CN102529976B - Vehicle running state nonlinear robust estimation method based on sliding mode observer - Google Patents

Abstract

The invention provides a vehicle running state nonlinear robust estimation method based on a sliding mode observer. The method comprises the following steps of: determining an automotive nonlinear kinetic model closer to the actual situation aiming at high mobility running conditions and complex variable road environments, designing the corresponding sliding mode observer, establishing external input quantity and measurement information of the sliding mode observer system by using low-cost vehicle-mounted wheel speed and steering wheel turning angle sensors, and thus implementing estimation of multiple running states of an automobile by a sliding mode observer estimation recursion algorithm. The method has the characteristics of strong anti-interference capacity, high precision, low cost and the like, and is easy to implement.

Description

A kind of travel condition of vehicle non linear robust method of estimation based on sliding mode observer

Technical field

The present invention relates to a kind of travel condition of vehicle non linear robust method of estimation based on sliding mode observer, its object is to pass through set up sliding mode observer and realizes the non-linear On-line Estimation to car running process, obtain automobile accurate, failure-free running state under the motor-driven operating condition of height and road environment complicated and changeable, these states can be used for the relevant control of automobile active safety, the inventive method has that antijamming capability is strong, precision is high, real-time is good and the distinguishing feature such as cost is low, belongs to automobile active safety and measures and control field.

Background technology

For preventing the generation of road traffic accident, automobile active safety technology has obtained swift and violent development in recent years.Automobile active safety technology can prevent trouble before it happens, and initiatively avoids the generation of accident, has become one of topmost developing direction of modern automobile.Common active safety technology mainly comprises anti-skid brake system (ABS) at present, vehicle electric stability program (ESP), anti-slip regulation (TCS), automatically controlled driving skid control system (ASR), four-wheel steering stabilizing control system (4WS) etc.These systems are usually directed to measurement or the estimation of the running statees such as the speed of motor tire, longitudinal speed of advance, side velocity, yaw velocity and the side slip angle of automobile, the measurement of these running statees be can be used for to follow-up automobile active safety control, therefore driving safety and the stability of its precision direct relation automobile, above-mentioned active safety control system can effectively be operated in depend on to a great extent travel condition of vehicle can be by real time, measure or estimate accurately.

In automobile active safety field, state of motion of vehicle is mainly measured or is estimated by three kinds of methods.The one, utilize onboard sensor (as inertial sensor and wheel speed sensors etc.) cheaply, the signal of its measurement is carried out to simple mathematics and calculate to obtain relevant travel condition of vehicle, this method cost is low, but because low cost sensor accuracy is poor and calculate that processing too simply exists larger measured error, thereby affected control effect.The 2nd, utilize high-precision sensor directly to measure (as utilized photoelectricity fifth wheel gauge or high-precision global navigation satellite system GNSS relevant travel condition of vehicle, especially high precision global position system GPS etc.), this method precision is high but expensive, cannot apply on a large scale.The third method is modelling tool, by the operational process of automobile is carried out to kinematics or Dynamic Modeling, simultaneously using relevant onboard sensor cheaply (as wheel speed sensors, gyroscope, accelerometer and GPS etc.) information as observation information, and then utilize suitable filtering algorithm for estimating to realize the estimation to motoring condition.The third method (being modelling tool) can realize being difficult to straight estimation of measuring, and the dimension that expanded state is estimated, also can improve the precision of measuring about straight, and cost is lower simultaneously.But the modelling tool having proposed is at present mainly the kinematics model based on automobile or car load or tire has been done compared with the kinetic model of polytenization supposition, and often the variation of outside road environment is considered not enough, at vehicle, compared with smooth running and external environment condition, disturb when less and can obtain good estimation effect and precision, but owing to being difficult to reflect that the real kinetic behavior of vehicle and situation cause that estimated accuracy is lower even disperses, and obstacle overcome ability has much room for improvement under the motor-driven operation conditions of height and road environment complicated and changeable.

Summary of the invention

For realize accurate, failure-free vehicle state estimation under the motor-driven operating mode of height and environment complicated and changeable, the present invention proposes a kind of travel condition of vehicle non linear robust method of estimation based on sliding mode observer.The method that the present invention proposes is established the Vehicle Nonlinear kinetic model of more approaching reality for the high motor-driven operation conditions of automobile and road environment complicated and changeable, and design corresponding sliding mode observer, utilize in addition vehicle-mounted wheel speed and steering wheel angle sensor cheaply to set up outside input and the metrical information of sliding mode observer system, and then realize automobile longitudinal speed of advance by the sliding mode observer estimation recursive algorithm proposing, yaw velocity, the estimation of the running state such as side velocity and side slip angle, there is antijamming capability strong, precision is high, the features such as easy realization and cost are low.

A kind of travel condition of vehicle non linear robust method of estimation based on sliding mode observer, the present invention is directed to the more front-wheel steering four-wheel automobile of current application, for adapt under high motor-driven operating condition and road conditions complicated and changeable automobile active safety control to the measurement of travel condition of vehicle with estimate needs, utilize cheaply vehicle-mounted wheel speed and steering wheel angle sensor to determine and set up outside input and metrical information, designed the sliding mode observer based on non-linear automobile dynamic quality model, by the estimation recursion of sliding mode observer, realize automobile longitudinal speed of advance, yaw velocity, the information such as side velocity and side slip angle accurately, Robust Estimation, concrete steps comprise:

1) set up the sliding mode observer of travel condition of vehicle:

The Vehicle Nonlinear kinetic model that adopts three degree of freedom, the state space equation form of this model is as follows:

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=f_1(x_1,x_2,x_3,{\mathrm{\δ}}_f)+g_1({\mathrm{\δ}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})\\ {\stackrel{\·}{x}}_2=f_2(x_1,x_2,x_3,{\mathrm{\δ}}_f)+g_2({\mathrm{\δ}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})\\ {\stackrel{\·}{x}}_3=f_3(x_1,x_2,x_3,{\mathrm{\δ}}_f)+g_3({\mathrm{\δ}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})\end{array}\right.---\left(1\right)$

In formula (1), x ₁, x ₂and x ₃for three states of sliding mode observer, and x ₁=v _x, x ₂=ω _z, x ₃=v _y, v _x, v _yand ω _zrespectively longitudinal speed of advance, side velocity and the yaw velocity of automobile;

δ _f, F _tfand F _trbe three outer input variables, wherein δ _ffront wheel steering angle, F _tfthe longitudinal force acting on single front-wheel, F _trit is the longitudinal force acting on single trailing wheel;

Each function f in formula (1) _j(j=1,2,3) and g _jthe value of (j=1,2,3) is as follows

$f_1(x_1,x_2,x_3,{\mathrm{\δ}}_f)=\frac{1}{m}[{\mathrm{mv}}_y{\mathrm{\ω}}_z+2\frac{v_y+{\mathrm{a\ω}}_z}{v_x}{C}_{\mathrm{\αf}}{\mathrm{\δ}}_f-\frac{1}{2}{C}_d{A}_f{\mathrm{\ρ}}_a{v}_x^2]$

$f_2(x_1,x_2,x_3,{\mathrm{\δ}}_f)=\frac{1}{I_z}[2a({\mathrm{\δ}}_f-\frac{(v_y+{\mathrm{a\ω}}_z)}{v_x})C_{\mathrm{\αf}}-2b{C}_{\mathrm{\αr}}\frac{({\mathrm{b\ω}}_z-v_y)}{v_x}]$

$f_3(x_1,x_2,x_3,{\mathrm{\δ}}_f)=\frac{1}{m}[-{\mathrm{mv}}_x{\mathrm{\ω}}_z+2({\mathrm{\δ}}_f-\frac{(v_y+a{\mathrm{\ω}}_z)}{v_x})C_{\mathrm{\αf}}+2C_{\mathrm{\αr}}\frac{{\mathrm{b\ω}}_z-v_y}{v_x}]$

$g_1({\mathrm{\δ}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})=\frac{2}{m}(F_{\mathrm{tf}}+F_{\mathrm{tr}})$

$g_2({\mathrm{\δ}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})=\frac{2a}{I_z}{F}_{\mathrm{tf}}{\mathrm{\δ}}_f$

$g_3({\mathrm{\δ}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})=\frac{2}{m}{F}_{\mathrm{tf}}{\mathrm{\δ}}_f$

At described f _j(j=1,2,3) and g _jin (j=1,2,3), m and I _zbe respectively the quality of vehicle and the rotor inertia of walking around the vertical axle of barycenter, a is the distance of vehicle front wheel shaft center to barycenter, and b is the distance of automobile back wheel wheel shaft center to barycenter, C _{α f}, C _{α r}represent respectively the cornering stiffness of forward and backward tire, C _drepresent aerodynamic drag factor, A _frepresent vehicle forward direction area, ρ _arepresent density of air;

For state x ₁and x ₂, corresponding v respectively _xand ω _z, on they and hind axle, the wheel speed of two non-steered wheels has following relation:

v _x＝(V _RL+V _RR)/2

ω _z＝(V _RL-V _RR)/T _W (2)

In formula (2), T _wthe wheelspan between two trailing wheels on hind axle, V _rLand V _rRrepresent respectively the linear velocity of left rear wheel and off hind wheel, i.e. the wheel linear velocity of two non-wheel flutters;

State x ₁and x ₂for the amount that can directly record, and state x ₃for the amount that cannot directly record need be estimated by sliding mode observer;

For the auto model shown in formula (1), propose and set up following sliding mode observer model:

$\left\{\begin{array}{c}{\stackrel{\stackrel{\&CenterDot;}{^}}{x}}_1=f_1({\hat{x}}_1,{\hat{x}}_2,{\hat{x}}_3,{\mathrm{\&delta;}}_f)+g_1({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})+l_1{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">s</figure-callout>}}_1+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\mathrm{sgn}\left(s_1\right)\\ {\stackrel{\stackrel{\&CenterDot;}{^}}{x}}_2=f_2({\hat{x}}_1,{\hat{x}}_2,{\hat{x}}_3,{\mathrm{\&delta;}}_f)+g_2({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})+l_2{s}_2+k_2\mathrm{sgn}\left(s_2\right)\\ {\stackrel{\stackrel{\&CenterDot;}{^}}{x}}_3=f_3({\hat{x}}_1,{\hat{x}}_2,{\hat{x}}_3,{\mathrm{\&delta;}}_f)+g_3({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})+{\mathrm{\&tau;}}_1\mathrm{sgn}\left(s_1\right)+{\mathrm{\&tau;}}_2\mathrm{sgn}\left(s_2\right)\end{array}\right.---\left(3\right)$

