CN105674804B - A kind of sky comprising normal acceleration derivative penetrates Cruise Missile downslide section multiple constraint method of guidance - Google Patents

Abstract

Cruise Missile downslide section multiple constraint method of guidance is penetrated the invention discloses a kind of sky comprising normal acceleration derivative, is specifically included：First, determine missile aerodynamic parameter and set up the parameter of Atmospheric models；2nd, the parameter according to missile aerodynamic parameter and Atmospheric models sets up motion equation of a missile group；3rd, linearization process is carried out to motion equation of a missile group；4th, normal direction acceleration derivative equation will be added in linearization process equation group, obtains Analytical Solution equation group；5th, the downslide section trajectory end-fixity and path constraint to be met is determined；6th, optimum control amount u is determined according to Analytical Solution equation group and optimal control index function；7th, solve optimum control amount u to obtain parsing Guidance Law, guided missile downslide section trajectory is met end-fixity and path constraint.Advantage is：Guided missile be ensure that while regulation end height is reached, the constraint of trajectory tilt angle, speed, normal acceleration and normal acceleration first derivative is met, the flight quality of guided missile is improved.

Description

A kind of sky comprising normal acceleration derivative penetrates Cruise Missile downslide section multiple constraint guidance Method

Technical field

The invention belongs to aeronautical and space technology, weapon field, it is related to ALCM downslide section multiple constraint Guidance Law Analytic solutions, specifically refer to a kind of sky comprising normal acceleration derivative and penetrate Cruise Missile downslide section multiple constraint method of guidance.

Background technology

ALCM refers to that, from air-launched cruise missile, its trajectory includes downslide section, flat winged section and underriding Section.Downslide section not only needs to meet multinomial end state constraint and (mainly including end height, trajectory tilt angle, speed and attacks normal direction Acceleration is constrained), the downslide section time is also shortened as far as possible, to improve disguise.

Traditional method of guidance is path tracking guidance, and normal trajectory used can be planned by project trajectory and trajectory is excellent The methods such as change are obtained；The design of Guidance Law of normal trajectory is tracked usually using methods such as PID control, feedback linearizations.Tracking The advantage of guidance is relatively easy Design of Guidance Law, and shortcoming is excessively to rely on normal trajectory, and the precision under large disturbances It is relatively poor.

The content of the invention

For in the prior art, guided missile downslide section is unable to rapid decrease to the present invention, and guided missile downslide section end can not put down The problems such as steady pull-up；Propose a kind of sky comprising normal acceleration derivative and penetrate Cruise Missile downslide section multiple constraint method of guidance, protect Guided missile has been demonstrate,proved while regulation end height is reached, trajectory tilt angle, speed, normal acceleration and normal acceleration one has been met The constraint of order derivative, improves the flight quality of guided missile.

Comprise the following steps that：

Step one, missile aerodynamic parameter is determined according to ALCM profile, while setting up the parameter of Atmospheric models；

Missile aerodynamic parameter includes：The lift coefficient C of guided missile_LWith the resistance coefficient C of guided missile_D

α is the guided missile angle of attack, and M is Mach number, and calculating formula isWherein V is the speed of guided missile relative atmospheric, V_sIt is to work as The ground velocity of sound.

The parameter of Atmospheric models includes atmospheric density ρ (units：kg/m³) and local velocity of sound V_s(unit：Metre per second (m/s))；

H is the flying height (unit residing for guided missile：Rice)；

Step 2, motion equation of a missile group is set up according to the parameter of missile aerodynamic parameter and Atmospheric models；

Motion equation of a missile group is as follows：

Wherein, t is the flight time of guided missile, and θ is trajectory tilt angle, and x is the horizontal flight distance of guided missile, and m is guided missile quality, G is acceleration of gravity；L and D are respectively the lift of guided missile and the resistance of guided missile, and calculating formula is as follows：

S is the area of reference of guided missile.

