CN102353301A - Guidance method with terminal restraint based on virtual target point - Google Patents

Abstract

The invention discloses a guidance method with terminal restraint based on a virtual target point and belongs to the technical field of aircraft guidance and control systems. By using a target virtualization method, an aircraft can fly in the optimum height to attack a target; a virtualized target is positioned, above a real target, in the height, namely a detonation height which is required for a damage element; by designing a guidance law, when the aircraft reaches an imaginary target, a warhead detonates; and the conventional mode of attacking the real target by flying is changed into a guidance mode of flying to the virtualized target and detonating according to a required attitude angle. By adoption of the guidance law of the method, the aim that the aircraft with the explosion-formed damage element flies and attacks in an expected height can be fulfilled, the requirement of the optimum detonation height of the damage element is met, the problem of the attitude angle of the aircraft with the explosion-formed damage element during end attacking can be solved, and the requirement of the detonation moment of the damage element on the attitude angle of the aircraft is met.

Description

The guidance method that has end conswtraint based on virtual target point

Technical field

The invention belongs to aircraft guidance and Control System Design field; Relate to and be furnished with the Design of Guidance Law problem that the aircraft of unit is injured in explosive forming; Specially refer to the mode that how in the flight guided procedure, adopts virtual target point and realize that aircraft is plunderred and fly to hit the top, and detonate and make aircraft keep the particular pose angle constantly injuring unit.

Background technology

Explosive forming is injured unit and is meant after the warhead activation under the effect of flexible linear-shaped charge; Metal liner is forged into a penetration body that is similar to bullet; This penetration body has stable flight characteristics, goes in the target with kinetic energy penetration at a high speed, realizes precisely strike mission efficiently.The Design of Guidance Law that is equipped with explosive forming to injure the aircraft of unit is faced with new problem: because the unidirectional narrow beam property of first outgoing is injured in explosive forming; Plunder the attack pattern that flies to hit the top, it is closely related with attitude of flight vehicle and position of aircraft that unit is injured in the feasible explosive forming constantly of detonating in addition; Particularly to different targets of attack; In order to obtain the best effect of injuring; Requirement has the different patterns of detonating to injure n-ary form n (like stock jet or explosive forming bullet) with what form expectation, and the different patterns of detonating has strict demand to warhead activation height constantly.Therefore, for bonding Shu Juneng injures the critical position that unit can pinpointing, require aircraft to have attitude angle and the flying height that helps injuring first hit constantly in warhead activation.

Injure the aircraft of unit and realize that maximum injures effect in order to realize having explosive forming, need end guiding stage of aircraft satisfy below 2 points:

(1) warhead activation constantly the aircraft transient posture be controlled, the attitude of promptly detonating constantly is by being required attitude;

(2) height of warhead activation moment aircraft is controlled, promptly reaches the requirement for height constantly of detonating.

Proportional guidance law of conventional aircraft guidance method and improved form thereof, optimal guidance law, differential Game Guidance Law, what technology was comparatively ripe at present is proportional guidance law or its improved form.Traditional guidance law does not add the end conswtraint angle and the limitation in height constantly of detonating.For the aircraft that unit is injured in explosive forming that has of this paper research,, just must consider in the guidance law design, to add and can realize that the satisfied unit of injuring detonates the moment to the attitude of aircraft and the part of flying height requirement in order to realize the best effect of injuring.

Summary of the invention

Have explosive forming and injure the aircraft of unit is realized precision strike in the guidance process requirement for satisfying, to the purpose of this invention is to provide a kind of guidance method that has end conswtraint based on virtual target point.

The present invention utilizes the method realization aircraft of virtual target to plunder at optimum height and flies target of attack; Promptly on real target, have an imaginary target in vain; Unit is desired detonates highly the height of imagination target in the spatial domain in order to injure; Preset guidance law makes aircraft warhead activation when flying to imaginary target; This moment with original employing plunder the mode that flies to attack real goal become fly to virtual target and by the guide mode that detonates of requirement attitude angle, when realizing that aircraft is constantly found target endways, the autonomous tracking target of aircraft also reduces flying height; When injuring unit's desired height of formation and attitude of flight vehicle according to expectation, ignite warhead, thereby accomplish the purpose that precisely strikes target.

