CN105242683A - Airship neural network terminal sliding mode track control method - Google Patents

Abstract

The invention provides an airship neural network terminal sliding mode track control method, and aims at the problem of airship track control under the condition of model uncertainty. Firstly error amount is calculated by the given instruction tracks and actual tracks, then the track control law is designed by adopting a terminal sliding mode control method through selecting a terminal sliding mode function, and a neural network is adopted for approximation to the uncertain model of an airship. A system controlled by the method can precisely track the instruction tracks under the condition of model uncertainty so that adaptability is wider and robustness is high, and an effective scheme is provided for engineering realization of airship track control.

Description

A kind of dirigible neural network terminal sliding mode flight tracking control method

Technical field

The present invention relates to a kind of flight control method of aviation field, it provides a kind of neural network TSM control method for dirigible, belongs to automatic control technology field.

Background technology

Dirigible fills buoyance lift gas (as helium, hydrogen etc.) in a kind of dependence to provide uplift, propulsion system and flight control system is relied on to realize handling the floating class aircraft of flight, there is the advantages such as airborne period is long, energy consumption is low, reusable, be particularly suitable as carrying platform, novel electronic information equipment is developed into by carrying multiple useful load, the fields such as land mapping, disaster monitoring, regional early warning, reconnaissance and surveillance can be widely used in, there is significant application value and wide application prospect, the current study hotspot having become aviation field.

Flight tracking control refers to the control to dirigible center of mass motion, usually requires that dirigible flies according to prebriefed pattern, to complete specific aerial mission.The spatial movement of dirigible has non-linear, passage coupling, the feature such as uncertain, and therefore, flight tracking control becomes the difficulties that airship flight controls.Existing document mostly based on inearized model, does not consider non-linear factor and the coupling longitudinally and between horizontal sideway movement to the research of dirigible flight tracking control, only effective near equilibrium state.Sliding-mode control has robustness to Parameter Perturbation and external interference, for dirigible flight tracking control provides a kind of effective means.But sliding formwork controls usually to adopt linear sliding mode, and after system arrives sliding-mode surface, status tracking error asymptotic convergence to zero, cannot at Finite-time convergence.For this problem, patent of invention " a kind of dirigible non-singular terminal sliding formwork flight tracking control method " (patent No.: ZL201410623630.4), propose a kind of non-singular terminal sliding formwork flight tracking control method, making flight tracking control error at Finite-time convergence to zero by choosing terminal sliding mode function, improve response speed and the control accuracy of flight tracking control.But the method does not consider dirigible model uncertain problem, therefore its adaptability and robustness need to be improved further.

Summary of the invention

For solving the problem, the invention provides a kind of dirigible neural network terminal sliding mode flight tracking control method.Neural network has the ability of approaching any complex nonlinear function, can be applied to the modeling problem of uncertain system, can significantly improve the control performance of system.The present invention, on the basis of non-singular terminal sliding formwork flight tracking control method, for dirigible model uncertain problem, adopts neural network to approach the ambiguous model of dirigible.

Flight path control system structured flowchart proposed by the invention as shown in Figure 1.As shown in Figure 1, first according to instruction flight path η _dwith actual flight path η error of calculation amount e; Then choose terminal sliding mode face according to margin of error e, adopt TSM control method design flight tracking control rule; For solving model uncertain problem, according to constructing neural network approaches device, the ambiguous model of On-line Estimation dirigible, thus obtains neural network terminal sliding mode flight tracking control rule.The system controlled by the method can under model condition of uncertainty high precision tracking instruction flight path, compared to patent of invention " a kind of dirigible non-singular terminal sliding formwork flight tracking control method ", there is wider adaptability and strong robustness, for the Project Realization of dirigible flight tracking control provides effective scheme.

A kind of dirigible neural network of the present invention terminal sliding mode flight tracking control method, first by given instruction flight path and actual flight path error of calculation amount, then by choosing terminal sliding mode function, adopt TSM control method design flight tracking control rule, and adopt neural network to approach the ambiguous model of dirigible.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.

A kind of dirigible neural network terminal sliding mode flight tracking control method, its concrete steps are as follows, as shown in Figure 2:

Step one: the given instruction flight path represented with generalized coordinate: η _d=[x _d, y _d, z _d, θ _d, ψ _d, φ _d] ^t;

Described instruction flight path is generalized coordinate η _d=[x _d, y _d, z _d, θ _d, ψ _d, φ _d] ^t, 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, subscript T represents vector or transpose of a matrix.

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

The margin of error between instruction flight path and actual flight path, its computing method are:

e＝η-η _d＝[x-x _d,y-y _d,z-z _d,θ-θ _d,ψ-ψ _d,φ-φ _d] ^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: TSM control rule design: choose terminal sliding mode function, adopts TSM control method design flight tracking control rule, and adopts neural network to approach the ambiguous model of dirigible, calculate flight tracking control amount;

1) mathematical model of dirigible spatial movement is set up

For ease of describing, coordinate system and the kinematic parameter of dirigible spatial movement are defined as follows.As shown in Figure 3, earth axes O is adopted _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, x _gfor centre of buoyancy is to the axial component of center of gravity, y _gfor centre of buoyancy is to the cross component of center of gravity, z _gfor centre of buoyancy is to the vertical component of center of gravity; 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 mathematical model of dirigible spatial movement 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, 0 _{3 × 3}represent the null matrix on 3 × 3 rank, represent the first differential of η, represent the first differential of V,

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

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

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

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

$\mathrm{\τ}=\left[\begin{array}{c}F\cos \mathrm{\μ}\cos \mathrm{\υ}\\ F\sin \mathrm{\μ}\\ F\cos \mathrm{\μ}\sin \mathrm{\υ}\\ {\mathrm{Fsin\υl}}_y\\ {\mathrm{Fcos\υl}}_z-{\mathrm{Fsin\υl}}_x\\ {\mathrm{Fcos\υl}}_z-{\mathrm{Fsin\υl}}_x\end{array}\right]---\left(8\right)$

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

Wherein, G represents the gravity suffered by dirigible, and B represents the buoyancy suffered by dirigible, and F represents the thrust suffered by dirigible,

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 ₁₁)]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; 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), $R\left(\mathrm{\η}\right)=\left[\begin{array}{cc}A& 0_{3\×3}\\ 0_{3\×3}& B\end{array}\right],$

