CN103552685A - Aircraft anti-skid brake control method - Google Patents

Abstract

The invention relates to an aircraft anti-skid brake control method. According to the aircraft anti-skid brake control method, an aircraft reduction rate signal is constructed by means of the robustness and anti-interference characteristics of a tracking differentiator or a second-order sliding mode differentiator, and is used for calculating slip force, and learning identification in a longitudinal dynamics process is avoided. According to the method, braking longitudinal force does not need to be assumed to be uniformly distributed, and the method can be used for a multi-wheel brake system. In addition, a safety operation range of the brake system is considered, the brake pressure is constrained, the anti-skid brake efficiency and anti-lock safety are considered simultaneously, and the method has certain advancement.

Description

A kind of Control Method for Airplane Antiskid Braking System

Technical field

A kind of Control Method for Airplane Antiskid Braking System the present invention relates to, belongs to the automatic control technology field of brake.

Background technology

Alighting gear-brake system is the subsystem on aircraft with relatively independent function, and its function is the kinetic energy while bearing the static weight of aircraft, dynamic impulsion load and absorption aircraft landing, realizes the landing of aircraft, the control of sliding and turn.Present generation aircraft is generally generally equipped anti-skid brake system (ABS) (Anti-skid Brake System, ABS), on the basis of traditional brake system, to adopt closed loop control method to realize the automatic adjusting of brake torque, the airplane wheel causing because brake pressure is excessive while preventing from braking locked.It can make full use of the maximum combined coefficient between aircraft tyre and ground, obtains higher braking efficiency, thereby effectively shortens braking distance, prevents the severe wear of the tire that causes because airplane wheel is locked, to improve the safety and reliability of aircraft.

In airplane brake system, there is many non-linear factors, as the attachment coefficient between aircraft tyre and ground, attachment coefficient and slip rate, brake torque and brake pressure etc.These non-linear factors directly have influence on the performance of airplane brake system.Aircraft likely runs into the mal-conditions such as crosswind, left and right main wheel and runway contact condition be asymmetric in take-off and landing process.These factors not only can reduce the efficiency of brake itself, even may cause aircraft to depart from even and gun off the runway.

Anti-skidding/anti-lock braking system, through the development of decades, at the large quantity research of land vehicle and mobile robot field, and is obtained good effect.

But there are some differences in above-mentioned these application and aircraft brake field: because the automobile kinetic factor of being bullied affects less, land vehicle brake system thinks that longitudinal deceleration is directly produced by brake, therefore longitudinal deceleration rate and the direct correlation of tire/road surface friction force, and for automobile application, can think that braking action is evenly distributed on each and takes turns.And for aircraft brake, these hypothesis are not too suitable, aircraft is in landing mission, and except wheel braking, the deceleration being caused by speed reduction gearing must take in, and aircraft may adopt differential brake simultaneously, and the braking action of left and right wheel is inconsistent.

Therefore, aircraft brake performance is also subject to the impact of aerodynamic force and moment.Just at present in general, ground flight force and moment is difficult to accurately obtain by existing sensor on aircraft.Meanwhile, these external force are easy to be subject to airport environment and wind direction impact.In addition, the uncertain load of vertical direction also has a great impact Landing Gear System and brake system, and the changes in weight of aircraft also can affect braking quality, therefore, for obtaining performance-oriented braking effect, when design anti-skid controller, must consider that aircraft longitudinal dynamics need to take into account.

Summary of the invention

For these problems, the present invention will adopt following method to improve this problem: first method is that the longitudinal dynamics process of aircraft is carried out to on-line identification by neural network, build straight skidding dynamic process, again by neural network control device, guarantee that brake system can exist tracking under vertical and longitudinal uncertain condition to set slip rate, consider that brake torque input exists physical constraint, excessive brake torque does not only have help to brake, easily making on the contrary system be absorbed in the degree of depth skids, the present invention retrains processing to control inputs, make controller can under input constraint situation, learn uncertain part and follow the tracks of and set sliding curve.Second method the present invention considers that aircraft dynamics model is difficult to accurately acquisition and causes longitudinal force calculating inaccurate, the present invention designs disturbance rejection differentiator structure aircraft moderating process, and by adaptive neural network identification tire-ground friction force curve, adopt extremum search algorithm to ask for optimal slip rate, the present invention has designed equally and has considered more new law of the adaptive control laws of input constraint and parameter, for learning uncertain part and follow the tracks of optimum sliding curve.

The Control Method for Airplane Antiskid Braking System that the present invention proposes, by the Robust interference characteristic of Nonlinear Tracking Differentiator or Second Order Sliding Mode differentiator, constructs aircraft moderating ratio signal, for skid force, calculates, and has avoided longitudinal dynamics process to carry out Learning Identification; Two kinds of methods are to be all uniformly distributed without hypothesis brake longitudinal force, can be used for many wheel brakes system; In addition, consider the range of safety operation of brake system, brake pressure is retrained, taken into account antiskid brake efficiency and anti-lock safety, there is certain advance.

The present invention relates to a kind of Control Method for Airplane Antiskid Braking System, it is characterized in that, comprise the steps:

The first step: obtain the speed V of aircraft by airborne-bus, resolve air speed by Second Order Sliding Mode differentiator and obtain airframe moderating ratio.Step is as follows:

$\mathrm{\σ}\left(t\right)=V-\hat{x}\left(t\right)$

${\mathrm{<figure-callout\; id="0"\; label="y"\; filenames="HDA0000418694800000021.png,HDA0000418694800000022.png"\; state="\{\{state\}\}">y</figure-callout>}}_0=\mathrm{\&alpha;}\sqrt{\left|\mathrm{\&sigma;}\right|}\mathrm{sign}\left(\mathrm{\&sigma;}\right)+y_1$

y ₁=βsign(σ)

Estimate that moderating ratio is

$\hat{V}=y_1$

Second step: by wheel speed sensors, obtain wheel cireular frequency Ω, computing machine wheel slip rate:

$\mathrm{\&lambda;}=\frac{V-r_w\mathrm{\&Omega;}}{V}$

Wherein, r _wit is wheel radius.

