CN110390070A - A pre-matrix-based method for iterative dynamic decoupling of multi-dimensional sensors - Google Patents

Abstract

The present invention is a kind of method of multidimensional sensor ofaiterative, dynamic decoupling based on pre- matrix, the ofaiterative, dynamic decoupling formula of sensor is constructed by introducing pre- matrix, reduce the spectral radius of Iterative Matrix, improve convergence, dynamic decoupling is carried out to the output of sensor with this, provides feasible improved method for the ofaiterative, dynamic decoupling of higher-dimension, close coupling sensor.Firstly, introducing the coupling output relation of pre- matrix and transformative transducer according to the output coupling transfer function matrix of sensor, the ofaiterative, dynamic decoupling formula based on pre- matrix is constructed；Secondly, corresponding pre- matrix is constructed for different ofaiterative, dynamic decoupling formulas, to reduce its Spectral Radii of Iterative Matrices；Then, the ofaiterative, dynamic decoupling formula for selecting the smallest iteration formula of Spectral Radii of Iterative Matrices final as sensor；Finally, carrying out dynamic decoupling according to the output coupling transfer function matrix of sensor to the reality output of sensor using the ofaiterative, dynamic decoupling formula of selection, reducing Dynamic Coupling error.

Description

A pre-matrix-based method for iterative dynamic decoupling of multi-dimensional sensors

technical field

The invention relates to a dynamic error correction technology of sensors, in particular to a dynamic decoupling technology for reducing the dynamic coupling error between dimensions of multi-dimensional sensors. Convergence performance of iterative dynamic decoupling of dimensional, strongly coupled sensors.

Background technique

Multi-dimensional sensors have multiple measurement input channels and multiple output channels, and there are often couplings between the measurement channels, including static coupling and dynamic coupling. Among them, dynamic coupling is an important source of sensor dynamic measurement error. Suppose the input vector of the n-dimensional sensor is U=[u ₁ , u ₂ , u ₃ ,...,u _n ] ^T , and the output vector of each measurement channel without coupling interference is X=[x ₁ ,x ₂ ,x ₃ ,... ,x _n ] ^T , the output vector after inter-dimensional cross-coupling interference is Y=[y ₁ , y ₂ , y ₃ ,...,y _n ] ^T , Y is the actual measurement output signal of the multi-dimensional sensor. Let the transfer function matrix between U and X be a diagonal matrix G, the transfer function matrix of the coupling channel between X and Y is H, and the matrices of G and H are as follows:

Among them, G _ii is the transfer function between the input ui of the _i -th channel of the multi-dimensional sensor and the uncoupled output xi of the _i -th channel, i=1, 2, 3, ..., n; H _ij is the i-th channel of the multi-dimensional sensor The coupling transfer function between the uncoupled output _xi and the coupled output generated for the jth channel, i,j=1,2,3,...,n,i≠j. Let I be the unit diagonal matrix, the input-output relationship of the multi-dimensional sensor is as follows:

The purpose of multi-dimensional sensor dynamic decoupling is to decouple the multi-dimensional sensor output vector Y through a certain decoupling algorithm to obtain the sensor output X without coupling interference, so as to remove its inter-dimensional dynamic coupling interference and improve its dynamic measurement accuracy. For the convenience of description, the multi-dimensional sensor is simply referred to as a sensor below. At present, the common sensor dynamic decoupling methods include invariant dynamic decoupling method (Xu Kejun, Yin Ming, Zhang Ying. A dynamic decoupling method for wrist force sensor. Journal of Applied Sciences, 1999), iterative dynamic decoupling method ( Xu Kejun, Li Cheng. Iterative dynamic decoupling method for multi-dimensional force sensors. China Mechanical Engineering, 1999) and diagonal dominance compensation decoupling method (Song Guomin, Zhang Weigong, Zhai Yujian. Sensor dynamic decoupling based on diagonal dominance compensation Research. Journal of Instrumentation, 2001) et al. These methods are based on the dynamic model of the sensor and use the decoupling algorithm to approximate the transfer function matrix of the sensor into a diagonal matrix or a diagonal dominant matrix to eliminate or weaken the coupling effect. Among them, the diagonal dominance compensation decoupling method belongs to the non-full decoupling method, which cannot completely eliminate the coupling error; the invariant decoupling method also belongs to the non-full decoupling method for systems with more than three dimensions, which has a principle error and is only suitable for In the case of weak coupling, the decoupling error is larger in the case of strong coupling; the iterative dynamic decoupling law can achieve full decoupling under the condition of satisfying iterative convergence, that is, in theory, the dynamic coupling error between sensor dimensions can be completely eliminated; but , the condition of iterative dynamic decoupling convergence is that the spectral radius of the iterative matrix in the entire measurement band of the sensor is less than 1, and the smaller the spectral radius, the faster the iteration convergence speed, and the iterative matrix depends on the sensor model, which makes the The iterative convergence of the method for high-dimensional or strongly coupled sensors is poor, and the existing iterative dynamic decoupling methods may not even converge, thus failing to achieve dynamic decoupling. We also proposed an iterative dynamic decoupling method for SOR based on frequency-dependent relaxation factors in the paper "Numerical Derivation-Based Serial Iterative Dynamic Decoupling-CompensationMethod for Multi-axis Force Sensors" (published in IEEE Transactions on Instrumentation and Measurement, 2014). , by constructing a frequency-dependent relaxation factor in the form of a second-order transfer function, the spectral radius of the iterative matrix in the entire measurement frequency band is less than 1 and as small as possible, thereby improving the convergence of iterative dynamic decoupling; however, this method is only based on An improvement of the SOR iterative dynamic decoupling method with constant relaxation factor, it can obtain better convergence performance than Jacobi, Gauss-Seidel, and the SOR iterative dynamic decoupling method based on constant relaxation factor under the premise of iterative convergence, but its The premise of iterative convergence is the convergence of the SOR iterative dynamic decoupling method based on a constant relaxation factor; this is often difficult to satisfy for high-dimensional and strongly coupled sensors, resulting in whether it is based on a constant relaxation factor or a frequency-dependent relaxation factor. The SOR iterative dynamic decoupling method fails to converge, and thus cannot reduce the inter-dimensional dynamic coupling error of the sensor.

