Full-range Adaptive Cruise Control Based on

Full-range Adaptive Cruise Control Based on Supervised Adaptive Dynamic
ProgrammingI
Dongbin Zhaoa , Zhaohui Hua,b , Zhongpu Xiaa,∗, Cesare Alippia,c , Yuanheng Zhua , Ding Wanga
a State
Key Laboratory of Management and Control for Complex Systems, Institute of Automation, Chinese Academy of Sciences, Beijing 100190,
China
b Electric Power Research Institute of Guangdong Power Grid Corporation, Guangzhou 510080, China
c Dipartimento di Elettronica e Informazione, Politecnico, di Milano, 20133 Milano, Italy
Abstract
The paper proposes a Supervised Adaptive Dynamic Programming (SADP) algorithm for a full-range Adaptive Cruise
Control (ACC) system, which can be formulated as a dynamic programming problem with stochastic demands. The
suggested ACC system has been designed to allow the host vehicle to drive both in highways and in Stop and Go
(SG) urban scenarios. The ACC system can autonomously drive the host vehicle to a desired speed and/or a given
distance from the target vehicle in both operational cases. Traditional adaptive dynamic programming (ADP) is a
suitable tool to address the problem but training usually suffers from low convergence rates and hardly achieves an
effective controller. A supervised ADP algorithm which introduces the concept of Inducing Region is here introduced
to overcome such training drawbacks. The SADP algorithm performs very well in all simulation scenarios and always
better than more traditional controllers. The conclusion is that the proposed SADP algorithm is an effective control
methodology able to effectively address the full-range ACC problem.
Keywords: adaptive dynamic programming, supervised reinforcement learning, neural networks, adaptive cruise
control, stop and go
1. Introduction
Nowadays, driving safety and driver-assistance systems are of paramount importance: by implementing
these techniques accidents reduce and driving safety
significantly improves [1]. There are many applications
derived from this concept, e.g., Anti-lock Braking Systems (ABS), Electronic Braking Systems (EBS), Electronic Brake-force Distribution systems (EBD), TracI This work was supported partly by National Natural Science
Foundation of China under Grant Nos. 61273136, 61034002, and
60621001), Beijing Natural Science Foundation under Grant No.
4122083, and Visiting Professorship of Chinese Academy of Sciences.
∗ Corresponding author at: State Key Laboratory of Management and Control for Complex Systems, Institute of Automation,
Chinese Academy of Sciences, Beijing, 100190, PR China. Tel.:
+8613683277856, fax:8610-8261-9580.
Email addresses: dongbin.zhao@ia.ac.cn (Dongbin Zhao),
huzhaohui27@foxmail.com (Zhaohui Hu),
zhongpu.xia@gmail.com (Zhongpu Xia),
alippi@elet.polimi.it (Cesare Alippi), zyh7716155@163.com
(Yuanheng Zhu), ding.wang@ia.ac.cn (Ding Wang)
Preprint submitted to Neurocomputing
tion Control Systems (TCS), Electronic Stability Program (ESP) [1].
1.1. Adaptive cruise control
Adaptive cruise control is surely another issue going
in the direction of safe driving and, as such, of particular relevance. Nowadays, ACC is mounted in some
luxury vehicles to increase both comfort and safety [2].
The system differentiates from the Cruise Control (CC)
system mostly used in highway driving, which controls the throttle position to maintain the constant speed as
set by the driver (eventually adjusted manually to adapt
to environmental changes). However, the driver has always to brake when approaching the target vehicle proceeding at a lower speed. Differently, an ACC system
equipped with a proximity radar [3] or sensors detecting the distance and the relative speed between the host
vehicle and the one in front of it, proceeding in the same
lane (target vehicle), can operate either on brake or the
engine throttle valve to keep a safe distance.
As a consequence, the ACC does not only free the
driver from frequent accelerations and decelerations but
September 17, 2012
also reduce the stress of the driver as pointed out in [4].
Interestingly, [5] showed that if 25% vehicles driving in
a highway were equipped with the ACC system, congestions could be avoided. The ACC problem could be
solved by considering different techniques, e.g., a PID
controller [12], a fuzzy controller as pointed out in [11],
a sliding mode approach [9] or a neural network [18].
ACC systems suggested in the literature, and currently implemented in vehicles, work nicely at a vehicle
speed over 40 km/h and in highways [1], but always
fail at a lower speed hence requiring accelerations (action on the throttle) and decelerations (mostly breaking)
to keep a safe clearance to the target vehicle in urban
areas. In this case, the driving activity increases significantly, even more within an urban traffic with an obvious impact on fuel consumption and pollutant emissions. To address the problem the literature suggested solutions like stop and go, collision warning and collision
avoidance [22]. When the ACC and the SG solutions are
considered together, we speak about a full-range ACC.
A full-range ACC system with collision avoidance was
proposed in [16]. There, driving situations were classified in three control modes based on the warning index
and the time-to-collision: comfort, large deceleration
and severe braking. Three controllers were proposed
and combined to provide the ultimate control strategy.
[16] pointed out how the full-range ACC problem was a
nonlinear process requesting a nonlinear controller, for
instance designed with reinforcement learning.
policy could be gained. The results showed that such
shaping method could be used also in dynamic models
by dramatically shortening the learning time.
Our team applied the SRL control strategy to the ACC problem first in [14]. There, we showed that the speed
and the distance control had enough accuracy and was
robust with respect to different drivers [14]. However,
since the state and the action needed to be discretized,
there are some drawbacks. Firstly, the discretization of
the distance, speed, and acceleration, introduces some
fluctuations in the continuous control problem. Secondly, the higher number of discretized states cause the
larger state and the action spaces. As a consequence,
there always exists a conflict between control accuracy
and required training time.
