Disturbance Observer Design with a Bode’s Ideal Filter for Sigma-Delta Modulators

Chi Xu, Yu Jin, Hang Liuand Duli Yu

(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)

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: DOB; ΣΔ modulator; BICO filter; noise-shaping; fractional-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 Simplified architecture of a DOB

(1)

According to Eq. (1), the equivalent disturbancedis 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:

2) The mathematic modelGp(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 andGn(s) can be used to solve the abovementioned problems. Fig.2 depicts the modified structure of the DOB.

Fig.2 Modified structure of the DOB

(2)

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):

(3)

whereω0denotes cut-off frequency,ηis the order of the BICO filter, andη∈(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 theβ(s) gain is -40(1-η) dB/dec. The Bode plot ofβ(s) (withω0=10 andη=0.5) is shown in 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 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) ann-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 A simplified architecture of the ΣΔ modulator

In Fig.5,

W(s)=K0M(s)H(s)

(4)

whereH(s) is the continuous time response of loop filterH(z). The transfer function of the mechanical sensor elementM(s) is expressed by Eq. (5):

(5)

wherem,b, andkare the proof mass, damping coefficient, and spring constant, respectively.bandkare 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 Principle of the proposed ΣΔ modulator

In Fig.6,W-1(s) is the inverse model of theW(s),β(s) is the BICO filter, and e-s·Tdis 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 filterH(z). In this paper, the BICO-DOB is used to eliminate the effect of the parameters’ derivation and the system noise

dtotal(dtotal=dBrown+dElectrical)

4 Simulation Results

The simulated model of the proposed ΣΔ modulator is shown in Fig.7.

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.

In this paper, we selected a 4th-order filter as the digital loop filterH(z), which can be written as

(6)

Fig.7 Simulated model of the proposed ΣΔ modulator

Proofmassm(mg)Dampingcoefficientb(Ns/m)Springconstantk(N/m)AFEgaink0(V/m)Comparatorgaink1Feedbackgaink2(m/s2)Externaldisturbancedtotal(ng/Hz)202.4×10-31004.55×107149.0534

The parameters of the digital loop filterH(z) have been adjusted empirically in our previous experimental work. Thus, we mainly focused on the validity of applying BICO-DOB inΣΔmodulator systems in this paper, and the design method of parameters of digital loop filterH(z) is not specifically described here. The time delay of theW(s) was about 0.1 ms, so theTdin Fig.7 was set as 0.1 ms. The cut-off frequencyω0was set as 1 000 rad/s, and without loss of generality, the gainKof BICO filterβ(s) was set as 1.

Therefore, the relative orderηofβ(s) was the only knob to tune. In order to rapidly obtain and optimize the relative orderη, PSO algorithm[17]was applied. In this paper, the SNR was set as the objective of theΣΔmodulator system, which can be calculated based on the power spectral density of the output bitstream produced by the proposedΣΔmodulator.

In the numerical experiment, the proposedΣΔmodulator with BICO-DOB achieved the highestSNR=148.409 dB of SNR by yielding the optimal value forη=-0.25. Therefore, here we tookη=-0.25 in Eq. (3):

(7)

It is worth noting that in the simulation the fractional order differentiators0.5of the system was approximated with Oustaloup approximation[18]in the frequency range [10-4,104] rad/s andN=5.

Takingη=-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 theΣΔmodulator without the BICO scheme. Compared with theΣΔmodulator without the BICO scheme, the proposedΣΔmodulator with the BICO-DOB achieved higher SNR and lower noise floor.

Fig.8 Noise floor of the proposed ΣΔ modulator

As discussed above, the damping coefficientband spring constantkare variables all the time. Therefore, we chose differentbandkvalues to verify the robustness of the proposed method. To be specific, we took spring constant asks1=60, 120, and 240 N/m. Fig.9 shows the PSD plot of the ΣΔ modulator with different spring constantkvalues.

Fig.9 Noise floor of the ΣΔ modulator with different spring constant k values

Similarly, we chose different damping coefficientbvalues to verify the robustness of the proposedΣΔmodulator. To be specific, we set thebvalues as 1.8×10-3Ns/m, 3.6×10-3Ns/m, and 5.4×10-3Ns/m. Fig.10 shows the PSD plot of theΣΔmodulator with differentbvalues.

Fig.10 PSD of the ΔΣ modulator with different b values

Fig.11 and Fig.12 demonstrate the weak robustness of theΣΔmodulator without the BICO-DOB scheme, which proves that the proposed method possesses better robustness than the pure 6th-orderΣΔmodulator does.

Fig.11 Noise floor of the 6th-order ΣΔ modulator with different k values

Fig.12 Noise floor of the 6th-order ΣΔ modulator with different b values

It can be seen from Figs.9-12 that the proposedΣΔmodulator 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 proposedΣΔmodulator is stronger than the pure 6th-orderΣΔmodulator.

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.