NL2025573B1

NL2025573B1 - Method for controlling path tracking of an autonomous vehicle with input saturation

Info

Publication number: NL2025573B1
Application number: NL2025573A
Authority: NL
Inventors: Chen Changfang; Shu Minglei; Liu Ruixia; Yang Yuanyuan; Wei Nuo; Xu Jiyong
Original assignee: Shandong Computer Science Ct Nat Supercomputer Ct Jinan
Priority date: 2020-03-04
Filing date: 2020-05-13
Publication date: 2021-04-22
Also published as: CN111176302A; NL2025573A; CN111176302B

Abstract

A method for controlling path tracking of an autonomous vehicle with input saturation is provided. The robust Hoo path tracking controller is designed to solve the 5 problems such as network delays and input saturation during the control of path tracking of the autonomous vehicle and improve the path tracking performance of the vehicle under extreme driving conditions. The lateral velocity and the yaw rate of the vehicle are adjusted to improve the vehicle steerability and stability as well as achieve path tracking control of the autonomous vehicle. The robust Hoo path tracking control lO gain matrix for the autonomous vehicle can be obtained by solving the linear matrix inequality, which is easy to compute. The uncertainty of the vehicle dynamics model and effects of external disturbances are considered in this path tracking control design, which improves the robustness of the path tracking control algorithm. A static output feedback controller is designed to realize ideal path tracking control as well as 15 significantly cut down the cost of the control system.

Description

-1-

METHOD FOR CONTROLLING PATH TRACKING OF AN AUTONOMOUS VEHICLE WITH INPUT SATURATION

TECHNICAL FIELD The present disclosure relates to the technical field of autonomous vehicles, and more particularly, to a method for controlling vehicle tracking with input saturation.

BACKGROUND With the rapid development of the new generation of information technology and improvements in people's requirements for vehicle safety and comfort, the path tracking control of autonomous vehicles has been an area of concentration in recent years and is widely used in mobile robots and automatic parking systems. Autonomous vehicles contribute to reducing labor intensity, improving driving safety, mitigating the accident, and improving road traffic efficiency. According to statistics from the automobile industry, driven by the goal of relieving road congestion and traffic accidents, most automobiles would be unmanned in the future and expected to dominate road traffic. For autonomous vehicles, one of the basic problems to be solved is to realize the path tracking control of vehicles, and the control goal thereof is achieved by allowing vehicles to track the ideal path and maintaining the steady-state path tracking error (i.e., the lateral offset and heading error) zero.

Path tracking control algorithms for autonomous vehicles include sliding mode control, adaptive control, robust Heo control, neural network control, model predictive control, linear matrix inequation (LMI) optimization control, and Lyapunov-function- based control. Most of these control methods only consider the vehicle steerability and stability. However, the problems of delays and data packet loss inevitably exist during the measurement of vehicle states and signal transmission. Moreover, in practical applications, there are physical limits to the actuator. For example, under extreme driving conditions, the tire force may reach saturation. When the system enters a

2- saturation state, the output of the controller and the input of the controlled object would not match, which greatly reduces the performance of the controller and even destabilizes the closed-loop system. Therefore, the method for realizing the path tracking control of autonomous vehicles with network delays and actuator saturation remains a challenging problem in the industrial and academic fields.

SUMMARY In order to overcome the shortcomings of the prior art, the present disclosure provides a method for controlling vehicle tracking with input saturation, which can realize excellent steering stability and path tracking performance of an autonomous vehicle under an extreme driving condition. The present disclosure adopts the following technical solutions to overcome the above-mentioned technical problems. A method for controlling path tracking of an autonomous vehicle with input saturation includes the following steps: a) establishing a vehicle dynamics model by formula (1): ! FE +F d Vv, = —( + Lv +d, (1) m ‚1 1 (1) j=—(I,F,—LF,)+—AM, +d)

