CN109506517B - Constraint-based medium guidance trajectory optimization method - Google Patents
Constraint-based medium guidance trajectory optimization method Download PDFInfo
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Abstract
The invention discloses a middle guidance trajectory optimization method with constraint, which is used for reducing the speed of terminal guidance shift change in an interception bullet and comprises the following steps: according to the set middle guidance optimized trajectory with constraints, establishing a full trajectory motion equation of the interception bullet longitudinal plane including trajectory constraints and terminal constraints, and constructing an interception bullet trajectory optimization model; and adopting an hp self-adaptive pseudo-spectrum method to solve and optimize the trajectory. The invention can reduce the terminal guidance shift-changing speed in the intercepting bullet, provides a good terminal guidance flight state, improves the intercepting precision, and is a ballistic method with great prospect.
Description
Technical Field
The invention belongs to the field of missile trajectory optimization, and particularly relates to a constrained medium guidance trajectory optimization method.
Background
When a traditional intercepting missile faces a high-speed target, the traditional intercepting missile is limited by the performances of a missile time constant, steering engine reaction time, guidance control system response speed and the like, and middle and end guidance shift switching failure can be caused under the condition of overlarge relative speed; even if the shift is successful, the guidance precision is reduced, thereby affecting the interception efficiency.
Aiming at the characteristics of large flying span and obvious characteristics of a high-speed target, the flight trajectory of the interception bullet in the guidance stage can be optimized, the speed of the interception bullet flying to a shift-passing area is reduced by designing a climbing track and utilizing gravity, so that the relative speed of the bullet eyes is reduced, and optimization of important indexes such as flying time, energy and the like can be realized while trajectory optimization is performed. Meanwhile, the initial state of the intercepting bullet entering the final guidance stage is crucial to the final intercepting effect, and the terminal state can be further restrained by optimizing the trajectory of the intermediate guidance stage, so that the intercepting bullet enters the final guidance stage at the optimal posture.
Disclosure of Invention
Aiming at the problem that middle and last guidance shift switching fails due to overlarge relative speed when a high-speed target is intercepted, the invention provides a middle guidance trajectory optimization method with constraint.
The invention is realized in such a way that a constrained middle guidance trajectory optimization method is used for reducing the terminal guidance shift-changing speed in an interception bullet, and the method comprises the following steps:
according to the set middle guidance optimized trajectory with constraints, establishing a full trajectory motion equation of the interception bullet longitudinal plane including trajectory constraints and terminal constraints, and constructing an interception bullet trajectory optimization model;
and adopting an hp self-adaptive pseudo-spectrum method to solve and optimize the trajectory.
Preferably, the constrained medium guidance optimized trajectory is set as: after the intercepting bullet finishes the cruise segment, the intercepting bullet flies according to the high-speed target information and the optimized trajectory in the middle guidance stage, the climbing and descending trajectory is adopted, the middle and last guidance shift change speed is reduced by gravity, the flying posture is adjusted, good flying conditions are provided for the last guidance, and the intercepting precision is improved.
