CN113625744A - Anti-saturation fixed time cooperative guidance law for attacking highly mobile target - Google Patents

Anti-saturation fixed time cooperative guidance law for attacking highly mobile target Download PDF

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CN113625744A
CN113625744A CN202110728719.7A CN202110728719A CN113625744A CN 113625744 A CN113625744 A CN 113625744A CN 202110728719 A CN202110728719 A CN 202110728719A CN 113625744 A CN113625744 A CN 113625744A
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CN113625744B (en
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戴可人
马翔
商玮宸
于航
张合
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Nanjing University of Science and Technology
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Abstract

The invention discloses an anti-saturation fixed time cooperative guidance law for attacking a high maneuvering target, which is designed by applying a distributed fixed time expansion state observer and an anti-saturation function. The present invention is set forth in the following background: with the rapid development of aerospace technology and complex network technology, the concept of multi-missile cooperative combat is more and more emphasized. Compared with the traditional single missile combat mode, the multi-missile cooperative combat mode has greater advantages, the aim of saturated attack can be achieved by adopting the explosion aggregation effect generated by the multi-point detonation mode, and the multi-missile cooperative combat mode has higher damage efficiency. The cooperative guidance law disclosed by the invention effectively solves the characteristic that multiple missiles cooperatively strike a high maneuvering target, and designs the anti-saturation function so that the striking efficiency of the whole system is improved and good guidance performance is obtained.

Description

Anti-saturation fixed time cooperative guidance law for attacking highly mobile target
Technical Field
The invention relates to the field of cooperative guidance law design, in particular to an anti-saturation fixed time cooperative guidance law for attacking a high maneuvering target.
Background
With the rapid development of aerospace technology and complex network technology, the challenge of adopting a single missile to strike a target is increased day by day, so that the concept of multi-missile cooperative combat is more and more emphasized. Compared with the traditional single missile combat mode, the multi-missile cooperative combat mode has greater advantages, the aim of saturated attack can be achieved by adopting the explosion aggregation effect generated by the multi-point detonation mode, and the multi-missile cooperative combat mode has higher damage efficiency. The design and research of the cooperative guidance law is one of the key technologies for realizing cooperative combat, and the research also becomes a hotspot problem of missile combat in recent years.
The simultaneous fire attack in the cooperative operation of multiple missiles is a typical operation form, and the requirement is that missiles from different positions need to hit a target simultaneously. The problem of consistency of guidance time mainly researches an information interaction mode between every two missiles in a missile cooperative combat system, so that the guidance time of all the missiles tends to be a uniform value. Many students have studied the cooperative guidance method of "leader-slave projectile", but the system of "leader-slave projectile" also has a great influence when the problem occurs in the "leader projectile". Therefore, when high-precision striking is carried out, a leaderless distributed cooperative guidance method can be adopted, so that more accurate striking efficiency and damage efficiency can be achieved for a target, and the method has more important significance for practical research.
In the following studies, the Time-controlled Guidance law (ITCG, Impact-Time Control Guidance) is introduced into multi-missile cooperative Guidance. On the basis, some scholars introduce the Finite-Time Cooperative Guidance law (FTCG) into multi-missile Cooperative Guidance, so that the Time for state consistency of multiple missiles is shortened to a great extent. Due to the short guidance time, the final guidance performance is greatly affected if the convergence time during the striking process is uncertain. Therefore, under the background, a Fixed-Time Cooperative Guidance law (FxTCG) is applied to the research of multi-missile Cooperative Guidance, and the Fixed-Time Cooperative Guidance law has the advantages that the consistency of each missile can be ensured in a short Time without depending on the initial state of each missile.
However, most of the above researches are directed to hitting static targets by multi-missile cooperative attack, and due to the diversity of battlefield conditions, multi-missile cooperative combat is often applied to interception or hitting maneuvering targets, and obviously, a cooperative guidance algorithm for static targets is far from meeting the requirements of current combat. The important sign of the maneuvering target is that the accurate value of the acceleration of the maneuvering target cannot be directly obtained, so that how to realize accurate striking of the target while lacking target acceleration information is concerned by wide scholars. Many studies have adopted methods such as Disturbance Observer (DOB), Extended State Observer (ESO), etc. to complement the error term of the cooperative guidance model. Because the consistency and the accuracy in the cooperative attack process of the moving target are high, the precision is often insufficient by only adopting a method of expanding a state observer, and a finite Time Disturbance Observer (TDO) is also provided for observing the disturbance in a control system. Then, another scholars have proposed a fixed time disturbance observer (FxDO) which is not dependent on the initial state of observation and is therefore suitable for use in the case of multi-missile attack maneuvering targets, but the above-mentioned research on multi-missile attack maneuvering targets is not sufficient for research on the limitation of driving force.
