CN107270892A - A kind of anti-interference fault-tolerant Initial Alignment Method of inertial navigation system - Google Patents

Abstract

The present invention relates to a kind of anti-interference fault-tolerant Initial Alignment Method of inertial navigation system.Invention construction interference observer is estimated and offsets the drift of the inertia device in many Unknown worm inertial navigation systems or inertia combined navigation system, design ART network thinks highly of structure navigation system, construct the wave filter with robust dissipation and guaranteed cost performance indications, wherein energy BOUNDED DISTURBANCES in robust dissipation and guaranteed cost performance indications suppression system, and the upper bound of energy optimal estimating error variance；Based on interference observer, ART network device, robust dissipation and guaranteed cost filter, anti-interference Fault Tolerant Filtering device is constructed；The anti-interference Fault Tolerant Filtering device gain array of robust is solved based on convex optimized algorithm.Instant invention overcomes the not high defect of prior art filtering accuracy.The present invention has the advantages that strong interference immunity, filtering accuracy height, good reliability, available for inertial navigation system or inertia combined navigation system etc..

Description

Anti-interference fault-tolerant initial alignment method for inertial navigation system

Technical Field

The invention relates to an inertial navigation system or an inertial integrated navigation system, in particular to an anti-interference fault-tolerant initial alignment method for the inertial navigation system.

Background

Due to the structural characteristics of the inertial navigation system and the variability of the environment where the inertial navigation system is located, the INS has various unknown inputs such as modeling uncertainty, measurement noise, external environment interference, sensor errors, system faults and the like.

Before the present invention was made, the conventional approach was to assume the interference as gaussian noise and then use an optimal random filtering method. However, this type of method is very conservative when the system contains non-gaussian noise and the like. System parameter changes due to INS operating environment or temperature changes; aging of the integrated circuit system can cause deviation or failure of the inertial device; in a special case, the system noise variance may be abruptly changed. These unpredictable variations greatly affect the accuracy and reliability of the inertial navigation system and can even lead to failure of the navigation system. Since various unknown inputs are widely present in systems and metrology processes, suppression, compensation and cancellation of interference has become a hot topic of research in the control community in recent years. For various energy-bounded interference signals or random noise, LQG (linear quadratic gaussian, such as kalman filter) and H have been proposed₂Filtering, H_∞Filtering, etc., the influence of interference on the performance of the filter can be minimized. For linear controlled objects, the above methods have theoretically achieved relatively perfect solutions. But a single interference suppression method (e.g. H)_∞Filtering, H₂Filtering, LQG filtering, etc.) or conservation is large, high-precision control and filtering performance are difficult to achieve; or the requirements on the controlled object and the interference model are strict, and the wide application is difficult. For the nonlinear model, the processing method mainly comprises EKF, UKF, particle filter and H_∞Filtering, H₂And H_∞Filtering, filtering methods based on a disturbance observer, etc. The EKF mainly linearizes a nonlinear model and requires that noise meets the Gaussian white noise assumption, and the model linearization generates a large model error to influence the filtering precision. The UKF, while it can directly use the nonlinear model of the system without linearization, still requires that the noise be white Gaussian noise. The particle filtering can solve the filtering problem under the nonlinear and non-Gaussian conditions, has strong tolerance to bad points in an observed value, is easy to fall into particle exhaustion, and has large calculation amount in the recursion process and real performanceThe timeliness cannot be guaranteed. H_∞The filtering method can solve the problem of filtering of a nonlinear system with unknown noise statistical characteristics, has certain robust performance, and is low in precision due to single performance index. For the filtering method based on the disturbance observer, only a linear disturbance observer aiming at constant-value disturbance is considered in the early work, the disturbance always has different types of time-varying values in practice, and in some systems, various types of disturbance such as harmonic disturbance, constant-value disturbance and other external model description disturbance, change rate bounded disturbance, model-free disturbance and the like exist at the same time, so that the research difficulty is further increased. H₂And H_∞The filtering method can solve the filtering problem of a system simultaneously containing Gaussian white noise and model uncertainty, but interference suppression is adopted for interference with known characteristics, so that the filtering accuracy is not high.

