Study on the control method about improving piezoelectric actuated photoelectric precision tracking system

Abstract

Fast steering mirrors (FSM) driven by Piezoelectric transducer (PZT) are widely used in various precision stable tracking systems. Aiming to counteract the hysteresis and non-linear interference in PZT, this work applies a radial basis function (RBF) neural network to approximate its nonlinearity. Adaptive backstepping sliding mode (ABSM) controller combinewith a sliding mode control method and backstepping control is designed. Combining the characteristics of PZT and voice coil motors (VCM), the FSM driven by VCM is designed as the power sub-system to ensure that the large-angle deflection of the FSM can match a wider field of view. The FSM driven by PZT is designed as a correction sub-system, which can adjust the system error within a small range. Finally, the power sub-system and the correction sub-system are combined into a two-level precision tracking system. The simulation results show that the maximum steady-state error of the system is about 15 $\mu$ rad, and the root mean square error is about 10 $\mu$ rad. Compared with the traditional PI controller, the error is reduced by about 75%, the response speed is up to 10 ms, and the output is smooth without some serious viberation.

Keywords

Fast steering mirror (FSM)voice coil motor (VCM)piezoelectric ceramic adaptive backstepping sliding mode control stable tracking

1. Introduction

In optical tracking and measurement systems, it is extremely difficult to directly finish fast and high-precision tracking due to the large inertia, narrow frequency band, and slow response of large-aperture telescopes [1, 2, 3]. In order to ensure a high-precision tracking of the target, a compound-axis control system is essential. FSM is a device developed in recent years, which is used to control beam deflection to achieve accurate optical axis pointing [4]. Micro-drivers such as VCM or PZT are often used to drive FSM to achieve precise control of the beam direction between the light source and the target. The VCM offers several advantages of simple structure, slightly larger volume, larger stroke, and larger specific thrust [5, 6]. Compared with the VCM, PZT is smaller in size, larger in thrust but short in travel distance, and it offers several advantages such as the convenient control, high displacement resolution, high resonance frequency, fast response speed, no heat, no magnetic interference, high control accuracy, etc. However, non-linear properties such as hysteresis of the piezoelectric material itself will limit the further improvement of positioning accuracy [7, 8].

In this paper, the precision tracking system is designed as a combination of two sub-systems, in which the FSM driven by VCM is designed as the power system, and the FSM driven by PZT is designed as the correction system. The structure of precision tracking system is shown in Fig. 1.

Figure 1.

Schematic diagram of fine tracking system.

This study uses the LuGre model to describe the hysteresis of PZT, and builds a mathematical model of the piezoelectric actuator, using RBF network to approximate the hysteresis effect. An adaptive backstepping sliding mode controller is designed for the piezoelectric driving FSM system, it can adjust the system errors within a small range. The deflection angle of the FSM driven by PZT is generally small, usually on the order of 10–20 urad, while the deflection angle of the FSM driven by VCM is generally larger than PZT. Therefore, in this paper, the FSM driven by VCM is used as the power system to ensure the large-angle deflection of the FSM. Finally, the numerical simulations are used to verify the performance of the designed ABSM controller as well as the tracking accuracy of the designed fine tracking system.

2. System design and modeling

Taking a certain type of FSM as the research object, FSM is driven by the same four PZT stack drivers, as shown in Fig. 2. A and C are used to push or pull the FSM to do pitching motion, while B and D are used to drive the FSM to do the similar azimuth motion. The driving principles of the azimuth axis and the pitch axis are the same.

Figure 2.

Actuator position.

The displacement increments of drives B and D are equal in magnitude, opposite in direction, which can be expressed as.

The drives B and D will conduct two movements with a equal magnitude and opposite direction.

$\displaystyle\left\{\begin{array}[]{l}x_{B}=R\cdot\tan(\Delta\theta/2)\\ x_{D}=-R\cdot\tan(\Delta\theta/2)\end{array}\right.$ (1)

$R$ is the FSM radius, $\Delta\theta$ is the deflection angle, $x_{B}$ and $x_{D}$ represent the displacements at points B and D. If the two drives are the same, the schematic diagram of the FSM drive system is shown in Fig. 3.

