Multi-missile formation flight control method based on sliding mode control

Abstract

The invention belongs to the field of formation flight control, and particularly relates to a multi-missile formation flight control method based on sliding mode control. The multi-missile formation flight control method comprises the following specific steps: defining a coordinate system of a relative motion equation; constructing a conversion relation between the coordinate systems; establishing a relative motion model function under a leading ballistic trajectory coordinate system; and designing a controller according to the error equation. Compared with the prior art, the position of the slave projectile in the leading projectile trajectory coordinate system and the speed of the leading projectile in the inertial coordinate system serve as state variables, the acceleration of the slave projectile serves as a control variable, a state equation is more visual, a sliding mode control method controller is adopted, an exponential function is set to serve as a parameter limiting reaching law, and the control precision is improved. And an ESO observer is designed to observe errors, so that the formation can still be kept stable when the leading bullet does curvilinear motion.

Description

Technical Field

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

[0002] With the development of information technology and the continuous improvement of anti-missile defense systems in various countries, a single missile used to strike a target gradually fails to meet combat requirements, and improving the penetration effectiveness of missiles has become an urgent problem to be solved. The coordinated combat of missile formations has achieved saturation strikes by launching multiple missiles at the same time, improving the penetration performance.

[0003] Many scholars are committed to studying formation control methods, and more research objects are vehicles and drones. There are fewer studies on high-speed moving missiles. At the same time, due to the faster speed, the control difficulty will increase. Existing literature mostly focuses on the design of formation controllers for leader-follower missiles in the ground system. This method requires the follower missile to master the global position information at all times to infer the relative position at each moment; in the ballistic coordinate system with the leader missile as the origin, less information is required to consider the change of the relative position of the follower missile. The position of the follower missile in the ballistic coordinate system of the leader missile and the speed of the leader missile in the inertial coordinate system are taken as state variables, and the acceleration of the follower missile is taken as the control variable. Combined with the error between the relative position of the follower missile and the expected position and the speed error of the leader missile, an intuitive state equation is constructed.

[0004] For the existing leader-follower model, most researchers will ignore the acceleration term of the leader, which has no effect on the tracking of the follower when the leader maintains a straight line motion; but when the leader moves in a curve, ignoring the acceleration term will cause the error to continue to accumulate and make it impossible to maintain the predetermined formation of the formation. Compensating 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 is robust to external noise interference and parameter perturbations, the present 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 the present invention is to provide a multi-missile formation flight control method based on sliding mode control, in which the leader missile acceleration term is compensated in the relative motion equation, the relative motion model is transformed into the leader missile ballistic coordinate system, the position of the follower missile in the ballistic coordinate system and the speed of the leader missile in the inertial coordinate system are taken as state variables, the acceleration of the follower missile is taken as the control quantity, the error between the relative position of the follower missile and the expected position and the speed error of the leader missile are combined, a sliding mode control method controller is adopted, and an exponential function is set as a parameter to limit the convergence law, and an ESO observer is designed to observe the error, so as to achieve that the formation can still remain stable when the leader missile makes a curved motion.

[0006] The technical solution to achieve the purpose of the present invention is: a multi-missile formation flight control method based on sliding mode control,

[0007] It is characterized in that the steps are specifically as follows: defining a coordinate system of the relative motion equation;

[0008] Construct the transformation relationship between various coordinate systems;

[0009] The relative motion model function is established in the ballistic coordinate system of the projectile;

[0010] Design the controller based on the error equation.

[0011] Compared with the prior art, the present invention has the following significant advantages:

[0012] (1) The present invention is based on the leader-follower relative motion model, compensates for the acceleration term of the leader, and transforms the relative motion model into the leader ballistic coordinate system. Compared with the ground coordinate system, the follower needs less position information and is easier to control.

[0013] (2) The position of the follower missile in the ballistic coordinate system of the leader missile and the velocity of the follower missile in the inertial coordinate system are taken as state variables, and the acceleration of the follower missile is taken as the control variable. The state equation is more intuitive.

