An Intelligent Fault-Tolerant Control Method for a Flexible-Link Manipulator with an Uncertain Dead-Zone and Intermittent Actuator Faults

Abstract

In this article, a new intelligent fault-tolerant control (FTC) is designed to control a flexible-link manipulator with uncertain dead-zone and intermittent actuator faults. Initially, a smooth dead-zone inverse model using a hyperbolic tangent function is introduced to handle dead-zone nonlinearity and suppress input chattering. An adaptive law is proposed to estimate an unknown coupling item, combining the upper bounds of compensation error and floating bias faults, achieving robust adaptive control of the system. A new FTC strategy is subsequently developed to address intermittent actuator faults. Finally, the bounded convergence of system state errors is proven using direct Lyapunov methods, and the effectiveness and superiority of the proposed controller are demonstrated through numerical simulation and experiment.

1. Introduction

With the rapid development of technology, modern industry is gradually transitioning to lighter and more flexible materials to accommodate the increasing complexity of work environments [1,2]. As a typical flexible structural system, a flexible-link manipulator is widely recognized for its outstanding flexibility, lightweight properties, and low energy consumption [3,4]. However, under harsh environmental conditions or heavy workloads, a flexible-link manipulator often encounters issues with vibrations and deformations [5,6,7]. These issues can reduce work efficiency, diminish production accuracy, and even adversely affect the system’s overall performance. Therefore, developing an effective vibration control strategy to steady the operation of a flexible-link manipulator is particularly crucial.

Since a flexible-link manipulator is part of a flexible structure system, it is also considered a typical distributed parameter system (DPS) with infinite-dimensional dynamic modes [8,9]. During the design process of the controller, modal simplification may lead to spillover effects [10,11]. Although distributed control can be implemented by installing multiple controllers on a single-link flexible manipulator to address this challenge, this method is costly and complex in design [12]. Therefore, many researchers have turned to boundary control (BC), which is regarded as a more efficient control approach. BC only requires interventions at the system’s boundaries, effectively managing the entire system’s dynamic behavior. This not only reduces the complexity of design and implementation but also helps to enhance the overall control efficiency of the system [13,14,15]. In [16], a new BC approach was presented to solve the angular position tracking problems of flexible-link manipulator systems. In [17], the authors designed a new BC scheme for active vibration suppression in flexible manipulator systems. However, these researches were restricted to vibration suppression or angular position tracking, and the methods used did not apply to flexible-link manipulator systems with input dead-zone nonlinearity.

Dead-zone nonlinearity is commonly found in the actuators of physical control systems, including motors, valves, and hydraulic systems [18,19]. Input dead-zones may decrease system performance, particularly in applications requiring high-precision control, potentially causing phase delays and inaccurate control. Therefore, effectively compensating for input dead-zones is the key to enhancing system performance. Recently, researchers have presented various approaches to compensate for the unknown dead-zone constraints in actuators. In [20], a method was proposed for high-performance motion control of a dual-valve hydraulic system experiencing an unknown input dead-zone. In [21], an improved prescribed performance-based adaptive control strategy was designed for uncertain nonlinear systems against the input dead-zone. Moreover, adaptive inverse compensation is an effective approach for addressing the input dead-zone [22]. In [23], a control algorithm using a dead-zone inverse dynamics model was introduced for managing sandwich nonsmooth nonlinear systems. However, as discussed in [23], this dynamics mode is nonsmooth and may induce chattering in the compensated control inputs. This can potentially reduce the lifespan of actuators and might even cause instability in the control system. To address this issue, a new inverse compensation control scheme was proposed for a nonlinear system subject to the input dead-zone [24]. However, it is worth mentioning that the above studies focused on adaptive inverse control of finite-dimensional systems with an input dead-zone and could not be directly applied to the control of infinite-dimensional systems with dead-zone constraints.

Due to the complexity and multiple interferences of the working environment, actuator faults inevitably occur in various industrial controls. In severe cases, sudden sticking or partial loss of effectiveness of an actuator may lead to system instability and even production accidents. FTC is considered an effective method to address actuator faults [25,26]. In recent years, FTC research based on DPSs has made significant progress. In [27], a new reinforcement learning strategy was proposed for interconnected nonlinear systems with actuator faults. In [28], a fault-tolerant controller was adopted for full control of a quadrotor UAV with motor faults. The FTC above schemes consider a finite number of actuator faults, which cannot be directly used to handle an infinite number of actuator faults. In actual industrial processes, actuator faults may occur more than once in a controlled system [29]. To address this problem, this study conducts research on adaptive FTC for flexible-link manipulator systems experiencing uncertain dead-zone and intermittent actuator faults.

