A multi-missile formation flight control method based on sliding mode control

Abstract

The application belongs to the field of formation flight control, and particularly relates to a multi-missile formation flight control method based on a sliding mode control, and the steps are specifically as follows: a coordinate system of a relative motion equation is defined; a conversion relationship between each coordinate system is constructed; a relative motion model function under a leading missile trajectory coordinate system is established; and a controller is designed for an error equation. Compared with the prior art, the position of a follower missile in the leading missile trajectory coordinate system and the velocity of the leading missile in an inertial coordinate system are taken as state variables, the acceleration of the follower missile is taken as a control variable, the state equation is more intuitive, a sliding mode control method is used for the controller, an exponential function is set as a parameter limit approaching law, and an ESO observer is used to observe errors, so that the formation can still be kept stable when the leading missile is doing curve motion.

Description

Technical Field

[0001] This invention belongs to the field of formation flight control, and in particular to a multi-missile formation flight control method based on sliding mode control. Background Technology

[0002] With the development of information technology and the continuous improvement of missile defense systems in various countries, using a single missile to strike a target is gradually no longer sufficient to meet operational requirements. Improving the penetration effectiveness of missiles has become an urgent problem to be solved. Missile formation coordinated operations enable multiple missiles to be launched simultaneously for saturation strikes, thus improving penetration performance.

[0003] Many scholars focus on formation control methods, primarily studying vehicles and drones, with less research on high-speed projectiles. Furthermore, the increased speed of high-speed projectiles increases the control complexity. Existing literature largely concentrates on leader-follower projectile formation controller design in a ground-based system. This approach requires follower projectiles to possess global position information at all times to calculate their relative positions at various moments. However, in a ballistic coordinate system with the leader projectile as the origin, even less information is needed to account for changes in the relative positions of follower projectiles. This paper constructs an intuitive state equation by using the follower projectile's position in the leader projectile's ballistic coordinate system and the leader projectile's velocity in the inertial coordinate system as state variables, and the follower projectile's acceleration as the control variable, combining the error between the follower projectile's relative position and the desired position, as well as the leader projectile's velocity error.

[0004] For existing leader-follower missile models, most researchers neglect the acceleration term of the leader missile. When the leader missile maintains linear motion, it has no impact on the tracking of the follower missiles. However, when the leader missile is moving in a curved path, neglecting the acceleration term will cause the error to accumulate continuously, making it impossible to maintain the predetermined formation. Compensating for this term in the dynamic model can have a good effect. Since the sliding mode control system has a simple algorithm, fast response speed, and robustness to external noise interference and parameter perturbations, this invention proposes a design method for a multi-missile formation flight controller based on sliding mode control. Summary of the Invention

[0005] The purpose of this invention is to provide a multi-missile formation flight control method based on sliding mode control. This method compensates for the leader missile's acceleration term in the relative motion equations, transforming the relative motion model into the leader missile's ballistic coordinate system. The position of the follower missile in the ballistic coordinate system and the velocity of the leader missile in the inertial coordinate system are used as state variables, and the acceleration of the follower missile is used as the control variable. Combining the error between the relative position of the follower missile and the desired position, as well as the velocity error of the leader missile, a sliding mode control method is employed. Furthermore, by setting an exponential function as a parameter to constrain the reaching law, a design is implemented. The observer measures the error to ensure that the formation remains stable when the lead missile is making a curved motion.

[0006] The technical solution to achieve the purpose of this invention is: a multi-missile formation flight control method based on sliding mode control, characterized in that the steps are as follows: defining a coordinate system for the relative motion equations;

[0007] Establish the transformation relationships between different coordinate systems;

[0008] A relative motion model function established in the missile trajectory coordinate system;

[0009] Design a controller based on the error equation.

[0010] The significant advantages of this invention compared to existing technologies are:

[0011] (1) Based on the relative motion model of the leader missile and the follower missile, the present invention compensates for the acceleration term of the leader missile and transforms the relative motion model into the trajectory coordinate system of the leader missile. Compared with the ground system, the follower missile requires less position information and is easier to control.

[0012] (2) The position of the projectile in the trajectory coordinate system of the lead projectile and the velocity of the lead projectile in the inertial coordinate system are used as state variables, and the acceleration of the projectile is used as the control variable, making the state equation more intuitive.

