#### Abstract

A digital mode-matching control system based on feedback calibration, where two pilot tones are applied to actuate the sense mode by the robust feedback controller, is presented for a MEMS gyroscope in this paper. A dual-mass decoupled MEMS gyroscope with the integrated electrostatic frequency tuning mechanisms, the quadrature correction electrode, and the feedback electrode is adopted to implement mode-matching control. Compared with the previous mode-matching method of forward excitation calibration, the proposed mode-matching scheme based on feedback calibration has better adaptability to the variation in the frequency of calibration pilot tones and the quality factor of the sense mode. The influences of calibration pilot tone frequency and the amplitude ratio on tuning performance are studied in theory and simulation. The simulation results demonstrate that the tuning error due to the amplitude asymmetry of the sense mode increases with a frequency split between pilot tones and the drive mode and is significantly reduced by the amplitude correction technology of pilot tones. In addition, the influence of key parameters on the stability of the mode-matching system is deduced by using the average analysis method. The MATLAB simulation of the mode-matching control system illustrates that simulation results have a good consistency with theoretical analysis, which verifies the effectiveness of the closed-loop mode-matching control system. The entire mode-matching control system based on a FPGA device is implemented combined with a closed-loop self-excitation drive, closed-loop force feedback control, and quadrature error correction control. Experimental results demonstrate that the mode-matching prototype has a bias instability of 0.63°/h and ARW of 0.0056°/h^{1/2}. Compared with the mode-mismatched MEMS gyroscope, the performances of bias instability and ARW are improved by 3.81 times and 4.20 times, respectively.

#### 1. Introduction

Over recent years, MEMS-based gyroscopes have found widespread application in the automotive and consumer fields, including rollover detection and electronic stability control, gaming consoles, and image stabilization in digital cameras [1–3]. However, MEMS gyroscopes have yet to satisfy the navigation-grade requirements needed for high-precision applications such as an inertial measurement unit for GPS-augmented navigation, unmanned surveillance vehicles, and aircraft and personal positioning systems [4–6]. A variety of means have been proposed to improve the accuracy of MEMS gyroscopes, such as array gyroscopes with multimasses [7], 3-D fabrication techniques with high-aspect ratio and effective volume utilization [8], digital control method based on Δ*Σ* technology [9], and mode-matching technology [1]. Mode-matching technology is widely concerned and becomes one of the most promising approaches to improve the accuracy of MEMS gyroscopes.

Under mode-matched conditions, the sense mode is designed to own the same (or nearly the same) resonant frequency with the drive mode. Due to Q-factor amplification, gyroscopes operated under mode-matched configuration offer higher sensitivity and better resolution [10]. It is hard to completely match the resonant frequencies between two modes through structural design due to the fabrication imperfections. Various approaches have been developed to reduce the frequency mismatch between two resonance modes in the literature. Postfabrication by depositing selective polysilicon [11] or laser trimming [12], localized thermal stress [13], and some other approaches were utilized in the early stage. However, these approaches, which require a manual tuning effort, are not suitable for mass production and temperature variation. Another effective approach based on electrostatic spring softening was adopted to tune the resonance mode frequencies by an adjustable DC potential [14]. Various means have been developed to acquire the appropriate tuning DC voltage. Evolutionary computation [15, 16] and extremum-seeking technique [17–19] are introduced to attain the tuning voltage by seeking the peak of the quadrature signal or sense mode. These approaches are either time-consuming or to be done offline, in which the dynamic requirements are difficult to meet. Another automatic mode-matching method owned high control precision in closed-loop control, which obtained continuous information about the resonance frequency of the sense mode by introducing a square wave dither signal or injecting out-of-band pilot tones [20–22]. The phase domain approach based on matching the phase relationship between two modes by PLL can be employed to fulfill the mode-matching control [23, 24]. However, the phase domain will suffer from phase coupling and uncertain parasitic phase shift stemmed from the readout circuits, which could lead to instability of the control system.

