CN104793629B

CN104793629B - A kind of contragradience neural network control method of dirigible Three-dimensional Track tracking

Info

Publication number: CN104793629B
Application number: CN201510221276.7A
Authority: CN
Inventors: 杨跃能; 闫野; 朱正龙; 刘二江; 徐博婷
Original assignee: National University of Defense Technology
Current assignee: National University of Defense Technology
Priority date: 2015-05-04
Filing date: 2015-05-04
Publication date: 2017-09-26
Anticipated expiration: 2035-05-04
Also published as: CN104793629A

Abstract

The present invention relates to a kind of contragradience neural network control method of dirigible Three-dimensional Track tracking.For the Track In Track control problem of dirigible, the present invention establishes the non-linear dynamic model of dirigible；As controll plant, non-linear dynamic model is decomposed into two subsystems, Backstepping is used to design Lyapunov functions and intermediate virtual controlled quentity controlled variable for each subsystem, by determining appropriate virtual feedback, so that the front position of system reaches Asymptotic Stability, " reversely deduce " to whole system always, so as to realize the Asymptotic Stability of whole system；For dirigible kinetic model uncertain problem, unknown dirigible kinetic model is accurately estimated using neural network approximator, to improve control accuracy and systematic function.The closed-loop system controlled by this method can any given parametrization of high precision tracking instruct flight path, and with good stability, adaptability, robustness and dynamic property, the Project Realization for dirigible flight tracking control provides effective scheme.

Description

A kind of contragradience neural network control method of dirigible Three-dimensional Track tracking

Technical field

The present invention relates to a kind of flight control method of field of aerospace, it provides a kind of contragradience for dirigible Track In Track Neural network control method, belongs to automatic control technology field.

Background technology

Dirigible refers to that a kind of dependence is lighter than the quiet buoyancy lift-off of gas (such as helium, hydrogen) offer of air, by automatic Flight control system realizes resident and low-speed maneuver the aircraft of fixed point, and with hang time length, energy consumption is low, efficiency-cost ratio is high and determines The advantages of point is resident, is widely used in the fields such as reconnaissance and surveillance, earth observation, environmental monitoring, emergency disaster relief, scientific exploration, has There is significant application value and wide application prospect, currently turn into the study hotspot of aviation field.

Track In Track refers to dirigible from given original state and tracks the boat of the instruction under given inertial coodinate system Mark.The spatial movement of dirigible have the coupling of non-linear, passage, it is uncertain, easily by external disturbance the features such as, therefore, flight tracking control One of key technology controlled as airship flight.Numerous researchers are directed to the Track In Track problem of dirigible, it is proposed that PID is controlled The methods such as system, feedback control, sliding formwork control, robust control, the technical side for being available for reference is provided for dirigible Track In Track Case.But above-mentioned flight tracking control method not yet effectively solves the problems, such as two categories below：One is that dirigible kinetic model is not known, and is existed Modeling error and Unmarried pregnancy；Two be that dirigible flight path control system is a complicated nonlinear multivariable systems, flight bag The stability of closed-loop control system is difficult to ensure that in line.

The content of the invention

In view of the deficienciess of the prior art, it is an object of the invention to provide a kind of contragradience god of dirigible Three-dimensional Track tracking Through network control method.

The present invention is directed to dirigible Three-dimensional Track tracking problem, establishes the non-linear dynamic model of dirigible；As by Object is controlled, non-linear dynamic model is decomposed into two subsystems, uses Backstepping to design Li Yapunuo for each subsystem Husband's (Lyapunov) function and intermediate virtual controlled quentity controlled variable, by determining appropriate virtual feedback so that the front position of system reaches To Asymptotic Stability, " reversely deduce " to whole system always, so as to realize the Asymptotic Stability of whole system；For dirigible dynamics Model uncertain problem, unknown dirigible kinetic model is accurately approached using neutral net, to improve control accuracy and system Performance.Advantages of the present invention is shown：1. designed using Backstepping and cause Liapunov (Lyapunov) function and control law Design process systematization, structuring, it is ensured that the stability of system；2. the uncertain of dirigible is accurately approached using neutral net Model so that Track In Track control system has strong adaptability and strong robustness.

The technical scheme is that：The flight tracking control margin of error is calculated by given instruction flight path and actual flight path first, Then using Backstepping techniques design flight tracking control rule, flight tracking control amount is calculated；To solve dirigible kinetic model uncertain problem, Unknown ambiguous model is accurately approached using neutral net.In practical application, dirigible flight path is measured by integrated navigation system Arrive, obtained controlled quentity controlled variable will be calculated by this method transmit to executing agency flight tracking control function can be achieved.

Specifically, a kind of contragradience neural network control method of dirigible Three-dimensional Track tracking, comprises the following steps：

Step one：Given instruction flight path：η_d=[x_d,y_d,z_d,θ_d,ψ_d,φ_d]^T；

Wherein：Described instruction flight path is generalized coordinates η_d=[x_d,y_d,z_d,θ_d,ψ_d,φ_d]^T, x_d、y_d、z_d、θ_d、ψ_dAnd φ_d Respectively instruct x coordinate, instruction y-coordinate, instruction z coordinate, the instruction angle of pitch, instruction yaw angle and instruct roll angle, subscript T tables Show the transposition of vector or matrix.

Step 2：The flight tracking control margin of error is calculated：The flight tracking control margin of error between computations flight path and actual flight path e；

The computational methods of the flight tracking control margin of error e are：

E=η_d- η=[x_d-x,y_d-y,z_d-z,θ_d-θ,ψ_d-ψ,φ_d-φ]^T (1)

Wherein：η=[x, y, z, θ, ψ, φ]^TFor actual flight path, x, y, z, θ, ψ, φ are respectively the x coordinate of actual flight path, y Coordinate, z coordinate, the angle of pitch, yaw angle and roll angle.

Step 3：Flight tracking control rule design：Lypaunov functions and intermediate virtual controlled quentity controlled variable are chosen, is designed using Backstepping Flight tracking control is restrained, and is calculated flight tracking control amount u, is specifically included following steps：

1) kinetic model of dirigible is set up

The coordinate system and kinematic parameter of dirigible spatial movement are defined as follows：Using earth axes o_ex_ey_ez_eSat with carrier Mark system o_bx_by_bz_bSpatial movement to dirigible is described, and CV is centre of buoyancy, and CG is center of gravity, and the vector of centre of buoyancy to center of gravity is r_G= [x_G,y_G,z_G]^T.Kinematic parameter is defined：Position P=[x, y, z]^T, x, y, z is respectively axial direction, the displacement of lateral and vertical direction； Attitude angle Ω=[θ, ψ, φ]^T, θ, ψ, φ are respectively the angle of pitch, yaw angle and roll angle；Speed v=[u, v, w]^T, u, v, w points Not Wei in carrier coordinate system axially, the speed of lateral and vertical direction；Angular velocity omega=[p, q, r]^T, p, q, r be respectively rolling, Pitching and yaw rate.Remember generalized coordinates η=[x, y, z, θ, ψ, φ]^T, generalized velocity is V=[u, v, w, p, q, r]^T。

The kinetic model of dirigible is described as follows：

In formula

Wherein

In formula, m is dirigible quality, m₁₁、m₂₂、m₃₃For additional mass, I₁₁、I₂₂、I₃₃For additional inertial, Λ is dirigible body Product；Q is dynamic pressure, and α is the angle of attack, and β is yaw angle, C_X、C_Y、C_Z、C_l、C_m、C_nFor Aerodynamic Coefficient；I_x、I_y、I_zRespectively around o_bx_b、 o_by_b、o_bz_bPrincipal moments；I_xy、I_xz、I_yzRespectively on plane o_bx_by_b、o_bx_bz_b、o_by_bz_bProduct of inertia；T is that thrust is big Small, μ is thrust vectoring and o_bx_bz_bAngle between face, provides it in o_bx_bz_bThe left side in face is just, υ is thrust vectoring in o_bx_bz_b The projection in face and o_bx_bAngle between axle, provides that it is projected in o_bx_bIt is just under axle；l_x、l_y、l_zRepresent thrust point away from Origin o_bDistance.