Preferably, neglecting the effects of earth rotation and non-spherical shape, the longitudinal plane full trajectory motion equation of the interceptor projectile is specifically as follows:
in the formula (1), the reaction mixture is,
respectively representing missile mass, speed, ground clearance, attack angle, pitch angle and trajectory inclination angle; l is a firing range; p is engine thrust; fNThe thrust of the rail-controlled engine; n isyNormal overload is available for the missile; q is dynamic pressure; s is a reference area; g is the acceleration of gravity; r is the mean radius of the earth; i isspThe specific impulse of the scramjet engine is super-combustion; cx,CyRespectively, a drag coefficient and a lift coefficient which are functions of an attack angle and a Mach number;
the ballistic constraints are:
(1) normal overload restraint
The interceptor projectiles are limited in their structural strength and capacity by the onboard equipment during flight,
its normal overload constraint is expressed as:
|ny|≤nymax (2)
in the formula (1), nymaxMaximum available overload for the missile;
(2) dynamic pressure restraint
In the formula (2), rho is air density, v is interception bullet speed, and q ismaxMaximum dynamic pressure limit for the missile;
(3) heat flow confinement
Heat flow constraint generally refers to the heat flow limit at a stagnation point on the surface of an aircraft, for
Representing the interceptor projectile heat flow, the constraint of which is expressed as:
in the formula (4), the reaction mixture is,
maximum heat flow limit for the missile;
the stagnation heat flux density is calculated by the following equation:
in the formula (5), kQIs a constant, k, related to missile configuration and materialQ＝3.08×10-5；RNFor the radius of curvature at the stagnation point, take RN0.02 m; ρ is the atmospheric density; v is the missile velocity;
(4) angle of attack restraint
In order to meet the control requirement of the intercepting bomb, the control quantity must be controlled within a certain range, and the control quantity cannot be changed violently, so that the attack angle serving as the control quantity needs to meet certain constraint conditions. On the other hand, the missile angle of attack during flight has corresponding constraints:
in the formula (6), α1And alpha2Representing maximum angle of attack, t0Initial moment, t, representing the beginning of trajectory optimization of the interceptor projectile1For the moment when the interceptor projectile flies to the highest point of the optimized trajectory, tfRepresenting the terminal guidance shift-changing time of the trajectory optimization terminal;
(5) height constraint
The optimization trajectory is divided into a climbing section and a descending section, so that according to the target position information and the requirement of terminal middle and terminal guidance shift for height, the height constraint of the intercepting bullet on the optimization trajectory can be expressed as follows:
in the formula (7), h1min h1maxRespectively representing the minimum value and the maximum value of the height of the intercepting bomb in the climbing section, h2min h2maxRespectively representing the minimum value and the maximum value of the height of the interception bomb in the descending section, t0Initial moment, t, representing the beginning of trajectory optimization of the interceptor projectile1For the moment when the interceptor projectile flies to the highest point of the optimized trajectory, tfRepresenting the terminal guidance shift-changing time of the trajectory optimization terminal;
the terminal constraints are:
in order to reduce the middle and last guidance shift-changing speed and adjust the intercepting bullet to enter the last guidance flight state, the speed v, the height h and the trajectory inclination angle theta of the intercepting bullet need to be restrained. In addition, since the height of the intercepted projectile after the trajectory optimization is different from the height of the high-speed target, the corresponding states, in particular the constraints of height and trajectory inclination angle, are different, specifically: when the interceptor missile height is above the target, the terminal constraint is expressed as:
in the formula (8), vf，vfminAnd vfmaxRespectively representing the terminal speed of the interceptor projectile, the minimum value and the maximum value of the terminal speed, hf1，hfmin1And hfmax1Respectively representing the height of the terminal of the interceptor projectile, the minimum value and the maximum value of the height of the terminal, thetaf1，θfmin1And thetafmax1Respectively representing the terminal trajectory inclination angle of the intercepted projectile, and the minimum value and the maximum value of the terminal trajectory inclination angle;
when the interceptor missile height is below the target, the terminal constraints are expressed as:
in the formula (9), vf，vfminAnd vfmaxRespectively representing the terminal speed of the interceptor projectile, the minimum value and the maximum value of the terminal speed, hf2，hfmin2And hfmax2Respectively representing the height of the terminal of the interceptor projectile, the minimum value and the maximum value of the height of the terminal, thetaf2，θfmin2And thetafmax2Respectively representing the terminal trajectory inclination angle of the interceptor projectile, and the minimum value and the maximum value of the terminal trajectory inclination angle.
Preferably, the intercepting bullet trajectory optimization model specifically comprises:
when the state variable x (t) meets the ballistic constraint condition, seeking an optimal control variable u (t) so that the performance index J takes a minimum value;
the performance index of the optimized trajectory should be
t0And tfRespectively representing the starting time and the ending time of ballistic optimization, and the physical meaning of the performance index J is flight time;
the state variable x (t) is a parameter in the equation of motion, i.e., x ═ v, θ, h, L]TWhen only the longitudinal plane motion is considered, the control variable is taken to be the angle of attack α, i.e., u ═ α.