This paper presents a digital mode-matching control loop based on feedback calibration for MEMS gyroscopes. Two pilot tones located on both sides of the drive signal are used to actuate the sense mode and estimate the frequency mismatch by the robust feedback controller, which enhances the adaptability to the variation in the frequency of calibration pilot tones and the quality factor of the sense mode. In Section 2, the structure principle of microgyroscopes is briefly reviewed. Then we describe the implementation of the control system of mode-matching and analyze the control characteristics of the system. The control system simulations are given in Section 3. We present experimental result demonstration in Section 4. Concluding remarks are finally given in the last section.

#### 2. Mode-Matching System Design and Analysis

##### 2.1. MEMS Gyroscope Design

Structural schematic of the dual-mass decoupled MEMS gyroscope is shown in Figure 1 [25]. The MEMS gyroscope structure consists of two identical substructures. Due to the decoupling effect of decoupled beams (including the drive and sense decoupled beams), the drive and sense frames can vibrate along only one direction (-axis or -axis), whereas the proof masses vibrate along two orthogonal directions to achieve Coriolis coupling between the drive and sense modes. The kinematic coupling of two substructures in the drive direction is implemented by drive-coupling suspension beams. Similarly, a complex lever system, which consists of the coupling crossbeams, sense-coupling suspension beams, and lever-supporting beams, is used to achieve the kinematic coupling of two substructures in the sense direction. The kinematic couplings in the drive or sense direction ensure that two substructures vibrate with the same frequencies in the in-phase and antiphase modes. Although the drive-decoupled beam is optimized to suppress the mechanical coupling between the drive and sense modes, fabrication imperfections can lead to an undesired mechanical cross-talk (quadrature error) between two modes. Quadrature correction electrodes, with the help of dedicated closed-loop quadrature cancellation electronics, are designed to cancel the quadrature error by applying the automatically adjusted DC potentials. In order to obtain the maximum mechanical sensitivity, the frequency of the driving mode is basically tuned to be identical with that of the sense mode by finite element simulation. However, unavoidable residual stress, fabrication tolerances, and temperature variation lead to mismatched mode frequencies. Therefore, the frequency tuning mechanisms are designed to automatically tune the sense mode frequency to approximate the drive modal frequency by using the electrostatic spring softening capability of the tune combs.

When the tuning voltage is applied on the frequency tuning electrodes, the subsequent electrostatic negative stiffness is presented as [9–11] where is the number of tuning combs, is permittivity, is comb thickness, is the length, and is the gap of the comb. The ideal mode matching can be achieved by changing the stiffness of the proof mass in the sense mode. When the tuning voltage is exerted on tuning combs, the resonant frequency of the sense mode can be described as where , , and are the original resonant frequency, stiffness, and proof mass of the sense structure, respectively. And . The equation can be simplified further as where is the mechanical parameter and . The structure parameters of the dual-mass decoupled MEMS gyroscope are shown in Table 1. In the dual-mass decoupled MEMS gyroscope with frequency tuning mechanism, a 35 Hz mode mismatch error between the drive mode and sense mode is designed in the structure.

##### 2.2. Proposed Mode-Matching Method

The proposed mode-matching control loop in the paper is shown in Figure 2. refers to the transfer function of the sense mode, represents the feedback gain, refers to the interface gain, represents the demodulation compensation term, LPF (low-pass filter) is used to filter the high-frequency signal after demodulation, and then the filtered signal is fed into a PI (proportional integral) controller to generate DC voltage for frequency tuning. Different from previous frequency calibration methods which inject directly pilot tones into the sense mode in the forward path and check the amplitude difference between the tones [21, 22], two pilot tones () in the proposed calibration means, whose frequencies are located at both sides of the drive mode symmetrically, are introduced after the front-end preamplifier and then exerted on the sense mode through force feedback path.

Disconnecting the feedback control loop of the tuning voltage and neglecting the influence of Coriolis signal, the transfer function of the mode-matching system in the open-loop conditions can be expressed as

Assuming that the open-loop gain is much greater than one, equation (4) can be simplified as where is resonant frequency of sense mode and is the Q-factor of the sense mode.

