Adaptive fuzzy dynamic surface tracking control for a class of nonlinear systems with unknown distributed time delays and backlash-like hysteresis

Abstract

In this article, an adaptive fuzzy backstepping dynamic surface control approach is developed for a class of nonlinear systems with unknown backlash-like hysteresis and unknown state discrete and distributed time-varying delays. Fuzzy logic systems are used to approximate the unknown nonlinear functions and a fuzzy state observer is designed for estimating the immeasurable states. Then, by combining the backstepping technique and the appropriate Lyapunov–Krasovskii functionals with the dynamic surface control approach, the output-feedback adaptive fuzzy tracking controller is designed. The main advantages of this article are (i) the existence of the state discrete and distributed time-varying delays such that the investigated systems are more general than that of the existing results, (ii) the proposed control scheme can eliminate the problem of “explosion of complexity” inherent in the backstepping design method, and (iii) for the nth nonlinear system, only one fuzzy logic system is used to approximate the unknown continuous time-varying delay functions since all of them are lumped into one unknown nonlinear function, which makes our design scheme easier to be implemented in practical applications. It is proven that the proposed design method is able to guarantee that all the signals in the closed-loop system are bounded and the tracking error can converge to a small neighborhood of origin with an appropriate choice of design parameters. Finally, the simulation results demonstrate the effectiveness of the proposed approach.

Introduction

In the past decades, many new control theories have been gradually established because of the controlled plants and the control objectives becoming more and more complex, such as the nonlinear sliding mode control, ¹ neural network control, ^2,3 fuzzy control, ⁴ and so on. It should be emphasized that fuzzy control with heuristic knowledge or linguistic information has been successfully applied to many nonlinear systems since it does not need an accurate mathematical model of the system. ^{5 –9} In a study by Qin et al., ⁵ a fuzzy adaptive robust control strategy was put forward for the trajectory tracking control of space robot, which can control the space manipulator to achieve a good trajectory tracking effect in different gravity environments without changing the structure or parameters. In a study by Bakdi et al., ⁶ the off-line path planning problem for mobile robots was dealt with by combining with the fuzzy adaptive control. And in a study by Cazarez-Castro et al., ^{7 –9} the output regulation problem for the servomechanisms with nonlinear backlash was solved using type-1 or type-2 fuzzy logic controllers. Moreover, in recent years, the control design of nonlinear systems with hysteresis has become a challenging yet rewarding problem. One of the main reasons is that the hysteresis phenomenon can be often encountered in a wide range of physical systems and devices. ¹⁰ On the other hand, since the hysteresis nonlinearity is non-differentiable, the system performance is often severely deteriorated and usually exhibits undesirable inaccuracies or oscillations and even instability. ^11,12 And in the literature, ^{13 –18} in order to control uncertain nonlinear systems with unknown backlash-like hysteresis, fuzzy logic systems (FLSs) of Mamdani type were used to approximate the uncertain smooth nonlinear functions, and then an adaptive backstepping technique was applied to design controllers. In a study by Boulkroune et al., ¹³ the authors investigated the fuzzy adaptive control design for uncertain multivariable systems with unknown backlash-like hysteresis and unknown control direction that possibly exhibited time delay. In a study by Wang et al., ¹⁴ in order to handle the nonlinear properties of hysteretic systems, an indirect adaptive fuzzy controller was proposed. Furthermore, in a study by Li et al., ^{15 –17} the proposed adaptive fuzzy control schemes with a state observer can be used to deal with those uncertain single-input single-output or multiple-input multiple-output (MIMO) nonlinear systems with unknown backlash-like hysteresis. And in a study by Cui et al., ¹⁸ an adaptive tracking control problem was studied for a class of switched stochastic nonlinear pure-feedback systems with unknown backlash-like hysteresis under arbitrary switching.

Moreover, it should be pointed out that a drawback with the backstepping technique is the problem of “explosion of complexity,” that is, the complexity of the controller grows drastically as the order n of the system increases. This “explosion of complexity” is caused by repeated differentiations of certain nonlinear functions. To overcome the “explosion of complexity,” dynamic surface control (DSC) was proposed in many existing results, for example, ^{19 –25} In a study by Swaroop et al., ¹⁹ a DSC technique was proposed to eliminate this problem by introducing a first-order filtering of the synthetic input at each step of the traditional backstepping approach. In a study by Swaroop et al., ²⁰ DSC was considered for the tracking problem of non-Lipschitz systems. In a study by Zhang and Ge, ²¹ the control scheme was developed using the DSC approach for pure-feedback nonlinear systems with an unknown dead zone. In a study by Wang et al., ²² the authors constructed an adaptive neural controller for a class of pure-feedback nonlinear time-delay systems through the DSC technique. Moreover, in the lierature, ^{23 –25} the adaptive fuzzy backstepping control approaches were proposed for nonlinear systems with unknown time delays and immeasurable states, MIMO nonlinear systems with immeasurable states, and uncertain stochastic nonlinear strict-feedback systems without the measurements of the states, respectively. And in a study by Zhang and Lin, ²⁶ for a class of nonlinear systems with unknown backlash-like hysteresis at the input, based on a high-gain observer, an adaptive DSC scheme was proposed which can be able to mitigate the effect of hysteresis, to eliminate the explosion of terms inherent in backstepping control. However, by combining the state observer with the FLS and the DSC technique, how to design an adaptive fuzzy output-feedback controller for a class of uncertain nonlinear systems with unknown backlash-like hysteresis is a challenging subject.

Since time-delay phenomena are often encountered in various engineering systems such as biological reactors, rolling mills, and so on, the existence of time delay is a significant cause of instability and deteriorative performance, so the control design of nonlinear time-delay systems has been receiving much attention. Recently, several fuzzy adaptive control schemes have been reported which combine the Lyapunov–Krasovskii functional with the adaptive backstepping fuzzy control for nonlinear systems with time delays. ^{2,13,17,27 –30} However, it is worth noting that the results in them were obtained in the context of continuous fuzzy systems with constant ^27,28 or time-varying delays. ^{2,13,17,29,30} When the number of summands in a system equation is increased and the differences between the neighboring argument values are decreased, systems with distributed delays will arise. Thus, the topic of distributed delay systems has been an attractive research topic in the past years. ^{31 –34} However, for a nonlinear system with unknown backlash-like hysteresis and unknown distributed time-varying delays, how to design an effective controller is worth studying.

Motivated by the above observations, an adaptive fuzzy DSC output-feedback control approach is presented in this article for a class of uncertain nonlinear systems, under the conditions of unknown backlash-like hysteresis, unmeasured states, unknown state discrete time-varying delays, and unknown state distributed time-varying delays. A fuzzy state observer is constructed to estimate the unmeasured states, the appropriate Lyapunov–Krasovskii functionals are used to deal with the unknown discrete and distributed time-varying delay terms, the FLS is employed to approximate the nonlinear functions, and finally, the adaptive backstepping approach is utilized to construct the fuzzy controller. The main contributions of this article are listed as follows:

Considering ^15,17,23 the distributed time-varying delays, the systems investigated in this article are more general. And the existence of the distributed time-varying delays such that the investigated systems are much more complex and difficult.

By combining the DSC approach, the proposed control scheme can eliminate the problem of “explosion of complexity” inherent in the backstepping design method.

All the unknown time-delay terms are lumped into one unknown nonlinear function which can be approximated by only one FLS such that the suggested adaptive fuzzy controller contains less adaptive parameters and this makes our design scheme easier to be implemented in practical applications. Moreover, in the studies by Li et al. and Tong et al., ^17,23 it was assumed that the time-delay nonlinear terms were bounded by known functions, but in this article, the assumption is relaxed.

It can be proven that all the signals in the closed-loop system are bounded and the tracking errors can converge to a small neighborhood around zero. The effectiveness of the developed scheme is illustrated by a simulation example.

Problem formulation and preliminaries

Problem formulation

Consider the system described by

$\{\begin{array}{l}{\dot{x}}_i=x_{i+1}+f_i\left({\overline{x}}_i\right)+h_i(y(t-d_{1i}(t)))\hfill \\ \hspace{1em}\hspace{1em}+{\displaystyle {\int }_{t-d_{2i}\left(t\right)}^t}{m}_i(y(s))\text{d}s\hfill \\ {\dot{x}}_n=\phi (\upsilon \left(t\right))+f_n\left({\overline{x}}_n\right)+h_n(y(t-d_{1n}\left(t\right)))\hfill \\ \hspace{1em}\hspace{1em}+{\displaystyle {\int }_{t-d_{2n}\left(t\right)}^t}{m}_n(y(s))\text{d}s\hfill \\ y=x_1\hfill \end{array}$

1

where $x=[x_1,\dots ,x_n{]}^T\in R^n$ is the state vector of the system with ${\overline{x}}_i=[x_1,\dots ,x_i{]}^T\in R^i$ and y ∈ R is the output signal. f_i(⋅), h_i(⋅), and m_i(⋅) are all unknown smooth nonlinear functions. $\upsilon (t)\in R$ is the control input and φ(υ) denotes the hysteresis type of nonlinearity. This article assumes that the states of the system (1) are unknown and only the output y(t) is available for measurement.

Assumption 1

The discrete and distributed time-varying delays $d_{1i}(t)$ and $d_{2i}(t)$ satisfy $0\le d_{1i}(t)\le d_1$ and $0\le d_{2i}\left(t\right)\le d_2$ with ${\dot{d}}_{1i}\left(t\right)\le d_1^{*}<1$ and ${\dot{d}}_{2i}\left(t\right)\le d_2^{*}<1$ , respectively. In particular, the constants d₁, d₂, $d_1^{*}$ , and $d_2^{*}$ may be unknown.

Assumption 2

There exists a known constant L_i such that

$\left|f_i\left({\overline{x}}_i\right)-f_i\left({\widehat{\overline{x}}}_i\right)\right|\le L_i\left(\left|x_1-{\widehat{x}}_1\right|+\cdots +\left|x_i-{\widehat{x}}_i\right|\right)$

2

where ${\widehat{x}}_i$ is the estimate of x_i.

Assumption 3

The desired trajectory y_r(t) is continuous and known. Moreover, y_r(t), ${\dot{y}}_r(t)$ , and ${\ddot{y}}_r(t)$ are all assumed to be bounded.

Assumption 4

There exists a large positive constant M such that $\left|\upsilon (t)\right|\le M$ .

Remark 1

Practically speaking, assumption 1 is common and relaxed, we can see it in many results, for example ^{2,17,29,34,35} and the references therein. Moreover, it should be emphasized that the constants d₁, d₂, $d_1^{*}$ , and $d_2^{*}$ are introduced only for the stability analysis and they are not used in the controller design, thus they can be unknown in the controller design procedure. Moreover, assumptions 3 and 4 are both common, which can be found in many existing literature. ^{25,15,17,23,36} Due to the assumption of $\left|\upsilon (t)\right|\le M$ (M may be a large constant), the stability of the closed-loop control system of this article is built in the sense of semi-globally uniformly ultimately bounded (SUUB). ^15,17

Moreover, since the states of the system (1) are unmeasurable, (1) can be rewritten as follows

$\{\begin{array}{l}{\dot{x}}_i=x_{i+1}+f_i\left({\widehat{\overline{x}}}_i\right)+\Delta F_i+h_i(y(t-d_{1i}(t)))\hfill \\ \hspace{1em}+{\displaystyle {\int }_{t-d_{2i}\left(t\right)}^t}{m}_i(y(s))\text{d}s\hfill \\ {\dot{x}}_n=\phi (\upsilon \left(t\right))+f_n\left({\widehat{\overline{x}}}_n\right)+\Delta F_n\hfill \\ \hspace{1em}+h_n(y(t-d_{1n}(t)))+{\displaystyle {\int }_{t-d_{2n}\left(t\right)}^t}{m}_n(y(s))\text{d}s\hfill \\ y=x_1\hfill \end{array}$

3

where $\Delta F_i=f_i({\overline{x}}_i)-f_i({\widehat{\overline{x}}}_i)$ , ${\widehat{\overline{x}}}_i$ is the estimate of ${\overline{x}}_i$ and the observer will be given later.

Control objective

Our control objective is to design the output-feedback controller υ(t) and parameter adaptive laws such that all the signals involved in the resulting closed-loop system are SUUB, and the output y(t) tracks the given reference signal y_r(t) as desired.

Fuzzy logic systems

In this article, the following rules are used to develop the adaptive fuzzy controller.

