Double closed-loop control of integrated optical resonance gyroscope with mean-square exponential stability

Hui Li; Liying Liu; Zhili Lin; Qiwei Wang; Xiao Wang; Lishuang Feng

doi:10.1364/OE.26.001145

1. Introduction

Optical gyros based on Sagnac effect are the key components of inertial measurement units which are widely used in the fields of inertial navigation and attitude determination [1,2]. Especially, the integrated optical resonance gyroscopes (IORGs) have many advantages such as high robustness, theoretical sensitivity and superior reliability due to its inherent characteristics of miniaturized structure, all-solid-state and the combination between integrated optic fabrication and MEMS fabrication [3,4]. Thus, it has been considered as the next generation of micro-sized optical gyro and a promising candidate in the field of inertial navigation [5,6]. However, the numerous noise resulted from complex mechanism of IORG causes the drift of gyro bias and influences the detection accuracy of gyro [7–9]. Therefore, the closed-loop detection technology is important to suppress noise and disturbance and further to optimize the performance of IORG.

Recently, a great deal of attention has been paid to closed-loop detection methods with the aim of achieving precise and reliable IORGs [10–16]. In order to lock the output frequency of laser, Hotate et al. proposed a feedback method that utilized digital serrodyne modulation to track the resonant frequency by LiNbO₃ integrated phase modulator [10]. However, the frequent resets of serrodyne waves bring high-frequency harmonics and deteriorate waveform quality of resonator output. The triangular phase modulation method was also designed to control the laser frequency and effectively suppress the backreflection error in the IORGs [12,13]. Feng et al. described a low-delay, high-bandwidth frequency-locking loop implemented on a single filed-programmable gate array with triangular phase modulation, and the signal processing delay-time was reduced to less than 1μs and the bandwidth of 10 kHz was obtained under the optimized loop parameters [14]. In order to reduce backscatter-induced errors, the integer period sampling method [15] and the three lasers scheme [16] using two phase-locked lasers were proposed to improve the scale factor of resonant optic gyro, respectively. These closed-loop detection schemes are achieved by locking the resonant frequency of one propagating lightwave at light source frequency, but the closed-loop tracking of angular velocity is not implemented which consequently leads to IORGs with the inherent limited linearity and dynamic range as well as the sensitivities of existing disturbance. Recently, a double closed-loop IORG is experimentally demonstrated by using a controlled hybrid digital phase modulation scheme [17]. However, we found that the influence mechanism of locking-frequency accuracy from laser frequency locking loop (LFLL) on main closed-loop performance is neglected, which is a challenging problem that confines the engineering process of IORG. Furthermore, other influence factors on angular rate tracking precision are unclear and neglected in the related researches on IORG. Therefore, one of our motivations behind this work is to propose a double closed-loop detection method for IORG to accurately extract the angular velocity while various noise sources are substantially suppressed.

In this work, a novel double closed-loop control scheme with mean-square exponential stability is proposed to optimize detection precision and dynamic performance of the integrated optical resonance gyroscope. Firstly, we investigate the influence mechanism of the nonlinearity of optical effect on detection sensitivity, and the maximum sensitivity of a gyro system is obtained through the optimization of modulation parameters. Secondly, we present the influence theory of laser locking-frequency noise on the performance of angular velocity tracking loop (AVTL), and obtain the stochastic disturbance model of the double closed-loop IORG. Thirdly, a control algorithm based on the double closed-loop model is proposed to ensure that the closed-loop IORG has mean-square exponential stability with a prescribed H∞ performance. Finally, the relevant experiment results are given to demonstrate the effectiveness of proposed closed-loop control method.

2. Problem description

The schematic diagram of the double closed-loop system of IORG with triangular phase modulation is shown in Fig. 1. When the IORG is in a rotary state, the CW and CCW lightwaves propagate different cycles in the resonant cavity, which brings in different resonant frequencies [12].

Fig. 1 The principle scheme of the double closed-loop IORG based on Sagnac effect. The triangular wave for phase modulation is applied on the upper arm and bottom arm of the IOPM. The feedback sawtooth wave is applied on the bottom arm of the IOPM. Meanwhile, the phase modulation triangular wave and digital feedback sawtooth wave are all differentially applied on the arms of the IOPM. The IOPM achieves the phase modulation and also enables the closed-loop control of AVTL.

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The rotation angular velocity is obtained by detecting the resonant frequency difference Δf_Ω. Proper laser frequency is chosen and locked at resonant frequency of CCW path through laser frequency locking loop (LFLL). Meanwhile, the frequency difference Δf_Ω can be demodulated through the difference between the demodulated signal of two loops, and compensated by optical phase modulator in the AVTL to lock the resonant frequency of CW path. The data output of AVTL controller is proportional to the slope coefficient of stair-like digital sawtooth wave imposed on IOPM. And the control update time of controller is constant, which is same as the period of phase modulation triangular wave. Therefore, the data output of AVTL controller and slope of sawtooth wave are equal to the frequency difference Δf_Ω.

The frequency difference Δf_Ω is proportional to the rotation angular velocity Ω according to the principle of Sagnac effect and is taken as the output of an IORG system, which can be presented as [18]

Δ f_{Ω} = \frac{D}{n_{e} λ} Ω

where D is the diameter of ring resonator, n_e is the refractive index of resonator host material, λ is the central wavelength of light, c is the speed of light in free space, and Ω is the rotation angular velocity.

However, the interference light intensity representing the difference frequency information is a weak signal containing numerous noise in practical applications [7], which is sensitive to external disturbance and unavoidable optical parameter fluctuation. Meanwhile, the detection accuracy of the IORG system often suffers from the frequency-locking noise of LFLL, but its influence mechanism has rarely been analyzed and reported yet. In this work, the double closed-loop detection model of IORG is analyzed that includes the laser frequency locking and angular velocity tracking. Besides, a robust control algorithm with mean-square exponential stability is also proposed to optimize the measurement accuracy and dynamic performance of IORGs and to promote the process of their engineering applications.

