Adaptive Control of a Flexible Manipulator With Unknown Hysteresis and Intermittent Actuator Faults

I. Introduction

IN the wake of technological developments, modern industry has seen an increasing demand for lightweight and flexible materials, and flexible structures have garnered increasing attention [1]. As a typical flexible structure system, a flexible manipulator has the advantages of high flexibility, lightweight, and low energy consumption [2]. Flexible manipulators often generate vibrations and deformations under harsh environments and working conditions, which may cause adverse effects, such as deterioration of system performance and limited production accuracy. Therefore, an effective vibration control methodology to stabilize flexible manipulators needs to be established.

A flexible manipulator is an infinite-dimensional distributed-parameter system (DPS). If the system is simplified to a finite-dimensional state-space model for control design, adverse spillover effects may occur [3]. To overcome these problems, boundary control (BC) is considered to be a more effective control method for DPSs due to the simultaneous suppression of all modes of simplification of infinite-dimensional system dynamics. In recent years, significant progress has been achieved in BC of flexible manipulators. In [4], an adaptive event-triggered BC scheme was developed for a flexible manipulator with communication constraints. In [5], a boundary controller for a flexible manipulator was proposed based on a partial differential equation (PDE) robust observer. However, these studies were limited to vibration suppression, and the methods used did not apply to single-link flexible manipulator systems with input hysteresis.

Hysteresis is a dynamic nonlinearity that typically exists in electronic, electromagnetic, mechanical actuators, and other areas [6]–[9]. For systems with high control accuracy, hysteresis can cause phase shift and harmonic distortion problems related to the input signal information, resulting in an unstable control system. Several scholars have proposed various methods to solve this problem. In [10], an innovative event-based adaptive decentralized output feedback control scheme was introduced. A passive adaptive neural network control problem was addressed for multi-input multi-output nonstrict-feedback nonlinear systems subject to unmeasurable states and actuator hysteresis in [11]. In [12], a novel adaptive fault-tolerant control (FTC) was designed for a flexible Timoshenko arm to prevent actuator failures, backlash-like hysteresis, and external disturbances. In [13], an adaptive BC was designed to suppress the vibrations of the flexible string, handle the uncertainties of the system parameters, and manage hysteresis nonlinearities. A novel adaptive control approach was introduced for the three-dimensional path-following of underactuated autonomous underwater vehicles, specifically designed to operate in environments affected by actuator hysteresis in [14]. In addition, the adaptive inverse compensation strategy is considered another effective method for handling hysteresis problems [15], [16]. In [17], perfect inverse dynamics were proposed for the Bouc-Wen hysteresis to compensate for hysteresis effects. The aforementioned approaches were conducted under the assumption that the hysteresis parameters were accurately known and were unavailable in the case of unknown hysteresis parameters. To eliminate this constraint, an adaptive hysteresis inverse output-feedback control scheme was developed in [18]. In [19], a novel indirect fuzzy control scheme with an alternative adaptive hysteresis inverse was proposed. However, the aforementioned unknown hysteresis inverse control was designed for systems represented by ordinary differential equations (ODEs) and cannot be applied to an infinite-dimensional flexible manipulator system with unknown hysteresis. To the best of our knowledge, although significant advancements in adaptive inverse control of ODE systems with unknown hysteresis have been achieved, limited studies have been conducted on addressing the simultaneous effects of unknown hysteresis and actuator faults in flexible manipulator systems; this motivates us to bridge the gap in research.