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

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

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

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

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

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

In formula

M _η＝R ^TMR(23)

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

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

$\stackrel{\‾}{u}=R^T\mathrm{\τ}---\left(26\right)$

In practical flight process, M _η, N _ηand G _ηall there is indeterminate, be respectively Δ M _η, Δ N _ηwith Δ G _η; Formula (22) can be written as thus:

$(M_{\mathrm{\η}}+{\mathrm{\ΔM}}_{\mathrm{\η}})\stackrel{\·\·}{\mathrm{\η}}+(N_{\mathrm{\η}}+{\mathrm{\ΔN}}_{\mathrm{\η}})\stackrel{\·}{\mathrm{\η}}+(G_{\mathrm{\η}}+{\mathrm{\ΔG}}_{\mathrm{\η}})=\stackrel{\‾}{u}---\left(27\right)$

Note $\mathrm{\Δ}f=-({\mathrm{\ΔM}}_{\mathrm{\η}}\stackrel{\·\·}{\mathrm{\η}}+{\mathrm{\ΔN}}_{\mathrm{\η}}\stackrel{\·}{\mathrm{\η}}+{\mathrm{\ΔG}}_{\mathrm{\η}}),$ Then formula (27) can be written as:

$M_{\mathrm{\η}}\stackrel{\·\·}{\mathrm{\η}}+N_{\mathrm{\η}}\stackrel{\·}{\mathrm{\η}}+G_{\mathrm{\η}}=\stackrel{\‾}{u}+\mathrm{\Δ}f---\left(28\right)$

With the mathematical model described by formula (28) for controlled device, adopt neural network TSM control method design flight tracking control rule.

2) sliding-mode surface design

Design terminal sliding-mode surface is:

$s=e+\mathrm{\λ}{\stackrel{\·}{e}}^{\mathrm{\κ}}---\left(29\right)$

Wherein, e ₁, e ₂, e ₃, e ₄, e ₅, e ₆be respectively the 1st to the 6th element of vectorial e, s=[s ₁, s ₂, s ₃, s ₄, s ₅, s ₆] ^t, s ₁, s ₂, s ₃, s ₄, s ₅, s ₆be respectively the 1st to the 6th element of vectorial s, λ=diag (λ ₁, λ ₂, λ ₃, λ ₄, λ ₅, λ ₆), λ ₁, λ ₂, λ ₃, λ ₄, λ ₅, λ ₆be respectively the 1st to the 6th element of vectorial λ, diag () represents diagonal matrix, and λ is positive definite matrix, and κ is arithmetic number and meets 1 < κ < 2.

3) design terminal sliding formwork control law:

$\begin{array}{c}\stackrel{\&OverBar;}{u}=M_{\mathrm{\&eta;}}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}}_d+N_{\mathrm{\&eta;}}\stackrel{\&CenterDot;}{\mathrm{\&eta;}}+G_{\mathrm{\&eta;}}-\frac{1}{\mathrm{\&kappa;}}{M}_{\mathrm{\&eta;}}{\mathrm{\&lambda;}}^{-1}diag\left({\stackrel{\&CenterDot;}{e}}^{2-\mathrm{\&kappa;}}\right)-\\ \frac{{\&lsqb;s^T\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}-1}\right)M_{\mathrm{\&eta;}}^{-1}\&rsqb;}^T}{\left|\right|s^T\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}-1}\right)M_{\mathrm{\&eta;}}^{-1}|{|}^2}\&CenterDot;\left|\right|\mathrm{\&Delta;}f\left|\right|\left|\right|s\left|\right|\left|\right|\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}-1}\right)M_{\mathrm{\&eta;}}^{-1}\left|\right|\end{array}---\left(30\right)$

In formula, λ ^-1represent the inverse matrix of λ, represent M _ηinverse matrix, || || represent euclideam norm.

The stability analysis of sliding-mode surface is as follows.

Be defined as follows Lyapunov function

Formula (29) utilizes to formula (31) differential, can obtain:

In formula, represent first differential, s ^trepresent the transposition of s.

Second-order differential asked to formula (1) and utilize formula (22) and formula (29), can obtain:

$\begin{array}{c}\stackrel{\&CenterDot;\&CenterDot;}{e}=\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}}_d=M_{\mathrm{\&eta;}}^{-1}(\stackrel{\&OverBar;}{u}-N_{\mathrm{\&eta;}}\stackrel{\&CenterDot;}{\mathrm{\&eta;}}-G_{\mathrm{\&eta;}})=M_{\mathrm{\&eta;}}^{-1}\&lsqb;-\frac{1}{\mathrm{\&kappa;}}{M}_{\mathrm{\&eta;}}{\mathrm{\&lambda;}}^{-1}diag\left({\stackrel{\&CenterDot;}{e}}^{2-\mathrm{\&kappa;}}\right)\&rsqb;+\\ M_{\mathrm{\&eta;}}^{-1}\&lsqb;-\frac{{\&lsqb;s^T\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}-1}\right)M_{\mathrm{\&eta;}}^{-1}\&rsqb;}^T}{\left|\right|s^T\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}-1}\right)M_{\mathrm{\&eta;}}^{-1}|{|}^2}\&CenterDot;\left|\right|\mathrm{\&Delta;}f\left|\right|\left|\right|s\left|\right|\left|\right|\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}-1}\right)M_{\mathrm{\&eta;}}^{-1}\left|\right|\&rsqb;\end{array}---\left(33\right)$

Formula (33) is substituted into formula (32), can obtain:

Formula (34) namely demonstrate,proves the stability of sliding-mode surface.

Model indeterminate Δ f in formula (30) is in fact not known, and therefore, through type (30) can not accurately provide flight tracking control amount.For this problem, the present invention adopts neural network approximate model indeterminate Δ f.

4) constructing neural network approaches device, design neural network TSM control rule:

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

Input layer: the input variable choosing network is $x=\&lsqb;e^T,{\stackrel{\&CenterDot;}{e}}^T,{\stackrel{\&CenterDot;\&CenterDot;}{e}}^T,{\mathrm{\&eta;}}^T,{\stackrel{\&CenterDot;}{\mathrm{\&eta;}}}^T,{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}}^T\&rsqb;,$ E ^tfor margin of error e transposition, for the first differential of margin of error e transposition, for transposition, the η of the second-order differential of margin of error e ^tfor η transposition, for the first differential of η transposition, for the transposition of the second-order differential of η.