Figure 3.

Schematic diagram of FSM driven system.

In Fig. 3, $\Delta\theta_{i}$ is the input signal namely the coarse tracking. ${x}^{\prime}_{B}$ and ${x}^{\prime}_{D}$ are the actual displacement output, and $\Delta\theta_{o}$ is the corresponding output angle.

2.1 FSM driven by VCM modeling

The VCM is also working based on the same principle as the speaker in which the conducting wire is woking in the magnetic field and driven by the electromagnetic force to push the load [9]. Its structure is shown in Fig. 4.

The dynamic equations of VCM are described as follows:

$\displaystyle\left\{{{\begin{array}[]{l}{u=L\frac{\text{d}i}{\text{d}t}+Ri+% \delta\textit{BLv}}\\ {F=\delta\textit{Bli}}\\ {F=\textit{Ma}+\textit{Kv}}\\ {v=\frac{\text{d}x}{\text{d}t}}\\ {a=\frac{\text{d}^{2}x}{\text{d}t^{2}}}\\ \end{array}}}\right.$ (2)

$u$ , $L$ , $R$ , $M$ , $x$ , $a$ , $F$ are respectively the voltage, inductance, resistance, mass, displacement, acceleration of the conducting coil and the ampere force received in the magnetic field. $\delta$ is the coefficient of inductance, $B$ is the electromagnetic field strength, and $K$ is the damping coefficient. Take the Laplace transform of the Eq. (2) and rearrange it:

$\displaystyle\textit{Ms}^{2}x+\textit{Ksx}=\frac{\delta\textit{Bl}(u-\delta% \textit{Bl}\frac{\text{d}x}{\text{d}t})}{\textit{Ls}+R}$ (3)

Figure 4.

Schematic diagram of the voice coil motor.

Namely:4

$\displaystyle\frac{X(s)}{U(s)}=\frac{\delta\textit{Bl}}{\textit{LMs}^{2}+(% \textit{KL}+\textit{RM})s+\textit{RK}+\delta^{2}B^{2}l^{2}}\cdot\frac{1}{s}$ (4)

As can be seen from the Eqs (3) and (4), the denominator of the system has the highest order 3, and the numerator is a constant. Third order polynomials must have a real root. In addition, the system is in one-dimensional direction, and two VCM drivers drive the FSM through push-pull method. So, the transfer function of FSM driven by VCM is shown as follows:

$\displaystyle G_{y}(s)=\frac{p_{1}}{p_{2}s^{2}+p_{3}s+1}\cdot\frac{1}{p_{4}s+1}$ (5)

$p_{1},p_{2},p_{3},p_{4}$ are undetermined parameters, and they can be identified by the least square method.

2.2 FSM driven by PZT modeling

PZT drivers are composed with the piezoelectric stacking. Hundreds of wafers are stacked top to bottom one by one to achieve a mechanical connecting, which makes it easier to use low voltage to produce large deformation. The equivalent model of piezoelectric stacking is shown in Fig. 5.

Figure 5.

Piezoelectric stack equivalent model diagram.

According to Kirchhoff’s voltage law:

$\displaystyle u_{i}(t)=rc_{p}\frac{\text{d}u_{o}(t)}{\text{d}t}+u_{o}(t)$ (6)

$r$ is the output resistance; $u_{o}$ is the output voltage; $u_{i}$ is the input voltage. Take the Laplace transform of the Eq. (6), assuming that the initial condition is zero, and we can get:

$\displaystyle U_{i}(s)=rc_{p}sU_{o}(s)+U_{o}(s)$ (7)

Assuming that $\tau=rc_{p}$ is the time constant, and its transfer function is

$\displaystyle G_{1}(s)=\frac{U_{o}(s)}{U_{i}(s)}=\frac{1}{\tau s+1}$ (8)