In formula (3),

with represent respectively x ₁, x ₂and x ₃calculating estimated valve, l _j(j=1,2) are the error convergence gains outside sliding-mode surface, and k _j(j=1,2) and τ _j(j=1,2) have represented the error convergence gain on sliding-mode surface, and sgn (.) is sign function, i.e. symbolic function; s ₁and s ₂what be defined as sliding mode observer can directly survey state x ₁, x ₂with estimated valve separately

between error,

${\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">s</figure-callout>}}_1=x_1-{\hat{x}}_1,$ $s_2=x_2-{\hat{x}}_2---\left(4\right)$

2) carry out the estimation recursion of discretization:

In actual estimation procedure, need to adopt the estimation recursive form of discretization, for this reason, by the sliding mode observer of above-mentioned design, be that formula (3) is carried out discretization processing,

${\hat{x}}_1\left(k\right)={\hat{x}}_1(k-1)+T{\hat{x}}_2(k-1){\hat{x}}_3(k-1)+2T{C}_{\mathrm{\&alpha;f}}{\mathrm{\&delta;}}_f(k-1)\frac{{\hat{x}}_3(k-1)+a{\hat{x}}_2(k-1)}{m{\hat{x}}_1(k-1)}-\frac{T}{2m}{C}_d{A}_f{\mathrm{\&rho;}}_a{{\hat{x}}_1}^2(k-1)$

$+\frac{2T}{m}[F_{\mathrm{tf}}(k-1)+F_{\mathrm{tr}}(k-1)]+l_{1}T[v_x^{\&prime;}(k-1)-{\hat{x}}_1(k-1)]+k_{1}T\&CenterDot;\tanh [\mathrm{\&lambda;}{v}_x^{\&prime;}(k-1)-\mathrm{\&lambda;}{\hat{x}}_1(k-1)]$

${\hat{x}}_2\left(k\right)={\hat{x}}_2(k-1)+\frac{2{\mathrm{TaC}}_{\mathrm{\&alpha;f}}}{I_z}[{\mathrm{\&delta;}}_f(k-1)+\frac{-{\hat{x}}_3(k-1)-a{\hat{x}}_2(k-1)}{{\hat{x}}_1(k-1)}]-2{\mathrm{TbC}}_{\mathrm{ar}}\frac{b{\hat{x}}_2(k-1)-{\hat{x}}_3(k-1)}{I_z{\hat{x}}_1(k-1)}$

$+\frac{2\mathrm{Ta}}{I_z}{F}_{\mathrm{tf}}(k-1){\mathrm{\&delta;}}_f(k-1)+l_{2}T[{\mathrm{\&omega;}}_z^{\&prime;}(k-1)-{\hat{x}}_2(k-1)]+k_{2}T\&CenterDot;\tanh [{\mathrm{\&lambda;\&omega;}}_z^{\&prime;}(k-1)-\mathrm{\&lambda;}{\hat{x}}_2(k-1)]---\left(5\right)$

${\hat{x}}_3(k-1)={\hat{x}}_3(k-1)-T{\hat{x}}_1(k-1){\hat{x}}_2(k-1)+\frac{{2\mathrm{TC}}_{\mathrm{\&alpha;f}}}{m}[{\mathrm{\&delta;}}_f(k-1)-\frac{({\hat{x}}_3(k-1)+a{\hat{x}}_2(k-1))}{{\hat{x}}_1(k-1)}]$

$+2{\mathrm{TC}}_{\mathrm{\&alpha;r}}\frac{b{\hat{x}}_2(k-1)-{\hat{x}}_3(k-1)}{m{\hat{x}}_1(k-1)}+\frac{2T}{m}{F}_{\mathrm{tf}}(k-1){\mathrm{\&delta;}}_f(k-1)$

In formula (5), k represents the discretization moment, and T represents the discrete cycle, and tanh (.) is hyperbolic tangent function, and λ is a design parameters that is used for adjusting hyperbolic tangent function inclined degree, v ' _xand ω ' _zrepresent respectively to measure the longitudinal direction of car speed of advance and the yaw velocity that obtain, i.e. v ' by wheel speed sensors _xand ω ' _zrepresent respectively v _xand ω _zcontaining noisy observed reading;

In above-mentioned filtering recursion computation process, can determine the longitudinal speed of advance v of automobile in each moment _x(k), yaw velocity ω _zand side velocity v (k) _y, and then according to following formula, can determine the side slip angle in each moment (k)

β(k)＝arctan[v _y(k)/v _x(k)] (6)。

Longitudinally speed of advance and yaw velocity are the amounts that can directly record, and by wheel speed sensors, are measured and are obtained longitudinal direction of car speed of advance and yaw velocity, are specially and utilize two left rear wheel angular velocity omegas that wheel speed sensors records on hind axle _rLwith off hind wheel angular velocity omega _rRbe multiplied by tire radius R and obtain V ' _rL=R ω _rLand V ' _rR=R ω _rR, V ' _rLand V ' _rRrepresent respectively V _rLand V _rRcontaining noisy observed reading and

wherein

with

the additivity that represents respectively the wheel linear velocity of left rear wheel and off hind wheel is measured noise, and then recycling formula (2) obtains longitudinal speed of advance and yaw velocity contains noisy observed reading v ' _xand ω ' _z;

For three outer input variables in formula (3), by method below, determine:

Front wheel steering angle δ _fthe steering wheel angle δ that can record by steering wheel angle sensor is divided by the steering gear ratio q from bearing circle to front-wheel _tdetermine, i.e. δ _f=δ/q _t;

Longitudinal force of tire F _tfand F _trcan determine according to the non-linear tire model of Dugoff;

For determining F _tfand F _tr, introduce longitudinal direction of car slip rate i _sj(j=f, r), i _sjbe divided into again front wheel spindle straight skidding rate i _sfwith hind axle straight skidding rate i _sr, in this method, subscript j gets f or r represents respectively front or rear wheel shaft, i _sjmethod of calculating is:

and j=f, r (7),

In formula (7), v _tfand v _trrepresent respectively the speed along tire direction on forward and backward wheel shaft, v _tfand v _trcan unify to be designated as v _tj(j=f, r); ω _fthe spin velocity on front wheel spindle is converted in the spin velocity equivalence that represents two wheels on front wheel spindle; ω _rrepresent that on hind axle, the spin velocity on hind axle, ω are converted in two rotation of wheel cireular frequency equivalences _fand ω _rcan unify to be designated as ω _j(j=f, r) and

${\mathrm{\&omega;}}_f=\frac{1}{2}({\mathrm{\&omega;}}_{\mathrm{fR}}+{\mathrm{\&omega;}}_{\mathrm{fL}})$

(8)

${\mathrm{\&omega;}}_r=\frac{1}{2}({\mathrm{\&omega;}}_{\mathrm{rR}}+{\mathrm{\&omega;}}_{\mathrm{rL}})$

In formula (8), ω _fL, ω _fR, ω _rLand ω _rRrepresent respectively the spin velocity of the near front wheel, off front wheel, left rear wheel and off hind wheel, by utilizing four wheel speed sensors to measure, obtain;

V _tj(j=f, r) can determine by formula (9):

v _tf＝v _xcosδ _f+(v _y+aω _z)sinδ _f

(9)

v _tr＝v _x

And then, longitudinal force of tire F _tfand F _trcan determine by through type (10)

$F_{\mathrm{tj}}=\frac{C_{\mathrm{tj}}{i}_{\mathrm{sj}}}{1-i_{\mathrm{sj}}}{f}_t\left(p_j\right)$ (j＝f，r) (10)

In formula (10), C _tfand C _trrepresent respectively the longitudinal rigidity of single forward and backward tire, the unified C that is designated as _tj(j=f, r); Variable p _j(j=f, r) and function f _t(p _j) (j=f, r) determined by following formula:

$p_j=\frac{{\mathrm{\&mu;F}}_{\mathrm{zj}}(1-{\mathrm{\&epsiv;}}_r{v}_x\sqrt{{i_{\mathrm{sj}}}^2+{\tan }^2{\mathrm{\&alpha;}}_j})(1-i_{\mathrm{sj}})}{2\sqrt{{C_{\mathrm{tj}}}^2{i_{\mathrm{sj}}}^2+{C_{\mathrm{\&alpha;j}}}^2{\tan }^2{\mathrm{\&alpha;}}_j}}$ (j＝f，r) (11)

$f_t\left(p_j\right)=\left\{\begin{array}{cc}{p}_j(2-p_j)& p_j<1\\ 1& p_j\&GreaterEqual;1\end{array}\right.$ (j＝f，r) (12)

In formula (11) and (12), μ represents the vertical friction coefficient between tire and ground; ε _rrepresent that road adheres to decay factor; α _f, α _rrepresent respectively the sideslip angle of forward and backward tire, the unified α that is designated as _j(j=f, r), can be calculated as follows

${\mathrm{\&alpha;}}_f={\mathrm{\&delta;}}_f-\frac{v_y+{\mathrm{a\&omega;}}_z}{v_x},$ ${\mathrm{\&alpha;}}_r=\frac{{\mathrm{b\&omega;}}_z-v_y}{v_x}---\left(13\right)$

And F _zj(j=f, r) represents be assigned to the vertical load on front or rear wheel shaft and can be calculated as follows

$F_{\mathrm{zf}}=\frac{\mathrm{mgb}}{2(a+b)},$ $F_{\mathrm{zr}}=\frac{\mathrm{mga}}{2(a+b)}---\left(14\right)$

In formula (14), g represents acceleration due to gravity;

For each gain of sliding mode observer in formula (3), according to the convergence stability principle of observer, design definitely, particularly can determine gain l according to formula below ₁, k ₁, l ₂, k ₂, τ ₁and τ ₂,

${\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">l</figure-callout>}}_1>\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{\&PartialD;x_1},$ ${\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">k</figure-callout>}}_1>\left|\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_2}{s}_2\right|+\left|\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_3}{\tilde{x}}_3\right|+{\mathrm{\&eta;}}_1---\left(15\right)$

$l_2>\frac{{\&PartialD;f}_2}{\&PartialD;x_2},$ $k_2>\left|\frac{{\&PartialD;f}_2}{{\&PartialD;x}_1}{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">s</figure-callout>}}_1\right|+\left|\frac{{\&PartialD;f}_2}{{\&PartialD;x}_3}{\tilde{x}}_3\right|+{\mathrm{\&eta;}}_2---\left(16\right)$

τ ₁＝τ ₂＝0 (17)

In formula (15) and formula (16),

(j=1,2,3) representative function f ₁for x _jthe partial derivative of (j=1,2,3),

(j=1,2,3) representative function f ₂for x _jthe partial derivative of (j=1,2,3), η _j(j=1,2) are given positive number, ${\tilde{x}}_3=x_3-{\hat{x}}_3;$

In actual recursion, sgn (.) function occurring in formula (3) sgn below _eq(.) function replaces:

sgn _eq(s _j)＝tanh(λ _sj)(j＝1，2) (18)。

The representative value of discrete period T is taken as 10 milliseconds, 20 milliseconds or 50 milliseconds.

