CN1314945C - Aerial in-flight alignment method for SINS/GPS combined navigation system - Google Patents

Abstract

The present invention relates to an air mobile alignment method of SINS/GPS combination navigation systems. The present invention is characterized in that the effect error of a lever arm of GPS observed quantity is compensated, and the precision of the observed quantity is improved; simultaneously, a mobile strategy of navigation change is adopted; the observed change of each state variate of a system is analyzed by a method of singular value decomposition when a carrier is mobile. Feedback factors of the state variate of the system are regulated by an adaptive feedback strategy according to the observed change, and corrected in a feedback mode. The present invention has the advantages of high precision and less possibility of emission; the present invention can be used for the navigation precision of integrated navigation systems used for airplanes, missiles, ships or land vehicles.

Description

A kind of air mobile alignment methods of SINS/GPS integrated navigation system

Technical field

The present invention relates to a kind of air mobile alignment methods of SINS/GPS integrated navigation system, can be used for improving aircraft, guided missile, naval vessel or surface car navigation accuracy with the SINS/GPS integrated navigation system.

Background technology

Strapdown inertial navigation system (SINS) is a kind of autonomous navigational system fully, position, speed and attitude information can be provided continuously, in real time, precision is very high in short-term for it, and has good concealment, be not subjected to advantages such as weather condition restriction, thereby be widely used in fields such as Aeronautics and Astronautics, navigation.But the SINS error increases in time, and therefore normal and GPS Global Positioning System (GPS) constitutes the SINS/GPS integrated navigation system.In the SINS/GPS integrated navigation system, GPS can provide position and velocity information, but can not provide attitude information, because site error and velocity error have direct observability, utilize Kalman Filter Technology can reduce the site error and the velocity error of system effectively, still, because attitude angle is not directly observed, Kalman filtering is difficult to convergence, promptly is difficult to suppress the accumulation of SINS attitude error.

The air mobile alignment methods is utilized the observability degree of the motor-driven raising SINS/GPS of carrier integrated navigation system state variable, effective then estimating system attitude error and inertia device error, and revise to improve the precision of system.For the air mobile technique of alignment of SINS/GPS integrated navigation system, many researchists have carried out a large amount of theoretical researches, have obtained a lot of progress.But, in the flight test of reality, but often can not get the desired result of theoretical analysis, behind the air mobile aligning, the precision of SINS/GPS integrated navigation system does not improve, even aerial motor-driven aligning has caused dispersing of system in some test, this mainly is owing to two reasons: the one, and the gps antenna phase center does not overlap with the measuring center of SINS, causes the relative velocity error between the two when there is angular motion in carrier; The 2nd, in the air mobile alignment procedures, the observability degree of each state variable variation and inequality in the system, the observability degree of some state variable improves very little, if carry out unity feedback this moment, must cause that system accuracy descends, even disperse.

Summary of the invention

Technology of the present invention is dealt with problems and is: overcome the deficiencies in the prior art, a kind of air mobile alignment methods of the SINS/GPS integrated navigation system of analyzing based on observed quantity lever arm effect error compensation and system state variables observability degree is provided, and this method has improved the convergence of system and the precision that air mobile is aimed at.

Technical solution of the present invention is: a kind of air mobile alignment methods of SINS/GPS integrated navigation system, its characteristics are to comprise the following steps:

(1), sets up GPS observed quantity lever arm effect error model and compensate the precision of raising observed quantity according to measuring center and the relation of the space geometry between the gps antenna phase center of IMU;

(2) analyze the variation of carrier each state variable observability degree of system when motor-driven again by singular value decomposition method,, adopt the feedback factor of self-adaptation feedback policy regulating system state variable, and carry out feedback compensation according to the variation of observability degree.

