Adaptive fuzzy finite-time fault-tolerant funnel control of nonlinear systems with actuators failures

Abstract

This article investigates adaptive fuzzy finite-time fault-tolerant funnel control of nonlinear systems with actuators failures, which contain outage, loss of effectiveness and stuck modes. The principal control objectives are ensure that the trajectory of the tracking error can be limited to the predefined area in finite time and whether the actuators are at faults or not. First, the original error equation is changed into a new one by introducing the revised funnel variable, which will be employed to design the controller later. Second, the required conditions for the existence of the adaptive fuzzy finite-time fault-tolerant controller are provided using backstepping technology, fuzzy logic system, and finite-time Lyapunov stability theory. Theoretical analysis verifies that all the signals of the closed-loop system are semi-globally practically finite-time stable. Finally, a numerical example illustrates that the proposed method can effectively improve the steady-state and transient-state performance of the system in finite time.

Introduction

In industrial applications, actuator or sensor failures usually lead to performance degradation or instability of the systems. In recent years, research interest in fault-tolerant control (FTC) has increased and an increasing number of related developments have been achieved. Wu and Yang ¹ designed a new robust adaptive FTC method to maintain the output tracking error in a small area of the origin. Fan and Song ² present a new FTC approach for a class of nonlinear systems in case of actuator failures, and the theoretical analysis shows that satisfactory performance can be obtained under the circumstance of actuator failures. Jin et al. ³ propose the robust FTC controller to make up for the influence of faults, which makes the closed-loop system asymptotically stable. However, among these achievements, most works address the issue of stability of closed-loop systems with/without failures, and the transient-state and steady-state performance, for instance, convergence rate, overshoot, and steady-state error, has not been entirely considered.

In recent years, a new control strategy, termed as prescribed performance control (PPC), was put forward, and its key idea is to improve transient-state and steady-state performance of the output tracking error signal using a prescribed performance function. At present, some valuable achievements have been achieved by Bechlioulis and Rovithakis^4,5 as well as by Chen et al. ⁶ However, one of the disadvantages of the traditional PPC is that it has the potential to undergo the singularity problem. Therefore, more and more scholars, such as Ilchman and colleagues,^7–9 have turned to the study of funnel control (FC). FC is a powerful algorithmic tool, which can bring the significant improvement of system performance, and its key feature is that a time-variable gain is introduced to guarantee the stability and tracking performance under acceptable measurement noises and parameter uncertainties. However, traditional FC method is mostly applied to some special systems; therefore, it is difficult to be extended to other type of systems.

The tracking control has become one of the most important schemes, and a lot of research achievements have been achieved in previous works.^10–16 Safa et al. ¹⁰ present a position-tracking control strategy using output feedback and adaptive sliding-mode approach. Sun et al. ¹¹ propose a novel hybrid coordinated control method, in which backstepping scheme and Hamilton control were utilized to improve the performance of the asymptotic position tracking. Another important issue is the problem of stability of the systems. As we know, stability is an important concept in dynamic systems, and some significant developments have been obtained.^17–21 However, most of the proposed methods were on account of Lyapunov stability theory or Lyapunov asymptotic stability, which means that the equilibrium or the small neighborhood of the equilibrium may be converged gradually and ultimately. However, it is not clear how long to approach the equilibrium or the small area of the equilibrium. Recently, the finite-time control has been studied generally, which is designed to maintain that the state variables are confined to the equilibrium or the small domain of the equilibrium in a finite time interval. Obviously, the finite-time control offers more advantages over the traditional stability in the improvement of rapidity and high-accuracy performance. Tang ²² proposed the finite-time control for linear systems using the terminal sliding-mode method. Inspired by the literature, some finite-time stable controllers for nonlinear systems were developed, in which the finite-time Lyapunov stable theories were widely developed and implemented in many other studies.^23–26 Throughout these achievements, one of the most remarkable development made is a semi-globally practically finite-time stability specification proposed by Wang et al.^27,28 Consequently, with the aid of the specification, a new adaptive fuzzy fault-tolerant tracking control will be devised by making use of backstepping method in this article.

As we know, the adaptive backstepping control has become an important method for nonlinear systems. This control approach has got great attention in the past few decades, and lots of new significant results have been obtained. Specifically, when the controlled system suffers from unknown nonlinear terms, adaptive backstepping technique together with fuzzy/neural control has become more and more important, in which neural network (NN) systems or fuzzy logic systems (FLSs) are employed to approximate the unknown functions.^29–33 NNs approximation approach is normally utilized to dispose of the control issue in nonlinear systems, which was put forward for a class of switched nonlinear systems by Cai and Xiang; ³⁴ two classes of uncertain multi-input multi-output (MIMO) nonlinear systems in block-triangular forms by Ge and Wang; ³⁵ and interconnected nonlinear systems with time delay by Hua and Guan. ³⁶ Apart from NNs, the fuzzy control, which is another alternative approximation technique, is also an effective method to approximate the unknown terms. Long and Zhao ³⁷ and Wang et al. ³⁸ investigagted a new adaptive fuzzy control strategy for a class of nonlinear systems, which can ensure that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded, meanwhile the tracking error can be limited to a small area of the origin. Another significant problem associated with the adaptive backstepping control is the backstepping technology, which is a valid design approach for nonlinear systems.

At present, some novel achievements about adaptive finite-time FTC have been made. Jin ³⁹ put forward an adaptive finite-time FTC approach for a class of MIMO nonlinear systems with constraint requirement on the system output tracking error. A finite-time FTC scheme was presented for robot manipulators and rigid spacecraft by Van et al. ⁴⁰ and Xing et al., ⁴¹ respectively. Among the aforementioned results, the transient and steady-state performance have not been addressed. Liu et al. ⁴² came up with an adaptive finite-time control method using prescribed performance, and the proposed controller can ensure that the system has good dynamic performance.

