Author NameAffiliation
Chi XuCollege of Information Science and Technology, Beijing University of Chemical Technology, Beijing 100029, China
Advanced Innovation Center for Soft Matter, Beijing University of Chemical Technology, Beijing 100029, China
Yu JinCollege of Information Science and Technology, Beijing University of Chemical Technology, Beijing 100029, China
Hang LiuCollege of Information Science and Technology, Beijing University of Chemical Technology, Beijing 100029, China
Duli YuAdvanced Innovation Center for Soft Matter, Beijing University of Chemical Technology, Beijing 100029, China
Abstract:
It is usually difficult to design a high performance Sigma-Delta (ΣΔ) modulator due to system noises. In this paper, a disturbance observer (DOB) is utilized to estimate the system noises and eliminate their effects on ΣΔ modulators. The applied DOB is introduced with a Bode's ideal cut-off (BICO) filter used for the Q-filter. The proposed DOB with the BICO filter used in ΣΔ modulators can achieve better noise-shaping ability, resulting from the less phase loss of the BICO filter. Finally, the simulation results show that the proposed BICO filter scheme is a useful additional tool for improving the performance of ΣΔ modulators.
Key words:DOBΣΔ modulatorBICO filternoise-shapingfractional-order system
DOI：10.11916/j.issn.1005-9113.18066
CLC NUMBER:TP212,TN432
Fund:
Chi Xu, Yu Jin, Hang Liu, Duli Yu. Disturbance Observer Design with a Bode's Ideal Filter for Sigma-Delta Modulators[J]. Journal of Harbin Institute of Technology (New Series), 2020, 27(1): 61-66. DOI: 10.11916/j.issn.1005-9113.18066
Fund Sponsored by the Top Scientific and Technological Innovation Team from Beijing University of Chemical Technology (Grant No. BUCTYLKJCX06) Corresponding author Yu Jin, E-mail: jiny@mail.buct.edu.cn Article history Received: 2018-06-16
ContentsAbstractFull textFigures/TablesPDF
Disturbance Observer Design with a Bode's Ideal Filter for Sigma-Delta Modulators
Chi Xu1,2, Yu Jin1
, Hang Liu1, Duli Yu2 1. College of Information Science and Technology, Beijing University of Chemical Technology, Beijing 100029, China;
2. Advanced Innovation Center for Soft Matter, Beijing University of Chemical Technology, Beijing 100029, China
Received: 2018-06-16
Sponsored by the Top Scientific and Technological Innovation Team from Beijing University of Chemical Technology (Grant No. BUCTYLKJCX06)
Corresponding author: Yu Jin, E-mail: jiny@mail.buct.edu.cn.
Abstract: It is usually difficult to design a high performance Sigma-Delta (ΣΔ) modulator due to system noises. In this paper, a disturbance observer (DOB) is utilized to estimate the system noises and eliminate their effects on ΣΔ modulators. The applied DOB is introduced with a Bode's ideal cut-off (BICO) filter used for the Q-filter. The proposed DOB with the BICO filter used in ΣΔ modulators can achieve better noise-shaping ability, resulting from the less phase loss of the BICO filter. Finally, the simulation results show that the proposed BICO filter scheme is a useful additional tool for improving the performance of ΣΔ modulators.
Keywords: DOBΣΔ modulatorBICO filternoise-shapingfractional-order system
1 Introduction Quantization is a fundamental process in analog to digital converters (ADCs). Emerging wireless standards require the ADCs to possess wide bandwidth, high dynamic range, high signal to noise rate (SNR), simple structure, and low power consumption[1-2]. Sigma-Delta (ΣΔ) ADCs are promising for achieving these features because of their simplified oversampling and noise-shaping features. ΣΔ modulator is the most important part in ΣΔ ADC systems so that high performance ΣΔ modulators are much desired to be researched.