The aircraft angle of attack is ignored in the aircraft flight process that the present invention relates to, and being approximately the trajectory inclination angle is the attitude of flight vehicle angle.

The present invention is divided into two stages with terminal trajectory, adopts the method design guide rule of control with changed scale coefficient.

The present invention adopts the method for optimum control to be write the conversion of trajectory equation as state equation, and realizes the constraint to the trajectory angle of fall by method for optimally controlling.

Described method for optimally controlling is realized following to the optimization detailed process of the angle of fall, Guidance Law terminal:

The first step: confirm the feasible moment point position of detonating;

Second step: the mathematical modeling of setting up aircraft and target relative motion relation;

The aircraft initial position is the M point, and target is positioned at the T point, and the distance of aircraft and target is r, and θ is the trajectory inclination angle;

r?sin?θ＝y (1)

(1) formula is carried out the secondary differentiate,

Adopt Taylor series that (2) formula is launched, carry out linearization process, in formula (2) Sin θ, cos θ exist respectively

θ ₀The place carries out the single order Taylor series expansion,

sinθ＝sinθ ₀+cosθ ₀×(θ-θ ₀)(3)

cosθ＝cosθ ₀-sinθ ₀×(θ-θ ₀)(4)

Then full scale equation (2) becomes

Formula (6) becomes through abbreviation

m＝r(cosθ ₀+θ ₀sinθ ₀)

The 3rd step: get two endways on the trajectory with reference to the stage: one for initial time

Point constantly; Another elects terminal θ constantly as ₀,

In the phase I, because θ,

Very little, cause q, k and other amount in the formula (7) to compare in a small amount, so formula (7) is reduced to

And, contain non-linear relation among the q in the formula (7) for second stage, because employing is the microvariations methods, the θ among the q can think the θ that second stage adopts ₀So formula (7) is reduced to

d＝r(cosθ ₀+θ ₀sinθ ₀)-θ ₀rsinθ ₀；

Adopt this kind variable element guidance law, promptly select proportionality coefficient K for use ₁, K ₂Carry out weights and distribute, make K ₁+ K ₂=1, draw,

a _my＝K ₁a _my1+K ₂a _my2

w＝rk ₁+dk ₂；

The 4th step: foundation has the optimal guidance law of terminal angle restriction;

θ is a trajectory tilt angle, order θ _xBe variable, θ _fTerminal trajectory tilt angle for expectation;

Order

column write the equation of state was

Abbreviation is

$A=\left[\begin{array}{cc}0& 1\\ p& n\end{array}\right]$

$B=\left[\begin{array}{c}0\\ \frac{1}{w}\end{array}\right]$

p＝-l/w

n＝-g/w

u＝a _my

Initial conditions t=t ₀The time, x ₁(t ₀)=θ (t ₀)+θ _f

End conswtraint condition t=t _fThe time, x ₁(t _f)=0; x ₂(t _f)=0; This moment θ=θ _f

Select quadratic performance index $J=X^T\mathrm{FX}+1/2{\∫}_{t_0}^{t_f}{u}^2\mathrm{dt},$

Then $x=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right];$ $F=\left[\begin{array}{cc}{f}_{11}& f_{12}\\ f_{21}& f_{22}\end{array}\right];$ $A=\left[\begin{array}{cc}0& 1\\ p& n\end{array}\right];$ $B=\left[\begin{array}{c}0\\ \frac{1}{w}\end{array}\right],$ R=1, Q=0.

Getting optimal control law by the theory of optimal control does

$u^{*}=-R^{-1}{B}^T{\mathrm{PX}}^{*}=-\left[\begin{array}{cc}0& \frac{1}{r}\end{array}\right]\left[\begin{array}{cc}{P}_{11}& P_{12}\\ P_{21}& P_{22}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=-\frac{1}{r}(P_{11}{x}_1+P_{22}{x}_2)$

P satisfies the matrix of Riccati equation in the formula, because of || F|| → ∝, separate Riccati equation and get

Order $P^{-1}=\left[\begin{array}{cc}{q}_{11}& q_{12}\\ q_{21}& q_{22}\end{array}\right],$ Q wherein ₁₂=q ₂₁

Solve $P=\left[\begin{array}{cc}2\mathrm{Bt}& b\\ b& -\mathrm{Nb}-\mathrm{Mbt}\end{array}\right],$ Wherein

