Adaptive Multi-Fault-Tolerant Boundary Control of an Euler–Bernoulli Beam System with Control-Matched Disturbances

Abstract

This article settles a new multi-fault-tolerant control problem of an Euler–Bernoulli beam system (EBBS) in the existence of multiplicative faults, additive faults, and control-matched disturbances simultaneously, using the direct adaptive learning control technique. Such an EBBS can be employed to model the vibration of flexible vehicles and flexible manipulators in the actual engineering control. An original hierarchical adaptive boundary control strategy is developed to compensate for multiplicative and additive faults and to reject control-matched disturbances. The classical Lyapunov direct method, together with the variation of Wirtinger’s inequality is utilized to demonstrate the closed-loop system performance. The modified ${\mathcal{C}}_0$-semigroup frame is exploited to certify the well-posedness of its solution under the designed controller. Simulation study on a simply supported beam is demonstrated to verify the validity of the evolved multi-fault-tolerant boundary control algorithm.

1. Introduction

Most physical phenomenon or processes in the real world, such as power systems, chemical processes, aerospace vehicles, have distributed features. This means that their energy or mass varies not only continuously in time but also distributedly in space. One system with such characteristics is usually called a distributed parameter system (DPS), for example, flexible manipulators [1], flexible flapping wings vehicles [2], multi-agent systems [3], etc. One of the most widely used DPSs in reality is EBBS, which describes some flexible vibrations [4,5,6]. Research on such a system has been conducted in recent decades. Various boundary control designs have been developed, such as the adaptive control framework [7], active disturbance rejection method [8,9], sliding mode control algorithm [10], bilinear matrix inequality-based control method [11], iterative learning control strategy [12,13], Riesz basis theory-based control approach [14,15], and backstepping control mechanism [16,17], to name a few. It should be worth noting that these works in the literature have dealt with the following control issues:

• Stabilization control designs without uncertainties;
• Active disturbance rejection control designs under boundary control-matched disturbances;
• Output tracking control designs under boundary mismatched disturbances;
• Adaptive control designs under system parameter uncertainties.

Also note that all the above results do not consider any faults, such as actuator faults, sensor faults, distributed faults, etc. This implies that achieving fault compensation of such a system is an open problem and needs further investigation.

In addition, fault-tolerant control (FTC) aims at keeping or maintaining the closed-loop system performance under some actuator/sensor/component faults. Such a control direction on the systems described by ordinary differential equations has attracted much attention for about the past fifty years, and fruitful FTC schemes have been proposed for different applications; see [18,19,20,21,22,23,24,25,26] and the references therein. However, these algorithms cannot be directly applied to the DPSs, especially for Euler–Bernoulli beam systems. This is because systems described by ordinary differential equations possess finite dimensional features, while DPSs have infinite dimensional properties. Such a difference implies that FTC designs for DPSs need to be payed some other attention. In the past twenty years, there have been some results on FTC designs of DPSs, such as [27,28,29,30,31]. These results aim at parabolic and hyperbolic DPSs, which are first/second-order partial differential equations. For EBBSs, there are also some achievements, such as [32,33,34,35,36,37]. In [32,34,36], the bilinear matrix inequality-based FTC algorithms and backstepping-based FTC scheme was proposed for EBBSs to compensate for distributed faults. In [35], a backstepping-based sliding mode FTC framework was put forward to accommodate the boundary actuator additive faults. In the above articles, FTC designs were developed to solve only the single fault compensation issue. Realizing boundary multi-fault compensation for EBBSs is still unclear and needs in-depth study.

In this article, an original adaptive fault-tolerant boundary control strategy is brought forward for EBBSs to settle both the boundary multiplicative and additive faults and matched disturbances, simultaneously. As far as we know, until now barely any results have been reported in this research field. This is because there is still no solution for dealing with the multiplicative uncertainties for DPSs, especially for the EBBS. Hence, the contributions of this article can be summarized as the following three aspects:

• A novel adaptive multi-fault-tolerant boundary control scheme is developed in the presence of both the boundary multiplicative and additive faults and control-matched disturbances, simultaneously.
• New parameter updating laws are designed to handle both the multiplicative and additive uncertainties.
• The complete performance analysis is provided by using the Lyapunov direct method to ensure the asymptotical stability and using the modified ${\mathcal{C}}_0$-semigroup theory to guarantee the well-posedness of the solution.

The organization of the rest of the article is as follows. The control plant with both the boundary multiplicative and additive faults and control-matched disturbances is described and control difficulties are further examined in Section 2. A solution to the adaptive multi-fault compensation problem is figured out in Section 3. Simulation study on a simply supported beam is presented in Section 4. Some discussions are concluded in Section 5.

Notation: In this paper, the following subscripts are used.

$$\begin{array}{ccc}\hfill w_t(x,t)& =& {\displaystyle \frac{\partial w(x,t)}{\partial t}},w_{tt}(x,t)={\displaystyle \frac{{\partial }^{2}w(x,t)}{\partial t^2}},w_x(x,t)={\displaystyle \frac{\partial w(x,t)}{\partial x}},\hfill \\ \hfill w_{xx}(x,t)& =& {\displaystyle \frac{{\partial }^{2}w(x,t)}{\partial x^2}},w_{xxx}(x,t)={\displaystyle \frac{{\partial }^{3}w(x,t)}{\partial x^3}},w_{xxxx}(x,t)={\displaystyle \frac{{\partial }^{4}w(x,t)}{\partial x^4}}.\hfill \end{array}$$

2. Problem Statement

This section provides the system model described by an EBBS, parameterizes multi-faults and control-matched disturbance, and condenses the difficulties in the fault-tolerant boundary control design.

2.1. Model Description

The specific system internal dynamics considered in this paper can be depicted by

$$\begin{array}{c}\hfill {\breve{\omega }}_{tt}(\breve{z},t)+\breve{m}{\breve{\omega }}_{\breve{z}\breve{z}\breve{z}\breve{z}}(\breve{z},t)=0,\breve{z}\in (0,\breve{L}),\end{array}$$

(1)

confined by three zero boundary conditions

$$\begin{array}{c}\hfill \breve{\omega }(0,t)={\breve{\omega }}_{\breve{z}}(0,t)={\breve{\omega }}_{\breve{z}\breve{z}}(\breve{L},t)=0,\end{array}$$

(2)

and the following terminal boundary control actuation:

$$\begin{array}{c}\hfill {\breve{\omega }}_{tt}(\breve{L},t)={\breve{\mathrm{\Omega }}}_o\left(t\right)+\breve{d}\left(t\right),\end{array}$$

(3)

where $\breve{\omega }(\breve{z},t)\in \Re$ represents the EBBS deflection, ${\breve{\mathrm{\Omega }}}_o\left(t\right)\in \Re$ denotes the actuator output signal, $\breve{d}\left(t\right)\in \Re$ means the control-matched disturbances, $\breve{m}>0$ is a known positive parameter related to the mass density and flexural rigidity of this considered Euler–Bernoulli beam, and $\breve{L}>0$ denotes the length of the considered Euler–Bernoulli beam. The initial conditions of (1)–(3) are $\breve{\omega }(\breve{z},0)={\breve{\omega }}_0\left(\breve{z}\right)$ and ${\breve{\omega }}_t(\breve{z},0)={\breve{\omega }}_1\left(\breve{z}\right)$.

Taking both the multiplicative faults and additive faults into consideration, the actuator fault model can be built by

$$\begin{array}{c}\hfill {\breve{\mathrm{\Omega }}}_o\left(t\right)=\breve{\rho }{\breve{v}}_o\left(t\right)+\breve{\mathrm{\Delta }}\left(t\right),\end{array}$$

(4)

where ${\breve{v}}_o\left(t\right)$ represents the actuator input signal which should be designed later, ${\breve{\rho }}_{min}\le \breve{\rho }\le 1$ denotes the uncertain actuator failure factor with ${\breve{\rho }}_{min}$ being a known positive parameter, which reflects the maximum fault reconfigurability capability, and $\breve{\mathrm{\Delta }}\left(t\right)\in \Re$ represents the uncertain actuator drift faults.

Generally speaking, the actuator fault model (4) has the following physical meaning:

• $\breve{\rho }=1$ and $\breve{\mathrm{\Delta }}\left(t\right)=0$: actuator is healthy;
• $\breve{\rho }=1$ and $\breve{\mathrm{\Delta }}\left(t\right)\ne 0$: actuator drift faults (additive faults);
• ${\breve{\rho }}_{min}\le \breve{\rho }<1$ and $\breve{\mathrm{\Delta }}\left(t\right)=0$: actuator partial loss of effectiveness faults (multiplicative faults);
• ${\breve{\rho }}_{min}\le \breve{\rho }<1$ and $\breve{\mathrm{\Delta }}\left(t\right)\ne 0$: actuator multi-faults (both multiplicative and additive faults).