At present, the industrial demand for ΣΔ modulators have been quickly increased since it was successfully used in Micro-Electromechanical System (MEMS) accelerometers[3-6]. Closed-loop ΣΔ modulators enable MEMS accelerometers to effectively suppress the quantization noise caused by ADCs or quantizers[7-10]. Based on these advantages, many researchers have been striving to enhance dynamic range, linearity, and bandwidth of ΣΔ modulators. So far, in order to improve both stability and noise-shaping ability, ΣΔ modulators are always designed by single loop and high-order architectures. Therefore, many latterly reported interfaces for ΣΔ modulators possess the features of single loop, high-order, and single-bit force feedback[11-13]. However, such architecture has to confront with loop instability. Therefore, it is necessary to compromise many aspects in circuit implementation so as to design a high performance ΣΔ modulator system with ideal robust stability. The stability of ΣΔ modulators will be weakened by unavoidable noises including Brownian noise, electrical noise, and quantization noise.In this paper, we applied a novel disturbance observer (DOB) with a Bode's ideal cut-off (BICO) filter to eliminate the system noises for ΣΔ modulators[14]. Specifically, a BICO filter was employed in the DOB to provide disturbance rejection performance for compensating the system noises and modeling mismatches.
In this paper, the proposed BICO with DOB was designed to make up mismatches of system parameters and external disturbances for ΣΔ modulator systems. It is normally impossible to obtain an accurate mathematic model for a ΣΔ modulator because of the Brownian noise, electrical noise, and quantization noise or deviations in the manufacturing process. However, the proposed BICO-DOB applied a fractional-order filter, which can provide extra freedom to improve performance by weakening the effect of system noises[15].
A BICO-DOB for the ΣΔ modulator was used in this paper. The rest of the paper is structured as follows: Section 2 demonstrates the concept of BICO-DOB and Section 3 presents the proposed BICO-DOB for the ΣΔ modulator. Then simulation results from the proposed high-order ΣΔ modulator with BICO is given in Section 4 with the conclusions drawn in Section 5.
2 Disturbance Observer The simplified architecture of a DOB is depicted in Fig. 1.
Fig.1
Fig.1 Simplified architecture of a DOB In Fig. 1, Gp(s) denotes the plant model, d and
$\tilde{d}_{f}=(u+d) \cdot G_{p}(s) \cdot G_{p}^{-1}(s)-u=d$ (1)
According to Eq. (1), the equivalent disturbance d is well observed by the DOB scheme shown in Fig. 1. DOB can also achieve independent tuning between the disturbance elimination and input command under the condition of external disturbances and modelling deviations. However, the realization of a DOB suffers three problems in a certain system[16] as follows:
1) Under normal conditions, the numerator's order of Gp(s) is less than or equal to the denominator's order so that Gp－1(s) cannot be implemented in practical systems;
2) The mathematic model Gp(s) is inaccurate due to the parameters' variation caused by external or internal disturbances;
3) The control quality of a practical system would be deteriorated by the unavoidable noises.
Fortunately, a Q-filter and Gn(s) can be used to solve the abovementioned problems. Fig. 2 depicts the modified structure of the DOB.
Fig.2
Fig.2 Modified structure of the DOB In Fig. 2, Q(s) is used to realize the Gp－1(s)·Q(s) and the Q-filter can be expressed as
$Q(s)=\frac{1}{(\tau s+1)^{n}}$ (2)
where τ is a time constant, n is the order, which is selected to make Gp－1(s)·Q(s) realizable. In other words, n is selected to make the numerator's order of Gp－1(s)·Q(s) less than or equal to denominator's order. Details about the BICO-DOB are discussed below.
In Ref. [12], the concept of the BICO filter was introduced by Bode, which provides an ideal trade-off between the system phase margin and the cut-off frequency. The BICO filter is expressed by Eq. (3):
$\beta(s)=\frac{K}{\left(\sqrt{1+\left(s / \omega_{0}\right)^{2}}+\left(s / \omega_{0}\right)\right)^{2(1-\eta)}}$ (3)
where ω0 denotes cut-off frequency, η is the order of the BICO filter, and η∈(0, 1) is a fractional order. BICO-DOB is provided with more flexibility than the traditional DOB for disturbance elimination. According to Eq. (3), the slope of the β(s) gain is －40(1－η) dB/dec. The Bode plot of β(s) (with ω0=10 and η=0.5) is shown in Fig. 3.
