The Estimation and Compensation of the Loop-Parameter-Drifting in the Digital Close-Loop Quartz Flexible Accelerometers

There have been increasing demands for the long-term performance retention of the Quartz Flexible Accelerometers (QFA), extensively employed in the Inertial Navigation System (INS). Considering the convenience of conducting the parameter identification and control algorithm, the Digital Close-Loop QFA (Digital-QFA) is established to improve the long-term performance. Based on the model of the entire close-loop system, the interfering factors relating to time are analyzed in detail, both in the forward channel and the feedback channel. Two schedules are proposed to estimate the gain variation, utilizing the modulating signal and the feedback signal separately. According to the reference gain settings in ideal conditions, the forward and the feedback gain drifting of the system are calibrated online through the adjustment of the digital controller and the reference voltage of the DA-chip, separately. The simulation experiments show that about 90.26% of the gain-drifting could be estimated and about 87.56% of the output error of the accelerate could be restrained. Consequently, the proposed schedules could promote the long-term stability of the Digital-QFA significantly.


I. INTRODUCTION
The Accelerometer is the primary component of Inertial Navigation System (INS), and it could provide the non-gravitational acceleration of the carrier relating to inertial space [1]. There are series accelerometers with different assembly processes, including Electrostatic accelerometer, Floated pendulous accelerometer, Quartz Flexible Accelerometer (QFA), MEMS Accelerometer, etc. They do have different accuracy indexes and application fields, and the typical characteristics of these accelerometers are stated in TABLE 1 [2]. The core problems of the accelerometer contain the improvement of the performance of its static accuracy (Resolution, Bias and Scale factor, etc.), dynamic characteristics (Range, Bandwidth, Vibration Rectification Error, etc.), etc. And there did pay much attention to the improvement of the accuracy of measurement. Different signal detection structures and temperature The associate editor coordinating the review of this manuscript and approving it for publication was Fabio Massaro. compensation were executed to improve the accuracy and achieve better temperature performance [3]- [5]. The detection error of the accelerometers resulting from vibrations or other dynamic environment was analyzed in detail. Simulation optimization approaches and several process constraints were proposed to improve the dynamic performance [6].
For the Navigation-level applications of accelerometers, the QFA is the most appropriate one considering the cost, size and weight, etc. With the extensive application in the Chariot, Warship, Aircraft, the QFA, satisfying the Navigation-level applications in INS, does start its high-speed development and the key problems tackling.
The Analog Close-Loop QFA (Analog-QFA) is extensively applied in the inertial field, and QA-series accelerometers from Honeywell are the typical high-performance products [7], [8]. The application of Analog-QFA does have its limitations attributing to the additional analog-to-digital conversion module when integrated in INS, which would increase the uncertainty of measurement. Benefiting from the development of digital chips and digital MEMS accelerometers [9]- [12], there developed Digital Close-Loop QFA (Digital-QFA) scheme as certain direction of exploration to improve its dynamic performance, and the Digital-QFA could also provide new ways to measure the inner parameters of the whole-loop.
There were series researches about the model of Analog-QFA to improve its performance and scale production capacity [13]- [15]. The Digital-QFA does have the same inner structure: Differential Capacitance, Electromagnetic Torque Feedback, etc. Whereas the difference is the signal detection and processing schedule. Applying embedded chip, there would be multiple detection methods of differential capacitance; for instance, Switched Capacitor Detection Method, Modulation/ Demodulation scheme [16]- [19], etc. And the Modulation/ Demodulation scheme based on AC-bridge is the most promising approach for micro capacitance measurement, which is extensively employed in Accelerometers, Capacitance tomography system [20]- [22], etc. Additionally, modern control theory could be employed in the digital system to improve its dynamic performance [23]. And the multiple control algorithms would provide new parameter identification methods to establish more accurate models [24], [25].
The long-term stability is also the core index of the accelerometers, which could be included in the static accuracy, and it has significant impact on the performance maintenance in storage period of most Weapons. The long-term stability of both Analog-QFA and Digital-QFA is still the weak indicator in INS. The drifting of the Bias and the Scale-factor of the accelerometers would reach 100 ug and 100ppm in one year [7], and the degradation of the performance should attribute to the loop parameters of the QFA be varying over time. The variation of loop parameters, as amplifier characteristics, resistors, structural changes, would bring about the gain of the close-loop be drifting. And it could lead to the deterioration of the controller and the drifting of the scale factor. The estimation and compensation of the loop-gain variation become significant works to improve the long-term performance of the accelerometers. There are several approaches to estimate the gain variation in Digital Close-loop systems [23], [26], [27], which is the significant advantages of Digital-QFA over Analog-QFA.
For the application of the accelerometer itself, the parameters (Bias, Scale factor, cross coupling factors, etc.) could be calibrated through several different known rotations [28], [29]. For the currently application of the accelerometers in the INS, the parameters should be calibrated through the High-Precision-Turntable. With the condition of long-time on service, the entire system must be disassembled from the carriers (weapons, vehicles, aircrafts, etc.) to re-calibrate the critical parameters precisely at fixed intervals, when there is not any external reference information [30], [31]. And it should be great consumption of manpower and material resources. For the application of accelerometer, the self-detection or self-estimation approaches without the turntable at sensors-level are lack of research. These new methods would improve the efficiency of calibration and provide new ways to judge the loop-parameter drifting. The estimation and compensation schedule in this paper is motivated by these factors, and it might be a new avenue to support the sensor-level-self-detection in the INS application. The main contributions of this paper are summarized as follows: 1) More perfect mathematical model is established for the Digital-QFA and most of the influence factors varying over time are analyzed detailly. About 90.92% of the feedback-gain drifting is estimated and compensated effectively. 4) The forward-gain drifting is compensated in the PID controller, and the feedback-gain drifting is compensated through adjusting the reference voltage of the DA chip in the feedback channel. The drifting of the parameters of the close-loop system would bring about two principal problems: the stability problem of the control system, and the output-drifting of the sensor.

