CN106382853B

CN106382853B - A kind of tape terminal trajectory tilt angle and the angle of attack constraint singularity perturbation suboptimum Guidance Law

Info

Publication number: CN106382853B
Application number: CN201610888077.6A
Authority: CN
Inventors: 陈万春; 赵鹏雷
Original assignee: Beihang University
Current assignee: Beihang University
Priority date: 2016-10-11
Filing date: 2016-10-11
Publication date: 2017-12-15
Anticipated expiration: 2036-10-11
Also published as: CN106382853A

Abstract

The present invention relates to the singular perturbation suboptimum Guidance Law that a kind of tape terminal trajectory tilt angle and the angle of attack constrain, comprise the following steps：Determine the time scale of system state variables；Establish singular perturbation system；Solve the slow system of zeroth order；Solve the fast system of single order；Single order residual non-uniformity τ_goCalculating；The optimum control command acceleration for meeting miss distance, terminal trajectory tilt angle, the angle of attack is calculated, carries out instruction renewal.The invention has the advantages that：Compared with traditional Terminal Guidance Laws, the parsing Guidance Law can meet miss distance, terminal trajectory tilt angle, angle of attack constraint, while meet that energy hole is optimal in flight course；The Guidance Law is solved based on singular perturbation theory, and the optimum control Guidance Law form of acquisition is more succinct, and physical significance is distincter；The Guidance Law provides a kind of new solution throughway for solving complexity high-order optimum control Guidance Law, can carry out the solution of more multiple constraint, has wide applicability.

Description

A kind of tape terminal trajectory tilt angle and the angle of attack constraint singularity perturbation suboptimum Guidance Law

Technical field

The present invention relates to the singular perturbation suboptimum Guidance Law that a kind of tape terminal trajectory tilt angle and the angle of attack constrain, belong to space flight skill Art, weapon technologies, Guidance and control field.

Background technology

With the continuous development of operational environment and use demand, present guided missile not only needs full when implementing end strike The constraint of sufficient miss distance, will also be with optimal posture hit, so as to play the fighting efficiency of warhead to greatest extent, with right Target causes optimal damage effectiveness, and this is required in terminal guidance, not only to consider miss distance, trajectory tilt angle, terminal-velocity etc. eventually End constraint, will also constrain the angle of attack, at utmost to reduce projectile penetrating resistance, prevent body unstability and curved trajectory from showing As occurring.

Based on different theories terminal trajectory tilt angle constraint Guidance Law largely study, precision guided weapon, More application is obtained on the tactical weapons such as antitank missile, anti-warship guided missle, but for meeting the guidance of terminal angle of attack constraint simultaneously Rule research is less.Even if guided missile meets that terminal trajectory tilt angle constrains, but if the larger terminal angle of attack be present, when guided missile and mesh Guided missile " ricochet " phenomenon or penetration trajectory bending can be easily caused during mark collision due to asymmetric impact, is invaded so as to weaken it Thorough ability.

Singular perturbation theory be 20th century mid-term progressively developed by hydromechanical boundary layer theory, it is hereafter wide It is general to be applied to solve various optimal control problems.Time scale (state variable of the singular perturbation theory according to system state variables The relative speed of change), original system is divided into lower order system and some high order systems, the solution of wherein lower order system is relatively simple It is single, then on the basis of low order solution, boundary layer correction is introduced to high order system, so as to obtain the analytic solutions of arbitrary accuracy.

Xing Qiang, Guidance Law analytic solutions research [D] the BJ University of Aeronautics ＆ Astronautics with terminal point and constraint of velocity, A kind of terminal point decomposed by markers is disclosed in 2015 chapter 3 content and constrains Guidance Law.In the following theory of the present invention Terminal angle leash law is referred to as in bright.

The content of the invention

The invention aims to solve the above problems, a kind of the unusual of tape terminal trajectory tilt angle and the angle of attack constraint is proposed Perturb suboptimum Guidance Law, to meet that the optimum control of miss distance, the constraint of terminal trajectory tilt angle and the constraint of the terminal angle of attack parses simultaneously Guidance Law.

The Guidance Law is based on singular perturbation theory, the time scale based on system state variables, by the vertical journey of guided missile, Highly, trajectory tilt angle is divided into slow variable, and acceleration, the angle of attack are divided into fast variable.Vertical journey, height, trajectory tilt angle composition depression of order The slow system of zeroth order, it is guided missile acceleration that it, which controls variable, by entering line to zero order system, solves slow system linearity two Secondary type optimal control problem, obtain the optimal control solution (desired optimal guided missile acceleration) of zero order system, the solution of zero order system Terminal miss distance and the trajectory tilt angle constraint of guided missile can be ensured；Acceleration, the angle of attack composition fast system of single order, in the base of zeroth order solution On plinth, time scale stretching is carried out, introduces boundary layer correction, be i.e. the fast variable angle of attack constrains, and obtains the optimal finger of revised guided missile Acceleration is made, this solution can ensure the constraint of Miss Distance, terminal trajectory tilt angle and the angle of attack simultaneously.

, it is necessary to calculate the residual non-uniformity of guided missile during the slow system solution of zeroth order, with remaining distance in the present invention The ratio of current speed carries out approximate substitution with guided missile；Residual non-uniformity in the fast system of single order, which solves, to be needed to slow system Residual non-uniformity carry out stretching conversion, the present invention proposes that the speed difference of first-order system and zero order system is due to system Caused by first-order lag, therefore first-order lag of the residual non-uniformity of first-order system also for the residual non-uniformity of zero order system rings Should.

The present invention is a kind of singular perturbation suboptimum Guidance Law of tape terminal trajectory tilt angle and the angle of attack constraint, first has to determine shape The time scale of state variable, provide Second-Order Singular perturbed system；Then, the optimal control solution of the slow system of zeroth order is calculated, is now solved Analysis solution can meet terminal miss distance and trajectory tilt angle constraint；Finally, substituted into the fast system of single order, carry out time scale drawing Stretch, and introduce boundary layer correction, obtain the revised higher precision analytic solutions for meeting terminal trajectory tilt angle and angle of attack constraint.It is whole Individual process includes following steps：

Step 1：Determine the time scale of system state variables

The computational methods of time scale are as follows

Wherein Δ x is state variable x constant interval, is typically calculated with the difference of maxima and minima,For state Variable x first derivative.The solution of the maximum of state variable, minimum value and first derivative, can be imitated by numerical method True or other conventional method approximations obtain, and because the time scale order of magnitude difference of state variable is larger, approximate calculation will not shadow The speed rung to state variable time scale divides.

Step 2：Establish singular perturbation system

Only consider the guidance problems of fore-and-aft plane, by numerical simulation and the calculating of time scale, can by the vertical journey of guided missile, Highly, trajectory tilt angle regards slow variable as, regards the acceleration of guided missile, the angle of attack as fast variable, establishes the Second-Order Singular perturbation of guided missile Model

X in formula, h, a_m,γ,α,a_cRespectively vertical journey, height, normal acceleration, trajectory tilt angle, the angle of attack and the normal direction of guided missile Command acceleration,For guided missile vertical journey, height, normal acceleration, trajectory tilt angle and the angle of attack to the derivative of time, V_mIt is constant value for the speed of guided missile, τ_m,τ_αRespectively the single order link time constant of guided missile and rate of turn time constant, ε are It is individual a small amount of and have 0 ＜ ε ＜＜ 1.

