Analysis method of guidance law in distributed cooperative air-to-air missile and its guidance system

Abstract

The present invention provides a method for analyzing the distributed cooperative mid-range guidance law of an air-to-air missile and a guidance system thereof, comprising: a mid-range and terminal guidance handover probability calculation model construction step: constructing a mid-range and terminal guidance handover probability calculation model by analyzing error sources and error transmission links; a multi-missile field of view splicing method establishment step: establishing a multi-missile field of view splicing method for a virtual guidance point; a relative motion model construction step: establishing a relative motion model of multiple air-to-air missiles and a target; a time-coordinated mid-range guidance law construction step: constructing a time-coordinated mid-range guidance law in the line of sight direction using the second-order multi-agent consistency theory; an angle-coordinated optimal mid-range guidance law construction step: constructing an angle-coordinated optimal mid-range guidance law in the line of sight normal direction using the Gaussian pseudo-spectral method. The present invention effectively improves the probability of mid-range and terminal guidance handover, ensures that the radar seeker can quickly identify and stably track the target at the mid-range and terminal handover moment, and verifies the convergence efficiency of the method.

Description

Technical Field

[0001] The present invention relates to the field of cooperative guidance technology, and in particular, to a method for analyzing a distributed cooperative mid-course guidance law of an air-to-air missile and a guidance system thereof, and in particular to a distributed cooperative mid-course guidance law of an air-to-air missile based on space-based information guidance. Background Art

[0002] With the introduction of the U.S. military's information network integrated combat thinking, the space-based system has demonstrated its unique advantage of occupying an absolute high position on the battlefield, becoming the key to solving the information guarantee of targets in long-range precision strikes, and the space-based guidance mode is the most promising combat mode in space-based information-supported long-range precision strike operations. The in-depth integration of space-based information and air-to-air missile weapon systems can further shorten the time spent on target discovery, enhance the ability to fight beyond visual range and strike time-sensitive targets, and thus establish an air superiority position.

[0003] At present, there is no relevant example of using the space-based mid-range guidance mode. The difficulty lies in the small search field of view of the seeker of a single air-to-air missile, and the detection error of the space-based radar is in the order of tens of kilometers, which is far from meeting the accuracy requirements of the guidance control system, resulting in difficulty in completing the mid-range and final handovers, and poor closed-loop capability of the guidance link. Therefore, it is urgent to develop key technologies for multi-missile coordination, especially the guidance law for time-space coordination, expand the search range of the seeker through field of view splicing and coordinated detection, and improve the handover probability. Related research has important engineering application value.

[0004] Patent document CN101832738A discloses a multi-platform cooperative guidance system for long-range air-to-air missiles and its implementation method. The multi-platform cooperative guidance system includes an early warning aircraft module, a ground-based radar detection module, a carrier aircraft module, an other aircraft module, a long-range air-to-air missile module, a target information time-space normalization module, and a target information fusion module; the implementation method includes step 1: the early warning aircraft and the ground-based radar guide the carrier aircraft and the other aircraft to fly to the combat area; step 2: the radar of the other aircraft is turned on to detect the target, and the carrier aircraft launches the air-to-air missile; step 3: the target information detected by the early warning aircraft radar, the ground-based radar and the other aircraft radar is integrated, and the other aircraft performs mid-range guidance on the air-to-air missile

[0005] However, there are the following problems in the coordinated guidance of air-to-air missiles in the existing technology. The first problem is that most of the existing time-cooperative guidance laws only have global asymptotic consistency, which cannot guarantee that the system state parameters converge to consistency quickly within a limited time, and cannot be applied to strike missions such as high-precision and rapid guidance. The second problem is that it is difficult to obtain an analytical solution for the optimal guidance law, and the calculation efficiency is low, making it difficult to meet the real-time requirements in engineering. Finally, the research on the coordination variables of the cooperative guidance law is limited to time and angle, and the combat mission of cooperative guidance and cooperative detection also imposes constraints on conditions such as the position of the cooperative formation and the angular rate of sight, and the existing research results cannot be directly applied. Summary of the invention

[0006] In view of the defects in the prior art, the purpose of the present invention is to provide a guidance law analysis method and a guidance system for air-to-air missiles in distributed collaborative control.

[0007] A method for analyzing the distributed cooperative guidance law of an air-to-air missile provided by the present invention includes any one or more of the following steps:

[0008] Steps for constructing a mid-terminal guidance handover probability calculation model: construct a mid-terminal guidance handover probability calculation model by analyzing the error source and error transmission link;

[0009] Steps for establishing a multi-projectile field of view stitching method: establishing a multi-projectile field of view stitching method for a virtual guidance point;

[0010] Steps for establishing a multi-missile coordinated mid-course guidance model: Establish a multi-missile coordinated mid-course guidance model through the relative motion relationship between multiple air-to-air missiles and targets;

[0011] The step of constructing the time-coordinated mid-course guidance law is as follows: the coordinated mid-course guidance rate is decoupled to the line of sight direction through the multi-missile coordinated mid-course guidance model, and the time-coordinated mid-course guidance law is constructed by using the second-order multi-agent consistency theory;

[0012] The angle coordinated optimal mid-range guidance law construction step is as follows: the coordinated mid-range guidance rate is decoupled to the line of sight normal direction through the multi-missile coordinated mid-range guidance model, and the angle coordinated optimal mid-range guidance law is constructed by using the Gaussian pseudo-spectral method;

[0013] Simulation verification step: verify the effectiveness of the mid-terminal guidance handover probability calculation model, the time coordinated mid-term guidance law and the angle coordinated optimal mid-term guidance law through simulation.