Structure RBF neural network Approximation of Continuous Functions non-linear friction force function

neural network configuration is

$\widehat{\mathrm{\&mu;}}\left(\mathrm{\&lambda;}\right)=W^T\mathrm{\&Phi;}\left(\mathrm{\&lambda;}\right)$

Here slip rate is inputted as neural network, weight vector neural network interstitial content 1 > 1; And Φ (λ)=[φ ₁(λ), φ ₂(λ) ..., φ _l(λ)] ^tfor radial basis function.

The 3rd step: neural network is obtained to friction force

extremum search, calculates maximum friction force and optimal slip rate corresponding to maximum frictional force

${\mathrm{\&lambda;}}_r={\arg \max }_{\mathrm{\&lambda;}\&Element;\left[\mathrm{0,1}\right]}\widehat{\mathrm{\&mu;}}\left(\mathrm{\&lambda;}\right)$

By second order filter, ask for optimal slip rate reference signal λ _d, the discountinuity bringing to eliminate numerical calculation

$\frac{{\mathrm{\&lambda;}}_d}{{\mathrm{\&lambda;}}_r}=\frac{{\mathrm{\&omega;}}_n^2}{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HDA0000418694800000021.png,HDA0000418694800000022.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+{2\mathrm{\&zeta;}}_n{\mathrm{\&omega;}}_n+{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000418694800000021.png,HDA0000418694800000022.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}$

The 4th step: design PI type tracking error

Calculate the linear velocity of wheel:

v _w=r _wΩ

Use optimal slip rate is λ _d, the linear velocity error of calculating wheel

e _w=v _w-v _xλ _d

Calculate PI type error

$s_w=e_w+\mathrm{\&alpha;}{\&Integral;}_0^t{e}_w\mathrm{dt}$

Calculate PI type wheel line of reference speed

$v_{\mathrm{wd}}=v_x{\mathrm{\&lambda;}}_D+\mathrm{\&alpha;}{\&Integral;}_0^t{e}_w\mathrm{dt}$

The 5th step: calculate adaptive neural network Feed forward Compensating Control Law;

Calculate PI control item

$T_{\mathrm{PI}}=\frac{J}{r_w}(-k_{\mathrm{PI}}(v_w-v_{\mathrm{wd}}))$

Wherein, J is wheel rotor inertia.K _pIit is gain factor.

Calculate feedforward compensation item:

$T_{\mathrm{ba}}=\frac{J}{r_w}(-\hat{v}+\frac{F_z\widehat{\mathrm{\&mu;}}{r}^2}{J})$

Wherein, F _zbe the suffered radial weight of wheel, by load born by engine body, distribute and determine.

The input of calculating controlling torque, input torque is above two sums:

T _b=T _bn+T _ba

The 6th step: calculating parameter is new law more;

Calculate more new law of auto-adaptive parameter

$W=-\mathrm{\&Gamma;}\frac{F_2{r}_w^2\mathrm{\&Phi;}}{J}s$

Wherein, Γ is that parameter is upgraded gain matrix.

Preferably: adopt Second Order Sliding Mode differentiator, be specially Super-twisting sliding formwork differentiator, sliding mode observer parameter value is:

$\mathrm{\&alpha;}=1.5\sqrt{L}$

β=1.1L

Wherein | F (V, t) |≤L, F (V, t) chooses and represents aircraft longitudinal dynamics process V=F (V, t) here.

Preferably: neural network adopts radial basis function

${\mathrm{\&phi;}}_i=\exp \left[\frac{{-(\mathrm{\&lambda;}-c_i)}^T(\mathrm{\&lambda;}-c_i)}{{\mathrm{\&eta;}}_i^2}\right],i=\mathrm{1,2},...,l$

C wherein _i=[c ₁, c ₂..., c _l] ^tthe center of tolerance domain, η _iit is the width of Gaussian function.

Wheel kinetics equation

Wheel sideslip friction force can be expressed as

F _yN=F _zNα _N F _yM=F _zMα _M (1)

Here F _zNand F _zMrepresent respectively the radial weight to wheel from main landing gear.α _nand α _mthe angle of side slip that represents respectively front-wheel and main wheel, can calculate by following expression

α _N=δ-β _N α _M=-β _M (2)

Here

${\mathrm{\&beta;}}_N=\mathrm{\&beta;}+\frac{L_{\mathrm{xN}}r}{v},{\mathrm{\&beta;}}_M=\mathrm{\&beta;}-\frac{L_{\mathrm{xM}}r}{v}---\left(3\right)$

The present invention does not discuss the driftage stable problem of aircraft floor manoeuvre, and the present invention supposes that the yaw rate r of aircraft and angle of side slip β can guarantee in more among a small circle, so aircraft floor control model can linearization.The longitudinal movement equation of aircraft can be expressed as

${m\stackrel{\&CenterDot;}{v}}_x=-F_x---\left(4\right)$

Here F _xrepresent longitudinal force sum, can be expressed as F _x=F _xaero+ F _xD, F here _xaerobe expressed as longitudinal aerodynamic force F _xDrepresent Tire Friction sum, can be expressed as

F _xD=F _xML+F _xMR+F _yN sinδ _N

Here F _xML, F _xMRbe expressed as the Tire Friction of left and right main landing gear, F _yNthe sideslip friction force that represents front-wheel.In actual brake is controlled, front-wheel can be thought free rotation, and its friction force is counted as enough I and ignores.