Preferably, the solving and optimizing the trajectory by using the hp adaptive pseudo-spectrum method comprises the following steps:
step 1: dividing network intervals as required, and setting the number of configuration points of each interval;
step 2: discretizing a state equation, a target function and constraint conditions by using a global Gaussian pseudo-spectrum method in each network interval, and converting an optimal control problem into a nonlinear programming problem;
and step 3: solving a nonlinear programming problem by using a sequential quadratic programming method;
and 4, step 4: judging whether the corresponding state quantity and the corresponding control quantity at the middle point of each grid interval meet the constraint precision requirement of the motion equation, if so, finishing iteration, and if not, jumping to the step 5 or the step 6;
and 5: if the magnitude of all elements in the residual vector beta is equivalent, increasing the number of matching points, namely increasing the times of the interpolation polynomial;
step 6: if the magnitude of some elements in the residual vector beta is obviously greater than that of other elements, refining the corresponding grid interval;
and 7: and after all grid intervals are adjusted, returning to the step 2 for next iteration.
Compared with the defects and shortcomings of the prior art, the invention has the following beneficial effects: the invention can reduce the terminal guidance shift-changing speed in the intercepting bullet, provides a good terminal guidance flight state, improves the intercepting precision, and is a ballistic method with great prospect.
Drawings
FIG. 1 is a schematic diagram of a basic trajectory of a constrained medium guidance optimized trajectory according to the present invention;
FIG. 2 is an optimized trajectory simulation curve of the present invention;
FIG. 3 is a graph of the height variation of the optimized trajectory of the present invention;
FIG. 4 is a velocity profile for an optimized trajectory according to the present invention;
FIG. 5 is a ballistic inclination variation curve for an optimized trajectory according to the present invention;
figure 6 is a graph of the angle of attack for the optimized trajectory of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The invention provides a constrained medium guidance trajectory optimization method, which is used for reducing the terminal guidance shift-changing speed of an interception bullet and comprises the following steps:
s1, according to the set middle guidance optimized trajectory with constraints, establishing a full trajectory motion equation of the interception bullet longitudinal plane including trajectory constraints and terminal constraints, and constructing an interception bullet trajectory optimization model.
In step S1, the constrained medium guidance optimized trajectory is set as follows, as shown in fig. 1: after the intercepting bullet finishes the cruise segment, the intercepting bullet flies according to the high-speed target information and the optimized trajectory in the middle guidance stage, the climbing and descending trajectory is adopted, the middle and last guidance shift change speed is reduced by gravity, the flying posture is adjusted, good flying conditions are provided for the last guidance, and the intercepting precision is improved.
In step S1, ignoring the effects of earth rotation and aspheric shape, the equation of motion of the full trajectory of the longitudinal plane of the interceptor projectile is specifically as follows:
in the formula (1), the reaction mixture is,
respectively representing missile mass, speed, ground clearance, attack angle, pitch angle and trajectory inclination angle; l is a firing range; p is engine thrust; fNThe thrust of the rail-controlled engine; n isyNormal overload is available for the missile; q is dynamic pressure; s is a reference area; g is the acceleration of gravity; r is the mean radius of the earth; i isspThe specific impulse of the scramjet engine is super-combustion; cx,CyRespectively, a drag coefficient and a lift coefficient which are functions of an attack angle and a Mach number;
the ballistic constraints are:
(1) normal overload restraint
The interception bullet is limited by the structural strength and the bearing capacity of airborne equipment in the flight process, and the normal overload constraint is expressed as follows:
|ny|≤nymax (2)
in the formula (1), nymaxMaximum available overload for the missile;
(2) dynamic pressure restraint
In the formula (2), rho is air density, v is interception bullet speed, and q ismaxMaximum dynamic pressure limit for the missile;
(3) heat flow confinement
Heat flow constraint generally refers to the heat flow limit at a stagnation point on the surface of an aircraft, for
Representing the interceptor projectile heat flow, the constraint of which is expressed as:
in the formula (4), the reaction mixture is,
maximum heat flow for missileLimiting;