$b=\frac{1}{(3p-2n^2-2\mathrm{Pnt}){\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000091628040000022.png,HDA0000091628040000023.png"\; state="\{\{state\}\}">r</figure-callout>}}^2}$

$P=\left[\begin{array}{cc}\frac{n+\mathrm{pt}}{b(2\mathrm{nt}+2\mathrm{<figure-callout\; id="2"\; label="p\; t"\; filenames="HDA0000091628040000022.png,HDA0000091628040000023.png"\; state="\{\{state\}\}">p</figure-callout>}{t}^{}{}^2+1)}& \frac{1}{b(2\mathrm{nt}+2\mathrm{<figure-callout\; id="2"\; label="p\; t"\; filenames="HDA0000091628040000022.png,HDA0000091628040000023.png"\; state="\{\{state\}\}">p</figure-callout>}{t}^{}{}^2+1)}\\ \frac{1}{b(2\mathrm{nt}+2\mathrm{<figure-callout\; id="2"\; label="p\; t"\; filenames="HDA0000091628040000022.png,HDA0000091628040000023.png"\; state="\{\{state\}\}">p</figure-callout>}{t}^{}{}^2+1)}& \frac{-2t}{b(2\mathrm{nt}+2\mathrm{<figure-callout\; id="2"\; label="p\; t"\; filenames="HDA0000091628040000022.png,HDA0000091628040000023.png"\; state="\{\{state\}\}">p</figure-callout>}{t}^{}{}^2+1)}\end{array}\right]$

Getting optimal control law does

In the formula

$b=\frac{1}{(3p-2n^2-2\mathrm{pnt}){\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000091628040000022.png,HDA0000091628040000023.png"\; state="\{\{state\}\}">r</figure-callout>}}^2};$ P=-l/w; N=-g/w;

$w=\mathrm{<figure-callout\; id="1"\; label="r\; k"\; filenames="HDA0000091628040000022.png,HDA0000091628040000023.png"\; state="\{\{state\}\}">r</figure-callout>}{k}_{}{}_1+\mathrm{<figure-callout\; id="2"\; label="d\; k"\; filenames="HDA0000091628040000022.png,HDA0000091628040000023.png"\; state="\{\{state\}\}">d</figure-callout>}{k}_{}{}_2;$

d＝r(cosθ ₀+θ ₀sinθ ₀)-θ ₀rsinθ ₀。

The optimal control law that finally obtains does

Good effect of the present invention is:

(1) solved the aircraft that the band explosive forming injures unit and realized plunderring the problem that flies to attack, satisfied the best of injuring unit requirement for height of detonating at Desired Height;

(2) solved aircraft the end game that the band explosive forming injures unit problem of attitude angle constantly, satisfied and injured unit and detonate constantly the requirement at attitude of flight vehicle angle.

Description of drawings

Fig. 1 is an aircraft flight trajectory sketch map.

Fig. 2 is based on the aircraft flight sketch map of virtual target point.

Fig. 3 is the relation of attitude of flight vehicle and flight position constantly of detonating.

Fig. 4 is aircraft and target relative motion geometrical relationship figure.

Fig. 5 a be when proportionality coefficient be K ₁=0.7, K ₂=0.3 makes, and flying height is change curve in time.

Fig. 5 b be when proportionality coefficient be K ₁=0.6; K ₂=0.4 o'clock, flying height is change curve in time.

Form 1 is the different proportion COEFFICIENT K ₁, K ₂The trajectory inclination effect is contrasted.

The specific embodiment

To combine accompanying drawing that the present invention is specified below, the specific embodiment is as follows:

Visible by Fig. 1, this aircraft is after presumptive area is thrown in, and wing launches, engine start, with the flight of about 200m, highly cruising of the speed of about 100m/s, ferret out.After aircraft is found target, adopt different attack patterns according to target characteristic, as: if treat target of attack is heavy armored target, then adopts the extension rod-type pattern of detonating; As treat that target of attack is the lightweight armor target, then adopts the pattern of detonating of explosive forming penetration body.The different patterns of injuring requires warhead activation to have the different height requirement constantly, therefore injures the aircraft of unit for having explosive forming, and in order to hit the mark accurately, the aircraft elevation information constantly that must will detonate adds in the design of Guidance Law.So the present invention proposes the notion of virtual target: promptly suppose the sky, a virtual target is arranged, need aircraft to hit virtual target with the angle of fall of design at realistic objective.