The method of the invention proposes a new iterative dynamic decoupling method for sensors based on a pre-matrix, which further improves the convergence and divergence of the iterative dynamic decoupling of the sensors, so as to reduce the inter-dimensional dynamic coupling errors of the sensors.

SUMMARY OF THE INVENTION

The invention solves the problem that the existing sensor iterative dynamic decoupling method may still have iterative divergence when it is aimed at high-dimensional and strongly coupled sensors, so that the dynamic coupling error between sensor dimensions cannot be reduced, and provides a pre-matrix-based sensor iterative dynamic decoupling. method to improve the convergence of iterative dynamic decoupling and reduce the dynamic coupling error between sensor dimensions.

The technical scheme adopted by the present invention is as follows: firstly, according to the output coupling transfer function matrix H of the sensor, a pre-matrix P is introduced to transform the coupling output relationship of the sensor, and an iterative dynamic decoupling formula based on the pre-matrix P is constructed; Influence on the spectral radius of the iterative matrix Construct a transfer function matrix P in diagonal form; then, compare the magnitude of the iterative matrix spectral radius of different iterative dynamic decoupling equations based on the pre-matrix P, and select the iterative equation with the smallest spectral radius as the sensor's The final iterative dynamic decoupling formula; finally, according to the output coupling transfer function matrix H of the sensor, the selected iterative dynamic decoupling formula and the constructed pre-matrix P are used to iteratively decouple the output signal of the sensor, thereby reducing the dimension of the sensor. Dynamic coupling error.

The technical process of the present invention is: iterative dynamic decoupling formula construction 1 → construction of pre-matrix P 2 → iterative dynamic decoupling formula selection 3 → sensor iterative dynamic decoupling 4, as shown in FIG. 1 .

The iterative dynamic decoupling formula construction 1 is to introduce the pre-matrix P into the relationship between the sensor's uncoupling interference output X and the actual output Y, and transform the coupling output relationship of the sensor, so as to construct the Jacobi, The Gauss-Seidel and SOR iterative dynamic decoupling formulas are as follows:

①Introduce the pre-matrix P: Y=[(I+H)P]·P ^-1 X

② Order then get

③ Let H=-(L+U), where L and U are the lower triangular matrix and the upper triangular matrix, respectively, as follows

④Construct three iterative dynamic decoupling equations based on the pre-matrix P, Jacobi, Gauss-Seidel, and SOR as follows:

Jacobi iterative dynamic decoupling formula:

Gauss-Seidel iterative dynamic decoupling formula:

SOR iterative dynamic decoupling formula:

where k is the number of iterations, μ is a constant relaxation factor or the second-order transfer described in the literature "Numerical Derivation-BasedSerial Iterative Dynamic Decoupling-Compensation Method for Multi-axis ForceSensors" (published in IEEE Transactions on Instrumentation and Measurement, 2014) frequency-dependent relaxation factor in functional form; the iterative matrices D of the above three iterative dynamic decoupling equations are:

Jacobi iteration matrix: D=I-P+LP+UP

Gauss-Seidel iteration matrix: D=(I-LP) ^-1 (I-P+UP)