[0014] Preferably, the step of constructing a mid-terminal guidance handover probability calculation model comprises:

[0015] The synthetic error calculation steps of the target indication position are as follows: the synthetic error of the target indication position is obtained according to the linear superposition method, wherein the error of the target position includes: the detection error of the high-orbit satellite, the combined navigation orientation error of the missile itself, the inertial navigation attitude error, the seeker angle measurement error and the position error caused by the delay time, etc. Since the error variables are independent of each other, the corresponding synthetic error is obtained by the following calculation formula:

[0016]

[0017]

[0018] Among them, μ() represents the mean, σ 2 () represents variance, ΔR2, ΔA2, ΔE2 represent the synthetic errors of the target indication position in terms of distance, elevation angle, and azimuth, respectively. The subscript position represents the target position detected by the high-orbit satellite, the subscript orientation represents the target direction determined by the integrated navigation, and the subscript delay represents the delay time of the signal transmission;

[0019] Calculation steps for the probability of mid-term and end-term handover: Since the target indication distance error is small compared to the target indication distance, the distance factor is ignored and only the angle interception condition is considered. The calculation formula for the probability of mid-term and end-term handover is as follows:

[0020]

[0021] Among them, A2 represents the target elevation angle, E2 represents the target azimuth angle, It represents the variance of the target indication position error in the line of sight height direction, represents the variance of the target indication position error in the line of sight azimuth direction, Δθ represents the angle measurement error of the seeker, P is the probability of mid-term handover, and θ0 is the available search range of the seeker.

[0022] Preferably, the step of establishing the multi-bullet field of view stitching method includes:

[0023] The steps of establishing virtual guidance points are as follows: each air-to-air missile is guided to its own virtual guidance point in the mid-range guidance phase. The virtual guidance points are distributed on a sphere with the target detection position as the center and the maximum detection distance of the radar seeker as the radius. The calculation formula is as follows:

[0024]

[0025] Where R represents the maximum detection range of the active radar seeker, R 误差 represents the radius of the 3σ synthetic error sphere, (x * (t),y * (t),z *(t)) represents the real-time position of the target detected by the space-based radar, (x i (t),y i (t),z i (t)) represents the dynamic virtual guidance point corresponding to the ith air-to-air missile, and x, y, and z represent the horizontal axis, vertical axis, and vertical axis in the three-dimensional spatial coordinate system, respectively.

[0026] Preferably, the relative motion model building step includes:

[0027] Step 1: Establish the geometric relationship equation of the relative motion between the missile and the target in the longitudinal plane as follows:

[0028]

[0029]

[0030] η i =θ i -q i

[0031] η T =θ T -q i

[0032]

[0033]

[0034] in, · represents the derivative of the variable with respect to time, M i and T represent the mass center positions of the ith missile and target respectively; V Mi and V T Represent the speed of the i-th missile and the target respectively; r i represents the relative distance between the i-th missile and the target; q i represents the sight angle of the i-th missile; θ i and θ T Respectively represent the ballistic inclination angles of the i-th missile and the target; η i and η T denote the velocity lead angles of the ith missile and the target respectively; a Mi and a T denote the normal acceleration of the i-th missile and the target respectively;

[0035] Step 2: According to the relative motion geometric relationship, a multi-missile coordinated mid-course guidance model is established. The calculation formula is as follows:

[0036]

[0037] Among them, u Mir =aMi sinη i represents the component of the tangential acceleration of the i-th missile in the line of sight direction, u Miq =a Mi cosη i represents the component of the normal acceleration of the ith missile in the normal direction of the line of sight, x 1i =r i , x 3i =q i , They all represent system state variables.

[0038] Preferably, the guidance law construction step in time coordination includes:

[0039] The guidance law has a two-layer coordination framework, including the top-level guidance law and the bottom-level guidance law. The top-level guidance law uses the remaining flight time of the missile as the coordination variable. According to the second-order multi-agent consistency convergence theory in finite time, the calculation formula of the top-level time coordination guidance law is established as follows:

[0040]

[0041] Among them, a ij represents the proportionality coefficient, x 1j =r j represents the relative distance between the jth missile and the target, represents the relative speed between the jth missile and the target, n represents the number of missiles, k1 and k2 represent gain coefficients, s1,s2∈(-∞,1) represent adjustment factors, g(x,s)=x·|x| exp(s-1)-1 .

[0042] Preferably, the step of constructing the angle coordinated optimal guidance law includes:

[0043]

[0044] Among them, u * (τ i ) represents τ i The optimal normal control command at time, represents the n×n dimensional positive definite symmetric time-varying control weight matrix, t goi represents the remaining flight time of the i-th missile, r i represents the relative distance between the i-th missile and the target, λ() represents the 2-dimensional Lagrange multiplier vector, and N represents the number of Gauss nodes.