In practical situations, input saturation constraints also have a significant influence on the design of the guidance law due to the limitation of driving force during the hitting process of multiple missiles on the target. In some researches, a method for constructing an auxiliary function is adopted to solve the problem of input saturation in the multi-missile cooperative guidance problem, but the guidance time can only be ensured to reach gradual consistency. The design of the cooperative guidance law is essentially the consistency problem of a control system, so that an expansion limit theory is introduced into the research of some cooperative theories to research the consistency problem of the multi-agent under the input saturation constraint. Subsequently, a bipolar limit homogeneous theory is applied in the cooperative guidance system, so that the fixed time consistency is obtained, but the influence of the quick movement of the target on the overall guidance law is not considered, so that the miss ratio is increased.
Disclosure of Invention
The invention aims to provide an anti-saturation fixed time cooperative guidance law for attacking a high maneuvering target. Firstly, a fixed time observer can be used to realize accurate recognition of the acceleration of the target within the limit of a fixed time, and then the locking of the high-mobility target is achieved. And secondly, by means of a fixed time theory and a saturation function design, the missile is cooperatively attacked under the condition that the driving force is limited, and the purpose of cooperative guidance is achieved. Finally, the fixed time convergence of the proposed guidance law is demonstrated through a Lyapunov function and a bipolar limit homogeneous theory, and the guidance law is verified through numerical simulation.
The technical solution for realizing the purpose of the invention is as follows: an anti-saturation fixed time cooperative guidance law for attacking a high maneuvering target comprises the following steps:
step 1, establishing a geometric mathematical model of the missile and the target, and turning to step 2.
And 2, designing a distributed extended state fixed time observer according to a motion geometric mathematical model of the missile and the target, and turning to the step 3.
And 3, designing a distributed anti-saturation cooperative guidance law according to the acceleration value observed by the distributed extended state observer and a geometric mathematical model of the motion of the missile and the target, and turning to the step 4.
And 4, performing stability verification by using a bipolar limit homogeneous theorem and a Lyapunov equation, and introducing a fixed time theory to obtain a fixed time stable result of the cooperative guidance system.
Compared with the prior art, the invention has the advantages that:
(1) compared with the traditional observer, the method can expand the multi-missile cooperative attack problem to the high maneuvering target, and can realize accurate identification on the acceleration of the target within a fixed time limit so as to achieve locking of the high maneuvering target.
(2) Compared with the traditional cooperative guidance law, the designed cooperative guidance law can realize the cooperative guidance under the limiting condition of input saturation, the input of each missile is within the limit of the maximum driving force through the design of the anti-saturation function, the consistency can be realized within a fixed time boundary, and the cooperative attack is realized.
(3) The adopted fixed time theory can converge the convergence time of the observed value of the observer and the consistent time of the cooperative guidance law within a fixed time limit, and does not depend on the initial state of each missile, so that the multiple missiles can accurately strike the target.
Drawings
FIG. 1 is a diagram of the consistent guidance method of the present invention.
FIG. 2 is a schematic diagram of a multi-missile cooperative attack target scenario.
FIG. 3 is a plan view of missile i in relation to the target.
Fig. 4 is a diagram of relative distances of eyes.
Fig. 5 is a diagram of relative projectile distance error.
Fig. 6 is a diagram of relative speed error of the eyes.
FIG. 7 is a graph of the change in the angle of view of the missile.
Fig. 8 is a view of the line-of-sight direction input diagram.
FIG. 9 is a line-of-sight normal direction input diagram.
Detailed Description
The invention is described in further detail below with reference to the figures and the embodiments.