Meet that the performance indications for the optimum control Guidance Law that miss distance, terminal trajectory tilt angle, the angle of attack constrain are described as

T in formula_fThe terminal juncture intersected for guided missile with target, x_f,h_f,γ_f,α_fJourney, height, trajectory are indulged for Missile Terminal to incline Angle and angle of attack constraint, x (t_f),h(t_f),γ(t_f),α(t_f) in the vertical journey of collision moment, height, trajectory tilt angle and attacked for guided missile Angle, a_cFor guided missile normal direction command acceleration, S_x,S_h,S_γ,S_αFor the weight coefficient of corresponding state variable bound.

Step 3：Solve the slow system of zeroth order

ε → 0 is made to obtain the reduced order system of original system

The amount of subscript 0 is expressed as zero order system parameter in formula and in full text.As can be seen that there is no attack in zero order system Horn shape state variable, accelerationAs control variable.Because the angle of attack, command acceleration have not occurred in zero order system, therefore select Take all the time in the acceleration of optimal value dynamic equilibriumControl variable as zero order system.

System performance index depression of order is

Hamilton functions are

In formulaFor association's state variable of corresponding state variable.

By optimality condition

Line is entered to this zero order system, obtains the system after line

Wherein y₁For the relative displacement in vertical direction of guided missile and target, y₂It is guided missile and target in vertical direction Relative velocity.

The system of linearisation can be designated as

Wherein

Performance indications are designated as

Subscript " T " represents the transposition of matrix, X (t in formula and in full text_f)=[y₁(t_f), y₂(t_f),γ(t_f)]^TFor system The state value of terminal juncture, X_f=[y_1f,y_2f,γ_f]^TConstrained for system terminal.For weight coefficient matrix, WhereinS_γRespectively state variable y₁, γ weight coefficient.

Hamilton functions are

Association state matrix be

NoteBy transversality condition

λ(t_f)=S [X (t_f)-X_f] (13)

Above formula substitution formula (12) is solved into association's state variable matrix is

λ=Φ (t_f, t) and S [X (t_f)-X_f] (14)

WhereinFor state-transition matrix, and have

T in formula_go=t_f- t is residual non-uniformity.In the present invention withApproximate substitution, R in formula_TMFor guided missile with The distance of target.

By optimality index

U=-B^Tλ=- B^T(Φ(t_f, t))^TS[X(t_f)-X_f] (16)

Due to it is determined that control under, Missile Terminal state X (t_f) determine, by the control solution substitution system side obtained by above formula Formula (9) simultaneously carries out State integral and obtained

Arrangement can obtain guided missile and be in the state of terminal juncture

Above formula is substituted into formula (16) to obtain

Weighting weight figure limitS_γ→ ∞ is obtained

Due to line-of-sight rate by lineSo it can obtain

This is optimum control Guidance Law after system linearization, is stood good in nonlinear system, now line-of-sight rate by line Calculate with the following methods

R in formula_TM1,R_TM2Respectively missile-target distance R_TMComponent in horizontally and vertically direction.

Step 4：Solve the fast system of single order

Carry out time scale stretching conversion, orderAnd allow ε → 0, first-order system equation can be obtained

The amount of subscript 1 is expressed as first-order system parameter in formula and in full text.As can be seen that in first-order system, journey, height are indulged The rate of change of degree and trajectory tilt angle is 0, because these three variant time yardsticks are much larger than the angle of attack and acceleration, relative to acceleration It is very slow with angle of attack variation, thus indulge journey in first-order system, height, trajectory tilt angle are approximately constant value, its value and accordingly assist state The value of variable is equal to the value solved in zero order system.

The performance indications of first-order system are

Hamilton functions are

In formulaλ_αRespectively state variable a_m, association's state variable corresponding to α.

By optimality conditionIt can obtain, revised command acceleration should be

As can be seen that being intended to obtain revised acceleration, then association's state variable must be calculatedλ_αValue.

The co-state equation of first-order system is

NoteThen

The transversality condition of first-order system is

τ in formula_fFor the terminal juncture value of first-order system.

It can be obtained by the co-state equation formula (30) on the angle of attack

λ_α(τ)=λ_α(τ_f)=S_α[α(τ_f)-α_f]=constant (33)

That is λ_αFor the constant value not changed over time, then the co-state equation formula (29) on acceleration can be regarded as on one Rank system time τ differential equation of first order, solve and substitute into the association's state variable tried to achieve in zero order systemFormula (7) and transversality condition Formula (31) can obtain association's state variable on acceleration

τ in formula_go=τ_f- τ is the residual non-uniformity of first-order system, for τ_goSolution can provide in steps of 5.

It is as follows on τ second time derivative that the angle of attack can be obtained to first-order system equation (25) derivation

The first derivative of command acceleration can be calculated by formula (28) in formula

Can obtain acceleration and the angle of attack by system equation (24) (25) has following relation

Formula (36) (37) substitution formula (35) then can be obtained into the second order differential equation containing only the angle of attack

Solve

Wherein C₁,C₂For constant value.

The current time angle of attack and angle of attack derivative state value α (τ)=α, α ' (τ)=α ' are substituted into, can calculate and try to achieve terminal juncture τ_fWhen angle of attack value

In addition, by transversality condition formula (33) and to weight coefficient S_αTake limit S_α→ ∞, the desired terminal juncture angle of attack can be obtained For

Then more than simultaneous two formulas (40) (41) can obtain is on association's state variable of the angle of attack

The association's state variable that will have been tried to achieveλ_αIt is updated to the satisfaction that can be obtained in revised command acceleration formula (28) finally The optimum control Guidance Law of institute's Prescribed Properties

It is abbreviated as

Wherein

Step 5：Single order residual non-uniformity τ_goCalculating

Single order residual non-uniformity has following relation with zeroth order residual non-uniformity

Wherein, τ_TFor time delay constant.

Substitute into τ_go=τ_fIt is as follows on zeroth order temporal t differential equation of first order that-τ can obtain single order time τ

Solve and substitute into the single order time τ (t that τ (t)=τ obtains terminal juncture_f) be

Due to τ (t_f)=τ_f, t_go=t_f-t₀, arrangement can obtain single order residual non-uniformity and be

Step 6：The optimum control command acceleration for meeting miss distance, terminal trajectory tilt angle, the angle of attack is calculated, is instructed more Newly.

Wherein

Due to differential term α ' in Guidance Law be present, Guidance Law can be caused to instruct the vibration in end by a relatively large margin, so as to make Into more energy expenditure.Therefore, the present invention is guidanceed command to this is modified, to weaken oscillation effect caused by differential term. It is observed that the factor of influence of differential term is its coefficient 1-k, it is a parameter changed with residual non-uniformity, there is (1-k) ＜ 1, and when residual non-uniformity level off to 0 when, 1-k also levels off to 0.If can largely it be weakened using its cubic form Influence of the differential term to guidanceing command, so as to effectively suppress end instruction vibration.Then it is revised guidance command for

Wherein

If ignoring influence of the angle of attack derivative to command acceleration amendment, the Guidance Law can be further simplified as

Wherein system

A kind of tape terminal trajectory tilt angle of the present invention and the singular perturbation suboptimum Guidance Law of angle of attack constraint, the advantage is that：

(1) miss distance, terminal trajectory tilt angle, the angle of attack can be met about compared with traditional Terminal Guidance Laws, the parsing Guidance Law Beam, while meet that energy hole is optimal in flight course；

(2) Guidance Law is solved based on singular perturbation theory, and the optimum control Guidance Law form of acquisition is more succinct, thing It is distincter to manage meaning.

(3) Guidance Law provides a kind of new solution throughway, Ke Yijin for solving complexity high-order optimum control Guidance Law The solution of row more multiple constraint, has wide applicability.