Owing to rolling, the friction force that resistance phase torque/mass ratio produces is very little, ignores here and rolls resistance, and the present invention obtains the tire dynamics process of main wheel

$J\stackrel{\&CenterDot;}{\mathrm{\&omega;}}=r_w{F}_w-T_b---\left(5\right)$

Here ω represents wheel cireular frequency, and J represents the active inertia of main wheel, and r represents wheel radius, T _brepresent brake torque input, F _wrepresent tire and the friction force of going to road surface, can be expressed as

F _w=F _zMμ(λ) (6)

Here λ represents slip rate, and slip rate is defined as λ=(V-ω r)/V _x.Here V airframe speed, considers airframe dynamic process

definition sliding velocity is v _w=V-ω r,

${\stackrel{\&CenterDot;}{v}}_w=\stackrel{\&CenterDot;}{V}-\frac{F_{\mathrm{zm}}\mathrm{\&mu;}\left(\mathrm{\&lambda;}\right)r_w^2}{J}+s\left(v_w\right)\frac{r_w{T}_b}{J}$

Here piecewise function s (V _w) be defined as

$s\left(v_w\right)=\left\{\begin{array}{cc}1& \mathrm{if}{v}_w>0\\ 0& \mathrm{else}\end{array}\right.---\left(7\right)$

Consider the tire dynamics process of main wheel

$\stackrel{\&CenterDot;}{\mathrm{J\&omega;}}{r}_w{F}_w-T_b,---\left(8\right)$

Here ω represents wheel cireular frequency, and J represents the active inertia of main wheel, and r represents wheel radius, T _brepresent brake torque input, F _wrepresent tire and the friction force of going to road surface, can be expressed as

F _w=F _zMμ(λ) (9)

Here λ represents slip rate, and slip rate is defined as λ=(V _x-ω r)/V _x.By λ, to time differentiate, the present invention can obtain following relation

$\stackrel{\&CenterDot;}{\mathrm{\&lambda;}}=-\frac{r_w}{v_x}\stackrel{\&CenterDot;}{\mathrm{\&omega;}}+\frac{r_w\mathrm{\&omega;}}{v_x^2}{\stackrel{\&CenterDot;}{v}}_x$

Note ω=(v _x/ r _w) (1-λ), can obtain

$\stackrel{\&CenterDot;}{\mathrm{\&lambda;}}=-\frac{r_w^2}{v_{x}J}{F}_{\mathrm{zM}}\mathrm{\&mu;}\left(\mathrm{\&lambda;}\right)-\frac{1-\mathrm{\&lambda;}}{{\mathrm{mv}}_x}{F}_x-\frac{r}{v_{x}J}{T}_b---\left(10\right)$

Tire Friction is easy to change because of the situation on road surface or runway road surface.In order accurately to describe friction force characteristic, in friction force model, need to have one group of adjustable parameters.In the present invention, compromise meeting between the application of precision needs and actual brake, suppose that these runway parameters can priori get, so adopted the brake model of a simplification, this model can be expressed as

$F_w={2F}_{w_{\max }}\frac{{\mathrm{\&lambda;}}_{\mathrm{opt}}\mathrm{\&lambda;}}{{\mathrm{\&lambda;}}_{\mathrm{opt}}^2+{\mathrm{\&lambda;}}^2}---\left(11\right)$

Profile frictional coefficient can be expressed as

$\mathrm{\&mu;}\left(\mathrm{\&lambda;}\right)=\frac{{2F}_{w_{\max }}}{F_{z_w}}\frac{{\mathrm{\&lambda;}}_{\mathrm{opt}}\mathrm{\&lambda;}}{{\mathrm{\&lambda;}}_{\mathrm{opt}}^2+{\mathrm{\&lambda;}}^2}---\left(12\right)$

Velocity contrast v between definition wheel and airframe speed _wfor

V _w=v _xone ω r _w(13)

Here 0≤v _w≤ v, by v _wto time differentiate, can obtain

${\stackrel{\&CenterDot;}{v}}_w=-\frac{F_X}{m}-\frac{F_{\mathrm{zm}}\mathrm{\&mu;}\left(\mathrm{\&lambda;}\right)r_w^2}{J}+s\left(v_w\right)\frac{r_w{T}_b}{J}---\left(14\right)$

Here functional relation s (V _w) be defined as

$s\left(v_w\right)=\left\{\begin{array}{cc}1& \mathrm{if}{v}_w>0\\ 0& \mathrm{else}\end{array}---\left(15\right)\right.$

Consider equation (10), when λ=0, μ (λ)=0, T _b=0 o'clock, wheel freely rotated, now

so need to add extra constraint condition.In the present invention, adopt v _w-dynamic process replaces λ-dynamic process to guarantee the robustness of state of a control.

In brake process, when tire is when the tangential velocity of earth point is less than the speed of aircraft, tire produces and slides.The slip of tire has produced friction force, thereby causes aircraft to slow down.While increasing when sliding, the friction coefficient between tire and runway increases, until reach maximum frictional force coefficient value at certain point, then along with the increase of sliding, reduces.If brake is operated in descent stage, thereby easily cause braking efficiency to reduce, cause wheel locking.If wheel locking, easily causing blows out skids, and causes aircraft loss of control ability.Therefore in brake system design and control design, generally allow brake system set and be operated in the friction coefficient ascent stage, guarantee braking efficiency and aircraft system safety.

Neural network extreme searching method based on sliding formwork differentiator

Above, the present invention has designed the brake controller method of designing of learning aircraft longitudinal dynamics process by Neural Network Online.It is to be noted that this method need to calculate body dynamic process, increased the burden of brake treater, simultaneously, by adaptive neural network on-line identification, can estimate that Longitudinal mathematic(al) parameter also can guarantee and actual value is restrained, but obtain enough high-precision estimated valves, need enough learning times, so especially in the starting stage.Solution is when working control designs, it to be used as control variable, and the present invention just can start to carry out on-line identification when antiskid brake is not worked after aircraft lands, and obtains the higher longitudinal force curve of precision.