As a result, inspired by the above observations, an adaptive fuzzy finite-time fault-tolerant FC scheme is discussed. Different from the the existing achievement, the proposed approach gives attention to stability, transient-state and steady-state performance of the fault system; at the same time, it also guarantees that the tracking error can be maintained in the prescribed region of the funnel boundary functions within finite-time, no matter if the actuators are in normal or abnormal conditions. Meanwhile, the FLSs are adopted to approach the unknown nonlinear functions in nonlinear systems. The main contribution of our work can be briefly characterized as follows:

1. First, a novel adaptive fuzzy finite-time fault-tolerant FC is proposed for nonlinear systems with actuators. An important kind of failure is considered, which often takes place in many practical systems in literature.^43,44

2. Most of the existing adaptive finite-time control approaches by Wang et al.,^27,28 those proposed in previous works^45–47 can guarantee that the tracking error can be limited to a small area of the equilibrium in a finite period of time. The transient-state performance, for instance the maximum overshoot, the convergence speed of the output tracking error, has not been thoroughly considered. A modified funnel variable is introduced to improve the performance of nonlinear systems to improve the performance of nonlinear systems with actuator failures.

3. Compared with most works in the PPC, in the note, the proposed control scheme is used to deal with the non-differentiable problem and the singularity that may exist in the design of backstepping method.

The remainder of this article is arranged as follows: some basic definitions and preliminaries are given in section “Problem formulation and preliminaries.” Section “Main results” presents an adaptive fuzzy finite-time fault-tolerant funnel controller of nonlinear systems with actuators failures. Section “Simulation results” shows the effectiveness of the proposed control method by the simulation analysis. Finally, the research is concluded in section “Conclusion.”

Problem formulation and preliminaries

Problem formulation

Consider the following nonlinear systems with actuator failures

$$\{\begin{array}{c}{\stackrel{·}{x}}_i(t)=x_{i+1}(t)+f_i({\overline{x}}_i),1⩽i⩽n-1\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_n(t)=B^{\mathrm{T}}{u}^{\mathrm{F}}(t)+f_n(x)}\\ \multicolumn{1}{c}{y(t)=x_1(t)}\end{array}$$ (1)

where $x(t)=[x_1,\dots ,x_n{]}^{\mathrm{T}}\in R^n$ are the state variables, ${\overline{x}}_i(t)=[x_1(t),\dots ,x_i(t){]}^{\mathrm{T}}$ , $y(t),f_i({\overline{x}}_i)$ stand for the output signal and the unknown system functions, respectively. $B_{m\times 1}\in R^m$ is a known control gain vector, and $u^{\mathrm{F}}(t)\in R^m$ denotes the actual control input vector.

The actuator failures to be considered conclude actuator outage, stuck, and loss of effectiveness. The following uniform actuator fault model has been widely recognized and applied by Yang et al.; ⁴⁸ the consistent actuator failure model is described as

$$u^{\mathrm{F}}(t)=\beta (t)u(t)+\varphi (t)$$ (2)

where $u(t)$ is the designed input signal; $\beta (t)=\mathrm{diag}[{\beta }_1(t),\dots ,{\beta }_m(t)],0⩽{\beta }_j(t)⩽1,j=1,\dots ,m$ is the multiplicative actuator fault that denotes the uncertain time-varying gain of the actuator with faults, and ${\beta }_j(t)$ is the actuator efficiency factor; and $\varphi (t)\in R^m$ is the additive actuator fault that occurs in the control input channel.

Thus, the system (1) is rewritten as

$$\{\begin{array}{c}{\stackrel{·}{x}}_i(t)=x_{i+1}(t)+f_i({\overline{x}}_i),1⩽i⩽n-1\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_n(t)=B^{\mathrm{T}}\beta (t)u(t)+B^{\mathrm{T}}\varphi (t)+f_n(x)}\\ \multicolumn{1}{c}{y(t)=x_1(t)}\end{array}$$ (3)

FC

Define the desired trajectory as $y_r(t)=x_{1r}(t)$ and the tracking error as $e(t)=x_1(t)-x_{1r}(t)$ . One of the objectives of the article is to guarantee that $e(t)$ converges to the prescribed region of the funnel boundary functions within finite-time no matter if the actuators are in normal or abnormal conditions. FC was proposed by Ilchman, which can guarantee a prescribed transient behavior. Its main feature is that a time-variable gain $\epsilon (t)$ is utilized to control class S systems with a relative degree $r=1$ or $2$ . The FC control law is described as

$$u(t)=\epsilon (\rho (t),\varphi (t),||e||)e$$ (4)

with $\rho (t)$ being the boundary constraint function; $\varphi (t)$ denotes the scaling function, which is bounded, continuous and positive; ||·|| denotes the Euclidian norm.

In general, the boundary constraint function $\rho (t)$ is expressed by the reciprocal of $\varphi (t)$ , that is

$$\rho (t)=\frac{1}{\varphi (t)}$$ (5)

Introduce a new function transformation as follows

$$D_{\upsilon }(t)=\rho (t)-||e||$$ (6)

where $D_{\upsilon }(t)$ represents the vertical distance between $\rho (t)$ and $||e(t)||$ . Thus, the time-varying gain $ϵ(t)$ is given as

$$ϵ(t)=\frac{\varphi (t)}{D_{\upsilon }(t)}=\frac{\varphi (t)}{\rho (t)-\Vert e(t)\Vert }$$ (7)

We choose the following function as the funnel boundary function

$$\rho (t)=({\rho }_{0-}{\rho }_{\infty })e^{-\kappa t}+{\rho }_{\infty }$$ (8)

with $\rho (t)$ being positive, strictly decreasing and $\underset{t\to \infty }{lim}\rho (t)={\rho }_{\infty }$ ; ${\rho }_0,$ ${\rho }_{\infty }$ , and $\kappa$ are design parameters.

