Adaptive fuzzy backstepping control for uncertain nonlinear systems with tracking error constraints

Abstract

This article deals with the design of adaptive fuzzy backstepping control for uncertain nonlinear systems in strict-feedback form with tracking error constraints. In this article, a fuzzy system is used to approximate the unknown nonlinear functions and the differential of virtual control law of each subsystem. In order to satisfy the limitation of tracking error constraints, the barrier Lyapunov function is introduced. Moreover, by applying the minimal learning parameters technique, the number of online parameters update for each subsystem is reduced to only 1. The control scheme not only ensures the tracking error is not to transgress the constraint bounds but also solves the problem of “explosion of complexity” and greatly reduces the initial control input and the number of the adaptive parameters; this provides the conditions for the practical application. The simulation results show the effectiveness of the proposed method.

Introduction

In nonlinear system modeling, due to modeling error, unknown physical phenomena (friction in mechanical system), load variation, and random disturbance, system uncertainty is unavoidable. Because of the existence of high non-linearity and uncertainty of the nonlinear system, it is very difficult for the design of the controller.^1–4 In the past few decades, the adaptive control technology for feedback linearization of nonlinear systems has made remarkable progress.^4–6 The feedback linearization requires the uncertainty to satisfy the linear parameterized condition. However, most of the systems in practice cannot be linearized.

LX Wang and JM Mendel ⁷ proposed adaptive fuzzy control using fuzzy system to approximate the unknown control law or the unknown nonlinear function. In the literature,^8–11 an adaptive neural network controller is designed using neural network to approximate uncertain continuous nonlinear functions. The advantage of sliding mode controller is that it has strong robustness to disturbances and unmodeled dynamics, so it has also been widely applied.^12–15 However, all the above methods require that the system meets an important condition, that is, the unknown nonlinearity and the control input appear in the same equation of the state space model, which is usually regarded as the matching condition.

In the actual system, there is a large class of nonlinear systems that do not meet the matching condition, such as the system of the mechanical hand which is driven by the motor. For mismatched uncertainty nonlinear systems, the backstepping control is very effective and has achieved great success.^16–20 Traditional backstepping control needs to do repeated differentiations of the virtual control law of the former subsystem. If there are nonlinear functions in the virtual controllers, repeated differentiations will lead to the problem of “explosion of complexity” with the increase in the order of the system. This makes high-order systems face great difficulties in controller implementation. If the system has parameters or structural uncertainties and external disturbances, it will further lead to the difficulty in the application of backstepping control.

Swaroop et al. ²¹ and Zhang and Ge ²² proposed a dynamic surface control (DSC) method, which can avoid the problem of repeated differentiations using n first-order low pass filters and has been widely used. But DSC cannot deal with the uncertainty problem. Since fuzzy systems and neural networks can approximate arbitrary nonlinear functions with arbitrary precision, many literatures have combined them with DSC or backstepping control in recent years.^23–30 Liu et al. ³¹ were the first to propose an adaptive backstepping finite-time fault-tolerant control for strictly feedback switched nonlinear systems based on neural networks. With the development of adaptive backstepping and DSC design in nonlinear systems, many fuzzy or neural adaptive control methods with the minimal-learning-parameters algorithm (MLPA) have been reported in Yang and colleagues. ³² , ³³ Previous works ²³ , ³⁴ described a direct adaptive fuzzy backstepping control with MLPA of uncertain nonlinear systems in the presence of input saturation.

Output constraints are important constraints for many industrial systems. Ignoring output constraints can lead to performance degradation, hazards, or system damage. In order to solve the output constraints problem, the barrier Lyapunov function (BLF) has received extensive attention, because this function grows to infinity when its related state is close to a certain limit. By maintaining the boundaries of BLF, any violation of output constraints can be prevented.^33,35–37 Liu et al., ³⁸ Li and Li, ³⁹ Li et al., ⁴⁰ and Gao et al. ⁴¹ extended the output constraints to full-state constraints based on BLFs for various nonlinear systems. Liu et al. ³⁸ described an adaptive control-based BLF for stochastic nonlinear systems with full-state constraints. Li and colleagues ³⁹ , ⁴⁰ described approximation-based neural control for multiple input multiple output (MIMO) or single input single output (SISO) nonlinear systems with state constraints and time-varying delays. Gao et al. ⁴¹ described an adaptive neural control for a class of nonlinear pure-feedback systems with time-varying full-state constraints.

Based on the above results, a novel adaptive fuzzy backstepping control method is studied in this article. The main advantages of this proposed control method are listed as follows:

1. An adaptive fuzzy backstepping control design is addressed for a class of strict-feedback nonlinear system which is more general for practical applications, in the presence of uncertain nonlinear function, unknown control gain, output constraints, and external disturbance.

2. In each subsystem, only one fuzzy system is used to approximate the unknown control gain, the unknown nonlinear function, and the differential of the virtual control of the previous subsystem. At the same time, compensate for all uncertainties and avoid inherent “explosion of complexity” problem. By applying MLPA, the number of online parameter updates of fuzzy logic system for each subsystem is reduced to only 1.

3. To prevent output from violating the constraints, we employ a BLF. The semi-globally uniformly ultimately boundedness (SGUUB) of all the signals of the closed-loop system are proven. The tracking error converges to an adequately small bound. The effectiveness of the proposed control is demonstrated by a simulation example.

This article is organized as follows: problem statement and preliminaries is described in section “Problem statement and preliminaries.” In section “Fuzzy system and its approximation,” fuzzy system and its approximation is presented. Control design and the adaptive law are presented in section “Adaptive fuzzy backstepping control design.” Stability analysis is proposed in section “Stability analysis.” The simulation results and conclusion are given in sections “Simulations” and “Conclusion,” respectively.

Problem statement and preliminaries

Many mechanical systems in practical engineering have the strict-feedback structure (1), such as robotic manipulators, autonomous quadrotor helicopters, hydraulic actuators, and induction motors

$$\{\begin{array}{c}{\stackrel{·}{x}}_1(t)=g_1(x_1(t))x_2(t)+f_1(x_1(t))+{\omega }_1(t)\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_2(t)=g_2(x_1(t),x_2(t))x_3(t)+f_2(x_1(t),x_2(t))+{\omega }_2(t)}\\ \multicolumn{1}{c}{\begin{array}{c}⋮\hfill \end{array}}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_i(t)=g_i(x_1(t),x_2(t),\dots ,x_i(t))x_{i+1}(t)+f_i(x_1(t),x_2(t),\dots ,x_i(t))+{\omega }_i(t)}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_n(t)=g_n(x_1(t),x_2(t),\dots ,x_n(t))u(t)+f_n(x_1(t),x_2(t),\dots ,x_n(t))+{\omega }_n(t)}\\ \multicolumn{1}{c}{y(t)=x_1(t)}\end{array}$$

(1)

where $1\le i\le n-1$ . $x_1,x_2,\dots ,x_n$ are the state variables. $u\in R$ , $y\in R$ are input and output of the system, respectively. ${\omega }_i(t)$ is external disturbance with unknown bound. $g_i(·)$ and $f_i(·)$ are unknown smooth functions.

Assumption 1

The signs of $g_i(·)$ are known, and there exist positive constants $g_{\mathrm{im}}$ and $g_{\mathrm{iM}}$ such that $0<g_{\mathrm{im}}\le |g_i(·)|\le g_{\mathrm{iM}}$ , $i=1,2,\dots ,n$ . Without losing generality, we can assume $0<g_{\mathrm{im}}\le g_i(·)\le g_{\mathrm{iM}}$ .

Assumption 2

There exist constants $g_i^d>0$ , $i=1,2,\dots ,n$ such that $|{\stackrel{·}{g}}_i(·)|\le g_i^d$ .

Assumption 3

The reference signal $y_d(t)$ is a sufficiently smooth function of t, and $y_d(t)$ , ${\stackrel{·}{y}}_d(t)$ are known and bounded.

Assumption 4

External disturbance ${\omega }_i(t)$ is bounded by the positive unknown constant ${\omega }_{\mathrm{iM}}$ , that is, $|{\omega }_i(t)|\le {\omega }_{\mathrm{iM}}$ .

Control objective

The control objective is to design an adaptive control scheme such that the output $y(t)$ tracks the desired trajectory $y_d(t)$ and the tracking error should always remain within given constraints while ensuring the boundedness of all the closed-loop signals.