LM I where, V, denotes the first derivative of V,, 7 denotes the first derivative of Y, V, denotes a longitudinal velocity of a center of mass CG of a vehicle, V, denotes a lateral velocity of the center of mass CG of the vehicle, } denotes a yaw rate of the vehicle, 77 denotes the mass of the vehicle, / - denotes a rotational inertia of the vehicle about the Z -axis, d, (£ ) and d 2 (¢ ) both denote unmodeled dynamics, Fy denotes a lateral force of a front tire of the vehicle, and F, denotes a lateral force of a rear tire of the vehicle; an external yaw moment AM ET calculated by formula (2) 2 4 v { . 0 i v AM, = YF [(=1)],c088, +1,sind,1+ > (DLE, 6 i=1 i=3 where, F, denotes a longitudinal force of the /" tire, l f denotes a distance from the center of mass CG of the vehicle to a front axle, / + denotes a distance from the center of mass CG of the vehicle to a rear axle, / + denotes a wheelbase, and 0 f denotes a steering angle of a front wheel, b) establishing a path tracking model by formula (3): & Ve) ¢ / 3) Ss where, l, denotes a horizontal distance between the center of mass CG of the vehicle and a sensor, Y e denotes a lateral offset [, distant from the center of mass CG of the vehicle, and Ó, denotes a heading error; an actual yaw angle @ of the vehicle is calculated by formula (4): $=9+%, (4) where, Ó, denotes a yaw angle of a tangent direction of a reference path relative to a global coordinate system, when the vehicle follows the reference path with a curvature of at the longitudinal velocity V,, 9, =V.0 ref , where Ó, denotes the first derivative of Ó, ; c) establishing a dynamic path tracking model by formula (5): x(t) = Ax(t)+ Bu(t) + d(t) (5)

4e where, X(Z) denotes the first derivative of X(£), X(#) denotes a state , variable, x(t ) = [v,., Vs 9. v,] ‚ 1" denotes a matrix transpose, u(t ) denotes ~ 7 an input variable, u(t) = [ò, AM, ] | and oo Tr d(t) = |, © d,() TV Pe AN ; the system matrix A and the system matrix B are calculated by formula (6); C +C Cl, -Cl | ! Sf Vv, 0 0 my, my, “1 “1 ~ 72 + 72 a= Cl, Cl, Cc A+ CL _ 0 1 0 0 1 [ v.

O i © Cr Cy 0 0 m a B= : 1 0 — 0 0 7 z d) a change in the longitudinal velocity V, of the vehicle is expressed by the formula 1/ Vv, = (1+ A) / Vv. , where A, denotes a time-varying parameter and | A IS 1, V. denotes a nominal value of V, | the system matrix A is expressed as A=A,+AA, where AA=EMF, M =A, and F denotes an identity matrix; A, is calculated by formula (7), and £ is calculated by formula (8);

C,+C, GN Lr ILO 00 MV.

MV, | A, =| — C, 1, ' Cr _ C, 1; Mead 0 0 IV, Iv, (7) 0 1 0 0 1 I Vv, 0 C,+C CI, CI AL rn tt ry v. 0 0 my.

MV, 1 Y TY 2 1 2 rol C ir Cl.

Cl +C I 0 0 Lv, Ly, (8) 0 0 0 0 0 0 Vv. 0 e) establishing a vehicle path tracking control system by formula (9): X(1) = (4, + ADx(1) + Bo(u(®)) + d(1) 9 z(t) = C.x(1) ©) 5 where, C, denotes a fourth-order identity matrix, u(t ) eR” where R” denotes an n-dimensional real space, out) =[o(u (1), o(u,(2)),---, ou, (6))] , U ars it u, (¢) > U max 0 (u, (0) 7 u, (£), if == U arr = u, (f) < Une i=1 2 Rh TU max? if U, (7) < “Ua where, Wimax denotes a maximum value of u, (¢ ) , and U, (¢ ) denotes the i element of U(?) : f) establishing a state feedback path tracking controller by formula (10): u(t) = Kx(t—7(1)) (10) where, T(t ) denotes a delay, T(t ) 7 +7,, T, denotes a transmission

-6- delay of a control signal from the sensor to the controller, 7, denotes a delay of a control signal from the controller to an actuator, and K denotes a control gain matrix to be designed, g) establishing a closed-loop system for controlling the path tracking of the autonomous vehicle by formula (11): (1) = (4, +AA)x(E) + Bo(Kx(t — (1) + d(t)

11 =(£) = Cx(1) (1h) wherein when d(#)=0 | the closed-loop system for controlling the path tracking of the autonomous vehicle established by formula (11) 1s asymptotically stable; when d(t) #0 the robust H „ disturbance suppression performance index }7 is calculated by formula (12); Lr 2f gr | 27 (Dz(Ddt Sy | d" (t)d(t)dt 12) h) calculating positive definite matrices X >0 and Q > 0, general matrices Y, , Y, , and N ; , and the quantity O > 0 to satisfy the linear matrix inequality shown in formula (13), where i =1,2,3;

7e — — — 5 ) ï ~T An En EB, { N, oF XI AC, + =, 2, 0 NM 0 0 0 * tE.