3. Analysis of closed-loop detection system

3.1 Optimization of sensitivity level

Because the resonant frequency difference containing the rotational angular velocity information is a weak signal, the high-frequency carrier technology is adopted in IORG to suppress noise based on weak signal detection theory. Since the interference light intensity is a nonlinear function of resonant frequency, we first analyze the optimal modulation frequency to achieve maximum signal to noise (SNR) and linear work region of IORG.

In the IORG system depicted in Fig. 1, the interference intensity detected by the photoelectric detector can be written as [19]

I_{T} = \frac{I_{0} η β R}{1 + q^{2} - 2 q \cos 2 π f τ}

where I₀ stands for the light intensity of laser source, f is the central frequency of lightwave outputting from the resonator, η is the total loss coefficient of optical path, β and R are the conversion efficiency and trans-impedance of photoelectric detector, τ is the resonator’s round-trip time given by τ = πn_eD/c, and q is the loss coefficient of ring resonator defined by

F = π / \cos^{- 1} (2 q / (1 + q^{2}))

[19] with F the resonator finesse in the experimental test.

As shown in Fig. 2(a), when the IORG is in the state of rotation, the output interference intensity of resonator I_T, is a function of light frequency. The resonant frequency containing information of rotational angular velocity is delivered into the high frequency carrier by triangular phase modulation shown as Fig. 2(b). The relationship between the resonator output and frequency of triangular wave f_T can be analyzed by the equivalent frequency bias method, where the equivalent square modulation frequency bias are ± f_bias. Then, the time-domain output of photoelectric detector is described in Fig. 2(c), and the closed error of tracking angular velocity is synchronizing demodulated by a digital square wave with same frequency and phase as those of modulation wave. Thus, the demodulated signal can be expressed as

\begin{matrix} I_{d} (f) = \frac{I_{0} η β R K_{q d}}{1 + q^{2} - 2 q \cos 2 π (f + f_{b i a s}) τ} - \frac{I_{0} η β R K_{q d}}{1 + q^{2} - 2 q \cos 2 π (f - f_{b i a s}) τ} \\ = \frac{- 2 m \sin 2 π f_{b i a s} τ}{1 + \frac{m^{2}}{2} \cos 4 π f_{b i a s} τ + \frac{m^{2}}{2} \cos 4 π f τ - 2 m \cos 2 π f τ \cos 2 π f_{b i a s} τ} \cdot \\ \frac{I_{0} η β R K_{q d}}{1 + q^{2}} \cdot \sin (2 π f τ) \end{matrix}

where K_qd is the circuit gain of the forward channel in the closed-loop IORG. Thus, we can simplify the Eq. (3) as follows

I_{d} (f) = (k_{1} + Δ k_{1}) \sin (2 π f τ)

where

f = f_{0} + Δ f

with f₀ the center frequency of resonator curve and is an integral multiple of 1/τ and Δf the frequency difference between real optical frequency and resonant frequency, and m = 2q/(1 + q²). Considering the closed-loop system operates at zero point, we can obtain

\cos 2 π Δ f τ \approx 1

. Then, k₁ is the gain of demodulation gain while

Δ f = 0

, and also the forward chain of double closed-loop system. Further, the optimal square frequency bias f_bias can be obtained by the Eq. (5).

Fig. 2 The flow diagram of detected signals during the modulation and demodulation processes including (a) the interference intensity I_T as a function of light frequency (b) the phase modulated carrier ϕ and equal frequency bias f_bias; (c) the time-domain output I_m of photoelectric detector after modulation; (d) the demodulated signal I_d .

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f_{b i a s} |_{\frac{\partial k_{1}}{\partial f_{b i a s}} = 0} = \frac{arc \cos \frac{\sqrt{8 m^{2} + 1} - 1}{2 m}}{2 π τ}

Therefore, based on the optimal square frequency bias f_bias, the optimal frequency of phase modulation triangular wave can be obtained. In our designed IORG system, a buried-type Ge-doped silica-based waveguide-type ring resonator with a diameter of 60mm and a finesse of 82 is employed by which we obtain q = 0.9624 and τ = 0.911ns. The integrated optical phase modulator (IOPM) made by LiNbO₃ waveguide, has two independent modulation arms, a bandwidth 200MHz and a half-wave voltage V_π 5.6V. The phase modulation triangular wave has peak voltage V_PP 22.4V.

Based on the above analysis, the optimized frequency bias 3.8657MHz is obtained, and then the optimal frequency of phase modulation triangular wave $f_{T} = f_{b i a s} V_{π} / V_{P P}$ is 0.966MHz. Therefore, the largest demodulation gain and the least nonlinearity can be realized as shown in Fig. 3(a). Meanwhile, as illustrated in Fig. 3(b), the multiple-beam interference effect of waveguide ring resonator brings in system nonlinearity, however the proposed modulation frequency guarantees the IORG system operates at largest work region and optimal detection sensitivity.

Fig. 3 (a) the relationship between demodulation gain k₁ and modulation frequency; (b)the demodulated signals I_d shown as functions of frequency deviation for different modulation frequencies.

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In this part, the influencing mechanism of optical effect nonlinearity on detection sensitivity of the IORG system is investigated, and the optimized modulation frequency f_bias is derived to obtain the maximum demodulation gain k₁. Consequently, the closed-loop error sensitivity before controller can be improved with a precise Δf_Ω. However, the IORG is still a nonlinear system, and it is necessary to design a control algorithm that takes nonlinearity into consideration for ensuring the fast dynamic response of the IORG system.

