Flexiblelink manipulators have been widely studied as a favorable behavior in industry or space. However, it is challenging to develop its controller because the flexible link is described by distributed parameter system which is infinite dimensional. The frequentlyused methods to obtain the system dynamics are finite dimensional ones such as finite element method and assumed mode method. The dynamic models are in the form of ordinary differential equations (ODEs) for which control algorithms can be designed in a relatively easy way^{[1, 2]}. But problems are, such as control and observation spillovers, and a relatively high order controller^{[3]}. In order to avoid the foregoing issues, distributed parameter model is used without any truncation and described by partial differential equations (PDEs) in which distributed states are integrally described. Many control methods have been developed based on this modeling approach^{[412]}.
Commonly, uncertain parameters or unknown disturbance in flexible manipulator systems can be solved by robust controller. Sliding mode control, as one of the robustest controls, has been studied on flexible manipulators^{[1317]}. In [13], a fuzzy sliding mode controller and a composite controller were developed to suppress the vibration of a flexible manipulator. After generating a twoflexiblelink robot model, an observerbased active sliding mode controller was implemented in [14]. Two nested independent sliding mode control loops were presented in [17] to solve the position tracking issue of an uncertain singlelink flexible robot arm.
To approximate the unknown parameters of robotic manipulators, adaptive neural networks^{[1823]} have been introduced by researchers. Adaptive neural network^{[18]} was developed to approach uncertainties of flexible manipulator which is a nonlinear nonminimum phase system. Formanbordar and Hoseini^{[19]} employed Lyapunov
Boundary control of flexible link has been paid more attention because of its easy implementation in engineering practice. The boundary control was improved with strain feedback in [26] based on PDEs model. Lee and Prevost^{[27]} developed a coupled sliding mode boundary control based on distributed parameter dynamic PDEs model. Boundary control based on an observer for a flexible twolink manipulator was designed in [28]. In [29], the end effector was regulated on the desired trajectory in task space by using a boundary control strategy, and vibrations were suppressed simultaneously. However, they did not consider the uncertainties and unknown disturbances in system.
In this paper, a flexiblelink manipulator with uncertain parameter and unknown disturbance is studied. We present an adaptive sliding mode boundary controller with neural network to deal with system uncertainties and unknown disturbances. The dynamics of the flexiblelink manipulators is described by PDEs. The proposed controller is directly derived from the PDEs model, which is challenging in control system design. The control objective is to drive the joint to a desired position and rapidly suppress elastic deflection on the flexible link. The contributions are summarized as follows:
1) The effects of both parameter uncertainty and unknown disturbances are compensated by sliding mode control and adaptive neural network, which is meaningful in engineering practice.
2) The stability of the closedloop system is validated based on infinite dimensional PDEs model, avoiding problems such as control spillovers caused by traditional finite dimensional truncated models.
3) The controller only requires measuring the boundary information, which facilitates implementation in engineering practice.
The rest of the paper is organized as follows. In Section 2, a PDE model is obtained by using the extended Hamilton
A singlelink flexible manipulator subjected to unknown disturbance is shown in Fig. 1. The flexible link can be taken as an EulerBernoulli beam for its intrinsic flexible properties. Its motor and tip load are regarded as material points. The motor serves as an actuator to drive the flexible manipulator. The case is taken into account that the disturbance existing in torque is unknown and timevarying. Frame
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Fig. 1. Geometric model of the singlelink flexible manipulator 
In order to facilitate the analysis of the modeling process, we shall give the following two assumptions:
Assumption 1. The flexible beam only moves in the horizontal plane and the effect of gravity is not taken into consideration.
Assumption 2. The manipulator is assumed flexible in transverse direction and rigid in other directions. The link elongations are small enough to be neglected.