Disclosure of Invention

The invention aims to overcome the defects and develop an anti-interference fault-tolerant initial alignment method for the inertial navigation system.

First, the disturbance observer estimation and cancellation multi-unknown input inertial navigation or inertial integrated navigation system is constructed ∑₁Inertial device drift, designing adaptive estimator to reconstruct navigation system, constructing filter with robust dissipation and cost performance index to suppress multiple unknown input inertial navigation or inertial integrated navigation system ∑₁Medium measurement noise and energy bounded interference; constructing a robust anti-interference fault-tolerant filter based on an interference observer, a self-adaptive estimator and a robust dissipation and cost-guaranteeing filter; finally, solving a robust anti-interference fault-tolerant filter gain array based on a convex optimization algorithm; the method comprises the following specific steps:

(1) the interference observer is constructed as follows:

wherein,in order to disturb the observer state variable,for the disturbance observer output variable, the gain array of the observer to be determined is K₂∈R^p×m，R^p×mRepresenting a p × m-dimensional real matrix space, wherein p and m are natural numbers;w ∈ R as robust anti-interference fault-tolerant filter output^p×pAndrespectively representing inertial device drift models ∑₂Of the system matrix and the output matrix q₁Is a natural number, y (t) ∈ R^mFor inertial navigation systems ∑₁The output variable of (1);

(2) the adaptive estimator is constructed as follows:

wherein,for adaptive estimator state variables, the gain array of the pending estimator is K₃；

(3) The robust dissipation and cost-preserving filter is constructed as follows:

wherein,to be robust H_∞And state variables of the cost filter, A ∈ R^n×nFor inertial navigation systems ∑₁System array of, K₁∈R^n×mGain array of undetermined filter;is a nonlinear term of the filter, and the gain array is F ∈ R^n×nAnd G ∈ R^m×n，For robust antijam fault-tolerant filter output, C ∈ R^m×nIs an output array of the inertial navigation system; is a calibration compensation term; b is₁，B₂，D₁，D₂Estimating gain arrays for the inertial device drift and the sensor, respectively; n and m are both natural numbers;

(4) constructing an anti-interference fault-tolerant filter based on an interference observer, a self-adaptive estimator and a robust dissipation and cost-guaranteed filter:

wherein,is a state variable of the antijam fault-tolerant filter,for the gain array of the anti-interference fault-tolerant filter to be determined,w and V respectively represent drift models of inertial devices∑₂The system matrix and the output matrix of (a),for the anti-jamming fault-tolerant filter output, A is the inertial navigation system ∑₁System array of R^{(n+2p)×(n+2p)}Denotes an (n +2p) × (n +2p) -dimensional real matrix space, B₁Is ∑₁Medium inertia device drift d₁(t)∈R^pGain array of (B)₂Is ∑₁Medium system fault w_f(t)∈R^pGain array of R^pRepresenting a p-dimensional real vector space, p being a natural number;D₁is ∑₁Inertial device drift d in intermediate measurement equation₁(t) gain array, D₂Is ∑₁System fault w in measurement equation_f(t) a gain array;

(5) solving for inertial navigation system ∑ based on convex optimization algorithm₁The anti-interference fault-tolerant filter obtains a gain array ofWherein P is₂、R₁The following convex optimization problem solved:

wherein, x (0), w_f(0) To a given initial value, C_d∈R^{(n+2p)×(n+2p)}、C_g∈R^{(n+2p)×(n+2p)}Closed loop system ∑ for estimation error₃α is an adjustable parameter, lambda₁、λ₂The value is [ 0.110 ] according to the nonlinear strength and weakness degree as the nonlinear weight parameter]Gamma is a undetermined interference suppression degree; s, Q, R is a given dissipation performance parameter matrix;

U₁、U₂are respectively ∑₁The Lipschitz parameter matrix of the nonlinear terms f (x (t)) and g (x (t)),in order for the gain array of the filter to be determined,respectively, system ∑₁Energy-bounded interference gain array for medium-state and output systems, B₄As an external interference model ∑₂Energy bounded interference gain array; symbols representing symmetrical blocks of the corresponding part of the symmetrical matrix, q₂Is a natural number.

