Abstract
This paper proposes a novel design of a high order closedloop SigmaDelta modulator using a fractionalorder disturbance observer, which can reduce the influence of possible dispersions of the sensing element and offset the effect of the external disturbance. The proof mass can maintain near its position of equilibrium by using the proposed fractionalorder disturbance observer, which ensures a better control of the proof mass displacement, and the small proof mass displacement can enhance the sensor resolution and the overall system performance. Fractionalorder disturbance observer possesses stronger robust stability of interference elimination, because the introduction of fractional calculus makes Qfilter expanding from an integer domain to a realnumber domain. Experimental results from the proposed architecture shows an increase of more than 2X improved in signal quantization noise ratio compared to the pure highorder closedloop SigmaDelta modulator, and also the proposed SigmaDelta modulator with a fractionalorder disturbance observer improves the performance of disturbance attenuation and system robustness.
1. Introduction
The fast development of microprocessing and integrated circuit technology were the main driver of the widely application for the MEMS (MicroElectromechanical System) accelerometer [1, 2]. MEMS accelerometers are commonly used in automotive applications, biomedical domain and navigation system [35] thanks to their small size, low cost, and low power consumption, etc. MEMS accelerometers using a capacitive sensing element incorporated in SigmaDelta ($\mathrm{\Delta}\mathrm{\Sigma}$) modulator closedloop control systems with electrostatic feedback. MEMS accelerometer systems consist of chips developed separately may not work properly together when they were integrated together even each chip might be designed and optimized in its own develop project. MEMS accelerometer requires system level integrated design and development. As we all know that an advanced MEMS accelerometer system contains several features such as (1) feedback system with MEMS sensor in the loop; (2) system with sensing and digitization in one step; (3) system with multiphysics principles; (4) $\mathrm{\Sigma}$$\mathrm{\Delta}$ closedloop system with force rebalance.
Among various techniques for implementing the MEMS accelerometer, $\mathrm{\Delta}\mathrm{\Sigma}$ modulator provides the combined benefits of force feedback and inherent analogtodigital conversion [6]. The sensing element of MEMS accelerometer can be modeled as a massspringdamper system, which provides a 2nd order dynamics. Therefore, a traditional MEMS accelerometer is mainly focused on using the sensing element as a lowpass filter to form a 2nd order $\mathrm{\Delta}\mathrm{\Sigma}$ modulator [7]. At present, many researchers have mainly focused on using the additional electronic integrator as a loop integrator to form the high order closedloop $\mathrm{\Delta}\mathrm{\Sigma}$ modulator to improve its SNR and noise performance [8]. There are a lot of advantages to the high order closedloop $\mathrm{\Delta}\mathrm{\Sigma}$ modulator, nevertheless such this architecture may suffer from the overall system instability. As discussed, a $\mathrm{\Delta}\mathrm{\Sigma}$ modulator, which can both achieve high SNR, outstanding noise performance and also ensure the system stable, is much difficulty to be designed. It is shown that the SigmaDelta modulator system can be studied as a nonlinear dynamical system with feedback control [9]. Representing a modulator as “plant” and “controller” transforms a modulator design problem into a control design problem. Therefore, many researchers introduced several advanced control techniques to SigmaDelta modulator. In ref [10, 11], a highorder SigmaDelta modulator design by slide mode control is proposed, which can precisely predict the modulator performs and reduce the nonlinear effect from the 1bit comparator; In ref [12], a Kalman state estimator is applied to a SigmaDelta microaccelerometer which can enhance the resolution of the SigmaDelta modulator. In this paper, we present an approach that ensures a better control of the proof mass displacement thanks to the proposed fractionalorder disturbance observer, which is able to improve the noise performance and system robust stability.
In practice, a physical motion system cannot be exactly the same as any mathematical model, no matter how the model is obtained [13]. In the system of a MEMS accelerometer, a precise mathematic model of the sensing element which may not be readily available because of the external disturbance (temperature, pressure) or dispersions in the manufacturing process. Thus, the mismatch between the real and ideal parameters can result in relatively large proof mass displacements and as a consequence in a decrease of the overall system performance. The concept of disturbance observer (DOB) was proposed in ref [14], which is able to improve the ability of disturbance rejection. The filter in DOB belongs to the lowpass filter called Qfilter. The socalled Qfilter has three adjustable parameters, including the order, the relative degree and the bandwidth. Therefore, compared to integral action, DOB offers more opportunities to improve the performance of the control system (rapidity, stability, phase shift etc.) through tuning the three parameters of Qfilter. In ref. [15], a fractionalorder disturbance observer (FODOB) based on a fractionalorder Qfilter was proposed and thus extending the integerorder disturbance observer to a new fractionalorder one. The fractionalorder Qfilter is employed to make a tradeoff between the robust stability and the disturbance suppressing force. Therefore, in this paper, FODOB is employed to observe and compensate the disturbances and model mismatch of the sensing element, which is able to make the overall system have better robust stability against variation of mathematic model of the sensing element and then ensure MEMS accelerometer achieve high SNR and satisfactory noise performance.
