CN103995540A - Method for rapidly generating finite time track of hypersonic aircraft - Google Patents

Abstract

The invention relates to finite time tracks of aircrafts, in particular to a method for rapidly generating a finite time track of a hypersonic aircraft. The method aims to solve the problems that the process of deriving an optimal solution according to a traditional method is complex, nonlinear programming problems with complex constraints cannot be effectively solved, the rapidity and real-time performance of track optimization cannot be met, and complex reentry environmental disturbance and environmental uncertainty cannot be handled. The method includes the steps that 1, the motion model of the hypersonic aircraft is obtained; 2, the nonlinear programming problem is formed; 3, the nonlinear programming problem is described to be a quadric form convex problem; 4, a high-speed resolver is generated; 5, the quadric form convex optimization problem is solved, and the solution is analyzed. The method is applied to the field of the finite time tracks of the aircrafts.

Description

A kind of finite time track rapid generation of hypersonic aircraft

Technical field

The present invention relates to the finite time track rapid generation of hypersonic aircraft.

Background technology

Hypersonic technology is the new technical field of the space flight of a collection, aeronautical technology advantage at present, be the strategic high ground of 21 century field of aerospace technology, its development will produce significant impact to following military development strategy, space technology, weapon system construction and even whole scientific and technological progress.The research of Efforts To Develop hypersonic technology has important strategic importance for comprehensive strength and the international status of consolidating and improve China.China lists Long-and Medium-term Development planning in the great special project of hypersonic aircraft science and technology engineering.Nearly ten years advancing by leaps and bounds aspect hi-tech, driven the development of hypersonic Push Technology, precision navigation and control technology etc., particularly under the drive of hypersonic cruise weapon and earth to orbit and return transportation system high investment research, emerge the compound propulsion system and the hypersonic aircraft new ideas that embody various countries' research characteristic, for hypersonic aircraft is laid a good foundation from contemplating to realize.

Hypersonic aircraft generally refers to that flight Mach number is greater than 5, and the hypersonic aircraft concept proposing at present, divides and mainly comprise Horizontal Take-off and Landing space launch vehicle, sky and space plane, Hypersonic Aircraft and hypersonic cruise missile etc. by function.The maximum difference of this aircraft and orthodox flight device is the uncertain of its flight environment of vehicle and the inaccuracy from mechanics Predicting Performance Characteristics under high-speed flight condition to aircraft.The state of flights such as the same Mach number of supersonic combustion punching engine performance that air suction type hypersonic aircraft was adopted that the X-43A of take is representative, height are closely related, and engine performance surplus is less, in order to obtain enough dynamic pressures and engine intake airflow, this class aircraft must fly in denser endoatmosphere all the time, and aircraft aeroperformance exists many characteristics that are difficult to prediction under hypersonic condition.Therefore the hypersonic flight closed loop guidance mode selection that will be inevitable in atmosphere, its guidance system must have independently, generate in real time the ability of guidanceing command that flight path and control system can realize.Particularly, guidance system not only needs to provide real-time, complete following flight path information to control system, and when control law is the flight path information based on planning in advance, requirement is necessary can be according to the current state of aircraft and aerial mission, online real-time update Trajectory Design, gives guidanceing command of making new advances.Because closed loop guidance is based on predicting the outcome of following flight path being provided to current TRAJECTORY CONTROL instruction, and in practical flight process, all there are the various uncertain factors that are difficult to Accurate Prediction in dummy vehicle and Atmospheric Condition.Therefore the rapid Optimum that how to realize flight path becomes the key issue of closed loop guidance.

Closed loop guidance instruction refers to the flight path steering order that arrives target in flight course according to the vectored flight device of current state and task generation, and it is the key issue that can decision hypersonic aircraft realize its flying method and using value.Therefore find and a kind ofly can either solve this class multiple constraint, large probabilistic nonlinear optimal problem can guarantee that again the optimization method in linearity, rapidity, real-time solving is very necessary.

Traditional track optimizing method comprises indirect method, direct method, dynamic programming, differential method etc.Indirect method is, based on Pontryagin minimal principle, optimal control problem is converted to Hamilton boundary value problem (Hamilton Boundary Value Problem-HBVP).Although the method has the advantage that solving precision is higher, optimum solution meets First Order Optimality Condition.But because the process of its derivation optimum solution is comparatively complicated and loaded down with trivial details and be not suitable for solving the optimal control problem of path constraint, so can not be by the track rapid Optimum problem that solves hypersonic aircraft.

Direct method is without solving necessary condition for optimality, but by Continuous Optimal Control Problems discrete and parametrization, directly apply numerical method to performance index optimizing.Its advantage comprises that the First Order Optimality Condition that do not need to derive, the relative indirect method of domain of convergence are broader, less demanding to initial estimate longitude.Its deficiency shows and can not guarantee that the nonlinear programming solution obtaining is the solution of former optimal control problem.Yet, in the track optimizing question essence of hypersonic aircraft, be constrained nonlinear optimal control problem, there is the non-linear and strict constraint condition of height, so direct method still cannot be competent at.

Dynamic programming is to solve the optimized a kind of mathematical method of multistage decision process, and its principle of optimization is: the substrategy of optimal strategy is always optimum.The advantage of the method is that Computing Principle is simple, computational accuracy is relatively high, there is strict theory support, but for the such complicated optimum problem of hypersonic aircraft track optimizing, its result of calculation needs a large amount of memory spaces, and the process of Finding Global Optimization is also very loaded down with trivial details, therefore cannot meet rapidity and the real-time of track optimizing.

The quick track optimizing of hypersonic aircraft has been the important guarantee of aerial mission, also be the necessary condition that realizes maneuvering flight, the existence of the non-linear and Complex Constraints condition of motion model, make hypersonic aircraft track optimizing problem become very complicated, in optimizing process, need the problem of considering to comprise: reentry environment and uncertainty, hypersonic aircraft reenters the wide variation of experience height, Mach number, to produce heat-flash stream, overload and dynamic pressure, atmospheric environmental parameters changes violent, forms complicated reentry environment interference and uncertain; Constraint condition is complicated, not only will consider reentry corridor constraint, also will consider to control constraint, end conswtraint etc.; To proposing higher requirement computing time, hypersonic aircraft is to be greater than the speed flight of 5Ma, therefore very strong to the time-constrain in line computation, particularly the in the situation that of needs change of flight track, need on-line optimization track, therefore need to propose high efficiency algorithm is controlled at Millisecond by the optimization time; The guidance precision of having relatively high expectations, for carrying out high-precision special duty (as hit dynamic object point), need to carry out zero-miss guidance.

Summary of the invention

The object of the invention is when solving the track rapid Optimum problem of hypersonic aircraft, comparatively complicated and loaded down with trivial details optimal control problem, the direct method that is not suitable for solving path constraint of the process of the derivation optimum solution of indirect method can not guarantee that the nonlinear programming solution obtaining is that former optimal control problem and dynamic programming cannot meet that the rapidity of track optimizing and real-time form that complicated reentry environment is disturbed and the problem such as environmental uncertainty, and the finite time track rapid generation of a kind of hypersonic aircraft of proposition.

Above-mentioned goal of the invention is achieved through the following technical solutions:

Step 1, hypersonic aircraft is reentered to the motion model that latter end motion analysis and modeling obtain hypersonic aircraft;

Step 2, the motion model that the hypersonic aircraft obtaining is reentered to final trajectory are optimized problem and describe, and form nonlinear optimal problem;

Step 3, hypersonic aircraft is reentered to latter end nonlinear optimal problem convexification processing, is the protruding problem of quadratic form by the constraint specification of the optimization index of optimization problem and optimization problem;

Step 4: write the protruding optimization problem of specification description quadratic form according to CVXGEN code, utilize compiled online device to compile and generate high speed solver code;

Step 5: high speed solver is embedded under MATALB environment, realizes the numerical solution of the protruding optimization problem of quadratic form, and logarithm value solving result is analyzed; Completed a kind of finite time track rapid generation of hypersonic aircraft.

Invention effect

The key issue that whole design process of the present invention be take in hypersonic aircraft closed loop guidance---track optimizing is design object fast; In modeling process, take its reentry stage carries out Dynamic Modeling as example; Optimization problem is described the state equation constraint of having considered aircraft in process, reentry corridor constraint, end conswtraint, controlled quentity controlled variable constraint and using energy function as optimizing index; The clear and definite theoretical premise of take in problem transfer process has realized the convexification of nonlinear optimal problem and has processed and introduced rolling Optimization of Time Domain algorithm to solve the Percussion Problems of maneuvering target point as basis; Finally utilize protruding Optimization Solution instrument CVXGEN to realize the rapid Optimum of hypersonic aircraft reentry stage track.