For continuous reinforcement learning problem, ADP was proposed in [8, 25] with neural networks mapping
the relationships between states and actions, and the relationships between states, actions and performance index. More in detail, the algorithm uses a single step
computation of the neural network to approximate the
performance index which will be obtained by iterating
the dynamic programming algorithm. The method provides us with a feasible and effective way to address
many optimal control problems; examples can be found
in the cart-pole control [13, 20], pendulum robot upswing control [26], urban intersection traffic signal control [15], freeway ramp metering [6, 27], play Go-Moku
[28], and so on. However, the learning inefficiency of RL is also inherited in ADP but can also be remedied with
a supervisor to formulate SADP.
1.2. Reinforcement learning and adaptive dynamic programming
Reinforcement Learning (RL) [21] is suited for the
ACC problem, because it can grant quasi-optimal control performance through a trial and error mechanism
in a changing environment. However, the convergence
rate of RL might be a problem [23] also leading to some
inefficiency. Most of the time, the agent (the software
implementing the controller) will learn the optimal policy after a relatively long training, especially when the
model is characterized by a large state space. This inefficiency can be fatal in some real time control systems.
Supervised Reinforcement Learning (SRL) can be introduced to mitigate the RL problem, by combining Supervised Learning (SL) and RL and, hence, taking advantage of both algorithms. Pioneering work has been
done in Rosenstein and Barto’s [7, 19] where SRL was
applied to solve the ship steering task and the manipulator control and the peg insertion task. All results
clearly showed how SRL outperforms RL. In [17], a
potential function was introduced to construct the shaping reward function; they proved that an optimal control
1.3. The idea
In this paper we propose a novel effective SADP algorithm able to deal with the full-range ACC problem.
The considered framework is as follows:
(1) There are two neural networks in SADP, the Action and the Critic networks. The Action network is
used to map the continuous state space to the control signal; the Critic network is used to evaluate
the goodness of the action signals generated by the
Action network and provides advice while training
both networks. In this way we avoid the curse of dimensionality caused by the large dimension of the
discrete state-action pairs.
(2) The supervisor can always provide information for
RL, hence speeding up the learning process.
In this paper, the ACC problem is described as a
Markov decision process. The main contributions are
as follows:
2
(1) A simple single neural network controller is proposed and optimized to solve the full-range adaptive
cruise control problem.
(2) An inducing region scheme is introduced as a supervisor, which is combined with ADP, provides an
effective learning algorithm.
(3) An extensive experimental campaign is provided to
show the effectiveness and robustness of the proposed algorithm.
Absolute speed: v H
Desired distance: d r
Radar distance: d d
Host speed:
vT
Bottom controller
Brake
Throttle
Actual acceleration
Host vehicle
Target vehicle
Figure 1: The SADP framework for the full-range ACC. The radar
detects the distance between the two vehicles and the target vehicle’s
speed. The host vehicle speed and the current acceleration come from
the mounted sensors. The upper controller generates the desired acceleration signal by combining the relative speed and the relative distance information. The bottom controller maps the acceleration to the
brake or the throttle control signals.
The bottom controller manages both the throttle and
the brake. A fuzzy gain scheduling scheme based on
a PID control is used to control the throttle. A hybrid
feed-forward & feedback control is applied to control
the brake. The throttle and the brake controllers are
coordinated by use of a proper switch logic. The control actions transfer the desired acceleration signal to the
corresponding throttle position or braking strength [10].
2.2. The driving habit function
As previously discussed, different drivers have different driving habits: an intelligent ACC controller should
learn the driving habit [29]. The host speed vH (t), the
desired distance d0 between the motionless host and target vehicles and the headway time index τ is adopted to
characterize the driving habit
dd (t) = d0 + vH (t)τ
(3)
It comes out that the headway time is high for conservative drivers, and low for sportive drivers.
(1)
Similarly, the relative distance ∆d(t) at step t is
∆d(t) = dr (t) − dd (t).
Target speed:
Hybrid feedback controller
2.1. The ACC model
The ACC model is shown in Figure 1 with the nomenclature give in Table 1.
During driving, the ACC system assists (or replaces)
the driver to control the host vehicle. In other words, ACC will control the throttle and the brake to drive
the vehicle safely despite the uncertainty scenarios we
might encounter. More in detail, there are two controllers in the ACC system: the upper and the bottom
ones. The upper controller generates the desired acceleration control signal according to the current driving
profile; the bottom controller transfers the desired acceleration signal to the brake or the throttle control action
according to the current acceleration of the host vehicle.
Denote as dr (t) the distance at step t between the host
and the target vehicles. Such a distance can be detected
by radar or other sensing devices, and it is used to compute the instant speed of the target vehicle vT (t) (refer to
Figure 1); the desired distance dd (t) between these vehicles is always set by the driver while the host vehicle
speed vH (t) can be read from the speed encoder.
The control goal is to keep the host vehicle within
a safety distance and maintain the safe relative speed
∆v(t)
∆v(t) = v (t) − v (t).
v
H
Desired acceleration
2. The adaptive cruise control
T
Driving habit
SADP
The paper is organized as follows. Section 2 formalizes the full-range ACC problem. Section 3 proposes the SADP algorithm based on the Inducing Region
concept and presents design details. Section 4 provides
experimental results based on typical driving scenarios.
Section 5 summarizes the paper.
H
Upper controller
2.3. Driving scenarios
In a full-range ACC the host vehicle driving conditions can be cast into five scenarios, as shown in Figure
2.
(2)
The upper controller goal is to simultaneously drive
variables (∆v(t), ∆d(t)) to zero by enforcing the most appropriate acceleration control action, more in detail, by
taking into account the different driving habits.
(1) The CC scenario: the host vehicle travels at a constant speed without any target vehicle in front of it.
