CN113359477B - Design method of vehicle longitudinal and lateral coupling trajectory tracking controller - Google Patents
Design method of vehicle longitudinal and lateral coupling trajectory tracking controller Download PDFInfo
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Abstract
The invention discloses a design method of a vehicle longitudinal and lateral coupling track tracking controller. The method comprises the following steps: firstly, establishing a track tracking model according to the kinematic and dynamic relation between a vehicle and an expected track; designing a track tracking controller coupled in the longitudinal direction and the lateral direction of the vehicle to obtain a complete system control law; the trajectory tracking controller consists of three parts of steady-state-like control, reference dynamic feedforward control and state-dependent error feedback control; compensating system uncertainty disturbance caused by vehicle motion state change by combining with an adaptive law; and step four, realizing the track tracking control of the whole vehicle through a bottom layer control distribution strategy. The invention reliably ensures that the automatic driving vehicle can overcome the disturbance caused by the problems of longitudinal and lateral nonlinear coupling dynamic characteristics, parameter uncertainty and the like of the system, stably and accurately controls the automatic driving vehicle to track the target motion track and expect the longitudinal motion speed, and completes the driving task.
Description
Technical Field
The invention belongs to the technical field of automobiles, and particularly relates to a design method of a vehicle longitudinal and lateral coupling track tracking controller.
Background
Due to breakthrough of key technologies such as perception, computer hardware, software and the like, in recent years, the automatic automobile driving technology has received great attention and development, and is considered to be an effective solution capable of effectively improving the driving safety of the automobile and reducing the fuel consumption rate. Among the related researches, trajectory tracking control is one of the most important core problems for realizing automatic driving of automobiles. The basic task of trajectory tracking is to ensure that the vehicle safely and stably tracks the desired trajectory automatically and accurately at a set speed.
However, the vehicle has many difficulties in tracking the track. First, the design of vehicle controllers relies on models that adequately reflect the key behavioral characteristics of trajectory tracking control. Too complex models will increase the design difficulty of the controller, and too simple models will result in a weak performance of the controller. Secondly, when the vehicle tracks, the system does not model the disturbance, the uncertainty of the parameters and other problems, which will affect the performance of the vehicle controller. Finally, in the tracking process, the longitudinal dynamics and the lateral dynamics of the vehicle have obvious nonlinear coupling dynamics characteristic relation due to factors such as tire force coupling and load transfer variation. This non-linear coupling dynamics characteristic relationship will deteriorate the robustness and accuracy of the controller.
Disclosure of Invention
The invention provides a design method of a vehicle longitudinal and lateral coupling track tracking controller, which reliably ensures that an automatic driving vehicle can overcome the disturbance caused by the problems of longitudinal and lateral nonlinear coupling dynamic characteristics, parameter uncertainty and the like of a system, stably and accurately controls the automatic driving vehicle to track a target motion track and expect longitudinal motion speed, and completes a driving task.
The technical scheme of the invention is described as follows by combining the attached drawings:
a design method for a vehicle longitudinal and lateral coupling trajectory tracking controller comprises the following steps:
step one, establishing a track tracking model containing system parameter uncertainty and vehicle longitudinal and lateral nonlinear coupling dynamics characteristic relation according to the kinematics and dynamics relation between a vehicle and an expected track;
designing a track tracking controller coupled in the longitudinal direction and the lateral direction of the vehicle to obtain a complete system control law; the trajectory tracking controller consists of three parts of steady-state-like control, reference dynamic feedforward control and state-dependent error feedback control;
compensating system uncertainty disturbance caused by vehicle motion state change by combining with an adaptive law;
and step four, realizing the track tracking control of the whole vehicle through a bottom layer control distribution strategy.
The specific method of the first step is as follows:
11) assuming that the left wheel and the right wheel are stressed symmetrically, a simplified vehicle dynamic model is established as follows:
in the formula, m represents the mass of the whole vehicle; v. ofxRepresenting a vehicle longitudinal speed; v. ofyRepresenting a vehicle lateral speed;
a differential representing a longitudinal speed of the vehicle;
a differential representing the lateral velocity of the vehicle; ω represents the yaw angular velocity of the vehicle;
represents the derivative of the yaw rate of the vehicle; fuxRepresenting an equivalent longitudinal force applied at the vehicle center of mass; cwxRepresenting a longitudinal wind resistance coefficient of the vehicle; cwyRepresenting a lateral wind resistance coefficient of the vehicle; i iszRepresenting the moment of inertia of the vehicle; lfRepresenting the distance of the center of mass to the front axle of the vehicle; lrRepresenting the distance of the center of mass to the rear axle of the vehicle; fyfIndicating the lateral force to which the front wheel tyre is subjected; fyrIndicating the lateral force to which the rear wheel tire is subjected;
12) during the tracking process of the vehicle, the tire is in a linear area, and the lateral force of the tire is approximately represented as:
in the formula, CfRepresenting tire front wheel side sheet stiffness; crRepresenting tire rear wheel cornering stiffness; alpha is alphafRepresenting a tire front wheel side slip angle; alpha is alpharIndicating a tire rear wheel side slip angle; fyfIndicating the lateral force to which the front wheel tyre is subjected; fyrIndicating the lateral force to which the rear wheel tire is subjected; beta represents the vehicle centroid slip angle; deltafIndicating a vehicle front wheel steering angle; lfRespectively representing the distances of the center of mass to the front axle of the vehicle; lrRepresenting the distance of the center of mass to the rear axis; v. ofxRepresenting vehicle longitudinal speed, ω representing vehicle yaw angular velocity;
13) assuming that the actual values of the vehicle system parameters during trajectory tracking are known, the complete vehicle dynamics model is represented as:
in the formula, m represents the mass of the whole vehicle; v. ofxRepresenting a vehicle longitudinal speed; v. ofyRepresenting a vehicle lateral speed;
a differential representing a longitudinal speed of the vehicle;
a differential representing the lateral velocity of the vehicle; ω represents the yaw angular velocity of the vehicle;
represents the derivative of the yaw rate of the vehicle; fuxRepresenting an equivalent longitudinal force applied at the vehicle center of mass; beta represents the vehicle centroid slip angle; deltafIndicating a vehicle front wheel steering angle;
ρiactual values representing system parameters;
ideal values representing the variation of system parameters with the vehicle motion state; rhooiStandard values for system parameters are expressed and are expressed as:
in the formula, CwyRepresenting a longitudinal wind resistance coefficient of the vehicle; cwyRepresenting a lateral wind resistance coefficient of the vehicle; i iszRepresenting the moment of inertia of the vehicle; m represents the mass of the whole vehicle; lfRepresenting the distance of the center of mass to the front axle of the vehicle; lrRepresenting the distance of the center of mass to the rear axle of the vehicle; cfRepresenting tire front wheel cornering stiffness; crRepresenting tire rear wheel cornering stiffness;
14) the vehicle kinematic model is:
in the formula, xeIndicating a vehicle longitudinal tracking error; y iseIndicating a vehicle lateral tracking error;
a differential representing a longitudinal tracking error of the vehicle;
a differential representing a vehicle lateral tracking error; v. ofxdRepresenting a desired vehicle longitudinal speed; v. ofxRepresenting an actual vehicle longitudinal speed;
a differential representing an actual vehicle yaw angle;
a derivative representing a desired vehicle yaw angle;
representing an error between the desired yaw angle and the actual yaw angle;
representing the differential error between the desired yaw angle and the actual yaw angle; v. ofyRepresenting a vehicle lateral speed; omegadRepresenting a desired vehicle yaw angular velocity; omega represents the true vehicle yaw angular velocity; rρA road curvature representing a target trajectory;