The remainder part of this paper is organized as follows: in Section 2, a basic architecture of FODOB and parameters design in FODOB are introduced; in Section 3, the proposed design of a high order closedloop $\mathrm{\Delta}\mathrm{\Sigma}$ modulator with a FODOB is presented. Then simulation results of the proposed design architecture are demonstrated and analyzed in Section 4. Finally, conclusions are given in Section 5.
2. Continuoustime fractionalorder disturbance observer
The basic idea of the conventional DOB is to apply a nominal inverse model of the plant to observe the disturbance which caused by external interference and parameters variation, and then an equivalent compensation is generated by DOB. The basic architecture of DOB is depicted in Fig. 1.
In Fig. 1, ${G}_{p}\left(s\right)$ is the transfer function of the plant model, $d$ is the equivalent disturbance, ${\stackrel{~}{d}}_{f}$ is the observing value of $d$, $c$ and $y$ are the input and output of DOB, respectively. By observation from Fig. 1, the observing value ${\stackrel{~}{d}}_{f}$ can be written as:
Unfortunately, the implementation of the conventional DOB encounters three key problems in a practical system [8] as follows:
1) In general cases, the relative order of ${G}_{p}\left(s\right)$ is not equal to zero, therefore the inverse of the plant model cannot be obtained.
2) The transfer function of the plant model cannot describe the practical system precisely, that is to say ${G}_{p}\left(s\right)$ is not totally accurate.
3) The existence of the measurement noise will deteriorate the control performance of control system.
In order to solve the above three problems, a Qfilter and a transfer function ${G}_{n}\left(s\right)$ are introduced, which is shown as in Fig. 2.
Fig. 1Basic architecture of DOB
Fig. 2Block diagram of DOB
In Fig. 2, $Q\left(s\right)$ is a lowpass filter, which is used to restrict the effective bandwidth of DOB. ${\stackrel{~}{d}}_{}$ and ${\stackrel{~}{d}}_{f}$ are the disturbance observe before and after by the Qfilter.
From the conventional DOB shown in Fig. 2, the signal transfer function ${G}_{cy}\left(s\right)$, the disturbance transfer function ${G}_{dy}\left(s\right)$ and the noise transfer function ${G}_{ny}\left(s\right)$ can be written as:
where, we assume that the cutoff frequency of the lowpass Qfilter is ${\omega}_{q}$. When $\omega <{\omega}_{q}$, $Q\left(s\right)\approx \text{1}$, taking $Q\left(s\right)\approx \text{1}$ into Eqs. (2), (3) and (4), the signal transfer function ${G}_{cy}\left(s\right)\approx {G}_{n}\left(s\right)$, the disturbance transfer function ${G}_{dy}\left(s\right)\approx \text{0}$ and the noise transfer function ${G}_{ny}\left(s\right)\approx \text{1}$; and when $\omega >{\omega}_{q}$, $Q\left(s\right)\approx \text{0}$, taking $Q\left(s\right)\approx \text{0}$ into Eqs. (2), (3) and (4), the signal transfer function ${G}_{cy}\left(s\right)\approx {G}_{p}\left(s\right)$, the disturbance transfer function ${G}_{dy}\left(s\right)\approx {G}_{p}\left(s\right)$ and the noise transfer function ${G}_{ny}\left(s\right)\approx \text{0}$. By observation from ${G}_{cy}\left(s\right)$, ${G}_{dy}\left(s\right)$ and ${G}_{ny}\left(s\right)$, we can find that the external disturbance can be suppressed by designing the lowpass Qfilter. In other words, the design of Qfilter is a key part in DOB design. In particular, the relative order of the lowpass Qfilter should be more than that of ${G}_{n}\left(s\right)$ to maintain $Q\left(s\right){G}_{n}^{1}\left(s\right)$ being a regular rational transfer function, and the bandwidth design of Qfilter should make a tradeoff between robust stability and disturbance elimination force of the overall system.
