2004 IEEE Intelligent Vehicles Symposium University of Parma Parma, Italy June 14-17,2004
Linear Parameter-varying Control and H-infinity synthesis dedicated to Lateral Driving Assistance M. T. Raharijaona
M. G. Duc
M. S. Mammar
Service Automatique SupClec 3, rue Joliot Curie 91 192 Gif-sur-Yvette, France thibaut.raharijaona@ supelec.fr
Service Automatique SupClec 3, rue Joliot Curie 91 192 Gif-sur-Yvette, France gilles.duc@ supelec.fr
LSC, CNRS-FRE 2494 UniversitC d’Evry Val d’Essonne 40 rue du Pelvoux CE1455 91025, Evry, Cedex, France
[email protected]
Abstract This paper presents the design of a linear parametervarying controller with guaranteed H, performance f o r lateral driving assistance. The considered uncertainties consist in road adhesion, the mass of the vehicle and driver dynamics. It used to add a steering torque to that of the driver in order to improve lane keeping and yaw dynamic under external disturbances (such as lateral wind) for a varying longitudinal velocity v. The controller, thus, changes with the operating parameter v so that it is gain-scheduled.
1 Introduction A great number of accidents are due to unexpected excessive yaw motion such as spin out and lane departure. Moreover such type of accidents generally occurs on rural road, and about 30% of fatalities in France imply vehicle alone. The study presented in this extended abstract considers a solution which consists in adding a supplementary torque using a motorized direction system to that of the driver, in situations of lane keeping or lane departure avoidance and ensuring the driving comfort criteria [4],[8],[9]. In the framework of the large french project named ARCoS (Action on safe driving) this paper is concerned in the development of an assistance which combines driver-vehicleroad interactions using individual wheel braking and active steering. By the mean of lateral dynamics improvement which consists in adding lateral generated forces, yaw and roll torque disturbances are rejected even on poor road adhesion conditions. Ground vehicles are complex systems subject to many parameter variations such as mass, velocity, adhesion, uncertain dynamics and disturbances such as lateral winds and
0-7803-8310-9/04/$20.00 0 2004 IEEE
road banking. Furthermore, the model dynamics can be affected by operating conditions. A linear parameter-varying (LPV) framework is used to represent structural dynamics. It represents the dynamics with dependency on the longitudinal velocity v as a set of state-space matrices that are affine functions of velocity. The controller output consists in an additional steering command which is continuously added to the driver steering one. Active steering can be implementable using only on-board internal vehicle sensors. Combination of video, INS, and DGPS system makes possible today the definition of a robust axial guide to be followed by a vehicle. The remainder of the paper is organized as follows. Section 2 briefly presents models and strategy for the control. For more information, one can refer to [12]. Section 3 gives the H , synthesis technique used for LPV systems. Next in section 4, the results of the synthesized linear parametervarying controller applied to lateral driving assistance are presented. Section 5 is dedicated to the results so obtained with the LPV controller for the lateral driving assistance. Finaly, section 6 aims at presenting an alternative control strategy which consists in adding a steering angle to that of the driver. It is the goal of our further investigations.
2 Model presentation and Control strategy As the study concerns lateral control, the classical fourth order linear model is used [1],[6]. The state vector is x, = [p, r , y ~ , y ~Its] components ~. are the vehicle side slip angle (p), the yaw rate (r),the relative yaw angle ( y ~and ) ) the centerline. The state space the lateral offset ( y ~from model is the following:
X, = A.x,
+B.e,
zv = C.x, +D.e,
407
where
0
Figure 2: Driver model (3)
and e, = [Sf,fw,preflT,where S f is the steering angle of the front wheels in radian, fiy is the disturbance wind force in Newton and pref is the road curvature in m-l. The output vector is z, = [ v ~ , y ~ , T The s ] ~last . output is the front wheels aligning torque Ts. The parameters are given below and in [12]:
The controller synthesis model is then obtained by embedding the driver, the steering column and the vehicle models. , and v are the only measurements It is assumed that y ~v~ which are available for feedback. y~ and Y Lare provided by two real time vision algorithms using a mono camera on board in the vehicle. Figure 3 presents the control strategy so obtained. Disturbances are mainly the lateral wind which is applied to the vehicle and the control input is the steering torque T, which is obtained on the basis of y ~W,L and v. Longltudhal relocltv v
I
T~ Control input
model
' vehicle model
Figure 3: Closed loop driving assistance
Figure 1: Steering column model An Electronic Power Steering (EPS) mechanism models the steering column (Figure 1). The inputs are the electric motor torque T,, the driver steering torque Td and the front wheels aligning torque T,. The output S f is the steering angle of the front wheels which is an input of the vehicle model. Finally, the synthesis model is completed with a driver model for lane keeping maneuver (Figure 2) [lo], [7]. It is constituted by a time delay representing inherent human processing time and neuromotor dynamics. This component represents the high frequency driver compensation component and is modelled by a dead time T~ = 0.2s. In order to keep a state space formulation, this time delay is approximated here by a first order Pad6 representation.