3.2 Analysis of detection accuracy

The high dynamic model of double closed-loop IORG is established and analyzed in the following. In order to obtain Δf_Ω, the input of AVTL controller is the differential mode signal of PD1 and PD2. In ideal conditions, the frequency difference Δf_Ω can be accurately demodulated through the digital closed-loop processor in AVTL. However, the unavoidable frequency-locking accuracy Δf_n existing in LFLL causes considerable difficulty in precisely extracting frequency difference Δf_Ω. We analyze the working principle of frequency-locking accuracy on AVTL. According to the frequency relationship, $Δ f = Δ f_{Ω} + Δ f_{n}$ , we can obtain the demodulated frequency difference in AVTL as

I_{d_{2}} = (k_{1} + Δ k_{1}) \sin (2 π (Δ f_{Ω} + Δ f_{n}) τ)

which can be also rewritten as

I_{d_{2}} = (k_{1} + Δ k_{1}) \cos 2 π Δ f_{n} τ \sin 2 π Δ f_{Ω} τ + (k_{1} + Δ k_{1}) \cos 2 π Δ f_{Ω} τ \sin 2 π Δ f_{n} τ

Noting that $\sin 2 π Δ f_{Ω} τ \approx 2 π Δ f_{Ω} τ$ , we find that the first term of demodulation result, $k_{1} \cos 2 π Δ f_{n} τ \sin 2 π Δ f_{Ω} τ$ , is the closed-loop error of AVTL. Due to the real-time locking of laser frequency, the closed-loop error in LFLL Δf_n is within a small range, so that we have $\cos 2 π Δ f_{n} τ \approx 1$ , which obviously leads to the parameter uncertainty in AVTL. As for the frequency-locking noise signal $g (x_{1} (k)) = \sin 2 π Δ f_{n} τ$ existing in the second term of demodulation result $k_{1} \cos 2 π Δ f_{Ω} τ \sin 2 π Δ f_{n} τ$ , we know that its mean $E (g (x_{1} (k)))$ is 0 and variance of noise σ satisfying $E (g^{2} (x_{1} (k))) = σ^{2}$ is finite. Therefore, the frequency-locking noise signal can be considered as the noise term with zero mean and finite variance. Meanwhile, the closed-loop mean error Δf_Ω in AVTL is 0, so that $v_{2} (k) = (k_{1} + Δ k_{1}) \cos 2 π Δ f_{Ω} τ$ can be supposed to be the finite-energy disturbance signal which belonging to L₂[0, + ∞).

If the error of laser frequency locking Δf_n is zero, we can see that the disturbing signal $g (x_{1} (k)) v_{2} (k)$ will disappear in AVTL. However, the error of laser frequency locking does exist in a practical system, and influences the AVTL of IORGs. Thus, the closed-loop model of AVTL is a stochastic dynamical system introduced by the frequency locking noise.

The system gain parameters can be defined as k₁ and k₂, which satisfy k₁ = k₂. We define the gain variation of the forward channels of two loops as Δk₁ and Δk₂ which satisfy $k_{2} + Δ k_{2} = (k_{1} + Δ k_{1}) \cos 2 π Δ f_{n} τ$ . Meanwhile, the output light power is considerably affected by environmental temperature fluctuation, which also leads to the gain variation of forward channels of two loops. Based on the above analysis, we suppose that Δk₁ and Δk₂ are less than ± 10%. Thus, the synchronizing demodulated closed-loop signal of AVTL can be obtained respectively as

I_{d_{2}} (Δ f) =(k_{2} + Δ k_{2}) \sin 2 π Δ f_{Ω} τ + g (x_{1} (k)) v_{2} (k)

4. Design of closed-loop detection

In this section, a robust control algorithm based on the closed-loop signal detection method discussed above is proposed to establish a fast tracking system with high detection accuracy.

Because the frequency locking accuracy can only be demodulated from PD1, the tracking result of the laser frequency is free from the influence of angular closed-loop error. However, the parameter uncertainty Δk₁ of LFLL exists based on Eq. (4). According to controllability canonical form of automatic control theory, the dynamic equation of LFLL of IORG system can be deduced as

x_{1} (k + 1) = A_{1} x_{1} (k) + B_{1} (k_{1} + Δ k_{1}) \sin (k_{f_{1}} \cdot K_{C_{1}} x_{1} (k)) + D_{1} w_{1} (k)

Meanwhile, the mathematic model for the AVTL of IORG system is given by

x_{2} (k + 1) = A_{2} x_{2} (k) + B_{2} (k_{2} + Δ k_{2}) \sin (k_{f_{2}} \cdot K_{C_{2}} x_{2} (k)) + B_{2} g (x_{1} (k)) \cdot v_{2} (k) + D_{2} w_{2} (k)

where

A_{2} = A_{1} = [\begin{matrix} 1 & 1 & 0 & \dots & 0 \\ 0 & 1 & 1 & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & ⋱ & 0 \\ 0 & \dots & 0 & 1 & 1 \\ 0 & \dots & 0 & 0 & 1 \end{matrix}]

,

B_{2} = B_{1} = [\begin{matrix} 0 \\ 0 \\ ⋮ \\ 0 \\ 1 \end{matrix}]

.

w_{1} (k)

and

w_{2} (k)

are the one-dimensional finite-energy disturbance inputs caused by optical error and noise, respectively.

D_{1}

and

D_{2} \in R^{n}

are the disturbance intensity constant vectors.

x_{1} (k)

and

x_{2} (k) \in R^{n}

are the state variables, k_f₁ and k_f₂ are the gain of feedback chain of LFLL and AVTL, K_C1 and K_C2∈R^{1 × n} are the feedback gain matrices of controller of LFLL and AVTL, respectively.