Owing to the complexity and multiple interferences of the working environment, actuator faults inevitably occur in various industrial controls [20]. In serious cases, sudden stucking or loss of partial effectiveness of an actuator may lead to an unstable system and even production accidents. FTC is considered an effective method for solving actuator faults. In recent years, research on FTC based on DPSs has progressed remarkably [21], [22]. In [23], an adaptive FTC scheme was developed to address actuator failures while suppressing the vibration of the system and achieving cooperative operation. The authors proposed an active adaptive FTC scheme for position tracking and vibration suppression of a constrained moving rigid-flexible manipulator system in [24]. The aforementioned FTC schemes consider a finite number of actuator faults, which cannot be directly used to handle infinite actuator faults. In an actual industrial process, actuator failure of a controlled system may occur more than once. To address this problem, a new adaptive failure/fault compensation control scheme was proposed for parametric strictly feedback nonlinear systems in [25]. In [26], a novel adaptive compensation strategy was proposed for a flexible manipulator in the presence of an infinite number of time-varying actuator faults. By contrast, the method in [26] was limited to eliminating the oscillation and compensating for infinite actuator faults, and intermittent actuator faults were not considered during the design. Although remarkable progress has been achieved in the FTC of flexible manipulators, no research has been conducted on adaptive inverse FTC to address the hybrid effects of unknown hysteresis nonlinearity and intermittent actuator faults in flexible manipulator systems.

Motivated by this background, we intend to develop a new adaptive compensation control for flexible manipulators with unknown hysteresis and intermittent actuator faults. The main contributions of this study are summarized as follows: 1) In contrast to previous studies [12], [13], [27], an adaptive inverse strategy is introduced and used to effectively counteract the input hysteresis effect in a flexible manipulator. In contrast to the control strategy in [17], this method can be applied to an unknown hysteresis problem. 2) A new neural network (NN)-based FTC is proposed to address intermittent actuator faults in a flexible manipulator system. In contrast to the literature [21], [22], the fault parameters in this study are time varying, and the faults occur frequently, rendering it more applicable to the actual situation. 3) The established control scheme can suppress the vibration of the controlled system without simplifying or discretizing the infinite-dimensional system dynamics, and the efficacy of the proposed controller is verified by numerical simulations and experiments.

Notations: The symbols in this article are standard. $\mathbb{Z}^+$ represents the set of all the positive integers. $\mathbb{R}$ represents all real numbers, with $\mathbb{R}^+=\{x\in\mathbb R:x>0\}$. $\mathbb{R}^{p \times q}$ denotes the set of all the real $p \times q$ matrices. $\|\boldsymbol{A}\|_F$ denotes the Frobenius norm of a matrix. $\boldsymbol{A} \succ 0$ represents a symmetric positive definite matrix. $\lambda_{\mathrm{max}}(\boldsymbol{A})$ represents the maximum eigenvalue of the symmetric matrix $\boldsymbol{A}$. For brevity, certain symbols are defined as $(\circ)=(\circ)(y,t)$, $(\dot{\circ})=\partial(\circ)/\partial t$, $(\ddot{\circ})=\partial^2(\circ)/\partial t^2$, $(\circ)'=\partial(\circ)/\partial y$, $(\circ)''=\partial^2(\circ)/\partial y^2$, $(\circ)'''=\partial^3(\circ)/\partial y^3$, $(\circ)''''=\partial^4(\circ)/\partial y^4$, $(\circ)_0= \circ(0,t)$, $(\circ)_L=\circ(L,t)$, and $\tilde\circ=\circ-\hat\circ$.

II. Proplem Statement

A System Model

Fig. 1 depicts a schematic diagram of a flexible manipulator system, where $YOZ$ and $yOq$ represent the global inertial coordinate system and the local rotating reference frames with the joint, respectively. In addition, y is a spatial variable with $y\in [0,L]$ and t is the time variable with $t\in\mathbb{R}^+$. $z(y,t)$ represents the elastic deflection, and $\theta(t)$ represents the angular position of the joint. $\tau(t)$ denotes torque input and $q(y,t)= y\theta(t)+z(y,t)$ denotes the total displacement of the link in coordinate system $YOZ$.

Figure  1.  Schematic of the considered flexible manipulator.

