CN113671826B - Rapid assessment method for accessibility of pneumatic auxiliary orbit of air vehicle crossing atmosphere - Google Patents
Rapid assessment method for accessibility of pneumatic auxiliary orbit of air vehicle crossing atmosphere Download PDFInfo
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Abstract
The invention discloses a method for rapidly evaluating accessibility of an air-assisted orbit of an aircraft crossing the atmosphere, and belongs to the technical field of aerospace. The implementation method of the invention comprises the following steps: on the basis of given initial parameters and system parameters, establishing a longitudinal dynamics equation related to the track size; establishing a pneumatic auxiliary rail maneuvering accessibility boundary problem description; through analyzing the pneumatic auxiliary maneuvering accessibility boundary, according to the fixed terminal speed, converting the accessibility assessment problem into a state maximum/minimum problem, and giving a boundary division discrete solving frame; aiming at the optimal control problem of discrete boundary points, a sequence convex optimization framework of the aircraft accessibility boundary problem is constructed through independent variable substitution and lossless convexity of nonlinear problems, and the pneumatic auxiliary orbit accessibility of the cross-domain aircraft is rapidly evaluated. The method has the advantages of strong robustness, high repeatability, high precision, strong reliability and high evaluation efficiency, and has no strict limitation on the initial state and the system of the aircraft.
Description
Technical Field
The invention relates to a method for rapidly evaluating the accessibility of pneumatic auxiliary orbits of an aircraft crossing the atmosphere, in particular to a method for determining the orbit number range of the aircraft after pneumatic synergism, and belongs to the technical field of aerospace.
Background
The cross-atmosphere aircraft is widely researched and applied to cross-domain orbit maneuvering tasks at present, and has space task execution capacity and a structure capable of controlling flight in atmosphere, so that the cross-atmosphere aircraft becomes an important direction for the development of future space and air integrated space tasks. In the task execution process of the aircraft crossing the atmosphere, the feasibility of the task is determined by the track maneuvering capability under the assistance of pneumatic cooperation, and task feasibility analysis is usually a key link and a first step of overall design, so that the task success or failure is played an important role. Because cross-domain orbit maneuver capability depends not only on the current state of the aircraft, but also on the system parameters and control capabilities of the aircraft, the accessibility range assessment problem belongs to the nonlinear optimal control problem of continuous control parameters. The existing accessibility assessment usually takes the limit of control parameters and then obtains the upper and lower boundaries of a certain number of accessibility, or converts the continuous optimal control problem into a single parameter solving problem by simplifying a problem model. However, none of the existing methods described above is suitable for accurate assessment of the ability of a flying vehicle to maneuver across the atmosphere with continuous control of roll. Therefore, the method for rapidly evaluating the accessibility of the pneumatic auxiliary orbit of the air vehicle crossing the atmosphere can not only realize the determination of the range of the maneuvering orbit under any initial parameters, but also ensure the evaluation calculation efficiency, thereby meeting the technical requirements of the design and application of future air vehicles crossing the atmosphere.
In the developed assessment method for the accessibility of the aerodynamic auxiliary orbit of an aircraft, the prior art [1] (see: linxiqiang, zhang Yolin. Determination method of the accessibility of aerodynamic auxiliary different surfaces [ J ]. National defense university of science and technology, 2000,22 (2): 7-10.) gives a calculation formula derived by a coordinate transformation method and comprising the change amount range of the inclination angle of a certain aircraft relative to an initial orbit (the orbit calculated on the assumption that the inclination angle of the initial orbit is 0), and the accessibility of aerodynamic auxiliary different surfaces at any initial inclination angle is obtained. However, there is no clear calculation method for the range of orbit reachable by the aircraft, and it is difficult to apply to the assessment of accessibility.
Prior art [2] (see: guelman m.planar Aeroassisted Attainability Domain [ J ]. Journal of Guidance, control, and Dynamics,1997,20 (3): 422-427.) studied the reach of planar aerodynamic auxiliary orbit maneuver, taking angle of attack as the aerodynamic auxiliary Control variable, and concluding that the boundary of the aerodynamic auxiliary reach depends on the initial angle of attack, thereby converting the problem into a univariate optimization problem, ultimately giving the boundary of the reach trajectory within the atmosphere and the terminal reach of the out-of-atmosphere location, however, the simplified strategy of modulating the angle of attack belongs to roll Control across the atmosphere, assuming that this simplified strategy of modulating the angle of attack is not suitable for assessment of the aerodynamic auxiliary orbit accessibility across the atmosphere.