Along with the continuous lifting of computing power, a kind of new optimization thought enters people's the visual field.It uses for reference the thought of PREDICTIVE CONTROL, aircraft track is carried out to the online limited period and optimize, and is called rolling Optimization of Time Domain (Receding Horizon-RH).Often there is uncertainty to a certain degree in actual path optimization problem, change, and the details of environmental parameter is also difficult to accurately know in advance such as aircraft arrives before destination perhaps target.And track is longer, uncertain just larger.Rolling optimization is optimized track in the limited period, has advantage on coping with uncertainty, is conducive to realize the real-time planning of track simultaneously.

The finite time track rapid generation of this kind of hypersonic aircraft, essence is the repeatedly iteration rolling Optimization of Time Domain algorithm of realizing based on protruding Optimization Solution tool box CVXGEN.This algorithm is converted into a series of protruding optimization problems by the nonlinear loci optimization problem under multiple constraint, by CVXGEN instrument, protruding optimization problem is described with a kind of higher level lanquage, and compiling forms the reliable high speed solver of such protruding optimization problem, with existing direct method, indirect method, the optimization methods such as dynamic programming are compared possesses following advantage: the one, and the rolling Optimization of Time Domain algorithm that the present invention proposes a kind of repeatedly iteration has large uncertainty to a class, the nonlinear optimal problem of multiple constraint solves, protruding optimization by nonlinear problem transforms the rapidity of having guaranteed algorithm, real-time (computation period is optimized in single step can be controlled at millisecond or Microsecond grade) and precision, and only considering under the prerequisite of the error of calculation, precision can reach in 1m, overcome the poor problem of the large real-time of existing track optimizing algorithm operation quantity, the 2nd, utilize solver that optimization tool CVXGEN generates can realize that high speed solves and reliable and stable, committed memory is few, and the conversion compilation process of problem carried out automatically by software completely, without specialized skills, use is simple and easy to left-hand seat, is beneficial to Project Realization, the 3rd, the method is close to hypersonic aircraft track rapid Optimum background, with strong points, with clearly defined objective, by realizing the online track optimizing of hypersonic aircraft, can further solve its closed loop guidance problem.

Accompanying drawing explanation

Fig. 1 is the finite time track rapid generation process flow diagram of a kind of hypersonic aircraft of proposition in embodiment one;

Fig. 2 is the programming interface schematic diagram that embodiment five proposes;

Fig. 3 is the code compilation interface schematic diagram that embodiment five proposes;

Fig. 4 be embodiment six propose hypersonic aircraft is reentered to the analysis of simulation result process flow diagram of final trajectory rapid Optimum problem;

Fig. 5 is the hypersonic aircraft trajectory schematic diagram that embodiment one proposes;

Fig. 6 is that the static monocular punctuate that embodiment proposes is optimized track schematic diagram, and wherein, horizontal ordinate is voyage, the flying height that ordinate is aircraft;

Fig. 7 is that the dynamic monocular punctuate that embodiment proposes is optimized track schematic diagram, and wherein, horizontal ordinate is voyage, the flying height that ordinate is aircraft.

Embodiment

Embodiment one: the finite time track rapid generation of a kind of hypersonic aircraft of present embodiment is by specifically preparing according to following steps:

Step 1, hypersonic aircraft is reentered to motion model that latter end motion analysis and modeling obtain hypersonic aircraft as Fig. 5;

Step 2, the motion model that the hypersonic aircraft obtaining is reentered to final trajectory are optimized problem and describe, and form nonlinear optimal problem;

Step 3, hypersonic aircraft is reentered to latter end nonlinear optimal problem convexification processing (Discrete Linear), is the protruding problem of quadratic form by the constraint specification of the optimization index of optimization problem and optimization problem;

Step 4: write the protruding optimization problem of specification description quadratic form according to CVXGEN code, utilize compiled online device to compile and generate high speed solver code;

Step 5: high speed solver is embedded under MATALB environment, realizes the numerical solution of the protruding optimization problem of quadratic form, and logarithm value solving result analyzes, as shown in Figure 1; Completed a kind of finite time track rapid generation of hypersonic aircraft.

The above; it is only preferably embodiment of the present invention; but protection scope of the present invention is not limited to this; anyly be familiar with those skilled in the art in the technical scope that the present invention discloses; according to technical scheme of the present invention and inventive concept thereof, be equal to replacement or changed, within all should being encompassed in protection scope of the present invention.

Present embodiment effect:

For hypersonic aircraft, because its flight environment of vehicle is complicated, all there is very large uncertainty in object model and parameter, aerial mission also may change according to current situation, and this requires hypersonic aircraft in flight course, will possess certain online track optimizing ability.And online track optimizing is just closed loop guidance the very corn of a subject, due to high speed feature and the advanced dynamic coupling characteristics of hypersonic aircraft, be keeping system stability and guidance precision, General Requirements is guidanceed command time update cycle 1s.The finite time track rapid generation of hypersonic aircraft proposed by the invention has been realized quick, the on-line optimization of its flight path, and its optimization time is controlled to Millisecond, reach the requirement of technical indicator, realized the precision strike to maneuvering target point simultaneously.

The key issue that the whole design process of present embodiment be take in hypersonic aircraft closed loop guidance---track optimizing is design object fast; In modeling process, take its reentry stage carries out Dynamic Modeling as example; Optimization problem is described the state equation constraint of having considered aircraft in process, reentry corridor constraint, end conswtraint, controlled quentity controlled variable constraint and using energy function as optimizing index; The clear and definite theoretical premise of take in problem transfer process has realized the convexification of nonlinear optimal problem and has processed and introduced rolling Optimization of Time Domain algorithm to solve the Percussion Problems of maneuvering target point as basis; Finally utilize protruding Optimization Solution instrument CVXGEN to realize the rapid Optimum of hypersonic aircraft reentry stage track.

Along with the continuous lifting of computing power, a kind of new optimization thought enters people's the visual field.It uses for reference the thought of PREDICTIVE CONTROL, aircraft track is carried out to the online limited period and optimize, and is called rolling Optimization of Time Domain (Receding Horizon-RH).Often there is uncertainty to a certain degree in actual path optimization problem, change, and the details of environmental parameter is also difficult to accurately know in advance such as aircraft arrives before destination perhaps target.And track is longer, uncertain just larger.Rolling optimization is optimized track in the limited period, therefore on coping with uncertainty, has advantage, is conducive to realize the real-time planning of track simultaneously.

The finite time track rapid generation of this kind of hypersonic aircraft, essence is the repeatedly iteration rolling Optimization of Time Domain algorithm of realizing based on protruding Optimization Solution tool box CVXGEN.This algorithm is converted into a series of protruding optimization problems by the nonlinear loci optimization problem under multiple constraint, by CVXGEN instrument, protruding optimization problem is described with a kind of higher level lanquage, and compiling forms the reliable high speed solver of such protruding optimization problem, with existing direct method, indirect method, the optimization methods such as dynamic programming are compared possesses following advantage: the one, and the rolling Optimization of Time Domain algorithm that present embodiment proposes a kind of repeatedly iteration has large uncertainty to a class, the nonlinear optimal problem of multiple constraint solves, protruding optimization by nonlinear problem transforms the rapidity of having guaranteed algorithm, real-time (computation period is optimized in single step can be controlled at millisecond or Microsecond grade) and precision, and only considering under the prerequisite of the error of calculation, precision can reach in 1m, overcome the poor problem of the large real-time of existing track optimizing algorithm operation quantity, the 2nd, utilize solver that optimization tool CVXGEN generates can realize that high speed solves and reliable and stable, committed memory is few, and the conversion compilation process of problem carried out automatically by software completely, without specialized skills, use is simple and easy to left-hand seat, is beneficial to Project Realization, the 3rd, the method is close to hypersonic aircraft track rapid Optimum background, with strong points, with clearly defined objective, by realizing the online track optimizing of hypersonic aircraft, can further solve its closed loop guidance problem.