R^l: if x₁ is $F_1^l$ , x₂ is $F_2^l$ and x_n is $F_n^l$ , then y is G^l, l = 1, 2…, Q

where $x={\left[x_1,\dots ,x_n\right]}^T$ and y are the FLS input and output, respectively. Fuzzy sets $F_i^l$ and G^l are associated with the membership functions ${\mu }_{F_i^l}\left(x_i\right)$ and ${\mu }_{G^l}\left(y\right)$ , respectively. Q is the rule number. Through singleton function, center average defuzzification, the FLS can be expressed as follows

$y\left(x\right)=\frac{{\displaystyle {\sum }_{l=1}^Q}{\mathit{\theta }}_l{\Pi }_{i=1}^n{\mu }_{F_i^l}\left(x_i\right)}{{\displaystyle {\sum }_{l=1}^Q}{\Pi }_{i=1}^n{\mu }_{F_i^l}\left(x_i\right)}$

4

where $x={\left[x_1,x_2,\dots ,x_n\right]}^T\in R^n,$ ${\mu }_{F_i^l}\left(x_i\right)$ is the membership function of $F_i^l$ , and ${\mathit{\theta }}_l=\text{arg}{\text{sup}}_{y\in R}{\mu }_{G^l}\left(y\right)$ . Define $\mathit{\theta }={\left[{\mathit{\theta }}_1,\dots ,{\mathit{\theta }}_Q\right]}^T$ and $\varphi \left(x\right)={\left[{\varphi }_1\left(x\right),\dots ,{\varphi }_Q\left(x\right)\right]}^T$ with the fuzzy basis function ξ_l given by

${\varphi }_l(x)=\frac{{\Pi }_{i=1}^n{\mu }_{F_i^l}\left(x_i\right)}{{\displaystyle {\sum }_{l=1}^Q}{\Pi }_{i=1}^n{\mu }_{F_i^l}\left(x_i\right)}$

5

Then, the FLS (4) can be rewritten as

$y\left(x\right)={\mathit{\theta }}^T\varphi \left(x\right)$

6

Our first choice for the membership function is the Gaussian function ${\mu }_{F_i^l}\left(x_i\right)=\text{exp}\left(-\frac{1}{2}{(\left(x_i-a_i^l\right)/{\sigma }_i^l)}^2\right)$ , where ${\sigma }_i^l$ and $a_i^l$ are fixed parameters. It has been proven that when the membership functions are chosen as Gaussian functions, the above FLS is capable of uniformly approximating any continuous nonlinear function over a compact set with any degree of accuracy. This property is shown by the following lemma

Lemma 1

Let f(x) be a continuous function defined on compact set Ω. ⁴ Then, for any constant $\overline{\overline{\iota }}>0$ , there exists an FLS (6) such that

$\underset{x\in \Omega }{\text{sup}}\left|f\left(x\right)-{\mathit{\theta }}^T\varphi \left(x\right)\right|\le \overline{\overline{\iota }}$

7

Lemma 2

Given a positive definite matrix $W\in R^{n\times n}$ and two scalars g > d ≥ 0 for any vector $\omega (t)\in R^n$ , we have

$\begin{array}{l}\left({\displaystyle {\int }_{t-g}^{t-d}}{\omega }^T\left(s\right)\text{d}s\right)W\left({\displaystyle {\int }_{t-g}^{t-d}}\omega \left(s\right)\text{d}s\right)\\ \le (d-g){\displaystyle {\int }_{t-g}^{t-d}}{\omega }^T\left(s\right)W\omega \left(s\right)\text{d}s\end{array}$

8

By lemma 1, the FLS (6) is a universal approximator, that is, it can approximate any unknown continuous function on a compact set. ³¹ Therefore, it can be assumed that the unknown function $f_i({\widehat{\overline{x}}}_i)$ in system (3) can be approximated by the following FLS. Define the optimal parameter vector ${\mathit{\theta }}_i^{*}$ as

${\mathit{\theta }}_i^{*}=\text{arg}\underset{{\widehat{\theta }}_i\in {\Xi }_{{ }_i}}{\text{min}}\left\{\underset{{\widehat{\overline{x}}}_i\in {\Omega }_i}{\text{sup}}\left|f_i\left({\widehat{\overline{x}}}_i\right)-{\widehat{\mathit{\theta }}}_i^T{\varphi }_i\left({\widehat{\overline{x}}}_i\right)\right|\right\}$

9

where Ξ _i and Ω _i are compact regions for ${\widehat{\mathit{\theta }}}_i$ and ${\widehat{\overline{x}}}_i$ , respectively. The fuzzy minimum approximation error and fuzzy approximation error ${\epsilon }_i({\overline{x}}_i)$ and ${\delta }_i({\overline{x}}_i)$ are defined as

$\begin{array}{r}{\epsilon }_i({\widehat{\overline{x}}}_i)=f_i\left({\widehat{\overline{x}}}_i\right)-{\mathit{\theta }}_i^{*}{ }^T{\varphi }_i\left({\widehat{\overline{x}}}_i\right)\\ {\delta }_i({\widehat{\overline{x}}}_i)=f_i\left({\widehat{\overline{x}}}_i\right)-{\widehat{\mathit{\theta }}}_i^T{\varphi }_i\left({\widehat{\overline{x}}}_i\right)\end{array}$

10

Assumption 5

$\left|{\epsilon }_i\right|\le {\epsilon }_i^{*}$ and $\left|{\delta }_i\right|\le {\delta }_i^{*}$ , where ${\epsilon }_i^{*}>0$ and ${\delta }_i^{*}>0$ are unknown constants, i = 1, …, n.

Denote ${\omega }_i={\epsilon }_i-{\delta }_i$ , by assumption 5, we have $\left|{\omega }_i\right|\le {\epsilon }_i^{*}+{\delta }_i^{*}={\omega }_i^{*}$ with ${\omega }_i^{*}$ being an unknown constant. By (10), system (3) can be rewritten as follows

$\{\begin{array}{l}{\dot{x}}_1=x_2+{\mathit{\theta }}_i^{*}{ }^T{\varphi }_1\left({\widehat{x}}_1\right)+{\epsilon }_1\left({\widehat{x}}_1\right)+\Delta F_1\hfill \\ +h_1(y(t-d_{11}\left(t\right)))+{\displaystyle {\int }_{t-d_{21}\left(t\right)}^t}{m}_1(y(s))\text{d}s\hfill \\ {\dot{x}}_i=x_{i+1}+{\mathit{\theta }}_i^{*}{ }^T{\varphi }_i\left({\widehat{\overline{x}}}_i\right)+{\epsilon }_i\left({\widehat{\overline{x}}}_i\right)+\Delta F_i\hfill \\ +h_i(y(t-d_{1i}\left(t\right)))+{\displaystyle {\int }_{t-d_{2i}\left(t\right)}^t}{m}_i(y(s))\text{d}s\hfill \\ {\dot{x}}_n=\phi (\upsilon )+{\mathit{\theta }}_n^{*}{ }^T{\varphi }_n\left({\widehat{\overline{x}}}_n\right)+{\epsilon }_n\left({\widehat{\overline{x}}}_n\right)+\Delta F_n\hfill \\ +h_n(y(t-d_{1n}\left(t\right)))+{\displaystyle {\int }_{t-d_{2n}\left(t\right)}^t}{m}_n(y(s))\text{d}s\hfill \\ y=x_1\hfill \end{array}$

11

From ^10,15,17 the control input υ and the hysteresis type of nonlinearity φ(υ), system (1) can be described by

$\frac{d\phi (\upsilon (t))}{dt}=\alpha \left|\frac{d\upsilon }{dt}\right|\left(c\upsilon -\phi \right)+B_1\frac{d\upsilon }{dt}$

12

where α, c, and B₁ are constants and c > 0 is the slope of the lines satisfying c > B₁ ¹² . In this article, the parameters of hysteresis in (12), that is, α, c, and B₁ are completely unknown. ¹⁷ Based on the analysis of the literature, ^10,11,15 (12) can be solved explicitly as

$\begin{array}{c}\phi \left(\upsilon \right)=c\upsilon \left(t\right)+d_1\left(\upsilon \right)\\ d_1\left(\upsilon \right)=\left[{\phi }_0-c{\upsilon }_0\right]e^{-\alpha \left(\upsilon -{\upsilon }_0\right){\text{sgn}}_{\dot{\upsilon }}}\\ \hspace{0.17em}\hspace{0.17em}+e^{-\alpha \upsilon {\text{sgn}}_{\dot{\upsilon }}}{\displaystyle {\int }_{{\upsilon }_0}^{\upsilon }}\left[B_1-c\right]e^{\alpha \eta {\text{sgn}}_{\dot{\upsilon }}}d\eta \end{array}$

13

where υ(0) = υ₀ and φ(0) = φ₀.

Remark 2. According to (12) and (13), the backlash-like hysteresis is dealt with successfully and then using the backstepping technique, the input signal υ(t) is designed finally

Based on the above solution, it is shown that d₁(υ) is bounded. ^10,11,15 In this article, we assume that $d_1(\upsilon )\le \overline{d}$ with $\overline{d}$ being a constant. Thus, using (11) and (13) gives that

$\{\begin{array}{l}{\dot{x}}_1=x_2+{\mathit{\theta }}_1^{*}{ }^T{\varphi }_1\left({\widehat{x}}_1\right)+{\epsilon }_1\left({\widehat{x}}_1\right)+\Delta F_1\\ \hspace{1em}+h_1(y(t-d_{11}\left(t\right)))+{\displaystyle {\int }_{t-d_{21}\left(t\right)}^t}{m}_1(y(s))\text{d}s\\ {\dot{x}}_i=x_{i+1}+{\mathit{\theta }}_i^{*}{ }^T{\varphi }_i\left({\widehat{\overline{x}}}_i\right)+{\epsilon }_i\left({\widehat{\overline{x}}}_i\right)+\Delta F_i\\ \hspace{1em}+h_i(y(t-d_{1i}\left(t\right)))+{\displaystyle {\int }_{t-d_{2i}\left(t\right)}^t}{m}_i(y(s))\text{d}s\\ {\dot{x}}_n=c\upsilon (t)+d_1(\upsilon )+{\mathit{\theta }}_n^{*}{ }^T{\varphi }_n\left({\widehat{\overline{x}}}_n\right)\\ \hspace{1em}+{\epsilon }_n\left({\widehat{\overline{x}}}_n\right)+\Delta F_n+h_n(y(t-d_{1n}\left(t\right)))\\ \hspace{1em}+{\displaystyle {\int }_{t-d_{2n}\left(t\right)}^t}{m}_n(y(s))\text{d}s\\ y=x_1\end{array}$

14

Fuzzy adaptive observer design

Note that the states ${\overline{x}}_n$ of system (1) are not available for feedback, thus a state observer must be designed to estimate the unmeasured states. In this article, a fuzzy adaptive observer is designed for (14) as follows

$\{\begin{array}{l}{\dot{\widehat{x}}}_1={\widehat{x}}_2+{\widehat{\mathit{\theta }}}_1^T{\varphi }_1\left({\widehat{x}}_1\right)+k_1\left(y-{\widehat{x}}_1\right)\hfill \\ {\dot{\widehat{x}}}_i={\widehat{x}}_{i+1}+{\widehat{\mathit{\theta }}}_i^T{\varphi }_i\left({\widehat{\overline{x}}}_i\right)+k_i\left(y-{\widehat{x}}_1\right)\hfill \\ {\dot{\widehat{x}}}_n=\widehat{c}\upsilon \left(t\right)+{\widehat{\mathit{\theta }}}_n^T{\varphi }_n\left({\widehat{\overline{x}}}_n\right)+k_n\left(y-{\widehat{x}}_1\right)\hfill \end{array}$

15

Rewriting (15) in the following form

$\begin{array}{l}\dot{\widehat{x}}=A\widehat{x}+Ky+F\left(\widehat{x}|{\widehat{\mathit{\theta }}}_n\right)+E_n\widehat{c}\upsilon \left(t\right)\\ \widehat{y}=E_1^T{\widehat{x}}_1\end{array}$

16

where $A=\left[\begin{array}{ccc}-k_1& & \\ ⋮& & I_{n-1}\\ -k_n& \dots & 0\end{array}\right]$ , $K={\left[k_1,\dots ,k_n\right]}^T$ , $F\left(\widehat{x}|{\widehat{\mathit{\theta }}}_n\right)={\left[{\widehat{\mathit{\theta }}}_1^T{\varphi }_1\left({\widehat{x}}_1\right),\cdots ,{\widehat{\mathit{\theta }}}_n^T{\varphi }_n\left({\widehat{\overline{x}}}_n\right)\right]}^T$ , $E_n^T=\left[0,\cdots \mathrm{,1}\right],E_1^T=\left[1,\cdots \mathrm{,0}\right]$ , and $\widehat{c}$ is the estimate of c.

Given a positive definite matrix Q = Q^T > 0, appropriate constants ψ and ε, choose the vector K in A such that the following matrix inequality is satisfied

$\begin{array}{l}\left(\psi {\displaystyle \sum _{i=1}^n}{L}_i^2+\frac{5}{8\epsilon }+\frac{1}{4\epsilon }\left(L_1^2+1\right)\right)I+A^{T}P+PA+\frac{1}{\psi }P{P}^T\le -Q\end{array}$

17

where P^T = P > 0 is a positive definite matrix.

Remark 3

To obtain the positive definite matrix P and vector K from (17), we can decompose A into $A=\overline{A}+KD$ with $\overline{A}=\left[\begin{array}{cc}0& I_{n-1}\\ 0& 0\end{array}\right]$ and $D=[-1\hspace{0.17em}0\dots 0]$ . Then, (17) can be transformed into a standard linear matrix inequality (LMI)

$\left[\begin{array}{cc}E& P\\ P& -\psi I\end{array}\right]<0$

18

where $E=\left(\frac{5}{8\epsilon }+\psi {\displaystyle \sum _{i=1}^n}{L}_i^2+\frac{1}{4\epsilon }\left(L_1^2+1\right)\right)I+{\overline{A}}^{T}P+P\overline{A}+D^T{N}^T+ND+Q$ and N = PK. By solving the LMI (18), one can obtain P and N, which are numerically efficient with a commercially available software. Finally, the matrix K can be computed as K = P⁻¹N.

Let $e=x-\widehat{x}$ be an observer error, then from (14) to (16), the observer error equation can be obtained as follows

$\dot{e}=Ae+\delta +\Delta F+E_n\tilde{c}\upsilon +E_n{d}_1\left(\upsilon \right)+H+M$

19

where $H={\left[\begin{array}{ccc}{h}_1& \dots & h_n\end{array}\right]}^T,\hspace{0.17em}M=[{\displaystyle {\int }_{t-d_{21}(t)}^t}{m}_1(y(s))\text{d}s\dots {{\displaystyle {\int }_{t-d_{2n}(t)}^t}{m}_n(y(s))\text{d}s]}^T$ , $\delta ={\left[{\delta }_1({\widehat{\overline{x}}}_1),\cdots ,{\delta }_i({\widehat{\overline{x}}}_i)\right]}^T$ , and $\tilde{c}=c-\widehat{c}$ .

