CN110929216B - Self-adaptive backstepping guidance law design method with anti-drag function - Google Patents

Abstract

The present invention proposes an adaptive backstepping guidance law design method with anti-drag. The method includes the following steps: (1), establishing a model based on the guidance ideas of the lead quantity method and the parallel approach method; (2), aiming at establishing model, considering the uncertainty in the system, and comprehensively utilizing the back-stepping sliding mode and adaptive methods to design an adaptive back-stepping sliding mode guidance law. Based on the guidance ideas of the lead quantity method and the parallel approach method, this invention proposes an adaptive back-step guidance law design method with anti-drag. It is a new type of back-step sliding mode guidance strategy. Its lead method ensures that the missile The guidance trajectory is an appropriate angle ahead of the towed decoy, and the parallel approach method ensures that the missile can accurately intercept the target.

Description

An adaptive back-step guidance law design method with anti-drag

Technical field

The invention relates to an adaptive back-step guidance law design method with anti-drag capability.

Background technique

In modern warfare, with the rapid development of various electronic jamming technologies, the survivability of carrier aircraft has been greatly improved. As a new type of self-defense jamming method, towed decoys have become an important means of combating air defense missile radar seekers and have been successfully used in wars. It makes use of radar seeker anti-aircraft missiles to intercept aircraft-borne targets with towed decoys facing severe challenges. At present, most of the literature mainly focuses on the interference mechanism analysis and performance analysis of towed decoys, but there are relatively few research results on the terminal guidance law of anti-drag decoys. Among nonlinear control theories, sliding mode control theory has good robustness to parameter perturbations inside the system and external disturbances, making it a rich research achievement in guidance law design.

Contents of the invention

Based on the guidance ideas of the lead quantity method and the parallel approach method, this invention proposes an adaptive back-step guidance law design method with anti-drag. It is a new type of back-step sliding mode guidance strategy. Its lead method ensures that the missile The guidance trajectory is an appropriate angle ahead of the towed decoy, and the parallel approach method ensures that the missile can accurately intercept the target.

In order to achieve the above objects, the technical solution of the present invention is:

An adaptive back-step guidance law design method with anti-drag, the method includes the following steps:

(1) Establish a model based on the guidance ideas of the pre-measurement method and the parallel approach method;

(2) Based on the established model, considering the uncertainty in the system, and comprehensively utilizing the back-stepping sliding mode and adaptive methods, an adaptive back-stepping sliding mode guidance law is designed.

The beneficial effects of the present invention are: the back-stepping sliding mode adaptive guidance law designed by the present invention has good stability and anti-interference properties, and has the ability to intercept maneuvering targets with high precision.

Description of drawings

Figure 1 is a relative motion relationship diagram with towed interception.

Figure 2 is a schematic diagram of the missile-target tracking trajectory.

Figure 3 is a schematic diagram of the relative distance between projectiles.

Figure 4 is a schematic diagram of the line of sight angular rate.

Figure 5 is a schematic diagram of the sight angle.

Figure 6 is a schematic diagram of the normal overload of the missile.

Figure 7 is a schematic diagram of the adaptive parameter change curve.

Detailed ways

The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

In the two-dimensional plane, the relative motion relationship between the missile and the target with drag interference is shown in Figure 1. The missile, target and drag interference are all regarded as particles, and are represented by M, T, and TRAD respectively. The line connecting the missile and the target That is the line of sight, and the other parameters are indicated in the figure.

Among them, r is the relative distance between the missile and the aircraft target, L is the length of the tow line, q is the line of sight angle, q _READ is the line of sight angle of the tow interference relative to the missile, Δq is the line between the tow interference and the missile and the tow interference and The angle between the aircraft-carrying missile line, and/> are the velocity direction angles of the missile and the aircraft target respectively, θ _m and θ _t are the lead angles of the missile and the aircraft target respectively, V _m and V _t represent the longitudinal speed of the missile and the longitudinal speed of the aircraft target respectively.

Assume that the missile longitudinal velocity V _m and the target longitudinal velocity V _t are both constant values. According to the geometric relationship in Figure 1, they can be described by the following differential equation:

Δq·r＝ _Lsinθt (1)

q＝q _TRAD -Δq (2)

By deriving formula (1), we can get

By deriving the derivative of equation (7) and substituting it into equation (6), we can get

By deriving equation (6), we can get

further organized into

in, and/> are the components of missile and target acceleration in the line of sight normal direction respectively.

According to the guidance method of preposition method and parallel approach method, if Δq and Satisfying the following relationship can ensure that the missile accurately attacks the target:

Among them, t _f is the time when the missile intercepts the target.

Define the state variable as x ₁ =Δq(t), Combining equation (7) and equation (10), we can get

in, u= _am is the control input, d=g(t) is the unknown disturbance in the system and satisfies g(t)≤d _M , and d _M is a positive constant.

3. Related mechanisms

Lemma 1: For x _i ∈R, i=1,...,n, and the real number p satisfies 0<p≤1, then the following inequality holds:

(|x ₁ |+…+|x _n |) ^p ≤|x ₁ | ^p +…+|x _n | ^p (14)

Lemma 2: For the system x(0)=0, f(0)=0, x∈R, assuming there is a continuous differentiable function V such that it satisfies the following conditions:

(1)V is a positive definite function.

(2) There are positive real numbers c>0 and α∈(0,1), and an open neighborhood including the origin. make established.