According to the deformation characteristics of piezoelectric stack, the relationship between the driving force $f$ and deformation $x$ can be obtained:

$\displaystyle x=au_{o}-\frac{1}{k_{p}}f=au_{o}-\frac{1}{k_{p}}\left(kx+c\frac{% \text{d}x}{\text{d}t}\right)$ (9)

$k_{p}$ is equivalent stiffness coefficient; $a$ is undetermined coefficient; $k$ is the connection stiffness; $c$ is damping coefficient. Take the Laplace transform of the Eq. (9), and we can get:

$\displaystyle G_{2}(s)=\frac{X(s)}{U_{o}(s)}=\frac{ak_{p}}{cs+(k+k_{p})}$ (10)

The transfer function of the system is:

$\displaystyle G(s)=G_{1}(s)G_{2}(s)=\frac{ak_{p}}{cc_{p}rs^{2}+(krc_{p}+k_{p}% rc_{p}+c)s+k+k_{p}}=\frac{a_{0}}{b_{0}s^{2}+b_{1}s+b_{2}}$ (11)

That $c_{p}$ is the piezoelectric stack equivalent capacitance.

3. Design of precision tracking system controller

3.1 Controller design for correction system

The mathematical model of the piezoelectric transducer can be described by differential equations as follows:

$\displaystyle m_{p}\ddot{x}=-K_{f}\dot{x}-K_{g}x+K_{u}(u-f)$ (12)

Where $m_{p}$ is effective mass; $u$ is driver terminal voltage that varies over time; $x$ is driver displacement; $K_{f}$ is damping coefficient; $K_{g}$ is mechanical stiffness; $K_{u}$ is input control coefficient; $f$ is system nonlinear interference.

It is defined as follows: $a=\frac{-K_{f}}{m_{p}}$ , $b=\frac{-K_{g}}{m_{p}}$ , $c=\frac{K_{u}}{m_{p}}$ , $x_{1}$ represents position; $x_{2}$ , represents the rate of change of position. The model can be simplified as follows:

$\displaystyle\left\{{\begin{array}[]{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=ax_{2}+bx_{1}+c(u-f)\\ \end{array}}\right.$ (13)

3.1.1 Description the hysteresis of piezoelectric ceramics

The hysteresis is the main interference of PZTcontrol, but it is actual a kind of friction, which can be described by certain friction model. At present, the main friction models including the LuGre model, bristle model, bliman-sorine model, Stribeck model, Reynold model and so on [10]. LuGre model includes the friction hysteresis, variable static friction force and viscous friction phenomenons, which can fully capture the dynamic hysteresis effect of piezoelectric ceramics. Its mathematical equations of the dynamic friction model can be expressed as:

$\displaystyle\left\{{\begin{array}[]{l}f=\sigma_{0}z+\sigma_{1}\frac{dz}{dt}+% \sigma_{2}\dot{\theta}\\ \frac{dz}{dt}=\dot{x}-\sigma_{0}\frac{\left|\dot{x}\right|}{g(\dot{x})}z\\ g(\dot{x})=f_{c}+(f_{s}-f_{c})\exp\left[{-(\dot{\theta}/\dot{x}_{s})^{2}}% \right]\\ \end{array}}\right.$ (14)

$\sigma_{0}$ is the stiffness of $z$ ; $\sigma_{1}$ is the dynamic damping; $\sigma_{2}$ is the coefficient of viscous friction. $f_{c}$ is coulomb friction torque; $f_{s}$ is the maximum static frictional force; $\dot{x}_{s}$ is the switching speed.In this chapter, LuGre model is used to describe the PZT hysteresis , and RBF neural network is used to adaptively approach the hysteresis effect $f$ . Then, combining the sliding mode control with backstepping design of excellent dynamic and static characteristics, the ABSM controller is designed for the correction system in this chapter.

