CN111027136A - Nonlinear ABS control method based on fractional extreme value search - Google Patents

Nonlinear ABS control method based on fractional extreme value search Download PDF

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CN111027136A
CN111027136A CN201911166318.6A CN201911166318A CN111027136A CN 111027136 A CN111027136 A CN 111027136A CN 201911166318 A CN201911166318 A CN 201911166318A CN 111027136 A CN111027136 A CN 111027136A
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tire
nonlinear
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CN111027136B (en
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何仁
苑磊
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Anhui Tiannuo Braking System Co ltd
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Jiangsu University
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B60—VEHICLES IN GENERAL
    • B60T—VEHICLE BRAKE CONTROL SYSTEMS OR PARTS THEREOF; BRAKE CONTROL SYSTEMS OR PARTS THEREOF, IN GENERAL; ARRANGEMENT OF BRAKING ELEMENTS ON VEHICLES IN GENERAL; PORTABLE DEVICES FOR PREVENTING UNWANTED MOVEMENT OF VEHICLES; VEHICLE MODIFICATIONS TO FACILITATE COOLING OF BRAKES
    • B60T8/00—Arrangements for adjusting wheel-braking force to meet varying vehicular or ground-surface conditions, e.g. limiting or varying distribution of braking force
    • B60T8/17—Using electrical or electronic regulation means to control braking
    • B60T8/176—Brake regulation specially adapted to prevent excessive wheel slip during vehicle deceleration, e.g. ABS

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Abstract

The invention discloses a nonlinear ABS control method based on fractional extremum search, which mainly comprises the steps of respectively establishing an automobile dynamic model, a tire model and a reference slip ratio model; designing a nonlinear ABS controller and a fractional extremum search controller; setting a filter and integral feedback in extremum search control into a fractional calculus algorithm, taking braking deceleration as output of search control, establishing a fractional coefficient optimizing function taking the braking deceleration as an index, and determining an optimal fractional coefficient by optimizing the fractional coefficient with the maximum fractional coefficient optimizing function. Has the advantages that: the invention can convert the advantages of robustness and stability of the fractional order system into the extreme value search controller, and can improve the response characteristic of the braking system, increase the braking deceleration of the vehicle and reduce the braking distance by matching with nonlinear control.

Description

Nonlinear ABS control method based on fractional extreme value search
Technical Field
The invention relates to a control method of an anti-lock brake system of an automobile, in particular to a nonlinear ABS control method based on fractional order extremum search, belonging to the field of automobile brake systems.
Background
The Antilock Brake System (ABS) is called as an Antilock Brake System (ABS). The function is that when the automobile brakes, the braking force of the brake is automatically controlled, so that the wheels are not locked and are in a state of rolling and sliding (the sliding rate is about 20 percent) to ensure that the adhesive force between the wheels and the ground is at the maximum.
However, the optimum slip ratio is different for different road surfaces. Most of the braking force control systems used in the prior vehicles adopt a road surface identification method based on a threshold, and only rough classification of road surface adhesion conditions can be obtained. Although the identification method is simple and effective, the estimation result is rough, and large uncertainty and error exist, and the braking force control strategy designed according to the road surface identification method cannot fully utilize the ground adhesion, so that the braking performance of the vehicle is reduced.
In order to solve the problem, an effective ABS control algorithm, namely extremum search control, is recently developed, road surface identification is not needed, and the control of the optimal slip ratio is completed through extremum search. The method has the advantages of high convergence rate and good steady-state performance within a certain parameter range. The method not only improves the performance of the ABS of the vehicle, but also has stronger robustness to parameter change. In this method, the gradient is calculated by perturbing the system with an external excitation signal. The structure of the extremum seeking control allows the use of unknown objective functions.
Disclosure of Invention
The purpose of the invention is as follows: in order to improve the convergence speed of the ABS control algorithm, the invention applies the algorithm of combining the fractional operator and the extremum search control to the control of the ABS system, and provides a nonlinear ABS control method based on the fractional extremum search.
The invention converts the advantages of robustness and stability of the fractional order system into the extreme value search controller, can improve the response characteristic of the vehicle braking system, increases the braking deceleration of the vehicle and reduces the braking distance.