Formula (3) is the expression formula on generalized velocity V, it is necessary to transform it into the expression formula on generalized coordinates η.

It can be obtained by formula (2)：

In formula, J^-1(η) is J (η) inverse matrix.

To formula (16) differential, it can obtain

In formula

Formula (19) premultiplicationIt can obtain

Composite type (3), formula (19) and formula (21) can be obtained:

In formula

M_η(η)=R^TMR (23)

Γ=R^Tτ (26)

Make x₁=η,Then kinetic simulation pattern (22) can be written as following form：

In formula,Representing matrix M_ηInverse matrix；M_ηIt is M_ηThe abbreviation of (η)；

Using the mathematical modeling described by formula (27) as controlled device, using Backstepping techniques design flight tracking control rule.

2) design flight tracking control rule

According to the flight tracking control margin of error e between instruction flight path and actual flight path, virtual amount is defined as follows：

In formula, α₁For virtual amount, k₁For adjustable control parameter.

The virtual amount α of definition₁With x₂Between error e '：

E '=x₂-α₁ (29)

Formula (29) to time diffusion and substitutes into formula (27), can obtain：

Order

Then formula (31) is represented by：

Choose liapunov function (Lyapunov functions) V₁

Formula (33) is substituted into time diffusion, and by formula (32), can be obtained：

According to formula (34), following flight tracking control rule is designed：

3) stability analysis

Flight tracking control rule formula (35) is substituted into formula (34), can be obtained：

Formula (36) shows：The stability that (35) ensure that closed-loop system is restrained using flight tracking control.

Step 4：Neural network approximator is designed：With flight tracking control error e and its rate of changeActual flight path η and its change RateFor the input variable of neutral net, with the estimate of dirigible kinetic modelSet for the output variable of neutral net Neural network approximator has been counted, function is infinitely approached using neutral net and estimates unknown ambiguous model, to improve control essence Degree, is comprised the following steps that：

1) due to being difficult to carry out dirigible Accurate Model during practical flight, f (x) is unknown function, it is difficult to according to Formula (35) is controlled rule and resolved, therefore, must use f (x) estimateFlight tracking control rule formula (35) is resolved；Adopt Unknown function f (x) is approached with neutral net, then is had：

F (x)=w^Th(x)+ε (37)

In formula, w is the weight vectors of neutral net, and ε is approximate error, h (x)=[h_i(x)]^T, h_i(x) it is gaussian basis letter Number, subscript i represents i-th of Gaussian bases；

2) input/output variable is selected

Make flight tracking control error e and its rate of changeActual flight path η and its rate of changeFor the input of neural network approximator Variable, makes estimateFor the output variable of neural network approximator.

3) neural network structure is designed

Neural network structure includes input layer, hidden layer and output layer.

Input layer：Choose neutral net input variable be

Hidden layer：Gaussian function is chosen as the basic function of hidden node

Wherein, c=[c_i]^T, c_iFor the intermediate value of i-th of Gaussian function, σ_iFor the base width parameter of i-th of node, | | | | Represent euclideam norm.

Output layer：Neutral net is output as

Wherein,For w estimate.

4) stability analysis

DefinitionWith w differences：

Choose Lyapunov functions：

In formula, ξ=[e e ']^T,Θ is adjustable positive definite matrix, λ^-1Representing matrix λ Inverse matrix.

To formula (41) differential, it can obtain：

Definition

Then formula (42) can be written as：

In formula,

Design following adaptive law：

In formula, γ ＞ 0 are adjustable parameter,

Adaptive law is substituted into formula (44), can be obtained：

According to Schwarz (Schwarz) inequality, have：

In formula, | | | |_FRepresent soft Bin Niusi (Frobenius) norm of volt.

Formula (47) is substituted into formula (46), can be obtained：

In formula, ε_NFor the upper bound of approximate error, k_minFor adjustable parameter matrix k minimal eigenvalue, W_MFor weight matrix W's Maximum element.

In view of following equation：

If so thatMust then have and be set up with lower inequality：

Or

Thus, uncertain dirigible nonlinear kinetics mould can accurately be estimated by above-mentioned neural network approximator Type.

The advantageous effects of the present invention：

1) this method is directly based upon the non-linear dynamic model design flight tracking control rule of dirigible, it is contemplated that every non-linear Factor and Multivariable Coupling effect, overcome the limitation that inearized model is only suitable to equilibrium state, being capable of high precision tracking times The given parametrization instruction flight path of meaning.

2) complicated non-linear flight path control system is resolved into two subsystems for being no more than systematic education by this method, so Lypaunov functions and intermediate virtual controlled quentity controlled variable are designed for each subsystem afterwards, by determining appropriate virtual feedback so that be The front position of system reaches Asymptotic Stability, and " pusher " is to whole system always, it is ensured that the Asymptotic Stability of whole system.

3) this method does not need accurately known to dirigible kinetic model, and unknown fly is estimated using neural network approximator Ship kinetic model, improves the adaptability and control accuracy of system.

Control engineer to give arbitrary instruction flight path according to actual dirigible in application process, and will be obtained by this method To controlled quentity controlled variable transmit to executing agency and realize flight tracking control function.

Brief description of the drawings

Fig. 1 is dirigible flight path control system structure chart of the present invention

Fig. 2 is dirigible Three-dimensional Track tracking and controlling method flow chart of steps of the present invention

Fig. 3 is that dirigible coordinate system of the present invention and kinematic parameter are defined

Fig. 4 is neural network structure figure of the present invention

Fig. 5 is dirigible Three-dimensional Track tracing control result of the present invention

Fig. 6 is dirigible Three-dimensional Track tracing control error of the present invention

Fig. 7 is neutral net Approaching Results

Symbol description is as follows in figure：

η η=[x, y, z, θ, ψ, φ]^TFor dirigible flight path, wherein x, y, z, θ, ψ, φ be respectively actual flight path x coordinate, Y-coordinate, z coordinate, the angle of pitch, yaw angle and roll angle；

η_d η_d=[x_d,y_d,z_d,θ_d,ψ_d,φ_d]^TFor instruction flight path, wherein x_d、y_d、z_d、θ_d、ψ_dAnd φ_dRespectively instruct x Coordinate, instruction y-coordinate, instruction z coordinate, the instruction angle of pitch, instruction yaw angle and instruction roll angle；

o_ex_ey_ez_e o_ex_ey_ez_eRepresent earth axes；

o_bx_by_bz_b o_bx_by_bz_bRepresent dirigible body coordinate system；

E e=[x_e,y_e,z_e,θ_e,ψ_e,φ_e]^TFor the flight tracking control margin of error, respectively the x coordinate error of flight tracking control, y Error of coordinate, z coordinate error；

U u are system control amount；

F (x) f (x) are the uncertain kinetic model of dirigible.