3.1.2 Design of ABSM controller

Because of its simple algorithm and good robustness, the sliding mode control is usually used in systems where model parameters are uncertain [11]. Firstly, the core idea of backstepping method is to decompose the complex nonlinear system. Then the virtual control variables and Lyapunov function are designed for each sub-system. Finally make each sub-system gradually “fallback” to the whole system to bring the system up to the desired performance metrics [12].

From the perspective of system control, the hysteresis is represented by an unknown nonlinear function $f$ , which cannot be directly applied to the design of system controller. Therefore, the RBF neural network, which can approach any nonlinear function, is used to approach the nonlinear function $f$ , which is used to describe the nonlinearity hysteresis phenomenon of PZT.

RBF neural network is generally composed of three layers, which are the input layer, hidden layer and output layer from left to right, as is shown in Fig. 6.

Figure 6.

RBF network structure diagram.

The hidden layer activation function is defined as the Gaussian activation function, and its algorithm for approximating $f(x)$ is:

$\displaystyle\left\{{\begin{array}[]{l}h_{i}(x)=\exp\left[{\frac{(x-c_{i})^{T}% (x-c_{i})}{2\sigma_{i}^{2}}}\right]\\ f=W^{T}h(x)+\varepsilon\\ \end{array}}\right.$ (15)

$x$ is an $n$ -dimensional input vector; $c_{i}$ is $n$ -dimensional central column vector; $\sigma_{i}$ is normalized constant; $h=[h_{i}]^{T}$ is the output of the gaussian basis function of RBF network; $W$ is ideal weight vector; $\varepsilon$ is approximate error. If the input is $x=[x_{1}\ \ x_{2}]^{T}$ , the output will be

$\displaystyle\hat{f}=\hat{W}^{T}h(x)$ (16)

Define $x_{d}$ as the desired position after the coordinate transformation, then the system position error will be $e_{1}=x_{1}-x_{d}$ , and its derivative will be $\dot{e}_{1}=x_{2}-\dot{x}_{d}$ . $x_{2}=\dot{x}_{1}$ .

Define the Lyapunov function:

$\displaystyle V_{1}=\frac{1}{2}e_{1}^{2}$ (17) $\displaystyle x_{2}=e_{2}+x_{d}-c_{1}e_{1}$ (18)

$c_{1}$ is a constant greater than zero, $e_{2}$ is virtual control variable, and they are defined as follows:

$\displaystyle e_{2}=x_{2}-\dot{x}_{d}+c_{1}e_{1}$ (19)

We can get

$\displaystyle\dot{e}_{1}=x_{2}-\dot{x}_{d}=e_{2}-c_{1}e_{1}$ (20) $\displaystyle\dot{V}_{1}=e_{1}\dot{e}_{1}=e_{1}e_{2}-c_{1}e_{1}^{2}$ (21)

Define the switch function:

$\displaystyle\sigma=k_{1}e_{1}+e_{2},{k_{1}>0}$ (22)

Take the Eq. (22) into Eq. (19), we can get

$\displaystyle\sigma=k_{1}e_{1}+\dot{e}_{1}+c_{1}e_{1}=(k_{1}+c_{1})e_{1}+\dot{% e}_{1}$ (23)

$k_{1}+c_{1}>0,$ if $\sigma=0$ , then $e_{1}=0$ , $e_{2}=0$ and $\dot{V}_{1}\leqslant 0$ . Therefore, it is necessary to further design and define the Lyapunov function:

$\displaystyle V_{2}=\frac{1}{2}e_{1}^{2}+\frac{1}{2}\sigma^{2}+\frac{1}{2% \gamma}\tilde{W}^{T}\tilde{W}$ (24)

$\tilde{W}=\hat{W}-W$ is the estimation error of, $\gamma$ is a constant greater than 0, so

