International Journal of Automation and Computing  2018, Vol. 15 Issue (2): 239-248   PDF    
Sliding Mode Control for Flexible-link Manipulators Based on Adaptive Neural Networks
Hong-Jun Yang, Min Tan     
State Key Laboratory of Management and Control for Complex Systems, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China
Abstract: This paper mainly focuses on designing a sliding mode boundary controller for a single flexible-link manipulator based on adaptive radial basis function (RBF) neural network. The flexible manipulator in this paper is considered to be an Euler-Bernoulli beam. We first obtain a partial differential equation (PDE) model of single-link flexible manipulator by using Hamiltons approach. To improve the control robustness, the system uncertainties including modeling uncertainties and external disturbances are compensated by an adaptive neural approximator. Then, a sliding mode control method is designed to drive the joint to a desired position and rapidly suppress vibration on the beam. The stability of the closed-loop system is validated by using Lyapunov s method based on infinite dimensional model, avoiding problems such as control spillovers caused by traditional finite dimensional truncated models. This novel controller only requires measuring the boundary information, which facilitates implementation in engineering practice. Favorable performance of the closed-loop system is demonstrated by numerical simulations.
Key words: Sliding mode control     adaptive control     neural network     flexible manipulator     partial differential equation (PDE)    
1 Introduction

Flexible-link 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 frequently-used 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[4-12].

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[13-17]. In [13], a fuzzy sliding mode controller and a composite controller were developed to suppress the vibration of a flexible manipulator. After generating a two-flexible-link robot model, an observer-based 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 single-link flexible robot arm.

To approximate the unknown parameters of robotic manipulators, adaptive neural networks[18-23] have been introduced by researchers. Adaptive neural network[18] was developed to approach uncertainties of flexible manipulator which is a nonlinear non-minimum phase system. Formanbordar and Hoseini[19] employed Lyapunov$'$s direct method to design adaptation laws for the adaptive gains and weights of neural network. The adaptive neural networks were also used for an uncertain $n$-link robot respectively with full-state constraints and input saturation in [24] and [25]. However, the foregoing studies were implemented based on ODEs model and have problems such as control and observation spillovers. As for the drawbacks of ODEs model, it is crucial to design a control law based on PDEs model which has a full description of the characteristics of distributed parameter systems.

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 two-link 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 flexible-link 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 flexible-link 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 closed-loop 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$'$s principle. In Section 3, we consider the problems of uncertain parameters and unknown disturbance and develop an adaptive sliding-mode controller with neural network. Section 4 uses simulation to illustrate the favorable performance of the closed-loop system. Finally, conclusions are provided in Section 5.

2 Model for a single-link flexible manipulator 2.1 Preliminaries

A single-link flexible manipulator subjected to unknown disturbance is shown in Fig. 1. The flexible link can be taken as an Euler-Bernoulli 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 time-varying. Frame ${{X}_{0}}{{O}_{0}}{{Y}_{0}}$ is the fixed inertial coordinate and $XOY$ is the local coordinate. The parameters needed in PDE model are listed in Table 1.

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Fig. 1. Geometric model of the single-link flexible manipulator

Table 1
Nomenclature of the system

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: $\dot{*}=\frac{\partial (*)}{\partial t}$, $\ddot{*}=\frac{{{\partial }^{2}}(*)}{\partial {{t}^{2}}}$, ${{(*)}_{x}}=\frac{\partial (*)}{\partial x}$, ${{(*)}_{xx}}=\frac{{{\partial }^{2}}(*)}{\partial {{x}^{2}}}$, ${{(*)}_{xxx}}=\frac{{{\partial }^{3}}(*)}{\partial {{x}^{3}}}$, and ${{(*)}_{xxxx}}=\frac{{{\partial }^{4}}(*)}{\partial {{x}^{4}}}$, where $x$ and $t$ respectively denote the position variable and time variable, and $(x, t)\in [0, L]\times [0, \infty)$.

2.2 Mathematical model

In order to derive the exact PDE model of the system, we first denote the inertial coordinate position of point $P$ as $r(x, t)=x\theta +w(x, t)$. Then, the kinetic energy ${{E}_{k}}$, the potential energy ${{E}_{p}}$, and the non-conservative work ${{W}_{nc}}$ of the system are obtained as follows.

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 non-conservative 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 $\delta {{E}_{k}}$, $\delta {{E}_{p}}$ and $\delta {{W}_{nc}}$ denote the variations of the total kinetic energy, the potential energy and the non-conservative work respectively. By some variational calculations, we give

$ \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 $A=B=C=D=0$ and obtain the following equations:

$ \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 Euler-Bernoulli 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 $t$ may be omitted to make the following contents more concise. For instance, the symbol $\theta $ denotes $\theta (t)$.