Disclosure of Invention
The invention discloses a method for rapidly evaluating the accessibility of a pneumatic auxiliary orbit of an aircraft crossing the atmosphere, which aims to solve the technical problems that: by constructing a sequence convex optimization framework of the aircraft accessibility boundary problem, the nonlinear optimal control problem under the tilting control is rapidly solved through the sequence convex optimization framework, the accessibility boundary taking the maximum and minimum ranges of the terminal track as the performance index is obtained, and the rapid evaluation of the accessibility of the pneumatic auxiliary track is realized. The invention has the following advantages: (1) strong robustness and high repeatability; (2) high precision and high reliability; (3) There is no strict limitation on the initial state and system of the aircraft; and (4) the evaluation efficiency is high, and the online evaluation capability is provided.
The invention aims at realizing the following technical scheme:
the invention discloses a rapid assessment method for the accessibility of a pneumatic auxiliary orbit of an aircraft crossing the atmosphere, which establishes a longitudinal dynamics equation related to orbit dimensions on the basis of given initial parameters and system parameters, and establishes a description of the boundary problem of the accessibility of the pneumatic auxiliary orbit on the basis of the longitudinal dynamics equation. The accessibility assessment problem is converted into a state maximum/minimum problem according to the fixed terminal speed by analyzing the pneumatic auxiliary maneuvering accessibility boundary, so that a discrete solving framework of the dividing boundary is provided. Aiming at the optimal control problem of discrete boundary points, a sequence convex optimization framework of the accessibility boundary problem of the aircraft is constructed through independent variable substitution and lossless convex of the nonlinear problem, so that the accessibility of the pneumatic auxiliary orbit of the cross-domain aircraft is rapidly evaluated.
The invention discloses a method for rapidly evaluating the accessibility of a pneumatic auxiliary orbit of an aircraft crossing the atmosphere, which comprises the following steps:
step one: a longitudinal dynamics equation is established that correlates to the track dimensions.
Because only the influence of the pneumatic auxiliary orbit maneuver on the orbit semi-long axis and the eccentricity is considered when the accessibility assessment of the pneumatic auxiliary orbit maneuver is carried out, the orbit dimension parameters and the terminal position vector diameter and the track angle in the spherical coordinate system are directly related to the speed, and a simplified model of dimensionless and small earth rotation removal is adopted when the accessibility assessment is carried out:
where μ represents the gravitational constant of the earth, r represents the position vector diameter, v represents the velocity, and γ represents the track angle. Sigma is the roll angle and is also the process control quantity for pneumatically assisted flight. The lift acceleration L and the drag acceleration D after dimensionless are as follows:
wherein R is M Is the earth radius, S is the reference area of the aircraft, C L And C D The lift coefficient and drag coefficient, respectively, p is the dimensional planetary atmospheric density, and for simplicity of calculation, an exponential model is taken here, i.e. ρ=ρ 0 exp(-R M h/h s ) Wherein ρ is 0 Is the earth surface atmospheric density, h s Is the atmospheric density coefficient.
Step two: and establishing a description of the maneuvering accessibility boundary problem of the pneumatic auxiliary track according to the state parameters related to the track after the aircraft is out of the atmosphere.
For pneumatically assisted motorized procedures, the end state is determined by the exit position vector magnitude r f Velocity V f And track angle gamma f Described, since the earth atmosphere is at a certain altitude, r is f Constant value. The definition of the pneumatically assisted maneuver reachable range is described herein as all possible V f And gamma f Set A of compositions (t f ). The reachable range boundary is defined herein, i.e. the reachable capability boundary is O 0 (t f ) I.e. O0 (t f )∈A(t f )。
Because the target track after pneumatic auxiliary track change cannot be tightly attached to the edge of the atmosphere, a minimum telecentric point height constraint h after the air is discharged needs to be added a,min Having a value greater than the atmospheric edge height h atm . The corresponding position vector diameter of the minimum telecentric point height constraint is r a,min ＝h a,min +R M The reachable range boundary corresponding to the constraint is marked as O L (t f ) Represented by curves B-C. The constraint equation corresponding to the constraint means that after the aircraft is out of the atmosphere, the height of the telecentric point of the corresponding kepler orbit is equal to h a,min In the specific form of
Wherein r is atm ＝h atm +R M Is the vector diameter of the atmospheric edge position. Since the number of dimensions of the orbit after the atmosphere is calculated by the end state variable, the equation (3) can be obtained by the energy square Cheng Tuidao of the elliptical orbit. Thus, the boundary O (t f ) Described by closed segment curves a-B-C.
Step three: and converting the accessibility assessment problem into a state maximum/minimum problem P1 according to the fixed terminal speed, thereby giving a discrete solving framework of the dividing boundary.
