CN111400907A - Unified modeling method for random errors of fiber-optic gyroscope - Google Patents

Unified modeling method for random errors of fiber-optic gyroscope Download PDF

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CN111400907A
CN111400907A CN202010180729.7A CN202010180729A CN111400907A CN 111400907 A CN111400907 A CN 111400907A CN 202010180729 A CN202010180729 A CN 202010180729A CN 111400907 A CN111400907 A CN 111400907A
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韩松来
罗世林
芦佳振
罗世彬
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Abstract

The invention discloses a unified modeling method for random errors of a fiber-optic gyroscope, which comprises the following steps: (1) analyzing the random error component of the fiber-optic gyroscope by adopting Allan variance, determining the random error type existing in the fiber-optic gyroscope to be analyzed, PSD corresponding to each random error type and the value of the noise parameter of each random error type, and deriving a differential equation of the random error based on the PSD of each random error; (2) the obtained differential equation is equivalent to a fiber optic gyroscope equivalent model established by a random process equivalent theorem driven by single white noise according to a plurality of random processes driven by independent white noise, wherein one random process is described by one differential equation driven by unit white noise. The unified modeling method provided by the invention can well compensate random errors in the fiber-optic gyroscope by deducing an equivalent model suitable for the fiber-optic gyroscope, and is superior to a Gaussian-Markov method in the aspects of initial alignment precision and convergence.

Description

Unified modeling method for random errors of fiber-optic gyroscope
Technical Field
The invention relates to the technical field of fiber optic gyroscopes, in particular to a unified modeling method for random errors of a fiber optic gyroscope.
Background
Traditional random error model building techniques fall into two categories: stochastic modeling based on differential equations and stochastic modeling based on differential equations. Based on the modeling of the difference equation, the constant coefficient linear difference equation driven by time series is also called Auto Regressive Moving Average (ARMA) Model. The ARMA model is a research hotspot in the field of random error modeling because it can describe random errors in a fiber-optic gyroscope, but the ARMA model has the disadvantage that the order and parameter determination of the model is empirical, and the ARMA model is model sensitive and is not suitable for handling odd power processes, higher order processes or large dynamic ranges. Based on the modeling of a differential equation, random errors in a gyroscope or an accelerometer are considered to be a mixture of a random constant process and a white noise process, the random constant process is modeled into the differential equation with a driving function of zero, and the white noise process is taken as driving noise and is directly incorporated into INS attitude and velocity error dynamics; or, the random error in the gyroscope or accelerometer is considered to be a mixture of the random walk process and the white noise process, the random walk process is described using a differential equation with white noise as the driving function, and the white noise process is incorporated directly into the INS attitude and velocity error dynamics as the driving noise. And random errors in gyros or accelerometers are often considered to be a smooth gaussian-markov process. In engineering applications, the gaussian-markov process can provide a good approximation for many stochastic processes by selecting appropriate parameters. However, according to the actual data analysis, the autocorrelation function of the random error of the inertial sensor may be far from the exponential function, so that the model establishment is biased.
Disclosure of Invention
In view of the above, the present invention provides a unified modeling method for random errors of a fiber optic gyroscope, which is a method for deriving differential equation description of random errors from PSD of random errors, and by proving an equivalent theorem, combines a plurality of differential equations into one differential equation, and finally derives an equivalent representation of multiple random errors of an inertial sensor by using the equivalent theorem, and finally applies the proposed modeling method to self-designed fiber optic gyroscope initial alignment, and verifies the effectiveness of the proposed method through experiments.
On one hand, the invention provides a unified modeling method for random errors of a fiber-optic gyroscope, which comprises the following steps:
(1) analyzing the random error component of the fiber-optic gyroscope by adopting Allan variance, determining the random error type existing in the fiber-optic gyroscope to be analyzed, PSD corresponding to each random error type and the value of the noise parameter of each random error type, and deriving a differential equation of the random error based on the PSD of each random error;
(2) the obtained differential equation is equivalent to a fiber optic gyroscope equivalent model established by a random process equivalent theorem driven by single white noise according to a plurality of random processes driven by independent white noise, wherein one random process is described by one differential equation driven by unit white noise.