Then the system is stable in finite time, and the convergence time T satisfies Where V ₀ is the initial value of V. If U=U ₀ =R ⁿ , the system is globally finite-time stable.

4. Adaptive back-stepping sliding mode guidance law design

For system models (12) and (13), considering the uncertainty in the system and comprehensively utilizing the back-stepping sliding mode and adaptive methods, an adaptive back-stepping sliding mode guidance law is designed. The specific process is as follows:

step 1

Define the error variable z ₁ as follows

z ₁ = x ₁ (15)

By deriving equation (15), we can get

According to equation (15), the virtual control design is

r ₁ =(2-γ)η ^γ-1 (19)

r ₂ =(γ-1)η ^γ-2 (20)

Among them, 0<γ<1, α, β, k ₁ , k ₂ and eta are positive constants.

Choose Lyapunov function

Derive V ₁ and substitute equations (15) and (17) into the equations to get

When |x ₁ |＞η, equation (22) can be organized as

When |x ₁ |≤η, equation (22) can be organized as

From the above proof, we can get that x ₁ converges to zero in a finite time.

Step 2

Define the error variable s as follows

By taking the derivative of (25), we can get

In order to deal with the disturbance d with an unknown upper bound, based on the back-stepping sliding mode control theory, an adaptive back-stepping sliding mode guidance law is designed:

in, is the estimated value of d _M , k ₃ , k ₄ and h are positive constants, δ>1.

Theorem 1: For systems (12)-(13), when d is bounded but the upper bound is unknown, the following conclusion can be obtained by using the adaptive back-stepping sliding mode guidance law (27)-(28).

(1) The sliding mode surface s converges in finite time.

(2) The states x ₁ and z ₂ of the system converge in finite time.

Proof: Select the Lyapunov function as

in,

By deriving equation (29) along the system trajectory, we can get

From formula (30) we can get V ₂ is not increasing, s and/> is bounded, therefore, it can be obtained that the adaptive estimation error d _M and s are both bounded.

Choose Lyapunov function

By deriving equation (31), we can get

Substituting controllers (27)-(28) into equation (32), we get

because and/> Yes/> Select/> The value of is large enough and σ satisfies

combine can be obtained

Case 1. When |x ₁ |＞η and s≠0, equation (33) can be organized as

Case 2. When |x ₁ |＞η and s≡0, equation (33) can be organized as

It can be seen from equation (37) that system x ₁ converges in finite time.

Case 3. When |x ₁ |＞η and When, substituting controller (27) into equation (13) can be organized as

so It is not an attractor, and the system sliding mode can be guaranteed to converge to zero in a limited time.

Case 4: When |x ₁ |≤η, and when s≠0, equation (33) can be organized as

Case 5: When |x ₁ |≤η, and s≡0

According to equation (40), system x ₁ converges in finite time.

Case 6: When |x ₁ |≤η and Substituting controller (27) into equation (13) can be organized as

so It is not an attractor, and the system sliding mode can be guaranteed to converge to zero in a limited time.

To sum up, the system converges in a finite time. It can further be obtained that the sliding mode surfaces s and x ₁ converge in finite time.

So far (1) has been proved.

Because the sliding mode surface s and x ₁ converge in finite time, according to equation (25), it can be obtained that z ₂ converges in finite time.

So far (2) is proved.

Theorem 1 is proved.

Simulation analysis

In order to verify the effectiveness of the adaptive finite-time controller (27) based on the back-stepping sliding mode, assume that a certain missile is flying at a certain altitude, the Mach number is 4.5, the speed of sound is 295.07m/s, and the target’s flight speed is 700m/s. s, the target and missile move in the vertical plane. Assume that at the initial moment of guidance, the missile's position in the inertial frame is x _m (0) = 0.5km, y _m (0) = 16km, and the missile's initial ballistic deflection angle is The initial position of the target is x _t (0) = 12km, y _t (0) = 17.5km, and the initial ballistic deflection angle of the target is/> The length of the towed decoy is L=100, assuming that the normal overload of the missile is limited to 40g, g=9.8m/s ² .

Assume that the measurement noise of the line of sight angular rate is Gaussian white noise with a mean value of 0 and a variance of 1×10 ^-4 . The target performs a cosine maneuver a _tε = 4gcos (πt/4) in the direction perpendicular to the line of sight. The controller parameters are selected as k ₁ =55, k ₂ =65, k ₃ =0.003, k ₄ =0.003, γ =0.85 and σ =2.1. The simulation results are shown in Figure 2-7.

As can be seen from Figure 2, the missile accurately intercepts the target in two-dimensional space. Figure 3 shows the relative distance curve of missile-target movement. It can be seen that the guidance and interception time is 15.21s, and the miss distance is 1.395m, which meets the needs of precise guidance. Figure 4 shows the line of sight angular rate curve of the missile in the plane. It can be seen that the line of sight angular rate curve converges to near zero during the entire guidance process, which ensures that the missile can accurately intercept the target. It can be seen from Figure 5 that the included angle Δq converges to near zero. Figure 6 shows the plane missile overload curve. It can be seen that after overload saturation in a short period of time, the curve is smooth and stable. Figure 7 tends to a steady-state value over a shorter period of time. In summary, it can be seen that the designed back-stepping sliding mode adaptive guidance law (27) has good stability and anti-interference performance, and has the ability to intercept maneuvering targets with high precision.

The above-described embodiments of the present invention do not limit the scope of the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention shall be included in the scope of protection 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.