Without loss of generality, we presume that the actuator drift faults $\breve{\mathrm{\Delta }}\left(t\right)$ and control-matched disturbances $\breve{d}\left(t\right)$ satisfy

$$\begin{array}{c}\hfill \breve{\mathrm{\Delta }}\left(t\right)={\theta }_a^{\top }{\varphi }_a\left(t\right),\breve{d}\left(t\right)={\theta }_d^{\top }{\varphi }_d\left(t\right),\end{array}$$

(5)

where ${\theta }_a\in {\Re }^a$ and ${\theta }_d\in {\Re }^d$ represent unknown constant vectors respectively, and ${\breve{\varphi }}_a\left(t\right)\in {\Re }^a$ and ${\breve{\varphi }}_d\left(t\right)\in {\Re }^d$ denote known basis functions respectively, which are generated from

$$\begin{array}{ccc}\hfill {\dot{\breve{\varphi }}}_a\left(t\right)& =& {\breve{G}}_a{\breve{\varphi }}_a\left(t\right),{\breve{\varphi }}_a\left(0\right)={\breve{\varphi }}_{a0},\hfill \\ \hfill {\dot{\breve{\varphi }}}_d\left(t\right)& =& {\breve{G}}_d{\breve{\varphi }}_d\left(t\right),{\varphi }_d\left(0\right)={\varphi }_{d0},\hfill \end{array}$$

with known matrices ${\breve{G}}_a\in {\Re }^{a\times a}$ and ${\breve{G}}_d\in {\Re }^{d\times d}$. It can be seen from condition (5) that the actuator drift faults and control-matched disturbances can be parameterized as

$$\begin{array}{c}\hfill \breve{\mathrm{\Delta }}\left(t\right)+\breve{d}\left(t\right)={\theta }_u^{\top }\breve{\varphi }\left(t\right),\end{array}$$

(6)

where ${\theta }_u\triangleq {[{\theta }_a^{\top },{\theta }_d^{\top }]}^{\top }$ represents the unknown constant vector and $\breve{\varphi }\left(t\right)\triangleq {[{\breve{\varphi }}_a^{\top }\left(t\right),{\breve{\varphi }}_d^{\top }\left(t\right)]}^{\top }$ denotes the known basis function, which meets

$$\begin{array}{c}\hfill \dot{\breve{\varphi }}\left(t\right)=\breve{G}\breve{\varphi }\left(t\right),\breve{\varphi }\left(0\right)={\breve{\varphi }}_0={\left[{\breve{\varphi }}_{a0}^{\top },{\breve{\varphi }}_{d0}^{\top }\right]}^{\top },\end{array}$$

(7)

with $\breve{G}=\mathrm{diag}\{{\breve{G}}_a,{\breve{G}}_d\}$.

Remark 1. 

The additive fault and control-matched disturbance considered in this paper is linear parametrized. However, these signals may be nonlinear parametrized or non-parametrized in practical engineer applications. Here we present basic ideas to compensate for the uncertainties described by nonlinear parametrization condition and non-parametrized condition.

• If the additive fault and control-matched disturbance with finite energy can be nonlinear parametrized, the inclusion principle developed in [38] can be used such that additive fault and control-matched disturbance satisfy

  $$\begin{array}{c}\hfill \breve{\mathrm{\Delta }}\left(t\right)\le {\theta }_a^{\top }{\varphi }_a\left(t\right),\breve{d}\left(t\right)\le {\theta }_d^{\top }{\varphi }_d\left(t\right),\end{array}$$

  (8)
  Under this nonlinear parametrized condition, the asymptotic stability can also be ensured by modifying the parameter update laws, which indicates the adaptive control scheme developed under linear parametrization remains valid.
• If the additive fault and control-matched disturbance with finite energy cannot be parametrized, the RBF neural network developed in [4] can be used such that additive fault and control-matched disturbance satisfy

  $$\begin{array}{c}\hfill \breve{\mathrm{\Delta }}\left(t\right)={W_a^{*}}^{\top }{S}_a\left(t\right)+{ϵ}_1,\breve{d}\left(t\right)={W_d^{*}}^{\top }{S}_d\left(t\right)+{ϵ}_2,\end{array}$$

  (9)

  where ${W_a^{*}}^{\top }\in {\Re }^{n_a}$ and ${W_d^{*}}^{\top }\in {\Re }^{n_d}$ are weight factors, $S_a\left(t\right)={[S_{a1}\left(t\right),S_{a2}\left(t\right),\dots ,S_{a{n}_a}\left(t\right)]}^{\top }$ and $S_d\left(t\right)={[S_{d1}\left(t\right),S_{d2}\left(t\right),\dots ,S_{d{n}_d}\left(t\right)]}^{\top }$ are Gaussian basic functions. Their components $S_{ai}\left(t\right)$ and $S_{di}\left(t\right)$ are as follows:

  $$\begin{array}{ccc}\hfill S_{ai}\left(t\right)& =& exp\left[{\displaystyle \frac{-{(t-{\eta }_{ai})}^{\top }(t-{\eta }_{ai})}{{\mu }_{ai}^2}}\right],i=1,2,\dots ,n_a,\hfill \end{array}$$

  (10)

  $$\begin{array}{ccc}\hfill S_{di}\left(t\right)& =& exp\left[{\displaystyle \frac{-{(t-{\eta }_{di})}^{\top }(t-{\eta }_{di})}{{\mu }_{di}^2}}\right],i=1,2,\dots ,n_d,\hfill \end{array}$$

  (11)

  where ${\eta }_{ai}$ and ${\eta }_{di}$ represent the center of the Gaussian basis functions, ${\mu }_{ai}$ and ${\mu }_{di}$ represent the width of Gaussian basis function, and ${ϵ}_1$ and ${ϵ}_2$ are bounded approximation errors within the compact set. Under this condition, the uniform ultimate boundness can be achieved by modifying the parameter update laws, which means that the developed adaptive control scheme is still valid.

2.2. Control Difficulties

In the existing literature, the internal system dynamics (1) has been well investigated [5,7,12,13,14,15]; however, different boundary conditions have not been fully studied, especially for the boundary condition (2)–(3). This indicates that the control design of an EBBS with boundary conditions (2)–(3) is still an open problem.

While taking the actuator fault into consideration, there are also some results, such as [32,33,34,35,36,37] to name a few. In [33,35,37], only the additive fault compensation issue is solved. In [32,34,36], only the distributed fault accommodation problem is settled. These imply that achieving the multi-fault compensation still needs further and in-depth discussion.

Based on the above overview of the stabilization control and FTC of the EBBS, the control goal of this work is to devise a new adaptive multi-fault-tolerant boundary compensator ${\breve{v}}_o\left(t\right)$ for (1)–(3) under the condition (4) to ensure the asymptotical stability in the norm ${\int }_0^{\breve{L}}\left({\breve{\omega }}^2(\breve{z},t)+{\breve{\omega }}_t^2(\breve{z},t)\right)d\breve{z}$, i.e.,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}{\int }_0^{\breve{L}}\left({\breve{\omega }}^2(\breve{z},t)+{\breve{\omega }}_t^2(\breve{z},t)\right)d\breve{z}=0.\end{array}$$

(12)

To fulfill this control objective, the following lemmas are quite essential:

Lemma 1 

([11]). Let $\breve{\underline{\omega }}(\breve{z},t)\in {\mathcal{L}}^2(0,\overline{L})$ be an absolutely continuous scalar function with square integrable derivative ${\breve{\underline{\omega }}}_{\breve{z}}(\breve{z},t)$. The following inequality

$$\begin{array}{c}\hfill {\int }_0^{\overline{L}}{\breve{\underline{\omega }}}^2(\breve{z},t)d\breve{z}\le {\displaystyle \frac{4{\overline{L}}^2}{{\pi }^2}}{\int }_0^{\overline{L}}{\breve{\underline{\omega }}}_{\breve{z}}^2(\breve{z},t)d\breve{z},\end{array}$$

(13)

holds for $\breve{\underline{\omega }}(0,t)=0$. This inequality is also called the variation of the Wirtinger’s inequality.

Lemma 2 

([11]). For any scalar function $\breve{\varpi }(\breve{z},t)$, the following variant Agmon’s inequality

$$\begin{array}{ccc}\hfill \underset{\breve{z}\in [0,\overline{L}]}{max}{\left|\breve{\varpi }(\breve{z},t)\right|}^2& \le & 2\sqrt{{\int }_0^{\overline{L}}{\breve{\varpi }}^2(\breve{z},t)d\breve{z}}\sqrt{{\int }_0^{\overline{L}}{\breve{\varpi }}_{\breve{z}}^2(\breve{z},t)d\breve{z}}\hfill \\ & \le & \breve{\varsigma }{\int }_0^{\overline{L}}{\breve{\varpi }}^2(\breve{z},t)d\breve{z}+{\displaystyle \frac{1}{2\breve{\varsigma }}}{\int }_0^{\overline{L}}{\breve{\varpi }}_{\breve{z}}^2(\breve{z},t)d\breve{z},\hfill \end{array}$$

(14)

holds for all $\breve{\varsigma }>0$, if $\breve{\varpi }(0,t)=0$ satisfies.

3. Adaptive Multi-Fault Compensation Strategy

This section is devoted to the adaptive learning control technique to achieve the multi-fault-tolerant control design. This controller, together with adaptive updating laws, guarantees the closed-loop asymptotical stability. In view of the obtained closed-loop system, the well-posedness of its solution is analyzed by adopting the modified ${\mathcal{C}}_0$-semigroup framework.

3.1. Adaptive Learning Control Design

In view of (4) and (5), the system (1)–(3) becomes

$$\begin{array}{ccc}\hfill {\breve{\omega }}_{tt}(\breve{z},t)& =& -\breve{m}{\breve{\omega }}_{\breve{z}\breve{z}\breve{z}\breve{z}}(\breve{z},t),\breve{z}\in (0,\breve{L}),\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill \breve{\omega }(0,t)& =& {\breve{\omega }}_{\breve{z}}(0,t)={\breve{\omega }}_{\breve{z}\breve{z}}(\breve{L},t)=0,\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill {\breve{\omega }}_{tt}(\breve{L},t)& =& \breve{\rho }{\breve{v}}_o\left(t\right)+{\theta }_u^{\top }\breve{\varphi }\left(t\right).\hfill \end{array}$$

(17)

In order to deal with the multiplicative faults, here we introduce an auxiliary signal, which is designed

$$\begin{array}{c}\hfill \breve{\alpha }\left(t\right)=a_1{\breve{\omega }}_t(\breve{L},t)+a_2{\breve{\omega }}_{\breve{z}}(\breve{L},t)+a_3{\breve{\omega }}_{\breve{z}\breve{z}\breve{z}}(\breve{L},t),\end{array}$$

(18)

where $a_i,i=1,2,3$ are pending parameters satisfying

$$\begin{array}{c}\hfill a_1\ne 0,a_2^2+a_3^2\ne 0.\end{array}$$

(19)