Step 3, linearization process is carried out to motion equation of a missile group, obtain linearization process equation group；

The linearization process of motion equation of a missile group is included：

(1) normal acceleration a is introduced_n：

(2) it is constant to set lift-drag ratio K：

(3) speed simplification is processed：The real-time speed V of guided missile in motion equation of a missile group is set to constant V₀；

(4) remaining time is introduced：The flight time t of guided missile replaces with residual non-uniformity t_go, t_goRelation with t is as follows：

t_go=t_f-t

t_fIt is the total flight time of guided missile；It is unknown constant；t_goCalculated by the correlation formula in step 7；

Obtain linearization process equation group：

Step 4, will in linearization process equation group add normal direction acceleration derivative equation, obtain Analytical Solution equation group；

Normal acceleration first derivative equation is：Normal acceleration second dervative equation is：

WhereinIt is the first derivative of normal acceleration,It is the second dervative of normal acceleration.

Analytical Solution equation group is as follows：

In last formulaRepresent handleAs controlled quentity controlled variable；

Step 5, guided missile determine the downslide section trajectory end-fixity and path constraint to be met when gliding；

End-fixity refers to guided missile in the downslide section distal point constraint to be met；Including end height constraint, constraint of velocity, Trajectory tilt angle constraint, normal acceleration constraint and end normal acceleration first derivative constraint；

It is specific to set as follows：

Wherein, h (t₀)=h₀Represent t=t₀The height at moment is h₀, h (t_f)=h_fRepresent in t=t_fThe height at moment is h_f, θ (t₀)=0 represents t=t₀The trajectory tilt angle at moment is 0；θ(t_f)=0 represents t=t_fThe trajectory tilt angle at moment is 0, x (t₀) =0 represents t=t₀The horizontal flight distance of moment guided missile is 0, x (t_f)=free represents t=t_fThe horizontal flight of moment guided missile away from From without constraint, V (t₀)=V₀Represent t=t₀The speed of moment guided missile relative atmospheric is constant V₀, V (t_f)=V_fRepresent t=t_fWhen The speed for carving guided missile relative atmospheric is V_f, a_n(t₀)=a_n0Represent t=t₀Moment guided missile normal acceleration is a_n0, a_n(t_f)=a_nf Represent t=t_fMoment guided missile normal acceleration is a_nf,Represent t=t₀The first derivative of moment normal acceleration is Represent t=t_fThe first derivative of moment normal acceleration is

Path constraint mainly speed and the normal acceleration constraint including guided missile, specific mathematical expression form is as follows：

Wherein V_min, V_max, a_nminAnd a_nmaxDesign parameter according to specific guided missile is determined.V_minRepresent missile flight speed appearance Perhaps minimum value, V_maxRepresent the maximum that missile flight speed is allowed, a_nminRepresent the minimum that guided missile normal acceleration is allowed Value, a_nmaxRepresent the maximum that guided missile normal acceleration is allowed.

Step 6, the target function for choosing optimum control, optimal control is determined according to Analytical Solution equation group and target function Amount u processed；

Selection energy hole is target function J：

Controlled quentity controlled variable u control guided missile downslide section trajectories are smoothed, and shorten the downslide section time.

Step 7, solution optimum control amount u obtain parsing Guidance Law, guided missile downslide section trajectory is met end-fixity and road Footpath constrains.

Based on the minimal principle in the theory of optimal control, Analytical Solution is carried out to optimal control problem, obtain parsing system Leading rule is：

U=16x₁-120x₂+480x₃-840x₄

Wherein,

Advantages of the present invention is with good effect：

(1) a kind of sky comprising normal acceleration derivative penetrates Cruise Missile downslide section multiple constraint method of guidance, and rule is led in proposition Compared with existing Guidance Law, the downslide section flight time is adjusted by changing initial normal acceleration, meet different task need Ask.

(2) a kind of sky comprising normal acceleration derivative penetrates Cruise Missile downslide section multiple constraint method of guidance, and rule is led in proposition Compared with existing Guidance Law, can artificially change initial normal acceleration, and constraint is applied to end normal acceleration derivative, So that trajectory downslide amplitude is larger in the early stage, end pull-up trajectory is gentler, and downslide section rapid decrease and end has been better achieved Hold the requirement of steady pull-up level with both hands.