the stagnation heat flux density is calculated by the following equation:
in the formula (5), kQIs a constant, k, related to missile configuration and materialQ＝3.08×10-5； RNFor the radius of curvature at the stagnation point, take RN0.02 m; ρ is the atmospheric density; v is the missile velocity;
(4) angle of attack restraint
In order to meet the control requirement of the intercepting bomb, the control quantity must be controlled within a certain range, and the control quantity cannot be changed violently, so that the attack angle serving as the control quantity needs to meet certain constraint conditions. On the other hand, the missile angle of attack during flight has corresponding constraints:
in the formula (6), α1And alpha2Representing maximum angle of attack, t0Initial moment, t, representing the beginning of trajectory optimization of the interceptor projectile1For the moment when the interceptor projectile flies to the highest point of the optimized trajectory, tfRepresenting the terminal guidance shift-changing time of the trajectory optimization terminal;
(5) height constraint
The optimization trajectory is divided into a climbing section and a descending section, so that according to the target position information and the requirement of terminal middle and terminal guidance shift for height, the height constraint of the intercepting bullet on the optimization trajectory can be expressed as follows:
in the formula (7), h1min h1maxRespectively representing the minimum value and the maximum value of the height of the intercepting bomb in the climbing section, h2min h2maxRespectively representing the minimum value and the maximum value of the height of the interception bomb in the descending section, t0initial moment, t, representing the beginning of trajectory optimization of the interceptor projectile1For the moment when the interceptor projectile flies to the highest point of the optimized trajectory, tfRepresenting the terminal guidance shift-changing time of the trajectory optimization terminal;
the terminal constraints are:
in order to reduce the middle and last guidance shift-changing speed and adjust the intercepting bullet to enter the last guidance flight state, the speed v, the height h and the trajectory inclination angle theta of the intercepting bullet need to be restrained. In addition, since the height of the intercepted projectile after the trajectory optimization is different from the height of the high-speed target, the corresponding states, in particular the constraints of height and trajectory inclination angle, are different, specifically: when the interceptor missile height is above the target, the terminal constraint is expressed as:
in the formula (8), vf，vfminAnd vfmaxRespectively representing the terminal speed of the interceptor projectile, the minimum value and the maximum value of the terminal speed, hf1，hfmin1And hfmax1Respectively representing the height of the terminal of the interceptor projectile, the minimum value and the maximum value of the height of the terminal, thetaf1，θfmin1And thetafmax1Respectively representing the terminal trajectory inclination angle of the intercepted projectile, and the minimum value and the maximum value of the terminal trajectory inclination angle;
when the interceptor missile height is below the target, the terminal constraints are expressed as:
in the formula (9), vf，vfminAnd vfmaxRespectively representing the terminal speed of the interceptor projectile, the minimum value and the maximum value of the terminal speed, hf2，hfmin2And hfmax2Respectively representing the height of the terminal of the interceptor projectile, the minimum value and the maximum value of the height of the terminal, thetaf2，θfmin2And thetafmax2Respectively representing the terminal trajectory inclination angle of the interceptor projectile, and the minimum value and the maximum value of the terminal trajectory inclination angle.
In step S1, the optimization model of the interceptor projectile trajectory specifically includes: when the state variable x (t) meets the ballistic constraint condition, seeking an optimal control variable u (t) so that the performance index J takes a minimum value;
the purpose of optimizing the trajectory in the guidance stage in the interceptor missile is to reduce the terminal speed, namely the middle and last guidance shift-changing speed, and considering the high-speed characteristic of the target, the performance index of the optimized trajectory is
t0And tfRepresenting the moments when ballistic optimization begins and ends, respectively, the physical meaning of this performance index J is time of flight.
The state variable x (t) is a parameter in the equation of motion, i.e., x ═ v, θ, h, L]T(ii) a In the case of only longitudinal plane movements, the control variable is taken to be the angle of attack α, i.e., u ═ α.
And S2, adopting an hp self-adaptive pseudo-spectrum method to solve and optimize the trajectory.