In summary, the objective of this study is to design a new control method. This method aims to enhance the system’s intelligence, enabling it to adaptively adjust its parameters in the face of unknown actuator dead-zones and intermittent faults. It addresses actuator nonlinearity and effectively suppresses system vibrations. The main contributions of this study are as follows:

(1)

As opposed to the previous study [30], this study introduces a smooth dead-zone inverse dynamics model that effectively compensates for the input dead-zone effect in a flexible-link manipulator and reduces actuator chattering.

(2)

A new FTC is proposed to address intermittent actuator faults in the flexible-link manipulator system. Unlike previous studies [31,32], the fault parameters in this study are time-varying, and the faults occur frequently, making this method more applicable to actual situations.

(3)

The established control method can suppress the vibration of the controlled system without the need to simplify or discretize infinite-dimensional system dynamics, and the effectiveness of the proposed controller is verified through numerical simulation and experiment.

Notation 1.

The symbols used in this study adhere to standard conventions. For the sake of conciseness, the following notations are defined: $\mathbb{R}$ denotes a collection of real numbers, $(★)=(★)(\nu ,t)$, $\dot{(★)}$ is the partial derivative of ★ with respect to t, ${(★)}^{\prime }$ is the partial derivative of ★ with respect to ν, ${\dot{(★)}}^{\prime }$ is the second partial derivative of ★ with respect to both ν and t, ${(★)}^{″}$ is the second partial derivative of ★ with respect to ν, ${(★)}^{‴}$ is the third partial derivative of ★ with respect to ν, ${(★)}^{⁗}$ is the fourth partial derivative of ★ with respect to ν, $\ddot{(★)}$ is the second partial derivative of ★ with respect to t, ${(★)}_l=(★)(l,t)$, and ${(★)}_0=(★)(0,t)$. $\mathbb{R}$ represents all real numbers. $Q\succ 0$ represents a symmetric positive definite matrix. ${\lambda }_{\max }\left(Q\right)$ represents the maximum eigenvalues of the symmetric matrix Q.

2. Problem Statement

The flexible-link manipulator system is described in Figure 1, where $\nu Op$ and $VOC$ represent the local rotating reference coordinate system connected to the joint and the global inertial coordinate system, respectively. Additionally, $\nu \in [0,l]$ is the spatial variable, and $t\in {\mathbb{R}}^{+}$ is the time variable. $c(\nu ,t)$, $\varpi \left(t\right)$, and $\xi \left(t\right)$ represent the elastic deflection, torque input, and angular position of the joint, respectively. The total displacement of the link in the coordinate system $VOP$ is denoted by $p(\nu ,t)$, where $p(\nu ,t)=\nu \theta \left(t\right)+c(\nu ,t)$.

First, we consider the dynamical model of the system as follows [33]:

$$EI{c}^{⁗}(\nu ,t)=T{c}^{″}(\nu ,t)-m\ddot{p}(\nu ,t)-C\dot{p}(\nu ,t),$$

(1)

$$\begin{array}{c}\hfill EI{c}^{‴}(l,t)-T{c}^{\prime }(l,t)=0,\end{array}$$

(2)

$$\begin{array}{cc}\hfill c(0,t)=c^{\prime }(0,t)& =c^{″}(l,t)=0,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill I\ddot{\xi }\left(t\right)-\varpi \left(t\right)=EI& c^{″}(0,t)+Tc(l,t),\hfill \end{array}$$

(4)

where I, $EI$, m, T, and C denote the inertia of the joint, bending stiffness, linear density, tension, and the coefficient of viscosity, respectively.

Remark 1.

In this study, we consider a flexible-link manipulator model characterized by a large ratio of length to cross-sectional diameter. Compared to its elastic deformation, the shear deformation induced by vibrations in this model is relatively negligible, allowing the link to be described by the classic Euler–Bernoulli beam. In recent years, many researchers have conducted control studies based on this model and have experimentally verified its effectiveness [33,34]. Therefore, this study also conducts control research based on this model (1)–(4).

2.1. Deadzone Characteristic Analysis

In this study, we account for the fact that the actuator input of the system is influenced by a dead-zone, which is represented as follows:

$$\begin{array}{c}\hfill U\left(t\right)=D\left(u\left(t\right)\right)=\left\{\begin{array}{cc}{d}_r(u\left(t\right)-{\varrho }_r),\hfill & \mathrm{if}u\left(t\right)\ge d_r\hfill \\ 0,\hfill & \mathrm{if}{d}_l<u\left(t\right)<d_r\hfill \\ d_l(u\left(t\right)-{\varrho }_l),\hfill & \mathrm{if}u\left(t\right)\le d_l\hfill \end{array}\right.,\end{array}$$

(5)

where $d_r>0$ and $d_l>0$ denote the slopes of the respective linear segments, with $|d_r|\ne |d_l|$. Here, ${\varrho }_r>0$ and ${\varrho }_l<0$ are the breakpoints with non-equal absolute values. $u\left(t\right)$ denotes the desired actuator input, while $U\left(t\right)$ represents the actual actuator output subject to dead-zone effects. Following this, a dead-zone inverse dynamics model is presented [24]:

$$\begin{array}{c}\hfill u\left(t\right)=\frac{U+d_r{\varrho }_r}{d_r}{\Sigma }_r\left(U\right)+\frac{U+d_l{\varrho }_l}{d_l}{\Sigma }_l\left(U\right),\end{array}$$

