CN104793629A - Method for controlling backstepping neural network for tracking three-dimensional flight path of airship - Google Patents

Abstract

The invention relates to a method for controlling a backstepping neural network for tracking a three-dimensional flight path of an airship. In order to control tracking of the flight path of the airship, a nonlinear kinetic model of the airship is established; the nonlinear kinetic model of the airship serves as a controlled object and is divided into two subsystems, a Lyapunov function and a middle virtual controlled quantity are designed for each subsystem by using a backstepping method, proper virtual feedback is determined, and the previous state of a system is gradually stable until the whole system is gradually stable in an inverse deducing mode; and in order to solve the problem that the kinetic model of the airship is uncertain, an unknown kinetic model of the airship is estimated accurately by a neural network approximator, and the control precision and the system performance are improved. A closed-loop system controlled by the method can track an optional parameterized command flight path precisely and has high stability, adaptability, robustness and dynamic performance, and an effective scheme is provided for an airship flight path control project.

Description

The contragradience neural network control method that a kind of dirigible Three-dimensional Track is followed the tracks of

Technical field

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

Background technology

Dirigible refers to that the gas (as helium, hydrogen etc.) that a kind of dependence is lighter than air provides quiet buoyancy to go up to the air, automatic flight control system is relied on to realize aircraft that is resident and low-speed maneuver of fixing a point, have that the hang time is long, energy consumption is low, efficiency-cost ratio is high and the advantage such as resident of fixing a point, be widely used in the fields such as reconnaissance and surveillance, earth observation, environmental monitoring, emergency disaster relief, scientific exploration, there is significant application value and wide application prospect, the current study hotspot having become aviation field.

Track In Track refers to that dirigible is from the instruction flight path under the inertial coordinates system of given original state tracing preset.The spatial movement of dirigible have non-linear, passage coupling, uncertain, be subject to the features such as external disturbance, therefore, flight tracking control becomes one of gordian technique that airship flight controls.Numerous researchist, for the Track In Track problem of dirigible, proposes the methods such as PID control, FEEDBACK CONTROL, sliding formwork control, robust control, for dirigible Track In Track provides can the technical scheme of reference for reference.But above-mentioned flight tracking control method not yet effectively solves following two class problems: one is that dirigible kinetic model is uncertain, there is modeling error and Unmarried pregnancy; Two is dirigible flight path control systems is a complicated nonlinear multivariable systems, and in flight envelope, the stability of closed-loop control system is difficult to ensure.

Summary of the invention

For the deficiency that prior art exists, the object of this invention is to provide the contragradience neural network control method that a kind of dirigible Three-dimensional Track is followed the tracks of.

The present invention is directed to dirigible Three-dimensional Track tracking problem, establish the non-linear dynamic model of dirigible; As controll plant, non-linear dynamic model is decomposed into two subsystems, Backstepping is adopted to be each subsystem design Liapunov (Lyapunov) function and intermediate virtual controlled quentity controlled variable, by determining suitable virtual feedback, the front position of system is made to reach Asymptotic Stability, " oppositely deduce " to whole system always, thus realize the Asymptotic Stability of whole system; For dirigible kinetic model uncertain problem, neural network is adopted accurately to approach unknown dirigible kinetic model, to improve control accuracy and system performance.Advantage of the present invention shows: 1. adopt Backstepping design to make design process systematization, the structuring of Liapunov (Lyapunov) function and control law, ensure that the stability of system; 2. adopt neural network accurately to approach the ambiguous model of dirigible, make Track In Track control system have strong adaptability and strong robustness.

Technical scheme of the present invention is: first calculate the flight tracking control margin of error by given instruction flight path and actual flight path, then adopts Backstepping techniques design flight tracking control rule, calculates flight tracking control amount; For solving dirigible kinetic model uncertain problem, neural network is adopted accurately to approach unknown ambiguous model.In practical application, dirigible flight path is obtained by integrated navigation system measurement, the controlled quentity controlled variable calculated is transferred to topworks can realize flight tracking control function by the method.

Particularly, the contragradience neural network control method that a kind of dirigible Three-dimensional Track is followed the tracks of, 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 coordinate 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 _dbe respectively instruction x coordinate, instruction y coordinate, instruction z coordinate, the instruction angle of pitch, instruction crab angle and instruction roll angle, subscript T represents vector or transpose of a matrix.

Step 2: the flight tracking control margin of error calculates: the flight tracking control margin of error e between computations flight path and actual flight path;

The computing method of described 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, crab angle and roll angle.

Step 3: flight tracking control rule design: choose Lypaunov function and intermediate virtual controlled quentity controlled variable, adopts Backstepping design flight tracking control rule, calculates flight tracking control amount u, specifically comprise the following steps:

1) kinetic model of dirigible is set up

Coordinate system and the kinematic parameter of dirigible spatial movement are defined as follows: adopt earth axes o _ex _ey _ez _ewith body coordinate system o _bx _by _bz _bbe described the spatial movement of dirigible, CV is centre of buoyancy, and CG is center of gravity, and centre of buoyancy is r to the vector of center of gravity _g=[x _g, y _g, z _g] ^t.Kinematic parameter defines: position P=[x, y, z] ^t, x, y, z is respectively the displacement of axis, side direction and vertical direction; Attitude angle Ω=[θ, ψ, φ] ^t, θ, ψ, φ are respectively the angle of pitch, crab angle and roll angle; Speed v=[u, v, w] ^t, u, v, w are respectively the speed of axis, side direction and vertical direction in body coordinate system; Angular velocity omega=[p, q, r] ^t, p, q, r are respectively rolling, pitching and yaw rate.Note generalized coordinate 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 below:

$\stackrel{\·}{\mathrm{\η}}=J\left(\mathrm{\η}\right)=\left[\begin{array}{cc}{J}_1& 0_{3\×3}\\ 0_{3\×3}& J_2\end{array}\right]V---\left(2\right)$

$M\stackrel{\·}{V}=\stackrel{\‾}{N}+\stackrel{\‾}{G}+\mathrm{\τ}---\left(3\right)$

In formula

$J_1=\left[\begin{array}{ccc}\cos \mathrm{\ψ}\cos \mathrm{\θ}& \cos \mathrm{\ψ}\sin \mathrm{\θ}\sin \mathrm{\φ}-\sin \mathrm{\ψ}\cos \mathrm{\φ}& \cos \mathrm{\ψ}\sin \mathrm{\θ}\cos \mathrm{\φ}+\sin \mathrm{\ψ}\sin \mathrm{\φ}\\ \sin \mathrm{\ψ}\cos \mathrm{\θ}& \sin \mathrm{\ψ}\sin \mathrm{\θ}\sin \mathrm{\φ}+\cos \mathrm{\ψ}\cos \mathrm{\φ}& \sin \mathrm{\ψ}\sin \mathrm{\θ}\cos \mathrm{\φ}-\cos \mathrm{\ψ}\sin \mathrm{\φ}\\ -\sin \mathrm{\θ}& \cos \mathrm{\θ}\sin \mathrm{\φ}& \cos \mathrm{\θ}\cos \mathrm{\φ}\end{array}\right]---\left(4\right)$

$J_2=\left[\begin{array}{ccc}0& \cos \mathrm{\φ}& -\sin \mathrm{\φ}\\ 0& \sec \mathrm{\θ}\sin \mathrm{\φ}& \sec \mathrm{\θ}\cos \mathrm{\φ}\\ 1& \tan \mathrm{\θ}\sin \mathrm{\φ}& \tan \mathrm{\θ}\cos \mathrm{\φ}\end{array}\right]---\left(5\right)$

$M=\left[\begin{array}{cccccc}m+m_{11}& 0& 0& 0& {\mathrm{mz}}_G& -{\mathrm{my}}_G\\ 0& m+m_{22}& 0& -{\mathrm{mz}}_G& 0& {\mathrm{mx}}_G\\ 0& 0& m+m_{33}& {\mathrm{my}}_G& -{\mathrm{mx}}_G& 0\\ 0& -{\mathrm{mz}}_G& {\mathrm{my}}_G& I_x+I_{11}& 0& 0\\ {\mathrm{mz}}_G& 0& -{\mathrm{mx}}_G& 0& I_x+I_{22}& 0\\ 0& {\mathrm{mx}}_G& 0& -I_{\mathrm{xz}}& 0& I_x+I_{33}\end{array}\right]---\left(6\right)$

$\stackrel{\‾}{G}=\left[\begin{array}{c}(B-G)\sin \mathrm{\θ}\\ (G-B)\cos \mathrm{\θ}\sin \mathrm{\φ}\\ (G-B)\cos \mathrm{\θ}\cos \mathrm{\φ}\\ y_{G}G\cos \mathrm{\θ}\cos \mathrm{\φ}-z_{G}G\cos \mathrm{\θ}\sin \mathrm{\φ}\\ -x_{G}G\cos \mathrm{\θ}\cos \mathrm{\φ}-z_{G}G\sin \mathrm{\θ}\\ x_{G}G\cos \mathrm{\θ}\sin \mathrm{\φ}+y_{G}G\sin \mathrm{\θ}\end{array}\right]---\left(7\right)$