Embodiment

The present invention " a kind of contragradience neural network control method of dirigible Three-dimensional Track tracking ", it is comprised the following steps that：

Step one：Given instruction flight path

Giving instruction flight path is：

η_d=[x_d,y_d,z_d,θ_d,ψ_d,φ_d]^T=[(3t) m, (0.93t) m, 10m, 0rad, 0.3rad, 0rad]^T, x_d、y_d、 z_d、θ_d、ψ_dAnd φ_dRespectively instruction x coordinate, instruction y-coordinate, instruction z coordinate, the instruction angle of pitch, instruction yaw angle and instruction rolling Corner；

Step 2：The flight tracking control margin of error is calculated

The flight tracking control margin of error between computations flight path and actual flight path：

E=η_d- η=[x_d-x,y_d-y,z_d-z,θ_d-θ,ψ_d-ψ,φ_d-φ]^T,

Wherein, η=[x, y, z, θ, ψ, φ]^TFor actual flight path, x, y, z, θ, ψ, φ are respectively the x coordinate of actual flight path, y Coordinate, z coordinate, the angle of pitch, yaw angle and roll angle, are consecutive variations value.

Initially flight path is：

η₀=[x₀,y₀,z₀,θ₀,ψ₀,φ₀]^T=[100m, -200m, 5m, 0.02rad, 0.02rad, 0.1rad]^T。

Initial velocity：

V₀=[u₀,v₀,w₀,p₀,q₀,r₀]^T=[8m/s, 0m/s, 0m/s, 0rad/s, 0rad/s, 0rad/s]^T

Step 3：Design flight tracking control rule：

1) dirigible kinetic model is set up

The mathematical modeling of dirigible spatial movement is represented by：

In formula

Wherein

In formula, m is dirigible quality, m₁₁、m₂₂、m₃₃For additional mass, I₁₁、I₂₂、I₃₃For additional inertial；Λ is dirigible body Product；Q is dynamic pressure, and α is the angle of attack, and β is yaw angle, C_X、C_Y、C_Z、C_l、C_m、C_nFor Aerodynamic Coefficient；I_x、I_y、I_zRespectively around o_bx_b、 o_by_b、o_bz_bPrincipal moments；I_xy、I_xz、I_yzRespectively on plane o_bx_by_b、o_bx_bz_b、o_by_bz_bProduct of inertia；T is that thrust is big Small, μ is thrust vectoring and o_bx_bz_bAngle between face, provides it in o_bx_bz_bThe left side in face is just, υ is thrust vectoring in o_bx_bz_b The projection in face and o_bx_bAngle between axle, provides that it is projected in o_bx_bIt is just under axle；l_x、l_y、l_zRepresent thrust point away from Origin o_bDistance.

Formula (52) is the expression formula on generalized velocity V, it is necessary to transform it into the expression formula on generalized coordinates η.

It can be obtained by formula (51)：

In formula, J^-1(η) is J (η) inverse matrix,

To formula (65) differential, it can obtain

In formula

Formula (68) premultiplicationIt can obtain

Composite type (52), formula (68) and formula (70) can be obtained:

In formula

M_η(η)=R^TMR (72)

Γ=R^Tτ (75)

Make x₁=η,Then kinetic model (71) can be written as following form：

In formula,Representing matrix M η inverse matrix；M_ηIt is M_ηThe abbreviation of (η)；

Using the mathematical modeling described by formula (76) as controlled device, using Backstepping techniques design flight tracking control rule.

Dirigible parameter in the present embodiment is shown in Table 1.

The dirigible parameter of table 1

2) design control law

In formula, α₁For virtual amount, k₁, k₂For adjustable control parameter.

The virtual amount α of definition₁With x₂Between error e '：

E '=x₂-α₁ (78)

Formula (78) to time diffusion and substitutes into formula (76), can obtain：

Order

Then formula (80) is represented by：

Wherein, k₁Value is 12.

Choose Lyapunov functions V₁

Formula (82) to time diffusion and substitutes into formula (81), can obtain：

According to formula (83), following control law is designed：

Wherein, k₂Value is 10.

Due to being difficult to that Accurate Model is carried out to dirigible during practical flight, f (x) is unknown function, it is difficult to according to formula (35) it is controlled rule to resolve, therefore, f (x) estimate must be usedFlight tracking control rule formula (35) is resolved；

Wherein, the design neural network approximator described in step 4, its design method is：

1) unknown function f (x) is approached using neutral net, then had：

F (x)=w^Th(x)+ε (37)

2) input/output variable is selected

3) neural network structure is designed

Neural network structure includes input layer, hidden layer and output layer, as shown in Figure 4.

Input layer：Choose neutral net input variable be

Hidden layer：Gaussian function is chosen as the basic function of hidden node

Wherein,

Output layer：Neutral net is output as

Wherein,Value be taken as

Dirigible Three-dimensional Track tracing control result in embodiment is as shown in Figure 5-Figure 7.Fig. 5 gives dirigible Three-dimensional Track Tracing control result, can be obtained by Fig. 5：Dirigible can be accurately tracked by instructing flight path, demonstrate the present invention by initial position The validity of the Track In Track control method proposed；Fig. 6 gives Track In Track control error, can be obtained by Fig. 6, institute of the present invention The flight tracking control method of proposition can accurately track given instruction flight path.Fig. 7 gives neutral net Approaching Results, by Fig. 7 can be obtained, and the neural network approximator designed by the present invention can accurately estimate uncertain dirigible kinetic model.

The explanation of the preferred embodiment of the present invention contained above, this be in order to describe the technical characteristic of the present invention in detail, and It is not intended to the content of the invention being limited in the concrete form described by embodiment, according to other of present invention purport progress Modifications and variations are also protected by this patent.The purport of present invention is to be defined by the claims, rather than by embodiment Specific descriptions are defined.