A. STABILITY PROBLEM OF THE CONTROL SYSTEM
The optimization of the control algorism would aim to improve the anti-interference performance, or the stability of the system. The parameters of structure, circuits, and VOLUME 8, 2020 the materials would vary when the surrounding environment changing. The perturbation range of these parameters should be analyzed first, and the System Sensitivity Function would be the design philosophy of the control system. The Anti-jamming ability, Gain margin, and the Phase margin would be the judgement-criterion of the effectiveness of the algorism.

B. OUTPUT-DRIFTING OF THE SENSOR
Core issues of this paper. The output-drifting of the sensor, more emphasizing on static characteristic preservation, would be the core point and self-detection or self-calibration is the key innovation.
The rest of this paper is organized as follows. In Chapter II, the entire model of the Digital-QFA would be represented. The mechanism of the entire loop-parameters changing would be analyzed, and the final presentation of the gain variation is descripted in detail. The estimation and compensation of the forward gain and the feedback gain variation are represented in Chapter III. The experiment verification and discussion are stated in Chapter IV. Finally, the summarize of this paper and the possible future researches are concluded in Chapter V.

II. THE LOOP-MODEL AND THE PARAMETER-VARIATION OF DIGITAL CLOSE-LOOP QFA
The Digital-QFA is composed of sense organ, signal detection and feedback circuit, electromagnetic torque. The sense organ is a flexible pendulum, shown in Fig. 1, which deviates away from the equilibrium position when sensing the external acceleration.  The model of the QFA had been researched in many articles [14], [32], and the loop model of the Digital Close-Loop QFA in this paper is shown in Fig. 2. The entire loop is divided into three sections: pendulosity (k 0 = mL); forward channel; feedback channel.