[0014] (3) The sliding mode control algorithm is simple, fast in response, and robust to external noise interference and parameter perturbations. By setting an exponential function as the parameter limit reaching law, the overshoot is reduced when the system reaches stability, and the ESO observer observation error is designed. Under the premise of ensuring flight accuracy, the formation can still remain stable when the leader is moving in a curve. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Figure 1 : The motion relationship between the leading projectile and the following projectile in the present invention;

[0016] Figure 2 : The three-dimensional model of the lead bullet in the present invention;

[0017] Figure 3 : The formation tracking curve trajectory simulation diagram provided by the present invention;

[0018] Figure 4 : The X-direction position curve diagram of the follower bullet in the lead bullet trajectory coordinate system provided by the present invention;

[0019] Figure 5 : The Y-direction position curve diagram of the follower bullet in the lead bullet trajectory coordinate system provided by the present invention;

[0020] Figure 6 : A Z-direction position curve diagram of a follower bullet in a lead bullet trajectory coordinate system provided by the present invention;

[0021] Figure 7: The X-direction acceleration curve of the follower projectile in the lead projectile trajectory coordinate system provided by the present invention;

[0022] Figure 8 : The Y-direction acceleration curve of the follower projectile in the lead projectile trajectory coordinate system provided by the present invention;

[0023] Fig. 9 : The Z-direction acceleration curve of the follower projectile in the lead projectile trajectory coordinate system provided by the present invention. DETAILED DESCRIPTION

[0024] The following will be combined with the attached embodiment of the present invention Figure 1-9 , the technical solutions in the embodiments of the present invention are clearly and completely described. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of them. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0025] S1: Coordinate system defining the relative motion equations:

[0026] S1.1: Define the inertial coordinate system Ox1y1z1: The origin O is the launch point of the missile, the Ox1 axis is the intersection of the ballistic plane and the horizontal plane, pointing to the target is positive, the Oy1 axis is upward along the vertical line, and the Oz1 axis is perpendicular to the other two axes to form a right-handed coordinate system.

[0027] S1.2: Define the ballistic coordinate system O l x l y l z l : Origin O l is the instantaneous center of mass of the lead projectile, O l x l The instantaneous velocity V of the shaft and the projectile l Coincidence, O l y l Located in the V l In the vertical plane and with O l x l Axis vertical, O l z l The axis is perpendicular to the other two axes to form a right-handed coordinate system.

[0028] S1.3: Define the trajectory coordinate system O f x f y f z f : Origin O f is the instantaneous center of mass of the lead projectile, O f x f The instantaneous velocity V of the shaft and the projectile f Coincidence, O f yf Located in the V f In the vertical plane and with O f x f Axis vertical, O f z f The axis is perpendicular to the other two axes to form a right-handed coordinate system.

[0029] S2: Construct the transformation relationship between the coordinate systems:

[0030] S2.1: Inertial coordinate system Ox1y1z1 and ballistic coordinate system O l x l y l z l The conversion relationship between

[0031]

[0032] In the formula, [x l y l z l ] T is the coordinate projection of the missile in the ballistic coordinate system, [x1 y1 z1] T is the coordinate projection of the missile in the inertial coordinate system. l Indicates the ballistic inclination angle of the lead projectile; θ l Indicates the ballistic deviation angle of the lead bullet.

[0033] The transformation matrix from the inertial coordinate system to the ballistic coordinate system can be obtained by two rotations. First, the inertial coordinate system is rotated around the Oy axis by a factor of ψ. l Angle, then rotate around the axis by θ l Angle, we can get the transformation matrix L(ψ l ,θ l )for:

[0034]

[0035] In the formula, L(ψ l ) represents the inertial coordinate system with angular velocity Rotation ψ around the Oy axis l The basic rotation matrix of the angle is then formed into the transition coordinate system Ox'yz, L(θ l ) represents the transition coordinate system Ox'yz with an angular velocity Rotate θ around the Oz axis l horn.

[0036] S2.2: Inertial coordinate system Ox1y1z1 and trajectory coordinate system O f x f y f z fThe conversion relationship between them is:

[0037]

[0038] In the formula, [x f y f z f ] T is the coordinate projection of the missile in the ballistic coordinate system. f represents the ballistic inclination angle of the projectile; θ f Indicates the ballistic deviation angle of the projectile.

[0039] Similarly, the transformation matrix from the inertial coordinate system to the ballistic coordinate system can be obtained by two rotations. The transformation matrix L(ψ f ,θ f )for:

[0040]

[0041] In the formula, L(ψ f ) represents the inertial coordinate system with angular velocity Rotation ψ around the Oy axis f The basic rotation matrix of the angle is then formed into the transition coordinate system Ox'yz, L(θ f ) represents the transition coordinate system Ox'yz with an angular velocity Rotate θ around the Oz axis f horn.