Claims

1. a kind of contragradience neural network control method of dirigible Three-dimensional Track tracking, it is characterised in that comprise the following steps：

Wherein：Instruction flight path is generalized coordinates η_d=[x_d,y_d,z_d,θ_d,ψ_d,φ_d]^T, x_d、y_d、z_d、θ_d、ψ_dAnd φ_dRespectively instruct X coordinate, instruction y-coordinate, instruction z coordinate, the instruction angle of pitch, instruction yaw angle and instruction roll angle, subscript T represent vector or square The transposition of battle array；

Step 2：The flight tracking control margin of error is calculated：Flight tracking control margin of error e between computations flight path and actual flight path；

Flight tracking control margin of error e computational methods are：

E=η_d- η=[x_d-x,y_d-y,z_d-z,θ_d-θ,ψ_d-ψ,φ_d-φ]^T (1)

η=[x, y, z, θ, ψ, φ]^TFor actual flight path, x, y, z, θ, ψ, φ are respectively that the x coordinate, y-coordinate, z of actual flight path are sat Mark, the angle of pitch, yaw angle and roll angle；

Step 3：Flight tracking control rule design：Lypaunov functions and intermediate virtual controlled quentity controlled variable are chosen, flight path is designed using Backstepping Control law, calculates flight tracking control amount u；

1) kinetic model of dirigible is set up

The coordinate system and kinematic parameter of dirigible spatial movement are defined as follows：Using earth axes o_ex_ey_ez_eAnd carrier coordinate system o_bx_by_bz_bSpatial movement to dirigible is described, and CV is centre of buoyancy, and CG is center of gravity, and the vector of centre of buoyancy to center of gravity is r_G=[x_G, y_G,z_G]^T；Kinematic parameter is defined：Position P=[x, y, z]^T, x, y, z is respectively axial direction, the displacement of lateral and vertical direction；Posture Angle Ω=[θ, ψ, φ]^T, θ, ψ, φ are respectively the angle of pitch, yaw angle and roll angle；Speed v=[u, v, w]^T, u, v, w are respectively Axial direction, the speed of lateral and vertical direction in carrier coordinate system；Angular velocity omega=[p, q, r]^T, p, q, r are respectively rolling, pitching And yaw rate；Remember generalized coordinates η=[x, y, z, θ, ψ, φ]^T, generalized velocity is V=[u, v, w, p, q, r]^T；

The kinetic model of dirigible is described as follows：

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<mrow> <mi>M</mi> <mover> <mi>V</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mover> <mi>N</mi> <mo>&OverBar;</mo> </mover> <mo>+</mo> <mover> <mi>G</mi> <mo>&OverBar;</mo> </mover> <mo>+</mo> <mi>&tau;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>

In formula

<mrow> <msub> <mi>J</mi> <mn>1</mn> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>cos</mi> <mi>&psi;</mi> <mi>cos</mi> <mi>&theta;</mi> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&theta;</mi> <mi>sin</mi> <mi>&phi;</mi> <mo>-</mo> <mi>sin</mi> <mi>&psi;</mi> <mi>cos</mi> <mi>&phi;</mi> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&theta;</mi> <mi>cos</mi> <mi>&phi;</mi> <mo>+</mo> <mi>sin</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&phi;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>sin</mi> <mi>&psi;</mi> <mi>cos</mi> <mi>&theta;</mi> </mrow> </mtd> <mtd> <mrow> <mi>sin</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&theta;</mi> <mi>sin</mi> <mi>&phi;</mi> <mo>+</mo> <mi>cos</mi> <mi>&psi;</mi> <mi>cos</mi> <mi>&phi;</mi> </mrow> </mtd> <mtd> <mrow> <mi>sin</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&theta;</mi> <mi>cos</mi> <mi>&phi;</mi> <mo>-</mo> <mi>cos</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&phi;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mi>sin</mi> <mi>&theta;</mi> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi>&theta;</mi> <mi>sin</mi> <mi>&phi;</mi> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi>&theta;</mi> <mi>cos</mi> <mi>&phi;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>J</mi> <mn>2</mn> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&phi;</mi> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mi>sin</mi> <mi>&phi;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mi>sec</mi> <mi>&theta;</mi> <mi>sin</mi> <mi>&phi;</mi> </mrow> </mtd> <mtd> <mrow> <mi>sec</mi> <mi>&theta;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&phi;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mi>&theta;</mi> <mi>sin</mi> <mi>&phi;</mi> </mrow> </mtd> <mtd> <mrow> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mi>&theta;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&phi;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mi>M</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>m</mi> <mo>+</mo> <msub> <mi>m</mi> <mn>11</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msub> <mi>mz</mi> <mi>G</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>my</mi> <mi>G</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mi>m</mi> <mo>+</mo> <msub> <mi>m</mi> <mn>22</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>mz</mi> <mi>G</mi> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msub> <mi>mx</mi> <mi>G</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mi>m</mi> <mo>+</mo> <msub> <mi>m</mi> <mn>33</mn> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>my</mi> <mi>G</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>mx</mi> <mi>G</mi> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>mz</mi> <mi>G</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>my</mi> <mi>G</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>I</mi> <mi>x</mi> </msub> <mo>+</mo> <msub> <mi>I</mi> <mn>11</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>mz</mi> <mi>G</mi> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>mx</mi> <mi>G</mi> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msub> <mi>I</mi> <mi>x</mi> </msub> <mo>+</mo> <msub> <mi>I</mi> <mn>22</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msub> <mi>mx</mi> <mi>G</mi> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msub> <mi>I</mi> <mi>x</mi> </msub> <mo>+</mo> <msub> <mi>I</mi> <mn>33</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow> 1

<mrow> <mover> <mi>G</mi> <mo>&OverBar;</mo> </mover> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mo>(</mo> <mi>B</mi> <mo>-</mo> <mi>G</mi> <mo>)</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&theta;</mi> </mtd> </mtr> <mtr> <mtd> <mo>(</mo> <mi>G</mi> <mo>-</mo> <mi>B</mi> <mo>)</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&theta;</mi> <mi>sin</mi> <mi>&phi;</mi> </mtd> </mtr> <mtr> <mtd> <mo>(</mo> <mi>G</mi> <mo>-</mo> <mi>B</mi> <mo>)</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&theta;</mi> <mi>cos</mi> <mi>&phi;</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mi>G</mi> </msub> <mi>G</mi> <mi> </mi> <mi>cos</mi> <mi>&theta;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&phi;</mi> <mo>-</mo> <msub> <mi>z</mi> <mi>G</mi> </msub> <mi>G</mi> <mi> </mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&theta;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&phi;</mi> </mtd> </mtr> <mtr> <mtd> <mo>-</mo> <msub> <mi>x</mi> <mi>G</mi> </msub> <mi>G</mi> <mi> </mi> <mi>cos</mi> <mi>&theta;</mi> <mi>cos</mi> <mi>&phi;</mi> <mo>-</mo> <msub> <mi>z</mi> <mi>G</mi> </msub> <mi>G</mi> <mi> </mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&theta;</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mi>G</mi> </msub> <mi>G</mi> <mi> </mi> <mi>cos</mi> <mi>&theta;</mi> <mi>sin</mi> <mi>&phi;</mi> <mo>+</mo> <msub> <mi>y</mi> <mi>G</mi> </msub> <mi>G</mi> <mi> </mi> <mi>sin</mi> <mi>&theta;</mi> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mover> <mi>N</mi> <mo>&OverBar;</mo> </mover> <mo>=</mo> <msup> <mrow> <mo>&lsqb;</mo> <msub> <mi>N</mi> <mi>u</mi> </msub> <mo>,</mo> <msub> <mi>N</mi> <mi>v</mi> </msub> <mo>,</mo> <msub> <mi>N</mi> <mi>w</mi> </msub> <mo>,</mo> <msub> <mi>N</mi> <mi>p</mi> </msub> <mo>,</mo> <msub> <mi>N</mi> <mi>q</mi> </msub> <mo>,</mo> <msub> <mi>N</mi> <mi>r</mi> </msub> <mo>&rsqb;</mo> </mrow> <mi>T</mi> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>

