CN112305919A - Design method of fixed time sliding mode guidance law with collision angle constraint - Google Patents

Design method of fixed time sliding mode guidance law with collision angle constraint Download PDF

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CN112305919A
CN112305919A CN202011272814.2A CN202011272814A CN112305919A CN 112305919 A CN112305919 A CN 112305919A CN 202011272814 A CN202011272814 A CN 202011272814A CN 112305919 A CN112305919 A CN 112305919A
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蔡远利
田野
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Xian Jiaotong University
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    • G05—CONTROLLING; REGULATING
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Abstract

The invention discloses a fixed time sliding mode guidance law design method with collision angle constraint, which is characterized in that a three-dimensional bullet mathematical model for a maneuvering target is established based on a ground launching coordinate system; according to the three-dimensional bullet mathematical model, a second-order dynamic equation of the bullet sight angle is obtained; designing a fixed-time nonsingular terminal sliding mode; and constructing a fixed-time sliding mode surface according to the controlled state quantity of the line-of-sight angle and the fixed-time nonsingular terminal sliding mode by adopting a terminal sliding mode control method based on the second-order dynamic equation of the line-of-sight angle of the bullet eye, and designing to obtain a fixed-time sliding mode guidance law with collision angle constraint. The convergence time upper bound of the guidance law designed by the invention is independent of the initial conditions of the missile, can be preset and is suitable for some battle scenes with unknown initial states. The guidance law designed by the invention has the advantages of simple structure, faster convergence speed of a guidance system, shorter missile interception time, high interception precision and good application prospect.

Description

Design method of fixed time sliding mode guidance law with collision angle constraint
Technical Field
The invention belongs to the technical field of missile guidance, and particularly relates to a fixed time sliding mode guidance law design method with collision angle constraint.
Background
With the continuous progress of military technology, the attack bullets in modern war develop towards high speed and large movement. In order to effectively destroy an incoming target, a kinetic energy killing technology taking direct collision as a requirement becomes a hotspot of research. Meanwhile, in the terminal guidance process, a guidance system is required to ensure higher guidance precision, limited time convergence and impact a target at a desired angle, so that the battle damage effect is maximized, and higher requirements are provided for the design of a guidance law. The classical proportional guidance law is widely applied due to the characteristic of convenient implementation, but the guiding effect on large maneuvering targets is not ideal. Aiming at the problem of intercepting maneuvering targets, in recent years, a method for designing a terminal guidance law by applying a modern control theory method becomes a research focus, such as an optimal guidance law, a sliding mode guidance law, a differential countermeasure guidance law and the like.
The sliding mode control has stronger robustness to system uncertainty and external interference, and has the advantages of simple structure, easy implementation, quick response and the like, so the sliding mode control is applied to guidance law design (Kumar SR, ghost D, Three-dimensional impact angle with coordinated environment dynamics, Proceedings of the institute of Mechanical Engineers Part G Journal of aeronautical Engineering 2017,231(G4):1-21) and some research results are obtained. The traditional sliding mode guidance law adopts a linear sliding mode surface, the line-of-sight angle and the angular rate of the missile eyes are gradually converged, and the convergence time tends to be infinite. However, the actual guidance time is usually short, and the designed guidance law should meet the limited time convergence to meet the actual requirement. According to the terminal sliding mode control method, the state of the system can be converged to be balanced in limited time by introducing the nonlinear sliding mode surface, and for the interception of a high-speed maneuvering target, a corresponding terminal sliding mode surface can be constructed for guidance law design, so that the limited time convergence is realized.
The existing sliding mode guidance law mostly focuses on finite time convergence. However, the upper limit of the convergence time of the finite-time terminal sliding mode control depends on the initial conditions of the system. The initial states of the intercepting missile and the target are usually difficult to accurately obtain in the actual guidance process, and the convergence time obtained by calculating different initial conditions is different, so that the actual application is influenced to a certain extent.
Disclosure of Invention
In order to solve the limitation of time-limited convergence in the prior art, the invention aims to provide a fixed-time sliding mode guidance law design method with collision angle constraint so as to realize accurate guidance.
The purpose of the invention is realized by the following technical scheme:
a fixed time sliding mode guidance law design method with collision angle constraint comprises the following processes:
establishing a three-dimensional bullet mathematical model aiming at the maneuvering target based on a ground launching coordinate system;
according to the three-dimensional bullet mathematical model, a second-order dynamic equation of the bullet sight angle is obtained;
designing a fixed-time nonsingular terminal sliding mode;
and constructing a fixed-time sliding mode surface according to the controlled state quantity of the line-of-sight angle and the fixed-time nonsingular terminal sliding mode by adopting a terminal sliding mode control method based on the second-order dynamic equation of the line-of-sight angle of the bullet eye, and designing to obtain a fixed-time sliding mode guidance law with collision angle constraint.