$\displaystyle\dot{V}_{2}=\dot{V}_{1}-\frac{1}{\gamma}\tilde{W}^{T}\dot{\hat{W}% }=e_{1}e_{2}-c_{1}e_{1}^{2}+\sigma\left[{\begin{array}[]{l}k_{1}(e_{2}-c_{1}e_% {1})+a(e_{2}+\dot{x}_{d}-c_{1}e_{1})\\ +bx_{1}+c(u+f)-\ddot{x}_{d}+c_{1}\dot{e}_{1}\\ \end{array}}\right]+\frac{1}{\gamma}\tilde{W}^{T}\dot{\hat{W}}=e_{1}e_{2}-c_{1% }e_{1}^{2}+\sigma\left[{\begin{array}[]{l}k_{1}(e_{2}-c_{1}e_{1})+a(e_{2}+\dot% {x}_{d}-c_{1}e_{1})\\ +bx_{1}+c(u+\hat{f})-\ddot{x}_{d}+c_{1}\dot{e}_{1}\\ \end{array}}\right]+\tilde{W}^{T}\left({\frac{1}{\gamma}\dot{\hat{W}}-c\sigma h% (X)}\right)$ (25)

Design control rule as follows:

$\displaystyle u=\frac{1}{c}\left\{{\begin{array}[]{l}-k_{1}(e_{2}-c_{1}e_{1})-% a(e_{2}+\dot{x}_{d}-c_{1}e_{1})-bx_{1}\\ +\ddot{x}_{d}-c_{1}\dot{e}_{1}-\alpha[\sigma+\beta sgn(\sigma)]\\ \end{array}}\right\}-\hat{f}$ (26)

Design adaptive law of RBF network:

$\displaystyle\dot{\hat{W}}=\gamma c\sigma h(X)$ (27)

Take the Eqs (27) and (26) into Eq. (25) and we can get

$\displaystyle\dot{V}_{2}=e_{1}e_{2}-c_{1}e_{1}^{2}-\alpha\sigma^{2}-\alpha% \beta\left|\sigma\right|$ (28)

Let

$\displaystyle Q=\left[{{\begin{array}[]{*{20}c}{c_{1}+\alpha k_{1}}\hfill&{% \alpha k_{1}-\frac{1}{2}}\hfill\\ {\alpha k_{1}-\frac{1}{2}}\hfill&\alpha\hfill\\ \end{array}}}\right],e=\left[{{\begin{array}[]{*{20}c}{e_{1}}\hfill\\ {e_{2}}\hfill\\ \end{array}}}\right],$

$\displaystyle e^{T}\textit{Qe}=\left[{{\begin{array}[]{*{20}c}{e_{1}}\hfill&{e% _{2}}\hfill\\ \end{array}}}\right]\left[{{\begin{array}[]{*{20}c}{c_{1}+\alpha k_{1}^{2}}% \hfill&{\alpha k_{1}-\frac{1}{2}}\hfill\\ {\alpha k_{1}-\frac{1}{2}}\hfill&\alpha\hfill\\ \end{array}}}\right]\left[{{\begin{array}[]{*{20}c}{e_{1}}\hfill&{e_{2}}\hfill% \\ \end{array}}}\right]^{T}=c_{1}e_{1}^{2}-e_{1}e_{2}+\alpha\sigma^{2}$ (29) $\displaystyle\left|Q\right|=\alpha(c_{1}+\alpha k_{1}^{2})-\left(\alpha k_{1}-% \frac{1}{2}\right)=\alpha(c_{1}+k_{1})-\frac{1}{4}$ (30)

It can be seen from Eq. (30) that we can make $\left|Q\right|$ be a positive definite matrix by adjusting the value of $\alpha_{1}$ , $c_{1}$ , and $k_{1}$ . So, it can be proved that:

$\displaystyle\dot{V}_{2}\leqslant-e^{T}Qe-\alpha\beta\left|\sigma\right|\leqslant 0$ (31)

The structural diagram of the controller can be deduced from above, as is shown in Fig. 7:

Figure 7.

Adaptive backstepping sliding mode controller structure.