Wherein

<mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <msub> <mi>N</mi> <mi>u</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mi>m</mi> <mo>+</mo> <msub> <mi>m</mi> <mn>22</mn> </msub> <mo>)</mo> </mrow> <mi>vr</mi> <mo>-</mo> <mrow> <mo>(</mo> <mi>m</mi> <mo>+</mo> <msub> <mi>m</mi> <mn>33</mn> </msub> <mo>)</mo> </mrow> <mi>wq</mi> <mo>+</mo> <mi>m</mi> <mo>[</mo> <msub> <mi>x</mi> <mi>G</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>p</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>y</mi> <mi>G</mi> </msub> <mi>pq</mi> <mo>-</mo> <msub> <mi>z</mi> <mi>G</mi> </msub> <mi>pr</mi> <mo>]</mo> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <msup> <mi>Q&Lambda;</mi> <mrow> <mn>2</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> <mrow> <mo>(</mo> <mo>-</mo> <msub> <mi>G</mi> <mi>X</mi> </msub> <mi>cos</mi> <mi></mi> <mi>&alpha;</mi> <mi>cos</mi> <mi>&beta;</mi> <mo>+</mo> <msub> <mi>G</mi> <mi>Y</mi> </msub> <mi>cos</mi> <mi></mi> <mi>&alpha;</mi> <mi>sin</mi> <mi>&beta;</mi> <mo>+</mo> <msub> <mi>C</mi> <mi>Z</mi> </msub> <mi>sin</mi> <mi>&alpha;</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <msub> <mi>N</mi> <mi>v</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mi>m</mi> <mo>+</mo> <msub> <mi>m</mi> <mn>33</mn> </msub> <mo>)</mo> </mrow> <mi>wp</mi> <mo>-</mo> <mrow> <mo>(</mo> <mi>m</mi> <mo>+</mo> <msub> <mi>m</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mi>ur</mi> <mo>+</mo> <mi>m</mi> <mo>[</mo> <msub> <mi>x</mi> <mi>G</mi> </msub> <mi>pq</mi> <mo>-</mo> <msub> <mi>y</mi> <mi>G</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>p</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>z</mi> <mi>G</mi> </msub> <mi>qr</mi> <mo>]</mo> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <msup> <mi>Q&Lambda;</mi> <mrow> <mn>2</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>X</mi> </msub> <mi>sin</mi> <mi>&beta;</mi> <mo>+</mo> <msub> <mi>C</mi> <mi>Y</mi> </msub> <mi>cos</mi> <mi>&beta;</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <msub> <mi>N</mi> <mi>w</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mi>m</mi> <mo>+</mo> <msub> <mi>m</mi> <mn>22</mn> </msub> <mo>)</mo> </mrow> <mi>vp</mi> <mo>-</mo> <mrow> <mo>(</mo> <mi>m</mi> <mo>+</mo> <msub> <mi>m</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mi>uq</mi> <mo>-</mo> <mi>m</mi> <mo>[</mo> <msub> <mi>x</mi> <mi>G</mi> </msub> <mi>pr</mi> <mo>+</mo> <msub> <mi>y</mi> <mi>G</mi> </msub> <mi>qr</mi> <mo>-</mo> <msub> <mi>z</mi> <mi>G</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>p</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>]</mo> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <mi>Q</mi> <msup> <mi>&Lambda;</mi> <mrow> <mn>2</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> <mrow> <mo>(</mo> <mo>-</mo> <msub> <mi>C</mi> <mi>X</mi> </msub> <mi>sin</mi> <mi></mi> <mi>&alpha;</mi> <mi>sin</mi> <mi>&beta;</mi> <mo>+</mo> <msub> <mi>C</mi> <mi>Y</mi> </msub> <mi>sin</mi> <mi></mi> <mi>&alpha;</mi> <mi>cos</mi> <mi>&beta;</mi> <mo>-</mo> <msub> <mi>C</mi> <mi>Z</mi> </msub> <mi>cos</mi> <mi>&alpha;</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <msub> <mi>N</mi> <mi>p</mi> </msub> <mo>=</mo> <mo>[</mo> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mi>y</mi> </msub> <mo>+</mo> <msub> <mi>I</mi> <mn>22</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mi>z</mi> </msub> <mo>+</mo> <msub> <mi>I</mi> <mn>33</mn> </msub> <mo>)</mo> </mrow> <mo>]</mo> <mi>qr</mi> <mo>+</mo> <msub> <mi>I</mi> <mi>xz</mi> </msub> <mi>pq</mi> <mo>-</mo> <msub> <mi>I</mi> <mi>xy</mi> </msub> <mi>pr</mi> <mo>-</mo> <msub> <mi>I</mi> <mi>yz</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>+</mo> </mtd> </mtr> <mtr> <mtd> <mo>[</mo> <msub> <mi>mz</mi> <mi>G</mi> </msub> <mrow> <mo>(</mo> <mi>ur</mi> <mo>-</mo> <mi>wp</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>y</mi> <mi>G</mi> </msub> <mrow> <mo>(</mo> <mi>uq</mi> <mo>-</mo> <mi>vp</mi> <mo>)</mo> </mrow> <mo>]</mo> <mo>+</mo> <mi>Q&Lambda;</mi> <msub> <mi>C</mi> <mi>l</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <msub> <mi>N</mi> <mi>q</mi> </msub> <mo>=</mo> <mo>[</mo> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mi>z</mi> </msub> <mo>+</mo> <msub> <mi>I</mi> <mn>33</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mi>x</mi> </msub> <mo>+</mo> <msub> <mi>I</mi> <mn>22</mn> </msub> <mo>)</mo> </mrow> <mo>]</mo> <mi>pr</mi> <mo>+</mo> <msub> <mi>I</mi> <mi>xy</mi> </msub> <mi>qr</mi> <mo>-</mo> <msub> <mi>I</mi> <mi>yz</mi> </msub> <mi>pq</mi> <mo>-</mo> <msub> <mi>I</mi> <mi>xz</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>p</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <mi>m</mi> <mo>[</mo> <msub> <mi>x</mi> <mi>G</mi> </msub> <mrow> <mo>(</mo> <mi>vp</mi> <mo>-</mo> <mi>uq</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>z</mi> <mi>G</mi> </msub> <mrow> <mo>(</mo> <mi>wp</mi> <mo>-</mo> <mi>vr</mi> <mo>)</mo> </mrow> <mo>]</mo> <mo>+</mo> <mi>Q&Lambda;</mi> <msub> <mi>C</mi> <mi>m</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <msub> <mi>N</mi> <mi>r</mi> </msub> <mo>=</mo> <mo>[</mo> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mi>y</mi> </msub> <mo>+</mo> <msub> <mi>I</mi> <mn>22</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mi>x</mi> </msub> <mo>+</mo> <msub> <mi>I</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>]</mo> <mi>pq</mi> <mo>+</mo> <msub> <mi>I</mi> <mi>xz</mi> </msub> <mi>qr</mi> <mo>-</mo> <msub> <mi>I</mi> <mi>xy</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mi>p</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>I</mi> <mi>yz</mi> </msub> <mi>pr</mi> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <mi>m</mi> <mo>[</mo> <msub> <mi>y</mi> <mi>G</mi> </msub> <mrow> <mo>(</mo> <mi>wq</mi> <mo>-</mo> <mi>vr</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>x</mi> <mi>G</mi> </msub> <mrow> <mo>(</mo> <mi>ur</mi> <mo>-</mo> <mi>wp</mi> <mo>)</mo> </mrow> <mo>]</mo> <mo>+</mo> <mi>Q&Lambda;</mi> <msub> <mi>C</mi> <mi>n</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>