Preferably, the three-dimensional bullet mathematical model is as follows:
Figure BDA0002778239500000021
Figure BDA0002778239500000022
Figure BDA0002778239500000023
Figure BDA0002778239500000024
Figure BDA0002778239500000025
Figure BDA0002778239500000026
Figure BDA0002778239500000027
in the formula (I), the compound is shown in the specification,
Figure BDA0002778239500000028
representing target-missile relative approach velocity, vmIndicating the velocity, v, of the interceptor projectiletRepresenting the speed of the target, r representing the relative distance of the eyes, thetamIn order to intercept the pitch angle of the velocity vector of the bomb relative to the sight line coordinate system,
Figure BDA0002778239500000029
is the pitch angle velocity of the interceptor projectile, phimFor the yaw angle of the velocity vector of the interceptor projectile with respect to the line-of-sight coordinate system,
Figure BDA00027782395000000210
is the angle velocity of the interceptor projectile yaw-rate, θtIs the pitch angle of the velocity vector of the target with respect to the line of sight coordinate system,
Figure BDA0002778239500000031
is the target spring pitch angle velocity, phitRespectively the yaw angle of the velocity vector of the target with respect to the line of sight coordinate system,
Figure BDA0002778239500000032
is the target missile yaw rate, θLIs the elevation angle of the line of sight with respect to the reference coordinate system,
Figure BDA0002778239500000033
is the line of sight pitch angle velocity, phiLRespectively the yaw angle of the line of sight with respect to the reference coordinate system,
Figure BDA0002778239500000034
is the line of sight yaw rate, aymTo intercept the y-direction acceleration of the projectile, azmTo intercept z-direction acceleration of the projectile, aytIs the y-direction acceleration of the target projectile, aztIs the z-direction acceleration of the target projectile.
Preferably, the second order dynamic equation of the line-of-sight angle of the bullet eye is as follows:
Figure BDA0002778239500000035
Figure BDA0002778239500000036
in the formula (I), the compound is shown in the specification,
Figure BDA0002778239500000037
is the line-of-sight pitch angular acceleration,
Figure BDA0002778239500000038
is the line of sight yaw angular acceleration.
Preferably, θ is selectedLAnd phiLFor the system state, the second order dynamic equation of the line-of-sight angle of the bullet eye is expressed as:
Figure BDA0002778239500000039
in the formula (I), the compound is shown in the specification,
Figure BDA00027782395000000310
Figure BDA00027782395000000311
Figure BDA00027782395000000312
Figure BDA00027782395000000313
where D is the external disturbance and u is the control acceleration.
Preferably, the external disturbance D is bounded, | | D | | ≦ Δ, and Δ is an unknown positive number.
Preferably, the design process of the fixed-time nonsingular terminal sliding mode is as follows:
and calculating the deviation amount of the current pitching sight angle, the yawing sight angle and the expected sight angle according to the required collision angle, taking the deviation amount as a controlled amount to design a sliding mode surface, and designing a non-singular terminal sliding mode with fixed time according to the controlled amount design sliding mode surface.
Preferably, the calculation formula of the controlled quantity design sliding mode surface x is as follows:
Figure BDA0002778239500000041
wherein x is1Representing the amount of angular deviation of the pitch line of sight, x2Representing the amount of angular deviation of the yaw line of sight, thetaLfIndicating a desired pitch line-of-sight angle, phiLfRepresenting a desired yaw line-of-sight angle;
the fixed-time nonsingular terminal sliding mode s is as follows:
Figure BDA0002778239500000042
wherein alpha is1、β1、m1、m2As a parameter of the slip form surface, α1＞0，β1＞0，
Figure BDA0002778239500000043
Figure BDA0002778239500000044
Preferably, the fixed-time sliding-form surface s is:
Figure BDA0002778239500000045
wherein the content of the first and second substances,
Figure BDA0002778239500000046
Figure BDA0002778239500000047
is the result of the derivation of x,
Figure BDA0002778239500000048
preferably, the fixed-time sliding-mode guidance law u with an impact angle constraint1The following were used:
Figure BDA00027782395000000412
wherein:
Figure BDA0002778239500000049
Figure BDA00027782395000000410
s＝[s1，s2]Txi is a constant larger than zero, xi is more than or equal to delta, delta is bounded external disturbance,
Figure BDA00027782395000000411
α2＞0，β2> 0, λ is a saturation function parameter, q1、n1、q2、n2、α2、β2Are the approach law parameters.