Visible by Fig. 2, aircraft treats that from the A point target of attack original position is the B point.Consider the mobility of target, hypothetical target moves to B _fThe time, require the aircraft A that should fly this moment _fPoint, A _fPoint is the virtual target point of design.Aircraft is at A _fChoosing of required attitude angle was to be determined by the relation between flying height h and the position of aircraft information when point detonated warhead.In order to discuss conveniently; Have below in the Design of Guidance Law of the angle of fall, terminal constraint; All supposition has explosive forming and injures the aircraft of unit and detonate at virtual impact point, is converted into and has the guidance law design problem that the angle of fall, terminal retrains so satisfy guidance law design that aircraft altitude and attitude require.

The present invention adopts method for optimally controlling to realize the optimization to the angle of fall, Guidance Law terminal.The practical implementation process is following:

1. the feasible moment point position of detonating is confirmed

Warhead activation is position, attitude, velocity magnitude and the direction of aircraft constantly, and the static first flying speed etc. of injuring all can influence final strike effect.In other parameters one regularly; Injure first point of impact ordinate and be monotone variation with the aircraft angle of pitch constantly that detonates; When promptly aircraft has the positive angle of pitch constantly when detonating, injure first point of impact and will be ahead of the body projected position on the ground constantly that detonates along the projecting direction of speed on ground.The ordinate of the point of impact that moment aircraft angle of pitch difference causes owing to detonate changes fairly obvious, so when selecting to detonate opportunity, need to consider that the aircraft angle of pitch is to injuring first vertically influence of accuracy at target.Fig. 3 expresses, and the radius of the feasible zone that detonates is R, the A if aircraft flies at this moment ₁Point, if the expectation hit, then the aircraft angle of pitch that must have is θ.

Among the present invention, ignore angle of attack influence, realize the task that target is precisely attacked through the constraint of trajectory tilt angle constantly that end is detonated.

2. the mathematical modeling of aircraft and target relative motion relation

Among Fig. 4, the aircraft initial position is the M point, and target is positioned at the T point, and the distance of aircraft and target is r, and θ is the trajectory inclination angle.

Know by geometrical relationship

r?sin?θ＝y(1)

(1) formula is carried out the secondary differentiate,

Owing to nonlinear elements such as sin θ, cos θ occurred in the model (2); Adopt conventional method to analyze to system; The present invention adopts Taylor series that (2) formula is launched, and carries out linearization process.

In formula (2)

Sin θ, cos θ exist respectively

θ ₀The place carries out the single order Taylor series expansion,

sinθ＝sinθ ₀+cosθ ₀×(θ-θ ₀)(3)

cosθ＝cosθ ₀-sinθ ₀×(θ-θ ₀)(4)

Then full scale equation (2) becomes

Formula (6) becomes through abbreviation

Here

m＝r(cosθ ₀+θ ₀sinθ ₀)

θ ₀,

Two Parameter selection have very big influence for guidance precision.

The trajectory that aircraft follows the trail of the objective can be divided into two stages: in the starting stage; Trajectory tilt angle θ ≈ 0; The variation of

is little; Can be similar to and be seen as

when aircraft during near target; The trajectory inclination angle changes greatly, and θ and

all can not think 0 °.But consider that the target to be attacked of discussion of the present invention is the ground maneuver target, be not one and make big maneuvering target, therefore can suppose that the trajectory inclination angle is to change within the specific limits.So get two endways on the trajectory with reference to the stage: one for initial time Point constantly; Another elects terminal θ constantly as ₀,

Should be noted that terminal reference point

θ ₀The precision of choosing for whole guidance have significant impact,, should look concrete condition and choose setting when using in the Design of Guidance Law of reality.

In the phase I; Because θ,

are very little; Cause q, k and other amount in the formula (7) to compare in a small amount, so formula (7) is reduced to

And, contain non-linear relation among the q in the formula (7) for second stage, because employing is the microvariations methods, the θ among the q can think the θ that second stage adopts ₀So formula (7) is reduced to

Here,

d＝r(cosθ ₀+θ ₀sinθ ₀)-θ ₀rsinθ ₀

In the derivation to said process, because the k value of formula (7) is very little, so it is ignored.