SOR iteration matrix: D=(I-μLP) ^-1 [(1-μ)I+μ(I-P+UP)]

To make the iterative dynamic decoupling converge, the spectral radius ρ(ω) of the iterative matrix D at different frequency points in the entire measurement frequency band of the sensor must be less than 1; where, the spectral radius ρ(ω) of the iterative matrix D at different frequency points is the maximum value of the eigenvalue of the amplitude matrix at the corresponding frequency point for its amplitude-frequency characteristic, that is, ρ(ω)=max(eig(|D(jω)|)); where, eig(|D(jω)|) is Find the eigenvalue of the matrix |D(jω)|, |D(jω)| is the amplitude matrix of the amplitude-frequency characteristic of the matrix D at the frequency ω, and max() means to find the maximum value.

The construction 2 of the pre-matrix P is the corresponding pre-construction of the three iterative dynamic decoupling formulas Jacobi, Gauss-Seidel and SOR constructed in the iterative dynamic decoupling formula construction 1 according to the coupling transfer function matrix H of the sensor. Matrix P, so that the spectral radius of the corresponding iterative matrix D in the entire measurement frequency band of the sensor is as small as possible. The pre-matrix P takes the form of a diagonal matrix with the same dimension as the matrix H, namely:

In the pre-matrix P, the element P _i = 1 in the i-th row on the diagonal is composed of one or more cascaded systems g _i,m (s) as shown in the following s-domain second-order model, i = 1, 2,3,…,n:

That is, P _i (s) = 1 or M is the number of cascaded second-order systems g _i,m (s).

Let ζ ₁ /ζ ₂ =λ, that is, we get

It can be seen from the above formula that when ω→0 or ω→+∞, |g _i,m (jω)|→1, and when ω→ _ωn , |g _i,m (jω)|→λ; then, When λ< ₁ , _{g i,m} (s) is the notch filter, the center frequency of the notch is ω _n , and the notch coefficient of the center frequency is λ. The notch frequency band is wider.

The construction of the pre-matrix P is to construct each element P _i on its diagonal, and determine whether it is 1 or the values of the parameters λ, ζ ₂ and ω _n of its basic composition g _i,m (s), so that P _i The amplitude-frequency characteristic of satisfies certain requirements so that the spectral radius of the iterative matrix D over the entire measurement band of the sensor is as small as possible.

For the three iterative dynamic decoupling equations based on the pre-matrix P constructed in the construction of the iterative dynamic decoupling equation 1, the construction methods of the pre-matrix P are the same, and the construction process is as follows:

① Initialize P=I, I is a unit diagonal matrix; set the preprocessing spectral radius threshold ρ _max , 0<ρ _max ＜1; initialize i=1;

② Calculate the spectral radius ρ(ω) of the iterative matrix D;

③ Determine the continuous frequency band of ρ(ω)>ρ _max The number M _i and The spectral radius within the peak frequency all continuous frequency bands is the preprocessing frequency band, m=1,2,...,M _i ;

④ Let P _i =0, calculate the spectral radius ρ ₀ (ω) of the iterative matrix D;

⑤Restore P _i =1, initialize m=1;

⑥If in the mth preprocessing frequency band If the inner ρ ₀ (ω) is not reduced compared to ρ(ω), then jump directly to the following step 8; otherwise, construct g _i,m (s) and obtain its parameters λ, ζ ₂ and ω _n :

First, directly let the natural frequency of g _i,m (s)

Second, calculate the frequency of all transfer functions in the i-th column of the sensor coupling matrix H the maximum gain at Find the optimal value λ ₀ with a step size Δλ in the interval [1/(10A), 1/A] such that the iterative matrix D calculated when λ ₀ ·P _i is substituted for P _i and substituted into the pre-matrix P Spectral radius exist The maximum value within is the smallest, then take the parameter λ=λ ₀ of g _i,m (s), namely:

Again, according to the aforementioned |g _i,m (jω)|, the formula for ζ ₂ is as follows:

Since g _i,m (s) is mainly used for preprocessing frequency bands The sensor coupling matrix H within is preprocessed, so let respectively and Substitute into the above formula, calculate the two solutions ζ _2,1 and ζ _2,2 of ζ ₂ ; take exist In the value interval of , take the value according to a certain step size Δζ and find the optimal value ζ ₂ , so that the parameters ω _n , λ, ζ ₂ are substituted into g _i,m (s), and g _i,m (s)·P _i The spectral radius of the iterative matrix D computed when replacing _Pi and substituting it into the pre-matrix P exist The maximum value within is the smallest, that is:

Finally, substitute ω _n , λ and ζ ₂ obtained above into the second-order model expression of gi, _m (s), that is, gi _,m (s).