Beneficial effect

1. the present invention proposes a kind of travel condition of vehicle non linear robust method of estimation based on sliding mode observer, can be used for about automobile active safety control is to the robust measure of travel condition of vehicle and estimation needs.

2. the inventive method proposes for the high motor-driven operation conditions of automobile and road environment complicated and changeable, can under the motor-driven operating mode of height and road environment complicated and changeable, realize to travel condition of vehicle accurately, reliably estimation.

3. the travel condition of vehicle non linear robust method of estimation based on synovial membrane observer that the present invention proposes not only can significantly improve the straight precision of measuring such as automobile longitudinal speed of advance and yaw velocity, and can realize the robust that side slip angle, side velocity etc. is difficult to directly measure and accurately estimate.

4. the method that the present invention proposes has that antijamming capability is strong, precision is high, cost is low and the feature such as real-time is good.

Accompanying drawing explanation

Fig. 1. vehicle dynamic model

Fig. 2. longitudinal speed of advance (meter per second-m/s) of setting and steering wheel angle (degree) temporal evolution figure

Fig. 3. side slip angle (radian-rad) curve and the partial enlarged drawing (vertical coefficientoffrictionμ=1 o'clock between tire and ground) over time of the inventive method and Carsim output

Fig. 4. the side slip angle that the inventive method obtains is with respect to the curve of error (vertical coefficientoffrictionμ=1 o'clock between tire and ground) of the side slip angle reference value of Carsim output

Fig. 5. side slip angle (radian-rad) curve and the partial enlarged drawing (vertical coefficientoffrictionμ=0.8 o'clock between tire and ground) over time of the inventive method and Carsim output

Fig. 6. the side slip angle that the inventive method obtains is with respect to the curve of error (vertical coefficientoffrictionμ=0.8 o'clock between tire and ground) of the side slip angle reference value of Carsim output

Fig. 7. side slip angle (radian-rad) curve and the partial enlarged drawing (vertical coefficientoffrictionμ=0.6 o'clock between tire and ground) over time of the inventive method and Carsim output

Fig. 8. the side slip angle that the inventive method obtains is with respect to the curve of error (vertical coefficientoffrictionμ=0.6 o'clock between tire and ground) of the side slip angle reference value of Carsim output

The specific embodiment

Embodiment 1

Current, between traffic safety, topic becomes increasingly conspicuous, and has become a global difficult problem.For preventing the generation of road traffic accident, automobile active safety technology has obtained swift and violent development in recent years.Automobile active safety technology can prevent trouble before it happens, and initiatively avoids the generation of accident, has become one of topmost developing direction of modern automobile.Common active safety technology mainly comprises anti-skid brake system (ABS) at present, vehicle electric stability program (ESP), anti-slip regulation (TCS), automatically controlled driving skid control system (ASR), four-wheel steering stabilizing control system (4WS) etc.These systems are usually directed to measurement or the estimation of the running statees such as the speed of motor tire, longitudinal speed of advance, side velocity, yaw velocity and the side slip angle of automobile, the measurement of these running statees can be used for follow-up automobile active safety control, therefore driving safety and the stability of its precision direct relation automobile, above-mentioned active safety control system can effectively be operated in depend on to a great extent travel condition of vehicle can be by real time, measure or estimate accurately.

In automobile active safety field, state of motion of vehicle is mainly measured or is estimated by three kinds of following methods at present:

The one, utilize onboard sensor (as inertial sensor and wheel speed sensors etc.) cheaply, the signal of its measurement is carried out to simple mathematics and calculate to obtain relevant travel condition of vehicle.For example, for automobile side slip angle, can utilize vertical and horizontal accelerometer first to record the acceleration/accel along both direction, then integral operation obtains respectively longitudinal speed of advance and side velocity, and then can try to achieve side slip angle.Although this method cost is low, because low cost sensor accuracy is poor and calculate that processing too simply exists larger measured error, thereby affected control effect.

The 2nd, utilize high-precision sensor directly to measure (as utilized photoelectricity fifth wheel gauge or high-precision global navigation satellite system GNSS relevant travel condition of vehicle, especially high precision global position system GPS etc.), this method precision is high but expensive, cannot apply on a large scale.

The third method is modelling tool, by the operational process of automobile is carried out to kinematics or Dynamic Modeling, simultaneously using relevant onboard sensor cheaply (as wheel speed sensors, gyroscope, accelerometer and GPS etc.) information as observation information, and then utilize suitable filtering algorithm for estimating (as Luenberger observer, nonlinear observer, Kalman filtering or neural network etc.) to realize the estimation to motoring condition.The third method (being modelling tool) can realize the estimation about being difficult to straight measurement, and the dimension that expanded state is estimated also can improve the precision of measuring about straight, and cost is lower simultaneously.But the modelling tool having proposed is at present mainly the kinematics model based on automobile or car load or tire has been done compared with the kinetic model of polytenization supposition, and often the variation of outside road environment is considered not enough, in vehicle smooth running and external environment condition, disturb when less and can obtain good estimation effect and precision, but owing to being difficult to reflect that the real kinetic behavior of vehicle and situation cause that estimated accuracy is lower even disperses, and obstacle overcome ability has much room for improvement under the motor-driven operation conditions of height and road environment complicated and changeable.

For realize accurate, failure-free vehicle state estimation under the motor-driven operating mode of height and environment complicated and changeable, the present invention proposes a kind of travel condition of vehicle non linear robust method of estimation based on sliding mode observer.The inventive method is accurate for the travel condition of vehicle under the high motor-driven operation conditions of automobile and road environment complicated and changeable, Robust Estimation proposes, have that antijamming capability is strong, precision is high, real-time is good and the feature such as cost is low, can be used for automobile active safety control field.In the present invention, high motor-driven operation refers to when automotive operation is during at common road traffic environment, need to turn to frequently and the Run-time scenario (within lateral acceleration 0.7g, g represents acceleration due to gravity) of acceleration and deceleration; Antijamming capability refer to the observer that (adheres to the variation of the environmental factors such as condition as road) under traffic environment complicated and changeable still can realize to travel condition of vehicle accurately, the estimation of robust.Concrete thought of the present invention is as follows:

For adapting to automobile active safety control under high motor-driven environment, to the measurement of travel condition of vehicle signal and estimation requirement, first automobile is carried out to suitable Dynamic Modeling.For application of the present invention, the present invention, for the front-wheel steering four wheeler (should have at present the widest situation, exemplary is as the car of front-wheel steering) travelling under common road traffic environment, can do following reasonable assumption:

1) ignore pitching, inclination and the upper and lower bounce motion of automobile.

2) ignore automotive suspension to the impact on tire axle.

3) ignore roll motion, can think that deflection angle, sideslip angle, longitudinal force and the side force of two tires in left and right are identical on automobile front axle; Similarly, on can assumed vehicle rear axle, sideslip angle, longitudinal force and the side force of two tires in left and right be identical.

According to above-mentioned application requirements and supposition, the present invention is directed to the more front-wheel steering four-wheel automobile of current application, adopt the vehicle dynamic model (being equivalent to forward and backward wheel by an imaginary Bicycle model that concentrates on respectively automobile axle mid point and form, as shown in Fig. 1 right side after equivalent-simplification) shown in accompanying drawing 1.This model has 3 degree of freedom, is respectively that longitudinal movement, sideway movement and yaw rotate.In Fig. 1, defined vehicle carrier coordinate system, its initial point o is positioned at barycenter place, and ox axle is along the longitudinal axis of vehicle consistent with vehicle forward direction, oz axle perpendicular to vehicle operating plane directed towards ground (downward, around the yaw velocity ω of oz axle _zpositive dirction define as shown), and oy axle can be determined by right-handed helix rule.Longitudinally speed of advance v _x, side velocity v _ywith yaw velocity ω _zall refer to vehicle barycenter.According to Newtonian mechanics, the kinetic model of vehicle can be described as

$m{\stackrel{\&CenterDot;}{v}}_x={\mathrm{mv}}_y{\mathrm{\&omega;}}_z+2F_{\mathrm{tf}}\cos {\mathrm{\&delta;}}_f-2F_{\mathrm{sf}}\sin {\mathrm{\&delta;}}_f+{2F}_{\mathrm{tr}}-\frac{1}{2}{C}_d{A}_f{\mathrm{\&rho;}}_a{v}_x^2$

$I_z{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_z=2{\mathrm{aF}}_{\mathrm{tf}}{\mathrm{\&delta;}}_f+2{\mathrm{aF}}_{\mathrm{sf}}\cos {\mathrm{\&delta;}}_f-{2\mathrm{bF}}_{\mathrm{sr}}---\left(1\right)$

$m{\stackrel{\&CenterDot;}{v}}_y=-{\mathrm{mv}}_x{\mathrm{\&omega;}}_z+2F_{\mathrm{tf}}\sin {\mathrm{\&delta;}}_f+{2F}_{\mathrm{sf}}\cos {\mathrm{\&delta;}}_f+{2F}_{\mathrm{sr}}$

In formula, v _x, v _yand ω _zrespectively longitudinal speed of advance, side velocity and the yaw velocity of automobile, m and I _zbe respectively the quality of vehicle and the rotor inertia around oz axle, a is the distance of vehicle front wheel shaft center to barycenter, and b is the distance of automobile back wheel wheel shaft center to barycenter, δ _ffront wheel steering angle, C _drepresent aerodynamic drag factor, A _frepresent vehicle forward direction area, ρ _arepresent density of air, F _tfthe longitudinal force acting on single front-wheel, F _trthe longitudinal force acting on single trailing wheel, F _sfthe side force acting on single front-wheel, F _srit is the side force acting on single trailing wheel.