$\mathrm{\τ}=\left[\begin{array}{c}{\mathrm{\τ}}_u\\ {\mathrm{\τ}}_v\\ {\mathrm{\τ}}_w\\ {\mathrm{\τ}}_l\\ {\mathrm{\τ}}_m\\ {\mathrm{\τ}}_n\end{array}\right]=\left[\begin{array}{c}T\cos \mathrm{\μ}\cos \mathrm{\υ}\\ T\sin \mathrm{\μ}\\ T\cos \mathrm{\μ}\sin \mathrm{\υ}\\ T\cos \mathrm{\μ}\sin \mathrm{\υ}{l}_y-T\sin {\mathrm{\μl}}_z\\ T\cos \mathrm{\μ}\cos \mathrm{\υ}{l}_z-T\cos \mathrm{\μ}\sin \mathrm{\υ}{l}_x\\ T\sin {\mathrm{\μl}}_x-T\cos \mathrm{\μ}\cos \mathrm{\υ}{l}_y\end{array}\right]---\left(8\right)$

$\stackrel{\‾}{N}={[N_u,N_v,N_w,N_p,N_q,N_r]}^T---\left(9\right)$

Wherein

N _u＝(m+m ₂₂)vr-(m+m ₃₃)wq+m[x _G(p ²+r ²)-y _Gpq-z _Gpr]

(10)

+QV ^2/3(-C _Xcosαcosβ+C _Ycosαsinβ+C _Zsinα)

N _v＝(m+m ₃₃)wp-(m+m ₁₁)ur-m[x _Gpq-y _G(p ²+r ²)+z _Gqr]

(11)

+QV ^2/3(C _Xsinβ+C _Ycosβ)

N _w＝(m+m ₂₂)vp-(m+m ₁₁)uq-m[x _Gpr+y _Gqr-z _G(p ²+q ²)]

(12)

+QV ^2/3(-C _Xsinαsinβ+C _Ysinαcosβ-C _Zcosα)

N _p＝[(I _y+m ₅₅)-(I _z+I ₆₆)]qr+I _xzpq-I _xypr-I _yz(r ²-q ²)+

(13)

[mz _G(ur-wp)+y _G(uq-vp)]+QVC _l

N _q＝[(I _z+m ₆₆)-(I _x+I ₄₄)]pr+I _xyqr-I _yzpq-I _xz(p ²-r ²)

(14)

+m[x _G(vp-uq)-z _G(wp-vr)]+QVC _m

N _r＝[(I _y+m ₅₅)-(I _x+I ₄₄)]pq-I _xzqr-I _xy(q ²-p ²)+I _yzpr

(15)

+m[y _G(wq-vr)-x _G(ur-wp)]+QVC _n

In formula, m is dirigible quality, m ₁₁, m ₂₂, m ₃₃for additional mass, I ₁₁, I ₂₂, I ₃₃for additional inertial, V is dirigible volume; ; 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 _zbe respectively around o _bx _b, o _by _b, o _bz _bprincipal moments; I _xy, I _xz, I _yzbe respectively about plane o _bx _by _b, o _bx _bz _b, o _by _bz _bproduct of inertia; T is thrust size, and μ is thrust vectoring and o _bx _bz _bangle between face, specifies that it is at o _bx _bz _bthe left side in face is just, υ is that thrust vectoring is at o _bx _bz _bthe projection in face and o _bx _bangle between axle, specifies that it is projected in o _bx _bjust be under axle; l _x, l _y, l _zrepresent that thrust point is apart from initial point o _bdistance.

Formula (3) is the expression formula about generalized velocity V, needs to be transformed to the expression formula about generalized coordinate η.

Can be obtained by formula (1):

$V=J^{-1}\left(\mathrm{\η}\right)\stackrel{\·}{\mathrm{\η}}=R\left(\mathrm{\η}\right)\stackrel{\·}{\mathrm{\η}}=\left[\begin{array}{cc}A& 0_{3\×3}\\ 0_{3\×3}& B\end{array}\right]\stackrel{\·}{\mathrm{\η}}---\left(16\right)$

In formula, J ^-1(η) be the inverse matrix of J (η).

$A=\left[\begin{array}{ccc}\cos \mathrm{\ψ}\cos \mathrm{\θ}& \sin \mathrm{\ψ}\cos \mathrm{\θ}& -\sin \mathrm{\θ}\\ \cos \mathrm{\ψ}\sin \mathrm{\θ}\sin \mathrm{\φ}-\sin \mathrm{\ψ}\cos \mathrm{\φ}& \sin \mathrm{\ψ}\sin \mathrm{\θ}\sin \mathrm{\φ}+\cos \mathrm{\ψ}\cos \mathrm{\φ}& \cos \mathrm{\θ}\sin \mathrm{\φ}\\ \cos \mathrm{\ψ}\sin \mathrm{\θ}\cos \mathrm{\φ}+\sin \mathrm{\ψ}\sin \mathrm{\φ}& \sin \mathrm{\ψ}\sin \mathrm{\θ}\cos \mathrm{\φ}-\cos \mathrm{\ψ}\sin \mathrm{\φ}& \cos \mathrm{\θ}\cos \mathrm{\φ}\end{array}\right]---\left(17\right)$

$B=\left[\begin{array}{ccc}0& -\sin \mathrm{\θ}& 1\\ \cos \mathrm{\φ}& \cos \mathrm{\θ}\sin \mathrm{\φ}& 0\\ -\sin \mathrm{\φ}& \cos \mathrm{\θ}\cos \mathrm{\φ}& 0\end{array}\right]---\left(18\right)$

To formula (16) differential, can obtain

$\stackrel{\·}{V}=\stackrel{\·}{R}\stackrel{\·}{\mathrm{\η}}+R\stackrel{\·\·}{\mathrm{\η}}---\left(19\right)$

In formula

$\stackrel{\·}{R}=\left[\begin{array}{cc}\stackrel{\·}{A}& 0_{3\×3}\\ 0_{3\×3}& \stackrel{\·}{B}\end{array}\right]---\left(20\right)$

Formula (19) premultiplication can obtain

$R^{T}M\stackrel{\·}{V}=R^{T}M\stackrel{\·}{R}\stackrel{\·}{\mathrm{\η}}+R^T\mathrm{MR}\stackrel{\·\·}{\mathrm{\η}}---\left(21\right)$

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

$M_{\mathrm{\η}}\left(\mathrm{\η}\right)\stackrel{\·\·}{\mathrm{\η}}+N_{\mathrm{\η}}(\mathrm{\η},\stackrel{\·}{\mathrm{\η}})\stackrel{\·}{\mathrm{\η}}+G_{\mathrm{\η}}\left(\mathrm{\η}\right)=u---\left(22\right)$

In formula

M _η(η)＝R ^TMR (23)

$N_{\mathrm{\η}}(\mathrm{\η},\stackrel{\·}{\mathrm{\η}})=R^{T}M\stackrel{\·}{R}---\left(24\right)$

$G_{\mathrm{\η}}\left(\mathrm{\η}\right)=-R^T(\stackrel{\‾}{N}+\stackrel{\‾}{G})---\left(25\right)$

u＝R ^Tτ (26)

Make x ₁=η, then kinetic simulation pattern (22) can be written as following form:

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=M_{\mathrm{\η}}^{-1}u-M_{\mathrm{\η}}^{-1}{N}_{\mathrm{\η}}\stackrel{\·}{\mathrm{\η}}-G_{\mathrm{\η}}\left(\mathrm{\η}\right)\end{array}\right.---\left(27\right)$

In formula, representing matrix M _ηinverse matrix;

With the mathematical model described by formula (27) for controlled device, adopt Backstepping techniques design flight tracking control rule.

2) flight tracking control rule is designed

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

$a_1=-k_{1}e+\stackrel{\·}{\mathrm{\η}}d$

(28)

In formula, α ₁for virtual amount, k ₁for adjustable controling parameters.

Defining virtual amount α ₁with x ₂between error e ':

e′＝x ₂-α ₁(29)

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

${\stackrel{\·}{e}}^{\′}={\stackrel{\·}{x}}_2-{\stackrel{\·}{a}}_1=M_{\mathrm{\η}}^{-1}u-M_{\mathrm{\η}}^{-1}{N}_{\mathrm{\η}}\left(\mathrm{\η}\right)\stackrel{\·}{\mathrm{\η}}-G_{\mathrm{\η}}\left(\mathrm{\η}\right)+k_1\stackrel{\·}{e}-{\stackrel{\·\·}{\mathrm{\η}}}_d---\left(30\right)$

Order

$f\left(x\right)=-M_{\mathrm{\η}}^{-1}{N}_{\mathrm{\η}}\left(\mathrm{\η}\right)\stackrel{\·}{\mathrm{\η}}-G_{\mathrm{\η}}\left(\mathrm{\η}\right)---\left(31\right)$

Then formula (31) can be expressed as:

${\stackrel{\·}{e}}^{\′}=f\left(x\right)+M_{\mathrm{\η}}^{-1}u+k_1\stackrel{\·}{e}-{\stackrel{\·\·}{\mathrm{\η}}}_d---\left(32\right)$

Choose Lyapunov function (Lyapunov function) V ₁

$V_1=\frac{1}{2}{e}^{T}e+\frac{1}{2}{e}^{\′T}{e}^{\′}---\left(33\right)$

Formula (32) to time diffusion, and substitutes into by formula (33), can obtain:

