CN107291986A - A kind of aircraft optimal sensor system of selection - Google Patents

A kind of aircraft optimal sensor system of selection Download PDF

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CN107291986A
CN107291986A CN201710370432.5A CN201710370432A CN107291986A CN 107291986 A CN107291986 A CN 107291986A CN 201710370432 A CN201710370432 A CN 201710370432A CN 107291986 A CN107291986 A CN 107291986A
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罗建军
靳锴
袁建平
马卫华
王明明
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Northwestern Polytechnical University
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Abstract

The invention discloses a kind of aircraft optimal sensor system of selection, first with the theoretical rapidity and the advantage of accuracy in track variance analysis of covariance, build covariance transmission and renewal equation, so as to give the relation between sensor parameters and final offset landings, and the sensor select permeability based on covariance is constructed using this relation.Then the sensor select permeability set up is converted into Second-order cone programming problem and solved, so as to be met the Optimum cost sensor of task accuracy requirement in the rapidity for solving optimization problem and the advantage of Global Optimality using Second-order cone programming.

Description

A kind of aircraft optimal sensor system of selection
【Technical field】
The present invention relates to a kind of aircraft optimal sensor system of selection.
【Background technology】
The optimal selection of sensor is the core technology for realizing high-performance Navigation, Guidance and Control.The performance of sensor is not The discrete distribution situation of navigation error and system time of day is only affected, so as to decide the completion feelings of every space mission Condition, equally also determine the cost of spacecraft, high performance sensor mean high-precision track with it is high into.How Enough select optimal sensor for spacecraft, can either ensure to complete every high-precision task, can effectively reduce again needed for into Originally it is the important technology in aerospace engineering.
In terms of sensor selection, scholar has carried out certain application study.Hovland have studied for robot system The real-time selection problem of variety classes sensor.Kincaid has carried out the research of sensor optimal location determination.Gupta is provided A kind of sensor selection strategy based on statistical theory, the engineering problem is solved from a new angle.Similarly for The selection of sensor also has certain methods research, and Welch solves the select permeability of sensor using branch and bound method.Yao is adopted Effective solution has been carried out to sensor selection with genetic algorithm.The method that Joshi then employs convex optimization first is passed Sensor is selected.All these applications have the emphasis of itself with method.In this patent, design employs a brand-new side Method, this method is based on linear covariance theory and Second-order cone programming method, realizes high-precision, quick, optimal sensor choosing Select.
Being applied to assess the analysis of covariance with analysis because the track that outside uncertain factor and disturbance are caused is discrete more Problem and the evaluated error problem caused due to modeling error.Maybeck gives general derivation, so as to establish true Real state is defenced jointly poor, filtering evaluated error and filter state covariance.Geller application covariance techniques devise a kind of new TRAJECTORY CONTROL and navigation solution method, this method can be applied to multiple systems.Christensen proposes a kind of for closing The linear Covariance Analysis Technique of loop system and nonlinear problem, and demonstrate its validity.
Recent years, convex optimization receives much concern in aerospace engineering application field.Second-order cone programming therein is most to send out Exhibition and the method for research potential value.Second-order cone programming because object function for it is linear, be constrained to the form of second order cone gain the name.It is convex The spacecraft of optimization is intersected, approached, the track optimizing of spacecraft, the power dropping section guidance of Mars landing.
【The content of the invention】
It is an object of the invention to the shortcoming for overcoming above-mentioned prior art, there is provided a kind of aircraft optimal sensor selecting party Method, sets up the optimization problem of the aircraft sensor selection based on linear covariance, then by convexification method, by optimization problem The form for meeting Second-order cone programming constraint is converted into, and is solved by Second-order cone programming, so as to realize quick accurate Sensor is selected.
To reach above-mentioned purpose, the present invention is achieved using following technical scheme：
A kind of aircraft optimal sensor system of selection, comprises the following steps：
1) the optimization problem covariance description of aircraft sensor selection；
2) matrix equation optimization problem vectorization；
3) by optimization problem convexification is Second-order cone programming form and solves；
4) based on the continuously close convex Optimization Solution of sequence.
Further improve of the invention is：
Step 1) aircraft sensor selection optimization problem covariance description specific method it is as follows：
Set up aircraft nonlinear model：
Wherein x (t) is flight state,For control instruction, w (t) is that dummy vehicle is uncertain, is met：
E[ω(t)ωT(τ)]=Sω(t)δ(t-τ) (2)
Set up the inertia measurement equation of aircraft：
WhereinFor continuous measurements, η is the measurement noise related to sensor；
Set up non-inertial measurement equation：
WhereinFor discrete measured values, νkFor the measurement noise related to sensor；
Transmission and the renewal equation of navigational state can be obtained after considering navigation algorithm：
With navigation error covariance equation：
Wherein
Control instruction using newest navigation information is：
Vehicle dynamics equation and navigation equation are linearized along standard trajectory：
Set up curved-edge polygons：
And transmission and the renewal equation for the state that is expanded：
Wherein
It is then able to obtain the transmission of covariance and with new equation：
The optimal sensor select permeability based on covariance can now be built：
Wherein z is sensor parameters, including Sη,Sω,Rν；K is sensor performance weighting function；For mission requirements.
