CN102073276B - Method for controlling flexible structure and self-adaptive changing structure by radial basis function (RBF) neural network - Google Patents

Abstract

The invention provides a method for controlling a flexible structure and a self-adaptive changing structure by a radial basis function (RBF) neural network, belonging to the field of aviation. The method aims at solving the problem that the existing method can not preferably solve the conflict between the shake of a solar sailboard and the high-precision control target of an attitude control system. The method comprises the following steps: an E1 input forming module is used for converting an inputted expected satellite attitude angle theta d into a response uE1, and outputting the response uE1 to a nominal system and a flexible spacecraft; the nominal system is used for outputting expected satellite attitude information xm (t), and the flexible spacecraft is used for outputting practical satellite attitude information x (t) to obtain an error e (t) by comparing the xm (t) with the x (t); a sliding film face control module is used for obtaining a proper sliding film face s according to the error e (t), and transmitting the s to the RBF neural network and a self-adaptive locoregional control module; the self-adaptive locoregional control module is used for outputting a self-adaptive locoregional control u* to the RBF neural network; and the RBF neural network is used for obtaining and adjusting a locoregional control un and an adding result between the un and the uE1 according to the s and the u* to control the satellite attitude of the flexible spacecraft to achieve an expected value.

Description

Adopt the oar property structure adaptive variable structure control method of RBF neural network

Technical field

The present invention relates to a kind of oar property structure adaptive variable structure control method of the RBF of employing neural network, belong to field of aerospace.

Background technology

Owing to have the simplification of spacecraft in modeling process and the complicacy of spacecraft working environment of flexible appendage; The spacecraft uncertain problem is very outstanding; Mainly show: (1) many flexible bodies spacecraft itself is a distributed parameter system; System state is the function in time and space, and infinite a plurality of degree of freedom is arranged.Be to come approximate description flexible body dynamics with the hypothesis modal method in the engineering design, the direct coupling between a plurality of flexible bodies is not considered in the modeling; (2) in whole flight course, because the consumption of fuel, phenomenons such as wearing out of components and parts are added in the variation of spacecraft mass and barycenter, all can cause model parameter to change, and have constituted the uncertainty of model parameter; Can receive the effect of various interference when (3) spacecraft orbits: the earth and lunar gravitation can be influential to spacecraft orbit and attitude; Sun optical pressure acts on solar array can produce disturbance torque; Rapid variation takes place in spacecraft temperature sometimes, thereby excites elasticity mode.The tradition control method is difficult to overcome the uncertainty of system.

The solar energy sailboard vibrations bring harm can for the satellite platform attitude control system, and existing method can not well solve the solar energy sailboard vibrations and to the contradiction between the High Accuracy Control target of attitude control system.

Summary of the invention

The present invention seeks to well to solve the solar energy sailboard vibrations and, a kind of oar property structure adaptive variable structure control method of the RBF of employing neural network is provided to the contradiction between the High Accuracy Control target of attitude control system in order to solve existing method.The RBF neural network is radial basis function neural network (Radical Basis Function).

The oar property structure adaptive variable structure control method of employing RBF neural network according to the invention; The related controller of this control method comprises EI input forming module, nominal system, sliding-mode surface control module, RBF neural network and adaptive control rate module, and this control method is:

EI input forming module receives the expectation attitude of satellite angle θ of oar property spacecraft _d, the expectation attitude of satellite angle θ that EI input forming module will be imported _dConvert response u into _EI, said response u _EIExport to nominal system, the nominal output expectation attitude of satellite information x of system _m(t), simultaneously, said response u _EIAlso export to oar property spacecraft, oar property spacecraft output real satellite attitude information x (t), expectation attitude of satellite information x _m(t) make comparisons and obtain its error e (t) with real satellite attitude information x (t); The sliding-mode surface control module is obtained suitable sliding-mode surface s according to error e (t); The sliding-mode surface s of sliding-mode surface control module output exports to RBF neural network and adaptive control rate module, adaptive control rate module output adaptive control rate u simultaneously ^*Give the RBF neural network, the RBF neural network is according to sliding-mode surface s and adaptive control rate u ^*Obtain adjustment control rate u _n, said adjustment control rate u _nWith said response u _EIThe attitude of satellite that the u as a result of addition is used to control oar property spacecraft reaches expectation value.