I N, OF 0 0 xk * * vl] 0 0 0 0 Q,= nev <0 * * * * TO 0 0 0 (13) * * * * * 9/7 0 0 x x * ze x * —0f 0 sk sk xk sk % xk % JI u? y _ N ae } . . ma IW <0, i=1,2,...,0 Vai —p X where, * in formula (13) denotes the transposition of the symmetric elements of the matrix, and V 1 denotes a performance index. =, = A,X +XA +N, +N], — : 17 UT 2, =BS(v, YY) N,+N,, _ T TT Sp 7 XA, + N, > _ 7 UT Ep = NN, _ ’ TT Soy = [BS (4, Y,,Y,)] —N;, =,=70-2X ViVi t+ (1 u VO Sk, ,)= : Vin + (1 = V, Vi HY += Hi) Su, Y, > Y, )] = . Hoy Vin + (1 = H, Von where, Vri denotes the i" line of Y. where i=12,..., n.

Vhi denotes the /" line of Y, , where I= L, 2, on: Vi denotes the i element of V ,

-8- where I =1,2,....1 : H, denotes the i™ element of MH ‚ where i=L2,....n.

T denotes an upper bound of delay T(t ) ‚ Pand Una both denote a positive constant, vel, uel’, V = {w eR" :w =lor 0} 1) calculating a gain matrix of the vehicle state feedback controller by formula (14):

_ ~1 K=Yx" 14 and solving the convex optimization problem in formula (15) to obtain the optimal robust H state feedback path tracking controller: min y, st.

Q, <0, X > 0, O0 >0, u (15) YY, Ni=12,3,ò>0 Further, in step a), the lateral force F yr of the front tire of the vehicle and the lateral force F w of the rear tire of the vehicle are calculated by the formulas Fy =2C f a, > Fy = -2C., , where C, denotes a cornering stiffness of the front wheel, C, denotes a cornering stiffness of the rear wheel, a, denotes a cornering angle of the front wheel, CX. denotes a cornering angle of the rear wheel, — 0. — Lr — Yy = Lr — Vy and Fr = 9 » Oy = : Vv, Vv, Vv, Vv Preferably, the curvature fs in step b) is obtained by a global positioning system (GPS) combined with a geographic information system (GIS). Further, after step g), the method further includes the following steps:

29. 7 ‚ 2 C X= } . h2) selecting an output vector J 2 7. 9. > el ‚ and calculating the positive definite matrices X, >0, X,>0, Q>0, general matrices Ve, +, ‚ and N i , and quantity 0>0 to satisfy the linear matrix inequality shown in formula (16), where 1 =1,2.3 Z, 8, Es I No XF' XC! + 5,082, 0 NM 0 0 0 + + 2. I NO 0 0 2 * * * vl 0 0 0 0 y 1 ml < 0 xk * % xk —7 . TO 0 0 0 16)

+k +k +k +k sk —0f 0 0

+k +k +k +k sk +k —0f 0 xk +k x +k xk x +k —7 u? 7 ma hi . a "<0, i=12,..,n Nui -p X where, * in formula (16) denotes the transposition of the symmetric elements of the matrix, and y | denotes the performance index;

-10- =, =A4,X+X4 +N, +N/, S BSV FT) NNT =p = BSV, YY) N, +N,, S= VAT NT =13 = XA, + N, 2 r— A7 AT Ey =—N,—N,, = vv UT ArT ZE, =[BS(4, 1 Y,)] — Ny, E,,=70-2X, X = NX NG +GX GG , U T v _ T Y,=YG JY, =YG ? ViVi + (1 VOY Sw, Yr) = , Vo Vi + (1 V, VV in hy, +1=p)y, S(p,Y, Y= : Ho Vin + (1 H, Vn where, Vi denotes the i" line of Y, where i=12,....n : Vhi denotes the i line of XY, , where i=1L2,...n : V, denotes the i element of V , where i=12,....n : H, denotes the i" element of H ‚ where i=12,....n : where, T denotes the upper bound of delay T({ ) , Pand Umar both denote a positive constant, vel, uel, V = fw eR":w =lor 0} . 12) calculating the gain matrix of the vehicle output feedback controller by formula (17): vv -1 K=rX (17) wherein the column of N, is a basis of a null space of the output matrix €, and

-11- the matrix G is calculated by formula (18): G=Cl+N,L L=NIXCT(C XC (1%) where, Cl = C, | (C,C, | 17! denotes the Moore-Penrose generalized inverse matrix of the matrix C,, and N J denotes the Moore-Penrose generalized inverse matrix of the matrix N,,.

