# Analysis and Control on a Specific Electromagnetic Micromirror

Authors: Wanyan Sun; Yonghong Tan; Ruili Dong; Wenjiang Shen
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In this paper, 10 resonant modes were divided based on the structure of a specific two-dimensional electromagnetic micromirror from Professor Shen’s team and Finite Element Method (FEM), and using as many as 10 resonant modes to do such analysis was the first time according to the best of our knowledge. The results showed that the slow axis can participate in several resonant motions under the signals of resonant frequencies. In particular, participating in Mode 1 for slow-scan axis was the key reason to the instability of Vertical Refreshing Scanning (VRS) in raster scanning. In addition, a piecewise PID control based on filters design for this electromagnetic micromirror was proposed to suppress the unexpected resonant oscillation and to improve the angular positioning accuracy in slow-scan axis control. Finally, the proposed method was applied to electromagnetic micromirror stages, and the experimental results showed that the proposed approach was reliable.


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## Body

## 1. Introduction

Micromirror is a MEMS (Micro-Electro-Mechanical System) device, which is widely used in optical communication, head-worn display, LIDAR (Lighting Detection and Ranging), projector, and medical imaging detection [1–9]. There are four main types of micromirrors in driving manners, including in electrostatic micromirror [3, 10], piezoelectric micromirror [11], electro-thermal micromirror [12], and electromagnetic micromirror [5, 6]. Among them, electrostatic micromirror is well developed for use by Thomas Bifano and Texas Instrument Inc. Thomas Bifano reported an array of electrostatic actuator shaping the mirror [4], while Texas Instrument Inc. was dedicated to developing an array of aluminum micrometer-sized mirrors, in an orthogonal layout for full HD projector and display [13]. Electromagnetic micromirrors are outstandingly of many advantages, such as lower power consumption, smaller size, and larger deflection angle [7, 14, 15]. So an electromagnetic micromirror was chosen to discuss in this paper.Ji et al. [16] proposed a two-dimensional electromagnetic micromirror using single turn coil and radial magnetic field from concentric magnet assembly in 2007, which was the first paper reported on this topic. Design and fabrication of this electromagnetic micromirror and 6 vibration modes were focused on in their paper, and the control was located in open loop control. Chen et al. [5] reported a two-dimensional scanning micromirror electromagnetically driven for slow-scan axis and harmonically driven for the fast-scan axis with larger mechanical angles up to ±7.5∘ and ±12.5∘, respectively. Structure and fabrication of the electromagnetic micromirror and the sensitivity of piezo-resistive sensor were emphasized to study, but control is still in open loop control. Steve et al. [17] focused on Newton’s method to determine the harmonic coefficients of the two-dimensional electromagnetic micromirror, but controlling the mirror deflection to correctly produce the desired angular motion presented a significant engineering challenge. Han et al. [18] proposed a new electromagnetic micromirror of unique current paths and gimbal geometry to provide independent 2-DOF (degree-of-freedom) actuation capability and large mechanical half scan angle up to ±6.43∘ and ±4.2∘, and PID control and a low-pass filter (LPF) were used in their research. However, the rings from driving coils of two axes were difficult to be eliminated.Therefore, based on the structure of an electromagnetic micromirror supported by patents [19–22] from Prof. Wenjiang Shen’s team, 10 resonant modes were divided to estimate the resonant frequencies for the first time in this paper. The results showed that the slow axis participated in several resonant motions under the signals of resonant frequencies, which were the key reasons for the instability in slow-scan axis control of this two-dimensional electromagnetic micromirror. Modal analysis of this electromagnetic micromirror was presented, and resonant frequencies were acquired. Then, a piecewise PID controller with filters was designed to improve the performances of this instability. Finally, the proposed methods were applied to an electromagnetic micromirror platform to determine their reliability.


