Design and Implementation of Safety Control for a
Class of Stochastic Order Preserving Systems with
Application to Collision Avoidance near
Intersections
MASSACHUSETTS MNB11ITE
OFTECHNOLOGY
by
OCT 16 2014
Mojtaba Forghani
LIBRARIES
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2014
@ Massachusetts Institute of Technology 2014. All rights reserved.
Signature redacted
A uthor .............................
Department of Mechanical Engineering
August 25, 2014
I
Certified by.......Signature
redacted ........
Domitilla Del Vecchio
Associate Professor
Thesis Supervisor
Signature redacted
Accepted by...........
David E. Hardt
Professor of Mechanical Engineering Department
Head of Graduate Office
2
Design and Implementation of Safety Control for a Class of
Stochastic Order Preserving Systems with Application to
Collision Avoidance near Intersections
by
Mojtaba Forghani
Submitted to the Department of Mechanical Engineering
on August 25, 2014, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
In this thesis, we have designed and implemented a safety control system for collision
avoidance near intersections. We have solved the corresponding control problems for
a general class of systems that also includes the scenario of the two consecutive vehicles approaching an intersection, which leads to the design of the collision avoidance
system. We have gathered the data of behavior of drivers as they approach intersections and have built a stochastic model for that through an optimization problem.
The model generates a non-deterministic profile for acceleration of a vehicle which
is not equipped with the collision avoidance system and it is used to estimate and
predict future stopping profiles of the vehicle in order to take the right control action
for avoidance or mitigation of accidents. First we have verified the consistency of
the theoretical model with its expected behavior after implementation and then we
have implemented the control system on the Prius vehicle in collaboration with TTC
(Toyota Technical Center), Ann Arbor, Michigan.
Thesis Supervisor: Domitilla Del Vecchio
Title: Associate Professor
3
4
Acknowledgments
I would like to express my appreciation to my advisor Professor Del Vecchio for her
priceless helps and continuous supports during my graduate study. I would like to
thank her for her motivation, patience, encouragement and caring about the students
and for teaching me how to do research and how to write my thesis. I am very glad
that I could have the opportunity to work under her supervision.
I would also like to thank Dr. John Michael McNew and Dr. Derek Caveney at
Toyota Technical Center (TTC), Ann Arbor, Michigan, for helping me to implement
the system on the vehicle. I would like to express my gratitude to Dr. McNew for his
invaluable helps during my attendance in Ann Arbor, summer 2013 and 2014.
I thank my friends at Professor Del Vecchio's research group, Control Networks
Group. I am very thankful to Dr. Daniel Hoehener at Control Network Group for
his helps and ideas. I also would like to thank the Graduate Office of MIT MechE.
My special thanks to my parents for all the helps, supports and sacrifices that
they have made for me. Words can never help me express how grateful I am for their
encouragements and prayers, that despite the far distance between us, have always
been a motivation for me.
I would also like to thank National Science Foundation (NSF) for supporting my
work under the award number 1161893.
5
6
Contents
1 Introduction
9
1.1
General Collision Scenario . . . . . . . . . . . . . . . . . . . . . . . .
10
1.2
Related Works . . . . . . ..
11
. . . . . . . . . . . . . . . . . . . . . . .
2 Deterministic vs Stochastic Systems
3
4
13
2.1
Deterministic System . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.2
Stochastic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
Stochastic Model
17
3.1
System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.2
Motivating Example
. . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.3
Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
3.4
Solution to Problem 1
. . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.5
Solution to Problem 2
. . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.6
Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.7
Simulations and Data Analysis . . . . . . . . . . . . . . . . . . . . . .
43
3.7.1
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . .
43
3.7.2
Experimental Results . . . . . . . . . . . . . . . . . . . . . . .
45
Conclusions and Future Works
53
7
8
Chapter 1
Introduction
The first recorded automobile fatality goes back to 1869 [1]. Today after almost 150
years, with the all developments in the safety technologies of vehicles, still number
of injuries and deaths caused by automobile accidents is significant.
In 2007 the
contribution of intersection related crashes among all types of accidents was reported
to be 40% [2]. Among all different possible intersection related crashes one is the
rear-end collision that takes place between two vehicles in the same lane. Drivers
may have wrong estimation of the decision that driver of their preceding vehicle is
making or going to make and this can put both vehicles in the dangerous situation
of rear-end collision. Obviously, a vehicle that is crossing an intersection with a high
velocity (namely a velocity higher than a maximum value) is the source of a different
type of collisions that occurs inside the intersection. Considering these two situations,
we are interested in design of a semi-autonomous control system that helps the driver
to avoid or mitigate collisions, with the basic assumption that only our vehicle is
equipped with the collision avoidance system.
In Section 1.1 we provide more details of the general collision scenario that we are
considering throughout the thesis and in Section 1.2 we review some of the related
works regarding the collision prevention and mitigation.
9
1.1
General Collision Scenario
We denote longitudinal position and velocity of the preceding vehicle (PV), if it
exists, by x, and v,, respectively.
The position and the velocity of the following
vehicle (FV), the vehicle that is equipped with the collision avoidance system, are Xf
and Vf, respectively. The longitudinal position of the intersection (stop sign) is also
denoted by St and the maximum allowable velocity (target velocity) is represented by
VT.
The minimum allowable distance between the two vehicles is 6. Mathematically,
(1) xp - xf < 6 or (2) Vf > VTand xf > St,
(1.1)
denotes the collision state. The scenario is depicted in Figure 1.1.
fs
xx
(a)
(b)
Figure 1.1: The collision is defined as (1)- If the distance between the two vehicles
becomes smaller than 6, or (2)- If FV passes the intersection with a velocity larger
than VT. In Figure (a) none of the two constraints are violated. In Figure (b), the top
figure, the first constraint is violated, and in the bottom figure the second constraint
is violated.
Since no collision avoidance system is implemented on PV, FV must be equipped
with a control system that has a reasonable estimation of the current and future
decisions of the driver of PV. Note that if any of the two constraints of equation (1.1)
is satisfied, at any time, the system will be in the collision state, and since we do
not have the information of the future states of PV and FV, the future estimation
is essential in order to design the control system. In the next section we focus on
different methods that have been employed for the estimation purposes.
10
1.2
Related Works
A popular tool to model a set of time series observations, which in our case is the past
behavior of drivers of PV as they approach intersections, e.g., x, and vP, collected
offline, is Hidden Markov Model (HMM). This method has been employed for similar driver behavior detection purposes in [1]-[11]. HMM captures different observed
behaviors through hidden states that their nature is not necessarily clear to us, and
consequently any of the hidden states affects the observation through multiple parameters. Although HMM is a powerful tool for estimation of the current state of
the system, but it is not good at long term predictions. The accurate predictions of
future states of the system are very essential. These predictions must be sufficiently
accurate in order that we can make the right decision based on what will happen in
future up to almost 30 sec, as the approximate maximum duration that the vehicle
is inside the intersection region. In general HMM constructs a model for the system
based on the available data and it does not consider the dynamic of the system. This
can make HMM a good choice for a highly unknown system, but we already know
the full dynamic of model of the two vehicles approaching an intersection.
In [3], [12] and [13] multiple noise driven linear systems have been considered as
different behaviors of drivers, which themselves are classified based on HMM. This
model can tackle the problem of using HMM solely, regarding the prediction of the
future behaviors, but this model does not capture the nature of the behavior of PV.
The position and the velocity of PV are generated by its acceleration and that is also
generated by a driver that even in his/her worst state he/she follows a set of logical
behaviors. Therefore a noise driven model while adds complexity to the problem, it
cannot capture the nature of PV well. Then the question is that "What is a good
model?" We will answer this question in the next chapter.
11
12
Chapter 2
Deterministic vs Stochastic
Systems
Since we are interested in having an estimation of the future profile of PV, we need
a model that outputs a profile until the intersection or until the time that FV stops.
The models mentioned in Chapter 1 consist of the discrete states that estimate or
predict the most probable action that the driver is making or going to make. These
models do not output any profiles for PV and in particular its acceleration, which
itself drives the velocity and the position of PV in turn. We present two approaches
to the collision avoidance problem. In Section 2.1 we present the deterministic model,
and in Section 2.2 we introduce the stochastic model.
2.1
Deterministic System
A simple solution to the estimation of the future decision of the driver of PV is to
consider a constant acceleration for it until any time that the vehicle stops or cross
the intersection. Since we are considering a constant acceleration for PV, we use the
term deterministic system for this model, versus the stochastic model in which we
do not consider a unique constant acceleration for PV and instead we assume it to
be a random variable. In order to guarantee that for the deterministic model none
of the inequalities of relation (1.1) are satisfied, we must consider an acceleration
13
value for PV that minimizes the distance between the vehicles, or in other words puts
the system in the most dangerous situation. If we can guarantee that the vehicles
are safe from the collision for this worst case scenario, then they are also safe for
any other inputs of the driver of PV. This acceleration value corresponds to the
minimum acceleration that PV can achieve. The minimum acceleration in vehicles
is generated by applying the maximum brake force, which can easily be provided for
any vehicles. In the deterministic model we check whether the profile corresponding
to the minimum acceleration violates the constraint x, - xf ;>
1, and based on that
we decide what control input must be provided to FV in order to guarantee the rearend collision avoidance. This method, in spite of being fast and simple, suffers big
problems which makes it almost inapplicable, in particular for the semi-autonomous
collision avoidance system.