3
Table 1: ACC nomenclatures
Parameter
vH (t)
vT (t)
dr (t)
dd (t)
∆v(t)
∆d(t)
d H (∆t)
∆dg (t)
∆vg (t)
d0
τ
Description
The speed of the host at step t
The speed of the target at step t
The distance between the host and the target vehicles at step t
The distance the host driver desires to maintain at step t
The relative speed at step t
The relative distance at step t
The distance the host vehicle travels in time interval ∆t
The maximum tolerable relative distance at step t
The maximum tolerable relative speed at step t
The zero-speed clearance between the two vehicles
The headway time
(2) The ACC scenario: both the target and host vehicles
are running at high speed and the host vehicle needs
to keep pace with the target vehicle or slow down to
keep a safe distance to a slower forerunner.
(3) The SG scenario: this case simulates the frequent
stop and go situations of the city traffic. The target
vehicle stops at first, then moves again; this profile
repeats frequently.
(4) The emergency braking scenario: the target vehicle
stops suddenly with a large abnormal deceleration,
the host vehicle must take an prompt braking action.
(5) The cut-in scenario: while the host car is operating
in a normal ACC or SG mode, another vehicle interferes with it. More in detail, the third vehicle,
coming from the neighboring lane, enters a position
between the host and the target vehicles. The entering vehicle becomes the new target vehicle.
3. The SADP control strategy
3.1. The ADP framework
The structure of the SADP system is shown in Figure
3. The system includes a basic ADP and a supervisor
(blue shadowed line). The Action and the Critic neural
networks are present to generate the ADP framework.
We recall that the Action network is used to model the
relationship between the state and the control signal. Instead, the Critic network is used to evaluate the performance of the control signal as coming from the Action
network. The Plant responds to the action and presents
new state to the agent; afterwards, the reward is given.
The dash lines represent the training process involving
the two neural networks. Some major notations are listed in Table 2.
The training process can be summarized by the following procedure: At first, the agent takes action u(t)
Figure 2: Different driving scenarios for the full-range ACC.
4
Shaping
Table 2: SADP nomenclatures
Parameter
x(t)
u(t)
r(t)
J(t)
R(t)
Uc (t)
γ
Nah
Nch
Ea (t)
Ec (t)
wa (t)
wc (t)
la (t)
lc (t)
Description
The current state
The control signal
The reward
The Critic network output
The return or the rewards-to-go
The desired objective
The discount factor
Number of hidden nodes, Action network
Number of hidden nodes, Critic network
Objective training function, Action network
Objective training function, Critic network
Weights matrix, Action network
Weights matrix, Critic network
Learning rate, Action network
Learning rate, Critic network
following the input state x(t) according to the Action
network indication; the plant moves then to the next state x(t+1) and the environment gives the agent a reward
r(t); then the Critic network output J(t) provides an approximate performance index (or return); the Critic and
the Action networks are then trained with error backpropagation based on the obtained reward [25]. These
procedures iterate until the networks weights converging.
The ADP control strategy is stronger than a procedure
based solely on RL. In fact, ADP possesses the common basic features of RL: state, action, transition, and
reward. However, in ADP the state and the action are
continuous values rather than discrete, and the method
used to gain the action and the state values is rather different.
=
+ J(t-1)
Supervisor
Z-1
Exploration
Inducing Region
x(t)
Nominal
Control
Critic
u(t)
Action
J(t)
x(t+1)
Plant
Z-1
Figure 3: The schematic diagram of the SADP framework: The Action network is used to generate the control signal; the Critic network
is used to evaluate the goodness of the control signal as generated by
the Action network. The dash lines represent the training of those neural networks. There are three types of supervisors: shaping, nominal
control, and exploration.
state value function V(s) or the state-action value function Q(s, a) is used to estimate R(t). The final goal is to
have a converged look-up Q-table in Q-learning [24]
Q(s, u) = Q(s, u)+α[r(t)+γmaxu′ Q(s′ , u′ )−Q(s, u)].(5)
where α is the step size parameter, u and s is the current
action and state, u′ and s′ is the next action and state,
respectively.
There are many strategies for action selection, e.g.,
those based on the Boltzmann action selection strategy,
the Softmax strategy and epsilon greedy strategy [21].
In ADP, the Critic network output J(t) is used to approximate the state-action value function Q(s, a). The
Critic network embeds the gained experience (through
trial and error) in the weights of the neural networks instead of relying on a look-up Q-table.
The definition of reward is somehow a tricky concept,
as it happens with human learning. A wrong definition
of reward will lead, with a high probability, to scarce
learning results.
3.1.1. The reward and the return
The return R(t), defined as “how good the situation
is”, is defined as the cumulated discounted rewards-togo
R(t) =
_
+
r(t)
r(t + 1) + γr(t + 2) + γ2 r(t + 3) + · · ·
T
∑
γk r(t + k + 1)
(4)
3.1.2. The Action network
The structures of the Action and the Critic networks are shown in Figure 4. Based on [20], simple three
layered feed-forward neural networks with hyperbolic
tangent activation function
k=0
where 0 ≤ γ ≤ 1 represents the discount factor, t the
step, r(t) the gained reward and T the terminal step.
The higher the cumulated discounted future rewardsto-go is, the better the agent performs. However, the
above definition needs the forward-in-time computation, hardly available. Therefore, in discrete RL, the
T h(y) =
1 − exp(−y)
1 + exp(−y)
is considered to solve the full-range ACC problem.
5
wa(1)
wa(2)
x1
wc(1)
pi (t) =
wc(2)
qi (t) =
x1
u
x2
Critic Network
Figure 4: The structure of the Action and the Critic networks. The
Action network has two inputs, namely, the relative distance and the
relative speed; the output is the acceleration control signal. The Critic
network has three inputs: the acceleration control signal, the relative
distance and the relative speed; its output is the rewarding value J(t).
