Modulation index stabilization technique of integrated optic phase modulator used in resonant integrated optic gyro

Qiwei Wang; Hui Li; Pengjie Wang; Keke Deng; Lishuang Feng

doi:10.1364/OE.27.003460

1. Introduction

The optical gyroscope based on Sagnac effect is an important component of inertial measurement system, which has wide application in inertial navigation field [1,2]. The resonant integrated optic gyro (RIOG) has been especially considered as the promising candidate of micro-sized optical gyro, due to its high integrated level, high theoretical sensitivity and superior reliability [3,4]. However, the RIOG’s complex error sources, particularly temperature resulted, constrain its wide application.

In the past few decades, a lot of work has been devoted to the research on reducing the various error sources to improve the detection accuracy of RIOG. The backreflection’s effect on resonant optic gyro, which can be effectively suppressed by hybrid phase-modulation technology (HPMT) [5] and phase difference traversal (PDT) technology [6,7], is theoretically and experimentally analyzed in [8] for the first time. The carrier-suppression phase modulation (CSPM) technology can reduce the backscattering error and ultra-high carrier suppression is crucial [9–11]. It can be seen from [5–11] that there are optimum modulation indexes for both HPMT, PDT and CSPM technologies to realize optimum efficiency. What’s more, all the schemes given above only lock the laser frequency at one propagation resonant frequency of the resonator, but the angular velocity signal, which is proportional to the difference between clockwise (CW) and counterclockwise (CCW) resonant frequencies caused by the Sagnac effect, is still detected in open-loop. Sanders et al. adopted the demodulated signal of CW to control the cavity length for locking the CW resonant frequency, and firstly exploited the demodulated signal of CCW to produce sawtooth wave on the integrated optic phase modulator (IOPM) with indirect frequency shift for locking the frequency of CCW lightwave and tracking the angular velocity [12]. Based on the theory, the angular velocity tracking loop of RIOG based on hybrid digital phase modulation is proposed to decrease the scale factor nonlinearity and improve the accuracy of RIOG [13–15]. The double closed-loop control schemes containing laser frequency locking loop and angular velocity tracking loop in the RIOG is the mainstream detection scheme for high precision RIOG. However, the modulation index fluctuation of IOPM, as a feedback unit of the angular velocity tracking loop, directly influences the closed-loop performance of RIOG. Thus, the modulation index stability is very important for both noise suppression and angular velocity tracking of the RIOG. Unfortunately, in practical engineering, the modulation index is variation at different temperatures since the half-wave voltage of IOPM changes with temperature [16,17]. Although many modulation schemes such as triangular wave modulation [18], serrodyne wave modulation [19], sinusoidal wave modulation [20], trapezoidal wave modulation [21] and their combinations [5,22–24] for RIOG have been proposed, none of them can acquire the modulation index effectively. The problem can be solved by double closed-loop using digital phase ramp [25] or four states wave modulation [26] in interferometric fiber optic gyroscope (IFOG). However, because of the short transit time and different interference principle of the RIOG, the half-wave voltage is hard to detect by using these methods. Thus, the traditional method cannot improve the full temperature performance of the RIOG, which is a key problem urgently to be solved.

In this paper, a novel modulation index stabilization technique (MIST), which includes modulation index detection technique (MIDT) and modulation index closed-loop control system (MICCS), is first proposed to detect and trace the modulation index of IOPM used in RIOG. First of all, definition of the modulation index is given and its influence mechanism on angular velocity tracking loop is analyzed in detail. Then, a novel structure of RIOG with the Mach-Zehnder Interferometer (MZI) added and a six-state wave modulation method based on this structure, which is MIDT, are proposed to synchronously demodulate the modulation index fluctuation, resonant frequency and angular velocity variation. Based on the MIDT, the mathematical model of the IOPM’s modulation index closed-loop tracking system is established considering the system nonlinearity and disturbance. Finally, the MICCS with H_∞ performance level to accurately track the IOPM’s modulation index is designed. The experimental results show that the MIST can detect and track the IOPM’s modulation index effectively. The IOPM’s modulation index fluctuation range is decreased from 5.3% to less than 0.1‰ and the gyro scale factor stability of RIOG resulted from IOPM’s modulation index stability is decreased to 189.26 ppm within −40°C to + 60°C, which, to the best of our knowledge, is the first time stabilizing the modulation index of IOPM in RIOG at full temperature.