Let $y_u\triangleq {\displaystyle \frac{1}{\breve{\rho }}}$. Let ${\widehat{y}}_u\left(t\right)$ and ${\widehat{\theta }}_u\left(t\right)$ denote the estimates of unknown parameter $y_u$ and unknown vector ${\theta }_u$, respectively. Defining the parameter estimate error as

$$\begin{array}{c}\hfill {\tilde{y}}_u\left(t\right)\triangleq {\widehat{y}}_u\left(t\right)-y_u,{\tilde{\theta }}_u\left(t\right)\triangleq {\widehat{\theta }}_u\left(t\right)-{\theta }_u,\end{array}$$

(20)

and we select the following Lyapunov functional candidate

$$\begin{array}{c}\hfill \breve{W}\left(t\right)={\breve{W}}_p\left(t\right)+{\breve{W}}_o\left(t\right)+{\breve{W}}_a\left(t\right)+{\breve{W}}_e\left(t\right),\end{array}$$

(21)

where

$$\begin{array}{ccc}\hfill {\breve{W}}_p\left(t\right)& =& {\displaystyle \frac{1}{2}}{\beta }_1{\int }_0^{\breve{L}}{\breve{\omega }}_t^2(\breve{z},t)d\breve{z}+{\displaystyle \frac{1}{2}}{\beta }_1{\beta }_2{\int }_0^{\breve{L}}{\breve{\omega }}_{\breve{z}}^2(\breve{z},t)d\breve{z}+{\displaystyle \frac{1}{2}}{\beta }_1\breve{m}{\int }_0^{\breve{L}}{\breve{\omega }}_{\breve{z}\breve{z}}^2(\breve{z},t)d\breve{z},\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill {\breve{W}}_o\left(t\right)& =& 2{\beta }_3{\int }_0^{\breve{L}}\breve{z}{\breve{\omega }}_t(\breve{z},t){\breve{\omega }}_{\breve{z}}(\breve{z},t)d\breve{z},\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill {\breve{W}}_a\left(t\right)& =& {\displaystyle \frac{1}{2}}{\breve{\alpha }}^2\left(t\right),\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill {\breve{W}}_e\left(t\right)& =& {\displaystyle \frac{1}{2}}\breve{\rho }{\gamma }_1^{-1}{\tilde{y}}_u^2\left(t\right)+{\displaystyle \frac{1}{2}}{\gamma }_2^{-1}{\tilde{\theta }}_u^{\top }\left(t\right){\tilde{\theta }}_u\left(t\right),\hfill \end{array}$$

(25)

where ${\beta }_i,i=1,2,3$ are positive parameters to be determined and ${\gamma }_i,i=1,2$ are arbitrary positive scalars.

First of all, the positive definiteness of functional (21) should be proven. It follows from (22) and (23) that

$$\begin{array}{ccc}\hfill {\breve{W}}_p\left(t\right)& \ge & {\beta }_{1}min\left\{1,{\beta }_2,\breve{m}\right\}\left({\displaystyle \frac{1}{2}}{\int }_0^{\breve{L}}{\breve{\omega }}_t^2(\breve{z},t)d\breve{z}+{\displaystyle \frac{1}{2}}{\int }_0^{\breve{L}}{\breve{\omega }}_{\breve{z}}^2(\breve{z},t)d\breve{z}+{\displaystyle \frac{1}{2}}{\int }_0^{\breve{L}}{\breve{\omega }}_{\breve{z}\breve{z}}^2(\breve{z},t)d\breve{z}\right),\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill |{\breve{W}}_o\left(t\right)|& \le & 4{\beta }_3\breve{L}\left({\displaystyle \frac{1}{2}}{\int }_0^{\breve{L}}{\breve{\omega }}_t^2(\breve{z},t)d\breve{z}+{\displaystyle \frac{1}{2}}{\int }_0^{\breve{L}}{\breve{\omega }}_{\breve{z}}^2(\breve{z},t)d\breve{z}+{\displaystyle \frac{1}{2}}{\int }_0^{\breve{L}}{\breve{\omega }}_{\breve{z}\breve{z}}^2(\breve{z},t)d\breve{z}\right).\hfill \end{array}$$

(27)

These inequalities, together with (24) and (25), lead to

$$\begin{array}{c}\hfill \breve{W}\left(t\right)\ge \left(1-{\displaystyle \frac{4{\beta }_3\breve{L}}{{\beta }_{1}min\left\{1,{\beta }_2,\breve{m}\right\}}}\right){\breve{W}}_p\left(t\right)+{\breve{W}}_a\left(t\right)+{\breve{W}}_e\left(t\right).\end{array}$$

The parameters ${\beta }_i,i=1,2,3$ are chosen to meet

$$\begin{array}{c}\hfill {\beta }_3\le {\displaystyle \frac{{\beta }_{1}min\{1,{\beta }_2,\breve{m}\}}{4\breve{L}}}.\end{array}$$

(28)

Such an inequality indicates that the functional (21) is positive definite.

Then under the condition (28), the derivative of (21) w.r.t. t is calculated along (15)–(17) as

$$\begin{array}{ccc}\hfill {\dot{\breve{W}}}_p\left(t\right)& =& -{\beta }_1\breve{m}{\breve{\omega }}_t(\breve{L},t){\breve{\omega }}_{\breve{z}\breve{z}\breve{z}}(\breve{L},t)+{\beta }_1{\beta }_2{\breve{\omega }}_{\breve{z}}(\breve{z},t){\breve{\omega }}_t(\breve{z},t)\hfill \\ & & -{\beta }_1{\beta }_2{\int }_0^{\breve{L}}{\breve{\omega }}_t(\breve{z},t){\breve{\omega }}_{\breve{z}\breve{z}}(\breve{z},t)d\breve{z},\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill {\dot{\breve{W}}}_o\left(t\right)& =& -2{\beta }_3\breve{m}\breve{L}{\breve{\omega }}_{\breve{z}}(\breve{L},t){\breve{\omega }}_{\breve{z}\breve{z}\breve{z}}(\breve{L},t)+{\beta }_3\breve{L}{\breve{\omega }}_t(\breve{L},t){\breve{\omega }}_{\breve{z}}(\breve{L},t)\hfill \\ & & -3{\beta }_3\breve{m}{\int }_0^{\breve{L}}{\breve{\omega }}_{\breve{z}\breve{z}}^2(\breve{z},t)d\breve{z}-{\beta }_3{\int }_0^{\breve{L}}{\breve{\omega }}_t^2(\breve{z},t)d\breve{z},\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill {\dot{\breve{W}}}_a\left(t\right)& =& \breve{\alpha }\left(t\right)\left(a_1{\breve{\omega }}_{tt}(\breve{L},t)+a_2{\breve{\omega }}_{\breve{z}t}(\breve{L},t)+a_3{\breve{\omega }}_{\breve{z}\breve{z}\breve{z}t}(\breve{L},t)\right)\hfill \\ & =& \breve{\alpha }\left(t\right)\left(a_1\left(\breve{\rho }{\breve{v}}_o\left(t\right)+{\theta }_u^{\top }\breve{\varphi }\left(t\right)\right)+a_2{\breve{\omega }}_{\breve{z}t}(\breve{L},t)+a_3{\breve{\omega }}_{\breve{z}\breve{z}\breve{z}t}(\breve{L},t)\right),\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill {\dot{\breve{W}}}_e\left(t\right)& =& {\gamma }_1^{-1}\breve{\rho }{\tilde{y}}_u\left(t\right){\dot{\widehat{y}}}_u\left(t\right)+{\gamma }_2^{-1}{\tilde{\theta }}_u^{\top }\left(t\right){\dot{\widehat{\theta }}}_u\left(t\right).\hfill \end{array}$$

(32)

Next, the multi-fault-tolerant boundary control law ${\breve{v}}_o\left(t\right)$ can be designed as

$$\begin{array}{ccc}\hfill {\breve{v}}_o\left(t\right)& =& -k{a}_1^{-1}\breve{\alpha }\left(t\right)-{\widehat{y}}_u\left(t\right){\widehat{\theta }}_u^{\top }\left(t\right)\breve{\varphi }\left(t\right)+a_1^{-1}{\widehat{y}}_u\left(t\right)\breve{\xi }\left(t\right)\hfill \\ & & -a_1^{-1}{\widehat{y}}_u\left(t\right)\left(a_2{\breve{\omega }}_{\breve{z}t}(\breve{L},t)+a_3{\breve{\omega }}_{\breve{z}\breve{z}\breve{z}t}(\breve{L},t)\right),\hfill \end{array}$$

(33)

where $\breve{\xi }\left(t\right)=b_1{\breve{\omega }}_t(\breve{L},t)+b_2{\breve{\omega }}_{\breve{z}}(\breve{L},t)+b_3{\breve{\omega }}_{\breve{z}\breve{z}\breve{z}}(\breve{L},t)$ with $b_i,i=1,2,3$ being pending design parameters, and $k>0$ is a positive scalar, which determines the convergence rate.