Brief description of the drawings

Fig. 1 is design flow diagram of the invention；

Fig. 2 is the outline drawing of battleax cruise missile；

Fig. 3 is ballistic curve schematic diagram of the analytic solutions under different release altitudes；

Fig. 4 is rate curve schematic diagram of the analytic solutions under different release altitudes；

Fig. 5 is trajectory tilt angle curve synoptic diagram of the analytic solutions under different release altitudes；

Fig. 6 is curve synoptic diagram of the analytic solutions under different release altitudes；

Fig. 7 is normal acceleration curve synoptic diagram of the analytic solutions under different release altitudes；

Fig. 8 is normal acceleration first derivative curve synoptic diagram of the analytic solutions under different release altitudes；

Fig. 9 is lift-drag ratio curve synoptic diagram of the analytic solutions under different release altitudes；

Figure 10 is the ballistic curve contrast schematic diagram of analytic solutions and optimal solution；

Figure 11 is the rate curve contrast schematic diagram of analytic solutions and optimal solution；

Figure 12 is the trajectory tilt angle curve comparison schematic diagram of analytic solutions and optimal solution；

Figure 13 is the normal acceleration curve comparison schematic diagram of analytic solutions and optimal solution；

Figure 14 is the angle of attack curve comparison schematic diagram of analytic solutions and optimal solution；

Figure 15 is the normal acceleration first derivative curve comparison schematic diagram of analytic solutions and optimal solution；

Figure 16 is the lift-drag ratio curve comparison schematic diagram of analytic solutions and optimal solution；

Figure 17 is the contrast schematic diagram of the present invention and the trajectory smoothness of prior art；

Figure 18 is the flight time contrast schematic diagram of the present invention and prior art.

Specific embodiment

Below in conjunction with accompanying drawing, the present invention is described in further detail.

Explicit Guidance strategy based on optimum control is the fresh approach for solving downslide section guidance problems, mainly including broad sense Explicit Guidance rule, segmentation optimal guidance law etc., with strong adaptability, robustness is good the characteristics of；The present invention is in the aobvious of optimum control On the basis of formula Guidance, using in the equation of motion add normal acceleration and its derivative equation, introduce normal acceleration and Its first derivative is constrained, and uses Analytic Method, based on the theory of optimal control, is solved sky using Optimal Control Model and is penetrated cruise Guided missile downslide section will meet the control problem of end-fixity, on the basis of multiple constraint Guidance Law, add normal acceleration single order to lead Number constraint, can simultaneously meet the constraint of terminal height, speed, trajectory tilt angle, normal acceleration and normal acceleration first derivative Multiple constraint Guidance Law.

The correlation theory of optimum control is as follows：

Optimum control belongs to the category of the calculus of variations, and the motion of system is described with following Nonlinear differential eguations：

Wherein, f [x (t), u (t), t] represents that f is a nonlinear function of x (t), u (t) and t.

Function of state x (t)=(x₁,x₂,…,x_n) it is n-dimensional vector function, control function u (t)=(u₁,u₂,…,u_n) it is m The vector function of dimension.

The optimum control problem to be solved is exactly to find optimal control function u (t) so that performance index function J is minimum：

Wherein, L [x (t), u (t), t] represents the Lagrangian item in target function, and its physical meaning is the shape to during The requirement of state amount x (t) and controlled quentity controlled variable u (t)；φ[x(t_f),t_f)] it is Mayer in target function, represent to end state With the requirement of end time.

The condition that can be derived optimum control function and should meet with reference to the relevant knowledge of the calculus of variations is

Wherein λ^TSubscript T represent transposition, H is Hamiltonian function, and its expression formula is：

H=L [x, u, t]+λ^T(t)f(x,u,t) (4)

It is the optimum control function necessary condition to be met, commonly known as impulse function (impulse Functions), λ is a vector function for n dimensions, is referred to as influence function (influence functions).

If x_iIn free terminal, then corresponding to boundary condition is,

If x_iIn terminal Constrained, then corresponding to boundary condition is,

x_i(t_f)=x_itf (6)

The subscript i=1,2 of x and λ ..., n；

Optimum control function u (t) is gone out according to above differential equation group and Boundary Condition for Solving.

