Optimal Distributed Controller Synthesis for Chain Structures
Applications to Vehicle Formations
Omid Khorsand, Assad Alam and Ather Gattami
Department of Automatic Control, EES, KTH-Royal Institute of Technology, Stockholm, Sweden
Keywords:
Optimal Control, Distributed Control, Linear Quadratic Gaussian Control (LQG), Intelligent Transportation
Systems (ITS), Heavy Duty Vehicle (HDV), Platooning.
Abstract:
We consider optimal distributed controller synthesis for an interconnected system subject to communication
constraints, in linear quadratic settings. Motivated by the problem of finite heavy duty vehicle platooning,
we study systems composed of interconnected subsystems over a chain graph. By decomposing the system
into orthogonal modes, the cost function can be separated into individual components. Thereby, derivation
of the optimal controllers in state-space follows immediately. The optimal controllers are evaluated under
the practical setting of heavy duty vehicle platooning with communication constraints. It is shown that the
performance can be significantly improved by adding a few communication links. The results show that the
proposed optimal distributed controller outperforms a suboptimal controller in terms of control input energy.
1 INTRODUCTION
The systems to be controlled are, in many applica-
tion domains, getting larger and more complex. When
there is interconnection between different dynamical
systems, conventional optimal control algorithms pro-
vide a solution where centralized state information is
required. However, it is often preferable and some-
times necessary to have a decentralized controller
structure, since in many practical problems, the phys-
ical or communication constraints often impose a spe-
cific interconnection structure. Hence, it is interesting
to design decentralized feedback controllers for sys-
tems of a certain structure and examine their overall
performance.
The control problem in this paper is motivated by
systems, generally referred to as vehicle platooning,
involving a chain of closely spaced heavy duty vehi-
cles (HDVs). Governing vehicle platoons by an auto-
mated control strategy, the overall traffic flow is ex-
pected to improve (Ioannou and Chien, 1993) and the
road capacity will increase significantly (De Schut-
ter et al., 1999). By traveling at a close intermediate
spacing, the air drag is reduced for each vehicle in the
platoon. Thereby, the control effort and inherently the
fuel consumption can be reduced significantly (Alam
et al., 2010). However, as the intermediate spacing is
reduced the control becomes tighter due to safety as-
pects; mandating an increase in control action through
additional acceleration and braking. Hence, it is of
vast interest for the industry to find a fuel optimal
control. Thus, with limited information and control
input constraints, the control objective is to maintain
a predefined headway to the vehicle ahead based upon
local state measurements, which makes it a decentral-
ized control problem.
Decentralized control problems are still in-
tractable in general. One approach has been to clas-
sify specific information patterns leading to linear op-
timal controllers. An important result was given in
(Ho and Chu, 1972) which showed that for a new in-
formation structure, referred to as partially nested,
the optimal policy is linear in the information set.
In (Rantzer, 2006), stochastic linear quadratic con-
trol problem was solved under the condition that all
the subsystems have access to the global informa-
tion from some time in the past. Control for chain
structures in the context of platoons has been stud-
ied through various perspectives, e.g., (Alam, 2011;
Bamieh et al., 2008; Barooah and Hespanha, 2005;
Swaroop and Hedrick, 1996; Varaiya, 1993). It has
been shown that control strategies may vary depend-
ing on the available information within the platoon.
However, communication constraints have not in gen-
eral been considered in control design for platooning
applications.
The aim of this study is to synthesize controllers
for a practical decentralized system composed of M
interacting systems over a chain. We minimize a
quadratic cost under the partially nested information
218
Khorsand O., Alam A. and Gattami A..
Optimal Distributed Controller Synthesis for Chain Structures - Applications to Vehicle Formations.