DownLoad: Full-Size Img PowerPoint

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

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

B Hysteresis Characteristic Analysis

In this study, we consider that the actuator input of the system is affected by the Bouc-Wen type of hysteresis, which is one of the most acceptable models for describing rate-dependent hysteresis [17], [28], [29], and it is represented as

where $0<o<1$ denotes the stiffness ratio and κ is a parameter related to the pseudo natural frequency. The constants $\mu_1$ and $\mu_2$ are unknown but share the same sign, with $\mu_1 = o\kappa$ and $\mu_2 = (1-o)\kappa$. $v(t)$ denotes the actual actuator inputs after inverse compensation. $u_f(t)$ represents the output of the input $v(t)$ after being affected by hysteresis. $\zeta\in\mathbb{R}$ is a rate-dependent variable that depends on input v with its derivative $\dot v$, expressed as

where $\beta>|\chi|$ and $n\geq 1$. β and χ are parameters describing the shape and amplitude of the hysteresis. n governs the smoothness of the transition from the initial slope to the slope of the asymptote. $f(\zeta,\dot v)$ is a function with two inputs ζ and $\dot v$ defined as

Remark 1: In [27], [30], the authors proposed a BC method for DPSs with hysteresis. The proposed hysteresis model is a special case of the hysteresis model (5).

For convenience of the control design, we parameterize the hysteresis model (5) as

where $\mu=[\mu_1,\mu_2]^T$ and $\aleph=[v,\zeta]^T$.

To address hysteresis parameter uncertainty, we introduce the following inverse dynamics [19]:

where $u_{fd}$ is the control signal to be designed, and we define the function $h_d(u_{fd},\zeta_d)$ as

where $\sigma_1$, $\sigma_2$, and $\epsilon$ are design parameters. Invoking (9), the expressions of $u_{fd}$ is given as

where $\hat\mu=[\hat\mu_1,\hat\mu_2]^T$ denotes the estimates of $\mu=[\mu_1,\mu_2]^T$, and $\aleph_d=[v,\zeta_d]^T$.

Additionally, the compensation error can be expressed as

where $d(t)=\mu_2(\zeta-\zeta_d)$, $\tilde\mu$ is the estimated error of μ defined as $\tilde\mu=\mu-\hat\mu$.

Remark 2: In contrast to previous studies [16], [17], [31], the inverse function can be designed directly without the use of parameter identification or knowledge of the specific hysteresis parameters. We assume that a positive constant D exists to ensure that the compensating error $d(t)$ satisfies the condition $d(t)\leq D=\sqrt[n]{1/(\beta+\chi)}+\sqrt[\epsilon]{1/(\sigma_1+\sigma_2)}.$

C Intermittent Actuator Fault Model

According to [32], the intermittent actuator fault model is described as

where h indicates the number of faults, $t\in [t_{h,s},t_{h,e})$, $\sigma_h$ is an unknown efficiency factor satisfying $0<\underline{\sigma}\leq\sigma_h\leq 1$. Note that $u_{f,h}(t)\leq \bar u_f$, and $\underline{\sigma}$, $u_f$ are unknown constants. (14) indicates that the actuator loses $(1-\sigma_h)\times 100\%$ of its effectiveness and produces the floating bias fault $u_{f,h}(t)$ from time $t_{h,s}$ till $t_{h,e}$. When $\sigma(t) = \sigma_h$, the system becomes a common fault model that considers both the efficiency partial loss fault and the floating bias fault [21], [33]. This model did not take into account the reoccurrence of fault.

Remark 3: In common fault models (see (1) in [23]), the parameter σ representing the degree of actuator fault is typically an unknown but fixed constant, which does not vary over time. In contrast, in (14), σ is set as an unknown and time-varying parameter. This design is inspired by the real-world observation that some actuator faults may exhibit intermittent characteristics. This means that an actuator may repeatedly and automatically switch from a faulty state to a normal operating state or to other different fault states. As expressed in (14), an actuator with a fault might experience multiple failures. Therefore, the actuator fault model proposed in our study is more reflective of real industrial environments, making the corresponding control design more complex and challenging.