In order to take the system nonlinearity, the inevitable parameter drift, the external disturbance and the frequency-locking noise into consideration, Eq. (9) and (10) are combined to establish the stochastic disturbance model of double closed loop system given by

x (k + 1) = A x (k) + \bar{B} f (k_{f_{1}} \cdot K_{C_{1}} x_{1} (k), k_{f_{2}} \cdot K_{C_{2}} x_{2} (k)) + I g (x_{1} (k)) v_{2} (k) + D w (k)

where

A = (\begin{matrix} A_{1} & 0 \\ 0 & A_{2} \end{matrix})

,

\bar{I} = (\begin{matrix} 0 \\ B_{2} \end{matrix})

,

D = (\begin{matrix} D_{1} \\ D_{2} \end{matrix})

and

\bar{B} = (\begin{matrix} B_{1} (k_{1} + Δ k_{1}) & 0 \\ 0 & B_{2} (k_{2} + Δ k_{2}) \end{matrix})

. The matrix

Δ B = (\begin{matrix} B_{1} Δ k_{1} & 0 \\ 0 & B_{2} Δ k_{2} \end{matrix})

can be transformed into

Δ B = H \cdot F (k) \cdot E

due to the bounded parameter Δk₁ and Δk₂, where H and E are constant matrices of appropriate dimensions that describe the variation intensity of the gain, F(k) is an uncertain matrix of appropriate dimension that satisfies the condition

F {(k)}^{T} F (k) \leq I

.

A desired control algorithm should be designed as robust as possible in practical engineering to overcome the previously-mentioned disadvantages. Therefore, it is of great significance to introduce the performance index for enhancing the disturbance rejection attenuation level of control algorithm. In addition to the detection accuracy, the exponential stability with desired H_∞ performance is necessary to ensure IORG system with a good tracking and a fast dynamic response. Thus, we present the method for designing the gain matrices, K_C1 and K_C2 of controller in the following.

In order to derive the main results, we first give some definitions and lemmas.

Definition 1 [20]:The system (11) with w(k) = 0 is said to be exponentially stable in a mean-square sense, if there exist some scalars $κ$ >0 and $0 < α < 1$ , such that the solution x(k) of system (11) satifies $E {‖ x (k) ‖} \leq κ α^{k - k_{0}} ‖ x (k_{0}) ‖$ for all any $x (k_{0}) \in R^{n}$ and $k \geq k_{0}$ .

Definition 2 [21]: For a given scalar γ>0, the system (11) is said to be exponentially stable in a mean-square sense with an expinential H_∞ performance γ, if it is exponentially stable and there holds $\sum_{s = 0}^{+ \infty} E {e^{T} (s) e (s)} \leq \sum_{s = 0}^{+ \infty} γ^{2} w^{T} (s) w (s)$ for all non-zero $w (k) \in l_{2} [0, \infty)$ under zero initial condition.

Lemma 1 (Schur Complement) [22]: For a given symmetric matrix S with the form S = [S_ij], $S_{11} \in R^{r \times r}$ , $S_{12} \in R^{r \times (n - r)}$ , S₂₂∈R⁽^n-r^{) × (}^n-r⁾, then, $S < 0$ if and only if S₁₁<0, $S_{22} - S_{21} S_{11}^{- 1} S_{12} < 0$ or $S_{22} < 0$ , $S_{11} - S_{12} S_{22}^{- 1} S_{21} < 0$ .

Lemma 2 [23]: Q, H and E are known constant matrixes with appropriate dimensions, where Q is a symmetrical matrix stisfying Q = Q^T, the following inequality Q + HFE + E^TF^TH^T <0 holds, for all satisfying $F^{T} F \leq I$ , if and only if there exists a scalar ε>0, such that $Q + ε^{- 1} H H^{T} + ε E^{T} E < 0$ .

Then, we first consider the exponential stability of system (11) with w(k) = 0 by a Lyapunov-based approach. The following theorem provides a sufficient condition to guarantee that IORG is mean-square exponentially stable for obtaining the desirable dynamic performance.

Theorem 1: For a given scalar $0 < α < 1$ , the system (11) is locally exponential stable in the mean square sense with w(k) = 0, if there exists positive definite matrix $P \in R^{n \times n}$ , feedback gain matrices $K_{c_{1}}, K_{c_{2}} \in R^{1 \times n}$ , and positive scalars ε₁, ε₂, ε₃ such that

(\begin{matrix} Φ + ε_{3} E^{T} E & ϕ_{1}^{T} P & ϕ_{2}^{T} P & 0 \\ * & - P & 0 & P H \\ * & * & - P & 0 \\ * & * & * & - ε_{3} I \end{matrix}) < 0

where

Φ = (\begin{matrix} - α P & - \frac{ε_{1} k_{f_{1}}}{2} H_{1} K_{C_{1}} & - \frac{ε_{2} k_{f_{2}}}{2} H_{2} K_{C_{2}} \\ * & - ε_{1} I & 0 \\ * & * & - ε_{2} I \end{matrix})

,

ϕ_{1} = (\begin{matrix} A & B \end{matrix})

,

ϕ_{2} = (\begin{matrix} σ G_{0} H_{1} & 0 \end{matrix})

.

Proof: See Appendix A.

Next, based on the results obtained in Theorem 1, we now consider the H_∞ performance for system(11) with noise w(k) to improve the detection accuracy of IORG. The following theorem guarantees that IORG is mean-square exponentially stabile with the desired H_∞ performance.

Theorem 2: For a given scalar $0 < α < 1$ and the H_∞ performance index γ, the system (11) is mean-square exponential stable with a prescribed H_∞ performance index γ, if there exist symmetric positive definite matrix $P, Q, R \in R^{n \times n}$ , feedback gain matrices $K_{c_{1}}, K_{c_{2}} \in R^{1 \times n}$ , and positive scalars ε₁, ε₂, ε_3, such that

(\begin{matrix} \tilde{Φ} + ε_{3} E^{T} E & {\hat{ϕ}}_{1}^{T} P & {\tilde{ϕ}}_{2} P & 0 \\ * & - P & 0 & P H \\ * & * & - P & 0 \\ * & * & * & - ε_{3} I \end{matrix}) < 0

where

\tilde{Φ} = (\begin{matrix} - α P + I & - \frac{ε_{1} k_{f_{1}}}{2} H_{1} K_{C_{1}} & - \frac{ε_{2} k_{f_{2}}}{2} H_{2} K_{C_{2}} & 0 \\ * & - ε_{1} I & 0 & 0 \\ * & * & - ε_{2} I & 0 \\ * & * & * & - γ^{2} I \end{matrix}), {\hat{ϕ}}_{1} = (\begin{matrix} A & B & D \end{matrix}), {\tilde{ϕ}}_{2} = (\begin{matrix} σ G_{0} H_{1} & 0 & 0 \end{matrix}) .