⑦Let P _i (s)=gi _,m (s)·P _i (s);

⑧ Let m=m+1; if m≤M _i , return to step ⑥; otherwise, the construction of P _i (s) ends, and P _i (s) is substituted into the pre-matrix P;

⑨ Let i=i+1; if i≤n, return to step ②; otherwise, the pre-matrix P construction ends.

Using the above construction method of the pre-matrix P, the corresponding pre-matrix P can be constructed for the three iterative dynamic decoupling equations constructed in the construction of the iterative dynamic decoupling equation 1, and the corresponding Jacobi iterative dynamic solution based on the pre-matrix P can be obtained accordingly. Coupling formula, Gauss-Seidel iterative dynamic decoupling formula based on pre-matrix P and SOR iterative dynamic decoupling formula based on pre-matrix P.

The selection 3 of the iterative dynamic decoupling formula is to select the spectral radius ρ(ω) of the iterative matrix D from the three iterative dynamic decoupling formulae based on the pre-matrix P, Jacobi, Gauss-Seidel and SOR obtained above. The iterative formula with the smallest maximum value in the entire measurement frequency band is used as the final iterative dynamic decoupling formula for the sensor measurement output signal.

The dynamic decoupling process of the sensor iterative dynamic decoupling 4 is divided into two steps:

Step 1: According to the output coupling transfer function matrix H of the sensor, use the iterative dynamic decoupling formula selected in the iterative dynamic decoupling formula selection 3 and the corresponding pre-matrix P constructed to perform an iterative dynamic solution on the actual output Y of the sensor. coupled, get

Step 2: According to Calculate the uncoupled output X of each channel of the sensor, so as to realize the dynamic decoupling of the measurement output of the sensor.

The advantages of the invention are: pre-processing the output coupling matrix H of the sensor by introducing the pre-matrix P, and constructing three iterative dynamic decoupling equations based on the pre-matrix P, Jacobi, Gauss-Seidel and SOR, by transforming the coupling output relationship of the sensor, Construct different pre-matrices P according to different iterative dynamic decoupling equations, compare and select the iterative dynamic decoupling equation with the smallest iterative matrix spectral radius to dynamically decouple the sensor, which can effectively improve the convergence characteristics of the iterative dynamic decoupling of the sensor and improve the performance of the sensor. The dynamic decoupling accuracy of the sensor, and provides a feasible improvement method for the dynamic decoupling of high-dimensional, strongly coupled sensors.

Description of drawings

Fig. 1 is the technical flow chart of the inventive method;

2 is a schematic structural diagram of a multi-dimensional sensor output coupling model according to a specific embodiment of the present invention;

3 is a schematic diagram of the amplitude-frequency characteristics of the constituent links g _i,m (s) of the diagonal elements P _i of the pre-matrix P according to a specific embodiment of the present invention;

4 is a schematic flow chart of the construction of a pre-matrix P according to a specific embodiment of the present invention;

FIG. 5 is a schematic diagram of an iterative matrix spectral radius curve containing three preprocessing frequency bands according to a specific embodiment of the present invention.

Detailed ways

The present invention will be further described below in conjunction with the accompanying drawings:

The design idea of the present invention is: for the iterative divergence problem and the slow convergence problem that may exist when the iterative dynamic decoupling method that can realize full decoupling is used to dynamically decouple the multi-dimensional force sensor, Jacobi, Gauss-Seidel, SOR Based on three iterative calculation methods, from the perspective of reducing the spectral radius of the iterative matrix, a pre-matrix P is introduced to preprocess the output coupling matrix H of the sensor, and the coupling-output relationship of the sensor is transformed to construct an iterative dynamic decoupling based on the pre-matrix P. formula to reduce the iterative matrix spectral radius through the pre-matrix P, thereby improving the convergence characteristics of the sensor iterative dynamic decoupling; the pre-matrix P adopts a diagonal transfer function matrix to simplify the calculation process of the iterative dynamic decoupling; in the pre-matrix P The diagonal elements of P _i = 1 or one or more second-order systems are cascaded to facilitate their construction one by one, so that each frequency band in the iterative matrix spectral radius curve that does not meet the requirements can be targeted. Preprocessing is used to reduce the spectral radius of the iterative matrix and improve the iterative convergence characteristics. Finally, according to the three iterative calculation methods of Jacobi, Gauss-Seidel and SOR, the pre-matrix P is constructed respectively. According to the iterative matrix spectral radius after the introduction of the pre-matrix P The iterative dynamic decoupling method with the smallest spectral radius is selected as the final iterative dynamic decoupling method of the sensor to perform iterative dynamic decoupling of the actual output signal of the sensor and reduce its inter-dimensional dynamic coupling error.