${\stackrel{\·}{V}}_1=-k_1{e}^{T}e+e^{\′T}(f\left(x\right)+M_{\mathrm{\η}}^{-1}u+k_1\stackrel{\·}{e}-{\stackrel{\·\·}{\mathrm{\η}}}_d)---\left(34\right)$

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

$u=M_{\mathrm{\η}}(-e-k_1\stackrel{\·}{e}-k_2{e}^{\′}+{\stackrel{\·\·}{\mathrm{\η}}}_d-f\left(x\right))---\left(35\right)$

3) stability analysis

Flight tracking control is restrained formula (35) and substitutes into formula (34), can obtain:

$\begin{array}{c}{\stackrel{\·}{V}}_1=-k_1{e}^{T}e+e^T{e}^{\′}+e^{\′T}\{f\left(x\right)+M_{\mathrm{\η}}^{-1}[M_{\mathrm{\η}}(-e-k_1\stackrel{\·}{e}-k_2{e}^{\′}+{\stackrel{\·\·}{\mathrm{\η}}}_d-f\left(x\right))]+k_1\stackrel{\·}{e}-k{\stackrel{\·\·}{\mathrm{\η}}}_d\}\\ =-k_1{e}^{T}e+e^{\′T}{e}^{\′}+e^{\′T}(-e-k_2{e}^{\′})=-k_1{e}^{T}e-k_2{e}^{\′T}{e}^{\′}\end{array}---\left(36\right)$

Formula (36) shows: adopt flight tracking control rule (35) can ensure the stability of closed-loop system.

Step 4: neural network approximator designs: with flight tracking control error e and rate of change thereof actual flight path η and rate of change thereof for the input variable of neural network, with the estimated value of dirigible kinetic model for the output variable of neural network devises neural network approximator, utilize neural network infinitely to approach function and estimate unknown ambiguous model, to provide control accuracy, concrete steps are as follows:

1) owing to being difficult to carry out Accurate Model to dirigible in practical flight process, f (x) is unknown function, is difficult to carry out control law according to formula (35) and resolves, and therefore, must adopt the estimated value of f (x) flight tracking control rule formula (35) is resolved; Adopt neural network to approach unknown function f (x), then have:

f(x)＝w ^Th(x)+ε (37)

In formula, w is the weight vectors of neural network, and ε is approximate error, h (x)=[h _i(x)] ^t, h _ix () is Gaussian bases, subscript i represents i-th Gaussian bases;

2) input/output variable is selected

Make flight tracking control error e and rate of change thereof actual flight path η and rate of change thereof for the input variable of neural network approximator, make estimated value for the output variable of neural network approximator.

3) neural network structure is designed

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

Input layer: the input variable choosing neural network is $x={\left[\begin{array}{cccc}e& \stackrel{\·}{e}& \mathrm{\η}& \stackrel{\·}{\mathrm{\η}}\end{array}\right]}^T.$

Hidden layer: choose the basis function of Gaussian function as hidden node

$h_i\left(x\right)=\exp \left(\frac{{\left|\right|x-c\left|\right|}^2}{{2\mathrm{\σ}}_i^2}\right)---\left(38\right)$

Wherein, c=[c _i] ^t, c _ibe the intermediate value of i-th Gaussian function, σ _ibe the base width parameter of i-th node, || || represent euclideam norm.

Output layer: the output of neural network is

$\hat{f}\left(x\right)={\hat{w}}^{T}h\left(x\right)---\left(39\right)$

Wherein, for the estimated value of w.

4) stability analysis

Definition with w difference:

$\tilde{w}=w-\hat{w}---\left(40\right)$

Choose Lyapunov function:

$V_2=\frac{1}{2}{\mathrm{\ξ}}^T\mathrm{\ξ}+\frac{1}{2}\mathrm{tr}\left({\tilde{W}}^T{Q}^{-1}\tilde{W}\right)---\left(41\right)$

In formula, $\mathrm{\ξ}={\left[\begin{array}{cc}{e}_1& e_2\end{array}\right]}^T,\tilde{W}=\left[\begin{array}{cc}0& 0\\ 0& \tilde{w}\end{array}\right],Q=\left[\begin{array}{cc}0& 0\\ 0& \mathrm{\Γ}\end{array}\right],$ Γ is adjustable positive definite matrix, Q ^-1the inverse matrix of representing matrix Q.

To formula (41) differential, can obtain:

$\begin{array}{c}{\stackrel{\·}{V}}_3={\mathrm{\ξ}}^T\mathrm{\ξ}+\mathrm{tr}\left({\tilde{W}}^T{Q}^{-1}\stackrel{\·}{\tilde{W}}\right)\\ =-k_1{e}_1^T{e}_1-k_2{e}_2^T{e}_2+e_2^T({\hat{w}}^{T}h\left(x\right)+\mathrm{\ϵ})+\mathrm{tr}\left({\tilde{W}}^T{Q}^{-1}\stackrel{\·}{\tilde{W}}\right)\end{array}---\left(42\right)$

Definition

$k=\left[\begin{array}{cc}{k}_1& 0\\ 0& k_2\end{array}\right]---\left(43\right)$

Then formula (42) can be written as:

$\begin{array}{c}{\stackrel{\·}{V}}_3={\mathrm{\ξ}}^T\stackrel{\·}{\mathrm{\ξ}}+\mathrm{tr}\left({\tilde{w}}^T{Q}^{-1}\stackrel{\·}{\tilde{w}}\right)\\ =-{\mathrm{\ξ}}^T\mathrm{k\ξ}+{\mathrm{\ξ}}^T\mathrm{\ϵ}+{\mathrm{\ξ}}^T\tilde{W}\mathrm{\Ψ}+\mathrm{tr}\left({\tilde{W}}^T{Q}^{-1}\stackrel{\·}{\tilde{W}}\right)\\ =-{\mathrm{\ξ}}^T\mathrm{k\ξ}+{\mathrm{\ξ}}^T\mathrm{\ϵ}+\mathrm{tr}(\tilde{W}\mathrm{\Ψ}{\mathrm{\ξ}}^T+{\tilde{W}}^T{Q}^{-1}\stackrel{\·}{\tilde{W}})\end{array}---\left(44\right)$

In formula, $\mathrm{\Ψ}=\left[\begin{array}{c}0\\ h\left(x\right)\end{array}\right].$

Design following adaptive law:

$\stackrel{\·}{\hat{W}}=\mathrm{Q\Ψ}{\mathrm{\ξ}}^T-\mathrm{\γQ}\left|\right|\mathrm{\ξ}\left|\right|\hat{W}---\left(45\right)$

In formula, γ > 0 is adjustable parameter, $\hat{W}=\left[\begin{array}{cc}0& 0\\ 0& \hat{w}\end{array}\right].$

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

$\begin{array}{c}{\stackrel{\·}{V}}_3={\mathrm{\ξ}}^T\stackrel{\·}{\mathrm{\ξ}}+\mathrm{tr}\left({\tilde{w}}^T{Q}^{-1}\stackrel{\·}{\tilde{w}}\right)\\ =-{\mathrm{\ξ}}^T\mathrm{k\ξ}+{\mathrm{\ξ}}^T\mathrm{\ϵ}+\mathrm{tr}(\tilde{W}\mathrm{\Ψ}{\mathrm{\ξ}}^T-{\tilde{W}}^T{Q}^{-1}(\mathrm{Q\Ψ}{\mathrm{\ξ}}^T-\mathrm{\γQ}|\left|\mathrm{\ξ}\right|\left|\hat{W}\right))\\ =-{\mathrm{\ξ}}^T\mathrm{k\ξ}+{\mathrm{\ξ}}^T\mathrm{\ϵ}+\mathrm{tr}\left(\mathrm{\γ}{\tilde{W}}^T\right|\left|\mathrm{\ξ}\right|\left|\hat{W}\right)\\ =-{\mathrm{\ξ}}^T\mathrm{k\ξ}+{\mathrm{\ξ}}^T\mathrm{\ϵ}+\mathrm{\γ}\left|\right|\mathrm{\ξ}\left|\right|\mathrm{tr}\left({\tilde{W}}^T(W-\tilde{W})\right)\end{array}---\left(46\right)$

According to Schwarz (Schwarz) inequality, have:

$\mathrm{TR}\left({\tilde{W}}^T(W-\tilde{W})\right)\≤{\left|\right|\tilde{W}\left|\right|}_F{\left|\right|W\left|\right|}_F-{\left|\right|\tilde{W}\left|\right|}_F^2---\left(47\right)$

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

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

$\begin{array}{c}{\stackrel{\·}{V}}_3\≤-k_{\min }{\left|\right|\mathrm{\ξ}\left|\right|}^2+{\mathrm{\ϵ}}_N\left|\right|\mathrm{\ξ}\left|\right|+\mathrm{\γ}\left|\right|\mathrm{\ξ}\left|\right|\left[{\left|\right|\tilde{W}\left|\right|}_F({\left|\right|W\left|\right|}_F-{\left|\right|\tilde{W}\left|\right|}_F)\right]\\ \≤-\left|\right|\mathrm{\ξ}\left|\right|\left[k_{\min }\right|\left|\mathrm{\ξ}\right||-{\mathrm{\ϵ}}_N+\mathrm{\γ}{\left|\right|\tilde{W}\left|\right|}_F({\left|\right|\tilde{W}\left|\right|}_F-W_M)]\end{array}---\left(48\right)$

In formula, ε _nfor the upper bound of approximate error, k _minfor the minimal eigenvalue of adjustable parameter matrix k, W _mfor the maximal value element of weight matrix W.