Brief description of the drawings

Fig. 1 is missile guidance schematic diagram.

Fig. 2 is Guidance Law instruction product process figure of the present invention.

Fig. 3 is that the Guidance Law contrasts with terminal angle leash law and the results of trajectory simulation of Gauss puppet spectrometry (numerical method).

Fig. 4 is the Guidance Law and terminal angle leash law and the trajectory tilt angle simulation result pair of Gauss puppet spectrometry (numerical method) Than.

Fig. 5 is that the Guidance Law contrasts with the angle of attack simulation result of terminal angle leash law and Gauss puppet spectrometry (numerical method).

Embodiment

Below in conjunction with drawings and examples, the present invention is described in further detail.

Step 1：Determine the time scale of system state variables

Time scale is defined as inverse of the state variable with maximum " speed " through the time in its maximum variable section, this In " speed " for state variable for the time derivative.Time scale is bigger, shows that " speed " of variable change is faster, if The order of magnitude difference of two time scales is larger, then can be classified as different time scales and carry out stepwise disposal.Time scale It is expressed as

Wherein Δ x is state variable x constant interval, is typically calculated with the difference of maxima and minima,For state First derivatives of the variable x to the time.The solution of the maximum of state variable, minimum value and first derivative, can pass through numerical value Method emulates or other conventional method approximations obtain, because the time scale order of magnitude of state variable differs larger, approximate calculation The speed division of state variable time scale is not interfered with.

Step 2：Establish singular perturbation system

The present invention only considers the guidance problems of fore-and-aft plane, by numerical simulation and the calculating of time scale, system variable Time scale size it is as shown in table 1 below, the vertical journey, height, trajectory tilt angle of guided missile can be divided into slow variable accordingly, by guided missile Acceleration, the angle of attack be divided into fast variable, establish the Second-Order Singular perturbation model of guided missile

X in formula, h, a_m,γ,α,a_cRespectively vertical journey, height, normal acceleration, trajectory tilt angle, the angle of attack and the normal direction of guided missile Command acceleration,For guided missile vertical journey, height, normal acceleration, trajectory tilt angle and the angle of attack to the derivative of time, V_mIt is a constant value for the speed of guided missile, τ_m,τ_αThe respectively single order link time constant and rate of turn time constant of guided missile, ε It is an a small amount of and 0 ＜ ε ＜＜ 1.

Table 1

Step 3：Solve the slow system of zeroth order

ε → 0 is made to obtain the reduced order system of original system

The amount of subscript 0 is expressed as zero order system parameter in formula and in full text.As can be seen that there is no attack in zero order system Horn shape state variable, accelerationAs control variable.This is due to the significantly larger than vertical journey of the time scale of the angle of attack and acceleration, height Degree and trajectory tilt angle, the response time to optimal value in the time scale of zero order system is that can ignore in a small amount, it is thus regarded that In the fast variable angle of attack and the acceleration dynamic equilibrium in optimal value all the time.Because the angle of attack, command acceleration be not in zeroth order system Occur in system, therefore choose all the time in the acceleration of optimal value dynamic equilibriumControl variable as zero order system.

System performance index depression of order is

Hamilton functions are

In formulaFor association's state variable of corresponding state variable.

By optimality condition

Line is entered to this zero order system, obtains the system after line

The system of linearisation can be designated as

Wherein

Performance indications are designated as

Subscript " T " represents the transposition of matrix, X (t in formula and in full text_f)=[y₁(t_f), y₂(t_f),γ(t_f)]^TFor system The state value of terminal juncture, X_f=[y_1f,y_2f,γ_f]^TConstrained for system terminal,.For weight coefficient matrix, WhereinS_γRespectively state variable y₁, γ weight coefficient.

Hamilton functions are

Association state matrix be

NoteBy transversality condition

λ(t_f)=S [X (t_f)-X_f] (13)

Above formula is substituted into formula (12), solving association's state variable matrix is

λ=Φ (t_f,t)S[X(t_f)-X_f] (14)

WhereinFor state-transition matrix, and have

By optimality index

U=-B^Tλ=- B^T(Φ(t_f, t))^TS[X(t_f)-X_f] (16)

Arrangement can obtain guided missile and be in the state of terminal juncture

Above formula is substituted into formula (16) to obtain

Weighting weight figure limitS_γ→ ∞ is obtained

Due to line-of-sight rate by lineSo it can obtain

Step 4：Solve the fast system of single order

The performance indications of first-order system are

Hamilton functions are

By optimality conditionIt can obtain, revised command acceleration should be

The co-state equation of first-order system is

NoteThen The transversality condition of first-order system is

τ in formula_fFor the terminal juncture value of first-order system.

It can be obtained by the co-state equation formula (30) on the angle of attack

λ_α(τ)=λ_α(τ_f)=S_α[α(τ_f)-α_f]=constant (33)

Solve

Wherein C₁,C₂For constant value.

It is abbreviated as

Wherein

Step 5：Single order residual non-uniformity τ_goCalculating

Caused by proposition first-order system and the speed difference of zero order system are due to the first-order lag of system in the present invention, therefore First-order lag of the residual non-uniformity of first-order system also for the residual non-uniformity of zero order system responds, i.e. the remaining flight of single order There is following relation time with zeroth order residual non-uniformity

Wherein, τ_TFor time delay constant.

Wherein

Wherein system

The Guidance Law has more succinct form and physical significance, and Guidance Law Section 1 is relevant with terminal angle of attack constraint, Section 2 is relevant with miss distance, and Section 3 is then relevant with terminal trajectory tilt angle constraint.The expression formula of Guidance Law is explicit Guidance, with Trajectory shaping Guidance Law has similar expression way, but ignores influence of the angle of attack derivative to end-fixity, can bring larger mistake Difference, therefore still emulated in the embodiment of the present invention using the command acceleration of formula (50) without reduced form.

Embodiment：

This implementation requires guided missile in Trajectory-terminal with desired trajectory tilt angle and the angle of attack and optimal energy expenditure hit mesh Mark, embodiment are emulated respectively using Guidance Law of the present invention with Gauss puppet spectrometry numerical method.Numerical solution is generally it can be thought that be Theoretical optimal solution, but typically because its is computationally intensive, needs off-line calculation without the real-time command solution frequently as guided missile.The present invention Guidance Law is parsing Guidance Law, by being contrasted with the Numerical Simulation Results of Gauss puppet spectrometry, can verify Guidance Law of the present invention with The Approximation effect of optimal solution and effect is met to end conswtraint, contrasted with terminal angle constraint Guidance Law simulation result, Ke Yifa Existing Guidance Law of the present invention has preferable guidance performance compared to conventional analytic Guidance Law.The initiation parameter and target component of guided missile It is as shown in table 2 below.

Table 2

By simulation result Fig. 3-5 as can be seen that the Guidance Law based on singular perturbation theory can be guided and led in the present invention Hit is played, and meets miss distance, terminal trajectory tilt angle, angle of attack constraint, demonstrates the validity of this Guidance Law；Simultaneously with height The optimum value solution similarity degree that this pseudo- spectrometry searches is higher, demonstrates this parsing Guidance Law and has to optimal solution and approaches well Property, with terminal angle constraint Guidance Law contrast, illustrate that the relatively conventional parsing Guidance Law of this parsing Guidance Law has preferably guidance Performance.The Missile Terminal state being given in Table 3 below under the effect of this Guidance Law, it can be seen that end error is smaller, can be very Good meets end conswtraint condition.

Table 3.