[0013] (3) Simultaneously, the sliding mode control algorithm is simple, has a fast response speed, and is robust to external noise interference and parameter perturbations. By setting an exponential function as the parameter limiting approach law, the overshoot is reduced when the system reaches stability. Observer observation error. While ensuring flight accuracy, the formation can remain stable even when the lead missile is making a curved motion. Attached Figure Description

[0014] Figure 1 The kinematic relationship between the leading and trailing projectiles in this invention;

[0015] Figure 2 The three-dimensional model of the missile leader in this invention;

[0016] Figure 3 The present invention provides a simulation diagram of the formation tracking curve ballistics.

[0017] Figure 4 The present invention provides a curve of the position of the projectile in the X direction in the lead projectile trajectory coordinate system;

[0018] Figure 5 The present invention provides a Y-direction position curve of the projectile in the lead projectile trajectory coordinate system;

[0019] Figure 6 The present invention provides a Z-direction position curve of the projectile in the lead projectile trajectory coordinate system;

[0020] Figure 7 The present invention provides the X-direction acceleration curve of the projectile in the lead projectile trajectory coordinate system;

[0021] Figure 8 The present invention provides the acceleration curve of the projectile in the Y direction in the lead projectile trajectory coordinate system;

[0022] Figure 9 The present invention provides the Z-direction acceleration curve of the projectile in the lead projectile trajectory coordinate system. Detailed Implementation

[0023] The following will refer to the appendices in the embodiments of the present invention. Figure 1-9 The technical solutions in the embodiments of the present invention are clearly and completely described herein. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0024] S1: Define the coordinate system for the equations of relative motion.

[0025] S1.1: Define the inertial coordinate system :origin Select the missile launch point. The axis is the line of intersection between the ballistic plane and the horizontal plane, pointing towards the target as positive. The axis points upward along the vertical line. The axis is perpendicular to the other two axes, forming a right-handed coordinate system.

[0026] S1.2: Define the missile trajectory coordinate system :origin For the instantaneous center of mass of the lead missile, Instantaneous velocity of the axis and the lead bullet coincide, Located in In the vertical plane and with The axis is perpendicular. The axis is perpendicular to the other two axes, forming a right-handed coordinate system.

[0027] S1.3: Define the ballistic coordinate system :origin For the instantaneous center of mass of the bullet, Instantaneous velocity of the axis and the lead bullet coincide, Located in In the vertical plane and with The axis is perpendicular. The axis is perpendicular to the other two axes, forming a right-handed coordinate system.

[0028] S2: Construct the transformation relationships between different coordinate systems:

[0029] S2.1: Inertial coordinate system With the ballistic coordinate system of the missile leader The conversion relationship between them is

[0030] (1)

[0031] In the formula, This represents the coordinate projection of the missile in the missile's trajectory coordinate system. This represents the coordinate projection of the missile in the inertial coordinate system. Indicates the trajectory angle of the missile; Indicates the trajectory deviation angle of the missile.

[0032] The transformation matrix from the inertial coordinate system to the ballistic coordinate system can be obtained through two rotations. First, rotate the inertial coordinate system around... The axis rotates by one Angle, then rotate around the axis by one From the angle, we can obtain the transformation matrix between the inertial coordinate system and the missile trajectory coordinate system. for:

[0033] (2)

[0034] In the formula, This indicates that the inertial coordinate system is used with angular velocity Around Axis rotation The basic rotation matrix of the angle is then used to form the transition coordinate system. , Indicates the transition coordinate system With angular velocity Around Axis rotation horn.

[0035] S2.2: Inertial coordinate system With the ballistic coordinate system The conversion relationship between them is:

[0036] (3)

[0037] In the formula, This represents the coordinate projection of the missile in the ballistic coordinate system. Indicates the trajectory angle of the bullet; This indicates the trajectory deviation angle of the bullet.

[0038] Similarly, the transformation matrix from the inertial coordinate system to the ballistic coordinate system can be obtained through two rotations. This yields the transformation matrix between the inertial coordinate system and the ballistic coordinate system. for:

[0039] (4)

[0040] In the formula, This indicates that the inertial coordinate system is used with angular velocity Around Axis rotation The basic rotation matrix of the angle is then used to form the transition coordinate system. , Indicates the transition coordinate system With angular velocity Around Axis rotation horn.