Proof: See Appendix B.

Theorem 2 provides the design rule of closed-loop feedback gain matrices of IORGs considering the nonlinearity, optical parameter uncertainty, external noise and frequency-locking disturbance of the LFLL. It guarantees that the system has mean-square exponential stability and optimal detection sensitivity with a prescribed H_∞ performance index γ. It is noted that, the perturbation parameters and frequency-locking disturbance of the LFLL in closed-loop control system of IORG vary dynamically due to the uncertain external environment, which cause the angular velocity tracking loop of IORG to be a parameter-uncertainty existing and noise-perturbed stochastic dynamical system.

We herewith obtain the model of double closed-loop detection system as shown in Fig. 4. When the control parameters of two loops are designed, the closed-loop error of AVTL does not affect the LFLL accuracy. Therefore, the control parameter K_C1 of the LFLL, whose design method of control parameter in LFLL is simple, can be designed first, and its closed-loop control model is similar with the model in [24]. Based on the obtained K_C1 referred to Theorems 1 in [24], we can calculate the control parameter K_C2 based on Theorems 2 with the designed K_C1.

Fig. 4 The block diagram of the double closed-loop detection scheme of IORG.

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The entire forward-path circuit consists of interference, photoelectric detector, analog amplifier, A/D converter and demodulation. The analog amplifier, drive circuit and linear amplifier are analog. The modulation, demodulation, controller and sawtooth generating are digital. During the demodulation, after the sampling and accumulating in the positive and negative half-period of the modulation triangular wave, the closed-loop error of each loop can be demodulated by subtraction of the positive and negative accumulation. And, Section 3.1 and 3.2 deduce that the forward-path circuit can be supposed as a proportional link multiplied by a sine function.

The circuit gain of the forward channel $K_{q d} = K_{G a} \cdot K_{a d} \cdot N_{d e m}$ where K_Ga is the gain of pre-amplifier, K_ad is the gain of A/D converter and N_dem is the sampling number of half demodulation period.

In accordance with the composition of forward channel of closed-loop system as shown in Fig. 4, the demodulation gain is given by

k_{1} = \frac{2^{1 + N_{a d}} m I_{0} η R β K_{G a} N_{d e m}}{(1 + q^{2}) V_{r e f}}

where

K_{a d} = 2^{N_{a d}} / V_{r e f a d}

and N_ad and V_refad are the resolution and reference voltage of A/D converter respectively. The values of the key parameters are shown in Table 1. Given that

n = 2

,

A_{1} = (\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})

,

B_{1} = (\begin{matrix} 0 \\ 1 \end{matrix})

,

D_{1} = (\begin{matrix} 0 \\ 1 \end{matrix})

,

P = I

, k₁ = 0.0265 and

k_{f_{1}} = k_{G d} \cdot V_{r e f d a} \cdot k_{l a} / 2^{N_{d a}} = 3.0469 \times 10^{- 4}

where

k_{G d}

is the gain of drive circuit, and k_la is the gain of linear amplifier. Then, the matrix parameter of K_C1 is calculated as

K_{C_{1}} = (\begin{matrix} 0.3125 & 0 .00488 \end{matrix})

by linear matrix inequality (LMI) [25]. The feedback gain of AVTL

k_{f_{2}} = k_{G d} f_{c l k} V_{r e f} V_{p p} 2^{- N_{d a}} / 4 V_{π} Δ H_{s a w}

where ΔH_saw is the digital amplitude of sawtooth wave, and f_clk is the working clock of FPGA. In our designed IORG system, k₂ = k₁ = 0.0265 and k_f₂ = 0.0571, Then given that α = 0.05, Q = I, the feedback gain matrix of AVTL can be calculated as

K_{C_{2}} = (\begin{matrix} 0.2 & 0 .00078 \end{matrix})

based on Theorem 2 with the designed K_C1.

Table 1. The value of parameters of block diagram

View Table

The realization of the control algorithm illustrated in Fig. 4 provides an efficient and robust method for optimizing the detection precision and fast tracking performance of IORG. Theorem 2 provides the method for designing the AVTL controller, which is robust for suppressing the influence of locking frequency accuracy and the main closed-loop disturbance.

5. Experiments and results

Some experiments are conducted in this section to verify the validity and effectiveness of the designed signal processing scheme of IORG. The optical configuration of the IORG experimental setup as previously presented in Fig. 1 is composed of the laser, the optical isolator (ISO), the IOPM and waveguide-type ring resonator. A piezoelectric low phase-noise fiber laser operating around 1550nm is utilized with linewidth less than 1kHz and outputting power 36.2mW. The type and parameter of IOPM and waveguide-type ring resonator is shown in Section 3.1.

The gyro output is at millisecond level and the closed-loop step of tracking angular velocity is at dozens of nano-second level, which brings difficulty in verifying the influence mechanism analysis of laser locking noise on main closed-loop performance and the validity of the double closed-loop controller. In this work, a high speed sampling design platform based on Fig. 4 is developed. The digital signal processing tasks, performed by the reconfigurable FPGA of Kintex-7 XC7K410T with working clock 100MHz, includes the modulation signal generation, the demodulation and control algorithm process. The FPGA is connected to host computer by PXI-e bus. Therefore, the designed platform not only monitors the nano-second state variables and analyze the frequency spectrum of the key signal, but also assist us in verifying the influence mechanism analysis of laser locking noise on the main closed-loop performance.

