CN114995495A - Time and angle constraint three-dimensional sliding mode guidance law design method under speed uncontrollable condition - Google Patents

Time and angle constraint three-dimensional sliding mode guidance law design method under speed uncontrollable condition Download PDF

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CN114995495A
CN114995495A CN202210600525.3A CN202210600525A CN114995495A CN 114995495 A CN114995495 A CN 114995495A CN 202210600525 A CN202210600525 A CN 202210600525A CN 114995495 A CN114995495 A CN 114995495A
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angle
missile
sliding mode
sight
guidance
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张科
游骏
孟源洁
张今
弋可行
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Northwestern Polytechnical University
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
    • G05D1/10—Simultaneous control of position or course in three dimensions
    • G05D1/107—Simultaneous control of position or course in three dimensions specially adapted for missiles
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
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Abstract

The invention relates to a time and angle constraint three-dimensional sliding mode guidance law design method under the condition of uncontrollable speed, and relates to the technical field of missile guidance. The guidance law designed by the invention is perpendicular to the speed direction, and the expected attack time and attack angle are realized only by changing the speed direction of the missile. In a pitching channel, a rapidly-converged nonsingular sliding mode surface and a sliding mode approach law which changes along with the distance of a missile are provided, and an expected sight elevation angle is realized; in a yaw channel, a special sliding mode surface is designed by analyzing the guidance conditions of the yaw channel, and the missile can be ensured to reach the sliding mode surface by the supercoiled sliding mode theory, so that the expected sight azimuth angle and the expected attack time are realized. Under the condition that the speed of the missile is uncontrollable, the missile strikes a target at the expected attack time and the attack angle in the three-dimensional space, the guidance law is small in miss distance, the expected attack time and the expected sight elevation angle can be accurately achieved, and the sight azimuth angle has small errors.