Lemma 1

Let $\mathrm{Z}:\{\xi \in R:\left|\xi \right|<1\}\subset R$ and $N=R^l\times Z\subset R^{l+1}$ be open sets. Consider the system $\stackrel{·}{\eta }=H(t,\eta )$ , where $\eta :[\omega ,\xi {]}^T\in N$ , and $H:R_{+}\times N\to R^{l+1}$ is piecewise continuous in t and locally lipschitz in $\eta$ , uniformly in t, on $R_{+}\times N$ . Suppose that there exist functions $U:R^l\times R_{+}\to R_{+}$ and $V_1:Z\to R_{+}$ , continuously differentiable and positive definite in their respective domains, such that

$$V_1(\xi )\to \infty \mathrm{as}\left|\xi \right|\to 1$$

(2)

$${\gamma }_1(\Vert \omega \Vert )\le U(\omega ,t)\le {\gamma }_2(\Vert \omega \Vert )$$

(3)

where ${\gamma }_1$ and ${\gamma }_2$ are class $K_{\infty }$ functions. Let $V(\eta ):V_1(\xi )+U(\omega )$ , and $\xi (0)\in Z$ . If the inequality holds

$$\stackrel{·}{V}=\frac{\partial V}{\partial \eta }H\le -CV+M$$

(4)

in the set $\eta \in N$ and C, M are positive constants then $\omega$ remains bounded and $\xi (t)\in Z$ , $\forall t\in [0,\infty )$ .

Lemma 2

For all $\left|\xi \right|<1$ , and any positive integer p, the following inequality holds

$$\log \frac{1}{1-{\xi }^{2p}}<\frac{{\xi }^{2p}}{1-{\xi }^{2p}}$$

(5)

Lemma 3

The following inequality holds for any $\varsigma >0$ and any $\rho \in R$

$$0\le \left|\rho \right|-\rho \tanh \left(\frac{\rho }{\varsigma }\right)\le 0.2758\varsigma =\varsigma \prime$$

(6)

Fuzzy system and its approximation

Fuzzy system with product inference, singleton fuzzifier, and center-average defuzzifier is a universal approximator. If the fuzzy rule has the following form:

IF $x_1$ is $F_1^j$ , and $x_2$ is $F_2^j$ , and … and $x_n$ is $F_n^j$ , THEN y is $B^j(j=1,2,\dots ,N)$ , where $\overline{x}=[x_1,x_2,\dots ,x_n{]}^T\in R^n$ is the system input, y represents the output of the system, $F_i^j(i=1,2,\dots ,n)$ and $B^j$ stand for fuzzy sets, N stands for the number of fuzzy rules, the output of fuzzy system can be expressed as

$$y(\overline{x})=\frac{\sum _{j=1}^N{\hat{\theta }}_j\underset{i=1}{\overset{n}{\Pi }}{\mu }_i^j(x_i)}{\sum _{j=1}^N\underset{i=1}{\overset{n}{\Pi }}{\mu }_i^j(x_i)}$$

(7)

where ${\hat{\theta }}_j=\underset{y\in R}{max}{B}^j(y)$ , ${\mu }_i^j(x_i)$ and $B^j(y)$ denote membership functions with respect to fuzzy sets $F_i^j$ and $B^j$ . Define fuzzy base functions as

$${\xi }_j(\overline{x})=\frac{\underset{i=1}{\overset{n}{\Pi }}{\mu }_i^j(x_i)}{\sum _{j=1}^N\underset{i=1}{\overset{n}{\Pi }}{\mu }_i^j(x_i)},j=1,2,\dots ,N$$

(8)

$\mathbf{\xi }\mathbf{(}\overline{\mathbf{x}}\mathbf{)}\mathbf{=}\mathbf{[}{\mathbf{\xi }}_1\mathbf{(}\overline{\mathbf{x}}\mathbf{)}\mathbf{,}{\mathbf{\xi }}_2\mathbf{(}\overline{\mathbf{x}}\mathbf{)}\mathbf{,}\dots \mathbf{,}{\mathbf{\xi }}_{\mathbf{N}}\mathbf{(}\overline{\mathbf{x}}\mathbf{)}{\mathbf{]}}^{\mathbf{T}}$ is the fuzzy basis function vector, and $\hat{\theta }=[{\hat{\theta }}_1,{\hat{\theta }}_2,\dots ,{\hat{\theta }}_N{]}^T$ is the weight parameter vector, thus the final output of the fuzzy system equation (7) can be rewritten as

$$y(\overline{x})=\widehat{{\theta }^T}\mathbf{\xi }\mathbf{(}\overline{\mathbf{x}}\mathbf{)}$$

(9)

According to the universal approximation theorem of fuzzy system, if $\hat{f}(x)$ is a continuous function defined on the compact set $\Omega$ , and using fuzzy system $y(\overline{x})=\widehat{{\theta }^T}\mathbf{\xi }\mathbf{(}\overline{\mathbf{x}}\mathbf{)}$ to approximate $\hat{f}(\overline{x})$ , there exist optimal parameter vector ${\mathbf{\theta }}^{\mathbf{*}}$ such that $\underset{\overline{x}\in \Omega }{sup}|\hat{f}(\overline{x})-{\mathbf{\theta }}^{\mathbf{*}}\mathbf{\xi }\mathbf{(}\overline{\mathbf{x}}\mathbf{)}\mathbf{|}\mathbf{\le }\mathbf{\epsilon }$ , for any given small constant $\epsilon$ , ⁷ where $0<\epsilon \le {\epsilon }_M$ .

Adaptive fuzzy backstepping control design

Backstepping control scheme

For convenience, symbols ${\overline{\mathit{x}}}_i$ are introduced, where ${\overline{\mathit{x}}}_i=[x_1,x_2,\dots ,x_i{]}^T\in R^i$ , $i=1,2,\dots ,n$ . The uncertain nonlinear system (1) can be reconstructed as follows:

For subsystem 1: ${\stackrel{·}{x}}_1\left(t\right)=g_1(x_1)x_2\left(t\right)+f_1(x_1)+{\omega }_1(t)$ , we can define $e_1=y-y_d$ . A virtual control signal ${\alpha }_1$ is introduced, and then we can have

$${\stackrel{·}{e}}_1=\stackrel{·}{y}-{\stackrel{·}{y}}_d=g_1(x_1)e_2+g_1(x_1){\alpha }_1+f_1(x_1)+{\omega }_1(t)-{\stackrel{·}{y}}_d$$

(10)

where $e_2=x_2-{\alpha }_1$ .

For subsystem 2, a virtual control signal ${\alpha }_2$ is introduced, and then we can have

$${\stackrel{·}{e}}_2={\stackrel{·}{x}}_2-{\stackrel{·}{\alpha }}_1=g_2({\overline{x}}_2)e_3+g_2({\overline{x}}_2){\alpha }_2-{\stackrel{·}{\alpha }}_1+f_2({\overline{x}}_2)+{\omega }_2$$

(11)

where $e_3=x_3-{\alpha }_2$ .

For subsystem k, a virtual control signal ${\alpha }_k$ is introduced, and then we can have

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{e}}_k={\stackrel{·}{x}}_k-{\stackrel{·}{\alpha }}_{k-1}=g_k({\overline{x}}_k)e_{k+1}+g_k({\overline{x}}_k){\alpha }_k}\\ -{\stackrel{·}{\alpha }}_{k-1}+f_k({\overline{x}}_k)+{\omega }_k\hfill \end{array}$$

(12)

where $e_{k+1}=x_{k+1}-{\alpha }_k$ .

Define $e_n=x_n-{\alpha }_{n-1}$ for the final subsystem, and then we can have

$${\stackrel{·}{e}}_n={\stackrel{·}{x}}_n-{\stackrel{·}{\alpha }}_{n-1}=g_n({\overline{x}}_n)u+f_n({\overline{x}}_n)+{\omega }_n-{\stackrel{·}{\alpha }}_{n-1}$$

(13)

As a result, system (1) can be rewritten as the following form

$$\{\begin{array}{c}{\stackrel{·}{e}}_k=g_k({\overline{x}}_k)e_{k+1}+g_k({\overline{x}}_k){\alpha }_k-{\stackrel{·}{\alpha }}_{k-1}+f_k({\overline{x}}_k)+{\omega }_k\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{e}}_n=g_n({\overline{x}}_n)u-{\stackrel{·}{\alpha }}_{n-1}+f_n({\overline{x}}_n)+{\omega }_n}\end{array}$$

(14)

where ${\alpha }_0=y_d$ , $1\le k\le n-1$ .