Pneumatic auxiliary maneuvering reachable range A (t) f ) Can be generalized to determine its boundary O (t f ) Solving for O (t) f ) The core of (2) is to solve the constraint h of dividing the minimum telecentricity height a,min Determined boundary O L (t f ) Outside boundaries, i.e. O 0 (t f ). For O 0 (t f ) By rapidly dividing the boundary points, the boundary point calculation is summarized as: fixed speed V f Solving the maximum and minimum track angles gamma by optimizing f 。
For a determinedk=1, …, N, straight line +.>With O 0 (t f ) Intersecting at two points, and the corresponding terminal track angle is +.>And->When solving them by optimization, the corresponding performance indexesIs that
Wherein solve forAnd->The corresponding optimal control problem comprises a control variable of roll angle sigma, the initial state is the state of the aircraft entering the atmosphere inlet, and the initial state is marked as x 0 ＝[r 0 ,V 0 ,γ 0 ] T . The optimal control problem is herein described as P1.
By fixing the terminal speed, the reachability boundary O (t f ) The evaluation problem of (2) is converted into a terminal track angle maximum/minimum problem as in formula (4), thereby giving a division boundary discrete solving framework.
Step four: aiming at the nonlinear optimal control problem P1, a sequence convex optimization framework of the aircraft accessibility boundary problem is constructed through independent variable substitution and lossless convex of the nonlinear problem.
The nonlinear optimal control problem P1 is a strong nonlinear optimal control problem because of the inclusion of nonlinear dynamical equations. Aiming at solving the nonlinear optimal control problem P1, by referring to the approximation theory of the nonlinear infinite dimensional system optimal control law, the sequence convex optimization framework of the aircraft accessibility boundary problem is constructed by independent variable replacement and lossless convex of the nonlinear problem, and nonlinear items in the optimal control problem P1 are respectively linearized near a given approximation solution by lossless convex, so that the linear optimal control sub-problem is obtained, and the sequence convex optimization framework of the aircraft accessibility boundary problem is obtained.
Step 4.1: the dimensionality reduction transformation of the nonlinear dynamics equation (1) is realized through independent variable substitution.
For the dynamics equation (1), consider the position vector r at the reachable ability boundary point determined by dimension-reduction division f Sum speed ofV f All are fixed values, so the section adopts variable substitution, and selects a new independent variable, namely e=1/r-V 2 The variable is actually equivalent to Kepler orbit energy, and the expression just comprises position vector diameter and speed, so that the terminal value is fixed when the accessibility boundary point is obtained. It is known from differentiation that de/dt=dv > 0 is constant, and therefore, this monotonically increasing variable can be used as an independent variable. By means of argument substitution, equation (1) is transformed into
From the above equation, the differential term of the velocity is eliminated from the dynamics equation, and the dynamics dimension is reduced.
Step 4.2: through variable transformation of the roll angle, the roll angle expression appearing in the dynamic equation in the form of cosine is converted into a linear form, and the linear constraint corresponding to the new control quantity is obtained.
For the control variables, a simple transformation is also required here. Since the control amount roll angle σ appears in the form of a cosine in the kinetic equation, it needs to be treated as linear as possible. Considering that cos σ monotonically increases between intervals [0 °,180 ° ], the new control amount u=cos σ is directly adopted. After the new control quantity is adopted, the constraint of the original roll angle is converted into
Wherein sigma min 0℃is the lower limit of the roll angle, and σ max The angle of less than or equal to 180 degrees is the upper limit of the tilting angle. In addition, the upper and lower bounds of the new control amount u after transformation are u min ＝cosσ max And u max ＝cosσ min . Obviously, the constraint corresponding to the new control amount is a linear constraint.
Step 4.3: and (3) gradually linearizing the dynamics equation after the dimension reduction in the step (4.1) to obtain a linear dynamics equation after the dimension reduction.
In order for a linear approximation method to be viable, the kinetic equation needs to be linearized around the approximation solution. First, the dynamics equation (5) is written in the form of a nonlinear affine control system, i.e
Wherein the nonlinear term of the state variable is
While the coefficient matrix of the control variable is
B(x,e)＝[0,(L/D)/V 2 ] T (9)
For the control affine system, the nonlinear term (8) of the state variable of the control affine system needs to be linearized. When there is an approximate solution of x z Where z represents the z-th linear successive approximation, then the nonlinear term (8) can be found at x z Vicinity linearization. Furthermore, although there is a velocity V in the control coefficient B (x, e), it was found by observation of V that during pneumatically assisted maneuvers, dimensionless r.apprxeq.1, therefore V at the time of the z+1th linear successive approximation z+1 By means of a beltAnd calculating, so that the accuracy of V calculation can be ensured when convergence is finally approximated. At this time, the corresponding linearized dynamic equation after dimension reduction is
Wherein the linear coefficient matrix a (x z ) Is f 0 Jacobian matrix of (x, e).
To this end, for the nonlinear optimal control problem P1, the dimension reduction is performed through independent variable substitution to form a dynamic equation (5), and the dynamic equation (10) is converted into a linear dynamic equation (10) after dimension reduction through successive linearization, and since the process constraint (6) and the dynamic equation (10) are linear constraints, the process constraint (6) and the dynamic equation (10) are the sequential convex optimization framework of the constructed aircraft accessibility boundary problem.