Step 2) matrix equation optimization problem vectorization specific method it is as follows：
The optimization problem that P0 is provided constitutes for matrix equation, in order to preferably solve optimization problem, need to turn matrix equation Turn to vector equation；
Kronecker Product definition and property is provided first：
Define 1：
Define 2：
Vec (), Vec (A)=[a11 a21…am1 a12…am2…amn]T (16)
Property 1：
Property 2：
It is then able to matrix equation optimization problem P0 being converted into vector equation optimization problem：
Step 3) optimization problem convexification is as follows for the specific method of Second-order cone programming form and solution：
In order to which problem is converted into Second-order cone programming canonical form, z=[S are definedω Sη Rν] optimize sensors for needed for Energy variable, and new intermediate variable t is introduced, inequality constraints is met, now performance indications are converted into Second-order cone programming canonical form Formula：
minimize t (20)
Wherein k is to have the weighting function designed by designer, shows the attention journeys different to different sensors performance parameter Degree；Inequality constraints (21) is still unsatisfactory for the requirement of Second-order cone programming, need to be standardized；, can be by formula by mathematical theory (21) inequality constraints is converted to second order cone constraint：
The problem of can now optimal sensor being selected completely sets up and is converted into Second-order cone programming problem P1, and utilizes Interior point method is solved；
Step 4) specific method based on the continuously close convex Optimization Solution of sequence is as follows：
In optimization problem P1, Kalman filtering coefficientSensor parameters R comprising required optimizationν, problem need into Row continuously close processing；
In order to express conveniently, the optimization problem P1 that optimal sensor parameter is selected is written as form P2：
Then by the A in every suboptimization problemk、BkIt is considered as parameter related to optimized variable in constant value matrix, matrix by upper Optimized parameter obtained by once calculating is with new；When loop iteration calculating meets following index, it is believed that result restrains：
|zk+1-zk|≤ε (25)。
Compared with prior art, the invention has the advantages that：
The present invention is based on the theoretical aircraft optimal sensor system of selection solved with Second-order cone programming theory of covariance, profit With the theoretical rapidity and the advantage of accuracy in track variance analysis of covariance, construct sensor parameters and final drop point is inclined Relation between difference, establishes the sensor select permeability based on covariance, is then solving optimization using Second-order cone programming The rapidity of problem and the advantage of Global Optimality, Second-order cone programming problem is converted into simultaneously by the sensor select permeability set up Solved, so as to quickly accurately be met the Optimum cost sensor of task accuracy requirement.
【Brief description of the drawings】
Fig. 1 is optimal velocity measurement noise figure of the present invention；
Fig. 2 is height velocity's change curve of the present invention；
Fig. 3 is covariance model Predictor-corrector guidance method of the present invention and conventional method comparison diagram；
Fig. 4 is covariance model Predictor-corrector guidance method of the present invention and conventional method comparison diagram.
【Embodiment】
The present invention is described in further detail below in conjunction with the accompanying drawings：
Referring to Fig. 1-4, aircraft optimal sensor system of selection of the present invention comprises the following steps：
Step 1: the optimization problem covariance description of aircraft sensor selection
First, the aircraft nonlinear model of general type is set up：
Wherein x (t) is flight state,For control instruction, w (t) is that dummy vehicle is uncertain, is met
E[ω(t)ωT(τ)]=Sω(t)δ(t-τ) (2)
Set up the inertia measurement equation of aircraft
WhereinFor continuous measurements, η is the measurement noise related to sensor；
Set up non-inertial measurement equation
WhereinFor discrete measured values, νkFor the measurement noise related to sensor；
Transmission and the renewal equation of navigational state can be obtained after considering navigation algorithm,
With navigation error covariance equation
Wherein
Control instruction using newest navigation information is
Vehicle dynamics equation and navigation equation are linearized along standard trajectory
Set up curved-edge polygons
And transmission and the renewal equation for the state that is expanded
Wherein
Then the transmission of covariance can be obtained and with new equation
The optimal sensor select permeability based on covariance can now be built
Wherein z is sensor parameters, including Sη,Sω,Rν；K is sensor performance weighting function；For mission requirements；
Step 2: the vectorization of matrix equation optimization problem
The optimization problem that P0 is provided constitutes for matrix equation, in order to preferably solve optimization problem, need to turn matrix equation Turn to vector equation；
Kronecker Product definition and property is provided first；
Define 1：
Define 2：
Vec (), Vec (A)=[a11 a21…am1 a12…am2…amn]T (16)
Property 1：
Property 2：
Then matrix equation optimization problem P0 can be converted into vector equation optimization problem
Step 3: by optimization problem convexification is Second-order cone programming form and solves
In order to which problem is converted into Second-order cone programming canonical form, z=[S are definedω Sη Rν] optimize sensors for needed for Energy variable, and new intermediate variable t is introduced, inequality constraints is met, now performance indications are converted into Second-order cone programming canonical form Formula
minimize t (20)
Wherein k is to have the weighting function designed by designer, shows the attention journeys different to different sensors performance parameter Degree；Inequality constraints (21) is still unsatisfactory for the requirement of Second-order cone programming, need to be standardized；, can be by formula by mathematical theory (21) inequality constraints is converted to second order cone constraint
The problem of can now optimal sensor being selected completely sets up and is converted into Second-order cone programming problem P1, and in Point method is solved；
Step 4: based on the continuously close convex Optimization Solution of sequence；
In optimization problem P1, Kalman filtering coefficientSensor parameters R comprising required optimizationν, problem need into Row continuously close processing；
In order to express conveniently, the optimization problem P1 that optimal sensor parameter is selected is written as form P2
Then by the A in every suboptimization problemk、BkIt is considered as parameter related to optimized variable in constant value matrix, matrix by upper Optimized parameter obtained by once calculating is with new；When loop iteration calculating meets following index, it is believed that result restrains；
|zk+1-zk|≤ε (25)。
Fig. 1 to Fig. 3 sets forth optimal velocity measurement noise in simulation process, accelerometer biasing, acceleration analysis Noise.All parameters can restrain after 15 iteration, and convergence process is dull and smooth.Convergence process and result, Optimal Parameters Meet constraint requirements.
