Volume 6
Issue 2
IEEE/CAA Journal of Automatica Sinica
| Citation: | Ruicheng Ma and Shuang An, "Minimum Dwell Time for Global Exponential Stability of a Class of Switched Positive Nonlinear Systems," IEEE/CAA J. Autom. Sinica, vol. 6, no. 2, pp. 471-477, Mar. 2019. doi: 10.1109/JAS.2018.7511264
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IN the recent years, there is a growing interest in microelectromechanical system (MEMS) due to the advantages like light weight, small size and low cost. The MEMS micromirror is one of the most important optical components which has been leveraged to benefit many industry fields, including high resolution display [1], optical switches [2], bar code reading [3] and bioimaging system [4]. Among several driving principles available for micromirror actuation, electrostatic actuation is the most widely used actuation scheme due to its simple implementation. However, electrostatic actuation suffers from nonlinear dynamics, leading to the pull-in phenomenon, which limits the stable operational range [5]. On the other hand, the presence of nonlinearity prevents the micromirror from providing the accurate positioning performance.
The works reported in the literatures have demonstrated the benefits of utilizing closed-loop control schemes to enhance the performance of MEMS micromirror [6], [7]. PID controller has been utilized to achieve fast response and precise positioning performance of a two-axis micromirror [8]. A PID controller augmented with approximate feedback linearization is proposed for a dual-axis optical mirror [9]. Nonlinear PD control and gain scheduling approach have been proposed to improve the performance of a 1-degree of freedom MEMS mirror, and enhance the robustness of the system to stochastic perturbations [10], [11]. A nonlinear servo control is utilized to stabilize the micromirror beyond the pull-in voltage for a large port count optical switch [12]. A nonlinear tracking control based on feedback linearization and trajectory planning has been developed to extend the stable operation range, enhance the system performance of an electrostatic comb-driven micromirror [13]. Several nonlinear feedback controllers, such as flatness-based control [14], closed-loop control based on input-output linearization technique [15], have been reported to control the tilt angle of optical micromirror. A robust adaptive control has been reported to improve the tracking performance of micromirror in presence of large parameter variations [16]. A closed-loop adaptive control scheme has been proposed to compensate online for parameter variations of the micromirror [17].
Sliding mode control has the ability to ensure good performance in the presence of various uncertainties and disturbances which has aroused interest among researchers for micro/nano manipulation recently [18]-[21]. However, the main disadvantage of sliding mode control is the chattering problem. One way for reducing chattering is to use higher order sliding mode control techniques. Second order sliding mode control maintains the robustness and performance as that of traditional sliding mode technique, and reduces the undesirable chattering phenomenon of sliding mode control [22].
The twisting algorithm and super-twisting algorithm are widely used second order sliding mode control schemes [23]. A robust technique using super-twisting algorithm is proposed for speed control application of variable reluctance motor [24]. A novel adaptive gain second order sliding mode algorithm combines the super-twisting algorithm and a linear term is proposed for switching power converters, this method is effective to reduce the number of sensors [25]. A super-twisting sliding mode observer is proposed to estimate the phase currents and load resistance which guarantees fast convergence rate of the observation error dynamics [26]. Super-twisting algorithm is utilized in adaptive gain second order sliding mode observer for a polymer electrolyte membrane fuel cell, this method extends the results of a class of nonlinear uncertain systems with Lipschitz nonlinearities [27].
The rapidly growing applications of MEMS micromirror have increased the demand for fast response and accurate positioning. For imaging applications, two-axis micromirror is rotated about two orthogonal axes at different frequencies to achieve raster scans. However, the cross-axis coupling effect degrades the tracking performance. The motivation of this study is to propose a simple and effective robust control scheme, which requires a very small computational effort, to improve the positioning precision, transient and tracking performance of a two-axis micromirror for imaging application.