The technical scheme is as follows: a nonlinear ABS control method based on fractional order extremum search mainly comprises the following steps:
step one, establishing a mathematical model: respectively establishing an automobile dynamic model, a tire model and a reference slip rate model;
step two, designing a nonlinear ABS controller: adopting an integral feedback method, taking the slip rate and the slip rate integral as control targets, minimizing a prediction tracking error in a continuous interval as an optimization target, adopting a Taylor series expansion method to predict the tire slip rate and the integral thereof in the next time interval lambda (t + h), taking a prediction period h as a prediction time domain, and calculating the current control quantity by ensuring the tire slip rate tracking error and the slip rate integral tracking error;
step three, designing a fractional order extremum search controller: setting a filter and integral feedback in extremum search control into a fractional calculus algorithm, taking braking deceleration as output of search control, establishing a fractional coefficient optimizing function taking the braking deceleration as an index, and determining an optimal fractional coefficient by optimizing the fractional coefficient with the maximum fractional coefficient optimizing function.
The vehicle dynamics model is as follows:
Figure BDA0002287552540000021
Figure BDA0002287552540000022
wherein R is the wheel radius, ItEquivalent moment of inertia for the tire;
Figure BDA0002287552540000023
in order to be the longitudinal acceleration of the vehicle,
Figure BDA0002287552540000024
as angular acceleration of the wheel, TbFor braking torque, FxIn order to apply a longitudinal force to the tire,
mtone quarter of the total mass of the vehicle:
Figure BDA0002287552540000025
wherein m isvsIs the sprung mass of the vehicle, mwIs the wheel mass;
1/4 tire vertical loads under the model car are:
Figure BDA0002287552540000026
wherein l is the wheelbase, hcgHeight of center of mass of sprung mass, FLSubstituting formula (1) into formula (4) to solve nonlinear algebraic equation to calculate F for braking transmitted dynamic loadz；
Defining the slip ratio λ of the tire:
Figure BDA0002287552540000027
wherein λ represents a tire slip ratio, v represents a vehicle longitudinal speed;
the derivation of equation (5) with respect to time:
Figure BDA0002287552540000028
in which is shown
Figure BDA0002287552540000029
The derivative of the slip rate is represented as,
substituting the formulas (1) and (5) into the formula (6) to obtain
Figure BDA00022875525400000210
The tire mathematical model is as follows:
using a non-linear Dugoff tire model:
Figure BDA0002287552540000031
wherein
Figure BDA0002287552540000032
Figure BDA0002287552540000033
Wherein, CiFor the longitudinal stiffness of the tire, CαIs the lateral stiffness of the tyre, u is the road friction coefficient, epsilonrFor road adhesion coefficient, α is the tire slip angle.
The reference slip ratio mathematical model is as follows:
Figure BDA0002287552540000034
wherein λopt0.15 represents the optimum slip ratio, and a 20 is a time constant; solving Laplace inverse transformation on two sides of the equation, and solving a first-order zero initial condition differential equation to obtain
λd(t)＝λopt-λopte-at。 (12)
Designing a nonlinear ABS controller in the second step:
a nonlinear vehicle system dynamic state equation under a braking working condition is constructed by using a formula (1) and a formula (7), a tire slip rate is considered as a system output, and the system output is written into a state space form
Figure BDA0002287552540000035
Figure BDA0002287552540000036
Selecting an output vector y1Is λ, i.e.