Remark 1. The derivative notations in this paper are defined as follows:
In order to derive the exact PDE model of the system, we first denote the inertial coordinate position of point
The kinetic energy of the joint is
$ \begin{equation}\label{eqn1} {{E}_{k1}}=\frac{1}{2}{{I}_{h}}{{\dot{\theta }}^{2}}(t). \end{equation} $  (1) 
The kinetic energy of the load is
$ \begin{equation}\label{eqn2} {{E}_{k2}}=\frac{1}{2}m{{\dot{r}}^{2}}(L, t). \end{equation} $  (2) 
The kinetic energy of the link is
$ \begin{equation}\label{eqn3} {{E}_{k3}}=\frac{\rho }{2}\int_{0}^{L}{{{{\dot{r}}}^{2}}(x, t)\textrm{d}x}. \end{equation} $  (3) 
Then, the total kinetic energy is
$ \begin{align}\label{eqn4} {{E}_{k}}&={{E}_{k1}}+{{E}_{k2}}+{{E}_{k3}}= \nonumber\\ & \quad\;\frac{1}{2}{{I}_{h}}{{{\dot{\theta }}}^{2}}(t)+\frac{\rho } {2}\int_{0}^{L}{{{{\dot{r}}}^{2}}(x, t)\textrm{d}x}+\frac{1}{2}m{{{\dot{r}}}^{2}}(L, t). \end{align} $  (4) 
The potential energy of the system is
$ \begin{equation}\label{eqn5} {{E}_{p}}=\frac{EI}{2}\int_{0}^{L}{{{[{{w}_{xx}}(x, t)]}^{2}}\textrm{d}x}. \end{equation} $  (5) 
The nonconservative work is
$ \begin{equation}\label{eqn6} {{W}_{nc}}=\left[\tau (t)+f(t) \right]\theta (t). \end{equation} $  (6) 
Then, we employ the extended Hamilton's principle as follows:
$ \begin{equation}\label{eqn7} \int_{{{t}_{1}}}^{{{t}_{2}}}{\left( \delta {{E}_{k}}\delta {{E}_{p}}+\delta {{W}_{nc}} \right)\textrm{d}t}=0. \end{equation} $  (7) 
where
$ \begin{align}\label{eqn8} & \int_{{{t}_{1}}}^{{{t}_{2}}}{A\delta \theta (t)\textrm{d}t}+\int_{{{t}_{1}}}^{{{t}_{2}}} {\int_{0}^{L}{B\delta w(x, t)\textrm{d}x\textrm{d}t}}+ \nonumber\\ & \int_{{{t}_{1}}}^{{{t}_{2}}}{C\delta w(L, t)\textrm{d}t}+\int_{{{t}_{1}}}^{{{t}_{2}}} {D\delta {{w}_{x}}(L, t)\textrm{d}t}=0 \end{align} $  (8) 
where
$ \begin{align} A =&\tau (t)+f(t)[{{I}_{h}}\ddot{\theta }(t)+\rho \frac{{{L}^{3}}}{3}\ddot{\theta }(t)+\rho \int_{0}^{L}{x\ddot{w}(x, t)\textrm{d}x}+ \nonumber\\ & m{{L}^{2}}\ddot{\theta }(t)+mL\ddot{w}(L, t)] \end{align} $  (9) 
$ \begin{align} B=&\left[\rho (x\ddot{\theta }(t)+\ddot{w}(x, t))+EI{{w}_{xxxx}}(x, t) \right] \end{align} $  (10) 
$ \begin{align} C=&EI{{w}_{xxx}}(L, t)\left[m\ddot{w}(L, t)+mL\ddot{\theta }(t) \right] \end{align} $  (11) 
$ \begin{align} D=&{{w}_{xx}}(L, t). \end{align} $  (12) 
By observing (8), we conclude
$ \begin{align} & \tau (t)=({{I}_{h}}+\rho \frac{{{L}^{3}}}{3})\ddot{\theta }(t)+\rho \int_{0}^{L}{x\ddot{w}(x, t)\rm{d}x}+ \\ & \ \ \ \ \ \ \ \ \ \ \ mL\ddot{r}(L, t)f(t) \\ \end{align} $  (13) 
$ \begin{align} &\rho \ddot{r}(x, t)=EI{{w}_{xxxx}}(x, t) \end{align} $  (14) 
$ \begin{align} &m\ddot{r}(L, t)=EI{{w}_{xxx}}(L, t) \end{align} $  (15) 
$ \begin{align} &{{w}_{xx}}(L, t)=0. \end{align} $  (16) 
The other boundary condition of the EulerBernoulli beam is
$ \begin{equation}\label{eqn17} w(0, t)=0, \rm{ }{{w}_{x}}(0, t)=0. \end{equation} $  (17) 
In fact, by integral calculations, (13) can be changed into the following format (see Appendix A):
$ \begin{equation}\label{eqn18} {{I}_{h}}\ddot{\theta }(t)=\tau (t)+f(t)+EI{{w}_{xx}}(0, t). \end{equation} $  (18) 
Thus, the dynamic model of the system is given by (14)(18). The characteristic of distributed parameters of the flexible manipulator is described by the partial differential equation (14). The other ODEs (15)(17) represent the boundary conditions.