Preferably, the y-direction acceleration a of the target projectile is taken into accountytAnd z-direction acceleration a of the target projectileztFixed-time sliding-mode guidance law u with collision angle constraint in the case ofmThe following were used:
Figure BDA0002778239500000051
wherein the content of the first and second substances,
Figure BDA0002778239500000052
Figure BDA0002778239500000053
s＝[s1，s2]Tdelta is the adaptive law coefficient, delta is diag (delta)1，δ2)，Q＝diag(sign(s1)，sign(s2))，
Figure BDA0002778239500000054
Figure BDA0002778239500000055
α2＞0，β2> 0, λ is a saturation function parameter, ε is an unknown constant,
Figure BDA0002778239500000056
is an estimate of ε, q1、n1、q2、n2、α2、β2Are the approach law parameters.
The invention has the following beneficial effects:
the invention relates to a fixed time sliding mode guidance law design method with an impact angle constraint, which aims at the problem of intercepting the final guidance of a maneuvering target, designs a guidance law on the basis of a coupled three-dimensional missile mathematical model and in combination with a nonsingular terminal sliding mode control method and a fixed time convergence theory, and enables a sliding mode surface, a line-of-sight angle and a line-of-sight angle rate to converge in fixed time and intercept the target at an expected impact angle. Different from the traditional limited time convergence guidance law, the guidance law is designed to have the convergence time upper bound independent of the bullet initial condition, can be preset and can be suitable for some battle scenes with unknown initial states. Compared with other typical guidance laws, the designed guidance law has the advantages of simple structure, higher convergence speed of a guidance system, shorter missile interception time, high interception precision and good application prospect.
Drawings
FIG. 1 is a schematic diagram of a missile-target three-dimensional relative motion model in the invention.
FIG. 2 is a comparison graph of interception tracks under different guidance laws.
FIG. 3 is a sliding mode surface variation diagram under different guidance laws.
FIG. 4 is a graph of changes in line-of-sight angles under different guidance laws.
FIG. 5 is a graph of changes in line-of-sight angular rate under different guidance laws.
Fig. 6 is a graph showing the change of acceleration in different guidance laws.
FIG. 7 shows guidance law u according to the present inventionmAnd next, the interception bombs use interception track graphs with different initial pitch angles.
FIG. 8 shows guidance law u according to the present inventionmAnd secondly, the intercepting bullet uses a sliding mode surface change diagram with different initial pitch angles.
FIG. 9 shows guidance law u according to the present inventionmNext, the interceptor projectiles use a line-of-sight angle variation map of different initial pitch angles.
FIG. 10 shows guidance law u according to the present inventionmNext, the interceptor projectiles use a plot of line-of-sight angular rate change for different initial pitch angles.
FIG. 11 shows guidance law u according to the present inventionmNext, the interceptor projectiles use a map of the acceleration change at different initial pitch angles.
Detailed Description
The technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings and examples, and it is to be understood that the described embodiments are only a part of the examples of the present invention, and are not intended to limit the present application.
The invention provides a fixed time sliding mode guidance law design method with an impact angle constraint. Firstly, establishing a three-dimensional relative motion model of the missile and a maneuvering target, and obtaining a second-order dynamic equation of the missile line-of-sight angle; then, in order to realize fixed time convergence, a novel nonsingular fast terminal sliding mode surface is designed for a class of second-order nonlinear systems, so that the system can be stable within fixed time; and finally, based on the fixed time terminal sliding mode control method, combining the line-of-sight angle tracking error of the missile, constructing a fixed time guidance law with angle constraint aiming at the maneuvering target, and proving through a Lyapunov stability theory that the designed guidance law can enable the guidance system to be converged to be balanced in limited time. The guidance strategy can intercept the maneuvering target within fixed time, and meets the requirements on the interception precision, the interception time and the collision angle constraint of the target.
The specific embodiment of the invention is as follows:
a collision angle constrained nonsingular terminal sliding mode fixed time convergence guidance law construction method comprises the following steps:
s01: establishing a three-dimensional missile target relative motion model of the missile interception maneuvering target based on a ground launching coordinate system;
s02: according to the three-dimensional relative movement model of the bullet in the step S01, a second-order dynamic equation of the visual line angle of the bullet is obtained;
s03: designing a fixed-time nonsingular terminal sliding mode;
s04: based on the second-order dynamic equation of the line-of-sight angle of the bullet eyes in the step S02, a terminal sliding mode control method is adopted, the controlled state quantity of the line-of-sight angle is constructed into a sliding mode surface according to the non-singular terminal sliding mode at the fixed time of S03, and a fixed time convergence sliding mode guidance law with collision angle constraint is designed.