The step response experiment is first conducted to demonstrate that the dynamic performance of IORG has fast exponential stability in Fig. 5. A ramp signal with digital stair height h_stair 93.75 is generated by FPGA with working clock f_clk 100MHz, then converted into digital amplitude 65536 with 16 bits D/A, and applied to phase modulator along with the feedback signal. The resonant frequency difference of ramp signal $Δ f_{Ω} = h_{s t a i}_{r} \cdot f_{c l k} / 2^{n_{D / A}}$ is obtained as 143.05kHz and the corresponding slope ratio is equivalent to 300°/s of the system input. The time domain output curves of CW, CCW loop and AVTL controller are observed by an oscilloscope in Fig. 5(a). It can be observed that the resulting output of digital controller is a standard sawtooth signal, which is imposed on the IOPM to feedback the resonant frequency difference Δf_Ω. The output light intensity of resonator is further converted into voltage signal by photoelectric detector, which is depicted in Fig. 5(b).

Fig. 5 Implementation of the closed-loop IORG system. (a) The oscilloscope display of output signals under 300°/s rotational speed measurement. Curve A is the CW output which detected by PD2 and displayed on oscilloscope, Curve B is the CCW output which detected by PD1, and curve C is AVTL controller output; (b) the PD1 curve of LFLL; (c) the frequency locking noise of LFLL after the frequency of laser is locked to the center frequency of curve resonator (d) the step response of IORG system.

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The control update time of frequency-locking loop is 0.2ms, and the experimental data depicted in Fig. 5(c) is the demodulated signal of LFLL. Since the demodulated signal of LFLL is mainly the laser's locking noise, so that we may estimate the maximum variance of the laser's locking noise by the experimental data shown in Fig. 5(c). The estimated variance of locking-frequency noise σ is 44.56Hz illustrated in Fig. 5(c), which is used to optimize the closed-loop control algorithm of AVTL.

In consideration of the influence of laser locking-frequency noise, the simulation and experiment results are presented in Fig. 5 about the dynamic performance of IORG by using the proposed control algorithm. And, the simulation result of the time-domain model shows a fast rise time of 73.265us. The experimental result demonstrates that the response time of IORG is 76us, which can be sampled by FPGA and illustrated in Fig. 5(d). The exponential stability coefficient α stays consistency with the theoretical value in Section 4, and the experimental result 76us is roughly consistent with the theoretical result 73.265us. It clearly shows that the improved dynamic performance of IORG with the influence of laser locking-frequency noise can meet the exponential stability using the proposed control algorithm.

Further, the measuring accuracy experiment is conducted to validate the effectiveness of our proposed closed-loop algorithm with a prescribed H_∞ performance. As shown in Fig. 6, a bias stability of 7.04°/h is demonstrated with an integration time of 10s over one-hour test, and the Allan deviation analysis of typical one-hour test shows that the bias stability has reached 1.841°/h. In contrast, without adopting our closed-loop detection method, the bias stability of the IORG system with an integration time of 10s is 46.8°/h adopting the same optical system [12]. Meanwhile, the IORG has the best long-term bias stability that ever been reported for an IORG based on a silica waveguide ring resonator. The experiment results verify that the proposed algorithm can effectively suppress the locking frequency noise, the system nonlinearity and the external disturbance, and therefore improve the detection accuracy and dynamic performance of the IORG system.

Fig. 6 The results of measuring accuracy experiment of the IORG system. a) 1h test of bias stability; c) Allan variance of the 1h test data.

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Finally, we conduct the measurement experiment of the scale factor to verify the linearity of IORG by using the proposed closed-loop controller, which is illustrated in Fig. 7. The obtained scale factor (SF) of IORG prototype is 7.6 (°/s)⁻¹ and the nonlinearity of scale factor is 344.71ppm. These results demonstrate that the proposed signal processing method of IORG successfully corrects the system nonlinearity.

Fig. 7 Results of scale factor experiment of the IORG system. a) the test of scale factor; b) The out curve and corresponding relative fitting error of the IORG system.

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6. Conclusions

In a practical IORG system, the performance of main angular velocity tracking loop is greatly affected by the locking-frequency accuracy of the laser frequency-locking loop. Meanwhile, the detection accuracy of IORG is degenerated by several adverse factors such as the nonlinear optical effect, the optical parameter fluctuation, and the unavoidable external noise existing in practical environment. In this paper, a novel closed-loop signal detection scheme is designed to extract the angular velocity of IORG system with high sensitivity and high precision of measurement. Firstly, we derive the optimal modulation frequency to obtain the maximum linear work region and demodulation gain for suppressing the nonlinearity of optical effect. Secondly, we explored the influence mechanism of the above four factors on system sensitivity and measurement accuracy. The mathematical model of the main angular rate tracking loop is also established. Thirdly, a robust control algorithm of the IORG for suppressing the frequency-locking disturbance and the existing optical noise is then proposed to ensure mean-square exponential stability with a prescribed H_∞ performance. The experimental results demonstrate the effectiveness and usefulness of our proposed method, by which the detection accuracy and dynamic performance of IORG have been greatly improved. The result of our research is of significance in promoting the wide application of IORGs in inertial navigation.

Appendix A

Proof of Theorem 1:

We select the following Lyapunov function for system (11) as $V (k) = x^{T} (k) P x (k)$ , where P is defined as that in Theorem 1. It is noted that $\sin (x)$ is a monotone increasing and differentiable function in the range $- π ≪ x ≪ π$ , and $\sin (\cdot)$ locally satisfies the Lipschitz condition $0 < \sin (x (k)) / x (k) \leq 1$ and $g (0) \equiv 0$ . It is easy to see that there exists positive scalar $0 < G_{0} \leq 1$ such that $\sin (x_{1} (k)) \leq G_{0} x_{1} (k)$ . And, we can obtain the following condition

f^{T} (K_{C_{i}} x_{i} (k)) \cdot (f (K_{C_{i}} x_{i} (k)) - K_{C_{i}} x_{i} (k)) \leq 0, where i = 1, 2.

