CN110929216B - Self-adaptive backstepping guidance law design method with anti-drag function - Google Patents
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Abstract
The invention provides a self-adaptive backstepping guidance law design method with anti-drag function, which comprises the following steps: firstly, establishing a model according to a prepositioning method and a parallel approaching method guiding thought; and secondly, aiming at the established model, taking uncertainty in the system into consideration, comprehensively utilizing a backstepping sliding mode and an adaptive method, and designing an adaptive backstepping sliding mode guidance law. The invention provides a self-adaptive backstepping guidance law design method with anti-dragging based on the guidance ideas of a preposition method and a parallel approach method, which is a novel backstepping sliding mode guidance strategy, wherein the preposition method ensures that the guided track of a missile is advanced by a proper angle compared with a towing bait, and the parallel approach method ensures that the missile can accurately intercept a target.
Description
Technical Field
The invention relates to a self-adaptive backstepping guidance law design method with anti-dragging function.
Background
In modern war, with the rapid development of various electronic interference technologies, the survivability of the carrier is greatly improved. The towing type bait is taken as a novel self-defense interference mode, has become an important means for resisting the air defense missile radar seeker at present, and is successfully applied to war. So that the interception of a carrier target with towed baits by means of the anti-air bombs of the radar seeker is a serious challenge. Most of the current literature is mainly focused on analysis of the interference mechanism and performance of the towed bait, but the end guidance law aiming at the towing-resistant bait has relatively few research results. In the nonlinear control theory, the sliding mode control theory has good robustness on parameter perturbation and external disturbance inside the system, so that the sliding mode control theory obtains rich research results in the law making design.
Disclosure of Invention
The invention provides a self-adaptive backstepping guidance law design method with anti-dragging based on the guidance ideas of a preposition method and a parallel approach method, which is a novel backstepping sliding mode guidance strategy, wherein the preposition method ensures that the guided track of a missile is advanced by a proper angle compared with a towing bait, and the parallel approach method ensures that the missile can accurately intercept a target.
In order to achieve the above purpose, the technical scheme of the invention is as follows:
an adaptive backstepping guidance law design method with drag resistance, the method comprising the steps of:
firstly, establishing a model according to a prepositioning method and a parallel approaching method guiding thought;
and secondly, aiming at the established model, taking uncertainty in the system into consideration, comprehensively utilizing a backstepping sliding mode and an adaptive method, and designing an adaptive backstepping sliding mode guidance law.
The beneficial effects of the invention are as follows: the backstepping sliding mode self-adaptive guidance law designed by the invention has good stability and anti-interference performance and high-precision capability of intercepting maneuvering targets.
Drawings
Fig. 1 is a graph of relative motion with drag interception.
Fig. 2 is a schematic diagram of a missile-target tracking trajectory.
FIG. 3 is a diagram of the relative distance between the shots.
Fig. 4 is a view angle rate diagram.
Fig. 5 is a view angle schematic diagram.
Fig. 6 is a schematic view of normal overload of a missile.
Fig. 7 is a schematic diagram of an adaptive parameter variation curve.
Detailed Description
The invention is further described below with reference to the drawings and detailed description.
In the two-dimensional plane, the relative movement relation of the missile and the target with the towing interference is shown in fig. 1, the missile, the target and the towing interference are regarded as mass points and are respectively represented by M, T and TRAD, the connecting line of the missile and the target is the line of sight, and the rest parameters are marked in the figure.
Wherein r is the relative distance between the missile and the carrier target, L is the length of the towing line, q is the angle of sight, and q READ For the angle of view of the towing disturbance relative to the missile, deltaq is the angle between the towing disturbance and the missile link and between the towing disturbance and the missile link of the carrier,and->Speed direction angle, θ, of missile and carrier targets respectively m And theta t Leading angles of missile and carrier targets respectively, V m And V t Respectively representing the missile longitudinal velocity and the target longitudinal velocity of the carrier.