Preferably, the random error types include white noise, quantization noise, angular random walk noise, zero-bias instability noise, rate random walk noise, and rate ramp noise.
Preferably, the Allan variance of the fiber optic gyroscope in step (1) is expressed by the following formula:
Figure BDA0002412433790000021
in the formula, σtotalDenotes the sum of Allan variances, σQRepresenting quantization noise, σNRepresenting random walk of angle, σBDenotes zero bias stability, σKRepresenting a random walk of the rate, σRRepresents a rate ramp, where the Allan variance value is approximately expressed by:
Figure BDA0002412433790000022
in the formula, σtotal(τ) denotes Allan variance root, AnRepresenting correlationsCoefficient, τ denotes different correlation times;
computing Allan variance roots of different correlation times through formulas (1) and (2), computing coefficients through a least square algorithm by utilizing Allan variance root values of different moments, and utilizing AnTo calculate the value of the noise parameter.
Preferably, the specific process of deriving the differential equation of the random error based on the PSD of the random error is as follows:
setting sx (w) as PSD of corresponding random process x, and g (jw) as transfer function of corresponding filter, then:
Sx(w)＝|G(jw)|2(3)
assuming that x (jw) is the fourier transform of the filter input and y (jw) is the fourier transform of the filter output, the following relationship holds according to the linear system theory:
Y(jw)＝G(jw)X(jw) (4)
wherein X (jw) is equal to the unit 1;
and (4) carrying out inverse Fourier transform on two sides of the formula (4) to obtain a differential equation of the random error.
Preferably, the angular random walk noise has a rational spectrum, and the differential equation thereof is obtained by the following process:
setting Srw(w) is the PSD of the angular random walk noise, then:
Srw(w)＝K2/w2(5)
wherein, K represents a noise parameter, and w represents a certain frequency;
setting Grw(j ω) is the transfer function of the corresponding filter, and Grw(j ω) is a rational function, then:
Grw(jω)＝K/jω (6)
wherein j ω represents a specific domain;
and (3) performing inverse Fourier transform on two sides of the above formula, and then expressing a differential equation of angle random walk as follows:
drw(t)＝Ku1(t) (7)
in the formula (d)rw(t) equation representing the random walk of angles, u1(t)Is a unit of white noise.
Preferably, the zero-bias instability has an irrational spectrum, the differential equation of which is obtained by:
setting Sbi(w) is the PSD of zero-bias instability noise, then:
Sbi(w)＝B2/w (8)
wherein B represents a noise parameter;
setting Gbi(j ω) is the transfer function of the corresponding filter to zero-bias instability noise, and Gbi(j ω) is an irrational function, then:
Figure BDA0002412433790000031
to pair
Figure BDA0002412433790000032
And performing Taylor series expansion, and then approximating a transfer function as:
Figure BDA0002412433790000033
in the formula (I), the compound is shown in the specification,
Figure BDA0002412433790000034
an approximate transfer function representing zero-bias instability noise, β, depends on the expansion point of the taylor series;
the differential equation for performing inverse fourier transform on both sides of equation (10) to obtain zero-bias instability noise is described as follows:
Figure BDA0002412433790000041
in the formula (d)bi(t) an expression for zero-bias instability noise,
Figure BDA0002412433790000042
representing a differential expression representing zero-bias instability noise, u2(t) represents a unit of gaussian white noise.
Preferably, the rate ramp noise has an irrational spectrum, the differential equation of which is obtained by:
setting Grn(j ω) is the transfer function of the corresponding filter to the rate ramp noise, and Grn(j ω) is an irrational function, then:
Grn(jω)＝R/(jω)1.5(13)
wherein R represents a noise parameter;
to three taylor series (jw)1.5And expanding to obtain an approximate transfer function as follows:
Figure BDA0002412433790000043
in the formula (I), the compound is shown in the specification,
Figure BDA0002412433790000044
an approximate transfer function, ω, representing the rate ramp noise0Represents a certain frequency;
inverse fourier transforms are performed on both sides of equation (13) to obtain a differential equation of the rate ramp noise, which is described as follows:
Figure BDA0002412433790000045
equation (15) represents a second order gaussian markov process.