15) deriving a vehicle kinematic model to obtain:
in the formula, xeIndicating a vehicle longitudinal tracking error; y iseIndicating a vehicle lateral tracking error;
a differential representing a longitudinal tracking error of the vehicle;
a differential representing a vehicle lateral tracking error;
a second derivative representing a longitudinal tracking error of the vehicle;
a second derivative representing a vehicle lateral tracking error; v. ofxdRepresenting a desired vehicle longitudinal speed;
a differential representing a desired vehicle longitudinal speed; v. ofxRepresenting a vehicle longitudinal speed;
a differential representing a longitudinal speed of the vehicle; v. ofyRepresenting a vehicle lateral speed;
a differential representing the lateral velocity of the vehicle;
a derivative representing a yaw angular velocity of the desired vehicle;
represents the derivative of true vehicle yaw angular velocity; ω represents the yaw angular velocity of the vehicle;
representing an error between the desired yaw angle and the actual yaw angle;
representing the differential error between the desired yaw angle and the actual yaw angle;
representing the error second derivative between the desired yaw angle and the actual yaw angle;
a second derivative representing an actual vehicle yaw angle;
a second derivative representing a desired vehicle yaw angle;
16) using longitudinal and lateral tracking errors as state variables of the system, i.e. x1＝[xe ye]T，
The equivalent longitudinal force applied at the vehicle center of mass and the front wheel steering angle serve as control variables of the system, i.e. u ═ Fuxδf]TSubstituting the vehicle dynamics model into the vehicle kinematics model to obtain a vehicle longitudinal and lateral coupling nonlinear trajectory tracking model containing system parameter uncertainty:
y＝x1 (18)
in the formula, x1And x2State variables representing the system; u represents a control variable of the system; y represents the system output, F1(x1)＝02×1Representing a first order system variable matrix; g1(x1)＝I2×2Representing a first order system state matrix;
representing a second-order system variable matrix;
representing a second order system control matrix; p1And P2Is a system parameter matrix, f2,1(x)、g2,1(x) Representing the local basis function of the system, f2(x)、g2(x) And represents the system overall basis function:
P1＝[ρ1 ρ2 ρ3ρ4 ρ6 ρ7]T,P2＝[ρ5 ρ8]T,g2(x)＝[g2,2(x) g2,3(x)]T,
f2(x)＝[f2,2(x) f2,3(x) f2,4(x) f2,5(x) f2,6(x) f2,7(x)]T,
in the formula, P1And P2Is a system parameter matrix, pi(i ═ 1,2, … 8) represents the true values of the system parameters; f. of2,j(x)(j＝1,2,…7)、g2,k(x) (k ═ 1,2,3) represents the system local basis function; f. of2(x)、g2(x) Representing the system overall basis function; v. ofxRepresenting a vehicle longitudinal speed; v. ofyRepresenting a vehicle lateral speed; v. ofxdRepresenting a desired vehicle longitudinal speed;
a differential representing a desired vehicle longitudinal speed;
representing an error between the desired yaw angle and the actual yaw angle;
representing the differential error between the desired yaw angle and the actual yaw angle; ω represents the yaw angular velocity of the vehicle; x is the number ofeIndicating a vehicle longitudinal tracking error; y iseIndicating a vehicle lateral tracking error; beta represents the vehicle centroid slip angle; and m represents the mass of the whole vehicle.
The specific method of the second step is as follows:
21) the system output y is derived until the occurrence of the control input u:
in the formula, x1And x2State variables representing the system;
representing a state variable differential of the system;
represents the differential of the system output;
represents the second derivative of the system output; f, F1(x1) Representing a first order system variable matrix; g1(x1) Representing a first order system state matrix; u represents a control variable of the system;
an output matrix representing the second order system output; b (x) ═ G1(x1)G2(x) A control matrix representing the second order system output; c (x) ═ G1(x1)F2(x) A parameter matrix representing the second order system output; f2(x) Representing a second-order system variable matrix; g2(x) Representing a second order system control matrix;
22) it is assumed that the system is already in steady state motion, at which point
Then the steady-state-like control law of the system is:
in the formula usRepresenting a steady-state-like control law of the system; g2(x) Representing a second order system control matrix; f2(x) Representing a second-order system variable matrix; f. of2,1(x)、g2,1(x) Representing a system local basis function; f. of2(x)、g2(x) Representing the system overall basis function; p1And P2Is a system parameter matrix;
23) adding a feedforward control law into a calibrated control law is used for improving the response performance of the system, namely:
u＝us+uf (22)
wherein u represents a system control law; u. ofsSteady state-like control law, u, representing the systemfRepresenting the dynamic reference feedforward control law of the system.
Let y be yd，
Substituting the control law (22) into the system (20) to obtain a feed forward control law as follows:
in the formula ufRepresenting a dynamic reference feedforward control law of the system; g2(x) Representing a second order system control matrix;
a differential representing a desired output of the system;
a second derivative representing the desired output of the system; a (x) represents an output matrix of the second order system output; a (x) represents an output matrix of the second order system output; p2Is a system parameter matrix; g2,1(x) Representing a system local basis function; g2(x) Representing the system overall basis function;
24) the introduced error feedback control law is as follows:
u＝us+uf+ue (24)
wherein u represents a system control law; u. ofsRepresenting a steady-state-like control law of the system; u. offRepresenting a dynamic reference feedforward control law of the system; u. ofeRepresenting the error feedback control law of the system;
defining a system tracking error as e1＝yd-y, substituting the control law (24) into the system (20) can determine:
in the formula,
represents the system tracking error differential;
a second derivative representing a system tracking error; a (x) represents an output matrix of the second order system output; g2(x) Representing a second order system control matrix; u. ofeRepresenting the error feedback control law of the system;
order to
The closed loop error system is then expressed as:
in the formula,
represents the system tracking error differential; e.g. of the type2Representing a virtual control input;
representing a virtual control input differential; a (x) represents an output matrix of the second order system output; g2(x) Representing a second order system control matrix, ueRepresenting the error feedback control law of the system;
25) option e2Defining Lyapunov functions for virtual control inputs
Then the corresponding derivative is:
in the formula,
representing a defined first lyapunov function differential; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; e.g. of the type2Representing a virtual control input;
26) to ensure stability of the closed-loop control system, i.e.
The selected virtual control inputs are:
in the formula,
representing a desired virtual control input; e.g. of the type1Indicating the system tracking error, k1Representing a controller parameter;
for k1>0，
It is possible to obtain:
in the formula,
representing a defined first lyapunov function differential; k is a radical of1Representing a controller parameter; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential;
27) to e is making e2Approach to
To ensure the system e1→ 0, we define
Then we get:
in the formula,
representing the first defined lyapunov function differential, k1Representing the three-step controller parameters, e1And
respectively representing the systematic tracking error and the systematic tracking error differential, e3Representing a virtual control input error.
With e3→ 0, obtaining
Then system e1Gradual stabilization to error e3The derivation is carried out to obtain:
in the formula,
representing a virtual control input error differential; e.g. of the type2Representing an actual virtual control input;
representing a desired virtual control input differential;
representing a virtual control input differential; k is a radical of1Representing a controller parameter;
represents the system tracking error differential; a (x) represents an output matrix of the second order system output; g2(x) Representing a second order system control matrix; u. ofeRepresenting the error feedback control law of the system;
28) definition includes error e3Lyapunov function of the interior
And (5) obtaining by derivation:
in the formula,
represents a defined second lyapunov function differential;
representing a defined first lyapunov function differential; e.g. of the type3Representing a virtual control input error;
representing a virtual control input error differential; k is a radical of1Representing a controller parameter; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; a (x) represents an output matrix of the second order system output; g2(x) Representing a second order system control matrix; u. ofeRepresenting the error feedback control law of the system.