DOI: 10.5220/0004039902180223
In Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2012), pages 218-223
ISBN: 978-989-8565-21-1
Copyright
c
2012 SCITEPRESS (Science and Technology Publications, Lda.)
structure. This problem is known to have a linear
optimal policy, (Ho and Chu, 1972) and (Voulgaris,
2000). However, most existing approaches do not
provide explicit optimal controller formulae and, the
order of the controllers can be large (Gattami, 2007),
which makes the implementation difficult. Some
work has been focused on finding numerical algo-
rithms to these problems, (Scherer, 2002) and (Ze-
lazo and Mesbahi, 2009). Recently, state-space so-
lutions to the so-called two-player state-feedback H
2
version of this problem have been given in (Swigart
and Lall, 2010). Also, in (Shah and Parrilo, 2010), us-
ing concepts from order theory, a control architecture
has been proposed for systems having the structure
of a partially ordered set. In contrast, we construct
conditional estimates based on the information shared
among the controllers. Thereby, we show how to de-
compose the cost function into independent terms and
hence to derive analytical forms for the controllers.
The main contribution of this paper is to intro-
duce a simple decomposition scheme to construct
optimal decentralized controllers with low computa-
tional complexity for chain structures which is appli-
cable to intelligent transportation systems in terms of
automated platooning. Derived from the character-
istics of actual Scania HDVs, we present a discrete
system model that includes physical coupling with a
preceding vehicle and consider interconnected sub-
systems over a chain structure. The proposed con-
trol scheme accounts for a constrained communica-
tion pattern among the vehicles and hence reduces the
communications compared to a centralized informa-
tion pattern where full state information is available
to each controller.
Notation. We denote a matrix partitioned into
blocks by A = [A
i j
], where A
i j
denotes the block ma-
trix of A in block position (i, j). The submatrix of A
formed by row partitions i through j and column par-
titions k through l will be denoted by A[i : j, k : l]:
A[i : j, k : l] =
A
ik
A
i(k+1)
··· A
il
A
(i+1)k
A
(i+1)(k+1)
··· A
(i+1)l
.
.
.
.
.
. . . .
.
.
.
A
jk
A
j(k+1)
··· A
jl
.
The expected value of a random variable x is denoted
by E{x}. The conditional expectation of x given y is
denoted by E{x|y}. The trace of a matrix A is denoted
by Tr{A}, and the sequence x(0), x(1), ..., x(t), is
denoted by x(0 : t).
Figure 1: The figure shows a platoon of M heavy duty vehi-
cles, where each vehicle is only able to communicate with
the preceding vehicles.
2 SYSTEM MODEL AND
PROBLEM STATEMENT
In this section we present the physical properties of
the system that we are considering, the linear discrete
system model structure for a heterogeneous HDV pla-
toon and its associated cost function. Finally, the
problem formulation is given.
2.1 System Model
We consider an HDV platoon as depicted in Figure 1.
The velocities do not deviate significantly for the ve-
hicles with respect to the lead vehicle’s velocity in an
automated HDV platoon. Thus, a linearized model
should give a sufficient description of the system be-
havior. The discrete HDV platoon model with respect
to a set reference velocity, an engine torque which
maintains the velocity, a fixed spacing between the
vehicles, and a constant slope is hence given by,
x(t +1) = Ax(t) + Bu(t) + w(t), (1)
where
A =
Θ
1
0 0 0 0 ··· 0 0 0
1 1 1 0 0 ··· 0 0 0
0 δ
2
Θ
2
0 0 ··· 0 0 0
0 0 1 1 1 · ·· 0 0 0
0 0 0 δ
3
Θ
3
··· 0 0 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 0 0 0 0 ··· Θ
M1
0 0
0 0 0 0 0 ··· 1 1 1
0 0 0 0 0 ··· 0 δ
M
Θ
M
,
B =
k
u
1
0 0 ··· 0
0 0 0 ··· 0
0 k
u
2
0 ··· 0
0 0 0 ··· 0
0 0 k
u
3
··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 0 0 ··· 0
0 0 0 ··· k
u
M
, x =
v
1
d
12
v
2
d
23
v
3
.
.
.
v
M1
d
(M1)M
v
M
,
u =
u
1
u
2
u
3
.
.
