CN113671826A - Method for rapidly evaluating accessibility of pneumatic auxiliary track of cross-atmosphere aircraft - Google Patents
Method for rapidly evaluating accessibility of pneumatic auxiliary track of cross-atmosphere aircraft Download PDFInfo
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Abstract
The invention discloses a method for rapidly evaluating the reachable ability of a pneumatic auxiliary orbit of a cross-atmospheric aircraft, and belongs to the technical field of aerospace. The implementation method of the invention comprises the following steps: establishing a longitudinal dynamic equation related to the size of the track on the basis of the given initial parameters and system parameters; establishing a boundary problem description of the maneuvering accessibility of the pneumatic auxiliary track; the reachable capacity evaluation problem is converted into a class of state maximum/minimum problem according to the fixed terminal speed by analyzing the reachable capacity boundary of the pneumatic auxiliary engine, and a divided boundary discrete solving frame is given; 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 replacement and non-destructive convex of a nonlinear problem, and the rapid assessment of the accessibility of the pneumatic auxiliary track of the cross-domain aircraft is realized. The method has the advantages of strong robustness, high repeatability, high precision, strong reliability and high evaluation efficiency, and does not have 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 reachable capacity of a pneumatic auxiliary orbit of an aerocraft across the atmospheric layer, in particular to a method suitable for determining the range of the number of orbits of the aerocraft after pneumatic synergistic action, and belongs to the technical field of aerospace.
Background
The air vehicle across the atmospheric layer is an air vehicle widely researched at present and applied to a cross-domain orbit maneuver task, and has the space task execution capacity and an atmospheric layer controllable flight structure, so that the air vehicle across the atmospheric layer becomes an important direction for the development of the air-space integrated space mission in the future. In the process of executing a task of an air vehicle across the atmosphere, the feasibility of the task is determined by the orbital maneuverability under the assistance of pneumatic coordination, and the analysis of the feasibility of the task is usually a key link and a primary step of the overall design, so that the method plays an important role in the success of the task. Since the cross-domain orbit maneuvering capability depends not only on the current state of the aircraft, but also on the system parameters and the control capability of the aircraft, the reachable capability range assessment problem belongs to the nonlinear optimal control problem of continuous control parameters. Currently, the existing reachable ability assessment usually refers to limiting a control parameter, then obtaining a certain number of reachable upper and lower boundaries, or converting a continuous optimal control problem into a single-parameter solution problem by simplifying a problem model. However, none of the above methods are suitable for accurate assessment of the maneuverability of an aircraft across the atmosphere with continuous control of roll. Therefore, the method for rapidly evaluating the accessibility of the pneumatic auxiliary orbit of the cross-atmospheric aircraft, which is provided by the patent, can not only realize the determination of the range of the maneuvering orbit under any initial parameter, but also ensure the calculation efficiency of evaluation, thereby meeting the technical requirements of the design and application of the future cross-region aircraft.
In the developed method for evaluating the reachable ability of the pneumatic auxiliary orbit of the aircraft, the prior art [1] (see Linxi Strength, Zhang Yanglin, discrimination method of the reachable range of the pneumatic auxiliary different surface orbital transfer [ J ]. academic newspaper of national defense science and technology university, 2000,22(2):7-10.) provides a calculation formula which is derived by a coordinate transformation method and contains the inclination angle change range of a certain aircraft relative to an initial orbit (an orbit which is calculated by assuming that the inclination angle of the initial orbit is 0), and the reachable range of the pneumatic auxiliary different surface at any initial inclination angle is obtained. However, there is no clear calculation method for the range of the orbit that can be reached by the aircraft, and it is difficult to apply the method to the evaluation of the reachable ability.
In the prior art [2] (see: Guelman M.planar Aeroassisted accessibility Domain [ J ]. Journal of Guidance, Control, and Dynamics,1997,20(3): 422-.
Disclosure of Invention
The invention discloses a method for quickly evaluating the reachable capacity of a pneumatic auxiliary track of a cross-atmospheric aircraft, which aims to solve the technical problems that: the method comprises the steps of constructing a sequence convex optimization framework of an aircraft reachable ability boundary problem, rapidly solving a nonlinear optimal control problem under the tilt control through the sequence convex optimization framework, obtaining a reachable ability boundary with the maximum and minimum range of a terminal track as a performance index, and achieving rapid evaluation of the reachable ability of the pneumatic auxiliary track. The invention has the following advantages: (1) the robustness is strong, and the repeatability is high; (2) the precision is high, and the reliability is strong; (3) the initial state and the system of the aircraft are not strictly limited; (4) the evaluation efficiency is high, and the online evaluation capability is achieved.