(6)

where smooth functions ${\Sigma }_r\left(U\right)$ and ${\Sigma }_l\left(U\right)$ are defined as follows:

$$\begin{array}{c}\hfill {\Sigma }_r\left(U\right)=\frac{1+tanh\left({\rho }_{h}U\right)}{2},{\Sigma }_l\left(U\right)=\frac{1-tanh\left({\rho }_{h}U\right)}{2},\end{array}$$

(7)

with ${\rho }_h>0$, and $tanh(·)$ denotes hyperbolic tangent function.

To facilitate the control design, we parameterize the dead-zone model as described in (5):

$$\begin{array}{c}\hfill U\left(t\right)={\sigma }^T\zeta ,\end{array}$$

(8)

where $\sigma ={[d_r,d_r{\varrho }_r,d_l,d_l{\varrho }_l]}^T$, and $\zeta ={[{\sigma }_r\left(t\right)u\left(t\right),-{\sigma }_r\left(t\right),{\sigma }_l\left(t\right)u\left(t\right),-{\sigma }_l\left(t\right)]}^T$. ${\sigma }_r$ and ${\sigma }_l$ are as follows:

$${\sigma }_r=\left\{\begin{array}{cc}1,& \mathrm{if}U\left(t\right)>0\\ 0,& \mathrm{otherwise},\end{array}\right.{\sigma }_l=\left\{\begin{array}{cc}1,& \mathrm{if}U\left(t\right)<0\\ 0,& \mathrm{otherwise}.\end{array}\right.$$

(9)

To manage uncertainties in dead-zone parameters, we implement the following inverse dynamics model:

$$\begin{array}{c}\hfill u\left(t\right)=\frac{U_d+\widehat{d_r{\varrho }_r}}{{\widehat{d}}_r}{\Sigma }_r\left(U_d\right)+\frac{U_d+\widehat{d_l{\varrho }_l}}{{\widehat{d}}_l}{\Sigma }_l\left(U_d\right),\end{array}$$

(10)

where $U_d$ is the control signal to be designed. Furthermore, $\widehat{\sigma }$ is an estimation of $\sigma$, and $\widehat{\sigma }$ and $\widehat{\zeta }$ are defined as follows:

$$\widehat{\sigma }={[{\widehat{d}}_r,\widehat{d_r{\varrho }_r},{\widehat{d}}_l,\widehat{d_l{\varrho }_l}]}^T,$$

(11)

$$\widehat{\zeta }={[{\Sigma }_r\left(u\right)u\left(t\right),-{\Sigma }_r\left(u\right),{\Sigma }_l\left(u\right)u\left(t\right),-{\Sigma }_l\left(u\right)]}^T.$$

(12)

Invoking (10)–(12), the expressions of $u_{fd}$ is given as follows:

$$U_d\left(t\right)={\widehat{\sigma }}^T\widehat{\zeta }.$$

(13)

Additionally, the compensation error can be expressed as follows:

$$U-U_d={(\sigma -\widehat{\sigma })}^T\widehat{\zeta }\left(t\right)+d\left(t\right),$$

(14)

where $d\left(t\right)={\sigma }^T(\zeta -\widehat{\zeta })$, and by referencing [24], we obtain that $\left|d\right(t\left)\right|\le D$, which is attainable as follows:

$$\begin{array}{c}\hfill D=\left\{\begin{array}{cc}\frac{1}{2}{e}^{-1}|d_r-d_l|/{\rho }_h+\frac{|d_r{\varrho }_r-d_l{\varrho }_l|}{e^{2{\varrho }_r/{\rho }_h}+1},\hfill & \mathrm{if}u\left(t\right)\ge {\varrho }_r\hfill \\ \frac{1}{2}{e}^{-1}|d_r-d_l|/{\rho }_h+\frac{|d_r{\varrho }_r-d_l{\varrho }_l|}{e^{-2{\varrho }_l/{\rho }_h}+1},\hfill & \mathrm{if}u\left(t\right)\le {\varrho }_l\hfill \\ max\left\{d_r,d_l\right\}|{\varrho }_r-{\varrho }_l|\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

(15)

Remark 2.

When ${\varrho }_l<u\left(t\right)<{\varrho }_r$, the variable $d\left(t\right)$ remains bounded. As v extends beyond the interval $({\varrho }_l,{\varrho }_r)$, the boundary of $d\left(t\right)$ diminishes with increasing ${\rho }_h$. Consequently, $d\left(t\right)$ remains bounded for all $t\ge 0$ and converges to zero as ${\rho }_h$ approaches infinity.