A. PENDULOSITY
The pendulosity is composed of the mass and the arm. On account of the sealing characteristics of the sense organ of the accelerometer, the mass of the pendulum could be assumed to maintain constant. The arm is affected by the linear expansion coefficient of the quartz material. Considering the smaller linear expansion coefficient and the mutual compensation of the accelerate torque and the feedback electromagnetism torque, it is rational to assume the pendulosity maintains constant.

B. FORWARD CHANNEL
The forward transfer function, G(s) · k 1 k CV k AD · k demo D(z), is composed of signal detecting and digital controller. Considering the demodulation module and the controller module are constant in the embedded chip, there would analysis the interfering factor of G(s) · k 1 k CV k AD .
. The function of the deflection over time is as follows.
When t → ∞, it can be simplified as: Therefore, the output deflection is only related to bending stiffness (D) at steady-state, when the input moment maintains constant. Considering that the bending stiffness is mainly determined by the elastic modulus of materials, and the elastic modulus of quartz would change little over time. And the little variation of elastic modulus can be further reduced through improvement of material and stressless processing technology. Consequently, it has little effect on the gain variation of the forward channel.
2) THE SECOND MODULE: [14], [32] This module is the function of arm, unilateral capacitance and plate spacing. It is mostly impacted by the coefficient of linear expansion of the quartz and it does change over time with great possibility. There assumes that the actual expression of the module ask 1 = k 1 · (1 + δk 1 ).

3) THE THIRD MODULE: k CV
Depending on the high precision reference capacitance, there employed AC-Bridge to measure C. The scheme block diagram is exhibited in Fig. 3. Suppose the amplification of the instrumentation amplifier is k amp , and the modulation signal is V IN , then it can be concluded: The reference capacitances are selected to compose AC bridge, and they must meet the conditions of considering the parasitic capacitance, input capacitance of the instrument amplifier, etc. When the accelerometer is in steady-state, there can be conducted C 1 = C 2 = C 0 . Then there can be obtained: The 2 N DA1 −1 · k tz is the amplitude of the modulation signal. Where N FPGA is the digital value produced by FPGA and it is constant over time; V DA1_ref is the reference voltage of the modulation DA-chip, and it would vary over time asV DA1_ref = V DA1_ref · (1 + δV DA1_ref ); k tz is the magnification of the follow-up amplifier, and it is the function of resistance, characteristic of amplifier, etc. Therefore, the actual magnification can be calculated aŝ Similarly, k amp is the magnification of amplifier and it could be computed ask amp = k amp · (1 + δk amp ).
In summary, the gain of CV converter is as follows.

4) THE FOURTH MODULE: k AD
Considering the interference factors of AD, the actual gain of the module is stated in equation (7). Where N AD is the digits of AD;V AD_ref = V AD_ref · (1 + δV AD_ref ) is the reference voltage of AD, and it is also changing over time.
In summary, the influence factors of the gain of the forward channel conclude the reference capacitance, operational amplifier, resistance, AD, DA, etc. The overall gain of the forward channel would be diverse when the close-loop system be started at different condition, and it would affect the performance of the control system. There can be conducted that the gain variation is approximately presented as linear relation, as stated in equation (8).

C. FEEDBACK CHANNEL
The gain of feedback channel does have direct correlation with the output scale factor of the control system. There contains three important modules in the feedback channel of the accelerometer system: DA; VI converter; Magnetic.

1) THE FIRST MODULE: k DA
Similar to the analysis in the forward channel, the gain of the DA module can be computed as equation (9).

2) THE SECOND MODULE: k VI
The VI converter is composed of amplifier, resistance, etc. The theoretically expression of the gain of VI converter is stated in equation (10), and the most significant interfering factor is the drifting of sampling resistor.
where V in is the voltage from DA; R is the sampling resistor. The actual gain of the VI converter isk VI = k VI · (1 + δk VI ).
3) THE THIRD MODULE: k t = BL coil · L In the expression, L coil is the length of coil winding; L is arm of ampere force. Both L coil and L are the function of the VOLUME 8, 2020 coefficient of linear expansion, and the actual expression of is stated in equation (11).
B is the magnetic induction intensity of permanent magnet. In the quartz flexible accelerometer, only the magnetic vector parallel to coil plane could conclude effective electromagnetic force. Whereas the minor structural variation would change the direction of magnetic vector, resulting in the change of effective magnetic induction intensity. In addition, demagnetization, magnetic flux leakage could also affect the value of magnetic induction intensity. The effective value of B can be simplified as equation (12), considering all the influencing factors.
Consequently, the gain of feedback could be computed as equation (13).