Figure 3 compares two tuning approaches based on pilot tone excitation. Obviously, there is a significant difference between the two open-loop systems. The open-loop system in the previous calibration method is similar to a bandpass system, while the open-loop system in the proposed calibration method is similar to a band elimination system. Two selected pilot tones ( and ), whose frequencies are referenced to the resonant frequency of the drive mode () and located symmetrically at the both sides of the drive mode, are applied to excite the sense mode. The difference of amplitude-frequency responses between two tones points out the direction of frequency tuning and is used to control the frequency tuning mechanism to implement the mode matching. In the proposed method shown in Figure 3(b), once the resonant frequency of the sense mode () is higher than that of the drive mode (), the tuning control circuit will be stimulated to increase the tuning voltage because the amplitude response of sideband is much greater than that of sideband . On the contrary, when the amplitude response of sideband is much less than that of sideband , which demonstrates that the resonant frequency of the sense mode is already less than that of the driving mode, the tuning control circuit will be promoted to decrease the tuning voltage. Finally, the frequencies of the drive mode and sense mode are matched when the amplitude responses in two sidebands of and are equal. The previous approach shown in Figure 3(a) has a similar control process.

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Compared with the previous tuning approach, the amplitude responses of two sidebands in the proposed approach, especially under high quality factor conditions, can obtain better signal-to-noise ratio due to the band elimination characteristic in the open-loop system. At the same time, the frequency difference between pilot tones and the drive mode can be increased to further improve the signal-to-noise ratio. The above characteristics are conducive to improving the control performance of the system. In addition, the proposed approach enhances the adaptability to the variation in the frequency of calibration pilot tones and quality factor of the sense mode, which will be discussed in detail in the next section.

In summary, compared with the previous open-loop tuning, offline tuning, and closed-loop tuning shown in the Figure 3(a) method, the proposed tuning approach, which is an in-run calibration technique and conducive to more precise real-time tuning control, is expected to implement a better SNR and get better bias instability and has better adaptability to the frequency of calibration pilot tones and quality factor of the sense mode.

##### 2.3. System Characteristics

###### 2.3.1. Frequency Limit of Calibration Pilot Tones

In order to analyze the pilot tone frequency limit of the proposed mode-matching technique based on feedback calibration, the open-loop amplitude phase response of the frequency tuning system is analyzed. Figure 4 illustrates the block diagram of open-loop frequency tuning. The PI controller is removed compared with the closed-loop mode-matching system shown in Figure 2. We focus on the open-loop output of LPF with varying tuning voltages which will be fed into the PI controller to generate the tuning voltage in the closed-loop mode-matching system. The open-loop tuning parameters are shown in Table 2.

Assuming that the open-loop gain is much greater than one, the transfer function of the open-loop tuning system can be expressed as

The responses in two pilot tones can be written as

The LPF module is applied to filter invalid high-frequency signals generated by multiplying and . Then the output of LPF can be derived. where

Figure 5 illustrates the open-loop LPF output with varying tuning voltages in different frequencies of calibration pilot tones according to the simulation parameters in Table 2.

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The resonant frequency of the sense mode is equal to that of the drive mode when the tuning voltage reaches 5.16 V under the given parameters. Five different frequency splits between calibration pilot tones and the drive mode vary from 10 Hz to 200 Hz as shown in Figure 5. The frequencies of five calibration pilot tones are representative and located in different locations, including those closing to the driving mode, those closing to the sense mode, those between the driving mode and the sense mode, or those away from the sense mode. The LPF outputs with different frequency splits shown in Figure 5(a) indicate good consistency during open-loop tuning procedure and are basically the same with the adjustment process in the closed-loop mode-matching control. And the tuning voltages are basically negatively correlated with the outputs of the LPF in different frequency splits. The monotonic characteristic facilitates the realization of closed-loop control, which proves that the proposed mode-matching method has good adaptability to the frequency change of calibration pilots.

However, simulation results shown in Figure 5(b) illustrate that the LPF outputs have a great correlation with the frequency location of the calibration pilot tones in the forward mode-matching method. The LPF outputs do not increase or decrease monotonically with the increase of tuning voltage in certain calibration pilot tones, especially in 10 Hz or 25 Hz frequency split between calibration pilot tones and the drive mode, which means that the ideal mode matching can hardly be achieved by the traditional closed-loop control theory. Simultaneously, the LPF outputs decrease monotonically with the increase of tuning voltage in 100 Hz or 200 Hz frequency split between calibration pilot tones and the drive mode; nevertheless, the output signal of LPF is extremely weak and is likely to be submerged in noise due to the large quality factor of the sense mode, which results in the subsequent signal not being processed. The above analysis shows that the forward mode-matching method is only suitable for low-quality factors.