Thus, combined with (13), the actual inputs $\tau(t)$ can be expressed as

Remark 4: Differently from [12] and [22], intermittent faults in the actuator is assumed to occur in this paper, which is more common than the common fault model with only one single fault in the FTC results. It is worth noting that the fault-free condition is a special case of (14) when $\sigma(t)=1$ and $u_{f,h}(t)=0$.

Remark 5: In this study, we consider a single-link flexible manipulator encountering both unknown Bouc-Wen type hysteresis and intermittent actuator faults, which are plausible concurrent issues in real-world environments. A quintessential example is the robotic arm systems used in the aerospace domain. In the space environment, due to extreme temperature variations and microgravity conditions, the actuators of these arms are likely to experience hysteresis. This hysteresis, typically resulting from internal mechanical friction or material thermal expansion, causes input hysteresis. Concurrently, in such environments, the actuators might also suffer intermittent faults. These faults could stem from radiation damage to electronic components, impacts from tiny meteorite fragments, or prolonged wear and tear. In such instances, actuators might exhibit partial efficiency loss faults and floating bias faults without warning, only to return to normal operation after some time. Both phenomena could coexist in aerospace robotic arm systems, posing dual challenges to their performance and reliability. Therefore, the design and control strategies for such systems need to comprehensively address both phenomena to ensure that the robotic arms operate stably and efficiently in these complex conditions.

D Radial Basis Function NN (RBFNN)

Because of its fixed and simple three-layer architecture, RBFNN is simpler in designing and training than other NNs [34]. In recent years, it has been widely used in the nonlinear control [35]. In this paper, the RBFNN is utilized to tackle coupling terms $\sigma(t)\aleph_d$ such that

where $W=[v,\zeta_d]^T$ and $p^{*}$ stand for the input and ideal weight vectors, respectively. $\varepsilon(W)$ denotes the approximation error and $\bar\varepsilon$ is an unknown constant. $q\in\mathbb{Z}^+$ denotes the node number in the hidden layer. $H(W)=[H_1(W),H_2(W), \dots, H_N(W)]\subset$ $\mathbb{R}^N$ represents the Gaussian function as

where $c_i$ is the center vector of the $i\mathrm{th}$ hidden layer neuron and $b_i\in\mathbb{R}^+$ is the width of the Gaussian function. Moreover, let $\hat p$ be the estimation of $p^{*}$, where $\|\tilde p\|=\|p^{*}-\hat p\|$.

E Control Objective and Related Lemmas

This paper aims at presenting an adaptive inverse compensation control strategy for a flexible manipulator with intermittent actuator faults and unknown input hysteresis.

To facilitate the subsequent analysis, the following assumptions and lemmas are provided.

Lemma 1 [36]: Consider the first-order differential equations as follows:

where $\gamma_s$ and $\kappa_s$ are positive constants. If $f_s,\; s(0)\geq 0$, the solution of (18) is rendered to be non-negative for all $t\geq 0$.

Lemma 2 [19]: For any piecewise continuous signals v and $\dot v$, the output $\zeta(t)$ is bounded and satisfies $|\zeta(t)|<\sqrt[n]{1/(\beta+\chi)}$. Similarly, we get $|\zeta_d(t)|<\sqrt[\epsilon]{1/(\sigma_1+\sigma_2)}$.

Lemma 3 [37]: If e and r are scalars and satisfy $e\in\mathbb{R}$, $r\in\mathbb{R}^+$, we have

If r is a positive uniformly bounded continuous function with $r(t)$ satisfying ${\mathrm{lim}}_{t\to\infty}\int_{t_0}^{t}r(t)dt\leq\bar r<\infty$, the above inequality is still satisfied, where $\bar r$ is a positive constant.

III. Control Design

In this section, an adaptive FTC is proposed to achieve control objectives. By combining the unknown compensation error with the time-varying actuator faults, an upper-bound adaptive law is proposed. Subsequently, the stability analysis is provided.