Equation (14) shows that the control object can be achieved as long as the appropriate virtual control law ${\alpha }_k$ and control law u are designed.

Control law and adaptive law design

Step 1. Define ${\hat{f}}_1=(f_1-{\stackrel{·}{y}}_d)/g_1$ . $\widehat{{\theta }_1^T}{\mathbf{\xi }}_1(x_1)$ is a fuzzy system for approximating nonlinear function ${\hat{f}}_1$ . ${\mathbf{\theta }}_1^{\mathbf{*}}$ is the optimal parameter vector and ${\epsilon }_1$ is the minimum approximation error. ${\mathrm{\Theta }}_1^{*}=\Vert {\theta }_1^{*}{\Vert }^2$ , ${\hat{\mathrm{\Theta }}}_1$ is the estimate of ${\mathrm{\Theta }}_1^{*}$ . Moreover, ${\tilde{\mathrm{\Theta }}}_1={\mathrm{\Theta }}_1^{*}-{\hat{\mathrm{\Theta }}}_1$ .

Let $D_1={\epsilon }_1+g_1^{-1}{\omega }_1$ . Since $\left|{\epsilon }_1\right|\le {\epsilon }_{1M}$ , $\left|g_1^{-1}\right|\le g_{1m}^{-1}$ and $\left|{\omega }_1\right|\le {\omega }_{1M}$ , there exist an unknown constant $D_1^{*}>0$ such that $\left|D_1\right|\le D_1^{*}$ . ${\hat{D}}_1$ is the estimate of $D_1^{*}$ . Moreover ${\tilde{D}}_1=D_1^{*}-{\hat{D}}_1$ .

Define $\left|e_1\right|=|y-y_d|<K$ , where $K>0$ is the tracking error constraint.

Now a BLF can be chosen as

$$V_1=\frac{1}{2g_1}\log \frac{K^2}{K^2-e_1^2}+\frac{1}{2{\gamma }_1}{\tilde{\mathrm{\Theta }}}_1^2+\frac{1}{2{\gamma \prime }_1}{\tilde{D}}_1^2$$

(15)

where ${\gamma }_1>0$ and $\gamma {\prime }_1>0$ are the positive design parameters.

Consider $\mu =(1/(K^2-e_1^2))$ . The time derivative of $V_1$ is equal to

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1=\frac{e_1{\stackrel{·}{e}}_1}{g_1(K^2-e_1^2)}-\frac{{\stackrel{·}{g}}_1}{2g_1^2}\log \frac{K^2}{K^2-e_1^2}-\frac{1}{{\gamma }_1}{\tilde{\mathrm{\Theta }}}_1\stackrel{·}{\widehat{{\mathrm{\Theta }}_1}}}\\ \multicolumn{1}{c}{-\frac{1}{{\gamma \prime }_1}{\tilde{D}}_1\stackrel{·}{\widehat{D_1}}}\\ \multicolumn{1}{c}{=\mu e_1(e_2+{\alpha }_1+{\hat{f}}_1+g_1^{-1}{\omega }_1)-\frac{{\stackrel{·}{g}}_1}{2g_1^2}\log \frac{K^2}{K^2-e_1^2}}\\ \multicolumn{1}{c}{-\frac{1}{{\gamma }_1}{\tilde{\mathrm{\Theta }}}_1\stackrel{·}{\widehat{{\mathrm{\Theta }}_1}}-\frac{1}{{\gamma \prime }_1}{\tilde{D}}_1\stackrel{·}{\widehat{D_1}}}\\ \multicolumn{1}{c}{=\mu e_1(e_2+{\alpha }_1+{\mathbf{\theta }}_1^{\mathbf{T}}{\mathbf{\xi }}_1\mathbf{+}{\mathbf{\epsilon }}_1\mathbf{+}{\mathbf{g}}_1^{\mathbf{-}1}{\mathbf{\omega }}_1\mathbf{)}}\\ \mathbf{-}\frac{{\stackrel{·}{\mathbf{g}}}_1}{2{\mathbf{g}}_1^2}\mathbf{log}\frac{{\mathbf{K}}^2}{{\mathbf{K}}^2\mathbf{-}{\mathbf{e}}_1^2}\mathbf{-}\frac{1}{{\mathbf{\gamma }}_1}{\tilde{\mathbf{\Theta }}}_1\stackrel{·}{\widehat{{\mathbf{\Theta }}_1}}\mathbf{-}\frac{1}{{\mathbf{\gamma }\mathbf{\text{'}}}_1}{\tilde{\mathbf{D}}}_1\stackrel{·}{\widehat{{\mathbf{D}}_1}}\hfill \\ \multicolumn{1}{c}{\mathbf{=}\mathbf{\mu }{\mathbf{e}}_1\mathbf{(}{\mathbf{e}}_2\mathbf{+}{\mathbf{\alpha }}_1\mathbf{+}{\mathbf{\theta }}_1^{\mathbf{T}}{\mathbf{\xi }}_1\mathbf{+}{\mathbf{D}}_1\mathbf{)}}\\ \multicolumn{1}{c}{\mathbf{-}\frac{{\stackrel{·}{\mathbf{g}}}_1}{2{\mathbf{g}}_1^2}\mathbf{log}\frac{{\mathbf{K}}^2}{{\mathbf{K}}^2\mathbf{-}{\mathbf{e}}_1^2}\mathbf{-}\frac{1}{{\mathbf{\gamma }}_1}{\tilde{\mathbf{\Theta }}}_1\stackrel{·}{\widehat{{\mathbf{\Theta }}_1}}\mathbf{-}\frac{1}{{\mathbf{\gamma }\mathbf{\text{'}}}_1}{\tilde{\mathbf{D}}}_1\stackrel{·}{\widehat{{\mathbf{D}}_1}}}\end{array}$$

(16)

By applying Young’s inequality, we have

$$\mu e_1{\mathbf{\theta }}_1^{\mathbf{T}}{\mathbf{\xi }}_1\mathbf{\le }\frac{1}{4\mathbf{\tau }}{\mathbf{\mu }}^2{\mathbf{e}}_1^2{\mathbf{\Theta }}_1^{\mathbf{*}}{\mathbf{\xi }}_1^{\mathbf{T}}{\mathbf{\xi }}_1\mathbf{+}\mathbf{\tau }$$

(17)

where $\tau >0$ is any given positive constant.

By applying Lemma 2, we have

$$\log \frac{K^2}{K^2-e_1^2}<\frac{e_1^2}{K^2-e_1^2},|e_1|<K$$

(18)

Substituting equations (17) and (18) into equation (16) results in

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1<\mu e_1(e_2+{\alpha }_1+\frac{1}{4\tau }\mu e_1{\mathrm{\Theta }}_1^{*}{\mathbf{\xi }}_1^{\mathbf{T}}{\mathbf{\xi }}_1\mathbf{+}{\mathbf{D}}_1)}\\ +\tau +\frac{g_1^d}{2g_{1m}^2}\mu e_1^2-\frac{1}{{\gamma }_1}{\tilde{\mathrm{\Theta }}}_1\stackrel{·}{\widehat{{\mathrm{\Theta }}_1}}-\frac{1}{{\gamma \prime }_1}{\tilde{D}}_1\stackrel{·}{\widehat{D_1}}\hfill \\ \le \mu e_1[e_2+{\alpha }_1+\frac{1}{4\tau }\mu e_1{\hat{\mathrm{\Theta }}}_1{\mathbf{\xi }}_1^{\mathbf{T}}{\mathbf{\xi }}_1\mathbf{+}{\hat{\mathbf{D}}}_1\mathbf{tanh}\mathbf{\left(}\frac{\mathbf{\mu }{\mathbf{e}}_1}{\mathbf{\delta }}\mathbf{\right)}]\hfill \\ +|\mu e_1|D_1^{*}-D_1^{*}\mu e_1\tanh \left(\frac{\mu e_1}{\delta }\right)\hfill \\ +\frac{g_1^d}{2g_{1m}^2}\mu e_1^2+\frac{1}{{\gamma }_1}{\tilde{\mathrm{\Theta }}}_1(\frac{{\gamma }_1}{4\tau }{\mu }^2{e}_1^2{\mathbf{\xi }}_1^{\mathbf{T}}{\mathbf{\xi }}_1\mathbf{-}\stackrel{·}{\widehat{{\mathbf{\Theta }}_1}})\hfill \\ +\frac{1}{{\gamma \prime }_1}{\tilde{D}}_1({\gamma \prime }_1\mu e_1\tanh (\frac{\mu e_1}{\delta })-\stackrel{·}{\widehat{D_1}})+\tau \hfill \end{array}$$