Consider following equation:

$\mathrm{\γ}{\left|\right|\tilde{W}\left|\right|}_F({\left|\right|\tilde{W}\left|\right|}_F-W_M)=\mathrm{\γ}{({\left|\right|\tilde{W}\left|\right|}_F-\frac{1}{2}{W}_M)}^2-\frac{\mathrm{\γ}}{4}{W}_M^2---\left(49\right)$

If make then must have and set up with lower inequality:

$\left|\right|\mathrm{\ξ}\left|\right|>\frac{{\mathrm{\ϵ}}_N+\frac{1}{4}\mathrm{\γ}{W}_M^2}{k_{\min }},$ Or ${\left|\right|\tilde{W}\left|\right|}_F>\frac{1}{2}{W}_M+\sqrt{\frac{W_M^2}{4}+\frac{{\mathrm{\ϵ}}_N}{\mathrm{\γ}}}---\left(50\right)$

If then have || ξ || with uniform ultimate bounded, from || ξ || convergence can obtain: Track In Track precision and neural network approximate error upper bound ε _n, k is relevant for adjustable parameter matrix.

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

Advantageous Effects of the present invention:

1) the method is directly based on the non-linear dynamic model design flight tracking control rule of dirigible, consider every non-linear factor and Multivariable Coupling effect, overcome the limitation that inearized model is only suitable for equilibrium state, can any given parametrization instruction flight path of high precision tracking.

2) the non-linear flight path control system of complexity is resolved into the subsystem that two are no more than systematic education by the method, then be each subsystem design Lypaunov function and intermediate virtual controlled quentity controlled variable, by determining suitable virtual feedback, the front position of system is made to reach Asymptotic Stability, " pusher " is to whole system always, ensure that the Asymptotic Stability of whole system.

3) the method does not need accurately known to dirigible kinetic model, adopts neural network approximator to estimate unknown dirigible kinetic model, improves adaptability and the control accuracy of system.

Control engineering teacher can according to the given arbitrary instruction flight path of actual dirigible in application process, and the controlled quentity controlled variable obtained by the method is transferred to topworks realizes flight tracking control function.

Accompanying drawing explanation

Fig. 1 is dirigible flight path control system structural drawing 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 dirigible coordinate system of the present invention and kinematic parameter definition

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 neural network Approaching Results

In figure, symbol description is as follows:

η η=[x, y, z, θ, ψ, φ] ^tfor dirigible flight path, wherein x, y, z, θ, ψ, φ are respectively the x coordinate of actual flight path, y coordinate, z coordinate, the angle of pitch, crab angle and roll angle;

η _dη _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 _dbe respectively instruction x coordinate, instruction y coordinate, instruction z coordinate, the instruction angle of pitch, instruction crab angle and instruction roll angle;

O _ex _ey _ez _eo _ex _ey _ez _erepresent earth axes;

O _bx _by _bz _bo _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, be respectively the x error of coordinate of flight tracking control, y error of coordinate, z coordinate error;

U u is system control amount;

The uncertain kinetic model that f (x) f (x) is dirigible.

Embodiment

The present invention's " contragradience neural network control method that a kind of dirigible Three-dimensional Track is followed the tracks of ", its concrete steps are as follows:

Step one: given instruction flight path

Given 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 _dbe respectively instruction x coordinate, instruction y coordinate, instruction z coordinate, the instruction angle of pitch, instruction crab angle and instruction roll angle;

Step 2: the flight tracking control margin of error calculates

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, crab angle and roll angle, are consecutive variations value.

Initial flight path is:

η ₀＝[x ₀,y ₀,z ₀,θ ₀,ψ ₀,φ ₀] ^T＝[100m,-200m,5m,0.02rad,0.02rad,0.1rad] ^T。

Initial velocity:

V ₀＝[u ₀,v ₀,w ₀,p ₀,q ₀,r ₀] ^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 model of dirigible spatial movement can be expressed as:

$\stackrel{\·}{\mathrm{\η}}=J\left(\mathrm{\η}\right)=\left[\begin{array}{cc}{J}_1& 0_{3\×3}\\ 0_{3\×3}& J_2\end{array}\right]V---\left(51\right)$

$M\stackrel{\·}{V}=\stackrel{\‾}{N}+\stackrel{\‾}{G}+\stackrel{\‾}{\mathrm{\τ}}---\left(52\right)$

In formula

$J_1=\left[\begin{array}{ccc}\cos \mathrm{\ψ}\cos \mathrm{\θ}& \cos \mathrm{\ψ}\sin \mathrm{\θ}\sin \mathrm{\φ}-\sin \mathrm{\ψ}\cos \mathrm{\φ}& \cos \mathrm{\ψ}\sin \mathrm{\θ}\cos \mathrm{\φ}+\sin \mathrm{\ψ}\sin \mathrm{\φ}\\ \sin \mathrm{\ψ}\cos \mathrm{\θ}& \sin \mathrm{\ψ}\sin \mathrm{\θ}\sin \mathrm{\φ}+\cos \mathrm{\ψ}\cos \mathrm{\φ}& \sin \mathrm{\ψ}\sin \mathrm{\θ}\cos \mathrm{\φ}-\cos \mathrm{\ψ}\sin \mathrm{\φ}\\ -\sin \mathrm{\θ}& \cos \mathrm{\θ}\sin \mathrm{\φ}& \cos \mathrm{\θ}\cos \mathrm{\φ}\end{array}\right]---\left(53\right)$

$J_2=\left[\begin{array}{ccc}0& \cos \mathrm{\φ}& -\sin \mathrm{\φ}\\ 0& \sec \mathrm{\θ}\sin \mathrm{\φ}& \sec \mathrm{\θ}\cos \mathrm{\φ}\\ 1& \tan \mathrm{\θ}\sin \mathrm{\φ}& \tan \mathrm{\θ}\cos \mathrm{\φ}\end{array}\right]---\left(54\right)$

$M=\left[\begin{array}{cccccc}m+m_{11}& 0& 0& 0& {\mathrm{mz}}_G& -{\mathrm{my}}_G\\ 0& m+m_{22}& 0& -{\mathrm{mz}}_G& 0& {\mathrm{mx}}_G\\ 0& 0& m+m_{33}& {\mathrm{my}}_G& -{\mathrm{mx}}_G& 0\\ 0& -{\mathrm{mz}}_G& {\mathrm{my}}_G& I_x+I_{11}& 0& 0\\ {\mathrm{mz}}_G& 0& -{\mathrm{mx}}_G& 0& I_x+I_{22}& 0\\ 0& {\mathrm{mx}}_G& 0& -I_{\mathrm{xz}}& 0& I_x+I_{33}\end{array}\right]---\left(55\right)$

$\stackrel{\‾}{G}=\left[\begin{array}{c}(B-G)\sin \mathrm{\θ}\\ (G-B)\cos \mathrm{\θ}\sin \mathrm{\φ}\\ (G-B)\cos \mathrm{\θ}\cos \mathrm{\φ}\\ y_{G}G\cos \mathrm{\θ}\cos \mathrm{\φ}-z_{G}G\cos \mathrm{\θ}\sin \mathrm{\φ}\\ -x_{G}G\cos \mathrm{\θ}\cos \mathrm{\φ}-z_{G}G\sin \mathrm{\θ}\\ x_{G}G\cos \mathrm{\θ}\sin \mathrm{\φ}+y_{G}G\sin \mathrm{\θ}\end{array}\right]---\left(56\right)$

$\mathrm{\τ}=\left[\begin{array}{c}{\mathrm{\τ}}_u\\ {\mathrm{\τ}}_v\\ {\mathrm{\τ}}_w\\ {\mathrm{\τ}}_l\\ {\mathrm{\τ}}_m\\ {\mathrm{\τ}}_n\end{array}\right]=\left[\begin{array}{c}T\cos \mathrm{\μ}\cos \mathrm{\υ}\\ T\sin \mathrm{\μ}\\ T\cos \mathrm{\μ}\sin \mathrm{\υ}\\ T\cos \mathrm{\μ}\sin \mathrm{\υ}{l}_y-T\sin {\mathrm{\μl}}_z\\ T\cos \mathrm{\μ}\cos \mathrm{\υ}{l}_z-T\cos \mathrm{\μ}\sin \mathrm{\υ}{l}_x\\ T\sin {\mathrm{\μl}}_x-T\cos \mathrm{\μ}\cos \mathrm{\υ}{l}_y\end{array}\right]---\left(57\right)$

$\stackrel{\‾}{N}={[N_u,N_v,N_w,N_p,N_q,N_r]}^T---\left(58\right)$

Wherein

N _u＝(m+m ₂₂)vr-(m+m ₃₃)wq+m[x _G(p ²+r ²)-y _Gpq-z _Gpr]

(59)

+QV ^2/3(-C _Xcosαcosβ+C _Ycosαsinβ+C _Zsinα)

N _v＝(m+m ₃₃)wp-(m+m ₁₁)ur-m[x _Gpq-y _G(p ²+r ²)+z _Gqr]

(60)