Remark 3.

Traditional methods for handling unknown dead-zones typically require identifying some or all dead-zone parameters before designing a controller for compensation [35]. However, the dead-zone inverse method used in this study addresses unknown dead-zones without parameter identification and avoids severe chattering in compensated control inputs, thus enhancing actuator lifespan and system stability [24]. Nevertheless, this model generates compensation errors $d\left(t\right)$. If these errors are not managed, the actual input may deviate significantly from the desired input, potentially affecting system stability.

Remark 4.

In [24], a new scheme was presented for designing adaptive inverse compensation controllers for uncertain systems affected by unknown dead-zone nonlinearity. However, the system studied in [24] is finite-dimensional, described using ordinary differential equations. In contrast, the flexible-link manipulator system examined in this study is infinite-dimensional and modeled using partial differential equations. As a result, the control strategy from [24] cannot be directly applied to the systems discussed in this study. Implementing the strategy proposed in [24] would necessitate reducing the infinite-dimensional dynamics (1)–(4) to a finite-dimensional model, potentially leading to spillover effects. Additionally, this study also considers intermittent actuator faults within the system, further increasing the complexity of the control design and enhancing its challenges.

2.2. Intermittent Actuator Fault Model

The intermittent actuator fault model is characterized as follows [36]:

$$\begin{array}{c}\hfill \varpi \left(t\right)=\varphi \left(t\right)U\left(t\right)+U_{f,j}\left(t\right)\varphi \left(t\right)={\varphi }_j,\end{array}$$

(16)

where j represents the fault instance, and t lies within the interval $[t_{j,s},t_{j,e})$. The parameter ${\varphi }_j$ denotes an unknown efficiency factor, constrained by $0<\underline{\varphi }\le {\varphi }_j\le 1$. It is noted that $U_{f,j}\left(t\right)\le {\overline{U}}_f$, where both $\underline{\varphi }$ and ${\overline{U}}_f$ are unknown constants. (16) describes a scenario in which the actuator loses $(1-{\varphi }_j)\times 100\%$ of its effectiveness and introduces a floating bias fault $U_{f,j}\left(t\right)$ that persists from time $t_{j,s}$ to $t_{j,e}$. When $\varphi \left(t\right)={\varphi }_j$, the system aligns with a standard fault model, which accounts for both partial loss of efficiency and the floating bias fault, as discussed in [37]. This formulation does not consider fault recurrence. Consequently, when combined with (14), the actual input $\varpi \left(t\right)$ can be described as follows:

$$\begin{array}{c}\hfill \varpi \left(t\right)=\varphi \left(t\right)U_d\left(t\right)+\varphi \left(t\right){\tilde{\sigma }}^T\widehat{\zeta }+\varphi \left(t\right)d\left(t\right)+U_{f,j}\left(t\right).\end{array}$$

(17)

2.3. Preliminaries

To support the forthcoming analysis, the subsequent assumptions and lemmas are presented.

Assumption 1.

For the dead-zone parameters $d_r$ and $d_l$, both are positive constants with $d_r\ge d_r>0$, and $d_l\ge d_l>0$ is satisfied.

Lemma 1

([38]). Consider the first-order differential equations as follows:

$$\begin{array}{c}\hfill \dot{q}\left(t\right)={\alpha }_q{f}_q-{\alpha }_q{\mu }_q(q\left(t\right)-q\left(0\right)),\end{array}$$

(18)

where ${\alpha }_q$ and ${\mu }_q$ are positive constants. If $f_q,q\left(0\right)\ge 0$, the solution of (18) is rendered to be non-negative for all $t\ge 0$.

Lemma 2

([39]). If $q_1$ and r are scalars and satisfy $q_1\in \mathbb{R}$, $r\in {\mathbb{R}}^{+}$, we have

$$\begin{array}{c}\hfill 0\le |q_1|-\frac{r^2}{\sqrt{q_1^2+r^2}}<\overline{r}.\end{array}$$

(19)

3. Control Design

This section proposes an adaptive FTC to meet the control objectives. This approach integrates compensation for unknown errors with adaptation to time-varying actuator faults, introducing an upper-bound adaptive law. A control block diagram is shown in Figure 2.