Fig. 4 gives the aberration curve of time of day medium velocity variable.The deviation initial value is zero, with the progress of task Gradually increase, maximum was reached at 53 seconds, be 0.876m/s.Then due to guidance and control algolithm, deviation with the time gradually Reduce, deviation is 0.0997m/s when reaching task terminal, meets mission requirements.
The principle of the present invention：
The present invention builds association side first with the theoretical rapidity and the advantage of accuracy in track variance analysis of covariance Difference transmission and renewal equation, so as to give the relation between sensor parameters and final offset landings, and apply this relation structure The sensor select permeability based on covariance is built.Then using Second-order cone programming solve the rapidity of optimization problem with it is complete The advantage of office's optimality, is converted into Second-order cone programming problem by the sensor select permeability set up and is solved, so that To the Optimum cost sensor for meeting task accuracy requirement.
Embodiment：
Using proposition based on the theoretical aircraft optimal sensor selecting party solved with Second-order cone programming theory of covariance Method, by taking the acceleration of one-dimensional rocket as an example, is emulated, verifies its feasibility, rapidity.Shown in model such as formula (26).
Wherein v is rocket speed, aactInstructed for acceleration, α is atmospheric drag coefficient, d is acceleration perturbation motion, and b is acceleration Degree biasing, s scale factors, τiFor time constant, ωiFor random perturbation.Uncertain factor is as shown in table 1 with disturbance parameter；Optimization Bound of parameter and task object, as shown in table 2.
1 uncertain factor of table/disturbance parameter
The Optimal Parameters border of table 2 and task object
The technological thought of above content only to illustrate the invention, it is impossible to which protection scope of the present invention is limited with this, it is every to press According to technological thought proposed by the present invention, any change done on the basis of technical scheme each falls within claims of the present invention Protection domain within.

Claims (5)

1. a kind of aircraft optimal sensor system of selection, it is characterised in that comprise the following steps：
1) the optimization problem covariance description of aircraft sensor selection；
2) matrix equation optimization problem vectorization；
3) by optimization problem convexification is Second-order cone programming form and solves；
4) based on the continuously close convex Optimization Solution of sequence.