Can not be for longitudinal frictional force nonlinear function optimizing in order to obtain estimated valve, the present invention wishes to obtain instant longitudinally load parameter, avoids adaptive learning process to bring derivative error to cause finding optimal working point.Also having a kind of mode is directly to obtain wheel acceleration/accel by acceleration pick-up, and current acceleration pick-up is difficult to meet the acceleration analysis requirement in motion process, and general non-accelerated sensor in current airborne sensor;

The present invention adopts numerical differentiation device on-line reorganization body restriction signal, avoids carrying out body dynamics calculation and parameter estimation.In Industry Control, if directly speed signal is carried out to derivation operation, because differentiator physics can not be realized, can only be similar to realization, may exist and disturb and cause interfering signal to be amplified because of signal, be difficult to meet the required precision of control.Same problem has also perplexed PID and has controlled design, and in traditional PID control, producing differential error signal de/dt does not have very good way.For example the transfer function of conventional approximate differential device is

$y=\frac{s}{\mathrm{\&tau;s}+1}v---\left(16\right)$

This transfer function can be launched into

$y=\frac{1}{\mathrm{\&tau;}}(1-\frac{1}{\mathrm{\&tau;s}+1})v---\left(17\right)$

It is approximate differential formula

$y=\frac{v\left(t\right)-v(t-\mathrm{\&tau;})}{\mathrm{\&tau;}}---\left(18\right)$

Realization.But when incoming signal v (t) is polluted by noise n (t), noise component n (t)/τ that approximate differential (v (t)-v (t-τ))/τ signal in output y is just exaggerated floods, and cannot utilize.PID controller, except special case, is in fact all PI controller, has also illustrated that traditional differentiator is difficult to meet Actual Control Effect of Strong.

In order to guarantee the antijamming capability of numerical differentiation device, the present invention adopts two kinds of numerical differentiation devices, and a kind of is steepest Nonlinear Tracking Differentiator, and another is Second Order Sliding Mode differentiator.

Body acceleration compensator

The present invention will take speed signal to carry out the mode of energy numerical differentiation, and therefore how suppressing mushing error amplification is the problem that must consider.Here two kinds of numerical differentiation devices of employing are obtained for acceleration signal.A kind of is based on Second Order Sliding Mode differentiator; Another is based on Nonlinear Tracking Differentiator.

Second Order Sliding Mode differentiator

Sliding mode observer can be designed as

$\stackrel{\stackrel{\&CenterDot;}{^}}{x}\left(t\right)=u\left(t\right),\hat{x}\left(0\right)=0---\left(19\right)$

If

$u\left(t\right)-(\mathrm{\&rho;}+L)\frac{\mathrm{\&sigma;}\left(t\right)}{\left|\right|\mathrm{\&sigma;}\left(t\right)\left|\right|},\mathrm{\&sigma;}\left(t\right)=y\left(t\right)-\hat{x}\left(t\right),\mathrm{\&rho;}>0---\left(20\right)$

In finite time,

meanwhile,

note discrete high frequency flutter characteristic that exists of u (t), by low-pass filter, carry out smothing filtering, can obtain

${\hat{u}}_{\mathrm{eq}}=\mathrm{LPF}\left(u\left(t\right)\right)---\left(21\right)$

Its evaluated error, and the timeconstantτ of low-pass filter is directly proportional

Because the precision of single order sliding formwork differentiator is

because Second Order Sliding Mode Control has not only guaranteed sliding variable σ (t) convergence in finite time, guaranteed its first derivative simultaneously

convergence.The present invention considers to adopt Second Order Sliding Mode differentiator, Super-twisting sliding formwork differentiator for the present invention here.

Sliding mode observer

$\mathrm{\&sigma;}\left(t\right)=y\left(t\right)-\hat{x}\left(t\right)---\left(23\right)$

u _i=α _i|σ _i| ^1/2signσ _i(t)+β _i∫signσ _i(t)dτ (24)

Here at finite time

$\begin{array}{c}{\mathrm{\&alpha;}}_i=1.5\sqrt{L_i}\\ {\mathrm{\&beta;}}_i=1.1L_i\\ \left|{\stackrel{\&CenterDot;}{F}}_i(x,t)\right|\&le;L_i\\ u\left(t\right)=\left\{u_i\left(t\right)\right\}\\ F\left(t\right)=\left\{F_i(x,t)\right\},i=1,...,n\end{array}$

Guaranteed in finite time,

simultaneously

here

${\stackrel{\widehat{\&CenterDot;}}{x}}_i\left(t\right)={\stackrel{\&CenterDot;}{x}}_i\left(t\right)=u_i\left(t\right)={\mathrm{\&alpha;}}_i{\left|{\mathrm{\&sigma;}}_i\right|}^{1/2}{\mathrm{sign\&sigma;}}_i\left(t\right)+{\mathrm{\&beta;}}_i\&Integral;{\mathrm{sign\&sigma;}}_i\left(t\right)\mathrm{d\&tau;}$

Or

${\stackrel{\widehat{\&CenterDot;}}{x}}_i\left(t\right)={\mathrm{\&beta;}}_i\&Integral;{\mathrm{sign\&sigma;}}_i\left(t\right)\mathrm{d\&tau;}$

Here due to high frequency differential term has been carried out to integration, u (t) is continuous, does not need that u (t) is carried out to LPF and obtains u _eq.