Substituting (33) into (31) yields

$$\begin{array}{ccc}\hfill {\dot{\breve{W}}}_a\left(t\right)& =& -k\breve{\rho }{\breve{\alpha }}^2\left(t\right)+a_1\breve{\alpha }\left(t\right)\left({\theta }_u^{\top }\breve{\varphi }\left(t\right)-\breve{\rho }{\widehat{y}}_u\left(t\right){\widehat{\theta }}_u^{\top }\left(t\right)\breve{\varphi }\left(t\right)\right)\hfill \\ & & +\breve{\alpha }\left(t\right)\left(a_2{\breve{\omega }}_{\breve{z}t}(\breve{L},t)+a_3{\breve{\omega }}_{\breve{z}\breve{z}\breve{z}t}(\breve{L},t)\right)\left(1-\breve{\rho }{\widehat{y}}_u\left(t\right)\right)+\breve{\alpha }\left(t\right)\breve{\rho }{\widehat{y}}_u\left(t\right)\breve{\xi }\left(t\right)\hfill \\ & =& -k\breve{\rho }{\breve{\alpha }}^2\left(t\right)-a_1\breve{\rho }\breve{\alpha }\left(t\right){\tilde{y}}_u\left(t\right){\widehat{\theta }}_u^{\top }\left(t\right)\breve{\varphi }\left(t\right)-a_1\breve{\alpha }\left(t\right){\tilde{\theta }}_u^{\top }\left(t\right)\breve{\varphi }\left(t\right)\hfill \\ & & +\breve{\alpha }\left(t\right)\breve{\rho }{\widehat{y}}_u\left(t\right)\breve{\xi }\left(t\right)-\breve{\rho }\breve{\alpha }\left(t\right){\tilde{y}}_u\left(t\right)\left(a_2{\breve{\omega }}_{\breve{z}t}(\breve{L},t)+a_3{\breve{\omega }}_{\breve{z}\breve{z}\breve{z}t}(\breve{L},t)\right).\hfill \end{array}$$

(34)

We select some proper parameters $a_i,i\in \{1,2,3\}$ and $b_i,i\in \{1,2,3\}$ such that

$$\begin{array}{c}\hfill \breve{\alpha }\left(t\right)\breve{\xi }\left(t\right)=-{\breve{\underline{\omega }}}_a\left(t\right)+{\breve{\underline{\omega }}}_b\left(t\right),\end{array}$$

(35)

where elements ${\breve{\underline{\omega }}}_a\left(t\right)$ and ${\breve{\underline{\omega }}}_b\left(t\right)$ are defined as

$$\begin{array}{ccc}\hfill {\breve{\underline{\omega }}}_a\left(t\right)& =& -{\beta }_1\breve{m}{\breve{\omega }}_t(\breve{L},t){\breve{\omega }}_{\breve{z}\breve{z}\breve{z}}(\breve{L},t)+({\beta }_3\breve{L}+{\beta }_1{\beta }_2){\breve{\omega }}_{\breve{z}}(\breve{L},t){\breve{\omega }}_t(\breve{L},t)\hfill \\ & & -2{\beta }_3\breve{m}\breve{L}{\breve{\omega }}_{\breve{z}}(\breve{L},t){\breve{\omega }}_{\breve{z}\breve{z}\breve{z}}(\breve{L},t),\hfill \end{array}$$

(36)

$$\begin{array}{ccc}\hfill {\breve{\underline{\omega }}}_b\left(t\right)& =& c_1{\breve{\omega }}_t^2(\breve{L},t)+c_2{\breve{\omega }}_{\breve{z}}^2(\breve{L},t)+c_3{\breve{\omega }}_{\breve{z}\breve{z}\breve{z}}^2(\breve{L},t),\hfill \end{array}$$

(37)

with parameters $c_i,i\in \{1,2,3\}$ being related to parameters $a_i,i\in \{1,2,3\}$, $b_i,i\in \{1,2,3\}$, and ${\beta }_i,i\in \{1,2,3\}$.

Under condition (35), (34) can be rewritten as

$$\begin{array}{ccc}\hfill {\dot{\breve{W}}}_a\left(t\right)& =& -k\breve{\rho }{\breve{\alpha }}^2\left(t\right)-a_1\breve{\alpha }\left(t\right){\tilde{\theta }}_u^{\top }\left(t\right)\breve{\varphi }\left(t\right)-\breve{\rho }{\widehat{y}}_u\left(t\right){\breve{\underline{\omega }}}_a\left(t\right)+\breve{\rho }{\widehat{y}}_u\left(t\right){\breve{\underline{\omega }}}_b\left(t\right)\hfill \\ & & -\breve{\rho }{\tilde{y}}_u\left(t\right)\breve{\alpha }\left(t\right)\left(a_1{\widehat{\theta }}_u^{\top }\left(t\right)\breve{\varphi }\left(t\right)+a_2{\breve{\omega }}_{\breve{z}t}(\breve{L},t)+a_3{\breve{\omega }}_{\breve{z}\breve{z}\breve{z}t}(\breve{L},t)\right),\hfill \end{array}$$

(38)

In order to deal with the term ${\underline{\omega }}_b\left(t\right)$, letting parameters $a_3=0$ and $b_1=0$, we have

$$\begin{array}{ccc}\hfill a_1{b}_2& =& {\beta }_3\breve{L}+{\beta }_1{\beta }_2,\hfill \end{array}$$

(39)

$$\begin{array}{ccc}\hfill a_1{b}_3& =& -{\beta }_1\breve{m},\hfill \end{array}$$

(40)

$$\begin{array}{ccc}\hfill a_2{b}_3& =& -2{\beta }_3\breve{m}\breve{L}.\hfill \end{array}$$

(41)

These three inequalities imply

$$\begin{array}{c}\hfill {\breve{\underline{\omega }}}_b\left(t\right)={\displaystyle \frac{2{\beta }_3\breve{L}\left({\beta }_3\breve{L}+{\beta }_1{\beta }_2\right)}{{\beta }_1}}{\breve{\omega }}_{\breve{z}}^2(\breve{L},t).\end{array}$$

(42)

Hereafter, the parameter updating laws are designed as

$$\begin{array}{ccc}\hfill {\dot{\widehat{y}}}_u\left(t\right)& =& {\gamma }_1\breve{\alpha }\left(t\right)\left(a_1{\widehat{\theta }}_u^{\top }\left(t\right)\breve{\varphi }\left(t\right)+a_2{\breve{\omega }}_{\breve{z}t}(\breve{L},t)\right)+{\gamma }_1{\breve{\underline{\omega }}}_a\left(t\right)\hfill \\ & & -{\displaystyle \frac{2{\beta }_3\breve{L}\left({\beta }_3\breve{L}+{\beta }_1{\beta }_2\right){\gamma }_1}{{\beta }_1}}{\breve{\omega }}_{\breve{z}}^2(\breve{L},t),\hfill \end{array}$$

(43)

$$\begin{array}{ccc}\hfill {\dot{\widehat{\theta }}}_u\left(t\right)& =& {\gamma }_2{a}_1\breve{\alpha }\left(t\right)\breve{\varphi }\left(t\right).\hfill \end{array}$$

(44)

Under (43) and (44), the derivative of $\breve{W}\left(t\right)$ w.r.t. t can be represented as

$$\begin{array}{ccc}\hfill \dot{\breve{W}}\left(t\right)& =& -k\breve{\rho }{\breve{\alpha }}^2\left(t\right)-3{\beta }_3\breve{m}{\int }_0^{\breve{L}}{\breve{\omega }}_{\breve{z}\breve{z}}^2(\breve{z},t)d\breve{z}\hfill \\ & & -{\beta }_3{\int }_0^{\breve{L}}{\breve{\omega }}_t^2(\breve{z},t)d\breve{z}+\breve{\mu }\left(t\right)+{\breve{\underline{\omega }}}_b\left(t\right),\hfill \end{array}$$

(45)

where

$$\begin{array}{ccc}\hfill \breve{\mu }\left(t\right)& =& -{\beta }_1{\beta }_2{\int }_0^{\breve{L}}{\breve{\omega }}_t(\breve{z},t){\breve{\omega }}_{\breve{z}\breve{z}}(\breve{z},t)d\breve{z}.\hfill \end{array}$$

(46)

According to Lemma 1, Lemma 2, and the well-known Young inequality, we have

$$\begin{array}{ccc}\hfill \breve{\mu }\left(t\right)& \le & {\displaystyle \frac{{\beta }_1{\beta }_2}{8}}{\int }_0^{\breve{L}}{\breve{\omega }}_t^2(\breve{z},t)d\breve{z}+4{\beta }_1{\beta }_2{\int }_0^{\breve{L}}{\breve{\omega }}_{\breve{z}\breve{z}}^2(\breve{z},t)d\breve{z},\hfill \end{array}$$

(47)

$$\begin{array}{ccc}\hfill {\breve{\underline{\omega }}}_b\left(t\right)& \le & {\displaystyle \frac{2{\beta }_3\breve{L}\left({\beta }_3\breve{L}+{\beta }_1{\beta }_2\right)}{{\beta }_1}}{\int }_0^{\breve{L}}{\breve{\omega }}_{\breve{z}\breve{z}}^2(\breve{z},t)d\breve{z}+{\displaystyle \frac{2{\beta }_3\breve{L}\left({\beta }_3\breve{L}+{\beta }_1{\beta }_2\right)}{{\beta }_1}}{\int }_0^{\breve{L}}{\breve{\omega }}_{\breve{z}}^2(\breve{z},t)d\breve{z}\hfill \\ \hfill & \le & \breve{\lambda }{\int }_0^{\breve{L}}{\breve{\omega }}_{\breve{z}\breve{z}}^2(\breve{z},t)d\breve{z}.\hfill \end{array}$$

(48)

where

$$\begin{array}{c}\hfill \breve{\lambda }={\displaystyle \frac{2{\beta }_3\breve{L}(4{\breve{L}}^2+{\pi }^2)({\beta }_3\breve{L}+{\beta }_1{\beta }_2)}{{\pi }^2{\beta }_1}}.\end{array}$$

(49)

Substituting these inequalities (47) and (48) into (45), we have

$$\begin{array}{ccc}\hfill \dot{\breve{W}}\left(t\right)& \le & -k\breve{\rho }{\breve{\alpha }}^2\left(t\right)-\left({\beta }_3-{\displaystyle \frac{{\beta }_1{\beta }_2}{8}}\right){\int }_0^{\breve{L}}{\breve{\omega }}_t^2(\breve{z},t)d\breve{z}\hfill \\ & & -\left(3{\beta }_3\breve{m}-4{\beta }_1{\beta }_2-\breve{\lambda }\right){\int }_0^{\breve{L}}{\breve{\omega }}_{\breve{z}\breve{z}}^2(\breve{z},t)d\breve{z}.\hfill \end{array}$$

(50)

On the basis of fulfilling the inequality (28), parameters ${\beta }_i,i=1,2,3$ should also satisfy

$$\begin{array}{c}\hfill 8{\beta }_3-{\beta }_1{\beta }_2>0,3{\beta }_3\breve{m}-4{\beta }_1{\beta }_2-\breve{\lambda }>0.\end{array}$$

(51)

Based on the above design and analysis, we can concentrate on the first significant outcome of this study.