Compared with the prior art, the invention has the advantages that:

(1) the robustness to interference is strong, under the condition that multiple unknown inputs such as energy bounded interference, external model description interference, sensor fault and the like exist at the same time, the interference observer in the constructed method counteracts the external model description interference, the robust dissipation and cost-saving filter suppresses and optimizes the energy bounded interference, and the constructed method has good interference cancellation and suppression capacity; overcomes the defects of EKF, UKF and H₂The filtering has higher requirements on the statistical characteristics of the noise; avoid H₂，H_∞And the filtering precision of the robust filtering method and the filtering method based on the observer is not high, so that the filtering precision of the system is improved.

(2) Tong (Chinese character of 'tong')The multi-objective design method based on convex optimization guarantees the mixed performance index requirements of the system, and the constructed method considers the interference cancellation, dissipation and cost performance index; overcomes the defects of EKF, UKF and H₂Filtering, H_∞The filtering method based on single performance indexes such as filtering and filtering based on the interference observer is insufficient, and the comprehensive performance of the system is improved; overcome H₂And H_∞The filtering method suppresses the external model description interference to cause accuracy reduction.

(3) The gain array of the composite layered anti-interference filter is obtained based on a convex optimization algorithm, and the defect of large operation amount in recursion processes of EKF, UKF, particle filtering and the like is overcome.

Drawings

FIG. 1 is a schematic flow chart of the present invention. The specific process is as follows: the method comprises the steps of starting, modeling a multi-unknown-input inertial navigation system, designing an interference observer, designing an adaptive estimator, designing a robust dissipation and cost-guaranteeing filter, designing an anti-interference fault-tolerant filter aiming at the multi-unknown-input inertial navigation system, and ending.

Detailed Description

The technical idea and principle of the invention are as follows:

H_∞filtering, H₂The single interference suppression methods such as filtering and dissipative filtering only aim at certain performance of the system, the conservatism is large, and high-precision filtering is difficult to realize; the filtering method based on the disturbance observer can counteract the disturbance with known characteristics, but when the system has more unknown input, the filtering precision is reduced. The invention aims at an inertial navigation system containing energy bounded interference, inertial device drift described by an external model and sensor faults; firstly, constructing a disturbance observer to estimate and counteract the drift of an inertial device in a system; second, construct filters with robust dissipation and guaranteed cost performance indicatorsThe robust dissipation performance index inhibits energy bounded interference in the system, and an upper bound of the variance of the cost performance index optimization estimation error is guaranteed; thirdly, constructing an adaptive estimator to estimate the sensor fault; then, constructing an anti-interference fault-tolerant filter with interference cancellation and suppression performance based on an interference observer, a robust dissipation and cost-guaranteed filter and a self-adaptive estimator; subtracting an inertial navigation system and an anti-interference fault-tolerant filter to obtain an estimated error closed-loop system, providing reference output of cost and dissipation performance indexes according to precision requirements, and converting the design problem of the composite layered anti-interference filter into a convex optimization problem based on a Linear Matrix Inequality (LMI) according to a robust control theory multi-objective optimization method; and finally, solving the convex optimization problem, and solving the gain array of the anti-interference fault-tolerant filter from the feasible solution of the convex optimization problem through corresponding algebraic transformation.

As shown in fig. 1, the specific implementation steps of the present invention are as follows: aiming at multiple unknown inputs of an inertial navigation system, establishing an inertial navigation system error model; designing a disturbance observer, a self-adaptive estimator and a robust dissipation and cost-guaranteeing filter; on the basis, an anti-interference fault-tolerant filter is designed for a multi-unknown-input inertial navigation system; and finally, realizing the anti-interference fault-tolerant initial alignment of the inertial navigation system. (the initial alignment of the platform inertial navigation system is taken as an example to illustrate the specific implementation of the method in the following):

1. establishing a platform inertial navigation system error state equation:

in order to improve the initial alignment precision, the small-angle assumption of the azimuth misalignment angle is cancelled, and then the attitude matrix from the navigation coordinate system to the platform coordinate system is navigatedComprises the following steps:

wherein,

(1) attitude error equation:

wherein, α_x、α_yAt a horizontal misalignment angle, α_zIs the azimuth misalignment angle;for two accelerometer dependent drifts (this embodiment takes the first order gaussian-markov process),for gyro-dependent drift (this embodiment takes a first order gaussian-markov process); omega is the rotational angular velocity of the earth; g is the local gravitational acceleration; r_M、R_NRespectively representing the curvature radius along the meridian circle and the prime circle; h is the local altitude; and L is the geographic latitude.