Ec (t) =
ec (t) =
m(t) =
hi (t) =
T h(hi (t)), i = 1, 2, · · · , Nah ,
2
∑
w(1)
ai j (t)x j (t), i = 1, 2, · · · , Nah ,
(6)
(7)
wc (t + 1) =
(8)
∆wc (t) =
(9)
ea (t) =
∆wa (t) =
wa (t) + ∆wa (t),
∂Ea (t) ∂J(t) ∂u(t)
−la (t)
,
∂J(t) ∂u(t) ∂wa (t)
(10)
(11)
(12)
(13)
3.1.3. The Critic network
The network receives as inputs both the state and the
control signal, and outputs the estimated return J(t),
Nch
∑
w(2)
ci (t)pi (t),
(21)
As mentioned above, ADP proposes a simple, feasible, and effective solution for the RL problem with continuous states and actions. Higher storage demand for
the Q-table in Q-learning can be avoided and the “curse
of dimensionality” problem in Dynamic Programming
(DP) can be solved with a single step computation by
using the above equations.
However, there are still some problems to be solved
with ADP. The first is associated with the choice of the
initial values of the network weights. Inappropriate configurations lead to poor Action and Critic networks (and
then it becomes interesting to know how likely we will
end in a good performing algorithm).
The second comes from Uc . This reward value is critical to the training phase. Usually, the reward is set 0
for encouragement and -1 for punishment and the return R(t) is zero if the action is an optimal one. Hence
the output J(t) of the Critic network converges to 0 if
optimal actions are always taken (and the induced value of Uc is 0). But in some complex cases a continuous reward would be a better choice. With error back
propagation, a large discrepancy on Uc might lead to a
where la (t) is the learning rate for the Action network.
J(t) =
(20)
3.2. The disadvantages of the ADP
where Uc is the desired objective.
Training is performed with error back propagation
wa (t + 1) =
wc (t) + ∆wc (t),
∂Ec (t) ∂J(t)
−lc (t)
,
∂J(t) ∂wc (t)
where lc (t) is the learning rate for the Critic network.
where Nah is the number of neurons in the hidden layer,
w(1)
ai j is the generic input weight of the Action network
and w(2)
ai is the generic output weight.
The Action network is trained to minimize the objective function
1 2
e (t),
2 a
J(t) − Uc ,
(18)
which is the same of return R(t). Therefore, the convergence of the Critic network output J(t) can be used to
evaluate the goodness of the control signal.
Again, training is modeled as
j=1
Ea (t) =
(17)
J(t − 1) = r(t) + γr(t + 1) + γ2 r(t + 2) + · · · , (19)
i=1
gi (t) =
1 2
e (t),
2 c
γJ(t) − J(t − 1) + r(t).
When the objective function Ec (t) approaches zero,
J(t − 1) can be derived from Eq. (18) as
The Action network’s input is state x(t) =
(∆d(t), ∆v(t)). The output is u(t) which can be derived
from
T h(m(t)),
Nah
∑
w(2)
ai (t)gi (t),
(16)
where Nch is the number of neurons in the hidden layer,
w(1)
ci j is the generic input weight of the Critic network to
be learned, and w(2)
ci the generic output weight.
The Critic network is trained by minimizing the objective function
x3 = u
u(t) =
(15)
j=1
J
x2
Action Network
T h(qi (t)), i = 1, 2, · · · , Nch ,
3
∑
w(1)
ci j (t)x j (t), i = 1, 2, · · · , Nch ,
(14)
i=1
6
large training error which will affect negatively the performance of the controller.
The above problems can be solved if we consider a
supervisor to guide the learning process.
3.4. SADP for the full-range ACC
3.3. The supervisor: Inducing Region
3.4.1. The state
The relative speed ∆v(t) and the relative distance
∆d(t) are the state variables x(t) = (∆v(t), ∆d(t)). The
aim of the full-range ACC is to achieve the final goal state with the minimum amount of time and an Inducing
Region characterized as
{
|∆v(t)| < 0.072 km/h
,
(23)
|∆d(t)| < 0.2 m
There are five components in the SADP framework:
the state, the action, the state transmission matrix, the
reward and the supervisor.
As shown in Figure 3, SADP combines the structure
of ADP and SL. Therefore, the agent learns from the interaction with the environment as well as benefits from
a feedback coming from the supervisor.
There are three ways to implement the supervisor in
SADP [7]: (1) shaping: the supervisor gives additional
reward, hence simplifying the learning process for the agent; (2) nominal control: the supervisor gives additional direct control signal to the agent; (3) exploration: the
supervisor gives hints that indicate which action should
be taken.
The exploration way gives the smallest supervisor information and is adopted here. Since the goal of the
control system is to drive the relative speed and the relative distance to zero, the desired target requires that
both v(t) and d(t) satisfy
{
|∆v(t)| < ϵv
,
|∆d(t)| < ϵd
Besides the goal state, a special “bump” state is introduced and reached when the host vehicle collides with
the target one, namely,
∆d(t) + dd (t) < 0.
(24)
3.4.2. Acceleration: the control variable
The full-range ACC problem can be intended as mapping different states to corresponding actions. Here, the
action is the acceleration of the host vehicle. In view
of the comfort of the driver and passengers, the acceleration should be bounded in the [−2, 2] m/s2 interval
in normal driving conditions, and to [−8, −2] m/s2 in
severe and emergency braking situations [16]. It is required to transfer u(t) which is within the [−1, 1] range
into the range [−8, 2] m/s2 , namely,
{
|amin | · u
u < amax /|amin |
a=
,
(25)
amax
u ≥ amax /|amin |
(22)
where ϵv and ϵd are feasible tolerable small positive values for ∆v(t) and ∆d(t), respectively.