Simulation results demonstrate that the proposed tuning method has good redundancy for the variation in calibration pilot tone frequencies and the quality factor of the sense mode, which is significantly better than the forward mode-matching method where the frequency of calibration pilot tones must be chosen to be within a certain frequency band [26], that is,

In addition, five zero crossing points shown in Figure 5(a) locate at different tuning voltages, which will result in frequency tuning errors shown in Table 3. The frequency tuning errors are mainly due to the amplitude asymmetry between two calibration pilot tones in the sense mode. This will be highlighted in the next section.

Theoretically, the change in the frequency of the calibration pilot tones does not affect the bias stability or Allan deviation measurement in the proposed mode-matching method. However, the actual circuit implementation may affect the final bias stability or Allan deviation measurement. If the frequencies of the calibration pilot tones are too close to that of the drive mode, in order to avoid the Coriolis signal and the quadrature signal coupling into the tuning loop shown in Figure 2, the cutoff frequency of the LPF in the tuning loop shown in Figure 2 must be much smaller than the frequency difference between the calibration pilot tones and the drive mode. Therefore, too low cutoff frequency of LPF will lead to difficulty in parameter designing of LPF and will also affect the dynamic performance and adjustment speed of the tuning control loop. On the contrary, if the frequencies of the calibration pilot tones are too far from that of the drive mode, in order to maintain a large output in the tuning loop, a larger signal needs to be fed back to the input through and the feedback signal may be much larger than the voltage limit of the power supply, which will lead to difficulties in circuit implementation. If the frequencies of the calibration tones are too close to or away from that of the drive mode, it may degrade the performance of the tuning loop system, eventually leading to the deterioration of the bias stability or Allan deviation measurement. Therefore, the frequency difference between the calibration pilot tones and the drive mode needs to be compromised and compatible with the above two situations.

###### 2.3.2. Amplitude Ratio of Calibration Pilot Tones

Different frequency splits between calibration pilot tones and drive mode result in different zero crossing points, which means how much the tuning voltage can reach in the closed-loop control system.

According to equation (8), the zero crossing points refer to

Resolving equation (11) under the perfect tuning voltage can figure out the amplitude ratio of two pilot tones.

Because is much smaller than , equation (13) can be simplified as where represents the frequency split between calibration pilot tones and the drive mode. Obviously, the amplitude ratio raises with the increase of frequency split.

Figure 6 verifies the function of equation (13). We set the frequency split . Obviously, when two calibration tones adopt the same amplitude, the amplitude difference between two calibration pilot tones in the sense mode results in error of the zero crossing point, which will eventually cause significant frequency tuning errors shown in Figure 7.

However, when the amplitudes of two calibration tones are corrected according to equation (14), a perfect zero crossing point shown in Figure 6 can be implemented. Figure 7 illustrates the change of frequency tuning error with a frequency split between calibration pilot tones and the drive mode. From Figure 7, the frequency split of 60 Hz between calibration pilot tones and the drive mode results in an error of 0.019% or 0.61 Hz frequency offset, while the frequency split of 200 Hz results in an error of 0.19% or 6.17 Hz offset. The frequency tuning errors mainly stem from imperfect symmetry of amplitude responses of calibration pilot tones in the sense mode. Simultaneously, the frequency tuning errors shown in Figure 7 can be significantly reduced by correcting the amplitude of two calibration tones.