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

$h_j\left(x\right)=\exp \left(\frac{\left|\right|x-c_j|{|}^2}{2b_j^2}\right)---\left(35\right)$

Wherein, c _j=[c _j1, c _j2..., c _jn] be the intermediate value of a jth Gaussian function, c _j1, c _j2..., c _jnbe respectively c _jthe 1st, the 2nd ..., the n-th component, n is the nodes of neural network; b _jfor the standard deviation of a jth Gaussian function, j=1,2 ..., n, n are the nodes of neural network, || || represent euclideam norm.

Output layer: the output of neural network approximator is

$\widehat{\mathrm{\&gamma;}}={\hat{W}}^{T}h\left(x\right)---\left(36\right)$

In formula, for || the estimated value of Δ f||, for optimal weights coefficient vector, h (x)=[h ₁(x), h ₂(x) ..., h _n(x)] ^t, h ₁(x), h ₂(x) ..., h _n(x) be respectively the 1st of vector function, the 2nd ..., the n-th component, n is the nodes of neural network.

Thus, neural network TSM control rule is:

$\begin{array}{c}{u}_N=M_{\mathrm{\&eta;}}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}}_d+N_{\mathrm{\&eta;}}\stackrel{\&CenterDot;}{\mathrm{\&eta;}}+G_{\mathrm{\&eta;}}-{\mathrm{\&kappa;M}}_{\mathrm{\&eta;}}{\mathrm{\&lambda;}}^{-1}diag\left({\stackrel{\&CenterDot;}{e}}^{2-\mathrm{\&kappa;}}\right)\\ -\frac{{\&lsqb;s^T\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}-1}\right)M_{\mathrm{\&eta;}}^{-1}\&rsqb;}^T}{\left|\right|s^T\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}-1}\right)M_{\mathrm{\&eta;}}^{-1}|{|}^2}\&CenterDot;\widehat{\mathrm{\&gamma;}}\left|\right|s\left|\right|\left|\right|\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}-1}\right)M_{\mathrm{\&eta;}}^{-1}\left|\right|\end{array}---\left(37\right)$

(3) advantage and effect:

Compared with prior art, its advantage is in the present invention:

1) the method directly designs based on the non-linear dynamic model of dirigible spatial movement, considers every non-linear factor and the coupling longitudinally and between horizontal sideway movement, overcomes the limitation that inearized model is only suitable for equilibrium state.

2) TSM control makes attitude control error at Finite-time convergence to zero by choosing terminal sliding mode function, overcome the asymptotic convergence problem that traditional sliding formwork controls, there is rapid dynamic response speed, finite time convergence control, steady-state tracking precision advantages of higher.

3) the method adopts neural network to approach the ambiguous model of dirigible, overcomes the flight model uncertain problem in practical flight process, has wider adaptability and strong robustness.

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 flight tracking control method step process flow diagram of the present invention

Fig. 3 is dirigible coordinate system of the present invention and kinematic parameter definition

Fig. 4 is neural network approximator structural drawing of the present invention

Fig. 5 is dirigible flight tracking control result of the present invention

Fig. 6 is dirigible flight tracking control error of the present invention

Fig. 7 is neural network Approaching Results of the present invention

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;

for the first differential of η;

for the second-order differential of η;

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;

Ee=[x _e, y _e, z _e, θ _e, ψ _e, φ _e] ^tfor flight tracking control error, x _e, y _e, z _e, θ _e, ψ _eand φ _ebe respectively the x error of coordinate of flight tracking control, y error of coordinate, z coordinate error, angle of pitch error, crab angle error and roll angle error;

for the first differential of margin of error e;

for the second-order differential of margin of error e; Ss is sliding-mode surface;

U _nu _nfor neural network terminal sliding mode flight tracking control amount;

represent first differential computing.

Embodiment

In order to make object of the present invention, technical scheme and beneficial effect clearly understand, below in conjunction with drawings and Examples, the present invention is further elaborated.It should be noted that specific embodiment described herein only in order to explain the present invention, be not intended to limit the present invention.

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=[(1.5t) m, 200sin (0.005t) m, 10m, 0rad, 0.02rad, 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 margin of error calculates

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

e＝η-η _d＝[x-x _d,y-y _d,z-z _d,θ-θ _d,ψ-ψ _d,φ-φ _d] ^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＝[50m,-100m,8m,0.01rad,0.01rad,0.01rad] ^T。

Initial velocity:

V ₀＝[u ₀,v ₀,w ₀,p ₀,q ₀,r ₀] ^T＝[5m/s,2.5m/s,0m/s,0rad/s,0rad/s,0rad/s] ^T

Step 3: design flight tracking control rule:

1) mathematical model of dirigible spatial movement is set up

The mathematical model of dirigible spatial movement can be expressed as:

$\stackrel{\&CenterDot;}{\mathrm{\&eta;}}=J\left(\mathrm{\&eta;}\right)=\left[\begin{array}{cc}{J}_1& 0_{3\&times;3}\\ 0_{3\&times;3}& J_2\end{array}\right]V---\left(38\right)$

$M\stackrel{\&CenterDot;}{V}=\stackrel{\&OverBar;}{N}+\stackrel{\&OverBar;}{G}+\mathrm{\&tau;}---\left(39\right)$

In formula

$J_1=\left[\begin{array}{ccc}\cos \mathrm{\&psi;}\cos \mathrm{\&theta;}& \cos \mathrm{\&psi;}\sin \mathrm{\&theta;}\sin \mathrm{\&phi;}-\sin \mathrm{\&psi;}\cos \mathrm{\&phi;}& \cos \mathrm{\&psi;}\sin \mathrm{\&theta;}cos\mathrm{\&phi;}+\sin \mathrm{\&psi;}\sin \mathrm{\&phi;}\\ \sin \mathrm{\&psi;}\cos \mathrm{\&theta;}& \sin \mathrm{\&psi;}\sin \mathrm{\&theta;}\sin \mathrm{\&phi;}+cos\mathrm{\&psi;}\cos \mathrm{\&phi;}& \sin \mathrm{\&psi;}\sin \mathrm{\&theta;}\cos \mathrm{\&phi;}-cos\mathrm{\&psi;}\sin \mathrm{\&phi;}\\ -\sin \mathrm{\&theta;}& \cos \mathrm{\&theta;}\sin \mathrm{\&phi;}& \cos \mathrm{\&theta;}\cos \mathrm{\&phi;}\end{array}\right]---\left(40\right)$