In the control method of the present invention, the vector equation of real satellite attitude information x (t) is expressed as:

$\stackrel{\·}{x}=\mathrm{Ax}+\mathrm{Bu}\left(t\right)+{\mathrm{Bu}}_{\mathrm{is}}+\mathrm{Bd}\left(t\right),$

Wherein, A, B are a coefficient once,

The expectation attitude of satellite information x of name system output _m(t) vector equation is expressed as:

${\stackrel{\·}{x}}_m=A_m{x}_m+B_m{u}_{\mathrm{is}}.$

In the control method of the present invention, error e (t) is obtained by following formula:

e(t)＝x(t)-x _m(t)。

In the control method of the present invention, sliding-mode surface s presses following formulae design:

$s\left(t\right)=\mathrm{\λ\; e}+\frac{d}{\mathrm{Dt}}e,$ λ＞0 wherein.

In the control method of the present invention, adaptive control rate u ^*Obtain by following formula:

$u^{*}\left(t\right)=-B^{-1}((A_m+\mathrm{\ΔA})e-\stackrel{\·}{e}+\mathrm{\Δ}{\mathrm{Ax}}_m$

$+\mathrm{\ΔB}{u}_{\mathrm{is}}+\stackrel{\·}{s}\left(t\right)+\mathrm{\λs}\left(t\right))-d\left(t\right).$

In the control method of the present invention, said RBF neural network is being obtained adjustment control rate u _nProcess in, according to the adaptive control rate u of adaptive control rate module output ^*Obtain the optimal weights coefficient W of RBF neural network ^*, make this coefficient and satisfy equation: max|u ^*(x, W ^*)-u _n|＜ε.

In the control method of the present invention, said RBF neural network is obtained adjustment control rate u by following formula _n:

u _n＝u ^*+ε＝W ^*Tφ+ε。

Advantage of the present invention: the equation of motion of flexible spacecraft, the vibration equation of flexible structure intercouple, and the present invention goes to zero systematic error through using RBF neural network adaptive variable structure control method, suppresses the vibration of solar energy sailboard.Control method of the present invention is simple.The present invention adopts RBF neural network (radial basis function neural network); Radial basis function neural network is a kind of neural network of feed forward type efficiently; It has optimal approximation performance and the global optimum's characteristic that other feedforward networks do not have, and simple in structure, and training speed is fast.Simultaneously, it also is a kind ofly can be widely used in the neural network model that pattern-recognition, nonlinear function such as approach at the field.

Description of drawings

Fig. 1 is the structural representation of flexible spacecraft;

Fig. 2 is a RBF neural network adaptive variable structure control block diagram.

Embodiment

Embodiment one: this embodiment is described below in conjunction with Fig. 1 and Fig. 2.

Following Fig. 1 is the approximate spacecraft model that has flexible appendage, and it is the Rigid Base of b by radius and even beam type flexible appendage constitutes, and flexible beam is for being symmetrically distributed.OXZ and oxz are respectively inertial coordinates system and body coordinate system among the figure, the dead in line of oz and undeformed flexible appendage, and its initial point is positioned at the junction of flexible appendage and Rigid Base, and what be used to describe the relation that relatively rotates between coordinate system is attitude angle.M is most advanced and sophisticated quality, and T is expressed as external control moment, wherein arbitrfary point [0 l] be deformed into w (x, t).

In order to simplify the kinetic model of the flexible spacecraft of deriving; Make following hypothesis: (1) flexible beam model hypothesis is Euler-Bernoulli beam (Euler's beam: ignored detrusion and moment of inertia; Think that initial cutting plane perpendicular to neutral axis still remains the plane perpendicular to neutral axis (Kirchhoff hypothesis) when being out of shape, think that promptly the rotation in cross section equals the sag curve tangent slope.Be applicable to that depth of beam is under the span situation), (x t), can only consider transverse bending vibration like this and ignores lengthwise movement to be defined as w with respect to the distortion of the flexible appendage of oxz coordinate system; (2) ignore action of gravity to simplify kinetic model.

Suppose any 1 x on the semi-girder, z is the vector of unit length of coordinate system oxz, and X, Z are the vector of unit length of coordinate system OXY, then the vectorial R from initial point to x _xCan be expressed as:

R _x＝xx+w(x，t)z (1)

Elastic displacement on the semi-girder can be expressed as:

$w(x,t)=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{\mathrm{\φ}}_i\left(x\right)q_i\left(t\right)---\left(2\right)$

φ _i(x) be the i first order mode function of flexible beam, q _i(t) be the broad sense modal coordinate.