The advantages of the present disclosure are as follows. The robust Hoo path tracking controller is designed to solve the problems such as network delays and input saturation during the control of path tracking of the autonomous vehicle and improve the path tracking performance of the vehicle under extreme driving conditions. The lateral velocity and the yaw rate of the vehicle are adjusted to improve the vehicle steerability and stability as well as achieve path tracking control of the autonomous vehicle. The robust Hoo path tracking control gain matrix for the autonomous vehicle can be obtained by solving the linear matrix inequality, which is easy to compute. The uncertainty of the vehicle dynamics model and effects of external disturbances are considered in this path tracking control design, which improves the robustness of the path tracking control algorithm. A static output feedback controller is designed to realize ideal path tracking control as well as significantly cut down the cost of the control system.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a schematic diagram showing the vehicle dynamics model of the present disclosure; and FIG. 2 1s a schematic diagram showing the vehicle path tracking of the present disclosure.

-12-

DETAILED DESCRIPTION OF THE EMBODIMENTS The present disclosure will be further described hereinafter with reference to FIG. 1 and FIG. 2. A method for controlling path tracking of an autonomous vehicle with input saturation includes the following steps: a) A vehicle dynamics model is established by formula (1), as shown in FIG. 1. 1 al = (FE) vd m . 1 1 1) y=(LF LF, )+—AM_+d,(1) LN: "IJ where, v, denotes the first derivative of Vv, , y denotes the first derrvative of Y, V. denotes a longitudinal velocity of a center of mass CG of a vehicle, V, denotes a lateral velocity of the center of mass CG of the vehicle, 2 denotes a yaw rate of the vehicle, 77? denotes the mass of the vehicle, 1 z denotes a rotational inertia of the vehicle about the Z -axis, d 1 (£ ) and d 2 (1 ) both denote unmodeled dynamics, F yr denotes a lateral force of a front tire of the vehicle, and FE, denotes a lateral force of a rear tire of the vehicle; an external yaw moment AM z IS calculated by formula (2); 2 | 4 : { . o i : AM, => Fl, cosó +1 sing 1+) CDF, 6) i=1 i=3 where, F, denotes a longitudinal force of the /" tire, l f denotes a distance from the center of mass CG of the vehicle to a front axle, / + denotes a distance from the center of mass CG of the vehicle to a rear axle, / + denotes a wheelbase, and 0 f denotes a steering angle of a front wheel; b) A path tracking model is established by formula (3), as shown in FIG. 2.

-13- ¢ Ve) e 0] (8) $ where, [ s denotes a horizontal distance between the center of mass CG of the vehicle and a sensor, y e denotes a lateral offset , distant from the center of mass CG of the vehicle, and 9, denotes a heading error.

An actual yaw angle @ of the vehicle is calculated by formula (4): ¢ = Ó, + Ó, (4) where, ¢, denotes a yaw angle of a tangent direction of a reference path relative to a global coordinate system, when the vehicle follows the reference path with a curvature of Pref at the longitudinal velocity V, , Ó, = vp ref , where Ó, denotes the first derivative of Ó, c) A dynamic path tracking model is established by formula (5).

X(t) = Ax(£) + Bu(t) + d(¢) (5) where, X(#) denotes the first derivative of X(£), X(f) denotes a state 7 variable, X(f)= [v,, V,9,, 3,1 | T denotes a matrix transpose, (1) denotes 7 an input variable, u(t) = [ò, AM] ‚ and 7 d(t) = [a © d,(t) VP AL oop | ‚the system matrix A and the system matrix B are calculated by formula (6),

-14- Cc, +C Cd, Cl dn ALE v‚ 0 0 my, my, } “1 1 ~ 72 -~v 72 gl Cd, Ct, 5 C+ 00 Iv, Iv, 0 1 0 O 1 [ v. 0 | . © m I B= : 1 0 — 0 0 I,

d) A change in the longitudinal velocity V, of the vehicle changes is expressed by the formula 1/ Vv. = (1+ A, )/ V_, where A, denotes a time-varying parameter and | A, |< 1, V. denotes a nominal value of V, ‚the system matrix A is expressedasd =A, + A4, where Ad=EMF, M =A, and F denotes an identity matrix; A, is calculated by formula (7), and £ is calculated by formula (8); CC +C.

CA Cl ee tt Oy, 0 0 MV, MV, . Cl, -Cl, C+, 4, = 44 rr 44 er 0 0 Iv, Iv, (7) 0 1 0 0 1 I Vv, 0 Cc, +C cl, Cl LA L450 0 my, MV, 1 7 p2 + 2 rol Cd, Cl.