$J_2=\left[\begin{array}{ccc}0& cos\mathrm{\&phi;}& -sin\mathrm{\&phi;}\\ 0& \sec \mathrm{\&theta;}sin\mathrm{\&phi;}& sec\mathrm{\&theta;}cos\mathrm{\&phi;}\\ 1& tan\mathrm{\&theta;}sin\mathrm{\&phi;}& tan\mathrm{\&theta;}cos\mathrm{\&phi;}\end{array}\right]---\left(41\right)$

$M=\left[\begin{array}{cccccc}m+m_{11}& 0& 0& 0& {\mathrm{mz}}_G& -{\mathrm{my}}_G\\ 0& m+m_{22}& 0& -{\mathrm{mz}}_G& 0& {\mathrm{mx}}_G\\ 0& 0& m+m_{33}& {\mathrm{my}}_G& -{\mathrm{mx}}_G& 0\\ 0& -{\mathrm{mz}}_G& {\mathrm{my}}_G& I_x+I_{11}& 0& 0\\ {\mathrm{mz}}_G& 0& -{\mathrm{mx}}_G& 0& I_x+I_{22}& 0\\ 0& {\mathrm{mx}}_G& 0& -I_{xz}& 0& I_x+I_{33}\end{array}\right]---\left(42\right)$

$\stackrel{\&OverBar;}{G}=\left[\begin{array}{c}(B-G)sin\mathrm{\&theta;}\\ (G-B)cos\mathrm{\&theta;}sin\mathrm{\&phi;}\\ (G-B)cos\mathrm{\&theta;}\cos \mathrm{\&phi;}\\ y_{G}G\cos \mathrm{\&theta;}cos\mathrm{\&phi;}-z_{G}Gcos\mathrm{\&theta;}sin\mathrm{\&phi;}\\ -x_{G}G\cos \mathrm{\&theta;}\cos \mathrm{\&phi;}-z_{G}Gsin\mathrm{\&theta;}\\ x_{G}G\cos \mathrm{\&theta;}\sin \mathrm{\&phi;}+y_{G}G\sin \mathrm{\&theta;}\end{array}\right]---\left(43\right)$

$\mathrm{\&tau;}=\left[\begin{array}{c}T\cos \mathrm{\&mu;}\cos \mathrm{\&upsi;}\\ T\sin \mathrm{\&mu;}\\ T\cos \mathrm{\&mu;}\sin \mathrm{\&upsi;}\\ {\mathrm{Tsin\&upsi;l}}_y\\ {\mathrm{Tcos\&upsi;l}}_z-{\mathrm{Tsin\&upsi;l}}_x\\ {\mathrm{Tcos\&upsi;l}}_z-{\mathrm{Tsin\&upsi;l}}_x\end{array}\right]---\left(44\right)$

$\stackrel{\&OverBar;}{N}={\&lsqb;N_u,N_v,N_w,N_p,N_q,N_r\&rsqb;}^T---\left(45\right)$

Wherein

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

(46)

+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]

(47)

+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)]

(48)

+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)+

(49)

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

N _q＝[(I _z+I ₃₃)-(I _x+I ₁₁)]pr+I _xyqr-I _yzpq-I _xz(p ²-r ²)

(50)

+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

(51)

+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 (39) is transformed to the expression formula about generalized coordinate η.

Can be obtained by formula (38):

$V=J^{-1}\left(\mathrm{\&eta;}\right)\stackrel{\&CenterDot;}{\mathrm{\&eta;}}=R\left(\mathrm{\&eta;}\right)\stackrel{\&CenterDot;}{\mathrm{\&eta;}}=\left[\begin{array}{cc}A& 0_{3\&times;3}\\ 0_{3\&times;3}& B\end{array}\right]\stackrel{\&CenterDot;}{\mathrm{\&eta;}}---\left(52\right)$

In formula, J ^-1(η) be the inverse matrix of J (η), $R\left(\mathrm{\&eta;}\right)=\left[\begin{array}{cc}A& 0_{3\&times;3}\\ 0_{3\&times;3}& B\end{array}\right],$

$A=\left[\begin{array}{ccc}\cos \mathrm{\&psi;}\cos \mathrm{\&theta;}& \sin \mathrm{\&psi;}\cos \mathrm{\&theta;}& -\sin \mathrm{\&theta;}\\ \cos \mathrm{\&psi;}\sin \mathrm{\&theta;}\sin \mathrm{\&phi;}-\sin \mathrm{\&psi;}\cos \mathrm{\&phi;}& \sin \mathrm{\&psi;}\sin \mathrm{\&theta;}\sin \mathrm{\&phi;}+\cos \mathrm{\&psi;}\cos \mathrm{\&phi;}& \cos \mathrm{\&theta;}\sin \mathrm{\&phi;}\\ \cos \mathrm{\&psi;}\sin \mathrm{\&theta;}\cos \mathrm{\&phi;}+\sin \mathrm{\&psi;}\sin \mathrm{\&phi;}& \sin \mathrm{\&psi;}\sin \mathrm{\&theta;}\cos \mathrm{\&phi;}-\cos \mathrm{\&psi;}\sin \mathrm{\&phi;}& \cos \mathrm{\&theta;}\cos \mathrm{\&phi;}\end{array}\right]---\left(53\right)$

$B=\left[\begin{array}{ccc}0& -sin\mathrm{\&theta;}& 1\\ cos\mathrm{\&phi;}& cos\mathrm{\&theta;}\sin \mathrm{\&phi;}& 0\\ -\sin \mathrm{\&phi;}& cos\mathrm{\&theta;}cos\mathrm{\&phi;}& 0\end{array}\right]---\left(54\right)$

To formula (52) differential, can obtain

$\stackrel{\&CenterDot;}{V}=\stackrel{\&CenterDot;}{R}\stackrel{\&CenterDot;}{\mathrm{\&eta;}}+R\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}---\left(55\right)$

