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

 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

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] =







i(k+1)

··· A

(i+1)k

(i+1)(k+1)

··· A

(i+1)l

. . . .

j(k+1)

··· A







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 =







0 0 0 0 ··· 0 0 0

1 1 −1 0 0 ··· 0 0 0

0 δ

0 0 ··· 0 0 0

0 0 1 1 −1 · ·· 0 0 0

0 0 0 δ

··· 0 0 0

0 0 0 0 0 ··· Θ

M−1

0 0

0 0 0 0 0 ··· 1 1 −1

0 0 0 0 0 ··· 0 δ







B =







0 0 ··· 0

0 0 0 ··· 0

0 k

0 ··· 0

0 0 0 ··· 0

0 0 k

··· 0

0 0 0 ··· 0

0 0 0 ··· k







, x =







M−1

(M−1)M







u =













= T

(1 − 2k

= −T

Φ(d

, i = 2, . . . , M,

= −T

(2)

OptimalDistributedControllerSynthesisforChainStructures-ApplicationstoVehicleFormations

219

where Φ(d) is the linearized air drag function, and

and Θ

denote physical dynamics constants (Alam,

2011). The derived model in (2) has a lower block

triangular structure, which can generally be stated as







(t + 1)













0 0 ··· 0

0 · ·· 0

0 A

··· 0

0 0 0 · ·· A













(t)













0 0 ··· 0

0 B

0 ·· · 0

0 0 B

··· 0

0 0 0 ·· · B













(t)













(t)







(3)

where the corresponding vehicle states for each sub-

system are

(t) = v

(t), x

(t) =



i−1,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

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

N−1

∑

t=0



∑

i=2

(i−1)i

(t) − τv

(t))

+ w

∆v

i−1

(t) − v

(t))

+ w

(i−1)i

(t) +

∑

i=1

(t) + w

(t)



=min

N−1

∑

t=0

∑

i=2





i−1

(t)

(i−1)i

(t)









i−1

(t)

(i−1)i

(t)





+ R

(t)

+ w

(t) + w

(t)

(4)

where





∆v

0 −w

∆v

0 w

+ w

−τw

−w

∆v

−τw

+ w

∆v

+ w





= w

(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

determines the importance

of not deviating from the desired time gap. Hence,

a large w

puts emphasis on safety. w

∆v

creates a

cost for deviating from the velocity of the preceding

vehicle, and w

punishes the control effort which is

proportional to the fuel consumption. The following

terms, w

, w

, 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

, w

∆v

and w

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 =





0 0

0 A





, B =





0 0

0 B

0 0 B





. (6)

Here, {w(t)} is a Gaussian disturbance vector with

covariance given by

E{w(k)w

(l)} =





0 0

0 W

0 0 W





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

has ac-

cess to the measurement history of subsystems, 1 and

2, while u

has access to its own measurements. Also,

to obtain partially nestedness u

must have access the

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220

global state measurements. Let I

denote the informa-

tion set of controller i at time t. Then

= {x

(0 : t)}, I

= {x

(0 : t),x

(0 : t)},

= {x

(0 : t),x

(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→∞

N−1

∑

t=0

(t)Qx(t) + u

(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

, B

) are

stabilizable,

ii. (Q, A), (Q[2 : 3, 2 : 3], A[2 : 3, 2 : 3]), and (Q

, A

)

are detectable.

Then, the optimal controller for the three-vehicle

chain is given by:



(t + 1)



= (A − BL

)[2 : 3, 1 : 3]





(t)





(t + 1) = (

A −

)[2, 1 : 2]



(t) − η

(t)







(t)





= −L





(t)





−







(t) − η

(t)







−





(t) − η

(t))





and the optimal cost is

Tr(X

) + Tr(X

The matrices X

, X

, and X

are the positive semidef-

inite stabilizing solutions to the Riccati equations

A + Q − A

B(B

B + R)

−1

A +

Q −

B +

−1

+ Q

−A

+ R

)

−1

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

is partitioned into blocks according to the partitions

of x as

= [X

i j

], i, j = 1, .., 3,

also, X

is partitioned according to the dimensions of

and x

= [X

i j

], i, j = 1, 2.

The gain matrices are given by

= (R + B

−1

= (

R +

−1

= (R

+ B

)

−1

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

] = [30000, 40000, 30000] kg. All the ve-

hicles are assumed to be traveling in the steady state

velocity v

= 19.44 m/s (70 km/h) at time gap τ = 1 s,

which gives an intermediate distance of d

= 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

Velocity [km/h]

1,D

2,D

3,D

1,S

2,S

3,S

50 100 150 200 250

time [s]

Intermediate spacing [m]

12,D

23,D

12,S

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

500

1000

1500

2000

time [s]

Input Torque [Nm]

1,D

2,D

3,D

1,S

2,S

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

222

Table 1: Table of the required control input (Torque) to han-

dle the disturbances in Figure 3.

i 1 2 3

||u

i,D

[kNm] 81.9 100.1 80.8

||u

i,S

[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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