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AMME 3500 DESIGN PROJECT

创建日期：2024-05-23 18:11:41

所属分类：物理Physical

标签： AMME3500

DESIGN PROJECT

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AMME 3500 DESIGN PROJECT
Abstract— This report focuses on designing the basic
components of an autonomous car such as cruise control and
lateral control by drawing directly on the knowledge of
linearization, second-order systems and second-order control
systems through numerical simulations performed using
MATLAB via Simulink.
Keywords—Linearisation, Controller Design, Validation,
Lane change, Disturbance, Feedback Gain
1. INTRODUCTION
The need for autonomous cars has increased in the recent
years due to the added comfort and safety. This is achieved by
installing cruise control and lateral control (lane-changing)
systems in the vehicle which help in reducing the fatigue
experienced by the drivers while travelling long distances. A
cruise control system is used to maintain a constant speed
under the effect of external disturbances like wind and grade
of the road by automatically controlling the speed of the
vehicle while the lateral control helps in automatic lane
changes. Hence, helping to improve travel efficacy.
This report emphases on designing the cruise control and
lateral control system for Toyota Camry Hybrid (Ascent)
which is currently in huge demand. It is a smooth front-wheel
drive with a four-cylinder engine, above average fuel
economy and a top speed of 180 kmph. The non-linear
dynamics for each system will be linearized and the controller
will be designed by choosing specific gains. Finally, the
feedback system with the controller will be validated by
running the system via Simulink against disturbances like
slope, additional mass and varying velocities. The relevant
physical properties and dimensions for Toyota Camry are
tabulated below as per the car module from the company [1].
Property Magnitude Unit
Curb weight (Including a 70kg
driver), m
1665 kg
Frontal Cross sectional Area, A 2.6588 m2
Drag Coefficient, CD 0.27 N.A
Wheelbase ( + ) 2.825 mm
Acceleration from 0-100kmph 7.8 s
Density of Air, ρ 1.225 kgm-3
Table 1: Physical Properties and Relevant Dimensions for Toyota
Camry
2. LONGITUDINAL CONTROLLER
2.1 Linearization
The car is considered to be moving along a straight line with
its velocity described by ‘v(t)’ at time ‘t’. Considering the
force demanded by the engine as the control input ‘u’, the non-
linear dynamics for the vehicle are follows:
̇ +
1
2
2 =
A drag coefficient of 0.27 is chosen which is a reasonable
assumption for an average automobile sedan. Considering the
equilibrium conditions for velocity and input force to be
(ve, ue), the system is linearized as follows:
̇ = (, ) =
−
2
With = 0.5, a positive constant.
At equilibrium, ̇ = ( , ) = 0
Hence,
=
2 = 0.5
2 (1)
Linearizing via Taylor Expansion at equilibrium points to
obtain:
(, ) = ( , ) + (
)
,
( − )
+ (
)
,
( − )
̇ = 0 −
2
( − ) +
−
Hence, the Linearized equation is:
̇ + = ()
For, ̇ = ̇ − ( , ), = − = −
Substituting equation (1) and values of ̇, , into
equation (2),
̇ =
1
( + 0.5
2 − ()) (3)
Hence, substituting known values from Table 1,
̇ =
( + .
− . () (4)
Considering the following conditions:
Equilibrium Condition (, )
1. (25kmph, 21.205 N)
2. (50kmph, 84.818 N)
3. (100kmph, 339.274 N)
Initial Condition v (0) = 0 m/s
Table 2 Equilibrium Pairs and Initial Conditions
AMME 3500 DESIGN PROJECT 1 ©2022 IEEE
The data in Table-2 is calculated using Equation (1) and the
car is assumed to start from rest (initial conditions).
The linearised equations obtained at the equilibrium pairs are
as follows:
Equilibrium Pair Linearised Equation
1. (25kmph, 21.205 N) ̇ + . − . =
2. (50kmph, 84.818 N) ̇ + . − . =
3. (100kmph, 339.274 N) ̇ + . − . =
Table 3 Linearized Equilibrium Equations for velocity in (m/s)
The three liner dynamics are simulated to obtain the
trajectories of velocities as follows:
Figure 1 Linearized Plant for a single equilibrium pair
Figure 2 Linearized v(t) trajectories for equilibrium pairs
Note: An additional trajectory for the trivial equilibrium pair
of (0m/s, 0N) has been added in figure 2 to compare with
positive velocities.
Similarities:
As per Figure 2, all the equilibrium velocities follow an
exponential trajectory and reach a stable value/equilibrium as
time goes to infinity. The magnitude of velocity is inversely
proportional to the time taken to reach the steady state which
implies that 100kmph reaches its equilibrium value the fastest
while 25kmph reaches the slowest. All the three trajectories
portray stable behaviour by achieving the respective steady
state values as time and increases.
Differences:
The initial growth/steepness is highest for 100kmph while it
decreases gradually with a decrease in the velocity and the
trajectory becomes flatter. The three trajectories start from an
initial value of zero and do not cross each other over the course
of time and become parallel as time tends to infinity.
The trivial (0m/s, 0N) trajectory is a continuously increasing
straight line which crosses all the three equilibrium
trajectories and goes to infinity as time increases. This is
makes it unstable and a poor choice for the equilibrium
conditions.
2.2 Controller Design and Effectiveness
2.2.1. Controller Design
A PI (Proportional Integral) controller is chosen for the car
which helps in achieving a desired reference speed. The PI
Controller makes use of a feedback loop to measure the errors
by obtaining the difference between the output (v) and the
reference (r). The reference to be a constant which ensures
that the zero in the system does not affect the output as the
transfer function for this controller does not have a zero.

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