3 LPV controller synthesis The synthesis problem investigated in this paper consists in determining a controller K ( 0 ) where 0 ( t ) is the varying parameter (longitudinal velocity in our application) and in ensuring closed loop stability of Figure 3. Moreover, the L2 gain between the exog.enousinputs w and the regulated outputs e is bounded by y. That is, Itell2
< YllWIl2
(5)
for any possible value of e@). The method of control design takes advantage of affine nature of the parameter dependency and assumes that the parameter can be measured in real time. If the state space
408
realization of the system to be controlled is affine with respect to the vector e ( t )of dimension p, each matrix is defined using a polytope whose vertices are obtained as follows: e ( t )= ni. i = 1, ...,2p. Thus for each t, e ( t ) is the barycentre of the vertices ni, such as:
and the state space matrices can be written as an affine combination of the matrices at the vertices.
Some terms in the matrices being dependent on v and %, they are approximated using first order developmentsin Experimentally,6, is deduced from the measurement of the longitudinal velocity v . The synthesis consists in choosing two vertices which correspond to two operating points v1 and v2 and in building ) each verthe matrices A ~ ( n i )B, ~ ( x j )C, ~ ( n i )D, ~ ( n ifor tex xi thanks to a system of linear matrix inequalities [2]. Similarly to H, synthesis, the control requirements are met using frequency-dependent weighting filters N.As shown on Figure 4, they are introduced to weight the control, the exogenous disturbances and the outputs in order to improve the stability margin of the system and to ensure good closed-loop performance [121. The resulting controller is
6,.
(7) v;
Let be the vector U the control inputs and y the measurements. The LPV controller has the following form:
Yr
Figure 4: Synthesis controller design where: time-varying and automatically gain-scheduled along parameter trajectories. The expression is the following:
Where A ~ ( n i )B, ~ ( n i C ) ,~ ( n i )D, ~ ( n i )are , the state space matrices of an H, controller at each vertice. The feasibility of the problem depends on the theorem explained in [2], [3]. Controller synthesis can be accomplished by considering a set of linear matrix inequalities (LMI) that may be solved using standard convex optimization algorithms [ 5 ] . This approach gives guarantees of both stability and performance; however, it may be conservative to use such a controller when the values of 8 have slow variations.
4 LPV controller dedicated to lateral driving assistance This section presents the application of the method to the lateral driving assistance. Only lateral motions are considered by the equations of the linear vehicle model. Considering that longitudinal velocity can be written as: 1 - = WO
+ w16,
where, 6, E [-1;+1], the plant state-space matrices are built to depend affinely on the time-varying parameter 6,.
(11) where 6, E [-1; $11. Considering that the longitudinal velocity of the vehicle varies between v1 = 10m.s-' and v2 = 20m.s-1 and road adhesion is v = 0.8, the algorithm leads to yopr = 2.87. The state space vector of the H, controller is of dimension 12. In the following, a frequency analysis is performed by comparing the responses of the LPV controller for the fixed velocity v = 18m.s-' to the responses of the vertices. Figure 5 presents the Bode diagrams of the LPV controller for the input y ~ . Figure 6 presents the Bode diagrams of closed loop transfers I & I and I [, These ones are particularly efficient at low fkquencies. Consequently, the LPV controller will ensure good rejection of disturbances such as lateral wind force. The Black diagram of the LPV controlled open-loop is shown on Figure 7. One can note that these responses vary continuously with respect to longitudinal velocity, the plots
%
409
.;lo,
s"
. .. ..... ,... ..
5.1 Tire-road nonlinear model
Bode DlagramlayL ,npn
. .. ............... ..
.., .
.... .
3;
m
f
i
..5p
Frequency (rads&
Figure 5: Bode diagrams of the LPV controller
As we are only concerned with lateral control, a simple nonlinear model of a vehicle is deirived by neglecting heave, roll and pitch motions. This inodel includes the two translational motions and the yaw motion. The vehicle wheels are numbered from l to 4. The interaction between the tire i (i = 1 , 2 , 3 , 4 ) and the road surface is decomposed into longitudinal forces fxi (A,.) and lateral forces fyi(ai). These forces will be detailed below. The nonlinear model is obtained by writing the translation and rotation equations, in the vehicle fixed frame : m (Gx - vy)=
+
m (Gy vx.)
=
Ji=
where vx and vr represent the velocity and : Figure 6: Closed loop frequency analysis (a)
121
I I and (b)
Black diagram
Table 1. Nonlinear formulas of tire s l h angles
Figure 7: Black diagram for the open loop
exhibit good phase and gain margins for all values of the velocity. Consequently, one can then expect good robustness properties, a more precise investigation of them will be performed using nonlinear vehicle models which integrate tirehoad interactions.