Step five: and rapidly solving the nonlinear optimal control problem P1 under the tilting control through the sequence convex optimization framework to obtain a maximum minimum track angle at any terminal speed, wherein the maximum minimum track angle corresponds to two points on the accessibility boundary, and the aerodynamic auxiliary accessibility boundary of the aircraft can be obtained by connecting all boundary points.
Step six: based on the pneumatic auxiliary accessibility boundary of the aircraft, the range of the feasible target orbit of the aircraft for the pneumatic auxiliary flight crossing the atmosphere is rapidly estimated, so that the reasonable design of the target orbit in the task implementation process is guided.
The beneficial effects are that:
1. according to the method for rapidly evaluating the accessibility of the pneumatic auxiliary orbit of the air vehicle crossing the atmosphere, disclosed by the invention, the accessibility evaluation problem is converted into the terminal track angle maximum/minimum problem through fixing the terminal speed, so that a discrete solving frame of a dividing boundary is provided, and the universality is realized.
2. According to the method for rapidly evaluating the accessibility of the pneumatic auxiliary orbit of the air-crossing aircraft, disclosed by the invention, the dimension-reducing transformation of the nonlinear dynamics equation is realized through independent variable replacement, the dynamics dimension is reduced, the number of calculated variables is reduced, so that the solving speed is high, the calculation time is short, and the evaluation efficiency is further improved.
3. The method for rapidly evaluating the accessibility of the pneumatic auxiliary orbit of the air vehicle crossing the atmosphere disclosed by the invention can be suitable for evaluating the accessibility of the pneumatic auxiliary maneuver in any planetary environment by setting different system parameters, so that the method has a wide application range to a target environment.
4. According to the method for rapidly evaluating the accessibility of the pneumatic auxiliary orbit of the air vehicle crossing the atmosphere, disclosed by the invention, the process constraint and the nonlinear dynamics formed by the control variables are converted into the linear constraint through lossless salinization, and the conversion process is not influenced by specific orbit parameters, so that the method has a wide application range on the target orbit.
5. The method for rapidly evaluating the accessibility of the pneumatic auxiliary orbit of the air vehicle crossing the atmosphere disclosed by the invention has the advantages of strong robustness and high repeatability due to the random given initial state and air vehicle parameters.
Drawings
FIG. 1 is a schematic diagram of the accessibility border in step 1 of the present invention;
FIG. 2 is a flow chart of a method of rapidly assessing the accessibility of a pneumatically assisted orbit of an aircraft across the atmosphere in accordance with the present invention;
FIG. 3 is a schematic diagram of the dimension reduction partitioning strategy in step 3 of the present invention;
FIG. 4 is a geometry of the reachability boundary in terms of end states in this embodiment;
fig. 5 is the geometry of the reachability boundary in this embodiment expressed in terms of near-centroid height and eccentricity.
Detailed Description
For a better illustration of the objects and advantages of the invention, a detailed explanation of the invention is provided below by performing a simulation analysis of the accessibility assessment problem of a pneumatic auxiliary orbit of a cross-atmosphere aircraft.
Example 1:
as shown in fig. 2, the method for rapidly evaluating the accessibility of the pneumatic auxiliary orbit of the aircraft crossing the atmosphere disclosed in the embodiment comprises the following steps:
step one: a longitudinal dynamics equation is established that correlates to the track dimensions.
Because only the influence of the pneumatic auxiliary orbit maneuver on the orbit semi-long axis and the eccentricity is considered when the accessibility assessment of the pneumatic auxiliary orbit maneuver is carried out, the orbit dimension parameters and the terminal position vector diameter and the track angle in the spherical coordinate system are directly related to the speed, and a simplified model of dimensionless and small earth rotation removal is adopted when the accessibility assessment is carried out:
where μ represents the gravitational constant of the earth, r represents the position vector diameter, v represents the velocity, and γ represents the track angle. Sigma is the roll angle and is also the process control quantity for pneumatically assisted flight. The lift acceleration L and the drag acceleration D after dimensionless are as follows:
wherein R is M Is the earth radius, S is the reference area of the aircraft, C L And C D The lift coefficient and drag coefficient, respectively, p is the dimensional planetary atmospheric density, and for simplicity of calculation, an exponential model is taken here, i.e. ρ=ρ 0 exp(-R M h/h s ) Wherein ρ is 0 Is the earth surface atmospheric density, h s Is the atmospheric density coefficient.
Step two: and establishing a description of the maneuvering accessibility boundary problem of the pneumatic auxiliary track according to the state parameters related to the track after the aircraft is out of the atmosphere.