Cl +C I 0 0 0 0 0 0 0 0 Vv. 0

-15-

e) In order to complete the task of controlling the path tracking of the autonomous vehicle, the lateral offset y, and the heading error ¢_ of the vehicle should be as small as possible.

Moreover, the lateral stability of the vehicle can be improved by adjusting the lateral velocity and yaw angle.

Further, in consideration of saturation characteristics of the actuator, a vehicle path tracking control system is established by formula (9):

x(t) = (A, + Ax (1) + Bo (u(t) + d(t)

9

=(t) = C,x(1) (9)

where, C, denotes a fourth-order identity matrix, % ({ ) e R" , where R” denotes an n-dimensional real space,

| 7 o(u(t)) = [o(%, (¢ )), o(u, (1)), tS o(u, (¢ ))] > Ua» if u, (1) > U nar o(u,®)=3u), if-u,, <u()<u,, [0 on TU ax > if ú, (7) < “Ux where, Umax denotes a maximum value of U, (¢ ) , and u, (¢ ) denotes the i element of 24(Z)

f) A state feedback path tracking controller is established by formula (10):

u(t) = Kx(t—7(t)) (10) where, T(t ) denotes a delay, in the vehicle path tracking control system based on network control, the vehicle status and control signals generally have varying degrees of delay and packet loss during the transmission process, and T(t ) —=7, +7,, where 7; denotes a transmission delay of a control signal from the sensor to the controller, 7, denotes a delay of a control signal from the controller to the actuator, and K denotes a control gain matrix to be designed.

-16- g) A closed-loop system for controlling the path tracking of the autonomous vehicle is established by formula (11). X(1)=(A4, + AA)x(1) + Bo(Kx(t —7(1))) + d(¢) 11 =(1) = Caf) (1) The goal of controlling the path tracking of the autonomous vehicle is achieved by designing the robust H state/output feedback controller.

When d()=0 the closed-loop system (11) is asymptotically stable.

When d(t) #0 the robust H oo disturbance suppression performance index 1 is calculated by formula (12). 2 (0z()de <7 [dT (Dd(Dd 2 (0)z(0)de < yd" (Wd (dt a h) The robust H state feedback controller and the static output feedback controller are designed to solve the problems of network delay and input saturation during the control of the path tracking of the autonomous vehicle.

When d(t)=0 | the closed-loop system is asymptotically stable and satisfies the given HA disturbance suppression performance index, and the control gain matrix can be obtained by solving the corresponding linear matrix inequality, which is easy to compute.

The positive definite matrices X >0 and Q > 0, general matrices Ye , Y, , and N, , and the quantity Ô> 0 are calculated to satisfy the linear matrix inequality shown in formula (13), where i =1,2.3;

-17- = = = NT \ r ~T An En EB, { N, oF XI AC, + =, 2, 0 NM 0 0 0 * tE.

I N, off 0 0 * * Sy 0 0 0 0 Q,= nev <0 * x x x TO 0 0 0 (13) * * * * *% 9/7 0 0 +k * + * * + 0/7 0 * * xk x* * xk * JI u? y ~ Himax hi . ; 1150, i=L2,...n Voi p X where, * in formula (13) denotes the transposition of the symmetric elements of the matrix, and V 1 denotes a performance index. _ TT 77 Zi =A4A,X+X4, +N, +N, _ 17: IT =p = BS(v.Y,.Y,)-N, +N, 5 =, =XA +N! _ x7 FT 92 = —N, 7 N, > _ T x37 Zs =[BS (1, Ys Yl — N; > ZE =70-2X Viva TU=v)y, Sv.

YY) = : V, J kn + (1 = V, )y hn Het A= U) Vi SY, Y= : Ho Yim + (1 == U ) Vu where, Vri denotes the i* line of Y, where i=12,..., n.: Vhi denotes

-18- the i® line of Y, ‚ where I= 1, 2, on: V, denotes the i element of V , where i=12,...n : LU, denotes the i™ element of ye , where i=12,....n : T denotes the upper bound of delay T(t ) ‚ Pand U,,,. both denote a positive constant, vel, uel’, V = {w eR" :w =lor 0} 1) The gain matrix of the vehicle state feedback controller is calculated by formula (14): _ ~1 K=YX (14) The convex optimization problem in formula (15) is solved to obtain the optimal robust 7 state feedback path tracking controller. min jy, st.