Preferably, the equivalent model of the mixed random error including at least the angular random walk noise, the zero-bias instability noise and the rate ramp noise is represented as follows:
Figure BDA0002412433790000046
in the formula, a1、a2、a3、a4、b0、b1、b2、b3Respectively representing the coefficients, ω (t) being a unit of white noise, ω(0)(t)、ω(1)(t)、ω(2)(t)、ω(3)(t) representing noise respectivelyDifferent orders, z (t) being a variable, z(0)(t)、z(1)(t)、z(2)(t)、z(3)(t) are respectively different orders, wherein:
z(t)＝drw(t)+dbi(t)+drn(t) (33)
Figure BDA0002412433790000047
Figure BDA0002412433790000051
in the formula, D is a differential operator;
Figure BDA0002412433790000052
Figure BDA0002412433790000053
the unified modeling method provided by the invention analyzes the random error component of the fiber-optic gyroscope by adopting an Allan variance analysis technology, and on the basis, an equivalent model suitable for the fiber-optic gyroscope is deduced by utilizing the random error component and related parameters, can well compensate the random error in the fiber-optic gyroscope, and is superior to a Gauss-Markov method in the aspects of initial alignment precision and convergence.
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The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate an embodiment of the invention and, together with the description, serve to explain the invention and not to limit the invention. In the drawings:
fig. 1 is a flowchart of a unified modeling method for a random error of a fiber optic gyroscope according to an embodiment of the present invention;
FIG. 2 is a diagram of a process for deriving a random error differential equation for the PSD of the random error;
FIG. 3 is a comparison graph of autocorrelation function data obtained by the equivalent model and measured data.
Detailed Description
It should be noted that the embodiments and features of the embodiments may be combined with each other without conflict. The present invention will be described in detail below with reference to the embodiments with reference to the attached drawings.
For a better understanding of the present invention, the following words are specifically explained:
a fiber optic gyroscope: based on the Sagnac effect, the optical fiber loop forms an angular velocity measuring device of the interferometer, and the change of the angular velocity is reflected by detecting the coherent intensity of output light.
Random error: random errors, also known as occasional and indeterminate errors, are errors that cancel each other out due to a series of small random fluctuations of the relevant factors during the measurement process.
Allan variance (Allen variance): the method is used for analyzing the phase and frequency instability of the oscillator, all the time domain representations of the frequency stability of the high-stability oscillator adopt Allan variance, and inertial sensors such as a gyroscope and the like also have the characteristics of the oscillator, so that the method is applied to random error identification of various inertial sensors.
PSD (Power SPectral Density): the power spectral density of a wave multiplied by a suitable coefficient will result in the power carried by the wave per unit frequency, which is called the power spectral density of the signal
Kalman filtering: an algorithm for performing optimal estimation on system state by using a linear system state equation and inputting and outputting observation data through a system.
Initial alignment: in the operating state before the inertial navigation system starts navigation, coordinate system alignment and initial parameter measurement are generally performed, and the platform is adjusted on a given navigation coordinate system.
The core idea of the invention is a method for deriving differential equation description of random errors from Power SPectral Density (PSD) of the random errors, a plurality of differential equations are combined into one differential equation by proving an equivalence theorem, an equivalent expression of multiple random errors of the fiber-optic gyroscope is derived by utilizing the equivalence theorem, and finally the provided modeling method is applied to initial alignment of the fiber-optic gyroscope and the effectiveness of the provided method is verified by experiments.
Fig. 1 is a flowchart of a unified modeling method for random errors of a fiber optic gyroscope according to an embodiment of the present invention. As shown in FIG. 1, the unified modeling method for the random error of the fiber-optic gyroscope of the invention comprises the following steps:
(1) analyzing the random error component of the fiber-optic gyroscope by adopting Allan variance, determining the random error type existing in the fiber-optic gyroscope to be analyzed, PSD corresponding to each random error type and the value of the noise parameter of each random error type, and deriving a differential equation of the random error based on the PSD of each random error;
specifically, the random error types include white noise, quantization noise, angular random walk noise, zero-bias instability noise, and rate ramp noise, and since the random errors are statistically independent, the Allan variance of the fiber-optic gyroscope can be expressed as follows:
Figure BDA0002412433790000071
in the formula, σtotalDenotes the sum of Allan variances, σQRepresenting quantization noise, σNRepresenting random walk of angle, σBDenotes zero bias stability, σKRepresenting a random walk of the rate, σRRepresents a rate ramp, where the Allan variance value is approximately expressed by:
the Allan root of variance approximation is expressed by:
Figure BDA0002412433790000072
in the formula, σtotal(τ) denotes Allan variance root, AnRepresents the correlation coefficient, and τ represents different correlation times;
the root Allan variance values of different correlation times can be calculated through the formulas (1) and (2), then the root Allan variance values of different moments are utilized, coefficients are calculated through a least square algorithm, and finally, the coefficients are calculated through the least square algorithmWith AnTo calculate the noise parameter.