[0042] S2.3: From formulas (1) and (3), the conversion relationship between the leading projectile trajectory coordinate system and the following projectile trajectory coordinate system can be obtained as follows:

[0043]

[0044] S3: Relative motion model function based on the ballistic coordinate system:

[0045] S3.1: Establish the relative motion model function in the inertial system;

[0046] Depend on Figure 1 The relative position relationship between the leading missile and the following missile gives the position vector of the leading missile in the inertial coordinate system, as shown in formula (6).

[0047] r=r f -r l (6)

[0048] The position vector of the lead projectile in the inertial coordinate system is differentiated, and the derivatives of both sides of the formula are obtained to obtain formula (7).

[0049]

[0050] In the formula, r l represents the position of the missile in the inertial coordinate system, r f represents the position of the follower projectile in the inertial coordinate system, r represents the displacement relationship between the follower projectile and the leader projectile in the inertial coordinate system; V f represents the velocity vector from the projectile, V l It represents the velocity vector of the lead projectile. Taking the inertial coordinate system as the fixed coordinate system and the lead projectile trajectory coordinate system as the moving coordinate system, according to the relationship between the absolute derivative and the relative reciprocal, we can get:

[0051]

[0052] In the formula, dr / dt represents the displacement relationship between the follower and the leader, r is the absolute derivative in the inertial coordinate system, It represents the relative derivative of r in the ballistic coordinate system of the leading projectile, and ω represents the angular velocity of rotation of the ballistic coordinate system of the leading projectile relative to the inertial coordinate system.

[0053] Combining formulas (7) and (8), we get the relative motion model function between the leading projectile and the following projectile:

[0054]

[0055] S3.2: Project the relative motion model function in the inertial system onto the projectile trajectory coordinate system;

[0056] According to formula (5), the velocity vector V f Projection onto the ballistic coordinate system:

[0057]

[0058] Velocity vector V of the lead projectile l Projection in the ballistic coordinate system:

[0059]

[0060] Where V is the velocity of the lead.

[0061] According to the conversion relationship between the inertial coordinate system and the leading ballistic coordinate system, the rotation angular velocity ω of the leading ballistic coordinate system relative to the inertial coordinate system is obtained as follows:

[0062]

[0063] In the formula, represents the derivative of the ballistic angle of the lead projectile, It represents the derivative of the ballistic inclination angle of the lead projectile.

[0064] Project the angular velocity ω to the ballistic coordinate system:

[0065]

[0066] Assume that the projections of the three axes of the missile on the trajectory coordinate system of the missile are x, y and z, then:

[0067]

[0068] Substituting formulas (10)-(15) into formula (9), we can obtain the relative motion model function between the leading projectile and the following projectile:

[0069]

[0070] In the formula, They are the derivatives of x, y, and z, respectively, and are the velocities in the three directions in the ballistic coordinate system.

[0071] S3.3: Simplification and arrangement of relative motion model functions;

[0072] The speed of the leading projectile and the following projectile is decomposed into the three axes of the inertial coordinate system. The decomposition method of the leading projectile is the same as that of the following projectile. The speed equation group of the following projectile in the three directions in the inertial coordinate system is:

[0073]

[0074] Where V fx represents the velocity decomposed from the projectile in the x direction in the inertial system, V fy represents the velocity decomposed from the projectile in the y direction in the inertial system, V fz Represents the velocity decomposed from the projectile in the z direction in the inertial system.

[0075] Derivative of the velocity equations obtained from formula (17):

[0076]

[0077] In the formula, represents the acceleration decomposed from the projectile in the x direction in the inertial system, represents the acceleration decomposed from the projectile in the y direction in the inertial system, Represents the acceleration decomposed from the projectile in the z direction in the inertial system.

[0078] Tangential acceleration from the projectile From the ballistic inclination acceleration Ballistic deflection acceleration and the acceleration a in the three directions of the projectile x 、a y 、a z The conversion equation is:

[0079]

[0080] Arrange formulas (16)-(19) to obtain the simplified relative motion model function:

[0081]

[0082] In the formula, They are the derivatives of X1, X2, and X3 respectively. X1 represents the coordinates of the follower in the ballistic coordinate system of the leader, X2 represents the three-direction components of the follower's velocity in the inertial coordinate system, and X3 represents the three-direction components of the leader's velocity in the inertial coordinate system. X1 = [xyz] T 、X2=[V fx V fy V fz ] T 、X3=[V lx V ly V lz ] T , U=[a x a y a z ] T is the control quantity, D is the error, a is the acceleration of the projectile in three directions in the ballistic coordinate system, and the matrices A, B, C, and E are as follows:

[0083]

[0084] Saturation function sat definition:

[0085]

[0086] Let the expected position of the projectile be X * =[x * y * z * ] T , let the position error e1=X1-X * , speed error e2 = X2-X3, rearrange formula (20) to get the error equation:

[0087]

[0088] S4: Design controller based on error equation:

[0089] Sliding mode variable structure control is used to design the controller of the formation control system.