A. THE ESTIMATION AND COMPENSATION OF THE FORWARD CHANNEL
As stated in chapter II, the gain variation is approximately linear changing. Rewrite the gain of forward channel: It can be conducted that k(D(z)) andk 1 ·k CV ·k AD do have identical effect on the overall gain. The variation of k pre , if estimated, could be compensated in k(D(z)), which could be adjusted in FPGA. Then the performance of the close-loop controller would maintain the same with the initial condition.
The complete expression of the forward gain is stated in equation (15). Andk pre represents the relation between moment and the signal detection, as N out =k pre · M in .
where k CV −T is the gain of CV converter excluding N FPGA . If the input moment, M in , maintains constant, there can be conducted: N out ∝ N FPGA . When the forward gain k pre varies over time, the relation between N out and N FPGA does have the identical variation. Therefore, the estimation of the scale factor drifting between N out and N FPGA is the forward gain drifting, simultaneously. When the quartz flexible accelerometer is in its open-loop state, the moving plate or the pendulum would be absorbed on the fixed plate, or the yoke iron. And the differential capacitance would be 0pF and C min , which maintains constant, shown in Fig. 4. The input moment maintains constant at this condition, and the output N out is varied with N FPGA . The circuit structure of the AC bridge is different from the close-loop state shown in Fig. 2. It is necessary to discuss the equivalence of the gain variation of AC bridge. The output of the instrument amplifier shown in Fig. 4. can be computed as equation (16). It indicates that there is just difference of the absolute value of gain, while the affection of the variation of C ref 1 , C ref 2 is identical with the normal AC bridge, relating to equation (5).
Consequently, the estimation of the gain variation between N out and N FPGA could represent the variation of the forward gain in the open-loop state. In order to ensure the accuracy of measurement and inhibit the effect of random errors, there employs the least square method to estimate the gain variation, through changing the value of N FPGA . The block diagram of the estimation schedule is shown in Fig. 5. The specific implementation steps are as follows. 1) In the initial status, the relation between N out and N FPGA is stated as follows: N (i) out = (N FPGA + N (i)) · k(D(z)) · k pre−eq · M in (17) where N (i) = {0, N 0 , 2 · N 0 , . . . n · N 0 }; k pre−eq = 1 D · k demo · k 1 · k CV −T · k AD , and it do have 5682 VOLUME 8, 2020 different absolute value but the same variation with the forward-gain, recorded as eq-forward-gain. (18) According to Least Square Criterion (19), shown at the bottom of this page. 2) In actual status, the relation between N out and N FPGA is stated in equation (20). (21) The gain variation (1 + δk pre ) could be conducted, relating to Least Square Criterion and the reference gain estimated in step 1).
3) The compensation of the gain variation can be conducted in the controller D(z), as shown in equation (22).