(19)

Choose the first virtual control law ${\alpha }_1$ , the parameter adaptive laws ${\hat{\mathrm{\Theta }}}_1$ and $\stackrel{·}{\widehat{D_1}}$ as follows

$${\alpha }_1=-{\lambda }_1{e}_1-\frac{1}{4\tau }\mu e_1{\hat{\mathrm{\Theta }}}_1{\mathbf{\xi }}_1^{\mathbf{T}}{\mathbf{\xi }}_1\mathbf{-}{\hat{\mathbf{D}}}_1\mathbf{tanh}\mathbf{\left(}\frac{\mathbf{\mu }{\mathbf{e}}_1}{\mathbf{\delta }}\mathbf{\right)}$$

(20)

$$\stackrel{·}{\widehat{{\mathrm{\Theta }}_1}}=\frac{{\gamma }_1}{4\tau }{\mu }^2{e}_1^2{\mathbf{\xi }}_1^{\mathbf{T}}{\mathbf{\xi }}_1\mathbf{-}{\mathbf{\sigma }}_1{\hat{\mathbf{\Theta }}}_1$$

(21)

$$\stackrel{·}{\widehat{D_1}}=\gamma {\prime }_1\mu e_1\tanh \left(\frac{\mu e_1}{\delta }\right)-\sigma {\prime }_1{\hat{D}}_1$$

(22)

where ${\lambda }_1>0$ , ${\sigma }_1>0$ , and $\sigma {\prime }_1>0$ are positive design parameters.

Based on Lemma 3, we have

$$|\mu e_1|D_1^{*}-D_1^{*}\mu e_1\tanh \left(\frac{\mu e_1}{\delta }\right)\le 0.2758\delta D_1^{*}=\delta \prime D_1^{*}$$

(23)

Substituting equations (20)–(23) into equation (19) results in

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1<-({\lambda }_1-\frac{g_1^d}{2g_{1m}^2})\mu e_1^2+\mu e_1{e}_2+D_1^{*}\delta \prime }\\ +\frac{{\sigma }_1}{{\gamma }_1}{\tilde{\mathrm{\Theta }}}_1{\hat{\mathrm{\Theta }}}_1+\frac{{\sigma \prime }_1}{{\gamma \prime }_1}{\tilde{D}}_1{\hat{D}}_1+\tau \hfill \end{array}$$

(24)

Completion of Young’ inequality in equations (25) and (26)

$$\frac{{\sigma }_1}{{\gamma }_1}{\tilde{\mathrm{\Theta }}}_1{\hat{\mathrm{\Theta }}}_1\le -\frac{{\sigma }_1}{2{\gamma }_1}{\tilde{\mathrm{\Theta }}}_1^2+\frac{{\sigma }_1}{2{\gamma }_1}{\mathrm{\Theta }}_1^{*2}$$

(25)

$$\frac{{\sigma \prime }_1}{{\gamma \prime }_1}{\tilde{D}}_1{\hat{D}}_1\le -\frac{{\sigma \prime }_1}{2{\gamma \prime }_1}{\tilde{D}}_1^2+\frac{{\sigma \prime }_1}{2{\gamma \prime }_1}{D}_1^{*2}$$

(26)

Substituting equations (25) and (26) into equation (24) leads to

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1<-({\lambda }_1-\frac{g_1^d}{2g_{1m}^2})\mu e_1^2+\mu e_1{e}_2-\frac{{\sigma }_1}{2{\gamma }_1}{\tilde{\mathrm{\Theta }}}_1^2}\\ -\frac{{\sigma \prime }_1}{2{\gamma \prime }_1}{\tilde{D}}_1^2+\frac{{\sigma }_1}{2{\gamma }_1}{\mathrm{\Theta }}_1^{*2}+\frac{{\sigma \prime }_1}{2{\gamma \prime }_1}{D}_1^{*2}+D_1^{*}\delta \prime +\tau \hfill \\ =-({\lambda }_1-\frac{g_1^d}{2g_{1m}^2})\mu e_1^2+\mu e_1{e}_2-\frac{{\sigma }_1}{2{\gamma }_1}{\tilde{\mathrm{\Theta }}}_1^2-\frac{{\sigma \prime }_1}{2{\gamma \prime }_1}{\tilde{D}}_1^2+M_1\hfill \end{array}$$

(27)

where $M_1=({\sigma }_1/2{\gamma }_1){\mathrm{\Theta }}_1^{*2}+(\sigma {\prime }_1/2\gamma {\prime }_1)D_1^{*2}+D_1^{*}\delta \prime +\tau$ .

Step 2. Define ${\hat{f}}_2=(f_2-{\stackrel{·}{\alpha }}_1)/g_2$ . $\widehat{{\theta }_2^T}{\mathbf{\xi }}_2\left({\overline{x}}_2\right)$ is a fuzzy system for approximating nonlinear function ${\hat{f}}_2$ . ${\mathbf{\theta }}_2^{\mathbf{*}}$ is the optimal parameter vector and ${\epsilon }_2$ is the minimum approximation error. ${\mathrm{\Theta }}_2^{*}=\Vert {\mathbf{\theta }}_2^{\mathbf{*}}{\Vert }^2$ , ${\hat{\mathrm{\Theta }}}_2$ is the estimates of ${\mathrm{\Theta }}_2^{*}$ , ${\tilde{\mathrm{\Theta }}}_2={\mathrm{\Theta }}_2^{*}-{\hat{\mathrm{\Theta }}}_2$ .

The Lyapunov function can be chosen as

$$V_2=V_1+\frac{1}{2g_2}{e}_2^2+\frac{1}{2{\gamma }_2}{\tilde{\mathrm{\Theta }}}_2^2$$

(28)

where ${\gamma }_2>0$ is the positive design parameter.

The time derivative of $V_2$ is equal to

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2={\stackrel{·}{V}}_1+\frac{1}{g_2}{e}_2{\stackrel{·}{e}}_2-\frac{{\stackrel{·}{g}}_2}{2g_2^2}{e}_2^2-\frac{1}{{\gamma }_2}{\tilde{\mathrm{\Theta }}}_2\stackrel{·}{\widehat{{\mathrm{\Theta }}_2}}}\\ ={\stackrel{·}{V}}_1+e_2(e_3+{\alpha }_2+g_2^{-1}(f_2-{\stackrel{·}{a}}_1)+g_2^{-1}{\omega }_2)\hfill \\ -\frac{{\stackrel{·}{g}}_2}{2g_2^2}{e}_2^2-\frac{1}{{\gamma }_2}{\tilde{\mathrm{\Theta }}}_2\stackrel{·}{\widehat{{\mathrm{\Theta }}_2}}\hfill \\ ={\stackrel{·}{V}}_1+e_2(e_3+{\alpha }_2+{\hat{f}}_2+g_2^{-1}{\omega }_2)\hfill \\ -\frac{{\stackrel{·}{g}}_2}{2g_2^2}{e}_2^2-\frac{1}{{\gamma }_2}{\tilde{\mathrm{\Theta }}}_2\stackrel{·}{\widehat{{\mathrm{\Theta }}_2}}\hfill \\ ={\stackrel{·}{V}}_1+e_2(e_3+{\alpha }_2+{\mathbf{\theta }}_2^{\mathbf{T}}{\mathbf{\xi }}_2\mathbf{+}{\mathbf{\epsilon }}_2\mathbf{+}{\mathbf{g}}_2^{\mathbf{-}1}{\mathbf{\omega }}_2\mathbf{)}\hfill \\ \mathbf{-}\frac{{\stackrel{·}{\mathbf{g}}}_2}{2{\mathbf{g}}_2^2}{\mathbf{e}}_2^2\mathbf{-}\frac{1}{{\mathbf{\gamma }}_2}{\tilde{\mathbf{\Theta }}}_2\stackrel{·}{\widehat{{\mathbf{\Theta }}_2}}\hfill \end{array}$$