With reference to fig. 1 to 3, the anti-saturation fixed-time cooperative guidance law for attacking a high maneuvering target according to the present invention includes the following steps:
step 1, establishing a geometric mathematical model of the movement of the missile and the target, specifically as follows:
step 1.1, considering a scene that a plurality of missiles cooperatively attack a high maneuvering target, and deducing a relative motion equation of the missiles and the target according to a kinetic model of the missiles and the target, wherein the relative motion equation is as follows:
Figure BDA0003138511690000041
Figure BDA0003138511690000042
ηT＝qi-θT (3)
ηi＝qi-θi (4)
Figure BDA0003138511690000043
Figure BDA0003138511690000044
in the formula: r isiIs the relative distance between the ith missile and the target,
Figure BDA0003138511690000045
the relative speed of the ith missile and the target; q. q.siIs the sight line direction angle of the ith missile,
Figure BDA0003138511690000046
the angular velocity of the sight line direction angle of the ith missile; viThe velocity, V, of the ith missile in the direction of the line of sightTThe speed of the target in the sight line direction; thetaiIs the trajectory inclination angle of the ith missile,
Figure BDA0003138511690000047
angular velocity, θ, of trajectory inclination of ith missileTThe target trajectory inclination angle is set as the target trajectory inclination angle,
Figure BDA0003138511690000048
angular velocity, which is the target ballistic dip; a isiNormal acceleration of the ith missile, aTA target normal acceleration; etaiIs the lead angle, eta, of the ith missileTIs the lead angle of the target.
And (3) obtaining the derivative of the formula (1) and the formula (2) after finishing:
Figure BDA0003138511690000049
in the formula: component u of acceleration of ith missile in sight line directionriComponent d of acceleration of the object in the direction of the line of sightr(ii) a Component u of normal angular velocity of missile in normal direction of sight lineqiComponent d of the normal angular velocity of the object in the normal direction of the line of sightq。
Step 1.2, setting the intermediate parameters as x respectively1i、x2i、x3i、x4iLet x1i＝ri，
Figure BDA00031385116900000410
x3i＝qi-q0，
Figure BDA0003138511690000051
q0Representing an ideal sight direction angle to obtain a relative movement model of the missile and the attack target, namely a movement geometric mathematical model of the missile and the target:
Figure BDA0003138511690000052
assume that 1: the moving acceleration of the task target is within a reasonable interval due to the limitation of the target driving force, so drAnd dqShould be bounded and meet the Lipschitz condition.
Step 2, designing a distributed extended state fixed time observer according to a motion geometric mathematical model of the missile and the target, wherein the observer is as follows:
step 2.1, because the acceleration of the moving target of the high maneuvering target is usually not measurable, designing a distributed fixed time observer in the direction of a line-of-sight angle according to a moving geometric mathematical model of the missile and the target as follows:
Figure BDA0003138511690000053
in the formula
Figure BDA0003138511690000054
Is x1iIs determined by the estimated value of (c),
Figure BDA0003138511690000055
is that
Figure BDA0003138511690000056
The derivative of (a) of (b),
Figure BDA0003138511690000057
is x2iIs determined by the estimated value of (c),
Figure BDA0003138511690000058
is that
Figure BDA0003138511690000059
The derivative of (a) of (b),
Figure BDA00031385116900000510
is drIs determined by the estimated value of (c),
Figure BDA00031385116900000511
is that
Figure BDA00031385116900000512
A derivative of (a); defining observer error
Figure BDA00031385116900000513
Respectively select kappa1，κ2And kappa3As a coefficient of observer, κ1＝3w0，
Figure BDA00031385116900000514
w0Is the gain bandwidth; with intermediate function sig (.)α＝sign(·)|·|αAnd alpha is an exponential coefficient; the observer index is chosen as follows1∈(2/3,1)，β1E (1,4/3) and let alpha2∈2α1-1，α3∈3α1-2；β2∈2β1-1，β3∈3β2-2。
Combining the equation (8) and the equation (9), the observer error state space equation is recorded as:
Figure BDA00031385116900000515
wherein the intermediate variable
Figure BDA00031385116900000516
Intermediate variables
Figure BDA00031385116900000517
According to the hypothesis 1, there must be a normal number ΔmaxSo that | dr|≤Δmax(ii) a Under the action of the above-mentioned observer, e1i，e2iAnd eriAt a fixed time TiI 1.. n converges to zero, and e is derived1,i＝e2,i＝e r,i0, i 1, n; and n is the number of missiles.
