Optimal Vehicle Dynamics – Yaw Rate and Side Slip Angle Control Using 4-Wheel Steering

88 Pages·2002·1.66 MB·English

ISSN 0280-5316
ISRN LUTFD2/TFRT--5697--SE
Optimal Vehicle Dynamics – Yaw Rate
and Side Slip Angle Control Using
4-Wheel Steering
Nenad Lazic

Department of Automatic Control
Lund Institute of Technology
October 2002

Department of Automatic Control Document name
MASTER THESIS
Lund Institute of Technology
Date of issue
Box 118
October 2002
SE-221 00 Lund Sweden
Document Number
ISRN LUTFD2/TFRT--5697--SE
Author(s) Supervisor
Nenad Lasic Anders Rantzer, LTH
Jens Kalkkuhl DaimlerChrysler AG, Stuttgart
Sponsoring organization
Title and subtitle
Optimal Vehicle Dynamics – Yaw Rate and Side Slip Angle Control Using 4-Wheel Steering.
(Optimal fordonsdynamik vid 4-hjulsstyrning).

Abstract
The development of steer-by-wire involves an increased amount of possibilities for control in
future cars. For example, the experimental vehicle available has both front and rear wheel
steering. This enables a simultaneous multivariable control of both the yaw rate and the side
slip angle. The today existing control approach is designed in the time domain and it has a
decoupled control structure, where the yaw rate is controlled by the front wheel steering angle,
and the side slip angle is controlled by the rear wheel steering angle. With this approach there
are some major difficulties, such as constraint handling and lack of robustness to plant
uncertainties, to time delays and to actuator or sensor failure. To solve these problems, a
multivariable control scheduling system can be used that has already been applied to robotics,
satellite attitude control and ship control but is a novelty for the field of automotive control. It is
a coupling control approach, based on Individual Channel Design in the frequency domain, ICD,
and considers the internal cross-coupling of the plant. It has been shown that meeting the
requirements is not trivial, and a tradeoff has to be done between high performance and entirely
meeting the specifications. However, the design shows a considerable degree of both robustness
and integrity, which has been verified in theory as well as in simulations, but has to be further
checked in real experiments.
Keywords
Classification system and/or index terms (if any)

Supplementary bibliographical information

ISSN and key title ISBN
0280-5316
Language Number of pages Recipient’s notes
English 90
Security classification
The report may be ordered from the Department of Automatic Control or borrowed through:
University Library 2, Box 3, SE-221 00 Lund, Sweden
Fax [phone] E-mail [email protected]

Optimal Vehicle Dynamics - Yaw Rate and
Side Slip Angle Control Using 4-Wheel
Steering
Nenad Lazic
October 23, 2002