If we were confronting a fully autonomous control system, which in particular did
not have human in the loop, we could take advantage of the deterministic system and
guarantee that no collision will take place as long as the control system is operating
properly. Existence of the driver of FV in the system (having human in the loop)
does not allow us to design a system that always considers PV as an adversarial
agent. While we are aware that the maximum brake or minimum acceleration is
applied in rare situations, assuming that PV has always the minimum acceleration
for the rest of its path leads to a very conservative system. Briefly, the deterministic
system has two major problems; (1): It takes control action very early, meaning that
from a considerably large distance to PV, which is not satisfactory for the driver of
FV; (2): Since the driver of PV rarely applies the maximum brake constantly, the
frequency of the false alarms increases significantly, meaning that the number of the
switches between the automatic control input and the driver input increases, which
again leads to the dissatisfaction of the driver of FV. From the application point of
view these two problems can make the model completely inapplicable and that is our
main motivation for considering the stochastic system which does not suffer the above
'Note that the violation of this constraint means satisfying the first inequality of relation (1.1)
which represents a collision state.
14
problems.
2.2
Stochastic System
The main difference between the deterministic system and the stochastic system is
that in the stochastic system, unlike the deterministic system, the acceleration of
PV is not assumed to always be its minimum possible value in order to estimate its
future profile. Since the decision that the driver of PV makes is related to its current
velocity and distance to the intersection 2 , we assume an acceleration as a function of
the position and the velocity of the PV3 . Moreover in order to capture the all possible
different behaviors, we assume a Gaussian distribution around this function. The
details of this model are provided in Chapter 3. With this model it is easier to relate
the profile to a safety value. The major problem in the stochastic model is that we
cannot guarantee 100% safety, and that is the reason that we have a safety level as
an input to the stochastic system.
We use stochastic systems to mitigate the rear-end collision instead of completely
preventing it from happening as we do in the deterministic model.
The driver's
satisfaction is the main reason for the transformation from the deterministic model
to the stochastic model. In the next chapter, first we introduce a general class of
systems and then prove that our collision avoidance scenario is also consistent with
this class of systems, and then we solve the corresponding control problems regarding
the expected safety of the model.
2
For instance when the speed is higher or the vehicle is closer to the stop sign, we expect a larger
required deceleration in order to stop the vehicle at the intersection.
3
1n our work we have assumed a linear function.
15
16
Chapter 3
Stochastic Model
In order to tackle the problems introduced in Chapter 2 regarding the conservativeness
of the deterministic model, which itself leads to the dissatisfaction of the driver of
FV, we take advantage of stochastic systems. In Chapter 3, we first introduce the
new model in Section 3.1. In Section 3.2 we prove that the collision avoidance system
can be modeled as a motivating example having the property of the class of systems
considered in Section 3.1. We then formulate the problems that need to be solved
based on the new model in Section 3.3. In Sections 3.4 and 3.5 we will solve the two
problems mentioned in Section 3.3. In Section 3.6 we present the general algorithm
to solve the relevant problems of Section 3.3, and we will introduce the discrete
algorithm for implementation purposes.
In the last section, Section 3.7, we will
present simulation results along with the required ools to build the model from the
available data.
3.1
System Model
Before introducing the model that we are considering, we define the strict and nonstrict order preserving properties, which we will use extensively throughout this chapter.
Definition 1. A non-strict partial order (or simply partial order) is a set P with
17
a partial order relation "<", which we denote by the pair (P, 5). The partial order
(Rn, <) with component-wise ordering is defined as follows. For all w, z ( RE we have
that w < z if and only if wi
zi for all i
E {1, 2,..., n},
in which wi denotes the i-th
component of w. We denote piecewise continuous signal on U by S(U) := PC(R+, U).
With this notation, for U C R' we define the partial order (S(U),
) by component-
wise ordering for all times, that is, for all w, z E S(U) we have that w < z provided
w(t)
z(t) for all t E R+. Moreover if (P,
p) and (Q, Q) are two partially ordered
sets, then the map f : P -+ Q is a non-strict order preserving (or simply order
preserving) map provided x <p y implies f(x) :Q f(y).
Definition 2. A strict partialorder is a set P with a partialorder relation "< ", which
we denote by the pair (P, <). The partial order (Rn, <) with component-wise ordering
is defined as follows. For all w, z E Rn we have that w < z if and only if wi < zi
for all i e {1, 2, ... , n}. We define the strict partial order (S(U), <) by componentwise orderingfor all times, that is, for all w, z E S(U) we have that w < z provided
w(t) < z(t) for all t E R+. Moreover if (P, <p) and (Q, <Q) are two strict partially
ordered sets, then the map f : P -+ Q is a strict order preserving map provided
x <p y implies f(x) <Q f(y).
We consider a class of continuous systems that have some order preserving properties.
Definition 3. A continuous system is a collection E = (X, U, A, 0, f, h), with state
x
E X c Rn, control input u E U c R', disturbance input d E A c
R", ouput
y E 0 C X, vector field in the form of f : X x U x A -+ X, and output map
h: X -+ 0.
Definition 4. Forthe systems E' = (X', U, A', 01,
f , h')
and E2
=
(X
2
U 2 , A 2 , 02
,f 2 , h 2 ) we define the parallel composition E = El E2 := (X,U,A,0 f, h), in which
X = X1 x
X 2, U :
XU 2, A:AXA
2
, 0:=01
X0 2,f:=(f
1, f 2) and
hW:= (hd, h 2f
We denote the flow of a system E at time t E R+ by 0(t, x, u, d), with initial
18
condition x E X, control input signal u E S(U) and disturbance input signal d E
S(A). We also denote the ith component of the flow by q5 (t, x, u, d).
Definition 5. A continuous system E = (X, U, A, 0, f, h) is called input/output
order preserving (or strict order preserving) with respect to the control input, if the
map h(q(t, x, u, d)) : U -+ 0 is an order preserving map (or strict order preserving
map).
Definition 6. A continuous system E = (X, U, A, 0, f, h) is called input/output
order preserving (or strict order preserving) with respect to the disturbance input, if
the map h(O(t, x, u, d)) : A -+ 0 is an orderpreserving map (or strict orderpreserving
map).
The system model that we are considering is defined as follows.
Definition 7. We consider the parallelcomposition of the systems E' = (X', U, 0, 0,
f', h') and
E2
=
(X 2 1,,
0
2
, f 2, h 2 ), where x1 E X1 C R , x 2
E
X2
c
Rn
u E U = [UM, uM] C Rm with um E R' and uM E Rm , the minimal and the maximal
control inputs for El, respectively, d E A = R, y' E 01, y 2 E 02, h'(x') : X'
and h2 (x 2)
X2
_+
02.
01
The vector fields are in the form of f 1(x',u) : X 1 x U -+X
.
and f 2 (X 2 , d) : X 2 x A -+ X 2
We impose the order preserving properties on the flow of the system as follows.
Assumption 1. System El2 has input/output order preserving property with respect
to the control input.
Assumption 2. System
EI2
has strict input/output order preserving property with
respect to the disturbance input.
Assumption 3. The disturbance input term of the system
E2
is a constant distur-
bance input that can be modeled as a Gaussian distribution, that is d = d
N(I, o2).
We solve the corresponding control problems for the general class of systems introduced in this section, in Sections 3.4 and 3.5.
19
3.2
Motivating Example
Throughout this section first we model the scenario of the two vehicles approaching
an intersection, and then we prove that it is consistent with the model that we have
defined in Definition 7 and satisfies Assumptions 1-3. We denote the position and
the speed of the following vehicle (FV) by Xf and Vf, respectively.
Similarly the
position and the speed of the preceding vehicle (PV) are represented by x, and Vp,
respectively. We denote the control input to the FV by u and the disturbance input
term of PV by d. The deceleration due to the road load (rolling resistance) and the
slope of the road on the FV are represented by a, and a, respectively, and the drag
coefficient is denoted by C. We also impose a condition that the speed of both PV
and FV must be non-negative. We use the superscript T to denote the transpose of
a vector or matrix, e.g., AT represents the transpose of matrix A. The dimension of
a vector space S is represented by dim S. Using these notations for the continuous
form of the state space model of the system we have
x E X C R 4 ,where x = (Xf, Vf , X,, V,) T ,
(3.1)
u E U c R, where U ={u I u E [um, u]},
(3.2)
d E R,
(3.3)
f : X xU
x R-+
X,where
= f(x, u, d),
(3.4)
with
f(x, u, d)
=
(f1 (X',u),f2 (X2, d)), X1 = (xf, vf)T, x 2
= (x
rvP)T, (3.5)
where
fl(x1 u){
f(xiu)
0
f22
d) =.
f (2 2 ,d)
0
20
if vf > 0
if Vf
0
if v, > 0
if vP 0
and
(3.6)
The functions f'(x', u) and f 2 (X2 , d) are also in the following forms:
(2,d)=
vand
f'(x',u)=
U - CV2, - a,. - a,
P
axp + bv, + d
(3.7)
The term ax,+ bvp+d is the acceleration of PV. We assume that d ~ N(p, O.2 ), which
is consistent with Assumption 3.
Since we cannot measure the acceleration of PV, we build a model that estimates
it from the states of the system, the position and the speed. The parameters a, b,
p and a can be extracted through an optimization problem. More details of the
acceleration model along with the optimization problem will be provided in Section
3.7.