2. aircraft optimal sensor system of selection according to claim 1, it is characterised in that step 1) aircraft sensing The specific method of the optimization problem covariance description of device selection is as follows：
Set up aircraft nonlinear model：
<mrow> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>,</mo> <mover> <mi>u</mi> <mo>^</mo> </mover> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>g</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mi>w</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
Wherein x (t) is flight state,For control instruction, w (t) is that dummy vehicle is uncertain, is met：
E[ω(t)ωT(τ)]=Sω(t)δ(t-τ) (2)
Set up the inertia measurement equation of aircraft：
<mrow> <mover> <mi>y</mi> <mo>~</mo> </mover> <mo>=</mo> <mi>c</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mover> <mi>u</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mo>+</mo> <mi>&amp;eta;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
WhereinFor continuous measurements, η is the measurement noise related to sensor；
Set up non-inertial measurement equation：
<mrow> <msub> <mover> <mi>z</mi> <mo>~</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>&amp;nu;</mi> <mi>k</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
WhereinFor discrete measured values, νkFor the measurement noise related to sensor；
Transmission and the renewal equation of navigational state can be obtained after considering navigation algorithm：
<mrow> <mtable> <mtr> <mtd> <mrow> <mover> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mover> <mi>f</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>,</mo> <mover> <mi>y</mi> <mo>~</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>+</mo> </msup> <mo>=</mo> <msup> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>-</mo> </msup> <mo>+</mo> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msub> <mover> <mi>z</mi> <mo>~</mo> </mover> <mi>k</mi> </msub> <mo>-</mo> <msub> <mover> <mover> <mi>z</mi> <mo>~</mo> </mover> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
With navigation error covariance equation：
<mrow> <mtable> <mtr> <mtd> <mrow> <mover> <mover> <mi>P</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mo>+</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>^</mo> </mover> </msub> <msub> <mover> <mi>C</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mo>)</mo> </mrow> <mover> <mi>P</mi> <mo>^</mo> </mover> <mo>+</mo> <mover> <mi>P</mi> <mo>^</mo> </mover> <msup> <mrow> <mo>(</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mo>+</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>^</mo> </mover> </msub> <msub> <mover> <mi>C</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>^</mo> </mover> </msub> <msub> <mover> <mi>S</mi> <mo>^</mo> </mover> <mi>&amp;eta;</mi> </msub> <msup> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>^</mo> </mover> </msub> <mi>T</mi> </msup> <mo>+</mo> <msub> <mover> <mi>S</mi> <mo>^</mo> </mover> <mi>&amp;omega;</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mover> <mi>P</mi> <mo>^</mo> </mover> <mi>k</mi> <mo>+</mo> </msubsup> <mo>=</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <msub> <mover> <mi>H</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>)</mo> </mrow> <msubsup> <mover> <mi>P</mi> <mo>^</mo> </mover> <mi>k</mi> <mo>-</mo> </msubsup> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <msub> <mover> <mi>H</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <msub> <mover> <mi>R</mi> <mo>^</mo> </mover> <mi>&amp;nu;</mi> </msub> <msubsup> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> <mi>T</mi> </msubsup> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
Wherein
Control instruction using newest navigation information is：
<mrow> <mover> <mi>u</mi> <mo>^</mo> </mover> <mo>=</mo> <mi>g</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
Vehicle dynamics equation and navigation equation are linearized along standard trajectory：
<mrow> <mi>&amp;delta;</mi> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>F</mi> <mi>x</mi> </msub> <mi>&amp;delta;</mi> <mi>x</mi> <mo>+</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>u</mi> <mo>^</mo> </mover> </msub> <msub> <mover> <mi>G</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mi>&amp;delta;</mi> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>+</mo> <mi>G</mi> <mi>w</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <mi>&amp;delta;</mi> <mover> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mo>+</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>~</mo> </mover> </msub> <msub> <mi>C</mi> <mover> <mi>u</mi> <mo>^</mo> </mover> </msub> <msub> <mover> <mi>G</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mo>)</mo> </mrow> <mi>&amp;delta;</mi> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>+</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>~</mo> </mover> </msub> <msub> <mi>C</mi> <mi>x</mi> </msub> <mi>&amp;delta;</mi> <mi>x</mi> <mo>+</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>~</mo> </mover> </msub> <mi>&amp;eta;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
Set up curved-edge polygons：
<mrow> <mi>X</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>&amp;delta;</mi> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> </mtd> </mtr> <mtr> <mtd> <mi>&amp;delta;</mi> <mover> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
And transmission and the renewal equation for the state that is expanded：
<mrow> <mtable> <mtr> <mtd> <mrow> <mover> <mi>X</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>F</mi> <mi>a</mi> </msub> <mi>X</mi> <mo>+</mo> <msub> <mi>G</mi> <mi>a</mi> </msub> <mi>&amp;eta;</mi> <mo>+</mo> <msub> <mi>W</mi> <mi>a</mi> </msub> <mi>w</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>X</mi> <mi>k</mi> <mo>+</mo> </msubsup> <mo>=</mo> <msub> <mi>A</mi> <mi>k</mi> </msub> <msubsup> <mi>X</mi> <mi>k</mi> <mo>-</mo> </msubsup> <mo>+</mo> <msub> <mi>B</mi> <mi>k</mi> </msub> <msub> <mi>&amp;nu;</mi> <mi>k</mi> </msub> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
Wherein