From the above equation, passing a unit of white gaussian noise through a pre-designed filter can produce random errors with a specific PSD. Correspondingly, if the PSD of the random error is already obtained, the corresponding filter can also be derived, and then the differential equation of the random error can be calculated according to the filter, and the general flow is shown in fig. 2. The above process is described in a random process x:
Sx(w) is the PSD of the corresponding random process x, let g (jw) be the transfer function of the corresponding filter, then:
Sx(w)＝|G(jw)|2(3)
Sx(w) can be determined by Allan analysis of variance, so G (jw) can be obtained from equation (3), let Y (jw) be the Fourier transform of the filter output, let X (jw) be the Fourier transform of the filter input, since the input to the shaping filter is unity white noise, and X (jw) is equal to unity 1, the following relationship holds true according to linear system theory:
Y(jw)＝G(jw)X(jw) (4)
g (jw) can be derived from equation (3), X (jw) is equal to unit 1, and Fourier inversion is performed on two sides of equation (4) to obtain a differential equation of random error.
Further, the following is a description of differential equations in turn deriving random errors. The white noise can be directly used as the driving noise of the inertial navigation system error dynamic state in the Kalman filtering process, so that modeling is not needed. The quantization noise can be dynamically converted to white noise by changing the INS error. Therefore, only zero-bias instability noise, angle random walk, and rate ramp noise are discussed subsequently.
In a further preferred embodiment, the angular random walk has a rational spectrum, so that the transfer function G of the corresponding shaping filterrw(j ω) is a rational function from which the corresponding differential equation description can be directly derived. The PSD of the angle random walk is:
Srw(w)＝K2/w2(5)
wherein, K represents a noise parameter, and w represents a certain frequency;
according to equation (3), the corresponding shaping filter transfer function is:
Grw(jω)＝K/jω (6)
and performing inverse Fourier transform on two sides of the above formula, and then describing a differential equation of angle random walk as follows:
drw(t)＝Ku1(t) (7)
in the formula (d)rw(t) equation representing the random walk of angles, u1(t) is a unit of white noise.
Zero-bias instability and rate-ramp noise have unreasonable spectra and therefore the transfer function of the corresponding shaping filter is an unreasonable function. Reasonable approximations should be made based on the above analysis. The approximation and the true transfer function are then compared to each other to evaluate the error introduced by the approximation.
The PSD of zero-bias instability noise is rewritten as follows:
Sbi(w)＝B2/w (8)
wherein B represents a noise parameter;
according to equation (3), the true transfer function of the corresponding shaping filter to zero-bias instability noise is:
Figure BDA0002412433790000081
to pair
Figure BDA0002412433790000082
And performing Taylor series expansion, and then approximating a transfer function as:
Figure BDA0002412433790000083
in the formula (I), the compound is shown in the specification,
Figure BDA0002412433790000084
approximate transfer function representing zero-bias instability noise, β, relies on Taylor stagesA number of flare points and will be determined later from an error analysis of the approximation.
Inverse fourier transform is performed on both sides of equation (10) to obtain a differential equation of zero-bias instability noise, which is described as follows:
Figure BDA0002412433790000091
in the formula (d)bi(t) an expression for zero-bias instability noise,
Figure BDA0002412433790000092
representing a differential expression representing zero-bias instability noise, u2(t) represents a unit of gaussian white noise.