The purpose of the invention is realized by the following technical scheme:
the invention discloses a method for quickly evaluating the reachable ability of a pneumatic auxiliary track of an air vehicle crossing the atmosphere. The reachable ability evaluation problem is converted into a class of state maximum/minimum problem according to the fixed terminal speed by analyzing the reachable ability boundary of the pneumatic auxiliary engine, so that a division boundary discrete solving framework is provided. Aiming at the optimal control problem of the discrete boundary points, a sequence convex optimization framework of the aircraft accessibility boundary problem is constructed through independent variable replacement and non-destructive convex of a nonlinear problem, and therefore the rapid assessment of the accessibility of the cross-domain aircraft pneumatic auxiliary track is achieved.
The invention discloses a method for quickly evaluating the reachable capacity of a pneumatic auxiliary track of a cross-atmospheric aircraft, which comprises the following steps:
the method comprises the following steps: and establishing a longitudinal dynamic equation related to the size of the rail.
Because the influence of the pneumatic auxiliary maneuvering on the semi-major axis and the eccentricity of the orbit is only considered when the accessibility of the pneumatic auxiliary orbit maneuvering is evaluated, and the dimensional parameters of the orbit are directly related to the terminal position radial, the track angle and the speed in a spherical coordinate system, a non-dimensionalized simplified model and a reduced model of going to earth rotation are adopted when the accessibility is evaluated:
where μ represents the gravitational constant of the earth, r represents the position vector, v represents the velocity, and γ represents the track angle. σ is the roll angle, which is also the process control quantity for the pneumatically assisted flight. The lift acceleration L and the drag acceleration D after dimensionless are as follows:
wherein R isMIs the radius of the earth, S is the reference area of the aircraft, CLAnd CDLift coefficient and drag coefficient, respectively, and ρ is the dimensional planetary atmospheric density, and for the sake of simplifying the calculation, an exponential model is taken here, that is, ρ ═ ρ0exp(-RMh/hs) Where ρ is0Is the atmospheric density, h, of the earth's surfacesIs the atmospheric density coefficient.
Step two: and establishing description of the boundary problem of the maneuvering accessibility of the pneumatic auxiliary track according to the state parameters associated with the track after the aircraft leaves the atmosphere.
For pneumatically assisted maneuvers, the terminal state passes through the exit positionRadius size rfVelocity VfAnd track angle gammafDescribed, since the earth has a constant atmospheric height, r isfIs constant. The definition of the reach of the pneumatic auxiliary maneuver is described herein as all possible VfAnd gammafSet of constituents A (t)f). The boundary of the reach range is defined herein, i.e., the reach capability boundary is O0(tf) I.e. O0 (t)f)∈A(tf)。
Because the target track after pneumatic auxiliary track transfer can not be tightly attached to the edge of the atmosphere, a minimum distance point height constraint h after the target track is out of the atmosphere needs to be addeda,minA value greater than the atmospheric edge height hatm. The minimum distance center point height constraint corresponds to a position radial of ra,min＝ha,min+RMThe reachable range boundary corresponding to the constraint is marked as OL(tf) Represented by curves B-C. The constraint equation corresponding to the constraint means that the corresponding Kepler orbit apocenter height is equal to h after the aircraft is out of the atmospherea,minIn a specific form of
Wherein r isatm＝hatm+RMIs the atmospheric edge position radial. Since the size number of the orbit after the atmosphere is released is calculated by the terminal state variable, the equation (3) can be derived by the energy equation of the elliptical orbit. Thus, the boundary O (t) of the reachable range is captured pneumaticallyf) Described by the closed piecewise curve a-B-C.
Step three: and converting the reachable capability evaluation problem into a type of state maximum/minimum problem P1 according to the fixed terminal speed, thereby providing a division boundary discrete solution framework.
Step two solved pneumatic auxiliary maneuvering reachable range A (t)f) Can be summarized as determining its boundary O (t)f) To solve for O (t)f) The core of (1) is to solve the height constraint h of the minimum apocenter pointa,minDetermined boundary OL(tf) Outer boundary, i.e. O0(tf). For O0(tf) By quickly dividing boundary points, the calculation of the boundary points is summarized as follows: constant velocity VfSolving for the maximum and minimum track angles gamma by optimizationf。
For a certain
k is 1, …, N, straight line
And O0(tf) Intersect at two points, and respectively correspond to tail end flight path angles of
And
when solving them by optimization, the corresponding performance index is
Wherein the solution is
And
the corresponding optimal control problem comprises a control variable which is an inclination angle sigma, and the initial state is a state of the aircraft entering an atmospheric inlet and is marked as x0＝[r0,V0,γ0]T. The optimal control problem is referred to herein as P1.
By fixing the terminal speed, the reachable capability bound O (t)f) The evaluation problem is converted into the terminal track angle maximum/minimum problem shown in the formula (4), and therefore a division boundary discrete solving framework is given.