Theorem 1. 

Regarding the internal and boundary dynamics described by (1)–(3) with multi-fault and disturbance satisfying (4)–(6), designing fault tolerant boundary control law as

$$\begin{array}{ccc}\hfill {\breve{v}}_o\left(t\right)& =& -k{\breve{\omega }}_t(\breve{L},t)-2k{\beta }_3{{\beta }_1}^{-1}\breve{L}{\breve{\omega }}_{\breve{z}}(\breve{L},t)-{\widehat{y}}_u\left(t\right){\widehat{\theta }}_u^{\top }\left(t\right)\breve{\varphi }\left(t\right)+({\beta }_3\breve{L}+{\beta }_1{\beta }_2)a_1^{-2}{\widehat{y}}_u\left(t\right)\hfill \\ & & \times {\breve{\omega }}_{\breve{z}}(\breve{L},t)-{\beta }_1\breve{m}{a}_1^{-2}{\widehat{y}}_u\left(t\right){\breve{\omega }}_{\breve{z}\breve{z}\breve{z}}(\breve{L},t)-2{\beta }_3{{\beta }_1}^{-1}\breve{L}{\breve{\omega }}_{\breve{z}t}(\breve{L},t),\hfill \end{array}$$

(52)

with $k>0$, $a_1\ne 0$, and ${\beta }_i>0,i\in \{1,2,3\}$ meeting (28) and (51), and parameter update laws as

$$\begin{array}{ccc}\hfill {\dot{\widehat{y}}}_u\left(t\right)& =& {\gamma }_1{a}_1^2{\breve{\omega }}_t(\breve{L},t){\widehat{\theta }}_u^{\top }\left(t\right)\breve{\varphi }\left(t\right)-{\gamma }_1{\beta }_1\breve{m}{\breve{\omega }}_t(\breve{L},t){\breve{\omega }}_{\breve{z}\breve{z}\breve{z}}(\breve{L},t)\hfill \\ & & +{\gamma }_1({\beta }_3\breve{L}+{\beta }_1{\beta }_2){\breve{\omega }}_{\breve{z}}(\breve{L},t){\breve{\omega }}_t(\breve{L},t)-2{\gamma }_1{\beta }_3\breve{m}\breve{L}{\breve{\omega }}_{\breve{z}}(\breve{L},t){\breve{\omega }}_{\breve{z}\breve{z}\breve{z}}(\breve{L},t)\hfill \\ & & +2{\beta }_2{{\beta }_1}^{-1}\breve{L}{a}_1^2\left({\breve{\omega }}_{\breve{xz}}(\breve{L},t){\widehat{\theta }}_u^{\top }\left(t\right)\breve{\varphi }\left(t\right)+{\breve{\omega }}_t(\breve{L},t){\breve{\omega }}_{\breve{z}t}(\breve{L},t)\right)\hfill \\ & & +4{\gamma }_1{\beta }_1^{-2}{\beta }_3^2{\breve{L}}^2{\breve{\omega }}_{\breve{z}}(\breve{L},t){\breve{\omega }}_{\breve{z}t}(\breve{L},t)-2{{\beta }_1}^{-1}{\beta }_3\breve{L}\left({\beta }_3\breve{L}+{\beta }_1{\beta }_2\right){\gamma }_1{\breve{\omega }}_{\breve{z}}^2(\breve{L},t),\hfill \end{array}$$

(53)

$$\begin{array}{ccc}\hfill {\dot{\widehat{\theta }}}_u\left(t\right)& =& {{\beta }_1}^{-1}{\gamma }_2{a}_1^2\left({\beta }_1+2{\beta }_3\breve{L}\right)\left({\breve{\omega }}_t(\breve{L},t)+{\breve{\omega }}_{\breve{z}}(\breve{L},t)\right)\breve{\varphi }\left(t\right),\hfill \end{array}$$

(54)

with ${\gamma }_1>0$ and ${\gamma }_2>0$, for any initial condition, the system states are asymptotically stable, i.e.,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}{\int }_0^{\breve{L}}\left({\breve{\omega }}^2(\breve{z},t)+{\breve{\omega }}_t^2(\breve{z},t)\right)d\breve{z}=0.\end{array}$$

(55)

Proof. 

It follows from (50) that the derivative of the Lyapunov functional (21) w.r.t t is a non-increasing function. This indicates that the states ${\breve{\omega }}_t(\breve{z},t)$ and ${\breve{\omega }}_{\breve{z}\breve{z}}(\breve{z},t)$ are asymptotically converging to zero when t goes to infinity in the sense of ${\mathcal{L}}^2$-norm, i.e.,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}{\int }_0^{\breve{L}}{\breve{\omega }}_t^2(\breve{z},t)d\breve{z}=\underset{t\to \infty }{lim}{\int }_0^{\breve{L}}{\breve{\omega }}_{\breve{z}\breve{z}}^2(\breve{z},t)d\breve{z}=0.\end{array}$$

(56)

In view of the variation of the Wirtinger’s inequality in Lemma 1, together with the zero boundary conditions $\breve{\omega }(0,t)=0$ and ${\breve{\omega }}_{\breve{z}}(0,t)=0$, we have

$$\begin{array}{ccc}\hfill {\int }_0^{\breve{L}}{\breve{\omega }}^2(\breve{z},t)d\breve{z}& \le & {\displaystyle \frac{4{\breve{L}}^2}{{\pi }^2}}{\int }_0^{\breve{L}}{\breve{\omega }}_{\breve{z}}^2(\breve{z},t)d\breve{z},{\int }_0^{\breve{L}}{\breve{\omega }}_{\breve{z}}^2(\breve{z},t)d\breve{z}\le {\displaystyle \frac{4{\breve{L}}^2}{{\pi }^2}}{\int }_0^{\breve{L}}{\breve{\omega }}_{\breve{z}\breve{z}}^2(\breve{z},t)d\breve{z}.\hfill \end{array}$$

(57)

These two inequalities imply that

$$\begin{array}{c}\hfill {\int }_0^{\breve{L}}{\breve{\omega }}^2(\breve{z},t)d\breve{z}\le {\displaystyle \frac{16{\breve{L}}^4}{{\pi }^4}}{\int }_0^{\breve{L}}{\breve{\omega }}_{\breve{z}\breve{z}}^2(\breve{z},t)d\breve{z}.\end{array}$$

(58)

Taking the limit when $t\to \infty$ and utilizing the condition (56), one has

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}{\int }_0^{\breve{L}}{\breve{\omega }}^2(\breve{z},t)d\breve{z}\le {\displaystyle \frac{16{\breve{L}}^4}{{\pi }^4}}\underset{t\to \infty }{lim}{\int }_0^{\breve{L}}{\breve{\omega }}_{\breve{z}\breve{z}}^2(\breve{z},t)d\breve{z}=0.\end{array}$$

(59)

This relationship, together with (56) evinces that the asymptotical stability of the system states $\breve{\omega }(\breve{z},t)$ and ${\breve{\omega }}_t(\breve{z},t)$ is guaranteed in the sense of ${\mathcal{L}}^2$-norm for any initial condition. This completes the proof. □

Remark 2. 

It should be noticed that all the signals used in the fault tolerant boundary controller (52) can be metered or calculated. The boundary velocity ${\breve{\omega }}_t(\breve{L},t)$ can be measured by velocity transducers. The boundary rotation ${\breve{\omega }}_{\breve{z}}(\breve{L},t)$ can be obtained by inclinometers. The parameter estimates ${\widehat{y}}_u\left(t\right)$ and ${\widehat{\theta }}_u\left(t\right)$ can be generated from (53) and (54), respectively. The boundary shear force ${\breve{\omega }}_{\breve{z}\breve{z}\breve{z}}(\breve{L},t)$ can be achieved by shear force transducers. The boundary angular velocity ${\breve{\omega }}_{\breve{z}t}(\breve{L},t)$ can be calculated by utilizing a backward difference algorithm due to the acquired boundary rotation angular ${\breve{\omega }}_{\breve{z}}(\breve{L},t)$.

Analysis of measurement noise in high-order partial differential terms. In the control signal (52) and parameter update law (53), the high-order partial deferential terms may be very sensitive to noise. In order to compensate for the effect caused by the measurement noise, an adaptive observer mechanism is designed as follows.