Fig.3
Fig.3 Bode plot of the BICO filter In this paper, the main topic is to validate the effect of using BICO-DOB scheme on ΣΔ modulators. As is well known, robustness is a crucial aspect to be considered when designing a ΣΔ modulator. As shown in Fig. 3, we can easily find that the BICO filter possesses insensitivity in pass-band frequency range and a sharp cut-off. Also it can be found that the phase of the BICO filter is relatively insensitive to uncertainties in stop-band frequency range because the phase property is almost constant. Therefore, we investigated the validity of applying BICO-DOB motivated by finding an effective way to design high performance and strong robust ΔΣ modulator systems.
3 High-Order ΣΔ Modulator with BICO-DOB The basic structure of the ΣΔ modulator is depicted in Fig. 4.
Fig.4
Fig.4 Basic structure of the ΣΔ modulator As shown in Fig. 4, the high-order ΣΔ modulator consists of six basic functional blocks: a) a mechanical sensor element; b) an AFE block; c) an n-bit ADC; d) a digital loop filter; e) a 1-bit quantizer; and f) a 1-bit DAC. ΣΔ modulator is a mixed nonlinear feedback system made up of components in both mechanical domain and electrical domain.
Fig. 5 shows a simplified architecture of the ΣΔ modulator system, where
Fig.5
Fig.5 A simplified architecture of the ΣΔ modulator In Fig. 5,
$W{\rm{ (}}s{\rm{) = }}{K_{\rm{0}}}M{\rm{ (}}s{\rm{) }}H{\rm{ (}}s{\rm{)}}$ (4)
where H(s) is the continuous time response of loop filter H(z). The transfer function of the mechanical sensor element M (s) is expressed by Eq. (5):
$M(s)=\frac{1}{s^{2}+\frac{b}{m} s+\frac{k}{m}}$ (5)
where m, b, and k are the proof mass, damping coefficient, and spring constant, respectively. b and k are not always the certain parameters, which is caused by the manufacturing process.
The proposed structure of the ΣΔ modulator with a BICO-DOB is shown in Fig. 6.
Fig.6
Fig.6 Principle of the proposed ΣΔ modulator In Fig. 6, W－1(s) is the inverse model of the W(s), β(s) is the BICO filter, and e－s·Td is the time delay. As mentioned above, Brownian noise, electrical noise, and quantization noise are the main noise sources in ΣΔ modulator systems. Brownian noise is caused by thermal motion of the air molecules surrounding the sensor element, and electrical noise is caused by amplifier circuit (AFE), whereas quantization noise comes from the 1-bit quantizer, which can be effectively eliminated by the digital loop filter H(z). In this paper, the BICO-DOB is used to eliminate the effect of the parameters' derivation and the system noise
$d_{\text {total }}\left(d_{\text {total }}=d_{\text {Brown }}+d_{\text {Electrical }}\right)$
4 Simulation Results The simulated model of the proposed ΣΔ modulator is shown in Fig. 7.
Fig.7
Fig.7 Simulated model of the proposed ΣΔ modulator In Fig. 7, the sample frequency is chosen as 128 kHz and the oversampling ratio is specified as 64. The system parameters in Fig. 7 are listed in Table 1.
表 1
Table 1 Parameters of the proposed ΣΔ modulator Proof mass m (mg) Damping coefficient b (Ns/m) Spring constant k(N/m) AFE gain k0 (V/m) Comparator gain k1 Feedback gain k2 (m/s2) External disturbance dtotal(ng/20 2.4×10-3 100 4.55×107 1 49.05 34
Table 1 Parameters of the proposed ΣΔ modulator
In this paper, we selected a 4th-order filter as the digital loop filter H(z), which can be written as
$H(z)=\frac{37 z^{4}-137.1 z^{3}+191.3 z^{2}-119 z+27.86}{z^{4}-2.995 z^{3}+2.99 z^{2}-0.995 z}$ (6)
The parameters of the digital loop filter H(z) have been adjusted empirically in our previous experimental work. Thus, we mainly focused on the validity of applying BICO-DOB in ΣΔ modulator systems in this paper, and the design method of parameters of digital loop filter H(z) is not specifically described here. The time delay of the W(s) was about 0.1 ms, so the Td in Fig. 7 was set as 0.1 ms. The cut-off frequency ω0 was set as 1 000 rad/s, and without loss of generality, the gain K of BICO filter β(s) was set as 1.