5 Siniulations and Results This section proposes to analyze the responses of the closed loop system using the LPV controller under external disturbances. The assistance strategy is tested in various situations such as lateral wind force disturbances, road adhesion loss and speed variations. To this end, a nonlinear vehicle model which is detailed in the following section is integrated.
The longitudinal forces depend directly on the tire slip coefficient (hi)while the lateral forces depend on the tire slip angles (ai).Table 1 summarizes the expressions of the tire slip angles. In this paper, the magic formula of Pacejka [l 11 is used for each tire in order to determine the lateral forces: f,i(ai) =
djsin[citan-'{bi(l --ei)ai+ ei tan-'(biai)}]
(13)
The coefficients bi, C i , di,ei depend on the tire characteristics, on the road conditions, and on the vehicle operational conditions.
5.2 Disturbance rejection A 500N lateral wind force disturbance, presented by Figure 8(a), is applied to the vehicle which is moving on a straight road at a varying velocity (Figure 8 (b)).
410
5.3 Disturbance rejection and Adhesion loss at
varying velocity
Figure 8: (a)Lateral gusts of wind (b)Velocity profile
01201.
:L;
. .. .: i.:. ; './,,,,...':
... .. :I :.. .:. .i...,,,/:
:
008-
.; ?. .... .... .. ..
-002-
-004 0
' 5
'
In this case, the vehicle is considered to perform a stationary cornering maneuver from continuously 20 m.s-' to 10 m.s-'. The driver steering angle is then constant. The vehicle is subject to two gusts of wind f w = 500N at t=O sec and t=20 sec and it is assumed that at t=10, 15,20,25 sec, the tires cross a low friction region of the road which corresponds to adhesion v = 0.1. The tires return to a dry road 2 seconds later. The lateral control assistance using the LPV controller is active and the results are compared with the ones obtained using the H, controller (in dashed line).
''
.. .. ... ... .. ..
'
'
'
~
'"
1 0 1 5 2 0 2 5 3 0 3 5 time in sec&
5
Figure 9: (a)yL with lateral gusts of wind f w = 500N (b)
10
15 20 lime in second
25
30
35
bme m s e d
WL
Figure 11: (a)yL (b) W L , with the gusts of wind and adhesion loss
On Figure 9, the solid lines represent the responses of the assisted vehicle with the LPV controller and the dotted lines stand for the closed loop system with the fixed H , controller obtained from Figure 4 at velocity v = 1Om.s-'. One can note from Figure 9 (a), that y~ is greatly reduced, the maximum is inferior to 7 cm. Thus, the vehicule remains closer to road centerline. Figure 9 (b), similarly presents the relative yaw angle which stays under 0.003 radian. The comparison to the fixed H , controller shows the effectiveness of the LPV one. Furthermore, the assistance torque is less than 3 N.m, as shown on Figure 10, which is compatible with the EPS actuator performances.
On Figure 11 (a), y~ obtained using the LPV controller in solid line, is inferior to 5 cm. Figure 11 (b) presents the relative yaw angle YL. It is satisfying, consequently the LPV assistance strategy is particularly efficient.
I
I
,
'0
5
10
I
I
15 20 lime In Second
/
.
I
25
30
35
Figure 10: assistance torque T, of the LPV controller
6 Perspective on a steering angle assistance Considering that the steer-by-wire systems will be in great deployment in a close future, this section proposes an intermediary solution which consists in adding a steering command which is continuously added to the driver steering one as shown on Figure 12. The design procedure adopted
Figure 12: Closed loop driving assistance in steering angle here will be based on H , and LPV control theory. The control input will be computed from y ~ W, L and v and
41 1
frequency-dependent filters Wi will weight the control and the outputs in order to satisfy expected performances. This strategy is the center of our further investigations.
[101 A. Modjtahedzadehand R. A. Hess. A model of driver steer-
’7
[l 11 Pacejka, H.B, “Tires factors and vehicle handling”, Int. J Vehicle Design, vol 1, 1-23, 1979.
Summary and Conclusions
In this paper, LPV control applied to lateral driving assistance has been investigated using a polytopic approach. It is advantageous over traditional controllers because the closed-loop system is stabilized for any value of the varying parameter. As given in the frequency analysis, the controller will ensure good rejection of disturbances and good robustness properties. Tn order to analyze more precisely the application of the LPV strategy, simulations and results are presented. They compare LPV and H, controllers. The efficiency of the 1,PV strategy is shown in various driving conditions. Moreover, a scheme to design a steering angle controller is presented. This control strategy will be the next center of investigations using experimental data in various situations, such as disturbance rejection, lane change maneuver and adhesion loss.
ing control behavior for use in assessing vehicle handling qualities. Transactions of the ASME, 15:456-464, 1993.
[12] Raharijaona T., Duc G. and Mamma S. ”Robust Control
and p-analysis with application to Lateral Driving Assistance”, IEEE Intelligent Transportation Systems Conference, 2003.
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