For the vehicle travelling at Ordinary road traffic environment, conventionally the side force acting on each wheel can be expressed as

F _sf＝C _αfα _f，F _sr＝C _αrα _r (2)

In formula (2), C _{α f}, C _{α r}represent respectively the cornering stiffness of forward and backward tire, α _f, α _rrepresent respectively the sideslip angle of forward and backward tire and can be expressed as

${\mathrm{\&alpha;}}_f={\mathrm{\&delta;}}_f-\frac{v_y+{\mathrm{a\&omega;}}_z}{v_x},$ ${\mathrm{\&alpha;}}_r=\frac{b{\mathrm{\&omega;}}_z-v_y}{v_x}---\left(3\right)$

By formula (2), (3) substitution formula (1), and consider δ _fnormally low-angle, i.e. sin δ _f≈ δ _f, cos δ _f≈ 1 and ignore second order and above high-order trace can obtain after arranging

${\stackrel{\&CenterDot;}{v}}_x=\frac{1}{m}[{\mathrm{mv}}_y{\mathrm{\&omega;}}_z+2\frac{v_y+a{\mathrm{\&omega;}}_z}{v_x}{C}_{\mathrm{\&alpha;f}}{\mathrm{\&delta;}}_f-\frac{1}{2}{C}_d{A}_f{\mathrm{\&rho;}}_a{v}_x^2]+\frac{2}{m}(F_{\mathrm{tf}}+F_{\mathrm{tr}})$

${\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_z=\frac{1}{I_z}[2a({\mathrm{\&delta;}}_f-\frac{(v_y+{\mathrm{a\&omega;}}_z)}{v_x})C_{\mathrm{\&alpha;f}}-2{\mathrm{bC}}_{\mathrm{\&alpha;r}}-\frac{({\mathrm{b\&omega;}}_z-v_y)}{v_x}]+\frac{2a}{I_z}{F}_{\mathrm{tf}}{\mathrm{\&delta;}}_f---\left(4\right)$

${\stackrel{\&CenterDot;}{v}}_y=\frac{1}{m}[{\mathrm{mv}}_x{\mathrm{\&omega;}}_z+2({\mathrm{\&delta;}}_f-\frac{(v_y+{\mathrm{a\&omega;}}_z)}{v_x})C_{\mathrm{\&alpha;f}}+{2C}_{\mathrm{\&alpha;r}}\frac{{\mathrm{b\&omega;}}_z-v_y}{v_x}]+\frac{2}{m}{F}_{\mathrm{tf}}{\mathrm{\&delta;}}_f$

For the front wheel steering angle δ in formula (4) _fthe steering wheel angle δ that can record by steering wheel angle sensor is divided by the steering gear ratio q from bearing circle to front-wheel _tdetermine (to be δ _f=δ/q _t).And for the longitudinal force of tire F in formula (4) _tfand F _trthe present invention adopts the non-linear tire model of Dugoff to estimate to determine [can list of references: Dugoff H., Fancher P.S., Segel L..An Analysis of Tire Traction Properties and TheirInfluence on Vehicle Dynamic Performance.SAE Transactions, 79:341-366,1970.SAE Paper No.700377].For this reason, introduce longitudinal direction of car slip rate i _sj(j=f, r) (can be divided into again front wheel spindle straight skidding rate i _sfwith hind axle straight skidding rate i _sr, in the present invention, subscript j gets f or r represents respectively front or rear wheel shaft), it calculates with the acceleration and deceleration situation of vehicle closely related, is specially

and j=f, r (5)

In formula (5), R represents wheel tyre radius (under normal circumstances, can think that the tire radius of four wheels is identical), v _tfand v _trrepresent that respectively the speed along tire direction on forward and backward wheel shaft (is mark convenience, v _tfand v _trcan unify to be designated as v _tj(j=f, r)), ω _fthe spin velocity on front wheel spindle, ω are converted in the spin velocity equivalence that represents two wheels on front wheel spindle _rrepresent that on hind axle, the spin velocity (ω on hind axle is converted in two rotation of wheel cireular frequency equivalences _fand ω _rcan unify to be designated as ω _j(j=f, r)), its computing formula is as follows

${\mathrm{\&omega;}}_f=\frac{1}{2}({\mathrm{\&omega;}}_{\mathrm{fR}}+{\mathrm{\&omega;}}_{\mathrm{fL}})$

(6)

${\mathrm{\&omega;}}_r=\frac{1}{2}({\mathrm{\&omega;}}_{\mathrm{rR}}+{\mathrm{\&omega;}}_{\mathrm{rL}})$

In formula (6), ω _fL, ω _fR, ω _rLand ω _rRrepresent respectively the spin velocity of the near front wheel, off front wheel, left rear wheel and off hind wheel, by utilizing four wheel speed sensors to measure, obtain.

In addition, according to the movement relation shown in Fig. 1, v _tj(j=f, r) can determine by following formula

v _tf＝v _xcosδ _f+(v _y+aω _z)sinδ _f

(7)

v _tr＝v _x

[can list of references: Dugoff H. according to Dugoff tire model, Fancher P.S., Segel L..AnAnalysis of Tire Traction Properties and Their Influence on Vehicle DynamicPerformance.SAE Transactions, 79:341-366,1970.SAE Paper No.700377], longitudinal force of tire F _tfand F _trcan determine by following formula

$F_{\mathrm{tj}}=\frac{C_{\mathrm{tj}}{i}_{\mathrm{sj}}}{1-i_{\mathrm{sj}}}{f}_t\left(p_j\right)$ (j＝f，r) (8)

In formula (8), C _tfand C _trthe longitudinal rigidity that represents respectively single forward and backward tire (can be united and be designated as C _tj(j=f, r)), variable p _j(j=f, r) and function f _t(pj) (j=f, r) determined or defined by following formula

$p_j=\frac{\mathrm{\&mu;}{F}_{\mathrm{zj}}(1-{\mathrm{\&epsiv;}}_r{v}_x)\sqrt{{i_{\mathrm{sj}}}^2+{\tan }^2{\mathrm{\&alpha;}}_j}(1-i_{\mathrm{sj}})}{2\sqrt{{C_{\mathrm{tj}}}^2{i_{\mathrm{sj}}}^2+{C_{\mathrm{\&alpha;j}}}^2{\tan }^2{\mathrm{\&alpha;}}_j}}$ j＝f，r (9)

$f_t\left(p_j\right)=\left\{\begin{array}{cc}{p}_j(2-p_j)& p_j<1\\ 1& p_j\&GreaterEqual;1\end{array}\right.$ j＝f，r (10)

In formula (9) and (10), μ represents the vertical friction coefficient between tire and ground, ε _rrepresent that road adheres to decay factor, F _zj(j=f, r) represents be assigned to the vertical load on front or rear wheel shaft and can be calculated as follows

$F_{\mathrm{zf}}=\frac{\mathrm{mgb}}{2(a+b)},$ $F_{\mathrm{zr}}=\frac{\mathrm{mga}}{2(a+b)}---\left(11\right)$

In formula (11), g represents acceleration due to gravity.

The model of describing for formula (4), it is a non-linear vehicle dynamic model with 3DOF, is different from the linear auto model of the 2DOF often adopting.In the linear auto model of the 2DOF often adopting, longitudinal speed of advance of vehicle is considered to permanent, and auto model is only the linear differential equation about side velocity and yaw velocity.Therefore, it is constant or change running condition (manoevreability is lower) slowly that the linear auto model of 2DOF is generally only suitable for forward speed, and for the motor-driven running condition of the high situation of acceleration and deceleration (need frequently to turn to and), there is larger modeling error in this model.And 3DOF nonlinear model of the present invention there is no permanent restriction to longitudinal speed of advance of vehicle, therefore can adapt to the accurate estimation that general motor-driven environment also can adapt to travel condition of vehicle under high motor-driven environment.

From kinematics angle, the vehicle movement shown in Fig. 1 is actually a Planar Compound motion (it is compound that longitudinal movement, sideway movement and yaw rotate), therefore according to Planar Compound movement relation, can obtain

$V_{\mathrm{RL}}=v_x+\frac{T_W}{2}{\mathrm{\&omega;}}_z$

(12)

$V_{\mathrm{RR}}=v_x-\frac{T_W}{2}{\mathrm{\&omega;}}_z$

In formula, V _rLand V _rRrepresent respectively the wheel linear velocity of left rear wheel and off hind wheel (i.e. two non-wheel flutters), T _wit is the wheelspan between two trailing wheels on hind axle.

Formula (12) is rearranged, can obtain

v _x＝(V _RL+V _RR)/2

ω _z＝(V _RL-V _RR)/T _W (13)

The wheel linear velocity that it is pointed out that left rear wheel and off hind wheel can obtain by two wheel speed sensors that are arranged on hind axle, and the cireular frequency that utilizes on hind axle two wheel speed sensors to record is multiplied by tire radius and obtains.Consider the measurement noise of wheel speed sensors, V ' _rL=R ω _rLand V ' _rR=R ω _rR, wherein V ' _rLand V ' _rRrepresent respectively V _rLand V _rRcontaining noisy observed reading; And then, V ' _rLand V ' _rRalso can be expressed as

wherein

with

the additivity that represents respectively the wheel linear velocity of left rear wheel and off hind wheel is measured noise (all can be modeled as average be 0 Gaussian white noise).

In addition, along with the development of automotive electronic technology, respectively wheel speed information also can be obtained by the CAN bus network in Automobile, so just does not need additionally on each wheel, to install wheel speed sensors additional, more economical.In being discussed below, longitudinally speed of advance and yaw velocity are using as the amount that can directly measure.

Set up after the non-linear vehicle dynamic model shown in formula (4), next discuss and how to design Sliding Mode Robust observer.

Compared with traditional observer, sliding mode observer has been proved to be a kind of processing with disturbing and the very effective method of estimation of model uncertainty nonlinear system.Due to the property complicated and changeable of road traffic environment, the manoevreability situation of vehicle self tends to occur to change frequently (as needed frequently acceleration or deceleration and turn to etc.), can be subject in addition the impact of various uncertain external disturbance, as the variation of road-adhesion coefficient and wind are disturbed etc.For realize under high road traffic environment motor-driven, complicated and changeable to travel condition of vehicle accurately, reliably estimate, the present invention will be for the non-linear vehicle dynamic model of formula (4), the sliding mode observer suitable according to sliding formwork Theoretical Design, detailed process is as follows:

(4) are turned to the form of describing with state space equation,

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=f_1(x_1,x_2,x_3,{\mathrm{\&delta;}}_f)+g_1({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})\\ {\stackrel{\&CenterDot;}{x}}_2=f_2(x_1,x_2,x_3,{\mathrm{\&delta;}}_f)+g_2({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})\\ {\stackrel{\&CenterDot;}{x}}_3=f_3(x_1,x_2,x_3,{\mathrm{\&delta;}}_f)+g_3({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})\end{array}\right.---\left(14\right)$

In formula (14), x ₁, x ₂and x ₃for three states of sliding mode observer, and x ₁=v _x, x ₂=ω _z, x ₃=v _y, δ _tf, F _tfand F _trbe three outer input variables, and

$f_1(x_1,x_2,x_3,{\mathrm{\&delta;}}_f)=\frac{1}{m}[{\mathrm{mv}}_y{\mathrm{\&omega;}}_z+2\frac{v_y+{\mathrm{a\&omega;}}_z}{v_x}{C}_{\mathrm{\&alpha;f}}{\mathrm{\&delta;}}_f-\frac{1}{2}{C}_d{A}_f{\mathrm{\&rho;}}_a{v}_x^2]$

$f_2(x_1,x_2,x_3,{\mathrm{\&delta;}}_f)=\frac{1}{I_z}[2a({\mathrm{\&delta;}}_f-\frac{(v_y+{\mathrm{a\&omega;}}_z)}{v_x})C_{\mathrm{\&alpha;f}}-2b{C}_{\mathrm{\&alpha;r}}\frac{({\mathrm{b\&omega;}}_z-v_y)}{v_x}]$

$f_3(x_1,x_2,x_3,{\mathrm{\&delta;}}_f)=\frac{1}{m}[-{\mathrm{mv}}_x{\mathrm{\&omega;}}_z+2({\mathrm{\&delta;}}_f-\frac{(v_y+a{\mathrm{\&omega;}}_z)}{v_x})C_{\mathrm{\&alpha;f}}+2C_{\mathrm{\&alpha;r}}\frac{{\mathrm{b\&omega;}}_z-v_y}{v_x}]$

$g_1({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})=\frac{2}{m}(F_{\mathrm{tf}}+F_{\mathrm{tr}})$

$g_2({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})=\frac{2a}{I_z}{F}_{\mathrm{tf}}{\mathrm{\&delta;}}_f$

$g_3({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})=\frac{2}{m}{F}_{\mathrm{tf}}{\mathrm{\&delta;}}_f$

In addition, notice in formula (14) x ₁and x ₂the amount that can directly record (cireular frequency that utilizes on hind axle two wheel speed sensors to record is multiplied by tire radius and obtains the wheel linear velocity of left rear wheel and off hind wheel, and then utilizes formula (13) to obtain), and x ₃be the amount that cannot directly record, need estimate by sliding mode observer.The sliding mode observer that the present invention discusses can improve the precision of straight survey state on the one hand, the estimation dimension of extendible state on the other hand, realization is estimated (being virtual observation) to the filtering of non-straight measurement, for active safety control provides more abundant and abundant travel condition of vehicle, be convenient to control the realization of target.