Embodiment two: present embodiment is different from embodiment one: the process of the addressing of fetching earth described in step 1 is: in step 1, hypersonic aircraft being reentered to the motion model detailed process that latter end motion modeling obtains hypersonic aircraft is:

(1) hypersonic aircraft is reentered to the work of process engine stop, be no longer subject to thrust and control, be mainly subject to gravity and aerodynamic effect, glide at a distance; In addition, the present invention do not consider aircraft around center of mass motion, think around center of mass motion in equilibrium,transient or trim condition, aircraft is considered as to particle and only studies its center of mass motion; Be simplified to barycenter kinetics equation:

$m\frac{d^{2}r}{{\mathrm{dt}}^2}=R+\mathrm{mg}---\left(1\right)$

Wherein, r---aircraft barycenter is with respect to geocentric altitude, vector representation;

M---reentry vehicle quality;

R---the suffered aerodynamic force of aircraft, vector representation;

Mg---the suffered universal gravitation of aircraft, vector representation;

T ∈ [t ₀, t _f] represent that hypersonic aircraft reenters process for the time of track optimizing;

T ₀represent that hypersonic aircraft reenters process initial time;

T _frepresent that hypersonic aircraft reenters process terminal time;

(2) on this basis, put aside the earth rotation, suppose that the earth is a ball, the process that reenters is without skid force, and yaw angle is always zero, obtains the motion model (the superb equation of motion that reenters that is Three Degree Of Freedom is) that hypersonic aircraft reenters latter end:

$\left\{\begin{array}{c}\stackrel{.}{r}=V\sin \mathrm{\γ}\\ \stackrel{.}{\mathrm{\γ}}=\frac{L\cos \mathrm{\σ}}{\mathrm{mV}}+(\frac{V}{r}-\frac{\mathrm{\μ}}{r^{2}V})\cos \mathrm{\γ}\\ \stackrel{.}{V}=-\frac{D}{m}-\frac{\mathrm{\μ}}{r^2}\sin \mathrm{\γ}\\ \stackrel{.}{\mathrm{\λ}}=\frac{V\cos \mathrm{\γ}\sin \mathrm{\ψ}}{r\cos \mathrm{\φ}}\\ \stackrel{.}{\mathrm{\φ}}=\frac{V\cos \mathrm{\γ}\cos \mathrm{\ψ}}{r}\\ \stackrel{.}{\mathrm{\ψ}}=\frac{L\sin \mathrm{\σ}}{\mathrm{mV}\cos \mathrm{\γ}}+\frac{V}{r}\cos \mathrm{\γ}\sin \mathrm{\ψ}\tan \mathrm{\φ}\end{array}\right.---\left(2\right)$

Wherein, V---the speed of hypersonic aircraft barycenter in half speed coordinate system, m/s;

---the derivative of the speed relative time of hypersonic aircraft barycenter in half speed coordinate system;

R---hypersonic aircraft barycenter is with respect to the height in the earth's core, m;

---hypersonic aircraft barycenter is with respect to the derivative of the height relative time in the earth's core;

γ---flight-path angle, rad;

---the derivative of flight-path angle relative time;

λ---longitude, rad;

---the derivative of longitude relative time;

φ---latitude, rad;

---the derivative of latitude relative time;

ψ---course angle, rad;

---the derivative of course angle relative time;

σ---speed oblique angle, rad;

D---resistance, N;

L---lift, N;

μ---Gravitational coefficient of the Earth;

M---reentry vehicle quality;

Wherein, state variable x=[r, γ, V, λ, φ, ψ] ^t, controlled quentity controlled variable u is the inclination angle of the angle of attack and speed, i.e. u=[α, σ] ^t;

So-called lengthwise movement, refers to aircraft movements parameter yaw angle β, roll angle γ, speed pitch angle γ _v, crab angle ψ, trajectory deflection angle ψ v, around x, the angular velocity of rotation ω of y axle _x, ω _y, the identically vanishing such as displacement on z axle motion; Be the translation motion in flight plane (or the longitudinal vertical guide of missile coordinate system) of the barycenter of aircraft and form around the rotational motion of z axle; So in lengthwise movement, aircraft flight speed V is at trajectory tilt angle θ, the angle of pitch and angle of attack is around the angular velocity of rotation of z axle x, the displacement on y is time dependent, wherein, flying speed V, trajectory tilt angle θ, the angle of pitch angle of attack and around the angular velocity of rotation of z axle and x, the displacement on y is commonly referred to the kinematics parameters of lengthwise movement, referred to as lengthwise movement parameter; According to the hypersonic aircraft of (2) formula, reenter latter end motion model, obtain the lengthwise movement model that hypersonic aircraft reenters latter end (latter end of reentry vehicle target of attack):

$\left\{\begin{array}{c}\stackrel{.}{h}=V\sin \mathrm{\γ}\\ \stackrel{.}{\mathrm{\γ}}=\frac{L}{\mathrm{mV}}+(\frac{V}{r}-\frac{\mathrm{\μ}}{r^{2}V})\cos \mathrm{\γ}\\ \stackrel{.}{V}=-\frac{D}{m}-\frac{\mathrm{\μ}}{r^2}\sin \mathrm{\γ}\end{array}\right.---\left(21\right)$

Wherein, h---aircraft is apart from the distance of earth surface;

---aircraft is apart from the derivative of the distance relative time of earth surface;

L---the lift that aircraft is suffered;

D---the resistance that aircraft is suffered;

Owing to only considering that hypersonic aircraft reenters the lengthwise movement of latter end (latter end of reentry vehicle target of attack), does not consider the impact of its sideway movement; Therefore σ=0, desirable speed inclination angle, it is u=α that controlled quentity controlled variable only has the angle of attack, and quantity of state is taken as x=[h γ V] ^tthe expression formula that obtains lift L and resistance D is as follows:

$L=\frac{1}{2}\mathrm{\ρ}{V}^2{\mathrm{SC}}_L,C_L=C_{L0}+C_{L1}\mathrm{\α}+C_{L2}{e}^{C_{L3}V}---\left(22\right)$

$D=\frac{1}{2}\mathrm{\ρ}{V}^2{\mathrm{SC}}_D,C_D=C_{D0}+C_{D1}{\mathrm{\α}}^2+C_{D2}{e}^{C_{D3}V}---\left(23\right)$

Wherein, C _l0, C _l1, C _l2, C _l3and C _d0, C _d1, C _d2, C _d3be constant; ρ is free air density, and the exponential form of ρ is ρ=ρ ₀e ^{-β h}, ρ wherein ₀be the atmospheric density at place, sea level, β is constant; S Chinese implication is Hypersonic Reentry Vehicles aerofoil area of reference, the truth of a matter that e is natural logarithm.Other step and parameter are identical with embodiment one.

Embodiment three: present embodiment is different from embodiment one or two: the process of the addressing of the field of fetching earth described in step 1 is: the motion model that in step 2, the hypersonic aircraft obtaining is reentered to final trajectory is optimized problem and describes, and the detailed process that forms nonlinear optimal problem is:

(1), hypersonic aircraft shown in (2) formula is reentered to the form that latter end motion model is expressed as the differential equation:

Wherein, (3) formula is that nonlinear equation is state equation constraint, x=[r, γ, V, λ, φ, ψ] ^tfor state variable, u=[α, σ] ^tfor controlled quentity controlled variable; This be must be satisfied in optimizing process state equation constraint; represent the real number field that state variable and controlled quentity controlled variable all belong to; M represents the dimension of controlled quentity controlled variable, and n represents the dimension of state variable;

In setting up hypersonic aircraft and reenter the motion model of latter end (the superb equation of motion that reenters of Three Degree Of Freedom), there is no thrust, controlled quentity controlled variable is angle of attack and speed inclination angle σ; The sidestep maneuver flight that the σ major control of speed inclination angle is superb, angle of attack is the superb Aerodynamic Coefficient of major effect, thus longitudinal track of change of flight device; Meet certain controlled quentity controlled variable constraint:

u _min≤u(t)≤u _max,t∈[t ₀,t _f](7)

(2) the end points constraint that flight path need to be satisfied comprises starting point constraint and end point constraint, establishes initial time t ₀with t terminal time _ffor condition, the initial time t that fixation locus is optimized ₀state be x ₀, fixed terminal is t constantly _fcorresponding quantity of state is x _f,

Corresponding end conswtraint is:

Wherein, q represents the dimension of end conswtraint; ψ end conswtraint function;

