CN102938002B - Aircraft modeling method based on adjustable parameter maximum information criterion - Google Patents

Abstract

The invention discloses an aircraft modeling method based on an adjustable parameter maximum information criterion. The method is used for solving a technical problem that an aerodynamic model and parameters given by a flight test are poor in correction due to the existing maximum information criterion. The technical scheme of the invention is as follows: introducing an adjustable parameter through a maximum information criterion, and correcting a modeling criterion according to flight test factors of different aircrafts; and carrying out U-D factorization on measurement variance estimations Rj and R(j+1) to acquire a scalar model selection discriminant. The aircraft modeling method is convenient to directly establish aerodynamic and moment models according to the flight test data, thus avoiding a technical problem that the aerodynamic model and parameters acquired by the flight test are incorrect based on the maximum information criterion due to increased fixed parameters.

Description

Based on the Modeling of Vehicle method of adjustable parameter maximum fault information criterion

Technical field

The present invention relates to a kind of Modeling of Vehicle method, particularly relate to a kind of Modeling of Vehicle method based on adjustable parameter maximum fault information criterion.

Background technology

Not only can determine the control stability of aircraft according to aircraft aerodynamic model and parameter, also can be ground and aerial emulator provides correct mathematical model; The wind tunnel experiment of checking aircraft aerodynamic parameter and the calculated results; For the design of aircraft control system and improvement provide master data; The flight quality of qualification sizing aircraft; The flight quality of research high performance airplane; Carry out crash analysis of aviation accident etc.; Set up aircraft mathematical model problem exactly with completely different by the theoretical method of the modelling by mechanism such as philosophy, theorem, the input and output data Modling model of main experimentally gained, its main theoretical basis of the reform of Chinese economic structure is Nonlinear Systems Identification and nonlinear flight dynamics; When low incidence microvariations flight done by aircraft, flight force and moment mould can launch to get once item, i.e. Bryan model representation with Tailor progression.When Mach number, height one timing, this model is Linear Time Invariant model, and this model, because form simply and is always used till today, becomes the foundation stone of pneumatic mathematical model; Adopt this model, aerocraft system identification have estimated the systematic parameter of known mathematical model with regard to having become; Modern combat aircraft, tactical missile are making wartime requirement comparatively high maneuver, fault speed even tailspin, and its angle of attack from tens degree, tens degree until spend 100, can not adopt linear model; The body-shedding vortex that Aircraft at High Angle of Attack is formed, unsteady flo w downwash flow field caused by separated vorticcs, steady model can not be suitable for again.The research unsteady flo w of aircraft, nonlinear aerodynamic model under At High Angle of Attack become current aircraft development in the urgent need to problem.But, the identification complex of Nonlinear Aerodynamic, it is general Nonlinear Systems Identification problem, and the funtcional relationship between input quantity and state is difficult to determine, needs to carry out identification to model; The key of Model Distinguish is modeling criterion and optimization algorithm, and for given version, application modeling criterion is carried out the optimum exponent number of Confirming model and select optimization model from candidate model; Because measured data contains noise, modeling criterion only can not investigate the error of fitting size to available data, and considers other factors, otherwise model will be made incorrect; Usually, modeling criterion should be able to make the model optimized have following characteristics: 1. the model existing flying quality of matching well; 2. model is every obvious physical significance; 3. model can predict the measured data under conditions of similarity; 4. under the condition that performance is suitable, order is minimum; The most frequently used identification Method is method of gradual regression, and its principle is selected into by the predictor affecting conspicuousness item by item, and the factor little for impact is rejected, and sets up the method for regression equation; This method calculates simple, practical; But this method has two obvious shortcomings: one is that choice criteria is determined by people, and does not provide the confidence level of result; Two is that the accumulation of error is large, easily leaks choosing and falsely drops; For this reason, people are to requiring that higher dummy vehicle identification problem usually adopts maximum fault information criterion AIC method, but the method processing speed is slow, Model Distinguish low precision when signal to noise ratio (S/N ratio) is less; Due under nonlinear situation; numerical integration can only be carried out to aircraft nonlinear equation; carry out sensitivity matrix calculating and iterative computation; thus make the complexity of calculating and calculated amount more much higher than Linear Estimation; the matching simultaneously also making model export between experimental data becomes more difficult, the aerodynamic model that the maximum fault information criterion of preset parameter number increment usually can cause flight test to provide and parameter incorrect.