A kind of sky comprising normal acceleration derivative penetrates Cruise Missile downslide section multiple constraint method of guidance, as shown in figure 1, specifically Step is as follows：

Step one, missile aerodynamic parameter is determined according to ALCM profile, while setting up the parameter of Atmospheric models；

Guided missile flies in atmosphere, and downslide section guided missile is in non-power state, and thrust is zero；First according to guided missile Profile determine the aerodynamic parameter of guided missile, resettle the model that atmospheric density and local velocity of sound change with flying height.

Atmospheric models use U.S.1976Standard Atmosphere model, describe atmospheric density and local sound The fast rule with height change.

The parameter of Atmospheric models includes atmospheric density ρ and local velocity of sound V_s；

Because cruise missile release altitude is general in below 10000m, in the altitude range, as height increases, air is close (unit is kg/m to degree ρ³) exponentially rule decline, local velocity of sound V_s(unit is m/s) linear decline；

H is the flying height residing for guided missile (unit is rice)；

Missile aerodynamic parameter

As shown in Fig. 2 with battleax cruise missile AGM-109 as prototype, its missile aerodynamic parameter is estimated with datcom softwares, And least square fitting obtains lift coefficient C_LWith resistance coefficient C_DApproximate formula：

α is the guided missile angle of attack (unit：Degree), M is Mach number (being dimensionless number), and the calculating formula of M isWherein V is The speed of guided missile relative atmospheric, V_sIt is local velocity of sound.

Step 2, motion equation of a missile group is set up according to the parameter of missile aerodynamic parameter and Atmospheric models；

The flight of guided missile is described by motion equation of a missile, guided missile is regarded as particle when Guidance Law is studied, for equation shape Formula it is succinct, the equation of motion of the guided missile under ballistic coordinate system is set up in perpendicular.

Consider motion of the guided missile in perpendicular, the equation of motion of downslide section is set up under ballistic coordinate system：

Wherein, t is the flight time of guided missile, and θ is trajectory tilt angle, and x is the horizontal flight distance of guided missile, and m is guided missile quality, G is acceleration of gravity；L and D are respectively the lift of guided missile and the resistance of guided missile, and calculating formula is as follows：

S is the area of reference of guided missile.

Step 3, linearization process is carried out to motion equation of a missile group, obtain linearization process equation group；

In order that equation can Analytical Solution, it is necessary to carry out a series of simplified treatment to equation, movement difference equations it is linear Change treatment and include the following steps：

1) normal acceleration a is introduced_n：

2) it is constant to set lift-drag ratio K：

The flight of guided missile downslide section is in free skating Xiang state, and the symbol of lift-drag ratio is constant, and size variation is not violent；For side Just solve and approximately regard constant K as, i.e.,

Bring formula (11) and (12) simultaneous into equation group (9), eliminate D and L.

3) speed simplification is processed

In solution procedure, the speed V in equation group (9) on the right of equation can approximately regard constant V as₀, in order to just In the Analytical Solution of controlled quentity controlled variable u.In fact, it is conceivable that, contain constant V in parsing solution's expression₀If, in final knot Real-time speed V is substituted in fruit, Guidance Law will progressively correct the error brought by solution procedure medium velocity simplification treatment.

4) remaining time is introduced：

The flight time t of guided missile replaces with residual non-uniformity t_go, t_goRelation with t is as follows：

t_go=t_f-t (13)

t_fIt is the total flight time of guided missile；It is unknown constant；t_goCalculated by the correlation formula in step 7；

Obtain linearization process equation group：

Step 4, will in linearization process equation group add normal direction acceleration derivative equation, obtain Analytical Solution equation group；

Supplemented in the movement difference equations obtained in step 3 and add two equations：Normal acceleration first derivative equation For：It is with normal acceleration second dervative equation：And willAs controlled quentity controlled variable u.

WhereinIt is the first derivative of normal acceleration,It is the second dervative of normal acceleration.

By in the equation group after normal acceleration derivative equation addition linearization process so that the single order of normal acceleration is led Number meets constraint.Second dervative as controlled quentity controlled variable, each equation correspondence one end-fixity, introduce second dervative equation be for The constraint of addition first derivative.