Description

Time and angle constraint three-dimensional sliding mode guidance law design method under speed uncontrollable condition
Technical Field
The invention relates to the technical field of missile guidance, in particular to a guidance law design for controlling the attack time and angle of a missile under the condition of uncontrollable speed.
Background
For the existing three-dimensional guidance law, two guidance models mainly exist, the first guidance model is established based on a sight line coordinate system, and after an obtained guidance instruction of the model is converted into a ballistic coordinate system, a component in the speed direction exists, so that an aircraft is required to be capable of controlling the missile speed. However, in the final guidance stage of the missile, the missile usually has no thrust and cannot control the speed of the missile. Aiming at the problem, a second model is established based on a missile preposed angular coordinate system, and a guidance instruction is perpendicular to the speed direction of the missile, so that the speed is not controlled.
Under the condition of uncontrollable speed, the three-dimensional guidance law of the missile can be divided into time constraint and angle constraint. The time constraint guidance law is based on the residual time estimation expression of the missile or is based on the consistency theory, and collaborative variables are designed for derivation. The angle-constrained guidance law is to deduce an expression of an overload instruction by a sliding mode control theory, an optimal control theory, an adaptive control theory and other methods according to the relationship between the angular acceleration of the view line in the model and the overload. However, at present, under the condition of uncontrollable speed, no guidance law capable of simultaneously realizing time constraint and angle constraint exists.
In the current three-dimensional guidance law capable of simultaneously realizing time constraint and angle constraint, the three-dimensional guidance law is based on speed control. In the final guidance stage of the missile, the missile usually has no thrust, the attitude of the missile is controlled only by pneumatic power, and the current guidance law can only realize single time constraint or angle constraint under the condition of uncontrollable speed.
In summary, the existing method cannot simultaneously constrain time and angle under the speed uncontrollable condition, and in order to meet the guidance requirement, a time and angle constrained three-dimensional sliding mode guidance law design method under the speed uncontrollable condition is urgently needed.
Disclosure of Invention
Technical problem to be solved
The invention provides a time and angle constraint three-dimensional sliding mode guidance law design method under an uncontrollable speed condition, aiming at the problem that the existing three-dimensional guidance law cannot simultaneously constrain attack time and attack angle under the uncontrollable speed condition.
Technical scheme
The invention designs the guidance laws of missile pitching and yawing channels respectively:
in a pitching channel, a rapidly-converged nonsingular sliding mode surface and an approach law which changes along with the distance of a missile are designed, and the derived guidance law can realize the height angle of an expected sight line.
In a yaw channel, a special sliding mode surface is designed by analyzing missile guidance conditions, and the missile can be ensured to reach the sliding mode surface by the supercoiled sliding mode theory, so that an expected sight azimuth angle and expected attack time are realized.
A time and angle constraint three-dimensional sliding mode guidance law design method under the condition of uncontrollable speed is characterized by comprising the following steps: respectively designing a based on a sliding mode control method after missile guidance conditions are analyzed my For controlling the missile to a desired angle of inclination of the line of sight, a mz The system is used for controlling the missile to reach a desired view declination and a desired attack time;
the missile guidance model can be expressed as:
Figure BDA0003668940690000021
Figure BDA0003668940690000022
Figure BDA0003668940690000023
Figure BDA0003668940690000024
Figure BDA0003668940690000025
Figure BDA0003668940690000026
wherein R is the relative distance of the bullet eyes, theta is the inclination angle of the trajectory,
Figure BDA0003668940690000027
is ballistic declination angle, θ L The high and low angles of the sight line are set,
Figure BDA0003668940690000028
is the line of sight azimuth angle, σ m Denotes the lead angle, θ m And
Figure BDA0003668940690000029
is σ m Decomposition of pitch and yaw channels in a line of sight coordinate system, representing the lead inclination and lead declination, respectively, gamma m Representing the forward roll angle, V m Representing the missile velocity;
further derivation of the elevation angle and azimuth angle of the line of sight can be obtained
Figure BDA0003668940690000031
Figure BDA0003668940690000032
The guidance law design method comprises the following steps:
(1) guidance law design for high and low angle directions of sight line
First, to achieve the desired view elevation angle, a rapidly converging nonsingular slip-form surface is designed:
Figure BDA0003668940690000033
wherein x is 1 ＝θ L -θ Ld ，
Figure BDA0003668940690000034
λ, β are both constants greater than 0, α ∈ [ -1,1](ii) a Designing an adaptive approach law along with the change of the distance of the missile:
Figure BDA0003668940690000035
wherein k is 1 ，k 2 Is a constant greater than 0, and k 1 And k is 2 Satisfies the following conditions:
Figure BDA0003668940690000036
the final guidance law of the elevation angle direction of the obtained sight line is
Figure BDA0003668940690000037
(2) Guidance law design for azimuth direction of sight line
Make xi ═ R/V m ，e 1 ＝θ L -θ Ld ，
Figure BDA0003668940690000039
φ＝t d T- ξ, and the following assumptions are made:
assume that 1: in pair e 2 When deriving, cos θ L This term varies less, and is considered to be a constant;
assume 2: in the process of guiding, a time error term phi is always greater than or equal to 0;
the design is as follows:
Figure BDA0003668940690000038