###### 2.3.3. The Impact of the Quality Factor in the Sense Mode

Figure 8 shows the open-loop LPF output with different quality factors of the sense mode in a 60 Hz frequency split between calibration pilot tones and the drive mode. Simulation results demonstrate that the open-loop LPF outputs in the proposed method are basically not affected by the quality factor change throughout the open-loop tuning process. However, the traditional forward mode-matching method is obviously affected by the quality factor due to the presence of resonance peaks in the tuning process. When the frequency deviation between calibration pilot tones and the drive mode is small and does not satisfy the formula shown in equation (10), the increase of the quality factor will lead to a sharper resonance peak shown in Figure 5(b), which will potentially cause significant amplitude fluctuations, and even lead to unsuccessful closed-loop control during the frequency tuning. When the frequency deviation between calibration tones and the drive mode is large and satisfies the formula shown in equation (10), the open-loop LPF output signals in the forward mode-matching method are weak under a large quality factor, which results in a poor signal-to-noise ratio. The above analysis indicates that the traditional forward mode-matching method is more suitable for low-quality factors. Compared with the traditional mode-matching method, the proposed mode-matching technique illustrates a good adaptability to the change of quality factor and can significantly improve the signal-to-noise ratio of the mode-matching loop under high quality factors and is especially suitable for large quality factor environments.

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##### 2.4. Stability Analysis

The proposed mode-matching technique based on feedback calibration guarantees good compatibility with the variation in the quality factor and frequency split. However, the feedback calibration with pilot tones introduced in the proposed mode-matching technique easily leads to system instability. A stability analysis of the mode-matching control loop shown in Figure 2 is implemented. To simplify the complexity of the analysis, the calibration signal mixed with two different frequencies tones is replaced by one tone and the reference point of the PI controller is also changed into a specific parameter instead of zero correspondingly.

The mode-matching system can be expressed as where represents the integration gain of the PI controller, represents the output of LPF, represents the cutoff frequency of the low-pass filter. Dynamic equation (15) constitutes the frequency tuning system. To analyze system transient properties, suppose that and , where , , and represent the transient amplitude, transient phase angle, and phase, respectively. Thus can be derived as

Relative to , , and are slowly varying variables, assuming that

Then the second-order derivative of voltage is

System transient characteristics can be derived from simultaneous equations (15)–(18) as

Equation (19) offers a transient relationship among the variables of amplitude , phase , LPF output , and tuning voltage . However, formulas in equation (19) are coupled reciprocally and nonlinear, which indicate the that characteristics of the system are implicit and difficult to analyze. Since phase angle is a quickly varying variable and , , , and are slowly varying variables, the averaging method can be applied. Setting the integral interval as , equation (19) can be rewritten as

The bar above the variables represents the average over one cycle. The average equation (20) is still complex and nonlinear. To further simplify the equation, local linearization can be done around its equilibrium point. Letting the formula on the right of equation (20) be zero, you can get the equilibrium points of each variables. Since four variables (, , , and ) are coupled to each other and the equations are complicated, it is difficult to directly obtain the analytical solutions of the equilibrium points. Therefore, we obtain the equilibrium points of the system by solving the numerical equations in the subsequent analysis. Numerical analysis shows that the equilibrium points are unique since the tuning voltage is always artificially set to positive, which can be further confirmed by subsequent numerical simulation results.

The relevant Jacobian matrix can be derived as where , ,, and are the only set of equilibrium points of each variables.

The characteristic equation of expression (21) is

The coefficients of characteristic variable are where represents the element of the Jacobian matrix and represents the coefficient of the characteristic variable. According to the Routh-Hurwitz stability criterion, the system is stable only when all characteristic roots have negative real components, that is, equation (24) must be satisfied. The impact of the key parameters on system stability, including , , , and , is indeterminate due to the coupled intricacy. By applying the control variable method, the relationships between analytical solutions of critical values , , , and are shown in Figure 9 according to stationary parameters shown in Table 2. Colored blocks in Figure 9 represent the stable area. Obviously, the critical value is in direct proportion to in general, but in inverse proportion to . And the change of the quality factor has no significant effect on the critical value, which confirms that the system has better redundancy for the change of quality factor, which is basically consistent with the previous theoretical analysis. The criterion inequality (24) only represents the stable condition near the equilibrium point. Since equilibrium points are unique, once the system is balanced, it must converge to the only equilibrium state, which will be confirmed in the subsequent system simulation.

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#### 3. Simulation

Figure 10 shows the block diagram of the closed-loop mode-matching system with feedback calibration constructed in Simulink. The model is designed to observe the transient responses of the mode-matching system and verify the analysis of open-loop system characteristics in the abovementioned chapter. The parameters of sense dynamics, pilot tones, and gain terms are kept the same as the open-loop parameters summarized in Table 2, while and of the PI controller are 20 and 10000, respectively, which is set to satisfy the constraints of stability illustrated in Figure 9. The frequency split between the calibration tone and the drive mode is set to 60 Hz. Figure 11 illustrates the transient responses of the mode-matching system.