In fact, equation (11) represents a first order Gaussian Markov process, which means that the zero-bias instability noise here approximates a first order Gaussian Markov process, G in dbbi(jw) and
Figure BDA0002412433790000093
the relative amplitude error of (d) can be calculated as:
Figure BDA0002412433790000094
similar processing can be done for rate ramp noise, resulting in a true and approximate transfer function as follows:
Grn(jω)＝R/(jω)1.5(13)
wherein R represents a noise parameter;
Figure BDA0002412433790000095
in the formula (I), the compound is shown in the specification,
Figure BDA0002412433790000096
an approximate transfer function, ω, representing the rate ramp noise0Represents a certain frequency;
here, as distinguished from the foregoingA three-term Taylor series (jw)1.5And (3) expanding to obtain an approximate transfer function, and obtaining a differential equation of the rate ramp noise from the equation (14) as follows:
Figure BDA0002412433790000097
it can be seen that the above formula represents a second order Gaussian Markov process, with db as the unit, and the relative amplitude error at Grn(jw) and
Figure BDA0002412433790000098
can be calculated as:
Figure BDA0002412433790000099
from equation (16), it can be found that the relative amplitude error is found in the entire frequency rangernAnd is also less than 0 db. The rate ramp noise is also a low frequency noise if ω is chosen00.01rad/s at a frequency band of 0.1Hz,10Hz]Internal relative amplitude errorrnLess than-0.5 dB. Therefore, differential equations of the angular random walk noise, the zero-bias instability noise, and the rate ramp noise are obtained from equations (7), (11), and (15).
(2) The obtained differential equation is equivalent to a fiber optic gyroscope equivalent model established by a random process equivalent theorem driven by single white noise according to a plurality of random processes driven by independent white noise, wherein one random process is described by one differential equation driven by unit white noise.
In order to prove that the invention reasonably bases on combining differential equations of different random errors in the fiber-optic gyroscope into a single differential equation for equivalent modeling, the following is demonstrated: a random process derived from the foregoing can be described by a differential equation driven by unity white noise. Let x (t) and y (t) represent two random processes, x (t) being modeled by an m-order differential equation driven by one unit of white noise, and y (t) being modeled by a P-order differential equation driven by another unit of white noise. They can be represented as follows:
a0x(t)+a1Dx(t)+L+amDmx(t)＝b0u(t)+b1Du(t)+L+bnDnu(t) (17)
in the formula, a0、a1、am，b0、b1、bnRespectively representing different coefficients, Dmx (t) represents the higher order term of the stochastic process, Du (t) represents noise, Dnu (t) higher order terms representing noise;
c0y(t)+c1Dy(t)+L+cpDpy(t)＝d0v(t)+d1Dv(t)+L+dqDqv(t) (18)
in the formula, c0、c1、cp，d0、d1、dqRespectively representing different coefficients, Dpy (t) represents the higher order terms of the stochastic process, dv (t) represents noise; dqv (t) represents the higher order terms of the noise.
In addition, D in the expressions (17) and (18) is a differential operator and satisfies Dmx(t)＝dmx(t)/dtmU (t) and v (t) are two independent units of white noise. Rearranging (17) and (18) the following equivalents can be obtained:
Figure BDA0002412433790000101
Figure BDA0002412433790000102
a new variable z (t) is introduced to equal the sum of x (t) and y (t). Rearranging the equation to obtain:
(a0+a1D+L+amDm)(c0+c1D+L+cpDp)z(t)
＝(b0+b1D+L+bnDn)(c0+c1D+L+cpDp)u(t)
+(d0+d1D+L+dqDq)(a0+a1D+L+amDm)v(t) (21)
the left side of the formula is the sum of z (t), and the right side of the formula is defined as:
s(t)＝(f0+f1D+L+fpnDpn)u(t)+(g0+g1D+L+gqmDqm)v(t) (22)
the following relationship exists:
f0+f1D+L+fpnDpn＝(b0+b1D+L+bnDn))(c0+c1D+L+cpDp)z (23)
g0+g1D+L+gqmDqm＝(d0+d1D+L+dqDq)(a0+a1D+L+amDm) (24)
as shown in equation (22), s (t) is a random process driven by two independent white noises. If it can be shown that s (t) is equivalent to a random process driven by a single white noise, the differential equation will represent a random process equivalent to the sum of two random processes x (t) and y (t). To demonstrate this, two arguments are introduced below.