+QV ^2/3(C _Xsinβ+C _Ycosβ)

N _w＝(m+m ₂₂)vp-(m+m ₁₁)uq-m[x _Gpr+y _Gqr-z _G(p ²+q ²)]

(61)

+QV ^2/3(-C _Xsinαsinβ+C _Ysinαcosβ-C _Zcosα)

N _p＝[(I _y+m ₅₅)-(I _z+I ₆₆)]qr+I _xzpq-I _xypr-I _yz(r ²-q ²)+

(62)

[mz _G(ur-wp)+y _G(uq-vp)]+QVC _l

N _q＝[(I _z+m ₆₆)-(I _x+I ₄₄)]pr+I _xyqr-I _yzpq-I _xz(p ²-r ²)

(63)

+m[x _G(vp-uq)-z _G(wp-vr)]+QVC _m

N _r＝[(I _y+m ₅₅)-(I _x+I ₄₄)]pq-I _xzqr-I _xy(q ²-p ²)+I _yzpr

(64)

+m[y _G(wq-vr)-x _G(ur-wp)]+QVC _n

In formula, m is dirigible quality, m ₁₁, m ₂₂, m ₃₃for additional mass, I ₁₁, I ₂₂, I ₃₃for additional inertial; 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 _zbe respectively around o _bx _b, o _by _b, o _bz _bprincipal moments; I _xy, I _xz, I _yzbe respectively about plane o _bx _by _b, o _bx _bz _b, o _by _bz _bproduct of inertia; T is thrust size, and μ is thrust vectoring and o _bx _bz _bangle between face, specifies that it is at o _bx _bz _bthe left side in face is just, υ is that thrust vectoring is at o _bx _bz _bthe projection in face and o _bx _bangle between axle, specifies that it is projected in o _bx _bjust be under axle; l _x, l _y, l _zrepresent that thrust point is apart from initial point o _bdistance.

Formula (52) is the expression formula about generalized velocity V, needs to be transformed to the expression formula about generalized coordinate η.

Can be obtained by formula (51):

$V=J^{-1}\left(\mathrm{\η}\right)\stackrel{\·}{\mathrm{\η}}=R\left(\mathrm{\η}\right)\stackrel{\·}{\mathrm{\η}}=\left[\begin{array}{cc}A& 0_{3\×3}\\ 0_{3\×3}& B\end{array}\right]\stackrel{\·}{\mathrm{\η}}---\left(65\right)$

In formula, J ^-1(η) be the inverse matrix of J (η),

$A=\left[\begin{array}{ccc}\cos \mathrm{\ψ}\cos \mathrm{\θ}& \sin \mathrm{\ψ}\cos \mathrm{\θ}& -\sin \mathrm{\θ}\\ \cos \mathrm{\ψ}\sin \mathrm{\θ}\sin \mathrm{\φ}-\sin \mathrm{\ψ}\cos \mathrm{\φ}& \sin \mathrm{\ψ}\sin \mathrm{\θ}\sin \mathrm{\φ}+\cos \mathrm{\ψ}\cos \mathrm{\φ}& \cos \mathrm{\θ}\sin \mathrm{\φ}\\ \cos \mathrm{\ψ}\sin \mathrm{\θ}\cos \mathrm{\φ}+\sin \mathrm{\ψ}\sin \mathrm{\φ}& \sin \mathrm{\ψ}\sin \mathrm{\θ}\cos \mathrm{\φ}-\cos \mathrm{\ψ}\sin \mathrm{\φ}& \cos \mathrm{\θ}\cos \mathrm{\φ}\end{array}\right]---\left(66\right)$

$B=\left[\begin{array}{ccc}0& -\sin \mathrm{\θ}& 1\\ \cos \mathrm{\φ}& \cos \mathrm{\θ}\sin \mathrm{\φ}& 0\\ -\sin \mathrm{\φ}& \cos \mathrm{\θ}\cos \mathrm{\φ}& 0\end{array}\right]---\left(67\right)$

To formula (65) differential, can obtain

$\stackrel{\·}{V}=\stackrel{\·}{R}\stackrel{\·}{\mathrm{\η}}+R\stackrel{\·\·}{\mathrm{\η}}---\left(68\right)$

In formula

$\stackrel{\·}{R}=\left[\begin{array}{cc}\stackrel{\·}{A}& 0_{3\×3}\\ 0_{3\×3}& \stackrel{\·}{B}\end{array}\right]---\left(69\right)$

Formula (68) premultiplication can obtain

$R^{T}M\stackrel{\·}{V}=R^{T}M\stackrel{\·}{R}\stackrel{\·}{\mathrm{\η}}+R^T\mathrm{MR}\stackrel{\·\·}{\mathrm{\η}}---\left(70\right)$

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

$M_{\mathrm{\η}}\left(\mathrm{\η}\right)\stackrel{\·\·}{\mathrm{\η}}+N_{\mathrm{\η}}(\mathrm{\η},\stackrel{\·}{\mathrm{\η}})\stackrel{\·}{\mathrm{\η}}+G_{\mathrm{\η}}\left(\mathrm{\η}\right)=u---\left(71\right)$

In formula

M _η(η)＝R ^TMR (72)

$N_{\mathrm{\η}}(\mathrm{\η},\stackrel{\·}{\mathrm{\η}})=R^{T}M\stackrel{\·}{R}---\left(73\right)$

$G_{\mathrm{\η}}\left(\mathrm{\η}\right)=-R^T(\stackrel{\‾}{N}+\stackrel{\‾}{G})---\left(74\right)$

u＝R ^Tτ (75)

Make x ₁=η, then kinetic model (71) can be written as following form:

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=M_{\mathrm{\η}}^{-1}u-M_{\mathrm{\η}}^{-1}{N}_{\mathrm{\η}}\left(\mathrm{\η}\right)\stackrel{\·}{\mathrm{\η}}-G_{\mathrm{\η}}\left(\mathrm{\η}\right)\end{array}\right.---\left(76\right)$

With the mathematical model described by formula (76) for controlled device, adopt Backstepping techniques design flight tracking control rule.

Dirigible parameter in the present embodiment is in table 1.

Table 1 dirigible parameter

2) design control law

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

$a_1=-k_{1}e+\stackrel{\·}{\mathrm{\η}}d$

(77)

In formula, α ₁for virtual amount, k ₁for adjustable controling parameters.

Defining virtual amount α ₁with x ₂between error e ':

e′＝x ₂-α ₁(78)

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

$e^{\′}={\stackrel{\·}{x}}_2-{\stackrel{\·}{a}}_1=M_{\mathrm{\η}}^{-1}u-M_{\mathrm{\η}}^{-1}{N}_{\mathrm{\η}}\left(\mathrm{\η}\right)\stackrel{\·}{\mathrm{\η}}-G_{\mathrm{\η}}\left(\mathrm{\η}\right)+k_1\stackrel{\·}{e}-{\stackrel{\·\·}{\mathrm{\η}}}_d---\left(79\right)$

Order

$f\left(x\right)=-M_{\mathrm{\η}}^{-1}{N}_{\mathrm{\η}}\left(\mathrm{\η}\right)\stackrel{\·}{\mathrm{\η}}-G_{\mathrm{\η}}\left(\mathrm{\η}\right)---\left(80\right)$

Then formula (80) can be expressed as:

${\stackrel{\·}{e}}^{\′}=f\left(x\right)+M_{\mathrm{\η}}^{-1}u+k_1\stackrel{\·}{e}-{\stackrel{\·\·}{\mathrm{\η}}}_d---\left(81\right)$

Wherein, k ₁value is 12.

Choose Lyapunov function V ₁

$V_1=\frac{1}{2}{e}^{T}e+\frac{1}{2}{e}^{\′T}{e}^{\′}---\left(82\right)$

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

${\stackrel{\·}{V}}_1=-k_1{e}^{T}e+e^{\′T}(f\left(x\right)+M_{\mathrm{\η}}^{-1}u+k_1\stackrel{\·}{e}-{\stackrel{\·\·}{\mathrm{\η}}}_d)---\left(83\right)$

According to formula (83), design following control law:

$u=M_{\mathrm{\η}}(-e-k_1\stackrel{\·}{e}-k_2{e}^{\′}+{\stackrel{\·\·}{\mathrm{\η}}}_d-f\left(x\right))---\left(84\right)$

Wherein, k ₂value is 10.

Owing to being difficult to carry out Accurate Model to dirigible in practical flight process, f (x) is unknown function, is difficult to carry out control law according to formula (35) and resolves, and therefore, must adopt the estimated value of f (x) flight tracking control rule formula (35) is resolved;

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

1) adopt neural network to approach unknown function f (x), then have:

f(x)＝w ^Th(x)+ε (37)

In formula, w is the weight vectors of neural network, and ε is approximate error, h (x)=[h _i(x)] ^t, h _ix () is Gaussian bases, subscript i represents i-th Gaussian bases;

2) input/output variable is selected

Make flight tracking control error e and rate of change thereof actual flight path η and rate of change thereof for the input variable of neural network approximator, make estimated value for the output variable of neural network approximator.