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As shown in Figure 11(a), the signal of the preamplifier is modulated by two calibration tones and mainly used to extract the excitation amplitude of the two calibration tones in the sense mode for subsequent evaluation of the tuning state. Simultaneously, the output of the LPF shown in Figure 11(b) is essentially zero after 0.5 s approximately. And the output of PI shown in Figure 11(c), that is, the tuning voltage, basically reaches the tuning point, which indicates that the amplitudes modulated by two calibration pilot tones in the sense mode are substantially equal and the mode matching is basically realized. We also alter the frequency split between the calibration pilot tone and the drive mode by adjusting the frequency of pilot tones, shown in Figure 11(d). Apparently, different frequency splits between the calibration tone and the drive mode lead to different frequency tuning errors. And the increase of frequency split enlarges the frequency tuning error, which is consistent with the analysis of equation (14). We compare the tuning effects at different quality factors, shown in Figure 11(e). Obviously, the frequency tuning voltage is basically the same under different quality factors. Finally, we analyze the frequency tuning error, shown in Figure 11(f). The frequency split between the calibration pilot tone and the drive mode is increased to 100 Hz in order to significantly demonstrate the effect of optimization. Simulation results show that a significant frequency tuning error of 3.29 Hz appears when the amplitude ratio of two pilot tones is 1. We use equation (14) to calibrate the amplitude ratio of two pilot tones. Then the frequency tuning error is significantly reduced to 0.14 Hz. Figures 11(d)–11(f) demonstrate that the closed-loop characteristics of the mode-matching system are in consistency with open-loop analysis results as shown in Figures 6–8. System stability comparison in different simulation parameters is shown in Figure 11(g). When simulation parameter , where the closed-loop mode-matching system is in the stable region according to the theoretical curve of Figure 9(a), the tuning voltage quickly reaches the steady state. The stability simulation results show that the whole system is stable and the equilibrium state is unique and the system converges to the only equilibrium stable after stabilization, which verifies that the stability analysis in Section 2.4 is correct. However, when the simulation parameters exceed the stable region shown in Figure 9(a), the tuning voltage reaches saturation and the entire system loses stability control, which confirms that theoretical analysis of stability is basically credible.

#### 4. Experiment

Experiments have been performed to verify the theoretical analysis and evaluate the characteristics of mode-matched MEMS gyroscopes compared to the former mode-mismatched state. Figure 12 shows the picture of the fabricated dual-mass decoupled MEMS gyroscope with a frequency tuning mechanism.

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In order to facilitate the realization of subsequent closed-loop circuits, the open-loop tuning capability of the tuning mechanism is first tested. Figure 13 shows the resonant frequency characteristics of the sense and drive modes. Preliminary open-loop sweep experiments shown in Figures 13(a) and 13(b) indicate that the drive mode has an original resonant frequency of 3057.5 Hz and a quality factor of 15760, while the sense mode has an original resonant frequency of 3083.5 Hz and a quality factor of 12230. As the tuning voltage increases from 0 V to 8 V, the resonant frequency of the sense mode shown in Figure 13(c) decreases from 3083.65 Hz to 3026.15 Hz due to the electrostatic tuning capability, while the resonant frequency of the drive mode remains essentially constant at 3057.51 Hz. The perfect mode matching is implemented approximately with the tuning voltage of 5.41 V for the MEMS gyroscope shown in Figure 1.