We can write (3.1)-(3.7) as the parallel composition of the two systems El
(X 1 ,U, 0, 01, f 1 , h') and E 2
and x 2
=
(X
2
, , A, 0 2 , f 2 , h 2 ), where x1 = (xf, vf)T E X C R 2
(xP,v,)T E X 2 C R 2 , with X = X1 xX 2 , u E U = [urn, uM] C R, d E A = R,
y =xf E R, y 2 = x, E R, hl(x') = (1, 0)x' and h2 (X2) = (1, 0)x 2 .The vector fields
f'(x1 ,u) :
x U -+ X1 and f 2 (2 2 , d) : X2 X A -+ X 2 will then take the forms
f'(xf1 ,)zU) =
f(x ,u)
1
0
if dhl(xl) > 0
(3.8)
,
if dhl(xl) < 0
and
P (X2
P2(X2,7
d)
0
if d2(2,d),
dh 2 (X2 dt2(X2> 0
2 (x2 )<0
if jh
dth~)<
(39
(3.9)
with fP(Xi, u) and f 2 (X 2 , d) as defined in (3.7). This model is consistent with Definition 7.
Since A' = 0 and U 2 = 0, we represent flow of the systems El and E2 with
q 1 (t, x 1 , u) and
# 2 (t, X 2 , d),
respectively. Throughout the rest of this section we prove
that Assumptions 1 and 2 are valid for the scenario of two consecutive vehicles approaching an intersection.
Assumption 1 states that El must have input/output
order preserving property with respect to the control input signal, meaning that
21
:= h'(#1(t, x, u)) = 01(t, x, u) must be order preserving with respect to the
Xf(t)
control input signal u. We prove that not only xf(t), but also Vf(t) :=
(t, x, u) has
order preserving property with respect to the control input signal. The order preserving property of vf(t) is not necessary for consistency of the model with Definition 7,
but it is required in order to satisfy another property that will be discussed in Section
3.3.
Proposition 1. The flows
1(t, x, u) = x 1 (t) and q1 (t, x, u) = vf(t) of the system
defined in (3.1)-(3.7) are order preserving with respect to-the control input signal u.
>
u2
,
Proof. If we consider two different control input signals ul and u 2 , such that ul
then for the velocity of PV at time t corresponding to these two control input signals,
with the same initial conditions xf,l(O)
Vf
(0), we have i'f,1(t)
=
=
Xf, 2 (0) =
x1 (0) and vf,1(0)
ui(t)-CVj,1(t) 2 -a,-aa and if,2 (t)
if both vf,1(t) > 0 and
Vf,2(t)
=
> 0. Let the function g(t)
u2 (t)-CV
=
2
Vf,2(O) =
(t)-ar-a,
Vf,2(t).
Vf,l(t) -
At an
arbitrary time t we have
= 7(t)
fn,1(t) -
')f,2(t) =
(ui(t) - u 2 (t))
-
C (vf(t)
-
vf, 2 (t))
.
(3.10)
-
Note that since we have chosen the same initial conditions, we have g(0) = vf,l(0)
vf,2(0)
= 0. Because of the continuity of flow of the system with respect to time, if
order in state vf is not preserved, we must have a time t' E R+ such that g(t') = 0,
since otherwise for all t E R+, either g(t) < 0 or g(t) > 0. Therefore we can define
t*
min{t E R+ Ig(t) = 0}. Since y(0) = u1 (0) -U 2 (0) > 0, 4(t*) = ui(t*) -u 2 (t*) >
0 and g(0) = g(t*) = 0, for the interval t E (0, t*) we have
lim g(h) - g(0)
NO)-
h-+O+
h- 0
-
lim g(h) >
h-+o+
h
since h > 0 : 3 h = hi E (0, t*) s.t. g(hi) > 0,
and similarly
.
gt)= lim
h-+-
g(t*) - g(t* + h) ==-lim g(t* + h) > 0 =
t* - (t* + h)
h-+0h
22
(3.11)
since h < 0 : - h = h 2 E (0, t*) s.t. g(h 2 ) <0,
(
(3.12)
and because of the continuity of the flow with respect to time, there is a t E [hi, h 2]
such that g(t) = 0, which is in contradiction with the initial assumption that t*
min{t E R+
I g(t)
that Vf,l(t) > 0 and
vf,1(t)
-
Vf,2(t)
= 0}. Therefore there is no such t*, and for all t E R+ such
> 0 we have either g(t) = vf, 1 (t)
Vf,2(t)
Vf, 2 (t)
> 0 or g(t) =
< 0. From (3.11) we conclude that the former is true.
We had assumed initially that vf,1(t) > 0 and
t' E R+ we have Vf,l(t') = 0 and
and f2 = min{t E R+
and vf,l(t)
-
-
Vf,2(t),
then g(t) =
Vf,l(t) -
I Vf,
2
(t)
=
vf,2(t') =
if t E [f4, oo), then g(t)
0, we let fi := min{t E R+ I Vf,1(t) = 0}
0}. Because of the non-negativity of Vf,l(t),
we must have
Vf,2(t) >
0. For a case that for some
Vf,2(t) >
f2
< fl. If an arbitrary time t such that t E (0, t 2 ),
0; If t E [f2, f4), then Vf,1(t)
= Vf,l(t) -
Vf,2(t)
Vf,2(t)
-
Vf,2(t)
= vf,i(t) > 0; And
= 0. Therefore, in any case the order of
the flow of the velocity is preserved with respect to the control input signal. Since
Xf,l(0)
= Xf,2(0)
= xf(0), then based on equation (3.7) Xf ,(t)
-
xf,2 (t)
=
f g(s)ds >
0, which implies that the order preserving property of the flow of xf is also satisfied
with respect to the control input signal.
In Proposition 2 we will prove that Assumption 2 is also valid for our motivating
example, meaning that x,(t)
h 2(q 2 (t, x, d))
=
#2 (t, x, d) is strictly order preserving
with respect to d.
Proposition 2. For the system in the form of (3.1)-(3.7) the flow
2(t, x,
d)
x,(t)
is strictly order preserving with respect to d.
Proof. Let Xo := (xf,of,o, XO,, vp,o)T be the initial condition, where vf,O > 0 and
v,,o > 0. According to Assumption 3 we have d(t) = d where d ~ (p, a2 ).
From
equations (3.6) and (3.7) we have that the velocity of PV, for v,(t) > 0, satisfies the
following differential equation:
VP - bp, - avp = 0 where v,(0) = v,,o and i),(0) = axp,o + bvp,o + d.
23
(3.13)
The above differential equation has the solution in the form
v,(t) = kieAlt
A,
=
+ k 2 eA2t
0.5(b + V2 +4a) and A2
=
where,
0.5(b - Vb2 +4a).
(3.14)
Since complex and real values of A, and A 2 reveal different behaviors for vp(t), we
consider different possible cases and analyze the behavior of xp(t) with respect to d
for each of them. We divide the problem into three different cases; (1): b 2 + 4a > 0,
(2): b 2 + 4a < 0 and (3): b 2 + 4a = 0. For each case we consider two disturbance
signals d' = d' and d2 = d2 such that d' > d2 and determine the relationship between
v (t) and vj(t) and then between
1(t) and x4(t), the velocity and the position of PV
at time t corresponding to d' and d2 , respectively.
Case (1): If b 2 + 4a > 0, then A, and A 2 in (3.14) are real numbers. The solution
of (3.13) then takes the form
v,(t ) =
A((vpo(
- b) - ax,o - d) e lt - (v,o(Al- b) - ax,O - d) eA2t)
(3.15)
If we replace d in equation (3.15) with d and d2 in order to obtain their corresponding
velocities at time t, represented by vo(t) and v (t), respectively, we have
_,(t
V2 v-(2t -=-
_exl3.) 16)t).
(eA2t
A
A
Note that (3.16) can become zero only when t = 0. Therefore because of the continuity
of flow of the system with respect to time, for all t E R+, either v,1(t)
-
V (t) > 0
or vi(t) - v (t) < 0. To determine which of these two cases holds, we note that in
general for any x E R - {0} we have that if x > 0, then ex - 1 > 0 and if x < 0, then
e' - 1 < 0. These two statements together imply that '-I
A2 -
Al #A 0 and we are considering t E R+, then (A 2 - Al)t
24
> 0. Since in Case (1)
#
0. Therefore we can
replace x with (A 2 - Al)t.
e(A2-A1)t
1
-
e(A2-A1)t
-> 0 => te*lt
d
(A2-~
1
-
> 0 =>
eA2
eA
eAt
> 0
=
IdA-2jtIA2-A
-j
d'- A2(eA2t
_
eXit) > 0 => V (t) > v (t),
(3.17)
where we have used the facts that t E R+, eAli > 0 and d' - d2 > 0. By integrating
both sides of (3.17) to determine the position of PV at time t, we obtain
t
t
fov2 (u) du=
0vP (u) du >
Xp,0
+
tv,(u)du > xp,, +
j0v(u)du
xpj(t)> x2(t).
(3.18)
Case (2): If b 2 + 4a < 0, then A1 and A2 in (3.14) are complex numbers. The
solution of (3.13) then takes the form
vP() = eat (ax,o + (b - a)v,o+d s
with a =.0.5b, and
#=
t + vO Cos
0.5N/-(b 2 + 4a).
(3.19)
If we replace d in equation (3.19) with d' and d2 in order to obtain their corresponding
velocities at time t, represented by v1(t) and vP (t), respectively, we have
ve(t) _v(t) =d at sin ,t.
