(𝑡(𝑡),𝑥̇)+0.5cos₂(𝑡)=−𝑥₁(𝑡)+𝐷(𝑢(𝑡))|𝐷(𝑢(𝑡))|+0.1−2𝑥²₁(𝑡)𝑥₂(𝑡),+𝑥²₁𝑡−𝜏₁(𝑡)𝑥₂𝑡−𝜏₂(𝑡)+0.2sin(𝑡),𝑦=𝑥₁, (78) where 𝜏₁(𝑡)=0.2(1+cos(𝑡)) and 𝜏₂(𝑡)=0.3(1+sin(𝑡)).
According to Theorem 12, the control laws and the adaptive laws are chosen as 𝛼₁=𝑞₁𝜒₁2𝑎²₁𝑁𝜍₁𝜒₁𝜗₁𝑞,𝑢=²𝜒₂2𝑎²₂𝑁𝜍₂𝜒₂𝜗₂,𝜍₁̇=𝑞₁𝜒₁2𝑎²₁𝜒²₁𝜗₁,̇𝜗₁=𝛾₁2𝑎²₁𝜒²₁−𝜂₁𝜗₁,𝜍₂̇=𝑞₂𝜒₂2𝑎²₂𝜒²₂𝜗₂,̇𝜗₂=𝛾₂2𝑎²₂𝜒²₂−𝜂₂𝜗₂. (79)
In the simulation, the design parameters are selected as 𝑎₁=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, 𝜗₁(0)=0.01, and 𝜗₂(0)=0.01. The simulation results are shown in Figures 1–5, respectively.

Figure 1 

Trajectories of system output 𝑦 and reference signal 𝑦𝑑.

Figure 2 

Trajectory of the system state 𝑥2.

Figure 3 

Control input 𝑢.

Figure 4 

The curve of adaptive parameters 𝜗1 and 𝜗2.

Figure 5 

The curve of adaptive parameters 𝜍1 and 𝜍2.

From Figure 1, 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 𝜗₁, 𝜗₂, and 𝜍₁, 𝜍₂, respectively.

Example 2. To further demonstrate the feasibility of the controller, we present the following nonlinear system: 𝑥̇₁(𝑡)=16𝑥₁(𝑡)𝑥₂(𝑡)+2𝑥₁𝑡−𝜏₁(𝑡)−0.5sin𝑥₁(𝑡),𝑥̇cos₂(𝑡)=−𝑥₁(𝑡)−𝑥₂(𝑡)+𝑥₁(𝑡)+𝑥₂(𝑡)𝑥₃(𝑡)+𝑥₁𝑡−𝜏₁(𝑡)𝑥₂𝑡−𝜏₂(𝑡)(),𝑥̇−0.5sin10𝑡₃(𝑡)=10𝑥₂(𝑡)−𝑥³₃(𝑡)+0.143𝑥₃(𝑡)+sin(5𝑡)+𝑥₁𝑡−𝜏₁(𝑡)+𝑥₃𝑡−𝜏₃(𝑡)+𝐷³(𝑢(𝑡+))1+𝑥²₂(𝑡)𝐷(𝑢(𝑡((𝑢(𝑡,𝑦(𝑡)))+sin0.1𝐷)))=𝑥₁(𝑡), (80) 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 Example 1, the control laws and the adaptive laws are chosen as 𝛼₁=𝑞₁𝜒₁2𝑎²₁𝑁𝜍₁𝜒₁𝜗₁,𝛼₂=𝑞₂𝜒₂2𝑎²₂𝑁𝜍₂𝜒₂𝜗₂,𝑞𝑢=³𝜒₃2𝑎²₃𝑁𝜍₃𝜒₃𝜗₃,𝜍₁̇=𝑞₁𝜒₁2𝑎²₁𝜒²₁𝜗₁,̇𝜗₁=𝛾₁2𝑎²₁𝜒²₁−𝜂₁𝜗₁,𝜍₂̇=𝑞₂𝜒₂2𝑎²₂𝜒²₂𝜗₂,̇𝜗₂=𝛾₂2𝑎²₂𝜒²₂−𝜂₂𝜗₂,𝜍₃̇=𝑞₃𝜒₃2𝑎²₃𝜒²₃𝜗₃,̇𝜗₃=𝛾₃2𝑎²₃𝜒²₃−𝜂₃𝜗₃. (81)
In this example, we choose the design parameters as 𝑎₁=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, 𝜗₁(0)=0.01, 𝜗₂(0)=0.01, and 𝜗₃(0)=0.01. The simulation results are shown in Figures 6–11, respectively.

Figure 6 

Trajectories of system output 𝑦 and reference signal 𝑦𝑑.

Figure 7 

Trajectory of the system state 𝑥2.

Figure 8 

Trajectory of the system state 𝑥3.

Figure 9 

Control input 𝑢.

Figure 10 

The curve of adaptive parameters 𝜗1, 𝜗2, and 𝜗3.

Figure 11 

The curve of adaptive parameters 𝜍1, 𝜍2, and 𝜍3.

From Figure 6, 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 𝜗₁, 𝜗₂, 𝜗₃, 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.