Consider a system described by

$$\begin{array}{c}\hfill \begin{array}{c}{\breve{\omega }}_{tt}(\breve{z},t)+\breve{m}{\breve{\omega }}_{\breve{z}\breve{z}\breve{z}\breve{z}}(\breve{z},t)=0,\breve{z}\in (0,\breve{L}),\hfill \\ \breve{\omega }(0,t)={\breve{\omega }}_{\breve{z}}(0,t)={\breve{\omega }}_{\breve{z}\breve{z}}(\breve{L},t)=0,\hfill \\ {\breve{\omega }}_{tt}(\breve{L},t)={\breve{\mathrm{\Omega }}}_o\left(t\right)+\breve{d}\left(t\right),\hfill \end{array}\end{array}$$

(60)

where the measurement signals are defined by

$$\begin{array}{c}\hfill y_1\left(t\right)={\breve{\omega }}_{\breve{z}\breve{z}\breve{z}}(\breve{L},t)+N\left(t\right),y_2\left(t\right)={\breve{\omega }}_t(\breve{L},t),\end{array}$$

(61)

with $\left|N\left(t\right)\right|\le \overline{N}$ being the bounded measurement noise in high-order partial differential term and $\overline{N}$ being a known scalar. Then an adaptive observer is designed as

$$\begin{array}{c}\hfill \begin{array}{c}{\widehat{\breve{\omega }}}_{tt}(\breve{z},t)+\breve{m}{\widehat{\breve{\omega }}}_{\breve{z}\breve{z}\breve{z}\breve{z}}(\breve{z},t)=0,\breve{z}\in (0,\breve{L}),\hfill \\ \widehat{\breve{\omega }}(0,t)={\widehat{\breve{\omega }}}_{\breve{z}}(0,t)={\widehat{\breve{\omega }}}_{\breve{z}\breve{z}}(\breve{L},t)=0,\hfill \\ {\widehat{\breve{\omega }}}_t(\breve{L},t)=y_2\left(t\right)-L_O(y_1\left(t\right)-{\widehat{\breve{\omega }}}_{\breve{z}\breve{z}\breve{z}}(\breve{L},t)+Z\left(t\right)),\hfill \end{array}\end{array}$$

(62)

where $L_o>0$ is the observer gain and

$$\begin{array}{c}\hfill Z\left(t\right)=\overline{N}\mathrm{sign}({\breve{\omega }}_{\breve{z}\breve{z}\breve{z}}(\breve{L},t)-{\widehat{\breve{\omega }}}_{\breve{z}\breve{z}\breve{z}}(\breve{L},t)).\end{array}$$

(63)

Define the estimate error as

$$\begin{array}{c}\hfill \zeta (\breve{z},t)=\breve{\omega }(\breve{z},t)-\widehat{\breve{\omega }}(\breve{z},t),\end{array}$$

(64)

then the error dynamics can be expressed by

$$\begin{array}{c}\hfill \begin{array}{c}{\zeta }_{tt}(\breve{z},t)+\breve{m}{\zeta }_{\breve{z}\breve{z}\breve{z}\breve{z}}(\breve{z},t)=0,\hfill \\ \zeta (0,t)={\zeta }_{\breve{z}}(0,t)={\zeta }_{\breve{z}\breve{z}}(\breve{L},t)=0,\hfill \\ {\zeta }_t(\breve{z},t)=L_o({\zeta }_{\breve{z}\breve{z}\breve{z}}(\breve{L},t)+N\left(t\right)+\overline{N}\mathrm{sign}\left({\zeta }_{\breve{z}\breve{z}\breve{z}}(\breve{L},t)\right)).\hfill \end{array}\end{array}$$

(65)

Select a positive definite function as

$$\begin{array}{c}\hfill E\left(t\right)={\int }_0^{\breve{L}}{\zeta }_t^2(\breve{z},t)d\breve{z}+\breve{m}{\int }_0^{\breve{L}}{\zeta }_{\breve{z}\breve{z}}^2(\breve{z},t)d\breve{z}.\end{array}$$

(66)

Taking the time derivative along the error dynamics yields

$$\begin{array}{c}\hfill \dot{E}\left(t\right)\le -2\breve{m}{L}_o{\zeta }_{\breve{z}\breve{z}\breve{z}}^2(\breve{L},t)\le 0.\end{array}$$

(67)

The positive definite property of $E\left(t\right)$, together with the negative definite property of its time derivative, indicates that $E\left(t\right)$ is decreasing. Taking the integration of $\dot{E}\left(t\right)$ with respect to time from zero to infinity and using the comparison principle, one has

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}{\zeta }_{\breve{z}\breve{z}\breve{z}}(\breve{L},t)=0,\end{array}$$

(68)

which indicate that the proposed adaptive observer can effectively compensate for the effect caused by the measurement noise.

3.2. Well-Posedness Analysis

After proving the closed-loop system stability, this section supplies a detailed well-posedness analysis of the solution under the control law (52). Define a Hilbert space

$$\begin{array}{c}\hfill \mathcal{H}\triangleq H_o^2(0,\breve{L})\times {\mathcal{L}}^2(0,\breve{L})\times {\Re }^{a+d+1},\end{array}$$

(69)

where $H_o^2(0,\breve{L})\triangleq \left\{ϵ|ϵ\in H^2(0,\breve{L}),ϵ\left(0\right)={ϵ}^{\prime }\left(0\right)=0\right\}$ and its inner product norm is equipped with

$$\begin{array}{ccc}\hfill \parallel \breve{\psi }\left(\breve{z}\right){\parallel }_{\mathcal{H}}^2& \triangleq & {\displaystyle \frac{1}{2}}\left({\int }_0^{\breve{L}}\breve{m}{\breve{\psi }}_2^2\left(\breve{z}\right)d\breve{z}+{\gamma }_1{\left({\breve{\psi }}_4-{\displaystyle \frac{1}{\breve{\rho }}}\right)}^2\right)+{\displaystyle \frac{1}{2}}\left({\int }_0^{\breve{L}}{\breve{\psi }}_1^2\left(\breve{z}\right)d\breve{z}+{\gamma }_2{({\breve{\psi }}_5-\theta )}^{\top }({\breve{\psi }}_5-\theta )\right),\hfill \end{array}$$

with $\breve{\psi }\left(\breve{z}\right)\triangleq {[\breve{\omega }\left(\breve{z}\right),{\breve{\omega }}_t\left(\breve{z}\right),{\breve{\varphi }}^{\top },{\widehat{y}}_u,{\widehat{\theta }}_u^{\top }]}^{\top }$.

In view of these presentations, the closed-loop form of (1)–(3) under the fault tolerant boundary control law (52) and parameter updating laws (53) and (54) can be expressed by

$$\begin{array}{c}\hfill \dot{\breve{\vartheta }}\left(t\right)\triangleq \breve{\mathcal{A}}\breve{\vartheta }\left(t\right),\breve{\vartheta }\left(0\right)\in \mathcal{H},\end{array}$$

(70)

where $\breve{\vartheta }\left(0\right)\triangleq {[{\breve{\omega }}_0\left(\breve{z}\right),{\breve{\omega }}_1\left(\breve{z}\right),{\breve{\varphi }}_0^{\top },{\widehat{y}}_{u0},{\widehat{\theta }}_{u0}^{\top }]}^{\top }$ and the system operator can be defined by

$$\begin{array}{c}\hfill \breve{\mathcal{A}}\breve{\psi }\triangleq \left[\begin{array}{c}{\breve{\psi }}_2\\ -\breve{m}{\breve{\psi }}_1^{\prime \prime \prime \prime }\\ \breve{G}{\breve{\psi }}_3\\ \breve{\kappa }\left(\breve{\psi }\right)\\ {\displaystyle \frac{{\gamma }_2{a}_1^2\left({\beta }_1+2{\beta }_3\breve{L}\right)}{{\beta }_1}}\left({\breve{\psi }}_2\left(\breve{L}\right)+{\breve{\psi }}_1^{\prime }\left(\breve{L}\right)\right){\breve{\psi }}_3\end{array}\right],\end{array}$$

(71)

with

$$\begin{array}{ccc}\hfill \breve{\kappa }\left(\breve{\psi }\right)& =& {\gamma }_1{a}_1^2{\breve{\psi }}_2\left(\breve{L}\right){\breve{\psi }}_5^{\top }{\breve{\psi }}_3-{\gamma }_1{\beta }_1\breve{m}{\breve{\psi }}_2\left(\breve{L}\right){\breve{\psi }}_1^{\prime \prime \prime }\left(\breve{L}\right)-2{\gamma }_1{\beta }_3\breve{m}\breve{L}{\breve{\psi }}_1^{\prime }\left(\breve{L}\right){\breve{\psi }}_1^{\prime \prime \prime }\left(\breve{L}\right)\hfill \\ & & +{\gamma }_1({\beta }_3\breve{L}+{\beta }_1{\beta }_2){\breve{\psi }}_1^{\prime }\left(\breve{L}\right){\breve{\psi }}_2\left(\breve{L}\right)+{\displaystyle \frac{2{\beta }_2\breve{L}{a}_1^2}{{\beta }_1}}\left({\breve{\psi }}_1^{\prime }\left(\breve{L}\right){\breve{\psi }}_5^{\top }{\breve{\psi }}_3+{\breve{\psi }}_2\left(\breve{L}\right){\breve{\psi }}_2^{\prime }\left(\breve{L}\right)\right)\hfill \\ & & +{\displaystyle \frac{4{\gamma }_1{\beta }_3^2{\breve{L}}^2}{{\beta }_1^2}}{\breve{\psi }}_1^{\prime }\left(\breve{L}\right){\breve{\psi }}_2^{\prime }\left(\breve{L}\right)-{\displaystyle \frac{2{\beta }_3\breve{L}\left({\beta }_3\breve{L}+{\beta }_1{\beta }_2\right){\gamma }_1}{{\beta }_1}}{\left({\breve{\psi }}_1^{\prime }\left(\breve{L}\right)\right)}^2.\hfill \end{array}$$

(72)

The domain of $\breve{\mathcal{A}}$ can be defined by

$$\begin{array}{ccc}\hfill \breve{\mathcal{D}}\left(\breve{\mathcal{A}}\right)& \triangleq & \left\{\breve{\psi }\right|{\breve{\psi }}_1\in {\mathcal{H}}_o^2(0,\breve{L})\cap H^4(0,\breve{L}),{\breve{\psi }}_2\in {\mathcal{H}}_o^2(0,\breve{L})\hfill \\ & & {\breve{\psi }}_3\in {\Re }^{a+d},{\breve{\psi }}_4\in \Re ,{\breve{\psi }}_5\in {\Re }^{a+d},{\breve{\psi }}_1^{\prime \prime }\left(\breve{L}\right)=0\hfill \\ & & {\dot{\breve{\psi }}}_2\left(\breve{L}\right)=-k{\breve{\psi }}_2\left(\breve{L}\right)-{\displaystyle \frac{2k{\beta }_3\breve{L}}{{\beta }_1}}{\breve{\psi }}_1^{\prime }\left(\breve{L}\right)-{\breve{\psi }}_4{\breve{\psi }}_5^{\top }{\breve{\psi }}_3\hfill \\ & & +{\displaystyle \frac{{\beta }_3\breve{L}+{\beta }_1{\beta }_2}{a_1^2}}{\breve{\psi }}_4{\breve{\psi }}_1^{\prime }\left(\breve{L}\right)-{\displaystyle \frac{{\beta }_1\breve{m}}{a_1^2}}{\breve{\psi }}_4{\breve{\psi }}_1^{\prime \prime \prime }\left(\breve{L}\right)-{\displaystyle \frac{2{\beta }_3\breve{L}}{{\beta }_1}}{\breve{\psi }}_2^{\prime }\left(\breve{L}\right)\}.\hfill \end{array}$$

(73)

Since the inner product norm of $\mathcal{H}$ is equivalent to Lyapunov functional $\breve{V}\left(t\right)$, this implies that the system operator $\breve{\mathcal{A}}$ is dissipative.