The aim of the full-range ACC is to satisfy the above
inequalities or “goal state” as soon as possible (promptness in action) and stay there during the operational
driving of the vehicle. However, at the beginning, the
agent is far away from the goal state, especially when
no priors are available. If the goal state region is too small the agent will always be penalized during learning
and the training process will hardly converge. Even if
a high number of training episodes are given there it is
not guaranteed that the ADP will learn an effective control strategy. On the contrary, if the goal state area is
too large, then the learning task might converge at the
expenses of a poor control performance.
It makes sense to have a large goal state area at the beginning to ease the agent entering into a feasible region
and reduce gradually afterwards the area, as learning
proceeds, to drive the learning towards the desired final state configuration. In other terms, it means that the
supervisor will guide the agent towards its goal through
a rewarding mechanism. This concept is at the base of
the Inducing Region where ϵv and ϵd evolve with time.
where amin is −8 m/s2 and amax is 2 m/s2 here.
3.4.3. The state transition
When the vehicle is in state x(t) = (∆v(t), ∆d(t)), and
takes action a = aH , the next state x(t + 1) is updated as



















vH (t + 1) = vH (t) + aH (t)∆t
d H (∆t) = vH (t) + aH ∆t2 /2
∆v(t + 1) = vH (t + 1) − vT (t + 1)
,
∆d(t + 1) = ∆d(t) − (d H (∆t) − (vT (t)
+vT (t + 1))∆t/2)
(26)
where ∆t represents the sampling time. It can be seen
that the next state x(t + 1) cannot be computed after taking an action, since the target speed of the next step or
the acceleration is unknown.
7
3.4.4. The reward
The reward is 0 when the agent reaches the goal state,
-2 when it reaches the bump state, -1 otherwise. The reward provides an encouragement for achieving the goal,
heavy penalty for collision, and a slight punishment for
having not reached the target state.
When the performance is satisfied, P s (S ) assumes value
1, 0 otherwise.
As SADP learns through a trial and error mechanism, it will explore exhaustively the state space provided
that the number of experiments is large enough. At the
end of the training process we can then test whether the
performance of the full-range ACC system P s (S ) is 1
or not. Of course, the performance satisfaction criterion
must be evaluated on a significant test set containing all
those operational modalities the host vehicle might encounter during its driving life. It is implicit that if the
full-range ACC system satisfies the performance satisfaction criterion it is also stable. The opposite does not
necessarily hold.
We observe that the unique randomization in the
training phase is associated with the process providing
the initial values for the network weights. Afterwards, SADP is a deterministic process that, given the same
initial configuration of weights and the fixed training data (experiment) provides the same final networks (not
necessarily satisfying the performance criterion).
Train now a generic system S i and compute the indicator function Id (S i ) defined as
3.4.5. Inducing Region
The updating rule for the Inducing Region is given by


∆dg (t) = ∆dg (t − 1) − Cd ,





0.2 < ∆dg (t) < ∆dg (0);





 ∆dg (t) = 0.2, 0.2 ≥ ∆dg (t);
.
(27)



∆vg (t) = ∆vg (t − 1) − Cv ,





0.072 < ∆vg (t) < ∆vg (0);



 ∆v (t) = 0.072, 0.072 ≥ ∆v (t);
g
g
where the ∆dg (t) and ∆vg (t) characterize the goal state
area for ∆d(t) and ∆v(t), respectively. Cd and Cv are
the constant shrinking length at each step for the goal
distance and the goal speed, set to 0.3 m and 0.36 km/h.
∆d(0) and ∆v(0) are the initial goal state ranges, set to
18 m and 18 km/h, respectively. As presented above,
the goal state area gradually shrinks to guide the Action
network towards the final goal.
{
Id (S i ) =
3.5. Learnability vs. stability
1,
0,
i f P s (S i ) = 1
.
otherwise
(28)
In fact, the indicator function Id (S i ) = P s (S i ) and
states whether the generic system satisfies the performance criterion or not for the full-range ACC for the
i-th training process.
Let ρ be the probability that a trained system S satisfies the performance criterion for the full-range ACC. ρ
is unknown but can be estimated with a randomization
process as suggested in [30, 31]. More specifically, we
can evaluate the estimate ρ̂N of ρ by drawing N initial
configurations for the Action and Critic networks, hence
leading to the N systems
It is very hard to prove the stability of the suggested
full-range ACC in a close form. However, we can make
a strong statement in probability by inspecting the learnability properties of the suggested full-range ACC problem. Since the suggested SADP algorithm is Lebesgue
measurable with respect to the weight spaces of the action and critical networks we can use Randomized Algorithms [30, 31] to assess the difficult of learning problem.
To do this, we define at first the “performance satisfaction” criterion P s (S ) and say that the performance
provided by the full-range ACC system S is satisfying
when:
ρ̂N =
(1) convergence: the Action and Critic networks converge, i.e., they reach a fixed configuration for the
weights at the end of the training process.
(2) comfortable: the acceleration of the host vehicle is
mostly within [−2, 2]m/s2 range and comes out of
that range only in emergency braking situations.
(3) accurate: the suggested full-range ACC system can
effectively control the host vehicle to achieve the
final goal state defined in Eq.(23) and, then, stay
there.
N
1 ∑
Id (S i ).
N i=1
(29)
To be able to estimate ρ we wish the discrepancy between ρ̂N and ρ to be small, say below a positive ε value, i.e., |ρ − p̂N | < ε. However, the satisfaction of the
inequality is a random variable, which depends on the
particular realization of the N systems. We then request
the inequality to be satisfied with high confidence and
require
Pr(|ρ − p̂N | < ε) ≥ 1 − δ,
8
(30)
where 1 − δ represents the confidence value. The
above equation holds for any value of ε and δ provided that N satisfies the Chernoff’s bound [32].