Above-mentioned GPS observed quantity lever arm effect error model is:

$\mathrm{\Δ}{V}_n=C_b^n\mathrm{\Δ}{V}_b=C_b^n\·\left[\begin{array}{c}\mathrm{\Δ}{V}_{\mathrm{bx}}\\ \mathrm{\Δ}{V}_{\mathrm{by}}\\ \mathrm{\Δ}{V}_{\mathrm{bz}}\end{array}\right]=C_b^n\·\left[\begin{array}{c}{\mathrm{\ω}}_{\mathrm{nby}}^b{r}_z-{\mathrm{\ω}}_{\mathrm{nbz}}^b{r}_y\\ {\mathrm{\ω}}_{\mathrm{bnz}}^b{r}_x-{\mathrm{\ω}}_{\mathrm{nbx}}^b{r}_z\\ {\mathrm{\ω}}_{\mathrm{nbx}}^b{r}_y-{\mathrm{\ω}}_{\mathrm{nby}}^b{r}_x\end{array}\right]$

Wherein, Δ V _nBe the lever arm effect error of GPS observed quantity, Δ V _bBe the projection of lever arm effect error in carrier coordinate system of GPS observed quantity, C _b ⁿBe the transition matrix of carrier coordinate system to navigation coordinate system, ω _Nb ^bBe the angular velocity of carrier, r is the measuring center of IMU and the distance between the gps antenna phase center.

The above-mentioned concrete steps based on singular value decomposition method are:

(1) calculates band may observe matrix (SOM) Q by the permanent method of piecewise linearity _s(r),

$Q_s\left(r\right)=\left[\begin{array}{c}{Q}_1\\ Q_2\\ \·\·\·\\ Q_r\end{array}\right]$

Wherein the may observe defined matrix of corresponding each time period j is:

(2) to strip-type may observe matrix Q _s(r) battle array is carried out svd:

Q _s(r)＝U*S*V ^T

U=[u wherein ₁, u ₂, Λ, u _m], V=[v ₁, v ₂, Λ, v _m] all be orthogonal matrix.

$S=\left[\begin{array}{cc}{\mathrm{\Λ}}_{r\×r}& 0\\ 0& 0\end{array}\right]$

Λ=diag (σ wherein ₁, σ ₂, Λ, σ _r), σ ₁＞σ ₂＞Λ＞σ _r＞0 is called matrix Q _s(r) singular value.

(3) establishing original state is X (t ₀) (n dimension), measuring value is Z (a m1 dimension), then

$Z=Q_s\left(r\right)*X\left(t_0\right)=\left({\mathrm{USV}}^T\right)X\left(0\right)=\left(\underset{i=1}{\overset{r}{\mathrm{\Σ}}}{\mathrm{\σ}}_i{u}_i{v}_i^T\right)X\left(0\right)$

Promptly

$Z=\underset{i=1}{\overset{r}{\mathrm{\Σ}}}{\mathrm{\σ}}_i\left(v_i^{T}X\left(0\right)\right)*u_i$

Can get:

$X\left(0\right)=\underset{i=1}{\overset{r}{\mathrm{\Σ}}}\left(\frac{u_i^T*Z}{{\mathrm{\σ}}_i}\right)v_i$

(4) to matrix u _iv _i ^TAnalyze, observe the size of its each column element, judge each singular value σ _iCorresponding initial state vector X _{0, i}, singular value σ _iSize directly shown state vector X _{0, i}The height of may observe degree.

Above-mentioned self-adaptation feedback policy is

α (i) is state variable Δ X _iThe feedback weights, β (i) is state variable Δ X _iFeedback factor, wherein feed back weights

Δ X _Real(i) true error of expression SINS/GPS integrated navigation system,

When the expression carrier is motor-driven, under the situation of state variable X observability degree maximum, the optimum valuation of state variable; Feedback factor $\mathrm{\β}\left(i\right)=\frac{\mathrm{\σ}\left(i\right)}{{\mathrm{\σ}}_{\mathrm{opti}}\left(i\right)},$ The observability degree of σ (i) expression state variable Δ X (i) in a certain period, σ _Opti(i) maximal value of the observability degree of expression state variable Δ X (i).