Q,<0,X>0,0>0, (15) 7 . _ X YY, N.,i=123,0>0 Embodiment 1: Preferably, in step a), the lateral force Fy of the front tire of the vehicle and the lateral force EF, of the rear tire of the vehicle are calculated by the formulas by = 2C, a 1 Lk, = -2C.a, , where C, denotes a cornering stiffness of the front wheel, C, denotes a cornering stiffness of the rear wheel, a denotes a cornering angle of the front wheel, CX, denotes a cornering angle of the rear wheel, and JS J > Mp Embodiment 2: the curvature ef in step b) is obtained by a global positioning system (GPS)

-19- combined with a geographic information system (GIS). Embodiment 3: The robust A state feedback controller and the static output feedback controller are designed to solve the problems of network delay and input saturation during the control of the path tracking of the autonomous vehicle.

When d(1)=0 | the closed- loop system is asymptotically stable and satisfies the given H, disturbance suppression performance index, and the control gain matrix can be obtained by solving the corresponding linear matrix inequality, which is easy to compute.

Therefore, after step g), the method further includes the following steps: h2) Since it is difficult to measure the lateral velocity v, of the vehicle by low- cost sensors, in order to diminish the cost of the control system, an output vector 7 =(C,x = : : : y 2 7. 9. ‚Jy A is selected, the static output feedback path tracking controller is designed, and the positive definite matrices X, >0, X,>0, 0>0, general matrices Y. +, , and N i , and quantity 0> 0 are calculated to satisfy the linear matrix inequality shown in formula (16), where i=1,2,3 ; ~ = = Yi . 7 7 HZ, EZ, 1 N, oF XF XC, + 2 E, 0 NM 0 0 0 + + =. I NE 0 0 2 xk xk * vl 0 0 0 0 y 1 mi < 0 * +k +k * T 70 0 0 0 (16) % +k +k +k +k —0f 0 0 +k +k +k +k +k +k —of 0 + xk x xk xk x xk —7

-20- u? y imax hi . 7 A7 <0, i=l2,...n Yi —pP where, * 1n formula (16) denotes the transposition of the symmetric elements of the matrix, and y | denotes the performance index;

= U TU AT A7 vz

HE, =A4,X+X4, +N, +N, ,

= _ CT UN AN WT

—12 = BS(v.Y..Y,)-N, +N, ’

= VAT A7

Ez = AA, +N,

= _ ONT ArT

—22 = —N 2 N 20

= TTT WT

3 = [BS(4, YY, )] —N, >

HE, =TO-24X,

X= Ny X (Ny + GX,G',

U T v _ T

Y, = VG ‚TY, = TG >

Via + (1 u VO) Vi S(11.7,.7,) = Va Ven + (1 = V, Von LV += HM) Vn Su, 1, ,Y,)]= : Hy Vi + (1 H, Von where, Vri denotes the f! line of Y, , where i=l2,...n : Vhi denotes the i line of T, , where i=12,...n : V; denotes the i" element of V ‚ where i=12,...,n : U, denotes the 7! element of H ‚ where i=12,....n ;

where, T denotes the upper bound of delay T(t ) ‚ Pand Umar both denote a positive constant, vel”, uel’, V = fw eR" :w =1lor 0}

21-

12) The gain matrix of the vehicle output feedback controller is calculated by formula (17): 1-1 K=} A (17) The column of N, is a basis of a null space of the output matrix C,, and the matrix G is calculated by formula (18): G=C,+N,L _ NT 7 yl (18) L=N!xclc,xclhy C=C (CC) ized where, Ly 1, ( 2% ) denotes the Moore-Penrose generalized inverse matrix of the matrix C,, and N 0 denotes the Moore-Penrose generalized inverse matrix of the matrix N,.

Claims

27 - Conclusions

A method for controlling a path finding of an autonomous vehicle with input saturation, comprising the following steps: a) establishing a dynamic model of a vehicle by formula (1): = F + E d, (= - V 1 Vy (yf +]) Vy + (8) nm. 1 1 (1) j = (LF, LF,) + - AM, + d, (t) LNV a where, V, the first derivative of V, 7 indicates the first derivative of 7, V, indicates a longitudinal speed of a center of mass CG of a vehicle, V, indicates a lateral speed of the center of mass CG of the vehicle,} indicates a yaw rate of the vehicle, # 7 indicates the mass of the vehicle, / z denotes a rotational slowdown of the vehicle about the Z axis, d, (t) and d, (7) denote both unmodulated dvnamica, Fy denotes a lateral force of a front tire of the vehicle and F +, reen indicates lateral force of a rear tire of the vehicle, where an external yaw moment AM z is calculated by means of formula (2); 2 4 _ INT a IV AM, => F [(= 1) 1, cos6, +1; sind 1+> (-V, F, (3 i = li = 3 where, F “indicates a longitudinal force of the i® tire, [f indicates a distance from the center of mass CG of the vehicle to a front axle, [a distance from the center of mass CG of the vehicle to a rear axle, 1, indicates a wheelbase and ò indicates a steering angle of a front wheel;