Fig. 1 shows the three-dimensional relative motion relationship of the interceptor projectile and the target.
Specifically, a missile-target relative motion model (namely a three-dimensional missile-target mathematical model) is established by taking the initial position of the intercepting missile in the terminal guidance stage as the origin of a reference coordinate system:
Figure BDA0002778239500000071
Figure BDA0002778239500000072
Figure BDA0002778239500000073
Figure BDA0002778239500000074
Figure BDA0002778239500000075
Figure BDA0002778239500000076
Figure BDA0002778239500000077
in the formula, vm，vtR represents the speed of the intercepting projectile and the target and the relative distance between the projectile and the target respectively, thetamAnd phimThe pitch angle and the yaw angle of the velocity vector of the intercepting bomb relative to the sight line coordinate system are respectively a leading angle of the velocity vector of the intercepting bomb. ThetatAnd phitThe pitch angle and the yaw angle of the velocity vector of the target with respect to the sight line coordinate system, i.e., the target velocity vector lead angle, respectively. ThetaLAnd phiLThe pitch angle and yaw angle of the line of sight with respect to the reference coordinate system, respectively. a isym，azm，ayt，aztThe lateral acceleration of the interceptor projectile and the target projectile, respectively.
The main goal of guidance is by controlling the normal acceleration a of the missileymAnd azmThe sight line is kept stable, so that the missile approaches to the target in a quasi-parallel mode and interception is carried out. A stable line of sight can be achieved by zeroing the target-missile tangential relative velocity. Once target-missile approaches velocity
Figure BDA0002778239500000078
And changing from a negative value to a positive value, and ending the guidance process. The relative distance between the target and the missile at this time is called the miss distance.
In the final guidance process, there are
Figure BDA0002778239500000079
0＜r(t)＜r(0)，
Figure BDA00027782395000000710
Representing the relative missile-target distance at the initial time.
The second order dynamic equation of the line-of-sight angle of the bullet is as follows:
Figure BDA0002778239500000081
Figure BDA0002778239500000082
selecting θ in consideration of the goals of the guidance law designLAnd phiLFor the system state, the bullet eye line-of-sight angle second order dynamic equation is written as:
Figure BDA0002778239500000083
wherein
Figure BDA0002778239500000084
Figure BDA0002778239500000085
The D comprises target acceleration information, and the condition that the target acceleration in the actual system is unknown but bounded is considered, if | | D | | | is less than or equal to delta, delta is an unknown positive number.
Calculating the deviation amount of the pitching sight angle, the yawing sight angle and the expected sight angle according to the required collision angle, wherein the formula is as follows:
Figure BDA0002778239500000086
wherein x1And x2Respectively representing the amount of angular deviation of pitching sight line, the amount of angular deviation of yawing sight line, thetaLfAnd phiLfRepresenting the desired impingement angle at the moment of intercepting the object. Derived from x
Figure BDA0002778239500000087
When the interceptor projectile hits the target at the desired impact angle,
Figure BDA0002778239500000088
θL＝θLf；φL＝φLf。
the invention mainly aims to design a guidance law so that the state of a system (a three-dimensional bullet mathematical model) converges to an expected value within fixed time, and the miss distance at the guidance ending moment meets the interception requirement. For ease of design, the following arguments and assumptions are first given.
Lesion 1(Bhat, S.P., Bernstein, D.S.Finite-time stability of continuous autonomous systems, SIAM J.control Optim, 2000, 38 (3): 751-
Figure BDA0002778239500000089
Wherein x ∈ RnAs a system state quantity, F (t, x): r+×D→RnFor a continuous non-linear function, D is the open area containing the origin. Given an initial time t0And initial state x0. If 1) the origin is an asymptotic stable equilibrium point, and there is an open area of the origin
Figure BDA00027782395000000913
Figure BDA00027782395000000914
For all x (0) e N \ 0, there is a positive definite function T (x)0): n → R, then the origin is the equilibrium point of finite time convergence.
Figure BDA0002778239500000091
Lemma 2(Polyakov A. nonlinear feedback design for fixed-time station stabiliz)and (8) of linear control systems IEEE Transactions on Automatic control.2011, 57: 2106-2110) if the nonlinear system (32) is globally time-limited stable and converges to a time function T (x)0) There is a real upper bound TmaxI.e. to any
Figure BDA0002778239500000092
With T (x)0)≤TmaxThe origin is the fixed time convergence balance point.