In formula, m is dirigible quality, m₁₁、m₂₂、m₃₃For additional mass, I₁₁、I₂₂、I₃₃For additional inertial, Λ is dirigible volume；Q is Dynamic pressure, α is the angle of attack, and β is yaw angle, C_X、C_Y、C_Z、C_l、C_m、C_nFor Aerodynamic Coefficient；I_x、I_y、I_zRespectively around o_bx_b、o_by_b、o_bz_b Principal moments；I_xy、I_xz、I_yzRespectively on plane o_bx_by_b、o_bx_bz_b、o_by_bz_bProduct of inertia；T is thrust size, and μ is thrust Vector and o_bx_bz_bAngle between face, provides it in o_bx_bz_bThe left side in face is just, υ is thrust vectoring in o_bx_bz_bThe projection in face with o_bx_bAngle between axle, provides that it is projected in o_bx_bIt is just under axle；l_x、l_y、l_zRepresent thrust point away from origin o_bAway from From；

Formula (3) is the expression formula on generalized velocity V, it is necessary to transform it into the expression formula on generalized coordinates η；

It can be obtained by formula (2)：

<mrow> <mi>V</mi> <mo>=</mo> <msup> <mi>J</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>)</mo> </mrow> <mover> <mi>&eta;</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mi>R</mi> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>)</mo> </mrow> <mover> <mi>&eta;</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <msub> <mn>0</mn> <mrow> <mn>3</mn> <mo>&times;</mo> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mn>0</mn> <mrow> <mn>3</mn> <mo>&times;</mo> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi>B</mi> </mtd> </mtr> </mtable> </mfenced> <mover> <mi>&eta;</mi> <mo>&CenterDot;</mo> </mover> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>

In formula, J^-1(η) is J (η) inverse matrix；

<mrow> <mi>A</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>cos</mi> <mi>&psi;</mi> <mi>cos</mi> <mi>&theta;</mi> </mrow> </mtd> <mtd> <mrow> <mi>sin</mi> <mi>&psi;</mi> <mi>cos</mi> <mi>&theta;</mi> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mi>sin</mi> <mi>&theta;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>cos</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&theta;</mi> <mi>sin</mi> <mi>&phi;</mi> <mo>-</mo> <mi>sin</mi> <mi>&psi;</mi> <mi>cos</mi> <mi>&phi;</mi> </mrow> </mtd> <mtd> <mrow> <mi>sin</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&theta;</mi> <mi>sin</mi> <mi>&phi;</mi> <mo>+</mo> <mi>cos</mi> <mi>&psi;</mi> <mi>cos</mi> <mi>&phi;</mi> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi>&theta;</mi> <mi>sin</mi> <mi>&phi;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>cos</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&theta;</mi> <mi>cos</mi> <mi>&phi;</mi> <mo>+</mo> <mi>sin</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&phi;</mi> </mrow> </mtd> <mtd> <mrow> <mi>sin</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&theta;</mi> <mi>cos</mi> <mi>&phi;</mi> <mo>-</mo> <mi>cos</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&phi;</mi> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi>&theta;</mi> <mi>cos</mi> <mi>&phi;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mi>B</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&theta;</mi> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&phi;</mi> </mrow> </mtd> <mtd> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&theta;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&phi;</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&phi;</mi> </mrow> </mtd> <mtd> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&theta;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&phi;</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>

To formula (16) differential, it can obtain

<mrow> <mover> <mi>V</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mover> <mi>R</mi> <mo>&CenterDot;</mo> </mover> <mover> <mi>&eta;</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <mi>R</mi> <mover> <mi>&eta;</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>

In formula

<mrow> <mover> <mi>R</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mover> <mi>A</mi> <mo>&CenterDot;</mo> </mover> </mtd> <mtd> <msub> <mn>0</mn> <mrow> <mn>3</mn> <mo>&times;</mo> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mn>0</mn> <mrow> <mn>3</mn> <mo>&times;</mo> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mover> <mi>B</mi> <mo>&CenterDot;</mo> </mover> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow>

Formula (19) premultiplicationIt can obtain

<mrow> <msup> <mi>R</mi> <mi>T</mi> </msup> <mi>M</mi> <mover> <mi>V</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <msup> <mi>R</mi> <mi>T</mi> </msup> <mi>M</mi> <mover> <mi>R</mi> <mo>&CenterDot;</mo> </mover> <mover> <mi>&eta;</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <msup> <mi>R</mi> <mi>T</mi> </msup> <mi>M</mi> <mi>R</mi> <mover> <mi>&eta;</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow>

Composite type (3), formula (19) and formula (21) can be obtained:

<mrow> <msub> <mi>M</mi> <mi>&eta;</mi> </msub> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>)</mo> </mrow> <mover> <mi>&eta;</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>N</mi> <mi>&eta;</mi> </msub> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>)</mo> </mrow> <mover> <mi>&eta;</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>G</mi> <mi>&eta;</mi> </msub> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>&Gamma;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mrow>

In formula

M_η(η)=R^TMR (23)

<mrow> <msub> <mi>N</mi> <mi>&eta;</mi> </msub> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>R</mi> <mi>T</mi> </msup> <mi>M</mi> <mover> <mi>R</mi> <mo>&CenterDot;</mo> </mover> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>24</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>G</mi> <mi>&eta;</mi> </msub> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <msup> <mi>R</mi> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mover> <mi>N</mi> <mo>&OverBar;</mo> </mover> <mo>+</mo> <mover> <mi>G</mi> <mo>&OverBar;</mo> </mover> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>25</mn> <mo>)</mo> </mrow> </mrow>

Γ=R^Tτ (26)

Make x₁=η,Then kinetic model (22) is written as following form：

<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <msubsup> <mi>M</mi> <mi>&eta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mi>&Gamma;</mi> <mo>-</mo> <msubsup> <mi>M</mi> <mi>&eta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>)</mo> </mrow> <mover> <mi>&eta;</mi> <mo>&CenterDot;</mo> </mover> <mo>-</mo> <msub> <mi>G</mi> <mi>&eta;</mi> </msub> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>27</mn> <mo>)</mo> </mrow> </mrow>

Using the mathematical modeling described by formula (27) as controlled device, using Backstepping techniques design flight tracking control rule；

2) design flight tracking control rule

<mrow> <msub> <mi>&alpha;</mi> <mn>1</mn> </msub> <mo>=</mo> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mi>e</mi> <mo>+</mo> <msub> <mover> <mi>&eta;</mi> <mo>&CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>28</mn> <mo>)</mo> </mrow> </mrow>

In formula, α₁For virtual amount, k₁For adjustable control parameter；

The virtual amount α of definition₁With x₂Between error e '：

E '=x₂-α₁ (29)