The flow chart of the technical solution of the present invention is shown in FIG. 1 . First, three iterative dynamic decoupling equations, Jacobi, Gauss-Seidel, and SOR after the introduction of the pre-matrix P are constructed by constructing the iterative dynamic decoupling formula 1; secondly, the Jacobi, Gauss-Seidel, and SOR are constructed respectively by constructing the pre-matrix P 2 The pre-matrix P of the three iterative dynamic decoupling equations, respectively, obtains three iterative dynamic decoupling equations, Jacobi, Gauss-Seidel, and SOR based on the pre-matrix P; Among the three iterative dynamic decoupling equations, Jacobi, Gauss-Seidel, and SOR, an equation with the smallest iterative matrix spectral radius is selected as the final iterative dynamic decoupling equation for the sensor; The formula decouples the actual output signal of the sensor dynamically to reduce the dynamic coupling error between dimensions.

The iterative dynamic decoupling formula construction 1 is to introduce the pre-matrix P and transform the coupling output relationship of the sensor into the relationship between the uncoupling interference output X and the actual output Y of the sensor according to the output coupling model of the sensor, so as to construct Jacobi, Gauss-Seidel and SOR iterative decoupling algorithms based on pre-matrix P. Figure 2 shows the output coupling model structure of the sensor. It can be known that:

Accordingly, the process of constructing the iterative dynamic decoupling formula is as follows:

①Introduce the pre-matrix P: Y=[(I+H)P]·P ^-1 X

② Order then get

③ Let H=-(L+U), where L and U are the lower triangular matrix and the upper triangular matrix, respectively, as follows

④Construct three iterative dynamic decoupling equations based on the pre-matrix P, Jacobi, Gauss-Seidel, and SOR as follows:

Jacobi iterative dynamic decoupling formula:

Gauss-Seidel iterative dynamic decoupling formula:

SOR iterative dynamic decoupling formula:

where k is the number of iterations, μ is a constant relaxation factor or the second-order transfer described in the literature "Numerical Derivation-BasedSerial Iterative Dynamic Decoupling-Compensation Method for Multi-axis ForceSensors" (published in IEEE Transactions on Instrumentation and Measurement, 2014) frequency-dependent relaxation factor in functional form; the iterative matrices D of the above three iterative dynamic decoupling equations are:

Jacobi iteration matrix: D=I-P+LP+UP

Gauss-Seidel iteration matrix: D=(I-LP) ^-1 (I-P+UP)

SOR iteration matrix: D=(I-μLP) ^-1 [(1-μ)I+μ(I-P+UP)]

To make the iterative dynamic decoupling converge, the spectral radius ρ(ω) of the iterative matrix D at different frequency points in the entire measurement frequency band of the sensor must be less than 1; where, the spectral radius ρ(ω) of the iterative matrix D at different frequency points is the maximum value of the eigenvalue of the amplitude matrix at the corresponding frequency point for its amplitude-frequency characteristic, that is, ρ(ω)=max(eig(|D(jω)|)); where, eig(|D(jω)|) is Find the eigenvalue of the matrix |D(jω)|, |D(jω)| is the amplitude matrix of the amplitude-frequency characteristic of the matrix D at the frequency ω, and max() means to find the maximum value.

The construction 2 of the pre-matrix P is the corresponding pre-construction of the three iterative dynamic decoupling formulas Jacobi, Gauss-Seidel and SOR constructed in the iterative dynamic decoupling formula construction 1 according to the coupling transfer function matrix H of the sensor. matrix P, so that the spectral radius of the corresponding iterative matrix D in the entire measurement frequency band is as small as possible. The pre-matrix P takes the form of a diagonal matrix with the same dimension as the matrix H, namely:

In the pre-matrix P, the element P _i = 1 in the i-th row on the diagonal is composed of one or more cascaded systems g _i,m (s) as shown in the following s-domain second-order model, i = 1, 2,3,…,n:

That is, P _i (s) = 1 or M is the number of cascaded second-order systems g _i,m (s).

Let ζ ₁ /ζ ₂ =λ, that is, we get

It can be seen from the above formula that when ω→0 or ω→+∞, |g _i,m (jω)|→1, and when ω→ _ωn , |g _i,m (jω)|→λ; then, When λ< ₁ , _{g i,m} (s) is a notch filter, the center frequency of the notch is ω _n , and the notch coefficient of the center frequency is λ. The notch frequency band is wider. Figure 3 shows the amplitude-frequency characteristics of g _i,m (s) when ζ ₂ is ζ ₂₁ , ζ ₂₂ , and ζ ₂₃ respectively, where ζ ₂₁ <ζ ₂₂ <ζ ₂₃ .