Utilize Hamilton principle (Hamilton principle: Lagrangian function

From moment t ₁To t ₂The variation of time integral equal zero.The conservative mechanical system that receives ideal constraint is from moment t ₁A certain position shape transfer to t constantly ₂Every motion of another one shape in, the actual motion that takes place is got the definite integral of Lagrangian function on this time interval of system and is stayed value, most minimalization.) equation of derivation system, then the kinetic energy of system can be expressed as

$T_e=\frac{1}{2}{I}_r{\stackrel{\·}{\mathrm{\θ}}}^2+\frac{1}{2}{\∫}_0^l\mathrm{\ρ}{[\stackrel{\·}{w}+(b+x)\stackrel{\·}{\mathrm{\θ}}]}^2\mathrm{dx}$

$+\frac{1}{2}\mathrm{\ρ}{\stackrel{\·}{\mathrm{\θ}}}^2{\∫}_0^l{w}^2\mathrm{dx}+\frac{1}{2}m{[\stackrel{\·}{w}(l,t)+(b+l)\stackrel{\·}{\mathrm{\θ}}]}^2+\frac{1}{2}{\mathrm{mw}}^2(l,t){\stackrel{\·}{\mathrm{\θ}}}^2---\left(3\right)$

I in the formula _rBe the moment of inertia of rigid body centrosome, l is the length of flexible appendage, and ρ is the density of flexible appendage.Next we introduce following variable I, M, Q (I, M, Q introduce for convenience of calculation):

$I=I_r+m{(b+l)}^2+{\∫}_0^l\mathrm{\ρ}{(b+x)}^2\mathrm{dx}$

$M={\∫}_0^l\mathrm{\ρ}{\mathrm{\Φ}}^T\left(x\right)\mathrm{\Φ}\left(x\right)\mathrm{dx}+m{\mathrm{\Φ}}^T\left(l\right)\mathrm{\Φ}\left(l\right)---\left(4\right)$

$Q={\∫}_0^l\mathrm{\ρ}(b+x)\mathrm{\Φdx}+m(b+l)\mathrm{\Φ}\left(l\right)$

Formula (4) substitution formula (3) is got:

$T=\frac{1}{2}J{\stackrel{\·}{\mathrm{\θ}}}^2+\frac{1}{2}{\stackrel{\·}{\mathrm{\θ}}}^2{q}^T\mathrm{Mq}+\stackrel{\·}{\mathrm{\θ}}Q\stackrel{\·}{q}+\frac{1}{2}{\stackrel{\·}{q}}^{T}M\stackrel{\·}{q}---\left(5\right)$

The total work W of system can be expressed as:

W＝W _h+W _s(6)

W wherein _sBe flexible structure institute work:

$W_s=-\frac{1}{2}{\∫}_0^l\mathrm{EI}{\left(\frac{{\∂}^{2}w}{\∂x^2}\right)}^2\mathrm{dx}=-\frac{1}{2}{q}^T\mathrm{Kq}---\left(7\right)$

In the formula

W _hBy control moment is done work:

W _h＝Tθ(8)

So the total work W that system is done is:

$W=\mathrm{T\θ}-\frac{1}{2}{q}^T\mathrm{Kq}---\left(9\right)$

And then push away to such an extent that the equation of motion of spacecraft of flexible appendage is:

$I\stackrel{\·\·}{\mathrm{\θ}}+\stackrel{\·\·}{\mathrm{\θ}}{q}^T\mathrm{Mq}+2\stackrel{\·}{\mathrm{\θ}}{q}^T\mathrm{Mq}+Q\stackrel{\·\·}{q}=T---\left(10\right)$

$Q^T\stackrel{\·\·}{\mathrm{\θ}}+M\stackrel{\·}{q}+(K-{\stackrel{\·}{\mathrm{\θ}}}^{2}M)q=0---\left(11\right)$

Ignore the second order coupled item, and, introduce like down conversion U according to the normalization and the property of orthogonality of the vibration shape ^TMU=I and U ^TKU=Λ U is an orthogonal matrix,

Be eigenvalue matrix, I is a unit matrix.Make q=U η, introduce simultaneously the modal damping item then the equation of motion become:

$I\stackrel{\·\·}{\mathrm{\θ}}+F\stackrel{\·\·}{\mathrm{\η}}=T---\left(12\right)$

$\stackrel{\·\·}{\mathrm{\η}}+C\stackrel{\·}{\mathrm{\η}}+\mathrm{\Λ\η}+F^T\stackrel{\·\·}{\mathrm{\θ}}=0---\left(13\right)$

Wherein, F=QU, modal damping matrix

(i=1,2 ..., n) ξ _iBe damping ratio,

Be the modal vibration frequency.