(3) to reenter latter end be that the latter end of reentry vehicle target of attack is the inflight phase of the tool feature of hypersonic aircraft return course to hypersonic aircraft, because aircraft utilizes this day heat condition of earth atmosphere, in return course, slow down and decline, follow huge energy consumption simultaneously; Therefore, the superb process that reenters meets strict rate of heat flow the constraint conditions such as dynamic pressure M and overload N:

$\left\{\begin{array}{c}\stackrel{.}{Q}=K_Q{\mathrm{\ρ}}^{0.5}{V}^3\≤{\stackrel{.}{Q}}_{\max }\\ M=\frac{1}{2}{\mathrm{\ρV}}^2\≤M_{\max }\\ N=\frac{\sqrt{L^2+D^2}}{{\mathrm{mg}}_0}\≤N_{\max }\end{array}\right.---\left(5\right)$

Wherein, ρ represents the atmospheric density that mass center of reentry vehicle height is corresponding; K _qrepresent heat flow density constant;

K _Q＝7.9686×10 ^-5Js ²m ^3.5kg ^0.5

(5) formula can be expressed as (6) formula, and being called reentry corridor constraint is path constraint: in addition, owing to reentering latter end (latter end of reentry vehicle target of attack), belong to endoatmosphere motion, therefore without considering reentry corridor constraint;

ζ (x (t), u (t))≤0, t ∈ [t ₀, t _f] (6), wherein, ζ is reentry corridor constraint;

(4) it is following form that the motion model that hypersonic aircraft reenters final trajectory is optimized problem representation:

Wherein, φ (x _f, t _f) and L (x, u, t) for optimizing target function, get energy function as optimizing index; J is for optimizing index, (x _f, t _f) be the quantity of state of terminal juncture.Other step and parameter are identical with embodiment one or two.

Embodiment four: present embodiment is different from one of embodiment one to three: in step 3, hypersonic aircraft being reentered to latter end nonlinear optimal problem convexification and process (Discrete Linear), is that the protruding problem detailed process of quadratic form is by the constraint specification of the optimization index of optimization problem and optimization problem:

What as previously mentioned, the convex optimized algorithm of wish utilization of the present invention based on CVXGEN solved hypersonic aircraft reenters final trajectory optimization problem; Yet (8) optimization problem shown in formula is not " protruding " problem, the conversion that therefore need to be optimized problem is that nonlinear optimal problem convexification is processed:

(1) theoretical premise, definition nonlinear system:

$\stackrel{.}{x}=f(x,t),x\left(0\right)=x_0---\left(9\right)$

Change into the form of State-dependent Coefficient (StateDependentCoefficient-SDC):

Wherein, for the derivative of quantity of state x relative time, the matrix of coefficients that A (x) determines for quantity of state x, initial point x=0 is that an equilibrium point and hypothesis A (x) meet Local Lipschitz Conditions, this is that former nonlinear system exists and separate unique minimum hypothesis; (10) nonlinear system that formula is described is approached by linear time varying system; The classical way that the method makes script be applied to Linear System Analysis is transplanted in nonlinear system and is used:

$\begin{array}{c}{\stackrel{.}{x}}^{\left[1\right]}\left(t\right)=A\left(x_0\right)x^{\left[1\right]}\left(t\right),x^{\left[1\right]}\left(0\right)=x_0\\ .\\ .\\ .\\ {\stackrel{.}{x}}^{[j-1]}\left(t\right)=A\left(x^{[j-2]}\left(t\right)\right)x^{[j-1]}\left(t\right),x^{[j-1]}\left(0\right)=x_0\\ {\stackrel{.}{x}}^{\left[j\right]}\left(t\right)=A\left(x^{[j-1]}\left(t\right)\right)x^{\left[j\right]}\left(t\right),x^{\left[j\right]}\left(0\right)=x_0\end{array}---\left(11\right)$

When j>=1, x ^[0](t) initial value is taken as original state x conventionally ₀; The solution that meets linear time varying system converges to the solution of the nonlinear system shown in (10), as follows:

lim _j→∞{x[j](t)}→x(t)(12)

Wherein, it should be noted that first Linear Estimation formula x ^[1](0)=x ₀first linear time invariant system, because x wherein (t) is by original state x ₀institute replaces, for first equation provides a constant coefficient battle array; { x ^[j](t) } _j>=1for j the linear time varying system obtaining after iterative computation j time, and

$\begin{array}{c}{\stackrel{.}{x}}^{\left[2\right]}\left(t\right)=A\left(x_0\right)x^{\left[2\right]}\left(t\right),x^{\left[2\right]}\left(0\right)=x_0\\ .\\ .\\ .\\ {\stackrel{.}{x}}^{[j-1]}\left(t\right)=A\left(x^{[j-2]}\left(t\right)\right)x^{[j-1]}\left(t\right),x^{[j-1]}\left(0\right)=x_0\\ {\stackrel{.}{x}}^{\left[j\right]}\left(t\right)=A\left(x^{[j-1]}\left(t\right)\right)x^{\left[j\right]}\left(t\right),x^{\left[j\right]}\left(0\right)=x_0\end{array}$ For Linear Estimation formula is linear time-varying equation;

Based on this theory, nonlinear optimal problem is converted into a series of linear optimization problem, the optimum solution of trying to achieve will converge to the solution of former nonlinear optimal problem;

(2) the track rapid Optimum problem that hypersonic aircraft reenters latter end (latter end of reentry vehicle target of attack) is still a nonlinear optimal problem, and wish utilizes convex optimized algorithm to solve must carry out the conversion of problem; The theoretical premise providing based on (10), (11), (12) formula is carried out convexification processing to optimizing index and constraint respectively; " protruding " that be optimization problem changes processing; Based on above-mentioned theory prerequisite, the described nonlinear optimal problem of (8) formula is converted into the protruding optimization problem that CVXGEN processes:

Steps A, by state constraint equation t ∈ [t ₀, t _f] be described as the form of State-dependent Coefficient (SDC) a given original state:

It is non-linear is coefficient matrices A ' (x), (x) non-linear of B '; According to the theoretical premise shown in formula (11), utilize linear system to approach this nonlinear system:

${\stackrel{.}{x}}^{\left[j\right]}\left(t\right)=A(x^{[j-1]}\left(t\right),u^{[j-1]}\left(t\right))x^{\left[j\right]}\left(t\right)+B(x^{[j-1]}\left(t\right),u^{[j-1]}\left(t\right))u^{\left[j\right]}\left(t\right)---\left(14\right)$

Brief note is:

${\stackrel{.}{x}}^{\left[j\right]}=A^{\′[j-1]}{x}^{\left[j\right]}+B^{\′[j-1]}{u}^{\left[j\right]},x^{\left[j\right]}\left(0\right)=x_0---\left(15\right)$

Wherein, (x), B ' is (x) matrix of coefficients of formula (15) to A ', thereby has realized the iterative approach of linear time varying system to former nonlinear system;

Step B, because rolling Optimization of Time Domain adopts discrete form, linear system (15) is carried out to discretize processing, the sampling period is taken as T, discretize step-length is P, adopts difference method to obtain final convexification form as follows:

x ^[j](k+1)＝A ^[j-1](k)x ^[j](k)+B ^[j-1](k)u ^[j](k)+C ^[j-1](k),k＝1,2,...,P(16)

Wherein, A ^[j-1](k)=T * A ' ^[j-1](k)+I, B ^[j-1](k)=T * B ' ^[j-1](k) C ^[j-1](k) represent j linear time varying system Discrete Linear error obtaining for j time;

Step C, for the reentry corridor constraint ζ (x (t), u (t))≤0 in (8) formula, t ∈ [t ₀, t _f] adopt identical disposal route to carry out linear discrete to obtain following convexification form:

C _a ^[j](k)x ^[j](k)+C _b ^[j](k)u ^[j](k)+C _c ^[j](k)≤0,k＝1,2...P

C _a(k), C _b(k), C _c(k) be that ζ (x (t), u (t))≤0 is carried out to the coefficient entry that Discrete Linear obtains;

Step D, the end conswtraint in (8) formula and controlled quentity controlled variable constraint convexification is processed, is obtained respectively following form:

x ^[j](P+1)-x _tar＝0

U _min≤ u ^[j](k)≤u _max, k=1,2...P wherein, x _tarfor target location dot information;

Step e, provide optimization index; Using energy function as optimizing target function, it should be noted that, in order to obtain better optimum results, terminal hard constraint is converted to soft-constraint (being that the end conswtraint that formula 4 provides is described as optimizing a part for index as a part for optimization index); Final optimization index form is:

$J=\frac{1}{P}\×\underset{k=1}{\overset{P}{\mathrm{\Σ}}}{\left|\right|u^{\left[j\right]}\left(k\right)\left|\right|}_2^2+{\left|\right|x^{\left[j\right]}(P+1)-x_{\mathrm{tar}}\left|\right|}_2^2$

Step F, hypersonic aircraft reenter final trajectory optimization problem and have converted by nonlinear optimal problem the protruding problem that protruding optimization tool CVXGEN can separate to; The final form of the protruding problem that can separate is:

$\begin{array}{c}\min J=\frac{1}{P}\×\underset{k=1}{\overset{P}{\mathrm{\Σ}}}{\left|\right|u^{\left[j\right]}\left(k\right)\left|\right|}_2^2+{\left|\right|x^{\left[j\right]}(P+1)-x_{\mathrm{tar}}\left|\right|}_2^2\\ s.t.\left\{\begin{array}{c}{x}^{\left[j\right]}(k+1)=A^{[j-1]}\left(k\right)x^{\left[j\right]}\left(k\right)+B^{[j-1]}\left(k\right)u^{\left[j\right]}\left(k\right)+C^{[j-1]}\left(k\right)\\ {C_a}^{\left[j\right]}\left(k\right)x^{\left[j\right]}\left(k\right)+{C_b}^{\left[j\right]}\left(k\right)u^{\left[j\right]}\left(k\right)+{C_c}^{\left[j\right]}\left(k\right)\≤0\\ u_{\min }\≤u^{\left[j\right]}\left(k\right)\≤u_{\max }\\ -e\≤x^{\left[j\right]}\left(k\right)-x_r\left(k\right)\≤e\end{array}\right.\\ k=\mathrm{1,2},...,P\end{array}---\left(19\right)$

(3) protruding optimization problem is static optimization problem, yet for the dynamic object point in this programme, in order to realize the quick dynamic optimization of hypersonic aircraft track, introduce rolling Optimization of Time Domain (Receding Horizon-RH) algorithm, finally obtain the rolling Optimization of Time Domain algorithm based on protruding Optimization Solution instrument CVXGEN, be used for solving hypersonic aircraft reentry vehicle target of attack reenter latter end be hypersonic aircraft reenter process the quick track optimizing problem of latter end to realize the strike to dynamic monocular punctuate; Using current state x (i) as original state, the model data in current i locates prediction optimization time domain (Optimal Step Size) P constantly, and be optimized accordingly solving of problem; According to (19) formula, to the description of optimization problem, can obtaining hypersonic aircraft, to reenter the quick track optimizing problem of latter end concrete form as follows:

$\begin{array}{c}\min J=\frac{1}{P}\×\underset{k=1}{\overset{P}{\mathrm{\Σ}}}{\left|\right|u^{\left[j\right]}(i+k|i)\left|\right|}_2^2+{\left|\right|x^{\left[j\right]}(i+P+1|i)-x_{\mathrm{tar}}\left|\right|}_2^2\\ s.t.\left\{\begin{array}{c}{x}^{\left[j\right]}(i+k+1|i)=A^{[j-1]}(i+k|i)x^{\left[j\right]}(i+k|i)+B^{[j-1]}(i+k|i)u^{\left[j\right]}(i+k|i)+C^{[j-1]}(i+k|i)\\ {C_a}^{\left[j\right]}(i+k|i)x^{\left[j\right]}(i+k|i)+{C_b}^{\left[j\right]}(i+k|i)u^{\left[j\right]}(i+k|i)+{C_c}^{\left[j\right]}(i+k|i)\≤0\\ u_{\min }\≤u^{\left[j\right]}(i+k|i)\≤u_{\max }\\ -e\≤x^{\left[j\right]}(i+k|i)-x_r(i+k|i)\≤e\end{array}\right.\\ k=\mathrm{1,2},...,P\end{array}---\left(20\right)$

Wherein, i+k|i is illustrated in the i predicted value to i+k step variable constantly, and j represents iterations; P is called control time domain, represents the time span of predict future, and u is controlled quentity controlled variable, C _a, C _band C _cthree factor arrays for reentry corridor constraint.Other step and parameter are identical with one of embodiment one to three.

Embodiment five: present embodiment is different from one of embodiment one to four: write the protruding optimization problem of specification description quadratic form according to CVXGEN code in step 4, utilize compiled online device to compile and generate high speed solver process and be code:

For (20) described hypersonic aircraft, reenter the quick track optimizing problem of latter end, in protruding Optimization Solution instrument CVXGEN, carry out corresponding single step iteration optimization problem code and write; The compiled online function of utilizing CVXGEN to provide, compiling generates the high speed solver based on code;

By (8) formula, can find out that the track optimizing that hypersonic aircraft reenters process is the nonlinear optimal problem of a multiple constraint; Flight characteristics based on superb, needs us to seek a kind of rapid solving algorithm; In addition,, due to the maneuverability of task object point, more need this optimized algorithm to have in linearity and real-time; This is the rolling Optimization of Time Domain method that solves instrument CVXGEN realization based on convex optimized algorithm provided by the present invention; Utilize CVXGEN instrument to be programmed into solver process from code to be:

(1) code is write, and as shown in Figure 2, the code of writing comprises dimension (Dimensions), parameter (Parameters), variable (Variables), objective function (Minimize) and constraint (Subjectto) at interface;

(2) after code is write, if when the code of writing is no problem, carry out the compiling of code, the interface after compiling as shown in Figure 3; Generate 5 main C source files, for the * .m file of MATLAB environment with for the CVX file of result verification; Wherein * .m file can be directly embedded under MATLAB, as the solver of optimization problem; Wherein, 5 main C source files comprise the solver.c of solved function and kernel program, the matrix_support.c of the assignment of the ldl.c that the factorization of KKT matrix solves, matrix-vector and vector phase multiplication, simply drive the testsolver.c of code and the util.c of trial function;

(3) use MATLAB interface; Making the simulation analysis of problem more efficient, is that self-defined solver is embedded under MATLAB environment by the solver of optimization problem, is extremely convenient to solving of optimization problem.Other step and parameter are identical with one of embodiment one to four.

Embodiment six: present embodiment is different from one of embodiment one to five: in step 5, high speed solver is embedded under MATALB environment, realize the numerical solution of the protruding optimization problem of quadratic form, and the detailed process that logarithm value solving result is analyzed is as follows:

(1) after code compilation success, self-defined high speed solver CVXGEN is embedded under MATLAB environment, the parameter reentering in final trajectory rapid Optimum problem is carried out to corresponding assignment;

(2) solve hypersonic aircraft and reenter final trajectory rapid Optimum problem, obtain the simulation result that final hypersonic aircraft reenters terminal flight track;

(3) simulation result that hypersonic aircraft is reentered to final trajectory rapid Optimum problem is analyzed: utilize CVXGEN to generate the high speed solver of optimization problem (20), to solve code is embedded in MATLAB, according to (11) formula, carry out repeatedly iterative, obtain the optimum control amount u* (i|i) in P time domain, ..., u* (i+P|i), according to rolling optimization, u* (i|i) is applied to longitudinal nonlinear motion model formula (21), calculates next state x (i+1) constantly; If do not reach assigned address, do not forward step 5 to and start to calculate, until aircraft hits the mark or aircraft arrives assigned address; The principle of solution procedure as shown in Figure 4; Wherein, u* (i+P|i) represents i+P the control variable (i.e. the predicted value to moment optimum control amount in the i moment) that i suboptimization obtains; Assigned address is that the hypersonic aircraft that meets every optimization index and constraint reenters terminal flight track.Other step and parameter are identical with one of embodiment one to five.