to slip form surface s 2 The two ends are derived and the formula is substituted to obtain:
Figure BDA0003668940690000041
designing an approximation rule based on a supercoiled sliding mode control method:
Figure BDA0003668940690000042
Figure BDA0003668940690000043
wherein l 1 And l 2 The constants are all constants larger than 0, and w is a self-adaptive parameter of the ultra-smooth die, so that the phenomenon of buffeting on the surface of the sliding die is prevented; substituting the above formula into a sliding form surface s 2 In (a), the controllable quantity a mz Comprises the following steps:
Figure BDA0003668940690000044
Figure BDA0003668940690000045
a computer system, comprising: one or more processors, a computer readable storage medium, for storing one or more programs, which when executed by the one or more processors, cause the one or more processors to implement the above-described method.
A computer-readable storage medium having stored thereon computer-executable instructions for, when executed, implementing the method described above.
A computer program comprising computer executable instructions which when executed perform the method described above.
Advantageous effects
The guidance law designed by the invention is perpendicular to the speed direction, and the expected attack time and attack angle are realized only by changing the speed direction of the missile. In a pitching channel, a rapidly-converged nonsingular sliding mode surface and a sliding mode approach law which changes along with the distance of a missile are provided, and an expected sight elevation angle is realized; in a yaw channel, a special sliding mode surface is designed by analyzing the guidance conditions of the yaw channel, and the missile can be ensured to reach the sliding mode surface by the supercoiled sliding mode theory, so that the expected sight azimuth angle and the expected attack time are realized.
Under the condition that the speed of the missile is uncontrollable, the missile strikes a target at the expected attack time and the attack angle in a three-dimensional space, the guidance law has small miss distance, the expected attack time and the expected sight elevation angle can be accurately achieved, and the sight azimuth angle has small error.
The innovation points of the invention are as follows:
(1) according to the sliding mode control theory, under the condition that the speed is uncontrollable, a sliding mode surface is designed according to the guiding condition, so that a guiding law capable of striking a target at a desired time and at a desired angle is deduced, which is not realized by other methods.
(2) The invention reasonably divides the three-dimensional guidance law of the missile, wherein a my Ensuring that the elevation angle of the visual line of the missile reaches the expected value, a mz And the sight azimuth angle and attack time of the missile are ensured to reach expected values.
(3) All results of the method are derived based on a three-dimensional nonlinear coupling dynamic model, small-angle hypothesis and linearization are not considered, the guidance law is high in precision and wide in application range, and omnibearing attack can be achieved.
Drawings
The drawings are only for purposes of illustrating particular embodiments and are not to be construed as limiting the invention, wherein like reference numerals are used to designate like parts throughout.
FIG. 1: a three-dimensional guidance model schematic diagram;
FIG. 2: a missile three-dimensional trajectory diagram;
FIG. 3: missile estimation residual time curve;
FIG. 4: lead angle change curve;
FIG. 5: a relative velocity profile;
FIG. 6: the variation curve of the high and low angles of the missile sight line;
FIG. 7: a missile sight azimuth change curve;
FIG. 8: overload change curves in the high and low angle directions of the sight line;
FIG. 9: and overload change curve of azimuth direction of sight.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the respective embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
According to fig. 1, M and T respectively represent the positions of the missile and the target, and in a coordinate system with the Y axis upward, a three-dimensional coupling guidance model of the missile can be described as follows:
Figure BDA0003668940690000061
Figure BDA0003668940690000062
Figure BDA0003668940690000063
Figure BDA0003668940690000064
Figure BDA0003668940690000065
Figure BDA0003668940690000066
wherein R is the relative distance of the bullet eyes, theta is the inclination angle of the trajectory,
Figure BDA0003668940690000067
is ballistic declination angle, θ L The high and low angles of the sight line are set,
Figure BDA0003668940690000068
is the azimuth of the line of sight, σ m Denotes the lead angle, θ m And
Figure BDA0003668940690000069
is σ m Decomposition of pitch and yaw channels in a line of sight coordinate system, representing the lead inclination and lead declination, respectively, gamma m Representing the forward roll angle, V m Indicating the missile velocity.
(1) Guidance law design for high and low angle directions of sight line
First, by deriving (5) and substituting formula (2), the following can be obtained
Figure BDA00036689406900000610
In order to realize the expected view elevation angle, a non-singular sliding mode surface with rapid convergence is designed:
s 1 ＝x 2 +λx 1 +βsig α (x 1 ) (8)
wherein x is 1 ＝θ L -θ Ld ，
Figure BDA0003668940690000071
θ Ld For the desired viewing angle, λ, β are each constants greater than 0, α ∈ [ -1,1]. In order to ensure that the elevation angle of the sight line can be quickly converged to a desired value and is kept stable, the invention provides a proximity law which changes along with the distance of a missile:
Figure BDA0003668940690000072
wherein k is 1 ，k 2 Is a constant greater than 0, and k 1 And k is 2 Satisfies the following conditions:
Figure BDA0003668940690000073
by taking the derivative of the formula (8) and combining the formulas (7) and (9), the available pitch channel guidance law is
Figure BDA0003668940690000074
To demonstrate that the missile can achieve the desired elevation angle of the line of sight under this guidance law, assume the following Lyapunov function:
Figure BDA0003668940690000075
derivation of the lyapunov function can result in:
Figure BDA0003668940690000076