Principle of the present invention is: the gps antenna phase center does not overlap with the SINS measuring center, there is one between the two apart from r (can be regarded as a lever arm), SINS has bigger angular velocity when carrier is motor-driven, cause the gps antenna phase center to produce relative velocity with respect to the SINS measuring center, if directly adopt this moment the GPS velocity information as observed quantity, must introduce the observed quantity error, the aerial effect of aiming at of influence, even can cause through the aerial error of back motion compensating system of aiming on the contrary greater than the error before the aerial aligning.Compensated after the GPS observed quantity lever arm effect error, improved the precision of observed quantity, also just improved the estimated accuracy of Kalman filtering, thereby improved the precision that air mobile is aimed at.

In addition, in the carrier mobile process, the observability degree of SINS/GPS integrated navigation system state variable is improved, but when adopting different motor-driven strategy, it is also inequality that the observability degree of each state variable improves situation, be not that whole state variables all is observable fully, should not carry out unity feedback for incomplete observable state variable, therefore, employing can obtain the raising situation of the observability degree of each state variable based on the observability degree analytical approach of svd, situation about changing then according to each state variable observability degree, regulate the feedback degree of each state variable, can improve the convergence and the precision of SINS/GPS integrated navigation system.

The present invention's advantage compared with prior art is: the present invention has compensated GPS observed quantity lever arm effect error, has improved the precision of observed quantity, has improved the estimated accuracy of SINS/GPS Kalman filter; The feedback policy of the automatic adjustment state variable of variation of system state variables observability degree when motor-driven according to carrier has improved the convergence of system, has improved the precision that air mobile is aimed at.

Description of drawings

Fig. 1 is a theory diagram of the present invention;

Fig. 2 is the process flow diagram that resolves of Kalman filtering rudimentary algorithm of the present invention;

Fig. 3 is a lever arm effect error synoptic diagram of the present invention.

Embodiment

As shown in Figure 1, 2, 3, concrete grammar of the present invention is as follows:

(1) foundation of the mathematical model of SINS/GPS integrated navigation system comprises system state equation and measurement equation, respectively suc as formula 1 and formula 2 shown in.

System state equation:

X&＝FX+GW (1)

Wherein, X is a system state vector, and W is the system noise vector, and F is system's transition matrix, and G is the noise transition matrix:

X＝[φ _x?φ _y?φ _z?δ _x?δ _y?δ _z?δL?δλ?δh?ε _x?ε _y?ε _z? _x? _y? _z] ^T

$F=\left[\begin{array}{cc}{F}_{\mathrm{INS}}& F_S\\ o_{6\×6}& F_M\end{array}\right],$ $F_S=\left[\begin{array}{cc}{C}_b^n& 0_{3\×3}\\ 0_{3\×3}& C_b^n\\ 0_{3\×3}& 0_{3\×3}\end{array}\right],$ FM＝[0 _6×15]， $G=\left[\begin{array}{cc}{C}_b^n& 0_{3\×3}\\ 0_{3\×3}& C_b^n\\ 0_{9\×3}& 0_{9\×3}\end{array}\right]$

The measurement equation of system

Z＝HX+η (2)

Wherein: Z is a measurement vector, and H is an observing matrix, and η is a measurement noise:

Z＝[δL?δλ?δh?δV _E?δV _N?δV _U] ^T

$H=\left[\begin{array}{ccc}{0}_{3\×6}& I_{3\×3}& 0_{3\×6}\\ 0_{3\×3}& I_{3\×3}& 0_{3\×9}\end{array}\right]$

$\mathrm{\η}={\left[{\mathrm{\η}}_L{\mathrm{\η}}_{\mathrm{\λ}}{\mathrm{\η}}_h{\mathrm{\η}}_{V_E}{\mathrm{\η}}_{V_N}{\mathrm{\η}}_{V_U}\right]}^T$

(2) Kalman filtering rudimentary algorithm layout, the process flow diagram of this algorithm as shown in Figure 2.