Analytical Solution equation group such as following formula：

In last formulaRepresent handleAs controlled quentity controlled variable；

Step 5, guided missile determine the downslide section trajectory end-fixity and path constraint to be met when gliding；

End-fixity is smoothly transferred to cruising phase for guided missile and provides advantage, and path constraint ensure that during downslide The reliability of missile flight.The end-fixity of downslide section refer to guided missile in downslide section distal point, to meet highly constrained, trajectory and incline Angle constraint, constraint of velocity, normal acceleration constraint and normal acceleration first derivative reach specific value.

Wherein, h (t₀)=h₀Represent t=t₀The height at moment is h₀, h (t_f)=h_fRepresent in t=t_fThe height at moment is h_f, θ (t₀)=0 represents t=t₀The trajectory tilt angle at moment is 0；θ(t_f)=0 represents t=t_fThe trajectory tilt angle at moment is 0, x (t₀) =0 represents t=t₀The horizontal flight distance of moment guided missile is 0, x (t_f)=free represents t=t_fThe horizontal flight of moment guided missile away from From without constraint, V (t₀)=V₀Represent t=t₀The speed of moment guided missile relative atmospheric is constant V₀, V (t_f)=V_fRepresent t=t_fWhen The speed for carving guided missile relative atmospheric is V_f, a_n(t₀)=a_n0Represent t=t₀Moment guided missile normal acceleration is a_n0, a_n(t_f)=a_nf Represent t=t_fMoment guided missile normal acceleration is a_nf,Represent t=t₀The first derivative of moment normal acceleration is Represent t=t_fThe first derivative of moment normal acceleration is

According to end normal acceleration for zero condition can solve needed for angle of attack_fValue, in movement difference equations (9) Made in second formula

That is,

α can be solved_f。

Path constraint mainly speed and the normal acceleration constraint including guided missile, specific mathematical expression form is as follows：

V_minAnd V_maxDesign parameter decision according to specific guided missile, V_minThe minimum value that missile flight speed is allowed is represented, V_maxRepresent the maximum that missile flight speed is allowed, a_nminRepresent the minimum value that guided missile normal acceleration is allowed, a_nmaxRepresentative is led Play the maximum that normal acceleration is allowed；Such as the subsonic cruise missile to, V is can use_max=300m/s, V_min=150m/ s；a_nminAnd a_nmaxAlso to be determined according to the design parameter of guided missile, typically can use a_nmin=-10g, a_nmax=10g, herein g attach most importance to Power acceleration.

Step 6, the target function for choosing optimum control, optimal control is determined according to Analytical Solution equation group and target function Amount u processed；

In order that trajectory is more smoothed, gliding speed is very fast, from energy hole as target function, i.e.,

Step 7, solution optimum control amount u obtain parsing Guidance Law, guided missile downslide section trajectory is met end-fixity and road Footpath constrains.

The equation of motion, constraints and the target function determined according to step 4 to step 6 are assured that controlled quentity controlled variable u.

Guidance Law concrete form is shown in formula (44) and (45)；The following is solution procedure：

The quantity of state of system is respectively：V, θ, x, h, a_n,

According to the solution procedure of the theory of optimal control, vector function λ is write as component form and is：

Hamiltonian function is,

WhereinRepresent Represent Represent Represent Represent RepresentTool Body expression formula is shown in the equation of motion (15) for Analytical Solution.

Euler-Lagrange equation：

Wherein c₁,c₂,c₃,c₄Represent constant.

By last formula in (22)Can obtain,

Due to end x freedom, then there is boundary condition,

With reference to the 3rd formula λ in (22)_x=c₃Can obtain,

λ_x=0 (25)

By the second formula in (22),

λ_θ=c₂t_go+C_θ(C_θIt is constant) (26)

Bring formula (26) into (22) the 5th formula

Integrate,

Bring formula (28) into (22) the 6th formulaControlled quentity controlled variable u can be obtained,

Integrate：

It is obvious that u is t_goCubic function, and the coefficient of each item is separate, therefore, u is written as form,

U=C₃t_go ³+C₂t_go ²+C₁t_go+C₀ (31)

C₃,C₂,C₁,C₀It is separate undetermined constant.