According to the Lyapunov direct method, the error feedback control law is selected as follows:
in the formula ueRepresenting the error feedback control law of the system; a (x) represents an output matrix of the second order system output; g2(x) Representing a second order system control matrix; k is a radical of1And k2Representing a controller parameter; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; e.g. of the type3Representing a virtual control input error;
selection of k2>0, then
In the formula,
represents a defined second lyapunov function differential; k is a radical of1And k2Representing a controller parameter; e.g. of the type1Representing a system tracking error; e.g. of the type3Representing a virtual control input error;
29) the error closed loop system is gradually stabilized, and (26), (29) and
and (36) obtaining the error feedback control law of practical application as follows:
in the formula ueRepresenting the error feedback control law of the system; a (x) an output matrix representing the output of the second order system, G2(x) Representing a second order system control matrix; k is a radical of1And k2Representing a three-step controller parameter; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; g2,1(x) Representing a system local basis function; g2(x) Representing the system overall basis function; p2Is a system parameter matrix;
the obtained complete system control law:
wherein u represents a system control law; u. ofsIs a system-like steady-state control law; u. offThe system dynamically refers to a feedforward control law; u. ofeRepresenting a system error feedback control law; f. ofP(x)＝[G2(x)]-1(1+k1k2) Representing a scale term parameter; f. ofD(x)＝[G2(x)]-1[k1+k2+A(x)]Representing a differential term parameter; a (x) represents an output matrix of the second order system output; g2(x) Representing a second order system control matrix; k is a radical of1And k2Representing a controller parameter; e.g. of the type1Representing a system tracking error;
representing the system tracking error differential.
The concrete method of the third step is as follows:
31) defining system parameter estimation values:
in the formula,
representing system parameter estimates;
an estimate representing a change in a system parameter with a vehicle motion state; rhooiStandard values representing system parameters;
32) the system control law is redefined as:
wherein u represents a system control input;
representing a second-order system variable matrix estimated value;
representing a second-order system control matrix estimated value;
P1and P2Is a system parameter matrix estimated value;
representing system parameter estimates;
a differential representing a desired output of the system;
a second derivative representing the desired output of the system;
an output matrix estimation value representing the output of the second-order system; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; k is a radical of1And k2Representing a controller parameter;
33) by substituting the obtained system control law (41) into the system (20), the following can be obtained:
in the formula,
represents the second derivative of the system output; k is a radical of1And k2Representing a controller parameter;
an output matrix estimation value representing the output of the second-order system; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; f2(x) The representation represents a second-order system variable matrix;
representing a second-order system variable matrix estimated value; g2(x) Representing a second order system control matrix;
representing a second-order system control matrix estimated value; u represents a system control input;
a second derivative representing the desired output of the system;
from the above equation, the closed-loop error system considering parameter uncertainty can be organized as:
wherein,
in the formula, k1And k2Representing a controller parameter;
an output matrix estimation value representing the output of the second-order system; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; Δ P represents the total error of the system parameter matrix; theta represents a system composition base; p1And P2Representing a system parameter matrix; delta P1And Δ P2Representing the system parameter matrix error;
an estimate representing a change in a system parameter with a vehicle motion state;
ideal values representing the variation of system parameters with the vehicle motion state; f. of2(x)、g2(x) Representing the system overall basis function; u represents a system control input;
34) selecting
Is the state variable of the error system, then:
in the formula, emA state variable representing an error system;
a state variable differential representing an error system;
a state variable parameter matrix representing an error system;
a parameter variable matrix representing an error system; theta represents a system composition base; Δ P represents the total error of the system parameter matrix; k is a radical of1And k2Representing a controller parameter;
35) defining a Lyapunov function including uncertainty of system parameters
Wherein Q>0 is a positive definite symmetric matrix, and the derivation is as follows:
in the formula,
represents a defined third lyapunov function differential; e.g. of the typemA state variable representing an error system; a. themA state variable parameter matrix representing an error system; q represents a positive definite symmetric matrix; b ismA parameter variable matrix representing an error system; theta represents a system composition base; Δ P represents the total error of the system parameter matrix;
representing the total error differential of the system parameter matrix; tau represents a parameter error variable parameter quantization parameter;
representing the system parameter adaptation law;
36) the system parameter self-adaptation law is taken as follows:
in the formula,
represents a system parameter ofAn adaptation law; tau represents a parameter error variable parameter quantization parameter; e.g. of the typemA state variable representing an error system; q represents a positive definite symmetric matrix; b ismA parameter variable matrix representing an error system, and theta represents a system combination base;
and substituting the adaptive law of the system parameters into (45) to obtain:
in the formula,
represents a defined third lyapunov function differential; e.g. of the typemA state variable representing an error system; q represents a positive definite symmetric matrix; a. themA state variable parameter matrix representing an error system.
The concrete method of the fourth step is as follows:
equivalent longitudinal force F applied to vehicle centroid and obtained based on solution of trajectory tracking controlleruxCan not be directly applied to vehicle control, and needs to be decomposed by a bottom-layer control distribution strategy to obtain a driving torque signal T which can be responded by the vehicletqAnd a brake pressure signal;
equivalent longitudinal force F when applied at the center of mass of the vehicleuxWhen the torque is greater than or equal to 0, the vehicle system is considered to be in the driving mode, and the corresponding driving torque is expressed as:
in the formula, TtqA signal indicative of a target drive torque of the vehicle; fuxRepresenting an equivalent longitudinal force applied at the vehicle center of mass; r represents a wheel rolling radius; i.e. i0Representing the whole equivalent transmission ratio of the whole vehicle; etaTThe whole equivalent transmission efficiency of the whole vehicle is represented;
equivalent longitudinal force F when applied at the center of mass of the vehicleuxLess than 0; consider a vehicle trainWhen the system is in the driving mode, the corresponding driving torque can be expressed as:
wherein P represents a vehicle target master cylinder brake pressure; fuxRepresenting an equivalent longitudinal force applied at the vehicle center of mass; r represents a wheel rolling radius; d represents the diameter of the brake master cylinder; rBRepresents the effective brake caliper radius; i istRepresenting the moment of inertia of the tire;
representing the differential of the tire speed.
The invention has the beneficial effects that:
1) the vehicle track tracking model built by the invention comprises the problems of longitudinal and lateral nonlinear coupling dynamic characteristics of the vehicle, uncertainty of system parameters and the like, and fully reflects the key behavior characteristics expressed in the track tracking control of the automatic driving vehicle;
2) the invention provides a vehicle longitudinal and lateral coupling track tracking controller based on a nonlinear three-step method control theory, and solves the nonlinear control problem brought to a system by the vehicle longitudinal and lateral nonlinear coupling dynamic characteristics;
3) the vehicle track tracking controller designed based on the nonlinear three-step method theory has a simple structure, has a structure similar to a PID algorithm applied to engineering, accords with the use habit calibrated by an algorithm engineer, and is convenient for practical application and popularization of engineering;
4) the invention provides a used parameter self-adaptive law which effectively compensates the uncertain disturbance of system parameters caused by the change of the vehicle motion state and improves the track tracking control precision and the system robustness;
5) based on the Lyapunov stability theory, the method deduces that the designed vehicle track tracking controller has good robustness to the problems of unmodeled disturbance, parameter uncertainty and the like of the system.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings needed to be used in the embodiments will be briefly described below, it should be understood that the following drawings only illustrate some embodiments of the present invention and therefore should not be considered as limiting the scope, and for those skilled in the art, other related drawings can be obtained according to the drawings without inventive efforts.
FIG. 1 is an architectural diagram of the present invention;
FIG. 2 is a schematic diagram of a trajectory tracking dynamics model of an autonomous vehicle;
FIG. 3 is a schematic diagram of a trajectory tracking kinematics model of an autonomous vehicle;
FIG. 4 is a schematic diagram of a sorting architecture for adaptive control law compensation of system parameter uncertainty;
FIG. 5 is a graph of longitudinal speed control performance;
FIG. 6 is a graph of longitudinal speed error control performance;
FIG. 7 is a trace-tracking graph;
FIG. 8 is a graph of trajectory tracking lateral error;
fig. 9 is a graph of yaw angle control performance;
FIG. 10 is a graph of yaw angle error control performance;
FIG. 11 is a graph of front wheel steering control performance;
fig. 12 is a graph of yaw-rate control performance.
Detailed Description
The technical solution in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.