Nonlinear Tracking Differentiator

For Second Order Integral system

$\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2=u\end{array}---\left(25\right)$

Here | u|≤r, v is x ₁reference value, the time optimal solution of u is

$u=-\mathrm{rsign}(x_1-v+\frac{x_2\left|x_2\right|}{2r})---\left(26\right)$

The present invention can obtain following dynamic process

$\begin{array}{c}{\stackrel{\&CenterDot;}{v}}_1=v_2\\ {\stackrel{\&CenterDot;}{v}}_2=-\mathrm{rsign}(v_1-v+\frac{{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="HDA0000418694800000021.png,HDA0000418694800000022.png"\; state="\{\{state\}\}">v</figure-callout>}}_2\left|{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="HDA0000418694800000021.png,HDA0000418694800000022.png"\; state="\{\{state\}\}">v</figure-callout>}}_2\right|}{2r})\end{array}---\left(17\right)$

Here v ₁reference locus, v ₂it is its differential.According to the physical restriction of concrete utilization, can be by selecting the size of r to accelerate or slowing down dynamic process.Bang-Bang characteristic due to function has chatter phenomenon when system enters stable state, for fear of this chatter phenomenon, adapts to the demand of numerical calculation, for discrete system

$\begin{array}{c}{v}_1(k+1)=v_1\left(k\right)+\mathrm{hv}2\left(k\right)\\ v_2(k+1)=v_2\left(k\right)-\mathrm{ru}\left(k\right),\left|u\left(k\right)\right|\&le;r\end{array}---\left(28\right)$

The present invention has

u=f _d(v ₁，v ₂，r ₀，h) (29)

Here h is the sampling period, r ₀and h ₀controller parameter, f _dcan be expressed as

$f_d={-r}_0(\frac{a}{d}-\mathrm{sign}\left(a\right))s_a-{\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="HDA0000418694800000021.png,HDA0000418694800000022.png"\; state="\{\{state\}\}">r</figure-callout>}}_0\mathrm{sign}\left(a\right)$

Here

$\begin{array}{c}d=h_0{\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="HDA0000418694800000021.png,HDA0000418694800000022.png"\; state="\{\{state\}\}">r</figure-callout>}}_0^2,a_0=h_0{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="HDA0000418694800000021.png,HDA0000418694800000022.png"\; state="\{\{state\}\}">v</figure-callout>}}_2,y={\mathrm{<figure-callout\; id="1"\; label="v"\; filenames="HDA0000418694800000022.png,HDA0000418694800000042.png"\; state="\{\{state\}\}">v</figure-callout>}}_1+a_0\\ a_1=\sqrt{d(d+8|y\left|\right)}\\ a_2=a_0+\mathrm{sign}\left(y\right)(a_1-d)/2\\ s_y=(\mathrm{sign}(y+d)-\mathrm{sign}(y-d))/2\\ a=(a_0+y-a_2)s_g+a_2\\ s_a=(\mathrm{sign}(y+d)-\mathrm{sign}(g-d))/2\end{array}$

Utilize function f _d(v ₁, v ₂, r ₀, h) setting up steepest feedback system, can obtain well following the tracks of and differential effect.

Adaptive neural network feedforward compensation

In the present invention, with following radial basis function (Radial Basis Function RBF) neural network Approximation of Continuous Functions non-linear friction force function

neural network configuration is

$\widehat{\mathrm{\&mu;}}=W^T\mathrm{\&Phi;}\left(\mathrm{\&lambda;}\right)$

Here slip rate is inputted as NN, weight vector

nN interstitial content l > 1;

And Φ (λ)=[φ ₁(λ), φ ₂(λ) ..., φ _l(λ)] ^t, wherein

${\mathrm{\&phi;}}_i=\exp \left[\frac{-{(\mathrm{\&lambda;}-c_i)}^T(\mathrm{\&lambda;}-c_i)}{{\mathrm{\&eta;}}_i^2}\right],i=\mathrm{1,2},...,l$

C wherein _i=[c ₁, c ₂..., c _l] ^tto hold Yu center, η _iit is the width of Gaussian function.Neural network is proved to be to compact

in can arbitrary accuracy follow the tracks of arbitrary smooth function

$\mathrm{\&mu;}\left(\mathrm{\&lambda;}\right)=W^{*}\mathrm{\&Phi;}\left(\mathrm{\&lambda;}\right)+\&Element;,\&ForAll;\mathrm{\&lambda;}\&Element;{\mathrm{\&Omega;}}_{\mathrm{\&lambda;}}$

Here W ^*be desirable constant weight vector, ∈ is approximate error.In ANN (Artificial Neural Network) Control, resulting stability conclusion is generally half overall situation, and input variable λ is some compacting of presetting

in, compact Ω here _λcan select as requested arbitrary size, as long as the node of nerve network controller is abundant, total energy guarantees closed loop system bounded.

Definition optimal weights W ^*for being defined as

Ω _λ，W ^*，|∈|＜∈ ^*，∈ ^*＞0λ∈Ω _λ

Neural network evaluated error is defined as

Friction force extremum search

By neural network, approach gained non-linear friction force function, the present invention can be in the hope of optimal slip rate function

${\mathrm{\&lambda;}}_r=\underset{\mathrm{\&lambda;}\&Element;\left[\mathrm{0,1}\right]}{\arg \max }{W}^T\mathrm{\&phi;}\left(\mathrm{\&lambda;}\right)---\left(31\right)$

The discountinuity bringing in order to eliminate numerical calculation, the present invention uses second order filter to obtain a smooth reference signal λ _d, for closed loop system, follow the tracks of

$\frac{{\mathrm{\&lambda;}}_d}{{\mathrm{\&lambda;}}_r}=\frac{{\mathrm{\&omega;}}_n^2}{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HDA0000418694800000021.png,HDA0000418694800000022.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+2{\mathrm{\&zeta;}}_n{\mathrm{\&omega;}}_n+{\mathrm{\&omega;}}_n^2}---\left(32\right)$