A Adaptive FTC

We construct the auxiliary signal $u_a(t)=\alpha\dot\theta(t)\;+\;\beta_2[\theta(t)\;- \theta_d]= \alpha\dot\theta(t)+\beta_2\theta_e(t)$, and $\theta_d$ denotes the reference signal. Then, the control signal $u_{fd}(t)$ is designed as

where $\beta_1(t)=-z_1(t)u_a(t)$. Let $\varrho=1/\underline{\sigma}$ and $\hat\varrho$ represent the estimate value of $\varrho$. $z_1$ is given as

where k, $k_\theta$, and $\beta_2$ are positive constants, $\delta\geq0.25$, and $k_1\in \mathbb{R}$. $\hat{\hbar}$ is the estimation of the upper bound $\hbar$ defined as $\hbar=\|p_M\mu^T\|+\mu_M\bar\varepsilon+D+\bar u_f$. The definitions of $p_M$ and $\mu_M$ can be found in Remark 6. Then, the parameter updating laws are designed as

where $\Gamma_\mu, \Gamma_p\succ0$, with $\gamma_\varrho$ and $\gamma_h$ being positive constants.

To avoid singularity, making the control inputs infinite, the projection mapping Proj$\left\{\circ\right\}$ is used to limit the range of adaptive parameters $\hat\mu$ and $\hat p$, defined as [38]

where $\Psi\succ 0$, $\bigtriangledown_{\xi}\mathcal{P}$ denotes an outward normal vector at $\bar\Omega$, $\breve\Omega$ and $\bar\Omega$ denote the interior and the boundary of the set $\Omega\in\mathbb{R}$, respectively.

Remark 6: In this research, we utilize the adaptive mapping operator (26) to design the adaptive laws (24) and (25) for unknown parameters μ and p. Based on the mapping operator (26), the estimates of the unknown parameters $\hat\mu$ and $\hat p$ can not exceed the established boundaries. Since the estimates are bounded and not divergent, the estimation errors $\|\tilde\mu\|$ and $\|\tilde p\|$ are also bounded. The proof process is similar to that of [26]. Here, we define the unknown constants $\mu_M$ and $p_M$ such that $\|\tilde\mu\|\leq\mu_M$ and $\|\tilde p\|\leq p_M$.

Remark 7: It should be emphasized that the FTC strategy proposed in [21], [22] and [39]–[41] cannot directly be applied to this system. If the actuator experiences multiple faults, the parameter $\dot\sigma(t)$ will be unbounded, which may lead to system instability. To avoid the unbounded term $\frac{\underline\sigma}{\gamma_{\varrho}}\tilde\varrho(\dot\varrho-\dot{\hat\varrho})$ appearing in the time derivative of subsequent Lyapunov functions $\dot V(t)$, we propose a novel FTC strategy, which is different from that in literature [26]. By introducing quadratic nonlinear damping terms $\beta_1$ and $z_1$, we can handle this issue effectively.

B Stability Analysis

Consider the Lyapunov function candidate as

with

where $q_e(y,t)=z(y,t)+y\theta_e(t)$. Obviously, $\dot q_e(y,t)=\dot q(y,t)$.

Lemma 4: The selected Lyapunnov function (27) satisfies the following property:

where $\lambda_2=1-\lambda_1$ and $\lambda_3=1+\lambda_1$ are two positive constants with $\lambda_1= \mathrm{max}\{\frac{\beta_2(1+L)}{\pi_1\alpha},\frac{\beta_2\rho L^2\pi_1}{k_{\theta}\alpha},\frac{\beta_2\rho L^2\pi_1}{T\alpha}\}$.

Proof: See Appendix A.■

Lemma 5: The time derivative of the Lyapunov function (27) is proved to be bounded as

where $\lambda, \iota\in\mathbb{R}^+$.

Proof: See Appendix B.■

Theorem 1: For the flexible manipulator system described by (1)−(4) in the presence of unknown hysteresis and intermittent actuator faults, under the proposed control (20) and (21) and the parameter updating laws (22)−(25), we can deduce the following:

1) The elastic displacement $z(y,t)$ is uniformly bounded, starts in a compact set $\Omega_1$, and converges eventually to another compact set $\Omega_2$.

where $\Phi_1=\sqrt{\frac{2L}{\alpha T\lambda_1}[V(0)+\frac{\iota}{\lambda}]}$ and $\Phi_2=\sqrt{\frac{2l\iota}{\alpha T\lambda_1\lambda}}$.