Consider the following Lyapunov candidate V₀

$V_0=\frac{1}{2}{e}^{T}Pe$

20

the time derivative of V₀ is given by

$\begin{array}{ll}{\dot{V}}_0\hfill & ={\dot{e}}^{T}Pe+e^{T}P\dot{e}\hfill \\ \hfill & =(Ae+\delta +\Delta F+E_n\tilde{c}\upsilon +E_n{d}_1\left(\upsilon \right){+H+M)}^{T}Pe+e^{T}P(Ae+\delta +\Delta F+E_n\tilde{c}\upsilon +E_n{d}_1\left(\upsilon \right)+H+M)\hfill \\ \hfill & =e^T\left(A^{T}P+PA\right)e+2e^{T}P(\Delta F+E_n\tilde{c}\upsilon \left(t\right)+\delta +H+M+E_n{d}_1\left(\upsilon \right))\hfill \end{array}$

21

Utilizing Young’s inequality, lemma 2 and assumptions 1, 2 and 4, one has

$2e^{T}P\Delta F\le \frac{1}{\psi }{e}^{T}P{P}^{T}e+\psi \Delta F^T\Delta F\le \frac{1}{\psi }{e}^{T}P{P}^{T}e+\psi {\displaystyle \sum _{i=1}^n}{L}_i^2{\Vert {\overline{x}}_i-{\widehat{\overline{x}}}_i\Vert }^2\le \frac{1}{\psi }{e}^{T}P{P}^{T}e+\psi {\displaystyle \sum _{i=1}^n}{L}_i^2{\Vert e\Vert }^{2}2e^{T}PH\le 8\epsilon {\Vert PH\Vert }^2+\frac{1}{8\epsilon }{\Vert e\Vert }^2\le 8\epsilon {\Vert P\Vert }^2{\displaystyle \sum _{j=1}^n}{h}_j^2\left(y\left(t-d_{1j}\left(t\right)\right)\right)+\frac{1}{8\epsilon }{\Vert e\Vert }^{2}2e^{T}PM\le 8\epsilon {\Vert PM\Vert }^2+\frac{1}{8\epsilon }{\Vert e\Vert }^2\le 8\epsilon {\Vert P\Vert }^2{\displaystyle \sum _{j=1}^n}{\left({\displaystyle {\int }_{t-d_{2j}(t)}^t}{m}_j\left(y\left(s\right)\right)\text{d}s\right)}^2+\frac{1}{8\epsilon }{\Vert e\Vert }^2\le 8\epsilon {\Vert P\Vert }^2{\displaystyle \sum _{j=1}^n}{d}_{2j}(t){\displaystyle {\int }_{t-d_{2j}(t)}^t}{m}_j^2\left(y\left(s\right)\right)\text{d}s+\frac{{\Vert e\Vert }^2}{8\epsilon }\le 8\epsilon {\Vert P\Vert }^2{\displaystyle \sum _{j=1}^n}{d}_2{\displaystyle {\int }_{t-d_{2j}(t)}^t}{m}_j^2\left(y\left(s\right)\right)\text{d}s+\frac{{\Vert e\Vert }^2}{8\epsilon }2e^{T}P\delta \le \frac{1}{8\epsilon }{\Vert e\Vert }^2+8\epsilon {\Vert P\Vert }^2{\Vert {\delta }^{*}\Vert }^{2}2e^{T}P{E}_n{d}_1\left(\upsilon \right)\le \frac{1}{8\epsilon }{\Vert e\Vert }^2+8\epsilon {\Vert P\Vert }^2{\overline{d}}^{2}2e^{T}P{E}_n\tilde{c}\upsilon \left(t\right)\le \frac{1}{8\epsilon }{\Vert e\Vert }^2+8\epsilon \tilde{c}\left(c-\widehat{c}\right){\Vert P\Vert }^2{\upsilon }^2\le \frac{1}{8\epsilon }{\Vert e\Vert }^2+\frac{\epsilon }{2}{\tilde{c}}^2{\Vert P\Vert }^2{\upsilon }^2+32\epsilon c^2{\Vert P\Vert }^2{\upsilon }^2-8\epsilon \tilde{c}\widehat{c}{\Vert P\Vert }^2{\upsilon }^2$

22

then, we have

$\begin{array}{l}{\dot{V}}_0\le e^T(A^{T}P+PA+\frac{1}{\psi }P{P}^T+\psi {\displaystyle \sum _{i=1}^n}{L}_i^{2}I+\frac{5}{8\epsilon }I)e\\ +8\epsilon {\Vert P\Vert }^2{\displaystyle \sum _{j=1}^n}{h}_j^2(y(t-d_{1j}\left(t\right)))\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+8\epsilon {\Vert P\Vert }^2{\Vert {\delta }^{*}\Vert }^2+8\epsilon {\Vert P\Vert }^2{\overline{d}}^2\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+8\epsilon {\Vert P\Vert }^2{\displaystyle \sum _{j=1}^n}{d}_2{\displaystyle {\int }_{t-d_{2j}(t)}^t}{m}_j^2(y\left(s\right))\text{d}s\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\left(\frac{\epsilon }{2}{\tilde{c}}^2{\Vert P\Vert }^2+32\epsilon c^2{\Vert P\Vert }^2-8\epsilon \tilde{c}\widehat{c}{\Vert P\Vert }^2\right){\upsilon }^2\end{array}$

23

Adaptive fuzzy control design and stability analysis

The adaptive fuzzy backstepping output-feedback control design is based on the following change of coordinates

$z_1=y-y_r,\hspace{1em}{z}_i={\widehat{x}}_i-{\alpha }_{i-1}$

24

where α_i−1 is called the intermediate control function, which will be defined later.

Step 1. The time derivative of z₁ is given by

$\begin{array}{l}{\dot{z}}_1=\dot{y}-{\dot{y}}_r=x_2+{\mathit{\theta }}_1^{*}{ }^T{\varphi }_1\left({\widehat{x}}_1\right)+{\epsilon }_1\left({\widehat{x}}_1\right)+\Delta F_1\\ \hspace{1em}+h_1(y(t-d_{11}\left(t\right)))+{\displaystyle {\int }_{t-d_{21}\left(t\right)}^t}{m}_1(y(s))\text{d}s-{\dot{y}}_r\\ \hspace{1em}=z_2+{\alpha }_1+e_2+{\mathit{\theta }}_1^{*}{ }^T{\varphi }_1\left({\widehat{x}}_1\right)+{\epsilon }_1\left({\widehat{x}}_1\right)+\Delta F_1\\ \hspace{1em}+h_1(y(t-d_{11}\left(t\right)))+{\displaystyle {\int }_{t-d_{21}\left(t\right)}^t}{m}_1(y(s))\text{d}s-{\dot{y}}_r\end{array}$

Consider the Lyapunov function candidate V₁ as follows

$V_1=\frac{1}{2}{z}_1^2$

25

The time derivative of V₁ can be given by

${\dot{V}}_1=z_1{\dot{z}}_1=z_1(z_2+{\alpha }_1+e_2+{\mathit{\theta }}_1^{*}{ }^T{\varphi }_1\left({\widehat{x}}_1\right)+{\epsilon }_1\left({\widehat{x}}_1\right)+\Delta F_1+h_1(y(t-d_{11}\left(t\right)))+{\displaystyle {\int }_{t-d_{21}\left(t\right)}^t}{m}_1(y(s))\text{d}s-{\dot{y}}_r)$

26

By using Young’s inequality under assumptions 1 and 2, one has

$\begin{array}{l}{z}_1{\epsilon }_1\left({\widehat{x}}_1\right)\le \left|z_1\right|{\epsilon }_1^{*},\hspace{1em}{z}_1\Delta F_1\le \epsilon z_1^2+\frac{1}{4\epsilon }{L}_1^2{\Vert e\Vert }^2\\ z_1{e}_2\le \epsilon z_1^2+\frac{1}{4\epsilon }{\Vert e\Vert }^2\\ z_1{h}_1(y(t-d_{11}\left(t\right)))\le \frac{1}{4}{z}_1^2+h_1^2(y(t-d_{11}\left(t\right)))\\ z_1{\displaystyle {\int }_{t-d_{21}\left(t\right)}^t}{m}_1(y(s))\text{d}s\le \frac{1}{4}{z}_1^2+d_2{\displaystyle {\int }_{t-d_{21}\left(t\right)}^t}{m}_1^2(y(s))\text{d}s\end{array}$

27

According to (26) and (27) results in

$\begin{array}{l}{\dot{V}}_1\le \frac{1}{2}{z}_1^2+2\epsilon z_1^2+\left|z_1\right|{\epsilon }_1^{*}-z_1{\epsilon }_1^{*}\text{tanh}\left(\frac{z_1}{\kappa }\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+z_1{\tilde{\epsilon }}_1\text{tanh}\left(\frac{z_1}{\kappa }\right)+z_1{\widehat{\epsilon }}_1\text{tanh}\left(\frac{z_1}{\kappa }\right)+z_1{z}_2\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\frac{1}{4\epsilon }\left(L_1^2+1\right){\Vert e\Vert }^2+z_1({\alpha }_1+{\mathit{\theta }}_1^{*}{ }^T{\varphi }_1\left({\widehat{x}}_1\right)-{\dot{y}}_r)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+h_1^2(y(t-d_{11}\left(t\right)))+d_2{\displaystyle {\int }_{t-d_{21}\left(t\right)}^t}{m}_1^2(y(s))\text{d}s\end{array}$

28

where the term $z_1{\epsilon }_1^{*}\text{tanh}\left(\frac{z_1}{\kappa }\right)$ is introduced to deal with the term $\left|z_1\right|{\epsilon }_1^{*}$ and ${\tilde{\epsilon }}_1={\epsilon }_1^{*}-{\widehat{\epsilon }}_1$ . Note that for any κ > 0, we have $\left|z_1\right|{\epsilon }_1^{*}-z_1{\epsilon }_1^{*}\text{tanh}\left(\frac{z_1}{\kappa }\right)\le 0.2785\kappa =\kappa \prime$ , then

$\begin{array}{l}{\dot{V}}_1\le \frac{1}{2}{z}_1^2+2\epsilon z_1^2+\kappa \prime +z_1{\tilde{\epsilon }}_1\text{tanh}\left(\frac{z_1}{\kappa }\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+z_1{\widehat{\epsilon }}_1\text{tanh}\left(\frac{z_1}{\kappa }\right)+\frac{1}{4\epsilon }\left(L_1^2+1\right){\Vert e\Vert }^2\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+z_1({\alpha }_1+{\mathit{\theta }}_1^{*}{ }^T{\varphi }_1\left({\widehat{x}}_1\right)-{\dot{y}}_r)+z_1{z}_2\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+h_1^2(y(t-d_{11}\left(t\right)))+d_2{\displaystyle {\int }_{t-d_{21}\left(t\right)}^t}{m}_1^2(y(s))\text{d}s\end{array}$

29

Step 2. The time derivative of z₂ is given by

$\begin{array}{l}{\dot{z}}_2={\dot{\widehat{x}}}_2-{\dot{\alpha }}_1={\widehat{x}}_3+{\widehat{\mathit{\theta }}}_2^T{\varphi }_2\left({\widehat{\overline{x}}}_2\right)+k_2\left(y-{\widehat{x}}_1\right)-{\dot{\alpha }}_1\\ \hspace{1em}={\widehat{x}}_3+{\mathit{\theta }}_2^{*}{ }^T{\varphi }_2\left({\widehat{\overline{x}}}_2\right)+f_2\left({\widehat{\overline{x}}}_2\right)-{\mathit{\theta }}_2^{*}{ }^T{\varphi }_2\left({\widehat{\overline{x}}}_2\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\widehat{\mathit{\theta }}}_2^T{\varphi }_2\left({\widehat{\overline{x}}}_2\right)-f_2\left({\widehat{\overline{x}}}_2\right)+k_2\left(y-{\widehat{x}}_1\right)-{\dot{\alpha }}_1\\ \hspace{1em}={\widehat{x}}_3+{\mathit{\theta }}_2^{*}{ }^T{\varphi }_2\left({\widehat{\overline{x}}}_2\right)+{\epsilon }_2-{\delta }_2+k_2\left(y-{\widehat{x}}_1\right)-{\dot{\alpha }}_1\\ \hspace{1em}={\widehat{x}}_3+{\widehat{\mathit{\theta }}}_2^T{\varphi }_2\left({\widehat{\overline{x}}}_2\right)+{\tilde{\mathit{\theta }}}_2^T{\varphi }_2\left({\widehat{\overline{x}}}_2\right)+{\omega }_2\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+k_2\left(y-{\widehat{x}}_1\right)-{\dot{\alpha }}_1\end{array}$

30

where ${\tilde{\mathit{\theta }}}_2={\mathit{\theta }}_2^{*}-{\widehat{\mathit{\theta }}}_2$ . Consider the Lyapunov function candidate V₂ as follows

$V_2=\frac{1}{2}{z}_2^2$

31

then, the time derivative of V₂ along the solutions of (30) is given by

${\dot{V}}_2=z_2{\dot{z}}_2=z_2({\widehat{x}}_3+{\widehat{\theta }}_2^T{\phi }_2\left({\widehat{\overline{x}}}_2\right)+{\tilde{\theta }}_2^T{\phi }_2\left({\widehat{\overline{x}}}_2\right)+{\omega }_2+k_2\left(y-{\widehat{x}}_1\right)-{\dot{\alpha }}_1)$

32

by using $z_2{\omega }_2\le \left|z_2\right|{\omega }_2^{*}$ , we have

$\begin{array}{l}{\dot{V}}_2=z_2{\dot{z}}_2\le z_2({\widehat{x}}_3+z_1+{\widehat{\mathit{\theta }}}_2^T{\varphi }_2\left({\widehat{\overline{x}}}_2\right)+{\tilde{\mathit{\theta }}}_2^T{\varphi }_2\left({\widehat{\overline{x}}}_2\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+k_2\left(y-{\widehat{x}}_1\right)-{\dot{\alpha }}_1)-z_2{z}_1+\left|z_2\right|{\omega }_2^{*}\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=z_2(z_3+{\alpha }_2+z_1+{\widehat{\mathit{\theta }}}_2^T{\varphi }_2\left({\widehat{\overline{x}}}_2\right)+{\tilde{\mathit{\theta }}}_2^T{\varphi }_2\left({\widehat{\overline{x}}}_2\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+k_2\left(y-{\widehat{x}}_1\right)-{\dot{\alpha }}_1)-z_2{z}_1+\left|z_2\right|{\omega }_2^{*}\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-z_2{\omega }_2^{*}\text{tanh}\left(\frac{z_2}{\kappa }\right)+z_2{\tilde{\omega }}_2\text{tanh}\left(\frac{z_2}{\kappa }\right)+z_2{\widehat{\omega }}_2\text{tanh}\left(\frac{z_2}{\kappa }\right)\end{array}$

