CN107291986B - Aircraft optimal sensor selection method - Google Patents

Aircraft optimal sensor selection method Download PDF

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CN107291986B
CN107291986B CN201710370432.5A CN201710370432A CN107291986B CN 107291986 B CN107291986 B CN 107291986B CN 201710370432 A CN201710370432 A CN 201710370432A CN 107291986 B CN107291986 B CN 107291986B
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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 an optimal sensor selection method for an aircraft, which comprises the steps of firstly, utilizing the advantages of the covariance theory in the rapidity and the accuracy of track deviation analysis to construct a covariance transfer and update equation, thereby providing the relationship between sensor parameters and final landing point deviation, and constructing a sensor selection problem based on covariance by applying the relationship. And then, the established sensor selection problem is converted into a second-order cone programming problem and solved by utilizing the advantages of rapidity and global optimality of the second-order cone programming in solving the optimization problem, so that the cost-optimized sensor meeting the task precision requirement is obtained.

Description

Aircraft optimal sensor selection method
[ technical field ] A method for producing a semiconductor device
The invention relates to an aircraft optimal sensor selection method.
[ background of the invention ]
The optimal selection of the sensor is the core technology for realizing high-performance navigation, guidance and control. The performance of the sensor not only influences the discrete distribution condition of the navigation error and the real state of the system, thereby determining the completion condition of each space mission, but also determines the cost of the spacecraft, and the high-performance sensor means high-precision track and high cost. How to select an optimal sensor for a spacecraft can ensure that each high-precision task is finished and effectively reduce the required cost is an important technology in aerospace engineering.
In terms of sensor selection, the scholars have developed certain application studies. Hovlan's studied the problem of real-time selection of different kinds of sensors for robotic systems. Kincaid developed a study of the optimal location determination of the sensors. Gupta presents a sensor selection strategy based on statistical theory, solving the engineering problem from a new perspective. Also, there are some methods for sensor selection, and Welch solves the problem of sensor selection by using branch-and-bound method. Yao employs a genetic algorithm to efficiently solve for sensor selection. Joshi then first adopted convex optimization for sensor selection. All of these applications and methods have their own emphasis. In the patent, a brand-new method is adopted for design, and the method is based on a linear covariance theory and a second-order cone programming method, so that high-precision, quick and optimal sensor selection is realized.
Covariance analysis is mostly applied to evaluating and analyzing the problem of trajectory dispersion caused by external uncertain factors and disturbance and the problem of estimation errors caused by modeling errors. Maybeck gives a general derivation, and thus establishes true state covariance, filter estimation error, and filter state covariance. The Geller applies covariance technology to design a new track control and navigation analysis method, and the method can be applied to various systems. Christensen proposed a linear covariance analysis method for closed-loop systems and nonlinear problems, and verified its validity.
In recent years, convex optimization has received much attention in the field of aerospace engineering applications. The second-order cone planning is the method with the most potential value for development and research. Second order cone planning is named because the objective function is linear and the constraint is in the form of a second order cone. Intersection and approach of the convex optimized spacecraft, trajectory optimization of the spacecraft and guidance of a dynamic descent section of Mars landing.
[ summary of the invention ]
The invention aims to overcome the defects of the prior art and provide an optimal sensor selection method for an aircraft, which is used for establishing an optimization problem of aircraft sensor selection based on linear covariance, converting the optimization problem into a form meeting second-order cone planning constraint through a convex method, and solving through second-order cone planning, so that the rapid and accurate sensor selection is realized.
In order to achieve the purpose, the invention adopts the following technical scheme to realize the purpose:
an aircraft optimal sensor selection method comprising the steps of:
1) optimization problem covariance description of aircraft sensor selection;
2) vectorizing a matrix equation optimization problem;
3) the optimization problem is transformed into a second-order cone programming form and solved;
4) and (4) solving the sequential convex optimization based on continuous approximation.
The invention further improves the following steps:
step 1) the detailed method for optimization problem covariance description for aircraft sensor selection is as follows:
establishing a non-linear model of the aircraft:
Figure BDA0001302644940000021
where x (t) is the aircraft state,
Figure BDA0001302644940000022
for control commands, w (t) is the aircraft model uncertainty, satisfying:
E[ω(t)ωT(τ)]＝Sω(t)(t-τ) (2)
establishing an inertia measurement equation of the aircraft:
Figure BDA0001302644940000031
wherein
Figure BDA0001302644940000032
For continuous measurements, η is the measurement noise associated with the sensor;
establishing a non-inertial measurement equation:
Figure BDA0001302644940000033
wherein
Figure BDA0001302644940000034
As discrete measured values, vkIs the measurement noise associated with the sensor;
the transfer and update equation of the navigation state can be obtained after the navigation algorithm is considered:
Figure BDA0001302644940000035
and navigation error covariance equation:
Figure BDA0001302644940000036
wherein
Figure BDA0001302644940000037
The control command applying the latest navigation information is:
Figure BDA0001302644940000038
linearizing an aircraft dynamics equation and a navigation equation along a standard track:
Figure BDA0001302644940000039
Figure BDA00013026449400000310
establishing an expansion equation:
Figure BDA00013026449400000311
and obtaining a transfer and update equation of the extended state:
Figure BDA0001302644940000041
wherein
Figure BDA0001302644940000042
The transfer and update equation for covariance can then be derived:
Figure BDA0001302644940000043
an optimal sensor selection problem based on covariance can be constructed at this time:
Figure BDA0001302644940000044
wherein z is a sensor parameter, including Sη,Sω,Rν(ii) a k is a sensor performance weighting function;
Figure BDA0001302644940000045
is a task requirement.