Step 2.2, adopting the same design principle as the direction of the sight angle for the design of the distributed fixed time observer in the direction of the sight normal, wherein the distributed fixed time observer in the direction of the sight normal is as follows:
Figure BDA0003138511690000061
in the formula
Figure BDA0003138511690000062
Is x3iIs determined by the estimated value of (c),
Figure BDA0003138511690000063
is that
Figure BDA0003138511690000064
The derivative of (a) of (b),
Figure BDA0003138511690000065
is x4iIs determined by the estimated value of (c),
Figure BDA0003138511690000066
is that
Figure BDA0003138511690000067
The derivative of (a) of (b),
Figure BDA0003138511690000068
is dqIs determined by the estimated value of (c),
Figure BDA0003138511690000069
is that
Figure BDA00031385116900000610
A derivative of (a); defining observer error
Figure BDA00031385116900000611
Step 3, designing a distributed anti-saturation cooperative guidance law according to the acceleration value observed by the distributed extended state observer and a geometric mathematical model of the motion of the missile and the target, wherein the method specifically comprises the following steps:
3.1, designing a distributed anti-saturation cooperative guidance law in the line-of-sight angular direction, wherein the guidance model in the line-of-sight angular direction is as follows:
Figure BDA00031385116900000612
in order to make the relative distance and speed from the target consistent for each missile, the following variables are introduced to characterize the cooperativity error:
Figure BDA00031385116900000613
in the formula, aijRepresenting communication parameters, a if there is information flowing from the ith missile to the jth missileijIs 1, otherwise is 0; x is the number of1jRepresents the relative distance, x, between the jth missile and the target2jRepresents the relative speed, v, of the jth missile and the target1iRepresents the distance difference between the ith missile and the jth missile from the target, v2iThe angular velocity difference of the current sight line between the ith missile and the jth missile from the target is shown, namely when v1iAnd v2iWhen the number of the missile targets approaches 0, the flight states of the ith missile and the jth missile are considered to be consistent, and the common attack can be realized when the target is finally attacked.
In this section, a consistent guidance law based on the fixed-time theory and the dual-pole homogeneous theorem is proposed: considering a guidance model (12) with input saturation constraints, the consistency guidance law can be designed as:
Figure BDA0003138511690000071
wherein σ1、σ’1、σ2、σ’2For consistency guidance law index, k1、k2、k’1、k’2For consistent guidance law gain and to satisfy:
Figure BDA0003138511690000072
wherein x is1maxThe maximum value of the relative distance between the ith missile and the target is represented; x is the number of4maxAngular velocity u representing the line-of-sight direction angle of the ith missilermaxA maximum value representing a component of the acceleration of the missile in the direction of the line of sight; wherein for any u rmax0 and satisfies:
(1) for all i ═ 1,2, …, n has | uri|≤urmax。
(2) The cooperative guidance law consistency for the system (11) can be completed in a fixed time, and the time upper limit is as follows:
Figure BDA0003138511690000073
in the formula TnIs composed of
Figure BDA0003138511690000074
And convergence parameter kv、k0、k∞、γ1、γ2、
Figure BDA0003138511690000075
And σ is designed as follows:
Figure BDA0003138511690000076
remarks 2: since the system can converge at a fixed time, i.e. within the convergence limit
Figure BDA0003138511690000077
Maximum value of (a), and x4iThe value of (a) is small and has a weak influence on the saturation characteristics of the whole.
And 3.2, adopting a design principle the same as the direction of the line-of-sight angle for the cooperative guidance law in the normal direction of the line-of-sight angle, wherein a guidance model in the normal direction of the line-of-sight angle is as follows:
Figure BDA0003138511690000078
in order to ensure that the included angle between the line-of-sight angle and the ideal line-of-sight angle approaches to 0, the normal design acceleration of the line-of-sight angle of each missile is designed, so that each missile is attacked according to a preset attack angle, and on the basis, the cooperative guidance law of the line-of-sight angle direction can be designed as follows:
Figure BDA0003138511690000081
wherein sigma3、σ’3、σ4、σ’4For consistency guidance law index, k3、k4、k3'、k'4For consistent guidance law gain and to satisfy:
(k3+k’3+k4+k’4)+Δmax+2x2maxx4max≤uqmax (20)
wherein x is2maxRepresenting the maximum value of the relative speed of the ith missile and the target; u. ofqmaxThe maximum value of the component of the acceleration of the missile in the normal line of the sight line direction is represented, and the proving process is also carried out in the manner of proving the angular direction of the sight line.