Remark 2. The time symbol
In industrial or space environment where temperature varies greatly with time, Young
First, radial basis function (RBF) neural network is developed to approximate the unknown disturbance
$ \begin{equation}\label{eqn19} f={{W}^{\rm T}}h(\chi )+\varepsilon \end{equation} $  (19) 
where
$ \begin{equation}\label{eqn20} {{h}_{j}}(\chi )=\exp \left( \frac{{{\left\ \chi {{c}_{j}} \right\}^{2}}}{b_{j}^{2}} \right), \rm{ }\chi \in \Psi \end{equation} $  (20) 
where
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Fig. 2. RBF neural network structure 
Then, we assume that the RBF NN approximation error is bounded as
$ \begin{equation}\label{eqn21} \left \varepsilon \right\le {{\varepsilon }_{0}}. \end{equation} $  (21) 
Let the output of the RBF network be
$ \begin{equation}\label{eqn22} \hat{f}={{\hat{W}}^{\rm T}}h(\chi ). \end{equation} $  (22) 
Thus, the vector of RBF weight error and the unknown disturbance estimate error could be defined as follows:
$ \begin{equation}\label{eqn23} \tilde{W}=W\hat{W} \end{equation} $  (23) 
$ \begin{equation}\label{eqn24} \tilde{f}=f\hat{f}. \end{equation} $  (24) 
In Section 4, we will design adaptive control laws to deal with the unknown functions
In this section, we propose a novel sliding mode controller based on adaptive method. The control objective is to drive the joint to a desired position and rapidly suppress vibration on the beam. One of the contributions is that the controller is employed based on the infinite dimensional dynamic model, rather than the traditional finite dimensional truncated model, avoiding problems such as control spillovers.
First, we design a sliding surface as follows:
$ \begin{equation}\label{eqn25} s=\dot{e}+\lambda e \end{equation} $  (25) 
where
Then, an adaptive sliding mode controller is presented as follows:
$ \begin{align}\label{eqn26} \tau = & \hat{f}\widehat{EI}\left( t \right){{w}_{xx}}\left( 0, t \right)\lambda {{I}_{h}}\dot{\theta }{{k}_{s}}s \nonumber \\ & {{k}_{\zeta }}w\left( \zeta, t \right)\int_{0}^{t}{s}w\left( \zeta, \sigma \right)\textrm{d}\sigma k \rm{sgn} \left( s \right) \end{align} $  (26) 
where
$ \begin{equation}\label{eqn27} \overset{\centerdot }{\mathop{\widehat{EI}}} \left( t \right)={{\gamma }_{1}}s{{w}_{xx}}\left( 0, t \right) \end{equation} $  (27) 
where
Remark 3. The term
To complete the controller design, the other adaptive law is provided by
$ \begin{equation}\label{eqn28} \overset{\centerdot }{\mathop{\widehat{W}}} \left( t \right)={{\gamma }_{2}}h\left( \chi \right)s \end{equation} $  (28) 
where
Theorem 1. The stability of the closedloop system is validated by the adaptive sliding mode controller (26) based on adaptive laws (27) and (28). The asymptotic behavior can be further verified by theoretical analysis, i.e., the control objectives