Slip form face design
Theorem 1 designs a fixed time convergence nonlinear system
Figure BDA0002778239500000093
In the formula I1＞0，l2＞0，
Figure BDA0002778239500000094
Figure BDA0002778239500000095
The convergence time of the system is limited to
Figure BDA0002778239500000096
And (3) proving that: general formula
Figure BDA0002778239500000097
z (0) ═ z0 is written as:
Figure BDA0002778239500000098
when the | z | is more than 1, making y equal to 1+ ln | z |; when | z | < 1, let
Figure BDA0002778239500000099
Formula (II)
Figure BDA00027782395000000910
Writing
Figure BDA00027782395000000911
Calculating an upper bound on convergence time
Figure BDA00027782395000000912
Figure BDA0002778239500000101
Order to
Figure BDA0002778239500000102
Is provided with
Figure BDA0002778239500000103
Figure BDA0002778239500000104
Theorem 1 proves the syndrome.
The three-dimensional fixed time-guidance law construction method with impact angle constraint according to claim 1, based on
Figure BDA0002778239500000105
Figure BDA0002778239500000106
Constructing a fixed-time nonsingular terminal sliding form as follows
Figure BDA0002778239500000107
Wherein alpha is1＞0，β1＞0，
Figure BDA0002778239500000108
Figure BDA0002778239500000109
Guidance law design
The fixed time sliding mode surface used by the guidance law is
Figure BDA00027782395000001010
Wherein
Figure BDA00027782395000001011
Figure BDA00027782395000001012
Figure BDA00027782395000001013
α1＞0，β1＞0。
The derivation can be obtained by the sliding mode surface s
Figure BDA00027782395000001014
Wherein
Figure BDA00027782395000001015
Figure BDA00027782395000001016
The guidance law is designed as follows:
Figure BDA0002778239500000111
wherein
Figure BDA00027782395000001113
ξ≥Δ，
Figure BDA00027782395000001114
Figure BDA00027782395000001115
α2＞0，β2＞0。
During actual guidance, the target acceleration ayt，aztOften not known accurately in advance. Therefore, the self-adaptive term is added into the guidance law to estimate the target acceleration so as to improve the guidance performance. In particular, theta is estimated more accuratelytAnd phitAdding an adaptive term epsilon to the brake-making law1，ε2Estimate target maneuver, set | d1|≤ε1，|d2|≤ε2。
Figure BDA0002778239500000112
Is epsilon ═ epsilon1 ε2]TWith an estimation error of
Figure BDA0002778239500000113
The obtained fixed time guidance law for intercepting the maneuvering target with collision angle constraint is
Figure BDA0002778239500000114
Wherein the content of the first and second substances,
Figure BDA0002778239500000115
Figure BDA0002778239500000116
Q＝diag(sign(s1)，sign(s2))，δ＝diag(δ1，δ2)，
Figure BDA0002778239500000117
Figure BDA0002778239500000118
α2＞0，β2＞0。
theorem 2 for guidance system
Figure BDA0002778239500000119
If the slip form surface constructed by the invention is adopted
Figure BDA00027782395000001110
And guidance law
Figure BDA00027782395000001111
In guidance law umUnder the action, the missile can successfully intercept the target and the state variable x of the guidance system1And x2At a fixed time TsInner convergence to 0. Wherein
Figure BDA00027782395000001112
And (3) proving that: the Lyapunov function was chosen as follows
Figure BDA0002778239500000121
Pair type V1Derivative to obtain
Figure BDA0002778239500000122
By
Figure BDA0002778239500000123
Can obtain V1Is bounded. Hence sliding mode variable s and estimation error
Figure BDA0002778239500000124
Are bounded.
Considering the Lyapunov function
Figure BDA0002778239500000125
Pair type V2Derivative to obtain
Figure BDA0002778239500000126
Figure BDA0002778239500000127
Selecting the appropriate
Figure BDA0002778239500000128
So that
Figure BDA0002778239500000129
And deltajSatisfy the requirement of
Figure BDA00027782395000001210
μ is a small normal number.
Is provided with
Figure BDA0002778239500000131
To obtain
Figure BDA0002778239500000132
When | | | s | | | is greater than or equal to 1, there are
Figure BDA0002778239500000133
When s < 1, there are
Figure BDA0002778239500000134
Known as a guidance systemThe system can be in a fixed time T2Inner convergence to s-0.
When the state quantity x of the guidance system1，x2After the system reaches the sliding form surface s is equal to 0, the system has
Figure BDA0002778239500000135
Consider the following Lyapunov function
Figure BDA0002778239500000136
Pair type V3Derivative to obtain
Figure BDA0002778239500000137
State variable x of guidance system1，x2Will be at time TsInner convergence to 0, i.e. the line-of-sight tracking error converges to zero.