y1＝x2(15)
x＝[v λ]TIs a system state vector, x1And x2Is a state vector, y1Output vector, T, for the systembRepresenting control inputs, the non-linear tyre model (8) having been incorporated into the function f1And f2Performing the following steps;
defining a new state variable x3Comprises the following steps:
Figure BDA0002287552540000037
the control system aims to control the wheel slip ratio x2λ andintegral x3═ λ dt converges to a target value, and the state variable y ═ x2x3]TAnd as the output of the system, constructing a performance index function for optimizing the tracking error at the next moment:
Figure BDA0002287552540000041
namely:
Figure BDA0002287552540000042
w1and w2Weighting coefficients which are respectively the tire slip ratio and the integral thereof;
the Taylor series of k orders at time t is approximated by
Figure BDA0002287552540000043
For x2Performing a first order Taylor series expansion on x3Performing second-order Taylor series expansion:
Figure BDA0002287552540000044
Figure BDA0002287552540000045
and performing Taylor series expansion on the reference slip rate and the integral of the reference slip rate:
Figure BDA0002287552540000046
Figure BDA0002287552540000047
thus, the control input T is obtained by substituting the formula (20) -the formula (23) into the formula (18)bPerformance index function as an independent variable, according to the optimal theory, the performance index function is the mostThe necessary condition for optimization is
Figure BDA0002287552540000048
To obtain
Figure BDA0002287552540000049
Wherein e is2And e3Tracking error e for current output vector2＝x2(t)-x2d(t)；e3＝x3(t)-x3d(t); β is the weight ratio:
Figure BDA0002287552540000051
designing a fractional extremum search controller in the third step comprises designing a fractional extremum search control scheme and determining a fractional coefficient of the fractional extremum search control;
the fractional order extremum search control scheme comprises:
the integer order extremum search controller scheme mathematical model is as follows:
Figure BDA0002287552540000052
where is a convolution operator, z represents the braking deceleration of the output, s represents the frequency domain unit, and L-1Is inverse Laplace transform, d is amplitude of sine exciting signal, k is extreme value search variation rate correction coefficient, omega is angular frequency of sine exciting signal, GHPF(s) is a first order high pass filter
Figure BDA0002287552540000053
Transfer function of, omegahFor high frequency filtering of values, GLPF(s) is a first order low pass filter
Figure BDA0002287552540000054
Transfer function of, omegalFor low frequency filtered values, z represents the braking deceleration,
Figure BDA0002287552540000055
estimating slip rate for an extremum search controller, wherein lambda is a target slip rate actually applied to a nonlinear control system, gamma represents the amplitude of a modulation signal of a high-pass filtered sinusoidal excitation signal, and ξ represents a low-pass filtered demodulation signal;
based on extremum seeking control of integer order output perturbations,
Figure BDA0002287552540000056
and λ*Is about the mean linearized model of the system
Figure BDA0002287552540000057
Wherein the content of the first and second substances,
Figure BDA0002287552540000058
is an error signal, and
Figure BDA0002287552540000059
assuming that the phase delay phi of the perturbation signal is set to 0, with respect to lambda and lambda*Is linearized by
Figure BDA00022875525400000510
The integer order integration is replaced by a fractional order integration,
Figure BDA00022875525400000511
and the high-pass and low-pass filters are fractional order filters
Figure BDA00022875525400000512
Wherein q is more than 0 and less than 1, and L(s) of the fractional order extremum search is
Figure BDA0002287552540000061
By definition of ρ ═ sqTheta and theta with respect to the fractional extremum search control*Can be expressed as
Figure BDA0002287552540000062
Designing a stable extremum searching controller by ensuring the gradual stability of the formula (31);
determining fractional order coefficients of the fractional order extremum search control:
establishing a fractional order coefficient optimization function of
Figure BDA0002287552540000063
Wherein the content of the first and second substances,
Figure BDA0002287552540000064
q is more than 0 and less than or equal to 1, J' is a fractional order coefficient optimizing function of a fractional order numerical value, T is total braking time, q value is optimized to obtain qmaxMaking the fractional order coefficient optimizing function have a minimum value of Jmax，qmaxI.e. the value of the fractional order sought.
Has the advantages that: the invention can convert the advantages of robustness and stability of the fractional order system into the extreme value search controller, and can improve the response characteristic of the braking system, increase the braking deceleration of the vehicle and reduce the braking distance by matching with nonlinear control.
Drawings
FIG. 1 is a schematic illustration of 1/4 vehicle braking models according to the present invention;
FIG. 2 is a schematic diagram of the fractional extremum based search integral nonlinear ABS control of the present invention;
FIG. 3 is a comparison of brake deceleration for different fractional order coefficients according to the present invention;
fig. 4 is a comparison of stopping distances according to the present invention.
Detailed Description
The invention will be further described with reference to the following figures and specific examples, but the scope of the invention is not limited thereto.