Self adaptation antiskid brake design of control law

Similar with previous methods, it is λ that the present invention first defines optimal slip rate _d, definition wheel velocity error is

e _w=v _w-Vλ _d (33)

Further, definition PI type error

$s_w=e_w+k_i{\&Integral;}_0^t{e}_w\mathrm{dt}---\left(34\right)$

Relative velocity reference signal can be expressed as

$v_{\mathrm{wd}}={\mathrm{V\&lambda;}}_d-k_i{\&Integral;}_0^t{e}_w\mathrm{dt}$

Drive slip rate to there being most set point λ _d, so that friction force F _wmaximize.In order to obtain smooth reference signal, wheel velocity error can be defined as e _w=v _w-v _xλ _d.Further, a PI type error can be expressed as

$s_w=e_w+\mathrm{\&alpha;}{\&Integral;}_0^t{e}_w\mathrm{dt}---\left(35\right)$

For the ease of analyzing, the present invention defines v _wdfor

$v_{\mathrm{wd}}=v_x{\mathrm{\&lambda;}}_D+\mathrm{\&alpha;}{\&Integral;}_0^t{e}_w\mathrm{dt}---\left(36\right)$

The robust nonlinear neural network control that the present invention proposes can be expressed as

T _b=T _bn+T _ba+T _br (37)

This control law comprises 3 parts: body acceleration compensation T _ba, neural network feedforward compensation T _bfwith robust item, wherein T _bnand T _babe expressed as

$T_{\mathrm{ba}}=\frac{J}{r_w}(-k_{\mathrm{PI}}(v_w-v_{\mathrm{wd}})-\stackrel{\widehat{\&CenterDot;}}{v}+\frac{F_z\widehat{\mathrm{\&mu;}}{r}^2}{J})---\left(38\right)$

$T_{\mathrm{bf}}=\frac{r_w}{J}(\frac{{\widehat{\mathrm{\&theta;}}}_x\mathrm{\&phi;}\left(v_w\right)}{m}+\frac{{\widehat{\mathrm{\&theta;}}}_z^T{\mathrm{\&phi;}}_z\left(v_w\right)\mathrm{\&mu;}\left(\mathrm{\&lambda;}\right)r_w^2}{J})---\left(39\right)$

Consider that after input constraint, design parameters is new law more

$\begin{array}{c}{\stackrel{\stackrel{\&CenterDot;}{^}}{\mathrm{\&theta;}}}_x={\mathrm{\&Gamma;}}_x{\mathrm{\&phi;}}_x(s_w-\mathrm{\&chi;})\\ \stackrel{\&CenterDot;}{\mathrm{\&chi;}}=-\mathrm{\&gamma;\&chi;}+s\left(v_w\right)\frac{r_w{T}_b}{J}\end{array},---\left(40\right)$

By defining an amended tracking error

robust item is designed to

$T_{\mathrm{br}}=-k_{r}q\left(\stackrel{\&OverBar;}{s}\right)---\left(41\right)$

Here k _r> 0 is a normal amount, q be one about odd function.How to choose

for well known in the art.

Accompanying drawing explanation

Fig. 1 is the kinetic model schematic diagram of airplane wheel;

Fig. 2 is that the self adaptation optimizing antiskid brake control method based on Second Order Sliding Mode differentiator realizes block diagram;

Slip rate curve when Fig. 3 is noiseless;

Attachment coefficient curve when Fig. 4 is noiseless;

Velocity curve when Fig. 5 is noiseless;

Moderating ratio curve when Fig. 6 is noiseless;

Fig. 7 is the slip rate curve having while disturbing;

Fig. 8 is the attachment coefficient curve having while disturbing;

Fig. 9 is the velocity curve having while disturbing;

Figure 10 is the moderating ratio curve having while disturbing.

The specific embodiment

In order to verify validity and the robustness of algorithm for design above, the present invention has considered two kinds of operating modes: a kind of is not consider the alighting gear vibration interference that identification brings to brake system friction force; Another situation thinks that alighting gear is spring-damp system, or brings periodic radial weight to wheel.Torque input constraint is at neighborhood

in, here

select

for emulation.The present invention has discussed two kinds of situations, and reference model setting parameter is at ω _n=10rad/s and ζ _n=1.The optimal slip rate of aircraft on dry runway and wet runway is respectively 0.16 and 0.14;

First discuss and there is no the periodically situation of radial weight of alighting gear, from Fig. 3, can observe, by the on-line identification of starting stage, the operation point of system can maintain near optimal slip rate λ=0.14, when aircraft enters wet runway, brake system is found again and has been chased after slip rate, and system works point maintains near new optimal slip rate λ=0.16.Fig. 4 is corresponding friction coefficient, can see, under two kinds of operating modes, wheel friction force all maintains higher level.Fig. 5 is wheel speed and body velocity curve, and because slip rate maintains more fixing level always, brake process is also very level and smooth, does not all occur locking process in starting stage and runway handoff procedure.Fig. 6 is corresponding moderating ratio curve.

Robustness for verification algorithm, the present invention simulates the spring damping characteristic of alighting gear, wheel is applied to sinusoidal load disturbance, from Fig. 7, can observe, when there is periodic longitudinally load, real work slip rate curve also has periodic variation, but scope is very little, when runway face is switched to wet runway from dry runway, control algorithm still can recognize the variation of runway condition, thereby selects optimal slip rate as system works point; From Fig. 8, can observe, although the variation of slip rate is little, but its corresponding friction resistance curve changes still greatly, this and friction force function are nonlinear relevant, thereby cause wheel moderating ratio to change also obvious (as shown in figure 10), but due to the periodic feature of load, speed is the integration of moderating ratio curve, therefore wheel velocity variations is pulsation-free very still, does not occur the even phenomenon of locking of large concussion.