Can find out that by above-mentioned derivation the equation of motion of flexible spacecraft, the vibration equation of flexible structure intercouple; The vibration of flexible structure is through coupling terms

and then influences the motion of spacecraft attitude; The control or the moment that act on the celestial body are the vibrations that item suppresses flexible structure through

.

This embodiment adopts the oar property structure adaptive variable structure control method of RBF neural network; Referring to shown in Figure 2; The related controller of this control method comprises EI input forming module 1, nominal system 2, sliding-mode surface control module 3, RBF neural network 4 and adaptive control rate module 5, and this control method is:

EI input forming module 1 receives the expectation attitude of satellite angle θ of oar property spacecraft _d, the expectation attitude of satellite angle θ that EI input forming module 1 will be imported _dConvert response u into _EI, said response u _EIExport to nominal system 2, the nominal 2 output expectation attitude of satellite information x of system _m(t), simultaneously, said response u _EIAlso export to oar property spacecraft, oar property spacecraft output real satellite attitude information x (t), expectation attitude of satellite information x _m(t) make comparisons and obtain its error e (t) with real satellite attitude information x (t); Sliding-mode surface control module 3 is obtained suitable sliding-mode surface s according to error e (t); The sliding-mode surface s of sliding-mode surface control module 3 outputs exports to RBF neural network 4 and adaptive control rate module 5, adaptive control rate module 5 output adaptive control rate u simultaneously ^*Give RBF neural network 4, RBF neural network 4 is according to sliding-mode surface s and adaptive control rate u ^*Obtain adjustment control rate u _n, said adjustment control rate u _nWith said response u _EIThe attitude of satellite that the u as a result of addition is used to control oar property spacecraft reaches expectation value.

Expectation attitude of satellite angle θ _dWith obtain after the series of pulses convolution its response u _EI, response u _EIMake the result who obtains have higher robustness through constraint to amplitude.

Concrete this embodiment two: this embodiment is to the further specifying of the oar property structure adaptive variable structure control method of embodiment one described employing RBF neural network, this embodiment be to real satellite attitude information x (t) with expect attitude of satellite information x _m(t) be described further.The vector equation of said real satellite attitude information x (t) is expressed as:

$\stackrel{\·}{x}=\mathrm{Ax}+\mathrm{Bu}\left(t\right)+{\mathrm{Bu}}_{\mathrm{is}}+\mathrm{Bd}\left(t\right),$

Wherein, A, B are a coefficient once,

The expectation attitude of satellite information x of name system 2 outputs _m(t) vector equation is expressed as:

${\stackrel{\·}{x}}_m=A_m{x}_m+B_m{u}_{\mathrm{is}}.$

Wherein, A _m, B _mBe coefficient once,

Concrete this embodiment three: this embodiment is that this embodiment is further specifying error e (t) to the further specifying of the oar property structure adaptive variable structure control method of embodiment one described employing RBF neural network.

The described error e of this embodiment (t) is obtained by following formula:

e(t)＝x(t)-x _m(t)。

Wherein, x (t) is the real satellite attitude information, x _m(t) be expectation attitude of satellite information.

Concrete this embodiment four: this embodiment is that this embodiment is further specifying sliding-mode surface s to the further specifying of the oar property structure adaptive variable structure control method of embodiment one described employing RBF neural network.

The described sliding-mode surface s of this embodiment presses following formulae design:

$s\left(t\right)=\mathrm{\λe}+\frac{d}{\mathrm{dt}}e,$

λ＞0 wherein.

Concrete this embodiment five: this embodiment is that this embodiment is to adaptive control rate u to the further specifying of the oar property structure adaptive variable structure control method of embodiment one described employing RBF neural network ^*Further specify.