And 4, performing stability verification by using a bipolar limit homogeneous theorem and a Lyapunov equation, and introducing a fixed time theory to obtain a result of stable fixed time of the system.
The cooperative guidance system composed of the formula (11) and the formula (13) is as follows:
Figure BDA0003138511690000082
wherein i is 1. Then can order the intermediate variable
Figure BDA0003138511690000083
Intermediate variables
Figure BDA0003138511690000084
And when the extended state is fixed for the time observer
Figure BDA0003138511690000085
Converge to drThe above formula can be written as:
Figure BDA0003138511690000086
wherein L is a Laplacian matrix; and let an intermediate variable
Figure BDA0003138511690000087
Intermediate variables
Figure BDA0003138511690000088
Intermediate vector INIs a feature vector, an intermediate variable, corresponding to a single feature value of 0
Figure BDA0003138511690000089
And (3) pushing out:
Figure BDA0003138511690000091
according to the definition of M, there is a if and only if x11＝x12＝…x1nAnd x21＝x22＝…x2nWhen p is 0 and q is 0; equation (21) can achieve fixed time agreement if and only if p and q converge to zero at fixed time.
From the mathematical formula tanh (sig (x)α)＝sig(x)α+o(sig(x)α) The system (12) may be changed to be near the origin
Figure BDA0003138511690000092
In the formula
Figure BDA0003138511690000093
If equation (12) satisfies the following condition, then a global fixed time stabilization can be deduced:
(1) equation (21) is globally asymptotically stable at the origin.
(2) Equation (21) is also substantially locally asymptotically stable, and the process function fl(q, p) is its degree klHomogeneous System < 0, Process function fg(q, p) is its degree kgHomogeneous system > 0:
Figure BDA0003138511690000094
(3) in formula (21)
Figure BDA0003138511690000095
Satisfy the requirement of
Figure BDA0003138511690000096
Wherein the convergence parameter k0Less than 0, epsilon is an intermediate variable, convergence parameter
Figure BDA0003138511690000097
And convergence parameters
Figure BDA0003138511690000098
Based on the above analysis, the fixed time stabilization of equation (22) can be achieved using the consistency guidance law designed by equation (19), and the controller parameters satisfy:
Figure BDA0003138511690000101
the following was demonstrated:
it is first demonstrated that the equilibrium point of equation (12) is globally asymptotically stable, considering the design of Lyapunov function V as follows1：
Figure BDA0003138511690000102
For formula (28), and only if
Figure BDA0003138511690000103
And
Figure BDA0003138511690000104
when there is V1> 0, when the following equation holds
Figure BDA0003138511690000105
Has a if and only if
Figure BDA0003138511690000106
And, likewise, if and only if p is 0
Figure BDA0003138511690000107
Thus, V1Is positive and is positive for V1And (5) obtaining time derivation:
Figure BDA0003138511690000108
furthermore, if and only if Lp is 0, then p is 0, then q is obtained, and the balance point of the formula (22) at the origin can be obtained according to the lasale invariant set principle, so that global gradual stabilization can be realized.
We will then demonstrate that equation (24) is locally fixed-time stable, and we design the nominal system for (24) as follows:
Figure BDA0003138511690000109
stable in fixed time and homogeneous in nature.
Therefore, the following Lyapunov equation V is selected for equation (30)l：
Figure BDA00031385116900001010
Then for VlAnd (5) obtaining time derivation:
Figure BDA0003138511690000111
we can thus demonstrate that equation (30) is globally asymptotically stable; in addition, the system (26) and k can be verified0-1 is homogeneous;
likewise, we can design a nominal system for equation (26), as follows:
Figure BDA0003138511690000112
therefore, the following Lyapunov equation V is selected for the system (30)g：
Figure BDA0003138511690000113
Similar to (27), the above formula is also asymptotically stable. And can verify the system (33) with k0-1 is homogeneous; the original formula (26) is locally fixed time stable; so far, it can be deduced that under the design of equation (14), equation (12) can achieve fixed time consistency; the guidance law in the normal direction of the line-of-sight angle is similar to the design result of the guidance law in the angle-of-sight direction, so that the process is proved to be in the formulas (21) - (34), and the system can be stabilized under the condition of input saturation.