the above formula is simplified to obtain:
Figure BDA0003668940690000077
integration of the above equation yields:
Figure BDA0003668940690000078
wherein s is 1 (0) And R (0) is the sliding mode surface at the starting moment and the relative distance between the missile and the target respectively. According to (15), there is a finite time t * So that the slip form surface s 1 ＝0。
After reaching the sliding surface, the following relational expression can be obtained from the expression (8)
x 2 ＝-λx 1 -βsig α (x 1 ) (16)
In order to prove that the elevation angle of the visual line of the missile can reach the expected value, a Lyapunov function is designed as follows:
Figure BDA0003668940690000081
the derivation of equation (17) and substitution of equation (16) can be obtained:
Figure BDA0003668940690000082
according to the formula (18), x 1 Can be converged to 0 within a limited time, the convergence time T 1 Satisfy the requirement of
Figure BDA0003668940690000083
In summary, the total time for the view angle to reach the desired value is the time t for the sliding mode surface to reach * And x after reaching the sliding form surface 1 Convergence time T of 1 Sum, T θ ═ T * +T 1 。
(2) Guidance law design for azimuth direction of sight line
Make xi ═ R/V m Representing the estimated time of flight remaining, e 1 ＝θ L -θ Ld And represents the elevation angle error of the line of sight,
Figure BDA0003668940690000084
indicating a line-of-sight azimuth error, wherein
Figure BDA0003668940690000085
For a desired azimuth of the line of sight, phi ═ t d -t- ξ. Representing the expected attack time error. The following assumptions are made:
assume that 1: in pair e 2 When deriving, cos θ L This term varies less and is considered to be a constant.
Assume 2: in the process of guiding, a time error term phi is always more than or equal to 0.
For xi, e in the above formula 1 ，e 2 Derivative and substituting equations (1) - (6) can yield:
Figure BDA0003668940690000086
Figure BDA0003668940690000087
Figure BDA0003668940690000088
Figure BDA0003668940690000089
designing an auxiliary variable omega, wherein the expression is as follows:
Figure BDA0003668940690000091
the two ends of omega are derived and substituted by the formulas (20) - (23), and the result can be obtained by sorting:
Figure BDA0003668940690000092
when a is my After the elevation angle of the line of sight of the missile is controlled to reach the expected value, the following conditions are met according to the formula (5):
Figure BDA0003668940690000093
in this case, the formula (25) can be rewritten as
Figure BDA0003668940690000094
According to (27), the following slip form is designed:
Figure BDA0003668940690000095
to slip form surface s 2 The two ends are derived and formula (3) is substituted to obtain:
Figure BDA0003668940690000096
in order to weaken the buffeting phenomenon of the sliding mode surface and ensure that the sliding mode surface can be converged quickly, the invention designs an approximation law based on a supercoiled sliding mode control method, and firstly writes a formula (29) as follows:
Figure BDA0003668940690000097
wherein the content of the first and second substances,
Figure BDA0003668940690000098
Figure BDA0003668940690000099
according to the supercoiled sliding mode theory, the approach law can be expressed as:
Figure BDA0003668940690000101
wherein l 1 And l 2 Are all constants greater than 0. The controlled variable a can be obtained by substituting the formula (33) into the formula (30) mz Comprises the following steps:
Figure BDA0003668940690000102
firstly, according to the theory of the supercoiled sliding mode, the sliding mode surface can reach s in a limited time 2 0. At the arrival of s 2 After 0, assume that a is present my And also to control the missile to the desired gaze elevation angle, according to equations (24) and (25):
Figure BDA0003668940690000103
due to cos theta m As 1, according to the trigonometric formula, the formula (20) and the formula (23) can be expressed as:
Figure BDA0003668940690000104
Figure BDA0003668940690000105
first, the following Lyapunov function was designed:
V 3 ＝0.5φ 2 (38)
to make derivatives at its two ends, have
Figure BDA0003668940690000106
Since phi is more than or equal to 0, therefore,
Figure BDA0003668940690000107
if and only if phi is 0,
Figure BDA0003668940690000108
so phi can converge progressively to 0. According to assumption 2, before the desired time is reached, Φ ≧ 0 is constantly established, and assuming Φ to be 0, it can be obtained from the above equation:
Figure BDA0003668940690000109
t+ξ＝t d (41)
if the missile can hit the target, when the missile hits the target, the + xi ═ R/V can be known by definition m When the expression (41) is satisfied, t is t ═ t d . In summary, as long as the missile can hit the target, the missile can hit the target at a desired time, and a desired azimuth of the line of sight can be achieved.
Binding a my And a mz It can be seen that the missile can achieve the following guidance goals:
Figure BDA0003668940690000111
that is, the missile can hit the target at a desired time and can achieve a desired inclination angle and declination angle of the line of sight.
In order to demonstrate feasibility and effectiveness of the time and angle constraint three-dimensional sliding mode guidance law design method under the condition of uncontrollable speed, the invention designs a simulation verification test. The initial conditions of the missiles are shown in Table 1, and the visual line height angle of four missiles is constrained to be theta Ld ＝[-35° -10° -15° -20°]The azimuth angle of the line of sight is constrained to
Figure BDA0003668940690000112
Expected attack time T of missile d The simulation time is 60s, the simulation termination condition is set to be r less than 0.5m, the maximum overload of the missile is 30g, and the simulation step length is 0.001 s. The simulation results are shown in Table 2 and FIGS. 2-9
TABLE 1 missile initial conditions
Figure BDA0003668940690000113
TABLE 2 simulation results
Figure BDA0003668940690000114
While the invention has been described with reference to specific embodiments, the invention is not limited thereto, and various equivalent modifications or substitutions can be easily made by those skilled in the art within the technical scope of the present disclosure.