Thus, we have

\begin{matrix} E {V (k + 1) - α V (k)} \leq E {x^{T} (k + 1) P x (k + 1) - α x^{T} (k) P x (k) \\ - ε_{1} \sin^{T} (K_{C_{1}} x_{1} (k)) (\sin (K_{C_{1}} x_{1} (k)) - K_{C_{1}} x_{1} (k)) \\ - ε_{2} \sin^{T} (K_{C_{2}} x_{2} (k)) (\sin (K_{C_{2}} x_{2} (k)) - K_{C_{2}} x_{2} (k))} \end{matrix}

In consideration of

\sin (x_{1} (k)) \leq G_{0} x_{1} (k)

and

E (v^{T} (k) v (k)) = σ^{2}

, the inequality holds:

E {{(\bar{I} g (x_{1} (k)) v (k))}^{T} P (\bar{I} g (x_{1} (k)) v (k))} \leq E {x^{T} (k) {(σ G_{0} H_{1})}^{T} P (σ G_{0} H_{1}) x (k)}

Noting that

E (v (k)) = 0

and the dynamics of System (11), we have

\begin{matrix} E {V (k + 1) - α V (k)} = E {ς_{1}^{T} (k) ({\bar{ϕ}}_{1}^{T} P {\bar{ϕ}}_{1}) ς_{1} (k) + \\ ς_{1}^{T} (k) Φ ς_{1} (k) + x^{T} (k) {(σ G_{0} H_{1})}^{T} P (σ G_{0} H_{1}) x (k)} \end{matrix}

where

ς_{1}^{T} (k) = (\begin{matrix} x^{T} (k) & \sin^{T} (K_{C_{1}} x_{1} (k)) & \sin^{T} (K_{C_{2}} x_{2} (k)) \end{matrix})

,

x_{1} (k) = H_{1} x (k) = (\begin{matrix} I & 0 \end{matrix}) x (k)

,

x_{2} (k) = H_{2} x (k) = (\begin{matrix} 0 & I \end{matrix}) x (k)

,

{\bar{ϕ}}_{1} = (\begin{matrix} A & \bar{B} \end{matrix})

and

Φ = (\begin{matrix} - α P & - \frac{ε_{1}}{2} H_{1} K_{C_{1}} & - \frac{ε_{2}}{2} H_{2} K_{C_{2}} \\ * & - ε_{1} I & 0 \\ * & * & - ε_{2} I \end{matrix})

.

Thus, we obtain

E {V (k + 1) - α V (k)} \leq E {ς_{1}^{T} (k) ({\bar{ϕ}}_{1}^{T} P {\bar{ϕ}}_{1} + ϕ_{2}^{T} P ϕ_{2} + Φ) ς_{1} (k)}

We can see that the sufficient condition for $E {V (k + 1) - α V (k)} < 0$ is if and only if ${\bar{ϕ}}_{1}^{T} P {\bar{ϕ}}_{1} + + ϕ_{2}^{T} P ϕ_{2} + Φ < 0$ .

Then by using Lemma 1, ${\bar{ϕ}}_{1}^{T} P {\bar{ϕ}}_{1} + + ϕ_{2}^{T} P ϕ_{2} + Φ < 0$ holds if and only if

(\begin{matrix} Φ & {\bar{ϕ}}_{1}^{T} P & ϕ_{2}^{T} P \\ P {\bar{ϕ}}_{1} & - P & 0 \\ P ϕ_{2} & 0 & - P \end{matrix}) < 0

Because the matrix $Δ B$ is bounded, from Lemma 2, one has

(\begin{matrix} Φ & {\bar{ϕ}}_{1}^{T} P & {\bar{ϕ}}_{2}^{T} P \\ P {\bar{ϕ}}_{1} & - P & 0 \\ P {\bar{ϕ}}_{2} & 0 & - P \end{matrix}) = (\begin{matrix} Φ & ϕ_{1}^{T} P & ϕ_{2}^{T} P \\ P ϕ_{1}^{T} & - P & 0 \\ P ϕ_{2} & 0 & - P \end{matrix}) + M^{T} F (k) N + N^{T} F^{T} (k) M < 0

where

ϕ_{1} = (\begin{matrix} A & B \end{matrix})

,

B = (\begin{matrix} {\bar{B}}_{1} k_{1} & 0 \\ 0 & {\bar{B}}_{2} k_{2} \end{matrix})

,

M = (\begin{matrix} 0 & H^{T} P & 0 \end{matrix})

,

N = (\begin{matrix} E & 0 & 0 \end{matrix})

.

By Lemma 1, Eq. (18) is equivalent to

(\begin{matrix} Φ + ε_{3} E^{T} E & ϕ_{1}^{T} P & ϕ_{2}^{T} P & 0 \\ * & - P & 0 & P H \\ * & * & - P & 0 \\ * & * & * & - ε_{3} I \end{matrix}) < 0

Then, we can see that $E {v (k)} \leq {(α)}^{k - k_{0}} V (k_{0})$ such that $E {‖ x (k) ‖} \leq \sqrt{\frac{β_{2}}{β_{1}}} {(α^{\frac{1}{2}})}^{k - k_{0}} ‖ x (k_{0}) ‖$ , where $β_{1} = \min λ (P)$ and $β_{2} = \max λ (P)$ are the minimum and maximum eigenvalues of matrix $P$ , respectivly. According to Definition 1, we conclude that the system(12) is exponentially stable in the mean-square sense with w(k)=0.