Step four: aiming at the nonlinear optimal control problem P1, a sequence convex optimization framework of the aircraft reachable ability boundary problem is constructed through independent variable replacement and lossless convex of the nonlinear problem.
The nonlinear optimal control problem P1 is a strong nonlinear optimal control problem due to the inclusion of nonlinear dynamical equations. Aiming at the solution of the nonlinear optimal control problem P1, a sequence convex optimization framework of the aircraft reachable ability boundary problem is constructed by using an approximation theory of the optimal control law of a nonlinear infinite dimensional system for reference, independent variable replacement and lossless convexity of the nonlinear problem, and nonlinear items in the optimal control problem P1 are linearized near a given approximate solution through the lossless convexity, so that a linear optimal control subproblem is obtained, and the sequence convex optimization framework of the aircraft reachable ability boundary problem is obtained.
Step 4.1: the dimension reduction transformation of the nonlinear dynamical equation (1) is realized through independent variable replacement.
For equation (1) of dynamics, the position vector r is considered at the reachable capability boundary point determined by dimension reduction partitioningfAnd velocity VfAre all fixed values, so this subsection uses variable substitution with a new argument, i.e., e-1/r-V2And/2, the variable is actually equivalent to Kepler orbit energy, and the expression of the variable just only comprises a position radius and a speed, so that the terminal value of the variable is fixed when the reachable capacity boundary point is obtained. By differentiating it, de/dt ═ DV > 0 is always true, and therefore this monotonically increasing variable can be used as an independent variable. By substituting the independent variable, equation (1) is transformed into
As seen from the above equation, the kinetic equation eliminates the differential term of the velocity, and reduces the kinetic dimension.
Step 4.2: and converting the roll angle expression which is in a cosine form and appears in a kinetic equation into a linear form through variable transformation of the roll angle, and obtaining the linear constraint corresponding to the new controlled variable.
For the control variables, here too, a simple transformation is required. Since the controlled variable roll angle σ appears in the form of cosine in the kinetic equation, it needs to be treated as linear as possible. Considering that cos σ monotonically increases between the intervals [0 °,180 ° ], the new control amount u is directly adopted as cos σ. After the new control quantity is adopted, the constraint of the original roll angle is converted into
Wherein σminNot less than 0 DEG is the lower limit of the roll angle, and sigmamaxThe inclination angle is not more than 180 degrees and is the upper limit of the inclination angle. In addition, the upper and lower bounds of the new control amount u after the transformation are u, respectivelymin＝cosσmaxAnd umax＝cosσmin. Obviously, the constraint corresponding to the new control amount is a linear constraint.
Step 4.3: and (4) carrying out successive linearization on the dynamic equation subjected to the dimension reduction in the step 4.1 to obtain a linear dynamic equation subjected to the dimension reduction.
In order to make the linear approximation method feasible, the kinetic equations need to be linearized in the vicinity of the approximation solution. The kinetic equation (5) is first written in the form of a nonlinear affine control system, i.e.
Wherein the state variable nonlinear term is
And the coefficient matrix of the control variable is
B(x,e)＝[0,(L/D)/V2]T (9)
For the above control affine system, its state variable nonlinear term (8) needs to be linearized. When there is an approximate solution of xzWhere z represents the z-th linear successive approximation, then the non-linear term (8) can be at xzAnd (4) carrying out adjacent linearization. Further, although there is a velocity V in the control coefficient B (x, e)However, it has been found by observing V that during the pneumatic assist maneuver, there is no dimension r ≈ 1, so V is the z +1 th linear successive approximationz+1By using the formula
And the calculation can further ensure that the calculation of V is correct when the convergence is approached finally. At this time, the corresponding linearized kinetic equation after dimension reduction is
Wherein the linear coefficient matrix A (x) of the state variablesz) Is f0The Jacobian matrix of (x, e).
At this point, for the nonlinear optimal control problem P1, the dimension is reduced into the kinetic equation (5) through independent variable substitution, and the linearized kinetic equation (10) after dimension reduction is converted through successive linearization, and since the process constraint (6) and the kinetic equation (10) are linear constraints, the process constraint (6) and the kinetic equation (10) are a sequence convex optimization framework of the constructed aircraft reachable ability boundary problem.
Step five: the nonlinear optimal control problem P1 under the tilt control is quickly solved through the sequence convex optimization framework, the maximum and minimum track angles under any terminal speed are obtained, the maximum and minimum track angles correspond to two points on the reachable capacity boundary, and the pneumatically-assisted reachable capacity boundary of the aircraft can be obtained by connecting all boundary points.
Step six: based on the pneumatic auxiliary reachable capacity boundary of the aircraft, the feasible target orbit range of the aircraft in the atmospheric layer pneumatic auxiliary flight is rapidly evaluated, so that the reasonable design of the target orbit in the task implementation process is guided.