[0045] Preferably, the Lyapunov equation is constructed to prove that the time-cooperative guidance law converges in a finite time, and the upper bound of the uniform convergence time is obtained, as follows:

[0046] Construct a Lyapunov functional of the following form:

[0047]

[0048] Among them, V r represents the Lyapunov functional, n represents the number of missiles, and t represents time;

[0049] When the control gain satisfies:

[0050]

[0051]

[0052] Among them, k, x 1k (0), x 2k (0) respectively represent the kth air-to-air missile, the missile-target distance of the kth air-to-air missile at the initial moment, and the missile-target relative speed of the kth air-to-air missile at the initial moment;

[0053] The derivative of the Lyapunov functional with respect to time satisfies:

[0054]

[0055] in, λ2 represents the second smallest eigenvalue, L n represents an n-order polynomial, s0=exp(s2-1);

[0056] Let K2 = min{K1λ2(L n )}, the upper bound of the uniform convergence time can be obtained as:

[0057]

[0058] Among them, t * represents the upper bound of the uniform convergence time, V r (0) represents the value of the Lyapunov functional at time 0.

[0059] Preferably, a Bolza-type performance index function is constructed that simultaneously satisfies the terminal sight angle constraint, position constraint and the maximum available overload constraint during flight, a Hamiltonian function is constructed to transform the functional extreme value problem into a two-point boundary value problem, and the Gauss pseudo-spectral method is used for discretization to obtain the optimal control input under the analytical situation.

[0060] Preferably, the optimal normal control instruction obtained by the calculation formula is an open-loop solution. Considering that the coefficient matrix in the system state equation and the weight matrix in the performance index function are time-varying, the time-varying matrix is ​​regarded as a constant in each step of the trajectory solution to solve the current control amount u* (τ0), and update the state quantity at the next moment, iterate continuously, and calculate the optimal guidance instruction at the current moment in real time.

[0061] A coordinated guidance system for multiple air-to-air missiles provided by the present invention adopts the distributed coordinated guidance law analysis method for air-to-air missiles.

[0062] Compared with the prior art, the present invention has the following beneficial effects:

[0063] 1. The collaborative guidance law proposed in the present invention can enable multiple air-to-air missiles to reach the designated mid-terminal guidance handover area at the same time with their respective desired line of sight angles, so as to meet the needs of field of view splicing and collaborative detection, thereby solving the problem of poor target indication accuracy under space-based information guidance and effectively improving the probability of mid-terminal guidance handover.

[0064] 2. Based on the distribution characteristics of the target position error, the present invention introduces a virtual guidance point to provide terminal constraints such as time, formation position, line of sight angle and line of sight angular rate for the guidance law, thereby ensuring that the radar seeker can quickly identify and stably track the target at the mid-term and final handover moments.

[0065] 3. The present invention establishes a time-cooperative guidance law based on the second-order multi-agent consistency theory in the line of sight, constructs a Lyapunov functional to derive the consistency time upper bound and its sufficient conditions, and fully verifies the convergence efficiency of this method.

[0066] 4. The present invention establishes an optimal guidance law with multi-terminal constraints in the line of sight normal direction, and uses the variational method and Gauss pseudo-spectral method to obtain an analytical solution in discrete form, which greatly improves the solution efficiency while ensuring the solution accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Other features, objects and advantages of the present invention will become more apparent from the detailed description of non-limiting embodiments made with reference to the following drawings:

[0068] Figure 1 This is a schematic diagram of the distribution of virtual guide points provided by the present invention.

[0069] Figure 2 A schematic diagram of the relative motion of multiple missiles and targets provided by the present invention.

[0070] Figure 3 This is a schematic diagram of the multi-missile communication topology structure provided by the present invention.

[0071] Figure 4 This is a schematic diagram of the speed curves of each missile provided by the present invention.

[0072] Figure 5 This is a schematic diagram of the relative distance curves between each missile and virtual guidance point provided by the present invention.

[0073] Figure 6 This is a schematic diagram of the elevation and depression angle curves of the sight lines of each missile provided by the present invention.

[0074] Figure 7 This is a schematic diagram of the line of sight deflection curves of each missile provided by the present invention.

[0075] Figure 8 This is a schematic diagram of the high and low angular rate curves of the line of sight of each missile provided by the present invention.

[0076] Fig. 9 This is a schematic diagram of the line of sight deflection rate curves of each missile provided by the present invention.

[0077] Fig.10 This is a schematic diagram of the normal overload curve of each missile provided by the present invention in the vertical plane.

[0078] Fig.11 This is a schematic diagram of the normal overload curve of each missile provided by the present invention in the horizontal plane.

[0079] Fig.12 This is a schematic diagram of the overload curve in the line of sight direction of each missile provided by the present invention.

[0080] Fig.13 This is a schematic diagram of the three-dimensional motion trajectory of each missile-target provided by the present invention.

[0081] Figures 4 to 12 In the equation, t represents time, v represents missile speed, r represents the relative distance between the missile and the virtual guidance point, ∈ represents the missile sight elevation angle, β represents the missile sight deflection angle, Indicates the missile's line of sight elevation angular rate, represents the missile line of sight deflection rate, u y Indicates the normal overload and u of the missile in the vertical plane z Indicates the normal overload of the missile in the horizontal plane, u r It indicates that the missile is overloaded in the line of sight direction. M1, M2, M3 and M4 represent 4 air-to-air missiles.