For the auto model shown in formula (14), propose and set up following sliding mode observer (Sliding ModeObserver, SMO) model

$\left\{\begin{array}{c}{\stackrel{\stackrel{\&CenterDot;}{^}}{x}}_1=f_1({\hat{x}}_1,{\hat{x}}_2,{\hat{x}}_3,{\mathrm{\&delta;}}_f)+g_1({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})+l_1{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">s</figure-callout>}}_1+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\mathrm{sgn}\left(s_1\right)\\ {\stackrel{\stackrel{\&CenterDot;}{^}}{x}}_2=f_2({\hat{x}}_1,{\hat{x}}_2,{\hat{x}}_3,{\mathrm{\&delta;}}_f)+g_2({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})+l_2{s}_2+k_2\mathrm{sgn}\left(s_2\right)\\ {\stackrel{\stackrel{\&CenterDot;}{^}}{x}}_3=f_3({\hat{x}}_1,{\hat{x}}_2,{\hat{x}}_3,{\mathrm{\&delta;}}_f)+g_3({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})+{\mathrm{\&tau;}}_1\mathrm{sgn}\left(s_1\right)+{\mathrm{\&tau;}}_2\mathrm{sgn}\left(s_2\right)\end{array}\right.---\left(15\right)$

In formula (15),

with

represent respectively x ₁, x ₂and x ₃estimated valve.For outer input variable δ _fthe steering wheel angle δ that can record by steering wheel angle sensor is divided by the steering gear ratio q from bearing circle to front-wheel _tobtain (is δ _f=δ/q _t), and for F _tfand F _tr, can estimate to obtain by above-mentioned Dugoff tire model (utilizing formula (5)-(11)).L _j(j=1,2) are the error convergence gains outside sliding-mode surface, and k _j(j=1,2) and τ _j(j=1,2) have represented the error convergence gain on sliding-mode surface, and sgn (.) is sign function (being symbolic function).In addition s, ₁and s ₂what be defined as sliding mode observer can directly survey state x ₁, x ₂with estimated valve separately

with

between error,

${\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">s</figure-callout>}}_1=x_1-{\hat{\mathrm{<figure-callout\; id="1"\; label="x\; ^"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">x</figure-callout>}}}_1,$ $s_2=x_2-{\hat{x}}_2---\left(16\right)$

Formula (14) is deducted to formula (15), can obtain the error dynamics equation of sliding mode observer state of the system

${\stackrel{\stackrel{\&CenterDot;}{~}}{x}}_1=\mathrm{\&Delta;}{f}_1-l_1{s}_1-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\mathrm{sgn}\left(s_1\right)$

${\stackrel{\stackrel{\&CenterDot;}{~}}{x}}_2=\mathrm{\&Delta;}{f}_2-l_2{s}_2-k_2\mathrm{sgn}\left(s_2\right)---\left(17\right)$

${\stackrel{\stackrel{\&CenterDot;}{~}}{x}}_3={\mathrm{\&Delta;f}}_3-{\mathrm{\&tau;}}_1\mathrm{sgn}\left(s_1\right)-{\mathrm{\&tau;}}_2\mathrm{sgn}\left(s_2\right)$

In formula, ${\tilde{x}}_1=x_1-{\hat{\mathrm{<figure-callout\; id="1"\; label="x\; ^"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">x</figure-callout>}}}_1={\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">s</figure-callout>}}_1,$ ${\tilde{x}}_2=x_2-{\hat{x}}_2=s_2,$ ${\tilde{x}}_3=x_3-{\hat{x}}_3,$ ${\mathrm{\&Delta;<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1=f_1(x_1,x_2,x_3,{\mathrm{\&delta;}}_f)-f_1({\hat{x}}_1,{\hat{x}}_2,{\hat{x}}_3,{\mathrm{\&delta;}}_f),$

${\mathrm{\&Delta;f}}_2=f_2(x_1,x_2,x_3,{\mathrm{\&delta;}}_f)-f_2({\hat{x}}_1,{\hat{x}}_2,{\hat{x}}_3,{\mathrm{\&delta;}}_f),$ ${\mathrm{\&Delta;f}}_3=f_3(x_1,x_2,x_3,{\mathrm{\&delta;}}_f)-f_3({\hat{x}}_1,{\hat{x}}_2,{\hat{x}}_3,{\mathrm{\&delta;}}_f).$

The key of Design of Sliding Mode Observer is how according to stability principle, to determine each gain of sliding mode observer in formula (17).Liapunov (Lyapunov) function being constructed as follows

$V({\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">s</figure-callout>}}_1,s_2)=\frac{1}{2}({\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">s</figure-callout>}}_1^2+s_2^2)---\left(18\right)$

For guaranteeing sliding formwork state, s in above formula _j(j=1,2) should meet (j=1,2),

$s_j{\stackrel{\&CenterDot;}{s}}_j<-{\mathrm{\&eta;}}_j\left|s_j\right|,$ $\&ForAll;{\mathrm{\&eta;}}_j>0(j=\mathrm{1,2})---\left(19\right)$

In formula (19), η _j(j=1,2) are given positive number (as positive decimal).

Below with state x ₁for example, how to confirm gain l is discussed ₁and k ₁.Δ f in first equation of formula (17) ₁a continuously differentiable function, therefore have

${\mathrm{\&Delta;<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1=f_1(x_1,x_2,x_3,{\mathrm{\&delta;}}_f)-f_1({\hat{x}}_1,{\hat{x}}_2,{\hat{x}}_3,{\mathrm{\&delta;}}_f)$

$=\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_1}{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">s</figure-callout>}}_1+\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_2}{s}_2+\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_3}{\tilde{x}}_3+{\&dtri;}_1---\left(20\right)$

$\&ap;\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_1}{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">s</figure-callout>}}_1+\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_2}{s}_2+\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_3}{\tilde{x}}_3$

In formula (20),

the high-order trace (can ignore) being caused by differential,

(j=1,2,3) representative function f ₁for x _jthe partial derivative of (j=1,2,3), can try to achieve by the method for asking local derviation, as

$\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_1}=\frac{-2C_{\mathrm{\&alpha;f}}(v_y+{\mathrm{\&alpha;\&omega;}}_z){\mathrm{\&delta;}}_f}{{{\mathrm{mv}}_x}^2}-\frac{C_d{A}_f{\mathrm{\&rho;}}_a{v}_x}{m}$

By first equation of formula (17) and formula (20) substitution formula (19), can obtain

$s_1[\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_1}{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">s</figure-callout>}}_1+\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_2}{s}_2+\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_3}{\tilde{x}}_3-l_1{s}_1-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\mathrm{sgn}\left(s_1\right)]<-{\mathrm{\&eta;}}_1\left|s_1\right|---\left(21\right)$

Arranging above-mentioned inequality can obtain

$(\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_1}-l_1){{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">s</figure-callout>}}_1}^2+\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_2}{s}_1{s}_2+\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_3}{s}_1{\tilde{x}}_3-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\left|s_1\right|<-{\mathrm{\&eta;}}_1\left|s_1\right|---\left(22\right)$

From formula (22), can find out, if below two not etc. condition meet simultaneously,

$\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_1}<{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">l</figure-callout>}}_1,$ $\left|s_1\right|\left(\right|\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_2}{s}_2|+|\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_3}{\tilde{x}}_3|-k_1)<-{\mathrm{\&eta;}}_1\left|s_1\right|---\left(23\right)$

Sliding mode just can be guaranteed, and can determine gain l according to the condition such as not below ₁and k ₁

${\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">l</figure-callout>}}_1>\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_1},$ ${\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">k</figure-callout>}}_1>\left|\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_2}{s}_2\right|+\left|\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{\&PartialD;x_3}{\tilde{x}}_3\right|+{\mathrm{\&eta;}}_1---\left(24\right)$

In like manner, utilize second equation and formula (19) in formula (17), and according to Δ f ₂continuous differentiability, can determine and state x ₂relevant gain l ₂and k ₂,

$l_2>\frac{{\&PartialD;f}_2}{{\&PartialD;x}_2},$ $k_2>\left|\frac{{\&PartialD;f}_2}{{\&PartialD;x}_1}{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">s</figure-callout>}}_1\right|+\left|\frac{{\&PartialD;f}_2}{\&PartialD;x_3}{\tilde{x}}_3\right|+{\mathrm{\&eta;}}_2---\left(25\right)$

In formula (25),

(j=1,2,3) representative function f ₂for x _jthe partial derivative of (j=1,2,3), can try to achieve by the method for asking local derviation, as

$\frac{{\&PartialD;f}_2}{{\&PartialD;x}_2}=\frac{-2(a^2{C}_{\mathrm{\&alpha;f}}+b^2{C}_{\mathrm{\&alpha;r}})}{I_z{v}_x}.$

When sliding mode reaches, have

first equation of substitution formula (17) can obtain

${\stackrel{\&CenterDot;}{s}}_1=\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_1}{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">s</figure-callout>}}_1+\frac{\&PartialD;{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_2}{s}_2+\frac{\&PartialD;{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{\&PartialD;x_3}{\tilde{x}}_3-l_1{s}_1-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\mathrm{sgn}\left(x_1\right)$

Through abbreviation, can obtain

$\mathrm{sgn}\left(s_1\right)=\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}\frac{\&PartialD;{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_3}{\tilde{x}}_3---\left(26\right)$

In like manner, when sliding mode reaches, utilize second equation of formula (17) to obtain

$\mathrm{sgn}\left(s_2\right)=\frac{1}{k_2}\frac{\&PartialD;f_2}{{\&PartialD;x}_3}{\tilde{x}}_3---\left(27\right)$