[0090] Select sliding surface:

[0091] s=k1e1+e2 (23)

[0092] Where s is the sliding surface of the system and k1 is the coefficient.

[0093] Derivative of the sliding surface:

[0094]

[0095] In the formula, are the derivatives of s, e1, and e2 respectively.

[0096] Use the exponential reaching law:

[0097]

[0098] In the formula, k2 is the coefficient.

[0099] After rearranging formulas (22)-(25), the initial control law is:

[0100]

[0101] Design the ESO observer to observe the error D and use the observed value Replace the error D.

[0102] The ESO observer model is:

[0103]

[0104] Where e is the observation error of the ESO observer on the system state quantity X2, z1 and z2 are the observation values ​​of the ESO observer on the system state quantity X2 and the observation values ​​of the expanded state quantity D (error D), respectively. is the observed value of D, and β1 and β2 are coefficients.

[0105] Considering that when the missile has not reached the expected position, the influence of the reaching law on the system is reduced, and an exponential function is added as the coefficient of the reaching law, so that the effect of the reaching law becomes smaller when the distance is far, and when the missile is close to the expected position, the reaching law takes effect to keep the system stable.

[0106]

[0107] In the formula, x, y, and z represent the relative positions of the follower in the ballistic coordinate system of the leader, R represents the relative distance of the follower in the ballistic coordinate system of the leader, and x * ,y * ,z * They respectively represent the expected position of the follower projectile in the lead projectile ballistic coordinate system, R' represents the expected distance of the follower projectile in the lead projectile ballistic coordinate system, and k and k3 both represent coefficients.

[0108] Combined with the ESO observer, the control law formula (25) is reorganized and the control law is finally designed as

[0109]

[0110] Example

[0111] Suppose there is a missile formation with one leader missile and two followers. The initial positions of the three missiles in the ground coordinate system are [0,5,0], [-50,0,300], [-100,0,200], and the expected positions of the two followers in the leader missile ballistic coordinate system are [200,400,-200], [200,100,-400], respectively. The leader missile is launched at an initial velocity of 900 m / s and a firing angle of 40°. The two followers are launched at the same initial velocity and firing angle at the corresponding positions. Figure 3 The flight simulation diagram of the entire missile formation shows the simulation of one of the missiles. Figure 4-9 As shown; Figure 4 It represents the X-direction position curve in the ballistic coordinate system of the missile, with an error within 5 meters; Figure 4 It represents the X-direction position curve in the ballistic coordinate system of the missile, with an error within 5 meters; Figure 5 It represents the Y-direction position curve in the ballistic coordinate system of the missile, with an error within 3 meters; Figure 6 It represents the Z-direction position curve in the ballistic coordinate system of the lead projectile, with an error within 0.5 meters; Figure 7-9 They respectively represent the acceleration curves in the X, Y, and Z directions in the ballistic coordinate system of the leading missile. Therefore, the above error values ​​are all within the error range of the formation flight control for the missiles, proving that the design method of the multi-missile formation flight controller based on sliding mode control of the present invention is effective.

Claims

1. A multi-missile formation flight control method based on sliding mode control, characterized in that: The steps are as follows: Define the coordinate system for the equations of relative motion; Construct the transformation relationship between various coordinate systems; The relative motion model function is established in the ballistic coordinate system of the projectile; Design a controller based on the error equation.