Claims (1)

1. neural network extreme based on sliding formwork differentiator Control Method for Airplane Antiskid Braking System search, that be applicable to many wheel brakes system, it is characterized in that, adopt numerical differentiation device on-line reorganization body restriction signal, avoid carrying out body dynamics calculation and parameter estimation;

In order to guarantee the antijamming capability of numerical differentiation device, take speed signal to carry out the mode of numerical differentiation; What adopt is to obtain for acceleration signal based on Nonlinear Tracking Differentiator or Second Order Sliding Mode differentiator;

When adopting Second Order Sliding Mode differentiator:

Design of Sliding Mode Observer is

$\stackrel{\stackrel{\&CenterDot;}{^}}{x}\left(t\right)=u\left(t\right),\hat{x}\left(0\right)=0---\left(1\right)$

If

$u\left(t\right)-(\mathrm{\&rho;}+L)\frac{\mathrm{\&sigma;}\left(t\right)}{\left|\right|\mathrm{\&sigma;}\left(t\right)\left|\right|},\mathrm{\&sigma;}\left(t\right)=y\left(t\right)-\hat{x}\left(t\right),\mathrm{\&rho;}>0---\left(2\right)$ In finite time,

meanwhile, there is discrete high frequency flutter characteristic in u (t), by low-pass filter, carries out smothing filtering, can obtain

${\hat{u}}_{\mathrm{eq}}=\mathrm{LPF}\left(u\left(t\right)\right)---\left(3\right)$

Its evaluated error, and the timeconstantτ of low-pass filter is directly proportional

Because the precision of single order sliding formwork differentiator is because Second Order Sliding Mode Control has not only guaranteed sliding variable σ (t) convergence in finite time, guaranteed its first derivative simultaneously

convergence; Adopt Second Order Sliding Mode differentiator, be specially Super-twisting sliding formwork differentiator:

Sliding mode observer

$\mathrm{\&sigma;}\left(t\right)=y\left(t\right)-\hat{x}\left(t\right)---\left(5\right)$

u _i=α _i|σi| ^1/2signσ _i(t)+β _i∫signσ _i(t)dτ (6)

Here at finite time

$\begin{array}{c}{\mathrm{\&alpha;}}_i=1.5\sqrt{L_i}\\ {\mathrm{\&beta;}}_i=1.1L_i\\ \left|{\stackrel{\&CenterDot;}{F}}_i(x,t)\right|\&le;L_i\\ u\left(t\right)=\left\{u_i\left(t\right)\right\}\\ F\left(t\right)=\left\{F_i(x,t)\right\},i=1,...,n\end{array}$

Guaranteed in finite time,

simultaneously here

${\stackrel{\widehat{\&CenterDot;}}{x}}_i\left(t\right)={\stackrel{\&CenterDot;}{x}}_i\left(t\right)=u_i\left(t\right)={\mathrm{\&alpha;}}_i{\left|{\mathrm{\&sigma;}}_i\right|}^{1/2}{\mathrm{sign\&sigma;}}_i\left(t\right)+{\mathrm{\&beta;}}_i\&Integral;{\mathrm{sign\&sigma;}}_i\left(t\right)\mathrm{d\&tau;}$

Or

${\stackrel{\widehat{\&CenterDot;}}{x}}_i\left(t\right)={\mathrm{\&beta;}}_i\&Integral;{\mathrm{sign\&sigma;}}_i\left(t\right)\mathrm{d\&tau;}$

Here due to high frequency differential term has been carried out to integration, u (t) is continuous, does not need that u (t) is carried out to LPF and obtains u _eq;

When adopting Nonlinear Tracking Differentiator:

For Second Order Integral system

$\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2=u\end{array}---\left(8\right)$

Here | u|≤r, v is x ₁reference value, the time optimal solution of u is

$u=-\mathrm{rsign}(x_1-v+\frac{x_2\left|x_2\right|}{2r})---\left(9\right)$

Can obtain following dynamic process:

$\begin{array}{c}{\stackrel{\&CenterDot;}{v}}_1=v_2\\ {\stackrel{\&CenterDot;}{v}}_2=-\mathrm{rsign}(v_1-v+\frac{v_2\left|v_2\right|}{2r})\end{array}---\left(10\right)$

Here v ₁reference locus, v ₂it is its differential; According to the physical restriction of concrete utilization, can be by selecting the size of r to accelerate or slowing down dynamic process; Bang-Bang characteristic due to function has chatter phenomenon when system enters stable state, for fear of this chatter phenomenon, adapts to the demand of numerical calculation, for discrete system

$\begin{array}{c}{v}_1(k+1)=v_1\left(k\right)+\mathrm{hv}2\left(k\right)\\ v_2(k+1)=v_2\left(k\right)-\mathrm{ru}\left(k\right),\left|u\left(k\right)\right|\&le;r\end{array}---\left(11\right)$

Have

u=fd(v ₁，v ₂，r ₀，h) (12)

Here h is the sampling period, r ₀and h ₀controller parameter, f _dcan be expressed as

$f_d={-r}_0(\frac{a}{d}-\mathrm{sign}\left(a\right))s_a-r_0\mathrm{sign}\left(a\right)$

Here

$\begin{array}{c}d=h_0{r}_0^2,a_0=h_0{v}_2,y=v_1+a_0\\ a_1=\sqrt{d(d+8|y\left|\right)}\\ a_2=a_0+\mathrm{sign}\left(y\right)(a_1-d)/2\\ s_y=(\mathrm{sign}(y+d)-\mathrm{sign}(y-d))/2\\ a=(a_0+y-a_2)s_g+a_2\\ s_a=(\mathrm{sign}(y+d)-\mathrm{sign}(g-d))/2\end{array}$