3) neural network structure is designed

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

Input layer: the input variable choosing neural network is $x={\left[\begin{array}{cccc}e& \stackrel{\·}{e}& \mathrm{\η}& \stackrel{\·}{\mathrm{\η}}\end{array}\right]}^T.$

Hidden layer: choose the basis function of Gaussian function as hidden node

$h_i\left(x\right)=\exp \left(\frac{{\left|\right|x-c\left|\right|}^2}{{2\mathrm{\σ}}_i^2}\right)---\left(86\right)$

Wherein,

$c=\left[\begin{array}{ccccccc}-1.5& -1& -0.5& 0& 0.5& 1& 1.5\\ -1.5& -1& -0.5& 0& 0.5& 1& 1.5\\ -1.5& -1& -0.5& 0& 0.5& 1& 1.5\\ -1.5& -1& -0.5& 0& 0.5& 1& 1.5\end{array}\right],{\mathrm{\σ}}_i=2.$

Output layer: the output of neural network is

$\hat{f}\left(x\right)={\hat{w}}^{T}h\left(x\right)---\left(87\right)$

Wherein, value be taken as $0.15\×\left[\begin{array}{ccccccc}1& 1& 1& 1& 1& 1& 1\\ 1& 1& 1& 1& 1& 1& 1\\ 1& 1& 1& 1& 1& 1& 1\\ 1& 1& 1& 1& 1& 1& 1\\ 1& 1& 1& 1& 1& 1& 1\\ 1& 1& 1& 1& 1& 1& 1\end{array}\right].$

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

Dirigible Three-dimensional Track tracing control result in embodiment 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 trace command flight path exactly by initial position, demonstrates the validity of Track In Track control method proposed by the invention; Fig. 6 gives Track In Track departure, can be obtained by Fig. 6, and flight tracking control method proposed by the invention can the instruction flight path of tracing preset accurately.Fig. 7 gives neural network Approaching Results, can be obtained by Fig. 7, and the neural network approximator designed by the present invention accurately can estimate uncertain dirigible kinetic model.

More than contain the explanation of the preferred embodiment of the present invention; this is to describe technical characteristic of the present invention in detail; be not want summary of the invention to be limited in the concrete form described by embodiment, other amendments carried out according to content purport of the present invention and modification are also protected by this patent.The purport of content of the present invention defined by claims, but not defined by the specific descriptions of embodiment.

Claims (3)

1. a contragradience neural network control method for dirigible Three-dimensional Track tracking, is characterized in that, comprise 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: instruction flight path is generalized coordinate 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 _dbe respectively instruction x coordinate, instruction y coordinate, instruction z coordinate, the instruction angle of pitch, instruction crab angle and instruction roll angle, subscript T represents vector or transpose of a matrix;

Step 2: the flight tracking control margin of error calculates: the flight tracking control margin of error e between computations flight path and actual flight path;

The computing method of 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)

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, crab angle and roll angle;

Step 3: flight tracking control rule design: choose Lypaunov function and intermediate virtual controlled quentity controlled variable, adopts Backstepping design flight tracking control rule, calculates flight tracking control amount u;

Step 4: neural network approximator designs: with flight tracking control margin of error e and rate of change thereof actual flight path η and rate of change thereof for the input variable of neural network, with the estimated value of dirigible kinetic model for the output variable design neural network approximator of neural network, utilize neural network infinitely to approach function and estimate unknown ambiguous model, to provide control accuracy.

2. the contragradience neural network control method of dirigible Three-dimensional Track tracking according to claim 1, is characterized in that: design flight tracking control rule in described step 3, calculate flight tracking control amount u, comprise the following steps:

1) kinetic model of dirigible is set up

Coordinate system and the kinematic parameter of dirigible spatial movement are defined as follows: adopt earth axes o _ex _ey _ez _ewith body coordinate system o _bx _by _bz _bbe described the spatial movement of dirigible, CV is centre of buoyancy, and CG is center of gravity, and centre of buoyancy is r to the vector of center of gravity _g=[x _g, y _g, z _g] ^t; Kinematic parameter defines: position P=[x, y, z] ^t, x, y, z is respectively the displacement of axis, side direction and vertical direction; Attitude angle Ω=[θ, ψ, φ] ^t, θ, ψ, φ are respectively the angle of pitch, crab angle and roll angle; Speed v=[u, v, w] ^t, u, v, w are respectively the speed of axis, side direction and vertical direction in body coordinate system; Angular velocity omega=[p, q, r] ^t, p, q, r are respectively rolling, pitching and yaw rate; Note generalized coordinate 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 below:

$\stackrel{\·}{\mathrm{\η}}=J\left(\mathrm{\η}\right)=\left[\begin{array}{cc}{J}_1& 0_{3\×3}\\ 0_{3\×3}& J_2\end{array}\right]V---\left(2\right)$

$M\stackrel{\·}{V}=\stackrel{\‾}{N}+\stackrel{\‾}{G}+\mathrm{\τ}---\left(3\right)$

In formula

$J=\left[\begin{array}{ccc}\cos \mathrm{\ψ}\cos \mathrm{\θ}& \cos \mathrm{\ψ}\sin \mathrm{\θ}\sin \mathrm{\φ}-\sin \mathrm{\ψ}\cos \mathrm{\φ}& \cos \mathrm{\ψ}\sin \mathrm{\θ}\cos \mathrm{\φ}+\sin \mathrm{\ψ}\sin \mathrm{\φ}\\ \sin \mathrm{\ψ}\cos \mathrm{\θ}& \sin \mathrm{\ψ}\sin \mathrm{\θ}\sin \mathrm{\φ}+\cos \mathrm{\ψ}\cos \mathrm{\φ}& \sin \mathrm{\ψ}\sin \mathrm{\θ}\cos \mathrm{\φ}-\cos \mathrm{\ψ}\sin \mathrm{\φ}\\ -\sin \mathrm{\θ}& \cos \mathrm{\θ}\sin \mathrm{\φ}& \cos \mathrm{\θ}\cos \mathrm{\φ}\end{array}\right]---\left(4\right)$

$J_2=\left[\begin{array}{ccc}0& \cos \mathrm{\φ}& -\sin \mathrm{\φ}\\ 0& \sec \mathrm{\θ}\sin \mathrm{\φ}& \sec \mathrm{\θ}\cos \mathrm{\φ}\\ 1& \tan \mathrm{\θ}\sin \mathrm{\φ}& \tan \mathrm{\θ}\cos \mathrm{\φ}\end{array}\right]---\left(5\right)$

$M=\left[\begin{array}{cccccc}m+m_{11}& 0& 0& 0& {\mathrm{mz}}_G& -{\mathrm{my}}_G\\ 0& m+m_{22}& 0& -{\mathrm{mz}}_G& 0& {\mathrm{mx}}_G\\ 0& 0& m+m_{33}& {\mathrm{my}}_G& -{\mathrm{mx}}_G& 0\\ 0& -{\mathrm{mz}}_G& {\mathrm{my}}_G& I_x+I_{11}& 0& 0\\ {\mathrm{mz}}_G& 0& -{\mathrm{mx}}_G& 0& I_x+I_{22}& 0\\ 0& {\mathrm{mx}}_G& 0& -I_{\mathrm{xz}}& 0& I_x+I_{33}\end{array}\right]---\left(6\right)$

$\stackrel{\‾}{G}=\left[\begin{array}{c}(B-G)\sin \mathrm{\θ}\\ (G-B)\cos \mathrm{\θ}\sin \mathrm{\φ}\\ (G-B)\cos \mathrm{\θ}\cos \mathrm{\φ}\\ y_{G}G\cos \mathrm{\θ}\cos \mathrm{\φ}-z_{G}G\cos \mathrm{\θ}\sin \mathrm{\φ}\\ -x_{G}G\cos \mathrm{\θ}\cos \mathrm{\φ}-z_{G}G\sin \mathrm{\θ}\\ x_G\cos \mathrm{\θ}\sin \mathrm{\φ}+y_{G}G\sin \mathrm{\θ}\end{array}\right]---\left(7\right)$

$\mathrm{\τ}=\left[\begin{array}{c}{\mathrm{\τ}}_u\\ {\mathrm{\τ}}_v\\ {\mathrm{\τ}}_w\\ {\mathrm{\τ}}_l\\ {\mathrm{\τ}}_m\\ {\mathrm{\τ}}_n\end{array}\right]=\left[\begin{array}{c}T\cos \mathrm{\μ}\cos \mathrm{\υ}\\ T\sin \mathrm{\μ}\\ T\cos \mathrm{\μ}\sin \mathrm{\υ}\\ T\cos \mathrm{\μ}\sin {\mathrm{\υl}}_y-T\sin {\mathrm{\μl}}_z\\ T\cos \mathrm{\μ}\cos {\mathrm{\υl}}_z-T\cos \mathrm{\μ}\sin {\mathrm{\υl}}_x\\ T\sin {\mathrm{\μl}}_x-T\cos \mathrm{\μ}\cos {\mathrm{\υl}}_y\end{array}\right]---\left(8\right)$

$\stackrel{\‾}{N}={[N_u,N_v,N_w,N_p,N_q,N_r]}^T---\left(9\right)$

Wherein

N _u＝(m+m ₂₂)vr-(m+m ₃₃)wq+m[x _G(p ²+r ²)-y _Gpq-z _Gpr]

(10)

+QV ^2/3(-C _Xcosαcosβ+C _Ycosαsinβ+C _Zsinα)