1. Establishing a mathematical model
1.1, establishing a vehicle model
The invention takes 1/4 vehicle model as the research object, and the vehicle brake model is shown in figure 1. The model has two degrees of freedom of longitudinal vehicle speed and wheel speed.
The dynamic mathematical model of the vehicle is as follows:
Figure BDA0002287552540000071
Figure BDA0002287552540000072
wherein R is the radius of the wheel and is the total inertia moment of the wheel,
Figure BDA0002287552540000073
in order to be the longitudinal acceleration of the vehicle,
Figure BDA0002287552540000074
as angular acceleration of the wheel, TbFor braking torque, FxIs the tire longitudinal force.
mtFor a given quarter of the total mass of the vehicle:
Figure BDA0002287552540000075
wherein m isvsIs the sprung mass of the vehicle, mwIs the wheel mass.
The longitudinal forces acting on the tire depend on the vertical load of the tire, which is composed of a static load due to the vehicle mass distribution and a dynamic load of the tire during braking. Thus, the tire vertical loads under the 1/4 vehicle model are:
Figure BDA0002287552540000076
wherein l is the wheelbase, hcgIs the height of the center of mass of the sprung mass. FLThe dynamic load transmitted for braking. In practice, the vertical load of the tire is calculated by measuring the longitudinal acceleration a of the tire. However, in simulation studies, substituting equation (1) into equation (4) solves the nonlinear algebraic equation to calculate Fz。
Defining the slip ratio of the tire:
Figure BDA0002287552540000077
derivation of equation (5) with respect to time yields the derivative of slip rate with respect to time as:
Figure BDA0002287552540000078
substituting the formula (1) and the formula (5) into the formula (6) to obtain
Figure BDA0002287552540000079
The formula (1) and the formula (7) constitute a control equation of a state space motion form. The vehicle speed v and the longitudinal slip ratio lambda of the wheel are state vectors and the braking torque TbAre control vectors. In the derivation of the equation, the influence of braking on the pitching and the yawing of the vehicle body is ignored, and only the linear braking condition in the non-steering state is considered.
1.2 tyre model building
Longitudinal force F of wheelxIs described as a function of the longitudinal slip ratio of the tire. When the longitudinal slip amount is smaller, the longitudinal force and the slip amount are in a linear relation, but the longitudinal force of the tire reaches a maximum saturation value at a certain position along with the increase of the slip rate, and the longitudinal force of the tire is reduced to some extent when the slip rate is increased. The saturated behavior of tire forces is a major cause of non-linear behavior of vehicle motion and is also a major cause of vehicle safety.
In order to take the saturation characteristic of the tire force into consideration, the invention adopts a nonlinear Dugoff tire model based on a friction ellipse idea. In this model, the tire longitudinal force is:
Figure BDA0002287552540000081
wherein
Figure BDA0002287552540000082
Figure BDA0002287552540000083
Wherein, CiFor the longitudinal stiffness of the tire, CαIs the lateral stiffness of the tyre, u is the road friction coefficient, epsilonrFor road adhesion coefficient, α is the tire slip angle.
The Dugoff tire model describes the tire characteristics of a tire during braking and steering. Thus, the model takes into account the interaction of the longitudinal and lateral forces of the tire. According to the friction ellipse concept used in the model, the lateral force gradually decreases when the braking force is applied due to the additional slip caused by the contact area. The invention only considers the straight line braking working condition without steering and applies pure longitudinal force to the tire.
1.3 reference slip Rate modeling
Based on the Dugoff tire model, in order to include the transient response of the wheel slip rate in the reference model, avoid larger tracking error and sudden increase of braking moment in the initial braking period, the reference model of the wheel slip rate is established
Figure BDA0002287552540000084
Wherein λopt0.15 represents the optimum slip ratio, and a 20 is a time constant. In fact, equation (11) assumes that the step response for the desired slip is a first order system. Solving Laplace inverse transformation on two sides of the equation, and solving a first-order zero initial condition differential equation to obtain
λd(t)＝λopt-λopte-at(12)
Equation (12) describes a wheel slip rate reference model in the time domain, based on which a non-linear optimal controller is designed to track the reference slip rate.