Remark 1

As shown in equation (7), while the error $e(t)$ converges to $\rho (t)$ , $ϵ(t)$ will increase and vice versa. Apparently, the error $e(t)$ can be guaranteed to evolve inside the funnel boundary function $\rho (t)$ . For convenience, the variable t is omitted in the following.

Based on the research results of the literature, ⁴⁹ an improved funnel variable is introduced

$$z_1=\frac{e^2}{{\rho }^2-e^2}$$ (9)

In the following chapters, $z_1$ will be used to design the controller. The time-derivative of $z_1$ can be derived

$${\stackrel{·}{z}}_1=2\mathrm{\Xi }({\stackrel{·}{x}}_1-{\stackrel{·}{y}}_r-e\frac{\stackrel{·}{\rho }}{\rho })$$ (10)

where $\mathrm{\Xi }={\rho }^{2}e/({\rho }^2-e^2{)}^2$

Basic definitions, lemmas, and assumptions

Definition 1

Consider a nonlinear system ⁵⁰

$$\stackrel{·}{x}=f(x,u),f(0,0)=0,x\in {\Re }^n,u\in \Re$$ (11)

where x and u represent the state variable and input variable, respectively; $f(·)$ is continuous on a neighborhood of the equilibrium point. Then the origin $x=0$ of the system (11) is semi-globally practically finite-time stable (SGPFS) if for all $x(t_0)=x_0,$ there exists positive real number $\delta$ and a settling time $T_s(\delta ,x_0)<\infty ,$ satisfying $||x||<\delta ,$ for all $t⩾t_0+T_s$ .

Lemma 1

The system (11) is SGPFS, if there exists a $V(x)>0,$ with its time derivative satisfying $\stackrel{·}{V}(x)⩽-\varsigma V^{\gamma }(x)+\rho$ for some real numbers $\varsigma >0$ , $\gamma \in (0,1)$ and $\rho >0$ . The settling time $T_s(\delta ,x_0)$ satisfies^27,28

$$T_s=\frac{1}{\varsigma (1-\gamma ){\varpi }_0}[V^{1-\gamma }(x(0))-{\left(\frac{\rho }{\varsigma (1-{\varpi }_0)}\right)}^{\frac{1-\gamma }{\gamma }}]$$ (12)

where $0<{\varpi }_0⩽1$ is a scalar.

Lemma 2

Define $f(Z):{\Re }^m\to \Re$ as a continuous function on a compact set ${\mathrm{\Omega }}_Z$ , accordingly, there exists an FLS $W^{\mathrm{T}}R(Z)$ , which is introduced to approximate $f(Z)$ for any constant $\epsilon >0$ such that ⁵¹

$$\underset{x\in {\mathrm{\Omega }}_Z}{sup}|f(Z)-W^{\mathrm{T}}R(Z)|⩽\epsilon$$ (13)

Lemma 3

For $x_j\in \Re ,j=1,\dots ,n,$ and $l\in (0,1)$ , then ⁵²

$${(\left|x_1\right|+\cdots +\left|x_n\right|)}^l⩽{\left|x_1\right|}^l+\cdots +{\left|x_n\right|}^l$$ (14)

Lemma 4

For $u\in \Re ,$ $v\in \Re ,$ and any real numbers $a,b,q,$ then ⁵¹

$${\left|u\right|}^a{\left|v\right|}^b⩽\frac{a}{a+b}q{\left|u\right|}^{a+b}+\frac{b}{a+b}{q}^{-\frac{a}{b}}{\left|v\right|}^{a+b}$$ (15)

Assumptions 1

Consider the additive fault $\varphi (t)$ and multiplicative actuator fault $\beta (t)$ in equation (2), we have $|\varphi (t)|<\overline{\varphi },$ and assume that

$$\begin{array}{c}{B}^{\mathrm{T}}\varphi =d,\left|d\right|<\overline{d}\hfill \\ \multicolumn{1}{c}{B^{\mathrm{T}}\beta B⩾\overline{b}}\end{array}$$

where $\overline{d}\mathrm{and}\overline{b}$ are unknown positive constants.

Next, we will offer an adaptive fuzzy finite-time FTC method. The control objective is that SGPFS of the closed-loop system with actuators failures can be ensured; at the same time, the tracking error can be kept in the predetermined regions in finite time.

Main results

The controller design

An adaptive fuzzy finite-time FTC funnel controller using backstepping technique will be designed. First, the following coordinate transformation is introduced

$$z_i=x_i-a_{i-1},i=2,\dots ,n$$ (16)

where ${\alpha }_{i-1}$ is the virtual control variable.

Thus, according to equations (10) and (16), the initial system can be transformed as follows

$$\{\begin{array}{c}{\stackrel{·}{z}}_1=2\mathrm{\Xi }(z_2+f_1(x_1)+{\alpha }_1-{\stackrel{·}{y}}_r-e\frac{\stackrel{·}{\rho }}{\rho })\hfill \\ {\stackrel{·}{z}}_i=z_{i+1}+f_i({\overline{x}}_i)+a_i-{\stackrel{·}{\alpha }}_{i-1},i=2,\dots ,n-1\hfill \\ {\stackrel{·}{z}}_n=f_n(x)+B^{\mathrm{T}}\beta u+B^{\mathrm{T}}\varphi -{\stackrel{·}{a}}_{n-1}\hfill \end{array}$$ (17)

In this section, the sufficient conditions for the existence of the proposed controller and design steps will be given for the system depicted by equation (17) using backstepping technology.