Assuming missile longitudinal velocity V m And a target longitudinal velocity V t All are constant values and can be described by the following differential equations according to the geometric relationship of fig. 1:
Δq·r＝Lsinθ t (1)
q＝q TRAD -Δq (2)
deriving (1)
Deriving the formula (7) and substituting the formula (6) to finish the product
Deriving (6) to obtain
Further arranged into
Wherein,and->The components of the missile and target accelerations in the line of sight directions, respectively.
Based on the theory of the pre-measurement method and the parallel approach method, if deltaq andthe following relation is satisfied, and the accurate attack target of the missile can be ensured:
wherein t is f The moment of intercepting the target for the missile.
Defining a state variable as x 1 ＝Δq(t)，Finishing of the combination of formula (7) and formula (10)
Wherein,u＝a m for control input, d=g (t) is an unknown disturbance in the system and satisfies g (t). Ltoreq.d M ，d M Is a positive constant.
3. Related mechanism
Lemma 1: for x i E R, i=1,..n, the real number p satisfies 0 < p.ltoreq.1, the following inequality holds:
(|x 1 |+…+|x n |) p ≤|x 1 | p +…+|x n | p (14)
and (4) lemma 2: for a systemx (0) =0, f (0) =0, x∈r, assuming that there is a continuous microtransaction V, so that it satisfies the following condition:
(1) V is a positive definite function.
(2) There are positive real numbers c > 0 and alpha epsilon (0, 1), and an open neighborhood containing the originSo thatThis is true.
The system is stable for a finite time and the convergence time T satisfiesWherein V is 0 For an initial value of V, if u=u 0 ＝R n The system is globally stable for a limited time.
4. Self-adaptive backstepping sliding mode guidance law design
Aiming at the system models (12) and (13), the uncertainty in the system is considered, a backstepping sliding mode and an adaptive method are comprehensively utilized, and an adaptive backstepping sliding mode guidance law is designed, and the specific process is as follows:
step 1
Defining an error variable z 1 The following are listed below
z 1 ＝x 1 (15)
Deriving (15) to obtain
According to equation (15), the virtual control is designed as
r 1 ＝(2-γ)η γ-1 (19)
r 2 ＝(γ-1)η γ-2 (20)
Wherein, 0 < gamma < 1, alpha, beta, k 1 ，k 2 And η is a positive constant.
Selecting lyapunov function
For V 1 Deriving and substituting the formula (15) and the formula (17) into the arrangement to obtain
When |x 1 When | > η, formula (22) may be arranged as
When |x 1 When the I is less than or equal to eta, the formula (22) can be arranged into
From the above evidence, x can be obtained 1 Converging to zero for a finite time.
Step 2
The error variable s is defined as follows
Deriving (25) to obtain
In order to process disturbance d with unknown upper bound, based on a backstepping sliding mode control theory, a self-adaptive backstepping sliding mode guidance law is designed:
wherein,is d M Estimate of k 3 ，k 4 And h is a normal number, delta > 1.
Theorem 1: with respect to systems (12) - (13), d being bounded but the upper bound unknown, the following conclusion can be drawn using adaptive backstepping sliding mode guidance laws (27) - (28).
(1) The slip plane s is convergent for a finite time.
(2) State x of the system 1 And z 2 Is of limited time convergence.
And (3) proving: selecting Lyapunov function as
Wherein,
deriving (29) along the system trajectory to obtain
Can be obtained from (30)V 2 Is not increased, s and +.>Is bounded and thus can be given an adaptive estimation error d M And s are all bounded.
Selecting lyapunov function
Deriving (31) to obtain
Substituting the controllers (27) - (28) into the database (32) to obtain
Because ofAnd->There is->Select->The value of the sum sigma is large enough to satisfy
Bonding ofCan obtain
Case 1 when |x 1 When | > η and s+.0, formula (33) can be arranged as
Case 2 when |x 1 When | > η and s≡0 formula (33) can be arranged as
From formula (37), system x 1 Is of limited time convergence.
Case 3 when |x 1 > eta andwhen the controller (27) is substituted into formula (13), the data can be organized into
So thatInstead of an attractor, the system sliding mode converges to zero in a limited time, and can be guaranteed.