And then, when sliding mode reaches, by the 3rd equation of formula (26) and formula (27) substitution formula (17), consider simultaneously ${\mathrm{\&Delta;f}}_3\&ap;\frac{{\&PartialD;f}_3}{{\&PartialD;x}_1}{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">s</figure-callout>}}_1+\frac{{\&PartialD;f}_3}{{\&PartialD;x}_2}{s}_2+\frac{{\&PartialD;f}_3}{{\&PartialD;x}_3}{\tilde{x}}_{3,}$ Can obtain

${\stackrel{\stackrel{\&CenterDot;}{~}}{x}}_3\&ap;\frac{{\&PartialD;f}_3}{{\&PartialD;x}_1}{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">s</figure-callout>}}_1+\frac{{\&PartialD;f}_3}{{\&PartialD;x}_2}{s}_2+\frac{{\&PartialD;f}_3}{{\&PartialD;x}_3}{\tilde{x}}_3-{\mathrm{\&tau;}}_1\mathrm{sgn}\left(s_1\right)-{\mathrm{\&tau;}}_2\mathrm{sgn}\left(s_2\right)$

$\frac{{\&PartialD;f}_3}{{\&PartialD;x}_3}{\tilde{x}}_3-{\mathrm{\&tau;}}_1\mathrm{sgn}\left(s_1\right)-{\mathrm{\&tau;}}_2\mathrm{sgn}\left(s_2\right)---\left(28\right)$

$=(\frac{{\&PartialD;f}_3}{{\&PartialD;x}_3}-\frac{{\mathrm{\&tau;}}_1}{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_3}-\frac{{\mathrm{\&tau;}}_2}{k_2}\frac{{\&PartialD;f}_2}{{\&PartialD;x}_3}){\tilde{x}}_3$

In formula (28),

(j=1,2,3) representative function f _j(j=1,2,3) are for x ₃partial derivative.According to formula (14), can try to achieve each local derviation item in formula (28),

$\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{{\&PartialD;x}_3}={\mathrm{\&omega;}}_z+\frac{2C_{\mathrm{\&alpha;f}}{\mathrm{\&delta;}}_f}{{\mathrm{mv}}_x}$

$\frac{{\&PartialD;f}_2}{{\&PartialD;x}_3}=\frac{2({\mathrm{bC}}_{\mathrm{\&alpha;r}}-a{C}_{\mathrm{\&alpha;f}})}{I_z{v}_x}---\left(29\right)$

$\frac{{\&PartialD;f}_3}{{\&PartialD;x}_3}=-\frac{2(C_{\mathrm{\&alpha;f}}+C_{\mathrm{\&alpha;r}})}{{\mathrm{mv}}_x}$

By (29) substitution formula (28) right side

product coefficient in, can obtain

$\frac{{\&PartialD;f}_3}{{\&PartialD;x}_3}-\frac{{\mathrm{\&tau;}}_1}{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}\frac{\&PartialD;{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{\&PartialD;x_3}-\frac{{\mathrm{\&tau;}}_2}{k_2}\frac{{\&PartialD;f}_2}{{\&PartialD;x}_3}=\frac{2(C_{\mathrm{\&alpha;f}}+C_{\mathrm{\&alpha;r}})}{{\mathrm{mv}}_x}-\frac{{\mathrm{\&tau;}}_1}{k_1}({\mathrm{\&omega;}}_z+\frac{{2C}_{\mathrm{\&alpha;f}}{\mathrm{\&delta;}}_f}{{\mathrm{mv}}_x})-\frac{{\mathrm{\&tau;}}_2}{k_2}\frac{2({\mathrm{bC}}_{\mathrm{\&alpha;r}}-a{C}_{\mathrm{\&alpha;f}})}{I_z{v}_x}---\left(30\right)$

By formula (30) and formula (28), can be found out, for guaranteeing about state x ₃the convergence of estimation procedure is stable, and the method for a simple possible is order gain τ ₁=τ ₂=0, because now following formula is set up

$\frac{\&PartialD;f_3}{\&PartialD;x_3}-\frac{{\mathrm{\&tau;}}_1}{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}\frac{\&PartialD;{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000120443810000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{\&PartialD;x_3}-\frac{{\mathrm{\&tau;}}_2}{k_2}\frac{{\&PartialD;f}_2}{{\&PartialD;x}_3}=-\frac{2(C_{\mathrm{\&alpha;f}}+C_{\mathrm{\&alpha;r}})}{{\mathrm{mv}}_x}<0---\left(31\right)$

Definite method of all gains in sliding mode observer formula (15) so far, has been discussed.In addition, be the excessive jitter of avoiding symbolic function to cause, in actual estimation procedure, sgn (.) function of above-mentioned various middle appearance replaces with sgneq (.) function below

sgn _eq(s _j)＝tanh(λs _j)(j＝1，2) (32)

In formula (32), tanh (.) is hyperbolic tangent function, and λ is a design parameters that is used for adjusting hyperbolic tangent function inclined degree.

In actual estimation procedure, need to adopt the estimation recursive form of discretization.For this reason, by the sliding mode observer of above-mentioned design, formula (15) is carried out discretization processing,

${\hat{x}}_1\left(k\right)={\hat{x}}_1(k-1)+T{\hat{x}}_2(k-1){\hat{x}}_3(k-1)+2T{C}_{\mathrm{\&alpha;f}}{\mathrm{\&delta;}}_f(k-1)\frac{{\hat{x}}_3(k-1)+a{\hat{x}}_2(k-1)}{m{\hat{x}}_1(k-1)}-\frac{T}{2m}{C}_d{A}_f{\mathrm{\&rho;}}_a{{\hat{x}}_1}^2(k-1)$

$+\frac{2T}{m}[F_{\mathrm{tf}}(k-1)+F_{\mathrm{tr}}(k-1)]+l_{1}T[v_x^{\&prime;}(k-1)-{\hat{x}}_1(k-1)]+k_{1}T\&CenterDot;\tanh [\mathrm{\&lambda;}{v}_x^{\&prime;}(k-1)-\mathrm{\&lambda;}{\hat{x}}_1(k-1)]$

${\hat{x}}_2\left(k\right)={\hat{x}}_2(k-1)+\frac{2{\mathrm{TaC}}_{\mathrm{\&alpha;f}}}{I_z}[{\mathrm{\&delta;}}_f(k-1)+\frac{-{\hat{x}}_3(k-1)-a{\hat{x}}_2(k-1)}{{\hat{x}}_1(k-1)}]-2{\mathrm{TbC}}_{\mathrm{\&alpha;r}}\frac{b{\hat{x}}_2(k-1)-{\hat{x}}_3(k-1)}{I_z{\hat{x}}_1(k-1)}$

$+\frac{2\mathrm{Ta}}{I_z}{F}_{\mathrm{tf}}(k-1){\mathrm{\&delta;}}_f(k-1)+l_{2}T[{\mathrm{\&omega;}}_z^{\&prime;}(k-1)-{\hat{x}}_2(k-1)]+k_{2}T\&CenterDot;\tanh [{\mathrm{\&lambda;\&omega;}}_z^{\&prime;}(k-1)-\mathrm{\&lambda;}{\hat{x}}_2(k-1)]$

${\hat{x}}_3(k-1)={\hat{x}}_3(k-1)-T{\hat{x}}_1(k-1){\hat{x}}_2(k-1)+\frac{{2\mathrm{TC}}_{\mathrm{\&alpha;f}}}{m}[{\mathrm{\&delta;}}_f(k-1)-\frac{({\hat{x}}_3(k-1)+a{\hat{x}}_2(k-1))}{{\hat{x}}_1(k-1)}]$

$+2{\mathrm{TC}}_{\mathrm{\&alpha;r}}\frac{b{\hat{x}}_2(k-1)-{\hat{x}}_3(k-1)}{m{\hat{x}}_1(k-1)}+\frac{2T}{m}{F}_{\mathrm{tf}}(k-1){\mathrm{\&delta;}}_f(k-1)$

(33)

In formula (33), k represents the discretization moment, and T represents discrete cycle (in the present invention, according to survey sensor characteristic, the representative value of T can be taken as 10 milliseconds, 20 milliseconds or 50 milliseconds etc.), v ' _xand ω ' _z(cireular frequency that utilizes on hind axle two wheel speed sensors to record is multiplied by tire radius and obtains the wheel linear velocity of left rear wheel and off hind wheel to represent respectively to measure the longitudinal direction of car speed of advance that obtains and yaw velocity by wheel speed sensors, and then utilize formula (13) to obtain), i.e. v ' _xand ω ' _zrepresent respectively v _xand ω _zcontaining noisy observed reading.Consider the measurement noise of wheel speed sensors, v ' _x, ω ' _zcan be expressed as

wherein

represent equivalence longitudinal speed of advance observation noise (can be modeled as average and be 0, variance is gaussian white noise),

represent equivalent yaw velocity observation noise (can be modeled as average and be 0, variance is

gaussian white noise).

In above-mentioned estimation recursive process, can determine the automobile longitudinal speed of advance v of automobile in each moment _x(k), yaw velocity ω _zand side velocity v (k) _y, and then according to following formula, can determine the side slip angle in each moment (k)

β(k)＝arctan[v _y(k)/v _x(k)] (34)

Embodiment 2

For the actual effect of the travel condition of vehicle non linear robust method of estimation based on sliding mode observer that check the present invention proposes, utilize professional vehicle dynamics simulation software CarSim to carry out simulating, verifying experiment.

CarSim is the special simulation software for vehicle dynamics by the exploitation of U.S. MSC (Mechanical Simulation Corporation) company, by numerous in the world automakers, components supplying business, adopted at present, be widely used in the business development of modern automobile control system, become the standard software of auto trade, enjoyed a very good reputation.The height modeling true to nature of each subsystems such as the vehicle dynamic model in Carsim is by the car body to automobile, suspension respectively, turn to, braking and each tire realizes, there is very high degree of freedom, the actual information of travel condition of vehicle accurately that approaches very much can be provided, therefore, the travel condition of vehicle information of Carsim output can be used as the reference output of vehicle.

The Robust Estimation effect of algorithm under the motor-driven operating mode of height and different road circumstance state proposing for check the present invention, longitudinal speed of advance that automobile is set in emulation experiment is constantly accelerating, braking deceleration and the variation such as at the uniform velocity, the steering wheel angle δ of automobile changes by the sinusoidal rule of 60 ° of amplitudes simultaneously, longitudinally specifically process is as shown in Figure 2 over time for speed of advance and steering wheel angle, and vertical coefficientoffrictionμ between tire and ground is pressed μ=1, μ=0.8 and μ=0.6 3 kind of situation is carried out respectively emulation, and (simulated roadway adheres to the variation of condition from being dried to wet and slippery road, be the variation of road conditions environment), the emulation duration that different roads adhere under condition is all set to 100 seconds (s).Vehicle used is the four-wheeled of a front-wheel steering, and principal parameter is as follows: m=960 (kilogram), I _z=1382 (kilogram-meters ²), a=0.948 (rice), b=1.422 (rice), C _{α f}=C _{α r}=25692 (newton/radians), T _w=1.390 (rice).The measurement noise of linear velocity (cireular frequency recording by wheel speed sensors is multiplied by tire radius and obtains) of setting four wheels is that average is 0, standard deviation is the Gaussian white noise of 0.05 (meter per second), and the measurement noise of steering wheel angle sensor is that average is 0, standard deviation is the Gaussian white noise of 0.0873 (radian).