The construction of the pre-matrix P is to construct each element P _i on its diagonal, and determine whether it is 1 or the values of the parameters λ, ζ ₂ and ω _n of its basic composition g _i,m (s), so that P _i The amplitude-frequency characteristic of satisfies certain requirements so that the spectral radius of the iterative matrix D over the entire measurement band of the sensor is as small as possible.

For the three iterative dynamic decoupling formulas based on the pre-matrix P constructed in the construction of the iterative dynamic decoupling formula 1, the construction methods of the pre-matrix P are the same, and the construction process is shown in Figure 4:

① Initialize P=I, I is a unit diagonal matrix; set the preprocessing spectral radius threshold ρ _max , 0<ρ _max ＜1; initialize i=1;

② Calculate the spectral radius ρ(ω) of the iterative matrix D;

③ Determine the continuous frequency band of ρ(ω)>ρ _max The number M _i and The spectral radius within the peak frequency all continuous frequency bands is the preprocessing frequency band, m=1,2,...,M _i ; as shown in Figure 5, it contains 3 preprocessing frequency bands The curve diagram of the iterative matrix spectral radius ρ(ω) of .

④ Let P _i =0, calculate the spectral radius ρ ₀ (ω) of the iterative matrix D;

⑤Restore P _i =1, initialize m=1;

⑥If in the mth preprocessing frequency band If the inner ρ ₀ (ω) is not reduced compared to ρ(ω), then jump directly to the following step 8; otherwise, construct g _i,m (s) and obtain its parameters λ, ζ ₂ and ω _n :

First, directly let the natural frequency of g _i,m (s)

Second, calculate the frequency of all transfer functions in the i-th column of the sensor coupling matrix H the maximum gain at Find the optimal value λ ₀ with a step size Δλ in the interval [1/(10A), 1/A] such that the iterative matrix D calculated when λ ₀ ·P _i is substituted for P _i and substituted into the pre-matrix P Spectral radius exist The maximum value within is the smallest, then take the parameter λ=λ ₀ of g _i,m (s), namely:

Again, according to the aforementioned |g _i,m (jω)|, the formula for ζ ₂ is as follows:

Since g _i,m (s) is mainly used for preprocessing frequency bands The sensor coupling matrix H within is preprocessed, so let respectively and Substitute into the above formula, calculate the two solutions ζ _2,1 and ζ _2,2 of ζ ₂ ; take exist In the value interval of , take the value according to a certain step size Δζ and find the optimal value ζ ₂ , so that the parameters ω _n , λ, ζ ₂ are substituted into g _i,m (s), and g _i,m (s)·P _i The spectral radius of the iterative matrix D computed when replacing _Pi and substituting it into the pre-matrix P exist The maximum value within is the smallest, that is:

Finally, substitute ω _n , λ and ζ ₂ obtained above into the second-order model expression of gi, _m (s), that is, gi _,m (s).

⑦Let P _i (s)=gi _,m (s)·P _i (s);

⑧ Let m=m+1; if m≤M _i , return to step ⑥; otherwise, the construction of P _i (s) ends, and P _i (s) is substituted into the pre-matrix P;

⑨ Let i=i+1; if i≤n, return to step ②; otherwise, the pre-matrix P construction ends.

Using the above construction method of the pre-matrix P, the corresponding pre-matrix P can be constructed for the three iterative dynamic decoupling equations constructed in the construction of the iterative dynamic decoupling equation 1, and the corresponding Jacobi iterative dynamic solution based on the pre-matrix P can be obtained accordingly. Coupling formula, Gauss-Seidel iterative dynamic decoupling formula based on pre-matrix P and SOR iterative dynamic decoupling formula based on pre-matrix P.

The selection 3 of the iterative dynamic decoupling formula is to select the spectral radius ρ(ω) of the iterative matrix D from the three iterative dynamic decoupling formulae based on the pre-matrix P, Jacobi, Gauss-Seidel and SOR obtained above. The iterative formula with the smallest maximum value in the entire measurement frequency band is used as the final iterative dynamic decoupling formula for the sensor measurement output signal.

The dynamic decoupling process of the sensor iterative dynamic decoupling 4 is divided into two steps:

Step 1: According to the output coupling transfer function matrix H of the sensor, use the iterative dynamic decoupling formula selected in the iterative dynamic decoupling formula selection 3 and the corresponding pre-matrix P constructed to perform an iterative dynamic solution on the actual output Y of the sensor. coupled, get

Step 2: According to Calculate the uncoupled output X of each channel of the sensor, so as to realize the dynamic decoupling of the measurement output of the sensor.