Summary of the invention

In order to overcome the deficiency of aerodynamic model that existing maximum fault information criterion causes flight test to provide and parameter correctness difference, the invention provides a kind of Modeling of Vehicle method based on adjustable parameter maximum fault information criterion.The method is revised maximum fault information criterion by introducing adjustable parameter, obtain new Model Distinguish criterion, index modeling is established by new criterion, directly may be used for the flight test modeling of aircraft, the technical matters set up according to flight test and checking aircraft at high angle of attack model exists can be avoided.

The technical solution adopted for the present invention to solve the technical problems is: a kind of Modeling of Vehicle method based on adjustable parameter maximum fault information criterion, is characterized in comprising the following steps:

The state equation of the aircraft candidate family that step one, flight test are to be determined is

$\stackrel{\·}{x}\left(t\right)=f\left\{f_0\right[x\left(t\right),{\mathrm{\Ω}}_0],f_1[x\left(t\right),{\mathrm{\θ}}_1],...,f_q[{\mathrm{\θ}}_q,x\left(t\right)],t\}---\left(1\right)$

Observation equation is

$\left\{\begin{array}{c}y\left(t\right)=g[x\left(t\right),\mathrm{\Ω},t]=g\left\{g_0\right[x\left(t\right),{\mathrm{\Ω}}_0],g_1[x\left(t\right),{\mathrm{\θ}}_1],...,g_q[{\mathrm{\θ}}_q,x\left(t\right)],t\}\\ z\left(t_k\right)=y\left(t_k\right)+v\left(k\right)\end{array}\right.---\left(2\right)$

(1), in (2) formula, x (t) is that n ties up state vector; Y (t) is m dimension observation vector;

F{f ₀[x (t), Ω ₀], f ₁[x (t), θ ₁] ..., f _q[θ _q, x (t)], t}, g{g ₀[x (t), Ω ₀], g _t[x (t), θ ₁] ..., g _q[θ _q, x (t)], t} is the known model structure function to be determined of expression formula, f ₀[x (t), Ω ₀], g ₀[x (t), Ω ₀] be the model that must be selected into according to physical concept, f _i[x (t), θ _i], g _i[x (t), θ _i] (i=1,2 ..., q) be candidate family, z (t _k) be at t _kmoment is to y (t _k) measured value; Ω is the parameter vector of unknown dimension, Ω ₀for the parameter vector of known dimension; V (k) is measurement noises, assuming that variance is R _kzero mean Gaussian white noise; f _i[x (t), θ _i], g _i[x (t), θ _i] (i=1,2 ..., q) whether to occur in a model and Ω ₀, θ _i(i=1,2 ..., value q) needs identification, and q is known candidate family number;

Usually higher to the model structure accuracy requirement of aircraft, the present invention provides following adjustable parameter maximum fault information criterion VAIC:

VAIC=-2lnL+2ap， (3)

In formula, the number of L to be maximum likelihood function: p be independent parameter in model, and a revises the adjustable parameter in maximum fault information criterion, according to different aircraft, surveying instrument, test condition, data length with actually to determine,

$\ln L=-\frac{1}{2}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{v}^T\left(k\right)R_k^{-1}v\left(k\right)-\frac{1}{2}N\ln \left(\frac{1}{N}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}\right|R_k\left|\right)+\mathrm{const}---\left(4\right)$

, const is constant, and N is data length, and ln is natural logarithm symbol;

In implementation process, desirable wherein: dem is the number of independent parameter in model, m is observation vector dimension, N is data length, and ln is natural logarithm symbol;

Step 2, according to supposition f ₀[x (t), Ω ₀], g ₀[x (t), Ω ₀], Ω ₀=Ω ₀be selected into model by optimization algorithm, and calculated by following algorithm iteration:

Make j=0,1,2 ..., q, assuming that f _j[x (t), θ _j], g _j[x (t), θ _j], Ω _jbe selected into model, select other candidate family in such a way:

Ask (4) formula maximum value, iterative computation:

$\mathrm{\Δ}{\mathrm{\Ω}}_j=A_j^{-1}{b}_j---\left(5\right)$

And

$R_j=\frac{1}{N}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{v}_j\left(k\right)v_j^T\left(k\right),$ v _j(k)=z(t _k)-g[x(t _k)，Ω _j，t _k] (6)

(5), in (6) formula: $\mathrm{\Δ}{\mathrm{\Ω}}_j={\mathrm{\Ω}}_j{\widehat{\mathrm{\Ω}}}_j,$ $b_j=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{\left(\frac{\∂y}{\∂{\mathrm{\Ω}}_j^T}\right)}^T{R}_j^{-1}[z\left(t_k\right)-y\left(t_k\right)],$

$A_j=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{\left(\frac{\∂y}{\∂{\mathrm{\Ω}}_j^T}\right)}^T{R}_j^{-1}\frac{\∂y}{\∂{\mathrm{\Ω}}_j^T}=B_j^T{P}_j^{-1}{B}_j,$ $B_j^T=[{\left(\frac{\∂y\left(t_1\right)}{\∂{\mathrm{\Ω}}_j^T}\right)}^T,{\left(\frac{\∂y\left(t_2\right)}{\∂{\mathrm{\Ω}}_j^T}\right)}^T,\·\·\·{,\left(\frac{\∂y\left(t_N\right)}{\∂{\mathrm{\Ω}}_j^T}\right)}^T]$

$P_j^{-1}=\mathrm{diag}\left[\begin{array}{cccc}{R}_j^{-1},& R_j^{-1},& ...& R_j^{-1}\end{array}\right],$

If ${\mathrm{\Ω}}_{j+1}=\left[\begin{array}{c}{\mathrm{\Ω}}_j\\ {\mathrm{\θ}}_{j+1}\end{array}\right],$ θ _j+1be selected into or kick-out condition is: when

$\left|R_j\right|/\left|R_{j+1}\right|>e^{\frac{2a}{N}}---\left(7\right)$

Time, θ _j+1, f _j+1[x (t), θ _j+1], g _j+1[x (t), θ _j+1] be selected into model, and ${\mathrm{\Ω}}_{j+1}=\left[\begin{array}{c}{\mathrm{\Ω}}_j\\ {\mathrm{\θ}}_{j+1}\end{array}\right];$ Otherwise reject f _j+1[x (t), θ _j+1], g _j+1[x (t), θ _j+1] candidate item, and Ω _j+1=Ω _j;

(7) in formula: $R_j=\frac{1}{N}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{v}_j\left(k\right)v_j^T\left(k\right),$ $R_{j+1}=\frac{1}{N}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{v}_{j+1}\left(k\right)v_{j+1}^T\left(k\right),$

v _j(k)=z(t _k)-g[x(t _k)，Ω _j，t _k]，v _j+1(k)=z(t _k)-g[x(t _k)，Ω _j+1，t _k]；

Step 3, because the dimension m of aircraft measurement vector y is comparatively large, adopt Gram-Schmidt orthogonalization method to R _jand R _j+1carry out U-D decomposition, R _jand R _j+1u-D be decomposed into:

$R_j=U_{\mathrm{Rj}}{D}_{\mathrm{Rj}}{U}_{\mathrm{Rj}}^T,$ $R_{j+1}=U_{R(j+1)}{D}_{R(j+1)}{U}_{R(j+1)}^T,$

In formula, U _rj, U _{r (j+1)}for unit upper triangular matrix; D _rj=diag [d _rj(1), d _rj(2) ..., d _rj(m)], D _{r (j+1)}=diag [d _{r (j+1)}(1), d _{r (j+1)}(2) ..., d _{r (j+1)}(m)]; Diag is diagonal angle symbol;

Adjustable parameter maximum fault information criterion is write as: when

$\underset{i=1}{\overset{m}{\mathrm{\Π}}}\left[\frac{d_{R\left(j\right)}\left(i\right)}{d_{R(j+1)}\left(i\right)}\right]>e^{\frac{2a}{N}}.---\left(8\right)$

During establishment, θ _j+1, f _j+1[x (t), θ _j+1], g _j+1[x (t), θ _j+1] be selected into model, and ${\mathrm{\Ω}}_{j+1}=\left[\begin{array}{c}{\mathrm{\Ω}}_j\\ {\mathrm{\θ}}_{j+1}\end{array}\right];$ Otherwise reject f _j+1[x (t), θ _j+1], g _j+1[x (t), θ _j+1] candidate item, and Ω _j+1=Ω _j.