[0082] Fig.13 In the figure, the horizontal axis represents the x-axis distance in the missile launch coordinate system, the vertical axis represents the z-axis distance in the missile launch coordinate system, and the vertical axis represents the y-axis distance in the missile launch coordinate system. DETAILED DESCRIPTION

[0083] The present invention is described in detail below in conjunction with specific embodiments. The following embodiments will help those skilled in the art to further understand the present invention, but are not intended to limit the present invention in any form. It should be noted that, for those of ordinary skill in the art, several changes and improvements can also be made without departing from the concept of the present invention. These all belong to the protection scope of the present invention.

[0084] First, by analyzing the error sources and their transmission modes, the invention derives the distribution characteristics of the target indication position error and establishes a mid-term handover probability calculation model. Secondly, a multi-missile field of view splicing method is established with the maximum search range and the minimum discovery time as comprehensive indicators, thereby determining the spatial distribution of the virtual guidance point. Finally, the coordinated guidance law is decoupled into two planes, the line of sight direction and the line of sight normal, to achieve the coordination of multiple coordination variables. The effectiveness of this method is verified by simulation, and it can meet the actual combat needs.

[0085] A method for analyzing the distributed collaborative mid-course guidance law of an air-to-air missile provided by the present invention comprises any one or more of the following steps: a mid-course and terminal guidance handover probability calculation model construction step, a multi-missile field of view splicing method establishment step, a relative motion model construction step, a time-coordinated mid-course guidance law construction step, and an angle-coordinated optimal mid-course guidance law construction step.

[0086] Among them, the step of constructing the probability calculation model of mid-terminal guidance handover is to construct the probability calculation model of mid-terminal guidance handover by analyzing the error source and error transmission link. This step includes:

[0087] The synthetic error calculation steps of the target indication position are as follows: the synthetic error of the target indication position is obtained according to the linear superposition method, wherein the error of the target position includes: the detection error of the high-orbit satellite, the combined navigation orientation error of the missile itself, the inertial navigation attitude error, the seeker angle measurement error and the position error caused by the delay time, etc. Since the error variables are independent of each other, the corresponding synthetic error is obtained by the following calculation formula:

[0088]

[0089]

[0090] Among them, μ() represents the mean of the synthetic error, σ 2 () represents the variance of the composite error, ΔR2, ΔA2, and ΔE2 represent the composite errors of the target indicated position in terms of distance, altitude, and azimuth, respectively. The subscript position represents the target position detected by the high-orbit satellite, the subscript orientation represents the target direction determined by the integrated navigation, and the subscript delay represents the delay time of the signal transmission.

[0091] Calculation steps for the probability of mid-term and end-term handover: Since the target indication distance error is small compared to the target indication distance, the distance factor is ignored and only the angle interception condition is considered. The calculation formula for the probability of mid-term and end-term handover is as follows:

[0092]

[0093] Among them, A2 represents the target elevation angle, E2 represents the target azimuth angle, It represents the variance of the target indication position error in the line of sight height direction, represents the variance of the target indication position error in the line of sight azimuth direction, Δθ represents the angle measurement error of the seeker, P is the probability of mid-term handover, and θ0 is the available search range of the seeker.

[0094] Steps for establishing a multi-projectile field of view stitching method: Establish a multi-projectile field of view stitching method for virtual guidance points. Figure 1 As shown, in order to maximize the multi-missile coordinated detection range, it is necessary to set a reasonable virtual guidance point, so this step includes: establishing a virtual guidance point step: guiding each air-to-air missile to its own virtual guidance point in the mid-guidance stage. The virtual guidance points are distributed on a sphere with the target detection position as the center and the maximum detection distance of the radar seeker as the radius. The calculation formula is as follows:

[0095]

[0096] Where R represents the maximum detection range of the active radar seeker, R 误差 represents the radius of the 3σ synthetic error sphere, (x * (t),y * (t),z * (t)) represents the real-time position of the target detected by the space-based radar, (x i (t),y i (t),z i (t)) represents the dynamic virtual guidance point corresponding to the ith air-to-air missile, and x, y, and z represent the horizontal axis, vertical axis, and vertical axis in the three-dimensional spatial coordinate system, respectively.

[0097] In order to further enhance the capability of multi-missile coordinated combat, the two links of coordinated guidance and coordinated detection are designed in an integrated manner. According to the 3σ principle, an elliptical envelope is used in two-dimensional space to describe the target position distribution characteristics. In order to make the search field of view after splicing cover the error distribution range and ensure a smooth transition during the mid-term and end-terminal handover process, the following terminal constraints are established:

[0098] t F1 =t F2 =t F3 =t F4

[0099] q1(t F )=q2(t F )=q3(t F )=q4(t F )=q F =0

[0100]

[0101] Among them, t Fi is the end time of the mid-range guidance of the i-th air-to-air missile; q i (t F ) is the sight angle between the ith air-to-air missile and the corresponding virtual target point at the end of the mid-range guidance; is the angular velocity of the line of sight between the ith air-to-air missile and the corresponding virtual target point at the end of the mid-range guidance.