Appendix B

Proof of Theorem 2

We consider the performance $Γ (k) = x^{T} (k) x (k) - γ^{2} w^{T} (k) w (k)$ for any nonzero $w (k) \in L_{2} [0, \infty)$ . Following the similar proof of Theorem 1 with the same Lyapunov-Krasovskii function, we have

\begin{array}{l} E {V (k + 1) - α V (k) + Γ (k)} \\ \leq E {V (k + 1) - α V (k) + Γ (k) \\ - ε_{1} f^{T} (K_{C_{1}} x_{1} (k)) (\sin (K_{C_{1}} x_{1} (k)) - K_{C_{1}} x_{1} (k)) \\ - ε_{2} f^{T} (K_{C_{2}} x_{2} (k)) (\sin (K_{C_{2}} x_{2} (k)) - K_{C_{2}} x_{2} (k))} \\ = E {ξ_{2}^{T} (k) ({\tilde{ϕ}}_{1}^{T} P {\tilde{ϕ}}_{1} + + {\tilde{ϕ}}_{2}^{T} P {\tilde{ϕ}}_{2} + \tilde{Φ}) ξ_{2} (k)} \end{array}

where

ξ_{2} (k) = {(\begin{matrix} x {(k)}^{T} & x {(k - 1)}^{T} & \sin {(Δ φ (k))}^{T} & \sin {(Δ φ (k - 1))}^{T} \begin{matrix} w {(k)}^{T} \end{matrix} \end{matrix})}^{T}

According to the above analysis, we can see that a sufficient condition for $E {V (k + 1) - α V (k)} < 0$ is equivalent to ${\tilde{ϕ}}_{1}^{T} P {\tilde{ϕ}}_{1} + + {\tilde{ϕ}}_{2}^{T} P {\tilde{ϕ}}_{2} + \tilde{Φ} < 0$ ,where ${\tilde{ϕ}}_{1} = (\begin{matrix} A & \bar{B} & D \end{matrix})$ , ${\tilde{ϕ}}_{2} = (\begin{matrix} σ G_{0} H_{1} & 0 & 0 \end{matrix})$ , $\tilde{Φ} = (\begin{matrix} - α P + I & - \frac{ε_{1}}{2} H_{1} K_{C_{1}} & - \frac{ε_{2}}{2} H_{2} K_{C_{2}} & 0 \\ * & - ε_{1} I & 0 & 0 \\ * & * & - ε_{2} I & 0 \\ * & * & * & - γ^{2} I \end{matrix})$

It follows from Lemma 1, ${\tilde{ϕ}}_{1}^{T} P {\tilde{ϕ}}_{1} + + {\tilde{ϕ}}_{2}^{T} P {\tilde{ϕ}}_{2} + \tilde{Φ} < 0$ holds if and only if

(\begin{matrix} \tilde{Φ} & {\tilde{ϕ}}_{1}^{T} P & {\tilde{ϕ}}_{2} P \\ * & - P & 0 \\ * & * & - P \end{matrix}) < 0

By using Lemma 2, Eq. (20) can be rewritten as

\begin{array}{l} (\begin{matrix} \tilde{Φ} & {\hat{ϕ}}_{1}^{T} P & {\tilde{ϕ}}_{2} P \\ * & - P & 0 \\ * & * & - P \end{matrix}) + M_{2}^{T} F_{2} (k) N_{2} + N_{2}^{T} F_{2} (k) M_{2} \\ \leq (\begin{matrix} \tilde{Φ} + ε_{3} E^{T} E & {\hat{ϕ}}_{1}^{T} P & {\tilde{ϕ}}_{2} P & 0 \\ * & - P & 0 & P H \\ * & * & - P & 0 \\ * & * & * & - ε_{3} I \end{matrix}) < 0 \end{array}

where $M_{2} = (\begin{matrix} 0 & H^{T} P & 0 \end{matrix})$ and $N_{2} = (\begin{matrix} E & 0 & 0 \end{matrix})$ .

Finally, we obtain

E {V (k + 1)} < E {α V (k) - Γ (k)} < E {α^{k - k_{0}} V (k_{0}) - \sum_{s = k_{0}}^{k} α^{k - s} Γ (s)}

On the other hand, considering the definition of the Lyapunov function $V (k + 1) > 0$ and the zero initial condition $x (k_{0}) = 0$ , we can see that $\sum_{s = k_{0}}^{k} α^{k - s} E (x^{T} (s) x (s)) < \sum_{s = k_{0}}^{k} α^{k - s} E (γ^{2} w^{T} (s) w (s))$ . By suming both sides of the equation from $k = k_{0}$ to $k = \infty$ , one gets $\sum_{s = k_{0}}^{+ \infty} E {x^{T} (s) x (s)} < γ^{2} \sum_{s = k_{0}}^{+ \infty} w^{T} (s) w (s)$ . Then, from Theorem 2, it follows that the closed-loop system (11) can satisfy $H_{\infty}$ performance. This completes the proof.

Funding

National Natural Science Foundation of China (No. 61405005).

References and links

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Parameter	Value	Parameter	Value	Parameter	Value
$I_{0}$	36.2µW	$β$	0.92	K_Ga	1
$R$	1117Ω	$N_{a d}$ , $N_{d a}$	16 bits	K_Gd	12.06
$V_{r e f a d}$ , $V_{r e f d a}$	2V	$N_{d e m}$	30	ΔH_saw	6.4e5

Double closed-loop control of integrated optical resonance gyroscope with mean-square exponential stability

Abstract

1. Introduction

2. Problem description

3. Analysis of closed-loop detection system

3.1 Optimization of sensitivity level

3.2 Analysis of detection accuracy

4. Design of closed-loop detection

5. Experiments and results

6. Conclusions

Appendix A

Proof of Theorem 1:

Appendix B

Proof of Theorem 2

Funding

References and links

Cited By

Figures (7)

Tables (1)

Equations (28)

Optics Express