[0041] S2.3: From formulas (1) and (3), the transformation relationship between the lead missile trajectory coordinate system and the follower missile trajectory coordinate system can be obtained as follows:

[0042] (5)

[0043] S3: Relative motion model function established in the missile trajectory coordinate system:

[0044] S3.1: Establish the relative motion model function in the inertial frame;

[0045] Depend on Figure 1 The relative positional relationship between the lead bullet and the follower bullet yields the position vector of the lead bullet in the inertial coordinate system, as shown in formula (6).

[0046] (6)

[0047] Differentiate the position vector of the missile in the inertial coordinate system, and differentiate both sides of the formula to obtain formula (7).

[0048] (7)

[0049] In the formula, This indicates the position of the missile leader in the inertial coordinate system. This indicates the position of the projectile in the inertial coordinate system. This represents the displacement relationship between the follower and leader projectiles in the inertial coordinate system. This represents the velocity vector of the projectile. This represents the velocity vector of the missile leader. Using the inertial coordinate system as the fixed coordinate system and the missile leader trajectory coordinate system as the moving coordinate system, based on the relationship between the absolute derivative and the relative reciprocal, we can obtain:

[0050] (8)

[0051] In the formula, This indicates the displacement relationship between the primary bullet and the lead bullet. The absolute derivative in an inertial coordinate system. express The relative derivative in the missile trajectory coordinate system This represents the angular velocity of the missile's trajectory coordinate system relative to the inertial coordinate system.

[0052] Combining equations (7) and (8), we obtain the relative motion model function between the leading and trailing projectiles:

[0053] (9)

[0054] S3.2: Project the relative motion model function in the inertial frame onto the missile trajectory coordinate system;

[0055] From formula (5), the velocity vector of the projectile will be... Projected onto the missile trajectory coordinate system:

[0056] (10)

[0057] velocity vector of the lead bullet Projection in the missile trajectory coordinate system:

[0058] (11)

[0059] In the formula, A numerical value representing the velocity of the projectile.

[0060] Based on the transformation relationship between the inertial coordinate system and the missile trajectory coordinate system, the rotational angular velocity of the missile trajectory coordinate system relative to the inertial coordinate system is obtained. for:

[0061] (12)

[0062] In the formula, The derivative representing the deflection angle of the missile trajectory. The derivative of the missile's trajectory angle.

[0063] angular velocity Projected onto the missile trajectory coordinate system:

[0064] (13)

[0065] Let the projection of the missile onto the three axes of the missile trajectory coordinate system be... , and Then we have:

[0066] (14)

[0067] (15)

[0068] Substituting equations (10)-(15) into equation (9), we obtain the relative motion model function between the leading and trailing projectiles:

[0069] (16)

[0070] In the formula, They are respectively The derivatives of are the velocities in the three directions in the missile trajectory coordinate system.

[0071] S3.3: Simplification and rearrangement of the relative motion model function;

[0072] The velocities of the lead and follower projectiles are decomposed onto the three axes of the inertial coordinate system. The decomposition method for the lead projectile is the same as that for the follower projectile. The velocity equations of the follower projectile in the three directions in the inertial coordinate system are as follows:

[0073] (17)

[0074] In the formula, Represents the impact of a projectile in an inertial frame. The speed of directional decomposition Represents the impact of a projectile in an inertial frame. The speed of directional decomposition Represents the impact of a projectile in an inertial frame. The speed of directional decomposition.

[0075] Differentiate the velocity equations obtained from equation (17):

[0076] (18)

[0077] In the formula, Represents the impact of a projectile in an inertial frame. Acceleration of directional decomposition Represents the impact of a projectile in an inertial frame. Acceleration decomposed by direction Represents the impact of a projectile in an inertial frame. Acceleration of directional decomposition.