The invention has the beneficial effects as follows: owing to passing through to introduce adjustable parameter in maximum fault information criterion, can according to the flight test factor correction modeling criterion of different aircraft; To measurement variance evaluation R _jand R _j+1u-D decompose, obtain Scalar Model and select discriminant, be convenient to directly set up flight vehicle aerodynamic power, moment model according to test flight data, avoiding maximum fault information criterion due to preset parameter number increment causes flight test to obtain aerodynamic model and the incorrect technical matters of parameter.

Below in conjunction with embodiment, the present invention is elaborated.

Embodiment

The Modeling of Vehicle method concrete steps that the present invention is based on adjustable parameter maximum fault information criterion are as follows:

1, many aircraft are commonly used candidate family form and are when the angle of attack is less than 60 degree:

$\stackrel{\·}{x}\left(t\right)=\mathrm{\Φ}\left({\mathrm{\Ω}}_0\right)f_0\left[x\left(t\right)\right]+{\mathrm{\θ}}_1{f}_1\left[x\left(t\right)\right]+...+{\mathrm{\θ}}_q{f}_q\left[x\left(t\right)\right]---\left(1\right)$

$\left\{\begin{array}{c}y\left(t\right)=g[x\left(t\right),\mathrm{\Ω}]=\mathrm{\Ψ}\left({\mathrm{\Ω}}_0\right)g_0\left[x\left(t\right)\right]+{\mathrm{\θ}}_1{g}_1\left[x\left(t\right)\right]+...+g_q[{\mathrm{\θ}}_q,x\left(t\right)]\\ z\left(t_k\right)=y\left(t_k\right)+v\left(k\right)\end{array}\right.---\left(2\right)$

(1), in (2) formula, x (t) is that n ties up state vector; Y (t) is m dimension observation vector; Φ (Ω ₀) f ₀[x (t)], Ψ (Ω ₀) g ₀[x (t)] model for being selected into according to physical concept, θ _if _i[x (t)], θ _ig _i[x (t)] (i=1,2 ..., q) be candidate family, z (t _k) be at t _kmoment is to y (t _k) measured value; Ω is the parameter vector of unknown dimension, Ω ₀for the parameter vector of known dimension; V (k) is measurement noises, assuming that variance is R _kzero mean Gaussian white noise; θ _if _i[x (t)], θ _ig _i[x (t)] (i=1,2 ..., q) whether to occur in a model and Ω ₀, θ _i(i=1,2 ..., value q) needs identification., q is known candidate family number;

Usually higher to the model structure accuracy requirement of aircraft, the present invention provides following adjustable parameter maximum fault information criterion VAIC:

VAIC=-2lnL+2ap， (3)

In formula, the number of L to be maximum likelihood function: p be independent parameter in model, and a revises the adjustable parameter in maximum fault information criterion is according to different aircraft, surveying instrument, test condition, data length with actually to determine, in implementation process, desirable dem is the number of independent parameter in model, and m is observation vector dimension, N is data length, and ln is natural logarithm symbol;

$\ln L=-\frac{1}{2}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{v}^T\left(k\right)R_k^{-1}v\left(k\right)-\frac{1}{2}N\ln \left(\frac{1}{N}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}\right|R_k\left|\right)+\mathrm{const}---\left(4\right)$

, const is constant;

2, according to supposition Φ (Ω ₀) f ₀[x (t)], Ψ (Ω ₀) g ₀[x (t)], Ω ₀=Ω ₀be selected into model by optimization algorithm, and calculated by following algorithm iteration:

Make j=0,1,2 ..., q, assuming that θ _jf _j[x (t)], θ _jg _j[x (t)], Ω _jbe selected into model, select other candidate family in such a way:

Ask (4) formula maximum value, iterative computation:

$\mathrm{\Δ}{\mathrm{\Ω}}_j=A_j^{-1}{b}_j---\left(5\right)$

And

$R_j=\frac{1}{N}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{v}_j\left(k\right)v_j^T\left(k\right),$ v _j(k)=z(t _k)-g[x(t _k)，Ω _j] (6)

(5), in (6) formula: $\mathrm{\Δ}{\mathrm{\Ω}}_j={\mathrm{\Ω}}_j{\widehat{\mathrm{\Ω}}}_j,$ $b_j=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{\left(\frac{\∂y}{\∂{\mathrm{\Ω}}_j^T}\right)}^T{R}_j^{-1}[z\left(t_k\right)-y\left(t_k\right)],$

$A_j=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{\left(\frac{\∂y}{\∂{\mathrm{\Ω}}_j^T}\right)}^T{R}_j^{-1}\frac{\∂y}{\∂{\mathrm{\Ω}}_j^T}=B_j^T{P}_j^{-1}{B}_j,$ $B_j^T=[{\left(\frac{\∂y\left(t_1\right)}{\∂{\mathrm{\Ω}}_j^T}\right)}^T,{\left(\frac{\∂y\left(t_2\right)}{{\∂\mathrm{\Ω}}_j^T}\right)}^T,...,{\left(\frac{\∂y\left(t_N\right)}{{\∂\mathrm{\Ω}}_j^T}\right)}^T]$

$P_j^{-1}=\mathrm{diag}\left[\begin{array}{cccc}{R}_j^{-1},& R_j^{-1},& ...& R_j^{-1}\end{array}\right],$

If ${\mathrm{\Ω}}_{j+1}=\left[\begin{array}{c}{\mathrm{\Ω}}_j\\ {\mathrm{\θ}}_{j+1}\end{array}\right];$ θ _j+1be selected into or kick-out condition is: when

$\left|R_j\right|/\left|R_{j+1}\right|>e^{\frac{2a}{N}}---\left(7\right)$

Time, θ _j+1f _j+1[x (t)], θ _j+1g _j+1[x (t)] is selected into model, and ${\mathrm{\Ω}}_{j+1}=\left[\begin{array}{c}{\mathrm{\Ω}}_j\\ {\mathrm{\θ}}_{j+1}\end{array}\right],$ Otherwise reject θ _j+1f _j+1[x (t)], θ _j+1g _j+1[x (t)] candidate item, and Ω _j+1=Ω _j;

(7) in formula: $R_j=\frac{1}{N}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{v}_j\left(k\right)v_j^T\left(k\right),$ $R_{j+1}=\frac{1}{N}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{v}_{j+1}\left(k\right)v_{j+1}^T\left(k\right),$

v _j(k)=z(t _k)-g[x(t _k)，Ω _j]，v _j+1(k)=z(t _k)-g[x(t _k)，Ω _j+1]；

3, the dimension m of aircraft measurement vector y is comparatively large usually, adopts Gram-Schmidt orthogonalization method to R _jand R _j+1carry out U-D decomposition, R _jand R _j+1u-D be decomposed into:

$R_j=U_{\mathrm{Rj}}{D}_{\mathrm{Rj}}{U}_{\mathrm{Rj}}^T,$ $R_{j+1}=U_{R(j+1)}{D}_{R(j+1)}{U}_{R(j+1)}^T,$

In formula, U _rj, U _{r (j+1)}for unit upper triangular matrix; D _rj=diag [d _rj(1), d _rj(2) ..., d _rj(m)], D _{r (j+1)}=diag [d _{r (j+1)}(1), d _{r (j+1)}(2) ..., d _{r (j+1)}(m)]; Diag is diagonal angle symbol;

Adjustable parameter maximum fault information criterion can be write as: when

$\underset{i=1}{\overset{m}{\mathrm{\Π}}}\left[\frac{d_{R\left(j\right)}\left(i\right)}{d_{R(j+1)}\left(i\right)}\right]>e^{\frac{2a}{N}}.---\left(8\right)$

During establishment, θ _j+1f _j+1[x (t)], θ _j+1g _j+1[x (t)] is selected into model, and ${\mathrm{\Ω}}_{j+1}=\left[\begin{array}{c}{\mathrm{\Ω}}_j\\ {\mathrm{\θ}}_{j+1}\end{array}\right],$ Otherwise reject θ _j+1f _j+1[x (t)], θ _j+1g _j+1[x (t)] candidate item, and Ω _j+1=Ω _j.

