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一个活跃的设计四轮转向系统的物理不确定性

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-个活跃的设计四轮转向系统的物理不确定性
Abstract: In this paper a new approach to the design of an active four-wheel-steered (4WS)
control system is introduced. The problems of the conventionally used LQ-controller-based
virtual model following control strategies are analyzed, and several problems are
addressed. Two main problems of the LQ controller are highlighted: the external
disturbance rejection, and the robustness of the controlled system in the face of parametric
uncertainties. In the paper a combined Robust Linear Quadratic Regulator (RLQR) and I-
controller design is presented for an active 4WS vehicle. The applicability of this kind of
controllers is proven in this particular design case.
Kcywords: vehicles, active steering, virtual model following control, disturbance rejection,
parametric uncertainty modeling, robust LQ control, minimax technique
1. INTRODUCTION
During the past decade extended research has been
performed by car manufacturers to improve the
steerability and stability of vehicles and to increase
the comfort of the driver and passengers. Several
controlled systems can be designed, acting on the
forces between the tires and the road surface. Fourwheel-
drive and the anti-lock braking systems have
already been applied in conventional cars, affecting
>
Fig. 1. Definition of forces acting on a vehicle tire
the longitudinal forces. (See the directions of the
forces in Fig. 1.) Using active or semi-active
suspension systems the vertical forces can be
regulated, resulting in better road-holding and
higher passenger comfort by minimizing the
dynamic wheel load and rejecting the vertical
acceleration of the vehicle body, respectively. To
achieve better lateral behaviour as well as higher
comfort for the passengers by decreasing the
unnecessarily large lateral acceleration and side-slip
angle of the vehicle, the lateral tire forces at the rear
wheels should also be modified. This leads to the
requirement of the application of 4-wheel-steering
(4WS) systems, where the rear wheels cornering
force is controlled directly by steering, and thus
becomes independent of the turning of the vehicles
body. The advantages of the 4WS car over a
conventional one with respect to the steering
behaviour have been presented in different studies by
many researchers. One of the earliest studies was
conducted by Nagai and Ohki (1988). A similar
approach can be found in more recent works by
Palkovics (1992), I-Iiguchi and Saito (1992), Lugner
and Mittermayr (1992), and Abe et al. (1992). A
1075
1076 L. Gianone et al.
comparison of the feedforward and feedback
compensation of a 4WS control system has been
conducted by Inoue and Sugasawa (1992). An
interesting approach to the problem is presented by
Shiotsuka et al. (1992), using neural networks in the
controller design.
So far, the solutions mentioned for the 4WS control
design are mainly based on a simplified, linearized
nominal model of the vehicle system. However, in
the studies conducted by Ackermann and Sienel
(1993) and Hirano et al. (1992) a robust controller is
designed to achieve predetermined performance and
stability.
In this paper the authors show by analyzing the
existing solutions that the use of a robust controller
is necessary. The paper presents a design method for
a controller that ensures stability against parametric
perturbation (such as varying cornering stiffness and
velocity of the vehicle) and that rejects external
disturbances (such as side wind gusts).
The outline of the paper is as follows: Section 2
deals with the modelling aspects of the vehicle
concerning the steering behaviour. In Section 3 a
new control structure is proposed, based on the
virtual model following control strategy and the
recently developed robust LQR control theory. The
control scheme and algorithm are considered on the
base of the performance requirements. Simulation
studies are presented in Section 4, comparing the
worst-case performance of the new controlled
scheme with the LQR and robust LQR control
strategies in two different experimental conditions,
namely under side wind disturbance and lanechange
motion.
into a single one. Furthermore, it is assumed that all
motions occur in the plane of the vehicle model.
This model describes the vehicle in-plane behaviour
sufficiently well for high speeds; however, it is not
able to consider the effect of load transfer between
the left and right side wheels. The schematic
diagram of the model is shown in Fig. 2.
The linearized equations of motion can be writtdn as
follows:
mv(fl I I t" 2 1", (1)
m,.,(,b ) /,; F2 Zw (2)
17; c, ot,, a, 6,-fl(-l) i I,T, i 1,2 (3a,b)
V
where ct i is the side-slip angle on the i th pair of tires,
j is the yaw angle at the centre of gravity (CG) of
the vehicle, [ is the side slip angle at the vehicle
C.G, F, is the side force acting on the ith tyre, v is
the forward speed of the vehicle, 1, w is the side wind
force acting on the vehicle body at a distance / w from
the C.G The model can be represented in a
conventional slate-space form:
Ax Brae w Bad B, sz," 2 (4)
where 6, are the steering angles of the wheels (i 1,2)
and the x is the state vector:
The measurable output of the system is the lateral
acceleration, and the stales are observable from this
output:
2. VEHICLE MODEL
To model the steering behaviour of the vehicle, the
so-called bicycle model is used with two degrees of
freedom. The body of the vehicle is assumed to be a
rigid beam, and the left and right tires are combined
(p t )v -
Fig. 2. Bicycle model of the vehicle
The system matrices of equations (4)-(5) can be
found in the Appendix. In the later sections the
following notations will be used:
G: The linear operator between all the inputs
[Fw 61 d2l and the states (see Eq.4),
¢;v- The linear operator between the inputs and the
lateral acceleration output,
Gp . The linear operator between the inputs and the
performance output (defined later),
GyM: The linear operator between the inputs of lhe
virtual model (defined later) and its states.
It can be seen that the velocity of the vehicle is a
parameter of the model. The investigation will be
performed at a constant forward speed: variation in
the speed can be considered in the control design
with the other highly time-varying parameters.

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