Figure BDA0002778239500000138
Theorem 2 proves the syndrome.
Therefore, the guidance law of the invention can intercept the maneuvering target within a fixed time, and guarantee the collision angle constraint, and has better adaptability and robustness.
For purposes of illustrating the objects, technical solutions and advantages of the embodiments of the present invention in detail, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. It is to be understood that the embodiments described are only a few embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention generally described and illustrated in the figures herein may be arranged and designed in a wide variety of different configurations. Thus, the following detailed description of the embodiments of the present invention, presented in the figures, is not intended to limit the scope of the invention, as claimed, but is merely representative of selected embodiments of the invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The invention designs a fixed time convergence guidance law with collision angle constraint by combining a fixed time convergence theory and a nonsingular terminal sliding mode method, and further analyzes the law by combining with a specific embodiment in order to verify the effectiveness and the practicability of the strategy.
Example 1
In the embodiment, a three-dimensional guidance simulation experiment is designed, an interception object is taken as a maneuvering target, and different typical guidance laws are respectively used for carrying out interception simulation comparison so as to prove the effectiveness and superiority of the method.
The target maneuver is designed as: assuming that the target makes uniform linear motion from the initial moment, and the turning maneuver is continuously started from t-4 s, the lateral maneuver acceleration is ayt＝azt＝30m/s2.
The initial conditions of the simulation are shown in Table 1
TABLE 1
Figure BDA0002778239500000141
Figure BDA0002778239500000151
The initial position of the interception bullet is (0, 0, 0) km, and the initial trajectory pitch angle is thetam(0) 15 deg. and initial ballistic yaw angle phim(0)＝5deg。
The parameter of the controller is designed to be m1＝n1＝9/7，m2＝n2＝7/9，α1＝0.8，β1＝0.8，α2＝200，β2＝800，η＝0.1，δ1＝δ2H is 100 and h is 0.1. Calculating the convergence time upper bound of T according to the selected parameterss＝7.45s。
In this embodiment, the proposed non-singular Terminal sliding mode fixed time convergent guidance law with collision angle constraint is used for comparison with the initial simulation conditions of the typical linear sliding mode guidance law (Kumar S.R., Ghos D.: Three-dimensional impact angle guidance with multiplied estimated dynamics, Proceedings of the instruction of Mechanical Engineers, Part G: Journal of Audio Engineering, 2017, 4 (231): 621-.
1) Linear sliding mode guidance law (CSMCGL):
the slip form surface is
Figure BDA0002778239500000152
Guidance law is
Figure BDA0002778239500000153
The parameters selected are C ═ diag (1.5 ), η ═ 40, M ═ diag (0.75 ), K ═ diag (1, 1)
2) Nonsingular terminal sliding mode guidance law (NTSMCGL):
the slip form surface is
Figure BDA0002778239500000154
Guidance law is
Figure BDA0002778239500000155
The parameters β ═ diag (2, 2), p ═ 9, q ═ 7, k0＝0.5，H0＝diag(1.5，1.5)，β＝0.5。
Fig. 2-6 show the graphs of the interception simulation parameters under different guidance law controls.
The interception results under different guidance laws are compared and shown in table 2.
Fig. 2 is an interception trajectory diagram in the simulation process, and as can be seen from fig. 2, all three guidance methods can successfully intercept a target, but the interception time of the method provided by the invention is shorter than that of the other two guidance laws.
Fig. 3 is a graph of the variation of the sliding mode surface during the simulation. The sliding mode surface of the guidance law has the fastest convergence speed, and the guidance law is kept stable after convergence without buffeting.
Fig. 4 and 5 are graphs of the line-of-sight angle and the rate of change of the line-of-sight angle during the simulation. Compared with CSMCGL and NTSMCGL methods, the method provided by the invention can more quickly enable the visual angle to approach the expected value, enables the visual angle rate to be converged to zero, and can keep the expected value until the target is hit.
Fig. 6 is a graph of acceleration during simulation. It can be seen that the acceleration value under the action of the guidance law is in a reasonable range, a saturation phenomenon appears in the initial stage of guidance, but the acceleration value is quickly converged to the reasonable range and is kept stable all the time in the later stage.
TABLE 2
Figure BDA0002778239500000161
Example 2
The initial conditions of the simulation are shown in table 1, and the moving mode of the maneuvering target is the same as that of example 1.
Initial trajectory pitch angle theta of intercepting projectilem(0) 5deg, 15deg and 30deg are respectively selected, and the initial ballistic yaw angle phi ism(0) The selection was 5 deg.