State one-step prediction equation

${\stackrel{\mathrm{\Λ}}{X}}_{k/k-1}={\mathrm{\φ}}_{k,k-1}{\stackrel{\mathrm{\Λ}}{X}}_{k-1}...\left(3\right)$

The State Estimation accounting equation

${\stackrel{\mathrm{\Λ}}{X}}_k={\stackrel{\mathrm{\Λ}}{X}}_{k/k-1}+K_k(Z_k-H_k{\stackrel{\mathrm{\Λ}}{X}}_{k/k-1})...\left(4\right)$

Filtering increment equation

${\stackrel{\mathrm{\Λ}}{K}}_k={\stackrel{\mathrm{\Λ}}{P}}_{k/k-1}{H}_k^T{(H_k{P}_{k/k-1}{H}_k^T+R_k)}^{-1}...\left(5\right)$

One-step prediction square error equation

${\stackrel{\mathrm{\Λ}}{P}}_{k/k-1}={\mathrm{\φ}}_{k,k-1}{P}_{k-1}{\mathrm{\φ}}_{k,k-1}^T+{\mathrm{\Γ}}_{k-1}{Q}_{k-1}{\mathrm{\Γ}}_{k-1}^T...\left(6\right)$

Estimate the square error equation

${\stackrel{\mathrm{\Λ}}{P}}_k=(I-K_k{H}_k)P_{k/k-1}{(I-K_k{H}_k)}^T+K_k{R}_k{K}_k^T...\left(7\right)$

(3) observed quantity lever arm effect compensation of error, lever arm effect error are meant that the gps antenna center does not overlap with the measuring center of SINS, have one between the two apart from OA (can be regarded as a lever arm), as shown in Figure 3.

During air maneuver, body movement angular velocity is ω _Nb ^b, Δ V represents the relative velocity that caused by the lever arm r between gps antenna phase center and the SINS measuring center, Δ V is projected as Δ V on the SINS body coordinate system _b:

${\mathrm{\ΔV}}_b=\left[\begin{array}{c}\mathrm{\Δ}{V}_{\mathrm{bx}}\\ \mathrm{\Δ}{V}_{\mathrm{by}}\\ \mathrm{\Δ}{b}_z\end{array}\right]=\left[\begin{array}{c}{\mathrm{\ω}}_{\mathrm{nby}}^b{r}_z-{\mathrm{\ω}}_{\mathrm{nbz}}^b{r}_y\\ {\mathrm{\ω}}_{\mathrm{nbz}}^b{r}_x-{\mathrm{\ω}}_{\mathrm{nbx}}^b{r}_z\\ {\mathrm{\ω}}_{\mathrm{nbx}}^b{r}_y-{\mathrm{\ω}}_{\mathrm{nby}}^b{r}_x\end{array}\right]...\left(8\right)$

With Δ V _bProject under the navigation coordinate system and be:

${\mathrm{\ΔV}}_n=C_b^n{\mathrm{\ΔV}}_b...\left(9\right)$

The observed quantity of motion compensating system is behind the compensation lever arm effect:

V _n＝B _gps-ΔV _n (10)

Wherein, V _GpsBe the GPS velocity, comprise east orientation speed, north orientation speed and sky to speed, Δ V _nBe the error of the GPS speed that causes by the carrier angular motion, V _nBe the GPS speed observed quantity after the compensation.

(4) based on the observability degree analytical approach of svd.

Piecewise linearity stational system (PWCS) Analysis on Observability method is based on the basis of the observability degree analytical approach of svd.The SINS/GPS integrated navigation system is a time-varying system, judges that the analytical approach of stational system observability is all inapplicable, and piecewise linearity stational system (PWCS) Analysis on Observability method is to be specifically designed to a kind of method of judging the time-varying system observability.In an enough little time interval, if the variation in coefficient matrix amount of linear time varying system can be ignored, in this time interval, just can be used as stational system to time-varying system so and handle, such system is called the sectional type stational system.