Design the virtual control law and the actual control input vector as

$${\alpha }_1={\mathrm{\Lambda }}_0(-\frac{1}{2a_1^2}{\hat{\theta }}_1{R}_1{(Z_1)}^{\mathrm{T}}{R}_1(Z_1)-\frac{1}{2}-k_1)$$ (18)

$${\alpha }_i=-\frac{1}{2a_i^2}{z}_i{\hat{\theta }}_i{R}_i(Z_i{)}^{\mathrm{T}}{R}_i(Z_i)-\frac{z_i}{2}-k_i{z}_i$$ (19)

$$\begin{array}{c}i=2,\dots ,n-1\hfill \\ u=-({\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2)z_n\hfill \end{array}$$ (20)

where $a_i\mathrm{and}{k}_i$ are postive design parameters; $R_i(Z_i)$ is the basis fuzzy function with $Z_i=[z_1,\dots ,z_i,{\hat{\theta }}_1,\dots ,{\hat{\theta }}_i,y_r,\dots ,y_r^{(i)},\rho ]$ , which is utlized to approximate the unknown nonlinear function at $i\mathrm{th}$ step; ${\mathrm{\Lambda }}_0=e({\rho }^2-e^2)/{\rho }^2,{\mathrm{\Lambda }}_1=B,{\mathrm{\Lambda }}_2=(B/2a_n^2){\hat{\theta }}_n{R}_n^{\mathrm{T}}{R}_n;{\hat{\theta }}_i$ is the estimation of ${\theta }_i,$ and ${\tilde{\theta }}_i={\theta }_i-{\hat{\theta }}_i$ is the estimation error.

The adaptive laws are designed as follows

$$\stackrel{·}{\widehat{{\theta }_i}}=\frac{r_i}{2a_i^2}{z}_i^2{R}_i^{\mathrm{T}}{R}_i-{\sigma }_i{\hat{\theta }}_i,i=1,\dots ,n-1$$ (21)

$$\stackrel{·}{\widehat{{\theta }_n}}=\frac{r_n}{2a_n^2}{z}_n^2{R}_n^{\mathrm{T}}{R}_n-{\overline{\sigma }}_n{\hat{\theta }}_n$$ (22)

where $r_i$ , ${\sigma }_i$ , and ${\overline{\sigma }}_n$ are the positive design parameters.

The realization of the control objective is ensured in the following theorem.

Theorem 1

Consider the nonlinear systems with actuators failures equation (1), the control laws as equations (18)–(20), and the adaptive laws as equation (21)–(22) ensure that, under the Assumptions 1, there exist the appropriate design parameters $a_i>0,k_i>0,r_i>0,{\sigma }_i>0,i=1,\dots ,n$ . All signals of the closed-loop system are SGPFS, and the trajectory of $e(t)$ can converge to the the given funnel boundary within finite time whether the actuators failures occur or not.

Proof

The design process concludes n steps. For each step, the unknown items will be approximated using the FLS approximators.

• Step 1.

Construct a Lyapunov function for the first subsystem in equation (17)

$$V_1=\frac{z_1^2}{4}+\frac{1}{2r_1}{\tilde{\theta }}_1^2$$ (23)

which yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1=\frac{1}{2}{z}_1{\stackrel{·}{z}}_1-\frac{1}{r_1}{\tilde{\theta }}_1{\stackrel{·}{\hat{\theta }}}_1}\\ =z_1(F_1+\mathrm{\Xi }{\alpha }_1)+k_1{z}_1^2-{\mu }_1{z}_1^{2\gamma }+\mathrm{\Xi }{z}_1{z}_2-\frac{1}{r_1}{\tilde{\theta }}_1{\stackrel{·}{\hat{\theta }}}_1\hfill \end{array}$$ (24)

where $F_1=\mathrm{\Xi }(f_1-{\stackrel{·}{y}}_r-(e\stackrel{·}{\rho }/\rho ))+{\mu }_1{z}_1^{2\gamma -1}-k_1{z}_1$ denotes a packaged unknown function, $0.5<\gamma <1$ .

Based on Lemma 2, the FLS is utilized to approximate $F_1(Z_1)$ for any accuracy ${\epsilon }_1>0$ , and we have

$$F_1(Z_1)=W_1^{\mathrm{T}}{R}_1(Z_1)+{\delta }_1(Z_1),|{\delta }_1(Z_1)|⩽{\epsilon }_1$$ (25)

where ${\delta }_1(Z_1)$ is the approximation error. Based on completion squares theorem, it can be obtained that

$$z_1{F}_1⩽\frac{1}{2a_1^2}{z}_1^2{\theta }_1{R}_1^{\mathrm{T}}{R}_1+\frac{a_1^2}{2}+\frac{z_1^2}{2}+\frac{{\epsilon }_1^2}{2}$$ (26)

where ${\theta }_1=||W_1|{|}^2.$ For convenience, the variable $Z_1$ is omitted in the following.

Substituting equation (26) into equation (24), we obtained

$$\begin{array}{c}{\stackrel{·}{V}}_1⩽\mathrm{\Xi }{z}_1{z}_2+\mathrm{\Xi }{z}_1{\alpha }_1-\frac{1}{r_1}{\tilde{\theta }}_1{\stackrel{·}{\hat{\theta }}}_1+\frac{1}{2a_1^2}{z}_1^2{\theta }_1{R}_1^{\mathrm{T}}{R}_1\hfill \\ +\frac{a_1^2}{2}+\frac{z_1^2}{2}+\frac{{\epsilon }_1^2}{2}-{\mu }_1{z}_1^{2\gamma }+k_1{z}^2\hfill \end{array}$$ (27)

On the basis of equation (18), the following equation can be obtained

$$\mathrm{\Xi }{z}_1{\alpha }_1=-\frac{1}{2a_1^2}{\hat{\theta }}_1{z}_1^2{R}_1^{\mathrm{T}}{R}_1-\frac{z_1^2}{2}-k_1{z}_1^2$$ (28)

Substituting equation (28) into equation (27) yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1⩽\mathrm{\Xi }{z}_1{z}_2+\frac{1}{2a_1^2}{\tilde{\theta }}_1{z}_1^2{R}_1^{\mathrm{T}}{R}_1-\frac{{\tilde{\theta }}_1{\stackrel{·}{\hat{\theta }}}_1}{r_1}}\\ +\frac{a_1^2}{2}+\frac{{\epsilon }_1^2}{2}-{\mu }_1{z}_1^{2\gamma }\hfill \end{array}$$ (29)