(29)

By applying Young’s inequality, we have

$$e_2{\mathbf{\theta }}_2^{\mathbf{T}}{\mathbf{\xi }}_2\mathbf{\le }\frac{1}{4\mathbf{\tau }}{\mathbf{e}}_2^2{\mathbf{\Theta }}_2^{\mathbf{*}}{\mathbf{\xi }}_2^{\mathbf{T}}{\mathbf{\xi }}_2\mathbf{+}\mathbf{\tau }$$

(30)

where $\tau >0$ is any given positive constant. Then, equation (29) becomes

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2\le {\stackrel{·}{V}}_1+e_2(e_3+{\alpha }_2+\frac{1}{4\tau }{e}_2{\mathrm{\Theta }}_2^{*}{\mathbf{\xi }}_2^{\mathbf{T}}{\mathbf{\xi }}_2\mathbf{+}{\mathbf{D}}_2)}\\ +\frac{g_2^d}{2g_{2m}^2}{e}_2^2-\frac{1}{{\gamma }_2}{\tilde{\mathrm{\Theta }}}_2\stackrel{·}{\widehat{{\mathrm{\Theta }}_2}}+\tau \hfill \\ ={\stackrel{·}{V}}_1+e_2(e_3+{\alpha }_2+\frac{1}{4\tau }{e}_2{\hat{\mathrm{\Theta }}}_2{\mathbf{\xi }}_2^{\mathbf{T}}{\mathbf{\xi }}_2)\hfill \\ +\frac{g_2^d}{2g_{2m}^2}{e}_2^2+e_2{D}_2+\frac{1}{{\gamma }_2}{\tilde{\mathrm{\Theta }}}_2(\frac{{\gamma }_2}{4\tau }{e}_2^2{\mathbf{\xi }}_2^{\mathbf{T}}{\mathbf{\xi }}_2\mathbf{-}\stackrel{·}{\widehat{{\mathbf{\Theta }}_2}})+\tau \hfill \end{array}$$

(31)

where $D_2={\epsilon }_2+g_2^{-1}{\omega }_2$ . Since $\left|{\epsilon }_2\right|\le {\epsilon }_{2M}$ , $\left|g_2^{-1}\right|\le g_{2m}^{-1}$ and $\left|{\omega }_2\right|\le {\omega }_{2M}$ , there exist an unknown constant $D_2^{*}>0$ such that $\left|D_2\right|\le D_2^{*}$ . ${\hat{D}}_2$ is the estimate of $D_2^{*}$ . Moreover, ${\tilde{D}}_2=D_2^{*}-{\hat{D}}_2$ .

Choose the virtual control law ${\alpha }_2$ and parameter adaptive law ${\hat{\mathrm{\Theta }}}_2$ as follows

$${\alpha }_2=-{\lambda }_2{e}_2-\mu e_1-\frac{1}{4\tau }{e}_2{\hat{\mathrm{\Theta }}}_2{\mathbf{\xi }}_2^{\mathbf{T}}{\mathbf{\xi }}_2$$

(32)

$$\stackrel{·}{\widehat{{\mathrm{\Theta }}_2}}=\frac{{\gamma }_2}{4\tau }{e}_2^2{\xi }_2^T{\xi }_2-{\sigma }_2{\hat{\mathrm{\Theta }}}_2$$

(33)

where ${\lambda }_2>0$ and ${\sigma }_2>0$ are positive design parameters.

By applying Young’s inequality, we have

$$-\frac{1}{2}{e}_2^2+|e_2{D}_2^{*}|\le \frac{1}{2}{D}_2^{*2}$$

(34)

Substituting equations (32)–(34) into equation (31) results in

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2\le {\stackrel{·}{V}}_1-({\lambda }_2-\frac{g_2^d}{2g_{2m}^2})e_2^2-\mu e_1{e}_2+e_2{e}_3}\\ +|e_2{D}_2^{*}|+\frac{{\sigma }_2}{{\gamma }_2}{\tilde{\mathrm{\Theta }}}_2{\hat{\mathrm{\Theta }}}_2+\tau \hfill \\ \le {\stackrel{·}{V}}_1-({\lambda }_2-\frac{1}{2}-\frac{g_2^d}{2g_{2m}^2})e_2^2-\mu e_1{e}_2\hfill \\ +e_2{e}_3+\frac{1}{2}{D}_2^{*2}-\frac{{\sigma }_2}{2{\gamma }_2}{\tilde{\mathrm{\Theta }}}_2^2+\frac{{\sigma }_2}{2{\gamma }_2}{\mathrm{\Theta }}_2^{*2}+\tau \hfill \\ \le {\stackrel{·}{V}}_1-({\lambda }_2-\frac{1}{2}-\frac{g_2^d}{2g_{2m}^2})e_2^2-\mu e_1{e}_2\hfill \\ +e_2{e}_3-\frac{{\sigma }_2}{2{\gamma }_2}{\tilde{\mathrm{\Theta }}}_2^2+M_2\hfill \\ \le -({\lambda }_1-\frac{g_1^d}{2g_{1m}^2})\mu e_1^2-({\lambda }_2-\frac{1}{2}-\frac{g_2^d}{2g_{2m}^2})e_2^2\hfill \\ +e_2{e}_3-\sum _{i=1}^2\frac{{\sigma }_i}{2{\gamma }_i}{\tilde{\mathrm{\Theta }}}_i^2+\sum _{i=1}^2{M}_i-\frac{{\sigma \prime }_1}{2{\gamma \prime }_1}{\tilde{D}}_1^2\hfill \end{array}$$

(35)

where $M_2=(1/2)D_2^{*2}+({\sigma }_2/2{\gamma }_2){\mathrm{\Theta }}_2^{*2}+\tau$ .

Step $k(k=3,\dots ,n-1)$ . Define ${\hat{f}}_k=(f_k-{\stackrel{·}{\alpha }}_{k-1})/g_k$ . $\widehat{{\theta }_k^T}{\mathbf{\xi }}_{\mathbf{k}}$ is a fuzzy system for approximating nonlinear function ${\hat{f}}_k$ . ${\mathbf{\theta }}_{\mathbf{k}}^{\mathbf{*}}$ is the optimal parameter vector and ${\epsilon }_k$ is the minimum approximation error. ${\mathrm{\Theta }}_k^{*}=\Vert {\mathbf{\theta }}_{\mathbf{k}}^{\mathbf{*}}{\Vert }^2$ , ${\hat{\mathrm{\Theta }}}_k$ is the estimates of ${\mathrm{\Theta }}_k^{*}$ , ${\tilde{\mathrm{\Theta }}}_k={\mathrm{\Theta }}_k^{*}-{\hat{\mathrm{\Theta }}}_k$ .

Let $D_k={\epsilon }_k+g_k^{-1}{\omega }_k$ . Since $\left|{\epsilon }_k\right|\le {\epsilon }_{\mathrm{kM}}$ , $\left|g_k^{-1}\right|\le g_{\mathrm{km}}^{-1}$ , and $\left|{\omega }_k\right|\le {\omega }_{\mathrm{kM}}$ , there exist an unknown constant $D_k^{*}>0$ such that $\left|D_k\right|\le D_k^{*}$ . ${\hat{D}}_k$ is the estimate of $D_k^{*}$ . Moreover, ${\tilde{D}}_k=D_k^{*}-{\hat{D}}_k$ .

Choose the following Lyapunov function

$$V_k=V_{k-1}+\frac{1}{2g_k}{e}_k^2+\frac{1}{2{\gamma }_k}{\tilde{\mathrm{\Theta }}}_k^2$$

(36)

The virtual control law ${\alpha }_k$ and parameter adaptive law ${\hat{\mathrm{\Theta }}}_k$ are chosen as

$${\alpha }_k=-{\lambda }_k{e}_k-e_{k-1}-\frac{1}{4\tau }{e}_k{\hat{\mathrm{\Theta }}}_k{\mathbf{\xi }}_{\mathbf{k}}^{\mathbf{T}}{\mathbf{\xi }}_{\mathbf{k}}$$

(37)

$$\stackrel{·}{\widehat{{\mathrm{\Theta }}_k}}=\frac{{\gamma }_k}{4\tau }{e}_k^2{\mathbf{\xi }}_{\mathbf{k}}^{\mathbf{T}}{\mathbf{\xi }}_{\mathbf{k}}\mathbf{-}{\mathbf{\sigma }}_{\mathbf{k}}{\hat{\mathbf{\Theta }}}_{\mathbf{k}}$$

(38)

where ${\lambda }_k>0$ , ${\gamma }_k>0$ , and ${\sigma }_k>0$ are positive design parameters.