For pneumatically assisted motorized procedures, the end state is determined by the exit position vector magnitude r f Velocity V f And track angle gamma f Described, since the earth atmosphere is at a certain altitude, r is f Constant value. The definition of the pneumatically assisted maneuver reachable range is described herein as all possible V f And gamma f Set A of compositions (t f ). The reachable range boundary is defined herein, i.e. the reachable capability boundary is O 0 (t f ) I.e. O 0 (t f )∈A(t f )。
Because the target track after pneumatic auxiliary track change cannot be tightly attached to the edge of the atmosphere, a minimum telecentric point height constraint h after the air is discharged needs to be added a,min Having a value greater than the atmospheric edge height h atm . The corresponding position vector diameter of the minimum telecentric point height constraint is r a,min ＝h a,min +R M The reachable range boundary corresponding to the constraint is marked as O L (t f ) Represented by curves B-C. The constraint equation corresponding to the constraint refers to the distance of the corresponding kepler orbit after the aircraft is out of the atmosphereThe height of the heart point is equal to h a,min In the specific form of
Wherein r is atm ＝h atm +R M Is the vector diameter of the atmospheric edge position. Since the number of dimensions of the orbit after the atmosphere is calculated by the end state variable, the equation (3) can be obtained by the energy square Cheng Tuidao of the elliptical orbit. Thus, the boundary O (t f ) Depicted by the closed segment curves a-B-C of fig. 1.
Step three: and converting the accessibility assessment problem into a state maximum/minimum problem P1 according to the fixed terminal speed, thereby giving a discrete solving framework of the dividing boundary.
Pneumatic auxiliary maneuvering reachable range A (t) f ) Can be generalized to determine its boundary O (t f ) Solving for O (t) f ) The core of (2) is to solve the constraint h of dividing the minimum telecentricity height a,min Determined boundary O L (t f ) Outside boundaries, i.e. O 0 (t f ). For O 0 (t f ) By rapidly dividing the boundary points, the boundary point calculation is summarized as: fixed speed V f Solving the maximum and minimum track angles gamma by optimizing f 。
As shown in fig. 2, for a certaink=1, …, N, straight line +.>With O 0 (t f ) Intersecting at two points, and the corresponding terminal track angle is +.>And->When solving them by optimization, the corresponding performance index is
Wherein solve forAnd->The corresponding optimal control problem comprises a control variable of roll angle sigma, the initial state is the state of the aircraft entering the atmosphere inlet, and the initial state is marked as x 0 ＝[r 0 ,V 0 ,γ 0 ] T . The optimal control problem is herein described as P1.
By fixing the terminal speed, the reachability boundary O (t f ) The evaluation problem of (2) is converted into a terminal track angle maximum/minimum problem as in formula (4), thereby giving a division boundary discrete solving framework.
Step four: aiming at the nonlinear optimal control problem P1, a sequence convex optimization framework of the aircraft accessibility boundary problem is constructed through independent variable substitution and lossless convex of the nonlinear problem.
The nonlinear optimal control problem P1 is a strong nonlinear optimal control problem because of the inclusion of nonlinear dynamical equations. Aiming at solving the nonlinear optimal control problem P1, by referring to the approximation theory of the nonlinear infinite dimensional system optimal control law, the sequence convex optimization framework of the aircraft accessibility boundary problem is constructed by independent variable replacement and lossless convex of the nonlinear problem, and nonlinear items in the optimal control problem P1 are respectively linearized near a given approximation solution by lossless convex, so that the linear optimal control sub-problem is obtained, and the sequence convex optimization framework of the aircraft accessibility boundary problem is obtained.
Step 4.1: the dimensionality reduction transformation of the nonlinear dynamics equation (1) is realized through independent variable substitution.
For the kinetic equation (1), consider the determinable by dimension-reduction partitioningReaching the capacity boundary point, the position vector diameter r f And velocity V f All are fixed values, so the section adopts variable substitution, and selects a new independent variable, namely e=1/r-V 2 The variable is actually equivalent to Kepler orbit energy, and the expression just comprises position vector diameter and speed, so that the terminal value is fixed when the accessibility boundary point is obtained. It is known from differentiation that de/dt=dv > 0 is constant, and therefore, this monotonically increasing variable can be used as an independent variable. By means of argument substitution, equation (1) is transformed into
From the above equation, the differential term of the velocity is eliminated from the dynamics equation, and the dynamics dimension is reduced.
Step 4.2: through variable transformation of the roll angle, the roll angle expression appearing in the dynamic equation in the form of cosine is converted into a linear form, and the linear constraint corresponding to the new control quantity is obtained.
For the control variables, a simple transformation is also required here. Since the control amount roll angle σ appears in the form of a cosine in the kinetic equation, it needs to be treated as linear as possible. Considering that cos σ monotonically increases between intervals [0 °,180 ° ], the new control amount u=cos σ is directly adopted. After the new control quantity is adopted, the constraint of the original roll angle is converted into
Wherein sigma min 0℃is the lower limit of the roll angle, and σ max The angle of less than or equal to 180 degrees is the upper limit of the tilting angle. In addition, the upper and lower bounds of the new control amount u after transformation are u min ＝cosσ max And u max ＝cosσ min . Obviously, the constraint corresponding to the new control amount is a linear constraint.