## 2. Modal Analysis of Electromagnetic Micromirror

Structure of this electromagnetic micromirror is shown in Figure1. The inner frame is connected to the outer frame through slow beams (Beam 1 and Beam 2). Ellipse micromirror and ellipse reinforce ring are connected to the inner frame through quick beams (Beam 3 and Beam 4). Four-Terminal Piezo-resistive Sensors (FTPSs) [23] are designed at the end of Beam 2 and Beam 4, respectively. In the beginning, a magnetic flux density B is applied externally with 45° to both the quick beams and the slow beams. Meanwhile, the beams, outer frame, inner frame, ellipse reinforce ring, ellipse micromirror, and FTPSs are in the same plane. Quick beams are perpendicular to slow beams. Beam 1 and Beam 2, Beam 3 and Beam 4 are symmetrical, respectively, and they are not in torsion. m turns of DACs (Driving Au Coils) are formed elliptically on the inner frame to drive this electromagnetic micromirror by current flow. Since current flow in a wire results from motion of charges in this wire, a wire with current placed in a magnetic field where directions of current and electromagnetic cannot parallel exerts magnetic forces. For a differential length dl, of a wire of current I⟶ placed in a magnetic field B⟶, the force dF⟶ [24] is given by(1)dF⟶=I⟶dl×dB⟶,where × is a cross product. Combining with (1), an electromagnetic torque Te that DACs generate under B⟶ is shown as follows:(2)Te=∑j1=1m∮Lj1r⟶j1×dF⟶j1=∑j1=1m∮Lj1r⟶j1×I⟶dlj1×B⟶=∑j1=1m∮Lj1rj1Idlj1Bsinθ=BI∑j1=1m∮Lj1rj1dlj1sinθ,where r⟶j1 is the distance between unit length of the coil and the rotation axis on the j1th coil (j1=1,…,m), the innermost elliptical Au coil is the first coil, and the outermost elliptical Au coil is the mth coil. Lj1 is the closed path where current is running through on the j1th coil (j1=1,…,m), and θ is the angle between I⟶ and B⟶. Elliptical ring 1 and elliptical ring 2 are under ellipse micromirror and inner frame, respectively.Figure 1
Structure of electromagnetic micromirror.Based on this structure (Figure1) together with the main properties of this electromagnetic micromirror shown in Table 1, 10 resonant modes (Figure 2) are divided through ANSYS Workbench [18] in FEM, and the characters of modal analysis are shown in Table 2, which shows a detailed description of the 10 kinds of movements corresponding to Figure 2.Table 1
Main properties of electromagnetic micromirror.
Mirror materialEffective mirror size (mm2)Coil materialCoil numberSi1.2 × 1.1Au10Figure 2
Ten resonant modes of electromagnetic micromirror. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6. (g) Mode 7. (h) Mode 8. (i) Mode 9. (j) Mode 10.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Table 2
Characters of modal analysis.
ModesResonant frequencies (Hz)MotionsMode 1770.97Inner frame, slow beams, ellipse micromirror, ellipse reinforce ring, quick beams, elliptical ring 1, and elliptical ring 2 were taken as a whole (W1) and rotate around slow beamsMode 21523.9Motion of W1 is perpendicular to the outer frame planeMode 32786.6Motion of W1 is horizontal in the outer frame plane and is perpendicular to original slow beamsMode 45150.2W1 is rocking around the quick beamsMode 55487.9W1 revolves around its center in outer frame planeMode 616309Ellipse micromirror, ellipse reinforce ring, quick beams, and elliptical ring 1 taken as a whole (W2) rotate around the quick beamsMode 721711W2 and inner frame with slow beams are vertical motions which are perpendicular to the outer frame, while their directions are oppositeMode 850074Motion of inner frame is horizontal along the original slow beams, while W2 rotates around quick beams and ellipse reinforce ring may cause some distortions which are not recoveredMode 952054W2 rotates around the slow beamsMode 1062000Edges along the slow beams and edges along the quick beams both in the inner frame are folding and unfolding at the same time, while in the opposite direction they may cause some distortions which are not expectedAmong motions in Figure2, Mode 1 and Mode 6 are usually used in Lissajous scanning as main operations, but they are hard to be controlled [13]. Different from Lissajous scanning, in this paper, raster scanning [1] is applied (Figure 3), which can avoid rings caused by driving coils for the fast axis and the slow axis. Sine is a signal of single frequency. Any signal, such as triangular waves, step signals, and so on, can be thought of as a composite of many sine signals of different amplitudes and frequencies [25]. If these signals of different frequencies as the input signals inspired the resonant motions shown in Figure 2, the triggered ringing effects could also occur. To avoid this phenomenon, sine signal is chosen as the input voltage. Sine signal of a high frequency in the horizontal direction called Horizontal Fast Scanning (HFS) is given; meanwhile, sine signal of a low frequency in the vertical direction called Vertical Refreshing Scanning (VRS) is supplied. Mode 6 is used as the quick-scan axis and is called HFS in raster scanning; DACs mentioned in structure description of the electromagnetic micromirror is used as the slow-scan axis and is called VRS in raster scanning. The analysis results show that the slow axis participates in many resonant motions under the signals of resonant frequencies (Figure 3 and Table 2). Combined with the FEM (Figure 2), in VRS control, when the slow-scan axis operates in a noisy environment with resonant frequencies shown in Table 2, ten resonant motions are inspired to destroy raster scanning, which means that the key to stable control of electromagnetic micromirror is the stability of controlling slow-scan axis. In this research, slow-scan axis control is focused on, and filters of the slow-scan axis are designed to give a solution to this issue.Figure 3
Principle of raster scanning.