Introduction 1: u (t) is the unit Gaussian white process, for integers i and j, the correlation function is as follows:
E[Diu(t1)Dju(t2)]＝(-1)jDi+j(t1-t2) (25)
in the formula, E represents the mathematical expectation, Dju(t1) Representing a differential equation, Dju(t2) Representing a second differential equation of the Gaussian process, Di+j(t1-t2) Representing the corresponding function of the system pulse.
Lesion 2 u (t) and v (t) are two independent Gaussian white processes, (22) where s (t) is a generalized stationary (WSS) process.
The equivalence theorem is given below and is expressed as follows:
s (t) is defined, wherein u (t) and v (t) are two independent white Gaussian noisesThen let r beis satisfies:
E[s(t)s(t-t)]＝r0d(t)+r1D2d(t)+L+rlD2ld(t) (26)
where d (t) is the Dirac function, r0、r1And rlRespectively representing different coefficients.
Defining w (t) as a Gaussian white noise unit, wherein if eis exists and satisfies
Figure BDA0002412433790000111
Figure BDA0002412433790000112
Figure BDA0002412433790000113
To obtain
s′(t)＝e0w(t)+e1Dw(t)+…+elDlw(t) (30)
In the formula, D represents a variance, ei and the like are calculated.
For WSS, s' (t) and s (t) are equal.
If two random processes are equivalent in terms of WSS, they will have the same autocorrelation function. This means that they also have the same PSD, since it is the fourier transform of the autocorrelation function. In kalman filtering, PSD is the only information needed to characterize random errors, so the equivalence of the stochastic process in WSS is sufficient for kalman filtering applications.
According to the theorem of equivalence, there is the following formula:
s(t)＝s′(t)＝e0w(t)+e1Dw(t)+…+elDlw(t) (31)
this means that the multiple differential equations are equivalent to differential equations driven by unity white noise. Thus, a differential equation driven by two independent white noises is equivalent to a single differential equation driven by another white noise. Thus, it is demonstrated that the mixture of two stochastic processes can be represented by a differential equation, and this conclusion can be generalized to the case of multiple stochastic processes.
In the equivalent modeling of multiple random errors in the fiber-optic gyroscope, quantization noise can be converted into white noise, the white noise can be directly used as driving noise of inertial navigation errors without modeling, so that the white noise and the quantization noise do not need modeling, an error source of rate random walk noise is not clear, and the modeling does not consider the random errors, so that the invention only discusses angle random walk, zero-bias instability noise and rate slope noise. These three types of noise may or may not be present in the same sensor at the same time and need to be determined by alan analysis of variance.
In a further technical scheme, the most complicated situation that three kinds of noise, namely angle random walk noise, zero-bias instability noise and rate slope noise, exist in the same sensor of the same fiber-optic gyroscope is set. The differential equations for angular random walk noise, zero-bias instability noise, and rate ramp noise are found in equations (7) (11) (15), and are rewritten here with the differential operator D as follows for convenience:
Figure BDA0002412433790000121
defining a new variable z (t) of the formula:
z(t)＝drw(t)+dbi(t)+drn(t) (33)
substituting formula (32) for formula (33) and rearranging to obtain a differential equation having
Figure BDA0002412433790000131
(34) The left side of the formula is expanded as:
z(4)(t)+a1z(3)(t)+a2z(2)(t)+a3z(1)(t)+a4z(0)(t) (35)
the coefficients are calculated as follows:
Figure BDA0002412433790000132
according to the previously proposed equivalence theorem, (34) the right equivalence is:
b0ω(3)(t)+b1ω(2)(t)+b2ω(1)(t)+b3ω(0)(t) (37)
ω (t) is a unit of white noise, and according to the equivalent theorem, the coefficients are calculated as follows:
Figure BDA0002412433790000133
combining (35) and (37), an equivalent differential equation that yields a mixture of angular random walk noise, zero-bias instability noise, and rate ramp noise is expressed as follows:
z(4)(t)+a1z(3)(t)+a2z(2)(t)+a3z(1)(t)+a4z(0)(t)＝
b0ω(3)(t)+b1ω(2)(t)+b2ω(1)(t)+b3ω(0)(t) (39)
the method is a fourth-order differential equation driven by unit white noise, namely an equivalent model of the obtained random error of the fiber-optic gyroscope.
