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Abstract
A novel demodulation method is presented to expand the measurement dynamic range for fiber optic strain sensors using PDH technology. The new control algorithm uses two integrators to form a 2nd order control, and the FFPI strain sensor can have a dynamic range of 20 dB/octave larger than the PID control method when the input signal frequency decreases a magnitude. A strain resolution of 4.7 pɛ Hz−1/2@10Hz, a 118 dB@10Hz dynamic range without consecution, and 158 dB with consecution is obtained. The experiment results show that the new control method can improve the sensing system’s dynamic range with the corner frequency unchanged and without the system noise level degradation.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Fiber optic sensors have been widely researched in the field of geophysics science due to their high sensitivity, large dynamic range, wide working frequency band, and can work in harsh environments [1]. With the development of fiber optic sensing technology, the fiber optic strainmeter has shown great potency in improving the strain observation ability [2–4]. To meet the earth tide and geological activity observation requirements, the strainmeter should cover the frequency range from DC to about 50 Hz and have a dynamic range above 160 dB [5]. Phase shifted fiber Bragg grating (PSFBG) and fiber Fabry-Perot interferometer (FFPI) with Pound-Drever-Hall (PDH) technique have been introduced to the instrument research. While the sensitivity depends on the choice of fiber grating, the system characteristic is also related to the design of the feedback control algorithm. By modifying the feedback control algorithm, the system sensitivity, working frequency band as well as the dynamic range will change synchronously.
The optical strainmeter using PDH technology can be divided into open-loop and close-loop types according to their demodulation methods. The open-loop type has two main structures: The one type drives the laser source [6], external cavity [7], intensity modulator [8], or optical single-sideband (SSB-SC) modulator [9] directly to generate the sweeping frequency signal and use PDH error signal as an identifier to get the frequency shift caused by the applied strain signal. The system has a noise level limited by the laser frequency noise level. To solve this problem, the other type is inspired by the laser locking structure [10], which introduces an extern feedback loop to compensate for the low-frequency noise of the laser source and uses the linear range of FFPI’s error signal to amplify the small frequency variation caused by the input strain signal. In 2005, Chow, J. H. et al. [11] introduced a locking structure and successfully achieved < 1 pɛ Hz−1/2 at the frequency band > 200Hz. In 2010, Timothy T.-Y. Lam et al. [12] used the gas cell to build a feedback loop further to suppress the laser noise floor at the low-frequency band. Both methods mentioned above have shortages mainly caused by the sweeping speed, sweeping range of the laser source, and the line-width of the used reference resonant cavity. These are limiting the system tracking speed and the system dynamic range.
To solve the problems of the open-loop type sensors, the close-loop strainmeter utilizes the feedback loop to read out the strain signal. In 2016, Jiageng Cheng et al. [13] introduced another feedback loop into the readout path, achieved a much larger dynamic range of about 120 dB (149 dB with resonance peak continue). To further improve the dynamic range of the locking system, they aimed to suppress the system output noise by using mode-locking laser (MLL) [14] and distributed feedback fiber lasers (DFB-FLs) [15]. The system shows the ability to sense the strain signal about 4 pɛ Hz−1/2@10s by using MLL, 580 fɛ Hz−1/2@1 kHz with a dynamic range of about 110 dB by using DFB-FLs, respectively. In 2018, Peide Liu et al. used random feedback fiber laser as the laser source and achieved nearly 132 fɛ Hz−1/2@>1 kHz noise level with a dynamic range of about 121 dB [16]. For the property of being easy to realize, nearly these articles chose the PID algorithm as their feedback control algorithm. Although such a high dynamic range can be achieved, they failed to take the influence of frequency and feedback control algorithm into consideration. With a higher frequency of the input signal, a less dynamic range is obtained, and the control algorithm also occupies a large proportion of dynamic range attenuation with frequency increase. When the PID control algorithm is chosen as the feedback loop in dynamic signal tracking, the system tracking range is limited by the traceability of the PID feedback loop. Using a higher-order control method, a minor tracking error and a larger maximum tracking range can be expected.
This paper proposes a novel control algorithm to enlarge the dynamic range using a higher-order control method. Using this new control algorithm, when the system is configured with the same corner frequency, a much more dynamic characteristic improvement can be obtained compared with the PID algorithm. With the frequency reduction, the dynamic range will act as 40 dB/octave expansion without the noise level degradation, which is 20 dB/octave better than the traditional PID control algorithm. This paper is arranged as follows: In section 2, the control model and the basic feedback principle are described as well as the 2nd order control method. In section 3, an experiment is illustrated to verify the validity of the novel control algorithm. The discussion and conclusion are given in Section 4 and Section 5, respectively.