Referring to fig. 1, a method for designing a vehicle longitudinal-lateral coupling trajectory tracking controller includes the following steps:
step one, establishing a track tracking model containing system parameter uncertainty and vehicle longitudinal and lateral nonlinear coupling dynamics characteristic relation according to the kinematics and dynamics relation between a vehicle and an expected track;
referring to fig. 2, 11), a complete vehicle dynamics model is established, and although the designed controller can be closer to the practical application, the design burden of the controller is also greatly increased. The relationship between the longitudinal motion and the lateral motion of the vehicle is more concerned in track tracking control, so that the roll, pitch and vertical motion of the vehicle are ignored, the left wheel and the right wheel are assumed to be stressed symmetrically, and a simplified vehicle dynamic model is established as follows:
in the formula, m represents the mass of the whole vehicle; v. ofxRepresenting a vehicle longitudinal speed; v. ofyRepresenting a vehicle lateral speed;
a differential representing a longitudinal speed of the vehicle;
a differential representing the lateral velocity of the vehicle; ω represents the yaw angular velocity of the vehicle;
represents the derivative of the yaw rate of the vehicle; fuxRepresenting an equivalent longitudinal force applied at the vehicle center of mass; cwxRepresenting a longitudinal wind resistance coefficient of the vehicle; cwyRepresenting a lateral wind resistance coefficient of the vehicle; i iszRepresenting the moment of inertia of the vehicle; lfRepresenting the distance of the center of mass to the front axle of the vehicle; lrRepresenting the distance of the center of mass to the rear axle of the vehicle; fyfIndicating the lateral force to which the front wheel tyre is subjected; fyrIndicating the lateral force to which the rear wheel tire is subjected;
12) during the tracking process of the vehicle, the tire is in a linear area, and the lateral force of the tire is approximately represented as:
in the formula, CfRepresenting tire front wheel side sheet stiffness; crRepresenting tire rear wheel cornering stiffness; alpha is alphafRepresenting a tire front wheel side slip angle; alpha is alpharIndicating a tire rear wheel side slip angle; fyfIndicating the lateral force to which the front wheel tyre is subjected; fyrIndicating the lateral force to which the rear wheel tire is subjected; beta represents the vehicle centroid slip angle; deltafIndicating a vehicle front wheel steering angle; lfRespectively representing the distances of the center of mass to the front axle of the vehicle; lrRepresenting the distance of the center of mass to the rear axis; v. ofxRepresenting vehicle longitudinal speed, ω representing vehicle yaw angular velocity;
13) in addition, vehicle system parameters can create uncertainties as certain changes occur in the motion state of the vehicle. Assuming that the actual values of the vehicle system parameters during trajectory tracking are known, the complete vehicle dynamics model is represented as:
in the formula, m represents the mass of the whole vehicle; v. ofxRepresenting a vehicle longitudinal speed; v. ofyRepresenting a vehicle lateral speed;
a differential representing a longitudinal speed of the vehicle;
a differential representing the lateral velocity of the vehicle; ω represents the yaw angular velocity of the vehicle;
represents the derivative of the yaw rate of the vehicle; fuxRepresenting an equivalent longitudinal force applied at the vehicle center of mass; beta represents the vehicle centroid slip angle; deltafIndicating a vehicle front wheel steering angle;
ρiactual values representing system parameters;
ideal values representing the variation of system parameters with the vehicle motion state; rhooiStandard values for system parameters are expressed and are expressed as:
in the formula, CwxRepresenting a longitudinal wind resistance coefficient of the vehicle; cwyRepresenting a lateral wind resistance coefficient of the vehicle; i iszRepresenting the moment of inertia of the vehicle; m represents the mass of the whole vehicle; lfRepresenting the distance of the center of mass to the front axle of the vehicle; lrRepresenting the distance of the center of mass to the rear axle of the vehicle; cfRepresenting tire front wheel cornering stiffness; crRepresenting tire rear wheel cornering stiffness;
referring to fig. 3, in order to obtain a trajectory tracking model, a kinematic model of the vehicle needs to be derived. FIG. 3 is a schematic view of trajectory tracking kinematics for an autonomous vehicle, wherein f1Represents the position, r, of the vehicle at the present moment3Representing the desired movement position, r, of the vehicle on the target trajectory2To a desired movement positionThe intersection of the normal of the velocity and the current motion position velocity direction.
14) The vehicle kinematic model is:
in the formula, xeIndicating a vehicle longitudinal tracking error; y iseIndicating a vehicle lateral tracking error;
a differential representing a longitudinal tracking error of the vehicle;
a differential representing a vehicle lateral tracking error; v. ofxdRepresenting a desired vehicle longitudinal speed; v. ofxRepresenting an actual vehicle longitudinal speed;
a differential representing an actual vehicle yaw angle;
a derivative representing a desired vehicle yaw angle;
representing an error between the desired yaw angle and the actual yaw angle;
representing the differential error between the desired yaw angle and the actual yaw angle; v. ofyRepresenting a vehicle lateral speed; omegadRepresenting a desired vehicle yaw angular velocity; omega represents the true vehicle yaw angular velocity; rρA road curvature representing a target trajectory;
15) deriving a vehicle kinematic model to obtain:
in the formula, xeIndicating a vehicle longitudinal tracking error; y iseIndicating a vehicle lateral tracking error;
a differential representing a longitudinal tracking error of the vehicle;
a differential representing a vehicle lateral tracking error;
a second derivative representing a longitudinal tracking error of the vehicle;
a second derivative representing a vehicle lateral tracking error; v. ofxdRepresenting a desired vehicle longitudinal speed;
a differential representing a desired vehicle longitudinal speed; v. ofxRepresenting a vehicle longitudinal speed;
a differential representing a longitudinal speed of the vehicle; v. ofyRepresenting a vehicle lateral speed;
a differential representing the lateral velocity of the vehicle;
a derivative representing a yaw angular velocity of the desired vehicle;
represents the derivative of true vehicle yaw angular velocity; ω represents the yaw angular velocity of the vehicle;
representing an error between the desired yaw angle and the actual yaw angle;
representing the differential error between the desired yaw angle and the actual yaw angle;
representing the error second derivative between the desired yaw angle and the actual yaw angle;
a second derivative representing an actual vehicle yaw angle;
a second derivative representing a desired vehicle yaw angle;
16) using longitudinal and lateral tracking errors as state variables of the system, i.e. x1＝[xe ye]T，
The equivalent longitudinal force applied at the vehicle's center of mass and the front wheel steering angle as control variables of the system, i.e.u＝[Fuxδf]TSubstituting the vehicle dynamics model into the vehicle kinematics model to obtain a vehicle longitudinal and lateral coupling nonlinear trajectory tracking model containing system parameter uncertainty:
y＝x1 (18)
in the formula, x1And x2State variables representing the system; u represents a control variable of the system; y represents the system output, F1(x1)＝02×1Representing a first order system variable matrix; g1(x1)＝I2×2Representing a first order system state matrix;
representing a second-order system variable matrix;
representing a second order system control matrix; p1And P2Is a system parameter matrix, f2,1(x)、g2,1(x) Representing the local basis function of the system, f2(x)、g2(x) And represents the system overall basis function:
P1＝[ρ1 ρ2 ρ3 ρ4 ρ6 ρ7]T,P2＝[ρ5 ρ8]T,g2(x)＝[g2,2(x) g2,3(x)]T,
f2(x)＝[f2,2(x) f2,3(x) f2,4(x) f2,5(x) f2,6(x) f2,7(x)]T,
in the formula, P1And P2Is a system parameter matrix, pi(i ═ 1,2, … 8) represents the true values of the system parameters; f. of2,j(x)(j＝1,2,…7)、g2,k(x) (k ═ 1,2,3) represents the system local basis function; f. of2(x)、g2(x) Representing the system overall basis function; v. ofxRepresenting a vehicle longitudinal speed; v. ofyRepresenting a vehicle lateral speed; v. ofxdRepresenting a desired vehicle longitudinal speed;
a differential representing a desired vehicle longitudinal speed;
representing an error between the desired yaw angle and the actual yaw angle;
representing the differential error between the desired yaw angle and the actual yaw angle; ω represents the yaw angular velocity of the vehicle; x is the number ofeIndicating a vehicle longitudinal tracking error; y iseIndicating a vehicle lateral tracking error; beta represents the vehicle centroid slip angle; and m represents the mass of the whole vehicle.