The relation between the real model ${G}_{p}\left(s\right)$ and ideal model ${G}_{n}\left(s\right)$ can be expressed by:
where $\mathrm{\Delta}\left(s\right)$ is a uncertain transfer function, which contains variable parameters of the plant model. The robust stability criterion of the uncertain system is discussed in [16], the sufficient criterion of robust stability for Qfilter is expressed as follows:
In order to simplify design process of the lowpass Qfilter, an equivalent block diagram of DOB is shown in Fig. 3.
Fig. 3Equivalent block diagram of DOB
It can be seen from the block $1/\left(1Q\left(s\right)\right)$ shown in Fig. 3, the disturbance observer is a high gain technique when $Q\left(s\right)\to \text{1}$. To simplify the analysis of uncertain transfer function $\mathrm{\Delta}\left(s\right)$, here we assume ${e}^{s{T}_{d}}$ is the only source of uncertain dynamics, so ${G}_{p}\left(s\right)$ can be rewritten as:
By observation from Eqs. (5) and (7), the uncertain transfer function $\mathrm{\Delta}\left(s\right)$ is written as:
In this paper, the lowpass filter $Q\left(s\right)$ can be expressed as:
where cutoff frequency ${\omega}_{q}=1/\tau $, here we set ${T}_{d}=$ 0.2 ms, $n=1\text{,}$${\omega}_{q1}=$ 50 rad/sec, ${\omega}_{q2}=$ 100 rad/sec, ${\omega}_{q3}=$ 500 rad/sec, so the uncertain transfer function $\mathrm{\Delta}\left(s\right)$ and three $Q\left(s\right)$ are rewritten as follows:
The robust stability of DOB with respect to the cutoff frequency ${\omega}_{q}$ of $Q\left(s\right)$ is depicted in Fig. 4.
It is obvious from Fig. 4 that the robust stability criterion Eq. (6) is violated when ${\omega}_{q}=$ 500 rad/sec. The magnitude responses of ${Q}_{2}\left(s\right)$ and ${Q}_{3}\left(s\right)$ are below the curve of $1/\mathrm{\Delta}$, so the robust stability criterion can be satisfied when ${\omega}_{q}\le $ 100 rad/sec. Also, here we set ${T}_{d}=$ 0.2 ms, ${\omega}_{q}=$ 100 rad/sec and $n=$1, 2, 3 to inspect the variations of the robust stability of DOB with respect to relative order $n$ of the $Q\left(s\right)$. Fig. 5 shows three curves with different relative order $n=\mathrm{}$1, 2, 3.
Fig. 4The robust stability of DOB with different cutoff frequency
Fig. 5The robust stability of DOB with different orders
By observation from Fig. 5, the higher relative order of $Q\left(s\right)$, the better robust stability of DOB. In ref. [14], the cutoff frequency ${\omega}_{q}$ is always determined according to the disturbance attenuation requirement in advance. Therefore, the relative order of the lowpass $Q\left(s\right)$ is the only knob to design. As discussed, if the relative order can be tuned in the realnumber domain instead of the integer domain, the tuning range will be broader dramatically, while the relative order is selected in realnumber domain, as a consequence, a traditional disturbance observer belongs to a fractionalorder disturbance observer. Therefore, FODOB possesses more flexibility than the conventional DOB for disturbance elimination in that the relative order of the lowpass Qfilter can be tuned in a realnumber domain. In the field of FODOB, the lowpass $Q\left(s\right)$ has become fractionalorder lowpass filter ${Q}_{\alpha}\left(s\right)$ and the fractional order $\alpha $ can be continuously tuned to improve the dynamic performance of FODOB. In this paper, the fractionalorder lowpass filter ${Q}_{\alpha}\left(s\right)$ is expressed as:
where $\alpha \in R$, $\tau =1/{\omega}_{q}$, in this paper, a continuous integerorder filter ${Q}_{I}\left(s\right)$ is used to approximate the fractionalorder filter ${Q}_{\alpha}\left(s\right)$ in a selected frequency bandwidth $[{\omega}_{b},{\omega}_{h}]$. General form of the integerorder filter ${Q}_{I}\left(s\right)$ is given as follows:
The steps of this approximation method are present in [17]. The fractionalorder filter ${Q}_{\alpha}\left(s\right)$ is approximated by a fifth order integerorder filter ${Q}_{I}\left(s\right)$ in bandwidth range [100, 1000], which is expressed as:
Fig. 6 shows the bode curves of the fractionalorder filter ${Q}_{\alpha}\left(s\right)$ with different $\alpha $.