Has the advantages that:
1. the quick accessibility evaluation method for the pneumatic auxiliary track of the cross-atmosphere aircraft disclosed by the invention has universality by fixing the terminal speed and converting the accessibility evaluation problem into the terminal track angle maximum/minimum problem so as to provide a division boundary discrete solving frame.
2. According to the method for rapidly evaluating the accessibility of the pneumatic auxiliary orbit of the cross-atmosphere aircraft, disclosed by the invention, the dimension reduction transformation of a nonlinear kinetic equation is realized through independent variable replacement, the kinetic dimension is reduced, the number of calculated variables is reduced, the solving speed is high, the calculation time is short, and the evaluation efficiency is further improved.
3. The quick evaluation method for the accessibility of the pneumatic auxiliary orbit of the cross-atmospheric aircraft disclosed by the invention can be suitable for evaluating the accessibility of the pneumatic auxiliary orbit in any planetary environment by setting different system parameters, so that the application range of the method to a target environment is wide.
4. The invention discloses a method for rapidly evaluating the reachable ability of a pneumatic auxiliary orbit of a cross-atmospheric aircraft, which converts process constraint and nonlinear dynamics formed by control variables into linear constraint through lossless convexity, and the conversion process is not influenced by specific orbit parameters, so that the method has wide application range on a target orbit.
5. The method for rapidly evaluating the accessibility of the pneumatic auxiliary orbit of the cross-atmospheric aircraft disclosed by the invention has the advantages of strong robustness and high repeatability because the initial state and the aircraft parameters are arbitrarily given.
Drawings
FIG. 1 is a schematic diagram of the reachable capability boundary in step 1 of the present invention;
FIG. 2 is a flow chart of a method for rapidly evaluating the reachable ability of a pneumatic auxiliary track of a cross-atmospheric aircraft according to the invention;
FIG. 3 is a diagram illustrating the dimension reduction partitioning strategy in step 3 according to the present invention;
FIG. 4 is a geometry of the reachable capability boundary in the terminal state representation in this embodiment;
FIG. 5 is a geometric structure of the achievable capability boundary in this embodiment in terms of the height of the isocenter and eccentricity.
Detailed Description
To better illustrate the objects and advantages of the present invention, the present invention is explained in detail below by simulation analysis of the accessibility assessment problem of a pneumatically assisted orbit of an air vehicle across the atmosphere.
Example 1:
as shown in fig. 2, the method for rapidly evaluating the reachable capacity of the pneumatic auxiliary track of the cross-atmosphere aircraft disclosed in the embodiment includes the following steps:
the method comprises the following steps: and establishing a longitudinal dynamic equation related to the size of the rail.
Because the influence of the pneumatic auxiliary maneuvering on the semi-major axis and the eccentricity of the orbit is only considered when the accessibility of the pneumatic auxiliary orbit maneuvering is evaluated, and the dimensional parameters of the orbit are directly related to the terminal position radial, the track angle and the speed in a spherical coordinate system, a non-dimensionalized simplified model and a reduced model of going to earth rotation are adopted when the accessibility is evaluated:
where μ represents the gravitational constant of the earth, r represents the position vector, v represents the velocity, and γ represents the track angle. σ is the roll angle, which is also the process control quantity for the pneumatically assisted flight. The lift acceleration L and the drag acceleration D after dimensionless are as follows:
wherein R isMIs the radius of the earth, S is the reference area of the aircraft, CLAnd CDLift coefficient and drag coefficient, respectively, and ρ is the dimensional planetary atmospheric density, and for the sake of simplifying the calculation, an exponential model is taken here, that is, ρ ═ ρ0exp(-RMh/hs) Where ρ is0Is the atmospheric density, h, of the earth's surfacesIs the atmospheric density coefficient.
Step two: and establishing description of the boundary problem of the maneuvering accessibility of the pneumatic auxiliary track according to the state parameters associated with the track after the aircraft leaves the atmosphere.
For pneumatically assisted maneuvers, the terminal state is through the exit position radiusSize rfVelocity VfAnd track angle gammafDescribed, since the earth has a constant atmospheric height, r isfIs constant. The definition of the reach of the pneumatic auxiliary maneuver is described herein as all possible VfAnd gammafSet of constituents A (t)f). The boundary of the reach range is defined herein, i.e., the reach capability boundary is O0(tf) I.e. O0(tf)∈A(tf)。
Because the target track after pneumatic auxiliary track transfer can not be tightly attached to the edge of the atmosphere, a minimum distance point height constraint h after the target track is out of the atmosphere needs to be addeda,minA value greater than the atmospheric edge height hatm. The minimum distance center point height constraint corresponds to a position radial of ra,min＝ha,min+RMThe reachable range boundary corresponding to the constraint is marked as OL(tf) Represented by curves B-C. The constraint equation corresponding to the constraint means that the corresponding Kepler orbit apocenter height is equal to h after the aircraft is out of the atmospherea,minIn a specific form of
Wherein r isatm＝hatm+RMIs the atmospheric edge position radial. Since the size number of the orbit after the atmosphere is released is calculated by the terminal state variable, the equation (3) can be derived by the energy equation of the elliptical orbit. Thus, the boundary O (t) of the reachable range is captured pneumaticallyf) Depicted by the closed piecewise curve a-B-C as in fig. 1.