In formula

$\stackrel{\&CenterDot;}{R}=\left[\begin{array}{cc}\stackrel{\&CenterDot;}{A}& 0_{3\&times;3}\\ 0_{3\&times;3}& \stackrel{\&CenterDot;}{B}\end{array}\right]---\left(56\right)$

Formula (55) premultiplication can obtain

$R^{T}M\stackrel{\&CenterDot;}{V}=R^{T}M\stackrel{\&CenterDot;}{R}\stackrel{\&CenterDot;}{\mathrm{\&eta;}}+R^{T}MR\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}---\left(57\right)$

Composite type (39), formula (55) and formula (57) can obtain:

$M_{\mathrm{\&eta;}}\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}+N_{\mathrm{\&eta;}}\stackrel{\&CenterDot;}{\mathrm{\&eta;}}+G_{\mathrm{\&eta;}}=\stackrel{\&OverBar;}{u}---\left(58\right)$

In formula

M _η＝R ^TMR(59)

$N_{\mathrm{\&eta;}}=R^{T}M\stackrel{\&CenterDot;}{R}---\left(60\right)$

$G_{\mathrm{\&eta;}}=-R^T(\stackrel{\&OverBar;}{N}+\stackrel{\&OverBar;}{G})---\left(61\right)$

$\stackrel{\&OverBar;}{u}=R^T\mathrm{\&tau;}---\left(62\right)$

Consider model indeterminate, formula (58) can be written as:

$M_{\mathrm{\&eta;}}\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}+N_{\mathrm{\&eta;}}\stackrel{\&CenterDot;}{\mathrm{\&eta;}}+G_{\mathrm{\&eta;}}=\stackrel{\&OverBar;}{u}+\mathrm{\&Delta;}f---\left(63\right)$

In formula, wherein, Δ M _η, Δ N _ηwith Δ G _ηbe respectively M _η, N _ηand G _ηunknown term and indeterminate.

Dirigible parameter in the present embodiment is in table 2.

Table 2 dirigible parameter

2) sliding-mode surface design

Design terminal sliding-mode surface is:

$s=e+\mathrm{\&lambda;}{\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}}---\left(64\right)$

Wherein, diag (2,2,2,2,2,2), diag () represents diagonal matrix, κ=5/3.

3) design terminal sliding formwork control law:

$\begin{array}{c}\stackrel{\&OverBar;}{u}=M_{\mathrm{\&eta;}}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}}_d+N_{\mathrm{\&eta;}}\stackrel{\&CenterDot;}{\mathrm{\&eta;}}+G_{\mathrm{\&eta;}}-\frac{1}{\mathrm{\&kappa;}}{M}_{\mathrm{\&eta;}}{\mathrm{\&lambda;}}^{-1}diag\left({\stackrel{\&CenterDot;}{e}}^{2-\mathrm{\&kappa;}}\right)-\\ \frac{{\&lsqb;s^T\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}-1}\right)M_{\mathrm{\&eta;}}^{-1}\&rsqb;}^T}{\left|\right|s^T\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}-1}\right)M_{\mathrm{\&eta;}}^{-1}|{|}^2}\&CenterDot;\left|\right|\mathrm{\&Delta;}f\left|\right|\left|\right|s\left|\right|\left|\right|\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}-1}\right)M_{\mathrm{\&eta;}}^{-1}\left|\right|\end{array}---\left(65\right)$

4) constructing neural network approaches device, design neural network TSM control rule:

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

Input layer: the input variable choosing network is $x=\&lsqb;e^T,{\stackrel{\&CenterDot;}{e}}^T,{\stackrel{\&CenterDot;\&CenterDot;}{e}}^T,{\mathrm{\&eta;}}^T,{\stackrel{\&CenterDot;}{\mathrm{\&eta;}}}^T,{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}}^T\&rsqb;.$

Hidden layer: choose the basis function of Gaussian function as hidden node, nodes n=7,

$h_j\left(x\right)=\exp \left(\frac{\left|\right|x-c_j|{|}^2}{2b_j^2}\right)---\left(66\right)$

Wherein, $c=\left[\begin{array}{ccccccc}-3& -2& -1& 0& 1& 2& 3\\ -3& -2& -1& 0& 1& 2& 3\\ -3& -2& -1& 0& 1& 2& 3\\ -3& -2& -1& 0& 1& 2& 3\\ -3& -2& -1& 0& 1& 2& 3\\ -3& -2& -1& 0& 1& 2& 3\end{array}\right],$ b _j＝0.12。

Output layer: the output of neural network approximator is

$\widehat{\mathrm{\&gamma;}}={\hat{W}}^{T}h\left(x\right)---\left(67\right)$

Wherein, be taken as that element is 0.01, dimension is the vector of 42.

Neural network TSM control rule is:

$\begin{array}{c}{u}_N=M_{\mathrm{\&eta;}}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}}_d+N_{\mathrm{\&eta;}}\stackrel{\&CenterDot;}{\mathrm{\&eta;}}+G_{\mathrm{\&eta;}}-{\mathrm{\&kappa;M}}_{\mathrm{\&eta;}}{\mathrm{\&lambda;}}^{-1}diag\left({\stackrel{\&CenterDot;}{e}}^{2-\mathrm{\&kappa;}}\right)\\ -\frac{{\&lsqb;s^T\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}-1}\right)M_{\mathrm{\&eta;}}^{-1}\&rsqb;}^T}{\left|\right|s^T\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}-1}\right)M_{\mathrm{\&eta;}}^{-1}|{|}^2}\&CenterDot;\widehat{\mathrm{\&gamma;}}\left|\right|s\left|\right|\left|\right|\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}-1}\right)M_{\mathrm{\&eta;}}^{-1}\left|\right|\end{array}---\left(68\right)$

Dirigible Three-dimensional Track tracking results in embodiment as shown in Figure 5-Figure 7.Fig. 5 gives dirigible flight tracking control result, can be obtained by Fig. 5: dirigible is by initial position (50m,-100m, 8m) set out and to reach home position (450m through 300s, 200m, 10m), can accurate trace command flight path, demonstrate the validity of flight tracking control method proposed by the invention; Fig. 6 gives flight tracking control error, can be obtained by Fig. 6: dirigible with zero steady-state error trace command flight path, can have higher control accuracy.Fig. 7 gives neural network Approaching Results, can be obtained by Fig. 7, and neural network approximator proposed by the invention can approach the ambiguous model of dirigible preferably.