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Combined with classic closed-loop self-excitation drive, closed-loop force feedback control, and quadrature error correction control, the entire control system scheme of the mode-matching MEMS gyroscope shown in Figure 14 is constructed according to the control scheme in Figure 10. All control loops are implemented in a FPGA device. The function and implementation of three classic control loops, which are indispensable in the control of the mode-matching MEMS gyroscope, are simplified as they have been expounded in our previous work [25]. The FPGA chip is EP3C55F484I7 from the Altera Corporation and has programmable logic units, RAM bits, 156 multipliers, and 4 PLL. The entire hardware platform of the FPGA is based on fixed-point arithmetic processing with a main clock frequency of 12.29 MHz. A chip of AD7767, which is a high-performance, 24-bit, oversampled SAR analog-to-digital converter (ADC) with a low-temperature drift and large dynamic range and input bandwidth, is utilized to implement analog to digital conversion with the sampling frequency of 48 kHz. A 24-bit Stereo D/A Converter of CS4344 based on a fourth-order multibit delta-sigma modulator with a linear analog low-pass filter is used to implement digital to analog conversion with the same sampling frequency of 48 kHz. The voltage-controlled oscillator (VCO) in the mode-matching controller is controlled by a frequency control word generated from the drive mode controller. The frequencies of two excitation pilot tones are designed to change with the drive mode resonant frequency that does not remain unchanged but vary with the external environment, which could enhance the adaptability of the mode-matching system. Figure 15 shows the prototype of the mode-matching MEMS gyroscope.

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The experiments are implemented to verify the theoretical analysis and simulation results about the characteristics of the mode-matching technique represented in this paper. The influences of the amplitude ratio of two pilot tones and the frequency split between pilot tones and the drive mode are analyzed. The open-loop tuning experiment of the mode-matching system is shown in Figure 16.

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Compared with the simulation results in Figure 5(a), the experiment results shown in Figure 16(a) indicate that the open-loop LPF output curves with different tuning voltages share the same tendency with the simulated result in different frequencies of calibration pilot tones. This illustrates that the frequency variation of calibration pilot tones will not substantially alter the tuning process but results in different tuning errors, which is basically consistent with the simulation results. However, a zero-crossing point of tuning voltage 5.09 V in the calibration pilot tone with a 30 Hz frequency spilt with respect to the drive mode results in 3.01 Hz tuning error, while a zero-crossing point of tuning voltage 4.33 V in the calibration pilot tones with a 40 Hz frequency spilt results in 9.42 Hz tuning error. Compared with the simulation results, the tuning errors in the experimental results are obviously large due to the difference of the quality factor, the asymmetry of the resonance curve, and the reduction of the feedback gain due to the clamping of the power supply voltage. Simultaneously, the amplitude ratio between two calibration pilot tones is corrected as 1.43 to further suppress tuning errors shown in Figure 16(b). Experimental results confirm that the tuning error is reduced from 9.42 Hz to almost zero, which verifies the effectiveness of the method. Nevertheless, the actual correction coefficient of the amplitude ratio has some differences with the theoretical value shown in equation (14), which is also related to the asymmetry of the resonance curve in the sense mode.

In order to verify the effect of mode matching, the signal waveforms of the mode-matching control system are measured as shown in Figure 17. Figure 17(a) shows the feedback signal of calibration pilot tones, which is primarily used to modulate the sense mode to extract amplitudes in the frequencies of calibration pilot tones and provides reference for subsequent mode-matching control. The preamplifier signal waveforms of the sensitive interface shown in Figure 17(b) contain the calibration signal modulated by two pilot tones, quadrature error signal and offset error signal, which shows an atypical modulated wave.

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In order to further analyze the signal components of the feedback calibration signal, the frequency spectrum of the feedback calibration signal is collected as shown in Figure 17(c). A significant amplitude difference in the frequencies of two pilot tones is observed before tuning. The main reason is that two pilot tones are located in an asymmetrical position on the open-loop curve shown in Figure 3(b). Since the left pilot tone is far away from the resonance frequency of the sense mode, the amplitude response in the left pilot tone is larger than that in the right pilot tone. However, the amplitude responses in the frequencies of two pilot tones are basically equivalent after tuning, which is consistent with the basic principle of the proposed mode-matching scheme shown in Figure 3(b). Similarly, Figure 17(d) shows the transient response of preamplifier signal waveforms and tuning voltage. Tuning control results indicate that the tuning voltage and the preamplifier signal are stable after 0.7 s. Since the frequencies of calibration pilot tones are different from those of the quadrature error and the Coriolis signal, the tuning control loop does not interfere with the normal operation of the gyroscope. The transition time in the experiment is slightly longer than that in the simulation. The main reason is that the gain in the tuning loop is appropriately reduced due to the compatibility stability in other control loops and the limitation of the circuit supply voltage. The output of the tuning voltage indicates that the final control system implements a mode matching within 0.29 Hz. The above signal detection fully proves the effectiveness of the closed-loop force feedback control, quadrature error correction control, and mode-matching control.