-23 - b} establishing a path finding model using formula (3): 4 = Ye JY «eT 3)

S where, A indicates a horizontal distance between the center of mass CG of the vehicle and a sensor, Ve indicates a lateral shift l distant from the center of mass CG of the vehicle and 9, indicates an upcoming fault; where a current yaw angle ¢ of the vehicle is calculated by formula (4): ¢ = p, + Ó, (4) where, d, indicates a yaw angle of a tangential direction of a reference path relative to a global coordinate system, if the vehicle indicates the follows reference path with curvature of ef at the longitudinal velocity V, Ó, = Ve Pres, where Ó indicates the first derivative of @; c) establishing a dynamic path finding model by formula (5): X (t) = Ax (t) + Bu (t) + d (1) (5) where, X (f) is the first derivative of X ( #), X (f) denotes a 7 state variable, x (t) = [v., 7,6, Ve], 7 denotes a 7 matrix transposition, u (t) an input variable, u (t) = [ò, AM, 1 and

T d (t) = 4, (t) d, (1) denotes VPs SAN; wherein the system matrix A and the system matrix B are calculated by formula (6);

24. C, + C, Cl, Cl fr LL Pry 000 my, my. 1 1 + 72 7 72 A Eh CC 0 1 0 0 1 [v. 0 | . (6) Cc Cy 0 0 m I.

B = 1 0 - 0 0 I,

d) a change in the longitudinal speed V, of the vehicle, is expressed by the formula 1 / Vr = (1 + 4,) / Vv, where A, a time-varying parameter and | A, <1 denotes, V, a nominal value of V,

denotes the system matrix A is expressed as 4A = A4 + AA, where AA = EMF, M = A, and F denotes an identity matrix; wherein A 1 is calculated by formula (7) and £ is calculated by formula (8); CC + C, Cd Cl + SL gy 0 0 my, pl. “1“ + “1 72“ 1 72 4 Er CHC 4 | 0 1 0 0 1 [v 0

25. CC, + C, cl CC tT rg 0 0 MV. my,. + Y ~ 32 ~ 72 Fe CC CA, 0 0 Lv, Iv, (8) 0 0 0 0 0 0 Vv. 0 e) establishing a vehicle path tracking control system by formula (9): (0) = (A, + AD) x (0) + Bo (u ()) + dt) u (9) z (t) = Cy x (1) where, C, denotes a fourth order identity matrix, u (t) € R ”. where R ”denotes an n-dimensional real space, 5 5 5 T out) = [o (u, (1), (4, (6), (u, (1))], U ax? if U; ( 1)> Ua o (4, (1)) 7 h, (1), if oo U nae 5 h, (2) SU par i = 1 2 Nn TU nere> if ú, ({) <TU ax where, Uimax denotes a maximum value of u, (£) and 4, (¢) denotes the i% element of # (?); F) establishing a state feedback path trace control by formula (10): u (t) = Kx (t -7 (1)) (10) where, (1) indicates a delay, 7 (1) = T, + 7, 7, indicates a transmission delay of a control signal from the sensor to the controller, 7; delay of a control signal from the controller to an actuator and K indicates a control gain matrix to be designed;

-26 - g) Establish a closed loop system for controlling the path finding of the autonomous vehicle by formula (11): X (t) = (A, + AA) x (1) + Bo (Kx (t - (1) + d (£): 11 = (1) = Cx (1) (th where if d (1) 7 = 0, the closed loop system for controlling the path finding of the autonomous vehicle established by formula (11), is asymptotically stable, if a (t) # 0, the robust MH, interference suppression performance index ”is calculated by formula (12); (znd <7: | d (0d (td oF (H) z (t) dt <p;, (1) d (t) dt (12) h) calculating positive finite matrices X> 0 and O> 0, general matrices Y,, Y ,, and N, and the quantity 0 > 0 to satisfy the linear matrix inequality shown in formula (13), where 1 = 1,2,3; _ - _ TF \ Tr ~ T ZE, Sp Sy / N, oF AF AC, + =, E, 0 NM, 0 0 0 * «= EZ, IN, oF 0 0« xx 1 0 0 0 0 Q, = nto vo <0 * * * * TO 0 0 0 (13) * * * * * —0f 0 0 * * * * * * —or 0 * * * * * * * / 2 U. ma Vn | <0, i = 12 ... n Vi -p AX where, * in formula (13) indicates the transposition of the symmetrical elements of the matrix and J'1 indicates a performance index,

-27 - 2, = A, X + XA + N, + N, Ey SBS, Vh) N + N, Eis - XA, + N, = _ UT 22 7 -N 2 N 25 Ey = [BS (4, V)] Ns, zE. = 70-2X Viva FLV) S (v, V.Y,) =: V, y kn + (1 = V,) y hn thy + = H) Vn SY.