2. Modulation index’s influence on angular velocity tracking loop

A double closed-loop control scheme containing laser frequency locking loop and angular velocity tracking loop in the RIOG is given in [12–15]. In the laser frequency locking loop, CCW lightwave detected by the photodetector is demodulated to lock the center frequency of the narrow linewidth laser. In the angular velocity tracking loop, difference between the resonant frequencies of CW and CCW lightwaves, which is proportion to angular velocity and taken as the final output of the RIOG, is demodulated and compensated by imposing the sawtooth wave on the IOPM. According to the principle of Sagnac effect, the resonant frequency difference can be expressed as [27]:

Δ f_{Ω} = \frac{D}{n_{e} λ} Ω,

where

D

is the diameter of waveguide ring resonator,

n_{e}

is an effective refractive index of the waveguide,

λ

is the central wavelength of light in vacuum, and

Ω

is the rotation angular velocity of gyro.

The resonant frequency difference resulting from the Sagnac effect can be feedback controlled by adjusting the slope coefficient of the sawtooth wave on the IOPM. According to the simplified block diagram of RIOG’s angular velocity tracking loop given in Fig. 1, it can be found that $Δ f = Δ f_{Ω} - f_{S W}$ , where $f_{S W}$ is the equivalent frequency shift of sawtooth wave used for the angular velocity tracing loop. Thus, when the error of angular velocity tracking loop equals to zero ( $Δ f = 0$ ), the gyro’s close-loop output can be expressed as:

G_{o u t} = \frac{2 V_{π} Δ f_{Ω}}{K_{S W} V_{p p}},

where

V_{p p}

is the peak voltage of the sawtooth wave,

V_{π}

is the half-wave voltage of the IOPM,

K_{S W}

is the gain of sawtooth generator. The gyro’s close-loop output changes with the ratio of the half-wave voltage to the peak voltage which is varies at different temperatures. Thus, the modulation index of IOPM is defined as:

Fig. 1 The simplified block diagram of RIOG’s angular velocity tracking loop. The K_FC refers the gain of forward channel which includes the gain of photodetectors and lock-in amplifier. K_SW refers to the gain of sawtooth generator

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ε_{r} = \frac{V_{p p}}{2 V_{π}} .

According to the Eqs. (2) to (3), the scale factor of the RIOG’s close-loop output can be expressed as:

K = \frac{D}{K_{S W} ε_{r} n_{e} λ} .

The expression indicates that the modulation index fluctuation of IOPM has an immediate impact on the scale factor of the RIOG’s close-loop output. It can be seen that the scale factor

K

of RIOG varies when the modulation index

ε_{r}

fluctuates with the temperature. The scale factor of the RIOG varies with the temperature to make the error mechanism of the RIOG system hard to accurately study.

3. Modulation index stabilization technique

The conventional RIOG [23,24] only focuses on the resonant frequency of two lightwaves propagating in the waveguide ring resonator (WRR) and cannot demodulate the modulation index of IOPM synchronously. Therefore, a novel MIST is proposed to demodulate and control the modulation index of IOPM in real time. The schematic diagram of RIOG with MIST is given in Fig. 2.

Fig. 2 The schematic diagram of RIOG with MIST. ISO, isolator; IOPM, integrated optic phase modulator; PD, photodetector; WRR, waveguide ring resonator; DEM, demodulator; C1, 99:1 optical coupler; C2, 99:1 optical coupler; C3, 50:50 optical coupler.