Therefore, the method analyzes the random error component of the fiber-optic gyroscope by adopting an Allan variance analysis technology, and deduces an equivalent model suitable for the fiber-optic gyroscope by utilizing the random error component and relevant parameters on the basis.
In order to verify the accuracy of the equivalent model proposed by the evaluation method, an autocorrelation function is introduced, and accuracy test is realized by comparing the autocorrelation function of the equivalent model with that of the test data set. One of the reasons for choosing an autocorrelation function to evaluate the accuracy of the model is that the autocorrelation function includes a random error contribution, whereas the Power Spectral Density (PSD) is a fourier transform of the autocorrelation function, which directly reflects the random error contribution. Another reason is that the autocorrelation function can be easily obtained from both model and experimental data, facilitating evaluation. The equivalent expression of the random error of the fiber optic gyroscope derived according to the previous method is as follows:
Figure BDA0002412433790000141
and comparing the autocorrelation function obtained by the model with the autocorrelation function obtained by calculating the actual measurement data, and collecting the test data on a fixed base for about 12 hours. As can be seen from fig. 3, the autocorrelation function obtained by the equivalent model representation is very similar to the autocorrelation function obtained by processing the real data. In the case that the correlation time is less than 30 minutes, the autocorrelation functions obtained by the two methods are substantially the same, and the autocorrelation function obtained by the equivalent expression slightly deviates from the autocorrelation function of the real data with the increase of time, and the equation (10) is explained. These numerical results verify that the equivalent representation in equation (39) is a good approximation to multiple random error combinations. Taking into account the differential equation form of (39), it is very convenient to incorporate it into INS (inertial navigation system) error dynamic calibration, initial alignment and integrated navigation applications.
In summary, the invention has the following advantages:
(1) the invention deduces a differential equation expression of the random error from the Power Spectral Density (PSD) of the random error, and provides a method for calculating the differential equation of the random error by using a rational approximation method aiming at the noise with unreasonable spectrum, and provides an error evaluation method for evaluating the error introduced by approximation;
(2) the generalized stationary process (WSS) of random errors is proved to be equivalently expressed by a differential equation through an equivalence theorem;
(3) based on the differential equation description and the equivalence theorem of random errors in the fiber optic gyroscope, the equivalent differential equation representation (equivalent model) of multiple random errors in the fiber optic gyroscope is established, the random errors in the fiber optic gyroscope are well compensated, and the method is superior to the Gauss-Markov method in the aspects of initial alignment precision and convergence.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.

Claims (8)

1. A unified modeling method for random errors of a fiber-optic gyroscope is characterized by comprising the following steps:
(1) analyzing the random error component of the fiber-optic gyroscope by adopting Allan variance, determining the random error type existing in the fiber-optic gyroscope to be analyzed, PSD corresponding to each random error type and the value of the noise parameter of each random error type, and deriving a differential equation of the random error based on the PSD of each random error;
(2) the obtained differential equation is equivalent to a fiber optic gyroscope equivalent model established by a random process equivalent theorem driven by single white noise according to a plurality of random processes driven by independent white noise, wherein one random process is described by one differential equation driven by unit white noise.
2. The method of claim 1, wherein the random error types include white noise, quantization noise, angular random walk noise, zero-bias instability noise, rate random walk noise, and rate ramp noise.
3. The method for unified modeling of fiber optic gyroscope random errors according to claim 2, characterized in that the Allan variance of the fiber optic gyroscope in step (1) is expressed by the following formula:
Figure FDA0002412433780000011
in the formula, σtotalDenotes the sum of Allan variances, σQRepresenting quantization noise, σNRepresenting angular random walk noise, σBTo representZero offset stability noise, σKRepresenting rate random walk noise, σRRepresenting the rate ramp noise, wherein the Allan variance value is approximately expressed by:
Figure FDA0002412433780000012
in the formula, σtotal(τ) denotes Allan variance root, AnRepresents the correlation coefficient, and τ represents different correlation times;
computing Allan variance roots of different correlation times through formulas (1) and (2), computing coefficients through a least square algorithm by utilizing Allan variance root values of different moments, and utilizing AnTo calculate the value of the noise parameter.