2) The angular error $\theta_e(t)$ is maintained in the compact set $\Omega_3$ and finally converges to the compact set $\Omega_4$.

where $\Phi_3=\sqrt{\frac{2L}{k_\theta\lambda_1}[V(0)+\frac{\iota}{\lambda}]}$ and $\Phi_4=\sqrt{\frac{2L\iota}{k_\theta\lambda_1\lambda}}$.

Proof: Multiplying (32) by $e^{\lambda t}$ yields

Substituting (35), we obtain

Using (28) and the Sobolev’s inequality (see Lemma 2 in [42]) yields

Invoking (37), (36) is rewritten as

Further, we have

Similarly, from (29), we have

Then, we arrive at

■

Remark 8: The control parameters are selected to satisfy Lemma 4 and constraint conditions (58)–(62) at the same time. The process of selecting parameters is described as follows. First, we choose appropriate parameters $\pi_1$ and $\beta_2$ to complete the proof of Lemma 4. Next, choosing appropriate parameters k, $k_{\theta}$, $k_1$, $\pi_1$, $\pi_2$, and $\pi_3$, the $\varpi_i$, $i=1,\dots,4$ and ι are satisfied. Thus, the proof of Lemma 5 is completed. Through Lemma 5, we can then derive (39) and (41), which define the convergence interval for the error.

IV. Numerical Simulation

To verify the effectiveness of the proposed control strategy, the finite difference method is used for the numerical simulation. The system parameters selected are $EI=0.157\; \mathrm{Nm^2}$, $T=0.1\; \mathrm{N}$, $L=0.419\; \mathrm{m}$, $\rho=0.1\; \mathrm{kg/m}$, $c=0.04\; \mathrm{NS/m}$, and $I=0.0038\; \mathrm{kgm^2}$. In this study, the simulation time and space steps are set as $\bigtriangleup t=3.75\times{10^{-5}}\; \mathrm{s}$ and $\bigtriangleup y=0.0221\; \mathrm{m}$, respectively. The desired trajectory $\theta_d$ is selected as $\frac{\pi}{12}$. The initial states of the system are $z(y,0)\;=\;0.5\; \mathrm{s^2\; m}$, $\dot z(y,0)\;=\;0\; \mathrm{m/s}$, $\theta(0)= 0\; \mathrm{rad}$, and $\dot\theta(0)=0\; \mathrm{rad/s}$. We consider a flexible manipulator with intermittent actuator faults.

where g is an odd positive number.

The parameters selected for the hysteresis inverse fault-tolerant controller (HIFTC) are $\alpha=1$, $\beta_2 = 0.05$, $\delta=0.8$, $k_{\theta}=4.5$, $r(t)=0.7+0.5e^{-0.2t}$, $k=1$, and $k_1=0.2$. The parameter-updating laws are selected as $\gamma_{\varrho}=0.1$, $\gamma_{h}=0.1$, $\hat\mu(0)= [0.32, 0.32]^T$, $\hat\hbar(0)=0.3$, $\hat p(0)=0$, and $\hat\varrho(0)=1$. This set is given by $\varOmega=\{\hat\mu\in\mathbb{R}^2|0.2\leq\hat\mu_1<0.4, 0.2\leq\hat\mu_2<0.4\}$. The hysteresis parameters are $\mu_1=0.3$, $\mu_2=0.3$, $\beta=1$, $n=1.2$, and $\chi=0.8$. Fig. 2 shows that the deflection $z(y,t)$ of the entire flexible link under the HIFTC approaches to zero within 1.5 s.

Figure  2.  Deflection of the flexible link under the HIFTC controller.

DownLoad: Full-Size Img PowerPoint

To further verify the effectiveness of the proposed control, we also provide simulation results with the following proportional differentiation controller (PDC) in [26] and BC in [43].