33

then, similar to (29) yields that

$\begin{array}{l}{\dot{V}}_2\le z_2{z}_3+z_2({\alpha }_2+z_1+{\widehat{\mathit{\theta }}}_2^T{\varphi }_2\left({\widehat{\overline{x}}}_2\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}{\tilde{\mathit{\theta }}}_2^T{\varphi }_2\left({\widehat{\overline{x}}}_2\right)+k_2\left(y-{\widehat{x}}_1\right)-{\dot{\alpha }}_1)-z_2{z}_1\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\kappa \prime +z_2{\tilde{\omega }}_2\text{tanh}\left(\frac{z_2}{\kappa }\right)+z_2{\widehat{\omega }}_2\text{tanh}\left(\frac{z_2}{\kappa }\right)\end{array}$

34

Step i. Similar to step 2, the time derivative of z_i is given by

$\begin{array}{l}{\dot{z}}_i={\widehat{x}}_{i+1}+{\widehat{\mathit{\theta }}}_i^T{\varphi }_i\left({\widehat{\overline{x}}}_i\right)+{\tilde{\mathit{\theta }}}_i^T{\varphi }_i\left({\widehat{\overline{x}}}_i\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{1em}+{\omega }_i+k_i\left(y-{\widehat{x}}_1\right)-{\dot{\alpha }}_{i-1}\end{array}$

35

Consider the Lyapunov function candidate V_i as follows

$V_i=\frac{1}{2}{z}_i^2$

36

the derivative of V_i is given by

$\begin{array}{l}{\dot{V}}_i=z_i{\dot{z}}_i\le z_i{z}_{i+1}+z_i({\alpha }_i+z_{i-1}+{\widehat{\mathit{\theta }}}_i^T{\varphi }_i\left({\widehat{\overline{x}}}_i\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\tilde{\mathit{\theta }}}_i^T{\varphi }_i\left({\widehat{\overline{x}}}_i\right)+k_i\left(y-{\widehat{x}}_1\right)-{\dot{\alpha }}_{i-1})-z_i{z}_{i-1}+\kappa \prime \\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+z_i{\tilde{\omega }}_i\text{tanh}\left(\frac{z_i}{\kappa }\right)+z_i{\widehat{\omega }}_i\text{tanh}\left(\frac{z_i}{\kappa }\right)\end{array}$

37

Step n: The time derivative of z_n is given by

${\dot{z}}_n=\widehat{c}\upsilon \left(t\right)+{\widehat{\mathit{\theta }}}_n^T{\varphi }_n\left({\widehat{\overline{x}}}_n\right)+k_n\left(y-{\widehat{x}}_1\right)-{\dot{\alpha }}_{n-1}$

38

Consider the Lyapunov function candidate V_i as follows

$V_n=\frac{1}{2}{z}_n^2$

39

the derivative of V_i is given by

${\dot{V}}_n=z_n{\dot{z}}_n\le z_n(\widehat{c}\upsilon \left(t\right)+z_{n-1}+{\widehat{\theta }}_n^T{\phi }_n\left({\widehat{\overline{x}}}_n\right)+{\tilde{\theta }}_n^T{\phi }_n\left({\widehat{\overline{x}}}_n\right)+k_n\left(y-{\widehat{x}}_1\right)-{\dot{\alpha }}_{n-1})-z_n{z}_{n-1}+{\kappa }^{\prime }+z_n{\tilde{\omega }}_n\tanh \left(\frac{z_n}{\kappa }\right)+z_n{\widehat{\omega }}_n\tanh \left(\frac{z_n}{\kappa }\right)$

40

Choosing the following Lyapunov function candidate

$V=\frac{1}{2}{e}^{T}Pe+{\displaystyle \sum _{i=1}^n}\frac{1}{2}{z}_i^2+\Lambda$

41

where

$\begin{array}{l}\Lambda =\frac{e^{-\gamma \left(t-d_1\right)}}{1-d_1^{*}}{\displaystyle {\int }_{t-d_{11}\left(t\right)}^t}{e}^{\gamma s}{h}_1^2(y\left(s\right))\text{d}s\\ +\frac{d_2}{1-d_2^{*}}{\displaystyle {\int }_{t-d_{21}\left(t\right)}^t}{\displaystyle {\int }_r^t}{e}^{-\gamma \left(t-d_2-s\right)}{m}_1^2(y\left(s\right))dsdr\\ +\frac{e^{-\gamma \left(t-d_1\right)}}{1-d_1^{*}}{\displaystyle \sum _{j=1}^n}8\epsilon {\Vert P\Vert }^2{\displaystyle {\int }_{t-d_{1j}\left(t\right)}^t}{e}^{\gamma s}{h}_j^2(y\left(s\right))\text{d}s\\ +\frac{8d_2\epsilon {\Vert P\Vert }^2}{1-d_2^{*}}{\displaystyle \sum _{j=1}^n}{\displaystyle {\int }_{t-d_{2j}\left(t\right)}^t}{\displaystyle {\int }_r^t}{e}^{-\gamma \left(t-d_2-s\right)}{m}_j^2(y\left(s\right))dsdr\end{array}$

42

then, we can get that

$\begin{array}{l}\dot{V}\le e^T(A^{T}P+PA+\frac{1}{\psi }P{P}^T+\psi {\displaystyle \sum _{i=1}^n}{L}_i^{2}I\\ +\frac{1}{4\epsilon }\left(L_1^2+1\right)I+\frac{5}{8\epsilon }I)e+{\displaystyle \sum _{i=2}^n}{z}_i{k}_i\left(y-{\widehat{x}}_1\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{i=2}^n}{z}_i\left({\widehat{\mathit{\theta }}}_i^T+{\tilde{\mathit{\theta }}}_i^T\right){\varphi }_i\left({\widehat{\overline{x}}}_i\right)+32\epsilon c^2{\Vert P\Vert }^2{\upsilon }^2+n\kappa \prime +{\displaystyle \sum _{i=2}^n}{z}_i\left({\tilde{\omega }}_i+{\widehat{\omega }}_i\right)\text{tanh}\left(\frac{z_i}{\kappa }\right)+\frac{\epsilon }{2}{\tilde{c}}^2{\Vert P\Vert }^2{\upsilon }^2\\ \hspace{0.17em}\hspace{0.17em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}-8\epsilon \tilde{c}\widehat{c}{\Vert P\Vert }^2{\upsilon }^2+z_1\left({\tilde{\epsilon }}_1+{\widehat{\epsilon }}_1\right)\text{tanh}\left(\frac{z_1}{\kappa }\right)+z_1\left({\alpha }_1+{\mathit{\theta }}_1^{*}{ }^T{\varphi }_1\left({\widehat{x}}_1\right)-{\dot{y}}_r+\frac{1}{2}{z}_1+2\epsilon z_1\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{i=2}^{n-1}}{z}_i\left({\alpha }_i+z_{i-1}-{\dot{\alpha }}_{i-1}\right)+z_n(\widehat{c}\upsilon \left(t\right)+z_{n-1}-{\dot{\alpha }}_{n-1})+h_1^2(y(t-d_{11}\left(t\right)))+8\epsilon {\Vert P\Vert }^2{\displaystyle \sum _{j=1}^n}{h}_j^2(y(t-d_{1j}\left(t\right)))\\ \hspace{0.17em}\hspace{0.17em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}+d_2{\displaystyle \underset{t-d_{21}\left(t\right)}{\overset{t}{\int }}}{m}_1^2(y(s))ds+8\epsilon {\Vert P\Vert }^2{d}_2{\displaystyle \sum _{j=1}^n}{\displaystyle \underset{t-d_{2j}(t)}{\overset{t}{\int }}}{m}_j^2\left(y\right)ds+\frac{e^{\gamma d_1}{h}_1^2\left(y\right)}{1-d_1^{*}}-\frac{e^{\gamma (d_1-d_{11}\left(t\right))}(1-{\dot{d}}_{11}\left(t\right))}{1-d_1^{*}}{h}_1^2(y(t-d_{11}\left(t\right)))\\ \hspace{0.17em}\hspace{0.17em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}-\frac{d_2(1-{\dot{d}}_{21}\left(t\right))}{1-d_2^{*}}{\displaystyle {\int }_{t-d_{21}\left(t\right)}^t}{e}^{-\gamma \left(t-d_2-s\right)}{m}_1^2(y\left(s\right))\text{d}s+\frac{d_2{d}_{21}\left(t\right)}{1-d_2^{*}}{e}^{\gamma d_2}{m}_1^2\left(y\right)+\frac{e^{\gamma d_1}}{\left(1-d_1^{*}\right)}{\displaystyle \sum _{j=1}^n}8\epsilon {\Vert P\Vert }^2{h}_j^2\left(y\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \sum _{j=1}^n}\frac{8\epsilon {\Vert P\Vert }^2{e}^{\gamma (d_1-d_{1j}\left(t\right))}(1-{\dot{d}}_{1j}\left(t\right))}{1-d_1^{*}}{h}_j^2(y(t-d_{1j}\left(t\right)))-{\displaystyle \sum _{j=1}^n}\frac{8d_2\epsilon {\Vert P\Vert }^2(1-{\dot{d}}_{2j}\left(t\right))e^{\gamma d_2}}{1-d_2^{*}}{\displaystyle \underset{t-d_{2j}\left(t\right)}{\overset{t}{\int }}}{e}^{-\gamma \left(t-s\right)}{m}_j^2\left(y\right)\text{d}s\\ \hspace{0.17em}\hspace{0.17em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{j=1}^n}\frac{d_2{d}_{2j}\left(t\right)8\epsilon {\Vert P\Vert }^2{e}^{\gamma d_2}}{1-d_2^{*}}{m}_j^2\left(y\right)+\Delta -\gamma \Lambda \end{array}$

43

where $\Delta =8\epsilon {\Vert P\Vert }^2{\Vert {\delta }^{*}\Vert }^2+8\epsilon {\Vert P\Vert }^2{\overline{d}}^2$ . From assumption 1 and (17) yields that

$\begin{array}{l}\dot{V}\le -e^{T}Qe+{\displaystyle \sum _{i=2}^n}{z}_i{k}_i\left(y-{\widehat{x}}_1\right)+{\displaystyle \sum _{i=2}^n}{z}_i\left({\widehat{\mathit{\theta }}}_i^T+{\tilde{\mathit{\theta }}}_i^T\right){\varphi }_i\left({\widehat{\overline{x}}}_i\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{i=2}^n}{z}_i{\tilde{\omega }}_i\text{tanh}\left(\frac{z_i}{\kappa }\right)+{\displaystyle \sum _{i=2}^n}{z}_i{\widehat{\omega }}_i\text{tanh}\left(\frac{z_i}{\kappa }\right)+n\kappa \prime \\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\frac{\epsilon }{2}{\tilde{c}}^2{\Vert P\Vert }^2{\upsilon }^2+32\epsilon c^2{\Vert P\Vert }^2{\upsilon }^2-8\epsilon \tilde{c}\widehat{c}{\Vert P\Vert }^2{\upsilon }^2\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+z_1{\tilde{\epsilon }}_1\text{tanh}\left(\frac{z_1}{\kappa }\right)+z_1{\widehat{\epsilon }}_1\text{tanh}\left(\frac{z_1}{\kappa }\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+z_1\left({\alpha }_1+{\mathit{\theta }}_1^{*}{ }^T{\varphi }_1\left({\widehat{x}}_1\right)-{\dot{y}}_r+\frac{1}{2}{z}_1^2+2\epsilon z_1\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{i=2}^{n-1}}{z}_i\left({\alpha }_i+z_{i-1}-{\dot{\alpha }}_{i-1}\right)+z_n(\widehat{c}\upsilon \left(t\right)-{\dot{\alpha }}_{n-1}+z_{n-1})\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\frac{e^{\gamma d_1}}{1-d_1^{*}}{h}_1^2\left(y\right)+\frac{d_2^2}{1-d_2^{*}}{e}^{\gamma d_2}{m}_1^2\left(y\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\frac{e^{\gamma d_1}}{1-d_1^{*}}{\displaystyle \sum _{j=1}^n}8\epsilon {\Vert P\Vert }^2{h}_j^2\left(y\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{j=1}^n}\frac{d_2^2}{1-d_2^{*}}8\epsilon {\Vert P\Vert }^2{e}^{\gamma d_2}{m}_j^2\left(y\right)+\Delta -\gamma \Lambda \end{array}$