3 Design of neural network

In industrial or space environment where temperature varies greatly with time, Young$'$s Modulus will change and thus leads to uncertainty in parameter $EI$. Then, we have to deal with the problem caused by the uncertainty of $EI$. In addition, the unknown disturbance $f$, consisting of unmodeled dynamics, error of motor and other factors, exists in torque. We intend to concurrently utilize sliding mode control and adaptive neural network to handle these issues.

First, radial basis function (RBF) neural network is developed to approximate the unknown disturbance $f$ in the following discuss. Suppose that the unknown disturbance $f$ is continuous on $\Psi $, it thus can be given by

$ \begin{equation}\label{eqn19} f={{W}^{\rm T}}h(\chi )+\varepsilon \end{equation} $ (19)

where $\chi ={{\left[{{\chi }_{1}}\rm{ }{{\chi}_{2}}\cdots\rm{ }{{\chi}_{n}} \right]}^{\rm T}}\in {{\bf R}^{n}}$ is the input vector, and $n$ represents the input neural nets number in the input layer. $h(\chi)={{\left[{{h}_{1}}(\chi)\rm{ }{{h}_{2}}(\chi)\cdots {{h}_{j}}(\chi)\cdots {{h}_{m}}(\chi) \right]}^{\rm T}}$, $j=1, 2, \cdots, m$ is the radial base function vector and represents the output of hidden layer, where $m$ is hidden neural nets number and ${{h}_{j}}(\chi)$ represents Gaussian function as

$ \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 ${{c}_{j}}\in {{\bf R}^{n}}$ denotes the center of the receptive field of Gaussian function, ${{b}_{j}}$ is a constant and represent the width of Gaussian function. $W={{\left[{{W}_{1}}\rm{ }{{W}_{2}}\cdots\rm{ }{{W}_{m}} \right]}^{\rm T}}$ is the vector of ideal weight for the output layer and is bounded as ${{\left\| W \right\|}_{F}}\le {{{w}}_{\max }}$, where ${{{w}}_{\rm\max }}>0$. $\varepsilon $ is the RBF NN approximation error. Fig. 2 shows the structure of RBF neural network.

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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 $EI$ and $W$.

4 Controller derivation

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 $e=\theta -{{\theta }_{d}}$, and $\lambda $ is the positive design constant. The desired joint angular position ${{\theta }_{d}}$ is a constant. We can accomplish the control task that $e\to 0$ and $\dot{e}\to 0$ by satisfying the condition $s\to 0$.

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 ${{k}_{s}}$, ${{k}_{\zeta }}$ and $k$ are positive design constants. $\zeta \in \left[0, L \right]$ is an arbitrary location on the flexible beam. $\widehat{EI}\left(t \right)$ is the adaptive estimate function of $EI$ and updated according to adaptive law as follows:

$ \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 ${{\gamma }_{1}}>0$ is the design constant.

Remark 3. The term $-{{k}_{\zeta }}w\left(\zeta, t \right)\int_{0}^{t}{s}w\left(\zeta, \sigma \right)\textrm{d}\sigma $ in (26) plays a significant role in the control law. This term will directly affect the elastic deflections of the beam and can be expected to suppress them, accordingly improving the performance of controller.

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 ${{\gamma }_{2}}>0$ is the design constant.

Theorem 1. The stability of the closed-loop 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 $\theta \to {{\theta }_{d}}$, $\dot{\theta }\to {{\dot{\theta }}_{d}}$ and $w(x, t)\to 0$ as $t\to \infty $ hold.

Proof. First, we choose a Lyapunov function candidate to prove the stability of the closed-loop 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 $\widetilde{EI}=EI-\widehat{EI}$ is the estimate error of adaptive parameter $EI$. Then, we arrive at the time derivative of (29) as follows:

$ \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 ${{I}_{h}}s\dot{s}$ of (30), we can further arrive at

$ \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 s-ks\rm{sgn} \left( s \right)-{{k}_{s}}{{s}^{2}}. \end{align} $ (35)

Further, we set $k\ge {{\varepsilon }_{0}}$ and have

$ \begin{equation}\label{eqn36} \dot{V}\left( t \right)\le {{\varepsilon }_{0}}s-k\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 $\dot{V}\left(t \right)\le -{{k}_{s}}{{s}^{2}}$, and the terms $s$, $\int_{0}^{t}{sw\left(\zeta, \sigma \right)\textrm{d}\sigma }$, $\widetilde{EI}$, $\tilde{W}$ are bounded. Then, we rewrite $\dot{s}$ as