Because stage one is chosen with the stage two, play control action for the trajectory of whole system, so adopt this kind variable element guidance law, promptly select proportionality coefficient K for use ₁, K ₂Carry out weights and distribute, make K ₁+ K ₂=1, draw,

a _my＝K ₁a _my1+K ₂a _my2

w＝rk ₁+dk ₂；

4. the optimal guidance law design that has the terminal angle restriction

θ is a trajectory tilt angle, in order to introduce the notion of optimum control, can make

θ＝θ _x+θ _f

θ _xBe variable, θ _fTerminal trajectory tilt angle for expectation.

Order

x ₁＝θ _x

Row are write state equation and are got

Abbreviation does

$A=\left[\begin{array}{cc}0& 1\\ p& n\end{array}\right]$

$B=\left[\begin{array}{c}0\\ \frac{1}{w}\end{array}\right]$

p＝-l/w

n＝-g/w

u＝a _my

Initial conditions t=t ₀The time,

x ₁(t ₀)＝θ(t ₀)+θ _f；

End conswtraint condition t=t _fThe time,

x ₁(t _f)＝0；

x ₂(t _f)＝0；

This moment θ=θ _f

Select quadratic performance index

$J=X^T\mathrm{FX}+1/2{\&Integral;}_{t_0}^{{\mathrm{<figure-callout\; id="2"\; label="t\; f\; u"\; filenames="HDA0000091628040000022.png,HDA0000091628040000023.png"\; state="\{\{state\}\}">t</figure-callout>}}_f}{u}^{}{}^2\mathrm{dt},$

Then $x=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right];$

$F=\left[\begin{array}{cc}{f}_{11}& f_{12}\\ f_{21}& f_{22}\end{array}\right]$

$A=\left[\begin{array}{cc}0& 1\\ p& n\end{array}\right];$

$B=\left[\begin{array}{c}0\\ \frac{1}{w}\end{array}\right],$ R＝1，Q＝0。

Getting optimal control law by the theory of optimal control does

$u^{*}=-R^{-1}{B}^{T}P{X}^{*}=-\left[\begin{array}{cc}0& \frac{1}{r}\end{array}\right]\left[\begin{array}{cc}{P}_{11}& P_{12}\\ P_{21}& P_{22}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=-\frac{1}{r}(P_{11}{x}_1+P_{22}{x}_2)$

P satisfies the matrix of Riccati equation in the formula.

Cause || F|| → ∝, separate Riccati equation and get

Order

$P^{-1}=\left[\begin{array}{cc}{q}_{11}& q_{12}\\ q_{21}& q_{22}\end{array}\right]$

Q wherein ₁₂=q ₂₁

Solve

$P=\left[\begin{array}{cc}2\mathrm{bt}& b\\ b& -\mathrm{nb}-\mathrm{mbt}\end{array}\right]$

Wherein

$b=\frac{1}{(3p-{2n}^2-2\mathrm{pnt}){\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000091628040000022.png,HDA0000091628040000023.png"\; state="\{\{state\}\}">r</figure-callout>}}^2}$

$P=\left[\begin{array}{cc}\frac{n+\mathrm{pt}}{b(2\mathrm{nt}+{2\mathrm{<figure-callout\; id="2"\; label="pt"\; filenames="HDA0000091628040000022.png,HDA0000091628040000023.png"\; state="\{\{state\}\}">pt</figure-callout>}}^2+1)}& \frac{1}{b(2\mathrm{nt}+{2\mathrm{<figure-callout\; id="2"\; label="pt"\; filenames="HDA0000091628040000022.png,HDA0000091628040000023.png"\; state="\{\{state\}\}">pt</figure-callout>}}^2+1)}\\ \frac{1}{b(2\mathrm{nt}+{2\mathrm{<figure-callout\; id="2"\; label="pt"\; filenames="HDA0000091628040000022.png,HDA0000091628040000023.png"\; state="\{\{state\}\}">pt</figure-callout>}}^2+1)}& \frac{-2t}{b(2\mathrm{nt}+{2\mathrm{<figure-callout\; id="2"\; label="pt"\; filenames="HDA0000091628040000022.png,HDA0000091628040000023.png"\; state="\{\{state\}\}">pt</figure-callout>}}^2+1)}\end{array}\right]$