And 4.3, gradually linearizing the dynamics equation after the dimension reduction in the step 4.1 to obtain a linear dynamics equation after the dimension reduction.
In order for a linear approximation method to be viable, the kinetic equation needs to be linearized around the approximation solution. First, the dynamics equation (5) is written in the form of a nonlinear affine control system, i.e
Wherein the nonlinear term of the state variable is
While the coefficient matrix of the control variable is
B(x,e)＝[0,(L/D)/V 2 ] T (9)
For the control affine system, the nonlinear term (8) of the state variable of the control affine system needs to be linearized. When there is an approximate solution of x z Where z represents the z-th linear successive approximation, then the nonlinear term (8) can be found at x z Vicinity linearization. Furthermore, although there is a velocity V in the control coefficient B (x, e), it was found by observation of V that during pneumatically assisted maneuvers, dimensionless r.apprxeq.1, therefore V at the time of the z+1th linear successive approximation z+1 By means of a beltAnd calculating, so that the accuracy of V calculation can be ensured when convergence is finally approximated. At this time, the corresponding linearized dynamic equation after dimension reduction is
Wherein the linear coefficient matrix a (x z ) Is f 0 Jacobian matrix of (x, e).
To this end, for the nonlinear optimal control problem P1, the dimension reduction is performed through independent variable substitution to form a dynamic equation (5), and the dynamic equation (10) is converted into a linear dynamic equation (10) after dimension reduction through successive linearization, and since the process constraint (6) and the dynamic equation (10) are linear constraints, the process constraint (6) and the dynamic equation (10) are the sequential convex optimization framework of the constructed aircraft accessibility boundary problem.
Step five: and rapidly solving the nonlinear optimal control problem P1 under the tilting control through the sequence convex optimization framework to obtain a maximum minimum track angle at any terminal speed, wherein the maximum minimum track angle corresponds to two points on the accessibility boundary, and the aerodynamic auxiliary accessibility boundary of the aircraft can be obtained by connecting all boundary points.
Step six: based on the pneumatic auxiliary accessibility boundary of the aircraft, the range of the feasible target orbit of the aircraft for the pneumatic auxiliary flight crossing the atmosphere is rapidly estimated, so that the reasonable design of the target orbit in the task implementation process is guided.
In the flow, the accessibility boundary can be obtained rapidly through linear approximation by setting any initial state, so that the universality and the high efficiency of the method are reflected.
To verify the feasibility of the method, taking pneumatic auxiliary maneuvering in a Mars environment as an example, the relevant physical parameters are all taken from a Mars environment model. Taking pneumatic auxiliary initial state x 0 ＝[r 0 ,V 0 ,γ 0 ] T As shown in table 1.
TABLE 1 pneumatic auxiliary initial State variable
Meanwhile, the upper and lower boundaries of the roll angle in the formula (6) are respectively sigma min =15°, and σ max =165°. In addition, the minimum telecentric point height constraint is h a,min ＝200km。
Since the solution method corresponding to the linear programming problem after direct conversion from the problem P1 is complete and efficient, there is no need to propose a new algorithm for the linear programming problem. Considering the robustness and rapidity of the original-dual interior point method in solving the linear programming problem, the patent directly adopts the method to solve the problem.
After the problem P1 is processed through the third step and the fourth step to obtain the problem, the complete accessibility boundary can be obtained through the calculation flow of the fifth step. Fig. 4 shows the geometry of the reachability boundary in terms of end states in this embodiment, and it can be seen from fig. 4 that the reachability range geometry represented by speed and track angle is in the form of a band. The area is wider in the region of lower speed, which also indicates that the low energy orbit is more accessible after the pneumatically assisted flight. Fig. 5 shows the geometry of the achievable capacity boundary in this embodiment in terms of the near-center altitude and eccentricity, and it can be seen from fig. 5 that a larger terminal track angle corresponds to a larger eccentricity, i.e. the larger the track angle, the smaller the energy attenuation of the track, the closer the track to a larger elliptical track for a pneumatically assisted atmospheric flight.
In order to demonstrate the effectiveness of the method presented in this patent, statistics of the calculation time are presented here. With MATLAB-R2018a on Intel i7-4790K CPU@4.00GHz desktop as the computing platform, the total time consumption of the reach envelope of fig. 4 was calculated at a single time to be 7.47s, wherein the total calculated envelope points were 57, which is equivalent to the time consumption of single point calculation to be 0.131s. Furthermore, the number of iteration steps generally required for single point calculation is about 4 to 7 steps, so the time required for each step of linear approximation iteration is about 0.02 to 0.03s. The time statistics fully verify the rapidity of the accessibility boundary assessment method provided by the patent.