Claims (1)

1. a dirigible neural network terminal sliding mode flight tracking control method, is characterized in that, comprise the following steps:

Step one: the given instruction flight path represented with generalized coordinate: η _d=[x _d, y _d, z _d, θ _d, ψ _d, φ _d] ^t;

Described instruction flight path is generalized coordinate η _d=[x _d, y _d, z _d, θ _d, ψ _d, φ _d] ^t, 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, subscript T represents vector or transpose of a matrix;

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

The margin of error between instruction flight path and actual flight path, its computing method are:

e＝η-η _d＝[x-x _d,y-y _d,z-z _d,θ-θ _d,ψ-ψ _d,φ-φ _d] ^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: TSM control rule design: choose terminal sliding mode function, adopts TSM control method design flight tracking control rule, and adopts neural network to approach the ambiguous model of dirigible, calculate flight tracking control amount;

1) mathematical model of dirigible spatial movement 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, x _gfor centre of buoyancy is to the axial component of center of gravity, y _gfor centre of buoyancy is to the cross component of center of gravity, z _gfor centre of buoyancy is to the vertical component of center of gravity; 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 mathematical model of dirigible spatial movement is described below:

$\stackrel{\&CenterDot;}{\mathrm{\&eta;}}=J\left(\mathrm{\&eta;}\right)=\left[\begin{array}{cc}{J}_1& 0_{3\&times;3}\\ 0_{3\&times;3}& J_2\end{array}\right]V---\left(2\right)$

$M\stackrel{\&CenterDot;}{V}=\stackrel{\&OverBar;}{N}+\stackrel{\&OverBar;}{G}+\mathrm{\&tau;}---\left(3\right)$

In formula, 0 _{3 × 3}represent the null matrix on 3 × 3 rank, represent the first differential of η, represent the first differential of V,

$J_1=\left[\begin{array}{ccc}\cos \mathrm{\&psi;}\cos \mathrm{\&theta;}& \cos \mathrm{\&psi;}\sin \mathrm{\&theta;}\sin \mathrm{\&phi;}-\sin \mathrm{\&psi;}\cos \mathrm{\&phi;}& \cos \mathrm{\&psi;}\sin \mathrm{\&theta;}\cos \mathrm{\&phi;}+\sin \mathrm{\&psi;}\sin \mathrm{\&phi;}\\ \sin \mathrm{\&psi;}\cos \mathrm{\&theta;}& \sin \mathrm{\&psi;}\sin \mathrm{\&theta;}\sin \mathrm{\&phi;}+\cos \mathrm{\&psi;}\cos \mathrm{\&phi;}& \sin \mathrm{\&psi;}\sin \mathrm{\&theta;}\cos \mathrm{\&phi;}-\cos \mathrm{\&psi;}\sin \mathrm{\&phi;}\\ -\sin \mathrm{\&theta;}& \cos \mathrm{\&theta;}\sin \mathrm{\&phi;}& \cos \mathrm{\&theta;}\cos \mathrm{\&phi;}\end{array}\right]---\left(4\right)$

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

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

$\stackrel{\&OverBar;}{G}=\left[\begin{array}{c}(B-G)sin\mathrm{\&theta;}\\ (G-B)cos\mathrm{\&theta;}\sin \mathrm{\&phi;}\\ (G-B)cos\mathrm{\&theta;}\cos \mathrm{\&phi;}\\ y_{G}G\cos \mathrm{\&theta;}cos\mathrm{\&phi;}-z_{G}Gcos\mathrm{\&theta;}sin\mathrm{\&phi;}\\ -x_{G}G\cos \mathrm{\&theta;}\cos \mathrm{\&phi;}-z_{G}Gsin\mathrm{\&theta;}\\ x_{G}G\cos \mathrm{\&theta;}\sin \mathrm{\&phi;}+y_{G}G\sin \mathrm{\&theta;}\end{array}\right]---\left(7\right)$

$\mathrm{\&tau;}=\left[\begin{array}{c}Fcos\mathrm{\&mu;}cos\mathrm{\&upsi;}\\ F\sin \mathrm{\&mu;}\\ Fcos\mathrm{\&mu;}sin\mathrm{\&upsi;}\\ {\mathrm{Fsin\&upsi;l}}_y\\ {\mathrm{Fcos\&upsi;l}}_z-{\mathrm{Fsin\&upsi;l}}_x\\ {\mathrm{Fcos\&upsi;l}}_z-{\mathrm{Fsin\&upsi;l}}_x\end{array}\right]---\left(8\right)$

$\stackrel{\&OverBar;}{N}={\&lsqb;N_u,N_v,N_w,N_p,N_q,N_r\&rsqb;}^T---\left(9\right)$

Wherein, G represents the gravity suffered by dirigible, and B represents the buoyancy suffered by dirigible, and F represents the thrust suffered by dirigible,

$\begin{array}{c}{N}_u=(m+m_{22})vr-(m+m_{33})wq+m\&lsqb;x_G(p^2+r^2)-y_{G}pq-z_{G}pr\&rsqb;\\ +{\mathrm{QV}}^{2/3}(-C_X\cos \mathrm{\&alpha;}\cos \mathrm{\&beta;}+C_Y\cos \mathrm{\&alpha;}\sin \mathrm{\&beta;}+C_Z\sin \mathrm{\&alpha;})\end{array}---\left(10\right)$

$\begin{array}{c}{N}_v=(m+m_{33})wp-(m+m_{11})ur-m\&lsqb;x_{G}pq-y_G(p^2+r^2)+z_{G}qr\&rsqb;\\ +{\mathrm{QV}}^{2/3}(C_X\sin \mathrm{\&beta;}+C_Y\cos \mathrm{\&beta;})\end{array}---\left(11\right)$

$\begin{array}{c}{N}_w=(m+m_{22})vp-(m+m_{11})uq-m\&lsqb;x_{G}pr+y_{G}qr-z_G(p^2+q^2)\&rsqb;\\ +{\mathrm{QV}}^{2/3}(-C_X\sin \mathrm{\&alpha;}\sin \mathrm{\&beta;}+C_Y\sin \mathrm{\&alpha;}\cos \mathrm{\&beta;}-C_Z\cos \mathrm{\&alpha;})\end{array}---\left(12\right)$