## 3. Filters Design

Although there are many noise removal methods such as m-derived filtering, Butterworth filters, and elliptical filters [26], two issues of filtering for electromagnetic micromirror in this paper should be briefly discussed. One is that the designed filtering must effectively eliminate noise with frequencies close to that in ten modes to avoid vibrations of the electromagnetic micromirror. In particular, environment noise of frequencies near mode 1 should be filtered to ensure the reliability of slow axis control in laser scanning. Another is that the effectiveness of the signal must be ensured under no phase distortion to realize closed-loop control in slow-scan axis control for electromagnetic micromirror better. So, secondary filtering is developed. First filtering, equal-ripple low-pass filtering, would filter these noises with resonating frequencies effectively. Second filtering, zero-phase filtering, would preserve the features of the feedback signal at the exact location where they occur and would avoid a phase distortion.In slow-scan axis control, noise with frequencies close to the resonant frequencies in ten resonant modes can inspire unexpected vibrations of the electromagnetic micromirror (Table2). The actual resonant frequency fac1 of electromagnetic micromirror in Mode 1 is derived from the ideal resonant frequency fid1=770.97Hz (see also Figure 2) as(3)fac1=Q1fid1,where the scaling factor (the ratio between the actual resonant frequency and the ideal resonant frequency) Q1=0.8432 in Mode 1 in this paper. It can be derived from (3) as follows:(4)fac1=Q1fid1=0.8432×770.97≈650Hz.


### 3.1. Equal-Ripple Low-Pass Filtering

Difference equation of Finite Impulse Response (FIR) filtering was described by Charles and Shenoi [27, 28]:(5)yi=∑i=0nhixn−i,i=0,1,…,n,where yi is the output, xn−i are the inputs, hi are the coefficients, and n is the filtering order.For slow-scan axis control in electromagnetic micromirror system, noise with frequencies close to resonant frequencies in ten modes can result in unexpected vibrations of the electromagnetic micromirror. Equal-ripple low-pass filtering, which has the advantage of controlling the boundary frequency strictly in the aiming frequency compared with other filters [29], is developed here.yf is the output of this equal-ripple low-pass filtering.Hdω is defined as the ideal magnitude of equal-ripple low-pass filtering, Hgω is the actual magnitude, N is the length of filtering, which is usually chosen as an odd number, and n is the order of filtering; the weighted error function Wω is constant. Then, the error Eω between Hdω and Hgω can be expressed as follows:(6)Eω=WωHdω−Hgω,with(7)Hgω=∑i=0M−12hicosωi+hMcosωM,(8)N=n+1,(9)M=N−12.Combined with (A.14), the unit impulse response of the designed filtering satisfies(10)hi=hN−i−1,i=0,1,…,n.When electromagnetic micromirror operates in slow-scan axis control with equal-ripple low-pass filtering, the constrained inequality about passband cut-off frequencyfp and stopband cut-off frequency fs should be satisfied:(11)fp<fs<fac1.From (4) and (11), it becomes(12)fp<fs<650Hz.Urey [9] reported that the slow-scan axis with the signal of the frequency 60Hz and the quick-scan axis with the signal of the frequency 20kHz, respectively, could support SVGA (800 × 600 pixels) for display. In this paper,(13)60Hz<650Hz.So, sine signal with a frequency of60Hz is used to drive the slow-scan axis and noise with resonant frequencies in ten resonant modes should be filtered by equal-ripple low-pass filter. We set fp=60Hz, fs=61Hz. From (7), (10), and (A.15), we get the following:(14)Eω=WωHdω−Hgω=WωHdω−∑i=012hicosωi−h2cos2ω.Combined with Chebyshev Best Consistent Approximation (CBCA) [29, 30], Eω on equal-ripple low-pass filtering in domain F=0,π could be represented as(15)minmaxω∈FEω.In the domainF, coefficients are chosen to make the maximum of Eω minimum.Forward coefficientheq of equal-ripple low-pass filtering on slow-scan axis control of electromagnetic micromirror, which is computed at Filter Design and Analysis (FDA) tool in Matlab (R2012a), is as follows:(16)heqT=1.13×10−16.65×10−46.64×10−46.65×10−41.13×10−1.This is the first filtering.