While the foregoing has been provided for the purpose of illustrating the general principles of the invention, it will be understood that the foregoing disclosure is only illustrative of the principles of the invention and is not intended to limit the scope of the invention, but is to be construed as limited to the specific principles of the invention.
Claims (2)
1. A method for rapidly evaluating the accessibility of a pneumatic auxiliary orbit of an aircraft crossing the atmosphere is characterized by comprising the following steps of: comprises the following steps of the method,
step one: establishing a longitudinal dynamics equation related to the track size;
the first implementation method of the step is that,
because only the influence of the pneumatic auxiliary orbit maneuver on the orbit semi-long axis and the eccentricity is considered when the accessibility assessment of the pneumatic auxiliary orbit maneuver is carried out, the orbit dimension parameters and the terminal position vector diameter and the track angle in the spherical coordinate system are directly related to the speed, and a simplified model of dimensionless and small earth rotation removal is adopted when the accessibility assessment is carried out:
wherein mu represents the gravitational constant of the earth, r represents the position vector diameter, v represents the speed, and gamma represents the track angle; sigma is a roll angle and is also a process control quantity of pneumatically assisted flight; the lift acceleration L and the drag acceleration D after dimensionless are as follows:
wherein R is M Is the earth radius, S is the reference area of the aircraft, C L And C D The lift coefficient and drag coefficient, respectively, p is the dimensional planetary atmospheric density, and for simplicity of calculation, an exponential model is taken here, i.e. ρ=ρ 0 exp(-R M h/h s ) Wherein ρ0 is the earth surface atmospheric density, h s Is the atmospheric density coefficient;
step two: establishing a description of a pneumatic auxiliary orbit maneuver accessibility boundary problem according to state parameters associated with the orbit after the aircraft is out of the atmosphere;
the implementation method of the second step is that,
for pneumatically assisted motorized procedures, the end state is determined by the exit position vector magnitude r f Velocity V f And track angle gamma f Description due to the earth's atmosphereThe height is fixed, thus r f Constant value; the definition of the pneumatically assisted maneuver reachable range is described herein as all possible V f And gamma f Set A of compositions (t f ) The method comprises the steps of carrying out a first treatment on the surface of the The reachable range boundary is defined herein, i.e. the reachable capability boundary is O 0 (t f ) I.e. O 0 (t f )∈A(t f )；
Because the target track after pneumatic auxiliary track change cannot be tightly attached to the edge of the atmosphere, a minimum telecentric point height constraint h after the air is discharged needs to be added a,min Having a value greater than the atmospheric edge height h atm The method comprises the steps of carrying out a first treatment on the surface of the The corresponding position vector diameter of the minimum telecentric point height constraint is r a,min ＝h a,min +R M The reachable range boundary corresponding to the constraint is marked as O L (t f ) Represented by curves B-C; the constraint equation corresponding to the constraint means that after the aircraft is out of the atmosphere, the height of the telecentric point of the corresponding kepler orbit is equal to h a,min In the specific form of
Wherein r is atm ＝h atm +R M Is the vector diameter of the edge position of the atmosphere; since the size number of the orbit after the atmosphere is calculated by the end state variable, the formula (3) can be obtained by the energy square Cheng Tuidao of the elliptical orbit; thus, the boundary O (t f ) Described by closed segment curves a-B-C;
step three: according to the speed of a fixed terminal, converting the accessibility assessment problem into a state maximum/minimum problem P1, thereby giving a discrete solving framework of a dividing boundary;
the implementation method of the third step is that,
pneumatic auxiliary maneuvering reachable range A (t) f ) Can be generalized to determine its boundary O (t f ) Solving for O (t) f ) The core of (2) is to solve the constraint h of dividing the minimum telecentricity height a,min Determined boundary O L (t f ) Outside boundaries, i.e. O 0 (t f ) The method comprises the steps of carrying out a first treatment on the surface of the For O 0 (t f ) By rapidly dividing the boundary points, the boundary point calculation is summarized as: fixed speed V f Solving the maximum and minimum track angles gamma by optimizing f ；
For a determinedStraight line->With O 0 (t f ) Intersecting at two points, and the corresponding terminal track angle is +.>And->When solving them by optimization, the corresponding performance index is
Wherein solve forAnd->The corresponding optimal control problem comprises a control variable of roll angle sigma, the initial state is the state of the aircraft entering the atmosphere inlet, and the initial state is marked as x 0 ＝[r 0 ,V 0 ,γ 0 ] T The method comprises the steps of carrying out a first treatment on the surface of the The optimal control problem is herein denoted as P1;
by fixing the terminal speed, the reachability boundary O (t f ) The evaluation problem of (2) is converted into a terminal track angle maximum/minimum problem as shown in the formula (4), so that a discrete solving framework of a dividing boundary is provided;
step four: aiming at the nonlinear optimal control problem P1, constructing a sequence convex optimization framework of the aircraft accessibility boundary problem through independent variable substitution and lossless convex of the nonlinear problem;