N≥
ln ( 2δ )
2ε2
(1) The host speed and the initial distance between the
two vehicles are 90 km/h and 60 m, respectively.
The target speed is 72 km/h and fixed in time interval [0, 90) s;
(2) The target speed then increases to 90 km/h in time
interval [90, 100) s with fixed acceleration;
(3) The target maintains the speed at 90 km/h in the
time interval [100, 150) s.
(31)
If we now select a high confidence, say 1 − δ then,
with probability at least 1 − δ inequality |ρ − p̂N | < ε
holds. In turn, that means that the unknown probability
ρ is bounded as
p̂N − ε ≤ ρ ≤ 1
In this case, ∆v(0) = 18 km/h and ∆d(0) = 18.36 m,
hence the agent starts from the initial state x(0) =
(18, 18.36), takes continuous action at each time instance and either ends in the bump state or in the goal
state. We have seen that if a collision occurs, a heavy
penalty is given and the training episode will restart.
Although the agent is trained in a simple scenario, the
training process is not trivial. SADP, through trial and
error, will force the agent to undergo many different states. The training phase is then exhaustive and the
trained SADP controller shows a good generalization
performance.
For comparison we also carried out training experiments with ADP which has the same final goal as SADP. The training episodes are increased until 3000 to give
the agent more time to learn. Table 3 shows the performance comparison between SADP and ADP. We say
that one experiment is successful when both the Action
and Critic networks weights keep fixed for the last 300
episodes and the performance of the system evaluated
on the test set satisfies the performance criterion defined
in section 3.5. As expected, the presence of the supervisor guarantees the training process convergence so that
the full-range ACC is always achieved.
(32)
Eq (32) must then be intended as follow: designed a
generic system S with the SADP method and the above
hypotheses, the system will satisfy the performance satisfaction criterion with at least probability p̂N − ε; the
statement holds with confidence 1 − δ. In other terms,
if p̂N assumes high values the learnability for a generic
system is granted with high probability and, as a consequence, the stability for the system satisfying the performance criterion is implicitly granted as well.
4. Experimental results
4.1. Longitudinal vehicle dynamic model
We adopt the complete all-wheel-drive vehicle
model present in the SimDriveline software of
Simulink/Matlab. The vehicle model is shown in Figure
5. It combines the Gasoline Engine, the Torque Convertor, the Differential, the Tire, the Longitudinal Vehicle
Dynamics and the Brake blocks. The throttle position,
the brake pressure and road slope act as input signals,
the acceleration and the velocity as output signals. Such
a model has been used to validate the performance of
the suggested controllers [10].
Table 3: Convergence comparison between SADP and ADP
SADP
ADP
ADP
4.2. Training process
In the SADP model, the discount factor γ is 0.9,
the initial learning rates for the Action and the Critic networks are set to 0.3, and decrease to 0.001 by
0.05 at each step. Both the Action and the Critic networks are three layered feed-forward neural networks
with 8 hidden neurons. The network weights are randomly generated initially, to test the SADP learning
efficiency and drawn from section 3.5. Here, we set
τ = 2 s, d0 = 1.64 m and ∆t = 1 s.
An experiment, e.g., a full training of a controller requires presentation of the same episode (training profile) 1000 times. Each episode is as follows:
Training
episodes
1000
1000
3000
Number of
experiments
1000
1000
1000
Number of
success
999
0
0
Analyzing the only one failed experiment from SADP, we obtain that the Action and Critic networks weights
keep fixed for the last 224 episodes. If the number of
episodes defining the success is smaller, e.g., 200, then
this experiment can also be thought of as a success one.
4.3. Generalization test with different scenarios
The effectiveness of the obtained SADP control strategy is tested in the driving scenarios of Section 2.
The driving habit parameters are changed as follows:
9
Figure 5: Longitudinal vehicle dynamic model suggested within Matlab/Simulink [10].
τ = 1.25 s and d0 = 4.3 m. Here, the CC scenario is
omitted for its simplicity. The test scenarios include the
normal ACC driving scenario, the SG scenario, the
emergency braking scenario, the cut-in scenario and
the changing driving habit scenario.
[16] proposed three different control strategies for the
full-range ACC problem, namely, the safe, the warning,
and the dangerous modes as the function of the warning
index and the time-to-collision. The outcome controller
provides an effective control strategy that we consider
here for comparison.
In this paper, only a single trained nonlinear controller is used to deal with the full-range ACC problem.
ates to a full stop. Results are shown in Figure 7. We
appreciate the fact that the host vehicle performs well
both in distance and speed control. In the first 10 s, the
host vehicle decelerates to a stop, then the host vehicle accelerates (constant acceleration) until time 80 s.
Afterwards, it keeps a constant speed for a period and,
finally, goes to a full stop. As in the case of the normal ACC scenario, the mixed control strategy [16] and
SADP both provide near-optimal control performance,
indicating the good learning ability of SADP.
4.3.3. The emergency braking scenario
This scenario is designed to test the control performance under extreme conditions to ensure that driving
safety is achieved. The target vehicle brakes suddenly
at time instant 60 s and passes from 80 km/h to 0 km/h
in 5 s.
Figure 8 shows the experimental results, clearly indicating that both the methods stop the vehicle successfully with similar clearances to the target vehicle, but the
SADP control strategy outperforms the mixed control strategy [16] with a smoother acceleration (e.g., see the
deceleration peak requested by the mixed approach). In
[16] the control signal was a combination of two control
strategies; as such it introduced frequent spikes in the
acceleration signal when prompt actions were requested.