Getting optimal control law does

In the formula

$b=\frac{1}{(3p-{2n}^2-2\mathrm{pnt}){\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000091628040000022.png,HDA0000091628040000023.png"\; state="\{\{state\}\}">r</figure-callout>}}^2};$

p＝-l/w；

n＝-g/w；

w＝rk ₁+dk ₂；

d＝r(cosθ ₀+θ ₀sinθ ₀)-θ ₀rsinθ ₀。

The optimal control law that finally obtains does

Realize angle from engineering, what the aircraft that the present invention relates to adopted is the laser radar target seeker, so r,

All can record; Remaining time t _GoThere is considerable influence in guidance precision, because r,

Can record, so according to

Can directly obtain t _Go, i.e. remaining time.Guidance precision is determined by the laser radar target seeker.Though the laser radar target seeker can not record, can the robust Kalman filter estimate

for

5. simulation example

In the pitching plane, the original position of aircraft, speed parameter are:

x _m0＝0m，y _m0＝100m；

v _m0＝120m/s；

Target initial position, speed parameter are:

x _t0＝2000m，y _t0＝0m；

v _t0＝30m/s；

θ _f＝0.3?rad；

(1)K ₁＝0.7；K ₂＝0.3

Adopt the above-mentioned described Guidance Law simulation result that follows the trail of the objective following, trajectory is shown in Fig. 5 a.

(2)K ₁＝0.6；K ₂＝0.4

Adopt the above-mentioned described Guidance Law simulation result that follows the trail of the objective following, trajectory is shown in Fig. 5 b.

Table 1 has provided the different proportion COEFFICIENT K ₁, K ₂The time, gained trajectory inclination angle transformation relation.

Table 1

Simulation result shows that the Guidance Law of the present invention's design has realized the control to flight terminal trajectory tilt angle; By the visible K of Fig. 5 ₁, K ₂Determined the shape of trajectory.

It is thus clear that the final designing requirement that this kind satisfied based on the method for designing that has the end conswtraint angle of virtual target point.

Claims (5)

1. based on the guidance method that has end conswtraint of virtual target point; It is characterized in that: utilize the method for virtual target to realize that aircraft is plunderred at optimum height and fly target of attack; Promptly on real target, have an imaginary target in vain; For injuring the desired height that detonates of unit, preset guidance law makes aircraft warhead activation when flying to imaginary target to the imagination target at the height in spatial domain, at this moment with original employing plunder the mode that flies to attack real goal become fly to virtual target and by the guide mode that detonates of requirement attitude angle; When realizing that aircraft is constantly found target endways; The autonomous tracking target of aircraft also reduces flying height, when injuring unit's desired height of formation and attitude of flight vehicle according to expectation, ignites warhead, thereby accomplishes the purpose that precisely strikes target.

2. the guidance method that has end conswtraint based on virtual target point as claimed in claim 1 is characterized in that: the aircraft angle of attack is ignored in the above-mentioned aircraft flight process, and being approximately the trajectory inclination angle is the attitude of flight vehicle angle.

3. the guidance method that has end conswtraint based on virtual target point as claimed in claim 1 is characterized in that: terminal trajectory is divided into two stages, adopts the method design guide rule of control with changed scale coefficient.

4. like claim 1 or the 2 or 3 described guidance methods that have end conswtraint based on virtual target point; It is characterized in that: adopt the method for optimum control to be write the conversion of trajectory equation as state equation, and realize constraint the trajectory angle of fall by method for optimally controlling.