### 3.2. Zero-Phase Filtering

For slow-scan axis of the electromagnetic micromirror, in order to preserve the features of feedback signal avoiding a phase distortion, the second filtering, zero-phase filtering, is developed, which would help preserve features appearing at the exact location to avoid the phase distortion [31]. Two traditional basic algorithms can express zero-phase filtering. One is FRR (Forward filter, Reverse filter, Reverse output), and the other is RRF (Reverse filter, Reverse filter, Forward output) [32]. Their principles are similar and the principle of FRR is given here:(17)y1i=yfi∗hi,i=0,1,…,n,(18)y2i=y1N−1−i,i=0,1,…,n,(19)y3i=y2i∗hi,i=0,1,…,n,(20)ysi=y3N−1−i,i=0,1,…,n.In (17)–(20), yfi is the signal sequence from first filtering, ∗ is the convolution algorithm, yi1i,i1=1,2,3 are the output sequences after every transformation, hi is the digital impulse response sequence, and ysi is the final output sequence after the FRR. In the frequency domain, (21)–(24) are deduced:(21)Y1ejω=YfejωHejω,(22)Y2ejω=e−jωN−1Y1e−jω,(23)Y3ejω=Y2ejωHejω,(24)Ysejω=e−jωN−1Y3e−jω.Referencing (B.1) and (B.2),(25)Ysejω=YfejωHejω2.In (25), there is zero phase between Ysejω and Yfejω. This is the second filtering.


### 3.3. Experiments on Filters

Experiments and validations completed on platform (see Section4) on these filters are shown in Figures 4 and 5.Figure 4
Validation of filters in aspect of signal.Figure 5
Validation of filters in aspect of spectrogram.Figure4 shows the slow-scan axis response from collecting circuit after equal-ripple low-pass filtering filtered by zero-phase filtering under external disturbance, especially under electromagnetic interference similar to cell phone calls. Equal-ripple low-pass filtering effectively eliminates the vibration caused by environmental noise with the frequency close to the resonant frequency 650Hz in Mode 1 (see the third rows in Figure 4). After the equal-ripple low-pass filtering, only the signal with the frequency of 60Hz is left, which is the same frequency of the input sine signal (see the first three rows in Figure 5), but this signal appears hairy (see the third row in Figure 4). Zero-phase filtering decreases this hairy appearance, especially in peaks and troughs (see the fourth row in Figure 4). Meanwhile, it preserves the frequency (see the fourth row in Figure 5). The forward coefficient hze of zero-phase filtering is hzeT=0.40.40.40.40.4.In a word, the first filtering, equal-ripple low-pass filtering, succeeds in filtering the signal with resonating frequency for slow-scan axis control of electromagnetic micromirror effectively, and the second filtering, zero-phase filtering, succeeds in preserving the features of feedback signal after equal-ripple low-pass filtering at the original location where they occurred as soon as possible.