$\begin{array}{c}{N}_p=\&lsqb;(I_y+I_{22})-(I_z+I_{33})\&rsqb;qr+I_{xz}pq-I_{xy}pr-I_{yz}(r^2-q^2)+\\ \&lsqb;{\mathrm{mz}}_G(ur-wp)+y_G(uq-vp)\&rsqb;+{\mathrm{QVC}}_l\end{array}---\left(13\right)$

$\begin{array}{c}{N}_q=\&lsqb;(I_z+I_{33})-(I_x+I_{11})\&rsqb;pr+I_{xy}qr-I_{yz}pq-I_{xz}(p^2-r^2)\\ +m\&lsqb;x_G(vp-uq)-z_G(wp-vr)\&rsqb;+{\mathrm{QVC}}_m\end{array}---\left(14\right)$

$\begin{array}{c}{N}_r=\&lsqb;(I_y+I_{22})-(I_x+I_{11})\&rsqb;pq-I_{xz}qr-I_{xy}(q^2-p^2)+I_{yz}pr\\ +m\&lsqb;y_G(wq-vr)-x_G(ur-wp)\&rsqb;+{\mathrm{QVC}}_n\end{array}---\left(15\right)$

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 (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):

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

J in formula ^-1(η) be the inverse matrix of J (η), $R\left(\mathrm{\&eta;}\right)=\left[\begin{array}{cc}A& 0_{3\&times;3}\\ 0_{3\&times;3}& B\end{array}\right];$

$A=\left[\begin{array}{ccc}\cos \mathrm{\&psi;}\cos \mathrm{\&theta;}& \sin \mathrm{\&psi;}\cos \mathrm{\&theta;}& -\sin \mathrm{\&theta;}\\ \cos \mathrm{\&psi;}\sin \mathrm{\&theta;}\sin \mathrm{\&phi;}-\sin \mathrm{\&psi;}\cos \mathrm{\&phi;}& \sin \mathrm{\&psi;}\sin \mathrm{\&theta;}\sin \mathrm{\&phi;}+\cos \mathrm{\&psi;}\cos \mathrm{\&phi;}& \cos \mathrm{\&theta;}\sin \mathrm{\&phi;}\\ \cos \mathrm{\&psi;}\sin \mathrm{\&theta;}\cos \mathrm{\&phi;}+\sin \mathrm{\&psi;}\sin \mathrm{\&phi;}& \sin \mathrm{\&psi;}\sin \mathrm{\&theta;}\cos \mathrm{\&phi;}-\cos \mathrm{\&psi;}\sin \mathrm{\&phi;}& \cos \mathrm{\&theta;}\cos \mathrm{\&phi;}\end{array}\right]---\left(17\right)$

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

To formula (16) differential, can obtain

$\stackrel{\&CenterDot;}{V}=\stackrel{\&CenterDot;}{R}\stackrel{\&CenterDot;}{\mathrm{\&eta;}}+R\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}---\left(19\right)$

In formula

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

Formula (19) premultiplication can obtain

$R^{T}M\stackrel{\&CenterDot;}{V}=R^{T}M\stackrel{\&CenterDot;}{R}\stackrel{\&CenterDot;}{\mathrm{\&eta;}}+R^{T}MR\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}---\left(21\right)$

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

$M_{\mathrm{\&eta;}}\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}+N_{\mathrm{\&eta;}}\stackrel{\&CenterDot;}{\mathrm{\&eta;}}+G_{\mathrm{\&eta;}}=\stackrel{\&OverBar;}{u}---\left(22\right)$

In formula

M _η＝R ^TMR(23)

$N_{\mathrm{\&eta;}}=R^{T}M\stackrel{\&CenterDot;}{R}---\left(24\right)$

$G_{\mathrm{\&eta;}}=-R^T(\stackrel{\&OverBar;}{N}+\stackrel{\&OverBar;}{G})---\left(25\right)$

$\stackrel{\&OverBar;}{u}=R^T\mathrm{\&tau;}---\left(26\right)$

In practical flight process, M _η, N _ηand G _ηall there is indeterminate, be respectively Δ M _η, Δ N _ηwith Δ G _η; Formula (22) can be written as thus:

$(M_{\mathrm{\&eta;}}+{\mathrm{\&Delta;M}}_{\mathrm{\&eta;}})\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}+(N_{\mathrm{\&eta;}}+{\mathrm{\&Delta;N}}_{\mathrm{\&eta;}})\stackrel{\&CenterDot;}{\mathrm{\&eta;}}+(G_{\mathrm{\&eta;}}+{\mathrm{\&Delta;G}}_{\mathrm{\&eta;}})=\stackrel{\&OverBar;}{u}---\left(27\right)$

Note $\mathrm{\&Delta;}f=-({\mathrm{\&Delta;M}}_{\mathrm{\&eta;}}\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}+{\mathrm{\&Delta;N}}_{\mathrm{\&eta;}}\stackrel{\&CenterDot;}{\mathrm{\&eta;}}+{\mathrm{\&Delta;G}}_{\mathrm{\&eta;}}),$ Then formula (27) can be written as:

$M_{\mathrm{\&eta;}}\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}+N_{\mathrm{\&eta;}}\stackrel{\&CenterDot;}{\mathrm{\&eta;}}+G_{\mathrm{\&eta;}}=\stackrel{\&OverBar;}{u}+\mathrm{\&Delta;}f---\left(28\right)$

With the mathematical model described by formula (28) for controlled device, adopt neural network TSM control method design flight tracking control rule;

2) sliding-mode surface design

Design terminal sliding-mode surface is:

$s=e+\mathrm{\&lambda;}{\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}}---\left(29\right)$

Wherein, e=[e ₁, e ₂, e ₃, e ₄, e ₅, e ₆] ^t, e ₁, e ₂, e ₃, e ₄, e ₅, e ₆be respectively the 1st to the 6th element of vectorial e, s=[s ₁, s ₂, s ₃, s ₄, s ₅, s ₆] ^t, s ₁, s ₂, s ₃, s ₄, s ₅, s ₆be respectively the 1st to the 6th element of vectorial s, λ=diag (λ ₁, λ ₂, λ ₃, λ ₄, λ ₅, λ ₆), λ ₁, λ ₂, λ ₃, λ ₄, λ ₅, λ ₆be respectively the 1st to the 6th element of vectorial λ, diag () represents diagonal matrix, and λ is positive definite matrix, and κ is arithmetic number and meets 1 < κ < 2;