4.3.1. The normal ACC scenario
The target vehicle runs with varied speeds and the
host vehicle has to either keep a safe distance or a relative speed with respect to the target.
Results are shown in Figure 6. We comment that
speed and distance requests are nicely satisfied. Moreover, the requested acceleration is more than acceptable.
More in detail, at time 20 s the host vehicle reaches the
goal state, and stays there. Whenever the target vehicle
slows down or increases its speed, the host vehicle reacts to the change by imposing the corresponding acceleration action. The normal ACC problem can be thought
as a linear process, while the mixed control strategy [16]
provides a near-optimal control. Experiments show that
the obtained SADP behaves as well as the mixed control
strategy.
4.3.4. The cut-in scenario
The host and target vehicles proceed at high speed, A
vehicle from the neighboring lane interferes and inserts
between the target and the host vehicle, which needs
to the host one brake. The distance to the new target
vehicle abruptly reduces up to 50%.
4.3.2. The SG scenario
Starting from 20 km/h the target vehicle accelerates
to reach a speed of about 40 km/h and, then, deceler10
80
Distance [m]
70
Distance [m]
25
SADP
Mixed [9]
Desired distance
60
50
40
30
20
0
25
50
75
100
125
150
175
20
15
5
0
0
200
SADP
Hybrid PD
Desired Distance
10
25
50
75
Time [s]
100
80
60
25
50
75
100
125
150
175
175
200
SADP
Hybrid PD
Target Velocity
20
10
25
50
75
100
125
2
0
−1
−2
100
150
30
0.5
Acceleration [m/s ]
2
Acceleration [m/s ]
1
75
200
40
0
0
200
SADP
Mixed [9]
50
175
Time [s]
2
25
150
50
Time [s]
−3
0
125
60
SADP
Mixed [9]
Target speed
Velocity [km/h]
Speed [km/h]
120
40
0
100
Time [s]
125
150
175
200
Time [s]
0
−0.5
−1
SADP
Hybrid PD
−1.5
−2
0
25
50
75
100
125
150
175
Time [s]
Figure 6: Experimental results with SADP and the mixed control strategies in the normal ACC scenario: (a) distance; (b) speed; (c) acceleration.
Figure 7: Experimental results with SADP and the mixed control strategies. SG scenario: (a) distance; (b) speed; (c) acceleration.
performances. In the following we consider sensing uncertainties by adding noise to the real values. Figure 10
shows an emergency braking situation. A random 2%
in magnitude uniform noise is added to the target speed.
Since the relative distance is derived from the speed uncertainty propagates. We see that SADP outperforms
the mixed control strategy [16], with a higher accuracy in the distance control and a smoother acceleration
requirements. We verified that SADP provides satisfactory performances when the noise increases up to 5% in
magnitude.
Other uncertainties may include the changing load of
the vehicle and the friction between the vehicle and the
road. They can be solved with the aforementioned bottom controller.
Figure 9 shows that both algorithms perform well. Since there is a significant reduction in the safety distance, the host brakes to avoid the crash. This is a normal action in current ACC systems. In our algorithm,
small driving habit parameters must be set to emulate
the behavior of a sportive driver, which might leave a
very small and safety distance for the neighboring vehicle to cut in.
4.3.5. The changing driving habit scenario
The above four scenarios are set with parameters
d0 = 4.3 m and τ = 1.25 s. In practical implementations, there could be several driving habits for the human
driver to choose from. We verify the proposed algorithm and it always meets the driver expectation.
4.5. Discussions
We can conclude that the SADP control strategy is
robust and effective in different driving scenarios.
Furthermore, the changing driving habit scenario immediately shows the generalization performance of the
4.4. Robustness
In real vehicles, measurement errors introduce uncertainties on the relative distance and the relative speed
measurements. Such uncertainties affect the controller
11
200
100
60
40
20
0
0
25
50
75
100
125
150
175
SADP
Mixed [9]
Desired Distance
80
Distance (m)
80
Distance [m]
100
SADP
Mixed [9]
Desired distance
60
40
20
0
0
200
25
50
75
100
50
25
50
75
100
125
150
175
175
200
SADP
Mixed [9]
Target Velocity
80
60
40
20
0
200
25
50
75
100
125
150
175
200
Time (s)
2
2
Acceleration (m/s )
1
2
Acceleration [m/s ]
150
100
Time [s]
0
SADP
Mixed
−2
−4
−6
0
125
120
SADP
Mixed [9]
Target speed
Speed (km/h)
Speed [km/h]
150
0
0
100
Time (s)
Time [s]
25
50
75
100
125
150
175
200
0
−1
−2
−3
SADP
Mixed [9]
−4
−5
0
25
50
75
100
125
150
175
Time (s)
Time [s]
Figure 8: Experimental results with SADP and the mixed control strategies. The emergency braking scenario: (a) distance; (b) speed;
(c) acceleration.
Figure 9: Experimental results with SADP and the mixed control strategies. The cut-in scenario: (a) distance; (b) speed; (c) acceleration.
control strategy: the controller performs well, especially
in its distance control, when the driving habit changes.
There are two reasons for the good performance of
the SADP control strategy:
given a training experiment the outcome controller is
effective. By having considered 1000 experiments (i.e.,
we have generated N = 1000 controllers) we discover
that only 1 out of 1000 does not provide the requested
performances.
(1) The training scenario only consists of changing the
speed in time. However, due to the trial and error
mechanism of SADP, the state space is exhaustively
explored during the training process. As a result,
most typical states are excited and used during the
training phases.
(2) The state of SADP is (∆v, ∆d) and not (∆v, dr ). As
such, different driving habits will solely lead to different states of SADP, which means that the Action network will provide corresponding action strategies.