Step 2) the specific method of vectorization of the matrix equation optimization problem is as follows:
the optimization problem given by P0 is composed of matrix equations, and in order to better solve the optimization problem, the matrix equations need to be converted into vector equations;
first, the definition and properties of the Kronecker Product are given:
definition 1:
Figure BDA0001302644940000051
definition 2:
Vec(·),Vec(A)＝[a11a21…am1a12…am2…amn]T(16)
properties 1:
Figure BDA0001302644940000052
properties 2:
Figure BDA0001302644940000053
the matrix equation optimization problem P0 can then be converted into a vector equation optimization problem:
Figure BDA0001302644940000054
step 3) the concrete method for raising the optimization problem into a second-order cone programming form and solving is as follows:
to convert the problem into a second order cone programming standard form, z ═ S is definedωSηRν]And introducing a new intermediate variable t to optimize the sensor performance variable required, and meeting inequality constraint, wherein the performance index is converted into a second-order cone programming standard form:
minimize t (20)
Figure BDA0001302644940000061
wherein k is a weight function designed by a designer and indicates different degrees of importance for different sensor performance parameters; the inequality constraint (21) does not meet the requirement of second-order cone planning, and standardization is needed; by mathematical theory, the inequality constraint of equation (21) can be converted into a second order cone constraint:
Figure BDA0001302644940000062
at the moment, the problem selected by the optimal sensor can be completely established and converted into a second-order cone programming problem P1, and an interior point method is utilized for solving;
Figure BDA0001302644940000063
step 4) the specific method of the sequential convex optimization solution based on continuous approximation is as follows:
in the optimization problem P1, Kalman Filter coefficients
Figure BDA0001302644940000064
Including the sensor parameter R to be optimizedνProblems require continuous proximity handling;
for convenience of expression, the optimization problem P1 for optimal sensor parameter selection is written as the following form P2:
Figure BDA0001302644940000065
a in the problem will then be optimized each timek、BkTaking the parameters as a constant matrix, and updating the parameters related to the optimized variables in the matrix by the optimal parameters obtained by the last calculation; when the loop iteration calculation meets the following indexes, the result is considered to be converged:
|zk+1-zk|≤ (25)。
compared with the prior art, the invention has the following beneficial effects:
the invention relates to an aircraft optimal sensor selection method based on solution of a covariance theory and a second-order cone programming theory, which constructs a relation between a sensor parameter and a final drop point deviation by utilizing the advantages of rapidity and accuracy of the covariance theory in track deviation analysis, establishes a sensor selection problem based on the covariance, converts the established sensor selection problem into a second-order cone programming problem and solves the problem by utilizing the advantages of rapidity and global optimality of the second-order cone programming in solution of an optimization problem, and thus, a cost optimal sensor meeting task accuracy requirements can be obtained quickly and accurately.
[ description of the drawings ]
FIG. 1 is a graph of the optimum speed measurement noise of the present invention;
FIG. 2 is a graph of the change in altitude speed according to the present invention;
FIG. 3 is a comparison of the covariance model predictive guidance method of the present invention with a conventional method;
FIG. 4 is a comparison of the covariance model predictive guidance method of the present invention with the conventional method.