Using $\stackrel{·}{\widehat{{\theta }_1}}$ , we have

$${\stackrel{·}{V}}_1⩽\mathrm{\Xi }{z}_1{z}_2-{\mu }_1{z}_1^{2\gamma }+\frac{a_1^2}{2}+\frac{{\epsilon }_1^2}{2}+\frac{{\sigma }_1}{r_1}{\tilde{\theta }}_1{\hat{\theta }}_1$$ (30)

that is

$${\stackrel{·}{V}}_1⩽\mathrm{\Xi }{z}_1{z}_2+{\tau }_1$$

where ${\tau }_1=-{\mu }_1{z}_1^{2\gamma }+1/2(a_1^2+{\epsilon }_1^2)+({\sigma }_1/r_1){\tilde{\theta }}_1{\hat{\theta }}_1$

• Step i. $(2⩽i⩽n-1)$

Define $V_i$ as

$$V_i=V_{i-1}+\frac{1}{2}{z}_i^2+\frac{1}{2r_i}{\tilde{\theta }}_i^2$$ (31)

Therefore, ${\stackrel{·}{V}}_i$ is estimated as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_i={\stackrel{·}{V}}_{i-1}+z_i{\stackrel{·}{z}}_i-\frac{1}{r_i}{\tilde{\theta }}_i{\stackrel{·}{\hat{\theta }}}_i}\\ ⩽z_i(z_{i-1}+f_i+z_{i+1}+h_{{\mu }_i}{\alpha }_i-{\stackrel{·}{\alpha }}_{i-1})+{\tau }_{i-1}-\frac{1}{r_i}{\tilde{\theta }}_i{\stackrel{·}{\hat{\theta }}}_i\hfill \end{array}$$ (32)

where $F_i=z_{i-1}+f_i-{\stackrel{·}{\alpha }}_{i-1}+{\mu }_i{z}_i^{2\gamma -1}-k_i{z}_i$

Similar to Step 1, we can get

$$\begin{array}{c}{F}_i=W_i^{\mathrm{T}}{R}_i+{\delta }_i,\left|{\delta }_i\right|⩽{\epsilon }_i\hfill \\ z_i{F}_i⩽\frac{z_i^2}{2a_i^2}{\theta }_i{R}_i^{\mathrm{T}}{R}_i+\frac{a_i^2}{2}+\frac{z_i^2}{2}+\frac{{\epsilon }_i^2}{2}\hfill \end{array}$$

Thus, we have

$$\begin{array}{c}{\stackrel{·}{V}}_i⩽z_i{z}_{i+1}+z_i{\alpha }_i+\frac{1}{2a_i^2}{z}_i^2{\theta }_i{R}_i^{\mathrm{T}}{R}_i\hfill \\ +\frac{1}{2}(a_i^2+{\epsilon }_i^2)+\frac{z_i^2}{2}+{\tau }_{i-1}-\frac{1}{r_i}{\tilde{\theta }}_i{\stackrel{·}{\hat{\theta }}}_i+{\mu }_i{z}_i^{2\gamma }-k_i{z}_i^2\hfill \end{array}$$ (33)

where ${\theta }_i=||W_i|{|}^2.$ Substituting equations (19) and (21) into equation (33), we obtain

$${\stackrel{·}{V}}_i⩽z_i{z}_{i+1}+{\tau }_i$$ (34)

where ${\tau }_i=-\sum _{j=1}^i{\mu }_j{z}_j^{2\gamma }+(1/2)\sum _{j=1}^i(a_j^2+{\epsilon }_j^2)+\sum _{j=1}^i({\sigma }_j/r_j){\tilde{\theta }}_j{\hat{\theta }}_j$

• Step $n.$

Choose a function

$$V_n=V_{n-1}+\frac{1}{2}{z}_n^2+\frac{\overline{b}}{2r_n}{\tilde{\theta }}_n^2$$ (35)

Then, ${\stackrel{·}{V}}_n$ is derived as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_n={\stackrel{·}{V}}_{n-1}+z_n{\stackrel{·}{z}}_n-\frac{\overline{b}}{r_n}{\tilde{\theta }}_n{\stackrel{·}{\hat{\theta }}}_n}\\ ⩽z_n(F_n+B^{\mathrm{T}}\beta u+d)-{\mu }_n{z}_n^{2\gamma }+\overline{b}{z}_n^2-z_n^2\hfill \\ -\frac{\overline{b}}{r_n}{\tilde{\theta }}_n{\stackrel{·}{\hat{\theta }}}_n+{\tau }_{n-1}\hfill \end{array}$$ (36)

where $F_n=z_{n-1}+f_n-{\stackrel{·}{\alpha }}_{n-1}+{\mu }_n{z}_n^{2\gamma -1}-\overline{b}{z}_n+z_n$

According to Assumption 1, we obtain

$$z_{n}d⩽\frac{1}{2}{z}_n^2+\frac{{\overline{d}}^2}{2}$$ (37)

Substituting $F_n=W_n^{\mathrm{T}}{R}_n+{\delta }_n,\left|{\delta }_n\right|⩽{\epsilon }_n,$ we have

$$z_n{F}_n⩽\frac{\overline{b}}{2a_n^2}{z}_n^2\frac{{\Vert W_n\Vert }^2}{\overline{b}}{R}_n^{\mathrm{T}}{R}_n+\frac{a_n^2}{2}+\frac{z_n^2}{2}+\frac{{\epsilon }_n^2}{2}$$ (38)

Let $||W_n|{|}^2/\overline{b}={\theta }_n,$ Substituting equations (37) and (38) into equation (36), we obtaine