Adopt following examples to verify beneficial effect of the present invention:

Embodiment mono-:

The finite time track rapid generation of a kind of hypersonic aircraft of the present embodiment, specifically according to following steps, prepare:

Step 1, hypersonic aircraft is reentered to latter end motion analysis and modeling obtains the motion model that hypersonic aircraft reenters final trajectory;

Hypersonic aircraft trajectory as shown in Figure 5; Study the end that it reenters process, i.e. the track optimizing problem that reenters latter end shown in figure; Its motion model is as follows:

$\left\{\begin{array}{c}\stackrel{.}{h}=V\sin \mathrm{\γ}\\ \stackrel{.}{\mathrm{\γ}}=\frac{L}{\mathrm{mV}}+(\frac{V}{r}-\frac{\mathrm{\μ}}{r^{2}V})\cos \mathrm{\γ}\\ \stackrel{.}{V}=-\frac{D}{m}-\frac{\mathrm{\μ}}{r^2}\sin \mathrm{\γ}\end{array}\right.---\left(1\right)$

In formula, h---aircraft is apart from the distance of earth surface;

---aircraft is apart from the derivative of the distance relative time of earth surface;

γ---flight-path angle, rad;

---the derivative of flight-path angle relative time;

V---the speed of hypersonic aircraft barycenter in half speed coordinate system, m/s;

---the derivative of the speed relative time of hypersonic aircraft barycenter in half speed coordinate system;

L---the lift that aircraft is suffered;

D---the resistance that aircraft is suffered;

The expression formula of lift and resistance is respectively:

$L=\frac{1}{2}\mathrm{\ρ}{V}^2{\mathrm{SC}}_L,C_L=C_{L0}+C_{L1}\mathrm{\α}+C_{L2}{e}^{C_{L3}V}---\left(2\right)$

$D=\frac{1}{2}\mathrm{\ρ}{V}^2{\mathrm{SC}}_D,C_D=C_{D0}+C_{D1}{\mathrm{\α}}^2+C_{D2}{e}^{C_{D3}V}---\left(3\right)$

Wherein, C _l0, C _l1, C _l2, C _l3and C _d0, C _d1, C _d2, C _d3be constant; ρ is free air density, and the exponential form of ρ is ρ=ρ ₀e ^{-β h}, ρ wherein ₀be the atmospheric density at place, sea level, β is constant; S Chinese implication is Hypersonic Reentry Vehicles aerofoil area of reference; Can find out, lift and resistance determine by the angle of attack, and therefore controlled quentity controlled variable being taken as to the angle of attack is u=α, and it is x=[h γ V that quantity of state is taken as flying height, flight-path angle, flying speed] ^t; The value of other parameter is as shown in the table:

Step 2, the motion model that the hypersonic aircraft obtaining is reentered to final trajectory are optimized problem and describe, and form nonlinear optimal problem;

Hypersonic aircraft reenters latter end in earth atmosphere, therefore need not consider that the reentry corridor being caused by atmospheric reentry retrains again; That considers state equation constraint, end conswtraint, controlled quentity controlled variable constraint and optimize that index obtains following form reenters final trajectory optimization problem

This step is the mathematical description to track optimization problem, does not therefore need the parameter of assignment;

Step 3: hypersonic aircraft is reentered to latter end nonlinear optimal problem convexification and process (Discrete Linear), optimization index and constraint are described as to the protruding problem of quadratic form;

The multiple constraint nonlinear loci optimization problem that formula (4) is described is carried out convexification processing, and introduces rolling Optimization of Time Domain strategy and obtain the protruding optimization problem that CVXGEN can separate, and specifically describes as follows:

$\begin{array}{c}\min J=\frac{1}{P}\×\underset{k=1}{\overset{P}{\mathrm{\Σ}}}{\left|\right|u^{\left[j\right]}(i+k|i)\left|\right|}_2^2+{\left|\right|x^{\left[j\right]}(i+P+1|i)-x_{\mathrm{tar}}\left(i\right|i)\left|\right|}_2^2\\ s.t.\left\{\begin{array}{c}{x}^{\left[j\right]}(i+k+1|i)=A^{[j-1]}(i+K|i)x^{\left[j\right]}(i+k|i)+B^{[j-1]}(i+k|i)u^{\left[j\right]}(i+k|i)+C^{[j-1]}(i+k|i)\\ u_{\min }\≤u^{\left[j\right]}(i+k|i)\≤u_{\max }\\ -e\≤x^{\left[j\right]}(i+k|i)-x_r(i+k|i)\≤e\end{array}\right.\\ k=\mathrm{1,2}...P\end{array}---\left(5\right)$

Wherein, i+k|i is illustrated in the i predicted value to i+k step variable constantly, and j represents iterations; P is called control time domain, represents the time span of predict future, and u is controlled quentity controlled variable; It is as follows that parameter is chosen result:

Step 4: write the protruding optimization problem of specification description quadratic form according to CVXGEN code, utilize compiled online device to compile and generate high speed solver code;

The code that utilizes the specification normative language of protruding Optimization Solution instrument CVXGEN to complete the protruding optimization problem of quadratic form that formula (5) is described is write, and compiling generates high speed solver; Because the compiling procedure of whole code is all parameterized, therefore this step does not relate to assignment problem;

Step 5: high speed solver is embedded under MATALB environment, realizes the numerical solution of the protruding optimization problem of quadratic form, and logarithm value solving result is analyzed;

The high speed solver that CVXGEN compiling is generated is embedded under MATLAB environment, completes finally solving of track optimizing problem, and parameter assignment situation is as follows:

The optimization track that finally to obtain take static monocular punctuate be target spot, as shown in Figure 6, its optimization time is 0.060272s; Utilize rolling Optimization of Time Domain method to realize the strike to motor-driven monocular punctuate, its optimum results as shown in Figure 7, during altogether rolling optimization 5 times, the accumulative total optimization time is 0.162109s.

Claims (6)

1. a finite time track rapid generation for hypersonic aircraft, is characterized in that: a kind of finite time track rapid generation of hypersonic aircraft specifically carries out according to following steps:

Step 1, hypersonic aircraft is reentered to the motion model that latter end motion analysis and modeling obtain hypersonic aircraft;

Step 2, the motion model that the hypersonic aircraft obtaining is reentered to final trajectory are optimized problem and describe, and form nonlinear optimal problem;

Step 3, hypersonic aircraft is reentered to latter end nonlinear optimal problem convexification processing, is the protruding problem of quadratic form by the constraint specification of the optimization index of optimization problem and optimization problem;

Step 4: write the protruding optimization problem of specification description quadratic form according to CVXGEN code, utilize compiled online device to compile and generate high speed solver code;

Step 5: high speed solver is embedded under MATALB environment, realizes the numerical solution of the protruding optimization problem of quadratic form, and logarithm value solving result is analyzed; Completed a kind of finite time track rapid generation of hypersonic aircraft.

2. the finite time track rapid generation of a kind of hypersonic aircraft according to claim 1, is characterized in that: in step 1, hypersonic aircraft being reentered to the motion model detailed process that latter end motion modeling obtains hypersonic aircraft is:

(1) hypersonic aircraft is reentered to process simplification and becomes barycenter kinetics equation:

$m\frac{d^{2}r}{{\mathrm{dt}}^2}=R+\mathrm{mg}---\left(1\right)$

Wherein, r---aircraft barycenter is with respect to geocentric altitude, vector representation;

M---reentry vehicle quality;

R---the suffered aerodynamic force of aircraft, vector representation;

Mg---the suffered universal gravitation of aircraft, vector representation;

T ∈ [t ₀, t _f] represent that hypersonic aircraft reenters process for the time of track optimizing;

T ₀represent that hypersonic aircraft reenters process initial time;

T _frepresent that hypersonic aircraft reenters process terminal time;

(2) suppose that yaw angle is always zero, obtains the motion model that hypersonic aircraft reenters latter end:

$\left\{\begin{array}{c}\stackrel{.}{r}=V\sin \mathrm{\γ}\\ \stackrel{.}{\mathrm{\γ}}=\frac{L\cos \mathrm{\σ}}{\mathrm{mV}}+(\frac{V}{r}-\frac{\mathrm{\μ}}{r^{2}V})\cos \mathrm{\γ}\\ \stackrel{.}{V}=-\frac{D}{m}-\frac{\mathrm{\μ}}{r^2}\sin \mathrm{\γ}\\ \stackrel{.}{\mathrm{\λ}}=\frac{V\cos \mathrm{\γ}\sin \mathrm{\ψ}}{r\cos \mathrm{\φ}}\\ \stackrel{.}{\mathrm{\φ}}=\frac{V\cos \mathrm{\γ}\cos \mathrm{\ψ}}{r}\\ \stackrel{.}{\mathrm{\ψ}}=\frac{L\sin \mathrm{\σ}}{\mathrm{mV}\cos \mathrm{\γ}}+\frac{V}{r}\cos \mathrm{\γ}\sin \mathrm{\ψ}\tan \mathrm{\φ}\end{array}\right.---\left(2\right)$