(2) With the outputs of the east and north accelerometers and east gyroscopes as measured values, the measurement equation of the system is:

wherein f is_E(t)、f_N(t)、w_E(t) outputs for east, north accelerometers and east gyroscopes, respectively.

(3) State space description of inertial navigation error equation of state:

and (3) connecting a speed error equation, an attitude error equation and an observation equation, and expressing the equations into a state space form:

wherein x is^T(t)＝[α_xα_yα_z]Is a transpose of the vector x (t), w_f(t) is a sensor failure of the system,is a vector d₁(t) the transposing of the first image, τ_i> 0(i ═ 1, …, 5) is the correlation time of the first order markov process.

2. For inertial device dependent drift d₁(t), constructing a disturbance observer:

wherein,for disturbance observer state variables, K₂In order to determine the gain array of the observer,is d₁(t), y (t) is a system measurement,outputting variables for the composite layered anti-interference filter;

3. aiming at the fault of the inertial navigation system, constructing an adaptive estimator as follows:

wherein,for adaptive estimator state variables, the gain array of the pending estimator is K₃；

4. Energy-bounded interference d in inertial navigation system₂(t) constructing a robust dissipation and cost-preserving filter as:

wherein,is a state variable of the filter and is,as the nonlinear term of the filter, K₁∈R^n×mGain array of undetermined filter;is a calibration compensation term; b is₁，B₂，D₁，D₂Estimating gain arrays for the inertial device drift and the sensor, respectively;

5. constructing an anti-interference fault-tolerant filter based on an interference observer, a self-adaptive estimator and a robust dissipation and cost-guaranteed filter:

wherein,is a state variable of the antijam fault-tolerant filter,is the output of the anti-interference fault-tolerant filter,in order for the gain array of the filter to be determined,w and V represent inertial device drift models ∑, respectively₂A is the inertial navigation system ∑₁System array of R^{(n +p)×(n+p)}Denotes an (n + p) × (n + p) -dimensional real matrix space, B₁Is ∑₁Medium inertia device driftThe gain array of (a) is set,denotes q₁Dimension real vector space, q₁Is a natural number;

6. solving a gain array of the composite layered anti-interference filter by using a convex optimization algorithm:

(1) the filtering estimation error closed-loop system obtained by subtracting the inertial navigation error state equation and the anti-interference fault-tolerant filter is as follows:

wherein,in order to filter the estimated error system state, z_d(t)、z_g(t) reference outputs for dissipation and cost performance, C_d、C_gAn output matrix is adjustable for dissipation and cost guarantee; solve to obtain a matrix P₂、R₁Then the filter gain array is

(2) Selecting a cost-guaranteeing and dissipation performance reference output matrix:

in order to restrain the influence of random noise on filtering precision, a dissipation performance reference matrix C of a filtering estimation error system_d∈R^10×10In this embodiment, take the value of I₁₀(ii) a To improve the filtering accuracy, the cost reference matrix C is guaranteed_g∈R^10×10In this embodiment, take the value of I₁₀。

(3) Tunable parameter α, nonlinear weight parameter λ₁、λ₂Selecting an interference suppression degree gamma:

alpha and r are respectively the circle center and radius of the pole point configuration of the region, alpha is more than 0, alpha is more than r, alpha is 3 in the embodiment, and r is 1; lambda is a nonlinear part adjusting parameter, the value of lambda is between [ 0.110 ], if the azimuth misalignment angle is more than 5 degrees, the nonlinearity is serious, and the lambda is close to 10; if the azimuth misalignment angle is less than 0.1 °, the nonlinearity is weak, the value of λ is close to 0.1, and the value of λ is 0.5 in this embodiment; γ is the suppression degree of interference, and the value is between [ 0.11 ], which can be determined according to the upper bound of energy bounded interference, and is 0.3 in this embodiment.