Substituting equations (37) and (38) into equation (36) results in

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_k\le -({\lambda }_1-\frac{g_1^d}{2g_{1m}^2})\mu e_1^2-\sum _{i=2}^k({\lambda }_k-\frac{1}{2}-\frac{g_k^d}{2g_{\mathrm{km}}^2})e_i^2}\\ \multicolumn{1}{c}{+e_k{e}_{k+1}-\sum _{i=1}^k\frac{{\sigma }_i}{2{\gamma }_i}{\tilde{\mathrm{\Theta }}}_i^2+\sum _{i=1}^k{M}_i-\frac{{\sigma \prime }_1}{2{\gamma \prime }_1}{\tilde{D}}_1^2}\end{array}$$

(39)

where $M_k=(1/2)D_k^{*2}+({\sigma }_k/2{\gamma }_k){\mathrm{\Theta }}_k^{*2}+\tau$ .

Step n, Define ${\hat{f}}_n=(f_n-{\stackrel{·}{\alpha }}_{n-1})/g_n$ . $\widehat{{\theta }_n^T}{\mathbf{\xi }}_{\mathbf{n}}({\overline{x}}_n)$ is a fuzzy system for approximating nonlinear function ${\hat{f}}_n$ . ${\mathbf{\theta }}_{\mathbf{n}}^{\mathbf{*}}$ is the optimal parameter vector and ${\epsilon }_n$ is the minimum approximation error. ${\mathrm{\Theta }}_n^{*}=\Vert {\mathbf{\theta }}_{\mathbf{n}}^{\mathbf{*}}{\Vert }^2$ , ${\hat{\mathrm{\Theta }}}_n$ is the estimates of ${\mathrm{\Theta }}_n^{*}$ . ${\tilde{\mathrm{\Theta }}}_n={\mathrm{\Theta }}_n^{*}-{\hat{\mathrm{\Theta }}}_n$ .

Let $D_n={\epsilon }_n+g_n^{-1}{\omega }_n$ . Since $\left|{\epsilon }_n\right|\le {\epsilon }_{\mathrm{nM}}$ , $\left|g_n^{-1}\right|\le g_{\mathrm{nm}}^{-1}$ , and $\left|{\omega }_n\right|\le {\omega }_{\mathrm{nM}}$ , there exist an unknown constant $D_n^{*}>0$ such that $\left|D_n\right|\le D_n^{*}$ . ${\hat{D}}_n$ is the estimate of $D_n^{*}$ . Moreover, ${\tilde{D}}_n=D_n^{*}-{\hat{D}}_n$ .

The Lyapunov function is chosen as

$$V_n=V_{n-1}+\frac{1}{2g_n}{e}_n^2+\frac{1}{2{\gamma }_n}{\tilde{\mathrm{\Theta }}}_n^2$$

(40)

where ${\gamma }_n>0$ is positive design parameter.

The time derivative of $V_n$ is equal to

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_n={\stackrel{·}{V}}_{n-1}+\frac{1}{g_n}{e}_n{\stackrel{·}{e}}_n-\frac{{\stackrel{·}{g}}_n}{2g_n^2}{e}_n^2-\frac{1}{{\gamma }_n}{\tilde{\mathrm{\Theta }}}_n\stackrel{·}{\widehat{{\mathrm{\Theta }}_n}}}\\ \multicolumn{1}{c}{={\stackrel{·}{V}}_{n-1}+e_n(u+{\hat{f}}_n+g_n^{-1}{\omega }_n)-\frac{{\stackrel{·}{g}}_n}{2g_n^2}{e}_n^2-\frac{1}{{\gamma }_n}{\tilde{\mathrm{\Theta }}}_n\stackrel{·}{\widehat{{\mathrm{\Theta }}_n}}}\\ \multicolumn{1}{c}{={\stackrel{·}{V}}_{n-1}+e_n(u+{\theta }_n^T{\xi }_n+{\epsilon }_n+g_n^{-1}{\omega }_n)}\\ \multicolumn{1}{c}{-\frac{{\stackrel{·}{g}}_n}{2g_n^2}{e}_n^2-\frac{1}{{\gamma }_n}{\tilde{\mathrm{\Theta }}}_n\stackrel{·}{\widehat{{\mathrm{\Theta }}_n}}}\\ \multicolumn{1}{c}{\le {\stackrel{·}{V}}_{n-1}+e_n(u+\frac{1}{4\tau }{e}_n{\mathrm{\Theta }}_n^{*}{\xi }_n^T{\xi }_n+D_n)}\\ \multicolumn{1}{c}{-\frac{{\stackrel{·}{g}}_n}{2g_n^2}{e}_n^2-\frac{1}{{\gamma }_n}{\tilde{\mathrm{\Theta }}}_n\stackrel{·}{\widehat{{\mathrm{\Theta }}_n}}+\tau }\\ \multicolumn{1}{c}{\le {\stackrel{·}{V}}_{n-1}+e_n(u+\frac{1}{4\tau }{e}_n{\hat{\mathrm{\Theta }}}_n{\xi }_n^T{\xi }_n)+\frac{g_n^d}{2g_{\mathrm{nm}}^2}{e}_n^2+e_n{D}_n}\\ \multicolumn{1}{c}{+\frac{1}{{\gamma }_n}{\tilde{\mathrm{\Theta }}}_n(\frac{{\gamma }_n}{4\tau }{e}_n^2{\xi }_n^T{\xi }_n-\stackrel{·}{\widehat{{\mathrm{\Theta }}_n}})+\tau }\end{array}$$

(41)

Choose the first virtual control law u and parameter adaptive law ${\hat{\mathrm{\Theta }}}_n$ as follows

$$u=-{\lambda }_n{e}_n-e_{n-1}-\frac{1}{4\tau }{e}_n{\hat{\mathrm{\Theta }}}_n{\xi }_n^T{\xi }_n$$

(42)

$$\stackrel{·}{\widehat{{\mathrm{\Theta }}_n}}=\frac{{\gamma }_n}{4\tau }{e}_n^2{\xi }_n^T{\xi }_n-{\sigma }_n{\hat{\mathrm{\Theta }}}_n$$

(43)

where ${\lambda }_n>0$ and ${\sigma }_n>0$ are positive design parameters.

Substituting equations (42) and (43) into equation (41) results in

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_n\le {\stackrel{·}{V}}_{n-1}-({\lambda }_n-\frac{g_n^d}{2g_{\mathrm{nm}}^2})e_n^2-e_{n-1}{e}_n+|e_n{D}_n^{*}|}\\ \multicolumn{1}{c}{+\frac{{\sigma }_n}{{\gamma }_n}{\tilde{\mathrm{\Theta }}}_n{\hat{\mathrm{\Theta }}}_n+\tau }\end{array}$$

(44)

Completion of Young’ inequality in equations (25) and (26)

$$\frac{{\sigma }_n}{{\gamma }_n}{\tilde{\mathrm{\Theta }}}_n{\hat{\mathrm{\Theta }}}_n\le -\frac{{\sigma }_n}{2{\gamma }_n}{\tilde{\mathrm{\Theta }}}_n^2+\frac{{\sigma }_n}{2{\gamma }_n}{\mathrm{\Theta }}_n^{*2}$$

(45)

$$-\frac{1}{2}{e}_n^2+|e_n{D}_n^{*}|\le \frac{1}{2}{D}_n^{*2}$$

(46)