3) design terminal sliding formwork control law:

$\begin{array}{c}\stackrel{\&OverBar;}{u}=M_{\mathrm{\&eta;}}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}}_d+N_{\mathrm{\&eta;}}\stackrel{\&CenterDot;}{\mathrm{\&eta;}}+G_{\mathrm{\&eta;}}-\frac{1}{\mathrm{\&kappa;}}{M}_{\mathrm{\&eta;}}{\mathrm{\&lambda;}}^{-1}diag\left({\stackrel{\&CenterDot;}{e}}^{2-\mathrm{\&kappa;}}\right)-\\ \frac{{\&lsqb;s^T\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{2-\mathrm{\&kappa;}}\right)M_{\mathrm{\&eta;}}^{-1}\&rsqb;}^T}{\left|\right|s^T\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{2-\mathrm{\&kappa;}}\right)M_{\mathrm{\&eta;}}^{-1}|{|}^2}\&CenterDot;\left|\right|\mathrm{\&Delta;}f\left|\right|\left|\right|s\left|\right|\left|\right|\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{2-\mathrm{\&kappa;}}\right)M_{\mathrm{\&eta;}}^{-1}\left|\right|\end{array}---\left(30\right)$

In formula, λ ^-1represent the inverse matrix of λ, represent M _ηinverse matrix, || || represent euclideam norm;

The stability analysis of sliding-mode surface is as follows;

Be defined as follows Lyapunov function

Formula (29) utilizes to formula (31) differential, can obtain:

In formula, represent first differential, s ^trepresent the transposition of s;

Second-order differential asked to formula (1) and utilize formula (22) and formula (29), can obtain:

$\begin{array}{c}\stackrel{\&CenterDot;\&CenterDot;}{e}=\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}}_d=M_{\mathrm{\&eta;}}^{-1}(\stackrel{\&OverBar;}{u}-N_{\mathrm{\&eta;}}\stackrel{\&CenterDot;}{\mathrm{\&eta;}}-G_{\mathrm{\&eta;}})=M_{\mathrm{\&eta;}}^{-1}\&lsqb;-\frac{1}{\mathrm{\&kappa;}}{M}_{\mathrm{\&eta;}}{\mathrm{\&lambda;}}^{-1}diag\left({\stackrel{\&CenterDot;}{e}}^{2-\mathrm{\&kappa;}}\right)\&rsqb;+\\ {\stackrel{\&CenterDot;\&CenterDot;}{M}}_{\mathrm{\&eta;}}^{-1}\&lsqb;-\frac{{\&lsqb;s^T\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}-1}\right)M_{\mathrm{\&eta;}}^{-1}\&rsqb;}^T}{\left|\right|s^T\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}-1}\right)M_{\mathrm{\&eta;}}^{-1}|{|}^2}\&CenterDot;\left|\right|\mathrm{\&Delta;}f\left|\right|\left|\right|s\left|\right|\left|\right|\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{\mathrm{\&kappa;}-1}\right)M_{\mathrm{\&eta;}}^{-1}\left|\right|\&rsqb;\end{array}---\left(33\right)$

Formula (33) is substituted into formula (32), can obtain:

Formula (34) namely demonstrate,proves the stability of sliding-mode surface;

For the model indeterminate Δ f in formula (30), adopt neural network approximate model indeterminate Δ f;

4) constructing neural network approaches device, design neural network TSM control rule:

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

Input layer: the input variable choosing network is e ^tfor margin of error e transposition, for the first differential of margin of error e transposition, for transposition, the η of the second-order differential of margin of error e ^tfor η transposition, for the first differential of η transposition, for the transposition of the second-order differential of η;

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

$h_j\left(x\right)=\exp \left(\frac{\left|\right|x-c_j|{|}^2}{2b_j^2}\right)---\left(35\right)$

Wherein, c _j=[c _j1, c _j2..., c _jn] be the intermediate value of a jth Gaussian function, c _j1, c _j2..., c _jnbe respectively c _jthe 1st, the 2nd ..., the n-th component, n is the nodes of neural network; b _jfor the standard deviation of a jth Gaussian function, j=1,2 ..., n, n are the nodes of neural network, || || represent euclideam norm;

Output layer: the output of neural network approximator is

$\widehat{\mathrm{\&gamma;}}={\hat{W}}^{T}h\left(x\right)---\left(36\right)$

In formula, for || Δ f| estimated value, for optimal weights coefficient vector, h (x)=[h ₁(x), h ₂(x) ..., h _n(x)] ^t, h ₁(x), h ₂(x) ..., h _n(x) be respectively the 1st of vector function, the 2nd ..., the n-th component, n is the nodes of neural network;

Thus, neural network TSM control rule is:

$\begin{array}{c}{u}_N{M}_{\mathrm{\&eta;}}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&eta;}}}_d+N_{\mathrm{\&eta;}}\stackrel{\&CenterDot;}{\mathrm{\&eta;}}+G_{\mathrm{\&eta;}}-{\mathrm{\&kappa;M}}_{\mathrm{\&eta;}}{\mathrm{\&lambda;}}^{-1}diag\left({\stackrel{\&CenterDot;}{e}}^{2-\mathrm{\&kappa;}}\right)\\ -\frac{{\&lsqb;s^T\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{2-\mathrm{\&kappa;}}\right)M_{\mathrm{\&eta;}}^{-1}\&rsqb;}^T}{\left|\right|s^T\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{2-\mathrm{\&kappa;}}\right)M_{\mathrm{\&eta;}}^{-1}|{|}^2}\&CenterDot;\widehat{\mathrm{\&gamma;}}\left|\right|s\left|\right|\left|\right|\mathrm{\&lambda;}diag\left({\stackrel{\&CenterDot;}{e}}^{2-\mathrm{\&kappa;}}\right)M_{\mathrm{\&eta;}}^{-1}\left|\right|\end{array}---\left(37\right).$