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Compared with the conventional RIOG’s optical structure, three optical couplers, whose split ratios are 99:1, 99:1 and 50:50, respectively, are added for detecting the modulation index of IOPM. The output of narrow linewidth laser is modulated and split equally into CW and CCW lightwaves by IOPM. Modulation signal is the six-state wave generated by the digital modulation generator for detecting the modulation index of IOPM. There are three controllers in the RIOG with MIST, which contains laser frequency locking loop, angular velocity tracking loop and IOPM’s modulation tracking loop. By using the first and second states of the six-state wave, the resonant frequency difference between CW and CCW lights caused by Sagnac effect is measured and compensated by angular tracking controller. In order to guarantee the gyro to work correctly, the frequency of laser needs to be controlled by frequency locking controller which can demodulate and control the difference between central light frequency of the laser and the resonant frequency of the CCW light for suppressing the frequency fluctuation of laser caused by the temperature. 99% of the CCW lightwave which is detected by photodetector PD1 for frequency locking controller to lock the center frequency of the narrow linewidth laser after demodulated by DEM1 is optically coupled into the WRR with the optical coupler C1. And 99% of the CW lightwave is also detected by photodetector PD2 after coupled into the WRR with the optical coupler C2. Its demodulation output is used for angular tracking controller generating sawtooth wave to track the angular velocity after subtracting the demodulation output of CW and CCW lights. Meanwhile, another critical independence loop in the RIOG with MIST is modulation index stabilization loop, composed of MIDT and MICCS, for detecting and tracing the modulation index of IOPM according to the last four states of the six-state wave in real time.

3.1 Modulation index detection technique (MIDT)

The MIDT is used for detect the modulation index of IOPM in real time. A Mach-Zehnder Interferometer (MZI) optical structure, which consists of IOPM, optical coupler C1, C2 and C3, is added into the conventional RIOG in Fig. 2. 1% of the CW and CCW lightwaves which are modulated by six-state wave interfere at the 50:50 optical coupler C3. The modulation index can be acquired by demodulating the interference light detected by photodetector PD3. The optical signal of the MIDT has no relationship with the resonator input signal, so it will not affect the laser frequency locking loop and angular velocity tracking loop. Although additional devices might introduce non-reciprocity and losses of the RIOG, by using couplers with the same splitting ratio (99:1) can reduce these negative influences.

For detect the modulation index of the IOPM, a six-state wave should be imposed on it. The 1st and 2nd states whose period is $τ_{1}$ are used for triangular phase modulation. The next four states, whose ideal phase amplitudes are $0$ , $π / 2$ , $3 π / 2$ and $2 π$ , respectively, are used for detecting the modulation index. Under an ideal condition, the modulation index $ε_{r}$ equals to 1, which means the peak voltage of DAC’s output being the same as twice the half-wave voltage of the IOPM. However, in practical engineering, the half-wave voltage changes because of the temperature variation, which may cause the modulation index $ε_{r}$ is no longer equivalent to 1. Because the half-wave voltage varies with temperature at millisecond level, it can be taken as fixed during one modulation period $τ_{2}$ which is around ten microseconds.

Thus, the phase amplitudes of the four states can be expressed as $0$ , $π ε_{r} / 2$ , $3 π ε_{r} / 2$ and $2 π ε_{r}$ , So the interference intensities of the adjacent four states detected by the photodetector can be separately written as:

{\begin{array}{l} I_{1} = α_{L} k_{C} I_{0} [1 + \cos (Δ φ)] & 0 \leq t < τ_{2} / 4 \\ I_{2} = α_{L} k_{C} I_{0} [1 + \cos (π ε_{r} / 2 + Δ φ)] & τ_{2} / 4 \leq t < τ_{2} / 2 \\ I_{3} = α_{L} k_{C} I_{0} [1 + \cos (3 π ε_{r} / 2 + Δ φ)] & τ_{2} / 2 \leq t < 3 τ_{2} / 4 \\ I_{4} = α_{L} k_{C} I_{0} [1 + \cos (2 π ε_{r} + Δ φ)] & 3 τ_{2} / 4 \leq t < τ_{2} \end{array}

where

I_{0}

is intensity of the CW or CCW lightwaves,

α_{L}

is total loss of the system,

k_{C}

is the split ratio of C1 and C2,

Δ φ

is the phase difference caused by the different transmission delay between the MZI’s two arms, which changes slowly with environment, and