Y =: Hp Via + (1 o U,) Vin where, Vii denotes the i line of Y, where [= L 2, nn: Voi denotes the je line of T, where I = L2, ..., n, Vi denotes the i element of V, where [= 1, 2, nn: HH; denotes the i% element of U, where i = 12, .... n T denotes an upper limit of retardation T (Ê), 2 and Ux both denote a positive constant, vel, "eV, V = {w eR": w = lof 0}; 1) calculating a gain matrix of the vehicle state feedback control by formula (14): -1 K = YX, (14) and solving the convex optimization problem in formula (15) to find the optimal robust H state feedback path tracing control to obtain:

28 - min /, st. 0, <0, X> 0.0> 0, -: (15) Y, .Y, .N.i = 123.0> 0

Method for controlling a path finding of the autonomous vehicle with the input saturation according to claim 1, characterized in that, in step a), the lateral force F or of the front tire of the vehicle and the lateral force F of the rear tire of the vehicle are calculated by the formulas Fy = 2C rps F, = -2C.a, where Cy denotes a bending stiffness of the front wheel, C, denotes a bending stiffness of the rear wheel, a, denotes a bend angle of the front wheel, © a bend angle of the rear wheel, and Ly Vy Ly V a zò La = Vv, Vv, Vv, Vy

Method for controlling the path finding of the autonomous vehicle with the input saturation according to claim 1, characterized in that the curvature Pros in step b) is obtained by means of a global positioning system (GPS) combined with a geographic information system (GIS).

A method for controlling the path finding of the autonomous vehicle with the input saturation according to claim 1, characterized in that, after step g), the method further comprises the following steps: = C, x =! h2) selecting an output vector y 2 7. Pe sheet and calculating the positive finite matrices X,> 0, X,> 0, 0> 0, general

-29. matrices TY, Y, and N, and quantity 0> 0 to satisfy the Co. ci ed. 2 ... i = 123 linear matrix disparity shown in formula (16), where I = 1, 2.3; z ZZ Vo Toxo! 2, 2, BE, [IN, oF XF 'XC * EZ, =, 0 N, 0 0 0 x ox 2.0] N, or 0 0 2 * * * vl 0 0 0 0 7 _ <0 * * * * Ti 0 0 0 0 (16)% * * * * —or 0 0 * * * * * * —0f 0% * * * +%% * - / 2 - U. oe y. } oe po i = 12, .... n Vai p X where, * in formula (16) denotes the transposition of the symmetric elements of the matrix and Y | indicates the performance index;

-30 - B, = A4, X + X4 + N + N, ~ vv.

TT we ONT Ep = BS (rY, .Y,) - N + N ,, = ~ - 7 TT Hy = A4, + N ,, a TT Sa = N, u N,> = TTT AT 5, = [BS (4 1, Y1 - Ny, Ey = 70 7 2X, X = NX, N, + GX, .G ", 7 _ TT _ TY = YG + = Y, G» Vin + (1 oM "Yin S ( v, Y..Y,) =:> V, Vi + (1 V, Yi Hy Vri + (1 MH Va S (u, YY,)] =: LV + (1 KH, Vo where, Vii de / * line of TY, where I = 1, 2, ..., n, Vi indicates the ith line of Y, where I = 1, 2, .... n; Vi indicates the i element of V, where [ = 1, 2, namely: MH, denotes the 7% element of H, where i = 12, .... n where, T denotes the upper limit of retardation T (Ê), Den ex both denote a positive constant, vel, Helv, = fw € R ”: Ww, = 1 or 0}; and 12) calculating the yield matrix of the vehicle output feedback control using formula (17): - vl K = ¥ rX (17)

231 -

where the column of N, is a basis of a zero space of the output matrix (’, and the matrix G is calculated by formula (18):

G = Cl + N, L.

L = N] XC] (C, XCy) "9)

where, C} = C, (CC)! denotes the Moore-Penrose generalized inverse matrix of the matrix C, and N; denotes the Moore-Penrose generalized inverse matrix of the matrix N.