2. The multi-missile formation flight control method based on sliding mode control according to claim 1 is characterized in that: The coordinate system is defined as follows: Define the inertial coordinate system Oxyz: The origin O is the launch point of the missile, the Ox axis is the intersection of the ballistic plane and the horizontal plane, pointing to the target is positive, the Oy axis is upward along the vertical line, and the Oz axis is perpendicular to the other two axes to form a right-handed coordinate system; Define the ballistic coordinate system O l x l y l z l : Origin O l is the instantaneous center of mass of the lead projectile, O l x l The instantaneous velocity V of the shaft and the projectile l Coincidence, O l y l Located in the V l In the vertical plane and with O l x l Axis vertical, O l z l The axis is perpendicular to the other two axes to form a right-handed coordinate system; Define the trajectory coordinate system O from the projectile f x f y f z f : Origin O f is the instantaneous center of mass of the lead projectile, O f x f The instantaneous velocity V of the shaft and the projectile f Coincidence, O f y f Located in the V f In the vertical plane and with O f x f Axis vertical, O f z f The axis is perpendicular to the other two axes to form a right-handed coordinate system.

3. The multi-missile formation flight control method based on sliding mode control according to claim 1 is characterized in that: In constructing the transformation relationship between various coordinate systems, The conversion relationship between the leading projectile trajectory coordinate system and the following projectile trajectory coordinate system is: In the formula, [x l y l z l ] T is the coordinate projection of the missile in the ballistic coordinate system, [x f y f z f ] T is the coordinate projection of the missile in the ballistic coordinate system; ψ l Indicates the ballistic inclination angle of the lead projectile; θ l represents the ballistic deviation angle of the lead projectile, L(ψ l ,θ l ) is the transformation matrix between the inertial coordinate system and the leading projectile trajectory coordinate system, and the transformation matrix L(ψ f ,θ f ).

4. The multi-missile formation flight control method based on sliding mode control according to claim 1 is characterized in that: The specific steps of establishing the relative motion model function in the ballistic coordinate system are as follows: S3.1: Establish the relative motion model function in the inertial system; S3.2: Project the relative motion model function in the inertial system onto the trajectory coordinate system; S3.3: Simplification of relative motion model functions.

5. The multi-missile formation flight control method based on sliding mode control according to claim 4 is characterized in that: S3.1: Establish the relative motion model function in the inertial system; The relative motion model function between the leader and follower: Where V f represents the velocity vector from the projectile, V l represents the velocity vector of the projectile, the absolute derivative of r in the inertial coordinate system, It represents the relative derivative of r in the ballistic coordinate system of the leading projectile, and ω represents the angular velocity of rotation of the ballistic coordinate system of the leading projectile relative to the inertial coordinate system.

6. The multi-missile formation flight control method based on sliding mode control according to claim 4 is characterized in that: The relative motion model function between the leader and follower: In the formula, is the relative velocity of the follower projectile in the ballistic coordinate system of the leader projectile, θ l Indicates the ballistic deviation angle of the lead projectile, θ f represents the ballistic deviation angle of the projectile, ψ l represents the ballistic inclination angle of the lead projectile, ψ f V represents the ballistic inclination angle of the projectile. f represents the velocity vector of the follower, V represents the value of the lead velocity, represents the derivative of the ballistic angle of the lead projectile, It represents the derivative of the ballistic inclination angle of the lead projectile.

7. The multi-missile formation flight control method based on sliding mode control according to claim 4 is characterized in that: The relative motion model function of the leading projectile and the following projectile: In the formula, X1 represents the coordinates of the follower in the ballistic coordinate system of the leader, X2 represents the three-direction components of the velocity of the follower in the inertial coordinate system, and X3 represents the three-direction components of the velocity of the leader in the inertial coordinate system. X1 = [xyz] T 、X2=[V fx V fy V fz ] T 、X3=[V lx V ly V lz ] T , U=[a x a y a z ] T , D is the error, A, B, C, E are matrices, sat is the saturation function, a is the acceleration of the projectile in the three directions in the ballistic coordinate system, U is the control amount, are the derivatives of X1, X2, and X3 respectively.

8. The multi-missile formation flight control method based on sliding mode control according to claim 1, characterized in that: In controller design based on the error equation, Design the ESO observer to observe the error D and use the observed value Replace the error D; The ESO observer model is: In the formula, e is the observation error of the ESO observer on the system state quantity X2, z1 and z2 are the observation values ​​of the ESO observer on the system state quantity X2 and the observation values ​​of the expanded state quantity D, respectively. is the observed value of D, and β1 and β2 are coefficients.

9. The multi-missile formation flight control method based on sliding mode control according to claim 8, characterized in that: The control law is finally designed as: In the formula, C, E are matrices, k, k1, k2 and k3 are coefficients, s is the sliding surface, is the derivative of the position error.

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