The contributions of this paper are: twisting algorithm with integral sliding surface is implemented to improve the transient and positioning performance of an electrostatic MEMS micromirror. To the best of our knowledge, the proposed scheme has not yet been proposed for an electrostatic MEMS micromirror; A closed-loop controlled micromirror-based laser scanning system is developed. The proposed control scheme is implemented to compensate the tracking error in the presence of cross-coupling effect, and thereby achieving accurate tracking performance and enhanced image quality.
The mathematical model of the micromirror is presented in Section Ⅱ. Section Ⅲ introduces the proposed controller for the MEMS micromirror. The experimental setup is presented, and the set-point regulation results of three control schemes are compared in Section Ⅳ. The operation performance of open loop control as well as the proposed closed loop control of MEMS micromirror inside a laser scanning system is illustrated in Section Ⅴ and experimental results are demonstrated to show the effectiveness the closed loop system with the proposed scheme. Section Ⅵ presents some concluding remarks.
The studied two-axis scanning micromirror with sidewall electrodes is shown in Fig. 1 [28], it can be seen that a double-gimbal structure is employed, the mirror plate (1000 ${\mathrm{\mu}}$m $\times$ 1000 $\mu$m in area and 35 $\mu$m in thickness) is sustained by a pair of inside torsional beams (3 $\mu$m in width, 12 $\mu$m in thickness) within a rigid gimbal. The gimbal is suspended by another pair of torsional beams which are connected to the supporting frame. There are four bottom electrodes 1060 $\mu$m $\times$ 1060 $\mu$m in area) underneath the mirror plate, and four bottom sidewall electrodes (250 $\mu$m in height) around the mirror plate. Voltages are applied to the bottom and sidewall electrodes to change the mirror plate's tilt angle on two axes.
We take the tilt angles $\theta _x $ and $\theta _y $ around the $x$-axis and $y$-axis respectively. The equation of motion of the two-axis micromirror is given by [29]
|
(J1+J2)¨θx+D1˙θx+K1θx=TxJ1¨θy+D2˙θy+K2θy=Ty |
(1) |
where $J$ is the mass moment of inertia, $D$ is the linear damping coefficient, $K$ is the mechanical stiffness coefficient. The total electrostatic torque is given by [30]
|
Tx=4∑i=1TEαei1+4∑i=13∑j=2TEsαeij+4∑i=1TEgαei2Ty=4∑i=1TEβei1+4∑i=13∑j=2TEsβeij. |
(2) |
The electrostatic torque contributed by the bottom electrodes on the mirror plate can be defined as [30]
|
4∑i=1TEαei1= 12ε04∑i=1V2i∬Sei1−y(sinφφ×1g−xcosθxsinθy+ysinθx)2dxdy4∑i=1TEβei1= 12ε04∑i=1V2i∬Sei1x(sinφφ×1g−xcosθxsinθy+ysinθx)2dxdy |
(3) |
with $\varphi =\cos ^{-1}\left( {\cos \theta _x \cos \theta _y } \right)$. $\varepsilon _0 $ is the permittivity of air, $g$ is the gap between the bottom electrodes and mirror plate. $S_{ei1} $ is the integral domain. $T_{\alpha eij}^{Es}, {\kern 1pt}{\kern 1pt}{\kern 1pt}T_{\beta eij}^{Es} $ and $T_{\alpha ei2}^{Eg} $ are the electrostatic torques from the sidewall electrodes on the mirror and gimbal frame respectively [30]. The actuated voltages $V_i $ are defined as [31]
|
V1=Vbias+0.5(−Vx+Vy)V2=Vbias+0.5(−Vx−Vy)V3=Vbias+0.5(Vx−Vy)V4=Vbias+0.5(Vx+Vy) |
(4) |
where a bias voltage of 50 volts is used to improve the linearity. $V_x $ and $V_y $ are control voltages applied to the bottom and sidewall electrodes for mirror actuation.