[0102] Relative motion model construction steps: Figure 2 As shown, a relative motion model between multiple air-to-air missiles and targets is established. This step includes: Step 1: For the problem of multi-missile coordinated attack, the relative motion geometric relationship equation between the missiles and the target in the longitudinal plane is established as follows:

[0103]

[0104]

[0105] η i =θ i -q i

[0106] η T =θ T -q i

[0107]

[0108]

[0109] in, · represents the derivative of the variable with respect to time, M i and T represent the mass center positions of the ith missile and target respectively; V Mi and V T Represent the speed of the i-th missile and the target respectively; r i represents the relative distance between the i-th missile and the target; q i represents the sight angle of the i-th missile; θ i and θ T Respectively represent the ballistic inclination angles of the i-th missile and the target; η i and η T denote the velocity lead angles of the ith missile and the target respectively; a Mi and a T represent the normal acceleration of the i-th missile and the target respectively.

[0110] Considering that the strike targets of the present invention include important air nodes such as tankers and early warning aircraft, which have low speed and low maneuverability compared to air-to-air missiles and can be regarded as uniform motion targets, the relative motion geometric relationship equation can be organized into the following form:

[0111]

[0112]

[0113] Among them, u Mir =a Mi sinη i is the component of the tangential acceleration of the i-th missile in the line of sight direction; u Miq =a Mi cosη i is the component of the normal acceleration of the i-th missile in the direction normal to the line of sight.

[0114] Step 2: According to the relative motion geometric relationship, a multi-missile coordinated mid-course guidance model is established. The calculation formula is as follows:

[0115]

[0116] Among them, u Mir =a Mi sinη i represents the component of the tangential acceleration of the i-th missile in the line of sight direction, u Miq =a Mi cosη i represents the component of the normal acceleration of the ith missile in the normal direction of the line of sight, x 1i =r i , x 3i =q i , They all represent system state variables.

[0117] Steps for constructing the time-coordinated mid-range guidance law: In the line of sight, the time-coordinated mid-range guidance law is constructed using the second-order multi-agent consistency theory. This step includes: the mid-range guidance law has a two-layer coordination framework including the top-level guidance law and the bottom-level guidance law, where the top-level guidance law uses the remaining flight time of the missile as the coordination variable; according to the second-order multi-agent finite time consistency convergence theory, the top-level time-coordinated guidance law calculation formula is established as follows:

[0118]

[0119] Among them, a ij represents the proportionality coefficient, x 1j =r j represents the relative distance between the jth missile and the target, represents the relative speed between the jth missile and the target, n represents the number of missiles, k1 and k2 represent gain coefficients, s1, s2∈(-∞, 1) represent adjustment factors, g(x, s) = x·|x| exp(s-1)-1 , x and s in g(x, s) are only used as independent variables to illustrate the form of the function. x and s can be assigned values ​​and have practical meanings after the assignment.

[0120] Specifically, Figure 3 As shown, the communication between the missile groups adopts an undirected graph The model is modeled in the following way, where v n ={1, 2, ..., n} is a node set, For edge sets.

[0121] Define the adjacency matrix A n =[a ij ]∈R n×n , where a ij is the connection weight coefficient between missile i and missile j, And i≠j, there is a ij =a ji , if a ij >0, it means that missiles i and j can communicate with each other. Unless otherwise specified, it is generally taken as a ii =0.

[0122] Define the Laplace matrix L n =[l ij ]∈R n×n , where l ii is the number of connections from which missile i receives information from other missiles, When i≠j, l ij =-a ij .

[0123] Decomposing the multi-missile coordinated mid-course guidance model into the line of sight direction, we get: Define the remaining flight time as: t goi =t Fi -t, control the remaining flight time of multiple missiles to be consistent, so as to ensure that they arrive at the designated target point at the same time and meet the handover distance constraint. The remaining flight time can be estimated by the following formula:

[0124] Definition 1: If for any initial state x 1i (0),x 2i (0), under the action of the set control quantity, there exists a finite time T∈[0,+∞) such that When t>T, there is x 1j (t) = x 1i (t), x2j (t) = x 2i (t), (i,j)∈{1,2,…,n}, then the system is said to be able to converge the cooperative variables to consistency within a finite time.

[0125] According to the second-order multi-agent finite-time consistency convergence theory, the top-level time coordination guidance law is:

[0126]

[0127] In addition, by constructing the Lyapunov equation, it is proved that the time-cooperative guidance law converges in a finite time, and the upper bound of the uniform convergence time is obtained, as follows:

[0128] The second-order dynamic multi-agent system model is:

[0129] Construct a Lyapunov functional of the following form:

[0130]

[0131] Among them, V r represents the Lyapunov functional, n represents the number of missiles, and t represents time.

[0132] When the control gain satisfies the following formula:

[0133]

[0134]

[0135] Among them, k, x 1k (0), x 2k (0) represent the kth air-to-air missile, the missile-target distance of the kth air-to-air missile at the initial moment, and the missile-target relative speed of the kth air-to-air missile at the initial moment, respectively.

[0136] The derivative of the Lyapunov functional with respect to time satisfies:

[0137]

[0138] in, λ2 represents the second smallest eigenvalue, L n Represents an n-order polynomial, s0=exp(s2-1).