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_n⩽\frac{\overline{b}}{2a_n^2}{z}_n^2{\theta }_n{R}_n^{\mathrm{T}}{R}_n+\frac{a_n^2}{2}+\frac{{\epsilon }_n^2}{2}+z_n{B}^{\mathrm{T}}\beta u}\\ \multicolumn{1}{c}{+\frac{{\overline{d}}^2}{2}-{\mu }_n{z}_n^{2\gamma }+\overline{b}{z}_n^2-\frac{\overline{b}}{r_n}{\tilde{\theta }}_n{\stackrel{·}{\hat{\theta }}}_n+{\tau }_{n-1}}\end{array}$$ (39)

Using the adaptive law $\stackrel{·}{\widehat{{\theta }_n}}$ , we have

$$\begin{array}{c}{\stackrel{·}{V}}_n⩽\frac{\overline{b}}{2a_n^2}{z}_n^2{\theta }_n{R}_n^{\mathrm{T}}{R}_n+\frac{a_n^2}{2}+\frac{{\epsilon }_n^2}{2}+z_n{B}^{\mathrm{T}}\beta u\hfill \\ +\frac{{\overline{d}}^2}{2}-{\mu }_n{z}_n^{2\gamma }+\overline{b}{z}_n^2-\frac{\overline{b}}{r_n}{\tilde{\theta }}_n{\stackrel{·}{\hat{\theta }}}_n+{\tau }_{n-1}\hfill \end{array}$$ (40)

In our final step, the control u is constructed. According to Assumption 1, we have

$$\begin{array}{c}{z}_n{B}^{\mathrm{T}}\beta u=-z_n{B}^{\mathrm{T}}\beta (B{z}_n+\frac{B}{2a_n^2}{\hat{\theta }}_n{R}_n^{\mathrm{T}}{R}_n{z}_n)\hfill \\ \multicolumn{1}{c}{⩽-\overline{b}{z}_n^2-\frac{\overline{b}}{2a_n^2}{\hat{\theta }}_n{R}_n^{\mathrm{T}}{R}_n{z}_n^2}\end{array}$$ (41)

Substituting equation (41) into equation (40), and let ${\sigma }_n=\overline{b}{\overline{\sigma }}_n,$ it can be derived that

$${\stackrel{·}{V}}_n⩽\sum _{j=1}^n\frac{{\sigma }_j}{r_j}{\tilde{\theta }}_j{\hat{\theta }}_j-\sum _{j=1}^n{\mu }_j{z}_j^{2\gamma }+\frac{1}{2}\sum _{j=1}^n(a_j^2+{\epsilon }_j^2)+\frac{{\overline{d}}^2}{2}$$ (42)

Since ${\tilde{\theta }}_j{\hat{\theta }}_j⩽-({\tilde{\theta }}_j^2/2)+({\theta }_j^2/2)$ , the following inequality can be derived

$$\begin{array}{c}{\stackrel{·}{V}}_n⩽\sum _{j=1}^n\frac{{\sigma }_j}{2r_j}{\theta }_j^2-\sum _{j=1}^n\frac{{\sigma }_j}{2r_j}{\tilde{\theta }}_j^2-{\left(\sum _{j=1}^n\frac{{\sigma }_j}{2r_j}{\tilde{\theta }}_j^2\right)}^{\gamma }\hfill \\ +{\left(\sum _{j=1}^n\frac{{\sigma }_j}{2r_j}{\tilde{\theta }}_j^2\right)}^{\gamma }-\sum _{j=1}^n{\mu }_j{z}_j^{2\gamma }+\frac{1}{2}\sum _{j=1}^n(a_j^2+{\epsilon }_j^2)+\frac{{\overline{d}}^2}{2}\hfill \end{array}$$ (43)

According to Lemma 4, we conclude that

$${\left(\sum _{j=1}^n\frac{{\sigma }_j}{2r_j}{\tilde{\theta }}_j^2\right)}^{\gamma }⩽(1-\gamma ){\gamma }^{\frac{\gamma }{1-\gamma }}+\left(\sum _{j=1}^n\frac{{\sigma }_j}{2r_j}{\tilde{\theta }}_j^2\right)$$ (44)

Substituting equation (44) into equation (43), we have

$${\stackrel{·}{V}}_n⩽-{\left(\sum _{j=1}^n\frac{{\sigma }_j}{2r_j}{\tilde{\theta }}_j^2\right)}^{\gamma }-\sum _{j=1}^n{\mu }_j{z}_j^{2\gamma }+\varrho$$

where $\varrho =({\overline{d}}^2/2)+(1/2)\sum _{j=1}^n(a_j^2+{\epsilon }_j^2)+\sum _{j=1}^n({\sigma }_j/2r_j){\theta }_j^2+(1-\gamma ){\gamma }^{(\gamma /(1-\gamma ))}$ . By Lemma 3, it can be derived that

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_n⩽-{\left(\sum _{j=1}^n\frac{{\sigma }_j}{2r_j}{\tilde{\theta }}_j^2\right)}^{\gamma }-4^{\gamma }{\mu }_1{\left(\frac{z_1^2}{4}\right)}^{\gamma }-\sum _{j=2}^n{2}^{\gamma }{\mu }_j{\left(\frac{z_j^2}{2}\right)}^{\gamma }+\rho }\\ ⩽-K_0{\left(\sum _{j=1}^n\frac{{\tilde{\theta }}_j^2}{2r_j}\right)}^{\gamma }-K_1{\left(\frac{z_1^2}{4}\right)}^{\gamma }-K_2\sum _{j=2}^n{\left(\frac{z_j^2}{2}\right)}^{\gamma }+\rho \hfill \end{array}$$ (45)

where $K_0=\min \{{\sigma }_j^{\gamma }\},K_1=4^{\gamma }{\mu }_1;K_2=2^{\gamma }\min \{{\mu }_j\}$

Considering $\varsigma =\min \{K_0,K_1,K_2\},$ we obtain

$${\stackrel{·}{V}}_n⩽-\varsigma [{\left(\sum _{j=1}^n\frac{{\tilde{\theta }}_j^2}{2r_j}\right)}^{\gamma }+{\left(\frac{z_1^2}{4}\right)}^{\gamma }+{\left(\sum _{j=2}^n\frac{z_j^2}{2}\right)}^{\gamma }]+\rho$$ (46)