2. Non-linear optimal controller design
A nonlinear vehicle system dynamic state equation under a braking working condition is constructed by using a formula (1) and a formula (7), a tire slip rate is considered as a system output, and the system output is written into a state space form
Figure BDA0002287552540000091
Figure BDA0002287552540000092
y1＝x2(15)
x＝[v λ]TIs a system state vector, y1Output vector, T, for the systembRepresenting a control input. The non-linear tire model (8) has been incorporated into the function f1And f2In (1).
The design idea of the tire slip rate tracking controller of the invention is as follows: in order to improve the robustness of the slip rate controller, an integral feedback technology is adopted, the integral of the slip rate is added to a control target, the tire slip rate and the tire slip rate integral of the next time interval lambda (T + h) are predicted by a Taylor series expansion method, the prediction period h is similar to the prediction time domain in prediction control, a nonlinear optimal controller is designed by ensuring the tracking error of the tire slip rate and the integral tracking error of the slip rate, and a control vector T is calculatedb(t)。
Defining a new state variable x3Comprises the following steps:
Figure BDA0002287552540000093
the purpose of the control system is to control the wheel slip ratio x2λ and its integral x3═ λ dt converges to the reference response. Changing the state variable y to [ x ]2x3]TAs output of the system, constructOptimizing a performance index function of the tracking error at the next moment:
Figure BDA0002287552540000094
namely:
Figure BDA0002287552540000095
w1and w2Weighting factors, respectively tyre slip rate and integral thereof, for optimal tracking, controlling the input TbThe weighted term of (2) is not included in the performance indicator.
The Taylor series of k orders at time t is approximated by
Figure BDA0002287552540000096
The key problem of taylor series prediction is the choice of order, the higher the approximation degree, but the control system energy will increase, and the lower order causes the prediction error to increase. Thus, the control order is considered a design parameter, making a compromise between performance and input energy requirements. The Taylor series prediction is performed under the condition that the order of the prediction vector is not lower than, and the first-order Taylor series is relative to x2Is sufficient for x3At least a second order taylor series is required for the expansion.
Figure BDA0002287552540000101
Figure BDA0002287552540000102
The above formula is used in the formula (15)
Figure BDA0002287552540000103
The relationship (2) of (c).
Similarly, the state vector of the reference slip rate can be expanded
Figure BDA0002287552540000104
Figure BDA0002287552540000105
Therefore, a performance index function with the control input as a variable is obtained by substituting (equation 18) with equation (20) to equation (23), and the necessary condition for optimizing the performance index function according to the optimization theory is
Figure BDA0002287552540000106
To obtain
Figure BDA0002287552540000107
Wherein e is2And e3Tracking error e for current output vector2＝x2(t)-x2d(t)；e3＝x3(t)-x3d(t); β is the weight ratio:
Figure BDA0002287552540000108
equation (25) is the output control of the nonlinear ABS control system.
Design of 3 fractional order extremum search controller
(1) Fractional extremum search control scheme
In the proposed fractional extremum control scheme, a fractional calculus algorithm is adopted for a filter and integral feedback in extremum search control, braking deceleration is taken as output of search control, and an integer integrator is replaced by the fractional integrator, so that the convergence speed of the extremum search controller can be improved.
The integer order extremum search controller scheme mathematical model is as follows:
Figure BDA0002287552540000111
where is a convolution operator, z represents the braking deceleration of the output, s represents the frequency domain unit, and L-1Is inverse Laplace transform, d is amplitude of sine exciting signal, k is extreme value search variation rate correction coefficient, omega is angular frequency of sine exciting signal, GHPF(s) is a first order high pass filter
Figure BDA0002287552540000112
Transfer function of, omegahFor high frequency filtering of values, GLPF(s) is a first order low pass filter
Figure BDA0002287552540000113
Transfer function of, omegalFor low frequency filtered values, z represents the braking deceleration,
Figure BDA0002287552540000114
the slip rate is estimated for the extremum search controller, λ is the target slip rate actually applied to the nonlinear control system, γ represents the amplitude of the modulation signal with the sinusoidal excitation signal after high-pass filtering, and ξ represents the demodulation signal after low-pass filtering.