[0139] Let K2 = min{K1λ2(L n )}, the upper bound of the uniform convergence time can be obtained as:

[0140]

[0141] Among them, t *represents the upper bound of the uniform convergence time, V r (0) represents the value of the Lyapunov functional at time 0.

[0142] Steps for constructing the angle-coordinated optimal mid-range guidance law: In the normal direction of the line of sight, the angle-coordinated optimal mid-range guidance law is constructed using the Gaussian pseudo-spectral method. This step includes:

[0143]

[0144] Among them, u * (τ i ) represents τ i The optimal normal control command at time, represents the n×n dimensional positive definite symmetric time-varying control weight matrix, t goi represents the remaining flight time of the i-th missile, r i represents the relative distance between the i-th missile and the target, λ() represents the 2-dimensional Lagrange multiplier vector, and N represents the number of Gauss nodes.

[0145] Specifically, firstly, the bottom guidance law of each air-to-air missile adopts the optimal guidance law. Considering that the ultra-long-range air-to-air missile has the longest flight time in the mid-guidance phase, in order to ensure the range and meet the conditions of terminal line of sight angle constraint, position constraint and maximum usable overload constraint during flight, the following Bolza-type performance index function is established:

[0146]

[0147] Among them, t0 and t F They are the initial and terminal moments of mid-course guidance respectively.

[0148] At the same time, the system state equation in the normal direction of the line of sight is satisfied:

[0149] Where X(t) = [x 3i (t) x 4i (t)] T ; u(t)=[u Mir u Miq ] T ;

[0150] Then, to avoid solving partial differential equations, the Bolza-type performance index is simplified to a quadratic performance index:

[0151]

[0152] Where F is a terminal weighting matrix of n×n dimensional positive definite symmetric constants; is an n×n dimensional positive definite symmetric time-varying control weight matrix.

[0153] Next, construct the Hamiltonian function: Where: λ(t) is the 2-dimensional Lagrange multiplier vector.

[0154] Next, the Pontryagin minimum principle is used to transform the functional extreme value problem into the following two-point boundary value problem:

[0155]

[0156] Then, the Gauss pseudo-spectral method is used for discretization, and Lagrange orthogonal interpolation polynomials are constructed at discrete points to approximate the state variables and Lagrange multiplier vectors. The optimal control input under analytical conditions is obtained by solving the discretized algebraic equations.

[0157] The original optimal control problem is guided into the time interval t∈[t0,t F ] is converted to the normalized time interval τ∈[-1,1]: The calculation formula for the final value of the state quantity and the initial value of the co-state quantity is obtained through Gauss integration:

[0158]

[0159] in, is the Gauss quadrature coefficient; ρ(τ) is the weight function on the interval [-1,1].

[0160] Finally, solve the optimal control quantity on each Gauss node:

[0161]

[0162] Among them, the optimal normal control command obtained by the calculation formula is an open-loop solution. Considering that the coefficient matrix in the system state equation and the weight matrix in the performance index function are time-varying, the time-varying matrix is ​​regarded as a constant in each step of the trajectory solution, and the current control amount u is solved. * (τ0), and update the state quantity at the next moment, iterate continuously, and solve the optimal guidance command at the current moment in real time. Construct a Bolza-type performance index function that simultaneously satisfies the terminal line of sight angle constraint, position constraint, and the maximum usable overload constraint during flight, construct a Hamiltonian function to transform the functional extreme value problem into a two-point boundary value problem, and use the Gauss pseudo-spectral method for discretization to obtain the optimal control input under the analytical situation.

[0163] Furthermore, the effectiveness of the method proposed in the present invention is verified by a specific example of multi-air-to-air missile coordinated guidance, which is specifically described as follows:

[0164] The simulation conditions are set as follows: the number of air-to-air missiles is 4, the radar seeker search distance is 80km, the initial distance of the target relative to the origin of the missile launch system is 200km, the flight altitude is 15km, the flight speed is 1.5Ma, and the target entry angle and off-axis angle are both 180°. The initial launch conditions of each air-to-air missile are shown in the following table:

[0165] Table 1

[0166]

[0167]

[0168] The line-of-sight guidance parameters are set as follows: k1 = -5, k2 = 4, s1 = 0.5, s2 = 0.8; the line-of-sight normal guidance parameters are set as follows: Gauss node number N = 3, performance index weighting matrix F = diag (500, 500, 500). The communication topology structure between air-to-air missiles is as follows: Figure 3 shown.