the realization method of the fourth step is that,
step 4.1: the dimensionality reduction transformation of the nonlinear dynamics equation (1) is realized through independent variable substitution;
for the dynamics equation (1), consider the position vector r at the reachable ability boundary point determined by dimension-reduction division f And velocity V f All are fixed values, so the section adopts variable substitution, and selects a new independent variable, namely e=1/r-V 2 The variable is actually equivalent to Kepler orbit energy, and the expression thereof just comprises position vector diameter and speed, so that the terminal value is fixed when the accessibility boundary point is obtained; it is known from differentiation that de/dt=dv > 0 is constant, and therefore, this monotonically increasing variable can be used as an independent variable; by means of argument substitution, equation (1) is transformed into
The equation shows that the differential term of the speed is eliminated in the dynamics equation, and the dynamics dimension is reduced;
step 4.2: converting a roll angle expression appearing in a dynamic equation in a cosine form into a linear form through variable transformation of the roll angle, and obtaining a linear constraint corresponding to the new control quantity;
for the control variables, a simple transformation is also required here; since the control amount roll angle σ appears in the form of a cosine in the kinetic equation, it needs to be processed as linearly as possible; considering that cos σ monotonically increases between intervals [0 °,180 ° ], the new control amount u=cos σ is directly adopted; after the new control quantity is adopted, the constraint of the original roll angle is converted into
Wherein sigma min 0℃is the lower limit of the roll angle, and σ max The angle of less than or equal to 180 degrees is the upper limit of the tilting angle; in addition, the upper and lower bounds of the new control amount u after transformation are u min ＝cosσ max And u max ＝cosσ min The method comprises the steps of carrying out a first treatment on the surface of the Obviously, the constraint corresponding to the new control quantity is a linear constraint;
step 4.3: gradually linearizing the dynamics equation after the dimension reduction in the step 4.1 to obtain a linear dynamics equation after the dimension reduction;
in order to make a linear approximation method viable, the kinetic equation needs to be linearized around the approximation solution; first, the dynamics equation (5) is written in the form of a nonlinear affine control system, i.e
Wherein the nonlinear term of the state variable is
While the coefficient matrix of the control variable is
B(x,e)＝[0,(L/D)/V 2 ] T (9)
For the affine control system, the nonlinear term (8) of the state variable of the affine control system needs to be linearized; when there is an approximate solution of x z Where z represents the z-th linear successive approximation, then the nonlinear term (8) can be found at x z Linearization of the vicinity; furthermore, although there is a velocity V in the control coefficient B (x, e), it was found by observation of V that during pneumatically assisted maneuvers, dimensionless r.apprxeq.1, therefore V at the time of the z+1th linear successive approximation z+1 By means of a beltThe calculation is carried out, so that the correct calculation of V can be ensured when convergence is approached finally; at this time, the corresponding dimension-reduced lineThe sexualization kinetic equation is that
Wherein the linear coefficient matrix a (x z ) Is f 0 A jacobian matrix of (x, e);
to this end, for the nonlinear optimal control problem P1, the dimensionality reduction is performed through independent variable substitution to form a dynamic equation (5), and the dynamic equation (10) is converted into a linear dynamic equation (10) after the dimensionality reduction through successive linearization, and as the process constraint (6) and the dynamic equation (10) are linear constraints, the process constraint (6) and the dynamic equation (10) are sequence convex optimization frames of the constructed aircraft accessibility boundary problem;
the nonlinear optimal control problem P1 is a strong nonlinear optimal control problem due to the inclusion of nonlinear dynamic equations; aiming at solving the nonlinear optimal control problem P1, constructing a sequence convex optimization framework of the aircraft accessibility boundary problem by referring to the approximation theory of the nonlinear infinite dimensional system optimal control law and through independent variable replacement and lossless convexity of the nonlinear problem, and linearizing nonlinear items in the optimal control problem P1 near a given approximation solution respectively through lossless convexity so as to obtain a linear optimal control sub-problem, namely, a sequence convex optimization framework of the aircraft accessibility boundary problem;
step five: and rapidly solving the nonlinear optimal control problem P1 under the tilting control through the sequence convex optimization framework to obtain a maximum minimum track angle at any terminal speed, wherein the maximum minimum track angle corresponds to two points on the accessibility boundary, and the aerodynamic auxiliary accessibility boundary of the aircraft can be obtained by connecting all boundary points.
2. A method for rapid assessment of accessibility of a pneumatically assisted orbit of an aircraft crossing the atmosphere according to claim 1, wherein: and step six, rapidly evaluating the feasible target track range of the aircraft for the aerodynamically assisted flight across the atmosphere based on the aerodynamically assisted accessibility boundary of the aircraft, thereby guiding the reasonable design of the target track in the task implementation process.
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