(3.20)
We observe that in Case (2), unlike Case (1), we cannot guarantee that for all t E
R+, v (t) - v2(t) # 0. Note that V1(t)
-
v2(t) = 0 for all t such that sin/#t = 0 or
alternatively, 3t = k7r, for all k E Z. The smallest t E R+ that satisfies sin #t = 0 is
C* = 0. The velocity of PV at time V* corresponding to d' and d 2 , based on equation
(3.19), is given by
v,'(t*) =
at* cos(3 vp,oe~t
=
-v,oe a* <0<for i E1,2}. (3.21)
25
Since for all t E R+, vp(t) > 0, we must have vp'(t*) = 0, or in other words, for
all t E [0,tt*] we have either vp(t) - v2(t) > 0 or vp(t)
-
v2(t) < 0.
determine which case holds, we note that for all t E [0, t*] we have
In order to
f
> 0 and
0 < sin ft < 1. Therefore in any case, for all t E [0, t*] we have 21n3t > 0. Also
eat(d' - d2) > 0. These two statements along with (3.20) imply that v (t) -v,(t) > 0.
Since in (3.19) we have VP(0) = v,,o > 0 and v2(t*) = -vp,oe*t* < 0, then because of
the continuity of flow of the system with respect to time, there is a f E (0, t*) such
that f:= min{t E (0, t*) I v (t) = O}. Then we have for all t E (0,), v, (t) - v (t) > 0.
For a t E (0, 0, we have
x (t) - x (t) = j(v(u) -
vp(u))du > 0;
(3.22)
For a t E [i, t*) we have
t
x (t)
-
(t) =
x
t
(v, (u)
0 + J(v (u)
-
-
(v (u) - v (u))du >
v (u))du +
v (u))du > 0 =>
4(t) -
x
(t) > 0;
(3.23)
and for a t E [t*, oo), we have
X4(t)
(t)
-
(v (u)
=J
-
v (u))du+
f(v (u) -
v (u))du =
J;*
(v (u)
-
Vo(u))du + 0 > 0 => X (t)
-
X (t) > 0.
(3.24)
Case (3): If b2 + 4a = 0, then A, = A2 = A, which is also a real number. The
solution of (3.13) then takes the form
vp(t) = eA't [vp,o + (axp,o + (b - A)vp,o + d) t],
(3.25)
and for v (t) - v2(t) we have
v (t)
-
v (t) = t(d' - d2)et > 0,
26
(3.26)
which implies
I(t)
4(t)
(v (u) - v2(u))du > 0.
(3.27)
We had assumed initially that v (t) > 0 and v2(t) > 0.
In general we may
-
=
have a time t* such that v (t*) = 0 and v,2(t*) = 0. In this case, because of the
continuity of flow of the system with respect to time, there are times fi and
that fi = sup{t E (0, t*) I v, (t) > 0} and
f2
f2
such
= sup{t E (0, t*) | v (t) > 0}. Since we
have proved through Cases (1)-(3) that as long as vo(t) > 0 and v (t) > 0 we have
v (t)
-
v2(t) > 0, then f2 < f. For an arbitrary time r E (0, f2 ) Cases (1)-(3) imply
that x (r) - x(r) > 0; If r E [2, [1),then
x
(r) - X (r) =
(v
(u) -
v2(u))du + J(v
(u) - 0)du > 0;
(3.28)
And if T E [f, oo), then
, (r)
- x (r)
=
j
(v (u) - vp(u))du+ J
(v (u) - 0)du+
L
and the proof is complete.
3.3
(0-0)du > 0, (3.29)
Problem Formulation
Before formulating the problem we define the bad set.
Definition 8. For a system with the states x E X, the bad set, B, is a subset of
the space of the states, B C X, that the system should never enter, that is, for all
t E R+, x(t)
B.
Because of the restrictions that we have on our control input, u E [um, UM] C Rm,
there is no guarantee that if an initial state of the system is outside of the bad set, it
will never enter it. Therefore we need to introduce a game between the control input
and the disturbance input such that the probability that the control input wins,
which means not entering the bad set, is a given value P. Our main goal is to design
27
a control strategy that guarantees success of the control input P% of the time. We use
Pr(.) to denote the probability and p(.) to denote the probability density function.
.
We represent signal of the states of the system by x E S(X), where X := X' x X 2
We denote a static feedback map with 7r : X -+ U. With these notations, the flow
of the system with feedback map 7r, initial condition x and disturbance signal d, is
represented by
#(t, x, u, d)
such that u = 7r(x). The complement of a set C C X
is denoted by Cc, defined as Cc := {x E X I x V C}.
The bad set that we are
considering has the following form:
Assumption 4. The bad set is in the form
1 1
B=U_ 1 {xEX | G3(x1) >g}U {xEX I Ch (x ) -
B1
2
h2
2)
> H},
(3.30)
B2
where C' and C2 are r x dim(01) and r x dim(02 ) matrices, respectively, with
ci,3,c?,3
0. hl(xl) and h2 (x 2 ) are as defined in Definition 7, H is a r-dimensional
vector, the functions Gj are such that Gi(x') : X
-+
RP' and g9s are p3-dimensional
vectors.
We impose one more assumption on function Gj before formulating the control
problems.
Assumption 5. The map G(xl) = G3(q1(t, x1(0), u)) : U -+ RP!, for
E {1, ... ,N},
is an order preserving map.
The two following problems concerned with the P% safety of the system introduced in Definition 7 must be solved.
Problem 1. For the system E = E1||E2, defined in Definition 7, with Assumptions
1-5 and P E (0,1), find the open loop maximal safe set given by
W := {x E X 13 u E S(U) s.t. Pr(#(t, x, u, d) V B, Vt E R+ and Vd E R)
Problem 2. For the system E =
>
P}.
1||E2 defined in Definition 7, with Assumptions
28
1-5 and P
E (0,1),
find the control map ir : X -+ U such that for all x E W we have
B,Vt E R+ and Vd E R) > P where u= ir(x).
Pr(<b(t,x, u, d)
We have proved in Section 3.2 that the scenario of the two consecutive vehicles
approaching an intersection is consistent with the system defined in Definition 7
and Assumptions 1-3. The bad set for the scenario of the two consecutive vehicles
approaching an intersection, based on equation (1.1), is in the form
B = {x
B = {x
E X I xf
> St and vf > vT} U {x E X I x
E X I (xf,vf )>
(St,vT)T} U {x E X
I xf
B1
-
xf
- x,
< } =>
> -- },
(3.31)
B2
for given St, VT and 6 representing the position of the intersection, the maximum
allowable velocity at the intersection and the minimum allowable distance between
vehicles, respectively. The set B1 corresponds to those states of the system that puts
FV at the intersection with a velocity higher than
VT
and the set B 2 corresponds
to those states of the system that leads to collision between PV and FV. If we let
C1 = C2
=
1, H = -6, G1(xl) = x 1 , g1 = (St, vT) T and N = 1, we observe that
the bad set can be written in the form assumed in Assumption 4. We have proved
in Proposition 2 that the flows of xf and vf are order preserving with respect to the
control input signal, therefore since G'(x')
=
X1
=
(x1 , vf)T, Assumption 5 is also
valid for our motivating example. In the next two sections we will solve Problems 1
and 2.
3.4
Solution to Problem 1
Before proposing the solution to Problem 1 we need to define the capture set.
Definition 9. For the system defined in Definition 7, with Assumptions 1-5, the Psafety capture set (P E (0,1)) for a given control input signal u
29
E S(U)
is the set of
states, x E X, defined as
C.(P) := {x E X I Pr((t,X, u, d) V B, Vt E R+ and Vd E R) < P}.
Lemma 1. The P-safety capture set of a given control input signal u E S(U), for
the bad set in the form of (3.30), can be written as
Cu(P) ={x E X I Pr (Vt E R+ and Vd E R,
C'h(#1(t, x 1, u))
-
C 2h 2 (
2 (t, x 2 , d))
; H) < P} U
Ix E X I 3t E R+, 3j E {l, ...
,7
N} s.t. Gi (#1(t, x1 , u)) > gj
Proof. The bad set based on (3.30) is B = B1 UB2 . According to Definition 9 P-safety
capture set for input signal u for this bad set is
Cu(P) = {x E X I Pr(#(t, x, u, d)
#(t, x, u, d)
B1 A
B2 , Vt E R+ and Vd E R) < P}.
(3.32)
Let the set S be defined as
S := {x E X
I 3t E
R+ and 3d E R s.t.
#(t,x,u,d)
E B1 } =
Ix E X I 3t E R+, 3j E {Il... N} s.t. Gj(# (t, x1, u)) > gi }
.
(3.33)
We can rewrite (3.32) in the following form in which Sc
{x
E X I x 0 S} represents
the complement of the set S.
Vt E R+ and Vd E R) < P} = {x E S
V B1 A #(t, x, u, d)
I Pr (#(t, x, u, d)
Vt E R+ and Vd E R) < P} U {x E Sc
30
VB
2
0 B1 A #(t, x, u, d) 0 B 2
I Pr (#(t, x, u, d)
,
I Pr(q(t, x, u, d)
,
Cu(P) = {x E S U SC
0 B1 A
#(t, x, u, d)
B2 , Vt E R+ and Vd E R)<P}.
(3.
(3.34)
If x E S, since from Assumption 4 for all j E {1, ... , N} the function Gi is not function
of the disturbance input d, then from (3.33) we have
Pr(#(t, x, u, d) ( B1 , Vt E R+ and Vd E R)
=
Pr(Vj E {1, ... , N}, Vt E R+, Gi(#(t, x, u)) < gi) = 0.