A discrete PWCS can use following model representation:

X(k+1)＝F _jX(k)+G _jU(k)+Г _jw(k) (11)

Z _j(k)＝H _jX(k)

X in the formula (k) ∈ R ⁿ, Fj ∈ R ^{N * n}, G _j∈ R ^{N * s}, U (k) ∈ R ^s, w (k) ∈ R ^l, Γ _j∈ R ^{N * l}, Z _j(k) ∈ R ^m, H _j∈ R ^{M * n}J=1,2, Λ Λ, r, the expression system segment is sequence number at interval.To each time period j, matrix F _j, G _jAnd H _jAll be constant, but the corresponding different time periods, each matrix can be different.Observability matrix (TOM) and striping observability matrix (SOM) that system is total are expressed as respectively:

$Q\left(r\right)=\left[\begin{array}{c}{Q}_1\\ Q_2{F}_1^{n-1}\\ \·\·\·\\ Q_r{F}_{r-1}^{n-1}{F}_{r-2}^{n-1}...F_1^{n-1}\end{array}\right]...\left(12\right)$

$Q_s\left(r\right)=\left[\begin{array}{c}{Q}_1\\ Q_2\\ \·\·\·\\ Q_r\end{array}\right]...\left(13\right)$

Wherein the may observe defined matrix of corresponding each time period j is:

According to the definition of system equation and measurement equation and above-mentioned may observe matrix, be output as by the system of first value representation:

Z=Q (r) * X (t ₀) (15) if the order of matrix Q (r) equals n, then as can be known by following formula, X (t ₀) well-determined separating arranged, show that system state is observable fully.Obviously, directly utilize the observability calculated amount of the discrete PWCS of Q (r) battle array research quite big, and adopt SOM to replace TOM to come the observability of analytic system, can make problem obtain simplifying.

Replace the TOM matrix with the SOM matrix, along with the increase of time period, the dimension of observability matrix is still very high, also is sizable to its workload of implementing svd.Therefore, adopt a kind of improved system state observability degree analytical approach here based on svd.

If the observability matrix of certain time period dynamic system is Q _{M * n}, original state is X (t ₀) (n dimension), measuring value is Z (a m1 dimension), then

Z＝Q*X(t ₀) (16)

The Q battle array is carried out svd:

Q＝U*S*V ^T (17)

U=[u wherein ₁, u ₂, Λ, u _m], V=[v ₁, v ₂, Λ, v _m] all be orthogonal matrix.

$S=\left[\begin{array}{cc}{\mathrm{\Λ}}_{r\×r}& 0\\ 0& 0\end{array}\right]...\left(18\right)$

Λ=diag (σ wherein ₁, σ ₂, Λ, σ _r), σ ₁＞σ ₂＞Λ＞σ _r＞0 is called the singular value of matrix Q.Formula (17) is brought in the formula (16):

$Z=\left({\mathrm{USV}}^T\right)X\left(0\right)=\left(\underset{i=1}{\overset{r}{\mathrm{\Σ}}}{\mathrm{\σ}}_i{u}_i{v}_i^T\right)X\left(0\right)...\left(19\right)$

Promptly

$Z=\underset{i=1}{\overset{r}{\mathrm{\Σ}}}{\mathrm{\σ}}_i\left(v_i^{T}X\left(0\right)\right)*u_i...\left(20\right)$

Can get according to formula (19):

$X\left(0\right)=\underset{i=1}{\overset{r}{\mathrm{\Σ}}}\left(\frac{u_i^T*Z}{{\mathrm{\σ}}_i}\right)v_i...\left(21\right)$

Traditional analytical approach is to calculate each singular value σ according to formula (21) _iCorresponding initial state vector X _{0, i}From numerical value, bigger singular value can obtain state estimation preferably, otherwise, for especially little singular value, may cause a plurality of X (t ₀) unusual, finally fall in the unobservable space.