$ \begin{align}\label{eqn37} \dot{s}=& \frac{1}{{{I}_{h}}}\left[\tilde{f}+\widetilde{EI}{{w}_{xx}}\left( 0, t \right)-{{k}_{s}}s-k\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}}s-k\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 $\dot{s}$ is also bounded, thus the uniformly continuity of $s$ is guaranteed. Utilizing Barbalat$'$s lemma, we arrive at the asymptotic behavior of $s$ such that $s\to 0$ as $t\to \infty $ and hence the control objectives $\theta \to {{\theta }_{d}}$, $\dot{\theta }\to {{\dot{\theta }}_{d}}$ as $t\to \infty $ are obtained. In the following, we have to guarantee that the elastic deflection tends to zero, i.e., $w(x, t)\to 0$ as $t\to \infty $.

Considering (35) and (36), we notice that it will maintain the condition $\dot{V}\left(t \right)=0$ only if the trajectory of system is restrict to $s=0$, i.e., $\theta ={{\theta }_{d}}$ and $\dot{\theta }={{\dot{\theta }}_{d}}=0$. However if $\ddot{\theta }(t)$ in (32) is not satisfying $\ddot{\theta }(t)\equiv 0$, it is impossible to achieve $\dot{\theta }=0$ and $\theta ={{\theta }_{d}}$. That is to say the system can maintain the condition $\dot{V}\left(t \right)=0$ only if

$ \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 $\eta =\int_{0}^{t}{s}w\left(\zeta, \sigma \right)\textrm{d}\sigma $ (this integral term is a constant after $s=0$). As the changes of disturbance $f$ and Young's Modulus $EI$ are random comparing with the system states ${{w}_{xx}}\left(0, t \right)$ and $w\left(\zeta, t \right)$, and the variations of $\tilde{f}$ and $\widetilde{EI}$ are independent with each other, the terms $\tilde{f}$, $\widetilde{EI}{{w}_{xx}}\left(0, t \right)$ and ${{k}_{\zeta }}\eta w\left(\zeta, t \right)$ are not linear correlative, they must be zero to satisfy (38). Thus, we obtain the asymptotic behavior of vibrations $w(\zeta, t)$. Since $\zeta $ is an arbitrary position on the beam, vibrations of the whole beam can be suppressed. Thus, the control objective $w(x, t)\to 0$ as $t\to \infty $ holds.

To conclude, the stability of the closed-loop system is validated by using Lyapunov$'$s method based on infinite dimensional model, avoiding the problems such as control spillovers caused by the traditional finite dimensional truncated models. The controller can drive the joint to a desired position and rapidly suppress the vibration on the beam, while solving the problems of the parameter uncertainty and unknown disturbance, achieving our control objective.

Remark 4. In practical engineering, we can set $\zeta =L$. Then, all the variables including $\dot{\theta }$, $s$, ${{w}_{xx}}\left(0, t \right)$ and $w\left(L, t \right)$ in the control input (26) can be measured by sensors installed at the borders of the beam. Thus, the proposed controller only requires the measurement of boundary information, which facilitates implementation in the engineering.

5 Simulation results

In this section, numerical simulations are carried out to demonstrate the stability of the closed-loop 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 $\Delta t=1\times {{10}^{-3}} {\rm s} $ and space step $\Delta x=0.1 \rm{m}$ are assumed, and the center finite difference approximations are used[30].

The physical parameter values of system are given as follows: $EI=2 {\rm N}\cdot{{\rm{m}}^{\rm{2}}}$, $\rho =0.2 {\rm kg}\cdot{{\rm{m}}^{-1}}$, ${{I}_{\rm{h}}}=0.5 {\rm kg}\cdot{{\rm{m}}^{\rm{2}}}$, $m=0.6 {\rm kg}$, $L=1 {\rm m}$. The disturbance is further chosen as $f=0.2+0.2\exp(-\dot{\theta })+0.02\dot{\theta }$. The initial position and desired position of joint angle are respectively selected as $\theta (0)=0.2 {\rm rad}$ and ${{\theta }_{d}}=0.5 \rm{rad}$. The initial elastic deflection of the whole beam is assumed to be zero.

From the technique of neural networks, we choose input vector of the network as $\chi={{[1, \dot{\theta }]}^{\rm T}}$, the parameters in Gaussian function is 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 ${{b}_{j}}=1$, $j=1, 2, \cdots, m$. Since $n=2$ and $m=6$, the RBF neural network structure is 2-6-1. The value of parameters ${{c}_{j}}$ and ${{b}_{j}}$ respectively denote the center of the receptive field and the width of Gaussian function, which should be chosen appropriately based on the range of input vector, otherwise the network will be ineffective on approaching the unknown disturbance.