Here we set $\tau =\mathrm{}$0.01 s, $\alpha =\mathrm{}$1.6, so the fractionalorder lowpass filter ${Q}_{\alpha}\left(s\right)$ is written as:
Fig. 6Bode curves of Qα(s) with different α
3. Implementation in the case of a $\mathbf{\Delta}\mathbf{\Sigma}$ modulator
The general system block diagram of $\mathrm{\Delta}\mathrm{\Sigma}$ modulator is shown in Fig. 7.
Fig. 7Block diagram of ΔΣ modulator system
The loop is consisted of an electromechanical sensing element, a charge amplifier, a digital integrator, and a 1bit quantizer. In Fig. 1, ${K}_{0}$ is the gain of AFE, $H\left(\mathrm{z}\right)$ is a digital integrator, ${K}_{1}$ is the quantizer gain, ${Q}_{1}$ is the quantization noise of 1bit quantizer, and ${K}_{2}$ is the equivalent linear model of 1bit DAC feedback. Unfortunately, the relationship of 1bit quantizer input and output is nonlinear, here, a quasilinear model of the 1bit quantizer is presented in Fig. 1, where the quantizer output is equal to the sum of quantization noise ${Q}_{1}$ and quantizer input with a quantization gain ${k}_{1}$.
The sensing element can be modeled as a massspringdamper system, a 2nd order dynamics with transfer function $M\left(s\right)$ that can be expressed as Eq. (18):
where $m$ is the proof mass, $b$ is the damping coefficient and $k$is the spring constant. Due to the parametric yield errors, $b$ and $k$ are not always the actual values caused by manufacturing process. So, FODOB is employed to observe and compensate the model mismatch of the proof mass. The proposed architecture of the closedloop $\mathrm{\Delta}\mathrm{\Sigma}$ modulator with a FODOB is given in Fig. 8.
In Fig. 8, ${M}^{1}\left(s\right)$ is the inverse model of the sensing element $M\left(s\right)$, ${Q}_{\alpha}\left(s\right)$is the fractionalorder Qfilter, ${e}^{s\cdot {T}_{d}}$is the time delay. By observation from Fig. 8, FODOB is employed to observe and compensate the disturbances and model mismatch of the sensing element. The effects of the employed FODOB are shown in the following.
Fig. 8Block diagram of the proposed ΔΣ modulator system
4. Simulation
Firstly, the Simulink model of the proposed $\mathrm{\Delta}\mathrm{\Sigma}$ modulator with a FODOB is developed shown in Fig. 9.
Fig. 9Simulink model of the proposed ΔΣ modulator with a FODOB
In Fig. 9, the oversampling ratio (OSR) needs to be specified in the $\mathrm{\Delta}\mathrm{\Sigma}$ modulator system. Here, we choose sample frequency as 128 kHz, and OSR = 64. The mechanical and system parameters in the proposed architecture of the closedloop $\mathrm{\Delta}\mathrm{\Sigma}$ modulator with a FODOB are listed in Table 1.
Table 1Parameters of the proposed architecture
Parameter  Value 
Proof mass $m$ [mg]  20 
Damping coefficient $b$ [Ns/m]  2.4×10^{3} 
Spring constant $k$ [N/m]  100 
AFE gain ${k}_{0}$ [V/m]  4.55×10^{7} 
Comparator gain ${k}_{1}$  1 
Feedback gain ${k}_{2}$ [m/s^{2}]  49.05 
External disturbance  3.4 ng/√Hz 
Two numerical experiments of the proposed architecture have been performed in the following.
Case one. A 5th order closedloop $\mathrm{\Delta}\mathrm{\Sigma}$ modulator with the FODOB.
In this case, we choose the digital integrator $H\left(z\right)$ for the closedloop $\mathrm{\Delta}\mathrm{\Sigma}$ modulator as follows:
Due to the system of sensing element responds quickly, ${T}_{d}$ in Fig. 8 is set as 0.2 ms, and $\alpha $ has been adjusted empirically, so finally $\alpha $ is selected as 2.4 to inspect the effects of FODOB.