Therefore, the relative order η of β(s) was the only knob to tune. In order to rapidly obtain and optimize the relative order η, PSO algorithm[17] was applied. In this paper, the SNR was set as the objective of the ΣΔ modulator system, which can be calculated based on the power spectral density of the output bitstream produced by the proposed ΣΔ modulator.
In the numerical experiment, the proposed ΣΔ modulator with BICO-DOB achieved the highest SNR=148.409 dB of SNR by yielding the optimal value for η=－0.25. Therefore, here we took η=－0.25 in Eq. (3):
$\begin{array}{c}\beta(s)=\frac{1}{(\sqrt{1+(s / 1000)^{2}}+(s / 1000))^{2.5}}= \\\frac{1}{(\sqrt{1+(s / 1000)^{2}+(s / 1000)})^{2}} \\\frac{1}{(\sqrt{1+(s / 1000)^{2}+(s / 1000)})^{0.5}}\end{array}$ (7)
It is worth noting that in the simulation the fractional order differentiator s0.5 of the system was approximated with Oustaloup approximation[18] in the frequency range [10－4, 104] rad/s and N=5.
Taking η=－0.25 into Fig. 7 and running the Simulink model (Fig. 7) again, the corresponding SNR was 148.409 dB. Fig. 8 shows the noise floor of the ΣΔ modulator without the BICO scheme. Compared with the ΣΔ modulator without the BICO scheme, the proposed ΣΔ modulator with the BICO-DOB achieved higher SNR and lower noise floor.
Fig.8
Fig.8 Noise floor of the proposed ΣΔ modulator As discussed above, the damping coefficient b and spring constant k are variables all the time. Therefore, we chose different b and k values to verify the robustness of the proposed method. To be specific, we took spring constant as ks1=60, 120, and 240 N/m. Fig. 9 shows the PSD plot of the ΣΔ modulator with different spring constant k values.
Fig.9
Fig.9 Noise floor of the ΣΔ modulator with different spring constant k values Similarly, we chose different damping coefficient b values to verify the robustness of the proposed ΣΔ modulator. To be specific, we set the b values as 1.8×10-3 Ns/m, 3.6×10-3 Ns/m, and 5.4×10-3 Ns/m. Fig. 10 shows the PSD plot of the ΣΔ modulator with different b values.
Fig.10
Fig.10 PSD of the ΔΣ modulator with different b values Fig. 11 and Fig. 12 demonstrate the weak robustness of the ΣΔ modulator without the BICO-DOB scheme, which proves that the proposed method possesses better robustness than the pure 6th-order ΣΔ modulator does.
Fig.11
Fig.11 Noise floor of the 6th-order ΣΔ modulator with different k values Fig.12
Fig.12 Noise floor of the 6th-order ΣΔ modulator with different b values It can be seen from Figs. 9-12 that the proposed ΣΔ modulator only shows slight fluctuation in noise floor and SNR, which indicates that the proposed BICO-DOB scheme enables the ΣΔ modulator to be robust to the variable parameters of the sensor element. We can also find that the robustness of the proposed ΣΔ modulator is stronger than the pure 6th-order ΣΔ modulator.
5 Conclusions A new design of a 6 th-order ΣΔ modulator using the proposed BICO-DOB is presented in this paper. The designed 6 th-order ΣΔ modulator achieved SNR=148.409 dB and noise floor under -190 dB in the simulation study. The results from the simulation by MATLAB platform show that the proposed BICO-DOB scheme for the ΣΔ modulator can achieve better robustness.
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