In step S2, the solution optimization of the trajectory by using the hp adaptive pseudo-spectrum method includes the following steps:
step 1: dividing network intervals as required, and setting the number of configuration points of each interval;
step 2: discretizing a state equation, a target function and constraint conditions by using a global Gaussian pseudo-spectrum method in each network interval, and converting an optimal control problem into a nonlinear programming problem;
and step 3: solving a nonlinear programming problem by using a sequential quadratic programming method;
and 4, step 4: judging whether the corresponding state quantity and the corresponding control quantity at the middle point of each grid interval meet the constraint precision requirement of the motion equation, if so, finishing iteration, and if not, jumping to the step 5 or the step 6;
and 5: if the magnitude of all elements in the residual vector beta is equivalent, increasing the number of matching points, namely increasing the times of the interpolation polynomial;
step 6: if the magnitude of some elements in the residual vector beta is obviously greater than that of other elements, refining the corresponding grid interval;
and 7: and after all grid intervals are adjusted, returning to the step 2 for next iteration.
In the practical application of the invention, the ballistic trajectory optimization is carried out aiming at ballistic trajectory schemes under different ignition modes, and the performance of different schemes is analyzed. The simulation conditions were as follows: initial condition of state variable: v. of0＝1800m/s，θ0＝0°，h0＝25km，L0＝0km，m0400 kg; ballistic constraints are shown in table 1:
TABLE 1 ballistic constraints
Considering that the height of the intercepted bullet at the end of trajectory optimization is higher than the target height, setting the terminal constraint as
The trajectory schemes of the air-absorbing hypersonic missile are optimized by using an hp self-adaptive pseudo-spectrum method, and simulation results are shown in figures 2 to 6.
As can be seen from the simulation results shown in fig. 2 to 6, the optimized trajectory adopts a trajectory scheme of climbing first and then descending, the highest climbing height is 42km, the maximum flying speed reaches 3000m/s, through the optimized trajectory, the height of the intercepted projectile at the end of intermediate guidance is 26km, the speed is 1082.43m/s, and the trajectory inclination angle is-30 degrees, so that terminal constraints are met; meanwhile, the main parameters of the attack angle, the trajectory inclination angle, the speed, the height and the like also meet the trajectory constraint.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents and improvements made within the spirit and principle of the present invention are intended to be included within the scope of the present invention.
Claims (2)
1. A constrained intermediate guidance trajectory optimization method is used for reducing the speed of terminal guidance shift change in an intercepting bullet, and is characterized by comprising the following steps:
according to the set middle guidance optimized trajectory with constraints, establishing a full trajectory motion equation of the interception bullet longitudinal plane including trajectory constraints and terminal constraints, and constructing an interception bullet trajectory optimization model;
adopting an hp self-adaptive pseudo-spectrum method to solve and optimize the trajectory;
the constrained medium guidance optimized trajectory is set as: after the intercepting bullet finishes the cruise segment, the intercepting bullet flies according to the high-speed target information and the optimized trajectory in the middle guidance stage, adopts the climbing and descending trajectory, reduces the middle and last guidance shift change speed by gravity, adjusts the flight attitude, provides good flight conditions for the last guidance and improves the intercepting precision;
neglecting the effects of earth rotation and non-spherical shape, the longitudinal plane full trajectory motion equation of the interceptor projectile is specifically as follows:
in the formula (1), the reaction mixture is,
respectively representing missile mass, speed, ground clearance, attack angle, pitch angle and trajectory inclination angle; l is a firing range; p is engine thrust; fNThe thrust of the rail-controlled engine; n isyNormal overload is available for the missile; q is dynamic pressure; s is a reference area; g is the acceleration of gravity; r is the mean radius of the earth; i isspThe specific impulse of the scramjet engine is super-combustion; cx,CyRespectively, a drag coefficient and a lift coefficient which are functions of an attack angle and a Mach number;
wherein the ballistic constraints comprise:
(1) normal overload restraint
Its constraints are expressed as:
|ny|≤nymax (2)
in the formula (2), nymaxMaximum available overload for the missile;
(2) dynamic pressure restraint
Its constraints are expressed as:
in the formula (3), rho is air density, v is interception bullet speed, and q ismaxMaximum dynamic pressure limit for the missile;
(3) heat flow confinement
By using
Representing the interceptor projectile heat flow, the constraint of which is expressed as:
formula (4) in
Maximum heat flow limit for the missile;
the stagnation heat flux density is calculated by the following equation:
in the formula (5), kQIs a constant, k, related to missile configuration and materialQ＝3.08×10-5；RNFor the radius of curvature at the stagnation point, take RN0.02 m; ρ is the atmospheric density; v is the missile velocity;
(4) angle of attack restraint
Its constraints are expressed as:
in the formula (6), α1And alpha2Representing maximum angle of attack, t0Initial moment, t, representing the beginning of trajectory optimization of the interceptor projectile1For the moment when the interceptor projectile flies to the highest point of the optimized trajectory, tfRepresenting the terminal guidance shift-changing time of the trajectory optimization terminal;
(5) height constraint
Its constraints are expressed as:
in the formula (7), h1min h1maxRespectively representing the minimum value and the maximum value of the height of the intercepting bomb in the climbing section, h2minh2maxRespectively representing the minimum value and the maximum value of the height of the interception bomb in the descending section, t0Initial moment, t, representing the beginning of trajectory optimization of the interceptor projectile1For the moment when the interceptor projectile flies to the highest point of the optimized trajectory, tfRepresenting the terminal guidance shift-changing time of the trajectory optimization terminal;
when the height of the interceptor missile is higher than the target, the terminal constraint is expressed as:
in the formula (8), vf，vfminAnd vfmaxRespectively representing the terminal speed of the interceptor projectile, the minimum value and the maximum value of the terminal speed, hf1，hfmin1And hfmax1Respectively representing the height of the terminal of the interceptor projectile, the minimum value and the maximum value of the height of the terminal, thetaf1，θfmin1And thetafmax1Respectively representing the terminal trajectory inclination angle of the intercepted projectile, and the minimum value and the maximum value of the terminal trajectory inclination angle;
when the height of the interception bullet is lower than the target, the terminal constraint is expressed as:
in the formula (9), vf，vfminAnd vfmaxRespectively representing the terminal speed of the interceptor projectile, the minimum value and the maximum value of the terminal speed, hf2，hfmin2And hfmax2Respectively representing the height of the terminal of the interceptor projectile, the minimum value and the maximum value of the height of the terminal, thetaf2，θfmin2And thetafmax2Respectively representing the terminal trajectory inclination angle of the intercepted projectile, and the minimum value and the maximum value of the terminal trajectory inclination angle;
the intercepting bullet trajectory optimization model specifically comprises the following steps:
when the state variable x (t) meets the ballistic constraint condition, seeking an optimal control variable u (t) so that the performance index J takes a minimum value;
the performance index of the optimized trajectory should be
t0And tfRespectively representing the starting time and the ending time of ballistic optimization, and the physical meaning of the performance index J is flight time;
the state variable x (t) is a parameter in the equation of motion, i.e., x ═ v, θ, h, L]TWhen only the longitudinal plane motion is considered, the control variable is taken to be the angle of attack α, i.e., u ═ α.
2. The constrained mid-guidance ballistic optimization method of claim 1, wherein the solution optimization of the trajectory using hp adaptive pseudo-spectroscopy comprises the steps of:
step 1: dividing network intervals as required, and setting the number of configuration points of each interval;
step 2: discretizing a state equation, a target function and constraint conditions by using a global Gaussian pseudo-spectrum method in each network interval, and converting an optimal control problem into a nonlinear programming problem;
and step 3: solving a nonlinear programming problem by using a sequential quadratic programming method;
and 4, step 4: judging whether the corresponding state quantity and the corresponding control quantity at the middle point of each grid interval meet the constraint precision requirement of the motion equation, if so, finishing iteration, and if not, jumping to the step 5 or the step 6;
and 5: if the magnitude of all elements in the residual vector beta is equivalent, increasing the number of matching points, namely increasing the times of the interpolation polynomial;
step 6: if the magnitude of some elements in the residual vector beta is obviously greater than that of other elements, refining the corresponding grid interval;
and 7: and after all grid intervals are adjusted, returning to the step 2 for next iteration.
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