Claims

1. a kind of tape terminal trajectory tilt angle and the singular perturbation suboptimum Guidance Law of angle of attack constraint, including following steps：

Step 1：Determine the time scale of system state variables

The computational methods of time scale are as follows

<mrow> <msub> <mi>S</mi> <mi>x</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> </mrow> <mrow> <mi>&Delta;</mi> <mi>x</mi> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>

Wherein Δ x is state variable x constant interval, is typically calculated with the difference of maxima and minima,For state variable X first derivative；

Step 2：Establish singular perturbation system

Only consider the guidance problems of fore-and-aft plane, by numerical simulation and the calculating of time scale, by the vertical journey of guided missile, height, Trajectory tilt angle regards slow variable as, regards the acceleration of guided missile, the angle of attack as fast variable, establishes the Second-Order Singular perturbation model of guided missile

<mrow> <mtable> <mtr> <mtd> <mrow> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>V</mi> <mi>m</mi> </msub> <mi>cos</mi> <mi>&gamma;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>h</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>V</mi> <mi>m</mi> </msub> <mi>sin</mi> <mi>&gamma;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>&epsiv;</mi> <msub> <mover> <mi>a</mi> <mo>&CenterDot;</mo> </mover> <mi>m</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mfrac> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>-</mo> <msub> <mi>a</mi> <mi>m</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>&gamma;</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mfrac> <msub> <mi>a</mi> <mi>m</mi> </msub> <msub> <mi>V</mi> <mi>m</mi> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>&epsiv;</mi> <mover> <mi>&alpha;</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mfrac> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> <mrow> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> <msub> <mi>V</mi> <mi>m</mi> </msub> </mrow> </mfrac> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>-</mo> <msub> <mi>a</mi> <mi>m</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>

X in formula, h, a_m,γ,α,a_cThe respectively vertical journey of guided missile, height, normal acceleration, trajectory tilt angle, the angle of attack and normal direction instruction Acceleration,For guided missile vertical journey, height, normal acceleration, trajectory tilt angle and the angle of attack to the derivative of time, V_mFor The speed of guided missile, is constant value, τ_m,τ_αThe respectively single order link time constant and rate of turn time constant of guided missile, ε are individual small Measure and there are 0 ＜ ε ＜＜ 1；

<mrow> <mtable> <mtr> <mtd> <mrow> <mi>J</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>S</mi> <mi>x</mi> </msub> <msup> <mrow> <mo>&lsqb;</mo> <mi>x</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>x</mi> <mi>f</mi> </msub> <mo>&rsqb;</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>S</mi> <mi>h</mi> </msub> <msup> <mrow> <mo>&lsqb;</mo> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>h</mi> <mi>f</mi> </msub> <mo>&rsqb;</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>S</mi> <mi>&gamma;</mi> </msub> <msup> <mrow> <mo>&lsqb;</mo> <mi>&gamma;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>&gamma;</mi> <mi>f</mi> </msub> <mo>&rsqb;</mo> </mrow> <mn>2</mn> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>S</mi> <mi>&alpha;</mi> </msub> <msup> <mrow> <mo>&lsqb;</mo> <mi>&alpha;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>&alpha;</mi> <mi>f</mi> </msub> <mo>&rsqb;</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <msub> <mi>t</mi> <mi>f</mi> </msub> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mi>d</mi> <mi>&tau;</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>

T in formula_fThe terminal juncture intersected for guided missile with target, x_f,h_f,γ_f,α_fFor Missile Terminal indulge journey, height, trajectory tilt angle and The angle of attack constrains, x (t_f),h(t_f),γ(t_f),α(t_f) for guided missile in vertical journey, height, trajectory tilt angle and the angle of attack of collision moment, a_c For guided missile normal direction command acceleration, S_x,S_h,S_γ,S_αFor the weight coefficient of corresponding state variable bound；

Step 3：Solve the slow system of zeroth order

ε → 0 is made to obtain the reduced order system of original system

<mrow> <mtable> <mtr> <mtd> <mrow> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>V</mi> <mi>m</mi> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&gamma;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>h</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>V</mi> <mi>m</mi> </msub> <mi>sin</mi> <mi>&gamma;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>&gamma;</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mfrac> <msubsup> <mi>a</mi> <mi>m</mi> <mn>0</mn> </msubsup> <msub> <mi>V</mi> <mi>m</mi> </msub> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>

The amount of subscript 0 is expressed as zero order system parameter in formula；There is no angle of attack state variable, acceleration in zero order systemInto To control variable；Because the angle of attack, command acceleration have not occurred in zero order system, therefore choose all the time in optimal value dynamic equilibrium AccelerationControl variable as zero order system；

System performance index depression of order is

<mrow> <msup> <mi>J</mi> <mn>0</mn> </msup> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>S</mi> <mi>x</mi> </msub> <msup> <mrow> <mo>&lsqb;</mo> <mi>x</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>x</mi> <mi>f</mi> </msub> <mo>&rsqb;</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>S</mi> <mi>h</mi> </msub> <msup> <mrow> <mo>&lsqb;</mo> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>h</mi> <mi>f</mi> </msub> <mo>&rsqb;</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>S</mi> <mi>&gamma;</mi> </msub> <msup> <mrow> <mo>&lsqb;</mo> <mi>&gamma;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>&gamma;</mi> <mi>f</mi> </msub> <mo>&rsqb;</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <msub> <mi>t</mi> <mi>f</mi> </msub> </msubsup> <msup> <mrow> <mo>(</mo> <msubsup> <mi>a</mi> <mi>m</mi> <mn>0</mn> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mi>d</mi> <mi>&tau;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>

Hamilton functions are

<mrow> <msup> <mi>H</mi> <mn>0</mn> </msup> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mrow> <mo>(</mo> <msubsup> <mi>a</mi> <mi>m</mi> <mn>0</mn> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>&lambda;</mi> <mi>x</mi> <mn>0</mn> </msubsup> <msub> <mi>V</mi> <mi>m</mi> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&gamma;</mi> <mo>+</mo> <msubsup> <mi>&lambda;</mi> <mi>h</mi> <mn>0</mn> </msubsup> <msub> <mi>V</mi> <mi>m</mi> </msub> <mi>sin</mi> <mi>&gamma;</mi> <mo>+</mo> <msubsup> <mi>&lambda;</mi> <mi>&gamma;</mi> <mn>0</mn> </msubsup> <mfrac> <msubsup> <mi>a</mi> <mi>m</mi> <mn>0</mn> </msubsup> <msub> <mi>V</mi> <mi>m</mi> </msub> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>

In formulaFor association's state variable of corresponding state variable；

By optimality condition

<mrow> <msubsup> <mi>&lambda;</mi> <mi>&gamma;</mi> <mn>0</mn> </msubsup> <mo>=</mo> <mo>-</mo> <msub> <mi>V</mi> <mi>m</mi> </msub> <msubsup> <mi>a</mi> <mi>m</mi> <mn>0</mn> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>

Line is entered to this zero order system, obtains the system after line

<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>y</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>y</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mo>-</mo> <msubsup> <mi>a</mi> <mi>m</mi> <mn>0</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>&gamma;</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mfrac> <msubsup> <mi>a</mi> <mi>m</mi> <mn>0</mn> </msubsup> <msub> <mi>V</mi> <mi>m</mi> </msub> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>

Wherein y₁For the relative displacement in vertical direction of guided missile and target, y₂It is guided missile and target in the relative of vertical direction Speed；

The system of linearisation is designated as

<mrow> <mover> <mi>X</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mi>A</mi> <mi>X</mi> <mo>+</mo> <mi>B</mi> <mi>u</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>