Shown in table 1 and Fig. 3～Fig. 4, provided the result of μ=1 o'clock.Table 1 has been listed the statistics contrast of directly surveying method and the inventive method reckoning travel condition of vehicle for whole process utilization, and the error in table is all (as longitudinal speed of advance error of straight survey method is utilized longitudinal speed of advance of straight survey method reckoning with respect to the error of longitudinal speed of advance reference value of Carsim output with regard to expression) for the corresponding reference value of Carsim output.Be pointed out that in addition, the concrete meaning of straight survey method and the inventive method is as follows: directly survey method refers to by direct measurement the convert longitudinal speed of advance and the yaw velocity that obtain, the cireular frequency that utilizes on hind axle two wheel speed sensors to record is multiplied by tire radius and obtains the wheel linear velocity of left rear wheel and off hind wheel, and then directly calculates according to the formula in embodiment 1 (13) the longitudinal speed of advance and the yaw velocity that obtain; The inventive method refers to the method for utilizing the travel condition of vehicle non linear robust method of estimation based on sliding mode observer that the present invention proposes to calculate each running state of vehicle.Fig. 3 has provided the result curve (indicating with SMO dot-dash dotted line in figure) of the side slip angle β of μ=1 o'clock the inventive method estimation, and the reference output valve of corresponding Carsim (indicating with the real black line of Carsim in figure).Fig. 4 has provided the β of μ=1 o'clock the inventive method estimation with respect to the curve of error of the β reference value of Carsim output.

Two kinds of methods of table 1 are calculated the contrast table of effect

In table, "--" represents the item that straight survey method cannot be calculated

Shown in table 2 and Fig. 5～Fig. 8, provided the related results of μ=0.8 and μ=0.6 o'clock.Wherein, table 2 provides the statistics (error in table is all for the corresponding reference value of Carsim output) that other two kinds of roads adhere to the relevant reckoning result under condition.Fig. 5 has provided the result curve (indicating with SMO dot-dash dotted line in figure) of the side slip angle β of μ=0.8 o'clock the inventive method estimation, and the reference output valve of corresponding Carsim (indicating with the real black line of Carsim in figure); Fig. 6 has provided the β of μ=0.8 o'clock the inventive method estimation with respect to the curve of error of the β reference value of Carsim output.And Fig. 7 has provided the result curve (indicating with SMO dot-dash dotted line in figure) of the side slip angle β of μ=0.6 o'clock the inventive method estimation, and the reference output valve of corresponding Carsim (indicating with the real black line of Carsim in figure); Fig. 8 has provided the β of μ=0.6 o'clock the inventive method estimation with respect to the curve of error of the β reference value of Carsim output.

Table 2 the inventive method is adhered to the reckoning Contrast on effect under condition at different roads

By contrast (especially standard deviation) and Fig. 3～Fig. 4 of table 1, can find out that the inventive method has had significantly and improved in precision aspect the reckoning of longitudinal speed of advance and yaw velocity with respect to straight survey method.In addition,, according to table 1 and Fig. 3～Fig. 4, it can also be seen that the inventive method also has very high precision aspect the estimation of side velocity and side slip angle.

In addition, according to the statistics of table 1～table 2 (especially standard deviation) and Fig. 3～Fig. 8, can find out that the inventive method is not almost changing aspect travel condition of vehicle estimated accuracy when larger variation occurs the road condition of adhering to, still kept higher precision, the inventive method has shown good robustness and antijamming capability.

To sum up, even when the motor-driven operating condition of height and road circumstance state generation significant change, the travel condition of vehicle non linear robust method of estimation based on sliding mode observer that the present invention proposes still can estimate the information such as longitudinal direction of car speed of advance, side velocity, yaw velocity and side slip angle exactly, and these information can meet the needs about automobile active safety control.

Claims (3)

1. the travel condition of vehicle non linear robust method of estimation based on sliding mode observer, it is characterized in that: this method is for the more front-wheel steering four-wheel automobile of current application, designed the sliding mode observer based on non-linear automobile dynamic quality model, by the estimation recursion of sliding mode observer, realize accurate, the Robust Estimation to automobile longitudinal speed of advance, yaw velocity, side velocity and side slip angle information, concrete steps comprise:

1) set up the sliding mode observer of travel condition of vehicle:

The Vehicle Nonlinear kinetic model that adopts three degree of freedom, the state space equation form of this model is as follows:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=f_1(x_1,x_2{,x}_3,{\mathrm{\&delta;}}_f)+g_1({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})\\ {\stackrel{\&CenterDot;}{x}}_2=f_2(x_1,x_2{,x}_3,{\mathrm{\&delta;}}_f)+g_2({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})\\ {\stackrel{\&CenterDot;}{x}}_3=f_3(x_1,x_2,x_3,{\mathrm{\&delta;}}_f)+g_3({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})\end{array}\right.---\left(1\right)$

In formula (1), x ₁, x ₂and x ₃for three states of sliding mode observer, and x ₁=v _x, x ₂=ω _z, x ₃=v _y, v _x, v _yand ω _zrespectively longitudinal speed of advance, side velocity and the yaw velocity of automobile;

δ _f, F _tfand F _trbe three outer input variables, wherein δ _ffront wheel steering angle, F _tfthe longitudinal force acting on single front-wheel, F _trit is the longitudinal force acting on single trailing wheel;

Each function f in formula (1) _j(j=1,2,3) and g _jthe value of (j=1,2,3) is as follows

$f_1(x_1,x_2,x_3,{\mathrm{\&delta;}}_f)=\frac{1}{m}[{\mathrm{mv}}_y{\mathrm{\&omega;}}_z+2\frac{v_y+{\mathrm{a\&omega;}}_z}{v_x}{C}_{\mathrm{\&alpha;f}}{\mathrm{\&delta;}}_f-\frac{1}{2}{C}_d{A}_f{\mathrm{\&rho;}}_a{v}_x^2]$

$f_2(x_1,x_2,x_3,{\mathrm{\&delta;}}_f)=\frac{1}{I_z}[2a({\mathrm{\&delta;}}_f-\frac{(v_y+{\mathrm{a\&omega;}}_z)}{v_x})C_{\mathrm{\&alpha;f}}-{2\mathrm{bC}}_{\mathrm{\&alpha;r}}\frac{({\mathrm{b\&omega;}}_z-v_y)}{v_x}]$

$f_3(x_1,x_2,x_3,{\mathrm{\&delta;}}_f)=\frac{1}{m}[{-\mathrm{mv}}_x{\mathrm{\&omega;}}_z+2({\mathrm{\&delta;}}_f-\frac{(v_y+{\mathrm{a\&omega;}}_z)}{v_x})C_{\mathrm{\&alpha;f}}+2C_{\mathrm{\&alpha;r}}\frac{{\mathrm{b\&omega;}}_z-v_y}{v_x}]$

$g_1=({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})=\frac{2}{m}(F_{\mathrm{tf}}+F_{\mathrm{tr}})$

$g_2=({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})=\frac{2a}{I_z}{F}_{\mathrm{tf}}{\mathrm{\&delta;}}_f$

$g_3=({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})=\frac{2}{m}{F}_{\mathrm{tf}}{\mathrm{\&delta;}}_f$

At described f _j(j=1,2,3) and g _jin (j=1,2,3), m and I _zbe respectively the quality of vehicle and the rotor inertia of walking around the vertical axle of barycenter, a is the distance of vehicle front wheel shaft center to barycenter, and b is the distance of automobile back wheel wheel shaft center to barycenter, C _{α f}, C _{α r}represent respectively the cornering stiffness of forward and backward tire, C _drepresent aerodynamic drag factor, A _frepresent vehicle forward direction area, ρ _arepresent density of air;

For state x ₁and x ₂, corresponding v respectively _xand ω _z, on they and hind axle, the wheel speed of two non-steered wheels has following relation:

v _x=(V _RL+V _RR)/2

（2）

ω _z=(V _RL-V _RR)/T _W

In formula (2), T _wthe wheelspan between two trailing wheels on hind axle, V _rLand V _rRrepresent respectively the linear velocity of left rear wheel and off hind wheel, i.e. the wheel linear velocity of two non-wheel flutters;

State x ₁and x ₂for the amount that can directly record, and state x ₃for the amount that cannot directly record need be estimated by sliding mode observer;

For the auto model shown in formula (1), propose and set up following sliding mode observer model:

$\left\{\begin{array}{c}{\stackrel{\stackrel{\&CenterDot;}{^}}{x}}_1=f_1({\hat{x}}_1,{\hat{x}}_2,{\hat{x}}_3,{\mathrm{\&delta;}}_f)+g_1({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})+l_1{s}_1+k_1\mathrm{sgn}\left(s_1\right)\\ {\stackrel{\stackrel{\&CenterDot;}{^}}{x}}_2=f_2({\hat{x}}_1,{\hat{x}}_2,{\hat{x}}_3,{\mathrm{\&delta;}}_f)+g_2({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})+l_2{s}_2+k_2\mathrm{sgn}\left(s_2\right)\\ {\stackrel{\stackrel{\&CenterDot;}{^}}{x}}_3=f_3({\hat{x}}_1,{\hat{x}}_2,{\hat{x}}_3,{\mathrm{\&delta;}}_f)+g_3({\mathrm{\&delta;}}_f,F_{\mathrm{tf}},F_{\mathrm{tr}})+{\mathrm{\&tau;}}_1\mathrm{sgn}\left(s_1\right)+{\mathrm{\&tau;}}_2\mathrm{sgn}\left(s_2\right)\end{array}\right.---\left(3\right)$

In formula (3),

represent respectively x ₁, x ₂and x ₃calculating estimated valve, l _j(j=1,2) are the error convergence gains outside sliding-mode surface, and k _j(j=1,2) and τ _j(j=1,2) have represented the error convergence gain on sliding-mode surface, and sgn (.) is sign function, i.e. symbolic function; s ₁and s ₂what be defined as sliding mode observer can directly survey state x ₁, x ₂with estimated valve separately

between error,

$s_1=x_1-{\hat{x}}_1,s_2=x_2-{\hat{x}}_2---\left(4\right)$

2) carry out the estimation recursion of discretization:

In actual estimation procedure, need to adopt the estimation recursive form of discretization, for this reason, by the sliding mode observer of above-mentioned design, be that formula (3) is carried out discretization processing,