44

then, (44) can be rewritten as

$\begin{array}{l}\dot{V}\le -e^{T}Qe+{\displaystyle \sum _{i=2}^n}{z}_i{k}_i\left(y-{\widehat{x}}_1\right)+{\displaystyle \sum _{i=2}^n}{z}_i{\widehat{\mathit{\theta }}}_i^T{\varphi }_i\left({\widehat{\overline{x}}}_i\right)\\ \hspace{1em}+{\displaystyle \sum _{i=2}^n}{z}_i{\tilde{\mathit{\theta }}}_i^T{\varphi }_i\left({\widehat{\overline{x}}}_i\right)+{\displaystyle \sum _{i=2}^n}{z}_i{\tilde{\omega }}_i\text{tanh}\left(\frac{z_i}{\kappa }\right)+n\kappa \prime \\ \hspace{1em}+{\displaystyle \sum _{i=2}^n}{z}_i{\widehat{\omega }}_i\text{tanh}\left(\frac{z_i}{\kappa }\right)+\frac{\epsilon }{2}{\tilde{c}}^2{\Vert P\Vert }^2{\upsilon }^2+32\epsilon c^2{\Vert P\Vert }^2{\upsilon }^2\\ \hspace{1em}-8\epsilon \tilde{c}\widehat{c}{\Vert P\Vert }^2{\upsilon }^2+z_1{\tilde{\epsilon }}_1\text{tanh}\left(\frac{z_1}{\kappa }\right)+z_1{\widehat{\epsilon }}_1\text{tanh}\left(\frac{z_1}{\kappa }\right)\\ \hspace{1em}+z_1({\alpha }_1+{\mathit{\theta }}_1^{*}{ }^T{\varphi }_1\left({\widehat{x}}_1\right)+J_1\left(Z_1\right)-{\dot{y}}_r)\\ \hspace{1em}+{\displaystyle \sum _{i=2}^{n-1}}{z}_i\left({\alpha }_i+z_{i-1}-{\dot{\alpha }}_{i-1}\right)\\ \hspace{1em}+z_n(\widehat{c}\upsilon \left(t\right)-{\dot{\alpha }}_{n-1}+z_{n-1})+\Delta -\gamma \Lambda \end{array}$

45

where

$J_1\left(Z_1\right)=\frac{\overline{H}\left(y\right)}{z_1}+\frac{1}{2}{z}_1+2\epsilon z_1$

46

$\begin{array}{l}\overline{H}\left(y\right)=\frac{e^{\gamma d_1}}{\left(1-d_1^{*}\right)}{h}_1^2\left(y\right)+\frac{e^{\gamma d_1}}{\left(1-d_1^{*}\right)}{\displaystyle \sum _{j=1}^n}8\epsilon {\Vert P\Vert }^2{h}_j^2\left(y\right)\\ +\frac{d_2^2{e}^{\gamma d_2}}{1-d_2^{*}}{m}_1^2\left(y\right)+{\displaystyle \sum _{j=1}^n}\frac{d_2^2}{1-d_2^{*}}8\epsilon {\Vert P\Vert }^2{e}^{\gamma d_2}{m}_j^2\left(y\right)\end{array}$

47

where $Z_1={\left[z_1,y\right]}^T\in {\Omega }_{Z_1}\subset R^2$ and ${\Omega }_{Z_1}$ is some known compact set in R². Notice that in (46), the term $\frac{\overline{H}\left(y\right)}{z_1}$ is discontinuous at z₁ = 0. Therefore, it cannot be approximated by the FLS. Similar to the study by Wang and Chen, we introduce hyperbolic tangent function $\text{tanh}\left(\frac{z_1}{\vartheta }\right)$ to deal with the term. Define

${\overline{J}}_1\left({\text{Z}}_1\right)=J_1\left({\text{Z}}_1\right)-\frac{\overline{H}(y)}{z_1}+\frac{16}{z_1}{\text{tanh}}^2\left(\frac{z_1}{\vartheta }\right)\overline{H}\left(y\right)$

where ϑ is a positive design parameter. Note that $\underset{z_1\to 0}{\text{lim}}\frac{16}{z_1}{\text{tanh}}^2\left(\frac{z_1}{\vartheta }\right)H\left(y\right)$ exists, thus the nonlinear function ${\overline{J}}_1\left(Z_1\right)$ can be approximated by an FLS ${\phi }_1^T{\varphi }_1\left(Z_1\right)$ such that

${\overline{J}}_1\left(Z_1\right)={\phi }_1^T{\varphi }_1\left(Z_1\right)+{\iota }_1\left(Z_1\right)$

then, by using

$\begin{array}{l}{z}_1\left({\phi }_1^T{\varphi }_1\left(Z_1\right)+{\iota }_1\left(Z_1\right)\right)\le \frac{{\beta }_1}{2{\eta }^2}{\varphi }_1^T\left(Z_1\right){\varphi }_1\left(Z_1\right)z_1^2\\ +\frac{z_1^2}{2\lambda }+\frac{{\eta }^2}{2}+\frac{\lambda {\overline{\iota }}_1^2}{2}\end{array}$

48

where ${\beta }_1={\phi }_1^T{\phi }_1$ is an unknown parameter and $\overline{\iota }$ is the upper bound of ${\iota }_1\left(Z_1\right)$ . We can have

$\begin{array}{l}\dot{V}\le -e^{T}Qe+{\displaystyle \sum _{i=2}^n}{z}_i{\widehat{\mathit{\theta }}}_i^T{\varphi }_i\left({\widehat{\overline{x}}}_i\right)+{\displaystyle \sum _{i=2}^n}{z}_i{\tilde{\mathit{\theta }}}_i^T{\varphi }_i\left({\widehat{\overline{x}}}_i\right)\\ \hspace{1em}+{\displaystyle \sum _{i=2}^n}{z}_i{\tilde{\omega }}_i\text{tanh}\left(\frac{z_i}{\kappa }\right)+{\displaystyle \sum _{i=2}^n}{z}_i{\widehat{\omega }}_i\text{tanh}\left(\frac{z_i}{\kappa }\right)\\ \hspace{1em}+n\kappa \prime +\frac{\epsilon }{2}{\tilde{c}}^2{\Vert P\Vert }^2{\upsilon }^2+32\epsilon c^2{\Vert P\Vert }^2{\upsilon }^2-8\epsilon \tilde{c}\widehat{c}{\Vert P\Vert }^2{\upsilon }^2\\ \hspace{1em}+z_1{\tilde{\epsilon }}_1\text{tanh}\left(\frac{z_1}{\kappa }\right)+z_1{\widehat{\epsilon }}_1\text{tanh}\left(\frac{z_1}{\kappa }\right)\\ \hspace{1em}+z_1\left({\alpha }_1+{\mathit{\theta }}_1^{*}{ }^T{\varphi }_1\left({\widehat{x}}_1\right)+\frac{{\beta }_1}{2{\eta }^2}{\varphi }_1^T{\varphi }_1{z}_1+\frac{z_1}{2\lambda }-{\dot{y}}_r\right)\\ \hspace{1em}+{\displaystyle \sum _{i=2}^{n-1}}{z}_i\left({\alpha }_i+z_{i-1}-{\dot{\alpha }}_{i-1}+k_i\left(y-{\widehat{x}}_1\right)\right)\\ \hspace{1em}+z_n\left(\widehat{c}\upsilon \left(t\right)-{\dot{\alpha }}_{n-1}+z_{n-1}+k_n\left(y-{\widehat{x}}_1\right)\right)+\frac{{\eta }^2}{2}+\frac{\lambda {\overline{\iota }}_1^2}{2}\\ \hspace{1em}+\left(1-16{\text{tanh}}^2\left(\frac{z_1}{\vartheta }\right)\right)\overline{H}+\Delta -\gamma \Lambda \end{array}$

49

in order to obtain the control laws α_i,d, now, define the boundary layer errors as

$\hspace{1em}{y}_i={\alpha }_i-{\alpha }_{i,d},\hspace{1em}i=1,\dots ,n-1$

50

where

$\begin{array}{l}{\tau }_i{\dot{\alpha }}_i+{\alpha }_i={\alpha }_{i,d},{\alpha }_i\left(0\right)={\alpha }_{i,d}\left(0\right),\hspace{1em}i=1,\dots ,n-1\end{array}$

51

using ${\tau }_i{\dot{\alpha }}_i+{\alpha }_i={\alpha }_{i,d}$ gives that

${\dot{y}}_i=-\frac{y_i}{{\tau }_i}-{\dot{\alpha }}_{i,d},\hspace{1em}i=1,\dots ,n-1$

52

substituting (50) into (49) yields that

$\begin{array}{l}\dot{V}\le -e^{T}Qe+{\displaystyle \sum _{i=2}^n}{z}_i{\widehat{\mathit{\theta }}}_i^T{\varphi }_i\left({\widehat{\overline{x}}}_i\right)+{\displaystyle \sum _{i=2}^n}{z}_i{\tilde{\mathit{\theta }}}_i^T{\varphi }_i\left({\widehat{\overline{x}}}_i\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{i=2}^n}{z}_i{\tilde{\omega }}_i\text{tanh}\left(\frac{z_i}{\kappa }\right)+{\displaystyle \sum _{i=2}^n}{z}_i{\widehat{\omega }}_i\text{tanh}\left(\frac{z_i}{\kappa }\right)\\ \hspace{0.17em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+n\kappa \prime +\frac{\epsilon }{2}{\tilde{c}}^2{\Vert P\Vert }^2{\upsilon }^2+32\epsilon c^2{\Vert P\Vert }^2{\upsilon }^2-8\epsilon \tilde{c}\widehat{c}{\Vert P\Vert }^2{\upsilon }^2\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+z_1({\tilde{\epsilon }}_1+{\widehat{\epsilon }}_1)\text{tanh}\left(\frac{z_1}{\kappa }\right)+z_1(y_1+{\alpha }_{\mathrm{1,}d}+{\mathit{\theta }}_1^{*}{ }^T{\varphi }_1\left({\widehat{x}}_1\right)\\ \hspace{0.17em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\frac{{\beta }_1}{2{\eta }^2}{\varphi }_1^T{\varphi }_1{z}_1+\frac{z_1}{2\lambda }-{\dot{y}}_r)\\ \hspace{0.17em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{i=2}^{n-1}}{z}_i(y_i+{\alpha }_{i\mathrm{,}d}+z_{i-1}-{\dot{\alpha }}_{i-1}+k_i\left(y-{\widehat{x}}_1\right))\\ \hspace{0.17em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+z_n(\widehat{c}\upsilon \left(t\right)-{\dot{\alpha }}_{n-1}+z_{n-1}+k_n\left(y-{\widehat{x}}_1\right))\\ \hspace{0.17em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\frac{{\eta }^2}{2}+\frac{\lambda {\overline{\iota }}_1^2}{2}+\left(1-16{\text{tanh}}^2\left(\frac{z_1}{\vartheta }\right)\right)\overline{H}+\Delta -\gamma \Lambda \end{array}$

53

then, choose the control laws defined as follows

$\begin{array}{c}{\alpha }_{1,d}=-l_1{z}_1-{\widehat{\mathit{\theta }}}_1^T{\varphi }_1\left({\widehat{x}}_1\right)-\frac{{\widehat{\beta }}_1}{2{\eta }^2}{\varphi }_1^T{\varphi }_1{z}_1\\ \hspace{0.17em}\hspace{0.17em}-\frac{z_1}{2\lambda }-{\widehat{\epsilon }}_1\text{tanh}\left(\frac{z_1}{\kappa }\right)+{\dot{y}}_r\end{array}$

54

$\begin{array}{c}{\alpha }_{i,d}=-l_i{z}_i-z_{i-1}+{\dot{\alpha }}_{i-1}-k_i\left(y-{\widehat{x}}_1\right)\\ \hspace{0.17em}\hspace{0.17em}-{\widehat{\omega }}_i\text{tanh}\left(\frac{z_i}{\kappa }\right)-{\widehat{\mathit{\theta }}}_i^T{\varphi }_i\left({\widehat{\overline{x}}}_i\right)\end{array}$

55

$\upsilon \left(t\right)=\frac{1}{\widehat{c}}\left(-l_n{z}_n-z_{n-1}+{\dot{\alpha }}_{n-1}-k_n\left(y-{\widehat{x}}_1\right)-{\widehat{\omega }}_n\text{tanh}\left(\frac{z_n}{\kappa }\right)-{\widehat{\mathit{\theta }}}_n{ }^T{\varphi }_n\left({\widehat{\overline{x}}}_n\right)\right)$

56

where l_i is a positive design parameter. Moreover, combining (53) to (56) gives that

$\begin{array}{l}\dot{V}\le -e^{T}Qe-{\displaystyle \sum _{i=1}^n}{l}_i{z}_i^2+{\displaystyle \sum _{i=1}^{n-1}}{z}_i{y}_i+{\displaystyle \sum _{i=1}^n}{z}_i{\tilde{\mathit{\theta }}}_i^T{\varphi }_i\left({\widehat{\overline{x}}}_i\right)\\ +{\displaystyle \sum _{i=2}^n}{z}_i{\tilde{\omega }}_i\text{tanh}\left(\frac{z_i}{\kappa }\right)+\frac{\epsilon }{2}{\tilde{c}}^2{\Vert P\Vert }^2{\upsilon }^2+32\epsilon c^2{\Vert P\Vert }^2{\upsilon }^2\\ \hspace{1em}-8\epsilon \tilde{c}\widehat{c}{\Vert P\Vert }^2{\upsilon }^2+z_1{\tilde{\epsilon }}_1\text{tanh}\left(\frac{z_1}{\kappa }\right)\\ +\frac{{\tilde{\beta }}_1}{2{\eta }^2}{\varphi }_1^T{\varphi }_1{z}_1^2+\frac{{\eta }^2}{2}+\frac{\lambda {\overline{\iota }}_1^2}{2}+\left(1-16{\text{tanh}}^2\left(\frac{z_1}{\vartheta }\right)\right)\overline{H}\\ +\Delta -\gamma \Lambda +n\kappa \prime \end{array}$

57

Remark 4

By (47) and (48), we can see that all the unknown time-delay terms are lumped into one unknown nonlinear function $\overline{H}\left(y\right)$ , thus only one FLS is used to approximate it. Furthermore, by using the technique ${\beta }_1={\phi }_1^T{\phi }_1$ such that only one unknown parameter β₁ needs to be adjusted, which reduces the online computation burden greatly and makes our design scheme easier to be implemented in practical applications.