Wherein

Performance indications are designated as

<mrow> <msup> <mi>J</mi> <mn>0</mn> </msup> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mrow> <mo>&lsqb;</mo> <mi>X</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>X</mi> <mi>f</mi> </msub> <mo>&rsqb;</mo> </mrow> <mi>T</mi> </msup> <mi>S</mi> <mo>&lsqb;</mo> <mi>X</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>X</mi> <mi>f</mi> </msub> <mo>&rsqb;</mo> <mo>+</mo> <msubsup> <mo>&Integral;</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <msub> <mi>t</mi> <mi>f</mi> </msub> </msubsup> <msup> <mi>u</mi> <mn>2</mn> </msup> <mi>d</mi> <mi>&tau;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>

Subscript " T " represents the transposition of matrix, X (t in formula_f)=[y₁(t_f),y₂(t_f),γ(t_f)]^TFor the state at system terminal moment Value, X_f=[y_1f,y_2f,γ_f]^TConstrained for system terminal；For weight coefficient matrix, wherein S_y1,S_γRespectively State variable y₁, γ weight coefficient；

Hamilton functions are

<mrow> <msup> <mi>H</mi> <mn>0</mn> </msup> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mi>u</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>&lambda;</mi> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mi>A</mi> <mi>X</mi> <mo>+</mo> <mi>B</mi> <mi>u</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>

Association state matrix be

<mrow> <mover> <mi>&lambda;</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mo>&part;</mo> <msup> <mi>H</mi> <mn>0</mn> </msup> </mrow> <mrow> <mo>&part;</mo> <mi>X</mi> </mrow> </mfrac> <mo>=</mo> <mo>-</mo> <msup> <mi>A</mi> <mi>T</mi> </msup> <mi>&lambda;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow> 2

NoteBy transversality condition

λ(t_f)=S [X (t_f)-X_f] (13)

λ=Φ (t_f,t)S[X(t_f)-X_f] (14)

WhereinFor state-transition matrix, and have

T in formula_go=t_f- t is residual non-uniformity；WithApproximate substitution, R in formula_TMFor guided missile and the distance of target；

By optimality index

U=-B^Tλ=- B^T(Φ(t_f,t))^TS[X(t_f)-X_f] (16)

Due to it is determined that control under, Missile Terminal state X (t_f) determine, the control solution obtained by above formula is substituted into system equation (9) and carry out State integral and obtain

<mrow> <mi>X</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>&Phi;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mi>X</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mo>&Integral;</mo> <mi>t</mi> <msub> <mi>t</mi> <mi>f</mi> </msub> </msubsup> <mi>&Phi;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>,</mo> <mi>&tau;</mi> <mo>)</mo> </mrow> <msup> <mi>BB</mi> <mi>T</mi> </msup> <msup> <mrow> <mo>(</mo> <mi>&Phi;</mi> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mi>S</mi> <mo>&lsqb;</mo> <mi>X</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>X</mi> <mi>f</mi> </msub> <mo>&rsqb;</mo> <mi>d</mi> <mi>&tau;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow>

Arrangement obtains guided missile and is in the state of terminal juncture

<mrow> <mi>X</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>&lsqb;</mo> <mi>I</mi> <mo>+</mo> <msubsup> <mo>&Integral;</mo> <mi>t</mi> <msub> <mi>t</mi> <mi>f</mi> </msub> </msubsup> <mi>&Phi;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>,</mo> <mi>&tau;</mi> <mo>)</mo> </mrow> <msup> <mi>BB</mi> <mi>T</mi> </msup> <msup> <mrow> <mo>(</mo> <mi>&Phi;</mi> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mi>S</mi> <mi>d</mi> <mi>&tau;</mi> <mo>&rsqb;</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>&lsqb;</mo> <mi>&Phi;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mi>X</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>X</mi> <mi>f</mi> </msub> <mo>&rsqb;</mo> <mo>+</mo> <msub> <mi>X</mi> <mi>f</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>

Above formula is substituted into formula (16) to obtain

<mrow> <mi>u</mi> <mo>=</mo> <mo>-</mo> <msup> <mi>B</mi> <mi>T</mi> </msup> <msup> <mrow> <mo>(</mo> <mi>&Phi;</mi> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mi>S</mi> <msup> <mrow> <mo>&lsqb;</mo> <mi>I</mi> <mo>+</mo> <msubsup> <mo>&Integral;</mo> <mi>t</mi> <msub> <mi>t</mi> <mi>f</mi> </msub> </msubsup> <mi>&Phi;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>,</mo> <mi>&tau;</mi> <mo>)</mo> </mrow> <msup> <mi>BB</mi> <mi>T</mi> </msup> <msup> <mrow> <mo>(</mo> <mi>&Phi;</mi> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mi>S</mi> <mi>d</mi> <mi>&tau;</mi> <mo>&rsqb;</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>&lsqb;</mo> <mi>&Phi;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mi>X</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>X</mi> <mi>f</mi> </msub> <mo>&rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>

Weighting weight figure limitS_γ→ ∞ is obtained

<mrow> <msubsup> <mi>a</mi> <mi>m</mi> <mn>0</mn> </msubsup> <mo>=</mo> <mi>u</mi> <mo>=</mo> <mfrac> <mn>6</mn> <msubsup> <mi>t</mi> <mrow> <mi>g</mi> <mi>o</mi> </mrow> <mn>2</mn> </msubsup> </mfrac> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <msub> <mi>t</mi> <mrow> <mi>g</mi> <mi>o</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>V</mi> <mi>m</mi> </msub> </mrow> <msub> <mi>t</mi> <mrow> <mi>g</mi> <mi>o</mi> </mrow> </msub> </mfrac> <mrow> <mo>(</mo> <mi>&gamma;</mi> <mo>-</mo> <msub> <mi>&gamma;</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow>

Due to line-of-sight rate by lineSo obtain

<mrow> <msubsup> <mi>a</mi> <mi>m</mi> <mn>0</mn> </msubsup> <mo>=</mo> <mn>6</mn> <msub> <mi>V</mi> <mi>m</mi> </msub> <mover> <mi>&lambda;</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>V</mi> <mi>m</mi> </msub> </mrow> <msub> <mi>t</mi> <mrow> <mi>g</mi> <mi>o</mi> </mrow> </msub> </mfrac> <mrow> <mo>(</mo> <mi>&gamma;</mi> <mo>-</mo> <msub> <mi>&gamma;</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow>

This be system linearization after optimum control Guidance Law, stood good in nonlinear system, now line-of-sight rate by line to Under type calculates

<mrow> <mover> <mi>&lambda;</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mfrac> <mrow> <msub> <mi>R</mi> <mrow> <mi>T</mi> <mi>M</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>V</mi> <mi>m</mi> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&gamma;</mi> <mo>-</mo> <msub> <mi>R</mi> <mrow> <mi>T</mi> <mi>M</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>V</mi> <mi>m</mi> </msub> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&gamma;</mi> </mrow> <msubsup> <mi>R</mi> <mrow> <mi>T</mi> <mi>M</mi> </mrow> <mn>2</mn> </msubsup> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mrow>

R in formula_TM1,R_TM2Respectively missile-target distance R_TMComponent in horizontally and vertically direction；

Step 4：Solve the fast system of single order

Carry out time scale stretching conversion, orderAnd allow ε → 0, obtain first-order system equation