(3.35)
Therefore if x E S, from (3.35) we have
Pr (#(t, x, u, d)
B1 A #(t, x, u, d)
B2 , Vt E R+ and Vd E R) = 0 < P,
(3.36)
which is true for all P E (0, 1). This implies that (3.34) can be written in the following
form:
C.(P) = S U {x E S' I Pr ((t, x, u, d)
0(t, x, u, d)
B1 A
B2, Vt E R+ and Vd E R) < P}.
(3.37)
If x E Sc, then from (3.33) we obtain
Pr(#(t, x, u, d)
B1, Vt E R+ and Vd E R) = 1,
(3.38)
which is independent of the event 0(t, x, u, d) E B 2 . This implies that if x E Sc, then
B 1 A #(t, x, u, d) V B 2 , Vt E R+ and Vd E R)
Pr (#(t, x, u, d) ( B 1 , Vt E R+ and Vd E R) .Pr(#(t, x, u, d)
=
B2
,
Pr (#(t, x, u, d)
,
Vt E R+ and Vd E R) = Pr (O(t, x, u, d) V B2 , Vt E R+ and Vd E R)
(3.39)
where to obtain the last equality we have used equation (3.38). From equations (3.37)
and (3.39) we have
Cu(P) = S U {x E S
I Pr(#(t, x, u, d)
V B 2 , Vt E R+ and Vd E R) < P}.
31
(3.40)
Since we know
{x E S
I Pr((t,x,u,d) V
B 2 , VtER+ and VdER) <P} CS,
(3.41)
and S U S' = X, then we can write (3.40) in the form
Cu(P) = S U {x E X
I Pr((t,
x, u, d) V B 2 , Vt E R+ and Vd E R) < P}.
(3.42)
If we replace S with its definition from (3.33) and use the definitions of B1 and B2 from
(3.30), we can write equation (3.42) in the form of the statement of the Lemma.
13
Lemma 2. Let
Ftxu(d) := Clh'(q1(t, x 1 , u)) -- C 2h 2 ( 2 (t, x 2 , d)),
and
(Ft",U)-'(s) := {d E R I Fj'x'u(d) = s}
with Ft,'U and (Fj'x')-1 denoting the ith component of Ftxu and (Ft*x'u)~, respec-
tively, and let the pair (t*, i*) (not necessarily unique) be
(t*, i*) = arg
min
VtER+
ViE{1,...,r}
Pr (Vd E R, d > (Fj'x)-1(Hj)),
then we have
{x E X
C2 h 2 (02 (t,x2 , d)) < H) < P} = {x E X
H. < F '"(p + -Q- (P))
.
Pr (Vt E R+ and Vd E R, Ch(1 (t, x , u))-
Proof. Since based on Assumption 2 the function h 2 (X 2 ) = 4 2 (t, x 2 , d) is strictly
order preserving with respect to d, then based on Assumption 4 C 2 h 2 (x 2 ) is also
strictly order preserving with respect to d and since h'(xl) is not function of d, then
Ftx'u(d) = Clhl(xl)
-
C 2 h2
(X 2 ) is a strictly decreasing function of d and therefore
32
invertible. Using this property we have
Pr (Vt E R+, Vd E R, C'h'(O'(t,x', u))
-
C 2 h2 (q 2 (t,x2 , d))
H) =
H) =
Pr (Vt E R+, Vd E R, Fxu(d)
Pr (Vt E R+, Vd E R, Vi E {1, ... , r}, Fxu(d) :H) =
Pr (Vt E R+,Vd E R,Vi E {1, ... ,r}, d > (Fit")- 1(Hi)) =
Pr (Vd E R, d >
=
(Fj'"tu)~1(H)
Pr (d E R, d > (Fit") 1 (Hi))
(3.43)
,
min
VtER+
ViE{1,...,r}
max
VtER+
and using the definition of (t*, i*) we have
min
VtER+
Pr (Vd E R,d
(Fjt'x'u)-1(H))=Pr VdERd>(Ft.''x')- (Hj)
ViE{1,...,r}
(3.44)
In order to find a relationship between the disturbance input and the desired safety
level P, we define the
Q function as
Q(z)
j
:=
Since based on Assumption 3 d = d
-
0
0.582 ds.
N(p, a2), using
(3.45)
Q notation and equation (3.44)
we have that if Pr(Vd E R, d > (Fj>.*'')-1(Hi.)) < P, then
Q (F.*''")-( ) - p < P.
(3.46)
01
Since Q(z) = 1 - 4b(z) where 4 (z) represents the c.d.f. (cumulative distribution
function) of the standard normal distribution, the
33
Q function
is strictly decreasing
and also invertible. Therefore equation (3.46) can be written as
(*''")-1(Hi*) - I > Q- 1 (P) => (F''"x)-1 (Hi.)
> p + oQ'(P),
a
s
(3.47)
and since FE.>''" is a strictly decreasing and invertible function, then
Hi* < F.t*'XU(p + -Q-1 (P)),
(3.48)
and the proof is complete.
The following theorem provides a solution to Problem 1.
Theorem 1. For the system defined in Definition 7, with Assumptions 1-5, x E W
if and only if x 0 Cum(P).
Proof. (<=) If x V Cum(P) then x E Cucm(P). Therefore
Pr(#(t,x, um, d) V B, Vt E R+, Vd E R) > P,
(3.49)
which implies that x E W.
(
(=>) If x E W, then there is control input signal u' E S(U) such that Pr(#(t, x, u', d)
B,Vt e R+,Vd e R) > P. If we replace the relation "<" in Definition 9 with the
relation ">" and use the results of Lemma 1 and Lemma 2, for x E W we have that
there is control input signal u' E S(U) such that
Pr (Vt E R+ and Vd E R, Clh(# 1 (t, xI, u'))
-
C2h 2 (
2
(t, x 2 , d))
and Vt E R+,Vj E {1, ... , N} G3(# 1 (t, x1, u'))
We prove that x
gi.
H) > P
(3.50)
C.m (P). Assume that by contradiction x E Cum (P), then we
have Pr(#(t, x, um, d)
B, Vt E R+, Vd E R) < P. Therefore x E Cum(P) based on
Lemma 1 implies
1
Pr (Vt E R+ and Vd E R, Ch (0(t,
x, um)) _ C 2 h2 ((t,
34
x2 , d))
H) < P
or tER+,j E l, ... N} s.t. Gi(ma(t, x1, um)) > gi,
(3.51)
which based on Lemma 2 implies that
or 3t E R+, 3j E {1, ...
,I
N} SAt. Gi (0'(t, x', um)) > gi,
(3.52)
Pr (Vd E R, d > (F"XzUm)-1(Hi))
(3.53)
where
min
.
(t*,i*) = arg
VtER+
ViE{1,...,r}
If x E Cum(P), then based on equation (3.52) we can consider two cases. Case
(1): Hj. < Fjt*,x,um(p + oQ-'(P)); Case (2): There is a time t E R+ and a j E
-
{1,) ...
,I
N} s.t. Gi (#1(t, x1,I um)) > gi
Case (1): If x E W then according to (3.50) and Lemma 2 there is a control input
signal u' E S(U) such that
Hit > F,'''U'(p+ aQ 1 (P)) where
min
VtER+
Pr (Vd E Rd > (Ft"u')-1(Hi))
(3.54)
.
(t', i') = arg
ViE{1,...,r}
Note that the pair (t*, i*) is not necessarily the same as (t', i'), but if Pr(d >
P (which is equivalent to RI / > F,'' '"'(p + aQ-'(P)) based
(F,'')-(Hi))
on Lemma 2), then according to (3.54) we also have Pr(d > (F*',x'')-1(Hi*)) > P
(which is equivalent to H
> F.*''"'(p + aQ'(P)) based on Lemma 2). This result
along with the equation Hj. < F.*,x,Um (p + aQ- 1 (P)), which is the main assumption
in Case 1, imply
+ aQ-(P)) <; H . < Fit.*:'(p+ oQ- 1 (P)) =>
dim(0 2
)
dim(Q')
*j,lh
C*,k)
35
02(* I2
-2
dim(0 1 )
)
dim(02
c,h2 (2(t*2 , 2 ,L + oQl(P)))
Cl.,*khk(#1 (t*, X1, Um)) k=1
1=1
dim(01)
C
.,k [h'(#1(t*, xl, u')) - h'(#1 (t*, xl, um))] < 0.
(3.55)
k=1
Since um is the minimal control input and based on Assumption 1 h' is an order
preserving function of the control input signal u, then for all k E {1, ... , dim(0 1 )} we
have h)i(l(t*,
1 , u')) -
h
h)(#1(t*,
x1, um)) > 0. In turn, from Assumption 4 we have
that cij > 0. These two statements together contradict (3.55). Therefore we must
have Hi. > Fb.*,xum(p+ OQ-1(P)).
Case (2): If x E Cum(P), then we must have a time r E R+ and a j E {1,
... ,
N}
such that Gi (#1 (r, x1, um)) > gi. Because of the order preserving property of the
function Gi (q1 (r, x 1 , u)) with respect to the control input signal based on Assumption 5, for all u E S(U) we have G (01 (r, x1, um))
Gi (1(T,
Gi(1(-, x1 , u)). Therefore if
x1, um)) > gi, then we also have G (#1(r, x1, u)) > gi for all u E S(U). Since
x E W, based on (3.50) there is also a control input signal u' E S(U) such tlhat
Vt E R+,Vj E {1, ... , N} : Gi(#1 (t, x1, u')) 5 g'.