Claims (1)

1., based on a Modeling of Vehicle method for adjustable parameter maximum fault information criterion, it is characterized in that comprising the following steps:

The state equation of the aircraft candidate family that step one, flight test are to be determined is

$\stackrel{\·}{x}\left(t\right)=f\left\{f_0\right[x\left(t\right),{\mathrm{\Ω}}_0],f_1[x\left(t\right),{\mathrm{\θ}}_1],...,f_q[{\mathrm{\θ}}_q,x\left(t\right)],t\}---\left(1\right)$

Observation equation is

$\left\{\begin{array}{c}y\left(t\right)=g[x\left(t\right),\mathrm{\Ω},t]=g\left\{g_0\right[x\left(t\right),{\mathrm{\Ω}}_0],g_1[x\left(t\right),{\mathrm{\θ}}_1],...,g_q[,{\mathrm{\θ}}_q,x\left(t\right)],t\}\\ z\left(t_k\right)=y\left(t_k\right)+v\left(k\right)\end{array}\right.---\left(2\right)$

(1), in (2) formula, x (t) is that n ties up state vector; Y (t) is m dimension observation vector; f{f ₀[x (t), Ω ₀], f ₁[x (t), θ ₁] ..., f _q[θ _q, x (t)], t}, g{g ₀[x (t), Ω ₀], g ₁[x (t), θ ₁] ..., g _q[θ _q, x (t)], t} is the known model structure function to be determined of expression formula, f ₀[x (t), Ω ₀], g ₀[x (t), Ω ₀] be the model that must be selected into according to physical concept, f _i[x (t), θ _i], g _i[x (t), θ _i]; I=1,2 ..., q is candidate family, z (t _k) be at t _kmoment is to y (t _k) measured value; Ω is the parameter vector of unknown dimension, Ω ₀for the parameter vector of known dimension; V (k) is measurement noises, assuming that variance is R _kzero mean Gaussian white noise; f _i[x (t), θ _i], g _i[x (t), θ _i]; I=1,2 ..., whether q occurs and Ω in a model ₀, θ _i; I=1,2 ..., the value of q needs identification, and q is known candidate family number;

Provide following adjustable parameter maximum fault information criterion VAIC:

VAIC＝-2lnL+2ap， (3)

In formula, the number of L to be maximum likelihood function: p be independent parameter in model, and a revises the adjustable parameter in maximum fault information criterion, according to different aircraft, surveying instrument, test condition, data length with actually to determine,

$\ln L=-\frac{1}{2}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{v}^T\left(k\right)R_k^{-1}v\left(k\right)-\frac{1}{2}N\ln \left(\frac{1}{N}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}\right|R_k\left|\right)+\mathrm{const}---\left(4\right)$

, const is constant, and N is data length, and ln is natural logarithm symbol;

In implementation process, get

Step 2, according to supposition f ₀[x (t), Ω ₀], g ₀[x (t), Ω ₀], Ω ₀=Ω ₀be selected into model by optimization algorithm, and calculate model structure by following algorithm iteration:

Make j=0,1,2 ..., q, assuming that f _j[x (t), θ _j], g _j[x (t), θ _j], Ω _jbe selected into model, select other candidate family in such a way:

Ask (4) formula maximum value, iterative computation:

${\mathrm{\Δ\Ω}}_j=A_j^{-1}{b}_j---\left(5\right)$

And

$R_j=\frac{1}{N}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{v}_j\left(k\right)v_j^T\left(k\right),v_j\left(k\right)=z\left(t_k\right)-g[x\left(t_k\right),{\mathrm{\Ω}}_j,t_k]---\left(6\right)$