τ_{2}

is the modulation period. By demodulating the interference intensities of the four-state square wave given in Eq. (5), it can be obtained that

\begin{matrix} Δ I_{2 π} = I_{1} - I_{4} \\ = α_{L} k_{C} I_{0} [\cos (Δ φ) - \cos (2 π ε_{r} + Δ φ)] \\ = 2 α_{L} k_{C} I_{0} \sin (π ε_{r}) \sin (π ε_{r} + Δ φ) \end{matrix}

and

\begin{matrix} Δ I_{d} = I_{2} - I_{3} \\ = α_{L} k_{C} I_{0} [\cos (π ε_{r} / 2 + Δ φ) - \cos (3 π ε_{r} / 2 + Δ φ)] \\ = 2 α_{L} k_{C} I_{0} \sin (π ε_{r} / 2) \sin (π ε_{r} + Δ φ) \end{matrix}

The simulation results which are divided by the normalization, $2 α_{L} k_{C} I_{0}$ , are given in Fig. 3 when the modulation indexes are not more than and not less than 1.00. It can be found that both $Δ I_{2 π}$ and $Δ I_{d}$ are not only related to the modulation index $ε_{r}$ but also relevant to the phase difference $Δ φ$ . Thus neither of them can be used as the demodulation result individually to detect the modulation index.

Fig. 3 The simulation results of the demodulation outputs ΔI_2π and ΔI_d at the different phase difference between the MZI’s two arms Δφ when (a) the modulation indexes are not more than 1.00 and (b) the modulation indexes are not less than 1.00.

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In order to combat the effects of phase difference $Δ φ$ , the demodulation result can be calculated by:

I_{D e m} = \frac{Δ I_{2 π}}{Δ I_{d}} = \frac{2 α_{L} k_{C} I_{0} \sin (π ε_{r}) \sin (π ε_{r} + Δ φ)}{2 α_{L} k_{C} I_{0} \sin (π ε_{r} / 2) \sin (π ε_{r} + Δ φ)} = 2 \cos (π ε_{r} / 2)

Equation (8) indicates that the

I_{D e m}

only depends on the modulation index

ε_{r}

which makes it a better option for MIDT. This is also the reason why four states, whose ideal phase amplitudes are

0

,

π / 2

,

3 π / 2

and

2 π

, respectively, are needed to detect the modulation index of IOPM. What’s more, it is worth reminding that if

π ε_{r}

plus

Δ φ

is multiples of

π

, the

I_{D e m}

should be ignored because the denominator equals zero. Hence, the proposed MIDT can realize the modulation and demodulation of the IOPM’s modulation index, which is very important to achieve better full temperature performance of the RIOG.

3.2 Modulation index closed-loop control system (MICCS)

In this section, a robust control algorithm based on the MIDT given above is proposed to establish a high accuracy tracking system for controlling the modulation index. In order to better design the MICCS, Eq. (8) can be rewritten as

I_{D e m} = - 2 \sin (π ε_{e} / 2)

where

ε_{e} = ε_{r} - ε_{0}

is the modulation index error which is the input of close-loop controller, the

ε_{0}

equals to 1. Thus, the dynamic equation of MICCS can be obtained as

x (k + 1) = A x (k) - B k f (K_{c} x (k)) + C w (k)

where

k = n

,

f (K_{c} x (k))

is the one-dimensional nonlinear function. According to the controllability canonical form of automatic control theory,

x (k)

,

x (k + 1)

is the Rⁿ state variable,

w (k)

is the unknown one-dimensional perturbation variable vector which belongs to

l_{2} [0, \infty)

,

A = [\begin{matrix} 1 & 1 & 0 & \dots & 0 \\ 0 & 1 & 1 & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & ⋱ & 0 \\ 0 & \dots & 0 & 1 & 1 \\ 0 & \dots & 0 & 0 & 1 \end{matrix}]

,

B = C = [\begin{matrix} 0 \\ 0 \\ ⋮ \\ 0 \\ 1 \end{matrix}]

,

K_{c} \in R^{1 \times n}

is the feedback gain matrix of controller in the IOPM gain tracking system,

n

is the order of the controller,

f (K_{c} x (k)) = \sin (ε_{e} π / 2)

which are the sinusoidal function of

K_{c} x (k)

, and

ε_{e} π = K_{c} x (k)

based on closed-loop feedback principle.

In order to derive the high-precision tracking method of the IOPM’s gain tracking system, we first introduce the following lemma.