Then, consider the following Lyapunov function

$\begin{array}{c}\overline{V}(t)=V(t)+{\displaystyle \sum _{i=1}^n}\frac{1}{2{\gamma }_i}{\tilde{\mathit{\theta }}}_i^T{\tilde{\mathit{\theta }}}_i+\frac{1}{4{\eta }^2}{\tilde{\beta }}_1^2+{\displaystyle \sum _{i=1}^{n-1}}\frac{1}{2}{y}_i^2\\ +\frac{1}{4\varsigma }{\tilde{c}}^2+{\displaystyle \sum _{i=2}^n}\frac{1}{2}{\tilde{\omega }}_i^T{\tilde{\omega }}_i+\frac{1}{2}{\tilde{\epsilon }}_1^T{\tilde{\epsilon }}_1\end{array}$

58

the time derivative of $\overline{V}$ is given by

$\begin{array}{l}\dot{\overline{V}}\le -e^{T}Qe-{\displaystyle \sum _{i=1}^n}{l}_i{z}_i^2+{\displaystyle \sum _{i=1}^{n-1}}{y}_i{\dot{y}}_i+{\displaystyle \sum _{i=1}^{n-1}}{z}_i{y}_i\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{i=1}^n}\frac{1}{{\gamma }_i}{\tilde{\mathit{\theta }}}_i^T\left({\gamma }_i{z}_i{\varphi }_i\left({\widehat{\overline{x}}}_i\right)-{\dot{\widehat{\mathit{\theta }}}}_i\right)+\frac{\epsilon }{2}{\tilde{c}}^2{\Vert P\Vert }^2{\upsilon }^2\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+32\epsilon c^2{\Vert P\Vert }^2{\upsilon }^2+\frac{{\tilde{\beta }}_1}{2{\eta }^2}({\varphi }_1^T(Z_1){\varphi }_1(Z_1)z_1^2-{\dot{\widehat{\beta }}}_1)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-\frac{\tilde{c}}{2\varsigma }\left(\dot{\widehat{c}}+16\varsigma \epsilon \widehat{c}{\Vert P\Vert }^2{\upsilon }^2\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\tilde{\epsilon }}_1^T\left(z_1\text{tanh}\left(\frac{z_1}{\kappa }\right)-{\dot{\widehat{\epsilon }}}_i\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{i=2}^n}{\tilde{\omega }}_i^T\left(z_i\text{tanh}\left(\frac{z_i}{\kappa }\right)-{\dot{\widehat{\omega }}}_i\right)+\frac{{\eta }^2}{2}+\frac{\lambda {\overline{\iota }}_1^2}{2}\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\left(1-16{\text{tanh}}^2\left(\frac{z_1}{\vartheta }\right)\right)\overline{H}+\Delta -\gamma \Lambda +n\kappa \prime \end{array}$

59

choose the adaptive laws as follows

$\begin{array}{l}{\dot{\widehat{\beta }}}_1={\varphi }_1^T(Z_1){\varphi }_1(Z_1)z_1^2-{\sigma }_1{\widehat{\beta }}_1\\ {\dot{\widehat{\epsilon }}}_1=z_1\text{tanh}\left(\frac{z_1}{\kappa }\right)-{\sigma }_3{\widehat{\epsilon }}_1\\ \dot{\widehat{c}}=-16\varsigma \epsilon \widehat{c}{\Vert P\Vert }^2{\upsilon }^2-{\sigma }_2(\widehat{c}-{\widehat{c}}_0)\end{array}$

60

$\begin{array}{l}{\dot{\widehat{\mathit{\theta }}}}_i={\gamma }_i{z}_i{\varphi }_i\left({\widehat{\overline{x}}}_i\right)-{\overline{\sigma }}_i{\widehat{\mathit{\theta }}}_i\\ {\dot{\widehat{\omega }}}_i=z_i\text{tanh}\left(\frac{z_i}{\kappa }\right)-{\overline{\overline{\sigma }}}_i{\widehat{\omega }}_i\end{array}$

61

where σ₁, σ₂, σ₃, ${\overline{\sigma }}_i$ , and ${\overline{\overline{\sigma }}}_i$ are all positive design parameters and ${\widehat{c}}_0=\widehat{c}(0)$ .

Remark 5

From (60), it can be seen that the function $S(t)=16\varsigma \epsilon {\Vert P\Vert }^2{\upsilon }^2+{\sigma }_2$ is nonnegative. This implies that if $\widehat{c}\left(t\right)\ge {\sigma }_2{\widehat{c}}_0/S\left(t\right)$ , then $\dot{\widehat{c}}\le 0$ . Consequently, $\widehat{c}$ decreases until $\widehat{c}\left(t\right)={\sigma }_2{\widehat{c}}_0/S\left(t\right)$ . So, for any given initial condition ${\widehat{c}}_0>0$ , $\widehat{c}>0$ holds for all t > t₀.

Then, using $z_i{y}_i\le \frac{1}{4{\nu }_i}{y}_i^2+{\nu }_i{z}_i^2$ yields that

$\begin{array}{l}\dot{\overline{V}}\le -e^{T}Qe-{\displaystyle \sum _{i=1}^n}{l}_i{z}_i^2+{\displaystyle \sum _{i=1}^{n-1}}\left(\frac{1}{4{\nu }_i}{y}_i^2-\frac{y_i^2}{{\tau }_i}\right)\\ -{\displaystyle \sum _{i=1}^{n-1}}{y}_i{\dot{\alpha }}_{id}+{\displaystyle \sum _{i=1}^n}\frac{{\overline{\sigma }}_i}{{\gamma }_i}{\widehat{\mathit{\theta }}}_i{\tilde{\mathit{\theta }}}_i^T+32\epsilon c^2{\Vert P\Vert }^2{\upsilon }^2\\ \hspace{1em}+\frac{{\sigma }_2}{2\varsigma }\tilde{c}\widehat{c}+\Delta +\frac{\epsilon }{2}{\tilde{c}}^2{\Vert P\Vert }^2{\upsilon }^2+\frac{{\sigma }_1}{2{\eta }^2}{\widehat{\beta }}_1{\tilde{\beta }}_1\\ +{\displaystyle \sum _{i=2}^n}{\overline{\overline{\sigma }}}_i{\tilde{\omega }}_i^T{\widehat{\omega }}_i+\left(1-16{\text{tanh}}^2\left(\frac{z_1}{\vartheta }\right)\right)\overline{H}\\ +{\sigma }_3{\tilde{\epsilon }}_1^T{\widehat{\epsilon }}_1+\frac{{\eta }^2}{2}+\frac{\lambda {\rho }_1^2}{2}-\gamma \Lambda +n\kappa \prime \end{array}$

62

moreover, we have

$\begin{array}{l}{\displaystyle \sum _{i=1}^n}\frac{{\overline{\sigma }}_i}{{\gamma }_i}{\widehat{\mathit{\theta }}}_i{\tilde{\mathit{\theta }}}_i^T\le {\displaystyle \sum _{i=1}^n}\frac{{\overline{\sigma }}_i}{2{\gamma }_i}{\mathit{\theta }}_i^T{\mathit{\theta }}_i-{\displaystyle \sum _{i=1}^n}\frac{{\overline{\sigma }}_i}{2{\gamma }_i}{\tilde{\mathit{\theta }}}_i^T{\tilde{\mathit{\theta }}}_i\\ \frac{{\sigma }_1}{2{\eta }^2}{\widehat{\beta }}_1{\tilde{\beta }}_1=\frac{{\sigma }_1}{2{\eta }^2}\left({\beta }_1-{\tilde{\beta }}_1\right){\tilde{\beta }}_1\\ \le \frac{{\sigma }_1}{4{\eta }^2}{\beta }_1^2-\frac{{\sigma }_1}{4{\eta }^2}{\tilde{\beta }}_1^2\\ {\displaystyle \sum _{i=2}^n}{\overline{\overline{\sigma }}}_i{\tilde{\omega }}_i^T{\widehat{\omega }}_i\le {\displaystyle \sum _{i=2}^n}\frac{{\overline{\overline{\sigma }}}_i}{2}{\omega }_i^T{\omega }_i-{\displaystyle \sum _{i=2}^n}\frac{{\overline{\overline{\sigma }}}_i}{2}{\tilde{\omega }}_i^T{\tilde{\omega }}_i\\ {\sigma }_3{\tilde{\epsilon }}_1^T{\widehat{\epsilon }}_1\le \frac{{\sigma }_3}{2}{\epsilon }_1^T{\epsilon }_1-\frac{{\sigma }_3}{2}{\tilde{\epsilon }}_1^T{\tilde{\epsilon }}_1\\ \frac{{\sigma }_2}{2\varsigma }\tilde{c}\left(\widehat{c}-{\widehat{c}}_0\right)=\frac{{\sigma }_2}{2\varsigma }\left(c-\tilde{c}\right)\tilde{c}-\frac{{\sigma }_2}{2\varsigma }\tilde{c}{\widehat{c}}_0\\ \le \frac{{\sigma }_2}{2\varsigma }{c}^2+\frac{{\sigma }_2}{8\varsigma }{\tilde{c}}^2-\frac{{\sigma }_2}{2\varsigma }{\tilde{c}}^2+\frac{{\sigma }_2}{2\varsigma }{\widehat{c}}_0{ }^2+\frac{{\sigma }_2}{8\varsigma }{\tilde{c}}^2\\ =-\frac{{\sigma }_2}{4\varsigma }{\tilde{c}}^2+\frac{{\sigma }_2}{2\varsigma }{\widehat{c}}_0{ }^2+\frac{{\sigma }_2}{2\varsigma }{c}^2\end{array}$

63

combining (62) and (63) yields that

$\begin{array}{l}\dot{\overline{V}}\le -e^{T}Qe-{\displaystyle \sum _{i=1}^n}{l}_i{z}_i^2-{\displaystyle \sum _{i=1}^{n-1}}{y}_i{\dot{\alpha }}_{id}\\ -{\displaystyle \sum _{i=1}^n}\frac{{\overline{\sigma }}_i}{2{\gamma }_i}{\tilde{\mathit{\theta }}}_i^T{\tilde{\mathit{\theta }}}_i+\frac{\epsilon }{2}{\tilde{c}}^2{\Vert P\Vert }^2{\upsilon }^2\\ -\frac{{\sigma }_2}{4\varsigma }{\tilde{c}}^2-\frac{{\sigma }_1}{4{\eta }^2}{\tilde{\beta }}_1^2-{\displaystyle \sum _{i=2}^n}\frac{{\overline{\overline{\sigma }}}_i}{2}{\tilde{\omega }}_i^T{\tilde{\omega }}_i-\frac{{\sigma }_3}{2}{\tilde{\epsilon }}_1^T{\tilde{\epsilon }}_1\\ \hspace{1em}+{\displaystyle \sum _{i=1}^{n-1}}\left(\frac{1}{4{\nu }_i}{y}_i^2-\frac{y_i^2}{{\tau }_i}\right)+32\epsilon c^2{\Vert P\Vert }^2{\upsilon }^2\\ +{\displaystyle \sum _{i=2}^n}\frac{{\overline{\overline{\sigma }}}_i}{2}{\omega }_i^T{\omega }_i+\frac{{\sigma }_3}{2}{\epsilon }_1^T{\epsilon }_1+\frac{{\sigma }_2}{4\varsigma }{c}^2+\frac{{\sigma }_2}{4\varsigma }{\widehat{c}}_0^2\\ +\frac{{\sigma }_1}{4{\eta }^2}{\beta }_1^2+{\displaystyle \sum _{i=1}^n}\frac{{\overline{\sigma }}_i}{2{\gamma }_i}{\mathit{\theta }}_i^T{\mathit{\theta }}_i+\frac{{\eta }^2}{2}+\frac{\lambda {\rho }_1^2}{2}\\ +\left(1-16{\text{tanh}}^2\left(\frac{z_1}{\vartheta }\right)\right)\overline{H}+\Delta -\gamma \Lambda +n\kappa \prime \end{array}$

64

According to assumption 4 gives that $32\epsilon c^2{\Vert P\Vert }^2{\upsilon }^2\le 32\epsilon c^2{\Vert P\Vert }^2{M}^2$ , thus we can obtain

$\begin{array}{l}\dot{\overline{V}}\le -e^{T}Qe-{\displaystyle \sum _{i=1}^n}{l}_i{z}_i^2-{\displaystyle \sum _{i=1}^{n-1}}{y}_i{\dot{\alpha }}_{id}-{\displaystyle \sum _{i=1}^n}\frac{{\overline{\sigma }}_i}{2{\gamma }_i}{\tilde{\mathit{\theta }}}_i^T{\tilde{\mathit{\theta }}}_i\\ +\frac{\epsilon }{2}{\tilde{c}}^2{\Vert P\Vert }^2{\upsilon }^2-\frac{{\sigma }_2}{4\varsigma }{\tilde{c}}^2-\frac{{\sigma }_1}{4{\eta }^2}{\tilde{\beta }}_1^2-{\displaystyle \sum _{i=2}^n}\frac{{\overline{\overline{\sigma }}}_i}{2}{\tilde{\omega }}_i^T{\tilde{\omega }}_i\\ -\frac{{\sigma }_3}{2}{\tilde{\epsilon }}_1^T{\tilde{\epsilon }}_1+{\displaystyle \sum _{i=1}^{n-1}}\left(\frac{1}{4{\nu }_i}{y}_i^2-\frac{y_i^2}{{\tau }_i}\right)\\ +\left(1-16{\text{tanh}}^2\left(\frac{z_1}{\vartheta }\right)\right)\overline{H}+C-\gamma \Lambda \end{array}$

65

where $C=32\epsilon c^2{\Vert P\Vert }^2{M}^2+\Delta +{\displaystyle \sum _{i=2}^n}\frac{{\overline{\overline{\sigma }}}_i}{2}{\omega }_i^T{\omega }_i+\frac{{\sigma }_3}{2}{\epsilon }_1^T{\epsilon }_1+\frac{{\sigma }_2}{4\varsigma }{c}^2+\frac{{\sigma }_2}{4\varsigma }{\widehat{c}}_0^2+\frac{{\sigma }_1}{4{\eta }^2}{\beta }_1^2+{\displaystyle \sum _{i=1}^n}\frac{{\overline{\sigma }}_i}{2{\gamma }_i}{\mathit{\theta }}_i^T{\mathit{\theta }}_i+\frac{{\eta }^2}{2}+\frac{\lambda {\rho }_1^2}{2}+n\kappa \prime$ .