Claims (5)

1. A method for iterative dynamic decoupling of multi-dimensional sensors based on pre-matrix. The pre-matrix is introduced to preprocess the output coupling matrix of the sensor, the coupling-output relationship of the sensor is transformed, and an iterative dynamic decoupling formula based on the pre-matrix is constructed to improve the The convergence characteristics of the sensor iterative dynamic decoupling provide a feasible improvement method for the iterative dynamic decoupling of high-dimensional and strongly coupled sensors, so as to perform iterative dynamic decoupling of the actual measurement output signal of the sensor and reduce its inter-dimensional dynamic coupling error. The process includes: iterative dynamic decoupling algorithm construction, pre-matrix P construction, iterative dynamic decoupling algorithm selection, sensor iterative dynamic decoupling, which is characterized by: First, according to the output coupling transfer function matrix H of the sensor, a pre-matrix P is introduced to transform the coupling-output relationship of the sensor, and the Jacobi, Gauss-Seidel, and SOR iterative dynamic decoupling equations based on the pre-matrix P are constructed; The influence of spectral radius to construct a transfer function matrix P in diagonal form; then, compare the size of the iterative matrix spectral radius of the Jacobi, Gauss-Seidel, and SOR iterative dynamic decoupling equations based on the pre-matrix P, and select the iterative equation with the smallest spectral radius as the The final iterative dynamic decoupling formula of the sensor; finally, according to the output coupling transfer function matrix H of the sensor, the selected sensor iterative dynamic decoupling formula and the constructed pre-matrix P are used to iteratively decouple the actual measurement output of the sensor, reducing the sensor The interdimensional dynamic coupling error of . 2. A method for iterative dynamic decoupling of multi-dimensional sensors based on a pre-matrix as claimed in claim 1, characterized in that: the construction of the iterative dynamic decoupling formula is the uncoupling interference output X and the actual output Y of the sensor. The pre-matrix P and the coupling output relationship of the transformation sensor are introduced into the relationship between the two, so as to construct the Jacobi, Gauss-Seidel, and SOR iterative dynamic decoupling equation based on the pre-matrix P, as follows: ①Introduce the pre-matrix P: Y=[(I+H)P]·P ^-1 X ② Order then get ③ Let H=-(L+U), where L and U are the lower triangular matrix and the upper triangular matrix, respectively, as follows ④Construct three iterative dynamic decoupling equations based on the pre-matrix P, Jacobi, Gauss-Seidel, and SOR as follows: Jacobi iterative dynamic decoupling formula: Gauss-Seidel iterative dynamic decoupling formula: SOR iterative dynamic decoupling formula: where k is the number of iterations, μ is a constant relaxation factor or the second-order transfer described in the literature "Numerical Derivation-Based SerialIterative Dynamic Decoupling-Compensation Method for Multi-axis ForceSensors" (published in IEEE Transactions on Instrumentation and Measurement, 2014) frequency-dependent relaxation factor in functional form; the iterative matrices D of the above three iterative dynamic decoupling equations are: Jacobi iteration matrix: D=I-P+LP+UP Gauss-Seidel iteration matrix: D=(I-LP) ^-1 (I-P+UP) SOR iteration matrix: D=(I-μLP) ^-1 [(1-μ)I+μ(I-P+UP)] The spectral radius ρ(ω) of the iterative matrix D at different frequency points is the maximum value of the eigenvalues of the magnitude matrix of the amplitude-frequency characteristics at the corresponding frequency points, that is, ρ(ω)=max(eig(|D(jω)| )); where eig(|D(jω)|) is the eigenvalue of the matrix |D(jω)|, |D(jω)| is the amplitude matrix of the amplitude-frequency characteristic of the matrix D at the frequency ω, max() means to find the maximum value. 3. A method for iterative dynamic decoupling of multi-dimensional sensors based on a pre-matrix as claimed in claim 1, characterized in that: the structure of the pre-matrix P is the coupling transfer function matrix H of the sensor for the constructed pre-matrix The three iterative dynamic decoupling equations of Jacobi, Gauss-Seidel and SOR of matrix P construct the corresponding pre-matrix P, so that the spectral radius of the corresponding iterative matrix D in the entire measurement frequency band of the sensor is as small as possible; the pre-matrix P adopts the The diagonal matrix form with the same dimension of matrix H, namely: In the pre-matrix P, the element P _i = 1 in the i-th row on the diagonal is composed of one or more cascaded systems g _i,m (s) as shown in the following s-domain second-order model, i = 1, 2,3,…,n: That is, P _i (s) = 1 or M is the number of cascaded second-order systems g _i,m (s); Let