The described adaptive control rate of this embodiment u ^*Obtain by following formula:

$u^{*}\left(t\right)=-B^{-1}((A_m+\mathrm{\ΔA})e-\stackrel{\·}{e}+\mathrm{\Δ}{\mathrm{Ax}}_m$

$+\mathrm{\ΔB}{u}_{\mathrm{is}}+\stackrel{\·}{s}\left(t\right)+\mathrm{\λs}\left(t\right))-d\left(t\right).$

Concrete this embodiment six: this embodiment is that this embodiment is according to sliding-mode surface s and adaptive control rate u to RBF neural network 4 to the further specifying of the oar property structure adaptive variable structure control method of embodiment one described employing RBF neural network ^*Obtain adjustment control rate u _nFurther specify.

The described RBF neural network 4 of this embodiment is being obtained adjustment control rate u _nProcess in, according to the adaptive control rate u of adaptive control rate module 5 output ^*, obtain the optimal weights coefficient W of RBF neural network 4 ^*, make this coefficient and satisfy equation: max|u ^*(x, W ^*)-u _n|＜ε.

Concrete this embodiment seven: this embodiment is that this embodiment is that RBF neural network 4 is obtained adjustment control rate u to the further specifying of the oar property structure adaptive variable structure control method of embodiment one described employing RBF neural network _nMethod further specify.

The described RBF neural network 4 of this embodiment is obtained adjustment control rate u by following formula _n:

u _n＝u ^*+ε＝W ^*Tφ+ε。

We are with the adaptive control laws u of s (t) and design ^*Send into RBF neural network 4, obtain its equivalent control device Can be similar to the character of any nonlinear function according to RBF neural network 4,

With optimum weight coefficient W ^*Make control rate u _nSatisfy following equation max|u ^*(x, W ^*)-u _n|＜ε, the output that obtains the RBF neural network is u as a result _n(4) u _nWith u _EICombine and obtain final controller u=u _n+ u _EI, purpose is to make actual output export consistent with nominal system as far as possible.

Its principle of following labor:

Can know that by becoming the structure control theory it is minimum through the value that following weight coefficient turnover rate makes that conditions must be fulfilled :

$w_j=-Y\frac{\∂s\left(t\right)\stackrel{\·}{s}\left(t\right)}{\∂w_j\left(t\right)}---\left(14\right)$

Then by RBF Neural Network Online adjustment weight.

RBF neural network (radial basis function neural network), radial basis function neural network are a kind of neural networks of feed forward type efficiently, and it has optimal approximation performance and the global optimum's characteristic that other feedforward networks do not have, and simple in structure, and training speed is fast.Simultaneously, it also is a kind ofly can be widely used in the neural network model that pattern-recognition, nonlinear function such as approach at the field.

The kinetics equation of system

can be converted into following form:

$\stackrel{\·}{x}=\mathrm{Ax}+\mathrm{Bu}\left(t\right)+{\mathrm{Bu}}_{\mathrm{is}}+\mathrm{Bd}\left(t\right)---\left(15\right)$

At first we suppose that nominal system is:

${\stackrel{\·}{x}}_m=A_m{x}_m+B_m{u}_{\mathrm{is}}---\left(16\right)$

If e (t)=x (t)-x _m(t) be error actual and nominal system, can derive the error dynamics equation:

$\stackrel{\·}{e}\left(t\right)=(A_m+\mathrm{\ΔA})e\left(t\right)+\mathrm{\ΔA}{x}_m+\mathrm{Bd}\left(t\right)+{\mathrm{\ΔBu}}_{\mathrm{is}}+\mathrm{Bu}\left(t\right)---\left(17\right)$

Δ A=A-A in the formula _m, Δ B=B-B _m, we design following control rate u ^*:

$u^{*}\left(t\right)=-B^{-1}((A_m+\mathrm{\ΔA})e-\stackrel{\·}{e}+\mathrm{\Δ}{\mathrm{Ax}}_m$

$+\mathrm{\ΔB}{u}_{\mathrm{is}}+\stackrel{\·}{s}\left(t\right)+\mathrm{\λs}\left(t\right))-d\left(t\right)---\left(18\right)$

Wherein $s\left(t\right)=\mathrm{\λ\; e}+\frac{d}{\mathrm{Dt}}e.$

In wushu (18) substitution (17), we can obtain:

$\stackrel{\·}{s}\left(t\right)+\mathrm{\λs}\left(t\right)=0---\left(19\right)$

Wherein λ＞0, so s can converge on zero, and systematic error also can go to zero then.