Following formula can be obtained by one, two, four formulas in equation group (22),

Integration is obtained,

Arrange,

T is obtained_goExpression formula on guided missile quantity of state；

t_goWith V₀It is relevant, it is designated as t_go=t_go(V₀)。

Bring u into full scale equation groups (22) to be integrated, respectively obtain,

Arrange,

Write as matrix form,

A·T_go=X (37)

Wherein,

Contain initial value V in X₀, X=X (V can be write₀)。

T is solved by formula (37)_go,

T_go=A^-1·X (41)

Wherein,

Formula (41) is brought into formula (31) to obtain,

By the V in X₀The current speed V of guided missile is replaced with, x=X (V) is designated as, by t_goIn V₀Replace with guided missile current Speed V, is designated as τ_go=t_go(V).Obtain,

U=16x₁-120x₂+480x₃-840x₄ (44)

Wherein,

Bringing formula (44) and (45) into following equation group carries out numerical simulation checking.

Numerical simulation checking is carried out respectively to different guided missile release altitudes (such as 4000,3000 and 2000 meters), as a result such as Shown in Fig. 3-Fig. 9, with increasing for distance, the velocity variations amplitude of guided missile increases, and the increase of trajectory tilt angle amplitude of variation such as exists Under 2000m height, guided missile maximal rate in below 245m/s, maximum trajectory tilt angle at 15 degree or so, and under 4000m height, , in more than 270m/s, maximum trajectory tilt angle is more than 20 degree for missile velocity.The tip speed of guided missile, trajectory tilt angle, normal direction accelerate The constraint of degree and normal acceleration first derivative is satisfied by.

Case study on implementation：

In order to check the feasibility and service behaviour of required Guidance Law, table 1 gives the initial strip for carrying out numerical simulation Part and end-fixity；

Table 1

The end-fixity emulated under 2000m release altitudes meets situation and is shown in Table 2, and remaining height is similar to therewith, end shape The deviation of state is very small, and this illustrates that the Guidance Law can well meet the requirement of end-fixity.

Table 2

In addition, the optimality in order to verify the analytic solutions, the optimal solution of numerical value is carried out using Gauss puppet spectrometry, and by number Value optimal solution is contrasted with analytic solutions, and comparing result is shown in Figure 10-Figure 16, it is seen then that the analytic solutions connect with numerical value optimal solution Short range degree is very good.The advantage of analytic solutions is that current state generation that can be in real time according to guided missile is guidanceed command, and is taken non- Often few, occupancy CyberSpace is also very small, is adapted to be used on bullet；And numerical method is time-consuming very long, lower online can only complete, no Optimum instruction can be in real time generated, there is certain limitation in practical application.

By the way that (Hu Jinchuan, Chen Wanchun have the ALCM downslide section multiple constraint Guidances of igniting window limit with prior art [J], flight mechanics, 2015) Guidance Law contrasted：As shown in Figure 17 and Figure 18, this Guidance Law is in identical initial method In the case of to acceleration, glided faster in initial segment, pull-up section in end is more steady, and the flight time is shorter, can more preferably expire Foot is quick to glide, the requirement of steady pull-up.

The present invention single order and second dervative equation supplemented with normal acceleration in traditional motion equation of a missile, introduce Constraint to normal acceleration and its first derivative；By rational linearization process, and based on the theory of optimal control, draw The constraint of downslide section end height, speed, trajectory tilt angle, normal acceleration and normal acceleration first derivative can simultaneously be met The Guidance Law of analytical form.