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>F</mi> <mi>a</mi> </msub> <mo>=</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <msub> <mi>F</mi> <mi>x</mi> </msub> </mtd> <mtd> <mrow> <msub> <mi>F</mi> <mover> <mi>u</mi> <mo>^</mo> </mover> </msub> <msub> <mover> <mi>G</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>~</mo> </mover> </msub> <msub> <mi>C</mi> <mi>x</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mo>+</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>~</mo> </mover> </msub> <msub> <mi>C</mi> <mover> <mi>u</mi> <mo>^</mo> </mover> </msub> <msub> <mover> <mi>G</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <msub> <mi>G</mi> <mi>a</mi> </msub> <mo>=</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <msub> <mn>0</mn> <mrow> <mi>n</mi> <mo>&amp;times;</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>~</mo> </mover> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <msub> <mi>W</mi> <mi>a</mi> </msub> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mn>0</mn> <mrow> <mover> <mi>n</mi> <mo>^</mo> </mover> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mi>w</mi> </msub> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>A</mi> <mi>k</mi> </msub> <mo>=</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <msub> <mi>I</mi> <mrow> <mi>n</mi> <mo>&amp;times;</mo> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <msub> <mn>0</mn> <mrow> <mi>n</mi> <mo>&amp;times;</mo> <mover> <mi>n</mi> <mo>^</mo> </mover> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <msub> <mi>H</mi> <mi>k</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>I</mi> <mrow> <mover> <mi>n</mi> <mo>^</mo> </mover> <mo>&amp;times;</mo> <mover> <mi>n</mi> <mo>^</mo> </mover> </mrow> </msub> <mo>-</mo> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <msub> <mover> <mi>H</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <msub> <mi>B</mi> <mi>k</mi> </msub> <mo>=</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <msub> <mn>0</mn> <mrow> <mi>n</mi> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mi>z</mi> </msub> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
It is then able to obtain the transmission of covariance and with new equation：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>C</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>a</mi> </msub> <mo>=</mo> <msub> <mi>F</mi> <mi>a</mi> </msub> <msub> <mi>C</mi> <mi>a</mi> </msub> <mo>+</mo> <msub> <mi>C</mi> <mi>a</mi> </msub> <msubsup> <mi>F</mi> <mi>a</mi> <mi>T</mi> </msubsup> <mo>+</mo> <msub> <mi>G</mi> <mi>a</mi> </msub> <msub> <mi>S</mi> <mi>&amp;eta;</mi> </msub> <msubsup> <mi>G</mi> <mi>a</mi> <mi>T</mi> </msubsup> <mo>+</mo> <msub> <mi>W</mi> <mi>a</mi> </msub> <msub> <mi>S</mi> <mi>&amp;omega;</mi> </msub> <msubsup> <mi>W</mi> <mi>a</mi> <mi>T</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mi>a</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mi>t</mi> <mi>k</mi> <mo>+</mo> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>A</mi> <mi>k</mi> </msub> <msub> <mi>C</mi> <mi>a</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mi>t</mi> <mi>k</mi> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <msubsup> <mi>A</mi> <mi>k</mi> <mi>T</mi> </msubsup> <mo>+</mo> <msub> <mi>B</mi> <mi>k</mi> </msub> <msub> <mi>R</mi> <mi>&amp;nu;</mi> </msub> <msubsup> <mi>B</mi> <mi>k</mi> <mi>T</mi> </msubsup> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>
The optimal sensor select permeability based on covariance can now be built：
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>P</mi> <mn>0</mn> <mo>:</mo> <mi>max</mi> <mi>i</mi> <mi>m</mi> <mi>i</mi> <mi>z</mi> <mi>e</mi> <mi> </mi> <mi>k</mi> <mi>z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mover> <mi>P</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mo>+</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>^</mo> </mover> </msub> <msub> <mover> <mi>C</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mo>)</mo> </mrow> <mover> <mi>P</mi> <mo>^</mo> </mover> <mo>+</mo> <mover> <mi>P</mi> <mo>^</mo> </mover> <msup> <mrow> <mo>(</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mo>+</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>^</mo> </mover> </msub> <msub> <mover> <mi>C</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>^</mo> </mover> </msub> <msub> <mover> <mi>S</mi> <mo>^</mo> </mover> <mi>&amp;eta;</mi> </msub> <msup> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>^</mo> </mover> </msub> <mi>T</mi> </msup> <mo>+</mo> <msub> <mover> <mi>S</mi> <mo>^</mo> </mover> <mi>&amp;omega;</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mover> <mi>P</mi> <mo>^</mo> </mover> <mi>k</mi> <mo>+</mo> </msubsup> <mo>=</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <msub> <mover> <mi>H</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>)</mo> </mrow> <msubsup> <mover> <mi>P</mi> <mo>^</mo> </mover> <mi>k</mi> <mo>-</mo> </msubsup> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <msub> <mover> <mi>H</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <msub> <mover> <mi>R</mi> <mo>^</mo> </mover> <mi>&amp;nu;</mi> </msub> <msubsup> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> <mi>T</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>C</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>a</mi> </msub> <mo>=</mo> <msub> <mi>F</mi> <mi>a</mi> </msub> <msub> <mi>C</mi> <mi>a</mi> </msub> <mo>+</mo> <msub> <mi>C</mi> <mi>a</mi> </msub> <msubsup> <mi>F</mi> <mi>a</mi> <mi>T</mi> </msubsup> <mo>+</mo> <msub> <mi>G</mi> <mi>a</mi> </msub> <msub> <mi>S</mi> <mi>&amp;eta;</mi> </msub> <msubsup> <mi>G</mi> <mi>a</mi> <mi>T</mi> </msubsup> <mo>+</mo> <msub> <mi>W</mi> <mi>a</mi> </msub> <msub> <mi>S</mi> <mi>&amp;omega;</mi> </msub> <msubsup> <mi>W</mi> <mi>a</mi> <mi>T</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mi>a</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mi>t</mi> <mi>k</mi> <mo>+</mo> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>A</mi> <mi>k</mi> </msub> <msub> <mi>C</mi> <mi>a</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mi>t</mi> <mi>k</mi> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <msubsup> <mi>A</mi> <mi>k</mi> <mi>T</mi> </msubsup> <mo>+</mo> <msub> <mi>B</mi> <mi>k</mi> </msub> <msub> <mi>R</mi> <mi>&amp;nu;</mi> </msub> <msubsup> <mi>B</mi> <mi>k</mi> <mi>T</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>z</mi> <mi>min</mi> </msub> <mo>&amp;le;</mo> <mi>z</mi> <mo>&amp;le;</mo> <msub> <mi>z</mi> <mi>max</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mi>a</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;le;</mo> <msub> <mi>C</mi> <msub> <mi>a</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>q</mi> </mrow> </msub> </msub> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>
Wherein z is sensor parameters, including Sη,Sω,Rν；K is sensor performance weighting function；For mission requirements.