Sliding mode control is one of the best choices for a number of engineering problems with disturbances and uncertainties. The closed-loop system is shown in Fig. 2. The basic design process of sliding mode control can be divided into two steps. First, design an adequate sliding surface according to the desired control objectives. Second, design a control law to drive and confine the system motion on this sliding surface.
The first stage in the control design procedure is to establish the sliding surface, which is expected to respond to desired control specifications and performance. The sliding surface in this case is defined by the equation
|
σ(t)=e(t)+c1˙e(t)+c2∫t0e(t)dτ |
(5) |
where $c_1$ and $c_2$ are independent positive coefficients. The tracking error $e\left( t \right)$ in a closed-loop system is defined as
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e(t)=θ(t)−θd(t) |
(6) |
where $\theta \left( t \right)$ is measured tilt angle, $\theta _d \left( t \right)$is the desired angle. The control objective is to force the tracking error converges to zero.
Then, define a control law to enforce the tracking error approach the sliding surface and move along the sliding surface to the origin, it implies that the tracking error will converge to zero. Twisting algorithm is utilized which is one of the most popular schemes among the second order sliding mode control. The main advantage of this scheme is that it maintains the robustness and performance and reduces the undesirable chattering phenomenon of sliding mode control. Consider a dynamic system [32]
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˙x=a(t,x)+b(t,x)uσ=σ(t,x) |
(7) |
where $x\in \mathbb{R}^n$ is the state, $u\in \mathbb{R}$ is control variable. $a, b, \sigma $ are smooth functions. The second time derivative of $\sigma $ along the trajectories of (7) is given [32] as
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¨σ=h(t,x)+g(t,x)uh=¨σ|u=0g=∂∂u¨σ≠0 |
(8) |
where $h$, $g$ are unknown smooth functions. Suppose that
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0<km≤∂∂u¨σ≤kM|¨σ|u=0|≤C |
(9) |
where $C$, $k_m $, $k_M $ are some known positive constants. Twisting algorithm is a discontinuous globally bounded controller which drives the system trajectory to converge in finite time to the origin $\sigma =\dot {\sigma }=0$ of the phase plane [23]
|
u=−r1sign(σ)−r2sign(˙σ) |
(10) |
where $r_1$ and $r_2 $ are positive control gains. sign$\left( \cdot \right)$ is the sign function. The finite-time convergence analysis has been performed by a geometrical setting, the convergence condition is [32]
|
(r1+r2)km−C>(r1−r2)kM+C, (r1−r2)km>C. |
(11) |
A generalization of the Zubov method of a Lyapunov function design is proposed for the Twisting algorithm stability analysis [33], [34]. A non-smooth strict Lyapunov function is proposed to prove the global finite time stability for Twisting algorithm [35].
The torques $T$ are used as the control variables which depend on control voltages $V_x $ and $V_y $. In the differential drive method, the electrostatic torques, $T_x $ and $T_y $, have the same sign as the corresponding control voltages, $V_x $ and $V_y $ [21]. The actual control inputs are the control voltages $V_x $ and $V_y $, which are applied to the bottom and sidewall electrodes for mirror actuation. For digital implementation, the discrete expressions of (5) and (10) are
|
σ(k)= σ(k−1)+[(1+c1Δt+c2Δt)e(k)−(1+2c1Δt)e(k−1)+c1Δte(k−2)] |
(12) |
|
u(k)=−r1sign(σ(k))−r2sign(σ(k)−σ(k−1)) |
(13) |
where $\Delta t$ is the sample time.
For optical switching applications, fast and precise switch are required in order to reduce the optical loss. To evaluate the transient and positioning performance of closed-loop controlled micromirror, the experimental setup is built as shown in Fig. 3. A laser beam is incident on the mirror plate and is deflected by mirror plate onto a position sensing detector (PSD), the displacement along the $x$ and $y$ axes is measured by PSD, then the tilt angles along each axis can be calculated from the PSD readouts. A field programmable gate array (FPGA) card is used to implement the closed-loop controller with Labview programming environment, the control voltages of FPGA are amplified by a four channel voltage amplifier for mirror actuation.