3.1. Adaptive FTC

We construct the auxiliary signal $U_b\left(t\right)=\eta \dot{\xi }\left(t\right)+\delta [\xi \left(t\right)-{\xi }_d]=\eta \dot{\theta }\left(t\right)+\delta {\xi }_e\left(t\right)$, and ${\xi }_d$ denotes the reference signal. Then, the control signal $U_d\left(t\right)$ is designed as follows:

$$\begin{array}{c}\hfill U_d\left(t\right)=-\widehat{\omega }\theta \left(t\right),\end{array}$$

(20)

where $\theta \left(t\right)=-\iota \left(t\right)U_b\left(t\right)$. Let $\omega =1/\underline{\varphi }$ and $\widehat{\omega }$ represent the estimated value of $\omega$. $\iota$ is given as follows:

$$\begin{array}{cc}\hfill \iota \left(t\right)=& k+\frac{1}{4\gamma }\left[T{c}_l^2+(k_1+I\delta ){\dot{\xi }}^2\left(t\right)+k_{\xi }{\xi }_e^2\left(t\right)+\frac{{\widehat{\aleph }}^2}{\sqrt{U_b^2\left(t\right){\widehat{\aleph }}^2+r^2}}\right],\hfill \end{array}$$

(21)

where k, $k_{\xi }$, and $\delta$ are positive constants, with $k_1\in \mathbb{R}$. $\widehat{\aleph }$ represents the estimation of ℵ, and ℵ is denoted as $\aleph =D+{\overline{U}}_f$. Then, the parameter updating laws are designed as follows:

$$\dot{\widehat{\omega }}={\phi }_{\omega }{\iota }_1\left(t\right)U_b^2\left(t\right)-{\phi }_{\omega }\widehat{\omega },$$

(22)

$$\dot{\widehat{\aleph }}={\phi }_{\aleph }{U}_b\left(t\right)-{\phi }_{\aleph }\widehat{\aleph },$$

(23)

$$\dot{\widehat{\sigma }}=-{\mathrm{Proj}}_{\sigma }\left\{{\Gamma }_{\sigma }{U}_b\left(t\right)\widehat{\zeta }\right\},$$

(24)

where ${\phi }_{\omega }$ and ${\phi }_{\aleph }$ are positive constants. ${\Gamma }_{\sigma }$ represents a symmetric positive definite matrix. To avoid singularity, making the control inputs infinite, projection mapping Proj$\left\{\circ \right\}$ is used to limit the range of adaptive parameters $\widehat{\sigma }$, as defined in [40].

Remark 5.

It is important to note that the FTC strategies outlined in [27,28,40] are not directly applicable to this system. If the actuator undergoes multiple abrupt changes, the parameter $\dot{\varphi }\left(t\right)$ could become unbounded, potentially leading to system instability [38]. To prevent the occurrence of the unbounded term $\frac{\underline{\varphi }}{{\phi }_{\omega }}\tilde{\omega }(\dot{\omega }-\dot{\widehat{\omega }})$ in the time derivative of subsequent Lyapunov functions $\dot{Z}\left(t\right)$, we introduce a new FTC approach. This strategy, distinct from those previously cited, incorporates quadratic nonlinear damping terms θ and ι, allowing us to effectively address this issue.

3.2. Stability Analysis

Consider the Lyapunov function candidate as

$$Z\left(t\right)=Z_a\left(t\right)+Z_b\left(t\right)+Z_c\left(t\right),$$

(25)

with

$$Z_a\left(t\right)=\frac{\eta }{2}m{\int }_0^l{\dot{p}}^{2}d\nu +\frac{\eta }{2}EI{\int }_0^l{c^{″}}^{2}d\nu +\frac{\eta }{2}T{\int }_0^l{c^{\prime }}^{2}d\nu ,$$

(26)

$$Z_b\left(t\right)=\frac{1}{2}I{U}_b^2+\frac{1}{2}{k}_{\xi }{\xi }_e^2+\frac{\underline{\varphi }}{2{\phi }_{\omega }}{\tilde{\omega }}^2+\frac{1}{2{\phi }_{\aleph }}{\tilde{\aleph }}^2+\frac{1}{2}{\tilde{\sigma }}^T{\Gamma }_{\sigma }^{-1}\tilde{\sigma },$$

(27)

$$Z_c\left(t\right)=\delta m{\int }_0^l\dot{p}(y,t)p_e(\nu ,t)d\nu ,$$

(28)

where $p_e(\nu ,t)=c(\nu ,t)+\nu {\xi }_e\left(t\right)$. Obviously, ${\dot{p}}_e(\nu ,t)=\dot{p}(\nu ,t)$.

Lemma 3.

The selected Lyapunov function (25) satisfies the following property:

$$0\le {\kappa }_2[Z_a\left(t\right)+Z_b\left(t\right)]\le Z\left(t\right)\le {\kappa }_3[Z_a\left(t\right)+Z_b\left(t\right)],$$

(29)

where ${\kappa }_2=1-{\kappa }_1$ and ${\kappa }_3=1+{\kappa }_1$ are two positive constants with ${\kappa }_1=\max \{\frac{\delta (1+l)}{{\chi }_1\eta },\frac{\delta m{l}^2{\chi }_1}{k_{\xi }\eta },\frac{\delta m{l}^2{\chi }_1}{T\eta }\}$.

Proof. 

See Appendix A. □

Lemma 4.

The time derivative of the Lyapunov function (25) is proved to be bounded as follows:

$$\dot{Z}\left(t\right)\le -\kappa Z\left(t\right)+\tau ,$$

(30)

where $\kappa ,\tau \in {\mathbb{R}}^{+}$.