Lemma 1 (Schur Complement) [28]. For a given symmetric matrix $S$ with the form $S = [S_{i j}]$ , $i, j \in 1, 2$ . $S_{11} \in R^{r \times r}$ , $S_{12} \in R^{r \times (n - r)}$ and $S_{22} \in R^{(n - r) \times (n - r)}$ . Then, $S < 0$ if and only if $S_{11} < 0$ , $S_{22} - S_{21} S_{11}^{- 1} S_{12} < 0$ or $S_{22} < 0$ , $S_{11} - S_{12} S_{22}^{- 1} S_{21} < 0$ .

To realize the high-precision tracking of IOPM’s gain, the feedback control matrix $K_{c}$ is designed to guarantee that the gain tracking system of IOPM can achieve the stability with a prescribed $H_{\infty}$ performance level.

Theorem 1. For a given scalar $γ > 0$ , the nonlinear dynamic system (10) is stable with a prescribed $H_{\infty}$ performance level $γ$ , if there exists symmetric positive definite matrix $P \in R^{n \times n}$ , matrix $G \in R^{n \times n}$ , and feedback gain matrix $K_{c} \in R^{1 \times n}$ , such that

[\begin{matrix} - P + I & K_{c}^{T} & 0 & A^{T} G \\ * & - 2 I & 0 & - k B^{T} G \\ * & * & - γ^{2} & C^{T} G \\ * & * & * & P - G - G^{T} \end{matrix}] < 0

Proof: See Appendix A.

Theorem 1 provides the design rule of MICCS considering the nonlinearity and external noise which guarantees the system achieving the optimal detection sensitivity with a prescribed $H_{\infty}$ performance index $γ$ . The principle diagram of closed-loop tracking of IOPM’s modulation index can be obtained as shown in Fig. 4.

Fig. 4 The principle diagram of closed-loop tracking of IOPM’s modulation index.

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The forward channel is composed of interference, photodetector, pre-amplifier and analog-to-digital converter (ADC). In demodulation process, after the sampled and accumulated according to the demodulation wave, the demodulation result, which can be supposed as a sine function of the modulation index error, can be demodulated by the MIDT. Because the demodulation result is calculated by division, the forward channel and demodulation gain have little impact on the demodulation result. The feedback channel is used to change the amplitude of the digital-to-analog converter’s (DAC) output according to the tracking system’s calculation result. The gain of feedback channel can be calculated by

K_{f} = \frac{K_{d} V_{r e f - d a c}}{2 V_{π} 2^{N_{d a c}}} = 2.27 \times 10^{- 5}

where

K_{d}

is the gain of drive circuit,

V_{r e f - d a c}

is the reference voltage of the DAC, and

N_{d a c}

is the resolution of the DAC. The key parameters are listed in Table 1. For

n = 2

, it can be given that

A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]

,

B = [\begin{matrix} 0 \\ 1 \end{matrix}]

,

C = [\begin{matrix} 0 \\ 1 \end{matrix}]

. Thus, the matrix parameter

K_{c}

can be calculated as

K_{c} = (0.25, 0.12)

according to Theorem 1 by liner matrix inequality (LMI). The realization of the control algorithm shown in Fig. 4 provides an efficient and robust method for eliminating the influence of IOPM’s gain fluctuation on the RIOG, which lays foundation for the research of the temperature error of the RIOG.

Table 1. Key parameters of the feedback channel

View Table

4. Experiments

The RIOG with MIST given in Fig. 2 is established and tested. The center wavelength and output intensity of the laser are 1550nm and 20mW, respectively. The half-wave voltage of the IOPM based on the lithium niobate waveguide is around 5.38V under room temperature of 25°C. Diameter, finesse and peak transmittance of the WRR are 60mm, 82 and 0.58, respectively. Thus, the modulation period of the triangular wave in the six-stare wave τ₁ is around 1μs [11]. The detector responsivity and dark current of the InGaSn PDs are 0.9A/W and 0.1pA, respectively. In order to monitor the state variable and perform the closed-loop tracking of the IOPM’s modulation index in real time, a high-speed sampling platform based on the virtual instrument which integrates high-speed ADCs, digital-to-analog converters (DACs) and field programmable gate array (FPGA) with 100MHz clock is developed. This designed platform has the ability of data collection, storage and processing at nanosecond level, which provides strong basis for the state monitor and research of the MIST. The digital signal processing designed on the platform can realize modulations, demodulations and control algorithms of the laser frequency locking loop, angular velocity tracking loop and MIST.