Taking $\alpha =$ 2.4 in Eq. (14):
From Eq. (20), ${Q}_{2.4}\left(s\right)\cdot {M}^{1}\left(s\right)$ can be obtained. Taking $H\left(\mathrm{z}\right)$, the linearized parameters ${k}_{0}$, ${k}_{1}$, ${k}_{2}$ and the major transfer function $M\left(s\right)$ into Fig. 9, The corresponding SNR is 124.454 dB which is 2X improved comparing to 5th order $\mathrm{\Delta}\mathrm{\Sigma}$ modulator only. The PSD plot of the proposed $\mathrm{\Delta}\mathrm{\Sigma}$ modulator is shown in Fig. 10. Comparing to the only 5th order $\mathrm{\Delta}\mathrm{\Sigma}$ modulator, the proposed $\mathrm{\Delta}\mathrm{\Sigma}$ modulator with the FODOB not only performs better SNR, but also lower noise floor.
Fig. 10PSD of the 5th order ΔΣ modulator using FODOB
Case two. A 6th order closedloop $\mathrm{\Delta}\mathrm{\Sigma}$ modulator with FODOB.
In this case, we choose the digital integrator $H\left(z\right)$ for the closedloop $\mathrm{\Delta}\mathrm{\Sigma}$ modulator as follows:
Taking $H\left(\mathrm{z}\right)$, the linearized parameters ${k}_{0}$, ${k}_{1}$, ${k}_{2}$ and the major transfer function $M\left(s\right)$ into Fig. 9, The corresponding SNR is 144.265 dB which is 2X improved comparing to 6th order $\mathrm{\Delta}\mathrm{\Sigma}$ modulator only. The PSD plot of the proposed $\mathrm{\Delta}\mathrm{\Sigma}$ modulator with FODOB is shown in Fig. 10. Comparing to the only 6th order $\mathrm{\Delta}\mathrm{\Sigma}$ modulator, the proposed $\mathrm{\Delta}\mathrm{\Sigma}$ modulator not only performs better SNR, but also lower noise floor.
Fig. 11PSD of the 6th order ΔΣ modulator using FODOB
Further simulation are also observed at system response with different spring constant k and damping coefficient b, For instance, taking different spring constants ${k}_{s1}=$ 60 N/m, ${k}_{s2}=$ 120 N/m, ${k}_{s3}=$ 240 N/m in MEMS transfer function to verify the robustness of the proposed $\mathrm{\Delta}\mathrm{\Sigma}$ modulator with the FODOB, Fig. 12 and Fig. 13 show the PSD plot of the proposed 5th order and 6th order $\mathrm{\Delta}\mathrm{\Sigma}$ modulator with different spring constant ${k}_{s1}$, ${k}_{s2}$, ${k}_{s3}$ using FODOB.
Fig. 12PSD of the 5th order ΔΣ modulator with different spring constant k
Fig. 13PSD of the 6th order ΔΣ modulator with different spring constant k
Also, taking ${b}_{1}=$1.8×10^{3}^{}Ns/m, ${b}_{2}=$ 3.6×10^{3}^{}Ns/m, ${b}_{3}=$ 5.4×10^{3}^{}Ns/m in MEMS transfer function, Fig. 13 and Fig. 14 show the PSD plot of the proposed 5th order and 6th order $\mathrm{\Delta}\mathrm{\Sigma}$ modulator with different damping coefficient $b$.
Simulation results show that SNR and the noise floor of the three sensing elements with different $k$ and three sensing elements with different $b$ only present an acceptable slight fluctuation which indicate that the proposed $\mathrm{\Delta}\mathrm{\Sigma}$ modulator using FODOB has strong robustness against to the sensitivity of MEMS devices.
Fig. 14PSD of the 5th order ΔΣ modulator with different damping coefficient b
Fig. 15PSD of the 6th order ΔΣ modulator with different damping coefficient b
5. Conclusions
A novel design of a high order closedloop SigmaDelta modulator using a fractionalorder disturbance observer is present in this paper. The proposed 5th order SigmaDelta modulator using FODOB achieved SNR = 124.454 dB in simulation and noise floor under –170 dB in frequency of [5150Hz] and the proposed 6th order SigmaDelta modulator using FODOB achieved SNR = 144.712 dB in simulation and noise floor under –190 dB in frequency of [5150 Hz]. The numerical experiments also show the improved robust stability of the proposed SigmaDelta modulator using fractionalorder disturbance observer comparing to the pure 5th order and 6th order SigmaDelta modulator. This study can promote the development of high performance of MEMS accelerometer, also provide a scientific and technical supports for the application of fractionalorder disturbance observer.
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About this article
This work was supported by the research fund to the top scientific and technological innovation team from Beijing University of Chemical Technology (No. buctylkjcx06).