The PDC in [26] is expressed as

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

The BC in [43] is given as

where $k=0.09$, $k_1=0.1$, $\beta_2=0.05$, and $k_{\theta}=5$.

We can observe from Figs. 3 and 4 that the HIFTC can maintain excellent control performance under the influence of input hysteresis and intermittent actuator faults. Compared with BC and PDC, the proposed controller can quickly track the reference signal $\theta_d$ while maintaining the system stability. In addition, the end-point offset $z(L,t)$ of the flexible manipulator can also approach to zero within 1.5 s under the HIFTC. Although the offset $z(L,t)$ is convergent under BC and PDC, after 1.5 s, the angle $\theta(t)$ still does not converge to the reference signal $\theta_d$ and some steady-state error occurs. Fig. 5 depicts the control inputs. By applying hysteresis inverse dynamics (9), the actual input $u_f$ of the system fits the expected input $u_{fd}$. Simultaneously, when the actuator loses 50% of its effectiveness, the stability of the control system is maintained. Therefore, it can be concluded that the HIFTC can better stabilize a flexible manipulator system with unknown hysteresis and intermittent actuator faults.

Figure  3.  Tracking of angle $\theta(t)$.

DownLoad: Full-Size Img PowerPoint

Figure  4.  End-point offset $z(L,t)$.

DownLoad: Full-Size Img PowerPoint

Figure  5.  Control input under the HIFTC controller.

DownLoad: Full-Size Img PowerPoint

V. Experiment

To further verify the effectiveness of the proposed control strategy, physical experiments are performed on the Quanser experimental platform, as shown in Fig. 6. The Quanser experimental platform is the most crucial component of this platform. It consists of a DC motor encased in a robust aluminum frame and a planetary gearbox. Additionally, the motor is equipped with an internal gearbox, enabling it to drive other gears. The basic unit of the Quanser experimental platform is equipped with a potentiometer sensor, which is used to measure the angular position of the load gear. The tip deflection angle is obtained using a strain gauge fixed on the rotary flexible beam. As illustrated in Fig. 6, the system also includes other key components such as a power amplifier, a filter device, a data acquisition board, a host computer, and a digital multimeter. The power amplifier and filter primarily serve to process signals from the aforementioned sensors. The data acquisition board is responsible for A/D conversion and is connected to both the filter and the host. Upon receiving these signals, the host executes the control algorithm and generates the control input signal.

Figure  6.  Rotary flexible link system.

DownLoad: Full-Size Img PowerPoint

The control strategies are also compared, and the parameters are selected by simulation. The experimental results are shown in Figs. 7-9. Table I summarizes the tracking performance under HIFTC, PDC, and BC. In terms of angular tracking performance, the tracking error of the system utilizing the control strategy proposed in this study decreased by 87.10% compared to BC, and by 97.84% compared to PDC. Moreover, under HIFTC, the system’s adaptation time is significantly reduced by 72.50% and 136.08% compared to BC and PDC, respectively, demonstrating a considerable enhancement in performance. As shown in Fig. 7, the HIFTC can track target angles $\theta_d$ in the face of system input hysteresis and intermittent actuator faults, whereas BC and PDC cannot meet the preset angle-tracking performance requirements. From Fig. 8, it can be observed that based on HIFTC, the angle error $\theta_e(t)$ converges well to zero, while other strategies exhibit certain steady-state errors. In addition, the deformation of the endpoint can converge uniformly under HIFTC, and the range of convergence depends on the selection of parameters, as shown in Fig. 9. In summary, the simulation and experimental results demonstrate that the proposed controller can better stabilize a flexible manipulator system with intermittent actuator faults and unknown hysteresis.

Figure  7.  Angle tracking $\theta(t)$.

DownLoad: Full-Size Img PowerPoint

Figure  9.  End-point offset $z(L,t)$.