4. The unified modeling method for random error of fiber optic gyroscope according to claim 3, wherein the specific process of deriving the differential equation of the random error based on the PSD of the random error is as follows:
setting Sx(w) is the PSD of the corresponding random process x, and g (jw) is the transfer function of the corresponding filter, then:
Sx(w)＝|G(jw)|2(3)
assuming that x (jw) is the fourier transform of the filter input and y (jw) is the fourier transform of the filter output, the following relationship holds according to the linear system theory:
Y(jw)＝G(jw)X(jw) (4)
wherein X (jw) is unit 1;
and (4) carrying out inverse Fourier transform on two sides of the formula (4) to obtain a differential equation of the random error.
5. The method of claim 4, wherein the angular random walk noise has a rational spectrum, and the differential equation is obtained by the following process:
setting Srw(w) is the PSD of the angular random walk noise, then:
Srw(w)＝K2/w2(5)
wherein, K represents a noise parameter, and w represents a certain frequency;
setting Grw(j ω) is the transfer function of the corresponding filter, and Grw(j ω) is a rational function, then:
Grw(jω)＝K/jω (6)
wherein j ω represents a specific domain;
and (3) performing inverse Fourier transform on two sides of the above formula, and then expressing a differential equation of angle random walk as follows:
drw(t)＝Ku1(t) (7)
in the formula (d)rw(t) equation representing the random walk of angles, u1(t) is a unit of white noise.
6. The unified modeling method for random error of fiber optic gyroscope according to claim 5, wherein the zero-bias unstable noise has unreasonable spectrum, and the differential equation is obtained by the following process:
setting Sbi(w) is the PSD of zero-bias instability noise, then:
Sbi(w)＝B2/w (8)
wherein B represents a noise parameter;
setting Gbi(j ω) is the transfer function of the corresponding filter to zero-bias instability noise, and Gbi(j ω) is an irrational function, then:
Figure FDA0002412433780000021
to pair
Figure FDA0002412433780000022
And performing Taylor series expansion, and then approximating a transfer function as:
Figure FDA0002412433780000023
in the formula (I), the compound is shown in the specification,
Figure FDA0002412433780000024
an approximate transfer function representing zero-bias instability noise, β, depends on the expansion point of the taylor series;
the differential equation for performing inverse fourier transform on both sides of equation (10) to obtain zero-bias instability noise is described as follows:
Figure FDA0002412433780000036
in the formula (d)bi(t) an expression for zero-bias instability noise,
Figure FDA0002412433780000037
representing a differential expression representing zero-bias instability noise, u2(t) represents a unit of gaussian white noise.
7. The method of claim 6, wherein the rate ramp noise has unreasonable spectrum, and the differential equation is obtained by the following process:
setting Grn(j ω) is the transfer function of the corresponding filter to the rate ramp noise, and Grn(j ω) is an irrational function, then:
Grn(jω)＝R/(jω)1.5(13)
wherein R represents a noise parameter;
to three taylor series (jw)1.5And expanding to obtain an approximate transfer function as follows:
Figure FDA0002412433780000031
in the formula (I), the compound is shown in the specification,
Figure FDA0002412433780000032
an approximate transfer function, ω, representing the rate ramp noise0Represents a certain frequency;
inverse fourier transforms are performed on both sides of equation (13) to obtain a differential equation of the rate ramp noise, which is described as follows:
Figure FDA0002412433780000033
equation (15) represents a second order gaussian markov process.
8. The method of claim 7, wherein the equivalent model of the mixed random error at least comprising angular random walk noise, zero-bias instability noise and rate ramp noise is represented as follows:
Figure FDA0002412433780000034
in the formula, a1、a2、a3、a4、b0、b1、b2、b3Respectively representing the coefficients, ω (t) being a unit of white noise, ω(0)(t)、ω(1)(t)、ω(2)(t)、ω(3)(t) represents the different orders of the noise, z (t) is a variable, z(0)(t)、z(1)(t)、z(2)(t)、z(3)(t) are respectively different orders, wherein:
z(t)＝drw(t)+dbi(t)+drn(t) (33)
Figure FDA0002412433780000035
in the formula, D is a differential operator;
Figure FDA0002412433780000041
Figure FDA0002412433780000042
in the formula, R represents a noise parameter.
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