Designing a vehicle longitudinal and lateral coupling track tracking controller based on a nonlinear three-step method theory to obtain a complete system control law; the three-step method track tracking controller consists of three parts of steady-state-like control, reference dynamic feedforward control and state-dependent error feedback control;
21) the system output y is derived until the occurrence of the control input u:
in the formula, x1And x2State variables representing the system;
representing a state variable differential of the system;
represents the differential of the system output;
represents the second derivative of the system output; f, F1(x1) Representing a first order system variable matrix; g1(x1) Representing a first order system state matrix; u represents a control variable of the system;
an output matrix representing the second order system output; b (x) ═ G1(x1)G2(x) A control matrix representing the second order system output; c (x) ═ G1(x1)F2(x) A parameter matrix representing the second order system output; f2(x) Representing a second-order system variable matrix; g2(x) Representing a second order system control matrix;
22) it is assumed that the system is already in steady state motion, at which point
Then the steady-state-like control law of the system is:
in the formula usRepresenting a steady-state-like control law of the system; g2(x) Representing a second order system control matrix; f2(x) Representing a second-order system variable matrix; f. of2,1(x)、g2,1(x) Representing a system local basis function; f. of2(x)、g2(x) Representing the system overall basis function; p1And P2Is a system parameter matrix;
23) adding a feedforward control law into a calibrated control law is used for improving the response performance of the system, namely:
u＝us+uf (22)
wherein u represents a system control law; u. ofsSteady state-like control law, u, representing the systemfRepresenting the dynamic reference feedforward control law of the system.
Let y be yd，
Substituting the control law (22) into the system (20) to obtain a feed forward control law as follows:
in the formula ufRepresenting a dynamic reference feedforward control law of the system; g2(x) Representing a second order system control matrix;
a differential representing a desired output of the system;
a second derivative representing the desired output of the system; a (x) represents the output moment of the second order system outputArraying; a (x) represents an output matrix of the second order system output; p2Is a system parameter matrix; g2,1(x) Representing a system local basis function; g2(x) Representing the system overall basis function;
24) considering modeling errors, external disturbance and other factors, the quasi-steady-state control law and the dynamic reference feedforward control law cannot completely eliminate the track tracking errors. Therefore, the error feedback control law continues to be introduced as:
u＝us+uf+ue (24)
wherein u represents a system control law; u. ofsRepresenting a steady-state-like control law of the system; u. offRepresenting a dynamic reference feedforward control law of the system; u. ofeRepresenting the error feedback control law of the system;
defining a system tracking error as e1＝yd-y, substituting the control law (24) into the system (20) can determine:
in the formula,
represents the system tracking error differential;
a second derivative representing a system tracking error; a (x) represents an output matrix of the second order system output; g2(x) Representing a second order system control matrix; u. ofeRepresenting the error feedback control law of the system;
order to
The closed loop error system is then expressed as:
in the formula,
represents the system tracking error differential; e.g. of the type2Representing a virtual control input;
representing a virtual control input differential; a (x) represents an output matrix of the second order system output; g2(x) Representing a second order system control matrix, ueRepresenting the error feedback control law of the system;
25) due to the introduction of the quasi-steady-state control law and the reference dynamic feedforward control law, the closed-loop error system is simpler in structure. In order to obtain an error feedback control law, the method adopts a classic backstepping method to deduce the error feedback control law, and the closed loop stability of a track tracking system is ensured. Option e2Defining Lyapunov functions for virtual control inputs
Then the corresponding derivative is:
in the formula,
representing a defined first lyapunov function differential; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; e.g. of the type2Representing a virtual control input;
26) to ensure stability of the closed-loop control system, i.e.
The selected virtual control inputs are:
in the formula,
representing a desired virtual control input; e.g. of the type1Indicating the system tracking error, k1Representing a controller parameter;
for k1>0，
It is possible to obtain:
in the formula,
representing a defined first lyapunov function differential; k is a radical of1Representing a controller parameter; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential;
27) however, in the tracking process, it is difficult to fully implement
To e is making e2Approach to
To ensure the system e1→ 0, we define
Then we get:
in the formula,
representing the first defined lyapunov function differential, k1Representing the three-step controller parameters, e1And
respectively representing the systematic tracking error and the systematic tracking error differential, e3Representing a virtual control input error.
With e3→ 0, obtaining
Then system e1Gradual stabilization to error e3The derivation is carried out to obtain:
in the formula,
representing a virtual control input error differential; e.g. of the type2Representing an actual virtual control input;
representing a desired virtual control input differential;
representing a virtual control input differential; k is a radical of1Representing a controller parameter;
represents the system tracking error differential; a (x) represents an output matrix of the second order system output; g2(x) Representing a second order system control matrix; u. ofeRepresenting the error feedback control law of the system;
28) definition includes error e3Lyapunov function of the interior
And (5) obtaining by derivation:
in the formula,
represents a defined second lyapunov function differential;
representing a defined first lyapunov function differential; e.g. of the type3Representing a virtual control input error;
representing a virtual control input error differential; k is a radical of1Representing a controller parameter; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; a (x) represents an output matrix of the second order system output; g2(x) Representing a second order system control matrix; u. ofeRepresenting the error feedback control law of the system.
According to the Lyapunov direct method, the error feedback control law is selected as follows:
in the formula ueRepresenting the error feedback control law of the system; a (x) represents an output matrix of the second order system output; g2(x) Representing a second order system control matrix; k is a radical of1And k2Representing a controller parameter; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; e.g. of the type3Representing a virtual control input error;
selection of k2>0, then
In the formula,
represents a defined second lyapunov function differential; k is a radical of1And k2Representing a controller parameter; e.g. of the type1Representing a system tracking error; e.g. of the type3Representing a virtual control input error;
29) this means that the error closed loop system is progressively stabilized, will (26), (29) and
and (36) obtaining the error feedback control law of practical application as follows:
in the formula ueRepresenting the error feedback control law of the system; a (x) an output matrix representing the output of the second order system, G2(x) Representing a second order system control matrix; k is a radical of1And k2Representing a three-step controller parameter; e.g. of the type1Presentation systemTracking errors;
represents the system tracking error differential; g2,1(x) Representing a system local basis function; g2(x) Representing the system overall basis function; p2Is a system parameter matrix;
thus far, the complete system control law obtained:
wherein u represents a system control law; u. ofsIs a system-like steady-state control law; u. offThe system dynamically refers to a feedforward control law; u. ofeRepresenting a system error feedback control law; f. ofP(x)＝[G2(x)]-1(1+k1k2) Representing a scale term parameter; f. ofD(x)＝[G2(x)]-1[k1+k2+A(x)]Representing a differential term parameter; a (x) represents an output matrix of the second order system output; g2(x) Representing a second order system control matrix; k is a radical of1And k2Representing a controller parameter; e.g. of the type1Representing a system tracking error;
representing the system tracking error differential.
Compensating system uncertainty disturbance caused by vehicle motion state change by combining with an adaptive law;
besides the need of processing the nonlinear coupling problem of the transverse and longitudinal dynamics, the vehicle track tracking controller has the system parameter true value rhoiUncertainty associated with ( i 1,2, …,8) changes in vehicle motion also interferes with trajectory tracking performance. Therefore, the adaptive control law is used for compensating the uncertainty problem of the system parameters, and the overall architecture is shown in FIG. 4.