Substituting equations (45) and (46) into equation (44) results in

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_n\le {\stackrel{·}{V}}_{n-1}-({\lambda }_n-\frac{1}{2}-\frac{g_n^d}{2g_{\mathrm{nm}}^2})e_n^2-e_{n-1}{e}_n-\frac{{\sigma }_n}{2{\gamma }_n}{\tilde{\mathrm{\Theta }}}_2^2}\\ \multicolumn{1}{c}{+\frac{{\sigma }_n}{2{\gamma }_n}{\mathrm{\Theta }}_n^{*2}+\frac{1}{2}{D}_n^{*2}+\tau }\\ \multicolumn{1}{c}{={\stackrel{·}{V}}_{n-1}-({\lambda }_n-\frac{1}{2}-\frac{g_n^d}{2g_{\mathrm{nm}}^2})e_n^2-e_{n-1}{e}_n-\frac{{\sigma }_n}{2{\gamma }_n}{\tilde{\mathrm{\Theta }}}_n^2+M_n}\\ \multicolumn{1}{c}{\le -({\lambda }_1-\frac{g_1^d}{2g_{1m}^2})\mu e_1^2-\sum _{i=2}^n({\lambda }_i-\frac{1}{2}-\frac{g_i^d}{2g_{\mathrm{im}}^2})e_i^2}\\ \multicolumn{1}{c}{-\sum _{i=1}^n\frac{{\sigma }_i}{2{\gamma }_i}{\tilde{\mathrm{\Theta }}}_i^2-\frac{{\sigma \prime }_1}{2{\gamma \prime }_1}{\tilde{D}}_1^2+\sum _{i=1}^n{M}_i}\end{array}$$

(47)

where $M_n=({\sigma }_n/2{\gamma }_n){\mathrm{\Theta }}_n^{*2}+(1/2)D_n^{*2}+\tau$ .

According to the virtual control laws and control laws (20), (32), (37), and (42) and the adaptive parameters laws (21), (22), (33), (38), and (43), not only all the uncertainties are compensated by fuzzy systems but also no repeated differentiation problems exist. Except for the first subsystem, other subsystem has only one parameter to learn online. Therefore, if the order of system is n, only $(n+1)$ adaptive learning parameters is needed to learn online.

Stability analysis

The positive Lyapunov candidate function of the closed-loop system is considered as

$$\begin{array}{c}\multicolumn{1}{c}{V=V_n=\frac{1}{2g_1}\log \frac{K^2}{K^2-e_1^2}+\sum _{i=2}^n\frac{1}{2g_i}{e}_i^2+\sum _{i=1}^n\frac{1}{2{\gamma }_i}{\tilde{\mathrm{\Theta }}}_i^2}\\ \multicolumn{1}{c}{+\frac{1}{2{\gamma \prime }_1}{\tilde{D}}_1^2}\end{array}$$

(48)

The derivation of V is as follows

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}\le -({\lambda }_1-\frac{g_1^d}{2g_{1m}^2})\mu e_1^2-\sum _{i=2}^n({\lambda }_i-\frac{1}{2}-\frac{g_i^d}{2g_{\mathrm{im}}^2})e_i^2}\\ \multicolumn{1}{c}{-\sum _{i=1}^n\frac{{\sigma }_i}{2{\gamma }_i}{\tilde{\mathrm{\Theta }}}_i^2-\frac{{\sigma \prime }_1}{2{\gamma \prime }_1}{\tilde{D}}_1^2+\sum _{i=1}^n{M}_i}\end{array}$$

(49)

Select the positive coefficients ${\lambda }_i$ as

$${\lambda }_1=a_1+\frac{g_1^d}{2g_{1m}^2}$$

(50)

$${\lambda }_i=a_i+\frac{1}{2}+\frac{g_i^d}{2g_{\mathrm{im}}^2},(i=2,3,\dots ,n)$$

(51)

where $a_1$ and $a_i$ are positive constants.

Substituting equations (50) and (51) into equation (49) results in

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}\le -a_1\mu e_1^2-\sum _{i=2}^n{a}_i{e}_i^2-\sum _{i=1}^n\frac{{\sigma }_i}{2{\gamma }_i}{\tilde{\mathrm{\Theta }}}_i^2}\\ \multicolumn{1}{c}{-\frac{{\sigma \prime }_1}{2{\gamma \prime }_1}{\tilde{D}}_1^2+\sum _{i=1}^n{M}_i}\\ \multicolumn{1}{c}{=-a_1\frac{e_1^2}{K^2-e_1^2}-\sum _{i=2}^n{a}_i{e}_i^2-\sum _{i=1}^n\frac{{\sigma }_i}{2{\gamma }_i}{\tilde{\mathrm{\Theta }}}_i^2}\\ \multicolumn{1}{c}{-\frac{{\sigma \prime }_1}{2{\gamma \prime }_1}{\tilde{D}}_1^2+\sum _{i=1}^n{M}_i}\\ \multicolumn{1}{c}{\le -a_1\log \frac{e_1^2}{K^2-e_1^2}-\sum _{i=2}^n{a}_i{e}_i^2-\sum _{i=1}^n\frac{{\sigma }_i}{2{\gamma }_i}{\tilde{\mathrm{\Theta }}}_i^2}\\ \multicolumn{1}{c}{-\frac{{\sigma \prime }_1}{2{\gamma \prime }_1}{\tilde{D}}_1^2+\sum _{i=1}^n{M}_i}\\ \multicolumn{1}{c}{\le -2a_1{g}_{1m}\frac{1}{2g_1}\log \frac{K^2}{K^2-e_1^2}-\sum _{i=2}^{n}2a_i{g}_{\mathrm{im}}\frac{1}{2g_i}{e}_i^2}\\ \multicolumn{1}{c}{-\sum _{i=1}^n\frac{{\sigma }_i}{2{\gamma }_i}{\tilde{\mathrm{\Theta }}}_i^2-\frac{{\sigma \prime }_1}{2{\gamma \prime }_1}{\tilde{D}}_1^2+\sum _{i=1}^n{M}_i}\end{array}$$

(52)

Define $C=min\{2a_i{g}_{\mathrm{im}},\sigma {\prime }_1,{\sigma }_i,i=1,2,\dots ,n\}$ and $M=\sum _{i=1}^n{M}_i$ .

So equation (52) can be rearranged as

$$\stackrel{·}{V}\le -CV+M$$

(53)

where C, M are positive constants. Moreover, based on the definition of $|e_1(t)|<K$ , the initial condition requirement is $-K<e_1(0)<K$ . This is equivalent to $\left|\frac{e_1(0)}{K}\right|<1$ . Therefore, Lemma 1 ensures that $\left|\frac{e_1(t)}{K}\right|<1$ , $\forall t>0$ and V is bounded on $[0,\infty )$ .

Multiply $\stackrel{·}{V}\le -CV+M$ by $e^{Ct}$ on both sides to infer that

$$e^{Ct}\stackrel{·}{V}\le (-CV+M)e^{Ct}$$

(54)

$$\frac{d}{dt}(V{e}^{Ct})\le M{e}^{Ct}$$

(55)

$$V{e}^{Ct}-V(0)\le \frac{M}{C}(e^{Ct}-1)$$

(56)

$$0\le V(t)\le V(0)e^{-Ct}+\frac{M}{C}(1-e^{-Ct})\le V(0)+\frac{M}{C}$$

(57)

That is, if $V(0)\le \nu$ , then $V(t)\le \nu +(M/C)$ , $\forall t>0$ . The boundedness of $e_1(t)/K$ and V guarantee that all signals of the closed-loop control system are semi-globally uniformly ultimately bounded. From the definitions of C and M, it is clear that $e_1(t)=y(t)-y_d(t)$ can be made arbitrarily small by appropriate design parameters.

Simulations

A motor-driven manipulator is used in the simulations, and the dynamic equation can be written as follows ⁴²

$$\{\begin{array}{c}{\stackrel{·}{x}}_1=x_2\hfill \\ {\stackrel{·}{x}}_2=f_2\left({\overline{x}}_2\right)+g_2\left({\overline{x}}_2\right)x_3\hfill \\ {\stackrel{·}{x}}_3=f_3\left({\overline{x}}_3\right)+g_3\left({\overline{x}}_3\right)u+\omega \hfill \\ y=x_1\hfill \end{array}$$

(58)

where $x_1=\theta$ , $x_2=\stackrel{·}{\theta }$ , $x_3=I$ , $N=\mathrm{mgl}+\mathrm{Mgl}$ , $M_t=J+(1/3)m{l}^2+(1/10)M{l}^{2}D$ . g is the gravity acceleration constant. $f_2$ and $f_3$ are unknown nonlinear functions. $\omega$ is the external disturbance. $\theta$ is connecting rod angle. I is electric current. $K_t$ is torque constant. $K_b$ is back electromotive force (EMF) coefficient. B is viscous friction coefficient of bearing. D is load diameter. l is connecting rod length. M is load quality. m is connecting rod weight. L is reactance. R is resistance. u is the control voltage of the motor. J is actuator torque. The desired trajectory is $y_d=\sin t$ . $f_2(·)=-(B/M_t)x_2+(N/M_t)\sin x_1$ , $g_2(·)=K_t/M_t$ , $f_3(·)=-(R/L)x_3-(K_b/L)x_2$ , $g_3(·)=(1/L)$ , $\omega (t)=-(4/L)\sin (t)$ .