(5), in (6) formula: ${\mathrm{\Δ\Ω}}_j={\mathrm{\Ω}}_j-{\widehat{\mathrm{\Ω}}}_j,$ $b_j=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{\left(\frac{\∂y}{{\∂\mathrm{\Ω}}_j^T}\right)}^T{R}_j^{-1}[z\left(t_k\right)-y\left(t_k\right)],$

$A_j=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{\left(\frac{\∂y}{{\∂\mathrm{\Ω}}_j^T}\right)}^T{R}_j^{-1}\frac{\∂y}{{\∂\mathrm{\Ω}}_j^T}=B_j^T{P}_j^{-1}{B}_j,$ $B_j^T=[{\left(\frac{\∂y\left(t_1\right)}{{\∂\mathrm{\Ω}}_j^T}\right)}^T,{\left(\frac{\∂y\left(t_2\right)}{{\∂\mathrm{\Ω}}_j^T}\right)}^T,...,{\left(\frac{\∂y\left(t_N\right)}{{\∂\mathrm{\Ω}}_j^T}\right)}^T]$

$P_j^{-1}=\mathrm{diag}\left[\begin{array}{cccc}{R}_j^{-1},& R_j^{-1},& ...& R_j^{-1}\end{array}\right],$

If θ _j+1be selected into or kick-out condition is: when

$\left|R_j\right|/\left|R_{j+1}\right|>e^{\frac{2a}{N}}---\left(7\right)$

Time, θ _j+1, f _j+1[x (t), θ _j+1], g _j+1[x (t), θ _j+1] be selected into model, and ${\mathrm{\Ω}}_{j+1}=\left[\begin{array}{c}{\mathrm{\Ω}}_j\\ {\mathrm{\θ}}_{j+1}\end{array}\right];$ Otherwise reject f _j+1[x (t), θ _j+1], g _j+1[x (t), θ _j+1] candidate item, and Ω _j+1=Ω _j;

(7) in formula: $R_j=\frac{1}{N}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{v}_j\left(k\right)v_j^T\left(k\right),$ $R_{j+1}=\frac{1}{N}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{v}_{j+1}\left(k\right)v_{j+1}^T\left(k\right),$

v _j(k)＝z(t _k)-g[x(t _k),Ω _j,t _k]，v _j+1(k)＝z(t _k)-g[x(t _k),Ω _j+1,t _k]；

Step 3, because the dimension m of aircraft measurement vector y is comparatively large, adopt Gram-Schmidt orthogonalization method to R _jand R _j+1carry out U-D decomposition, R _jand R _j+1u-D be decomposed into:

$R_j=U_{\mathrm{Rj}}{D}_{\mathrm{Rj}}{U}_{\mathrm{Rj}}^T,$ $R_{j+1}=U_{R(j+1)}{D}_{R(j+1)}{U}_{R(j+1)}^T,$

In formula, U _rj, U _{r (j+1)}for unit upper triangular matrix; D _rj=diag [d _rj(1), d _rj(2) ..., d _rj(m)],

D _{r (j+1)}=diag [d _{r (j+1)}(1), d _{r (j+1)}(2) ..., d _{r (j+1)}(m)]; Diag is diagonal angle symbol;

Adjustable parameter maximum fault information criterion is write as: when

$\underset{i=1}{\overset{m}{\mathrm{\Π}}}[\frac{d_{R\left(j\right)}\left(i\right)}{d_{R(j+1)}\left(i\right)}>e^{\frac{2a}{N}}.---\left(8\right)$

During establishment, θ _j+1, f _j+1[x (t), θ _j+1], g _j+1[x (t), θ _j+1] be selected into model, and ${\mathrm{\Ω}}_{j+1}=\left[\begin{array}{c}{\mathrm{\Ω}}_j\\ {\mathrm{\θ}}_{j+1}\end{array}\right];$ Otherwise reject f _j+1[x (t), θ _j+1], g _j+1[x (t), θ _j+1] candidate item, and Ω _j+1=Ω _j.