Then the invertibility of system operator $\breve{\mathcal{A}}$ is proved. Letting $\overline{\psi }\in \mathcal{H}$ in view of (71)

$$\begin{array}{c}\hfill \breve{\mathcal{A}}\breve{\psi }=\overline{\psi },\breve{\psi }\in \breve{\mathcal{D}}\left(\breve{\mathcal{A}}\right),\end{array}$$

(74)

we can acquire the following relationship:

$$\begin{array}{ccc}\hfill {\overline{\psi }}_1\left(\breve{z}\right)& =& {\breve{\psi }}_2\left(\breve{z}\right),\hfill \end{array}$$

(75)

$$\begin{array}{ccc}\hfill {\overline{\psi }}_2\left(\breve{z}\right)& =& -\check{m}{\breve{\psi }}_1^{\prime \prime \prime \prime }\left(\breve{z}\right),\hfill \end{array}$$

(76)

$$\begin{array}{ccc}\hfill {\overline{\psi }}_3& =& \breve{G}{\breve{\psi }}_3,\hfill \end{array}$$

(77)

$$\begin{array}{ccc}\hfill {\overline{\psi }}_4& =& {\gamma }_1{a}_1^2{\breve{\psi }}_2\left(\breve{L}\right){\breve{\psi }}_5^{\top }{\breve{\psi }}_3-{\gamma }_1{\beta }_1\breve{m}{\breve{\psi }}_2\left(\breve{L}\right){\breve{\psi }}_1^{\prime \prime \prime }\left(\breve{L}\right)-2{\gamma }_1{\beta }_3\breve{m}\breve{L}{\breve{\psi }}_1^{\prime }\left(\breve{L}\right){\breve{\psi }}_1^{\prime \prime \prime }\left(\breve{L}\right)\hfill \\ & & +{\gamma }_1({\beta }_3\breve{L}+{\beta }_1{\beta }_2){\breve{\psi }}_1^{\prime }\left(\breve{L}\right){\breve{\psi }}_2\left(\breve{L}\right)+{\displaystyle \frac{2{\beta }_2\breve{L}{a}_1^2}{{\beta }_1}}\left({\breve{\psi }}_1^{\prime }\left(\breve{L}\right){\breve{\psi }}_5^{\top }{\breve{\psi }}_3+{\breve{\psi }}_2\left(\breve{L}\right){\breve{\psi }}_2^{\prime }\left(\breve{L}\right)\right)\hfill \\ & & +{\displaystyle \frac{4{\gamma }_1{\beta }_3^2{L}^2}{{\beta }_1^2}}{\psi }_1^{\prime }\left(L\right){\psi }_2^{\prime }\left(L\right)-{\displaystyle \frac{2{\beta }_3\breve{L}\left({\beta }_3\breve{L}+{\beta }_1{\beta }_2\right){\gamma }_1}{{\beta }_1}}{\left({\breve{\psi }}_1^{\prime }\left(\breve{L}\right)\right)}^2,\hfill \end{array}$$

(78)

$$\begin{array}{ccc}\hfill {\overline{\psi }}_5& =& {\displaystyle \frac{{\gamma }_2{a}_1^2\left({\beta }_1+2{\beta }_3\breve{L}\right)}{{\beta }_1}}\left({\breve{\psi }}_2\left(\breve{L}\right)+{\breve{\psi }}_1^{\prime }\left(\breve{L}\right)\right){\breve{\psi }}_3,\hfill \end{array}$$

(79)

with respect to the boundary conditions

$$\begin{array}{ccc}\hfill {\breve{\psi }}_1\left(0\right)& =& {\breve{\psi }}_1^{\prime }\left(0\right)={\breve{\psi }}_1^{\prime \prime }\left(\breve{L}\right)=0,\hfill \end{array}$$

(80)

$$\begin{array}{ccc}\hfill {\dot{\breve{\psi }}}_2\left(\breve{L}\right)& =& -k{\breve{\psi }}_2\left(\breve{L}\right)-{\displaystyle \frac{2k{\beta }_3\breve{L}}{{\beta }_1}}{\breve{\psi }}_1^{\prime }\left(\breve{L}\right)-{\breve{\psi }}_4{\breve{\psi }}_5^{\top }{\breve{\psi }}_3+{\displaystyle \frac{{\beta }_3\breve{L}+{\beta }_1{\beta }_2}{a_1^2}}{\breve{\psi }}_4{\breve{\psi }}_1^{\prime }\left(\breve{L}\right)\hfill \\ & & -{\displaystyle \frac{{\beta }_1\breve{m}}{a_1^2}}{\breve{\psi }}_4{\breve{\psi }}_1^{\prime \prime \prime }\left(\breve{L}\right)-{\displaystyle \frac{2{\beta }_3\breve{L}}{{\beta }_1}}{\breve{\psi }}_2^{\prime }\left(\breve{L}\right),\hfill \end{array}$$

(81)

Solving the differential Equations (75)–(79) subject to (80) and (81) yields

$$\begin{array}{ccc}\hfill {\breve{\psi }}_1\left(\breve{z}\right)& =& -{\int }_0^{\breve{z}}{\int }_0^{\breve{y}}{\int }_{\breve{L}}^{\breve{x}}{\int }_0^{\breve{s}}{\breve{m}}^{-1}{\overline{\psi }}_2\left(\breve{e}\right)d\breve{e}d\breve{s}d\breve{x}d\breve{y},\hfill \end{array}$$

(82)

$$\begin{array}{ccc}\hfill {\breve{\psi }}_2\left(\breve{z}\right)& =& {\overline{\psi }}_1\left(\breve{z}\right),\hfill \end{array}$$

(83)

$$\begin{array}{ccc}\hfill {\breve{\psi }}_3& =& {\breve{G}}^{-1}{\overline{\psi }}_3,\hfill \end{array}$$

(84)

$$\begin{array}{ccc}\hfill {\breve{\psi }}_4& =& {\breve{\mathrm{\Omega }}}_3^{-1}({\overline{\psi }}_2\left(\breve{L}\right)+k{\overline{\psi }}_1\left(\breve{L}\right)+2{\beta }_1^{-1}{\beta }_3\breve{L}{\overline{\psi }}_1^{\prime }\left(\breve{L}\right)\hfill \\ & & -2k{\breve{m}}^{-1}{\beta }_1^{-1}{\beta }_3\breve{L}{\int }_0^{\breve{L}}{\int }_{\breve{L}}^{\breve{x}}{\int }_0^{\breve{s}}{\overline{\psi }}_2\left(\breve{e}\right)d\breve{e}d\breve{s}d\breve{x}),\hfill \end{array}$$

(85)

$$\begin{array}{ccc}\hfill {\breve{\psi }}_5& =& {\breve{\mathrm{\Omega }}}_2^{-1}{\breve{\mathrm{\Omega }}}_1{\left({\breve{G}}^{-1}{\overline{\psi }}_3{\overline{\psi }}_3^{\top }{\breve{G}}^{-\top }\right)}^{-1}{\breve{G}}^{-1}{\overline{\psi }}_3,\hfill \end{array}$$

(86)

where

$$\begin{array}{ccc}\hfill {\overline{\psi }}_2\left(\breve{L}\right)& =& -k{\breve{\psi }}_2\left(\breve{L}\right)-{\displaystyle \frac{2k{\beta }_3\breve{L}}{{\beta }_1}}{\breve{\psi }}_1^{\prime }\left(\breve{L}\right)-{\breve{\psi }}_4{\breve{\psi }}_5^{\top }{\breve{\psi }}_3\hfill \\ & & +{\displaystyle \frac{{\beta }_3\breve{L}+{\beta }_1{\beta }_2}{a_1^2}}{\breve{\psi }}_4{\breve{\psi }}_1^{\prime }\left(\breve{L}\right)-{\displaystyle \frac{{\beta }_1\breve{m}}{a_1^2}}{\breve{\psi }}_4{\breve{\psi }}_1^{\prime \prime \prime }\left(\breve{L}\right)-{\displaystyle \frac{2{\beta }_3\breve{L}}{{\beta }_1}}{\breve{\psi }}_2^{\prime }\left(\breve{L}\right),\hfill \end{array}$$

(87)

$$\begin{array}{ccc}\hfill {\breve{\mathrm{\Omega }}}_1& =& {\overline{\psi }}_4-{\gamma }_1{\beta }_1{\overline{\psi }}_1\left(\breve{L}\right){\int }_0^{\breve{L}}{\overline{\psi }}_2\left(\breve{e}\right)d\breve{e}-{\breve{\zeta }}_1{\overline{\psi }}_1\left(\breve{L}\right){\overline{\psi }}_1^{\prime }\left(\breve{L}\right)\hfill \\ & & +{\breve{\zeta }}_2{\int }_0^{\breve{L}}{\int }_{\breve{L}}^{\breve{x}}{\int }_0^{\breve{s}}{\overline{\psi }}_2\left(\breve{e}\right)d\breve{e}d\breve{s}d\breve{x}+2{\gamma }_1{\breve{m}}^{-2}{\beta }_1^{-1}{\beta }_3\breve{L}\hfill \\ & & \times ({\beta }_3\breve{L}+{\beta }_1{\beta }_2){\left({\int }_0^{\breve{L}}{\int }_{\breve{L}}^{\breve{x}}{\int }_0^{\breve{s}}{\overline{\psi }}_2\left(\breve{e}\right)d\breve{e}d\breve{s}d\breve{x}\right)}^2,\hfill \end{array}$$