[ detailed description ] embodiments
The invention is described in further detail below with reference to the accompanying drawings:
referring to fig. 1-4, the aircraft optimal sensor selection method of the invention comprises the following steps:
step one, optimization problem covariance description of aircraft sensor selection
First, a general form of nonlinear model of the aircraft is built:
Figure BDA0001302644940000071
where x (t) is the aircraft state,
Figure BDA0001302644940000072
for control commands, w (t) is the aircraft model uncertainty, satisfied
E[ω(t)ωT(τ)]＝Sω(t)(t-τ) (2)
Establishing an inertial measurement equation for an aircraft
Figure BDA0001302644940000081
Wherein
Figure BDA0001302644940000082
For continuous measurements, η is the measurement noise associated with the sensor;
establishing a non-inertial measurement equation
Figure BDA0001302644940000083
Wherein
Figure BDA0001302644940000084
As discrete measured values, vkIs the measurement noise associated with the sensor;
the transfer and update equations of the navigation state can be obtained after the navigation algorithm is considered,
Figure BDA0001302644940000085
and navigation error covariance equation
Figure BDA0001302644940000086
Wherein
Figure BDA0001302644940000087
The control command using the latest navigation information is
Figure BDA0001302644940000088
Linearizing aircraft dynamics equation and navigation equation along standard trajectory
Figure BDA0001302644940000089
Figure BDA00013026449400000810
Establishing an expansion equation
Figure BDA00013026449400000811
And obtaining the transmission and update equation of the extended state
Figure BDA00013026449400000812
Wherein
Figure BDA0001302644940000091
Thus, a covariance transfer and update equation can be obtained
Figure BDA0001302644940000092
At this point, an optimal sensor selection problem based on covariance can be constructed
Figure BDA0001302644940000093
Wherein z is a sensor parameter, including Sη,Sω,Rν(ii) a k is a sensor performance weighting function;
Figure BDA0001302644940000094
is a task requirement;
step two, vectorization of matrix equation optimization problem
The optimization problem given by P0 is composed of matrix equations, and in order to better solve the optimization problem, the matrix equations need to be converted into vector equations;
firstly, the definition and the property of a Kronecker Product are given;
definition 1:
Figure BDA0001302644940000101
definition 2:
Vec(·),Vec(A)＝[a11a21…am1a12…am2…amn]T(16)
properties 1:
Figure BDA0001302644940000102
properties 2:
Figure BDA0001302644940000103
the matrix equation optimization problem P0 can then be converted into a vector equation optimization problem
Figure BDA0001302644940000104
Step three, the optimization problem is emphasized into a second-order cone programming form and solved
To convert the problem into a second order cone programming standard form, z ═ S is definedωSηRν]Introducing a new intermediate variable t to optimize the sensor performance variable required, satisfying inequality constraint, and converting the performance index into a second-order cone programming standard form
minimize t (20)
Figure BDA0001302644940000111
Wherein k is a weight function designed by a designer and indicates different degrees of importance for different sensor performance parameters; the inequality constraint (21) does not meet the requirement of second-order cone planning, and standardization is needed; by mathematical theory, the inequality constraint of equation (21) can be converted into a second-order cone constraint
Figure BDA0001302644940000112
At the moment, the problem selected by the optimal sensor can be completely established and converted into a second-order cone programming problem P1, and an interior point method is utilized for solving;
Figure BDA0001302644940000113
fourthly, solving the sequence convex optimization based on continuous approximation;
in the optimization problem P1, Kalman Filter coefficients
Figure BDA0001302644940000114
Including the sensor parameter R to be optimizedνProblems require continuous proximity handling;
for convenience of presentation, the optimization problem P1 for optimal sensor parameter selection is written as the following form P2
Figure BDA0001302644940000115
A in the problem will then be optimized each timek、BkViewed as a constant matrix, the parameters of the matrix relating to the optimization variablesUpdating the number by the optimal parameter obtained by the last calculation; when the loop iteration calculation meets the following indexes, the result is considered to be converged;
|zk+1-zk|≤ (25)。
fig. 1 to 3 show the optimal velocity measurement noise, accelerometer bias, and acceleration measurement noise in the simulation process, respectively. All parameters can converge after 15 iterations, and the convergence process is monotonic and smooth. And in the convergence process and the result, the optimized parameters meet the constraint requirement.
Fig. 4 gives the deviation curve of the speed variation in the real state. The deviation is initially zero and increases gradually as the task progresses, reaching a maximum of 0.876m/s at 53 seconds. And then, due to a guidance and control algorithm, the deviation is gradually reduced along with the time, and the deviation value is 0.0997m/s when the task terminal point is reached, so that the task requirement is met.
The principle of the invention is as follows:
the method firstly utilizes the advantages of the covariance theory in rapidity and accuracy of track deviation analysis to construct a covariance transfer and update equation, thereby providing a relation between sensor parameters and final drop point deviation, and applying the relation to construct a sensor selection problem based on covariance. And then, the established sensor selection problem is converted into a second-order cone programming problem and solved by utilizing the advantages of rapidity and global optimality of the second-order cone programming in solving the optimization problem, so that the cost-optimized sensor meeting the task precision requirement is obtained.
Example (b):
the method for selecting the optimal sensor of the aircraft based on the solution of the covariance theory and the second-order cone programming theory is utilized, and the one-dimensional rocket acceleration is taken as an example to carry out simulation and verify the feasibility and the rapidity of the method. The model is shown in formula (26).