Examples
In order to assess the performance of the designed guidance law, the effectiveness of the algorithm is verified by using an example of a maneuvering target attacked by multiple missiles in a coordinated mode. The number of the bullets is five, and the maneuvering target is one. The initial parameters of its maneuvering target were set to 15000m in the x-direction, 15000m in the y-direction, and 100m/s in speed.
TABLE 1 initial parameters for each missile
Figure BDA0003138511690000114
The parameter design of the observer for the fixed time of the distributed extended state is as follows: w is a0＝5，α1＝0.8，α2＝0.6，α3＝0.4；β1＝1.2，β2＝1.4，β31.6. The parameters of the distributed fixed time guidance law are designed as follows: k is a radical of1＝k’1＝k2＝k’2＝50，k3＝k’3＝k4＝k’4＝50；α1＝α3＝0.67，α2＝α4＝0.8，α’1＝α’3＝1.33，α’2＝α’4＝1.14。
FIG. 4 is a diagram of relative distances of the eyes; FIGS. 5 and 6 are error plots of projectile relative distance and approach velocity; it can be seen from the figure that the starting distance of each missile is different in the process of flying each missile, but after flying for 20s, the missile target distance and speed are consistent, and then the purpose of cooperative guidance is completed, and the missiles collectively concentrate on the target at 43 s. FIG. 7 is a graph of line of sight angle and line of sight angular velocity of the missile; according to the figure, each missile completes a preset attack angle within 15s, the maximum efficiency of missile attack is ensured, and the aim of attacking targets from different angles is fulfilled.
Fig. 8-9 are input diagrams of the line-of-sight angle direction and the line-of-sight angle normal direction, respectively, and it can be known from the diagrams that the anti-saturation strategy proposed herein has good anti-saturation characteristics, so that the control input of the anti-saturation strategy can be stabilized within a certain limit, and the overload requirement of the missile can be met in the actual engineering.

Claims (6)

1. An anti-saturation fixed time cooperative guidance law for attacking a high maneuvering target is characterized by comprising the following steps:
step 1, establishing a geometric mathematical model of the movement of the missile and the target, and turning to step 2;
step 2, designing a distributed extended state fixed time observer according to a motion geometric mathematical model of the missile and the target, and turning to step 3;
step 3, designing a distributed anti-saturation cooperative guidance law according to an acceleration value observed by the distributed extended state observer and a geometric mathematical model of the motion of the missile and the target, and turning to step 4;
and 4, performing stability verification by using a bipolar limit homogeneous theorem and a Lyapunov equation, and introducing a fixed time theory to obtain a fixed time stable result of the cooperative guidance system.
2. The anti-saturation fixed-time cooperative guidance law for attacking high maneuvering targets according to claim 1, characterized in that a geometrical model of the missile and the target motion is established in step 1, specifically as follows:
step 1.1, considering a scene that a plurality of missiles cooperatively attack a high maneuvering target, and deducing a relative motion equation of the missiles and the target according to a kinetic model of the missiles and the target, wherein the relative motion equation is as follows:
Figure FDA0003138511680000011
Figure FDA0003138511680000012
ηT＝qi-θT (3)
ηi＝qi-θi (4)
Figure FDA0003138511680000013
Figure FDA0003138511680000014
in the formula: r isiIs the relative distance between the ith missile and the target,
Figure FDA0003138511680000015
the relative speed of the ith missile and the target; q. q.siIs the sight line direction angle of the ith missile,
Figure FDA0003138511680000016
the angular velocity of the sight line direction angle of the ith missile; viThe velocity, V, of the ith missile in the direction of the line of sightTThe speed of the target in the sight line direction; thetaiIs the trajectory inclination angle of the ith missile,
Figure FDA0003138511680000017
angular velocity, θ, of trajectory inclination of ith missileTThe target trajectory inclination angle is set as the target trajectory inclination angle,
Figure FDA0003138511680000018
angular velocity, which is the target ballistic dip; a isiNormal acceleration of the ith missile, aTA target normal acceleration; etaiIs the lead angle, eta, of the ith missileTIs the lead angle of the target;
and (3) obtaining the derivative of the formula (1) and the formula (2) after finishing:
Figure FDA0003138511680000021
in the formula: input u of acceleration of ith missile in sight directionriComponent d of acceleration of the object in the direction of the line of sightr(ii) a Input u of normal angular velocity of ith missile in normal direction of sight lineqiComponent d of the normal angular velocity of the object in the normal direction of the line of sightq；
Step 1.2, setting the intermediate parameters as x respectively1i、x2i、x3i、x4iLet x1i＝ri，
Figure FDA0003138511680000022
x3i＝qi-q0，
Figure FDA0003138511680000023
q0Representing an ideal sight direction angle to obtain a relative movement model of the missile and the attack target, namely a movement geometric mathematical model of the missile and the target:
Figure FDA0003138511680000024
assume that 1: the moving acceleration of the task target is within a reasonable interval due to the limitation of the target driving force, so drAnd dqShould be bounded and meet the Lipschitz condition.