(3.56)
Since equation (3.56) is for all t E R+, then Gi(# 1 (r, x 1 , u') 5 g3 , which contradicts our previous statement that for all control input signals u E S(U) we have
G(1(r, x1, u)) > gi. Therefore there is no j E {1,...,N}
and t E R+ such that
Gi (1 (-r, x1, u)) > gi.
Since none of the assumptions of Case 1 or Case 2 are valid, then we must have
Hi. ;> F.*,xum(p + o-Q-1(P))
and At E R+, Aj E {1, ... , N} s.t. Gi(#1 (t, xl, Um)) > gi,
therefore x 0 Cum (P).
(3.57)
E
36
Solution to Problem 2
3.5
We consider the feedback control map
r(x)=
Cum(P)U
..
OCum(P)
U
if x
UM
if x E Cum(P)U &Cum(P)
and state the following theorem.
Theorem 2. For the system defined in Definition 7, with Assumptions 1-5, for all
x
EW
the feedback map r : X -+ U, as defined in equation (3.58), guarantees that
Pr(#(t, x, u, d)
BVt E R+,Vd E R)
P, where u= u([O, t]) =ir(x([O,t))).
Proof. We consider two different cases.
Case (1): If for all t E R+ we have
#(t,
x, u, d)
Cum (P), where u E S(U) is
an arbitrary control input signal, then based on Theorem 1 for all t E R+, x(t) =
#(t, x, u, d) E W, and since W n B = 0, then Pr(#(t, x, w(x([O, t))), d)
B, Vt E
R+, Vd E R) = 1 > P and the statement of the theorem is satisfied.
Case (2): If there is a time t* E R+ such that x(t*) =
#(t*,
x, u, d) E Cum(P),
then because of the continuity of flow of the system with respect to time, there is a
time f := sup{t E (0, t*)
I
4(t, x, u, d)
Cum (P)}, where we have also used the fact
that based on Theorem 1 x = x(0) E W implies that x
#(f,
X, U, d) E OCum(P). Assume that by contradiction O(f, x, u, d)
Cum(P). We prove that
Cu,(P). Since
C = Cl(C) n Cl(Cc), where Cl(C) represents the closure of the set C, O(f, x, u, d)
Cum (P),
#(t*,
x, u, d) E Cum (P), and the flow is continuous with respect to time,
then there is a time t' E (f, t*) such that
Cum (P) is an open set, then
t:= sup{t E (0, t*)
I
#(t', x, u, d)
(t, x, u, d)
#(t',
x, u, d) E Oum (P). Since the set
Cum (P), which contradicts the fact that
Cum(P)}, therefore
#(,
x, u, d) E OCum(P)-
In order to guarantee that in Case (2) the control feedback map (3.58) provides
the minimum P%, we divide Case (2) into two different subcases which we refer to
as Subcase (2-a) and (2-b). In Subcase (2-a), for all t > f we have x(t) E Cum(P) U
9Cum(P).
Therefore according to (3.58) for all t > i we have u(t) = um. Since x() =
O(, x, u, d) 0 Cum(P), then we have Pr(#(t, x(), um, d) V B, Vt > f, Vd E R) > P
37
and for all t < f we have x(t) E W or alternatively Pr(#(t, x, u, d)
V B, Vt
< t) > P.
Therefore for the Subcase (2-a), the following control signal
U
if t E [0,7t
ifte[O=)(39)
Uum) U if t E [f, oo)
will guarantee the minimum P% safety.
In Subcase (2-b), we assume that there is a t > f such that i
(f, oo]
I #(t, x(t), um, d)
inf{t E
V Cum(P) U OCum(P)}- According to (3.58), u(i) E U. If
for all t > i we have that #(t, x(i), u, d) V Cum(P) U aCum(P) in which u is an
arbitrary control input, then based on similar analysis as in Case (1), the minimum P% safety for t > t is guaranteed. Also if there is a time t >
t := inf{t >
i
such that
1O
#(f, x(i), u, d) E Cum(P) U aCum(P)}, then based on the similar
analysis as in Subcase (2-a) we conclude that the minimum P% safety for t E [t, t) is
satisfied and the control map for t E [0, t) will be
u(t)
U
if t E [0, 0~
um
if t E [f,)
(3.60)
U if t E [i, i)
Also for t > t we can divide the problem into two subcases as in Subcases (2-a) and
(2-b) and then we can guarantee that the minimum P% is satisfied.
3.6
Algorithms
In this section we propose the algorithms to calculate the control map suggested in
Section 3.5.
Theorem 3. Let
dim(01)
2
)
dim(0
Ckh1(#1(tx 1 ,um)) -
max
ViE{1,...,r}
L k1
ci,h2(# 2 (t, x 2 ,d))
1
38
-
Hi
(3.61)
where d = L + uQ- 1(P), then necessary and sufficient conditions for state x E X
such that x
0 and GiQ#k(t,x 1 ,um))
Cum (P) is to have Fl''*'"(d)
gi for all
j E {1, ... , N} andt E R+Proof. From Lemma 2 and Theorem 1 we know that a necessary condition for x
Fit*,,um(, + oQ-1(P)), which can be expanded in the form
Cum(P) is H .
dim(0 2
)
dim(O)
-Hi. <
ShJ(#1(t*,
x 1, um)). (3.62)
k=1
If
dim(0 2
)
dim(O')
mi,
VtER+
l2 (02
Ci,kh(1(t , xuum))
2
+ Hi
> 0,
(3.63)
k=1
ViE{1,...,r}
then because of the strictly increasing property of the function -Ft,X,um (d) + Hi (the
expression inside the bracket) with respect to the disturbance input signal d, we have
Vt E R+, Vi E Il, ...
,r}, V d > j:
]
dim(0 2
)
dim(02)
2
c 1h (# (t,x , d))
chh1(#1(t, x 1, um)) + Hi
-
2
k=1
Therefore if we choose the pair (t*, i*) such that
> 0.
(3.64)
dim(O1)
(t*, i*) = arg
max
ci h1(O1(t,X1, uM))-
VtER+
ViE{ 1,...,r}
k=1
)
dim(0 2
S,1
C=1 1(02(t, X2,L+ O-Q-
(P))) - Hi
(3.65)
then a necessary condition for an initial condition x to be outside of the capture set
is i.*,Xm () = F.*,xUm(d)
-
Hi* < 0.
Another necessary condition for a x E X such that x 0
C um (P),
based on Lemma
1, is that for all j E {1, ... , N} and t E R+, we have G (#1(t, x 1, um)) < gi. According
39
to Lemma 1 and Lemma 2, this condition along with Ft*,x,m (d)
0 are also sufficient
conditions for a state x e X in order that x 0 Cum(P)-
Algorithm 1: Control Feedback Computation
+ -Q-o1(P) corresponding to P safety using Gaussian z-table [17].
2: For i E {1, ... , r} calculate F''x,um (d) = maxtER+ [Zdimo
=(
2
)
c11 hf(# 2 (t, x 2 , d))
Hi], where x
-
=
C;,khk(# (t, 1 , Ur))
-
1: Calculate d=
(x 1 , x 2 )T is the current state of the sys-
tem.
3: For j E {1, ... , N} calculate G = Gi(#1 (t, x, um)) - gi.
4: If maxiE{1,...,r} F
X
(d)
>
0 or there is a j E {1, ..., N} and there is a time
t E R+ such that Gj > 0 return u = um, otherwise u E [UM, UM].
The domain t E R+ in steps 2 and 4 of Algorithm 1 can be replaced with a finite
domain. This replacement will be helpful, in particular for termination of discrete
algorithm that we will introduce in Algorithm 2. Assumption 6 states the required
condition for this replacement.
Assumption 6. The functions f 1 (x', u) and f 2 (x2 , d) of the system model in Definition 7 are in the following form.
f 1(X1 , U) =
f1( X7d1xU)
if lh({)
o
if dhl(xl)
> 0
0
and
2
f (x , d)F (
O
and there is a finite time
T
2
if h 2 (x 2 ) > 0
,d)
- if Ah2 (x 2 )
0
E (0, oo) such that r = min{t E R+
I Ah1 (# 1 (t, X,
un)) =
0}.
Note that the functions f1 (x', u) and
f
2
(X 2
, d) of system model of the motivating
example, the scenario of two consecutive vehicles approaching an intersection, as
40
defined in (3.6), is in the above form. Moreover for a sufficiently small urn such that
for all t E R+, urn - Cvf(t) - a, - a, < 0, since Vf(t) =
iVU(t) = Ur
-
h 1((k(t, x 1 , urn)) and
CVo(t) - a, - a,, we can guarantee that r is a finite time.
Proposition 3. The interval t E R+ in the second and fourth steps of Algorithm 1
can be replaced with t E (0,r], where r = min{t E R+ | A h($1(t, x1 ,um)) = 01.
Proof. Note that based on the system model in Definition 7 along with Assumption
6, for all t > r we have t'(t) = 0 1(t, x 1 , um) = 0, therefore
C1-d hl(x(t)) - C2- h2(X2(t))
dt
dt
=
(3.66)
C1 JAi(X1 (t))'(t) - C2- h2(X2(t)) = -C2-h2(X2(t)),
dt
dt
. We
where Jhl(xl(t)) is the Jacobian of function h' defined as [Jhl(xl(t))]i, =
3
have that jh
2 (x 2
(t)) > 0 and also based on Assumption 4, C2 is a non-negative
matrix. This implies that for all t > r, we have
djt,,um
(d) =-C
2
h2 (2 (t))X2 0
where FtxU- (d) is as defined in (3.61). Therefore Pt,X,um (d) takes its global maximum
at a time t < -r and based on definition (3.61), that is what we are looking for to
check whether the state of the system is inside the capture set. Also we have
Vt > r, Vj E {1, ... , N}: dGi(xl(t)) = JGj(Xl(t))-' = 0,
dt
(3-67)
where JGi (xl(t)) is the Jacobian of function Gj. Equation (3.67) implies that the
function Gj, for all j E
{1, ...