[0078] From the tangential acceleration acceleration from the trajectory angle Ballistic deflection acceleration acceleration in three directions from the projectile , , The transformation equation is:

[0079] (19)

[0080] Simplifying equations (16)-(19), we obtain the simplified relative motion model function:

[0081] (20)

[0082] In the formula, They are respectively The derivative of This represents the coordinates of the missile in the missile trajectory coordinate system. This represents the three directional components of the projectile's velocity in the inertial coordinate system. This represents the three-directional components of the missile's velocity in the inertial coordinate system. , , , To control the quantity, For error, Let the acceleration of the missile in the three directions in the inertial coordinate system be denoted by the matrix. as follows:

[0083]

[0084]

[0085] Saturation function definition:

[0086] (twenty one)

[0087] Let the expected position of the missile be... Assume position error Speed ​​error Simplifying formula (20), we obtain the error equation:

[0088] (twenty two)

[0089] S4: Controller design based on the error equation:

[0090] The controller design for the formation control system is carried out using sliding mode variable structure control.

[0091] Selecting a sliding surface:

[0092] (twenty three)

[0093] In the formula, For the system's sliding surface, is a coefficient.

[0094] Differentiate with respect to the sliding surface:

[0095] (twenty four)

[0096] In the formula, They are respectively The derivative of .

[0097] Choose the exponential approach law:

[0098] (25)

[0099] In the formula, is a coefficient.

[0100] Rearranging formulas (22)-(25), we get the initial control law as follows:

[0101] (26)

[0102] design Observer error Conduct observations and use the observed values Substitution error .

[0103] The observer model is as follows:

[0104] (27)

[0105] In the formula, for Observer for system state variables The observation error, and They are respectively Observer for system state variables Observations and extended state variables (error ) observations, for The observed values, and All are coefficients.

[0106] Considering that the impact of the approaching law on the system is reduced when the projectile has not reached the desired position, an exponential function is added as the coefficient of the sliding surface and the approaching law. This makes the approaching law less effective when the distance is greater, while it makes the approaching law effective when the projectile is closer to the desired position, so that the system remains stable.

[0107] (28)

[0108] In the formula, These represent the relative positions of the missile and the lead missile in the missile trajectory coordinate system. This represents the relative distance between the missile and the target missile in the missile's trajectory coordinate system. These represent the desired positions of the missile in the lead missile trajectory coordinate system, respectively. This represents the expected distance from the missile in the lead missile trajectory coordinate system. and All represent coefficients.

[0109] Knot The observer rearranges the control law formula (25), and the final control law is designed as follows:

[0110] (29)

[0111] Example

[0112] Assume there exists a missile formation with one lead missile and two follower missiles, and the initial positions of the three missiles in the ground coordinate system are as follows: The expected positions of the two follower missiles in the lead missile trajectory coordinate system are respectively Leading the missile with 900 initial velocity, The missiles are launched at the same angle of attack and fly in the same direction. The two missiles are launched from their respective positions with the same initial velocity and angle of attack. Figure 3 A flight simulation diagram showing the overall missile formation, with the simulation of one of the missiles as shown below. Figure 4-9 As shown; Figure 4 This represents the X-direction position curve in the missile trajectory coordinate system, with an error within 5 meters. Figure 4 This represents the X-direction position curve in the missile trajectory coordinate system, with an error within 5 meters. Figure 5 This represents the position curve in the Y direction of the missile trajectory coordinate system, with an error within 3 meters. Figure 6 This represents the position curve in the Z-direction of the missile trajectory coordinate system, with an error within 0.5 meters. Figure 7-9 These represent the acceleration curves in the X, Y, and Z directions of the lead missile trajectory coordinate system, respectively. Therefore, the above error values ​​are all within the error range of the missile formation flight control, proving that the multi-missile formation flight controller design method based on sliding mode control of this invention is effective.