Contents
1 Introduction 2
1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 The Car Model 4
2.1 Wheel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The One-track Bicycle Model . . . . . . . . . . . . . . . . . . 8
2.3 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Control Strategies and Speci(cid:12)cations 14
3.1 Speci(cid:12)cations . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 The Feedforward . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.1 Linear system . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.2 Nonlinear system . . . . . . . . . . . . . . . . . . . . . 17
3.2.3 Combining the nonlinear feedforward and the linear
feedback . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 The Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Feedback - Individual Channel Design 21
4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Adding Actuators . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 The Main Advantages and Disadvantages . . . . . . . . . . . . 25
4.4 The Design Procedure . . . . . . . . . . . . . . . . . . . . . . 27
5 Feedback Design - Results 28
5.1 Controller Scheduling . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 System Integrity and Robustness . . . . . . . . . . . . . . . . 41
6 Simulation Results 56
6.1 Introducing Time Delay . . . . . . . . . . . . . . . . . . . . . 56
6.2 Keeping the Time Delay and Breaking the Second Loop . . . . 72
7 Conclusions and Future Work 76
A Matlab Commands 77
A.1 State Space Modelling . . . . . . . . . . . . . . . . . . . . . . 77
A.2 An Iteration Step . . . . . . . . . . . . . . . . . . . . . . . . . 78
A.3 Making a PID-controller . . . . . . . . . . . . . . . . . . . . . 78
A.3.1 Converting into PID-parameters . . . . . . . . . . . . . 78
A.3.2 Scheduling with respect to speed . . . . . . . . . . . . 79
A.3.3 Discretising the controller . . . . . . . . . . . . . . . . 80
i
Acknowledgements
This thesis was done in cooperationbetween DaimlerChrysler AG, Stuttgart,
and the Department of Automatic Control, Lund Institute of Technology. It
has been carried out at the Department of Driver Assistant Systems, Daim-
lerChrysler, in Esslingen, just outside Stuttgart, from March to September
2002.
IamgratefultoallthepeopleatthedepartmentinEsslingen. EspeciallyI
would like to thank Dr. Jens Kalkkuhl for his great supervising and support
throughout my work. I also would like to thank Prof. Anders Rantzer
for giving me the opportunity to do this thesis and for giving me valuable
comments on the report.
Lund, October 2002
Nenad Lazic
ii
1 INTRODUCTION 2
1 Introduction
1.1 Problem Formulation
Today’s cars are becoming more and more sophisticated. They contain more
electrical hardware components and less mechanical and hydraulic ones. Ex-
amples of this are the actuators used for control such as in brake-by-wire,
steer-by-wire and active suspension. In future cars there might be a very
large number of possible con(cid:12)gurations for these actuators. This means that
forces and torque for control action can be applied in many di(cid:11)erent ways.
What we want is to optimally employ these actuators such that
(cid:15) control is achieved with minimum control action
(cid:15) the solution is robust to plant uncertainties, and a failure of one actu-
ator can be compensated by the remaining ones
(cid:15) the controller can be adapted to any speci(cid:12)c actuator con(cid:12)guration
To achieve all three objectives, a multivariable control scheduling system
has to be used that has already been applied to robotics, satellite attitude
control and ship control but is a novelty for the (cid:12)eld of automotive control.
It requires the solution of problems unique to automotive control such as
(cid:15) handling uncertain nonlinear tyre forces
(cid:15) time varying constraints on the maximum available tyre forces due to
changing road conditions
Using existing modelled vehicle dynamics, reference yaw rateandside slip
angle are generated from a driver command. With proper model matching,
and considering disturbances and uncertainties, control commands should be
given such that the reference is followed.
In this work 4 wheel steering (front and rear axle) is used for control,
with yaw rate and side slip angle as outputs from the car, and front and
rear wheel steering angles as the input to the car. The problem is to control
the yaw rate and side slip angle simultaneously (normally, the side slip angle
has to be kept under a certain threshold), and at the same time ful(cid:12)l the
speci(cid:12)cations that are set up for the control system.
The internal structure of the system involves a certain degree of cross-
coupling between inputs and outputs. An approach with a decoupled con-
troller structure, as well as a model based state space design in the time
domain, where one controller is used for yaw rate and another for side slip
angle, constructed independently of each other, already exists. The front
1 INTRODUCTION 3
wheel steering angle is generated using both feedforward and feedback, and
for the rear wheel steering angle only feedforward is used. This decoupling
approach has some major disadvantages. One is the lack of robustness, both
toplantuncertainties andtimedelays. Thelimitationsinconstrainthandling
are another disadvantage.
By also taking the dependency between yaw rate and side slip angle con-
trol into account, and using a coupling approach, a more robust solution can
be obtained that meets the theory and speci(cid:12)cations in a better way. The
intention is to use feedforward and feedback for both rear and front wheel
steering. However, feedback design will be the central point.
1.2 Thesis Outline
This work consists of three main parts. The (cid:12)rst part, system modelling,
problem speci(cid:12)cation and survey of di(cid:11)erent design strategies, is discussed
in chapter 2 to chapter 4. In chapter 2 the fundamental concepts of ve-
hicle dynamics are explained. The general ideas and speci(cid:12)cations of the
multivariable controller design are discussed in chapter 3. Some ideas about
feedforward will be presented, but focus will be put on feedback design. The
feedback design approach chosen for this work, Individual Channel Design,
is explained in chapter 4. Beyond that, a rough structure for the controller
will be discussed in the chapter.
The second part is the controller design. This will be described in chapter
5. The software environment used for the design is Matlab, and in particular
the SISO root-locus design tool in Matlab.
Simulations and experimental results are the third part of this work.
The simulations will be carried out in Matlab Simulink. A complete car
model, containing the same dynamics as a real car, will be used to verify
the performance and robustness of the controller. The simulation results are
discussed in chapter 6. The most important parts of the Matlab code used
during the design are given in the Appendix.