5. the guidance method that has end conswtraint based on virtual target point as claimed in claim 4 is characterized in that: described method for optimally controlling is realized following to the optimization detailed process of the angle of fall, Guidance Law terminal:

The first step: confirm the feasible moment point position of detonating;

Second step: the mathematical modeling of setting up aircraft and target relative motion relation;

The aircraft initial position is the M point, and target is positioned at the T point, and the distance of aircraft and target is r, and θ is the trajectory inclination angle;

r?sin?θ＝y(1)

(1) formula is carried out the secondary differentiate,

Adopt Taylor series that (2) formula is launched, carry out linearization process, in formula (2) Sin θ, cos θ exist respectively

θ ₀The place carries out the single order Taylor series expansion,

sinθ＝sinθ ₀+cosθ ₀×(θ-θ ₀)(3)

cosθ＝cosθ ₀-sinθ ₀×(θ-θ ₀)(4)

Then full scale equation (2) becomes

Formula (6) becomes through abbreviation

m＝r(cosθ ₀+θ ₀sinθ ₀)

The 3rd step: get two endways on the trajectory with reference to the stage: one for initial time

Point constantly; Another elects terminal θ constantly as ₀,

In the phase I, because θ,

Very little, cause formula

(7) q in, k and other amount are compared in a small amount, so formula (7) is reduced to

And, contain non-linear relation among the q in the formula (7) for second stage, because employing is the microvariations methods, the θ among the q can think the θ that second stage adopts ₀So formula (7) is reduced to

d＝r(cosθ ₀+θ ₀sinθ ₀)-θ ₀rsinθ ₀；

Adopt this kind variable element guidance law, promptly select proportionality coefficient K for use ₁, K ₂Carry out weights and distribute, make K ₁+ K ₂=1, draw,

a _my＝K ₁a _my1+K ₂a _my2

w＝rk ₁+dk ₂；

The 4th step: foundation has the optimal guidance law of terminal angle restriction;

θ is a trajectory tilt angle, order

θ _xBe variable, θ _fTerminal trajectory tilt angle for expectation;

Order

column write the equation of state was

Abbreviation is

$A=\left[\begin{array}{cc}0& 1\\ p& n\end{array}\right]$

$B=\left[\begin{array}{c}0\\ \frac{1}{w}\end{array}\right]$

p＝-l/w

n＝-g/w

u＝a _my

Initial conditions t=t ₀The time, x ₁(t ₀)=θ (t ₀)+θ _f

End conswtraint condition t=t _fThe time, x ₁(t _f)=0; x ₂(t _f)=0; This moment θ=θ _f

Select quadratic performance index $J=X^T\mathrm{FX}+1/2{\&Integral;}_{t_0}^{t_f}{u}^2\mathrm{dt},$

Then $x=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right];$ $F=\left[\begin{array}{cc}{f}_{11}& f_{12}\\ f_{21}& f_{22}\end{array}\right];$ $A=\left[\begin{array}{cc}0& 1\\ p& n\end{array}\right];$ $B=\left[\begin{array}{c}0\\ \frac{1}{w}\end{array}\right],$ R=1, Q=0.

Getting optimal control law by the theory of optimal control does

$u^{*}=-R^{-1}{B}^T{\mathrm{PX}}^{*}=-\left[\begin{array}{cc}0& \frac{1}{r}\end{array}\right]\left[\begin{array}{cc}{P}_{11}& P_{12}\\ P_{21}& P_{22}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=-\frac{1}{r}(P_{11}{x}_1+P_{22}{x}_2)$

P satisfies the matrix of Riccati equation in the formula, because of || F|| → ∝, separate Riccati equation and get

Order $P^{-1}=\left[\begin{array}{cc}{q}_{11}& q_{12}\\ q_{21}& q_{22}\end{array}\right],$ Q wherein ₁₂=q ₂₁

Solve $P=\left[\begin{array}{cc}2\mathrm{Bt}& b\\ b& -\mathrm{Nb}-\mathrm{Mbt}\end{array}\right],$ Wherein

$b=\frac{1}{(3p-2n^2-2\mathrm{Pnt})r^2}$

$P=\left[\begin{array}{cc}\frac{n+\mathrm{pt}}{b(2\mathrm{nt}+2p{t}^2+1)}& \frac{1}{b(2\mathrm{nt}+2p{t}^2+1)}\\ \frac{1}{b(2\mathrm{nt}+2p{t}^2+1)}& \frac{-2t}{b(2\mathrm{nt}+2p{t}^2+1)}\end{array}\right]$

Getting optimal control law does

In the formula $b=\frac{1}{(3p-2n^2-2\mathrm{pnt})r^2};$ P=-l/w; N=-g/w;

w＝rk ₁+dk ₂；

d＝r(cosθ ₀+θ ₀sinθ ₀)-θ ₀rsinθ ₀。

The optimal control law that finally obtains does

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