Case 4: when |x 1 When +.eta.and s+.0, formula (33) can be organized as
Case 5: when |x 1 When |is less than or equal to eta and s.ident.0
From (40), system x 1 Is of limited time convergence.
Case 6: when |x 1 The I is less than or equal to etaSubstitution of the controller (27) into the group (13) can be done as
So thatInstead of an attractor, the system sliding mode converges to zero in a limited time, and can be guaranteed.
In summary, the system is time-constrained. Further the slip planes s and x can be obtained 1 Is of limited time convergence.
The evidence is obtained in the step (1).
Because of the slip-form surfaces s and x 1 Is of finite time convergence, z can be obtained according to equation (25) 2 Is of limited time convergence.
And (3) obtaining the evidence in the step (2).
Theorem 1 is proved.
Simulation analysis
In order to verify the effectiveness of a self-adaptive finite time controller (27) based on a backstepping sliding mode, a certain missile flies at a certain height, mach number is 4.5, sonic speed is 295.07m/s, the flying speed of a target is 700m/s, and the target and the missile move in a vertical plane. Assuming the initial moment of terminal guidance, the position of missile under inertial system is x m (0)＝0.5km，y m (0) =16 km, initial ballistic deflection of missile ofTarget initial position x t (0)＝12km，y t (0) =17.5 km, initial ballistic deflection of target +.>Towing bait length l=100, assuming a normal overload of the missile defined as 40g, g=9.8 m/s 2 。
Assume that the measurement noise of the line-of-sight angular rate is 0 as the mean and 1×10 as the variance -4 Is a cosine motor a of the target in the direction perpendicular to the line of sight tε =4gcos (pi t/4), the controller parameter is chosen to be k 1 ＝55、k 2 ＝65、k 3 ＝0.003、k 4 =0.003, γ=0.85, and σ=2.1, and simulation results are shown in fig. 2 to 7.
As can be seen from fig. 2, the missile intercepts the target precisely in a two-dimensional space. Fig. 3 is a graph of relative distance of movement of the bullet mesh, and the interception time of guidance is 15.21s, the off-target amount is 1.395m, so that the requirement of accurate guidance is met. Fig. 4 shows the angular velocity of the missile in the plane, and it can be seen that the angular velocity of the missile converges to near zero during the entire guidance process, which ensures that the missile can accurately intercept the target. As can be seen from fig. 5, the angles Δq all converge to around zero. Fig. 6 shows a planar missile overload curve, which can be seen to be smooth and steady after a short period of overload saturation. Fig. 7 tends to a steady state value in a shorter time. In summary, the designed backstepping sliding mode self-adaptive guidance law (27) has good stability and anti-interference performance and high-precision capability of intercepting maneuvering targets.
The embodiments of the present invention described above do not limit the scope of the present invention. Any modifications, equivalent substitutions and improvements made within the spirit principles of the present invention should be included in the scope of the claims of the present invention.
Claims (4)
1. An adaptive backstepping guidance law design method with anti-drag function, which is characterized by comprising the following steps:
firstly, establishing a model according to a prepositioning method and a parallel approaching method guiding thought;
secondly, aiming at the established model, considering uncertainty in the system, comprehensively utilizing a backstepping sliding mode and an adaptive method, and designing an adaptive backstepping sliding mode guidance law;
the implementation process of the step (I) is as follows:
in a two-dimensional plane, the missile, the target and the towing disturbance are regarded as particles, and are respectively represented by M, T and TRAD, and the connecting line of the missile and the target is the sight;
wherein r is the relative distance between the missile and the carrier target, L is the length of the towing line, q is the angle of sight, and q READ For the angle of view of the towing disturbance relative to the missile, deltaq is the angle between the towing disturbance and the missile link and between the towing disturbance and the missile link of the carrier,and->Speed direction angle, θ, of missile and carrier targets respectively m And theta t Leading angles of missile and carrier targets respectively, V m And V t Respectively representing the missile speed and the target speed of the carrier;
assuming that the longitudinal speed of the missile and the target longitudinal speed are constant, the model is described by the following differential equation:
Δq·r＝Lsinθ t (1)
q＝q TRAD -Δq (2)
deriving (1)
Deriving the formula (7) and substituting the formula (6) for finishing
Deriving (6)
Further arranged into
Wherein,and->The components of the acceleration of the missile and the target in the normal direction of the sight line are respectively;
based on the theory of the pre-measurement method and the parallel approach method, if deltaq andthe following relation is satisfied, and the accurate attack target of the missile is ensured:
wherein t is f The moment of intercepting the target for the missile;
defining a state variable as x 1 ＝Δq(t)，Combining the formula (7) and the formula (10)
Wherein,u＝a m for control input, d=g (t) is an unknown disturbance in the system and satisfies g (t). Ltoreq.d M ，d M Is a positive constant.