, N}, takes its global maximum at a time smaller than
r. Therefore we can replace the interval t E R+ with t E (0, r].
In order to implement the algorithm on a computer, we need the discretized version
of Algorithm 1. We use the forward Euler approximation for discretization purpose.
We use At to denote the time step size. We also denote the state of the system at
step k by x[k] = (xl[k], x 2 [k])T. Therefore x(kAt) = x[k] where x(kAt) is the state
of the system in the continuous-time model. All other notations are similar to the
continuous-time model. The discrete-time model of the system defined in Definition
41
7, with Assumptions 1-6, is in the following form.
x[k + 1] = x[k] + At(f(x[k], u[k], d[kj)), where
f(x[k], u[k], d[k]) = (f'(xl[k], u[k]), f 2 (x2 [k], d[k]))
T
,
(3.68)
and
f1 (x', u)
=
f 2 (xf2(X22 , d)
d)
=
and
ff(xi,
u)
0
if h'(xl[k + 1]) - h1 (xl[k]) > 0
if h'(xl[k +1])- h1 (xl[k])
0
If
if h21(2[k +
+ 1]) - h 2(2[k])
[]> > 0
2
2
if h (X [k + 1]) - h 2 (x 2 [k])
0
M2,,d)d)
0
-
(369)
(-0
(3.70)
.
The discrete-time algorithm is in the following form:
Algorithm 2: Control Feedback Computation (discrete version)
1: Calculate d = p+uQ-1(P) corresponding to P safety, using Gaussian z-table [17].
2: For i E {1, ... , r} set Pf''X[01'U"' =
C(,hl(xl[0]) - EZim(,02 , c2h2(x
dim
2
[0])
-
H
and for j E {1, ... , N} set Gi = Gi(x 1 [0]) - gi, where x[0] = (Xl[0], X2[0])T is the
current state of the system. Set k = 0.
3: Calculate x 1 [k + 1] = x1[k] + At(f1(xl[k], u[0])).
4: While -h1 (x'[k])
=
dt
[1(xAlk+1
> 0,
At-h1(x1[k])
do:
4.1: Calculate x[k + 1] = x[k] + At(f(x[k], urn, j))
4.2: For i E {1, ..., r}, if
'"t
+1]) - Hi, then Pf't*'x[O],U' + E
=1< k k
c1=1 hI(x2[k
(01)Ci, hl (X1[k+ 1]) -
I
2)
c
2 (X2[k+1])-
Hi.
4.3: For j E {1, ... , N}, if G = Gi(xl[k +1]) - gi > 0, return u = um and STOP.
4.4: k +- k + 1
5: If maxiE{1,...,,1P x*l',xI'," > 0, return u = ur, otherwise u E U. STOP.
42
3.7
Simulations and Data Analysis
This section consists of two subsections. In Subsection 3.7.1, we provide details of
the experimental setup including measurements, data gathering, details of location
of experiments and etc. In Subsection 3.7.2 we provide details of data analysis and
consistency of experimental results with theoretical model.
3.7.1
Experimental Setup
In Figure 3.1 the path that is used for gathering data is depicted. This path is located
Figure 3.1: The path that is used for experiment.
in Ann Arbor, Michigan. The length of this path is 11 km and it consists of 30 study
areas. The study area is defined as an area, approximately 300-400 m of the road, that
the driver frequently reduces his/her speed in. A study area can be an intersection, a
speed bump, or any region that requires reduction in speed. The data of any of these
regions is used to construct the model. We use a computer software that after at least
five courses of the path depicted in Figure 3.1, learns and saves the study regions in
collaboration with GPS measurements, and then it ouputs the relative distance to the
43
stop sign St - xf and the target velcoity
VT,
whenever the system is inside a study
area.
We build a model for the disturbance input, the acceleration of PV, a,, as it
+
approaches an intersection, based on a linear function of the form ap(t) = ax,(t)
bvp(t) + d with d ~ N(p, a'). We measure xf from GPS. Vf, a, and a, are also
available from on-board sensors of FV. The value of a,, the deceleration due to rolling
resistance, is constant and the value of a,, the slope of the road, is the average slope
from the current position of FV until the stop sign. If we denote the difference
between altitude (height) of the,current position of FV and the position of the stop
sign by Ah, then a, is
a. = -mg sin arctan
St
-
Xf
),
where mg is the weight of the vehicle. The signals Xrel = x, - Xf and
(3.71)
Vrel = VP -
Vf
are obtained from the radar. The control input urn is provided through the automatic
brake in FV.
There are two options to calculate the parameters a, b, p and a. We can either
calculate these parameters from the data of FV (our vehicle) as it approaches an
intersection and then use this model as the acceleration of PV, or we can use the data
of radar, which provides the data of random preceding vehicles as they appear in
front of FV. The drawback of using the first set of data is that we have less variety of
drivers as opposed to the data of PV obtained directly from the radar which includes
more assortments of driving styles. But the advantage of using the data of FV (first
option) is that more data of FV are available and more importantly, the measurements
of signals of FV are less noisy and more accurate. Since eventually the signals of PV,
obtained from the radar, must be used for on-line computations, we will also compare
the model built based on the data of FV with the signals of PV. This comparison is
necessary, in particular, since the source of the two signals are different.
44
3.7.2
Experimental Results
Profiles of FV near the stop signs are depicted in Figure 3.2. From the plots we
can conclude that the assumption d
-
N(p, .2 ) (Assumption 3), the acceleration
profile with a constant time independent variance, is reasonable, since the interval of
observed accelerations (Figure 3.2-c) does not have a significant dependence on time.
I
I
I
I
*mama
*mowuipVm.Wip
(a)
(b)
(c)
Figure 3.2: Plots of profiles of the position, speed and acceleration versus time near
stop signs for FV. From Figure (3.2-c) we conclude that the assumption of constant
and time independent variance for the acceleration (Assumption 3) is reasonable,
while for the position (3.2-a) and the speed (3.2-b) it is not a reasonable assumption,
since the dependence of the interval of observed positions and velocities on time, is
more significant than the accelerations.
We have used the least square method to calculate the parameters a, b, p and o-.
If we discretize the state space model of PV defined in f 2 (x 2 , d) of equation (3.7), for
vp[k] > 0 we have
xp[k
+ 1]
vp[k + 1]
1
[
xp[k] + Atvp[k]
vp[k] + At(axp[k] + bv,[k] + d)
j
(3.72)
If we replace xp[k] in the second equation of (3.72) with xp[k - 1] + Atv,[k - 1], we
obtain
vp[k + 1]
=
a (Atxp[k - 1] + At 2 v [k - 1]) + (1 + bAt) vp[k] + dAt.
45
(3.73)
We define the new parameters a' = a, b' = 1+ bAt and d' = d. Minimizing the mean
square error for the speed leads to the following optimization problem:
minlICX - D1 2 , where X = (a', b', c)T,
(3.74)
and C[1, 1] = Atx,[1], C(1, 2) = v,[1], C(1, 3) = At, for k > 2 : C(k, 1) =
Atx,[k - 1] + At 2 v [k - 1], C(k, 2) = v,[k), C(k, 3) = At and D(k - 1)
=
vp[k].
The algorithm that we have implemented on the vehicle is as the following:
Algorithm 3: Collision Avoidance Feedback Map (discrete version)
1: Calculate d= I + aQ- 1(P) corresponding to P safety, using Gaussian z-table [17].
2: Set Pt*,xO],Um = xf[0] - x,[0] + 6 and set 6 = (x1 [0], vf[0})T
x[0] = (xf[0], Vf[0], x,[0], VP[0])T
-
(St, VT)T, where
is the current state of the system. Set k = 0.
3: Calculate (Xf[1], vf[1])T = (x[0] + Atvf [0], vf[0] + At(af [0]
-
CV2[0] - a, - a,))T,
where af[0] is the current acceleration of FV.
4: While vf[k] > 0, do:
4.1: Calculate (xf [k+1], vj [k+1])T = (Xf [k]+Atvf [k], vf [k+At(Um-CV [k] -a,a,))T and (xp[k+1],vp[k 1]) T = (x,[k]+Atv,[k],vp[k]+
At(axp[k]+bvp[k]+j))T.
4.2: If Pt*,xIO],Um < xf[k+1]-xp[k+1]+3,then
+- xf[k+ 1] -xp[k+1] +6.
Ft*,x[O],Um
4.3: If 0= (xf[k + 1], vf [k + 1])T - (St, vT) T > 0, return
4.4: k
+-
= um and STOP.
k+ 1
5: If Pt*,x[O],um
>
0, return u = um, otherwise u E U. STOP.
The data of FV as it approaches a stop sign is depicted in Figure 3.2. There
are 421 trajectories plotted in each sub-figure. We use these data and the optimization problem (.3.74) to construct the model for PV, which includes the calculation of
the parameters a, b, 1 and a.