<mrow> <mtable> <mtr> <mtd> <mrow> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>h</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>&gamma;</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>23</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mover> <mi>a</mi> <mo>&CenterDot;</mo> </mover> <mi>m</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msubsup> <mi>a</mi> <mi>c</mi> <mn>1</mn> </msubsup> <mo>-</mo> <msub> <mi>a</mi> <mi>m</mi> </msub> </mrow> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>24</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mover> <mi>&alpha;</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mfrac> <mrow> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mi>a</mi> <mi>c</mi> <mn>1</mn> </msubsup> <mo>-</mo> <msub> <mi>a</mi> <mi>m</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> <msub> <mi>V</mi> <mi>m</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>25</mn> <mo>)</mo> </mrow> </mrow>

The amount of subscript 1 is expressed as first-order system parameter in formula；In first-order system, the rate of change of journey, height and trajectory tilt angle is indulged It is very slow relative to acceleration and angle of attack variation because these three variant time yardsticks are much larger than the angle of attack and acceleration for 0, Therefore it is approximately constant value that journey, height, trajectory tilt angle are indulged in first-order system, the value of its value and corresponding association's state variable is equal in zeroth order system The value solved in system；

The performance indications of first-order system are

<mrow> <mtable> <mtr> <mtd> <mrow> <msup> <mi>J</mi> <mn>1</mn> </msup> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>S</mi> <mi>x</mi> </msub> <msup> <mrow> <mo>&lsqb;</mo> <mi>x</mi> <mrow> <mo>(</mo> <msub> <mi>&tau;</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>x</mi> <mi>f</mi> </msub> <mo>&rsqb;</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>S</mi> <mi>h</mi> </msub> <msup> <mrow> <mo>&lsqb;</mo> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>&tau;</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>h</mi> <mi>f</mi> </msub> <mo>&rsqb;</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>S</mi> <mi>&gamma;</mi> </msub> <msup> <mrow> <mo>&lsqb;</mo> <mi>&gamma;</mi> <mrow> <mo>(</mo> <msub> <mi>&tau;</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>&gamma;</mi> <mi>f</mi> </msub> <mo>&rsqb;</mo> </mrow> <mn>2</mn> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>S</mi> <mi>&alpha;</mi> </msub> <msup> <mrow> <mo>&lsqb;</mo> <mi>&alpha;</mi> <mrow> <mo>(</mo> <msub> <mi>&tau;</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>&alpha;</mi> <mi>f</mi> </msub> <mo>&rsqb;</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <msub> <mi>&tau;</mi> <mi>f</mi> </msub> </msubsup> <msup> <mrow> <mo>(</mo> <msubsup> <mi>a</mi> <mi>c</mi> <mn>1</mn> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mi>d</mi> <mi>&tau;</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>26</mn> <mo>)</mo> </mrow> </mrow>

Hamilton functions are

<mrow> <mtable> <mtr> <mtd> <mrow> <msup> <mi>H</mi> <mn>1</mn> </msup> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mrow> <mo>(</mo> <msubsup> <mi>a</mi> <mi>c</mi> <mn>1</mn> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>&lambda;</mi> <mi>x</mi> <mn>0</mn> </msubsup> <msub> <mi>V</mi> <mi>m</mi> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&gamma;</mi> <mo>+</mo> <msubsup> <mi>&lambda;</mi> <mi>h</mi> <mn>0</mn> </msubsup> <msub> <mi>V</mi> <mi>m</mi> </msub> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&gamma;</mi> <mo>+</mo> <msub> <mi>&lambda;</mi> <msub> <mi>a</mi> <mi>m</mi> </msub> </msub> <mfrac> <mrow> <msubsup> <mi>a</mi> <mi>c</mi> <mn>1</mn> </msubsup> <mo>-</mo> <msub> <mi>a</mi> <mi>m</mi> </msub> </mrow> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msubsup> <mi>&lambda;</mi> <mi>&gamma;</mi> <mn>0</mn> </msubsup> <mfrac> <msub> <mi>a</mi> <mi>m</mi> </msub> <msub> <mi>V</mi> <mi>m</mi> </msub> </mfrac> <mo>+</mo> <msub> <mi>&lambda;</mi> <mi>&alpha;</mi> </msub> <mfrac> <mrow> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mi>a</mi> <mi>c</mi> <mn>1</mn> </msubsup> <mo>-</mo> <msub> <mi>a</mi> <mi>m</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> <msub> <mi>V</mi> <mi>m</mi> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>27</mn> <mo>)</mo> </mrow> </mrow>

In formulaλ_αRespectively state variable a_m, association's state variable corresponding to α；

By optimality conditionDraw, revised command acceleration should be

<mrow> <msubsup> <mi>a</mi> <mi>c</mi> <mn>1</mn> </msubsup> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <msub> <mi>V</mi> <mi>m</mi> </msub> <msub> <mi>&lambda;</mi> <msub> <mi>a</mi> <mi>m</mi> </msub> </msub> <mo>+</mo> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> <msub> <mi>&lambda;</mi> <mi>&alpha;</mi> </msub> </mrow> <mrow> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> <msub> <mi>V</mi> <mi>m</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>28</mn> <mo>)</mo> </mrow> </mrow>

It is intended to obtain revised acceleration, then must calculates association's state variableλ_αValue；

The co-state equation of first-order system is

<mrow> <msub> <mover> <mi>&lambda;</mi> <mo>&CenterDot;</mo> </mover> <msub> <mi>a</mi> <mi>m</mi> </msub> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mo>&part;</mo> <msup> <mi>H</mi> <mn>1</mn> </msup> </mrow> <mrow> <mo>&part;</mo> <msub> <mi>a</mi> <mi>m</mi> </msub> </mrow> </mfrac> <mo>=</mo> <mfrac> <msub> <mi>&lambda;</mi> <msub> <mi>a</mi> <mi>m</mi> </msub> </msub> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mfrac> <mo>-</mo> <mfrac> <msubsup> <mi>&lambda;</mi> <mi>&gamma;</mi> <mn>0</mn> </msubsup> <msub> <mi>V</mi> <mi>m</mi> </msub> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> <msub> <mi>&lambda;</mi> <mi>&alpha;</mi> </msub> </mrow> <mrow> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> <msub> <mi>V</mi> <mi>m</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>29</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mover> <mi>&lambda;</mi> <mo>&CenterDot;</mo> </mover> <mi>&alpha;</mi> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mo>&part;</mo> <msup> <mi>H</mi> <mn>1</mn> </msup> </mrow> <mrow> <mo>&part;</mo> <mi>&alpha;</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>30</mn> <mo>)</mo> </mrow> </mrow>

NoteThen

The transversality condition of first-order system is

<mrow> <msub> <mi>&lambda;</mi> <msub> <mi>a</mi> <mi>m</mi> </msub> </msub> <mrow> <mo>(</mo> <msub> <mi>&tau;</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>31</mn> <mo>)</mo> </mrow> </mrow> 4

<mrow> <msub> <mi>&lambda;</mi> <mi>&alpha;</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&tau;</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mo>&part;</mo> <msup> <mi>&phi;</mi> <mn>1</mn> </msup> </mrow> <mrow> <mo>&part;</mo> <mi>&alpha;</mi> <mrow> <mo>(</mo> <msub> <mi>&tau;</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>=</mo> <msub> <mi>S</mi> <mi>&alpha;</mi> </msub> <mo>&lsqb;</mo> <mi>&alpha;</mi> <mrow> <mo>(</mo> <msub> <mi>&tau;</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>&alpha;</mi> <mi>f</mi> </msub> <mo>&rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>32</mn> <mo>)</mo> </mrow> </mrow>

τ in formula_fFor the terminal juncture value of first-order system；

Obtained by the co-state equation formula (30) on the angle of attack

λ_α(τ)=λ_α(τ_f)=S_α[α(τ_f)-α_f]=constant (33)