ζ ₁ /ζ ₂ =λ, when λ<1, g _i,m (s) is the notch filter, the notch center frequency is ω _n , the center frequency notch coefficient is λ, the smaller ζ ₂ is, the more notch The narrower the frequency band, the larger the ζ ₂ and the wider the notch frequency band; The construction of the pre-matrix P is to construct each element P _i on its diagonal, and determine whether it is 1 or the values of the parameters λ, ζ ₂ and ω _n of its basic composition g _i,m (s), so that P _i The amplitude-frequency characteristic of satisfies specific requirements so that the spectral radius of the iterative matrix D over the entire measurement frequency band of the sensor is as small as possible; For the three iterative dynamic decoupling equations constructed based on the pre-matrix P, Jacobi, Gauss-Seidel and SOR, the construction method of the pre-matrix P is the same, and the construction process is as follows: ① Initialize P=I, I is a unit diagonal matrix; set the preprocessing spectral radius threshold ρ _max , 0<ρ _max ＜1; initialize i=1; ② Calculate the spectral radius ρ(ω) of the iterative matrix D; ③ Determine the continuous frequency band of ρ(ω)>ρ _max The number M _i and The spectral radius within the peak frequency all continuous frequency bands is the preprocessing frequency band, m=1,2,...,M _i ; ④ Let P _i =0, calculate the spectral radius ρ ₀ (ω) of the iterative matrix D; ⑤Restore P _i =1, initialize m=1; ⑥If in the mth preprocessing frequency band If the inner ρ ₀ (ω) is not reduced compared to ρ(ω), then jump directly to the following step 8; otherwise, construct g _i,m (s) and obtain its parameters λ, ζ ₂ and ω _n : First, directly let the natural frequency of g _i,m (s) Second, calculate the frequency of all transfer functions in the i-th column of the sensor coupling matrix H the maximum gain at Find the optimal value λ ₀ with a step size Δλ in the interval [1/(10A), 1/A] such that the iterative matrix D calculated when λ ₀ ·P _i is substituted for P _i and substituted into the pre-matrix P has Spectral radius exist The maximum value within is the smallest, then take the parameter λ=λ ₀ of g _i,m (s), namely: Again, the formula for ζ ₂ is as follows: Since g _i,m (s) is mainly used for preprocessing frequency bands The sensor coupling matrix H within is preprocessed, so let respectively and Substitute into the above formula, calculate the two solutions ζ _2,1 and ζ _2,2 of ζ ₂ ; take exist In the value interval of , take the value according to a certain step size Δζ and find the optimal value ζ ₂ , so that the parameters ω _n , λ, ζ ₂ are substituted into g _i,m (s), and g _i,m (s)·P _i The spectral radius of the iterative matrix D calculated when replacing _Pi and substituting it into the pre-matrix P exist The maximum value within is the smallest, that is: Finally, substitute ω _n , λ and ζ ₂ obtained above into the second-order model expression of g i, _m (s), that is, g _i,m (s); ⑦Let P _i (s)=gi _,m (s)·P _i (s); ⑧ Let m=m+1; if m≤M _i , return to step ⑥; otherwise, the construction of P _i (s) ends, and P _i (s) is substituted into the pre-matrix P; ⑨ Let i=i+1; if i≤n, return to step ②; otherwise, the pre-matrix P construction ends; By using the above construction method of the pre-matrix P, the corresponding pre-matrix P can be constructed for the three iterative dynamic decoupling equations constructed, and the corresponding Jacobi iterative dynamic decoupling equation based on the pre-matrix P and the Gaussian dynamic decoupling equation based on the pre-matrix P are obtained. -Seidel iterative dynamic decoupling formula and SOR iterative dynamic decoupling formula based on pre-matrix P. 4. The method for iterative dynamic decoupling of multi-dimensional sensors based on pre-matrix as claimed in claim 1, characterized in that: the iterative dynamic decoupling formula selection is obtained from Jacobi, Gauss-Seidel based on pre-matrix P obtained Among the three iterative dynamic decoupling equations of SOR and SOR, an iterative equation with the smallest maximum value of the spectral radius ρ(ω) of the iterative matrix D in the entire measurement frequency band of the sensor is selected as the final iterative dynamic decoupling equation of the sensor measurement output signal. 5. The method for iterative dynamic decoupling of multi-dimensional sensors based on pre-matrix as claimed in claim 1, wherein the process of iterative dynamic decoupling of sensors is divided into two steps: Step 1: According to the output coupling transfer function matrix H of the sensor, use the final iterative dynamic decoupling formula of the selected sensor and the corresponding pre-matrix P constructed to perform iterative dynamic decoupling on the actual output Y of the sensor, and obtain: Step 2: According to Calculate the uncoupled output X of each channel of the sensor, so as to realize the dynamic decoupling of the measurement output of the sensor.