31) Defining system parameter estimation values:
in the formula,
representing system parameter estimates;
an estimate representing a change in a system parameter with a vehicle motion state; rhooiStandard values representing system parameters;
32) the system control law is redefined as:
wherein u represents a system control input;
representing a second-order system variable matrix estimated value;
representing a second-order system control matrix estimated value;
P1and P2Is a system parameter matrix estimated value;
representing system parameter estimates;
a differential representing a desired output of the system;
a second derivative representing the desired output of the system;
an output matrix estimation value representing the output of the second-order system; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; k is a radical of1And k2Representing a controller parameter;
33) substituting the obtained system control law (41) into the system (20) to obtain:
in the formula,
represents the second derivative of the system output; k is a radical of1And k2Representing a controller parameter;
an output matrix estimation value representing the output of the second-order system; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; f2(x) The representation represents a second-order system variable matrix;
representing a second-order system variable matrix estimated value; g2(x) Representing a second order system control matrix;
representing a second-order system control matrix estimated value; u represents a system control input;
a second derivative representing the desired output of the system;
from the above equation, the closed-loop error system considering parameter uncertainty can be organized as:
wherein,
in the formula, k1And k2Representing a controller parameter;
an output matrix estimation value representing the output of the second-order system; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; Δ P represents the total error of the system parameter matrix; theta represents a system composition base; p1And P2Representing a system parameter matrix; delta P1And Δ P2Representing the system parameter matrix error;
an estimate representing a change in a system parameter with a vehicle motion state;
ideal values representing the variation of system parameters with the vehicle motion state; f. of2(x)、g2(x) Representing the system overall basis function; u represents a system control input;
34) selecting
Is the state variable of the error system, then:
in the formula, emIndicating the error systemA state variable of the system;
a state variable differential representing an error system;
a state variable parameter matrix representing an error system;
a parameter variable matrix representing an error system; theta represents a system composition base; Δ P represents the total error of the system parameter matrix; k is a radical of1And k2Representing a controller parameter;
35) defining a Lyapunov function including uncertainty of system parameters
Wherein Q>0 is a positive definite symmetric matrix, and the derivation is as follows:
in the formula,
represents a defined third lyapunov function differential; e.g. of the typemA state variable representing an error system; a. themA state variable parameter matrix representing an error system; q represents a positive definite symmetric matrix; b ismA parameter variable matrix representing an error system; theta represents a system composition base; Δ P represents the total error of the system parameter matrix;
representing the total error differential of the system parameter matrix; tau represents a parameter error variable parameter quantization parameter;
representing the system parameter adaptation law;
36) the system parameter self-adaptation law is taken as follows:
in the formula,
representing the system parameter adaptation law; tau represents a parameter error variable parameter quantization parameter; e.g. of the typemA state variable representing an error system; q represents a positive definite symmetric matrix; b ismA parameter variable matrix representing an error system, and theta represents a system combination base;
substituting the system parameter adaptation law into (45) to obtain:
in the formula,
represents a defined third lyapunov function differential; e.g. of the typemA state variable representing an error system; q represents a positive definite symmetric matrix; a. themA state variable parameter matrix representing an error system;
thus, only a suitable a needs to be selectedmAnd Q guarantee
Can make
Thus proving that the closed loop system is consistently bounded, i.e., limt→∞e1→0。
37) The following performs a system robustness analysis. In order to simplify problem analysis, factors such as parameter uncertainty, unmodeled disturbance and the like faced by the system are uniformly defined as xi, so that the original system control law is changed into:
u＝uim+ξ (1)
in the formula, xi represents parameter uncertainty and unmodeled disturbance factors faced by the system; u. ofimRepresenting the original system control input; u represents a system control input;
substitution of (48) into (20) in combination with e1＝yd-y is reduced to yield:
in the formula, e1Representing a system tracking error;
represents the system tracking error differential;
representing a system tracking error second derivative; k is a radical of1And k2Representing a controller parameter;
an output matrix estimation value representing the output of the second-order system; Δ P represents the total error of the system parameter matrix; delta P2Representing the system parameter matrix error; theta represents a system composition base; ξ represents the parameter uncertainty, unmodeled perturbation factor faced by the system.
Let d be Δ P2ξ, then the closed-loop error system is represented as:
in the formula, emA state variable representing an error system;
a state variable differential representing an error system; a. themState variable parameter moment representing error systemArraying; b ismA parameter variable matrix representing an error system; theta represents a system composition base; Δ P represents the total error of the system parameter matrix; d represents the overall disturbance faced by the system;
38) definition of
And gamma is>0, then Lyapunov function
The derivative of (c) is reduced to:
in the formula,
represents a defined third lyapunov function differential; e.g. of the typemA state variable representing an error system; a. themA state variable parameter matrix representing an error system; b ismA parameter variable matrix representing an error system; theta represents a system composition base; Δ P represents the total error of the system parameter matrix; d represents the overall disturbance faced by the system;
representing the system parameter adaptation law; tau represents a parameter error variable parameter quantization parameter; q represents a positive definite symmetric matrix; l represents a defined comparison matrix; γ represents a defined comparison matrix parameter quantization parameter.
In view of
Is a continuous positive function, and the error system (48) is input state stable for the comprehensive disturbance d faced by the system. Therefore, the system is robust to the uncertainty of the system parameters considered, unmodeled disturbances, etc.
And step four, realizing the track tracking control of the whole vehicle through a bottom layer control distribution strategy.
Equivalent longitudinal force F applied to vehicle centroid and obtained based on solution of trajectory tracking controlleruxCan not be directly applied to vehicle control, and needs to be decomposed by a bottom-layer control distribution strategy to obtain a driving torque signal T which can be responded by the vehicletqAnd a brake pressure signal.
Equivalent longitudinal force F when applied at the center of mass of the vehicleuxWhen the torque is greater than or equal to 0, the vehicle system is considered to be in the driving mode, and the corresponding driving torque is expressed as:
in the formula, TtqA signal indicative of a target drive torque of the vehicle; fuxRepresenting an equivalent longitudinal force applied at the vehicle center of mass; r represents a wheel rolling radius; i.e. i0Representing the whole equivalent transmission ratio of the whole vehicle; etaTThe whole equivalent transmission efficiency of the whole vehicle is represented;
equivalent longitudinal force F when applied at the center of mass of the vehicleuxLess than 0; considering the vehicle system in the drive mode, the corresponding drive torque can be expressed as:
wherein P represents a vehicle target master cylinder brake pressure; fuxRepresenting an equivalent longitudinal force applied at the vehicle center of mass; r represents a wheel rolling radius; d represents the diameter of the brake master cylinder; rBRepresents the effective brake caliper radius; i istRepresenting the moment of inertia of the tire;
representing the differential of the tire speed.
Examples
A joint simulation platform is built based on MATLAB/Simulink and vehicle dynamics software CarSim and used for testing the trajectory tracking controller designed by the application. The target vehicle was subjected to a double lane test at a varying longitudinal movement speed on a road surface having a road surface adhesion coefficient of 0.5. The target vehicle speed is between 45km/h and 85km/h, and is continuously transformed in a sine form. Fig. 5, 6, 7, 8, 9, 10, 11, and 12 are a longitudinal speed control performance curve, a longitudinal speed error control performance curve, a trajectory tracking coordinate axis control performance curve, a trajectory tracking lateral error control performance curve, a yaw angle error control performance curve, a front wheel corner control performance curve, and a yaw angle control performance curve, respectively. It can be seen from the experimental curve that the longitudinal and lateral coupling track tracking controller based on the adaptive three-step method can ensure that the longitudinal speed of the vehicle can follow well, and the error of the longitudinal speed is within 0.66 km/h. In the aspect of lateral track tracking, the lateral track tracking error is controlled within 0.05m, and the cross shoving error is controlled within 0.72 deg. The front wheel angle control in the whole track tracking process is stable, and the smoothness of the transverse swing angular speed of the whole vehicle is good. Therefore, the designed track tracking controller can realize stable, accurate and smooth control of the longitudinal and lateral track tracking motion of the vehicle.