Following the derivation given in section 3.5 and the
Chernoff’s bound, let δ = 0.01 and ε = 0.05, then
N ≥ 1060 is obtained. With pˆN = 0.999 obtained from
the experiment results, we can state that the probability
that our controller satisfies the performance criterion is
above 0.95: the statement holds with confidence 0.99.
In other terms, the learning process is particularly efficient. Since performance validation is carried out on a
significant test set covering the functional driving conditions for our vehicle, stability is implicitly granted, at
least for the considered conditions.
The Demonstration of stability for the obtained controller in a close form is not an easy task. However, as
shown in section 3.5. We can estimate how the learning
process is difficult. Such a complexity can be intended in terms of learnability, namely, the probability that
Future analysis might consider a double form of randomization where driving habits are also drawn randomly and provided to the vehicle so as to emulate its
lifetime behavior.
12
200
Distance (m)
40
6. Acknowledgments
SADP
Mixed [9]
Desired distance
30
We strongly acknowledge Prof. Derong Liu for valuable discussions and Mr. Yongsheng Su for the assistance with the experimental campaign.
20
10
0
0
5
10
15
20
Time (s)
Speed (km/h)
100
SADP
Mixed [9]
Target speed
80
60
40
20
0
0
5
10
15
20
Time (s)
2
Acceleration (m/s )
2
0
−2
SADP
Mixed [9]
−4
−6
−8
0
5
10
15
20
Time (s)
Figure 10: Robust experiments with SADP and the mixed control strategies in an emergency braking scenario (Moon et al., 2009): (a)
distance; (b) speed; (c) acceleration.
5. Conclusions
The major contribution of this paper is the suggestion
of a simple and effective learning control strategy for the
full-range ACC problem. The control action is based on
SADP and introduces the concept of Inducing Region
to speed up the learning efficiency.
The trained SADP is applied to different driving scenarios including normal ACC, SG, emergency braking,
cut-in and driver habits changing. The SADP control strategy performs well in all encountered scenarios. The
method shows to be particularly effective in the emergency braking case.
We also show, by using randomized algorithms, how
the proposed SADP is particularly effective to provide
good control performance on our test scenarios at least
with probability 0.95 and confidence 0.99.
13
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14
Dongbin Zhao (M’06, SM’10): received the
B.S., M.S., Ph.D. degrees in Aug. 1994, Aug.
1996, and Apr. 2000 respectively, in materials
processing engineering from Harbin Institute of
Technology, China. Dr. Zhao was a postdoctoral fellow in humanoid robot at the Department
of Mechanical Engineering, Tsinghua University, China, from May 2000 to Jan. 2002.
Yuanheng Zhu received the B.S. degree in
school of management and engineering from
Nanjing University, Nanjing, China, in July
2010. He is currently a PhD candidate at the
State Key Laboratory of Management and
Control for Complex Systems, Institute of
Automation, Chinese Academy of Sciences,
China. His current research interests lies in
the area of adaptive dynamic programming and
fuzzy system.
He is currently an associate professor at the State Key Laboratory
of Management and Control for Complex Systems, Institute of Automation, Chinese Academy of Sciences, China. He has published one
book and over thirty international journal papers. His current research
interests lies in the area of computational intelligence, adaptive dynamic programming, robotics, intelligent transportation systems, and
process simulation.
Ding Wang received the B.S. degree in mathematics from Zhengzhou University of Light Industry, Zhengzhou, China, the M.S. degree in operational research and cybernetics from Northeastern University, Shenyang, China, and the
Ph.D. degree in control theory and control engineering from Institute of Automation, Chinese
Academy of Sciences, Beijing, China, in 2007,
2009, and 2012, respectively. He is currently
an assistant professor with the State Key Laboratory of Management and Control for Complex Systems, Institute of
Automation, Chinese Academy of Sciences. His research interests
include adaptive dynamic programming, neural networks, and intelligent control.
Dr. Zhao is an Associate Editor of the IEEE Transactions on Neural Networks and Learning Systems, and Cognitive Computation.
Zhaohui Hu received the B.S. degree in mechanical engineering from the University of Science & Technology Beijing, Beijing, China and
the M.S. degree in Institute of Automation, Chinese Academy of Sciences, Beijing, China, in
2008 and 2010, respectively. He is now with
the Electric Power Research Institute of Guangdong Power Grid Corporation, Guangzhou, China. His main research interests include the area
of computational intelligence, adaptive dynamic
programming, power grids, and intelligent transportation systems.
Zhongpu Xia received the B.S. degree in automation control from China University of Geosciences, Wuhan, China in 2011. He is currently working toward the M.S. degree in the State
Key Laboratory of Management and Control for
Complex Systems, Institute of Automation, Chinese Academy of Sciences, Beijing, China. His
research interests include computational intelligence, adaptive dynamic programming and intelligent transportation systems.
Cesare Alippi received the degree in electronic engineering cum laude in 1990 and the PhD
in 1995 from Politecnico di Milano, Italy. Currently, he is a Full Professor of information processing systems with the Politecnico di Milano.
He has been a visiting researcher at UCL (UK),
MIT (USA), ESPCI (F), CASIA (CN). Alippi is
an IEEE Fellow, Vice-President education of the
IEEE Computational Intelligence Society (CIS),
Associate editor (AE) of the IEEE Computational Intelligence Magazine, past AE of the IEEE-Tran. Neural Networks, IEEE-Trans Instrumentation and Measurements (2003-09) and member and chair of
other IEEE committees including the IEEE Rosenblatt award.
In 2004 he received the IEEE Instrumentation and Measurement
Society Young Engineer Award; in 2011 has been awarded Knight of
the Order of Merit of the Italian Republic.Current research activity
addresses adaptation and learning in non-stationary environments and
Intelligent embedded systems.
He holds 5 patents and has published about 200 papers in international journals and conference proceedings.
15
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