Proof. 

See Appendix B. □

Theorem 1.

In this study, the convergence of system state errors is defined as uniformly bounded convergence, meaning that the state errors will ultimately converge within a small and acceptable error range [41]. This is sufficient for many control systems in practical applications. After completing the analysis of Lemmas 3 and 4, we can further demonstrate the convergence of system state errors. Since we desire the system to remain vibration-free, the deformation error is expressed as $c_e(\nu ,t)=c(\nu ,t)-0=c(\nu ,t)$. The specific convergence bounds can be described as follows:

(1) The elastic displacement $c_e(\nu ,t)$ is uniformly bounded and starts in a compact set ${\vartheta }_1$ and converges eventually to another compact set ${\vartheta }_2$.

$$\begin{array}{cc}\hfill & {\vartheta }_1:=\{c_e(\nu ,t)\in \mathbb{R}|\mid c_e(\nu ,t)\mid \le {\Lambda }_1\}\hfill \\ \hfill & {\vartheta }_2:=\{c_e(\nu ,t)\in \mathbb{R}|\underset{t\to \infty }{lim}\mid c_e(\nu ,t)\mid \le {\Lambda }_2\},\hfill \end{array}$$

(31)

where ${\Lambda }_1=\sqrt{\frac{2l}{\eta T{\kappa }_1}[V\left(0\right)+\frac{\tau }{\kappa }]}$ and ${\Lambda }_2=\sqrt{\frac{2l\tau }{\eta T{\kappa }_1\kappa }}$.

(2) The angular error ${\xi }_e\left(t\right)$ is maintained in the compact set ${\vartheta }_3$ and finally converges to the compact set ${\vartheta }_4$.

$$\begin{array}{cc}\hfill & {\vartheta }_3:=\{{\xi }_e\left(t\right)\in \mathbb{R}|\mid {\xi }_e\left(t\right)\mid \le {\Lambda }_3\}\hfill \\ \hfill & {\vartheta }_4:=\{{\xi }_e\left(t\right)\in \mathbb{R}|\underset{t\to \infty }{lim}\mid {\xi }_e\left(t\right)\mid \le {\Lambda }_4\},\hfill \end{array}$$

(32)

where ${\Lambda }_3=\sqrt{\frac{2l}{k_{\xi }{\kappa }_1}[V\left(0\right)+\frac{\tau }{\kappa }]}$ and ${\Lambda }_4=\sqrt{\frac{2l\tau }{k_{\xi }{\kappa }_1\kappa }}$.

Proof of Theorem 1.

Multiplying (30) by $e^{\kappa t}$ yields

$$\begin{array}{c}\hfill \frac{\partial }{\partial t}\left[Z\left(t\right)e^{\kappa t}\right]\le {\tau }^{\kappa t}.\end{array}$$

(33)

Substituting (33), we obtain the following:

$$\begin{array}{c}\hfill Z\left(t\right)\le [Z\left(0\right)-\frac{\tau }{\kappa }]e^{-\kappa t}+\frac{\tau }{\kappa }.\end{array}$$

(34)

Using (26) and Sobolev’s inequality (see Lemma 2 in [42]) yields

$$\begin{array}{c}\hfill \frac{1}{2l}\eta T{c}^2(\nu ,t)\le \frac{\eta }{2}T{\int }_0^l{c^{\prime }}^2(\nu ,t)d\nu \le Z_a\left(t\right)\le \frac{1}{{\kappa }_1}Z\left(t\right).\end{array}$$

(35)

Invoking (35) and (34) is rewritten as follows:

$$\begin{array}{cc}\hfill \left|c\right(\nu ,t\left)\right|& \le {\Lambda }_1=\sqrt{\frac{2l}{\eta T{\kappa }_1}[Z\left(0\right)+\frac{\tau }{\kappa }]}.\hfill \end{array}$$

(36)

Further, we have

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}|c_e(\nu ,t)|=\underset{t\to \infty }{lim}\left|c(\nu ,t)\right|\le {\Lambda }_2=\sqrt{\frac{2l\tau }{\eta T{\kappa }_1\kappa }},\forall \nu \in [0,l].\end{array}$$

(37)

Similarly, from (27), we obtain

$$\begin{array}{c}\hfill |{\xi }_e\left(t\right)|\le {\Lambda }_3=\sqrt{\frac{2l}{k_{\xi }{\kappa }_1}[Z\left(0\right)+\frac{\tau }{\kappa }]},\forall t\in [0,+\infty ).\end{array}$$

(38)

Then, we arrive at

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\left|{\xi }_e\left(t\right)\right|\le {\Lambda }_4=\sqrt{\frac{2l\tau }{k_{\xi }{\kappa }_1\kappa }},\forall t\in [0,+\infty ).\end{array}$$

(39)

The proof is complete. □

Remark 6.