2. Principle
To form the tracking system, the close-loop PDH configuration requires the tracking error within the narrow linear range of the PDH error signal, or else the nonlinear distortion will increase, and even the failure of the locking process will happen when the tracking error exceeds the linear range of the PDH error signal. The feedback algorithm is designed to generate a compensation signal from the tracking error signal in a much larger frequency shift input, track, follow the resonance peak wavelength variation and limit the tracking error within the linear range of the PDH error signal. The characteristic of the feedback loop decides the system’s total traceability.
Figure 1(a) gives the feedback PDH tracking system principle. The error signal is generated from the difference between input frequency shift νin from FFPI and the tracking signal νf from the locking system. As shown in Fig. 1(b), the generated frequency tracking error Δν is amplified by the PDH system and converted to the error voltage signal ΔV. The control loop generates the feedback voltage, drives the controlled system, and updates the tracking signal to track the input frequency shift. Figure 1(c) gives the two different control algorithms of PID control and 2nd order control. The 2nd order control algorithm has an additional integrator than the PID control algorithm, which makes the control system have a larger dynamic error signal gain with the same input signal frequency. The equivalent error signal curves using different control algorithms are illustrated in Fig. 1(d).
Fig. 1. Configuration of the locking system and fundamental principle of dynamic range expansion. a) Diagram of the close loop PDH tracking system. b) The generation of PDH error signal. c) The block diagram of the feedback algorithm Gc(s) (blue: PID, red: 2nd order control). d) The equivalent gain of the different control algorithm
At the linear range of the PDH error signal, the relationship between frequency shift and voltage offset can be regarded as a linear function. So, we can simplify the error signal as it has a linear slope of Kslope. Then, if we assume the controlled system has a transfer function GL(s) and the control system has a transfer function Gc(s), the whole system transform function can be inferred. It has the following form:
Typically, the control system has a much broader response passband than the whole system tracking passband. Then, the GL(s) can be simplified as an all-pass filter with passband gain K.From the working principle described above, we know that the feedback range of the system is correlated with the tracking error and the setup of the control loop. For FFPI, the linear region is approximately equal to the resonance Full Width at Half Maximum (FWHM). Given a known FWHM, the maximum tracking range, as same as the maximum measurable signal input, can be clearly defined from Fig. 1(a) and has the form of:
The function Herr(s) is the tracking error function of the system, and it represents the residual frequency deviation for a given input signal. Formula 2 also indicates that the maximum measurable signal is related to the input signal frequency.
From formula 2, we know that the system’s maximum measurable signal is determined by the order of control system Gc(s). For traditional PDH tracking system usage, the PI control is always chosen rather than the whole PID control structure [10]. Due to only one pole existing in the PID control system, the PID system is worked as a 1st order system, and the feedback range will be 20 dB larger when the input signal frequency decreases a magnitude. If we change the control loop to 2nd order system, the feedback range should also be extended. Under this analysis, the 2nd control function is expressed as follows:
Table 1. Maximum measurable signal gain and tracking error delay
Figure 2 gives the theoretical curves generated from the formula in Table 1. The solid lines in Fig. 2(a) give the relative gain of the maximum input signal compared with FWHM, and the dash lines are the reference gains of different feedback control algorithms. From formula 2, at the frequency band ω < ωn, the Gc(s) dominates the leading position. So, the system’s maximum measurable input is mainly decided by the feedback output range of the control algorithm. At the frequency band ω > ωn, the control output's feedback signal is small and negligible compared to the input signal. In this case, the input signal amplitude should not exceed the linear range of FFPI.
Fig. 2. Theory results of the maximum measurable signal input using two different control methods. a) Relative gain of the maximum input signal compared with FWHM and control algorithm’s gain as reference. b) The phase characteristic of the tracking error signal.
The phase characteristic shown in Fig. 2(b) is the phase lag of the tracking error signal compared with the input signal. From the theory curves, we know that the maximum tracking error will appear at the peak of the input sine wave when using the 2nd order control method, while PID’s maximum tracking error will be at the zero-crossing point. From the theory analysis, it’s clear that the system has a much larger feedback range when using the 2nd order control algorithm with the same passband configuration.