Analyzing the fixed time guidance law designed by the invention at different initial pitch angles thetamThe interception effect of the guidance law and the result of fixed time convergence under the condition.
Fig. 7-11 show graphs of the interception parameter under different initial conditions.
As can be seen from the interception trajectory graph of FIG. 7, the missile successfully intercepts the target at different initial pitch line-of-sight angles.
Fig. 8 shows that the slip-form face is able to converge rapidly to 0 in a fixed time at different initial pitch line-of-sight angles.
Fig. 9 and 10 show that, although the simulated initial angles are different, each of the line-of-sight angles and the line-of-sight angular rates converge to zero within the set upper time bound.
Fig. 11 is a graph of acceleration during simulation. It can be seen that the acceleration values are in a reasonable range, saturation occurs in the initial stage of guidance, but the acceleration values are quickly converged to the reasonable range and are kept stable.
The curves shown in the figures are smooth, the convergence rate is high, and the control performance is good. The correctness and the effectiveness of the nonsingular terminal sliding mode surface at the fixed time designed by the invention are verified.
Although the present invention has been described herein with reference to the illustrated embodiments thereof, which are intended to be preferred embodiments of the present invention, many other modifications and embodiments can be devised by those skilled in the art, and it is intended that all such modifications and variations fall within the scope of the appended claims.
Aiming at the problem of intercepting the terminal guidance of a maneuvering target, the invention designs a guidance law based on a coupled three-dimensional missile and target relative motion model and in combination with a nonsingular terminal sliding mode control method and a fixed time convergence theory, so that the sliding mode surface, the line-of-sight angle and the line-of-sight angle rate converge in fixed time and intercept the target at an expected collision angle. Different from the traditional limited time convergence guidance law, the guidance law is designed to have the convergence time upper bound independent of the bullet initial condition, can be preset and can be suitable for some battle scenes with unknown initial states. Compared with other typical guidance laws, the designed guidance law has the advantages of simple structure, higher convergence speed of a guidance system, shorter missile interception time, high interception precision and good application prospect.
Compared with the prior art, the fixed time guidance law designed by the invention has the remarkable advantages that (1) the guidance law is designed based on the terminal sliding mode surface, a maneuvering target can be intercepted well, and the guidance precision is higher; (2) the convergence speed is high, and the upper bound of the convergence time is irrelevant to the initial state of the system; (3) the impact angle constraint can be satisfied, and the missile can hit the target at the required impact angle. Has wide application prospect.

Claims (10)

1. A fixed time sliding mode guidance law design method with collision angle constraint is characterized by comprising the following processes:
establishing a three-dimensional bullet mathematical model aiming at the maneuvering target based on a ground launching coordinate system;
according to the three-dimensional bullet mathematical model, a second-order dynamic equation of the bullet sight angle is obtained;
designing a fixed-time nonsingular terminal sliding mode;
and constructing a fixed-time sliding mode surface according to the controlled state quantity of the line-of-sight angle and the fixed-time nonsingular terminal sliding mode by adopting a terminal sliding mode control method based on the second-order dynamic equation of the line-of-sight angle of the bullet eye, and designing to obtain a fixed-time sliding mode guidance law with collision angle constraint.
2. The fixed-time sliding-mode guidance law design method with impact angle constraint according to claim 1, characterized in that the three-dimensional bullet mathematical model is as follows:
Figure FDA0002778239490000011
Figure FDA0002778239490000012
Figure FDA0002778239490000013
Figure FDA0002778239490000014
Figure FDA0002778239490000015
Figure FDA0002778239490000016
Figure FDA0002778239490000017
in the formula (I), the compound is shown in the specification,
Figure FDA0002778239490000018
representing target-missile relative approach velocity, vmIndicating the velocity, v, of the interceptor projectiletRepresenting the speed of the target, r representing the relative distance of the eyes, thetamIn order to intercept the pitch angle of the velocity vector of the bomb relative to the sight line coordinate system,
Figure FDA0002778239490000019
is the pitch angle velocity of the interceptor projectile, phimFor the yaw angle of the velocity vector of the interceptor projectile with respect to the line-of-sight coordinate system,
Figure FDA00027782394900000110
is the angle velocity of the interceptor projectile yaw-rate, θtIs the pitch angle of the velocity vector of the target with respect to the line of sight coordinate system,
Figure FDA00027782394900000111
is the target spring pitch angle velocity, phitRespectively the yaw angle of the velocity vector of the target with respect to the line of sight coordinate system,
Figure FDA00027782394900000112
is the target missile yaw rate, θLIs the elevation angle of the line of sight with respect to the reference coordinate system,
Figure FDA00027782394900000113
is the line of sight pitch angle velocity, phiLRespectively, the line of sight with respect to a reference coordinate systemThe angle of yaw of (a) is,
Figure FDA00027782394900000114
is the line of sight yaw rate, aymTo intercept the y-direction acceleration of the projectile, azmTo intercept z-direction acceleration of the projectile, aytIs the y-direction acceleration of the target projectile, aztIs the z-direction acceleration of the target projectile.