In addition, based on Lemma 3, we have

$$\begin{array}{c}\multicolumn{1}{c}{V_n^{\gamma }={(\frac{1}{4}{z}_1^2+\frac{1}{2}\sum _{j=2}^n{z}_j^2+\sum _{j=1}^n\frac{{\tilde{\theta }}_j^2}{2r_j})}^{\gamma }}\\ ⩽{\left(\sum _{j=1}^n\frac{{\tilde{\theta }}_j^2}{2r_j}\right)}^{\gamma }+{\left(\frac{z_1^2}{4}\right)}^{\gamma }+{\left(\frac{1}{2}\sum _{j=2}^n{z}_j^2\right)}^{\gamma }\hfill \end{array}$$ (47)

Combining equation (46) and equation (47) results in

$${\stackrel{·}{V}}_n⩽-\varsigma V_n^{\gamma }+\rho$$ (48)

Tracking error analysis

According to Definition 1 and Lemma 1, the inequality equation (48) indicates that the system satisfies the sufficient condition of SGPFS. Let

$$T_s=\frac{1}{\varsigma (1-\gamma ){\varpi }_0}[V^{1-\gamma }(\chi (0))-{\left(\frac{\rho }{\varsigma (1-{\varpi }_0)}\right)}^{\frac{1-\gamma }{\gamma }}]$$ (49)

where $\chi (0)=[z(0),\overline{\theta }(0){]}^{\mathrm{T}},z(0)=[z_1(0),\dots ,z_n(0){]}^{\mathrm{T}},\overline{\theta }(0)=[{\theta }_1(0),\dots ,{\theta }_n(0){]}^{\mathrm{T}}$ with $0<{\varpi }_0⩽1.$ Obviously, the inequality equation (48) can be rewritten as

$${\stackrel{·}{V}}_n(\chi )⩽-{\varpi }_0\varsigma V_n^{\gamma }(\chi )-(1-{\varpi }_0)\varsigma V_n^{\gamma }(\chi )+\rho$$ (50)

Let ${\mathrm{\Omega }}_{\chi }=\{\chi |V_n^{\gamma }(\chi )>(\rho /(\varsigma (1-{\varpi }_0)))\}$ and ${\overline{\mathrm{\Omega }}}_{\chi }=\{\lambda |V_n^{\gamma }(\chi )⩽(\rho /(\varsigma (1-{\varpi }_0)))\}$ . Now, we consider the first condition. If $\chi (t)\in {\mathrm{\Omega }}_{\chi },$ we have

$${\stackrel{·}{V}}_n⩽-{\varpi }_0\varsigma V_n^{\gamma }(\chi )$$ (51)

According to equation (51), we have

$$\underset{0}{\overset{T}{\int }}\frac{{\stackrel{·}{V}}_n(\chi )}{V_n^{\gamma }(\chi )}\mathrm{dt}⩽-\underset{0}{\overset{T}{\int }}{\varpi }_0\varsigma \mathrm{dt}$$ (52)

Solving equation (52) gives

$$\frac{1}{1-\gamma }{V}_n^{1-\gamma }(\chi (T))-\frac{1}{1-\gamma }{V}_n^{1-\gamma }(\chi (0))⩽-{\varpi }_0\varsigma T$$ (53)

Furthermore, we can obtain

$$\begin{array}{c}0⩽V_n^{1-\gamma }(\chi (T))\hfill \\ ⩽-{\varpi }_0\varsigma (1-\gamma )T+V_n^{1-\gamma }(\chi (0))\hfill \\ ⩽V_n^{1-\gamma }(\chi (0))\hfill \end{array}$$ (54)

According to $V_1$ , it has

$$0⩽\frac{1}{4}{z}_1^2=\frac{1}{4}\frac{e^4}{{({\rho }^2-e^2)}^2}⩽V_1$$ (55)

Based on equation (54), it can be derived as follows

$${\left[\frac{e^4}{{({\rho }^2-e^2)}^2}\right]}^{1-\gamma }⩽{[4V_n(\chi (0))]}^{1-\gamma }$$ (56)

Equation (56) can be rewritten

$$e^{4(1-\gamma )}⩽{[4V_n(\chi (0))({\rho }^4-2{\rho }^2{e}^2+e^4)]}^{(1-\gamma )}$$ (57)

which implies that

$$e^4⩽4V_n(\chi (0))({\rho }^4-2{\rho }^2{e}^2+e^4)$$ (58)

Equation (58) can be expressed as

$$(1-4V_n\chi (0))e^4⩽V_n(\chi (0))(4{\rho }^4-8{\rho }^2{e}^2)$$ (59)

If $0<V_n(\chi (0))<(1/4)$ , then

$${\rho }^4-2{\rho }^2{e}^2⩾0$$ (60)

Thus, we can obtain

$${\rho }^2⩾2e^2$$ (61)

It can be derived that

$$\left|e\right|⩽\frac{\left|\rho \right|}{\sqrt{2}}<\left|\rho \right|$$ (62)

Therefore, the tracking error e can converge to the predefined region based on the funnel boundary function after $T_s$ .

For inequality equation (48), based on Lemma 1, the setting time $T_s$ can be derived as follows

$$T_s=\frac{1}{\varsigma (1-\gamma ){\varpi }_0}[V^{1-\gamma }(z(0))-{\left(\frac{\rho }{\varsigma (1-{\varpi }_0)}\right)}^{\frac{1-\gamma }{\gamma }}]$$ (63)

where $0<{\varpi }_0<1;z(0)=[z_1,\dots ,z_n]$

Remark 2

According to Definition 1, Lemma 1 and inequality equation (48), it is indicated that the sufficient condition of SGPFS for the closed-loop system is satisfied in finite time. Through analysis of the tracking error, inequality equation (62) shows that the tracking error can be confined to the predetermined region as well no matter if the actuators are in normal or abnormal conditions.