3. The anti-saturation fixed-time cooperative guidance law for attacking high maneuvering targets according to claim 1 or 2, characterized in that in step 2, a distributed extended-state fixed-time observer is designed according to a geometrical model of the missile and the target, specifically as follows:
since the acceleration of a moving target of a high maneuvering target is usually not measurable, a distributed fixed time observer of the visual angle direction is designed according to a geometrical model of the movement of the missile and the target as follows:
Figure FDA0003138511680000025
in the formula
Figure FDA0003138511680000026
Is x1iIs determined by the estimated value of (c),
Figure FDA0003138511680000027
is that
Figure FDA0003138511680000028
The derivative of (a) of (b),
Figure FDA0003138511680000029
is x2iIs determined by the estimated value of (c),
Figure FDA00031385116800000210
is that
Figure FDA00031385116800000211
The derivative of (a) of (b),
Figure FDA00031385116800000212
is drIs determined by the estimated value of (c),
Figure FDA00031385116800000213
is that
Figure FDA00031385116800000214
A derivative of (a); defining observer error
Figure FDA00031385116800000215
Respectively select kappa1，κ2And kappa3As a coefficient of observer, κ1＝3w0，
Figure FDA0003138511680000031
w0Is the gain bandwidth; with intermediate function sig (.)α＝sign(·)|·|αAnd alpha is an exponential coefficient; alpha is alpha1、β1、α2、β2、α3、β3Are all indexes of observer, and are selected as follows1∈(2/3,1)，β1E (1,4/3) and let alpha2∈2α1-1，α3∈3α1-2；β2∈2β1-1，β3∈3β2-2；
Combining the equation (8) and the equation (9), the observer error state space equation is recorded as:
Figure FDA0003138511680000032
wherein the intermediate variable
Figure FDA0003138511680000033
Intermediate variables
Figure FDA0003138511680000034
According to hypothesis 1, there must be one normalNumber deltamaxSo that | dr|≤Δmax(ii) a Under the action of the above-mentioned observer, e1i，e2iAnd eriAt a fixed time TiI 1.. n converges to zero, and e is derived1,i＝e2,i＝er,i0, i 1, n; and n is the number of missiles.
4. The anti-saturation fixed-time cooperative guidance law for attacking highly maneuverable targets according to claim 3, characterized in that: the design of the distributed fixed time observer in the normal direction of the sight line adopts the same design principle as the angular direction of the sight line, and the distributed fixed time observer in the normal direction of the sight line is as follows:
Figure FDA0003138511680000035
in the formula
Figure FDA0003138511680000036
Is x3iIs determined by the estimated value of (c),
Figure FDA0003138511680000037
is that
Figure FDA0003138511680000038
The derivative of (a) of (b),
Figure FDA0003138511680000039
is x4iIs determined by the estimated value of (c),
Figure FDA00031385116800000310
is that
Figure FDA00031385116800000311
The derivative of (a) of (b),
Figure FDA00031385116800000312
is dqIs estimated byThe value of the one or more of the one,
Figure FDA00031385116800000313
is that
Figure FDA00031385116800000314
A derivative of (a); defining observer error
Figure FDA00031385116800000315
5. The anti-saturation fixed-time cooperative guidance law for attacking highly maneuverable targets according to claim 4, characterized in that: in step 3, a distributed anti-saturation cooperative guidance law is designed according to the acceleration value observed by the distributed extended state observer and a geometric mathematical model of the motion of the missile and the target, and the method specifically comprises the following steps:
3.1, designing a distributed anti-saturation cooperative guidance law in the line-of-sight angular direction, wherein the guidance model in the line-of-sight angular direction is as follows:
Figure FDA0003138511680000041
in order to make the relative distance and speed from the target consistent for each missile, the following variables are introduced to characterize the cooperativity error:
Figure FDA0003138511680000042