In order to verify that Algorithm 3 is consistent with our expectation, meaning
that an algorithm designed based on a P% safety can save the vehicles from collisions
46
P% of time, we take advantage of the law of large numbers, according to which for
an event x with mean M we have
lim Xn
Pr(n-+oo
p
1 where Xn
number of times event x observed in n trials
n
(3.75)
We run an algorithm that generates an initial condition for FV and chooses a trajectory for PV among our available data, randomly, for n times. Based on the equation
(3.75) for a large n we expect to observe approximately (100 - P)n number of collisions. The logic diagram of the algorithm is demonstrated in Figure 3.3. The result
of running the algorithm for 10000 times is shown in Tables 3.1 and 3.2. In order to
analyze the effect of the length of study areas on the performance of the model, we
run the algorithm for different effective distances to stop signs, which we refer to as
the activation region. The activation region has the unit of length and corresponds
to the distance to the stop sign that the collision avoidance system is activated from.
We observe that changing the activation region does not affect the safety very much.
The average safety for all three cases, 70%, 80% and 90%, for the 100 m activation
region model is slightly larger than the corresponding safety level of 400 m activation
region.
Safety level
70%
80%
90%
Number of simulations
10000
10000
10000
Number of collisions
2777
1769
832
Empirical safety level
72.2%
82.3%
91.7%
Table 3.1: Result of running the algorithm for 10000 times with activation region
= 100 m, meaning that the control system is activated from 100 m before stop signs.
Safety level
70%
80%
90%
Number of simulations
10000
10000
10000
Number of collisions
2836
1992
1038
Empirical safety level
71.6%
80.1%
89.6%
Table 3.2: Result of running the algorithm for 10000 times with activation region
400 m, meaning that the control system is activated from 400 m before stop signs.
-
In order to verify that the parameters are not overfitted, we must check how a
47
N
Yes
I
Y
No
-
N=0, number of collisions
M=O, number of runs
n: Number of tests
igYs
Yes
No
Figure 3.3: The algorithm to verify safety of the system. The algorithm does not
count those randomly generated initial conditions that are inside the capture set,
since this is not consistent with x E W which according to Lemma 1 is equivalent to
x Cm (P). Also if the control input is never applied during a test, we exclude it,
since the performance of the control map is not reflected in such situations that we
do not use it.
model that is constructed based on a limited set of data will respond to a new dataset.
We take advantage of the k-fold cross validation method [18] for this purpose. In
particular we use 10-fold cross validation method. In this method we partition our
available data into 10 groups and solve the minimization problem of (3.74) for a pool
of data that consists of 9 groups and then run the algorithm of Figure 3.3 for the
data of the 10th group. We repeat this for all 10 groups and compare their average
safety level with the expected safety level as we have done in Tables 3.1 and 3.2. The
result is shown in Table 3.3.
Since we have built the disturbance model based on the data of FV, we need to
verify that it captures the behaviors of PV. Profiles of PV near stop signs are plotted
in Figure 3.4. In Tables 3.4 and 3.5 we have shown the result of using the disturbance
model constructed from the data of FV on the profiles of PV.
From Tables 3.4 and 3.5 we observe that the behavior of PV cannot be captured
by the model built based on the data of FV, for farther distances from stop signs.
The reason for this can be either lack of sufficient data of PV in farther distances from
stop signs or inability of the model to capture different behaviors of PV in farther
distances. If the reason for the undesired behavior of the model at farther distances is
48
Safety level
Group 1
Group 2
Group 3
Group 4
Group 5
Group 6
Group 7
Group 8
Group 9
Group 10
Average
70%
67.8%
69%
68.8%
72.1%
71.3%
69.8%
71.1%
68.3%
70.4%
70.3%
69.89%
80%
80.3%
79.8%
80.1%
81.3%
80.2%
78.3%
80.3%
79.5%
80%
80.3%
80.1%
90%
88.9%
90.3%
89.2%
92.1%
91.1%
90.5%
93%
92.1%
91.2%
88.9%
90.73%
Table 3.3: Result of running the algorithm for 10000 times for all of the 10 groups
within the available data. The activation region is 400 m, meaning that the control
system is activated from 400 m before stop signs. We observe that the average safety
level is close to the expected safety level.
I
I
a ap
(a)
I
(b)
(c)
Figure 3.4: Plots of profiles of position, speed and acceleration near stop signs for
PV.
Safety level
70%
80%
90%
Number of simulations
5000
5000
5000
Number of collisions
1493
1008
447
Empirical safety level
70.2%
79.8%
91.1%
Table 3.4: Result of running the algorithm for 5000 times for PV with the activation
region = 100 m, meaning that the control system is activated from 100 m before stop
signs.
insufficient data, we can tackle this problem by obtaining more data of PV, otherwise
the control system must be activated at a distance lower than 400 m from stop signs.
49
Safety level
70%
80%
90%
Number of simulations
5000
5000
5000
Number of collisions
1984
1530
1156
Empirical safety level
60.3%
69.4%
76.9%
Table 3.5: Result of running the algorithm for 5000 times for PV with the activation
region = 400 m, meaning that the control system is activated from 400 m before stop
signs.
Figure 3.5 shows the plots of the plane ap = axp + bvp + p for 100 mi and 400 m
activation region. The significant difference between the two planes indicates that the
linear function ap = axp + bvp + d cannot cover all different stopping behaviors for a
distance as large as 400 m. Therefore a smaller activation region must be considered
for implementation. The diversity of stopping profiles at larger distances to stop signs
in Figure 3.4-c also supports this fact.
position (i)d
Figure 3.5: The plots of the plane a,
region.
=
ax,+ by,+p
1 for 100 m and 400 m activation
We have implemented a hybrid control system on the vehicle that apart from Algorithm 3, it guides the driver to take the right control action using multiple levels
of warnings and different stopping profiles (other than U =Urn). An important difference between this case and Algorithm 3 is that we must also consider a reaction time
Menigthat both the system is activated from 100 m to the stop sign and the parameters are
calculated with the data of 100 m before the stop sign.
50
to issue warnings, which consequently leads to larger capture sets and earlier actions
(meaning that the warnings will be issued earlier than applying the automatic brake).
With this hybrid model we can decrease the frequencies of applying automatic brake
and let the driver collaborate with the control system through appropriate HMIs (Human Machine Interface).
The frequency of switches between u E U and u = urn (the driver input and the
automatic brake), in deterministic model is larger than the stochastic model, and
moreover in the stochastc model the average number of switches decreases by decreasing the safety level. Since we are interested in having a higher safety level, we
need to find a solution to the problem of frequent switches. We have used a hysteresis time for taking the control action. At any time that the system exits the capture
set, instead of setting the control input to u E U, we turn on a counter that keeps
the input u = ur, the automatic brake, on, until a certain interval of time is passed.
Figure 3.6 shows the control input for the stochastic model with 98.8% safety without
the hysteresis counter, and with the hysteresis counter, for the same initial condition
and profile of PV.
CL
0
5
10
15
20
time (sec)
25
0
30
10
15
20
time (sec)
2
30
Figure 3.6: The plots of the control input for the stochastic model with 98.8% safety
without the hysteresis counter (left), and with the hysteresis counter (right), for the
same initial condition and profile of PV.
We observe that using the hystersis time we could reduce number of switches significantly. In the case of Figure 3.6, we could reduce the number of switches from 46 to
8.
51
52
Chapter 4
Conclusions and Future Works
We have implemented Algorithm 3 on the vehicle and we have set the activation
region= 100 m. The main advantage of the model that we have introduced in this
work over the deterministic model is that it is more desirable for drivers, since it does
not take control action from a large distance to the preceding vehicle. Here the large
distance corresponds to a distance that is sufficiently larger than what drivers choose
to apply brake. The basis of this model is mitigation of collisions in contrast to the
deterministic model which is prevention of collisions.
The plots of Figure 3.5 imply that some modifications of the model is necessary
if we are interested in activating the system from a large distance to stop signs. One
modification is changing the disturbance function a, = ax , + by , + d, from a linear
model to a more sophisticated function. A piecewise linear function is a good option which can capture a variety of profiles that may not be captured using higher
order functions. Unfortunately, in general, the strict order preserving property of x,
with respect to the disturbance input d (Assumption 2) cannot be guaranteed with a
piecewise linear model. A part of the future works can be investigating the additional
constraints on the model that makes the piecewise linear function consistent with
Assumption 2, or looking into other possible profiles for acceleration of PV instead
of the linear function.
According to the model that we have considered in this work, future possible
profiles of PV can be uniquely determined of we know the current state of PV, its
53
position x, and velocity v,. In reality we know that there is also a third factor that
determines the future profiles of PV, and that is the state of the driver. Not all
drivers who are at St - x, distance to the stop sign and have the velocity vp follow
the same behaviors for t E R+. A part of future works can be adding an estimator
to the model that estimates current state of the driver of PV using the data of the
state of the system from a couple of seconds ago until the current time, and according
to that predicts the future profiles of PV. There are varieties of Bayesian estimators
available that can be employed for this purpose.
The study areas that we are considering in this work which we refer to as stop
signs, consist of any regions that drivers frequently reduces their speed. A useful
modification to the model is to branch the data based on the type of the stop sign.
For instance we can build separate models for turning right at intersections, turning
left at intersection, going straight at intersections, stopping at intersection, reducing
speed due to speed bump and etc. A necessary condition to employ this model is the
ability to communicate with stop signs through the available infrastructures.
A useful improvement is to solve Problems 1 and 2 for a system that two vehicles
are approaching intersections from two different branches of the road. This problem
has been solved for deterministic case in [19] and [20]. For this scenario, both the system model and the assumptions will be different from the scenario of two consecutive
vehicles approaching a stop sign.
54
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