That is λ_αFor the constant value not changed over time, then when the co-state equation formula (29) on acceleration is regarded as on first-order system Between τ differential equation of first order, solve and substitute into the association's state variable tried to achieve in zero order systemFormula (7) obtains with transversality condition formula (31) To association's state variable on acceleration

<mrow> <msub> <mi>&lambda;</mi> <msub> <mi>a</mi> <mi>m</mi> </msub> </msub> <mo>=</mo> <mo>-</mo> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> <msub> <mi>&lambda;</mi> <mi>&alpha;</mi> </msub> </mrow> <msub> <mi>V</mi> <mi>m</mi> </msub> </mfrac> <mo>+</mo> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> <msubsup> <mi>a</mi> <mi>m</mi> <mn>0</mn> </msubsup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <msub> <mi>&tau;</mi> <mrow> <mi>g</mi> <mi>o</mi> </mrow> </msub> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mfrac> </mrow> </msup> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>34</mn> <mo>)</mo> </mrow> </mrow>

τ in formula_go=τ_f- τ is the residual non-uniformity of first-order system, for τ_goSolution can provide in steps of 5；

It is as follows on τ second time derivative that the angle of attack is obtained to first-order system equation (25) derivation

<mrow> <mover> <mi>&alpha;</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mo>=</mo> <mfrac> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> <mrow> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> <msub> <mi>V</mi> <mi>m</mi> </msub> </mrow> </mfrac> <mrow> <mo>(</mo> <msubsup> <mover> <mi>a</mi> <mo>&CenterDot;</mo> </mover> <mi>c</mi> <mn>1</mn> </msubsup> <mo>-</mo> <msub> <mover> <mi>a</mi> <mo>&CenterDot;</mo> </mover> <mi>m</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>35</mn> <mo>)</mo> </mrow> </mrow>

The first derivative of command acceleration is calculated by formula (28) in formula

<mrow> <msubsup> <mover> <mi>a</mi> <mo>&CenterDot;</mo> </mover> <mi>c</mi> <mn>1</mn> </msubsup> <mo>=</mo> <mo>-</mo> <mfrac> <msub> <mover> <mi>&lambda;</mi> <mo>&CenterDot;</mo> </mover> <msub> <mi>a</mi> <mi>m</mi> </msub> </msub> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mfrac> <mo>=</mo> <mo>-</mo> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> <msub> <mi>&lambda;</mi> <mi>&alpha;</mi> </msub> </mrow> <mrow> <msubsup> <mi>&tau;</mi> <mi>m</mi> <mn>2</mn> </msubsup> <msub> <mi>V</mi> <mi>m</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <msubsup> <mi>a</mi> <mi>m</mi> <mn>0</mn> </msubsup> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mfrac> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <msub> <mi>&tau;</mi> <mrow> <mi>g</mi> <mi>o</mi> </mrow> </msub> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mfrac> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>36</mn> <mo>)</mo> </mrow> </mrow>

Obtaining acceleration and the angle of attack by system equation (24) (25) has following relation

<mrow> <msub> <mover> <mi>a</mi> <mo>&CenterDot;</mo> </mover> <mi>m</mi> </msub> <mo>=</mo> <mfrac> <msub> <mi>V</mi> <mi>m</mi> </msub> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> </mfrac> <mover> <mi>&alpha;</mi> <mo>&CenterDot;</mo> </mover> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>37</mn> <mo>)</mo> </mrow> </mrow>

Formula (36) (37) is then substituted into formula (35) and obtains the second order differential equation containing only the angle of attack

<mrow> <mover> <mi>&alpha;</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mfrac> <mover> <mi>&alpha;</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mo>-</mo> <mrow> <mo>(</mo> <mfrac> <mrow> <msubsup> <mi>&tau;</mi> <mi>&alpha;</mi> <mn>2</mn> </msubsup> <msub> <mi>&lambda;</mi> <mi>&alpha;</mi> </msub> </mrow> <mrow> <msubsup> <mi>&tau;</mi> <mi>m</mi> <mn>3</mn> </msubsup> <msubsup> <mi>V</mi> <mi>m</mi> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> <msubsup> <mi>a</mi> <mi>m</mi> <mn>0</mn> </msubsup> </mrow> <mrow> <msubsup> <mi>&tau;</mi> <mi>m</mi> <mn>2</mn> </msubsup> <msub> <mi>V</mi> <mi>m</mi> </msub> </mrow> </mfrac> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <msub> <mi>&tau;</mi> <mrow> <mi>g</mi> <mi>o</mi> </mrow> </msub> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mfrac> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>38</mn> <mo>)</mo> </mrow> </mrow>

Solve

<mrow> <mi>&alpha;</mi> <mo>=</mo> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>C</mi> <mn>2</mn> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <mi>&tau;</mi> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mfrac> </mrow> </msup> <mo>-</mo> <mrow> <mo>(</mo> <mfrac> <mrow> <msubsup> <mi>&tau;</mi> <mi>&alpha;</mi> <mn>2</mn> </msubsup> <msub> <mi>&lambda;</mi> <mi>&alpha;</mi> </msub> </mrow> <mrow> <mn>2</mn> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> <msubsup> <mi>V</mi> <mi>m</mi> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> <msubsup> <mi>a</mi> <mi>m</mi> <mn>0</mn> </msubsup> </mrow> <mrow> <mn>2</mn> <msub> <mi>V</mi> <mi>m</mi> </msub> </mrow> </mfrac> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <msub> <mi>&tau;</mi> <mrow> <mi>g</mi> <mi>o</mi> </mrow> </msub> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mfrac> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>39</mn> <mo>)</mo> </mrow> </mrow>

Wherein C₁,C₂For constant value；

The current time angle of attack and angle of attack derivative state value α (τ)=α, α ' (τ)=α ' are substituted into, terminal juncture τ is tried to achieve in calculating_fWhen attack Angle value

<mrow> <mi>&alpha;</mi> <mrow> <mo>(</mo> <msub> <mi>&tau;</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>&alpha;</mi> <mo>+</mo> <msup> <mi>&alpha;</mi> <mo>&prime;</mo> </msup> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <msub> <mi>&tau;</mi> <mrow> <mi>g</mi> <mi>o</mi> </mrow> </msub> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mfrac> </mrow> </msup> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <mfrac> <mrow> <msubsup> <mi>&tau;</mi> <mi>&alpha;</mi> <mn>2</mn> </msubsup> <msub> <mi>&lambda;</mi> <mi>&alpha;</mi> </msub> </mrow> <mrow> <mn>2</mn> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> <msubsup> <mi>V</mi> <mi>m</mi> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> <msubsup> <mi>a</mi> <mi>m</mi> <mn>0</mn> </msubsup> </mrow> <mrow> <mn>2</mn> <msub> <mi>V</mi> <mi>m</mi> </msub> </mrow> </mfrac> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <msub> <mi>&tau;</mi> <mrow> <mi>g</mi> <mi>o</mi> </mrow> </msub> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mfrac> </mrow> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>40</mn> <mo>)</mo> </mrow> </mrow>

In addition, by transversality condition formula (33) and to weight coefficient S_αTake limit S_α→ ∞, obtaining the desired terminal juncture angle of attack is

<mrow> <mi>&alpha;</mi> <mrow> <mo>(</mo> <msub> <mi>&tau;</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mi>lim</mi> <mrow> <msub> <mi>S</mi> <mi>&alpha;</mi> </msub> <mo>&RightArrow;</mo> <mi>&infin;</mi> </mrow> </munder> <mrow> <mo>(</mo> <mfrac> <msub> <mi>&lambda;</mi> <mi>&alpha;</mi> </msub> <msub> <mi>S</mi> <mi>&alpha;</mi> </msub> </mfrac> <mo>+</mo> <msub> <mi>&alpha;</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>&alpha;</mi> <mi>f</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>41</mn> <mo>)</mo> </mrow> </mrow>