When the system noise N(s) is determined, the system dynamic range will also be distinct and can be written as:
From the working principle and formula 1, the system output noise is close to the input noise at the frequency band below the corner frequency. If laser noise is compensated and the environment temperature is stable enough, the system noise will be mainly limited by the circuit noise in the high-frequency band and the heat exchange between sensing FFPI and the environment at the lower frequency band. Hence, the total dynamic range will mainly be related to the maximum measurable signal, which is influenced by the control method we have chosen in Table 1.
3. Experiment
The experiment setup is shown in Fig. 3. An NKT-E15 laser is used with wavelength adjustment mode enabled as the laser source. A Y-type waveguide is used as a phase-modulator (PM). A single-sideband modulator (SSB) works as the laser frequency shifter. The modulation frequency is set to 15 MHz, and the modulation signal is demodulated by the locking-in amplifier (LIA). The demodulated signal is captured by a NI-7856R DAQ system with embedded FPGA onboard. The feedback frequency is set to 10 kHz so that the system can function the vibration signal up to a few hundred hertz. The optical fiber is clamped on a PZT piezo nano-positioner (PI P363-3cd) with a clamped position distance of about 40 cm. The FFPI used in the experiment has a 22 cm resonant cavity, and the free spectral range of the FFPI is about 465 MHz. The non-linear output of the voltage controlled oscillator (VCO) is compensated by the control software with a pre-scanned voltage-frequency mapping table.
Fig. 3. Experiment Configuration. (CIR: Circulator, PD: Photonic Detector). a) Block diagram of the experiment setup. b) Photos of FFPIs setup and the demodulation box.
The lower part in Fig. 3(a) is the extra PDH locking loop working in a laser stabilization configuration and locking the laser output wavelength onto one of the resonance peaks of reference FFPI. Both two FFPIs are with the exact specifications. The reference FFPI is used to compensate for the frequency fluctuations caused by environment thermal disturbance [13]. Figure 3(b) gives the photos of the FFPIs setup and the demodulation box.
In our test, we scan the PDH system's error curve first. The result is shown in Fig. 4(a). The FFPI used in our experiment has a linear range of about 1.3 MHz. The other parameters tested for calculating the theory result are given in Table 2:
Fig. 4. The error signal and tracking output with a given input. a) The scanned error signal of the PDH system. b) The system output with a given strain input.
Table 2. Parameters used in calculation
In the experiment, we set the integral parameter to 30 and the proportion parameter to 0.0001 for the PID control system, so that both the two control systems have the same corner frequency of 200 Hz. Using the nano-positioner to give a known stimulation, the output of the sensing system is provided in Fig. 4(b). The tracking system shows a good tracing ability with the input strain signal, and the system has a sensitivity of about 3.235 V/µɛ.
To test the feedback gain of the system, an external signal is applied to the sensing FFPI to generate the error voltage ΔV. The input error voltage and the system output (VCO Driving Voltage) are captured and recorded by the FPGA. By correcting the input and output phase delay, the absolute output gain can be then illustrated in Fig. 5.
Fig. 5. Feedback Gain of the feedback system. a) The output gain using different control algorithms with a 40 Hz input signal. b) The output gain using different control algorithms with a 60 Hz input signal.
From Fig. 5, it can be seen the 2nd order control algorithm has a larger gain factor compared with the PID algorithm. With different input signal frequencies, the feedback gain is changed along with the frequency increasing. And the 2nd order control algorithm gain is about 0.65 @ 40 Hz and 0.30 @ 60 Hz compared with those of PID, which are about 0.12 @ 40 Hz and 0.08 @ 60 Hz, respectively. The test results fit the theory-predicted curves well using the above formulas and parameters. The difference is caused by the out-of-band signal distraction. The distraction is mainly caused by the harmonic wave generated from the piezo nano-positioner. The larger signal generated comes with the larger harmonic distraction, which also mains the larger error compared with the simulated result.
Figure 6 shows the experiment result of the whole feedback loop with different frequencies of input signals. The input signal is tuned to the amplitude that the system reached its ultimate tracking range. The blue dash line in Fig. 6(a) shows the theoretical result using the PID control algorithm and the yellow triangles indicate the real test result of the sensing system. The brown dash line shows the theoretical result using the 2nd order control algorithm, and the purple triangles show the real test result. From the theory and test results, when both the two control methods were configured with the same corner frequency of 200 Hz, the new control algorithm can be about 11.1 dB at 60 Hz, 15.0 dB at 40 Hz, and 17.7 dB at 20 Hz larger than that using the PID control algorithm. Figure 6(b) is the corresponding phase lag of the tracking error signal of the system, and the dashed lines show the theory results using the formulas given in Table 1. The difference between the test and theory results is caused by the inaccuracy of the estimation parameters relative to the real system and the system readout resolution.