According to Eq. (3), it can be found that variations of the peak voltage $V_{p p}$ and the half-wave voltage $V_{π}$ is equivalent in changing the modulation index $ε_{r}$ . Thus different modulation voltages are imposed on the IOPM under room temperature of 25°C to verify the MIDT’s correctness. Figure 5 shows the change trend of the demodulation result when modulation voltages are around 8.2V and 13.0V, respectively. When the modulated voltage is less than $V_{2 π}$ which means the modulation index is less than 1, the change trend of two demodulation values $Δ P_{d}$ and $Δ P_{2 π}$ is same, and the value of demodulation result $oe----g00$ is positive. On the contrary, when the modulated voltage is more than $V_{2 π}$ which means the modulation index is more than 1, the change trend of two demodulation values $Δ P_{d}$ and $Δ P_{2 π}$ is the opposite, and the value of demodulation result $I_{D e m}$ is negative. What’s more, although the demodulated values $Δ P_{d}$ and $Δ P_{2 π}$ show random changes during the measurement because of the uncertain phase difference between the MZI’s two arms, the demodulation result $I_{D e m}$ hold constant for the fixed modulation index, which agrees well with the theoretical analysis.

Fig. 5 The demodulation values ∆P_d and ∆P_2π, and the demodulation result I_Dem when the IOPM’s modulation index is around (a) less than 1, and (b) more than 1.

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To further analyze the MIDT, the demodulation result $I_{D e m}$ is measured with different modulation indexes. The test time and sampling interval for each modulation indexes are 100s and 0.1s, respectively. The measurement and simulation results are both given in Fig. 6. The measurement points refer to the mean values and the whiskers show the standard deviation (SD) of the measurements. It can be found that the measurement results are in great agreement with the simulation results, which demonstrates the MIDT’s validity.

Fig. 6 The measurement and simulation results of the demodulation result I_Dem with different modulation indexes.

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The IOPM is then put into the Clima Temperature System T-40/25 for full temperature experiment to demonstrate the performance of MICCS. First of all, temperature of the incubator is cooled to −40°C. After holding the temperature for around 120 minutes, the IOPM is heated to + 60°C with the speed of 1°C/min. Finally, the temperature is kept for another 120min. The temperature is monitored by DS18B20 digital thermometer whose thermometer error and drift are ± 2°C and ± 0.2°C, respectively. The integral time of the sampling is 10 seconds. As shown in Fig. 7(a), there is obviously changed in modulation index with the various temperature. And the variation range of modulation index at full temperature is approximately 5.3%, which is unacceptable for RIOG. According to the measurement result with the MICCS given in Fig. 7(b), this detrimental influence is reduced efficiently. The variation range is reduced to less than 0.1‰ which is around 27dB reduction. And the gyro scale factor stability [29] can be calculated by:

K_{s} = \frac{1}{\bar{K}} {[\frac{1}{Q - 1} \sum_{i = 1}^{Q} {(K_{i} - \bar{K})}^{2}]}^{1 / 2}

where

K_{i}

is sampling data of the scale factor

K

at intervals of 1s,

\bar{K}

is the average of

K_{i}

,

Q

is the sampling number. Thus, after ignoring the influence of the other factors, the gyro scale factor stability strongly correlated with modulation index stability is 189.26 ppm from −40°C to + 60°C, which is calculated by Eqs. (4) and (13). It can be found that the IOPM’s gain fluctuation with temperature is accurately tracked, thus it’s influence on the RIOG can be effectively suppressed. The experimental results validate that the MIST can be applied to the closed-loop detection system for achieving the high-precision RIOG in practical engineering. The proposed modulation index stabilization technique can real-timely demodulate and control the modulation index of IOPM, which is first reported for RIOG. It is of significance in explaining the error mechanism to promote the wide application of RIOGs at full temperature.