## 3.1. Equal-Ripple Low-Pass Filtering

Difference equation of Finite Impulse Response (FIR) filtering was described by Charles and Shenoi [27, 28]:(5)yi=∑i=0nhixn−i,i=0,1,…,n,where yi is the output, xn−i are the inputs, hi are the coefficients, and n is the filtering order.For slow-scan axis control in electromagnetic micromirror system, noise with frequencies close to resonant frequencies in ten modes can result in unexpected vibrations of the electromagnetic micromirror. Equal-ripple low-pass filtering, which has the advantage of controlling the boundary frequency strictly in the aiming frequency compared with other filters [29], is developed here.yf is the output of this equal-ripple low-pass filtering.Hdω is defined as the ideal magnitude of equal-ripple low-pass filtering, Hgω is the actual magnitude, N is the length of filtering, which is usually chosen as an odd number, and n is the order of filtering; the weighted error function Wω is constant. Then, the error Eω between Hdω and Hgω can be expressed as follows:(6)Eω=WωHdω−Hgω,with(7)Hgω=∑i=0M−12hicosωi+hMcosωM,(8)N=n+1,(9)M=N−12.Combined with (A.14), the unit impulse response of the designed filtering satisfies(10)hi=hN−i−1,i=0,1,…,n.When electromagnetic micromirror operates in slow-scan axis control with equal-ripple low-pass filtering, the constrained inequality about passband cut-off frequencyfp and stopband cut-off frequency fs should be satisfied:(11)fp<fs<fac1.From (4) and (11), it becomes(12)fp<fs<650Hz.Urey [9] reported that the slow-scan axis with the signal of the frequency 60Hz and the quick-scan axis with the signal of the frequency 20kHz, respectively, could support SVGA (800 × 600 pixels) for display. In this paper,(13)60Hz<650Hz.So, sine signal with a frequency of60Hz is used to drive the slow-scan axis and noise with resonant frequencies in ten resonant modes should be filtered by equal-ripple low-pass filter. We set fp=60Hz, fs=61Hz. From (7), (10), and (A.15), we get the following:(14)Eω=WωHdω−Hgω=WωHdω−∑i=012hicosωi−h2cos2ω.Combined with Chebyshev Best Consistent Approximation (CBCA) [29, 30], Eω on equal-ripple low-pass filtering in domain F=0,π could be represented as(15)minmaxω∈FEω.In the domainF, coefficients are chosen to make the maximum of Eω minimum.Forward coefficientheq of equal-ripple low-pass filtering on slow-scan axis control of electromagnetic micromirror, which is computed at Filter Design and Analysis (FDA) tool in Matlab (R2012a), is as follows:(16)heqT=1.13×10−16.65×10−46.64×10−46.65×10−41.13×10−1.This is the first filtering.


## 3.2. Zero-Phase Filtering

For slow-scan axis of the electromagnetic micromirror, in order to preserve the features of feedback signal avoiding a phase distortion, the second filtering, zero-phase filtering, is developed, which would help preserve features appearing at the exact location to avoid the phase distortion [31]. Two traditional basic algorithms can express zero-phase filtering. One is FRR (Forward filter, Reverse filter, Reverse output), and the other is RRF (Reverse filter, Reverse filter, Forward output) [32]. Their principles are similar and the principle of FRR is given here:(17)y1i=yfi∗hi,i=0,1,…,n,(18)y2i=y1N−1−i,i=0,1,…,n,(19)y3i=y2i∗hi,i=0,1,…,n,(20)ysi=y3N−1−i,i=0,1,…,n.In (17)–(20), yfi is the signal sequence from first filtering, ∗ is the convolution algorithm, yi1i,i1=1,2,3 are the output sequences after every transformation, hi is the digital impulse response sequence, and ysi is the final output sequence after the FRR. In the frequency domain, (21)–(24) are deduced:(21)Y1ejω=YfejωHejω,(22)Y2ejω=e−jωN−1Y1e−jω,(23)Y3ejω=Y2ejωHejω,(24)Ysejω=e−jωN−1Y3e−jω.Referencing (B.1) and (B.2),(25)Ysejω=YfejωHejω2.In (25), there is zero phase between Ysejω and Yfejω. This is the second filtering.


## 3.3. Experiments on Filters

Experiments and validations completed on platform (see Section4) on these filters are shown in Figures 4 and 5.Figure 4
Validation of filters in aspect of signal.Figure 5
Validation of filters in aspect of spectrogram.Figure4 shows the slow-scan axis response from collecting circuit after equal-ripple low-pass filtering filtered by zero-phase filtering under external disturbance, especially under electromagnetic interference similar to cell phone calls. Equal-ripple low-pass filtering effectively eliminates the vibration caused by environmental noise with the frequency close to the resonant frequency 650Hz in Mode 1 (see the third rows in Figure 4). After the equal-ripple low-pass filtering, only the signal with the frequency of 60Hz is left, which is the same frequency of the input sine signal (see the first three rows in Figure 5), but this signal appears hairy (see the third row in Figure 4). Zero-phase filtering decreases this hairy appearance, especially in peaks and troughs (see the fourth row in Figure 4). Meanwhile, it preserves the frequency (see the fourth row in Figure 5). The forward coefficient hze of zero-phase filtering is hzeT=0.40.40.40.40.4.In a word, the first filtering, equal-ripple low-pass filtering, succeeds in filtering the signal with resonating frequency for slow-scan axis control of electromagnetic micromirror effectively, and the second filtering, zero-phase filtering, succeeds in preserving the features of feedback signal after equal-ripple low-pass filtering at the original location where they occurred as soon as possible.