Claims (4)

1. A time and angle constraint three-dimensional sliding mode guidance law design method under the condition of uncontrollable speed is characterized by comprising the following steps: respectively designing a based on a sliding mode control method after missile guidance conditions are analyzed my For controlling the missile to a desired angle of inclination of the line of sight, a mz The system is used for controlling the missile to reach a desired view declination and a desired attack time;
the missile guidance model can be expressed as:
Figure FDA0003668940680000011
Figure FDA0003668940680000012
Figure FDA0003668940680000013
Figure FDA0003668940680000014
Figure FDA0003668940680000015
Figure FDA0003668940680000016
wherein R is the relative distance of the bullet eyes, theta is the inclination angle of the trajectory,
Figure FDA0003668940680000017
is ballistic declination angle, θ L The high and low angles of the sight line are set,
Figure FDA0003668940680000018
is the azimuth of the line of sight, σ m Denotes the lead angle, θ m And
Figure FDA0003668940680000019
is σ m Decomposition of pitch and yaw channels in a line of sight coordinate system, representing the lead inclination and lead declination, respectively, gamma m Representing the forward roll angle, V m Representing the missile velocity;
further derivation of the elevation angle and azimuth angle of the line of sight can be obtained
Figure FDA00036689406800000110
Figure FDA00036689406800000111
The guidance law design method comprises the following steps:
(1) guidance law design for high and low angle directions of sight line
Firstly, in order to realize the desired view elevation angle, a rapidly converging nonsingular sliding mode surface is designed:
Figure FDA00036689406800000112
wherein x is 1 ＝θ L -θ Ld ，
Figure FDA00036689406800000113
λ, β are both constants greater than 0, α ∈ [ -1,1](ii) a Designing an adaptive approach law along with the change of missile distance:
Figure FDA0003668940680000021
wherein k is 1 ，k 2 Is a constant greater than 0, and k 1 And k is 2 Satisfies the following conditions:
Figure FDA0003668940680000022
the final guidance law of the elevation angle direction of the obtained sight line is
Figure FDA0003668940680000023
(2) Guidance law design for azimuth direction of sight line
Make xi ═ R/V m ，e 1 ＝θ L -θ Ld ，
Figure FDA0003668940680000024
φ＝t d -t- ξ, and the following assumptions are made:
assume that 1: in pair e 2 When deriving, cos θ L This term varies less, and is considered to be a constant;
assume 2: in the process of guiding, a time error term phi is always greater than or equal to 0;
the design is as follows:
Figure FDA0003668940680000025
to slip form surface s 2 The two ends are derived and the formula is substituted to obtain:
Figure FDA0003668940680000026
designing an approximation rule based on a supercoiled sliding mode control method:
Figure FDA0003668940680000027
Figure FDA0003668940680000028
wherein l 1 And l 2 The constants are all constants larger than 0, and w is a self-adaptive parameter of the ultra-smooth die, so that the phenomenon of buffeting on the surface of the sliding die is prevented; substituting the above formula into the sliding form surface s 2 In (a), the controllable amount a mz Comprises the following steps:
Figure FDA0003668940680000031
Figure FDA0003668940680000032
2. a computer system, comprising: one or more processors, a computer readable storage medium, for storing one or more programs, wherein the one or more programs, when executed by the one or more processors, cause the one or more processors to implement the method of claim 1.
3. A computer-readable storage medium having stored thereon computer-executable instructions for, when executed, implementing the method of claim 1.
4. A computer program comprising computer executable instructions which when executed perform the method of claim 1.
CN202210600525.3A 2022-05-30 2022-05-30 Time and angle constraint three-dimensional sliding mode guidance law design method under speed uncontrollable condition Pending CN114995495A (en)

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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN117891271A (en) * 2024-03-18 2024-04-16 西北工业大学 Three-dimensional collaborative guidance method for high-speed aircraft in consideration of time and angle constraints
CN117891271B (en) * 2024-03-18 2024-05-31 西北工业大学 Three-dimensional collaborative guidance method for high-speed aircraft in consideration of time and angle constraints

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN117891271A (en) * 2024-03-18 2024-04-16 西北工业大学 Three-dimensional collaborative guidance method for high-speed aircraft in consideration of time and angle constraints
CN117891271B (en) * 2024-03-18 2024-05-31 西北工业大学 Three-dimensional collaborative guidance method for high-speed aircraft in consideration of time and angle constraints

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