Then, we define the following initial parameter values of adaptive sliding mode controller (26) as ${{\gamma }_{1}}=50$, ${{\gamma }_{2}}=5$, $\lambda =5$, ${{k}_{s}}=5$, $k=2$. In the simulation, we choose $\zeta =L$. The initial values of adaptive function are given by $\widehat{EI}\left(0 \right)=0$ and $\hat{W}\left(0 \right)={{[\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]}^{\rm T}}$. Then, we divide the controller term ${{k}_{\zeta }}$ into two cases, ${{k}_{\zeta }}=0$ and ${{k}_{\zeta }}\ne 0$, to demonstrate the significant effect of this term.

Case 1. ${{k}_{\zeta }}=0$.

Simulation results are shown in Figs. 3-9 with the absence of the term $-{{k}_{\zeta }}w\left(\zeta, t \right)\int_{0}^{t}{s}w\left(\zeta, \sigma \right)\textrm{d}\sigma $ in (26). Figs. 3 and 4 display the estimates of unknown bending stiffness $EI$ and disturbance $f$. Fig. 5 shows the relationship among the estimate error $\widetilde{EI}$, system term ${{w}_{xx}}\left(0, t \right)$ and their product $\widetilde{EI}{{w}_{xx}}\left(0, t \right)$. From (38), we conclude that if ${{k}_{\zeta }}=0$ happens, the terms $\tilde{f}$, $\widetilde{EI}{{w}_{xx}}\left(0, t \right)$ will tend to zero as $t\to \infty $, which are demonstrated by Figs. 3-5. Notice that since ${{w}_{xx}}\left(0, t \right)$ is not approaching zero, $\widetilde{EI}$ must tend to zero to guarantee the product term $\widetilde{EI}{{w}_{xx}}\left(0, t \right)\to 0$ as $t\to \infty $. In addition, Fig. 4 implies the favorable performance of approximation of RBF neural networks.

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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 $\widetilde{EI}$ and system term ${{w}_{xx}}\left(0, t \right)$

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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 $\dot{w}(x, t)$ on the whole beam

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Fig. 9. Control torque

Figs. 6-9 illustrate the effectiveness of the controller (26) with ${{k}_{\zeta }}=0$. We could see that only the joint angular position and speed are regulated to the desired values in Fig. 6, but the vibrations on the beam displayed in Figs. 7 and are not suppressed. So our objective is not achieved if we lose the term $-{{k}_{\zeta }}w\left(\zeta, t \right)\int_{0}^{t}{s}w\left(\zeta, \sigma \right)\textrm{d}\sigma $.

Case 2: ${{k}_{\zeta }}\ne 0$.

We set ${{k}_{\zeta }}=5$ and the simulation results are presented in Figs. 10-16. From (38) and due to the presence of the term $-{{k}_{\zeta }}w\left(\zeta, t \right)\int_{0}^{t}{s}w\left(\zeta, \sigma \right)\textrm{d}\sigma $, we conclude the result that $\tilde{f}$, $\widetilde{EI}{{w}_{xx}}\left(0, t \right)$ and $w\left(\zeta, t \right)$ tend to zero as $t\to \infty $, which were illustrated by Figs. 11, 12 and 14. In Fig. 12, the term $\widetilde{EI}{{w}_{xx}}\left(0, t \right)$ tends to zero but $\widetilde{EI}$ does not tend to zero as well, which is also shown in Fig. 10. It is reasonable because we can only guarantee the product term tend to zero. Further, it does not affect the stability of the closed-loop system. To further improve the control performance, viscous damping is added in system and the effectiveness of closed-loop system is demonstrated by Figs. 13-16. In Fig. 13, the joint angular position and speed tracking are shown. Figs. 14 and 15 respectively display the elastic deflection and its speed of the beam. Fig. 16 shows the control input.

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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 $\widetilde{EI}$ and system term ${{w}_{xx}}\left(0, t \right)$

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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 $\dot{w}(x, t)$ on the whole beam

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Fig. 16. Control torque

Figs. 10-16 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 Conclusions

In this paper, a novel adaptive sliding mode controller is designed based on neural networks for a single flexible-link manipulator. The stability of the closed-loop 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.

Appendix

First, we multiply both sides of (14) by $x$, integrate from 0 to $L$, and obtain

$ \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).

Acknowledgement

The authors would like to thank the Editor-in-Chief, 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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