Assumption 6

²³ For all initial conditions, there exists positive constant p satisfying $\overline{V}(0)\le p$ .

According to $\frac{1}{2}{\lambda }_{\text{min}}\left(P\right)e^{T}e\le \frac{1}{2}{e}^{T}Pe$ and assumption 6 gives that

$\begin{array}{l}\frac{1}{2}{\lambda }_{\min }\left(P\right){\displaystyle \sum _{i=1}^n}{e}_i^2+{\displaystyle \sum _{i=1}^n}\frac{1}{2}{z}_i^2+{\displaystyle \sum _{i=1}^n}\frac{1}{2{\gamma }_i}{\tilde{\theta }}_i^T{\tilde{\theta }}_i+\frac{1}{4{\eta }^2}{\tilde{\beta }}_1^2\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{i=1}^{n-1}}\frac{1}{2}{y}_i^2+\Lambda +{\displaystyle \sum _{i=2}^n}\frac{1}{2}{\tilde{\omega }}_i^T{\tilde{\omega }}_i+\frac{1}{4\varsigma }{\tilde{c}}^2+\frac{1}{2}{\tilde{\epsilon }}_1^T{\tilde{\epsilon }}_1\\ \hspace{1em}\hspace{1em}\le \frac{1}{2}{e}^{T}Pe+{\displaystyle \sum _{j=1}^n}\frac{1}{2}{z}_i^2+{\displaystyle \sum _{i=1}^n}\frac{1}{2{\gamma }_i}{\tilde{\theta }}_i^T{\tilde{\theta }}_i+\frac{1}{4{\eta }^2}{\tilde{\beta }}_1^2\\ +{\displaystyle \sum _{i=1}^{n-1}}\frac{1}{2}{y}_i^2+\Lambda +{\displaystyle \sum _{i=2}^n}\frac{1}{2}{\tilde{\omega }}_i^T{\tilde{\omega }}_i+\frac{1}{4\varsigma }{\tilde{c}}^2+\frac{1}{2}{\tilde{\epsilon }}_1^T{\tilde{\epsilon }}_1\le p\end{array}$

66

then, from (54) and (55) yields that

$\begin{array}{l}{\dot{\alpha }}_{1d}=-l_1{\dot{z}}_1-{\widehat{\mathit{\theta }}}_1^T{\dot{\varphi }}_1\left({\widehat{x}}_1\right)-{\dot{\widehat{\mathit{\theta }}}}_1^T{\varphi }_1\left({\widehat{x}}_1\right)\\ \hspace{1em}\hspace{1em}\hspace{0.17em}-k_1\left(\dot{y}-{\dot{\widehat{x}}}_1\right)-\frac{{\dot{\widehat{\beta }}}_1}{2{\eta }^2}{\varphi }_1^T{\varphi }_1{z}_1\\ \hspace{1em}\hspace{1em}\hspace{0.17em}-\frac{{\dot{\widehat{\beta }}}_1}{{\eta }^2}{\dot{\varphi }}_1^T{\varphi }_1{z}_1-\frac{{\widehat{\beta }}_1}{2{\eta }^2}{\varphi }_1^T{\varphi }_1{\dot{z}}_1\\ \hspace{1em}\hspace{1em}\hspace{0.17em}-\frac{{\dot{z}}_1}{2\lambda }-{\dot{\widehat{\epsilon }}}_1\text{tanh}\left(\frac{z_1}{\kappa }\right)-{\widehat{\epsilon }}_1{\left(\text{tanh}\left(\frac{z_1}{\kappa }\right)\right)}^{\text{'}}+{\ddot{y}}_r\\ \hspace{1em}\hspace{0.17em}={\chi }_1\left(e_1\mathrm{,}{e}_2\mathrm{,}{z}_1\mathrm{,}{z}_2\mathrm{,}{y}_1\mathrm{,}{\widehat{\mathit{\theta }}}_1\mathrm{,}{\widehat{\beta }}_1\mathrm{,}{\widehat{\epsilon }}_1\mathrm{,}{y}_r\mathrm{,}{\dot{y}}_r\mathrm{,}{\ddot{y}}_r\right)\\ {\dot{\alpha }}_{id}=-l_i{\dot{z}}_i-{\dot{z}}_{i-1}-\frac{{\dot{y}}_{i-1}}{{\tau }_{i-1}}-k_i\left(\dot{y}-{\dot{\widehat{x}}}_1\right)\\ \hspace{1em}\hspace{1em}\hspace{0.17em}-{\dot{\widehat{\omega }}}_i\text{tanh}\left(\frac{z_i}{\kappa }\right)-{\widehat{\omega }}_i{\left(\text{tanh}\left(\frac{z_i}{\kappa }\right)\right)}^{\text{'}}\\ \hspace{1em}\hspace{1em}\hspace{0.17em}-{\dot{\widehat{\mathit{\theta }}}}_i^T{\varphi }_i\left({\widehat{\overline{x}}}_i\right)-{\widehat{\mathit{\theta }}}_i^T{\dot{\varphi }}_i\left({\widehat{\overline{x}}}_i\right)\\ \hspace{1em}\hspace{0.17em}={\chi }_i(e_1\mathrm{,}\dots \mathrm{,}{e}_{i+1}\mathrm{,}{z}_1\mathrm{,}\dots \mathrm{,}{z}_{i+1}\mathrm{,}{y}_1\mathrm{,}\dots \mathrm{,}{y}_i\mathrm{,}\\ \hspace{1em}\hspace{1em}\hspace{0.17em}{\widehat{\mathit{\theta }}}_1\mathrm{,}\dots \mathrm{,}{\widehat{\mathit{\theta }}}_i\mathrm{,}{\widehat{\epsilon }}_1\mathrm{,}{\widehat{\omega }}_2\mathrm{,}\dots \mathrm{,}{\widehat{\omega }}_i\mathrm{,}{y}_r\mathrm{,}{\dot{y}}_r\mathrm{,}{\ddot{y}}_r)\end{array}$

67

where χ_i is a continuous function. For any B₀ > 0 and p > 0, the sets $\Pi :=\{(y_r,{\dot{y}}_r,$ ${\ddot{y}}_r):y_r^2+{\dot{y}}_r^2+{\ddot{y}}_r^2\le B_0\}$ and ${\Pi }_1:=\frac{{\lambda }_{\text{min}}\left(P\right)}{2}{\displaystyle \sum _{j=1}^2}{e}_j^2+{\displaystyle \sum _{j=1}^2}\frac{1}{2}{z}_j^2+\frac{1}{2{\gamma }_1}{\tilde{\mathit{\theta }}}_1^T{\tilde{\mathit{\theta }}}_1+\frac{1}{4{\eta }^2}{\tilde{\beta }}_1^2+\frac{1}{2{\iota }_1}{\tilde{\epsilon }}_1^T{\tilde{\epsilon }}_1+\frac{1}{2}{y}_1^2\le p$ are compact in R³ and R⁸, respectively; therefore, Π × Π₁ is also compact in R¹¹, and χ₁ has a maximum M₁ on Π × Π₁. Similarly, the sets $\Pi :=\left\{\left(y_r,{\dot{y}}_r,{\ddot{y}}_r\right):y_r^2+{\dot{y}}_r^2+{\ddot{y}}_r^2\le B_0\right\}$ and ${\Pi }_i:=\{\frac{1}{2}{\lambda }_{\text{min}}\left(P\right){\displaystyle \sum _{j=1}^{i+1}}{e}_j^2+{\displaystyle \sum _{j=1}^{i+1}}\frac{1}{2}{z}_j^2+{\displaystyle \sum _{j=1}^i}\frac{1}{2{\gamma }_j}{\tilde{\mathit{\theta }}}_j^T{\tilde{\mathit{\theta }}}_j+\frac{1}{4{\eta }^2}{\tilde{\beta }}_1^2+{\displaystyle \sum _{j=1}^i}\frac{1}{2}{y}_j^2+{\displaystyle \sum _{j=2}^i}\frac{1}{2{\iota }_j}{\tilde{\omega }}_j^T{\tilde{\omega }}_j+\frac{1}{2{\iota }_1}{\tilde{\epsilon }}_1^T{\tilde{\epsilon }}_1\le p\}$ are compact in R³ and $R^{5i+3}$ , respectively; therefore, Π₁ × Π _i is also compact in $R^{5i+6}$ and χ_i has a maximum M_i on Π₁ × Π _i . Using $\left|y_i{\chi }_i\right|\le \frac{\tau }{2}+\frac{y_i^2{M}_i^2}{2\tau }$ with τ being a positive design parameter gives that

$\begin{array}{l}\dot{\overline{V}}\le -e^{T}Qe-{\displaystyle \sum _{i=1}^n}{l}_i{z}_i^2-\left(\frac{{\sigma }_2}{4\varsigma }-\frac{\epsilon }{2}{\Vert P\Vert }^2{\upsilon }^2\right){\tilde{c}}^2\\ -{\displaystyle \sum _{i=1}^n}\frac{{\overline{\sigma }}_i}{2{\gamma }_i}{\tilde{\mathit{\theta }}}_i^T{\tilde{\mathit{\theta }}}_i-\frac{{\sigma }_1}{4{\eta }^2}{\tilde{\beta }}_1^2-{\displaystyle \sum _{i=2}^n}\frac{{\overline{\overline{\sigma }}}_i}{2}{\tilde{\omega }}_i^T{\tilde{\omega }}_i\\ -\frac{{\sigma }_3}{2}{\tilde{\epsilon }}_1^T{\tilde{\epsilon }}_1+{\displaystyle \sum _{i=1}^{n-1}}\left(\frac{1}{4{\nu }_i}-\frac{1}{{\tau }_i}+\frac{M_i^2}{2\tau }\right)y_i^2\\ +\left(1-16{\text{tanh}}^2\left(\frac{z_1}{\vartheta }\right)\right)\overline{H}+{\displaystyle \sum _{i=1}^{n-1}}\frac{\tau }{2}+C-\gamma \Lambda \\ \le -e^{T}Qe-{\displaystyle \sum _{i=1}^n}{l}_i{z}_i^2-\left(\frac{{\sigma }_2}{4\varsigma }-\frac{\epsilon }{2}{\Vert P\Vert }^2{M}^2\right){\tilde{c}}^2\\ -{\displaystyle \sum _{i=1}^n}\frac{{\overline{\sigma }}_i}{2{\gamma }_i}{\tilde{\mathit{\theta }}}_i^T{\tilde{\mathit{\theta }}}_i-\frac{{\sigma }_1}{4{\eta }^2}{\tilde{\beta }}_1^2-{\displaystyle \sum _{i=2}^n}\frac{{\overline{\overline{\sigma }}}_i}{2}{\tilde{\omega }}_i^T{\tilde{\omega }}_i\\ -\frac{{\sigma }_3}{2}{\tilde{\epsilon }}_1^T{\tilde{\epsilon }}_1+{\displaystyle \sum _{i=1}^{n-1}}\left(\frac{1}{4{\nu }_i}-\frac{1}{{\tau }_i}+\frac{M_i^2}{2\tau }\right)y_i^2+C\\ +\left(1-16{\text{tanh}}^2\left(\frac{z_1}{\vartheta }\right)\right)\overline{H}+{\displaystyle \sum _{i=1}^{n-1}}\frac{\tau }{2}-\gamma \Lambda \end{array}$

68

choosing $\rho =\frac{{\sigma }_2}{4\varsigma }-\frac{\epsilon }{2}{\Vert P\Vert }^2{M}^2>0$ yields that

$\begin{array}{l}\dot{\overline{V}}\le -e^{T}Qe-{\displaystyle \sum _{i=1}^n}{l}_i{z}_i^2-\rho {\tilde{c}}^2-{\displaystyle \sum _{i=1}^n}\frac{{\overline{\sigma }}_i}{2{\gamma }_i}{\tilde{\mathit{\theta }}}_i^T{\tilde{\mathit{\theta }}}_i\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}-\frac{{\sigma }_1}{4{\eta }^2}{\tilde{\beta }}_1^2-{\displaystyle \sum _{i=2}^n}\frac{{\overline{\overline{\sigma }}}_i}{2}{\tilde{\omega }}_i^T{\tilde{\omega }}_i-\frac{{\sigma }_3}{2}{\tilde{\epsilon }}_1^T{\tilde{\epsilon }}_1\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{i=1}^{n-1}}\left(\frac{1}{4{\nu }_i}-\frac{1}{{\tau }_i}+\frac{M_i^2}{2\tau }\right)y_i^2+{\displaystyle \sum _{i=1}^{n-1}}\frac{\tau }{2}\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}+\left(1-16{\text{tanh}}^2\left(\frac{z_1}{\vartheta }\right)\right)\overline{H}(y)+C-\gamma \Lambda \end{array}$

69

Stability analysis

Before proposing the main theorem, we first introduce the following lemma

Lemma 3

Consider the set Ω _ϑ defined by ${\Omega }_{\vartheta }:=\text{{}z|\left|z\right|<0.2554\vartheta \}$ . Then, for any $z\notin {\Omega }_{\vartheta }$ , the inequality $1-16{\text{tanh}}^2\left(\frac{z}{\vartheta }\right)\le 0$ is satisfied. ²⁸

The main results are summarized as follows.

Theorem 1

Consider the closed-loop system consisting of system (1) under assumptions 1 to 6, the control laws (54) to (56), and the parameter adaptive laws (60) to (61). Suppose that the packaged uncertain function ${\overline{J}}_1(Z_1)$ can be approximated by the FLSs in the sense that approximation errors are bounded. Then, there exists l_i, τ_i, σ₁, σ₂, σ₃, ${\overline{\sigma }}_i$ , and ${\overline{\overline{\sigma }}}_i$ such that the solution of the closed-loop system is SUUB, and the steady-state tracking error is smaller than a prescribed error bound.