Wherein, V---the speed of hypersonic aircraft barycenter in half speed coordinate system, m/s;

---the derivative of the speed relative time of hypersonic aircraft barycenter in half speed coordinate system;

R---hypersonic aircraft barycenter is with respect to the height in the earth's core, m;

---hypersonic aircraft barycenter is with respect to the derivative of the height relative time in the earth's core;

γ---flight-path angle, rad;

---the derivative of flight-path angle relative time;

λ---longitude, rad;

---the derivative of longitude relative time;

φ---latitude, rad;

---the derivative of latitude relative time;

ψ---course angle, rad;

---the derivative of course angle relative time;

σ---speed oblique angle, rad;

D---resistance, N;

L---lift, N;

μ---Gravitational coefficient of the Earth;

M---reentry vehicle quality;

Wherein, state variable x=[r, γ, V, λ, φ, ψ] ^t, controlled quentity controlled variable u is the inclination angle of the angle of attack and speed, i.e. u=[α, σ] ^t;

According to the hypersonic aircraft of (2) formula, reenter latter end motion model, obtain the lengthwise movement model that its hypersonic aircraft reenters latter end:

$\left\{\begin{array}{c}\stackrel{.}{h}=V\sin \mathrm{\γ}\\ \stackrel{.}{\mathrm{\γ}}=\frac{L}{\mathrm{mV}}+(\frac{V}{r}-\frac{\mathrm{\μ}}{r^{2}V})\cos \mathrm{\γ}\\ \stackrel{.}{V}=-\frac{D}{m}-\frac{\mathrm{\μ}}{r^2}\sin \mathrm{\γ}\end{array}\right.---\left(21\right)$

Wherein, h---aircraft is apart from the distance of earth surface;

---aircraft is apart from the derivative of the distance relative time of earth surface;

L---the lift that aircraft is suffered;

D---the resistance that aircraft is suffered;

Get σ=0, speed inclination angle, it is u=α that controlled quentity controlled variable only has the angle of attack, and quantity of state is taken as x=[h γ V] ^tthe expression formula that obtains lift L and resistance D is as follows:

$L=\frac{1}{2}\mathrm{\ρ}{V}^2{\mathrm{SC}}_L,C_L=C_{L0}+C_{L1}\mathrm{\α}+C_{L2}{e}^{C_{L3}V}---\left(22\right)$

$D=\frac{1}{2}\mathrm{\ρ}{V}^2{\mathrm{SC}}_D,C_D=C_{D0}+C_{D1}{\mathrm{\α}}^2+C_{D2}{e}^{C_{D3}V}---\left(23\right)$

Wherein, C _l0, C _l1, C _l2, C _l3and C _d0, C _d1, C _d2, C _d3be constant; ρ is free air density, and the exponential form of ρ is ρ=ρ ₀e ^{-β h}, ρ wherein ₀be the atmospheric density at place, sea level, β is constant; S Chinese implication is Hypersonic Reentry Vehicles aerofoil area of reference, the truth of a matter that e is natural logarithm.

3. the finite time track rapid generation of a kind of hypersonic aircraft according to claim 1, it is characterized in that: the motion model that in step 2, the hypersonic aircraft obtaining is reentered to final trajectory is optimized problem and describes, and the detailed process that forms nonlinear optimal problem is:

(1), hypersonic aircraft shown in (2) formula is reentered to the form that latter end motion model is expressed as the differential equation:

Wherein, (3) formula is that nonlinear equation is state equation constraint, x=[r, γ, V, λ, φ, ψ] ^tfor state variable, represent the real number field that state variable and controlled quentity controlled variable all belong to, m represents the dimension of controlled quentity controlled variable, and n represents the dimension of state variable; Controlled quentity controlled variable is angle of attack and speed inclination angle σ; U=[α, σ] ^tfor controlled quentity controlled variable, meet certain controlled quentity controlled variable constraint:

u _min≤u(t)≤u _max,t∈[t ₀,t _f] (7)

(2) establish initial time t ₀with t terminal time _ffor condition, the initial time t that fixation locus is optimized ₀state be x ₀, fixed terminal is t constantly _fcorresponding quantity of state is x _f, corresponding end conswtraint is:

Wherein, q represents the dimension of end conswtraint; ψ end conswtraint function;

(3) hypersonic aircraft reenters the inflight phase that latter end is the tool feature of hypersonic aircraft return course, and the superb process that reenters meets rate of heat flow dynamic pressure M and overload N constraint condition:

$\left\{\begin{array}{c}\stackrel{.}{Q}=K_Q{\mathrm{\ρ}}^{0.5}{V}^3\≤{\stackrel{.}{Q}}_{\max }\\ M=\frac{1}{2}{\mathrm{\ρV}}^2\≤M_{\max }\\ N=\frac{\sqrt{L^2+D^2}}{{\mathrm{mg}}_0}\≤N_{\max }\end{array}\right.---\left(5\right)$

Wherein, ρ represents the atmospheric density that mass center of reentry vehicle height is corresponding; K _qrepresent heat flow density constant;

(5) formula can be expressed as (6) formula, and being called reentry corridor constraint is path constraint:

ζ (x (t), u (t))≤0, t ∈ [t ₀, t _f] (6), wherein, ζ is reentry corridor constraint;

(4) it is following form that the motion model that hypersonic aircraft reenters final trajectory is optimized problem representation:

Wherein, φ (x _f, t _f) and L (x, u, t) for optimizing target function, get energy function as optimizing index; J is for optimizing index, (x _f, t _f) be the quantity of state of terminal juncture.

4. the finite time track rapid generation of a kind of hypersonic aircraft according to claim 1, it is characterized in that: in step 3, hypersonic aircraft being reentered to latter end nonlinear optimal problem convexification and process, is that the protruding problem detailed process of quadratic form is by the constraint specification of the optimization index of optimization problem and optimization problem:

The conversion of optimization problem is that nonlinear optimal problem convexification is processed:

(1) definition nonlinear system:

$\stackrel{.}{x}=f(x,t),x\left(0\right)=x_0---\left(9\right)$

Change into the form of State-dependent Coefficient:

Wherein, derivative for quantity of state x relative time, the matrix of coefficients that A (x) determines for quantity of state x, initial point x=0 is that an equilibrium point and hypothesis A (x) meet Local Lipschitz Conditions, and the nonlinear system that (10) formula is described is approached and is applied to nonlinear system by linear time varying system:

$\begin{array}{c}{\stackrel{.}{x}}^{\left[1\right]}\left(t\right)=A\left(x_0\right)x^{\left[1\right]}\left(t\right),x^{\left[1\right]}\left(0\right)=x_0\\ .\\ .\\ .\\ {\stackrel{.}{x}}^{[j-1]}\left(t\right)=A\left(x^{[j-2]}\left(t\right)\right)x^{[j-1]}\left(t\right),x^{[j-1]}\left(0\right)=x_0\\ {\stackrel{.}{x}}^{\left[j\right]}\left(t\right)=A\left(x^{[j-1]}\left(t\right)\right)x^{\left[j\right]}\left(t\right),x^{\left[j\right]}\left(0\right)=x_0\end{array}---\left(11\right)$

When j>=1, x ^[0](t) initial value is taken as original state x conventionally ₀; The solution that meets linear time varying system converges to the solution of the nonlinear system shown in (10), as follows:

lim _j→∞{x ^[j](t)}→x(t)(12)

Wherein, first Linear Estimation formula first linear time invariant system, { x ^[j](t) } _j>=1for j the linear time varying system obtaining after iterative computation j time, and

$\begin{array}{c}{\stackrel{.}{x}}^{\left[2\right]}\left(t\right)=A\left(x_0\right)x^{\left[2\right]}\left(t\right),x^{\left[2\right]}\left(0\right)=x_0\\ .\\ .\\ .\\ {\stackrel{.}{x}}^{[j-1]}\left(t\right)=A\left(x^{[j-2]}\left(t\right)\right)x^{[j-1]}\left(t\right),x^{[j-1]}\left(0\right)=x_0\\ {\stackrel{.}{x}}^{\left[j\right]}\left(t\right)=A\left(x^{[j-1]}\left(t\right)\right)x^{\left[j\right]}\left(t\right),x^{\left[j\right]}\left(0\right)=x_0\end{array}$ For Linear Estimation formula is linear time-varying equation;