Finally, the performance experiments of the prototype are implemented. Figure 18 shows the input-output relationship curve. The prototype with mode-matching and mode-mismatching conditions has scale factors of −10.90 mV/°/s and 25.52 mV/°/s, respectively, under an input rate range of ±100^{o}/s. Nonlinear analysis shows that the nonlinearities of the prototype under the condition of mode matching and mode mismatching are 478.73 ppm and 669.81 ppm, respectively. Simultaneously, the drift characteristics of prototype are evaluated under mode-matching and mode-mismatching conditions. Figure 19 compares the Allan variance under the mode-matched and mode-mismatched conditions. Compared with those in the mode-mismatched state, the bias instabilities in the mode-matching state decrease from 2.40°/h to 0.63°/h and ARWs decrease from 0.0235°/h^{1/2} to 0.0056°/h^{1/2}, respectively. Experimental results demonstrate the bias instability and ARW performance are substantially improved by 3.81 times and 4.20 times, respectively, due to the mode-matched control. The Allan variance curve is not an ideal curve, especially when the Allan variance time is less than 10 s. The main reason is that some slowly variational interference signals with low-frequency or tiny periodic signals, which come possibly from the unsatisfactory layout of the chip or interface circuit, fabrication error, and so on, are coupled into the circuit.

Indeed, the decrease of the frequency difference between the drive mode and sense mode by tuning means is beneficial in improving the mechanical sensitivity and bias stability or Allan deviation measurement. However, it will lead to an increase of nonlinearity due to the enhancement in the mechanical motion amplitude of the sense mode, which will decrease the dynamic range, since the closed-loop detection technology used in the system will increase the dynamic range of the system and the entire detection range is not large. Therefore, the tuning technique does not have a significant impact on the measurement range.

#### 5. Conclusions

This paper proposes a digital mode-matching control loop based on feedback calibration for a MEMS gyroscope. The structure of a dual-mass decoupled MEMS gyroscope with integrated electrostatic frequency tuning mechanisms, quadrature correction electrode, and feedback electrode is presented briefly. A digital mode-matching control scheme, where two pilot tones are applied to actuate the sense mode and extract tuning voltage by the robust feedback controller, is designed. Compared with the previous mode-matching method of forward excitation calibration, the proposed mode-matching scheme based on feedback calibration has better adaptability to the variation in the frequency of calibration pilot tones and the quality factor of the sense mode. The influences of the calibration pilot tone frequency and amplitude ratio on tuning performance are studied in theory and simulation. Theoretical analysis and simulation results demonstrate that the tuning error increases with a frequency split between pilot tones and the drive mode, which is mainly due to the amplitude asymmetry of the sense mode. Simultaneously, the tuning error is significantly reduced by the amplitude correction technology of pilot tones. And the proposed mode-matching method has better robustness to changes in the quality factor of the sense mode. The relationship between the stability of a mode-matching control system and key parameters is deduced by using the average analysis method. The MATLAB simulation model of the mode-matching control system is constructed. The simulation results have a good consistency with theoretical analysis, which verifies the effectiveness of the closed-loop mode-matching control system. The entire mode-matching control system combined with the closed-loop self-excitation drive, closed-loop force feedback control, and quadrature error correction control is implemented in a FPGA device. Experimental results demonstrate that the mode-matching prototype has a bias instability of 0.63°/h and ARW of 0.0056°/h^{1/2}. Compared with those of the mode-mismatched MEMS gyroscope, the performance of bias instability and ARW of the mode-matched prototype are improved by 3.81 times and 4.20 times, respectively.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors wish to thank the support from the National Natural Science Foundation of China (Grant nos. 61571126 and 61874025), the Aviation Science Foundation (Grant no. 20150869005), Equipment Pre-research Field Foundation (Grant no. 6140517010316JW06001), the Fundamental Research Funds for the Central Universities (Grant no. 2242018k1G017), and the Eleventh Peak Talents Programme Foundation in the Six New Industry Areas.