According to the requirement of extremum search controller to meet omega < omegah＜ωlAnd k and ξ must be sufficiently small if the mean model is asymptotically stable
Figure BDA0002287552540000115
Is sufficiently small that, where the initial conditions are appropriate, there will be a dependence on k, d and
Figure BDA0002287552540000116
exponentially stable periodic solutions.
Extremum search control based on integer order output disturbances, slip ratio obtained with respect to extremum search controller
Figure BDA0002287552540000117
And an optimum target slip ratio lambda*Is a mean linearized model of
Figure BDA0002287552540000118
Wherein the content of the first and second substances,
Figure BDA0002287552540000119
is an error signal, and
Figure BDA00022875525400001110
the model can be used for stability analysis of an integer order extremum search control average model. If the phase delay φ of the perturbation signal is set to 0, then equation (28) is asymptotically stable for any d > 0. For a single-input single-output extremum search system, by assuming φ as 0, with respect to λ and λ*Is linearized by
Figure BDA00022875525400001111
The integer order integration is replaced by a fractional order integration,
Figure BDA00022875525400001112
and the high-pass and low-pass filters are fractional order filters
Figure BDA00022875525400001113
Wherein 0 < q < 1, as shown in FIG. 2, L(s) of the fractional order extremum search is
Figure BDA0002287552540000121
By definition of ρ ═ sqTheta and theta with respect to the fractional extremum search control*Can be expressed as
Figure BDA0002287552540000122
The average integer order extremum search model has a pair of poles near the imaginary axis that are slightly damped, and thus the settling time of the system is long. And no pole is arranged near the stable boundary of the fractional extremum search average model, so that the system has very high convergence speed.
(2) Fractional coefficient determination for fractional extremum search control
And establishing a fractional order parameter optimizing objective function with the braking deceleration as an index, determining an optimal fractional order parameter by optimizing the fractional order parameter with the maximum fractional order parameter of the fractional order parameter optimizing objective function, and realizing the nonlinear ABS control of the optimal fractional order extremum search.
The selection of the fractional order can cause the overshoot of the braking deceleration and the amplitude variation of the convergence value, so how to select the value of the fractional order has a very important influence on the ABS system. In emergency braking, the greatest possible braking deceleration is achieved, and therefore the objective function of the fractional-order parameter optimization is selected to be
Figure BDA0002287552540000123
Wherein the content of the first and second substances,
Figure BDA0002287552540000124
0＜q≤1。
optimizing the q value to obtain qmaxMake the fractional order parameter optimizing objective function have the minimum value Jmax，qmaxI.e. the value of the fractional order sought.
Example (b):
in order to verify the effectiveness of the control method, the ABS braking working condition of the road surface with the high adhesion coefficient is simulated, and the fractional order extremum based search nonlinear ABS controller and the integer order extremum based search nonlinear ABS controller are compared and analyzed. Parameters required for practical application: u is road friction coefficient selected from 0.8, R is 0.326m, v is 30m/s, Ci＝30000，Cα＝50000N/rad，mw＝40kg,It＝1.7kgm2，mvs＝415kg，a＝20，h＝0.01，β＝20，εr＝0.015s/m，hcg＝0.5m, d 0.002, k 45, λ is set to 0.
FIG. 3 is a brake deceleration comparison graph for different fractional order coefficients of the integral non-linear ABS control based on a fractional order extremum search. As can be seen from the figure, the lower the value of the fractional order q is, the larger the initial overshoot is, and the larger the absolute value of the converged steady-state value of the braking deceleration is; as q increases, the initial overshoot becomes smaller, and the absolute value of the converged steady-state value of the braking deceleration becomes smaller. Importantly, the braking deceleration obtained by the fractional order extremum search control is obviously larger than the numerical value of the traditional integer order extremum search control, and the effectiveness of the fractional order extremum search controller provided by the invention is demonstrated.
Fig. 4 is a braking distance comparison graph of the fractional order extremum-based search nonlinear ABS control and the integer order extremum-based search nonlinear ABS control. When the initial speed of emergency braking is 30m/s, the braking distance of the nonlinear ABS control searched by adopting the integer order extremum is 55.69m, the braking distance of the nonlinear ABS control searched by adopting the fractional order extremum is 54.94m, and the braking distance is reduced by 1.35 percent relative to the integer order, so that the effectiveness of the nonlinear ABS controller searched based on the fractional order extremum provided by the invention is demonstrated.