In the process of proofing stability, we derived some constraint conditions (A11)–(A13) that the parameter settings must satisfy. We first select an appropriate set of parameters, η, β, and ${\chi }_3$, to ensure that (A11) holds. Then, we further adjust k, $k_1$, γ, ${\chi }_2$, and $k_{\xi }$ to meet the inequalities in (A12). After selecting the relevant parameters, it is clear that (A13) is satisfied. Finally, during the simulation’s parameter adjustment, we appropriately tune the values of η, β, k, $k_1$, γ, and $k_{\xi }$ until excellent transient and steady-state performance is achieved.

4. Numerical Simulation

To assess the effectiveness of the proposed control strategy, numerical simulations are conducted using the finite difference approach. The selected system parameters are outlined below: $EI=0.157{\mathrm{Nm}}^2$, $T=0.01\mathrm{N}$, $l=0.419\mathrm{m}$, $m=0.1\mathrm{kg}/\mathrm{m}$, $C=0.04\mathrm{NS}/\mathrm{m}$, and $I=0.0038{\mathrm{kgm}}^2$. The simulation employs time and space increments of $△t=3.75\times {10}^{-5}\mathrm{s}$ and $△\nu =0.0221\mathrm{m}$, respectively. The target trajectory ${\xi }_d$ is set to $\frac{\pi }{4}$. Initially, the system conditions are $c(\nu ,0)=0.5{\nu }^2\mathrm{m}$, $\dot{c}(\nu ,0)=0\mathrm{m}/\mathrm{s}$, $\xi \left(0\right)=0\mathrm{rad}$, and $\dot{\xi }\left(0\right)=0\mathrm{rad}/\mathrm{s}$. We consider a flexible-link manipulator with intermittent actuator faults. The intermittent actuator faults are given as follows: [38]:

$$\varpi \left(t\right)=\left\{\begin{array}{cc}0.5U\left(t\right)+0.1e^{-0.2t}\hfill & \mathrm{if}t\in (o-1,o),\hfill \\ U\left(t\right)\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(40)

where o is an odd positive number.

The parameters specified for the dead-zone inverse fault-tolerant controller (DIFTC) include $\eta =1$, $\delta =0.05$, $\gamma =0.8$, $k_{\xi }=4.5$, and $r=0.17$, and control gains $k=1$ and $k_1=0.2$. The parameter updating laws are defined with ${\phi }_{\aleph }=0.1$, ${\phi }_{\omega }=0.1$, initial estimates $\widehat{\sigma }\left(0\right)={[1.1,0.22,1.2,-0.23]}^T$, $\widehat{\aleph }\left(0\right)=0.3$, and $\widehat{\omega }\left(0\right)=1$. The feasible set for these parameters is $\Omega =\{\widehat{\sigma }\in {\mathbb{R}}^{4\times 1}|0.8\le {\widehat{d}}_r<1.2,0.17\le \widehat{d_r{\varrho }_r}<0.23,0.8\le {\widehat{d}}_l<1.2,-0.25\le \widehat{d_l{\varrho }_l}<-0.18\}$. The dead-zone parameters set are $d_r=1$, $d_l=1.1$, ${\varrho }_r=0.2$, and ${\varrho }_l=-0.2$. To further validate the effectiveness of the proposed control strategy, we provide simulation results employing the proportional differentiation controller (PDC), the BC discussed in [33], and the FTC discussed in [40].

The PDC is expressed as follows:

$$\begin{array}{c}\hfill \varpi \left(t\right)=-k_d\dot{\xi }\left(t\right)-k_p{\xi }_e\left(t\right),\end{array}$$

(41)

where $k_d=2$ and $k_p=8$.

The BC in [33] is given as follows:

$$\begin{array}{c}\hfill \varpi \left(t\right)=-k{U}_b\left(t\right)-(k_1+T)c_l-I\delta \dot{\xi }\left(t\right)-k_{\xi }{\xi }_e\left(t\right),\end{array}$$

(42)

where $k=0.09$, $k_1=0.1$, $\delta =0.05$, and $k_{\xi }=5$.

The FTC in [40] is given as follows:

$$\begin{array}{cc}\hfill \varpi \left(t\right)& =-\widehat{\omega }(k{U}_b\left(t\right)+(k_1+T)c_l+I\delta \dot{\xi }\left(t\right)+k_{\xi }{\xi }_e\left(t\right)),\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill \dot{\widehat{\omega }}& ={\phi }_{\omega }{U}_b\left(t\right)(k{U}_b\left(t\right)+(k_1+T)c_l+I\delta \dot{\xi }\left(t\right)+k_{\xi }{\xi }_e\left(t\right))-{\phi }_{\omega }\widehat{\omega }.\hfill \end{array}$$

(44)

As shown in Figure 3, Figure 4 and Figure 5, the deflection $c(\nu ,t)$ of the entire flexible-link under DIFTC control approaches zero within 1.5 s, whereas the deflection converges more slowly under PDC and BC control. From Figure 6, it can be observed that around 2 s, the deflection of the system begins to increase, indicating that traditional FTC is unable to handle intermittent actuator faults. The specific reason for this is provided in Remark 5.