3.1.3 Simulation analysis of ABSM controller

Select the PZT driver produced by PI company, and its run length is 80 $\mu$ m. The radius of the FSM and the distance of the driver relative to the center fulcrum are both equal to $R=$ 30 mm. The PZT parameters are as follows: $a=$ $-$ 108.16, $b=-$ 5.979 $\times$ 10 ${}^{5}$ , $c=$ 1.293 $\times$ 10 ${}^{6}$ . LuGre model parameters are set as follows: $f_{c}=$ 0.6, $f_{s}=$ 0.01, $\dot{x}_{s}=$ 0.013, $\sigma_{0}=$ 200, $\sigma_{1}=$ 0.02. Set the parameters of the controller as follows: $c_{1}=$ 100, $k_{1}=$ 15, $\alpha=$ 20, $\gamma=$ 30. Set the number of input nodes of RBF network to 2, the number of hidden nodes to 8, and the number of output nodes to 1. Set the initial value of $c_{i}$ and $\sigma_{i}$ to 3 and 0.75.

Taking $\Delta\theta_{i}=$ 2 mrad as the step input signal; $\Delta\theta_{i}=$ 2sin(2 $\pi t$ ) mrad as the sinusoidal input signal, aiming at the correction system driven by PZT, PI controller and ABSM controller are used for simulation comparison in this chapter. Simulation results are shown in Figs 8 and 9.

Figure 8.

Response curve of step signal.

It can be seen from Fig. 8 that the system has a quick response due to its simple structure. However, the driver is seriously disturbed by the hysteresis and the steady state error is increased. Although the response speed of ABSM controller is not quick enough, it can effectively suppress hysteretic nonlinear problems, and adjusting time is only about 0.2 ms. It can be seen from Fig. 9 that, the maximum tracking error is about 57.58 $\mu$ rad and the root mean square error is 20.334 $\mu$ rad when using PI controller, but the overshoot is too high. While using ABSM controller, the maximum tracking error is only 14.632 $\mu$ rad, and the root mean square error is 5.1732 $\mu$ rad, which are reduced by 74.6% and 74.56% compared with the PI controller.

Figure 9.

Sinusoidal signal tracking error.

3.2 Power system controller design

From Section 2 we know that the transfer function of FSM driven by VCM is:

$\displaystyle G_{y}(s)=\frac{p_{1}}{p_{2}s^{2}+p_{3}s+1}\cdot\frac{1}{p_{4}s+1}$ (32)

Design the PI controller as $C_{\textit{Fy}}(s)=a_{1}(p_{4}s+1)$ , then the transfer function will be:

$\displaystyle G_{f}(s)=C_{\textit{Fy}}(s)G_{y}(s)=\frac{a_{1}p_{1}}{p_{2}p_{4}% s^{3}+(p_{3}p_{4}+p_{2})s^{2}+(p_{3}+p_{4})s+1}$ (33)

Generally, $p_{2},p_{3},p_{4}$ are much less than 1, thus

$\displaystyle G_{f}(s)\approx a_{1}p_{1}$ (34)

satisfying $a_{1}=1/p_{1}$ , we can achieve that the error is approximately 0.

4. Simulation and numerical analysis of precision tracking system

According to the established and designed mathematical model of precision tracking system controller, we can get the precision tracking system structure as shown in Fig. 10.

Figure 10.

Precision tracking system structure.

Figure 11.

Response curve of step signal.

Figure 12.

Controlled quantity of correction sub-system.

Figure 13.

Sinusoidal input.

Figure 14.

Response error.

Figure 15.

Controlled quantity of correction sub-system.

The deflection angle of FSM driven by PZT is generally small. Its tracking accuracy is high but there is hysteresis in PZT. However, the deflection angle of FSM driven by VCM is big, while its tracking accuracy is low. Therefore, VCM is selected as the driver of power sub-system to ensure that the large-angle deflection of the FSM can match the field of view. The FSM driven by PZT is designed as the correction sub-system, which adjusts system errors in a small range.

The parameters of VCM driver are set as follows:

$\displaystyle p_{1}=0.5495,p_{2}=2.51\times 10^{-6},p_{3}=2.33\times 10^{-3},p% _{4}=4.813\times 10^{-4},a_{1}=1.82$

The parameters of PZT driver are the same as that in Section 3.1.3. Take the azimuth axis as example, $\Delta\theta_{i}=$ 2 mrad is the system step input, the simulation results as Figs 11 and 12 show.