Claims (4)
1. A design method for a vehicle longitudinal and lateral coupling trajectory tracking controller is characterized by comprising the following steps:
step one, establishing a track tracking model containing system parameter uncertainty and vehicle longitudinal and lateral nonlinear coupling dynamics characteristic relation according to the kinematics and dynamics relation between a vehicle and an expected track;
designing a track tracking controller coupled in the longitudinal direction and the lateral direction of the vehicle to obtain a complete system control law; the trajectory tracking controller consists of three parts of steady-state-like control, reference dynamic feedforward control and state-dependent error feedback control;
compensating system uncertainty disturbance caused by vehicle motion state change by combining with an adaptive law;
fourthly, track tracking control of the whole vehicle is realized through a bottom layer control distribution strategy;
the specific method of the first step is as follows:
11) assuming that the left wheel and the right wheel are stressed symmetrically, a simplified vehicle dynamic model is established as follows:
in the formula, m represents the mass of the whole vehicle; v. ofxRepresenting a vehicle longitudinal speed; v. ofyRepresenting a vehicle lateral speed;
a differential representing a longitudinal speed of the vehicle;
a differential representing the lateral velocity of the vehicle; ω represents the vehicle yaw rate;
a differential representing a yaw rate of the vehicle; fuxRepresenting an equivalent longitudinal force applied at the vehicle center of mass; cwxRepresenting a longitudinal wind resistance coefficient of the vehicle; cwyRepresenting a lateral wind resistance coefficient of the vehicle; i iszRepresenting the moment of inertia of the vehicle; lfRepresenting the distance of the center of mass to the front axle of the vehicle; lrRepresenting the distance of the center of mass to the rear axle of the vehicle; fyfIndicating the lateral force to which the front wheel tyre is subjected; fyrIndicating the lateral force to which the rear wheel tire is subjected;
12) during the tracking process of the vehicle, the tire is in a linear area, and the lateral force of the tire is approximately represented as:
in the formula, CfRepresenting tire front wheel side sheet stiffness; crRepresenting tire rear wheel cornering stiffness; alpha is alphafRepresenting a tire front wheel side slip angle; alpha is alpharIndicating a tire rear wheel side slip angle; beta represents the vehicle centroid slip angle; deltafIndicating a vehicle front wheel steering angle;
13) assuming that the actual values of the vehicle system parameters during trajectory tracking are known, the complete vehicle dynamics model is represented as:
in the formula,
ρiactual values representing system parameters;
ideal values representing the variation of system parameters with the vehicle motion state; rhooiStandard values for system parameters are expressed and are expressed as:
in the formula, CwyRepresenting a lateral wind resistance coefficient of the vehicle;
14) the vehicle kinematic model is:
in the formula, xeIndicating a vehicle longitudinal tracking error; y iseIndicating a vehicle lateral tracking error;
a differential representing a longitudinal tracking error of the vehicle;
a differential representing a vehicle lateral tracking error; v. ofxdRepresenting a desired vehicle longitudinal speed;
a differential representing an actual vehicle yaw angle;
a derivative representing a desired vehicle yaw angle;
representing an error between the desired yaw angle and the actual yaw angle;
representing the differential error between the desired yaw angle and the actual yaw angle; omegadRepresenting a desired vehicle yaw rate; rρA road curvature representing a target trajectory;
15) deriving a vehicle kinematic model to obtain:
in the formula,
a second derivative representing a longitudinal tracking error of the vehicle;
a second derivative representing a vehicle lateral tracking error;
a differential representing a desired vehicle longitudinal speed;
a derivative representing a desired vehicle yaw rate;
representing the second derivative of the error between the desired yaw angle and the actual yaw angle;
a second derivative representing an actual vehicle yaw angle;
a second derivative representing a desired vehicle yaw angle;
16) using longitudinal and lateral tracking errors as state variables of the system, i.e. x1＝[xe ye]T，
The equivalent longitudinal force applied at the vehicle center of mass and the front wheel steering angle serve as control variables of the system, i.e. u ═ Fux δf]TSubstituting the vehicle dynamics model into the vehicle kinematics model to obtain a vehicle longitudinal and lateral coupling nonlinear trajectory tracking model containing system parameter uncertainty:
y＝x1 (18)
in the formula, x1And x2State variables representing the system; u represents a control variable of the system; y represents the system output, F1(x1)＝02 ×1Representing a first order system variable matrix; g1(x1)＝I2×2Representing a first order system state matrix;
representing a second-order system variable matrix;
representing a second order system control matrix; p1And P2Is a system parameter matrix, f2，1(x)、g2，1(x) Representing the local basis function of the system, f2(x)、g2(x) And represents the system overall basis function:
P1＝[ρ1 ρ2 ρ3 ρ4 ρ6 ρ7]T，P2＝[ρ5 ρ8]T，g2(x)＝[g2，2(x) g2，3(x)]T，
f2(x)＝[f2，2(x) f2，3(x) f2，4(x) f2，5(x) f2，6(x) f2，7(x)]T，
in the formula, P1And P2Is a system parameter matrix, pi1,2, … 8, representing the true value of the system parameter; f. of2，j(x)，j＝1，2，…7，、g2，k(x) K is 1,2,3, representing the system local basis function; f. of2(x)、g2(x) Representing the overall basis function of the system；
Representing the derivative of the desired vehicle longitudinal speed.
2. The design method of the vehicle longitudinal and lateral coupling track following controller according to claim 1, wherein the specific method of the second step is as follows:
21) the system output y is derived until the occurrence of the control input u:
in the formula, x1And x2State variables representing the system;
representing a state variable differential of the system;
represents the differential of the system output;
represents the second derivative of the system output; f, F1(x1) Representing a first order system variable matrix; g1(x1) Representing a first order system state matrix; u represents a control variable of the system;
representing outputs of a second order systemMatrix generation; b (x) ═ G1(x1)G2(x) A control matrix representing the second order system output; c (x) ═ G1(x1)F2(x) A parameter matrix representing the second order system output; f2(x) Representing a second-order system variable matrix; g2(x) Representing a second order system control matrix;
22) it is assumed that the system is already in steady state motion, at which point
Then the steady-state-like control law of the system is:
in the formula usRepresenting a steady-state-like control law of the system; g2(x) Representing a second order system control matrix; f2(x) Representing a second-order system variable matrix; f. of2，1(x)、g2，1(x) Representing a system local basis function; f. of2(x)、g2(x) Representing the system overall basis function; p1And P2Is a system parameter matrix;
23) adding a feedforward control law into a calibrated control law is used for improving the response performance of the system, namely:
u＝us+uf (22)
wherein u represents a system control law; u. ofsSteady state-like control law, u, representing the systemfRepresenting a dynamic reference feedforward control law of the system;
let y be yd，
Substituting the control law (22) into the system (20) to obtain a feed forward control law as follows:
in the formula ufRepresenting a dynamic reference feedforward control law of the system; g2(x) Representing a second order system control matrix;
a differential representing a desired output of the system;
a second derivative representing the desired output of the system; a (x) represents an output matrix of the second order system output; a (x) represents an output matrix of the second order system output; p2Is a system parameter matrix; g2，1(x) Representing a system local basis function; g2(x) Representing the system overall basis function;
24) the introduced error feedback control law is as follows:
u＝us+uf+ue (24)
wherein u represents a system control law; u. ofsRepresenting a steady-state-like control law of the system; u. offRepresenting a dynamic reference feedforward control law of the system; u. ofeRepresenting the error feedback control law of the system;
defining a system tracking error as e1＝yd-y, substituting the control law (24) into the system (20) can determine:
in the formula,
represents the system tracking error differential;
a second derivative representing a system tracking error; a (x) represents an output matrix of the second order system output; g2(x) Representing a second order system control matrix; u. ofeRepresenting the error feedback control law of the system;
order to
The closed loop error system is then expressed as:
in the formula,
represents the system tracking error differential; e.g. of the type2Representing a virtual control input;
representing a virtual control input differential; a (x) represents an output matrix of the second order system output; g2(x) Representing a second order system control matrix, ueRepresenting the error feedback control law of the system;
25) option e2Defining Lyapunov functions for virtual control inputs
Then the corresponding derivative is:
in the formula,
representing a defined first lyapunov function differential; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; e.g. of the type2Representing a virtual control input;
26) to ensure stability of the closed-loop control system, i.e.