(88)

$$\begin{array}{ccc}\hfill {\breve{\mathrm{\Omega }}}_2& =& {\gamma }_1{a}_1^2{\overline{\psi }}_1\left(\breve{L}\right)-2{\breve{m}}^{-1}{\beta }_1^{-1}{\beta }_2\breve{L}{a}_1^2{\int }_0^{\breve{L}}{\int }_{\breve{L}}^{\breve{x}}{\int }_0^{\breve{s}}{\overline{\psi }}_2\left(\breve{e}\right)d\breve{e}d\breve{s}d\breve{x},\hfill \end{array}$$

(89)

$$\begin{array}{ccc}\hfill {\breve{\mathrm{\Omega }}}_3& =& a_1^{-2}{\beta }_1{\int }_0^{\breve{L}}{\overline{\psi }}_2\left(\breve{e}\right)d\breve{e}-{\breve{m}}^{-1}{a}_1^{-2}({\beta }_3\breve{L}+{\beta }_1{\beta }_2){\int }_0^{\breve{L}}{\int }_{\breve{L}}^{\breve{x}}{\int }_0^{\breve{s}}{\overline{\psi }}_2\left(\breve{e}\right)d\breve{e}d\breve{s}d\breve{x}\hfill \\ & & -{\breve{\mathrm{\Omega }}}_2^{-1}{\breve{\mathrm{\Omega }}}_1{\overline{\psi }}_3^{\top }{\breve{G}}^{-\top }{\left({\breve{G}}^{-1}{\overline{\psi }}_3{\overline{\psi }}_3^{\top }{\breve{G}}^{-\top }\right)}^{-\top }{\breve{G}}^{-1}{\overline{\psi }}_3,\hfill \end{array}$$

(90)

with ${\breve{\zeta }}_1=2{\beta }_1^{-1}{\beta }_2\breve{L}{a}_1^2$ and ${\breve{\zeta }}_2={\gamma }_1{\breve{m}}^{-1}({\beta }_3\breve{L}+{\beta }_1{\beta }_2){\overline{\psi }}_1\left(\breve{L}\right)+4{\gamma }_1{\beta }_1^{-2}{\beta }_3^2{\breve{L}}^2{\breve{m}}^{-1}-2{\gamma }_1{\beta }_3\breve{L}$.

The relationship (82)–(86) declares that the system operator $\breve{\mathcal{A}}$ is invertible. It follows from the contraction mapping theory that the operator ${\breve{\lambda }}_{o}I-\breve{\mathcal{A}}=\breve{\mathcal{A}}({\breve{\lambda }}_o{\breve{\mathcal{A}}}^{-1}-I)$ is also invertible for all $0<{\breve{\lambda }}_o<\parallel {\breve{\mathcal{A}}}^{-1}\parallel$. This indicates that there exists an independent positive scalar $\breve{\eta }$ such that

$$\begin{array}{c}\hfill \parallel \breve{\psi }{\parallel }_{\mathcal{H}}^2\le \breve{\eta }{\parallel \overline{\psi }\parallel }_{\mathcal{H}}^2.\end{array}$$

(91)

This inequality implies that the invertible system operator ${\breve{\mathcal{A}}}^{-1}$ is bounded and $0\in \rho \left(\breve{\mathcal{A}}\right)$. It follows from the Lumer–Phillips theorem, the invertible and dissipative properties of $\breve{\mathcal{A}}$, that the system operator $\breve{\mathcal{A}}$ generates a ${\mathcal{C}}_0$-semigroup on the Hilbert space $\mathcal{H}$.

Based on the above analysis, we can concentrate on the second significant outcome of this study.

Theorem 2. 

Consider the closed-loop system (70) with the system operator shown in (71), for any initial condition ${\left({\breve{\omega }}_0\left(\breve{z}\right),{\breve{\omega }}_1\left(\breve{z}\right),{\breve{\varphi }}_0^{\top },{\widehat{y}}_{u0},{\widehat{\theta }}_{u0}^{\top }\right)}^{\top }\in \mathcal{H}$, the closed-loop system (70) has a unique classical solution. Moreover, the system operator $\breve{\mathcal{A}}$ of the closed-loop system (70) generates a ${\mathcal{C}}_0$-semigroup.

Remark 3. 

Note that the well-posedness analysis for a DPS is as important as its closed-loop system stability analysis. They explain the essential characteristics of the closed-loop system from two different perspectives. The well-posedness analysis aims at certifying the existence and uniqueness of the closed-loop system solution, while closed-loop system stability analysis shows the convergence property of the closed-loop system from the energy point of view.

4. Simulation Study

This section provides a simulation study on a simply supported beam to illustrate the validity of the developed adaptive multi-fault-tolerant boundary control algorithm.

The system parameters are shown as $\breve{m}=100$ and $\breve{L}=1$. The multi-fault signals are selected as $\breve{\rho }=0.8$ and

$$\breve{\mathrm{\Delta }}\left(t\right)=0.5+0.13sin\left(0.16\pi t\right)+0.18sin\left(0.28\pi t\right)+0.55sin\left(0.33\pi t\right)={\theta }_a^{\top }{\breve{\varphi }}_a\left(t\right)$$

with ${\theta }_a={(0.5,0.13,0.18,0.55)}^{\top }$ and ${\breve{\varphi }}_a\left(t\right)={(1,sin\left(0.16\pi t\right),sin\left(0.28\pi t\right),sin\left(0.33\pi t\right))}^{\top }$.

The disturbance signal is chosen as

$$\breve{d}\left(t\right)=1+0.82sin\left(0.74\pi t\right)+0.23sin\left(0.53\pi t\right)+0.27sin\left(0.91\pi t\right)$$

with ${\theta }_d={(1,0.82,0.23,0.27)}^{\top }$ and ${\breve{\varphi }}_d\left(t\right)={(1,sin\left(0.74\pi t\right),sin\left(0.53\pi t\right),sin\left(0.91\pi t\right))}^{\top }$.

In order to maintain conditions (19), (39)–(41), and inequalities (28), (51), the parameters in the controller (52) and updating laws (53)–(54) are designed as ${\beta }_1=12$, ${\beta }_2=0.5$, ${\beta }_3=1$, ${\gamma }_1={\gamma }_2=1$, $a_1=1$, $a_2=100$, $a_3=0$, $b_1=0$, $b_2=7$, $b_3=-1$, $k=10$. The initial conditions are presented as ${\breve{\omega }}_0\left(\breve{z}\right)=0$, ${\breve{\omega }}_1\left(\breve{z}\right)=0$, ${\widehat{\theta }}_{a0}=0.9{\theta }_a$, ${\widehat{\theta }}_{d0}=0.9{\theta }_d$, ${\widehat{y}}_0=0.9{\breve{\rho }}^{-1}$.

Applying this adaptive multi-fault tolerant control law to the system (1)–(3), the closed-loop system responses are shown in Figure 1 and Figure 2. It can be easily perceived from these two curves that the displacement $\breve{\omega }(\breve{z},t)$ and velocity ${\breve{\omega }}_t(\breve{z},t)$ are asymptotically converging to zero. This phenomenon is consistent with our theoretical results.

Figure 1. The closed-loop system state response $\breve{\omega }(\breve{z},t)$ under the developed control scheme.

Figure 2. The closed-loop system state response ${\breve{\omega }}_t(\breve{z},t)$ under the developed control scheme.

In order to show the advantages of the proposed control scheme compared with the existing works, the following simulation results are provided. It follows from Figure 3 and Figure 4 that the system states $\breve{\omega }(\breve{z},t)$ and ${\breve{\omega }}_t(\breve{z},t)$ cannot achieve the asymptotical stability under the backstepping control scheme. Figure 5 and Figure 6 shows that the system states $\breve{\omega }(\breve{z},t)$ and ${\breve{\omega }}_t(\breve{z},t)$ also cannot achieve the asymptotical stability under the sliding mode control scheme.

Figure 3. The closed-loop system state response $\breve{\omega }(\breve{z},t)$ under backstepping control scheme.

Figure 4. The closed-loop system state response ${\breve{\omega }}_t(\breve{z},t)$ under backstepping control scheme.

Figure 5. The closed-loop system state response $\breve{\omega }(\breve{z},t)$ under sliding mode control scheme.

Figure 6. The closed-loop system state response ${\breve{\omega }}_t(\breve{z},t)$ under sliding mode control scheme.

In order to show the robustness of the developed control scheme under the measurement noise, two simulation results are also provided. It follows from the Figure 7 and Figure 8 that the system states $\breve{\omega }(\breve{z},t)$ and ${\breve{\omega }}_t(\breve{z},t)$ remain the uniform ultimate bounded under the measurement noise.

Figure 7. The closed-loop system state response $\breve{\omega }(\breve{z},t)$ under measurement noise.

Figure 8. The closed-loop system state response ${\breve{\omega }}_t(\breve{z},t)$ under measurement noise.

5. Conclusions

This article has found a complete solution to a new boundary multi-fault compensation problem of an EBBS in the presence of matched disturbances, simultaneously. A novel direct adaptive fault-tolerant boundary control frame has been developed, in which a new auxiliary control signal has been constructed to settle the multiplicative boundary fault. The closed-loop system stability has been analyzed by using the Lyapunov direct method with the help of the devised parameter updating laws. The well-posedness of the closed-loop system solution has been certificated by using the modified ${\mathcal{C}}_0$-semigroup framework. A numerical simulation study has also been provided to show the effectiveness of the developed adaptive multi-fault-tolerant boundary control algorithm.