.
u
M
,
Θ
1
= T
s
(1 2k
d
v
0
),
Θ
i
= T
s
2k
d
Φ(d
0
)v
0
, i = 2, . . . , M,
δ
i
= T
s
κ
1
k
d
v
2
0
,
(2)
OptimalDistributedControllerSynthesisforChainStructures-ApplicationstoVehicleFormations
219
where Φ(d) is the linearized air drag function, and
δ
i
and Θ
i
denote physical dynamics constants (Alam,
2011). The derived model in (2) has a lower block
triangular structure, which can generally be stated as
x
1
(t + 1)
x
2
(t + 1)
x
3
(t + 1)
.
.
.
x
M
(t + 1)
=
A
11
0 0 ··· 0
A
21
A
22
0 · ·· 0
0 A
32
A
33
··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 0 0 · ·· A
MM
x
1
(t)
x
2
(t)
x
3
(t)
.
.
.
x
M
(t)
+
B
1
0 0 ··· 0
0 B
2
0 ·· · 0
0 0 B
3
··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 0 0 ·· · B
M
u
1
(t)
u
2
(t)
u
3
(t)
.
.
.
u
M
(t)
+
w
1
(t)
w
2
(t)
w
3
(t)
.
.
.
w
M
(t)
(3)
where the corresponding vehicle states for each sub-
system are
x
1
(t) = v
1
(t), x
i
(t) =
d
i1,i
v
i
, i = 2, . . . , M.
2.2 Performance Criteria
The performance criteria of an HDV platoon can be
mapped into quadratic costs. Hence, we formulate
the weight parameters for a quadratic cost function
based upon performance and safety objectives. The
objective of the lead vehicle is to minimize the fuel
consumption and control input, while maintaining a
set reference velocity. The objective of the follower
vehicles in addition, is to follow the preceding ve-
hicles velocity, while maintaining a set intermediate
spacing. The intermediate spacing reference could be
constant or, as in this case, time varying. It is deter-
mined by setting a desired time gap τ s, which in turn
determines the spacing policy as
d
re f
(t) = τv(t).
Thereby, the vehicles will maintain a larger interme-
diate spacing at higher velocities. Hence, the weights
for a discrete M HDV platoon can be set up as
J(u
) =min
u
N1
t=0
M
i=2
w
τ
i
(d
(i1)i
(t) τv
i
(t))
2
+ w
v
i
(v
i1
(t) v
i
(t))
2
+ w
d
i
d
2
(i1)i
(t) +
M
i=1
w
v
i
v
2
i
(t) + w
u
i
i
u
2
i
(t)
=min
u
N1
t=0
M
i=2
v
i1
(t)
d
(i1)i
(t)
v
i
(t)
T
Q
i
v
i1
(t)
d
(i1)i
(t)
v
i
(t)
+ R
i
u
2
i
(t)
+ w
v
1
v
2
1
(t) + w
u
1
1
u
2
1
(t)
(4)
where
Q
i
=
w
v
i
0 w
v
i
0 w
d
i
+ w
τ
i
τw
τ
i
w
v
i
τw
τ
i
τ
2
w
τ
i
+ w
v
i
+ w
v
i
,
R
i
= w
u
i
i
.
(5)
The weights in (4) give a direct interpretation of
how to enforce the objectives for a vehicle traveling in
a platoon. The value of w
τ
i
determines the importance
of not deviating from the desired time gap. Hence,
a large w
τ
i
puts emphasis on safety. w
v
i
creates a
cost for deviating from the velocity of the preceding
vehicle, and w
u
i
i
punishes the control effort which is
proportional to the fuel consumption. The following
terms, w
d
i
, w
v
i
, put a cost on the deviation from the lin-
earized states. Note that the main objective is to main-
tain a set intermediate distance, while maintaining a
fuel efficient behavior. Therefore, w
τ
i
, w
v
i
and w
u
i
i
must be set larger than the remaining weights. The
weights are chosen such that Q is positive semidefi-
nite and R is positive definite.
2.3 Problem Formulation
Although the approach used in this paper is applicable
for systems over general acyclic graphs, for simplicity
we will concentrate on a simple chain structure, which
we refer to as the three-vehicle chain. The aim is to
synthesize controllers under imposed communication
constraints.