Fig. 6. Test result of close loop tracking system. a) Maximum input frequency shift and theory curves. b) The phase lag of tracking error signal.
In Fig. 7, the system noise level and system power spectrum with signal input are given. As shown in Fig. 7(a), when the whole system is placed in a noise-free environment, the test result shows that the system noise floor can be up to 4.7 pɛ Hz−1/2@10Hz and share the same noise level compared with the PID control method. In Fig. 7(b), a 1 µɛpp signal is injected into the system, and the blue line is the power spectrum of the tracking output. Considering the system sensitivity is 127.4 MHz/µɛ and the total tuning range of RF-VCO is about 500 MHz, the dynamic range of 118 dB can be guaranteed at the frequency band < 1 Hz with only one resonance peak of FFPI used. The system dynamic range can be further expanded if the system can jump to the nearest resonance peak when the input signal is too large for the system to track, and it will be discussed in section 4. Figure 7 also shows the influence of environment disturb. As shown in Fig. 7(b), the rapid noise level raising below 1 Hz comes from the not well isolated experimental condition and nano-positioner low-frequency noise. The noise is much smaller with appropriate environment isolation, as shown in Fig. 7(a).
Fig. 7. The system noise level and the power spectrum of the system with a 20 Hz, 1 µɛpp signal input. a) System noise level. b) The system power spectrum with a 1 µɛpp signal input.
4. Discussion
It can be inferred from the formula described in Section 2 that by modifying the control algorithm to a higher-order control method, the system can get a greater dynamic range improvement. However, the instability of the system will also be promoted. When the algorithm is deployed on the real tracking system, the slight difference between the model used to design the control algorithm with the real physical system may have a great impact. For example, the system control delay or the uncertainty of the controlled system phase characteristic, any that exists in the real system would make the system much more complex than the ideal model. The higher-order control algorithm will also mean a higher requirement of system precision of numeric operations. For the FPGA control system using a fixed-point calculation structure, the integrator saturation problem should also be considered in designing the control system. It will also need more calculation resources and more specific algorithm optimization. The control algorithm used in this article is also facing the same problem. Increasing the feedback speed, setting the control system with a higher passband frequency and a larger damping ratio, will be helpful in controlling the system instability.
For the lower frequency (< 1 Hz) band, the dynamic range will not be mainly limited by the control algorithm but limited by the adjustment range of the laser source or the SSB-SC. In our experiment, the RF-VCO’s adjustable range is about 500 MHz, which is about 4 pm @1550 nm. Although the total dynamic range can be continued by switching the tracking peak to the nearest resonance peak of FFPI, the discontinuity point at the jump point caused by the rapid release-lock action of the control system still cannot be neglectable in some high precision, high working frequency application. Even so, some inference or prediction algorithms can be used to compensate for the jump point in some low working frequency applications. Then the dynamic range will only be limited by the reflection range of FFPI. The system dynamic range can achieve more than 140 dB with the peak-switching continue method. For a typical FFPI, the total working range is the sum of the two FFPI’s reflection ranges (about 640 pm). With a frequency resolution of about 1 kHz, the entire dynamic range can be up to 158 dB.
Table 3 shows the comparison with others' works.
Table 3. Comparison with other works
5. Conclusion
This paper builds a feedback model for a PDH-based sensing system and analyses the dynamic range characteristic with different feedback algorithms. By adding an additional integrator into the control system to form the 2nd order control algorithm, the tracking error is compensated by the new algorithm and thus improves the system tracking range. The system noise and the frequency performance were tested by the experiment. Compared with the PID control system, both control algorithms meet the same noise floor below 1 kHz Hz−1/2, but the 2nd order control algorithm can track a much larger signal at the same frequency input. Experimental results show that the dynamic range of the new system is about 15.0 dB at 40 Hz and 17.7 dB at 20 Hz larger than the PID control algorithm with the same corner frequency. And the 2nd order control system also performs a 40 dB/octave slope expansion with the frequency decreasing.
Funding
National Science Fund for Distinguished Young Scholars (61925501); National Natural Science Foundation of China (61975040, 62127815); Guangdong Introducing Innovative and Entrepreneurial Teams (2019ZT08X340); Introducing Leading Talents of “The Pearl River Talent Recruitment Program” of Guangdong Province (2019CX01X010).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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