The selected virtual control inputs are:
in the formula,
representing a desired virtual control input; e.g. of the type1Indicating the system tracking error, k1Representing a controller parameter;
for k1＞0，
It is possible to obtain:
in the formula,
representing a defined first lyapunov function differential; k is a radical of1Representing a controller parameter; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential;
27) to e is making e2Approach to
To ensure the system e1→ 0, we define
Then we get:
in the formula,
representing the first defined lyapunov function differential, k1Representing the three-step controller parameters, e1And
respectively representing the systematic tracking error and the systematic tracking error differential, e3Representing a virtual control input error;
with e3→ 0, obtaining
Then system e1Gradual stabilization to error e3The derivation is carried out to obtain:
in the formula,
representing virtual control input errorsDifferentiating; e.g. of the type2Representing an actual virtual control input;
representing a desired virtual control input differential;
representing a virtual control input differential; k is a radical of1Representing a controller parameter;
represents the system tracking error differential; a (x) represents an output matrix of the second order system output; g2(x) Representing a second order system control matrix; u. ofeRepresenting the error feedback control law of the system;
28) definition includes error e3Lyapunov function of the interior
And (5) obtaining by derivation:
in the formula,
represents a defined second lyapunov function differential;
representing a defined first lyapunov function differential; e.g. of the type3Representing a virtual control input error;
representing a virtual control input error differential; k is a radical of1Representing a controller parameter; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; a (x) represents an output matrix of the second order system output; g2(x) Representing a second order system control matrix; u. ofeRepresenting the error feedback control law of the system;
according to the Lyapunov direct method, the error feedback control law is selected as follows:
in the formula ueRepresenting the error feedback control law of the system; a (x) represents an output matrix of the second order system output; g2(x) Representing a second order system control matrix; k is a radical of1And k2Representing a controller parameter; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; e.g. of the type3Representing a virtual control input error;
selection of k2> 0, then
In the formula,
represents a defined second lyapunov function differential; k is a radical of1And k2Representing a controller parameter; e.g. of the type1Representing a system tracking error; e.g. of the type3Representing a virtual control input error;
29) the error closed loop system is gradually stabilized, and (26), (29) and
and (36) obtaining the error feedback control law of practical application as follows:
in the formula ueRepresenting the error feedback control law of the system; a (x) an output matrix representing the output of the second order system, G2(x) Representing a second order system control matrix; k is a radical of1And k2Representing a three-step controller parameter; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; g2，1(x) Representing a system local basis function; g2(x) Representing the system overall basis function; p2Is a system parameter matrix;
the obtained complete system control law:
wherein u represents a system control law; u. ofsIs a system-like steady-state control law; u. offThe system dynamically refers to a feedforward control law; u. ofeRepresenting a system error feedback control law; f. ofP(x)＝[G2(x)]-1(1+k1k2) Representing a scale term parameter; f. ofD(x)＝[G2(x)]-1[k1+k2+A(x)]Representing a differential term parameter; a (x) represents an output matrix of the second order system output; g2(x) Representing a second order system control matrix; k is a radical of1And k2Representing a controller parameter; e.g. of the type1Representing a system tracking error;
representing the system tracking error differential.
3. The design method of the vehicle longitudinal and lateral coupling track following controller according to claim 1, wherein the specific method of the third step is as follows:
31) defining system parameter estimation values:
in the formula,
representing system parameter estimates;
an estimate representing a change in a system parameter with a vehicle motion state; rhooiStandard values representing system parameters;
32) the system control law is redefined as:
wherein u represents a system control input;
representing a second-order system variable matrix estimated value;
representing a second-order system control matrix estimated value;
P1and P2Is a system parameter matrix estimated value;
representing system parameter estimatesA value;
a differential representing a desired output of the system;
a second derivative representing the desired output of the system;
an output matrix estimation value representing the output of the second-order system; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; k is a radical of1And k2Representing a controller parameter;
33) by substituting the obtained system control law (41) into the system (20), the following can be obtained:
in the formula,
represents the second derivative of the system output; k is a radical of1And k2Representing a controller parameter;
an output matrix estimation value representing the output of the second-order system; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; f2(x) The representation represents a second-order system variable matrix;
representing a second-order system variable matrix estimated value; g2(x) Representing a second order system control matrix;
representing a second-order system control matrix estimated value; u represents a system control input;
a second derivative representing the desired output of the system;
from the above equation, the closed-loop error system considering parameter uncertainty can be organized as:
wherein,
in the formula, k1And k2Representing a controller parameter;
an output matrix estimation value representing the output of the second-order system; e.g. of the type1Representing a system tracking error;
represents the system tracking error differential; Δ P represents the total error of the system parameter matrix; theta represents a system composition base; p1And P2Representing a system parameter matrix; delta P1And Δ P2Representing the system parameter matrix error;
an estimate representing a change in a system parameter with a vehicle motion state;
ideal values representing the variation of system parameters with the vehicle motion state; f. of2(x)、g2(x) Representing the system overall basis function; u represents a system control input;
34) selecting
Is the state variable of the error system, then:
in the formula, emA state variable representing an error system;
a state variable differential representing an error system;
a state variable parameter matrix representing an error system;
a parameter variable matrix representing an error system; theta represents a system composition base; Δ P represents the total error of the system parameter matrix; k is a radical of1And k2Representing a controller parameter;
35) defining a Lyapunov function including uncertainty of system parameters
Wherein Q > 0 is a positive definite symmetric matrix, and the derivation is as follows:
in the formula,
represents a defined third lyapunov function differential; e.g. of the typemA state variable representing an error system; a. themA state variable parameter matrix representing an error system; q represents a positive definite symmetric matrix; b ismA parameter variable matrix representing an error system; theta represents a system composition base; Δ P represents the total error of the system parameter matrix;
representing the total error differential of the system parameter matrix; tau represents a parameter error variable parameter quantization parameter;
representing the system parameter adaptation law;
36) the system parameter self-adaptation law is taken as follows:
in the formula,
representing the system parameter adaptation law; tau represents a parameter error variable parameter quantization parameter; e.g. of the typemA state variable representing an error system; q represents a positive definite symmetric matrix; b ismA parameter variable matrix representing an error system, and theta represents a system combination base;
substituting the system parameter adaptation law into (45) to obtain:
in the formula,
represents a defined third lyapunov function differential; e.g. of the typemA state variable representing an error system; q represents a positive definite symmetric matrix; a. themA state variable parameter matrix representing an error system.
4. The design method of the vehicle longitudinal and lateral coupling track following controller according to claim 1, wherein the specific method of the fourth step is as follows:
equivalent longitudinal force F applied to vehicle centroid and obtained based on solution of trajectory tracking controlleruxCan not be directly applied to vehicle control, and needs to be decomposed by a bottom-layer control distribution strategy to obtain a driving torque signal T which can be responded by the vehicletqAnd a brake pressure signal;
equivalent longitudinal force F when applied at the center of mass of the vehicleuxWhen the torque is greater than or equal to 0, the vehicle system is considered to be in the driving mode, and the corresponding driving torque is expressed as:
in the formula, TtqA signal indicative of a target drive torque of the vehicle; fuxRepresenting an equivalent longitudinal force applied at the vehicle center of mass; r represents a wheel rolling radius; i.e. i0Representing the whole equivalent transmission ratio of the whole vehicle; etaTThe whole equivalent transmission efficiency of the whole vehicle is represented;
equivalent longitudinal force F when applied at the center of mass of the vehicleuxLess than 0; considering the vehicle system in the drive mode, the corresponding drive torque can be expressed as:
wherein P represents a vehicle target master cylinder brake pressure; fuxRepresenting an equivalent longitudinal force applied at the vehicle center of mass; r represents a wheel rolling radius; d represents the diameter of the brake master cylinder; rBRepresents the effective brake caliper radius; i istRepresenting the moment of inertia of the tire;
representing the differential of the tire speed.
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