N _v＝(m+m ₃₃)wp-(m+m ₁₁)ur-m[x _Gpq-y _G(p ²+r ²)+z _Gqr]

(11)

+QV ^2/3(C _Xsinβ+C _Ycosβ)

N _w＝(m+m ₂₂)vp-(m+m ₁₁)uq-m[x _Gpr+y _Gqr-z _G(p ²+q ²)]

(12)

+QV ^2/3(-C _Xsinαsinβ+C _Ysinαcosβ-C _Zcosα)

N _p＝[(I _y+I ₂₂)-(I _z+I ₃₃)]qr+I _xzpq-I _xypr-I _yz(r ²-q ²)+

(13)

[mz _G(ur-wp)+y _G(uq-vp)]+QVC _l

N _q＝[(I _z+I ₃₃)-(I _x+I ₂₂)]pr+I _xyqr-I _yzpq-I _xz(p ²-r ²)

(14)

+m[x _G(vp-uq)-z _G(wp-vr)]+QVC _m

N _r＝[(I _y+I ₂₂)-(I _x+I ₁₁)]pq-I _xzqr-I _xy(q ²-p ²)+I _yzpr

(15)

+m[y _G(wq-vr)-x _G(ur-wp)]+QVC _n

In formula, m is dirigible quality, m ₁₁, m ₂₂, m ₃₃for additional mass, I ₁₁, I ₂₂, I ₃₃for additional inertial, V is dirigible volume; 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 _zbe respectively around o _bx _b, o _by _b, o _bz _bprincipal moments; I _xy, I _xz, I _yzbe respectively about plane o _bx _by _b, o _bx _bz _b, o _by _bz _bproduct of inertia; T is thrust size, and μ is thrust vectoring and o _bx _bz _bangle between face, specifies that it is at o _bx _bz _bthe left side in face is just, υ is that thrust vectoring is at o _bx _bz _bthe projection in face and o _bx _bangle between axle, specifies that it is projected in o _bx _bjust be under axle; l _x, l _y, l _zrepresent that thrust point is apart from initial point o _bdistance;

Formula (3) is the expression formula about generalized velocity V, needs to be transformed to the expression formula about generalized coordinate η;

Can be obtained by formula (1):

$V=J^{-1}\left(\mathrm{\η}\right)\stackrel{\·}{\mathrm{\η}}=R\left(\mathrm{\η}\right)\stackrel{\·}{\mathrm{\η}}=\left[\begin{array}{cc}A& 0_{3\×3}\\ 0_{3\×3}& B\end{array}\right]\stackrel{\·}{\mathrm{\η}}---\left(16\right)$

In formula, J ^-1(η) be the inverse matrix of J (η);

$A=\left[\begin{array}{ccc}\cos \mathrm{\ψ}\cos \mathrm{\θ}& \sin \mathrm{\ψ}\cos \mathrm{\θ}& -\sin \\ \cos \mathrm{\ψ}\sin \mathrm{\θ}\sin \mathrm{\φ}-\sin \mathrm{\ψ}\cos \mathrm{\φ}& \sin \mathrm{\ψ}\sin \mathrm{\θ}\sin \mathrm{\φ}+\cos \mathrm{\ψ}\cos \mathrm{\φ}& \cos \mathrm{\θ}\sin \mathrm{\φ}\\ \cos \mathrm{\ψ}\sin \mathrm{\θ}\cos \mathrm{\φ}+\sin \mathrm{\ψ}\sin \mathrm{\φ}& \sin \mathrm{\ψ}\sin \mathrm{\θ}\cos \mathrm{\φ}-\cos \mathrm{\ψ}\sin \mathrm{\φ}& \cos \mathrm{\θ}\cos \mathrm{\φ}\end{array}\right]---\left(17\right)$

$B=\left[\begin{array}{ccc}0& -\sin \mathrm{\θ}& 1\\ \cos \mathrm{\φ}& \cos \mathrm{\θ}\sin \mathrm{\φ}& 0\\ -\sin \mathrm{\φ}& \cos \mathrm{\θ}\cos \mathrm{\φ}& 0\end{array}\right]---\left(18\right)$

To formula (16) differential, can obtain

$\stackrel{\·}{V}=\stackrel{\·}{R}\stackrel{\·}{\mathrm{\η}}+R\stackrel{\·\·}{\mathrm{\η}}---\left(19\right)$

In formula

$\stackrel{\·}{R}=\left[\begin{array}{cc}\stackrel{\·}{A}& 0_{3\×3}\\ 0_{3\×3}& \stackrel{\·}{B}\end{array}\right]---\left(20\right)$

Formula (19) premultiplication can obtain

$R^{T}M\stackrel{\·}{V}=R^{T}M\stackrel{\·}{R}\stackrel{\·}{\mathrm{\η}}+R^T\mathrm{MR}\stackrel{\·\·}{\mathrm{\η}}---\left(21\right)$

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

$M_{\mathrm{\η}}\left(\mathrm{\η}\right)\stackrel{\·\·}{\mathrm{\η}}+N_{\mathrm{\η}}(\mathrm{\η},\stackrel{\·}{\mathrm{\η}})\stackrel{\·}{\mathrm{\η}}+G_{\mathrm{\η}}\left(\mathrm{\η}\right)=u---\left(22\right)$

In formula

M _η(η)＝R ^TMR (23)

$N_{\mathrm{\η}}(\mathrm{\η},\stackrel{\·}{\mathrm{\η}})=R^{T}M\stackrel{\·}{R}---\left(24\right)$

$G_{\mathrm{\η}}\left(\mathrm{\η}\right)=-R^T(\stackrel{\‾}{N}+\stackrel{\‾}{G})---\left(25\right)$

u＝R ^Tτ (26)

Make x ₁=η, then kinetic model (22) is written as following form:

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=M_{\mathrm{\η}}^{-1}u-M_{\mathrm{\η}}^{-1}{N}_{\mathrm{\η}}\left(\mathrm{\η}\right)\stackrel{\·}{\mathrm{\η}}-G_{\mathrm{\η}}\left(\mathrm{\η}\right)\end{array}\right.---\left(27\right)$

In formula, representing matrix M _ηinverse matrix;

With the mathematical model described by formula (27) for controlled device, adopt Backstepping techniques design flight tracking control rule;

2) flight tracking control rule is designed

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

${\mathrm{\α}}_1=-k_{1}e+{\stackrel{\·}{\mathrm{\η}}}_d---\left(28\right)$

In formula, α ₁for virtual amount, k ₁for adjustable controling parameters;

Defining virtual amount α ₁with x ₂between error e ':

e′＝x ₂-α ₁(29)

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

${\stackrel{\·}{e}}^{\′}={\stackrel{\·}{x}}_2-{\stackrel{\·}{\mathrm{\α}}}_1=M_{\mathrm{\η}}^{-1}u-M_{\mathrm{\η}}^{-1}{N}_{\mathrm{\η}}\left(\mathrm{\η}\right)\stackrel{\·}{\mathrm{\η}}-G_{\mathrm{\η}}\left(\mathrm{\η}\right)+k_1\stackrel{\·}{e}-{\stackrel{\·\·}{\mathrm{\η}}}_d---\left(30\right)$

Order

$f\left(x\right)=-M_{\mathrm{\η}}^{-1}{N}_{\mathrm{\η}}\left(\mathrm{\η}\right)\stackrel{\·}{\mathrm{\η}}-G_{\mathrm{\η}}\left(\mathrm{\η}\right)---\left(31\right)$

Then formula (31) can be expressed as:

${\stackrel{\·}{e}}^{\′}=f\left(x\right)+M_{\mathrm{\η}}^{-1}u+k_1\stackrel{\·}{e}-{\stackrel{\·\·}{\mathrm{\η}}}_d---\left(32\right)$

Choose Lyapunov function V ₁

$V_1=\frac{1}{2}{e}^{T}e+\frac{1}{2}{e}^{\′T}{e}^{\′}---\left(33\right)$

Formula (32) to time diffusion, and substitutes into by formula (33), can obtain:

${\stackrel{\·}{V}}_1=-k_1{e}^{T}e+e^{\′T}(f\left(x\right)+M_{\mathrm{\η}}^{-1}u+k_1\stackrel{\·}{e}-{\stackrel{\·\·}{\mathrm{\η}}}_d)---\left(34\right)$

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

$u=M_{\mathrm{\η}}(-e-k_1\stackrel{\·}{e}-k_2{e}^{\′}+{\stackrel{\·\·}{\mathrm{\η}}}_d-f\left(x\right))---\left(35\right)$

3) stability analysis

Flight tracking control is restrained formula (35) and substitutes into formula (34), can obtain:

$\begin{array}{c}{\stackrel{\·}{V}}_1=-k_1{e}^{T}e+e^T{e}^{\′}+e^{\′T}\{f\left(x\right)+M_{\mathrm{\η}}^{-1}[M_{\mathrm{\η}}(-e-k_1\stackrel{\·}{e}-k_2{e}^{\′}+{\stackrel{\·\·}{\mathrm{\η}}}_d-f\left(x\right)))]+k_1\stackrel{\·}{e}k{\stackrel{\·\·}{\mathrm{\η}}}_d\}\\ =-k_1{e}^{T}e+e^{\′T}{e}^{\′}+e^{\′T}(-e-k_2{e}^{\′})=-k_1{e}^{T}e-k_2{e}^{\′T}{e}^{\′}\end{array}---\left(36\right)$

Formula (36) shows: adopt flight tracking control rule (35) can ensure the stability of closed-loop system.