For this problem, the system matrices are given by
A =
A
11
0 0
A
21
A
22
0
0 A
32
A
33
, B =
B
1
0 0
0 B
2
0
0 0 B
3
. (6)
Here, {w(t)} is a Gaussian disturbance vector with
covariance given by
E{w(k)w
T
(l)} =
W
1
0 0
0 W
2
0
0 0 W
3
δ(k l).
In this system, the dynamics of subsystem i (Vehi-
cle i) propagates to subsystem j (Vehicle j) only if
i < j. If all subsystems have access to the global
state measurements the information structure would
be classical, and the optimal linear controller could
be obtained from the linear quadratic control theory.
However, in the practical setting of HDV platooning
the lead vehicle only has its own state information,
whereas the follower vehicle can also measure the
states of the preceding vehicle through radar sensors.
Therefore, we consider the case in which u
2
has ac-
cess to the measurement history of subsystems, 1 and
2, while u
1
has access to its own measurements. Also,
to obtain partially nestedness u
3
must have access the
ICINCO2012-9thInternationalConferenceonInformaticsinControl,AutomationandRobotics
220
global state measurements. Let I
t
i
denote the informa-
tion set of controller i at time t. Then
I
t
1
= {x
1
(0 : t)}, I
t
2
= {x
1
(0 : t),x
2
(0 : t)},
I
t
3
= {x
1
(0 : t),x
2
(0 : t),x
3
(0 : t)}.
(7)
This information pattern is not classical anymore and
is a simple case of a partially nested information
structure. This is one of a few non-classical informa-
tion patterns for which the optimal policy is known to
be unique and linear in the information set.
Here only one communication link is needed from
vehicle 1 to vehicle 3, since vehicle 2 and 3 can mea-
sure the preceding vehicle states with on-board radar
sensors.
Thus, the problem that we solve is finding an
analytical formulation for optimal controllers con-
strained to specified information sets that minimize
the infinite-horizon quadratic cost
lim
N
1
N
E
N1
t=0
(x
T
(t)Qx(t) + u
T
(t)Ru(t)), (8)
subject to the given system dynamics and perfor-
mance objectives.
3 MAIN RESULT:
THREE-VEHICLE CHAIN
The aim of this section is to present an outline of
the optimal controller structures for the three vehicles
chain problem. A detailed version and intuition for
the general case is given in (Khorsand et al., 2012).
Theorem 1. Assume that
i. (A, B), (A[2 : 3, 2 : 3], B[2 : 3, 2 : 3]), and (A
33
, B
3
) are
stabilizable,
ii. (Q, A), (Q[2 : 3, 2 : 3], A[2 : 3, 2 : 3]), and (Q
33
, A
33
)
are detectable.
Then, the optimal controller for the three-vehicle
chain is given by:
η
1
(t + 1)
η
2
(t + 1)
= (A BL
1
)[2 : 3, 1 : 3]
x
1
(t)
η
1
(t)
η
2
(t)
η
3
(t + 1) = (
˜
A
˜
BL
2
)[2, 1 : 2]
x
2
(t) η
2
(t)
η
3
(t)
u
1
(t)
u
2
(t)
u
3
(t)
= L
1
x
1
(t)
η
1
(t)
η
2
(t)
0
L
2
x
2
(t) η
1
(t)
η
3
(t)
0
0
L
3
(x
3
(t) η
2
(t) η
3
(t))
,
and the optimal cost is
Tr(X
1
11
W
1
) + Tr(X
2
11
W
2
) + Tr(X
3
W
3
).
The matrices X
1
, X
2
, and X
3
are the positive semidef-
inite stabilizing solutions to the Riccati equations
X
1
=A
T
X
1
A + Q A
T
X
1
B(B
T
X
1
B + R)
1
B
T
X
1
A
X
2
=
˜
A
T
X
2
˜
A +
˜
Q
˜
A
T
X
2
˜
B(
˜
B
T
X
2
˜
B +
˜
R)
1
˜
B
T
X
2
˜
A
X
3
=A
T
33
X
3
A
33
+ Q
33
A
T
33
X
3
B
3
(B
T
3
X
3
B
3
+ R
33
)
1
B
T
3
X
3
A
33
where
˜
A = A[2 : 3, 2 : 3],
˜
B = B[2 : 3, 2 : 3],
˜
Q =
Q[2 : 3, 2 : 3] and
˜
R = R[2 : 3,2 : 3]. The matrix X
1
is partitioned into blocks according to the partitions
of x as
X
1
= [X
1
i j
], i, j = 1, .., 3,
also, X
2
is partitioned according to the dimensions of
x
2
and x
3
as
X
2
= [X
2
i j
], i, j = 1, 2.