3. aircraft optimal sensor system of selection according to claim 1, it is characterised in that step 2) matrix equation is excellent The specific method of the vectorization of change problem is as follows：
The optimization problem that P0 is provided constitutes for matrix equation, in order to preferably solve optimization problem, need to be converted into matrix equation Vector equation；
Kronecker Product definition and property is provided first：
Define 1：
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>A</mi> <mo>=</mo> <mo>&amp;lsqb;</mo> <msub> <mi>a</mi> <mrow> <mi>m</mi> <mo>&amp;times;</mo> <mi>n</mi> </mrow> </msub> <mo>&amp;rsqb;</mo> <mo>,</mo> <mi>B</mi> <mo>=</mo> <mo>&amp;lsqb;</mo> <msub> <mi>b</mi> <mrow> <mi>i</mi> <mo>&amp;times;</mo> <mi>j</mi> </mrow> </msub> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>A</mi> <mo>&amp;CircleTimes;</mo> <mi>B</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>a</mi> <mn>11</mn> </msub> <mi>B</mi> </mrow> </mtd> <mtd> <mrow> <msub> <mi>a</mi> <mn>12</mn> </msub> <mi>B</mi> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <msub> <mi>a</mi> <mrow> <mn>1</mn> <mi>n</mi> </mrow> </msub> <mi>B</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>a</mi> <mn>21</mn> </msub> <mi>B</mi> </mrow> </mtd> <mtd> <mrow> <msub> <mi>a</mi> <mn>22</mn> </msub> <mi>B</mi> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <msub> <mi>a</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mi>B</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>a</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mi>B</mi> </mrow> </mtd> <mtd> <mrow> <msub> <mi>a</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mi>B</mi> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <msub> <mi>a</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mi>B</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>
Define 2：
Vec (), Vec (A)=[a11 a21 … am1 a12 … am2 … amn]T (16)
Property 1：
<mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <mi>A</mi> <mi>X</mi> <mi>B</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <msup> <mi>B</mi> <mi>T</mi> </msup> <mo>&amp;CircleTimes;</mo> <mi>A</mi> <mo>)</mo> </mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow> 2
Property 2：
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>A</mi> <mi>X</mi> <mo>+</mo> <mi>X</mi> <mi>B</mi> <mo>=</mo> <mi>C</mi> <mi> </mi> <mi>A</mi> <mo>&amp;Element;</mo> <msup> <mi>R</mi> <mrow> <mi>n</mi> <mo>&amp;times;</mo> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mi>B</mi> <mo>&amp;Element;</mo> <msup> <mi>R</mi> <mrow> <mi>m</mi> <mo>&amp;times;</mo> <mi>m</mi> </mrow> </msup> <mo>,</mo> <mi>X</mi> <mo>&amp;Element;</mo> <msup> <mi>R</mi> <mrow> <mi>n</mi> <mo>&amp;times;</mo> <mi>m</mi> </mrow> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mi>m</mi> </msub> <mo>&amp;CircleTimes;</mo> <mi>A</mi> <mo>+</mo> <msup> <mi>B</mi> <mi>T</mi> </msup> <mo>&amp;CircleTimes;</mo> <msub> <mi>I</mi> <mi>n</mi> </msub> <mo>)</mo> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mo>(</mo> <mi>X</mi> <mo>)</mo> <mo>=</mo> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>
It is then able to matrix equation optimization problem P0 being converted into vector equation optimization problem：
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>P</mi> <mn>0</mn> <mo>:</mo> <mi>max</mi> <mi>i</mi> <mi>m</mi> <mi>i</mi> <mi>z</mi> <mi>e</mi> <mi> </mi> <mi>k</mi> <mi>z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>s</mi> <mi>u</mi> <mi>b</mi> <mi>j</mi> <mi>e</mi> <mi>c</mi> <mi>t</mi> <mi> </mi> <mi>t</mi> <mi>o</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <mover> <mover> <mi>P</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mi>F</mi> </msub> <mo>&amp;CircleTimes;</mo> <mo>(</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mo>+</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>^</mo> </mover> </msub> <msub> <mover> <mi>C</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mo>+</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>^</mo> </mover> </msub> <msub> <mover> <mi>C</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mo>)</mo> </mrow> <mo>&amp;CircleTimes;</mo> <msub> <mi>I</mi> <mi>F</mi> </msub> <mo>)</mo> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <mover> <mi>P</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mover> <mi>P</mi> <mo>^</mo> </mover> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>^</mo> </mover> </msub> <mo>&amp;CircleTimes;</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>^</mo> </mover> </msub> <mo>)</mo> </mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>S</mi> <mo>^</mo> </mover> <mi>&amp;eta;</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>S</mi> <mo>^</mo> </mover> <mi>&amp;omega;</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msubsup> <mover> <mi>P</mi> <mo>^</mo> </mover> <mi>k</mi> <mo>+</mo> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <msub> <mover> <mi>H</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;CircleTimes;</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <msub> <mover> <mi>H</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>)</mo> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msubsup> <mover> <mi>P</mi> <mo>^</mo> </mover> <mi>k</