Step three: and converting the reachable capability evaluation problem into a type of state maximum/minimum problem P1 according to the fixed terminal speed, thereby providing a division boundary discrete solution framework.
Step two solved pneumatic auxiliary maneuvering reachable range A (t)f) Can be summarized as determining its boundary O (t)f) To solve for O (t)f) The core of (1) is to solve the height constraint h of the minimum apocenter pointa,minDetermined boundary OL(tf) Outer boundary, i.e. O0(tf). For O0(tf) By quickly dividing boundary points, the calculation of the boundary points is summarized as follows: constant velocity VfSolving for the maximum and minimum track angles gamma by optimizationf。
As shown in fig. 2, for a certain
k is 1, …, N, straight line
And O0(tf) Intersect at two points, and respectively correspond to tail end flight path angles of
And
when solving them by optimization, the corresponding performance index is
Wherein the solution is
And
the corresponding optimal control problem comprises a control variable which is an inclination angle sigma, and the initial state is a state of the aircraft entering an atmospheric inlet and is marked as x0＝[r0,V0,γ0]T. The optimal control problem is referred to herein as P1.
By fixing the terminal speed, the reachable capability bound O (t)f) The evaluation problem is converted into the terminal track angle maximum/minimum problem shown in the formula (4), and therefore a division boundary discrete solving framework is given.
Step four: aiming at the nonlinear optimal control problem P1, a sequence convex optimization framework of the aircraft reachable ability boundary problem is constructed through independent variable replacement and lossless convex of the nonlinear problem.
The nonlinear optimal control problem P1 is a strong nonlinear optimal control problem due to the inclusion of nonlinear dynamical equations. Aiming at the solution of the nonlinear optimal control problem P1, a sequence convex optimization framework of the aircraft reachable ability boundary problem is constructed by using an approximation theory of the optimal control law of a nonlinear infinite dimensional system for reference, independent variable replacement and lossless convexity of the nonlinear problem, and nonlinear items in the optimal control problem P1 are linearized near a given approximate solution through the lossless convexity, so that a linear optimal control subproblem is obtained, and the sequence convex optimization framework of the aircraft reachable ability boundary problem is obtained.
Step 4.1: the dimension reduction transformation of the nonlinear dynamical equation (1) is realized through independent variable replacement.
For equation (1) of dynamics, the position vector r is considered at the reachable capability boundary point determined by dimension reduction partitioningfAnd velocity VfAre all fixed values, so this subsection uses variable substitution with a new argument, i.e., e-1/r-V2And/2, the variable is actually equivalent to Kepler orbit energy, and the expression of the variable just only comprises a position radius and a speed, so that the terminal value of the variable is fixed when the reachable capacity boundary point is obtained. By differentiating it, de/dt ═ DV > 0 is always true, and therefore this monotonically increasing variable can be used as an independent variable. By substituting the independent variable, equation (1) is transformed into
As seen from the above equation, the kinetic equation eliminates the differential term of the velocity, and reduces the kinetic dimension.
Step 4.2: and converting the roll angle expression which is in a cosine form and appears in a kinetic equation into a linear form through variable transformation of the roll angle, and obtaining the linear constraint corresponding to the new controlled variable.
For the control variables, here too, a simple transformation is required. Since the controlled variable roll angle σ appears in the form of cosine in the kinetic equation, it needs to be treated as linear as possible. Considering that cos σ monotonically increases between the intervals [0 °,180 ° ], the new control amount u is directly adopted as cos σ. After the new control quantity is adopted, the constraint of the original roll angle is converted into
Wherein σminNot less than 0 DEG is the lower limit of the roll angle, and sigmamaxThe inclination angle is not more than 180 degrees and is the upper limit of the inclination angle. In addition, the upper and lower bounds of the new control amount u after the transformation are u, respectivelymin＝cosσmaxAnd umax＝cosσmin. Obviously, the constraint corresponding to the new control amount is a linear constraint.
And 4.3, carrying out successive linearization on the dynamic equation subjected to the dimension reduction in the step 4.1 to obtain a linear dynamic equation subjected to the dimension reduction.
In order to make the linear approximation method feasible, the kinetic equations need to be linearized in the vicinity of the approximation solution. The kinetic equation (5) is first written in the form of a nonlinear affine control system, i.e.