$\begin{array}{c}{\hat{x}}_1\left(k\right)={\hat{x}}_1(k-1)+T{\hat{x}}_2(k-1){\hat{x}}_3(k-1)+2T{C}_{\mathrm{\&alpha;f}}{\mathrm{\&delta;}}_f(k-1)\frac{{\hat{x}}_3(k-1)+a{\hat{x}}_2(k-1)}{m{\hat{x}}_1(k-1)}-\frac{T}{2m}{C}_d{A}_f{\mathrm{\&rho;}}_a{{\hat{x}}_1}^2(k-1)\\ +\frac{2T}{m}[F_{\mathrm{tf}}(k-1)+F_{\mathrm{tr}}(k-1)]+l_{1}T[v_x^{\&prime;}(k-1)-{\hat{x}}_1(k-1)]+k_{1}T\&CenterDot;\tanh [\mathrm{\&lambda;}{v}_x^{\&prime;}(k-1)-\mathrm{\&lambda;}{\hat{x}}_1(k-1)]\end{array}$

$\begin{array}{c}{\hat{x}}_2\left(k\right)={\hat{x}}_2(k-1)+\frac{{2\mathrm{TaC}}_{\mathrm{\&alpha;f}}}{I_z}[{\mathrm{\&delta;}}_f(k-1)+\frac{-{\hat{x}}_3(k-1)-{a\hat{x}}_2(k-1)}{{\hat{x}}_1(k-1)}]-2\mathrm{Tb}{C}_{\mathrm{\&alpha;r}}\frac{{b\hat{x}}_2(k-1)-{\hat{x}}_3(k-1)}{I_z{\hat{x}}_1(k-1)}\\ +\frac{2\mathrm{Ta}}{I_z}{F}_{\mathrm{tf}}(k-1){\mathrm{\&delta;}}_f(k-1)+l_{2}T[{\mathrm{\&omega;}}_z^{\&prime;}(k-1)-{\hat{x}}_2(k-1)]+k_{2}T\&CenterDot;\tanh [\mathrm{\&lambda;}{\mathrm{\&omega;}}_z^{\&prime;}(k-1)-\mathrm{\&lambda;}{\hat{x}}_2(k-1)]\end{array}---\left(5\right)$

$\begin{array}{c}{\hat{x}}_3(k-1)={\hat{x}}_3(k-1)-T{\hat{x}}_1(k-1){\hat{x}}_2(k-1)+\frac{{2\mathrm{TC}}_{\mathrm{\&alpha;f}}}{m}[{\mathrm{\&delta;}}_f(k-1)-\frac{({\hat{x}}_3(k-1)+a{\hat{x}}_2(k-1))}{{\hat{x}}_1(k-1)}]\\ +{2\mathrm{TC}}_{\mathrm{\&alpha;r}}\frac{b{\hat{x}}_2(k-1)-{\hat{x}}_3(k-1)}{m{\hat{x}}_1(k-1)}+\frac{2T}{m}{F}_{\mathrm{tf}}(k-1){\mathrm{\&delta;}}_f(k-1)\end{array}$

In formula (5), k represents the discretization moment, and T represents the discrete cycle, and tanh (.) is hyperbolic tangent function, and λ is a design parameters that is used for adjusting hyperbolic tangent function inclined degree, v ' _xand ω ' _zrepresent respectively to measure the longitudinal direction of car speed of advance and the yaw velocity that obtain, i.e. v ' by wheel speed sensors _xand ω ' _zrepresent respectively v _xand ω _zcontaining noisy observed reading;

In the estimation recursion computation process of above-mentioned discretization, can determine the longitudinal speed of advance v of automobile in each moment _x(k), yaw velocity ω _zand side velocity v (k) _y, and then according to following formula, can determine the side slip angle in each moment (k)

β(k)=arctan[v _y(k)/v _x(k)] （6）。

2. the travel condition of vehicle non linear robust method of estimation based on sliding mode observer according to claim 1, it is characterized in that longitudinal speed of advance and yaw velocity are the amounts that can directly record, by wheel speed sensors, measure and obtain longitudinal direction of car speed of advance and yaw velocity, be specially and utilize two left rear wheel angular velocity omegas that wheel speed sensors records on hind axle _rLwith off hind wheel angular velocity omega _rRbe multiplied by tire radius R and obtain V ' _rL=R ω _rLand V ' _rR=R ω _rR, V ' _rLand V ' _rRrepresent respectively V _rLand V _rRcontaining noisy observed reading and $V_{\mathrm{RL}}^{\&prime;}=V_{\mathrm{RL}}+n_{V_{\mathrm{RL}}},V_{\mathrm{RR}}^{\&prime;}=V_{\mathrm{RR}}+n_{V_{\mathrm{RR}}},$ Wherein with

the additivity that represents respectively the wheel linear velocity of left rear wheel and off hind wheel is measured noise, and then recycling formula (2) obtains longitudinal speed of advance and yaw velocity contains noisy observed reading v ' _xand ω ' _z;

For three outer input variables in formula (3), by method below, determine:

Front wheel steering angle δ _fthe steering wheel angle δ that can record by steering wheel angle sensor is divided by the steering gear ratio q from bearing circle to front-wheel _tdetermine, i.e. δ _f=δ/q _t;

Longitudinal force of tire F _tfand F _trcan determine according to the non-linear tire model of Dugoff;

For determining F _tfand F _tr, introduce longitudinal direction of car slip rate i _sj(j=f, r), i _sjbe divided into again front wheel spindle straight skidding rate i _sfwith hind axle straight skidding rate i _sr, in this method, subscript j gets f or r represents respectively front or rear wheel shaft, i _sjmethod of calculating is:

In formula (7), v _tfand v _trrepresent respectively the speed along tire direction on forward and backward wheel shaft, v _tfand v _trcan unify to be designated as v _tj(j=f, r); ω _fthe spin velocity on front wheel spindle is converted in the spin velocity equivalence that represents two wheels on front wheel spindle; ω _rrepresent that on hind axle, the spin velocity on hind axle, ω are converted in two rotation of wheel cireular frequency equivalences _fand ω _rcan unify to be designated as ω _j(j=f, r) and

${\mathrm{\&omega;}}_f=\frac{1}{2}({\mathrm{\&omega;}}_{\mathrm{fR}}+{\mathrm{\&omega;}}_{\mathrm{fL}})$

（8）

${\mathrm{\&omega;}}_r=\frac{1}{2}({\mathrm{\&omega;}}_{\mathrm{rR}}+{\mathrm{\&omega;}}_{\mathrm{rL}})$

In formula (8), ω _fL, ω _fR, ω _rLand ω _rRrepresent respectively the spin velocity of the near front wheel, off front wheel, left rear wheel and off hind wheel, by utilizing four wheel speed sensors to measure, obtain;

V _tj(j=f, r) can determine by formula (9):

v _tf=v _xcosδ _f+(v _y+aω _z)sinδ _f

（9）

v _tr=v _x

And then, longitudinal force of tire F _tfand F _trcan determine by through type (10)

$F_{\mathrm{tj}}=\frac{C_{\mathrm{tj}}{i}_{\mathrm{sj}}}{1-i_{\mathrm{sj}}}{f}_t\left(p_j\right),(j=f,r)---\left(10\right)$

In formula (10), C _tfand C _trrepresent respectively the longitudinal rigidity of single forward and backward tire, the unified C that is designated as _tj(j=f, r); Variable p _j(j=f, r) and function f _t(pj) (j=f, r) determined by following formula:

$p_j=\frac{{\mathrm{\&mu;F}}_{\mathrm{zj}}(1-{\mathrm{\&epsiv;}}_r{v}_x\sqrt{{i_{\mathrm{sj}}}^2+{\tan }^2{\mathrm{\&alpha;}}_j})(1-i_{\mathrm{sj}})}{2\sqrt{{C_{\mathrm{tj}}}^2{i_{\mathrm{sj}}}^2+{C_{\mathrm{\&alpha;j}}}^2{\tan }^2{\mathrm{\&alpha;}}_j}},(j=f,r)---\left(11\right)$

$f_t\left(p_j\right)=\left\{\begin{array}{cc}{p}_j(2-p_j)& p_j<1\\ 1& p_j\&GreaterEqual;1\end{array}\right.,(j=f,r)---\left(12\right)$

In formula (11) and (12), μ represents the vertical friction coefficient between tire and ground; ε _rrepresent that road adheres to decay factor; α _f, α _rrepresent respectively the sideslip angle of forward and backward tire, the unified α that is designated as _j(j=f, r), can be calculated as follows

${\mathrm{\&alpha;}}_f={\mathrm{\&delta;}}_f-\frac{v_y+{\mathrm{a\&omega;}}_z}{v_x},{\mathrm{\&alpha;}}_r=\frac{{\mathrm{b\&omega;}}_z-v_y}{v_x}---\left(13\right)$ And F _zj(j=f, r) represents be assigned to the vertical load on front or rear wheel shaft and can be calculated as follows

$F_{\mathrm{zf}}=\frac{\mathrm{mgb}}{2(a+b)},F_{\mathrm{zr}}=\frac{\mathrm{mga}}{2(a+b)}---\left(14\right)$

In formula (14), g represents acceleration due to gravity;

For each gain of sliding mode observer in formula (3), according to the convergence stability principle of observer, design definitely, particularly can determine gain l according to formula below ₁, k ₁, l ₂, k ₂, τ ₁and τ ₂,

$l_1>\frac{\&PartialD;f_1}{\&PartialD;x_1},k_1>\left|\frac{\&PartialD;f_1}{\&PartialD;x_2}{s}_2\right|+\left|\frac{\&PartialD;f_1}{\&PartialD;x_3}{\tilde{x}}_3\right|+{\mathrm{\&eta;}}_1---\left(15\right)$

$l_2>\frac{\&PartialD;f_2}{\&PartialD;x_2},k_2>\left|\frac{\&PartialD;f_2}{\&PartialD;x_1}{s}_1\right|+\left|\frac{\&PartialD;f_2}{\&PartialD;x_3}{\tilde{x}}_3\right|+{\mathrm{\&eta;}}_2---\left(16\right)$

τ ₁=τ ₂=0 （17）

In formula (15) and formula (16),

representative function f ₁for x _jthe partial derivative of (j=1,2,3),

representative function f ₂for x _jthe partial derivative of (j=1,2,3), η _j(j=1,2) are given positive number, ${\tilde{x}}_3=x_3-{\hat{x}}_3;$

In actual recursion, sgn (.) function occurring in formula (3) sgn below _eq(.) function replaces:

sgn _eq(s _j)=tanh(λs _j)(j=1,2) （18）。

3. the travel condition of vehicle non linear robust method of estimation based on sliding mode observer according to claim 1, is characterized in that the representative value of discrete period T is taken as 10 milliseconds, 20 milliseconds or 50 milliseconds.

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