The present invention is not limited to the above-described embodiments, and any obvious improvements, substitutions or modifications can be made by those skilled in the art without departing from the spirit of the present invention.

Claims (6)

1. A nonlinear ABS control method based on fractional order extremum search is characterized by mainly comprising the following steps:
step one, establishing a mathematical model: respectively establishing an automobile dynamic model, a tire model and a reference slip rate model;
step two, designing a nonlinear ABS controller: adopting an integral feedback method, taking the slip rate and the slip rate integral as control targets, minimizing a prediction tracking error in a continuous interval as an optimization target, adopting a Taylor series expansion method to predict the tire slip rate and the integral thereof in the next time interval lambda (t + h), taking a prediction period h as a prediction time domain, and calculating the current control quantity by ensuring the tire slip rate tracking error and the slip rate integral tracking error;
step three, designing a fractional order extremum search controller: setting a filter and integral feedback in extremum search control into a fractional calculus algorithm, taking braking deceleration as output of search control, establishing a fractional coefficient optimizing function taking the braking deceleration as an index, and determining an optimal fractional coefficient by optimizing the fractional coefficient with the maximum fractional coefficient optimizing function.
2. The fractional extremum search based nonlinear ABS control method of claim 1, wherein the vehicle dynamics model is:
Figure FDA0002287552530000011
Figure FDA0002287552530000012
wherein R is the wheel radius, ItEquivalent moment of inertia for the tire;
Figure FDA0002287552530000013
in order to be the longitudinal acceleration of the vehicle,
Figure FDA0002287552530000014
as angular acceleration of the wheel, TbFor braking torque, FxIn order to apply a longitudinal force to the tire,
mtone quarter of the total mass of the vehicle:
Figure FDA0002287552530000015
wherein m isvsIs the sprung mass of the vehicle, mwIs the wheel mass;
1/4 the tire vertical loads under the vehicle model are:
Figure FDA0002287552530000016
wherein l is the wheelbase, hcgHeight of center of mass of sprung mass, FLSubstituting formula (1) into formula (4) to solve nonlinear algebraic equation to calculate F for braking transmitted dynamic loadz；
Defining the slip ratio λ of the tire:
Figure FDA0002287552530000017
wherein λ represents a tire slip ratio, v represents a vehicle longitudinal speed;
the derivation of equation (5) with respect to time:
Figure FDA0002287552530000021
in which is shown
Figure FDA0002287552530000022
The derivative of the slip rate is represented as,
substituting the formulas (1) and (5) into the formula (6) to obtain
Figure FDA0002287552530000023
3. The fractional extremum search based nonlinear ABS control method of claim 2, wherein the tire mathematical model is:
using a non-linear Dugoff tire model:
Figure FDA0002287552530000024
wherein
Figure FDA0002287552530000025
Figure FDA0002287552530000026
Wherein, CiFor the longitudinal stiffness of the tire, CαIs the lateral stiffness of the tyre, u is the road friction coefficient, epsilonrFor road adhesion coefficient, α is the tire slip angle.
4. The fractional extremum search based nonlinear ABS control method of claim 3 wherein the reference slip ratio mathematical model is:
Figure FDA0002287552530000027
wherein λopt0.15 represents the optimum slip ratio, and a 20 is a time constant; solving Laplace inverse transformation on two sides of the equation, and solving a first-order zero initial condition differential equation to obtain
λd(t)=λopt-λopte-at(12)。
5. The fractional extremum search based nonlinear ABS control method according to claim 4, wherein the step two designs the nonlinear ABS controller:
a nonlinear vehicle system dynamic state equation under a braking working condition is constructed by using a formula (1) and a formula (7), a tire slip rate is considered as a system output, and the system output is written into a state space form
Figure FDA0002287552530000031
Figure FDA0002287552530000032
Selective infusionOutput vector y1Is λ, i.e.