Observations from Figure 7 and Figure 8 reveal that despite challenges such as input dead-zones and intermittent actuator faults, the DIFTC consistently achieves excellent control performance. Compared to BC and PDC, the proposed controller demonstrates a superior ability to rapidly follow the reference signal ${\xi }_d$ while ensuring system stability. Additionally, under the influence of DIFTC, the endpoint displacement $c_l$ of the flexible-link manipulator approaches zero within 1.5 s. Although the displacement $c_l$ also converges under BC and PDC, these controls exhibit slower convergence. As shown in Figure 9, the system state ${\xi }_e$ can be adjusted through parameter tuning to bring the error close to zero, whereas the comparative controllers cannot reduce ${\xi }_e$ to zero, leaving a steady-state error. This proves the superiority of the designed controller. As shown in Figure 7, Figure 8 and Figure 9, it can be observed that systems under traditional FTC are prone to instability. This not only confirms the descriptions in Remark 5 but also highlights the superiority of DIFTC in handling intermittent actuator faults. Figure 10 demonstrates the control inputs, where the actual input $U\left(t\right)$ closely matches the expected input $U_d$ through the application of dead-zone inverse dynamics (10), showcasing the effectiveness of the designed inverse compensation control in handling unknown dead-zone nonlinearity. Compared to the simulation results in [43], the control input’s oscillations are effectively suppressed, enhancing the actuator’s lifespan. The fault model (40) reveals that within 1s, the actuator fails, and under BC, the input is nearly halved compared to $U\left(t\right)$. However, under DIFTC, the input effectively compensates for the fault, aligning $\varpi \left(t\right)$ closely with $U\left(t\right)$. From 1–2 s, the actuator usually operates, and faults occur again from 2–3 s. Despite these faults, the system states progressively converge, unaffected by the actuator faults, confirming the efficacy of the designed DIFTC.

5. Experiment

The simulation results demonstrate the effectiveness of the designed control strategy. In this section, we conduct experiments based on the Quanser experimental platform to further validate the practical effectiveness of the control strategy. The experimental platform is shown in Figure 11.

In the experimental section, the control strategies are also compared, and the controller parameters are kept consistent with the simulation. The experimental results are shown in Figure 12 and Figure 13. From Figure 12, it can be observed that the DIFTC gradually reduces the end-point offset to near 0. In contrast, the performance of other controllers is poorer, particularly the traditional FTC controller, which fails to stabilize the system after 2 s. Similarly, from Figure 13, it can be seen that the DIFTC enables the flexible-link manipulator to rapidly approach the desired angle ${\xi }_d$, while systems controlled by BC and PDC exhibit some steady-state error in the angle. Additionally, under FTC, the flexible-link manipulator fails to maintain the desired angle ${\xi }_d$ during multiple faults. These experimental results further demonstrate the effectiveness and superiority of the proposed controller.

Remark 7.

The simulation and experiment results indicate that the proposed control strategy effectively addresses the control issues of a flexible-link manipulator system with unknown dead-zone and intermittent actuator faults. However, this method has limitations. Due to the lack of redundancy in the number of controllers, it cannot resolve control issues in systems where actuators are stuck, posing a challenge to system stability. Meanwhile, as shown in Figure 7 and Figure 10, the adoption of a robust adaptive controller designed to compensate for the coupling term ℵ, composed of two unknown upper bounds D and $U_f$, causes chattering in the controller input, thus causing slight chattering in the system end-point offset $c_l$ near the zero domain. Using an observer to effectively observe the unknown terms $d\left(t\right)$ and $U_{f,j}\left(t\right)$ can address this issue. In the future, we plan to consider using a new observer-based finite-time FTC to tackle these problems [44,45].

6. Conclusions

This study introduces a novel adaptive FTC strategy to stabilize a flexible-link manipulator system subject to unknown dead-zone and intermittent actuator faults. By employing the dead-zone inverse dynamics model, the detrimental effects of dead-zones are effectively mitigated. Furthermore, we devise an adaptive strategy to offset the upper bound of unknown terms, including compensation error and floating bias fault. Simulation and experiment analysis compares the proposed control strategy with PDC, BC, and traditional FTC, demonstrating that the state errors under the proposed strategy achieve consistent bounded convergence rapidly. The control input simulations show that the dead-zone inverse dynamics model effectively compensates for unknown dead-zone, and the proposed FTC maintains system stability despite intermittent actuator faults, proving its effectiveness and superiority.

However, there are some areas that need improvement. In the future, we intend to develop a neural network-based FTC to handle systems with stuck actuators [44].