B. THE ESTIMATION AND COMPENSATION OF THE FEEDBACK CHANNEL
When the Digital-QFA is working in open-loop state, the output would be proportional to the excitation at the feedbackchannel, shown in Fig. 6. and equation (23), where a feedback would balance the input acceleration. The entire loop-gain, k pre · k fb , could be estimated through a in2 and N out . Actually, when the Digital-QFA is at its open-loop state, it is unstable, and the electrostatic adsorption force would have great impact on the pendulum. Therefore, the testing repeatability could not achieve the precision for gain compensating.  Through aging screening and temperature cyclic test, the variation of magnetic induction and the length of coil over time could be restrained at 5 × 10 −6 per year, which means the transfer module k t could be constant compared with the rest sections. The constant current source could achieve high precision, about 5 × 10 −6 per year, and it is identical with the acceleration. Therefore, another acceleration reference, constant current source, could be introduced to the Digital-QFA, shown in Fig. 7. The analog switch M1 is controlled by FPGA, and it could be closed when conducting the estimation schedule, and it would be opened when in working-state, which do not impact the original close-loop working.
Therefore, the output of the entire close-loop system is stated in equation (24) and (25) separately when the switch on or off.
The loop-gain of the system would be much larger than 1, as k t · k VI k DA · k pre 1. Then there can be concluded: With the exact value of I ref , the variation of could be estimated through N out−on and N out−off . And the practical operation steps are stated as follows: 1) In initial status, the relation between N out and I ref : 2) In actual status, the relation between N out and I ref is stated in equation (28).
The gain variation (1 + δk fb ) would be computed, relating to the reference gain k VI k DA in initial status, and the estimation error could be restrained in 10 × 10 −6 .
VOLUME 8, 2020 3) Considering (1 + δk fb ) could only be compensated in the feedback channel, the compensation would be completed at the DA segment. The expression of the output of DA is stated in equation (29). Where N PI is the result of the digital controller; N DA2 is the digits of DA; V DA2_ref is the reference voltage of DA.
Actual, the reference voltage V DA2_ref is provided through another DA, recorded as DA0. Therefore, the expression of V DA2_ref , the output of DA0, is stated in equation (30). The variation of K DA0 is also embodied in (1 + δk fb ). (30) where N VREF is the digits transferring to DA0 from FPGA and it could be adjusted online. Consequently, the feedback gain variation could be compensated with N VREF = N VREF · 1 (1+δk fb ) ≈ N VREF · (1 − δk fb ), controlled through FPGA. And the gain of feedback channel would be in consistent condition with the initial status.
In summary, the overall loop gain variation over time is compensated rationally through the separation of the gain of the forward and the feedback channel. The control structure diagram with compensation procession is shown in Fig. 8.

A. THE PROCEDURE OF EXPERIMENT
The compensation schedule is aiming at the gain variation over time. In order to simplify the test process, the loop-gain would be varied through changing the resistance value of the forward and feedback channel. And the absolute variation of the gain would be predictable and the estimation results could be judged precisely.
The estimation and compensation procedure are integrated in the accelerometer system, and the block diagram is presented in Fig. 9.
The testing procedure are conducted as follows: 1) At initial state, estimate the reference of the Eq-forward-gain and the entire-loop gain, recorded as k pre−eq0 and k VI 0 k DA0 . 2) Fix the Digital-QFA in the stabilized platform, and conduct static testing. The average output of the accelerometer is recorded as N out (0). 3) Maintaining the install condition of accelerometer, the variation of the loop-gain is fixed to (−200ppm, +200ppm), separately for the forward and feedback channel. Conduct static testing without gain compensation. The average output of the accelerometer is recorded as N out (1). 4) Similar to step 3), conduct static testing after the compensation of loop gain. The average output of the accelerometer is recorded as N out (1). And the new Eq-forward-gain, the gain of VI module and the DA module are recorded as k pre−eq1 and k VI 1 k DA1 . 5) Execute comparative analysis with N out (0), N out (1) and N out (1).

B. THE RESULT OF ESTIMATION TEST
The estimation test is conducted as stated in section A. And the fundamental static performance indicators of Digital-QFA are shown in TABLE 2. The structure and the circuit of the Digital-QFA tested are shown in Fig. 10.