From the angle analysis of lineary system theory, state variable X (t ₀) observability should only depend on system architecture, and irrelevant with observed quantity Z, according to formula (19) to matrix u _iv _i ^TAnalyze, observe the size of its each column element, just can judge each singular value σ _iCorresponding initial state vector X _{0, i}, the not only calculating of this improved observability degree analytical approach is simple, and the more important thing is can be at the observability degree that does not have to test analytic system state under the situation that records metric data.

(5) the self-adaptation feedback policy that changes based on the system state variables observability degree.

The air mobile alignment methods of traditional SINS/GPS integrated navigation system simply is divided into may observe and unobservable two classes to system state variables, implements unity feedback for observable state variable, and unobservable state variable is implemented not feed back.If represent feedback vector in the SINS/GPS integrated navigation system with Δ Y,

Be the valuation of the state variable of system, the i.e. valuation of the every error of system.In traditional air mobile alignment methods, the feedback policy of system can be expressed as so:

Wherein, α is the column vector of one 15 dimension, and if the weights of expression system state variables feedback are certain state variable Δ X in the system _iMay observe, then α is 1; If Δ X _iUnobservable, then α is 1.That is to say, if certain state variable may observe just feed back, if unobservablely just do not feed back.

Observability degree analysis by the SINS/GPS integrated navigation system is as can be known: the may observe degree that the may observe of state variable could quantitative, definite each state variable of expression, the observability degree of some state variable is little, but does not represent unobservable fully., and in the mobile process of carrier, the may observe degree of some state variables is also constantly changing.Therefore, should just adopt the self-adaptation feedback policy of analyzing based on the state variable observability degree, can be expressed as according to the feedback policy of the variation adjustment state variable of state variable observability degree:

Wherein, α (i) is state variable Δ X _iThe feedback weights, when optimum motor-driven strategy is adopted in expression, in whole mobile process, Δ X _iObservability degree when maximum, Δ X _iThe best feedback weights; β (i) is state variable Δ X _iFeedback factor, in the expression carrier mobile process, Δ X _iObservability degree and the ratio of maximum observability degree.

Feedback weights α (i) need determine by method of computer simulation.In Computer Simulation, data are produced by computer simulation, the actual position of carrier, speed, attitude and inertia device error all are known, result of calculation by SINS/GPS Kalman filter, can obtain having the navigation information of error and the valuation of every error, comparison by result of calculation and real information, can obtain the true error of system, then, error estimator and true error that the SINS/GPS Kalman filter is calculated compare, and then feeding back weights α (i) can be expressed as:

Wherein, Δ X _Real(i) true error of expression SINS/GPS integrated navigation system, When the expression carrier is motor-driven, under the situation of state variable X observability degree maximum, the optimum valuation of state variable.

The method following formula of determining of feedback factor β (i):

$\mathrm{\β}\left(i\right)=\frac{\mathrm{\σ}\left(i\right)}{{\mathrm{\σ}}_{\mathrm{opti}}\left(i\right)}...\left(25\right)$

Wherein, the observability degree of σ (i) expression state variable Δ X (i) in a certain period, σ _Opti(i) maximal value of the observability degree of expression state variable Δ X (i).

Claims (2)

1, a kind of air mobile alignment methods of SINS/GPS integrated navigation system is characterized in that comprising the following steps:

(1) at first according to measuring center and the relation of the space geometry between the gps antenna phase center of IMU, set up GPS observed quantity lever arm effect error compensation model, and compensate, the GPS observed quantity lever arm effect error model of being set up is:

${\mathrm{\ΔV}}_n=C_b^n{\mathrm{\ΔV}}_b=C_b^n\·\left[\begin{array}{c}{\mathrm{\ΔV}}_{\mathrm{bx}}\\ {\mathrm{\ΔV}}_{\mathrm{by}}\\ {\mathrm{\ΔV}}_{\mathrm{bz}}\end{array}\right]=C_b^n\·\left[\begin{array}{c}{\mathrm{\ω}}_{\mathrm{nby}}^b{r}_z-{\mathrm{\ω}}_{\mathrm{nbz}}^b{r}_y\\ {\mathrm{\ω}}_{\mathrm{nbz}}^b{r}_x-{\mathrm{\ω}}_{\mathrm{nbx}}^b{r}_z\\ {\mathrm{\ω}}_{\mathrm{nbx}}^b{r}_y-{\mathrm{\ω}}_{\mathrm{nby}}^b{r}_x\end{array}\right]$

Wherein, Δ V _nBe the lever arm effect error of GPS observed quantity, Δ V _bBe the projection of lever arm effect error in carrier coordinate system of GPS observed quantity, C _b ⁿBe the transition matrix of carrier coordinate system to navigation coordinate system, ω _Nb ^bBe the angular velocity of carrier, r is the measuring center of IMU and the distance between the gps antenna phase center;

(2) variation by observability degree analytical carrier each state variable observability degree of system when motor-driven;

(3) according to the variation of observability degree, adopt

The feedback factor of the self-adaptation feedback policy regulating system state variable of (i=1, Λ, 15), and carry out feedback compensation, wherein,

Be state variable Δ X _iThe feedback weights, Δ X _Real(i) true error of expression SINS/GPS integrated navigation system,

Represent under the situation of state variable X observability degree maximum when carrier is motor-driven the optimum valuation of state variable; $\mathrm{\β}\left(i\right)=\frac{\mathrm{\σ}\left(i\right)}{{\mathrm{\σ}}_{\mathrm{opti}}\left(i\right)},$ Be Δ X _iFeedback factor, the observability degree of σ (i) expression state variable Δ X (i) in a certain period, σ _Opti(i) maximal value of the observability degree of expression state variable Δ X (i).

2, the air mobile alignment methods of SINS/GPS integrated navigation system according to claim 1, it is characterized in that: described observability degree analytical approach is a kind of improved observability degree analytical approach based on svd, and the concrete steps of this method are:

(1) calculates band may observe matrix (SOM) Q by the permanent method of piecewise linearity _S(r),

$Q_S\left(r\right)=\left[\begin{array}{c}{Q}_1\\ Q_2\\ \·\·\·\\ Q_r\end{array}\right]$

Wherein the may observe defined matrix of corresponding each time period j is:

(2) to strip-type may observe matrix Q _S(r) battle array is carried out svd:

Q _S(r)＝U*S*V ^T

U=[u wherein ₁, u ₂, Λ, u _m], V=[v ₁, v ₂, Λ, v _m] all be orthogonal matrix,

$S=\left[\begin{array}{cc}{\mathrm{\Λ}}_{r\×r}& 0\\ 0& 0\end{array}\right]$

Λ=diag (σ wherein ₁, σ ₂, Λ, σ _r), σ ₁＞σ ₂＞Λ＞σ _r＞0 is called matrix Q _S(r) singular value;

(3) establishing original state is X (t ₀) (n dimension), measuring value is Z (a m1 dimension), then

$Z=Q_S\left(r\right)*X\left(t_0\right)=\left({\mathrm{USV}}^T\right)X\left(0\right)=\left(\underset{i=1}{\overset{r}{\mathrm{\Σ}}}{\mathrm{\σ}}_i{u}_i{v}_i^T\right)X\left(0\right)$

Promptly

$Z=\underset{i=1}{\overset{r}{\mathrm{\Σ}}}{\mathrm{\σ}}_i\left(v_i^{T}X\left(0\right)\right)*u_i$

Can get:

$X\left(0\right)=\underset{i=1}{\overset{r}{\mathrm{\Σ}}}\left(\frac{u_i^T*Z}{{\mathrm{\σ}}_i}\right)v_i$

(4) to matrix u _iv _i ^TAnalyze, observe the size of its each column element, judge each singular value σ _iCorresponding initial state vector X _{0, i}, singular value σ _iSize directly shown state vector X _{0, i}The height of may observe degree.

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