DownLoad: Full-Size Img PowerPoint

Table  I.  Comparison of System Performance Based on Different Approaches in Experiment

Tracking performance
Methods	Overshoot (%)	Accommodation time (s)	Steady-state error (rad)
HIFTC	0.0063	0.6261	0.0293
BC	0	1.0800	0.2272
PDC	0	1.4781	1.3582

 | Show Table

DownLoad: CSV

Figure  8.  Angle tracking error $\theta_e(t)$.

DownLoad: Full-Size Img PowerPoint

VI. Conclusion

In this study, a new adaptive NN-based FTC strategy is proposed to stabilize a single-link flexible manipulator system affected by an unknown Bouc-Wen type of hysteresis and intermittent actuator faults. Using the hysteresis inverse dynamics model, the input hysteresis effect of the system was effectively counteracted. Furthermore, an unknown upper-bound adaptive compensation strategy was proposed to offset the side effects caused by the compensation error and produce an uncontrollable additive actuation fault. The designed control ensured the consistency and boundedness of the controlled system, and the control performance was further evaluated through simulation and experimental analysis.

Appendix A. Proof of Lemma 4

Based on the Young’s inequality (see Lemma 1 in [42]), the inequality holds

where $\pi_1\in\mathbb{R}^+$, $\lambda_1=\mathrm{max}\{\frac{\beta_2(1+L)}{\pi_1\alpha},\frac{\beta_2\rho L^2\pi_1}{k_{\theta}\alpha},\frac{\beta_2\rho L^2\pi_1}{T\alpha}\}$.

Let $\lambda_1$ satisfy $0<\lambda_1<1$, we obtain

where $\lambda_2=1-\lambda_1$ and $\lambda_3=1+\lambda_1$ are positive constants. ■

Appendix B. Proof of Lemma 5

By differentiating (28) and substituting (1)–(3), we obtain

Applying Lemma 1, (4), (20) and (21) to $\dot V_2(t)$ generates the following:

where $\dot V_k$ is defined as

Let $Y(t) =\sigma(t)\tilde\mu^T\aleph_d+\sigma(t)d(t)+u_{f,h}(t)$, according (15) and (16), we derive

Thus, we get

According to Young’s inequality (see Lemma 1 in [42]), (51) can be expressed as

where $\delta_0=\frac{\delta}{4}[I\beta_2+T+k_{\theta}+1]+\bar r$.

Using the adaptive laws (22)–(25), Lemma 3, and Young’s inequality (see Lemma 1 in [42]), we then obtain

Invoking the PDE model (1)–(4) and Young’s inequality (see Lemma 1 in [42]), $\dot V_3(t)$ is calculated as

By substituting (47), (53), and (54) into $\dot V(t)$, we derive

Invoking $u_a(t)\;=\;\alpha\dot\theta(t)+\beta_2\theta_e(t)\Rightarrow \dot\theta(t)\;=\;\frac{u_a(t)-\beta_2\theta_e(t)}{\alpha(t)}$, we obtain

Furthermore, we get

where $\varpi_0=k+\frac{k_1}{4\delta\alpha^2}$, $\pi_2, \pi_3\in\mathbb{R}^+$, and $\iota=\frac{1}{2}\hbar^2+\frac{1}{2}\|p^*\|^2_F\;+ \frac{\underline\sigma}{2}\varrho^2+\frac{1}{2}\mu^2+\delta_0$.

The constraint conditions that allow (57) to hold are as follows:

In addition, by employing Lemma 4 and (58)–(62), we obtain

where $\lambda\leq\frac{1}{\lambda_3}\mathrm{min}(\frac{2\varpi_1}{\rho\alpha}, \frac{2\varpi_2}{T\alpha}, \frac{2\varpi_3}{I\alpha}, \frac{2\varpi_4}{k_{\theta}\alpha}, \frac{1}{\lambda_{{\mathrm{max}}}(\Gamma^{-1}_{\mu})}, \frac{1}{\lambda_{{\mathrm{max}}}(\Gamma^{-1}_p)},\nonumber \gamma_{\varrho}, \gamma_h)$.

■