2. The method of claim 1, wherein a correlation mechanism is introduced:
lemma 1: for x i E R, i=1..n, real number p satisfies 0<p.ltoreq.1, the following inequality holds:
(|x 1 |+…+|x n |) p ≤|x 1 | p +…+|x n | p (14)
and (4) lemma 2: for a systemx (0) =0, f (0) =0, x∈r, assuming that there is a continuous microtransaction V, so that it satisfies the following condition:
(1) V is a positive definite function;
(2) There is a positive real number c>0 and alpha E (0, 1) and an open neighborhood containing the originMake->Establishment;
the system is stable for a finite time and the convergence time T satisfiesWherein V is 0 For an initial value of V, if u=u 0 ＝R n The system is globally stable for a limited time.
3. The method of claim 2, wherein the step (ii) is performed by:
aiming at the system models (12) and (13), the uncertainty in the system is considered, a backstepping sliding mode and an adaptive method are comprehensively utilized, and an adaptive backstepping sliding mode guidance law is designed, and the specific process is as follows:
step 1
Defining an error variable z 1 The following are listed below
z 1 ＝x 1 (15)
Deriving (15)
According to equation (15), the virtual control is designed to:
r 1 ＝(2-γ)η γ-1 (19)
r 2 ＝(γ-1)η γ-2 (20)
wherein 0 is<γ<1，α，β，k 1 ，k 2 And η is a positive constant;
selecting lyapunov function
For V 1 Deriving and substituting the formula (15) and the formula (17) into the order
When |x 1 |>Eta, formula (22) is arranged as
When |x 1 When the I is less than or equal to eta(22) Is arranged into
From the above demonstration, x is obtained 1 Converging to zero in a finite time;
step 2
The error variable s is defined as follows
Deriving (25)
In order to process disturbance d with unknown upper bound, based on a backstepping sliding mode control theory, a self-adaptive backstepping sliding mode guidance law is designed:
wherein,is d M Estimate of k 3 ，k 4 And h is a normal number, delta>1。
4. A method according to claim 3, characterized in that for systems (12) - (13), d is bounded but the upper bound is unknown, the following conclusion is reached with the adaptive backstepping sliding mode guidance laws (27) - (28);
(1) The slip plane s is convergent for a finite time;
(2) State x of the system 1 And z 2 Is of limited time convergence.
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| CN112346474B (en) * | 2020-10-20 | 2021-12-07 | 南京航空航天大学 | Design method of differential game guidance law with limited time convergence |
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| CN106934120A (en) * | 2017-02-23 | 2017-07-07 | 哈尔滨工业大学 | Three-dimensional guidance based on preceding from the interception hypersonic aircraft to guidance restrains method for designing |
| CN108416098A (en) * | 2018-02-02 | 2018-08-17 | 北京航空航天大学 | A kind of attack time constraint Design of Guidance Law method intercepting maneuvering target |
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| CN106934120A (en) * | 2017-02-23 | 2017-07-07 | 哈尔滨工业大学 | Three-dimensional guidance based on preceding from the interception hypersonic aircraft to guidance restrains method for designing |
| CN108416098A (en) * | 2018-02-02 | 2018-08-17 | 北京航空航天大学 | A kind of attack time constraint Design of Guidance Law method intercepting maneuvering target |
| CN109557814A (en) * | 2018-12-04 | 2019-04-02 | 四川航天***工程研究所 | A kind of finite time Integral Sliding Mode Terminal Guidance Laws |
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