1) THE ESTIMATION OF THE EQ-FORWARD-GAIN VARIATION
The estimation of the forward-gain is conducted with 6-times. The results are shown in TABLE 3, where k pre-eq0 could be the benchmark value of the forward-gain, and k pre-eq1  is the testing result of the real forward-gain when varying it by (-200ppm). Therefore, the relative variation between ( k pre-eq1 −k pre-eq0 k pre-eq0 · 10 6 )(ppm) and (−200ppm) could be the judgement of the effectiveness of the estimation procedure. As shown in TABLE 2, the estimated residuals of the procedure are about 9.74%, which means that about 90.26% of the error could be estimated and compensated. One of the original data is shown in Fig. 11. , where (a) Blue-line: the Eq-forward-gain at initial status, or the reference forward-gain; (b) Red-line: the Eq-forward-gain when varying the forward parameter by (−200ppm).

2) THE ESTIMATION OF THE FEEDBACK-GAIN VARIATION
The estimation of the feedback-gain is also conducted with 6-times. The results are shown in TABLE 4, where k VI 0 k DA0 could be the benchmark value of the feedback-gain, and k VI 1 k DA1 is the testing result of the real feedback-gain when varying it by (+200ppm). Therefore, the relative variation between ( k VI 1 k DA1 −k VI 0 k DA0 k VI 0 k DA0 · 10 6 )(ppm) and (+200ppm) could be the judgement of the effectiveness of the estimation procedure. As shown in TABLE 3, the estimated residuals of the procedure are about 9.08%, which means that about 90.92% of the error could be estimated and compensated. One of the original data is shown in Fig. 12., where (a) Blue-line: k VI 0 k DA0 at initial status, or the reference gain; (b) Red-line: k VI 1 k DA1 when varying the feedback parameter by (+200ppm). There are additional fluctuating errors when the constant current source be introduced to the system, and it does have strong correlation with the temperature. Also, the magnetic induction would be impacted by the temperature. These factors would degrade the estimation results, and enough testing time and stable environment temperature are necessary for high-precision estimation.

3) THE OUTPUT OF THE DIGITAL-QFA
One of the results of N out (0), N out (1) and N out (1) are integrated in Fig. 13 : • The average of N out (0) could be the accurate value of the accelerate; • The average of N out (1) could be the output of the accelerometer without compensation; VOLUME 8, 2020 • The average of N out (1) could be the output of the accelerometer with compensation. Without the gain compensation, there is difference 193ug between N out (0) and N out (1), approximately. Through the online compensation of the loop gain, the difference between N out (0) and N out (1) is declined to 24ug, close to the static precision of the testing Digital-QFA. The estimation effect could be computed as equation (31).
The residual error of the estimation procedure is about 12.44%, which means that about 87.56% of the output error is restrained. The residual error would be the inaccurate measurement introduced by the magnetic induction, the length of the coil, etc. These factors were neglected rationally in the estimation model and the testing error is restrained in 10ppm.
There must be noticed that the entire testing procedure is conducted in laboratory and the fluctuation of room temperature is within 3 • C. The large temperature fluctuation must be taken into consideration when applied in practical engineering, which could be calibrated in laboratory. Additionally, the estimation schedules are verified through changing the electrical parameters, and there are other elements which would vary over time. There should be more verification tests to check the effectiveness of the method, which would be the succeeding activities in our researches.

V. CONCLUSION
The entire model and the confounding factors in the Digital-QFA system over time are analyzed detailly. Most of the influence factors would bring about the variation of the loop gain, and ultimately, they would embody as the drifting of the output accelerate. Considering the gain of forward and feedback channel must be compensated separately, we propose two estimation schedules to distinguish the forward and the feedback gain variation, utilizing the modulating and the feedback signal. Through the simulation experiment, the estimation and compensation schedules are verified to be effective. And the testing results illustrate that about 87.56% of the variation of the output accelerate could be compensated. The long-term stability of the Digital-QFA would be promoted significantly.
Nevertheless, there still exist neglecting items in the compensation model, which lead to the residual of the compensation. And the long-term incentive mode is not identical with the temperature changing. To overcome this, we plan to establish more accurate models and pay more attention to the equivalence of the compensation mode and working mode. New detection circuits with equivalent topologies should also be included to the research content. Furthermore, long-term demonstration test is in progress to optimize the schedule.