Fig. 7 The measurement results of modulation indexes at full temperature (−40°C to + 60°C) experiment (a) without the MICCS and (b) with the MICCS.

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5. Conclusions

In the practical engineering, the integrated optic phase modulator (IOPM) as the key components of the noise suppression technique and angular velocity tracking loop is inevitably affected by the temperature drift which degenerates the performance of RIOG. Therefore, a novel modulation index stabilization technique (MIST) including modulation index detection technique (MIDT) and modulation index closed-loop control system (MICCS) are imposed to detect and track the modulation index of the IOPM in real time, which, to the best of our knowledge, is the first time stabilizing the modulation index of IOPM in RIOG at full temperature. By using the MIST, the experimental results demonstrate that the fluctuation of modulation index is reduced by 27dB. What’s more, the gyro scale factor stability related to the modulation index at full temperature (−40°C to + 60°C) is 189.26 ppm, which is very important to improve the temperature performance of RIOG. The result of our research is of significance in promoting the wide application of RIOG s in inertial navigation.

Appendix A

Proof of Theorem 1: Choosing the following Lyapunov function

\begin{array}{l} V (k + 1) - V (k) \\ \leq x^{T} (k + 1) P x (k + 1) - x^{T} (k) P x (k) - 2 f^{T} (K_{c} x (k)) [f^{T} (K_{c} x (k)) - K_{c} x (k)] \\ + [x^{T} (k) x (k) - γ^{2} w^{T} (k) w (k)] - [x^{T} (k) x (k) - γ^{2} w^{T} (k) w (k)] \\ = η^{T} (k) (Φ + Ω^{T} P Ω) η (k) - [x^{T} (k) x (k) - γ^{2} w^{T} (k) w (k)] \end{array}

where

η^{T} (k) = [x^{T} (k) f_{1} (k) w (k)]

,

Ω (k) = [A - k B C]

,

Φ = [\begin{matrix} - P + I & K_{c}^{T} & 0 \\ * & - 2 I & 0 \\ * & 0 & - γ^{2} \end{matrix}]

.

By Lemma 1, the sufficient condition for $Φ + Ω^{T} P Ω < 0$ is if and only if the following inequality holds

[\begin{matrix} - P + I & K_{c}^{T} & 0 & A^{T} \\ * & - 2 I & 0 & - k B^{T} \\ * & * & - γ^{2} & C^{T} \\ * & * & * & - P^{- 1} \end{matrix}] < 0

Pre and post multiplying by diag ${I, I, I, G^{T}}$ and its transpose, and by invoking the inequality $- G^{T} P^{- 1} G \leq P - G^{T} - G$ , it can be seen that Eq. (15) holds if Eq. (11) holds.

For Eq. (15), getting sum from $k = 0$ to $k = N$ , it can be obtained that

\sum_{k = 0}^{N - 1} {V (k + 1) - V (k)} = V (N) - V (0) \leq \sum_{k = 0}^{k - 1} η^{T} (k) φ η (k) - \sum_{k = 0}^{k - 1} [x^{T} (k) x (k) - γ^{2} w^{T} (k) w (k)]

If Eq. (11) holds, due to the zero initial condition and the fact $V (N) > 0$ , it can be obtained that

\sum_{k = 0}^{k - 1} [x^{T} (k) x (k) - γ^{2} w^{T} (k) w (k)] \leq 0

Thus, Theorem 1 guarantee that the closed-loop system has a prescribed $H_{\infty}$ performance level. This completes the proof.

Funding

National Natural Science Foundation of China (61875006) and the Defense Industrial Technology Development Program (JCKY201601C006).

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Parameter	Value	Parameter	Value
K_d	8	N_dac	16bits
V_ref-dac	2V	V_π	5.38V

Modulation index stabilization technique of integrated optic phase modulator used in resonant integrated optic gyro

Abstract

1. Introduction

2. Modulation index’s influence on angular velocity tracking loop

3. Modulation index stabilization technique

3.1 Modulation index detection technique (MIDT)

3.2 Modulation index closed-loop control system (MICCS)

4. Experiments

5. Conclusions

Appendix A

Funding

References

Cited By

Figures (7)

Tables (1)

Equations (17)

Optics Express