Nonlinear optimal problem is converted into a series of linear optimization problem, and the optimum solution of trying to achieve will converge to the solution of former nonlinear optimal problem;

(2) the described nonlinear optimal problem of (8) formula is converted into the protruding optimization problem that CVXGEN processes:

Steps A, by state constraint equation be described as form a given original state of State-dependent Coefficient:

Utilize linear system to approach this nonlinear system:

${\stackrel{.}{x}}^{\left[j\right]}\left(t\right)=A(x^{[j-1]}\left(t\right),u^{[j-1]}\left(t\right))x^{\left[j\right]}\left(t\right)+B(x^{[j-1]}\left(t\right),u^{[j-1]}\left(t\right))u^{\left[j\right]}\left(t\right)---\left(14\right)$

Brief note is:

${\stackrel{.}{x}}^{\left[j\right]}=A^{\′[j-1]}{x}^{\left[j\right]}+B^{\′[j-1]}{u}^{\left[j\right]},x^{\left[j\right]}\left(0\right)=x_0---\left(15\right)$

Wherein, (x), B ' is (x) matrix of coefficients of formula (15) to A ', has realized the iterative approach of linear time varying system to former nonlinear system;

Step B, linear system (15) is carried out to discretize processing, the sampling period is taken as T, and discretize step-length is P, adopts difference method to obtain final convexification form as follows:

x ^[j](k+1)＝A ^[j-1](k)x ^[j](k)+B ^[j-1](k)u ^[j](k)+C ^[j-1](k),k＝1,2,...,P(16)

Wherein, A ^[j-1](k)=T * A ' ^[j-1](k)+I, B ^[j-1](k)=T * B ' ^[j-1](k) C ^[j-1](k) represent j linear time varying system Discrete Linear error obtaining for j time;

Step C, for the reentry corridor constraint ζ (x (t), u (t))≤0 in (8) formula, t ∈ [t ₀, t _f] adopt linear discrete to obtain following convexification form:

C _a ^[j](k)x ^[j](k)+C _b ^[j](k)u ^[j](k)+C _c ^[j](k)≤0,k＝1,2...P

C _a(k), C _b(k), C _c(k) be that ζ (x (t), u (t))≤0 is carried out to the coefficient entry that Discrete Linear obtains;

Step D, the end conswtraint in (8) formula and controlled quentity controlled variable constraint convexification is processed, is obtained respectively following form:

x ^[j](P+1)-x _tar＝0

u _min≤u ^[j](k)≤u _max,k＝1,2...P

Wherein, x _tarfor target location dot information;

Step e, provide optimization index; Using energy function as optimizing target function, terminal hard constraint is converted to soft-constraint; Final optimization index form is:

$J=\frac{1}{P}\×\underset{k=1}{\overset{P}{\mathrm{\Σ}}}{\left|\right|u^{\left[j\right]}\left(k\right)\left|\right|}_2^2+{\left|\right|x^{\left[j\right]}(P+1)-x_{\mathrm{tar}}\left|\right|}_2^2$

Step F, hypersonic aircraft reenter final trajectory optimization problem and have converted by nonlinear optimal problem the protruding problem that protruding optimization tool CVXGEN can separate to; The final form of the protruding problem that can separate is:

$\begin{array}{c}\min J=\frac{1}{P}\×\underset{k=1}{\overset{P}{\mathrm{\Σ}}}{\left|\right|u^{\left[j\right]}\left(k\right)\left|\right|}_2^2+{\left|\right|x^{\left[j\right]}(P+1)-x_{\mathrm{tar}}\left|\right|}_2^2\\ s.t.\left\{\begin{array}{c}{x}^{\left[j\right]}(k+1)=A^{[j-1]}\left(k\right)x^{\left[j\right]}\left(k\right)+B^{[j-1]}\left(k\right)u^{\left[j\right]}\left(k\right)+C^{[j-1]}\left(k\right)\\ {C_a}^{\left[j\right]}\left(k\right)x^{\left[j\right]}\left(k\right)+{C_b}^{\left[j\right]}\left(k\right)u^{\left[j\right]}\left(k\right)+{C_c}^{\left[j\right]}\left(k\right)\≤0\\ u_{\min }\≤u^{\left[j\right]}\left(k\right)\≤u_{\max }\\ -e\≤x^{\left[j\right]}\left(k\right)-x_r\left(k\right)\≤e\end{array}\right.\\ k=\mathrm{1,2},...,P\end{array}---\left(19\right)$

(3) introduce the quick track optimizing problem of latter end that reenters strike to dynamic monocular punctuate with realization that rolling Optimization of Time Domain solves hypersonic aircraft reentry vehicle target of attack; Using current state x (i) as original state, the model data in current i locates prediction optimization time domain P constantly, and be optimized accordingly solving of problem; According to (19) formula, to the description of optimization problem, can obtaining hypersonic aircraft, to reenter the quick track optimizing problem of latter end concrete form as follows:

$\begin{array}{c}\min J=\frac{1}{P}\×\underset{k=1}{\overset{P}{\mathrm{\Σ}}}{\left|\right|u^{\left[j\right]}(i+k|i)\left|\right|}_2^2+{\left|\right|x^{\left[j\right]}(i+P+1|i)-x_{\mathrm{tar}}\left|\right|}_2^2\\ s.t.\left\{\begin{array}{c}{x}^{\left[j\right]}(i+k+1|i)=A^{[j-1]}(i+k|i)x^{\left[j\right]}(i+k|i)+B^{[j-1]}(i+k|i)u^{\left[j\right]}(i+k|i)+C^{[j-1]}(i+k|i)\\ {C_a}^{\left[j\right]}(i+k|i)x^{\left[j\right]}(i+k|i)+{C_b}^{\left[j\right]}(i+k|i)u^{\left[j\right]}(i+k|i)+{C_c}^{\left[j\right]}(i+k|i)\≤0\\ u_{\min }\≤u^{\left[j\right]}(i+k|i)\≤u_{\max }\\ -e\≤x^{\left[j\right]}(i+k|i)-x_r(i+k|i)\≤e\end{array}\right.\\ k=\mathrm{1,2},...,P\end{array}---\left(20\right)$

Wherein, i+k|i is illustrated in the i predicted value to i+k step variable constantly, and j represents iterations; P is called control time domain, represents the time span of predict future, and u is controlled quentity controlled variable, C _a, C _band C _cthree factor arrays for reentry corridor constraint.

5. the finite time track rapid generation of a kind of hypersonic aircraft according to claim 1, it is characterized in that: in step 4, according to CVXGEN code, write the protruding optimization problem of specification description quadratic form, utilize compiled online device to compile and generate high speed solver process and be code:

(1) code is write, and the code of writing comprises dimension, parameter, variable, objective function and constraint;

(2) code is write the compiling of laggard line code; Generate 5 main C source files, for the * .m file of MATLAB environment with for the CVX file of result verification; Wherein * .m file can be directly embedded under MATLAB, as the solver of optimization problem;

(3) using MATLAB interface, is that self-defined solver is embedded under MATLAB environment by the solver of optimization problem.

6. the finite time track rapid generation of a kind of hypersonic aircraft according to claim 1, it is characterized in that: in step 5, high speed solver is embedded under MATALB environment, realize the numerical solution of the protruding optimization problem of quadratic form, and the detailed process that logarithm value solving result is analyzed is as follows:

(1) after code compilation success, self-defined high speed solver CVXGEN is embedded under MATLAB environment, the parameter that aircraft is reentered in final trajectory rapid Optimum problem is carried out corresponding assignment;

(2) solve hypersonic aircraft and reenter final trajectory rapid Optimum problem, obtain the simulation result that final hypersonic aircraft reenters terminal flight track;

(3) simulation result that hypersonic aircraft is reentered to final trajectory rapid Optimum problem is analyzed: utilize CVXGEN to generate the high speed solver of optimization problem (20), to solve code is embedded in MATLAB, according to (11) formula, carry out repeatedly iterative, obtain the optimum control amount u* (i|i) in P time domain, ..., u* (i+P|i), according to rolling optimization, u* (i|i) is applied to longitudinal nonlinear motion model formula (21), calculates next state x (i+1) constantly; If do not reach assigned address, do not forward step 5 to and start to calculate, until aircraft hits the mark or aircraft arrives assigned address; Wherein, u* (i+P|i) represents i+P the control variable that i suboptimization obtains; Assigned address is that the hypersonic aircraft that meets every optimization index and constraint reenters terminal flight track.