Claims (3)

1. a kind of sky comprising normal acceleration derivative penetrates Cruise Missile downslide section multiple constraint method of guidance, it is characterised in that including Following steps：

Step one, missile aerodynamic parameter is determined according to ALCM profile, while setting up the parameter of Atmospheric models；

Missile aerodynamic parameter includes：The lift coefficient C of guided missile_LWith the resistance coefficient C of guided missile_D；

$\left\{\begin{array}{c}{C}_L=0.07907\mathrm{\α}+0.08512\\ C_D=7.852\×{10}^{-4}{\mathrm{\α}}^2+9.029\×{10}^{-4}\mathrm{\α}+0.2818M^2-0.2636M+0.08394\end{array}\right.$

α is the guided missile angle of attack, and M is Mach number, and calculating formula isWherein V is the speed of guided missile relative atmospheric, V_sIt is local sound Speed；

The parameter of Atmospheric models includes atmospheric density ρ and local velocity of sound V_s：

$\left\{\begin{array}{c}\mathrm{\ρ}=1.2353\×e^{-1.06\×{10}^{-4}h}\\ V_s=340.63-4.076\×{10}^{-3}h\end{array}\right.$

H is the flying height residing for guided missile；

Step 2, motion equation of a missile group is set up according to the parameter of missile aerodynamic parameter and Atmospheric models；

Motion equation of a missile group is as follows：

$\left\{\begin{array}{c}\frac{dV}{dt}=-gsin\mathrm{\θ}-\frac{D}{m}\\ \frac{d\mathrm{\θ}}{dt}=-\frac{g\cos \mathrm{\θ}}{V}+\frac{L}{mV}\\ \frac{dx}{dt}=V\cos \mathrm{\θ}\\ \frac{dh}{dt}=V\sin \mathrm{\θ}\end{array}\right.$

Wherein, t is the flight time of guided missile, and θ is trajectory tilt angle, and x is the horizontal flight distance of guided missile, and m is guided missile quality, and g is Acceleration of gravity；L and D are respectively the lift of guided missile and the resistance of guided missile, and calculating formula is as follows：

$\left\{\begin{array}{c}L=L(V,\mathrm{\α},h)=\frac{1}{2}\mathrm{\ρ}{V}^2{C}_{L}S\\ D=D(V,\mathrm{\α},h)=\frac{1}{2}\mathrm{\ρ}{V}^2{C}_{D}S\end{array}\right.$

S is the area of reference of guided missile；

Step 3, linearization process is carried out to motion equation of a missile group, obtain linearization process equation group；

Specially：

The linearization process of motion equation of a missile group is included：

(1) normal acceleration a is introduced_n：

(2) it is constant to set lift-drag ratio K：

(3) speed simplification is processed：The real-time speed V of guided missile in motion equation of a missile group is set to constant V₀；

(4) remaining time is introduced：The flight time t of guided missile replaces with residual non-uniformity t_go, t_goRelation with t is as follows：t_go= t_f-t；

t_fIt is the total flight time of guided missile, is unknown constant；t_goCalculated by the correlation formula in step 7；

Obtain linearization process equation group：

$\left\{\begin{array}{c}\frac{dV}{{\mathrm{dt}}_{go}}=\frac{a_n+g}{K}+g\mathrm{\θ}\\ \frac{d\mathrm{\θ}}{{\mathrm{dt}}_{go}}=-\frac{a_n}{V_0}\\ \frac{dx}{{\mathrm{dt}}_{go}}=-V_0\\ \frac{dh}{{\mathrm{dt}}_{go}}=-V_0\mathrm{\θ}\end{array}\right.$

Step 4, will in linearization process equation group add normal direction acceleration derivative equation, obtain Analytical Solution equation group；

Normal acceleration first derivative equation is：Normal acceleration second dervative equation is：

WhereinIt is the first derivative of normal acceleration,It is the second dervative of normal acceleration；

Analytical Solution equation group is as follows：

$\left\{\begin{array}{c}\frac{dV}{{\mathrm{dt}}_{go}}=\frac{a_n+g}{K}+g\mathrm{\θ}\\ \frac{d\mathrm{\θ}}{{\mathrm{dt}}_{go}}=-\frac{a_n}{V_0}\\ \frac{dx}{{\mathrm{dt}}_{go}}=-V_0\\ \frac{dh}{{\mathrm{dt}}_{go}}=-V_0\mathrm{\θ}\\ \frac{{\mathrm{da}}_n}{{\mathrm{dt}}_{go}}=-{\stackrel{\·}{a}}_n\\ \frac{d{\stackrel{\·}{a}}_n}{{\mathrm{dt}}_{go}}=-{\stackrel{\·\·}{a}}_n=-u\end{array}\right.$