Simulation results

A numerical simulation example will be provided to illustrate the feasibility and effectiveness of the presented approach. Consider the following system

$$\{\begin{array}{c}{\stackrel{·}{x}}_1=x_1+x_2+{\sin }^2(x_1)x_1\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_2=B^{\mathrm{T}}{u}^{\mathrm{F}}(t)-2.5x_2+x_1{x}_2^2}\\ \multicolumn{1}{c}{y=x_1}\end{array}$$ (64)

According to Theorem 1, the adaptive fuzzy finite-time FTC for equation (64) is devised.

It is necessary to select some appropriate parameters to design the controller in the process of the proposed controller implementation. It is found that the choice of the design parameters has great influence on system performance. For example, for the design of parameter $k_i$ , if it is too large or too small, the system could oscillate and even diverge. Furthermore, if design parameter $a_i$ is too small, it will produce a large control output. Accordingly, other design parameters such as $r_i\mathrm{and}{\sigma }_i$ also needed to be regulated appropriately.

During the simulation process, using the trial and error approach, the design parameters are chosen as follows: $k_1=k_2=60,a_1=a_2=0.01$ , $r_1=r_2=150$ , ${\sigma }_1={\sigma }_2=0.8$ . Other parameters for the proposed controller are selected as $y_r=0.5\sin (t)+0.5\sin (2t)$ as the desired trajectory, and the control gain vector is $B^{\mathrm{T}}=[\begin{array}{cc}1\hfill & 0.5\hfill \end{array}].$ Choose the boundary constraint function as $\rho (t)=(0.4-0.03)e^{-t}+0.03$ ; the initial conditions are $x(0)=[0.3,0.2{]}^{\mathrm{T}}$ , and ${\hat{\theta }}_1(0)={\hat{\theta }}_2(0)=0.1$ .

Assume that the possible actuator faults occur after $t=10\mathrm{s}$ . The additive actuator fault that occurs in the control channel is selected as $\varphi (t)=[\begin{array}{cc}0.1\sin (0.3t)\hfill & 0.15\cos (0.6t)\hfill \end{array}{]}^{\mathrm{T}},$ and the actuator efficiency factors are given as follows

$${\beta }_1(t)=\{\begin{array}{c}1,0⩽t<10\hfill \\ \multicolumn{1}{c}{0,10⩽t<11}\\ \multicolumn{1}{c}{0.3,11⩽t<15}\\ \multicolumn{1}{c}{0.5,15⩽t<20}\\ \multicolumn{1}{c}{0.6,t⩾20}\end{array}$$

$${\beta }_2(t)=\{\begin{array}{c}1,0⩽t<10\hfill \\ \multicolumn{1}{c}{0.01,10⩽t<11}\\ \multicolumn{1}{c}{0.2,11⩽t<15}\\ \multicolumn{1}{c}{0.6,15⩽t<20}\\ \multicolumn{1}{c}{0.85,t⩾20}\end{array}$$

It is noted that, when ${\beta }_i(t)=1,$ there is no fault. ${\beta }_i(t)=0$ means the $i\mathrm{th}$ actuator is outage, and $0<{\beta }_i(t)<1$ represents that the actuator $u_i$ loses its effectiveness.

Next, we choose seven fuzzy sets stated during the interval $[-3,+3]$ , and the fuzzy membership functions are given by

$$r_j(Z_i)=\exp [-0.5{(Z_i+4-j)}^2],j=1,2,\dots ,7$$

The simulation response curves are shown in Figures 1–7.

Figure 1.

Trajectories of y and $y_r$ of the proposed method.

Figure 2.

Trajectories of the tracking error of the proposed method without actuators failures.

Figure 3.

Trajectories of the tracking error with actuators failures without funnel control.

Figure 4.

Trajectories of the tracking error of the proposed method with actuators failures.

Figure 5.

Trajectories of $x_2$ of the proposed method.

Figure 6.

Trajectory of variable ${\hat{\theta }}_1$ .

Figure 7.

Trajectory of control input $u_1$ .

The trajectories of the output signal using the proposed method are exhibited in Figure 1. It is obviously shown that the system possesses the superior output tracking performance within finite time. Figure 2 shows the tracking error of the proposed method in the absence of failure. In order to further verify the effectiveness of the design method, the simulation curves of the tracking error with actuators failures with/withiout the proposed method are illustrated in Figures 3 and 4. Apparently, it is found that the tracking error can be limited to the predefined area whether the actuators failures occur or not using the proposed method in Figure 4. Figures 5 and 6 present the trajectories of $x_2$ and ${\hat{\theta }}_1$ , which show apparently that the state $x_2$ and ${\hat{\theta }}_1$ are bounded stable in finite time. Figures 7 and 8 show the trajectories of $u_1$ and $u_2$ , respectively.

Figure 8.

Trajectory of control input $u_2$ .

It can be derived from the above simulation results that the system is SGPFS, at the meantime, the tacking error can be confined to the prescribed region of the funnel boundary functions no matter if the actuators is in normal or abnormal conditions. Therefore, it should be pointed out that the transient-state and steady-state performance of the system with actuators failures has been greatly improved using the proposed method.

Conclusion

A new method of the adaptive fuzzy finite-time fault-tolerant FC has been put forward for nonlinear systems with actuators failures. Using backstepping technology, the new adaptive fuzzy finite-time FTC strategy is investigated, which can maintain that the system is SGPFS, in the meantime, the output tracking error can be confined to the predefined region within finite-time no matter if the actuators are in normal or abnormal circumstances. It is noted that the proposed method relies on the complete information about the states of the system based on the premise that the system states are completely measurable. If this restriction is removed, the output feedback control based on the proposed observer can solve the problem of non-measurable states. Next, we are embarking on considering this problem within the framework of the existing research findings.