3. The fixed-time sliding-mode guidance law design method with impact angle constraint according to claim 2, characterized in that the second-order dynamic equation of the line-of-sight angle of the bullet eye is as follows:
Figure FDA0002778239490000021
Figure FDA0002778239490000022
in the formula (I), the compound is shown in the specification,
Figure FDA0002778239490000023
is the line-of-sight pitch angular acceleration,
Figure FDA0002778239490000024
is the line of sight yaw angular acceleration.
4. The fixed-time sliding-mode guidance law design method with impact angle constraints as set forth in claim 3, wherein θ is selectedLAnd phiLFor the system state, the second order dynamic equation of the line-of-sight angle of the bullet eye is expressed as:
Figure FDA0002778239490000025
in the formula (I), the compound is shown in the specification,
Figure FDA0002778239490000026
Figure FDA0002778239490000027
Figure FDA0002778239490000028
Figure FDA0002778239490000029
where D is the external disturbance and u is the control acceleration.
5. The fixed-time sliding-mode guidance law design method with collision angle constraint according to claim 4, characterized in that external disturbance D is bounded, | | D | | ≦ Δ, Δ is an unknown positive number.
6. The fixed-time sliding-mode guidance law design method with collision angle constraint according to claim 4, characterized in that the design process of the fixed-time nonsingular terminal sliding mode is as follows:
and calculating the deviation amount of the current pitching sight angle, the yawing sight angle and the expected sight angle according to the required collision angle, taking the deviation amount as a controlled amount to design a sliding mode surface, and designing a non-singular terminal sliding mode with fixed time according to the controlled amount design sliding mode surface.
7. The fixed-time sliding-mode guidance law design method with the collision angle constraint according to claim 6, characterized in that a controlled-quantity design sliding-mode surface x is calculated according to the following formula:
Figure FDA0002778239490000031
wherein x is1Representing the amount of angular deviation of the pitch line of sight, x2Representing the amount of angular deviation of the yaw line of sight, thetaLfIndicating a desired pitch line-of-sight angle, phiLfRepresenting a desired yaw line-of-sight angle;
the fixed-time nonsingular terminal sliding mode s is as follows:
Figure FDA0002778239490000032
wherein alpha is1、β1、m1、m2As a parameter of the slip form surface, α1＞0，β1＞0，
Figure FDA0002778239490000033
Figure FDA0002778239490000034
8. The fixed-time sliding-mode guidance law design method with impact angle constraints according to claim 7, wherein the fixed-time sliding-mode surface s is:
Figure FDA0002778239490000035
wherein the content of the first and second substances,
Figure FDA0002778239490000036
Figure FDA0002778239490000037
is the result of the derivation of x,
Figure FDA0002778239490000038
9. the retainer of claim 8 with an impingement angle restraintThe method for designing the time sliding mode guidance law is characterized in that the fixed time sliding mode guidance law u with the constraint of an impact angle1The following were used:
Figure FDA0002778239490000039
wherein:
Figure FDA00027782394900000310
Figure FDA00027782394900000311
s＝[s1，s2]Txi is a constant greater than zero, xi is equal to or greater than delta, delta is a bounded external disturbance,
Figure FDA00027782394900000312
α2＞0，β2> 0, λ is a saturation function parameter, q1、n1、q2、n2、α2、β2Are the approach law parameters.
10. The fixed-time sliding-mode guidance law design method with impact angle constraints as set forth in claim 8, wherein the y-direction acceleration a of the target projectile is consideredytAnd z-direction acceleration a of the target projectileztFixed-time sliding-mode guidance law u with collision angle constraint in the case ofmThe following were used:
Figure FDA0002778239490000041
wherein the content of the first and second substances,
Figure FDA0002778239490000042
Figure FDA0002778239490000043
s＝[s1，s2]Tdelta adaptive term coefficient, delta diag (delta)1，δ2)，Q＝diag(sign(s1)，sign(s2))，
Figure FDA0002778239490000044
Figure FDA0002778239490000045
α2＞0，β2> 0, λ is a saturation function parameter, ε is an unknown constant,
Figure FDA0002778239490000046
is an estimate of ε, q1、n1、q2、n2、α2、β2Are the approach law parameters.
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