mi> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>&amp;CircleTimes;</mo> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>R</mi> <mo>^</mo> </mover> <mi>&amp;nu;</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>C</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>a</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>I</mi> <msub> <mi>F</mi> <mi>a</mi> </msub> </msub> <mo>&amp;CircleTimes;</mo> <msub> <mi>F</mi> <mi>a</mi> </msub> <mo>+</mo> <msub> <mi>F</mi> <mi>a</mi> </msub> <mo>&amp;CircleTimes;</mo> <msub> <mi>I</mi> <msub> <mi>F</mi> <mi>a</mi> </msub> </msub> <mo>)</mo> </mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>a</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>a</mi> </msub> <mo>&amp;CircleTimes;</mo> <msub> <mi>C</mi> <mi>a</mi> </msub> <mo>)</mo> </mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>S</mi> <mi>&amp;eta;</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>W</mi> <mi>a</mi> </msub> <mo>&amp;CircleTimes;</mo> <msub> <mi>W</mi> <mi>a</mi> </msub> <mo>)</mo> </mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>S</mi> <mi>&amp;omega;</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>a</mi> </msub> <mo>(</mo> <msubsup> <mi>t</mi> <mi>k</mi> <mo>+</mo> </msubsup> <mo>)</mo> </mrow> <mo>)</mo> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>k</mi> </msub> <mo>&amp;CircleTimes;</mo> <msub> <mi>A</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>a</mi> </msub> <mo>(</mo> <msubsup> <mi>t</mi> <mi>k</mi> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <mo>)</mo> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mi>k</mi> </msub> <mo>&amp;CircleTimes;</mo> <msub> <mi>B</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>&amp;nu;</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mi>min</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;le;</mo> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>&amp;le;</mo> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mi>max</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>a</mi> </msub> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>)</mo> <mo>&amp;le;</mo> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>C</mi> <msub> <mi>a</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>q</mi> </mrow> </msub> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow>
4. aircraft optimal sensor system of selection according to claim 3, it is characterised in that step 3) by optimization problem Convexification is as follows for the specific method of Second-order cone programming form and solution：
In order to which problem is converted into Second-order cone programming canonical form, z=[S are definedω Sη Rν] optimize sensor performance change for needed for Amount, and new intermediate variable t is introduced, inequality constraints is met, now performance indications are converted into Second-order cone programming canonical form：
minimize t (20)
<mrow> <mi>t</mi> <mo>&amp;GreaterEqual;</mo> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mi>z</mi> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow>
Wherein k is to have the weighting function designed by designer, shows the attention degrees different to different sensors performance parameter；No Equality constraint (21) is still unsatisfactory for the requirement of Second-order cone programming, need to be standardized；, can be by formula (21) by mathematical theory Inequality constraints is converted to second order cone constraint：
<mrow> <mo>|</mo> <mo>|</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mi>t</mi> <mo>-</mo> <mi>k</mi> <mi>z</mi> </mtd> </mtr> </mtable> </mfenced> <mo>|</mo> <mo>|</mo> <mo>&amp;le;</mo> <mi>t</mi> <mo>+</mo> <mi>k</mi> <mi>z</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mrow>
The problem of can now optimal sensor being selected completely sets up and is converted into Second-order cone programming problem P1, and utilizes interior point Method is solved；
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>P</mi> <mn>1</mn> <mo>:</mo> <mi>min</mi> <mi>i</mi> <mi>m</mi> <mi>i</mi> <mi>z</mi> <mi>e</mi> <mi> </mi> <mi>t</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mrow> <mi>s</mi> <mi>u</mi> <mi>b</mi> <mi>j</mi> <mi>e</mi> <mi>c</mi> <mi>t</mi> <mi> </mi> <mi>t</mi> <mi>o</mi> </mrow> </mtd> <mtd> <mrow> <mo>|</mo> <mo>|</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mi>t</mi> <mo>-</mo> <mi>k</mi> <mi>z</mi> </mtd> </mtr> </mtable> </mfenced> <mo>|</mo> <mo>|</mo> <mo>&amp;le;</mo> <mi>t</mi> <mo>+</mo> <mi>k</mi> <mi>z</mi> </mrow> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <mover> <mover> <mi>P</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mi>F</mi> </msub> <mo>&amp;CircleTimes;</mo> <mo>(</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mo>+</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>^</mo> </mover> </msub> <msub> <mover> <mi>C</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mo>+</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>^</mo> </mover> </msub> <msub> <mover> <mi>C</mi> <mo>^</mo> </mover> <mover> <mi>x</mi> <mo>^</mo> </mover> </msub> <mo>)</mo> </mrow> <mo>&amp;CircleTimes;</mo> <msub> <mi>I</mi> <mi>F</mi> </msub> <mo>)</mo> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <mover> <mi>P</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mover> <mi>P</mi> <mo>^</mo> </mover> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>^</mo> </mover> </msub> <mo>&amp;CircleTimes;</mo> <msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mover> <mi>y</mi> <mo>^</mo> </mover> </msub> <mo>)</mo> </mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>S</mi> <mo>^</mo> </mover> <mi>&amp;eta;</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>S</mi> <mo>^</mo> </mover> <mi>&amp;omega;</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msubsup> <mover> <mi>P</mi> <mo>^</mo> </mover> <mi>k</mi> <mo>+</mo> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <msub> <mover> <mi>H</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;CircleTimes;</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <msub> <mover> <mi>H</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>)</mo> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msubsup> <mover> <mi>P</mi> <mo>^</mo> </mover> <mi>k</mi> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>&amp;CircleTimes;</mo> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>R</mi> <mo>^</mo> </mover> <mi>&amp;nu;</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>C</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>a</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>I</mi> <msub> <mi>F</mi> <mi>a</mi> </msub> </msub> <mo>&amp;CircleTimes;</mo> <msub> <mi>F</mi> <mi>a</mi> </msub> <mo>+</mo> <msub> <mi>F</mi> <mi>a</mi> </msub> <mo>&amp;CircleTimes;</mo> <msub> <mi>I</mi> <msub> <mi>F</mi> <mi>a</mi> </msub> </msub> <mo>)</mo> </mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>a</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>a</mi> </msub> <mo>&amp;CircleTimes;</mo> <msub> <mi>C</mi> <mi>a</mi> </msub> <mo>)</mo> </mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>S</mi> <mi>&amp;eta;</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>W</mi> <mi>a</mi> </msub> <mo>&amp;CircleTimes;</mo> <msub> <mi>W</mi> <mi>a</mi> </msub> <mo>)</mo> </mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>S</mi> <mi>&amp;omega;</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>a</mi> </msub> <mo>(</mo> <msubsup> <mi>t</mi> <mi>k</mi> <mo>+</mo> </msubsup> <mo>)</mo> </mrow> <mo>)</mo> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>k</mi> </msub> <mo>&amp;CircleTimes;</mo> <msub> <mi>A</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>a</mi> </msub> <mo>(</mo> <msubsup> <mi>t</mi> <mi>k</mi> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <mo>)</mo> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mi>k</mi> </msub> <mo>&amp;CircleTimes;</mo> <msub> <mi>B</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>&amp;nu;</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mi>min</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;le;</mo> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>&amp;le;</mo> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mi>max</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>a</mi> </msub> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>)</mo> <mo>&amp;le;</mo> <mi>V</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>C</mi> <msub> <mi>a</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>q</mi> </mrow> </msub> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>23</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow>
5. aircraft optimal sensor system of selection according to claim 4, it is characterised in that step 4) it is based on continuously connecing The specific method of the convex Optimization Solution of near sequence is as follows：
In optimization problem P1, Kalman filtering coefficientSensor parameters R comprising required optimizationν, problem needs to be connected The processing of continued access closely；
In order to express conveniently, the optimization problem P1 that optimal sensor parameter is selected is written as form P2：
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>P</mi> <mn>2</mn> <mo>:</mo> <mi>min</mi> <mi>i</mi> <mi>m</mi> <mi>i</mi> <mi>z</mi> <mi>e</mi> <mi> </mi> <mi>t</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mrow> <mi>s</mi> <mi>u</mi> <mi>b</mi> <mi>j</mi> <mi>e</mi> <mi>c</mi> <mi>t</mi> <mi> </mi> <mi>t</mi> <mi>o</mi> </mrow> </mtd> <mtd> <mrow> <msup> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>A</mi> <mi>k</mi> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msup> <mi>x</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>B</mi> <mi>k</mi> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msup> <mi>z</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>z</mi> <mi>min</mi> </msub> <mo>&amp;le;</mo> <msup> <mi>z</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>&amp;le;</mo> <msub> <mi>z</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>24</mn> <mo>)</mo> </mrow> </mrow>
Then by the A in every suboptimization problemk、BkIt is considered as parameter related to optimized variable in constant value matrix, matrix by the last time Optimized parameter obtained by calculating is with new；When loop iteration calculating meets following index, it is believed that result restrains：
|zk+1-zk|≤ε (25)。
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