The gain matrices are given by
L
1
= (R + B
T
X
1
B)
1
B
T
X
1
A,
L
2
= (
˜
R +
˜
B
T
X
2
˜
B)
1
˜
B
T
X
2
˜
A,
L
3
= (R
33
+ B
T
3
X
3
B
3
)
1
B
T
3
X
3
A
33
.
Proof 1. See (Khorsand et al., 2012).
4 NUMERICAL RESULTS
In this section, we implement the proposed controller
on an M = 3 HDV platoon (Figure 1) and evaluate
the performance through a realistic scenario that HDV
platoons often face on the road. We assume that the
vehicles in the platoon can only measure the veloc-
ity and relative distance of the preceding vehicle and
only receive state information through wireless com-
munication from all the preceding vehicles. This as-
sumption is made to evaluate if a small addition in
communication links could improve the system per-
formance. Hence, a numerical comparison is made
between the proposed controller and a suboptimal
controller specifically designed for HDV platooning
(Alam et al., 2011). The suboptimal controller uses
local information, namely it only accounts for the dy-
namics of the preceding vehicle. Finally, we compare
the proposed controller with the fully centralized lin-
ear quadratic controller.
In practice, many random disturbances such as
wind variation, changing topology, or varying road
properties are inflicted upon the HDV platoon. These
disturbances are modeled as disturbances in state
measurements. An additional disturbance of interest
is a mandated deviation in the lead vehicles velocity.
This often occurs due to varying traffic events that the
lead vehicle must adhere to. Hence, integral action for
OptimalDistributedControllerSynthesisforChainStructures-ApplicationstoVehicleFormations
221
the lead vehicle is also added as a state to the system
presented in (2), to model such disturbances.
The modeled HDVs are described as traveling in
a longitudinal direction on a flat road. We consider
a heterogeneous platoon, where the masses are set to
[m
1
, m
2
, m
3
] = [30000, 40000, 30000] kg. All the ve-
hicles are assumed to be traveling in the steady state
velocity v
0
= 19.44 m/s (70 km/h) at time gap τ = 1 s,
which gives an intermediate distance of d
0
= 19.44.
The maximum engine and braking torque for a com-
mercial HDV varies based upon vehicle configura-
tion but can be approximated to be 2500 Nm and
60000 Nm/Axle respectively.
State disturbances as well as several lead vehicle
deviation disturbances are imposed on the system, see
Figure 2. The lead vehicle deviation disturbances can
be explained by the following scenario. The platoon
travels along a road where the road speed is 70 km/h.
Suddenly a slower vehicle enters the lane through a
shoulder path (at the 45 s time marker). The lead
vehicle must therefore reduce its speed to 60 km/h,
in turn forcing the follower vehicles to reduce their
speed and adapt their relative distance accordingly.
After a while, the slower vehicle increases its speed
to the road speed of 70 km/h and no longer inhibits
the platoon (120 s time marker). Hence, the lead ve-
hicle again resumes the road speed and the follower
vehicles adapt the speed and distance automatically as
well. Finally, the platoon arrives at a point where the
road speed is changed to 80 km/h (180 s time marker).
Figure 2 shows the velocity trajectories of three
HDV platoon in the top plot and the corresponding in-
termediate spacings in this scenario. The trajectories
obtained through the optimal decentralized controller
are bold. The trajectories are also plotted, with thin-
ner lines, for the suboptimal decentralized controller.