(4) The existence condition of the anti-interference fault-tolerant filter is as follows:

since the initial state x (0) is unknown but its covariance matrix is cov (x)₀)，w(0)＝0，w_f(0) Optimization of cost performance can translate into 0 ═ cAdjustable output matrix C_d，C_gAdjustable parameter α, nonlinear weight parameter λ₁、λ₂And the interference suppression degree gamma is obtained by solving the following convex optimization problem:

wherein,

U₁、U₂are respectively ∑₁The Lipschitz parameter matrix of the nonlinear terms f (x (t)) and g (x (t)),in order for the gain array of the filter to be determined,respectively, system ∑₁Energy-bounded interference gain array for medium-state and output systems, B₄As an external interference model ∑₂Energy bounded interference gain array; symbols representing symmetrical blocks of the corresponding part of the symmetrical matrix, P₁Is a positive definite LMI matrix variable; p₂Is LMI matrix variable; r₁Is an LMI matrix variable.

(5) Solving an anti-interference fault-tolerant filter gain array:

solving a convex optimization problem to obtain a matrix P₂、R₁Then the filter gain array is

Those skilled in the art will appreciate that the invention may be practiced without these specific details.

Claims (1)

1. An anti-interference fault-tolerant initial alignment method for an inertial navigation system is characterized by comprising the following steps:

first, the disturbance observer estimation and cancellation multi-unknown input inertial navigation or inertial integrated navigation system is constructed ∑₁Inertial device drift, designing adaptive estimator to reconstruct navigation system, constructing filter with robust dissipation and cost performance index to suppress multiple unknown input inertial navigation or inertial integrated navigation system ∑₁Medium measurement noise and energy bounded interference; self-adaptation based on interference observerThe method comprises the steps of estimating, constructing a robust anti-interference fault-tolerant filter by using a robust dissipation and cost-protection filter; finally, solving a robust anti-interference fault-tolerant filter gain array based on a convex optimization algorithm; the method comprises the following specific steps:

(1) the interference observer is constructed as follows:

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wherein,in order to disturb the observer state variable,for the disturbance observer output variable, the gain array of the observer to be determined is K₂∈R^p×m，R^p×mRepresenting a p × m-dimensional real matrix space, wherein p and m are natural numbers;w ∈ R as robust anti-interference fault-tolerant filter output^p×pAndrespectively representing inertial device drift models ∑₂Of the system matrix and the output matrix q₁Is a natural number, y (t) ∈ R^mFor inertial navigation systems ∑₁The output variable of (1);

(2) the adaptive estimator is constructed as follows:

<mrow> <msub> <mover> <mover> <mi>w</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>K</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>-</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow>

wherein，For adaptive estimator state variables, the gain array of the pending estimator is K₃；

(3) The robust dissipation and cost-preserving filter is constructed as follows:

<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mover> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>F</mi> <mi>f</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>K</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>-</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>u</mi> <mrow> <mi>c</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>C</mi> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>G</mi> <mi>g</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>u</mi> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>

wherein,to be robust H_∞And state variables of the cost filter, A ∈ R^n×nFor inertial navigation systems ∑₁System array of, K₁∈R^n×mGain array of undetermined filter;is a nonlinear term of the filter, and the gain array is F ∈ R^n×nAnd G ∈ R^m×n，For robust antijam fault-tolerant filter output, C ∈ R^m×nIs an output array of the inertial navigation system; is a calibration compensation term; b is₁，B₂，D₁，D₂Estimating gain arrays for the inertial device drift and the sensor, respectively; n and m are both natural numbers;

(4) constructing an anti-interference fault-tolerant filter based on an interference observer, a self-adaptive estimator and a robust dissipation and cost-guaranteed filter:

<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mover> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mover> <mi>w</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mover> <mi>w</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mover> <mi>A</mi> <mo>&amp;OverBar;</mo> </mover> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>w</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>w</mi> <mo>^</mo> </mover> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mi>F</mi> <mi>f</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mi>K</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>-</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mover> <mi>C</mi> <mo>&amp;OverBar;</mo> </mover> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>w</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>w</mi> <mo>^</mo> </mover> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mi>G</mi> <mi>g</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>1

wherein,is a state variable of the antijam fault-tolerant filter,for the gain array of the anti-interference fault-tolerant filter to be determined,w and V represent inertial device drift models ∑, respectively₂The system matrix and the output matrix of (a),for the anti-jamming fault-tolerant filter output, A is the inertial navigation system ∑₁System array of R^{(n+2p)×(n+2p)}Denotes an (n +2p) × (n +2p) -dimensional real matrix space, B₁Is ∑₁Medium inertia device drift d₁(t)∈R^pGain array of (B)₂Is ∑₁Medium system fault w_f(t)∈R^pGain array of R^pRepresenting a p-dimensional real vector space, p being a natural number;D₁is ∑₁Inertial device drift d in intermediate measurement equation₁(t) gain array, D₂Is ∑₁System fault w in measurement equation_f(t) a gain array;

(5) solving for inertial navigation system ∑ based on convex optimization algorithm₁The anti-interference fault-tolerant filter obtains a gain array ofWherein P is₂、R₁The following convex optimization problem solved:

<mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <mrow> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <msup> <mi>x</mi> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msup> <mi>w</mi> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>w</mi> <mi>f</mi> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <msub> <mi>P</mi> <mn>1</mn> </msub> <msup> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <msup> <mi>x</mi> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msup> <mi>w</mi> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>w</mi> <mi>f</mi> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> </mrow> <mo>)</mo> </mrow> </mrow>

<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>&amp;Phi;</mi> <mn>11</mn> </msub> </mtd> <mtd> <msub> <mi>&amp;Phi;</mi> <mn>12</mn> </msub> </mtd> <mtd> <mrow> <msub> <mi>P</mi> <mn>2</mn> </msub> <msub> <mi>F</mi> <mn>1</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>R</mi> <mn>1</mn> </msub> <mi>G</mi> </mrow> </mtd> <mtd> <msub> <mi>&amp;Phi;</mi> <mn>15</mn> </msub> </mtd> <mtd> <mrow> <msubsup> <mi>C</mi> <mi>d</mi> <mi>T</mi> </msubsup> <mi>M</mi> </mrow> </mtd> <mtd> <msubsup> <mi>C</mi> <mi>g</mi> <mi>T</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <msub> <mi>&amp;Phi;</mi> <mn>22</mn> </msub> </mtd> <mtd> <mrow> <msub> <mi>&amp;alpha;P</mi> <mn>2</mn> </msub> <msub> <mi>F</mi> <mn>1</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;alpha;R</mi> <mn>1</mn> </msub> <mi>G</mi> </mrow> </mtd> <mtd> <msub> <mi>&amp;Phi;</mi> <mn>25</mn> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;lambda;</mi> <mn>1</mn> </msub> <mi>I</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;lambda;</mi> <mn>2</mn> </msub> <mi>I</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mrow> <mi>&amp;gamma;</mi> <mi>I</mi> <mo>-</mo> <mi>R</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mrow> <mo>-</mo> <mi>I</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mrow> <mo>-</mo> <mi>I</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>&lt;</mo> <mn>0</mn> </mrow>

wherein, x (0), w_f(0) To a given initial value, C_d∈R^{(n+2p)×(n+2p)}、C_g∈R^{(n+2p)×(n+2p)}Closed loop system ∑ for estimation error₃α is an adjustable parameter, lambda₁、λ₂The value is [ 0.110 ] according to the nonlinear strength and weakness degree as the nonlinear weight parameter]Gamma is a undetermined interference suppression degree; s, Q, R is a given dissipation performance parameter matrix;

U₁、U₂are respectively ∑₁The Lipschitz parameter matrix of the nonlinear terms f (x (t)) and g (x (t)),in order for the gain array of the filter to be determined,respectively, system ∑₁Energy-bounded interference gain array for medium-state and output systems, B₄As an external interference model ∑₂Energy bounded interference gain array; symbols representing symmetrical blocks of the corresponding part of the symmetrical matrix, q₂Is a natural number.