Then more than simultaneous two formulas (40) (41) obtain be on association's state variable of the angle of attack

<mrow> <msub> <mi>&lambda;</mi> <mi>&alpha;</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> <msubsup> <mi>V</mi> <mi>m</mi> <mn>2</mn> </msubsup> <mo>&lsqb;</mo> <mi>&alpha;</mi> <mo>-</mo> <msub> <mi>&alpha;</mi> <mi>f</mi> </msub> <mo>+</mo> <msup> <mi>&alpha;</mi> <mo>&prime;</mo> </msup> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <msub> <mi>&tau;</mi> <mrow> <mi>g</mi> <mi>o</mi> </mrow> </msub> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mfrac> </mrow> </msup> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <msub> <mi>&tau;</mi> <mrow> <mi>g</mi> <mi>o</mi> </mrow> </msub> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mfrac> </mrow> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>-</mo> <mfrac> <mrow> <msub> <mi>V</mi> <mi>m</mi> </msub> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> <msubsup> <mi>a</mi> <mi>m</mi> <mn>0</mn> </msubsup> </mrow> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>42</mn> <mo>)</mo> </mrow> </mrow>

The association's state variable that will have been tried to achieveλ_αIt is updated to and obtains final satisfaction in revised command acceleration formula (28) and own The optimum control Guidance Law of constraints

<mrow> <msubsup> <mi>a</mi> <mi>c</mi> <mn>1</mn> </msubsup> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>V</mi> <mi>m</mi> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <msub> <mi>&tau;</mi> <mrow> <mi>g</mi> <mi>o</mi> </mrow> </msub> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mfrac> </mrow> </msup> <mo>&lsqb;</mo> <msub> <mi>&alpha;</mi> <mi>f</mi> </msub> <mo>-</mo> <mi>&alpha;</mi> <mo>-</mo> <msup> <mi>&alpha;</mi> <mo>&prime;</mo> </msup> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <msub> <mi>&tau;</mi> <mrow> <mi>g</mi> <mi>o</mi> </mrow> </msub> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mfrac> </mrow> </msup> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> <mrow> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <msub> <mi>&tau;</mi> <mrow> <mi>g</mi> <mi>o</mi> </mrow> </msub> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mfrac> </mrow> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <msubsup> <mi>a</mi> <mi>m</mi> <mn>0</mn> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>43</mn> <mo>)</mo> </mrow> </mrow>

It is abbreviated as

<mrow> <msubsup> <mi>a</mi> <mi>c</mi> <mn>1</mn> </msubsup> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>V</mi> <mi>m</mi> </msub> <mi>k</mi> <mo>&lsqb;</mo> <msub> <mi>&alpha;</mi> <mi>f</mi> </msub> <mo>-</mo> <mi>&alpha;</mi> <mo>-</mo> <msup> <mi>&alpha;</mi> <mo>&prime;</mo> </msup> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> <mrow> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>k</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <msubsup> <mi>a</mi> <mi>m</mi> <mn>0</mn> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>44</mn> <mo>)</mo> </mrow> </mrow>

Wherein

Step 5：Single order residual non-uniformity τ_goCalculating

Wherein, τ_TFor time delay constant；

Substitute into τ_go=τ_fIt is as follows on zeroth order temporal t differential equation of first order that-τ obtains single order time τ

Due to τ (t_f)=τ_f, t_go=t_f-t₀, arrangement obtains single order residual non-uniformity and is

Step 6：The optimum control command acceleration for meeting miss distance, terminal trajectory tilt angle, the angle of attack is calculated, carries out instruction renewal；

<mrow> <msubsup> <mi>a</mi> <mi>c</mi> <mn>1</mn> </msubsup> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>V</mi> <mi>m</mi> </msub> <mi>k</mi> <mo>&lsqb;</mo> <msub> <mi>&alpha;</mi> <mi>f</mi> </msub> <mo>-</mo> <mi>&alpha;</mi> <mo>-</mo> <msup> <mi>&alpha;</mi> <mo>&prime;</mo> </msup> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> <mrow> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>k</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mn>6</mn> <msub> <mi>V</mi> <mi>m</mi> </msub> <mover> <mi>&lambda;</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>V</mi> <mi>m</mi> </msub> </mrow> <msub> <mi>t</mi> <mrow> <mi>g</mi> <mi>o</mi> </mrow> </msub> </mfrac> <mrow> <mo>(</mo> <mi>&gamma;</mi> <mo>-</mo> <msub> <mi>&gamma;</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>49</mn> <mo>)</mo> </mrow> </mrow>

Wherein

Due to differential term α ' in Guidance Law be present, Guidance Law can be caused to instruct vibration in end by a relatively large margin, so as to cause compared with More energy expenditures；Therefore, this is guidanceed command and be modified, to weaken oscillation effect caused by differential term；The shadow of differential term It is its coefficient 1-k to ring the factor, is a parameter changed with residual non-uniformity, has (1-k) ＜ 1, and work as residual non-uniformity Level off to 0 when, 1-k also levels off to 0；Differential term can largely be weakened to the shadow guidanceed command using its cubic form Ring, so as to effectively suppress end instruction vibration；Then it is revised guidance command for

<mrow> <msubsup> <mi>a</mi> <mi>c</mi> <mn>1</mn> </msubsup> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>V</mi> <mi>m</mi> </msub> <mi>k</mi> <mo>&lsqb;</mo> <msub> <mi>&alpha;</mi> <mi>f</mi> </msub> <mo>-</mo> <mi>&alpha;</mi> <mo>-</mo> <msup> <mi>&alpha;</mi> <mo>&prime;</mo> </msup> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>k</mi> <mo>)</mo> </mrow> <mn>3</mn> </msup> <mo>&rsqb;</mo> </mrow> <mrow> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>k</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mn>6</mn> <msub> <mi>V</mi> <mi>m</mi> </msub> <mover> <mi>&lambda;</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>V</mi> <mi>m</mi> </msub> </mrow> <msub> <mi>t</mi> <mrow> <mi>g</mi> <mi>o</mi> </mrow> </msub> </mfrac> <mrow> <mo>(</mo> <mi>&gamma;</mi> <mo>-</mo> <msub> <mi>&gamma;</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>50</mn> <mo>)</mo> </mrow> </mrow>

Wherein

Ignore influence of the angle of attack derivative to command acceleration amendment, the Guidance Law is further simplified as

<mrow> <msubsup> <mi>a</mi> <mi>c</mi> <mn>1</mn> </msubsup> <mo>=</mo> <mi>N</mi> <mrow> <mo>(</mo> <msub> <mi>&alpha;</mi> <mi>f</mi> </msub> <mo>-</mo> <mi>&alpha;</mi> <mo>)</mo> </mrow> <mo>+</mo> <mn>6</mn> <msub> <mi>V</mi> <mi>m</mi> </msub> <mover> <mi>&lambda;</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>V</mi> <mi>m</mi> </msub> </mrow> <msub> <mi>t</mi> <mrow> <mi>g</mi> <mi>o</mi> </mrow> </msub> </mfrac> <mrow> <mo>(</mo> <mi>&gamma;</mi> <mo>-</mo> <msub> <mi>&gamma;</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>51</mn> <mo>)</mo> </mrow> </mrow>

Wherein system