Proof. First, we choose a Lyapunov function candidate to prove the stability of the closedloop system as follows:
$ \begin{align}\label{eqn29} V\left( t \right)=& \frac{1}{2}{{I}_{h}}{{s}^{2}}+\frac{{{k}_{f}}}{2}{{\left[\int_{0}^{t}{sw\left( \zeta, \sigma \right){\rm d}\sigma } \right]}^{2}}+ \nonumber \\ & \frac{1}{2{{\gamma }_{1}}}{{\widetilde{EI}}^{2}}+\frac{1}{2{{\gamma }_{2}}}{{{\tilde{W}}}^{\rm T}}\tilde{W} \end{align} $  (29) 
where
$ \begin{align}\label{eqn30} \dot{V}\left( t \right)= & {{I}_{h}}s\dot{s}+{{k}_{\zeta }}sw\left( \zeta, t \right)\int_{0}^{t} {sw\left( \zeta, \sigma \right)\textrm{d}\sigma } +\nonumber\\ & \frac{1}{{{\gamma }_{1}}}\widetilde{EI}\overset{\centerdot } {\mathop{\widetilde{EI}}} +\frac{1}{{{\gamma }_{2}}}{{{\tilde{W}}}^{\rm T}}\dot{\tilde{W}}. \end{align} $  (30) 
Taking the derivative of (25) with respect to time, we obtain
$ \begin{equation}\label{eqn31} \dot{s}=\ddot{e}+\lambda \dot{e}. \end{equation} $  (31) 
From (18), the following equation is obtained:
$ \begin{equation}\label{eqn32} \ddot{\theta }=\frac{1}{{{I}_{h}}}\left[\tau +f+EI{{w}_{xx}}(0, t) \right]. \end{equation} $  (32) 
By applying (25), (26), (31) and (32) to the term
$ \begin{align}\label{eqn33} {{I}_{h}}s\dot{s}=&{{I}_{h}}s\left( \ddot{e}+\lambda \dot{e} \right)= \nonumber\\ & {{I}_{h}}s\left\{ \frac{1}{{{I}_{h}}}\left[\tau +f+EI{{w}_{xx}}(0, t) \right]+\lambda \dot{\theta } \right\}= \nonumber\\ & s\tilde{f}+s\widetilde{EI}{{w}_{xx}}\left( 0, t \right) {{k}_{\zeta }}sw\left( \zeta, t \right)\int_{0}^{t}{s}w\left( \zeta, \sigma \right)\textrm{d}\sigma \nonumber\\ & {{k}_{s}}{{s}^{2}}ks\rm{sgn} \left( s \right). \end{align} $  (33) 
Substituting (33) into (30), and using adaptive control laws (27) and (28), we obtain
$ \begin{align}\label{eqn34} \dot{V}\left( t \right) =& s\tilde{f}+s\widetilde{EI}{{w}_{xx}}\left( 0, t \right){{k}_{\zeta }}sw\left( \zeta, t \right)\int_{0}^{t}{s}w\left( \zeta, \sigma \right)\textrm{d}\sigma \nonumber\\ & {{k}_{s}}{{s}^{2}}ks\rm{sgn} \left( s \right)+{{k}_{\zeta }}sw\left( \zeta, t \right)\int_{0}^{t}{sw\left( \zeta, \sigma \right)\textrm{d}\sigma } \nonumber\\ & \frac{1}{{{\gamma }_{1}}}\widetilde{EI}\overset{\centerdot }{\mathop{\widehat{EI}}} \frac{1}{{{\gamma }_{2}}}{{{\tilde{W}}}^{\rm T}}\overset{\centerdot }{\mathop{\widehat{W}}}= \nonumber\\ & s\tilde{f}+s\widetilde{EI}{{w}_{xx}}\left( 0, t \right){{k}_{s}}{{s}^{2}}ks\rm{sgn} \left( s \right) \nonumber\\ & \widetilde{EI}s{{w}_{xx}}\left( 0, t \right){{{\tilde{W}}}^{\rm T}}h\left( \chi \right)s= \nonumber\\ & s\tilde{f}{{k}_{s}}{{s}^{2}}ks\rm{sgn} \left( s \right) {{{\tilde{W}}}^{\rm T}}h\left( \chi \right)s. \end{align} $  (34) 
Considering (19), (21)(24), we could rearrange (34) as follows:
$ \begin{align}\label{eqn35} \dot{V}&\left( t \right)= s\tilde{f}{{k}_{s}}{{s}^{2}}ks\rm{sgn} \left( s \right){{{\tilde{W}}}^{\rm T}}h\left( \chi \right)s= \nonumber\\ & s\left(\! f\!\!\hat{f}\! \right){{k}_{s}}{{s}^{2}}\!\!ks\rm{sgn} \left(\! s \right)\!\!\left(\!{{W}^{\rm T}}\!\!{{{\hat{W}}}^{\rm T}} \!\right)h\left( \chi \right)s= \nonumber\\ & s\left[{{W}^{\rm T}}h(x)+\varepsilon{{{\hat{W}}}^{\rm T}}h(\chi ) \right] {{k}_{s}}{{s}^{2}} ks\rm{sgn} \left( s \right) \nonumber\\ & \left( {{W}^{\rm T}}{{{\hat{W}}}^{\rm T}} \right)h\left( \chi \right)s = \nonumber\\ & \varepsilon sks\rm{sgn} \left( s \right){{k}_{s}}{{s}^{2}}. \end{align} $  (35) 
Further, we set
$ \begin{equation}\label{eqn36} \dot{V}\left( t \right)\le {{\varepsilon }_{0}}sk\left s \right{{k}_{s}}{{s}^{2}} \le {{k}_{s}}{{s}^{2}}. \end{equation} $  (36) 
According to (36), we conclude that the Lyapunov function (29) is decreasing because of
$ \begin{align}\label{eqn37} \dot{s}=& \frac{1}{{{I}_{h}}}\left[\tilde{f}+\widetilde{EI}{{w}_{xx}}\left( 0, t \right){{k}_{s}}sk\rm{sgn} \left( s \right)\right. \nonumber\\ & \left. {{k}_{\zeta }}w\left( \zeta, t \right)\int_{0}^{t}{s}w\left( \zeta, \sigma \right)\textrm{d}\sigma \right] = \nonumber\\ & \frac{1}{{{I}_{h}}}\left[{{{\tilde{W}}}^{\rm T}}h(x)+\varepsilon +\widetilde{EI}{{w}_{xx}}\left( 0, t \right) \right.{{k}_{s}}sk\rm{sgn} \left( s \right)\nonumber\\ & \left.{{k}_{\zeta }}w\left( \zeta, t \right)\int_{0}^{t}{s}w\left( \zeta, \sigma \right)\textrm{d}\sigma \right]. \end{align} $  (37) 
Using the above bounded terms, it is not difficult to observe that
Considering (35) and (36), we notice that it will maintain the condition
$ \begin{align}\label{eqn38} & \tau (t)+f(t)+EI{{w}_{xx}}(0, t)= \nonumber\\ & \tilde{f}+\widetilde{EI}{{w}_{xx}}\left( 0, t \right){{k}_{\zeta }}\eta w\left( \zeta, t \right) \equiv 0. \end{align} $  (38) 
where
To conclude, the stability of the closedloop system is validated by using Lyapunov
Remark 4. In practical engineering, we can set
In this section, numerical simulations are carried out to demonstrate the stability of the closedloop system as well as the suppression of elastic vibration by using the proposed adaptive sliding controller.
We utilize finite difference approximation method to discretize time and space variables in the partial differential equations, and arrive at finite difference equations. We employ Matlab tools to achieve the simulation, in which the time step
The physical parameter values of system are given as follows:
From the technique of neural networks, we choose input vector of the network as
$ \begin{align} c & =\left[{{c}_{1}}, {{c}_{2}}, \cdots, {{c}_{j}}, \cdots, {{c}_{m}} \right]= \nonumber\\ & \quad\left[\begin{matrix} 0.1 & 0.1 & 0.3 & 0.5 & 0.7 & 0.9 \nonumber\\ 0.1 & 0.1 & 0.3 & 0.5 & 0.7 & 0.9 \nonumber\\ \end{matrix} \right] \end{align} $ 
and
Then, we define the following initial parameter values of adaptive sliding mode controller (26) as
Case 1.