Wherein the state variable nonlinear term is
And the coefficient matrix of the control variable is
B(x,e)＝[0,(L/D)/V2]T (9)
For the above control affine system, its state variable nonlinear term (8) needs to be linearized. When there is an approximate solution of xzWhere z represents the z-th linear successive approximation, then the non-linear term (8) can be at xzAnd (4) carrying out adjacent linearization. In addition, although controlThere is a velocity V in the coefficient B (x, e), but it is found by observing V that during the pneumatic assist maneuver, there is no dimension r ≈ 1, so V is the z +1 th linear successive approximationz+1By using the formula
And the calculation can further ensure that the calculation of V is correct when the convergence is approached finally. At this time, the corresponding linearized kinetic equation after dimension reduction is
Wherein the linear coefficient matrix A (x) of the state variablesz) Is f0The Jacobian matrix of (x, e).
At this point, for the nonlinear optimal control problem P1, the dimension is reduced into the kinetic equation (5) through independent variable substitution, and the linearized kinetic equation (10) after dimension reduction is converted through successive linearization, and since the process constraint (6) and the kinetic equation (10) are linear constraints, the process constraint (6) and the kinetic equation (10) are a sequence convex optimization framework of the constructed aircraft reachable ability boundary problem.
Step five: the nonlinear optimal control problem P1 under the tilt control is quickly solved through the sequence convex optimization framework, the maximum and minimum track angles under any terminal speed are obtained, the maximum and minimum track angles correspond to two points on the reachable capacity boundary, and the pneumatically-assisted reachable capacity boundary of the aircraft can be obtained by connecting all boundary points.
Step six: based on the pneumatic auxiliary reachable capacity boundary of the aircraft, the feasible target orbit range of the aircraft in the atmospheric layer pneumatic auxiliary flight is rapidly evaluated, so that the reasonable design of the target orbit in the task implementation process is guided.
In the process, the reachable capacity boundary can be quickly obtained 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, take the example of a pneumatic assistance maneuver in a Mars environment, a correlationThe physical parameters are all taken from a Mars environment model. Taking pneumatic auxiliary initial state x0＝[r0,V0,γ0]TAs shown in table 1.
TABLE 1 pneumatically assisted initial State variables
Meanwhile, the upper and lower boundaries of the roll angle in equation (6) are σmin15 ° and σmax165 deg.. Furthermore, the minimum distance center point height constraint is taken as ha,min＝200km。
Since the solution method corresponding to the linear programming problem directly converted from the problem P1 is complete and efficient, it is not necessary to provide a new algorithm for the linear programming problem. In consideration of the robustness and the rapidity of the original-dual interior point method in the aspect of solving the linear programming problem, the method is directly adopted to solve the problem.
After the problem P1 is processed through the third step and the fourth step, a complete reachable capacity boundary can be obtained through the calculation process of the fifth step. The geometry of the reachable capability boundaries in the terminal state representation in this embodiment is given in fig. 4, and it can be seen from fig. 4 that the reachable range geometry represented by speed and track angle appears as a band. The region is wider in the region of lower velocity, which also indicates that the low energy orbit is more accessible after the pneumatically assisted flight. Fig. 5 shows the geometry of the achievable capability boundary in the embodiment in terms of the height of the isocenter and the eccentricity, and it can be seen from fig. 5 that a larger terminal track angle corresponds to a larger eccentricity, i.e., when the aircraft is out of the atmosphere, the larger the track angle, the smaller the energy attenuation of the orbit, and for the pneumatically assisted atmospheric flight process, the closer the orbit is to the large elliptical orbit.
In order to show the high efficiency of the method proposed by the patent, statistics of the computation time are given here. Taking MATLAB-R2018a on an Intel i7-4790K CPU @4.00GHz desktop as a computing platform, the total time consumption of the reach envelope of FIG. 4 for a single computation is 7.47s, wherein the total computed envelope points are 57, which is equivalent to the time consumption computed for a single point of 0.131 s. In addition, the number of iteration steps required by single-point calculation is about 4-7, so that the time required by each step of linear approximation iteration is about 0.02-0.03 s. The time statistics fully verify the rapidity of the reachable capability boundary evaluation method provided by the patent.
The above detailed description is intended to illustrate the objects, aspects and advantages of the present invention, and it should be understood that the above detailed description is only exemplary of the present invention, and is not intended to limit the scope of the present invention, and any modifications, equivalents, improvements, etc. made within the spirit and principle of the present invention should be included in the scope of the present invention.