In the following, similar to the above discussion, the proof process is also divided into two cases.

Case 1. $z_1\in {\Omega }_{{\vartheta }_1}$ . In this case, $\left|z_1\right|\le 0.2554{\vartheta }_1\mathrm{.}$ Since $z_1=x_1-y_r$ , the boundedness of y_r and z₁, we can obtain that x₁ is bounded. Then, according to lemma 3 yields that $0<\left(1-16{\text{tanh}}^2\left(\frac{z_1}{\vartheta }\right)\right)\le 1$ . Moreover, since $\overline{H}(y)$ is a continuous function of the variable x₁, we obtain that $\overline{H}(y)$ is bounded. Let $\left|\overline{H}(y)\right|\le {\Delta }_1$ , thus

$0<\left(1-16{\text{tanh}}^2\left(\frac{z_1}{\vartheta }\right)\right)\overline{H}\le {\Delta }_1$

70

combining (69) with (70) gives that

$\begin{array}{l}\dot{\overline{V}}\le -e^{T}Qe-{\displaystyle \sum _{i=1}^n}{l}_i{z}_i^2-\rho {\tilde{c}}^2-{\displaystyle \sum _{i=1}^n}\frac{{\overline{\sigma }}_i}{2{\gamma }_i}{\tilde{\mathit{\theta }}}_i^T{\tilde{\mathit{\theta }}}_i\\ -\frac{{\sigma }_1}{4{\eta }^2}{\tilde{\beta }}_1^2-{\displaystyle \sum _{i=2}^n}\frac{{\overline{\overline{\sigma }}}_i}{2}{\tilde{\omega }}_i^T{\tilde{\omega }}_i-\frac{{\sigma }_3}{2}{\tilde{\epsilon }}_1^T{\tilde{\epsilon }}_1\\ -{\displaystyle \sum _{i=1}^{n-1}}\left(\frac{1}{{\tau }_i}-\frac{1}{4{\nu }_i}-\frac{M_i^2}{2\tau }\right)y_i^2\\ +C+{\Delta }_1+{\displaystyle \sum _{i=1}^{n-1}}\frac{\tau }{2}-\gamma \Lambda \end{array}$

71

Now, choose $l_1=l_2=\dots =l_n=l^{*}$ and $\frac{1}{{\tau }_i}=\frac{1}{4{\nu }_i}+\frac{M_i^2}{2\tau }+{\tau }_i{ }^{*}$ with l* and ${\tau }_i{ }^{*}$ being positive constants, then we have

$\begin{array}{l}\dot{\overline{V}}\le -e^{T}Qe-{\displaystyle \sum _{i=1}^n}{l}^{*}{z}_i-\rho {\tilde{c}}^2-{\displaystyle \sum _{i=1}^n}\frac{{\overline{\sigma }}_i}{2{\gamma }_i}{\tilde{\mathit{\theta }}}_i^T{\tilde{\mathit{\theta }}}_i\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}-\frac{{\sigma }_1}{4{\eta }^2}{\tilde{\beta }}_1^2-{\displaystyle \sum _{i=2}^n}\frac{{\overline{\overline{\sigma }}}_i}{2}{\tilde{\omega }}_i^T{\tilde{\omega }}_i-\frac{{\sigma }_3}{2}{\tilde{\epsilon }}_1^T{\tilde{\epsilon }}_1\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \sum _{i=1}^{n-1}}{\tau }_i{ }^{*}{y}_i^2+\overline{C}-\gamma \Lambda \le -2{\alpha }_0\overline{V}+\overline{C}\end{array}$

72

where $\overline{C}=C+{\Delta }_1+{\displaystyle \sum _{i=1}^{n-1}}\frac{\tau }{2}$ and ${\alpha }_0=\text{min}\{\frac{{\lambda }_{\text{min}}(Q)}{{\lambda }_{\text{max}}\left(P\right)},l^{*},$ $2\varsigma \rho ,\frac{{\overline{\sigma }}_i}{2},\frac{{\sigma }_1}{2},\frac{{\overline{\overline{\sigma }}}_i}{2},\frac{{\sigma }_3}{2},{\tau }_i{ }^{*}\}\mathrm{.}$ Let ${\alpha }_0>\frac{\overline{C}}{2p},$ , then $\overline{V}\le 0$ on $\overline{V}(0)=p$ . Thus, $\overline{V}\le p$ is an invariant set, that is, if $\overline{V}(0)\le p$ , then $\overline{V}(t)\le p$ for all t ≥ 0. Therefore, (72) holds for all $\overline{V}(t)\le p$ and all t ≥ 0. Solving (72) gives that $0\le \overline{V}\left(t\right)\le \frac{\overline{C}}{2{\alpha }_0}+\left(\overline{V}\left(0\right)-\frac{\overline{C}}{2{\alpha }_0}\right)e^{-2{\alpha }_{0}t},\forall t\ge 0,$ which means that $\overline{V}(t)$ eventually is bounded by $\frac{\overline{C}}{2{\alpha }_0}$ . Furthermore, the boundedness of ${\tilde{\mathit{\theta }}}_i$ , ${\tilde{\beta }}_1$ , z_i(t), e_i(t), $\tilde{c}$ , ${\tilde{\epsilon }}_1,$ ${\tilde{\omega }}_i$ , and y_i are obtained. In addition, according to the boundedness of the signals θ_i, β₁, c, ε₁, ω_i gives that ${\widehat{\mathit{\theta }}}_i$ , ${\widehat{\beta }}_1$ , c, ${\widehat{\epsilon }}_1$ , and ${\widehat{\omega }}_i$ are all bounded, and from (54) to (56) yields that ${\alpha }_{1,d},\dots ,{\alpha }_{n-1,d}$ and υ are all bounded. Finally, from (51), the boundedness of α_i is ensured for $i=1,\dots ,n-1.$

Case 2. $z_1\notin {\Omega }_{{\vartheta }_1}$ . In this case, $\left|z_1\right|>0.2554{\vartheta }_1\mathrm{.}$ According to lemma 3 gives that $\left(1-16{\text{tanh}}^2\left(\frac{z_1}{\vartheta }\right)\right)\overline{H}\le 0.$ Similar to (72), we obtain

$\begin{array}{l}\dot{\overline{V}}\le -e^{T}Qe-{\displaystyle \sum _{i=1}^n}{l}^{*}{z}_i-\rho {\tilde{c}}^2-{\displaystyle \sum _{i=1}^n}\frac{{\overline{\sigma }}_i}{2{\gamma }_i}{\tilde{\mathit{\theta }}}_i^T{\tilde{\mathit{\theta }}}_i\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}-\frac{{\sigma }_1}{4{\eta }^2}{\tilde{\beta }}_1^2-{\displaystyle \sum _{i=2}^n}\frac{{\overline{\overline{\sigma }}}_i}{2}{\tilde{\omega }}_i^T{\tilde{\omega }}_i-\frac{{\sigma }_3}{2}{\tilde{\epsilon }}_1^T{\tilde{\epsilon }}_1\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \sum _{i=1}^{n-1}}{\tau }_i{ }^{*}{y}_i^2+C_1-\gamma \Lambda \le -2{\alpha }_0\overline{V}+C_1\end{array}$

73

where ${\alpha }_0=\text{min}\left\{\frac{{\lambda }_{\text{min}}(Q)}{{\lambda }_{\text{max}}\left(P\right)},l^{*}\mathrm{,2}\varsigma \rho ,\frac{{\overline{\sigma }}_i}{2},\frac{{\sigma }_1}{2},\frac{{\overline{\overline{\sigma }}}_i}{2},\frac{{\sigma }_3}{2},{\tau }_i{ }^{*}\right\}$ and $C_1=C+{\displaystyle \sum _{i=1}^{n-1}}\frac{\tau }{2}$ . Then, similar to case 1, the boundedness of all the signals in the closed-loop can be proven.

This concludes the proof.

Simulation

In this section, to illustrate the validity of the proposed scheme, consider the following nonlinear time-varying delay system

$\{\begin{array}{l}{\dot{x}}_1=x_2-e^{-x_1^2}-2\text{sin}\left(x_1\left(t-d_{11}\left(t\right)\right)\right)\hfill \\ \hspace{1em}-{\displaystyle {\int }_{t-d_{21}\left(t\right)}^t}{x}_1\left(s\right)\text{d}s\hfill \\ {\dot{x}}_2=\phi \left(\upsilon \left(t\right)\right)-\text{cos}(x_1{x}_2)-\frac{1}{2}{x}_2\left(t-d_{12}\left(t\right)\right)\hfill \\ \hspace{1em}-{\displaystyle {\int }_{t-d_{22}\left(t\right)}^t}\text{sin}\left(x_1\left(s\right)\right)\text{d}s\hfill \\ y=x_1\hfill \end{array}$

74

In this simulation, we choose $d_{11}(t)=1+0.4\text{sin}(t)$ , $d_{21}(t)=1-0.4\text{sin}(t)$ , $d_{12}(t)=0.2-0.5\text{s}in(t)$ , and $d_{22}(t)=0.2\text{sin}(t)+0.5$ . The upper bounds of them are d₁ = 1.4 and d₂ = 1.4. The simulation objective is to apply the developed adaptive fuzzy controller such that (1) boundedness of all the signals in the closed-loop system is guaranteed and (2) the system output y follows the reference signal y_r to a small neighborhood of zero with $y_r=0.5\text{sin}(0.5t)+0.5\text{sin}(t)$ .

Choose the first intermediate control function a_1,d as (54) and the input of the backlash-like υ(t) as (56) and the parameter update laws as (60) and (61), respectively. Given the symmetric positive matrix $Q=0.02I_{2\times 2}$ , ε = 75, ψ = 12.5 and L₁ = L₂ = 0.02, by solving LMI (18), the symmetric positive matrix P and observer gain matrix K are obtained as $P=\left[\begin{array}{cc}\text{0}\mathrm{.}\text{1046}& -\text{0}\mathrm{.}\text{0692}\\ -\text{0}\mathrm{.}\text{0692}& \text{0}\mathrm{.}\text{0704}\end{array}\right]$ and $K=\left[8.2356,8.4658\right]$ . The initial states are chosen as $x_1\left(0\right)=x_2\left(0\right)={\widehat{x}}_1\left(0\right)={\widehat{x}}_2\left(0\right)=0$ , y₁(0) = 0, ${\widehat{\beta }}_1=0.001$ , ε₁(0) = 0.001, $\widehat{c}(0)=5$ , ${\widehat{\mathit{\theta }}}_1\left(0\right)=0$ , and ${\widehat{\mathit{\theta }}}_2\left(0\right)=0$ . Then, select the design parameters as η = 0.5, κ = 0.1, λ = 0.5, l₁ = 299, l₂ = 300, ς = 0.0028, ${\gamma }_1={\gamma }_2=1,$ ${\overline{\sigma }}_1=0.1$ , ${\overline{\sigma }}_2=0.1$ , ${\overline{\overline{\sigma }}}_2=0.1$ , σ₁ = 0.2, and σ₃ = 0.2. Furthermore, when $t\in \left[-d_i\mathrm{,0}\right]$ , for i = 1, 2, j = 1, 2, choose x₁(t) = x₂(t) = 0. Membership functions are specified as ${\mu }_{F_{i,j}^l}\left(x_{i,j}\right)=\text{exp}\left(-0.5{\left(x_{i,j}+1.5-0.5(l-1)\right)}^2\right)\mathrm{,1}\le l\le 7,i=1,2,j=1,2$ .

$\phi (\upsilon (t))$ represents an output of the following backlash-like hysteresis

$\frac{d\phi \left(\upsilon (t)\right)}{dt}=\alpha \left|\frac{d\upsilon }{dt}\right|\left(c\upsilon -\phi \right)+B_1\frac{d\upsilon }{dt}$

75

where α = 6, c = 3.1635, and B₁ = 0.345.

The simulation results are shown in Figures 1 to 7, where Figure 1 illustrates the trajectories of the tracking error, the output, and tracking signals; Figures 2 and 3 exhibit the boundedness of the trajectories of ${\widehat{x}}_1$ , ${\widehat{x}}_2$ , and υ(t); and Figure 4 shows that $\Vert {\widehat{\mathit{\theta }}}_1\Vert$ and $\Vert {\widehat{\mathit{\theta }}}_2\Vert$ are all bounded. Finally, in Figures 5 to 7, the boundedness of ${\widehat{\beta }}_1$ , $\widehat{c}$ , and $\phi (\upsilon (t))$ are illustrated, respectively.

Figure 1.

The trajectories of x₁, y_r (solid line) and z₁ (dot line).

Figure 2.

The signals ${\widehat{x}}_1$ (dash–dot line) and ${\widehat{x}}_2$ (solid line).

Figure 3.

The signal υ(t).

Figure 4.

The signals $\Vert {\widehat{\theta }}_1\Vert$ (dashed line) and $\Vert {\widehat{\theta }}_2\Vert$ (solid line).

Figure 5.

The trajectory of ${\widehat{\beta }}_1$ .

Figure 6.

The trajectory of $\widehat{c}$ .

Figure 7.

The trajectory of $\phi (\upsilon (t))$ .

Conclusion

In this article, based on an appropriate observer, an adaptive fuzzy DSC control scheme is presented for a class of nonlinear time-varying delay systems with unknown backlash-like hysteresis. By choosing appropriate Lyapunov–Krasovskii functionals, the adaptive output-feedback fuzzy controller is designed. The proposed adaptive fuzzy controller guarantees that the closed-loop system is stable in the sense of SUUB. Moreover, all the unknown time-delay terms are lumped into one unknown nonlinear function which can be approximated by only one FLS such that the suggested adaptive fuzzy controller contains less adaptive parameters and this makes our design scheme easier to be implemented in practical applications. The simulation results have been given to illustrate the effectiveness of the proposed scheme.