Represent handleAs controlled quentity controlled variable；

Step 5, guided missile determine the downslide section trajectory end-fixity and path constraint to be met when gliding；

End-fixity specifically sets as follows：

$\left\{\begin{array}{cc}h\left(t_0\right)=h_0& h\left(t_f\right)=h_f\\ \mathrm{\θ}\left(t_0\right)=0& \mathrm{\θ}\left(t_f\right)=0\\ x\left(t_0\right)=0& x\left(t_f\right)=free\\ V\left(t_0\right)=V_0& V\left(t_f\right)=V_f\\ a_n\left(t_0\right)=a_{n0}& a_n\left(t_f\right)=a_{nf}\\ {\stackrel{\·}{a}}_n\left(t_0\right)={\stackrel{\·}{a}}_{n0}& {\stackrel{\·}{a}}_n\left(t_f\right)={\stackrel{\·}{a}}_{nf}\end{array}\right.$

Wherein, h (t₀)=h₀Represent t=t₀The height at moment is h₀, h (t_f)=h_fRepresent in t=t_fThe height at moment is h_f, θ (t₀)=0 represents t=t₀The trajectory tilt angle at moment is 0；θ(t_f)=0 represents t=t_fThe trajectory tilt angle at moment is 0, x (t₀The table of)=0 Show t=t₀The horizontal flight distance of moment guided missile is 0, x (t_f)=free represents t=t_fThe horizontal flight of moment guided missile is apart from nothing Constraint, V (t₀)=V₀Represent t=t₀The speed of moment guided missile relative atmospheric is constant V₀, V (t_f)=V_fRepresent t=t_fMoment leads The speed for playing relative atmospheric is V_f, a_n(t₀)=a_n0Represent t=t₀Moment guided missile normal acceleration is a_n0, a_n(t_f)=a_nfRepresent t =t_fMoment guided missile normal acceleration is a_nf,Represent t=t₀The first derivative of moment normal acceleration is Represent t=t_fThe first derivative of moment normal acceleration is

Path constraint includes speed and the normal acceleration constraint of guided missile, as follows：

$\left\{\begin{array}{c}{V}_{\min }\≤V\≤V_{\max }\\ a_{n\min }\≤a_n\≤a_{n\max }\end{array}\right.$

Wherein V_min, V_max, a_nminAnd a_nmaxDesign parameter according to specific guided missile is determined；V_minRepresent what missile flight speed was allowed Minimum value, V_maxRepresent the maximum that missile flight speed is allowed, a_nminRepresent the minimum value that guided missile normal acceleration is allowed, a_nmax Represent the maximum that guided missile normal acceleration is allowed；

Step 6, the target function for choosing optimum control, optimum control amount is determined according to Analytical Solution equation group and target function u；

Selection energy hole is target function J：

$J={\∫}_0^{t_f}\frac{1}{2}{u}^2{\mathrm{dt}}_{go}$

Step 7, solution optimum control amount u obtain parsing Guidance Law, guided missile downslide section trajectory is met end-fixity and path about Beam；

Based on the minimal principle in the theory of optimal control, Analytical Solution is carried out to optimal control problem, obtain parsing Guidance Law For：

U=16x₁-120x₂+480x₃-840x₄

Wherein,

$t_{go}=\frac{K}{g}(V-V_f)+\frac{V}{g}\mathrm{\θ}+\frac{K}{V}(h-h_f).$

2. a kind of sky comprising normal acceleration derivative as claimed in claim 1 penetrates Cruise Missile downslide section multiple constraint guidance side Method, it is characterised in that end-fixity refers in the step 5：Guided missile is in the downslide section distal point constraint to be met；Including end Hold highly constrained, constraint of velocity, trajectory tilt angle constraint, normal acceleration constraint and end normal acceleration first derivative constraint.

3. a kind of sky comprising normal acceleration derivative as claimed in claim 1 penetrates Cruise Missile downslide section multiple constraint guidance side Method, it is characterised in that in the step 5：The speed V of guided missile_max=300m/s, V_min=150m/s；Normal acceleration a_nmin =-10g, a_nmax=10g.