Claims (6)
1. A method for rapidly evaluating the reachable capacity of a pneumatic auxiliary track of a cross-atmosphere aircraft is characterized by comprising the following steps: comprises the following steps of (a) carrying out,
the method comprises the following steps: establishing a longitudinal dynamic equation related to the size of the rail;
step two: establishing description of the boundary problem of the maneuvering accessibility of the pneumatic auxiliary track according to state parameters related to the track after the aircraft leaves the atmosphere;
step three: converting the reachable capability evaluation problem into a class of state maximum/minimum problem P1 according to the fixed terminal speed, thereby providing a division boundary discrete solving frame;
step four: aiming at the nonlinear optimal control problem P1, constructing a sequence convex optimization framework of the aircraft reachable ability boundary problem through independent variable replacement and lossless convex of the nonlinear problem;
the nonlinear optimal control problem P1 is a strong nonlinear optimal control problem due to the inclusion of nonlinear dynamical equations; aiming at the solution of the nonlinear optimal control problem P1, a sequence convex optimization framework of the aircraft reachable ability boundary problem is constructed by using an approximation theory of the optimal control law of a nonlinear infinite dimensional system for reference, independent variable replacement and lossless projection of the nonlinear problem, and nonlinear items in the optimal control problem P1 are linearized near a given approximate solution through the lossless projection, so that a linear optimal control subproblem is obtained, and the sequence convex optimization framework of the aircraft reachable ability boundary problem is obtained;
step five: the nonlinear optimal control problem P1 under the tilt control is quickly solved through the sequence convex optimization framework, the maximum and minimum track angles under any terminal speed are obtained, the maximum and minimum track angles correspond to two points on the reachable capacity boundary, and the pneumatically-assisted reachable capacity boundary of the aircraft can be obtained by connecting all boundary points.
2. The method for rapidly evaluating the accessibility of the pneumatic auxiliary track of the cross-atmospheric aircraft according to claim 1, characterized in that: and step six, rapidly evaluating the feasible target orbit range of the aircraft in the atmospheric layer pneumatic auxiliary flight based on the aircraft pneumatic auxiliary reachable capacity boundary, thereby guiding the reasonable design of the target orbit in the task implementation process.
3. The method for rapidly evaluating the accessibility of the pneumatic auxiliary track of the cross-atmospheric aircraft according to claim 1 or 2, characterized in that: the first implementation method comprises the following steps of,
because the influence of the pneumatic auxiliary maneuvering on the semi-major axis and the eccentricity of the orbit is only considered when the accessibility of the pneumatic auxiliary orbit maneuvering is evaluated, and the dimensional parameters of the orbit are directly related to the terminal position radial, the track angle and the speed in a spherical coordinate system, a non-dimensionalized simplified model and a reduced model of going to earth rotation are adopted when the accessibility is evaluated:
wherein mu represents the gravity constant of the earth, r represents the position vector, v represents the velocity, and gamma represents the track angle; sigma is a roll angle and is also a process control quantity of pneumatic auxiliary flight; the lift acceleration L and the drag acceleration D after dimensionless are as follows:
wherein R isMIs the radius of the earth, S is the reference area of the aircraft, CLAnd CDLift coefficient and drag coefficient, respectively, and ρ is the dimensional planetary atmospheric density, and for the sake of simplifying the calculation, an exponential model is taken here, that is, ρ ═ ρ0exp(-RMh/hs) Where ρ is0Is the atmospheric density, h, of the earth's surfacesIs the atmospheric density coefficient.
4. The method for rapidly evaluating the accessibility of the pneumatic auxiliary track of the cross-atmospheric aircraft according to claim 3, wherein: the second step is realized by the method that,
for pneumatically assisted maneuvers, the terminal state passes through the exit position radius size rfVelocity VfAnd track angle gammafDescribed, since the earth has a constant atmospheric height, r isfA constant value; the definition of the reach of the pneumatic auxiliary maneuver is described herein as all possible VfAnd gammafSet of constituents A (t)f) (ii) a The boundary of the reach range is defined herein, i.e., the reach capability boundary is O0(tf) I.e. O0(tf)∈A(tf)；
Because the target track after pneumatic auxiliary track transfer can not be tightly attached to the edge of the atmosphere, a minimum distance point height constraint h after the target track is out of the atmosphere needs to be addeda,minA value greater than the atmospheric edge height hatm(ii) a The minimum distance center point height constraint corresponds to a position radial of ra,min＝ha,min+RMThe reachable range boundary corresponding to the constraint is marked as OL(tf) Represented by curves B-C; the constraint equation corresponding to the constraint means that the corresponding Kepler orbit apocenter height is equal to h after the aircraft is out of the atmospherea,minIn a specific form of
Wherein r isatm＝hatm+RMThe vector diameter is the edge position of the atmosphere; since the size number of the orbit after the atmosphere is released is calculated through the terminal state variable, the formula (3) can be obtained through the derivation of an energy equation of the elliptical orbit; thus, the boundary O (t) of the reachable range is captured pneumaticallyf) Described by the closed piecewise curve a-B-C.