[0169] The simulation results are analyzed as follows: Figure 4 and Figure 5 It can be seen that the speed and relative distance of the four air-to-air missiles are quite inconsistent at the initial moment, but under the effect of the line of sight overload command, the two reach consistency before 45 seconds, and the convergence time is less than the upper bound of the calculated consistency time, which is reflected in Fig.12 In the initial moment, the overload instruction of the sight direction is large so that the speed and relative distance can converge quickly to the same. Figure 6 and Figure 7 It can be seen that the sight angles of the four air-to-air missiles gradually decrease and converge to 0 before the mid-term handover, meeting the design goal of the underlying optimal guidance law. Figure 8 As shown in Figure 9, the line-of-sight angular rate of the four air-to-air missiles reaches a peak value at 10s and then converges to 0 rapidly. Finally, the line-of-sight angular rate error stabilizes at 10 -6 Within 10 seconds, the smooth transition of the mid-terminal guidance handover is guaranteed. Fig.10 and Fig.11 It can be seen that the normal overload of the four air-to-air missiles is always within 2g, and the peak value is caused by the end of the pulse ignition of the missile engine at the 4th second, resulting in a large jump in thrust. Finally, under the action of the time-varying weight function in the optimal performance index, the overload tends to 0, and the full-range overload is less than the actual available overload of the air-to-air missile, so it is feasible in engineering. As shown in Figure 13, the four air-to-air missiles arrived at the designated virtual guidance point at the same time under different initial launch conditions, and the arrival times were: t1 = 67.11s, t2 = 67.10s, t3 = 67.11s, t4 = 67.11s, and the arrival time difference did not exceed 0.01s, meeting the design requirements of simultaneous arrival and coordinated detection.

[0170] In order to further verify the advantages of the method proposed in the present invention, the mid-term and end-term handover probabilities are calculated and compared for different working conditions and different seeker search ranges. The calculation results are shown in Table 2. It can be seen that the use of the distributed collaborative mid-term guidance law of the present invention increases the mid-term and end-term handover probability to more than 97%, expands the search range by more than 3.6 times, and reduces the search time by about 80%, achieving the design goals of collaborative detection and wide-area target rapid search, and has certain technical advantages and engineering application value. The mid-term and end-term handover probabilities under different working conditions are as follows:

[0171] Table 2

[0172]

[0173]

[0174] A coordinated guidance system for multiple air-to-air missiles provided by the present invention adopts the distributed coordinated guidance law analysis method for air-to-air missiles.

[0175] Those skilled in the art know that, in addition to implementing the system, device and its various modules provided by the present invention in a purely computer-readable program code, it is entirely possible to implement the same program in the form of logic gates, switches, application-specific integrated circuits, programmable logic controllers and embedded microcontrollers by logically programming the method steps. Therefore, the system, device and its various modules provided by the present invention can be considered as a hardware component, and the modules included therein for implementing various programs can also be considered as structures within the hardware component; the modules for implementing various functions can also be considered as both software programs for implementing the method and structures within the hardware component.

[0176] The above describes the specific embodiments of the present invention. It should be understood that the present invention is not limited to the above specific embodiments, and those skilled in the art can make various changes or modifications within the scope of the claims, which does not affect the essence of the present invention. In the absence of conflict, the embodiments of the present application and the features in the embodiments can be combined with each other arbitrarily.

Claims

1. A method for analyzing guidance law in distributed collaborative air-to-air missiles, characterized in that: include: Steps for constructing a mid-terminal guidance handover probability calculation model: construct a mid-terminal guidance handover probability calculation model by analyzing the error source and error transmission link; The step of establishing a multi-missile field of view stitching method is to establish a multi-missile field of view stitching method for a virtual guidance point; the step of establishing a multi-missile field of view stitching method includes the step of establishing a virtual guidance point: guiding each air-to-air missile to its own virtual guidance point in the mid-guidance stage, the virtual guidance points are distributed on a spherical surface with the target detection position as the center and the maximum detection distance of the radar seeker as the radius, and the calculation formula is as follows: Where R represents the maximum detection range of the active radar seeker, R 误差 represents the radius of the 3σ synthetic error sphere, (x * (t),y * (t),z * (t)) represents the real-time position of the target detected by the space-based radar, (x i (t),y i (t),z i (t)) represents the dynamic virtual guidance point corresponding to the i-th air-to-air missile, and x, y, and z represent the horizontal axis, vertical axis, and vertical axis in the spatial coordinate system, respectively; Steps for establishing a multi-missile coordinated mid-course guidance model: Establish a multi-missile coordinated mid-course guidance model through the relative motion relationship between multiple air-to-air missiles and targets; Steps for constructing the guidance law in time coordination: decoupling the coordination guidance rate to the line of sight direction through the multi-missile coordination guidance model, and constructing the guidance law in time coordination using the second-order multi-agent consistency theory; the steps for constructing the guidance law in time coordination include: The guidance law has a two-layer coordination framework, including the top-level guidance law and the bottom-level guidance law. The top-level guidance law uses the remaining flight time of the missile as the coordination variable. According to the second-order multi-agent consistency convergence theory in finite time, the calculation formula of the top-level time coordination guidance law is established as follows: Among them, a ij represents the proportionality coefficient, x 1j =r j represents the relative distance between the jth missile and the target, represents the relative speed between the jth missile and the target, n represents the number of missiles, k1 and k2 represent gain coefficients, s1,s2∈(-∞,1) represent adjustment factors, g(x,s)=x·x exp(s-1)-1 ; The angle coordinated optimal mid-range guidance law construction step is as follows: the coordinated mid-range guidance rate is decoupled to the line of sight normal direction through the multi-missile coordinated mid-range guidance model, and the angle coordinated optimal mid-range guidance law is constructed by using the Gaussian pseudo-spectral method; the angle coordinated optimal mid-range guidance law construction step includes: Among them, u * (τ i ) represents τ i The optimal normal control command at time, represents the n×n dimensional positive definite symmetric time-varying control weight matrix, t goi represents the remaining flight time of the i-th missile, r i represents the relative distance between the ith missile and the target, λ() represents the 2-dimensional Lagrange multiplier vector, and N represents the number of Gauss nodes; Simulation verification step: verify the effectiveness of the mid-terminal guidance handover probability calculation model, the time coordinated mid-term guidance law and the angle coordinated optimal mid-term guidance law through simulation.