## 4. Piecewise PID Control and Experiments

Feedback control of MEMS is complicated due to the limited availability of sensor and disturbing characteristics on MEMS device. In this paper, FTPS is used to acquire the output signal to indirectly detect the angle of slow-scan axis [1], and the resolution of FTPS on the end of slow axis in VRS control is 1V@3deg. Slow axis operates in open loop control with input sine signal voltage of a frequency 60Hz and an amplitude 2V, respectively. The maximum absolute angle error 0.67∘ appears in sines peaks and troughs (shown in Figure 6). Discrete PID control with filters design is firstly considered to be utilized for angle error reduction in slow-scan axis control. Its algorithm is presented by(26)uk=Kpek+Ki∑j2=0kej2+Kdek−ek−1,with an integral gain Ki:(27)Ki=KpTTI,and a derivative gain Kd:(28)Kd=KpTDT,where T is a sampling time, ek is a current error at k point, ek−1 is a previous error at k−1 point, Kp is a proportional gain, and e0 is set as 0.Figure 6
Slow-scan axis control of electromagnetic micromirror in open-loop control.Figure7 shows its control results. Indirectly measured real angle from feedback voltage can track the desired angle to some extent (in the 2nd and 3rd row of Figure 7). The maximum absolute angle error in sines peaks and troughs is reduced, compared with that in open loop control (Figure 6). But this maximum absolute angle error is still big. Here, Kp=3.685, Ki=0.002144, and Kd=35.175, respectively.Figure 7
Slow-scan axis control of electromagnetic micromirror in discrete PID control.Due to the accumulation in an integral partKi∑j2=0kej2, the angle error may be bigger for slow-scan axis control of electromagnetic micromirror in discrete PID control. Therefore, a discrete piecewise PID control with filters design is developed and its algorithm can be denoted as follows:(29)uk=Kpek+βKi∑j2=0kej2+Kdek−ek−1,with a switching coefficient β:(30)β=1,ek≤e0,0,ek>e0.Figure8 shows its structure diagram in slow-scan axis control. When ek>e0, discrete Proportional Derivative (PD) control is adopted to avoid the accumulation of integral part to reduce the angle error; when ek≤e0, discrete PID control is used to achieve the desired angle of slow-scan axis for the electromagnetic micromirror. Figure 9 shows that its control advantage is that the maximum absolute error is limited in 0.30∘.Figure 8
Structure diagram in slow-scan axis control.Figure 9
Slow-scan axis control of electromagnetic micromirror in discrete piecewise PID control.Notably,β=1ek≤e00ek>e0, e0=0.009, and sampling frequency FS in experiments is 5kHz. Screens were captured, when the slow-scan axis was in open control, in discrete PID control, and in discrete piecewise PID control, respectively. The ringing effect occurred in open control seen as in Figure 10(a). A tiny fraction of this ringing effect was removed in PID control as shown in Figure 10(b). Substantial amount of ringing effect was removed in discrete piecewise PID control (Figure 10(c)).Figure 10
Screens captured in open control, in discrete PID control, and in discrete piecewise PID control. (a) Screen captured in open control; (b) screen captured in discrete PID control; (c) screen captured in discrete piecewise PID control.
(a)(b)(c)Figure11 shows the raster scanning realized by the single red laser beam projected perpendicularly to this electromagnetic micromirror of two dimensions and reflected to the screen. In Figure 11, the slow axis operates in piecewise PID control; meanwhile, the quick axis is harmonically driven by sine signal with the resonant frequency 22.113kHz and the amplitude 1.8V supplied by the function waveform generator (Agilent 33250A). This raster scanning pixel resolution is SVGA (800 × 600 pixels).Figure 11
Raster scanning realized by the electromagnetic micromirror of two dimensions: 1. function waveform generator (Agilent 33250A); 2. electromagnetic micromirror with FTPS, driving circuits, and data acquisition circuits; 3. red laser; 4. screen; 5. dSPACE and PC; 6.Vin1; 7. Vout1; 8. power.


## 5. Conclusions

In this paper, motions of electromagnetic micromirror succeed in being analyzed through FEM. Mode 1 is emphasized to be eliminated to improve the stability in the slow-scan axis control of raster scanning. The discrete piecewise PID control with secondary filters put forward here was efficiently verified by experiment to remove the ringing effect caused by the instability of slow-scan control.



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*Source: 4612376-2020-09-10.xml*