Claims

1. A multi-missile formation flight control method based on sliding mode control, characterized in that, The steps are specifically as follows: S1: defining a coordinate system of a relative motion equation; S2: constructing a conversion relationship between the coordinate systems; S3: establishing a relative motion model function under a leader missile trajectory coordinate system; The specific steps are as follows: S3.1: establishing a relative motion model function under an inertial system; S3.2: projecting the relative motion model function under the inertial system onto the leader missile trajectory coordinate system; S3.3: simplifying the relative motion model function; The relative motion model function between the leader missile and the follower missile: ， wherein denotes the coordinates of the target missile in the target missile coordinate system, denotes the three directional components of the velocity of the target missile in the inertial coordinate system, denotes the three directional components of the velocity of the target missile in the inertial coordinate system, , , , , is an error, is a matrix, is a saturation function, is the three directional accelerations of the target missile in the inertial coordinate system, is a control variable, are the derivatives of , respectively; S4: designing a controller for an error equation; The error equations for position error and velocity error are: ; wherein is a position error, is a velocity error; position error velocity error wherein represents the coordinates of the target missile in the target missile coordinate system, represents the three-directional components of the velocity of the target missile in the inertial coordinate system, represents the three-directional components of the velocity of the target missile in the inertial coordinate system, is the desired position of the target missile; Design Observer on error Observation is made with observation value Substitute error ; The observer model is given by: ， wherein is the observation error of the observer for the system state quantity , and are the observation value of the observer for the system state quantity and the observation value of the observer for the extended state quantity , is the observation value of the observer for the extended state quantity and are coefficients; The control law is finally designed as: ； wherein is a matrix, , , and each represent a coefficient, is a sliding surface, is a derivative of a position error, wherein , represents a relative distance from the leader missile in the leader missile's body coordinate system, represents a desired distance from the leader missile in the leader missile's body coordinate system. 2.The multi-missile formation flight control method based on sliding mode control according to claim 1, wherein, The coordinate system is defined as follows: The inertial coordinate system is defined : origin The launch point of the missile is selected, The axis is the intersection line of the trajectory plane and the horizontal plane, and points to the target, The axis is upward along the vertical line, The axis is perpendicular to the other two axes to form a right-handed coordinate system; Define the missile trajectory coordinate system :origin For the instantaneous center of mass of the lead missile, Instantaneous velocity of the axis and the lead bullet coincide, Located in In the vertical plane and with The axis is perpendicular. The axis is perpendicular to the other two axes, forming a right-handed coordinate system; The origin is defined from the ballistic coordinate system The origin is defined from the ballistic coordinate system The z-axis coincides with the instantaneous velocity of the projectile The z-axis coincides with the instantaneous velocity of the projectile The z-axis coincides with the instantaneous velocity of the projectile The z-axis coincides with the instantaneous velocity of the projectile The z-axis coincides with the instantaneous velocity of the projectile The z-axis coincides with the instantaneous velocity of the projectile​ 3.The multi-missile formation flight control method based on sliding mode control according to claim 1, wherein, In the construction of the conversion relationship between the coordinate systems, The conversion relationship between the leader missile trajectory coordinate system and the follower missile trajectory coordinate system is: ， wherein is the coordinate projection of the missile in the leader trajectory coordinate system, is the coordinate projection of the missile in the follower trajectory coordinate system; denotes the trajectory inclination angle of the leader missile; denotes the trajectory azimuth angle of the leader missile, denotes the trajectory inclination angle of the follower missile; denotes the trajectory azimuth angle of the follower missile; is the transformation matrix between the inertial coordinate system and the leader trajectory coordinate system, and the transformation matrix between the inertial coordinate system and the follower trajectory coordinate system .

4. The multi-missile formation flight control method based on sliding mode control according to claim 1, characterized in that, S3.1: establishing a relative motion model function between the leader missile and the follower missile under an inertial system; ， wherein denotes the velocity vector of the projectile, denotes the velocity vector of the leading projectile, the absolute derivative in the inertial coordinate system, denotes the relative derivative in the leading projectile's trajectory coordinate system, denotes the angular velocity of the leading projectile's trajectory coordinate system relative to the inertial coordinate system.

5. The multi-missile formation flight control method based on sliding mode control according to claim 4, characterized in that, S3.2: projecting the relative motion model function under the inertial system onto the leader missile trajectory coordinate system to obtain a relative motion model function between the leader missile and the follower missile: ， wherein is the relative velocity of the target projectile in the target trajectory coordinate system, denotes the trajectory angle of the target projectile, denotes the trajectory angle of the target projectile, denotes the trajectory angle of the target projectile, denotes the trajectory angle of the target projectile, denotes the velocity vector of the target projectile, denotes the value of the target projectile velocity, denotes the derivative of the trajectory angle of the target projectile, denotes the derivative of the trajectory angle of the target projectile.

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