Figure BDA0001302644940000131
Where v is rocket velocity, aactFor acceleration command, α is the atmospheric drag coefficient, d is the acceleration disturbance, b is the acceleration offset, and s is the ratioExample factor, τiIs a time constant, ωiIs a random perturbation. The uncertainty factors and disturbance parameters are shown in table 1; the parameter boundaries and task objectives are optimized as shown in table 2.
TABLE 1 uncertainty factors/disturbance parameters
Figure BDA0001302644940000132
TABLE 2 optimization of parameter boundaries and task goals
Figure BDA0001302644940000133
The above-mentioned contents are only for illustrating the technical idea of the present invention, and the protection scope of the present invention is not limited thereby, and any modification made on the basis of the technical idea of the present invention falls within the protection scope of the claims of the present invention.

Claims (1)

1. An aircraft optimal sensor selection method, characterized by comprising the steps of:
1) the optimization problem covariance description of aircraft sensor selection is specifically as follows:
establishing a non-linear model of the aircraft:
Figure FDA0002500704860000011
where x (t) is the aircraft state,
Figure FDA0002500704860000012
for control commands, w (t) is the aircraft model uncertainty, satisfying:
E[ω(t)ωT(τ)]＝Sω(t)(t-τ) (2)
establishing an inertia measurement equation of the aircraft:
Figure FDA0002500704860000013
wherein
Figure FDA0002500704860000014
For continuous measurements, η is the measurement noise associated with the sensor;
establishing a non-inertial measurement equation:
Figure FDA0002500704860000015
wherein
Figure FDA0002500704860000016
As discrete measured values, vkIs the measurement noise associated with the sensor;
the transfer and update equation of the navigation state can be obtained after the navigation algorithm is considered:
Figure FDA0002500704860000017
and navigation error covariance equation:
Figure FDA0002500704860000018
wherein
Figure FDA0002500704860000019
The control command applying the latest navigation information is:
Figure FDA00025007048600000110
linearizing an aircraft dynamics equation and a navigation equation along a standard track:
Figure FDA00025007048600000111
Figure FDA0002500704860000021
establishing an expansion equation:
Figure FDA0002500704860000022
and obtaining a transfer and update equation of the extended state:
Figure FDA0002500704860000023
wherein
Figure FDA0002500704860000024
The transfer and update equation for covariance can then be derived:
Figure FDA0002500704860000025
an optimal sensor selection problem based on covariance can be constructed at this time:
Figure FDA0002500704860000026
wherein z is a sensor parameter, including Sη,Sω,Rν(ii) a k is a sensor performance weighting function;
Figure FDA0002500704860000027
is a task requirement;
2) the matrix equation optimization problem vectorization method comprises the following specific steps:
the optimization problem given by P0 is composed of matrix equations, and in order to better solve the optimization problem, the matrix equations need to be converted into vector equations;
first, the definition and properties of the Kronecker Product are given:
definition 1:
Figure FDA0002500704860000031
definition 2:
Vec(·),Vec(A)＝[a11a21…am1a12…am2…amn]T(16)
properties 1:
Figure FDA0002500704860000032
properties 2:
Figure FDA0002500704860000033
the matrix equation optimization problem P0 can then be converted into a vector equation optimization problem:
Figure FDA0002500704860000034
3) the optimization problem is transformed into a second-order cone programming form and solved, and the specific method comprises the following steps:
to convert the problem into a second order cone programming standard form, z ═ S is definedωSηRν]And introducing a new intermediate variable t to optimize the sensor performance variable required, and meeting inequality constraint, wherein the performance index is converted into a second-order cone programming standard form:
minimize t (20)
Figure FDA0002500704860000041
wherein k is a weight function designed by a designer and indicates different degrees of importance for different sensor performance parameters; the inequality constraint (21) does not meet the requirement of second-order cone planning, and standardization is needed; by mathematical theory, the inequality constraint of equation (21) can be converted into a second order cone constraint:
Figure FDA0002500704860000042
at the moment, the problem selected by the optimal sensor can be completely established and converted into a second-order cone programming problem P1, and an interior point method is utilized for solving;
Figure FDA0002500704860000043
4) the sequential convex optimization solution based on continuous approximation comprises the following specific steps:
in the optimization problem P1, Kalman Filter coefficients
Figure FDA0002500704860000044
Including the sensor parameter R to be optimizedνProblems require continuous proximity handling;
for convenience of expression, the optimization problem P1 for optimal sensor parameter selection is written as the following form P2:
Figure FDA0002500704860000051
a in the problem will then be optimized each timek、BkTaking the parameters as a constant matrix, and updating the parameters related to the optimized variables in the matrix by the optimal parameters obtained by the last calculation; when the loop iteration calculation meets the following indexes, the result is considered to be converged:
|zk+1-zk|≤ (25)。
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CN101458259A (en) * 2007-12-14 2009-06-17 西北工业大学 Sensor setting method for supporting failure prediction
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