It can be seen from Fig. 11 that the response time is about 0.016s, and the overshoot is about 5%. Furthermore, the output curve is smooth without viberation. When the input is $\theta_{i}=0.01\sin(2\pi t)$ ; rad, the simulation results are as Fig. 13 to Fig. 15.

It can be seen from above that the system maximum steady-state error is about 15 $\mu$ rad, and the root mean square error is about 10.607 $\mu$ rad. It is concluded that the ABSM controller can effectively suppress viberation.

5. Conclusions

The hysteresis nonlinearity of the piezoelectric ceramic actuator will reduce the control accuracy of FSM. Aiming at this problem, this article uses RBF neural network to approximate its hysteresis nonlinearity, and designs ABSM controller, combing sliding mode control and backstepping method. Meanwhile, the FSM driven by VCM driver is designed as the power sub-system, and the FSM driven by PZT driver is designed as the correction sub-system to reduce the error. The simulation results of ABSM Controller show that the error is reduced by 75% compared to PI controller, and chattering phenomenon in sliding mode control is effectively suppressed. The simulation results also show that the response speed of the system is fast, and the response time is only 0.016 s. The correction system can quickly correct the output error of the power system, and the tracking error is on the order of 10 $\mu$ rad, which can be used in shipboard photoelectric tracking equipment.

In the further discussion more attentions should be paid on the refine the math model from both the FSM and the VCM. Experiment platforms and comparisons are essential to verify the improvement numerical algorithms.

References

Fan

and Zhang

, Theoretical and experimental determination of bandwidth for a two-axis fast steering mirror, Optik 124(16) (2013), 2443–2449.

Liu

Sun

et al., Analysis of complex axis control loop in satellite laser communications, Optik 123(5) (2012), 458–461.

Huisman

Aalders

J.W.G.

Eggens

M.J.

et al., Cryogenic mechatronic design of the HIFI Focal Plane Chopper, Mechatronics 21(8) (2011), 1259–1271.

Tang

Huang

Y.M.

C.Y.

and Liu

S.F.

, Acceleration feedback of a CCD-based tracking loop for fast steering mirror, Optical Engineering 48(1) (2009), 013001.

Chen

S.Y.

and Chia

C.S.

, Precision position control of a voice coil motor using self-tuning fractional order proportional- integral- derivative control, Micromachines 7(11) (2016), 207.

Kluk

D.J.

Boulet

M.T.

and Trumper

D.L.

, A high-bandwidth, high-precision, two-axis steering mirror with moving iron actuator, Mechatronics 22(3) (2012), 257–270.

Ghafarirad

Rezaei

S.M.

Zareinejad

et al., Robust control with unknown dynamic estimation for multi-axial piezoelectric actuators with coupled dynamics, Comptes Rendus Mécanique 340(9) (2012), 646–660.

Almeida

Donadon

M.V.

De Faria

A.R.

et al., The effect of piezoelectrically induced stress stiffening on the aeroelastic stability of curved composite panels, Composite Structures 94(12) (2012), 3601–3611.

Hsu

C.F.

and Kao

W.F.

, Perturbation wavelet neural sliding mode position control for a voice coil motor driver, Neural Computing and Applications 31(10) (2019), 5975–5988.

10.

De Wit

C.C.

Olsson

Astrom

K.J.

et al., A new model for control of systems with friction, IEEE Transactions on Automatic Control 40(3) (1995), 419–425.

11.

Utkin

V.I.

, Sliding mode control design principles and applications to electric drives, IEEE Transactions on Industrial Electronics 40(1) (1993), 23–36.

12.

Shen

P.H.

and Lin

F.J.

, Intelligent backstepping sliding-mode control using RBFN for two-axis motion control system, IEE Proceedings-Electric Power Applications, 152(5) (2005), 1321–1342.