Formula (29) to time diffusion and substitutes into formula (27), can obtain：

<mrow> <msup> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>&prime;</mo> </msup> <mo>=</mo> <msub> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <msub> <mover> <mi>&alpha;</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <msubsup> <mi>M</mi> <mi>&eta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mi>r</mi> <mo>-</mo> <msubsup> <mi>M</mi> <mi>&eta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>N</mi> <mi>&eta;</mi> </msub> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>)</mo> </mrow> <mover> <mi>&eta;</mi> <mo>&CenterDot;</mo> </mover> <mo>-</mo> <msub> <mi>G</mi> <mi>&eta;</mi> </msub> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>-</mo> <msub> <mover> <mi>&eta;</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>30</mn> <mo>)</mo> </mrow> </mrow>

Order

<mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <msubsup> <mi>M</mi> <mi>&eta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>N</mi> <mi>&eta;</mi> </msub> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>)</mo> </mrow> <mover> <mi>&eta;</mi> <mo>&CenterDot;</mo> </mover> <mo>-</mo> <msub> <mi>G</mi> <mi>&eta;</mi> </msub> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>31</mn> <mo>)</mo> </mrow> </mrow>

Then formula (31) is represented by：

<mrow> <msup> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>&prime;</mo> </msup> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>M</mi> <mi>&eta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mi>r</mi> <mo>+</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>-</mo> <msub> <mover> <mi>&eta;</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>32</mn> <mo>)</mo> </mrow> </mrow>

Choose Lyapunov functions V₁

<mrow> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mi>e</mi> <mi>T</mi> </msup> <mi>e</mi> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mi>e</mi> <mrow> <mo>&prime;</mo> <mi>T</mi> </mrow> </msup> <msup> <mi>e</mi> <mo>&prime;</mo> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>33</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mover> <mi>V</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msup> <mi>e</mi> <mi>T</mi> </msup> <mi>e</mi> <mo>+</mo> <msup> <mi>e</mi> <mrow> <mo>&prime;</mo> <mi>T</mi> </mrow> </msup> <mrow> <mo>(</mo> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>M</mi> <mi>&eta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mi>&Gamma;</mi> <mo>+</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>-</mo> <msub> <mover> <mi>&eta;</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>)</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>34</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mi>&Gamma;</mi> <mo>=</mo> <msub> <mi>M</mi> <mi>&eta;</mi> </msub> <mrow> <mo>(</mo> <mo>-</mo> <mi>e</mi> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>-</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <msup> <mi>e</mi> <mo>&prime;</mo> </msup> <mo>+</mo> <msub> <mover> <mi>&eta;</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>-</mo> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>)</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>35</mn> <mo>)</mo> </mrow> </mrow>

In formula, k₂For adjustable control parameter；

3) stability analysis

<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>V</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msup> <mi>e</mi> <mi>T</mi> </msup> <mi>e</mi> <mo>+</mo> <msup> <mi>e</mi> <mi>T</mi> </msup> <msup> <mi>e</mi> <mo>&prime;</mo> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow> <mo>&prime;</mo> <mi>T</mi> </mrow> </msup> <mo>{</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>M</mi> <mi>&eta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>&lsqb;</mo> <msub> <mi>M</mi> <mi>&eta;</mi> </msub> <mrow> <mo>(</mo> <mo>-</mo> <mi>e</mi> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>-</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <msup> <mi>e</mi> <mo>&prime;</mo> </msup> <mo>+</mo> <msub> <mover> <mi>&eta;</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>-</mo> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>)</mo> <mo>&rsqb;</mo> <mo>+</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>-</mo> <mi>k</mi> <msub> <mover> <mi>&eta;</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msup> <mi>e</mi> <mi>T</mi> </msup> <mi>e</mi> <mo>+</mo> <msup> <mi>e</mi> <mrow> <mo>&prime;</mo> <mi>T</mi> </mrow> </msup> <msup> <mi>e</mi> <mo>&prime;</mo> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow> <mo>&prime;</mo> <mi>T</mi> </mrow> </msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>e</mi> <mo>-</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <msup> <mi>e</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msup> <mi>e</mi> <mi>T</mi> </msup> <mi>e</mi> <mo>-</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <msup> <mi>e</mi> <mrow> <mo>&prime;</mo> <mi>T</mi> </mrow> </msup> <msup> <mi>e</mi> <mo>&prime;</mo> </msup> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>36</mn> <mo>)</mo> </mrow> </mrow>

Formula (36) shows：The stability that (35) ensure that closed-loop system is restrained using flight tracking control；

Step 4：Neural network approximator is designed：With flight tracking control margin of error e and its rate of changeActual flight path η and its change RateFor the input variable of neutral net, with the estimate of dirigible kinetic modelDesigned for the output variable of neutral net Neural network approximator, function is infinitely approached using neutral net and estimates unknown ambiguous model, to improve control accuracy.

2. the contragradience neural network control method of dirigible Three-dimensional Track tracking according to claim 1, it is characterised in that：Institute The neural network approximator described in step 4 is stated, its design method is：

1) due to being difficult to carry out dirigible Accurate Model during practical flight, f (x) is unknown function, it is difficult to according to formula (35) it is controlled rule to resolve, therefore, f (x) estimate must be usedFlight tracking control rule formula (35) is resolved；Using Neutral net approaches unknown function f (x), then has：

F (x)=w^Th(x)+ε (37)

In formula, w is the weight vectors of neutral net, and ε is approximate error, h (x)=[h_i(x)]^T, h_i(x) it is Gaussian bases, under Mark i represents i-th of Gaussian bases；

2) input/output variable is selected

Make flight tracking control margin of error e and its rate of changeActual flight path η and its rate of changeBecome for the input of neural network approximator Amount, makes estimateFor the output variable of neural network approximator；

3) neural network structure is designed

Neural network structure includes input layer, hidden layer and output layer；

Input layer：Choose neutral net input variable be

Hidden layer：Gaussian function is chosen as the basic function of hidden node

<mrow> <msub> <mi>h</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>exp</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <mo>|</mo> <mo>|</mo> <mi>x</mi> <mo>-</mo> <mi>c</mi> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> </mrow> <mrow> <mn>2</mn> <msubsup> <mi>&sigma;</mi> <mi>i</mi> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>38</mn> <mo>)</mo> </mrow> </mrow>

Wherein, c=[c_i]^T, c_iFor the intermediate value of i-th of Gaussian function, σ_iFor the base width parameter of i-th of node, | | | | represent Euclideam norm；

Output layer：Neural network approximator is output as

Wherein,For w estimate；

4) stability analysis

DefinitionWith w differences：

Choose Lyapunov functions：

<mrow> <msub> <mi>V</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mi>&xi;</mi> <mi>T</mi> </msup> <mi>&xi;</mi> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msup> <mover> <mi>W</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <msup> <mi>&lambda;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mover> <mi>W</mi> <mo>~</mo> </mover> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>41</mn> <mo>)</mo> </mrow> </mrow>

In formula, ξ=[e e ']^T,Θ is adjustable positive definite matrix, λ^-1Representing matrix λ's is inverse Matrix；