y1＝x2(15)
x＝[v λ]TIs a system state vector, x1And x2Is a state vector, y1Output vector, T, for the systembRepresenting control inputs, the non-linear tyre model (8) having been incorporated into the function f1And f2Performing the following steps;
defining a new state variable x3Comprises the following steps:
Figure FDA0002287552530000033
the control system aims to control the wheel slip ratio x2λ and its integral x3═ λ dt converges to a target value, and the state variable y ═ x2x3]TAnd as the output of the system, constructing a performance index function for optimizing the tracking error at the next moment:
Figure FDA0002287552530000034
namely:
Figure FDA0002287552530000035
w1and w2Weighting coefficients which are respectively the tire slip ratio and the integral thereof;
the Taylor series of k orders at time t is approximated by
Figure FDA0002287552530000036
For x2Performing a first order Taylor series expansion on x3Performing second-order Taylor series expansion:
Figure FDA0002287552530000037
Figure FDA0002287552530000038
and performing Taylor series expansion on the reference slip rate and the integral of the reference slip rate:
Figure FDA0002287552530000041
Figure FDA0002287552530000042
thus, the control input T is obtained by substituting the formula (20) -the formula (23) into the formula (18)bFor the performance index function of the independent variable, the necessary condition for optimizing the performance index function according to the optimal theory is
Figure FDA0002287552530000043
To obtain
Figure FDA0002287552530000044
Wherein e is2And e3Tracking error e for current output vector2＝x2(t)-x2d(t)；e3＝x3(t)-x3d(t); β is the weight ratio:
Figure FDA0002287552530000045
6. the fractional extremum search based nonlinear ABS control method according to claim 4 or 5, wherein designing the fractional extremum search controller in the third step comprises designing a fractional extremum search control scheme and determining fractional coefficients of the fractional extremum search control;
the fractional order extremum search control scheme comprises:
the integer order extremum search controller scheme mathematical model is as follows:
Figure FDA0002287552530000046
where is a convolution operator, z represents the braking deceleration of the output, s represents the frequency domain unit, and L-1Is inverse Laplace transform, d is amplitude of sine exciting signal, k is extreme value search variation rate correction coefficient, omega is angular frequency of sine exciting signal, GHPF(s) is a first order high pass filter
Figure FDA0002287552530000047
Transfer function of, omegahFor high frequency filtering of values, GLPF(s) is a first order low pass filter
Figure FDA0002287552530000048
Transfer function of, omegalFor low frequency filtered values, z represents the braking deceleration,
Figure FDA0002287552530000049
estimating slip rate for an extremum search controller, wherein lambda is a target slip rate actually applied to a nonlinear control system, gamma represents the amplitude of a modulation signal of a high-pass filtered sinusoidal excitation signal, and ξ represents a low-pass filtered demodulation signal;
based on extremum seeking control of integer order output perturbations,
Figure FDA0002287552530000051
and λ*Is about the mean linearized model of the system
Figure FDA0002287552530000052
Wherein the content of the first and second substances,
Figure FDA0002287552530000053
is an error signal, and
Figure FDA0002287552530000054
assuming that the phase delay phi of the perturbation signal is set to 0, with respect to lambda and lambda*Is linearized by
Figure FDA0002287552530000055
The integer order integration is replaced by a fractional order integration,
Figure FDA0002287552530000056
and the high-pass and low-pass filters are fractional order filters
Figure FDA0002287552530000057
Wherein q is more than 0 and less than 1, and L(s) of the fractional order extremum search is
Figure FDA0002287552530000058
By definition of ρ ═ sqTheta and theta with respect to the fractional extremum search control*Can be expressed as
Figure FDA0002287552530000059
Designing a stable extremum searching controller by ensuring the gradual stability of the formula (31);
determining fractional order coefficients of the fractional order extremum search control:
establishing a fractional order coefficient optimization function of
Figure FDA00022875525300000510
Wherein the content of the first and second substances,
Figure FDA00022875525300000511
j' is a fractional order value fractional order coefficient optimizing function, T is total braking time, q value is optimized to obtain qmaxMaking the fractional order coefficient optimizing function have a minimum value of Jmax，qmaxI.e. the value of the fractional order sought.
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