A comparison between traditional sliding mode control, twisting control with proportional-derivative (PD) sliding surface and twisting control with PID sliding surface for single axis switching is shown in Fig. 4. The experiment results show that the closed-loop system has small overshoot, short response time and accurate positioning using the proposed scheme. When comparing with the traditional sliding mode control and twisting control with PD sliding surface, it is shown that the proposed scheme with PID sliding surface has enhanced positioning performance.
We use the closed-loop controlled two-axis micromirror to build a laser scanning system. A schematic diagram of the system is shown in Fig. 5. The key component of the imaging system is a two-axis micromirror. In this experiment, the micromirror is held on an optical table which is actuated to make the laser beam achieve raster scans on the target. A beam splitter is used to split the reflect laser beam into two parts, one part is collected by PSD to measure the position of micromirror and to obtain the tilt angle [21]. At the same time, the other part is used to scan over the target. A beam expander is placed between the beam splitter and objective lens, which is used to increase the effective numerical aperture for objective lens. The light intensity is collected by a photon detector which is used to rebuild the image. The optical power signals and corresponding position signals are collected by a FPGA card, and processed to reconstruct the image of scanned target, the data process and image reconstruction system is programmed under LABVIEW environment.
To evaluate the tracking performance of the closed-loop system, triangular waves with frequencies of 14, 21, 35 and 42 Hz are used as the desired traces of the $x$-axis trajectories, the scanning range is from 0 mm to 3 mm. Fig. 6 shows the desired and actual trajectories of $x$-axis in the closed-loop system. It can be seen that the proposed controller is able to achieve high quality scan with fast and enhanced tracking performance. To produce the raster scan for imaging application, a triangular waveform is used to drive the fast axis ($x$-axis) of the micromirror, a ramp signal is utilized to drive the slow axis ($y$-axis) of the micromirror simultaneously. The system response under open-loop control is demonstrated in Fig. 7. The excitation effect can be seen at the $x$-axis and $y$-axis due to the cross-coupling effect.
To demonstrate the improvement achieved in the positioning of the scanner by implementing the proposed scheme. The proposed controller is applied to the $x$-axis of the scanner for tracking a triangular reference signal, the $y$-axis is actuated by the same ramp signal as shown in open-loop control system. In Fig. 8, the response of $x$-axis shows that the scanner precisely follows the reference signal in presence of the cross-coupling effect. From Figs. 9 and 10, it can be seen that the comparison of the raster scans is obtained using open-loop control and the proposed scheme. A significant improvement can be seen due to introducing the proposed controller to $x$-axis. The distortion of raster scan is reduced by using the proposed scheme, one important contribution to high quality scan is that the proposed scheme is able to reduce the cross coupling effect and achieve precise tracking performance. The scanned target is shown in Fig. 11. As shown in Fig. 12, the 2D image of the target is distorted due to the distortion of raster scan pattern under open loop control. In Fig. 13, it can be seen that the image performance is improved with the proposed controller.
In this paper, a twisting algorithm incorporating a PID sliding surface is implemented for a two-axis electrostatic MEMS micromirror. The experiment results of set-point tracking demonstrated that the proposed scheme is able to achieve fast and accurate positioning performance. The proposed scheme has been implemented in a laser scanning system for fast image scanning. The proposed scheme is able to reduce the tracking error in presence of cross-coupling effect. From comparison of scanning images, it can be noted that the raster scans and images with the proposed scheme are better than open loop system.
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Figures(2)
Hui Chen, Zhendong Sun, Weijie Sun and John Tze Wei Yeow, "Twisting Sliding Mode Control of an Electrostatic MEMS Micromirror for a Laser Scanning System," IEEE/CAA J. Autom. Sinica, vol. 6, no. 4, pp. 1060-1067, June 2019. doi: 10.1109/JAS.2016.7510223
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