We see that the proposed optimal controller displays
a good performance. The suboptimal controller dis-
plays a slightly harsher behavior with a faster speed
change, since it does not take follower vehicles into
account. Hence, the required relative control input
energy is much higher for the suboptimal controller
compared to our proposed controller, as can been seen
in Figure 3. It can also be seen that the required con-
trol input to handle the disturbances are well within
the feasible physical range. By estimating the states
of the follower vehicles, the mandated control input
to handle the presented disturbances can be reduced
significantly. Both the maximum and minimum val-
ues are lowered in the control input requirement for
the optimal decentralized controller. The first two
rows in Table 1 states the total control input energy
required to handle the imposed disturbances. We can
see that the optimal distributed controller reduces the
control input energy by 10.4 % for the lead vehicle,
50 100 150 200 250
60
65
70
75
80
Velocity [km/h]
v
1,D
v
2,D
v
3,D
v
1,S
v
2,S
v
3,S
50 100 150 200 250
16
18
20
22
time [s]
Intermediate spacing [m]
d
12,D
d
23,D
d
12,S
d
23,S
Figure 2: Three HDV platoon, where a disturbance in ve-
locity of the lead vehicle is imposed. The top plot shows
the velocity trajectories for the M = 3 HDV platoon and
the bottom plot shows the intermediate spacings. The tra-
jectories obtained through the optimal decentralized con-
troller are bold and subindexed with i, D and the trajectories
obtained through the suboptimal controller are subindexed
with i, S, where i = 1, 2, 3 denote the platoon position index.
50 100 150 200 250
−1500
−1000
−500
0
500
1000
1500
2000
time [s]
Input Torque [Nm]
u
1,D
u
2,D
u
3,D
u
1,S
u
2,S
u
3,S
Figure 3: Corresponding input torque to handle the im-
posed disturbances in Figure 2. Similarly, the trajectories
obtained through the optimal decentralized controller are
bold and subindexed with i, D and the trajectories obtained
through the suboptimal controller are subindexed with i, S,
i = 1, 2, 3.
by 16.3 % for the second vehicle, and by 15.5 % for
the third vehicle. By estimating the states of the fol-
lower vehicles, the proposed controller mimics a cen-
tralized control strategy and displays a smoother be-
havior. However, the reduced control energy is ob-
tained at the cost of adding a communication link be-
tween vehicle 1 and vehicle 3, since vehicle 1’s state
cannot be measured through the mounted radar on ve-
hicle 3. Furthermore, the average velocity is reduced
by 2.6 %. Travel time is equally important for fleet
operators. However, it is clear that there is a consid-
erable saving in the fuel consumption at the cost of
additional communication links and a much smaller
ICINCO2012-9thInternationalConferenceonInformaticsinControl,AutomationandRobotics
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Table 1: Table of the required control input (Torque) to han-
dle the disturbances in Figure 3.
i 1 2 3
||u
i,D
||
2
[kNm] 81.9 100.1 80.8
||u
i,S
||
2
[kNm] 91.4 112.6 89.0
reduction in travel time.
The proposed controller also accounts for all the
states in the platoon by estimating the states of the
follower vehicles, the behavior is close to the central-
ized controller. The computed relative differences in
the cost function as well as the difference in required
control inputs to handle the disturbances are minimal.
5 CONCLUSIONS
We have presented a quadratic optimal distributed
control method for chain structures with applications
to heterogeneous vehicle platooning under commu-
nication constraints. A procedure has been given
for constructing low order optimal decentralized con-
trollers through a simple decomposition scheme. A
discrete HDV platoon model has been derived that in-
cludes physical coupling between the vehicles upon
which the controllers are evaluated. The results show
that the total control input energy required for the pro-
posed controller is very close to a centralized con-
troller where communication is needed among all the
vehicles, and is significantly lower compared to a sub-
optimal controller which only accounts for the im-
mediate preceding vehicle. In particular, by estimat-
ing the interaction with the follower vehicles, per-
formance can be improved by adding a communica-
tion link from the first to the third vehicle in a three-
vehicle platoon. Thus, considering preceding vehicles
as well as follower vehicles is significant for fuel op-
timality.
A natural extension to the presented work is to de-
rive explicit solutions for the problem of M-HDVs. It
is planned for future work.
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