In order to improve the performance of controller, we introduce the RBF neural network and come the Approximate Equivalent controller

$\stackrel{\·}{s}\left(t\right)+\mathrm{\λs}\left(t\right)-b[u_n-u\left(t\right)]=0---\left(20\right)$

In theory, the RBF neural network can be similar to any nonlinear function.Suppose to exist optimum weight coefficient W ^*Make control rate u satisfy following equation

max|u ^*(x，W ^*)-u _n|＜ε(21)

U wherein ^*=W ^{* T}φ so u _n=W ^{* T}φ+ε.

Next we define weighted error coefficient vector

equation (20) and are:

$\stackrel{\·}{s}\left(t\right)=-\mathrm{\λs}\left(t\right)+b({\tilde{W}}^T\mathrm{\φ}+\mathrm{\ϵ})---\left(22\right)$

We consider following Lyapunov function, and V is the Liapunov candidate functions:

$V=\frac{1}{2}{s}^2+\frac{b}{2Y}{\tilde{W}}^T\tilde{W}---\left(23\right)$

is the time-derivative of Liapunov candidate functions V:

$\stackrel{\·}{V}=s\stackrel{\·}{s}+\frac{b}{Y}{\tilde{W}}^T\tilde{W}---\left(24\right)$

Substitution (22) and (14)

$\stackrel{\·}{V}=s[-\mathrm{\λs}+b({\tilde{W}}^T\mathrm{\φ}+\mathrm{\ϵ})]-b{\tilde{W}}^T\mathrm{s\φ}$

(25)

$=-\mathrm{\λ}{s}^2+\mathrm{sb\ϵ}$

Obtain at last

$\stackrel{\·}{V}\≤\left|s\right|(-\mathrm{\λ}|s|+\mathrm{b\ϵ})---\left(26\right)$

If following condition is arranged | s|＞b ε/λ sets up, and can release

so sliding-mode surface can converge on zero.

The present invention goes to zero systematic error through using RBF neural network adaptive variable structure control method, suppresses the vibration of solar energy sailboard.Control method of the present invention is simple.The present invention adopts RBF neural network (radial basis function neural network); Radial basis function neural network is a kind of neural network of feed forward type efficiently; It has optimal approximation performance and the global optimum's characteristic that other feedforward networks do not have, and simple in structure, and training speed is fast.Simultaneously, it also is a kind ofly can be widely used in the neural network model that pattern-recognition, nonlinear function such as approach at the field.

Claims (1)

1. adopt the oar property structure adaptive variable structure control method of RBF neural network; It is characterized in that; The related controller of this control method comprises EI input forming module (1), nominal system (2), sliding-mode surface control module (3), RBF neural network (4) and adaptive control rate module (5), and this control method is:

EI input forming module (1) receives the expectation attitude of satellite angle θ of oar property spacecraft _d, the expectation attitude of satellite angle θ that EI input forming module (1) will be imported _dConvert response u into _EI, said response u _EIExport to nominal system (2), the output expectation attitude of satellite information x of nominal system (2) _m(t), simultaneously, said response u _EIAlso export to oar property spacecraft, oar property spacecraft output real satellite attitude information x (t), expectation attitude of satellite information x _m(t) make comparisons and obtain its error e (t) with real satellite attitude information x (t); Sliding-mode surface control module (3) is obtained suitable sliding-mode surface s according to error e (t); The sliding-mode surface s of sliding-mode surface control module (3) output exports to RBF neural network (4) and adaptive control rate module (5), adaptive control rate module (5) output adaptive control rate u simultaneously ^*Give RBF neural network (4), RBF neural network (4) is according to sliding-mode surface s and adaptive control rate u ^*Obtain adjustment control rate u _n, said adjustment control rate u _nWith said response u _EIThe attitude of satellite that the u as a result of addition is used to control oar property spacecraft reaches expectation value,

Error e (t) is obtained by following formula:

e(t)＝x(t)-x _m(t)，

Sliding-mode surface s presses following formulae design:

$s\left(t\right)=\mathrm{\λ\; e}+\frac{d}{\mathrm{Dt}}e,$ λ＞0 wherein.

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