To formula (41) differential, it can obtain：

<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>V</mi> <mo>&CenterDot;</mo> </mover> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>&xi;</mi> <mi>T</mi> </msup> <mover> <mi>&xi;</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msup> <mover> <mi>W</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <msup> <mi>&lambda;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mover> <mover> <mi>W</mi> <mo>~</mo> </mover> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msubsup> <mi>e</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mi>e</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <msubsup> <mi>e</mi> <mn>2</mn> <mi>T</mi> </msubsup> <msub> <mi>e</mi> <mn>2</mn> </msub> <mo>+</mo> <msubsup> <mi>e</mi> <mn>2</mn> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <msup> <mover> <mi>w</mi> <mo>^</mo> </mover> <mi>T</mi> </msup> <mi>h</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>+</mo> <mi>&epsiv;</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msup> <mover> <mi>W</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <msup> <mi>&lambda;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mover> <mover> <mi>W</mi> <mo>~</mo> </mover> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>42</mn> <mo>)</mo> </mrow> </mrow>

Definition

Then formula (42) can be written as：

<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>V</mi> <mo>&CenterDot;</mo> </mover> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>&xi;</mi> <mi>T</mi> </msup> <mover> <mi>&xi;</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msup> <mover> <mi>w</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <msup> <mi>&lambda;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mover> <mover> <mi>w</mi> <mo>~</mo> </mover> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <msup> <mi>&xi;</mi> <mi>T</mi> </msup> <mi>k</mi> <mi>&xi;</mi> <mo>+</mo> <msup> <mi>&xi;</mi> <mi>T</mi> </msup> <mi>&epsiv;</mi> <mo>+</mo> <msup> <mi>&xi;</mi> <mi>T</mi> </msup> <mover> <mi>W</mi> <mo>~</mo> </mover> <mi>&Psi;</mi> <mo>+</mo> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msup> <mover> <mi>W</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <msup> <mi>&lambda;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mover> <mover> <mi>W</mi> <mo>~</mo> </mover> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>-</mo> <msup> <mi>&xi;</mi> <mi>T</mi> </msup> <mi>k</mi> <mi>&xi;</mi> <mo>+</mo> <msup> <mi>&xi;</mi> <mi>T</mi> </msup> <mi>&epsiv;</mi> <mo>+</mo> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <mover> <mi>W</mi> <mo>~</mo> </mover> <msup> <mi>&Psi;&xi;</mi> <mi>T</mi> </msup> <mo>+</mo> <msup> <mover> <mi>W</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <msup> <mi>&lambda;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mover> <mover> <mi>W</mi> <mo>~</mo> </mover> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>44</mn> <mo>)</mo> </mrow> </mrow>

In formula,

Design following adaptive law：

<mrow> <mover> <mover> <mi>W</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <msup> <mi>&lambda;&Psi;&xi;</mi> <mi>T</mi> </msup> <mo>-</mo> <mi>&gamma;</mi> <mi>&lambda;</mi> <mo>|</mo> <mo>|</mo> <mi>&xi;</mi> <mo>|</mo> <mo>|</mo> <mover> <mi>W</mi> <mo>^</mo> </mover> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>45</mn> <mo>)</mo> </mrow> </mrow>

In formula, γ ＞ 0 are adjustable parameter,

Adaptive law is substituted into formula (44), can be obtained：

<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>V</mi> <mo>&CenterDot;</mo> </mover> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>&xi;</mi> <mi>T</mi> </msup> <mover> <mi>&xi;</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msup> <mover> <mi>w</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <msup> <mi>&lambda;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mover> <mover> <mi>w</mi> <mo>~</mo> </mover> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>-</mo> <msup> <mi>&xi;</mi> <mi>T</mi> </msup> <mi>k</mi> <mi>&xi;</mi> <mo>+</mo> <msup> <mi>&xi;</mi> <mi>T</mi> </msup> <mi>&epsiv;</mi> <mo>+</mo> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <mover> <mi>W</mi> <mo>~</mo> </mover> <msup> <mi>&Psi;&xi;</mi> <mi>T</mi> </msup> <mo>-</mo> <msup> <mover> <mi>W</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <msup> <mi>&lambda;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <mrow> <msup> <mi>&lambda;&Psi;&xi;</mi> <mi>T</mi> </msup> <mo>-</mo> <mi>&gamma;</mi> <mi>&lambda;</mi> <mo>|</mo> <mo>|</mo> <mi>&xi;</mi> <mo>|</mo> <mo>|</mo> <mover> <mi>W</mi> <mo>~</mo> </mover> </mrow> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>-</mo> <msup> <mi>&xi;</mi> <mi>T</mi> </msup> <mi>k</mi> <mi>&xi;</mi> <mo>+</mo> <msup> <mi>&xi;</mi> <mi>T</mi> </msup> <mi>&epsiv;</mi> <mo>+</mo> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <mi>&gamma;</mi> <msup> <mover> <mi>W</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <mo>|</mo> <mo>|</mo> <mi>&xi;</mi> <mo>|</mo> <mo>|</mo> <mover> <mi>W</mi> <mo>~</mo> </mover> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>-</mo> <msup> <mi>&xi;</mi> <mi>T</mi> </msup> <mi>k</mi> <mi>&xi;</mi> <mo>+</mo> <msup> <mi>&xi;</mi> <mi>T</mi> </msup> <mi>&epsiv;</mi> <mo>+</mo> <mi>&gamma;</mi> <mo>|</mo> <mo>|</mo> <mi>&xi;</mi> <mo>|</mo> <mo>|</mo> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msup> <mover> <mi>W</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <mo>(</mo> <mrow> <mi>W</mi> <mo>-</mo> <mover> <mi>W</mi> <mo>~</mo> </mover> </mrow> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>46</mn> <mo>)</mo> </mrow> </mrow>

According to Schwarz inequality, have：

In formula, | | | |_FRepresent Frobenius norms；

Formula (47) is substituted into formula (46), can be obtained：

In formula, ε_NFor the upper bound of approximate error, k_minFor adjustable parameter matrix k minimal eigenvalue, W_MFor weight matrix W maximum It is worth element；

In view of following equation：

<mrow> <mi>&gamma;</mi> <mo>|</mo> <mo>|</mo> <mover> <mi>W</mi> <mo>~</mo> </mover> <mo>|</mo> <msub> <mo>|</mo> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mo>|</mo> <mo>|</mo> <mover> <mi>W</mi> <mo>~</mo> </mover> <mo>|</mo> <msub> <mo>|</mo> <mi>F</mi> </msub> <mo>-</mo> <msub> <mi>W</mi> <mi>M</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>&gamma;</mi> <msup> <mrow> <mo>(</mo> <mo>|</mo> <mo>|</mo> <mover> <mi>W</mi> <mo>~</mo> </mover> <mo>|</mo> <msub> <mo>|</mo> <mi>F</mi> </msub> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>W</mi> <mi>M</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <mfrac> <mi>&gamma;</mi> <mn>4</mn> </mfrac> <msubsup> <mi>W</mi> <mi>M</mi> <mn>2</mn> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>49</mn> <mo>)</mo> </mrow> </mrow>

If so thatMust then have and be set up with lower inequality：

Thus, uncertain dirigible non-linear dynamic model can accurately be estimated by neural network approximator.