Utilize function f _d(v ₁, v ₂, r ₀, h) set up steepest feedback system, obtained good tracking and differential effect;

Adaptive neural network feedforward compensation:

With following RBF neural network Approximation of Continuous Functions non-linear friction force function

neural network configuration is

$\widehat{\mathrm{\&mu;}}=W^T\mathrm{\&Phi;}\left(\mathrm{\&lambda;}\right)$

Here slip rate is inputted as NN, weight vector nN interstitial content l > 1; And Φ (λ)=[φ ₁(λ), φ ₂(λ) ..., φ _l(λ)] ^t, wherein

${\mathrm{\&phi;}}_i=\exp \left[\frac{-{(\mathrm{\&lambda;}-c_i)}^T(\mathrm{\&lambda;}-c_i)}{{\mathrm{\&eta;}}_i^2}\right],i=\mathrm{1,2},...,l$

C wherein _i=[c ₁, c ₂..., c _l] ^tto hold Yu center, η _iit is the width of Gaussian function; Neural network is proved to be to compact

in can arbitrary accuracy follow the tracks of arbitrary smooth function;

$\mathrm{\&mu;}\left(\mathrm{\&lambda;}\right)=W^{*}\mathrm{\&Phi;}\left(\mathrm{\&lambda;}\right)+\&Element;,\&ForAll;\mathrm{\&lambda;}\&Element;{\mathrm{\&Omega;}}_{\mathrm{\&lambda;}}$

Here W ^*be desirable constant weight vector, ∈ is approximate error; In ANN (Artificial Neural Network) Control, resulting stability conclusion is generally half overall situation, and input variable λ is some compacting of presetting

in, compact Ω here _λcan select as requested arbitrary size, as long as the node of nerve network controller is abundant, total energy guarantees closed loop system bounded;

Definition optimal weights W ^*for being defined as

Ω _λ，W ^*，|∈|＜∈ ^*，∈ ^*＞0λ∈Ω _λ

Neural network evaluated error is defined as

Friction force extremum search

By neural network, approach gained non-linear friction force function, try to achieve optimal slip rate function

${\mathrm{\&lambda;}}_r=\underset{\mathrm{\&lambda;}\&Element;\left[\mathrm{0,1}\right]}{\arg \max }{W}^T\mathrm{\&phi;}\left(\mathrm{\&lambda;}\right)---\left(14\right)$

The discountinuity bringing in order to eliminate numerical calculation, is used second order filter to obtain a smooth reference signal λ _d, for closed loop system, follow the tracks of

$\frac{{\mathrm{\&lambda;}}_d}{{\mathrm{\&lambda;}}_r}=\frac{{\mathrm{\&omega;}}_n^2}{s^2+2{\mathrm{\&zeta;}}_n{\mathrm{\&omega;}}_n+{\mathrm{\&omega;}}_n^2}---\left(15\right)$

Self adaptation antiskid brake design of control law:

First defining optimal slip rate is λ _d, definition wheel velocity error is

e _w=v _w-Vλ _d (16)

Further, definition PI type error

$s_w=e_w+k_i{\&Integral;}_0^t{e}_w\mathrm{dt}---\left(17\right)$

Relative velocity reference signal can be expressed as

$v_{\mathrm{wd}}={\mathrm{V\&lambda;}}_d-k_i{\&Integral;}_0^t{e}_w\mathrm{dt}$

Drive slip rate to there being most set point λ _d, so that friction force F _wmaximize; In order to obtain smooth reference signal, wheel velocity error can be defined as e _w=v _w-v _xλ _d; Further, a PI type error can be expressed as

$s_w=e_w+\mathrm{\&alpha;}{\&Integral;}_0^t{e}_w\mathrm{dt}---\left(18\right)$

For the ease of analyzing, definition v _wdfor

$v_{\mathrm{wd}}=v_x{\mathrm{\&lambda;}}_D+\mathrm{\&alpha;}{\&Integral;}_0^t{e}_w\mathrm{dt}---\left(19\right)$

The robust nonlinear neural network control proposing can be expressed as

T _b=T _bn+T _ba+T _br (20)

This control law comprises 3 parts: body acceleration compensation T _ba, neural network feedforward compensation T _bfwith robust item, wherein T _bnand T _babe expressed as

$T_{\mathrm{ba}}=\frac{J}{r_w}(-k_{\mathrm{PI}}(v_w-v_{\mathrm{wd}})-\stackrel{\widehat{\&CenterDot;}}{v}+\frac{F_z\widehat{\mathrm{\&mu;}}{r}^2}{J})---\left(21\right)$

$T_{\mathrm{bf}}=\frac{r_w}{J}(\frac{{\widehat{\mathrm{\&theta;}}}_x\mathrm{\&phi;}\left(v_w\right)}{m}+\frac{{\widehat{\mathrm{\&theta;}}}_z^T{\mathrm{\&phi;}}_z\left(v_w\right)\mathrm{\&mu;}\left(\mathrm{\&lambda;}\right)r_w^2}{J})---\left(22\right)$

Consider that after input constraint, design parameters is new law more

$\begin{array}{c}{\stackrel{\stackrel{\&CenterDot;}{^}}{\mathrm{\&theta;}}}_x={\mathrm{\&Gamma;}}_x{\mathrm{\&phi;}}_x(s_w-\mathrm{\&chi;})\\ \stackrel{\&CenterDot;}{\mathrm{\&chi;}}=-\mathrm{\&gamma;\&chi;}+s\left(v_w\right)\frac{r_w{T}_b}{J}\end{array},---\left(23\right)$

By defining an amended tracking error

robust item is designed to

$T_{\mathrm{br}}=-k_{r}q\left(\stackrel{\&OverBar;}{s}\right)---\left(24\right)$

Here k _r> 0 is a normal amount, q be one about

odd function.

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