The parameters of the manipulator: $B=0.015$ , $L=0.0008$ , $D=0.05$ , $R=0.075$ , $m=0.01$ , $J=0.05$ , $l=0.6$ , $K_b=0.085$ , $M=0.05$ , $K_t=1$ , and $g=9.8$ . The initial state of the manipulator is $\mathbf{x}(0)=[0,0,0{]}^T$ .

The modified adaptive fuzzy backstepping control (MAFBC) scheme described above is summed up as equation (59). In order to verify the effectiveness, we compare the control performance with that of conventional adaptive fuzzy backstepping control (CAFBC) without considering the output constraints which is described as equation (60). The number of adaptive parameters in traditional fuzzy backstepping control is 27, and the number of adaptive parameters in our control method is only 4

$$\{\begin{array}{c}{\alpha }_1=-{\lambda }_1{e}_1-\frac{1}{4\tau }\mu e_1{\hat{\mathrm{\Theta }}}_1{\xi }_1^T{\xi }_1-{\hat{D}}_1\tanh \left(\frac{\mu e_1}{\delta }\right)\hfill \\ {\alpha }_2=-{\lambda }_2{e}_2-\mu e_1-\frac{1}{4\tau }{e}_2{\hat{\mathrm{\Theta }}}_2{\xi }_2^T{\xi }_2\hfill \\ u=-{\lambda }_3{e}_3-e_2-\frac{1}{4\tau }{e}_3{\hat{\mathrm{\Theta }}}_3{\xi }_3^T{\xi }_3\hfill \\ \stackrel{·}{\widehat{{\mathrm{\Theta }}_1}}=\frac{{\gamma }_1}{4\tau }{\mu }^2{e}_1^2{\xi }_1^T{\xi }_1-{\sigma }_1{\hat{\mathrm{\Theta }}}_1,\stackrel{·}{\widehat{D_1}}={\gamma \prime }_1\mu e_1\tanh \left(\frac{\mu e_1}{\delta }\right)-{\sigma \prime }_1{\hat{D}}_1\hfill \\ \stackrel{·}{\widehat{{\mathrm{\Theta }}_i}}=\frac{{\gamma }_i}{4\tau }{e}_i^2{\xi }_i^T{\xi }_i-{\sigma }_i{\hat{\mathrm{\Theta }}}_i,i=1,2\hfill \\ e_1=x_1-y_d,e_2=x_2-{\alpha }_1,e_3=x_3-{\alpha }_2\hfill \end{array}$$

(59)

$$\{\begin{array}{c}{\alpha }_1=-{\lambda }_1{e}_1-{\mathbf{\theta }}_1^T{\xi }_1({\overline{x}}_1)\hfill \\ {\alpha }_2=-{\lambda }_2{e}_2-e_1-{\mathbf{\theta }}_2^T{\xi }_2({\overline{x}}_2)\hfill \\ u=-{\lambda }_3{e}_3-e_2-{\mathbf{\theta }}_3^T{\xi }_3({\overline{x}}_3)\hfill \\ \stackrel{·}{{\theta }_i}={\gamma }_i{\mathbf{e}}_i{\mathbf{\xi }}_i({\overline{x}}_i)-2{\sigma }_i{\mathbf{\theta }}_i,i=1,2,3\hfill \\ e_1=x_1-y_d,e_2=x_2-{\alpha }_1,e_3=x_3-{\alpha }_2\hfill \end{array}$$

(60)

The control parameters in the two control methods are designed as follows: ${\lambda }_1=3$ , ${\lambda }_2=8.5$ , ${\lambda }_3=8.5$ , ${\gamma }_1=\gamma {\prime }_1={\gamma }_2={\gamma }_3=2$ , ${\sigma }_1=\sigma {\prime }_1={\sigma }_2={\sigma }_3=3$ , $K=0.15$ , and $\tau =0.5$ .

Choose fuzzy membership functions as ${\mu }_{F_i^l}(x_i)=\exp [-(x_i+2.5-l/2{)}^2/2]$ , where $l=1,2,\dots ,9$ .

Then

$$\begin{array}{c}\multicolumn{1}{c}{{\xi }_{2j}({\overline{x}}_2)=\frac{{\mu }_{F_1^j}(x_1)\times {\mu }_{F_2^j}(x_2)}{\sum _{j=1}^9{\mu }_{F_1^j}(x_1)\times {\mu }_{F_2^j}(x_2)},}\\ \multicolumn{1}{c}{{\xi }_{3j}({\overline{x}}_3)=\frac{{\mu }_{F_1^j}(x_1)\times {\mu }_{F_2^j}(x_2)\times {\mu }_{F_3^j}(x_3)}{\sum _{j=1}^9{\mu }_{F_1^j}(x_1)\times {\mu }_{F_2^j}(x_2)\times {\mu }_{F_3^j}(x_3)}}\end{array}$$

Define the fuzzy base function vectors as: ${\mathbf{\xi }}_1({\overline{x}}_1)=[{\xi }_{11}({\overline{x}}_1),{\xi }_{12}({\overline{x}}_1),\dots ,{\xi }_{19}({\overline{x}}_1){]}^T$ , ${\mathbf{\xi }}_2({\overline{x}}_2)=[{\xi }_{21}({\overline{x}}_2),{\xi }_{22}({\overline{x}}_2),\dots ,{\xi }_{29}({\overline{x}}_2){]}^T$ , and ${\mathbf{\xi }}_3({\overline{x}}_3)=[{\xi }_{31}({\overline{x}}_2),{\xi }_{32}({\overline{x}}_3),\dots ,{\xi }_{39}({\overline{x}}_3){]}^T$ .

The simulation results are shown in Figures 1–5. Figures 1 and 2 show the performance of the position tracking of the control methods. From Figure 2, we can see that the tracking error provided by MAFBC satisfies the constraints, while the tracking error provided by CAFBC violates the error constraints. Figures 3 and 4 show the performance of the speed tracking. Figure 5 shows the control input signal. It can be seen that the initial control input of MAFBC is much less than that of CAFBC.

Figure 1.

The position tracking.

Figure 2.

The error of position tracking.

Figure 3.

The speed tracking.

Figure 4.

The error of speed tracking.

Figure 5.

The control input.

Finally, for a better illustration, the simulation results are summarized in Table 1. From Table 1, it is clearly shown that the proposed control method MAFBC has much less learning parameters, but its control performance is better than that of CAFBC.

Table 1.

The performance comparison of the control schemes.

Performance comparison	CAFBC	MAFBC
The number of adaptive parameters	27	4
$max(|e(t)|)$ of position tracking	0.33	0.05
${\int }_0^t{e}^{2}dt$ of position tracking	1200	6.8
${\int }_0^t{e}^{2}dt$ of speed tracking	1700	150
$u_{max}$	11.0860	3.9831

CAFBC: conventional adaptive fuzzy backstepping control; MAFBC: modified adaptive fuzzy backstepping control.

Conclusion

In this article, a novel adaptive fuzzy backstepping control scheme has been proposed for a class of nonlinear systems in strict-feedback form. Using fuzzy systems and backstepping control, not only uncertainties, such as unknown functions and unknown control gains, are identified, but also the problem of “explosion of complexity” is avoided. By introducing the BLF, the tracking error satisfies the restriction conditions. Moreover, in order to reduce the number of adaptive learning parameters, MLPA is used such that only $(n+1)$ parameter is needed to learning online for n order system. Finally, the stability is proved and all the signals are guaranteed to be bounded. Some simulation results have demonstrated the effectiveness of the proposed control scheme. Future research directions are the extension of the results to nonstrict-feedback stochastic MIMO nonlinear systems with uncertainties and unmeasured states.