3. the contragradience neural network control method of dirigible Three-dimensional Track tracking according to claim 2, it is characterized in that: the neural network approximator described in described step 4, its method for designing is:

1) owing to being difficult to carry out Accurate Model to dirigible in practical flight process, f (x) is unknown function, is difficult to carry out control law according to formula (35) and resolves, and therefore, must adopt the estimated value of f (x) flight tracking control rule formula (35) is resolved; Adopt neural network to approach unknown function f (x), then have:

f(x)＝w ^Th(x)+ε (37)

In formula, w is the weight vectors of neural network, and ε is approximate error, h (x)=[h _i(x)] ^t, h _ix () is Gaussian bases, subscript i represents i-th Gaussian bases;

2) input/output variable is selected

Make flight tracking control margin of error e and rate of change thereof actual flight path η and rate of change thereof for the input variable of neural network approximator, make estimated value for the output variable of neural network approximator;

3) neural network structure is designed

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

Input layer: the input variable choosing neural network is $x={\left[\begin{array}{cccc}e& \stackrel{\·}{e}& \mathrm{\η}& \stackrel{\·}{\mathrm{\η}}\end{array}\right]}^T;$

Hidden layer: choose the basis function of Gaussian function as hidden node

$h_i\left(x\right)=\exp \left(\frac{{\left|\right|x-c\left|\right|}^2}{2{\mathrm{\σ}}_i^2}\right)---\left(38\right)$

Wherein, c=[c _i] ^t, c _ibe the intermediate value of i-th Gaussian function, σ _ibe the base width parameter of i-th node, || || represent euclideam norm;

Output layer: the output of neural network approximator is

$\hat{f}\left(x\right)={\hat{w}}^{T}h\left(x\right)---\left(39\right)$

Wherein, for the estimated value of w;

4) stability analysis

Definition with w difference:

$\tilde{w}=w-\hat{w}---\left(40\right)$

Choose Lyapunov function:

$V_2=\frac{1}{2}{\mathrm{\ξ}}^T\mathrm{\ξ}+\frac{1}{2}\mathrm{tr}\left({\tilde{W}}^T{Q}^{-1}\tilde{W}\right)---\left(41\right)$

In formula, $\mathrm{\ξ}={\left[\begin{array}{cc}{e}_1& e_2\end{array}\right]}^T,\tilde{W}=\left[\begin{array}{cc}0& 0\\ 0& \tilde{w}\end{array}\right],Q=\left[\begin{array}{cc}0& 0\\ 0& \mathrm{\Γ}\end{array}\right],$ Γ is adjustable positive definite matrix, Q ^-1the inverse matrix of representing matrix Q;

To formula (41) differential, can obtain:

$\begin{array}{c}{\stackrel{\·}{V}}_3={\mathrm{\ξ}}^T\stackrel{\·}{\mathrm{\ξ}}+\mathrm{tr}\left({\tilde{W}}^T{Q}^{-1}\stackrel{\stackrel{\·}{~}}{W}\right)\\ =-k_1{e}_1^T{e}_1-k_2{e}_2^T{e}_2+e_2^T({\hat{w}}^{T}h\left(x\right)+\mathrm{\ϵ})+\mathrm{tr}\left({\tilde{w}}^T{Q}^{-1}\stackrel{\stackrel{\·}{~}}{W}\right)\end{array}---\left(42\right)$

Definition

$k=\left[\begin{array}{cc}{k}_1& 0\\ 0& k_2\end{array}\right]---\left(43\right)$

Then formula (42) can be written as:

$\begin{array}{c}{\stackrel{\·}{V}}_3={\mathrm{\ξ}}^T\stackrel{\·}{\mathrm{\ξ}}+\mathrm{tr}\left({\tilde{w}}^T{Q}^{-1}\stackrel{\stackrel{\·}{~}}{w}\right)\\ =-{\mathrm{\ξ}}^T\mathrm{k\ξ}+{\mathrm{\ξ}}^T\mathrm{\ϵ}+{\mathrm{\ξ}}^T\tilde{W}\mathrm{\Ψ}+\mathrm{tr}\left({\tilde{W}}^T{Q}^{-1}\stackrel{\stackrel{\·}{~}}{W}\right)\\ =-{\mathrm{\ξ}}^T\mathrm{k\ξ}+{\mathrm{\ξ}}^T\mathrm{\ϵ}+\mathrm{tr}(\tilde{W}\mathrm{\ψ}{\mathrm{\ξ}}^T+{\tilde{W}}^T{Q}^{-1}\stackrel{\stackrel{\·}{~}}{W})\end{array}---\left(44\right)$

In formula, $\mathrm{\Ψ}=\left[\begin{array}{c}0\\ h\left(x\right)\end{array}\right];$

Design following adaptive law:

$\stackrel{\stackrel{\·}{^}}{W}=\mathrm{Q\Ψ}{\mathrm{\ξ}}^T-\mathrm{\γQ}\left|\right|\mathrm{\ξ}\left|\right|\hat{W}---\left(45\right)$

In formula, γ > 0 is adjustable parameter, $\hat{W}=\left[\begin{array}{cc}0& 0\\ 0& \hat{w}\end{array}\right];$

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

$\begin{array}{c}{\stackrel{\·}{V}}_3={\mathrm{\ξ}}^T\stackrel{\·}{\mathrm{\ξ}}+\mathrm{tr}\left({\tilde{w}}^T{Q}^{-1}\stackrel{\stackrel{\·}{~}}{w}\right)\\ =-{\mathrm{\ξ}}^T\mathrm{k\ξ}+{\mathrm{\ξ}}^T\mathrm{\ϵ}+\mathrm{tr}(\tilde{W}\mathrm{\ψ}{\mathrm{\ξ}}^T-{\tilde{W}}^T{Q}^{-1}(\mathrm{Q\Ψ}{\mathrm{\ξ}}^T-\mathrm{\γQ}|\left|\mathrm{\ξ}\right|\left|\hat{W}\right))\\ =-{\mathrm{\ξ}}^T\mathrm{k\ξ}+{\mathrm{\ξ}}^T\mathrm{\ϵ}+\mathrm{tr}\left(\mathrm{\γ}{\tilde{W}}^T\right|\left|\mathrm{\ξ}\right|\left|\hat{W}\right)\\ =-{\mathrm{\ξ}}^T\mathrm{k\ξ}+{\mathrm{\ξ}}^T\mathrm{\ϵ}+\mathrm{\γ}\left|\right|\mathrm{\ξ}\left|\right|\mathrm{tr}\left({\tilde{W}}^T(W-\tilde{W})\right)\end{array}---\left(46\right)$

According to Schwarz inequality, have:

$\mathrm{tr}\left({\tilde{W}}^T(W-\tilde{W})\right)\≤{\left|\right|\tilde{W}\left|\right|}_F{\left|\right|W\left|\right|}_F-{\left|\right|\tilde{W}\left|\right|}_F^2---\left(47\right)$

In formula, || || _frepresent Frobenius norm;

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

$\begin{array}{c}{\stackrel{\·}{V}}_3\≤-k_{\min }{\left|\right|\mathrm{\ξ}\left|\right|}^2+{\mathrm{\ϵ}}_N\left|\right|\mathrm{\ξ}\left|\right|+\mathrm{\γ}\left|\right|\mathrm{\ξ}\left|\right|\left[{\left|\right|\tilde{W}\left|\right|}_F({\left|\right|W\left|\right|}_F-{\left|\right|\tilde{W}\left|\right|}_F)\right]\\ \≤-\left|\right|\mathrm{\ξ}\left|\right|\left[k_{\min }\right|\left|\mathrm{\ξ}\right||-{\mathrm{\ϵ}}_N+\mathrm{\γ}{\left|\right|\tilde{W}\left|\right|}_F({\left|\right|\tilde{W}\left|\right|}_F-W_M)]\end{array}---\left(48\right)$

In formula, ε _nfor the upper bound of approximate error, k _minfor the minimal eigenvalue of adjustable parameter matrix k, W _mfor the maximal value element of weight matrix W;

Consider following equation:

$\mathrm{\γ}{\left|\right|\tilde{W}\left|\right|}_F({\left|\right|\tilde{W}\left|\right|}_F-W_M)=\mathrm{\γ}{({\left|\right|\tilde{W}\left|\right|}_F-\frac{1}{2}{W}_M)}^2-\frac{\mathrm{\γ}}{4}{W}_M^2---\left(49\right)$

If make then must have and set up with lower inequality:

$\left|\right|\mathrm{\ξ}\left|\right|>\frac{{\mathrm{\ϵ}}_N+\frac{1}{4}\mathrm{\γ}{W}_M^2}{k_{\min }},$ Or ${\left|\right|\tilde{W}\left|\right|}_F>\frac{1}{2}{W}_M+\sqrt{\frac{W_M^2}{4}+\frac{{\mathrm{\ϵ}}_N}{\mathrm{\γ}}}---\left(50\right)$

If then have uniform ultimate bounded, from || ξ || convergence can obtain: Track In Track precision and neural network approximate error upper bound ε _n, k is relevant for adjustable parameter matrix;

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