Simulation results are shown in Figs. 39 with the absence of the term
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Fig. 3. Estimate of unknown bending stiffness EI 
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Fig. 4. Estimate of unknown disturbance f 
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Fig. 5. Estimate error 
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Fig. 6. Angular position and speed tracking 
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Fig. 7. Elastic deflection w(x, t) on the whole beam 
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Fig. 8. Elastic deflection speed 
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Fig. 9. Control torque 
Figs. 69 illustrate the effectiveness of the controller (26) with
Case 2:
We set
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Fig. 10. Estimate of unknown bending stiffness EI 
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Fig. 11. Estimate of unknown disturbance f 
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Fig. 12. Estimate error 
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Fig. 13. Angular position and speed tracking 
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Fig. 14. Elastic deflection w(x, t) on the whole beam 
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Fig. 15. Elastic deflection speed 
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Fig. 16. Control torque 
Figs. 1016 illustrate that the proposed controller can regulate the joint angular position and suppress the vibration of the beam actively, while dealing with the issues caused by uncertain parameters and unknown disturbance.
6 ConclusionsIn this paper, a novel adaptive sliding mode controller is designed based on neural networks for a single flexiblelink manipulator. The stability of the closedloop system is proven based on infinite dimensional model. With the unknown disturbance and uncertain parameters in the system, the joint position is regulated and the elastic deflection of the flexible link is suppressed, which reached our control objective. However, this controller cannot deal with the issue of input saturation. In practical engineering, the control torque has the maximum upper bound. The motor will be damaged if the control input exceeds this bound. Thus, future study demands development of controller with input constraint.
AppendixFirst, we multiply both sides of (14) by
$ \begin{equation}\label{eqn39} \rho \int_{0}^{L}{x\ddot{r}(x, t)\textrm{d}x}=\int_{0}^{L}{EIx{{w}_{xxxx}}(x, t)\textrm{d}x}. \end{equation} $  (39) 
Then, the left side of (39) can be rewritten by
$ \begin{align}\label{eqn40} \qquad\rho \int_{0}^{L}{x\ddot{r}(x, t)\textrm{d}x}&=\rho \int_{0}^{L}{x\left[ x\ddot{\theta }(t)+\ddot{w}(x, t) \right]\textrm{d}x}= \nonumber\\ & \rho \left[\frac{1}{3}{{x}^{3}}\ddot{\theta }(t)\left _{0}^{L} \right.+\int_{0}^{L}{x\ddot{w}(x, t)\textrm{d}x} \right]=\qquad\qquad\qquad\qquad \nonumber\\ & \frac{1}{3}\rho {{L}^{3}}\ddot{\theta }(t)+\rho \int_{0}^{L}{x\ddot{w}(x, t)\textrm{d}x}\qquad\qquad\qquad\qquad \end{align} $  (40) 
and the right side
$ \begin{align}\label{eqn41} & \int_{0}^{L}{EIx{{w}_{xxxx}}(x, t)}\textrm{d}x= \nonumber\\ & EI\left[x{{w}_{xxx}}(x, t){{w}_{xx}}(x, t) \right]\left _{0}^{L} \right.= \nonumber\\ & EI\left[L{{w}_{xxx}}(L, t){{w}_{xx}}(L, t)+{{w}_{xx}}(0, t) \right]. \end{align} $  (41) 
Thus, we have
$ \begin{align}\label{eqn42} & \frac{1}{3}\rho {{L}^{3}}\ddot{\theta }(t)+\rho \int_{0}^{L} {x\ddot{w}(x, t)\textrm{d}x}= \nonumber\\ & EI\left[L{{w}_{xxx}}(L, t){{w}_{xx}}(L, t)+{{w}_{xx}}(0, t) \right]. \end{align} $  (42) 
Substituting (13) and (16) into (42), we get (18).
AcknowledgementThe authors would like to thank the EditorinChief, the Associate Editor, and the anonymous reviewers for their constructive comments, which helped to improve the quality and presentation of this paper.
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