in the formula, aijRepresenting communication parameters, a if there is information flowing from the ith missile to the jth missileijIs 1, otherwise is 0; x is the number of1jRepresents the relative distance, x, between the jth missile and the target2jRepresents the relative speed, v, of the jth missile and the target1iRepresents the distance difference between the ith missile and the jth missile from the target, v2iThe angular velocity difference of the current sight line between the ith missile and the jth missile from the target is shown, namely when v1iAnd v2iWhen the number of the missile targets approaches 0, the flight states of the ith missile and the jth missile are considered to be consistent, namely, the common attack can be realized when the target is finally attacked;
considering equation (12), the consistency guidance law based on the fixed time theory and the dual-limit homogeneous theorem is designed as follows:
Figure FDA0003138511680000043
wherein the disturbance estimator
Figure FDA0003138511680000044
Is drEstimate of, σ1、σ'1、σ2、σ'2Are all consistent guidance law indexes, k1、k2、k'1、k'2For consistent guidance law gain and to satisfy:
Figure FDA0003138511680000045
wherein x is1maxThe maximum value of the relative distance between the ith missile and the target is represented; x is the number of4maxAngular velocity u representing the line-of-sight direction angle of the ith missilermaxA maximum value representing a component of the acceleration of the missile in the direction of the line of sight;
for arbitrary urmax0 and satisfies:
1) for all i 1,2ri|≤urmax；
2) The cooperative guidance law consistency of the formula (12) can be completed in a fixed time, and the time upper limit T issComprises the following steps:
Figure FDA0003138511680000046
in the formula TnIs composed of
Figure FDA0003138511680000051
Convergence time of kv、k0、k∞、γ1、γ2、dV0、dV∞σ is designed as follows:
Figure FDA0003138511680000052
kv、k0、k∞、γ1、γ2、
Figure FDA0003138511680000053
sigma is a convergence parameter;
and 3.2, adopting a design principle the same as the direction of the line-of-sight angle for the cooperative guidance law in the normal direction of the line-of-sight angle, wherein a guidance model in the normal direction of the line-of-sight angle is as follows:
Figure FDA0003138511680000054
in order to ensure that the included angle between the line-of-sight angle and the ideal line-of-sight angle approaches to 0, the normal design acceleration of the line-of-sight angle of each missile is designed, so that each missile is attacked according to a preset attack angle, and on the basis, the cooperative guidance law of the line-of-sight angle direction is designed as follows:
Figure FDA0003138511680000055
wherein sigma3、σ'3、σ4、σ'4For consistency guidance law index, k3、k4、k'3、k'4For consistent guidance law gain and to satisfy:
(k3+k'3+k4+k'4)+Δmax+2x2maxx4max≤uqmax (20)
wherein x is2maxRepresenting the maximum value of the relative speed of the ith missile and the target; u. ofqmaxThe maximum value of the component of the acceleration of the missile in the normal line of the sight line direction is represented, and the proving process is also carried out in the manner of proving the angular direction of the sight line.
6. The anti-saturation fixed-time cooperative guidance law for attacking a high maneuvering target according to claim 5, characterized in that stability is proved by applying bipolar limit homogeneous theorem and Lyapunov equation in step 4, and a fixed-time theory is introduced to obtain a result of the fixed-time stability of the cooperative guidance system, specifically as follows:
the cooperative guidance system composed of the formula (12) and the formula (14) is as follows:
Figure FDA0003138511680000061
wherein i is 1., n; then order the intermediate variable
Figure FDA0003138511680000062
Intermediate variables
Figure FDA0003138511680000063
And when the extended state is fixed for the time observer
Figure FDA0003138511680000064
Converge to drThe above formula is written as:
Figure FDA0003138511680000065
wherein L is a Laplacian matrix; and let an intermediate variable
Figure FDA0003138511680000066
Intermediate variables
Figure FDA0003138511680000067
Intermediate vector INIs a feature vector, an intermediate variable, corresponding to a single feature value of 0
Figure FDA0003138511680000068
And (3) pushing out:
Figure FDA0003138511680000069
if and only if x11＝x12＝…x1nAnd x21＝x22＝…x2nWhen p is 0 and q is 0; equation (21) can achieve fixed time agreement if and only if p and q converge to zero at fixed time.
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