5. The method for rapidly evaluating the accessibility of the pneumatic auxiliary track of the cross-atmospheric aircraft according to claim 4, wherein: the third step is to realize the method as follows,
step two solved pneumatic auxiliary maneuvering reachable range A (t)f) Can be summarized as determining its boundary O (t)f) To solve for O (t)f) The core of (1) is to solve the height constraint h of the minimum apocenter pointa,minDetermined boundary OL(tf) Outer boundary, i.e. O0(tf) (ii) a For O0(tf) By quickly dividing boundary points, the calculation of the boundary points is summarized as follows: constant velocity VfSolving for the maximum and minimum track angles gamma by optimizationf；
For a certain
Straight line
And O0(tf) Intersect at two points, and respectively correspond to tail end flight path angles of
And
when solving them by optimization, the corresponding performance index is
Wherein the solution is
And
the corresponding optimal control problem comprises a control variable which is an inclination angle sigma, and the initial state is a state of the aircraft entering an atmospheric inlet and is marked as x0＝[r0,V0,γ0]T(ii) a The optimal control problem is referred to herein as P1;
by fixing the terminal speed, the reachable capability bound O (t)f) The evaluation problem is converted into the terminal track angle maximum/minimum problem shown in the formula (4), and therefore a division boundary discrete solving framework is given.
6. The method for rapidly evaluating the accessibility of the pneumatic auxiliary track of the cross-atmospheric aircraft according to claim 5, wherein: the implementation method of the fourth step is that,
step 4.1: realizing the dimension reduction transformation of the nonlinear dynamical equation (1) through independent variable replacement;
for equation (1) of dynamics, the position vector r is considered at the reachable capability boundary point determined by dimension reduction partitioningfAnd velocity VfAre all fixed values, so this subsection uses variable substitution with a new argument, i.e., e-1/r-V2The variable is actually equivalent to Kepler orbit energy, and the expression of the variable just only comprises a position vector and a speed, so that a terminal value is fixed when a reachable capacity boundary point is solved; by differentiating the variable, de/dt ═ DV > 0 is always true, so the monotonically increasing variable can be used as an independent variable; by substituting the independent variable, equation (1) is transformed into
As seen from the above equation, the differential term of the speed is eliminated in the kinetic equation, and the kinetic dimension is reduced;
step 4.2: converting an inclination angle expression which is in a cosine form and appears in a kinetic equation into a linear form through variable transformation of the inclination angle to obtain linear constraint corresponding to the new controlled variable;
for the control variables, here too, a simple transformation is required; since the controlled variable roll angle σ appears in the dynamic equation in the form of cosine, it needs to be treated as linear as possible; considering that cos σ monotonically increases between the intervals [0 °,180 ° ], the new control amount u is directly adopted as cos σ; after the new control quantity is adopted, the constraint of the original roll angle is converted into
Wherein σminNot less than 0 DEG is the lower limit of the roll angle, and sigmamaxThe upper limit of the inclination angle is not more than 180 degrees; in addition, the upper and lower bounds of the new control amount u after the transformation are u, respectivelymin＝cosσmaxAnd umax＝cosσmin(ii) a Obviously, the constraint corresponding to the new control quantity is a linear constraint;
step 4.3: successively linearizing the dynamic equation subjected to the dimension reduction in the step 4.1 to obtain a linear dynamic equation subjected to the dimension reduction;
in order to make the linear approximation method feasible, the kinetic equation needs to be linearized near the approximate solution; the kinetic equation (5) is first written in the form of a nonlinear affine control system, i.e.
Wherein the state variable nonlinear term is
And the coefficient matrix of the control variable is
B(x,e)＝[0,(L/D)/V2]T (9)
For the control affine system, the nonlinear term (8) of the state variable is linearized; when there is an approximate solution of xzWhere z represents the z-th linear successive approximation, then the non-linear term (8) can be at xzCarrying out nearby linearization; furthermore, although there is a velocity V in the control coefficient B (x, e), it is found by observing V that during the pneumatic assist maneuver, there is no dimension r ≈ 1, so V is the z +1 th linear successive approximationz+1By using the formula
Calculating, and further ensuring that V is calculated correctly when convergence is approached finally; at this time, the corresponding linearized kinetic equation after dimension reduction is
Wherein the linear coefficient matrix A (x) of the state variablesz) Is f0A Jacobian matrix of (x, e);
at this point, for the nonlinear optimal control problem P1, the dimension is reduced into the kinetic equation (5) through independent variable substitution, and the linearized kinetic equation (10) after dimension reduction is converted through successive linearization, and since the process constraint (6) and the kinetic equation (10) are linear constraints, the process constraint (6) and the kinetic equation (10) are a sequence convex optimization framework of the constructed aircraft reachable ability boundary problem.
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