2. The method for analyzing guidance law in distributed collaborative air-to-air missile according to claim 1 is characterized in that: The mid-terminal guidance handover probability calculation model construction step includes: The synthetic error calculation steps of the target indication position are as follows: the synthetic error of the target indication position is obtained according to the linear superposition method, wherein the error of the target position includes: the detection error of the high-orbit satellite, the combined navigation orientation error of the missile itself, the inertial navigation attitude error, the seeker angle measurement error and the position error caused by the delay time. Since the error variables are independent of each other, the corresponding synthetic error is obtained by the following calculation formula: Among them, μ() represents the mean of the synthetic error, σ 2 () represents the variance of the synthetic error, ΔR2, ΔA2, and ΔE2 represent the synthetic errors of the target indication position in terms of distance, elevation angle, and azimuth, respectively. The subscript position represents the target position detected by the high-orbit satellite, the subscript orientation represents the target direction determined by the integrated navigation, and the subscript delay represents the delay time of the signal transmission; Calculation steps for the probability of mid-term and end-term handover: Since the target indication distance error is small compared to the target indication distance, the distance factor is ignored and only the angle interception condition is considered. The calculation formula for the probability of mid-term and end-term handover is as follows: Among them, A2 represents the target elevation angle, E2 represents the target azimuth angle, It represents the variance of the target indication position error in the line of sight height direction, represents the variance of the target indication position error in the line of sight azimuth direction, Δθ represents the angle measurement error of the seeker, P is the probability of mid-term handover, and θ0 is the available search range of the seeker.

3. The method for analyzing guidance law in distributed collaborative air-to-air missile according to claim 1, characterized in that: The steps of building the relative motion model include: Step 1: Establish the geometric relationship equation of the relative motion between the missile and the target in the longitudinal plane as follows: Among them, represents the derivative of the variable with respect to time, M i and T represent the mass center positions of the ith missile and target respectively; V Mi and V T Represent the speed of the i-th missile and the target respectively; r i represents the relative distance between the i-th missile and the target; q i represents the sight angle of the i-th missile; θ i and θ T Respectively represent the ballistic inclination angles of the i-th missile and the target; η i and η T denote the velocity lead angles of the ith missile and the target respectively; a Mi and a T denote the normal acceleration of the i-th missile and the target respectively; Step 2: According to the relative motion geometric relationship, a multi-missile coordinated mid-course guidance model is established. The calculation formula is as follows: Among them, u Mir =a Mi sinη i represents the component of the tangential acceleration of the i-th missile in the line of sight direction, u Miq =a Mi cosη i represents the component of the normal acceleration of the ith missile in the normal direction of the line of sight, x 1i =r i , x 3i =q i , They all represent system state variables.

4. The method for analyzing guidance law in distributed collaborative air-to-air missile according to claim 1 is characterized in that: By constructing the Lyapunov equation, it is proved that the time-cooperative guidance law converges in a finite time, and the upper bound of the uniform convergence time is obtained, as follows: Construct a Lyapunov functional of the following form: Among them, V r represents the Lyapunov functional, n represents the number of missiles, and t represents time; When the control gain satisfies: Among them, k, x 1k (0), x 2k (0) respectively represent the kth air-to-air missile, the missile-target distance of the kth air-to-air missile at the initial moment, and the missile-target relative speed of the kth air-to-air missile at the initial moment; The derivative of the Lyapunov functional with respect to time satisfies: in, λ2 represents the second smallest eigenvalue, L n represents an n-order polynomial, s0=exp(s2-1); Let K2 = min{K1λ2(L n )}, the upper bound of the uniform convergence time can be obtained as: Among them, t* represents the upper bound of the uniform convergence time, V r (0) represents the value of the Lyapunov functional at time 0.

5. The method for analyzing guidance law in distributed collaborative air-to-air missile according to claim 1 is characterized in that: A Bolza-type performance index function is constructed to simultaneously satisfy the terminal line of sight angle constraint, position constraint and the maximum usable overload constraint during flight. A Hamiltonian function is constructed to transform the functional extreme value problem into a two-point boundary value problem. The Gauss pseudo-spectral method is used for discretization to obtain the optimal control input under the analytical situation.

6. The method for analyzing guidance law in distributed collaborative air-to-air missile according to claim 1 is characterized in that: The optimal normal control command obtained by the calculation formula is an open-loop solution. Considering that the coefficient matrix in the system state equation and the weight matrix in the performance index function are time-varying, the time-varying matrix is ​​regarded as a constant in each step of the trajectory solution, and the current control amount u is solved. * (τ0), and update the state quantity at the next moment, iterate continuously, and solve the optimal guidance instruction at the current moment in real time.

7. A coordinated guidance system for multiple air-to-air missiles, characterized in that: The method for analyzing the guidance law of air-to-air missile distributed coordination according to any one of claims 1 to 6 is adopted.

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